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{"url":"https:\/\/mathoverflow.net\/questions\/241754\/equivariant-sheaves-over-affine-schemes","text":"# Equivariant sheaves over affine schemes\n\nLet $k$ be a field, let $G$ be a linear algebraic group over $k$ and let $A$ be a commutative $k$-algebra which is acted on by $G$.\n\nWe say that an $A$-module $M$ is a $(G,A)$-module if it satisfies the following two properties:\n\n1) $M$ is a rational $G$-module (over $k$)\n\n2) The multiplication map $A \\otimes M \\rightarrow M$ is a morphism of rational $G$-modules.\n\nIf we let $X=\\text{Spec}(A)$ and view $M$ as a quasi-coherent sheaf on $X$, then giving $M$ the structure of a $(G,A)$-module should be equivalent to giving it the structure of a $G$-equivariant sheaf on $X$ (see https:\/\/en.wikipedia.org\/wiki\/Equivariant_sheaf). Is there an easy, explict way to see why these two definitions coincide?\n\n\u2022 No, because you only give one definition. What you could do is give a definition of $G$-equivariant sheaf that matches. Jun 9, 2016 at 7:18\n\u2022 @WilberdvanderKallen : en.wikipedia.org\/wiki\/Equivariant_sheaf Jun 9, 2016 at 10:28\n\u2022 @Wilberdvanderkallen: I apologize, I should linked the Wikipedia article... Jun 9, 2016 at 12:53\n\u2022 Here are some thoughts, which are not entirely rigorous. I would like to write these as a comment but I do not have enough reputation. The kernel of the surjective multiplication map $A\\otimes M\\rightarrow M$ is both a $G$-module and a sub $A$-module. Therefore, the kernel is a $(G,A)$-module (The $(G,A)$-module structure on $M\\otimes A$ comes from pulling back the $(G,k)$-module $M$ over Spec $k$). In my understanding we should be able to give a quotient of $(G,A)$-modules a natural structure of a $(G,A)$-module. I gathered these based on ch3 in d-nb.info\/98519670X\/34. Jun 13, 2022 at 19:38","date":"2023-04-02 08:28:42","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\": 0.9595276713371277, \"perplexity\": 183.60044249061232}, \"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\/1679296950422.77\/warc\/CC-MAIN-20230402074255-20230402104255-00055.warc.gz\"}"}
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\section{Introduction}
Experimental realization of Bose-Einstein condensates (BEC) in ultracold
trapped gases opened up a rapidly expanding field of studies of quantum-degenerate
systems \cite{Dalfovo:1999,Giorgini:2008,Bloch:2008}. Among the others,
statistical properties of the condensate, such as its fluctuations and
correlations, have attracted a wide attention. While the theory on this subject
is well developed (see e.g. \cite{Idziaszek3} and references therein), there are
only few experiments that address this issue. To date, only the fluctuations of the
total number of atoms in a condensed gas have been measured \cite{Raizen}, and the
subpoissonian scaling has been observed. The second order correlation functions,
that are directly connected to the condensate atom number fluctuations, have been
investigated experimentally in the collisions of metastable helium condensates
\cite{Westbrook} and in the expanding rubidium condensate \cite{Schmiedmayer}.
One of the potential tools to measure the statistics of quantum-degenerate gases is
based on atom-light interactions.This possibility has been noticed already some
time ago, and has been proposed for a detection of the Bose condensed phase
\cite{Lewenstein:1993,Javanainen:1995,Saito:1999,Moore:1999}, superfluidity in
Fermi gases \cite{Zhang:1999,Ruostekoski:1999,Torma:2000,Wong:2000} and, quite
recently, for a detection of quantum phases in ultracold gases in optical lattices
\cite{Mekhov:2007,Mekhov:2007a,Chen:2007,Eckert:2007}. The optical imaging techniques
have already been used to measure coherence properties of BEC in the Raman superradiant
scattering \cite{sadler:2007} and the second order correlation functions
\cite{Schmiedmayer}.
The light scattered from a quantum gas carries information on atoms statistics,
and thus can be used to measure the condensate fluctuations \cite{IdziaszekScatt}.
In the case of BEC in a trap the profile of the scattered light is dominated by a
component, which depends on the mean occupation number of the condensate.
In order to detect a much weaker component resulting from fluctuations,
one has to resort to the variance of the number of scattered photons,
which can be difficult to measure.
The situation changes, however, in the presence of a periodic potential.
In this case the dominating classical component exhibits interference
pattern characteristic for the Bragg scattering, and the quantum component
can be measured at the angles corresponding to destructive interference,
where the large classical component vanishes. This property has been first
noticed by Mekhov {\it et al.} \cite{Mekhov:2007}, and these authors have
proposed a method of probing the statistics of an ultracold gas in a lattice
\cite{Mekhov:2007b}, based on the relatively strong coupling between atoms
and light modes of a cavity. In this case one should be able to perform
non-demolition measurement allowing to distinguish between superfluid
and Mott-insulator (MI) quantum phases at temperature $T=0$.
In this paper we study a less complex situation of measuring a quantum gas
statistics based on the far-off-resonance light scattering from a Bose gas
in a lattice, focusing on the effects of statistics at finite temperatures.
In order to avoid atom losses and suppress a possibility of perturbing the
quantum state by the probing laser, we assume that the probing light is
sufficiently weak and far-detuned. We show that the mean number of photons
detected at some special angles, carries enough information not only to
distinguish between different thermodynamic phases of the gas but also to
directly measure the effects of the on-site atom statistics driven by
quantum and thermal fluctuations. Hence, it allows to verify the validity
of some well-grounded literature approaches, such as the Bogoliubov method,
to describe higher order correlation functions in an interacting Bose gas.
Our paper is organized as follows. In section \ref{sec:atom-light} we develop
a model for scattering of light from ultracold atoms, showing that the number
of scattered photons is directly related to the second-order correlation function.
In section \ref{sec:opt-latt} we tailor our model to the external potential created
by an optical lattice. The scattering from atoms in one dimensional (1D) lattice
is considered in section \ref{sec:oneDimTzeroScatt}, where for simplicity we focus
only on the zero-temperature statistics discussing the effects of different approximations.
The finite temperature statistics of a Bose gas in a lattice is analyzed in
section \ref{sec:statProperties}. Section \ref{sec:3D} investigates scattering
from atoms in three-dimensional (3D) optical lattice at finite temperatures. We
conclude in section \ref{sec:conclusions}, and finally two appendixes present
technical details related to the influence of non-local Franck-Condon coefficients
(Appendix A) and optimal configuration of a probing laser and a photon detector
in the 3D lattice case (Appendix B).
\section{Interaction of light with many atoms}
\label{sec:atom-light}
In this section we consider a general problem of light scattering from a gas
of bosons in an arbitrary external potential. We assume that the trapped atoms
are illuminated with a weak, and far-detuned laser light. The angularly resolved
scattered light is measured by detectors in the far-field region. The full
Hamiltonian of the system consists of the following parts:
\begin{equation}
\mathcal{H}=\mathcal{H}_{a} + \mathcal{H}_{f} + \mathcal{H}_{al} + \mathcal{H}_{af}
\end{equation}
where $\mathcal{H}_{a}$ is the atomic Hamiltonian, $\mathcal{H}_{f}$ represents vacuum modes
of the electromagnetic field (EM), $\mathcal{H}_{al}$ describes interaction of atoms
with the laser light
and $\mathcal{H}_{af}$ interaction of atoms with vacuum modes.
The atomic Hamiltonian can be split into two parts
\begin{equation}
\mathcal{H}_{a}=\mathcal{H}_{0} + \mathcal{H}_{int}
\end{equation}
where $\mathcal{H}_{0}$ describes the system of two-level atoms in the second-quantization
formalism \cite{Lewenstein}
\begin{equation}
\mathcal{H}_{0}=\sum\limits_{\mbf{n}} \hbar \omega^{g}_{\mbf{n}} g^{\dagger}_{\mbf{n}}
g_{\mbf{n}} + \sum\limits_{\mbf{m}} \hbar \left( \omega_{\mbf{m}}^{e}
+\omega_{0}\right) e^{\dagger}_{\mbf{m}} e_{\mbf{m}},
\end{equation}
and the part including the atom-atom interactions reads
\begin{equation}
\mathcal{H}_{int} = \frac{1}{2} \sum\limits_{\mbf{n}, \mbf{m}, \mbf{p}, \mbf{q}}
U_{\mbf{n} \mbf{m} \mbf{p} \mbf{q}}
\,g_{\mbf{n}}^{\dagger} g_{\mbf{m}}^{\dagger} g_{\mbf{p}} g_{\mbf{q}}.
\end{equation}
Here, $g_{\mbf{n}}$ ($g^{\dagger}_{\mbf{n}}$) is the annihilation (creation)
operator of an atom in the ground electronic state and state $\mbf{n}$ of the
center-of-mass (COM) motion, and $e_{\mbf{m}}$ ($e^{\dagger}_{\mbf{m}}$) is
the annihilation (creation) operator of an atom in an electronic excited
state and state $\mbf{m}$ of COM motion. The operators obey the standard
bosonic commutation relations:
$[g_{\mbf{n}},g_{\mbf{m}}^{\dagger}]=\delta_{\mbf{n},\mbf{m}}$ and
$[e_{\mbf{n}},e_{\mbf{m}}^{\dagger}]=\delta_{\mbf{n},\mbf{m}}$.
The corresponding eigenenergies of atom COM motion are denoted by
$\hbar \omega^{g}_{\mbf{n}}$ and $\hbar \omega_{\mbf{m}}^{e}$ for atoms in the
ground and excited electronic states, respectively. The matrix elements
$U_{\mbf{n} \mbf{m} \mbf{p} \mbf{q}}$ of the interaction Hamiltonian read
\begin{align}
U_{\mbf{n} \mbf{m} \mbf{p} \mbf{q}} \equiv \frac{4 \pi a_s \hbar^2}{m}
\int d^{3}r \phi^{\ast}_{\mbf{n}}(\mbf{r})
\phi^{\ast}_{\mbf{m}}(\mbf{r}) \phi_{\mbf{p}}(\mbf{r}) \phi_{\mbf{q}}(\mbf{r})
\label{eqn:Hamint}
\end{align}
where we model short-range interactions through a contact potential with $s$-wave
scattering length $a_s$ and a mass of the atom $m$. We neglect ground-excited
and excited-excited atom interactions, assuming that for a weak and far-detuned
probing light excited atoms constitute only a small fraction of the whole sample.
The Hamiltonian of the EM field takes a standard form:
\begin{equation}
\mathcal{H}_{f}=\sum\limits_{\lambda} \int
d^{3}k \ \hbar \omega_{\mbf{k}} a^{\dagger}_{\mbf{k} \lambda} a_{\mbf{k} \lambda}
\label{eqn:HamEM}
\end{equation}
with $a_{\mbf{k} \lambda}$ ($a^{\dagger}_{\mbf{k} \lambda}$) being an annihilation
(creation) operator of a photon with a wave vector $\mbf{k}$ and a polarization
$\lambda$.
The interaction of atoms with a laser beam is described as follows:
\begin{equation}
\mathcal{H}_{al}=\frac{\hbar \Omega}{2} \sum\limits_{\mbf{n},\mbf{m}} \bra{\mbf{n},g}
u_{\mbf{k}_L}(\mbf{r}) \ket{\mbf{m},e} e^{\imath \omega_{L} t} g^{\dagger}_{\mbf{n}}
e_{\mbf{m}} + h.c.
\label{eqn:Hamal}
\end{equation}
where we treat the macroscopically occupied laser mode classically. Here,
$u_{\mbf{k}_L}(\mbf{r})$ characterizes a laser mode with a wave vector $\mbf{k}_L$, $\omega_{L}$
is the laser frequency, $\Omega$ is a Rabi frequency of the atomic transition,
and the Franck-Condon coefficients $\bra{\mbf{n},g} u_{\mbf{k}_L}(\mbf{r}) \ket{\mbf{m},e}$
describe a transition amplitude between COM motion states $\mbf{n}$ and $\mbf{m}$
of the atoms in the ground and excited electronic states, respectively. Typically,
for a single probing laser, we have $u_{\mbf{k}_L}(\mbf{r}) = e^{\imath \mbf{k}_L \mbf{r}}$
(running wave), and for the two counter-propagating probing beams
$u_{\mbf{k}_L}(\mbf{r}) = \cos\left( \mbf{k}_L \mbf{r}\right)$ (standing wave).
In general $u_{\mbf{k}_L}(\mbf{r})$ can also represent the modes of an optical
cavity \cite{Mekhov:2007}.
The part of the Hamiltonian that describes coupling of atoms with quantized EM
field is given by:
\begin{align}
\mathcal{H}_{af} & = \imath \sum\limits_{\lambda} \int d^{3}k \, \hbar c_{\mbf{k}\lambda}
a^{\dagger}_{\mbf{k} \lambda} \sum\limits_{\mbf{n}, \mbf{m}} \bra{\mbf{n},g}
u_{\mbf{k}}(\mbf{r}) \ket{\mbf{m},e} g^{\dagger}_{\mbf{n}} e_{\mbf{m}} \nonumber \\
& \phantom{=} + h.c.
\end{align}
in which $c_{\mbf{k}\lambda} = \sqrt{\omega_{\mbf{k}}/(16 \pi^3 \epsilon_{0} \hbar)}
\left(\textbf{d} \cdot \mbox{\boldmath$\epsilon$}_{\mbf{k} \lambda}\right)$, $\mbf{d}$ is a dipole
moment of the atomic transition, $u_{\mbf{k}}(\mbf{r})$ is a mode function of the
EM field with a wavevector $\mbf{k}$ and frequency $\omega_{\mbf{k}}$, and
$\mbox{\boldmath$\epsilon$}_{\textbf{k}\lambda}$ is a unit vector perpendicular to $\mbf{k}$
describing the mode of light with a polarization $\lambda$.
We solve the quantum equations of motion in the Heisenberg picture under the
following approximation: i) we assume that the atomic operators are driven only
by the dominating laser mode of the EM field, neglecting the back action of
atoms on the laser mode, ii) the quantum dynamics of the vacuum modes is
determined by the evolution of atomic operators, ignoring the back-action
of the vacuum modes, which is equivalent to neglecting the process of
spontaneous emission, iii) for the weak and far-detuned laser field we
perform adiabatic elimination of the weakly populated excited state. Our
approximations are analogous to those used in \cite{IdziaszekScatt}, with
the only difference that here we perform the adiabatic elimination of the
excited state, instead of assuming short probing pulses. We carry out our
derivation neglecting the interactions between atoms and we comment on the
generalization to the interacting gas case at the end of this section.
The equations of motion for the atomic operators in the interaction picture
with respect to $\mathcal{H}_{0}$:~$\tilde{g}_\mbf{m}(t) = g_\mbf{m}(t) e^{-\imath \omega_\mbf{m}^g t}$
and $\tilde{e}_\mbf{n}(t) = e_\mbf{n}(t) e^{-\imath (\omega_\mbf{n}^e + \omega_0) t}$, read
\begin{align}
\label{eq:MotionAtomic1}
\frac{d \tilde{g}_\mbf{m}}{d \tau} & = - \imath \frac{\Omega}{2 \Delta} \sum_{\mbf{n}}
\eta_{\mbf{n}\mbf{m}}^\ast (\mbf{k}_L) \exp\left[\imath \frac{\omega_\mbf{m}^g
- \omega_\mbf{n}^e + \Delta}{\Delta}\tau\right] \tilde{e}_\mbf{n}(\tau), \\
\label{eq:MotionAtomic2}
\frac{d \tilde{e}_\mbf{n}}{d \tau} & = - \imath \frac{\Omega}{2 \Delta} \sum_{\mbf{m}}
\eta_{\mbf{n}\mbf{m}} (\mbf{k}_L) \exp\left[\imath \frac{\omega_\mbf{n}^e
- \omega_\mbf{m}^g - \Delta}{\Delta} \tau\right] \tilde{g}_\mbf{m}(\tau),
\end{align}
where $\eta_{\mbf{n}\mbf{m}} (\mbf{k}) = \bra{\mbf{n},e} u_{\mbf{k}}(\mbf{r}) \ket{\mbf{m},g}$,
$\Delta = \omega_{L} - \omega_{0}$ and we have introduced rescaled time variable
$\tau = \Delta t$. We solve Eqs.~\eqref{eq:MotionAtomic1} and \eqref{eq:MotionAtomic2}
by applying the Laplace transformation $F^{\cal L}(s)
= \int_0^\infty d\tau \, e^{-s \tau} F(\tau)$. The Laplace transformed equations take the form
\begin{align}
\label{eq:Lapl1}
s \tilde{g}_\mbf{m}^{\cal L}(s) - \tilde{g}_\mbf{m}(0) & = \nonumber \\
- \imath \frac{\Omega}{2 \Delta} \sum_{\mbf{n}} & \eta_{\mbf{n}\mbf{m}}^\ast (\mbf{k}_L)
\tilde{e}_\mbf{n}^{\cal L} (s) \left[s -\imath \frac{\omega_\mbf{m}^g
- \omega_\mbf{n}^e + \Delta}{\Delta} \right],\\
\label{eq:Lapl2}
s \tilde{e}_\mbf{n}^{\cal L}(s) - \tilde{e}_\mbf{n}(0) & = \nonumber \\
- \imath \frac{\Omega}{2 \Delta} \sum_{\mbf{m}} & \eta_{\mbf{n}\mbf{m}} (\mbf{k}_L)
\tilde{g}_\mbf{m}^{\cal L} (s) \left[s -\imath \frac{\omega_\mbf{n}^e
- \omega_\mbf{m}^g + \Delta}{\Delta}\right].
\end{align}
For a far-detuned light the prefactor on the right-hand-side of the equations is small:
$\Omega/\Delta \ll 1$, and, in principle, the equations can be solved by iterations
in a perturbative manner. Here, however, we proceed with solving Eq.~\eqref{eq:Lapl2}
for $\tilde{e}_\mbf{n}^{\cal L}(s)$ and then substituting the result into Eq.~\eqref{eq:Lapl1}.
For a far-detuned light we apply $\Delta \gg \omega_\mbf{n}^e, \omega_\mbf{m}^g$ and
we use the identity $\displaystyle \sum_{\mbf{n}} \eta_{\mbf{n}\mbf{m}}^\ast (\mbf{k}_L) \eta_{\mbf{n}\mbf{m}^\prime}
(\mbf{k}_L) = \delta_{\mbf{m}\mbf{m}^\prime}$, which results in
\begin{equation}
\label{eq:Lapl3}
\tilde{g}^{\cal L}_\mbf{m} (s) \approx \frac{1}{s + \imath \frac{\Omega^2}{\Delta^2}}
\left( \tilde{g}_\mbf{m}(0) - \imath\frac{\Omega}{2 \Delta} \sum_\mbf{n}
\eta_{\mbf{n}\mbf{m}}^\ast (\mbf{k}_L) \frac{\tilde{e}_\mbf{n}(0)}{s-\imath} \right).
\end{equation}
By substituting back this result into Eq.~\eqref{eq:Lapl2}, and performing the
inverse Laplace transformation we obtain the following time-dependence of the
atomic operators
\begin{align}
\label{eq:sol1}
\tilde{g}_\mbf{m}(t) & = \tilde{g}_\mbf{m}(0) e^{- \imath \omega_\mathrm{AC} t}
+ {\cal O}\left(\frac{\Omega}{\Delta}\right), \\
\tilde{e}_\mbf{n}(t) & = \tilde{e}_\mbf{n}(0) \nonumber \\
& \, + \frac{\Omega}{2 \Delta} \sum_\mbf{m}
\eta_{\mbf{n}\mbf{m}}(\mbf{k}_L) \tilde{g}_\mbf{m}(0) \left[
e^{\imath(\omega_\mbf{n}^e - \omega_\mbf{m}^g - \Delta - \omega_\mathrm{AC}) t} -1 \right] \nonumber \\
& \, + {\cal O}\left(\frac{\Omega^2}{\Delta^2}\right),
\label{eq:sol2}
\end{align}
Here, $\omega_\mathrm{AC} = \frac{\Omega^2}{4 \Delta}$ denotes AC Stark shift of atomic
levels in the field of the probing laser. In Eq.~\eqref{eq:sol1} we have not included
terms of the order of $\Omega/\Delta$ , which are proportional to $\tilde{e}_\mbf{n} (0)$,
since they do not give any contribution to the mean number of photons, assuming that
there are no excited atoms at the beginning.
We substitute Eqs.~\eqref{eq:sol1} and \eqref{eq:sol2} into equation of motion of the
E-M field operators $\tilde{a}_\mbf{k \lambda}(t) = a_\mbf{k \lambda}(t) e^{-\imath \omega_\mbf{k} t}$
in the interaction picture. In the lowest order in $\Omega/\Delta$ this yields
\begin{multline}
\label{eq:sola}
\tilde{a}_\mbf{k \lambda}(t) - \tilde{a}_\mbf{k \lambda}(0) = \\
= c_{\mbf{k} \lambda} \frac{\Omega}{\Delta} \sum_{\mbf{n}\mbf{n}^\prime\mbf{m}}
\eta_{\mbf{m}\mbf{n}^\prime}(\mbf{k}) \eta_{\mbf{m}\mbf{n}}^\ast(\mbf{k}_L)
\tilde{g}^\dagger_\mbf{n}(0) \tilde{g}_{\mbf{n}^\prime}(0) \\
\times \frac{e^{\imath (\omega_\mbf{k} -\omega^L_{\mbf{n} \mbf{n}^\prime})t/2}}
{\omega_\mbf{k} -\omega^L_{\mbf{n} \mbf{n}^\prime}} \sin
\left( \frac{\omega_\mbf{k} -\omega^L_{\mbf{n} \mbf{n}^\prime}}{2} t \right)
\end{multline}
where $\omega^L_{\mbf{n} \mbf{n}^\prime} \equiv \omega_L + \omega_{\mbf{n}^\prime}^g -\omega_\mbf{n}^g$.
At $t \rightarrow \infty$ the sine term will produce a term proportional to the delta
function, describing the energy conservation in the process of a single photon scattering:
$\omega_\mbf{k} = \omega_{\mbf{k}_L} + \omega_{\mbf{n}^\prime}^g -\omega_\mbf{n}^g$. However, in our
case we are interested in the total number of photons scattered into a given solid angle,
and not in the spectrum of the scattered light. Hence, we use the approximation
$\omega^L_{\mbf{n} \mbf{n}^\prime} \approx \omega_L$. This condition is also applicable in the
physical systems where the natural linewidth $\Gamma$ associated with the atomic transition
is broader than frequencies of atom COM motion: $\Gamma \gg \omega_\mbf{n}^g$.
Now, by using Eq.~\eqref{eq:sola} and approximation
$\omega^L_{\mbf{n} \mbf{n}^\prime} \approx \omega_L$ we calculate the mean number
of photons with a wavevector $\mbf{k}$ and a polarization $\lambda$
\begin{align}
\label{MeanPhotonNumber}
\!\!\left\langle a^{\dagger}_{\mbf{k} \lambda} \left(t\right) a_{\mbf{k} \lambda} \left(t\right)
\right\rangle = \frac{\Omega^2 c_{\mbf{k} \lambda}^2}{\Delta^2} \frac{\sin^{2}
\left( \left( \omega_{\mbf{k}} - \omega_{L} \right) t/2 \right) }{\left( \omega_{\mbf{k}}
- \omega_{L} \right)^2} F(\mbf{k}, \mbf{k}_{L}),
\end{align}
where the function $F(\mbf{k}, \mbf{k}_{L})$ is defined as follows:
\begin{align}
F(\mbf{k}, \mbf{k}_{L}) \equiv & \sum_{\substack{ {\mbf{n}},{\mbf{n}}' \\ {\mbf{m}},{\mbf{m}}'}}
\bra{\mbf{n}} u_{\mbf{k}}^{\ast}({\mbf{r}}) u_{\mbf{k}_L}({\mbf{r}}) \ket{\mbf{n}'} \bra{\mbf{m}}
u_{\mbf{k}}(\mbf{r}) u_{\mbf{k}_L}^{\ast}(\mbf{r}) \ket{\mbf{m}'} \nonumber \\
& \times \aver{g_{\mbf{n}}^{\dagger}(0) g_{\mbf{n}'}(0) g_{\mbf{m}}^{\dagger}(0) g_{\mbf{m}'}(0)}.
\label{eqn:Fq}
\end{align}
Notice that, in Eq.~\eqref{eqn:Fq} all the
matrix elements are calculated between COM states of ground-state atoms
$|\mbf{n},g \rangle$ and to shorten the notation $|\mbf{n}\rangle \equiv |\mbf{n},g\rangle$.
In the particular case when the mode functions $u_{\mbf{k}}(\mbf{r})$
and $u_{\mbf{k}_{L}}(\mbf{r})$ are the plane waves, $F(\mbf{k}, \mbf{k}_{L})$
reduces to the Fourier transform of the second-order correlation function
in atomic field operators $\hat{\Psi}_{g}(\mbf{x})$ of the atoms in the electronic ground state
\begin{align}
F(\mbf{q}) = \int \!\!d^{3}x \int \!\!d^{3}y\, e^{\imath \mbf{q}(\mbf{x}-\mbf{y})}
\left\langle \hat{\Psi}_{g}^{\dagger}(\mbf{x}) \hat{\Psi}_{g}(\mbf{x})
\hat{\Psi}_{g}^{\dagger}(\mbf{y}) \hat{\Psi}_{g}(\mbf{y}) \right\rangle
\end{align}
where $\mbf{q} = \mbf{k} - \mbf{k}_L$ is the wave vector of the momentum transfer.
In the rest of the paper we will use the $F(\mbf{q})$ function only.
An analogous result is obtained when considering the scattering of neutrons from liquid
helium \cite{vanHove}. In that case the number of scattered particles associated with
the momentum transfer $\mbf{q}$ and the energy transfer to the system $\hbar \omega$ is
described by the dynamic structure factor
\begin{multline}
S(\mbf{q},\omega) \equiv
\frac{1}{N} \int \!\!d^{3}x \!\! \int \!\!d^{3}y \,\, e^{\imath \mbf{q}(\mbf{x}-\mbf{y})} \\
\times \langle \Psi_E|
\hat{\rho}(\mbf{x}) \delta (H - E -\hbar \omega) \hat{\rho}(\mbf{x}) | \Psi_E \rangle
\end{multline}
where $\hat{\rho}(\mbf{x}) = \hat{\Psi}^{\dagger}(\mbf{x}) \hat{\Psi}(\mbf{x})$,
$H$ is the Hamiltonian of the system, and $|\Psi_E\rangle$ is an eigenstate with energy $E$.
By integrating over energies of the scattered particles one obtains the static structure factor
\begin{equation}
S(\mbf{q}) = \hbar \int_{-\infty}^{\infty} d\omega\, S(\mbf{q},\omega)
\end{equation}
which is equivalent to our function $F(\mbf{q})$ describing an amplitude of scattered
photons integrated over photon frequencies \cite{rem1}. We will refer to $F(\mbf{q})$
as the structure function in the rest of the paper.
For evolution time $t$ much longer than the time scale determined by
the optical frequencies $\omega_L$, we can apply the
following identity
\begin{equation}
\lim_{t \rightarrow \infty}
\frac{\sin^{2} \left( \left( \omega_{\mbf{k}} - \omega_{L} \right) t/2 \right)}
{t \left( \omega_{\mbf{k}} - \omega_{L} \right)^2}
= \frac{\pi}{2} \,\delta \left( \omega_{\mbf{k}} - \omega_{L} \right),
\label{lim}
\end{equation}
to show that for the weak and far-detuned laser the scattered light
described by Eq.~\eqref{MeanPhotonNumber} has spectrum centered around
elastic component.
The total number of photons scattered into a solid angle $d\Omega$ is equal to
\begin{equation}
\frac{d N_{ph}}{d \Omega} (\hat{\mbf{k}}) = \sum\limits_{\lambda} \int\!\! dk\; k^2
\aver{a^{\dagger}_{\mbf{k} \lambda} \left(t\right) a_{\mbf{k} \lambda} \left(t\right)}
\label{eqn:distrib}
\end{equation}
where $\hat{\mbf{k}}=\mbf{k}/|\mbf{k}|$ represents the direction of
measurement. Since, according to Eq.~\eqref{lim}, the number of photons
is proportional to pulse length as expected, it is more convenient to calculate
the number of photons scattered into $d\Omega$ per unit of time
\begin{align}
\frac{d^2 N_{ph}}{d\Omega dt} (\mbf{k},\mbf{k}_L) & = \frac{\Omega^2 \omega_{L}^{3} d^2}
{32 \pi^2 \Delta^2 \epsilon_{0} \hbar c^3} {\cal W}(\hat{\mbf{k}}) F(\mbf{k}-\mbf{k}_L)
\nonumber \\
& = \left[ \frac{d^2 N_{ph}}{d\Omega dt} (\mbf{k},\mbf{k}_L) \right]_{
\substack{\!\!\mathrm{one}\\\mathrm{atom}}} F(\mbf{q})
\label{eqn:scaling}
\end{align}
where ${\cal W}(\hat{\mbf{k}})= \left( 1-( \mbox{\boldmath$\epsilon$}_{\textbf{d}}
\cdot\mbox{\boldmath$\epsilon$}_{\textbf{k}})^2 \right)$ is the dipole pattern of the
emitted light and $\mbf{\mbox{\boldmath$\epsilon$}}_{\textbf{d}}$ is a unit vector in the
direction of the dipole moment $\textbf{d}$ that is determined by a
polarization of the probing laser. Eq.~\eqref{eqn:scaling}
shows that the angular distribution of the scattered light, apart from the
contribution from the dipole pattern, is determined only by $F(\mbf{q})$.
In addition, for a single atom $F(\mbf{q})~=~1 $ and thus all the information
about scattering from the system of $N$ atoms is contained in $F(\mbf{q})$.
Therefore, in the subsequent sections, we can focus solely on the properties
of $F(\mbf{q})$, keeping in mind that the remaining contribution is the same
as for the scattering from a single atom.
The generalization of our derivation to the case of interacting atoms can be
performed in an analogy to the problem of neutron scattering from liquid helium
\cite{vanHove,Pitaevskii}. If one applies the Born approximation, and eliminates
adiabatically the excited state, one ends up with the result identical to the one
presented here. A similar approach has been applied to the study of Raman scattering
in the superradiant regime \cite{uys:2008}.
Finally we note that our perturbative treatment neglects the effects of
the momentum transfer resulting from the photon recoil in the process of
light scattering. We assume, however, that the scattered light is weak and
far-detuned, therefore we expect that the fraction of atoms which experience
the photon recoil is sufficiently small, such that the atom statistics is not
significantly affected. Moreover, in the presence of a tight trapping potential,
such as a deep optical lattice, one finds that the scattering is recoilless
\cite{gajda:1996}, which requires the trap size smaller than a wave length of
the scattered light.
\section{Scattering from ultracold gas of bosons in an optical lattice}
\label{sec:opt-latt}
\begin{figure}[b]
\includegraphics[scale=0.8]{1.eps}
\caption{(Color online) Setup. An ultracold gas of bosons confined in an
optical lattice is illuminated with a probing laser beam (yellow arrow)
characterized by the wavevector $\mbf{k}_{L}$. The photons scattered into
a selected direction (green arrow) are collected by a detector.}
\label{fig:setup}
\end{figure}
The setup we consider is schematically plotted in Fig.~\ref{fig:setup}.
It consists of an ultracold gas of $N$ bosons confined in an optical cubic
lattice of $M$ sites. We assume a homogeneous system with an equal average
number of atoms $n = N/M$ in each site of a lattice. The periodic potential
of the lattice reads \cite{Bloch:2008}:
\begin{equation}
V_{p}(x,y,z) = V_{0} \left( \sin^2 k_{p} x + \sin^2 k_{p} y
+ \sin^2 k_{p} z \right)
\label{eqn:lattPotential}
\end{equation}
where $\mbf{k}_{p}$ is a wave vector of laser beams that are used to form the
lattice and $V_{0}$ is the potential depth. The exact configuration of the
probing beam and detectors will depend on a dimensionality of the lattice and
will be discussed later. In order to use the results of the previous section,
we need to specify a single-particle basis. In the case of atoms confined in
an optical lattice it is convenient to choose the basis of Wannier functions
$w_\mbf{m}(\mbf{r})$ that represent wave functions localized at single lattice
sites $\mbf{m}$ and are linear combinations of Bloch states. In our approach we
consider only excitations within the lowest Bloch band, so in a limit of deep
optical lattices the Wannier functions describe only the ground state wave
functions in local potential wells.
\subsection{Deep lattice regime}
For a deep optical lattice, the Wannier states are well localized within
the sites of the lattice, and in equation (\ref{eqn:Fq}) we can restrict to
optical transitions between the states localized at the same lattice sites:
$\mbf{n} = \mbf{n}'$ and $\mbf{m} = \mbf{m}'$. In this case
$F(\mbf{q})$ simplifies to the following expression
\begin{align}
F(\mbf{q}) &= \sum\limits_{\mbf{n},\mbf{m}} \bra{\mbf{n}} e^{\imath \mbf{q} \mbf{r}}
\ket{\mbf{n}} \bra{\mbf{m}} e^{-\imath \mbf{q} \mbf{r}} \ket{\mbf{m}}
\aver{g_{\mbf{n}}^{\dagger} g_{\mbf{n}} g_{\mbf{m}}^{\dagger} g_{\mbf{m}}} \nonumber \\
&= \left| f_{\textbf{0},\textbf{0}} (\mbf{q}) \right|^2 \sum\limits_{\mbf{n},\mbf{m}}
e^{ \imath \mbf{q} \left( \mbf{r}_{\mbf{n}} - \mbf{r}_{\mbf{m}} \right) }
\aver{n_{\mbf{n}} n_{\mbf{m}}}
\label{eqn:FdeepLatt}
\end{align}
where $n_{\mbf{m}} \equiv g_{\mbf{m}}^{\dagger} g_{\mbf{m}}$ and
\begin{equation}
f_{\mbf{n}, \mbf{m}}(\mbf{q}) \equiv \bra{\mbf{n}} e^{\imath \mbf{q} \mbf{r}}
\ket{\mbf{m}} = \int d^{3}r \,w^{\ast}_{\mbf{n}}(\mbf{r}) e^{\imath \mbf{q} \mbf{r}}
w_{\mbf{m}}(\mbf{r}).
\end{equation}
In analogy to the scattering of light into an optical cavity
\cite{Mekhov:2007b}, we can define the classical part $F^{clas}(\mbf{q})$
and the quantum part $F^{quant}(\mbf{q})$ of the function $F(\mbf{q})$
\begin{align}
F^{clas}(\mbf{q}) & \equiv n^2 \left| f_{\textbf{0},\textbf{0}} (\mbf{q}) \right|^2
\left| \sum\limits_{\mbf{m}} e^{\imath \mbf{q} \mbf{r}_{\mbf{m}}} \right|^{2},
\label{eqn:FClasDef} \\
F^{quant}(\mbf{q}) & \equiv F(\mbf{q}) - F^{clas}(\mbf{q}) \nonumber \\
& = \left| f_{\textbf{0},\textbf{0}} (\mbf{q}) \right|^2 \sum\limits_{\mbf{n},\mbf{m}}
e^{ \imath \mbf{q} \left( \mbf{r}_{\mbf{n}} - \mbf{r}_{\mbf{m}} \right) }
\left( \aver{n_{\mbf{n}} n_{\mbf{m}}}- n^2 \right).
\label{eqn:FQuantDef}
\end{align}
The former yields the classical amplitude of the scattered light
$\left| \aver{a_{\mbf{k} \lambda}} \right|^{2}$, whereas the latter
represents the remaining quantum contribution that together with $F^{clas}(\mbf{q})$
sum up to the total number of photons $\aver{a_{\mbf{k} \lambda}^\dagger a_{\mbf{k} \lambda}}$.
We note that $F^{clas}(\mbf{q})$ has a form characteristic for a Bragg scattering
and it is not affected by any statistical properties of the ultracold gas of bosons.
On the contrary, $F^{quant}(\mbf{q})$ is sensitive to the atom number statistics and
thus enables us to investigate statistical properties of different quantum states.
\section{Scattering from a Bose gas in one-dimensional optical lattice at zero temperature}
\label{sec:oneDimTzeroScatt}
The geometry of the system we investigate in this section is depicted in
Fig. \ref{fig:System}. We consider one-dimensional homogeneous optical
lattice generated by two overlapping and counterpropagating laser beams
characterized by the wavelength $\lambda_p$. Atoms confined to the periodic
potential are illuminated with a single laser beam with the wavelength $\lambda_{L}$.
For the single particle basis that we have chosen the states
\begin{equation}
\psi_{m}(\mbf{r}) = w_{m}(z) \,\psi_{\scriptscriptstyle{\perp}}(x,y)
\end{equation}
are products of a Wannier function $w_{m}(z)$ localized at lattice site $m$
along $z$-direction, and a Gaussian function $\psi_{\scriptscriptstyle{\perp}}(x,y)$
in tightly confined, perpendicular direction. For simplicity we assume the cylindrical
symmetry $\psi_{\scriptscriptstyle{\perp}}(x,y) = \psi_{\scriptscriptstyle{\perp}}(\rho)$.
\begin{figure}[b]
\centering
\includegraphics[scale=0.803]{2.eps}
\caption{(Color online) Setup. A quasi one-dimensional optical lattice is
illuminated with a probing laser set at an angle $\alpha\,\epsilon\,[-\pi,\pi[$.
A detector is set at an angle $\beta\,\epsilon\,[-\pi,\pi[$.}
\label{fig:System}
\end{figure}
At zero temparature a gas of bosons in a periodic potential appears in two distinct
quantum phases \cite{fisher:1989,jaksch:1998}. When the tunneling process dominates
over the on-site atom repulsion the system is found in the superfluid (SF) phase that
is characterized by the presence of a global coherence and a non-zero order parameter.
In contrast, for the on-site interactions stronger than the tunneling rate, the system
exhibits the Mott-insulator phase. In the latter case the global coherence is lost,
while the on-site particle number is fixed and the on-site fluctuations are suppressed.
For a Bose gas at zero temperature and deep in the MI regime, the on-site fluctuations
and correlations vanish: $\aver{n_m n_{m'}} - \aver{n_m}\aver{n_{m'}}=0$. Hence, the
quantum part $F^{quant}(\mbf{q})$ is zero identically, and the scattering is described
by the standard Bragg pattern with characteristic set of maxima and minima, corresponding
to the directions of constructive and destructive interference. In contrast, SF phase at
$T=0$ exhibits nonzero fluctuations and correlations:
$\aver{n_m n_{m'}} - \aver{n_m}\aver{n_{m'}} = n \delta_{m m'}- \frac{n^2}{N}$. Hence,
apart from the similar behavior of the classical part $F^{clas}(\mbf{q})$ as for MI phase,
the SF phase also gives rise to nonzero quantum component $F^{quant}(\mbf{q})$ which, within
the deep lattice approximation (Eq.~\eqref{eqn:FdeepLatt}), is given by
$F^{quant}(\mbf{q}) = N \left| f_{0,0} (\mbf{q}) \right|^2 \label{eqn:FSF0rd}$.
This offers a unique possibility of a non-destructive measurement that allows one
to distinguish between SF and MI phases \cite{Mekhov:2007}.
Fig.~\ref{fig:DiffA} and Fig.~\ref{fig:DiffL} compare the scattering patterns from
the SF and MI phases for the systems of $M=55$ sites with different configurations
of the probing laser and different ratios of $\lambda_p$ to $\lambda_{L}$. At some
characteristic angles corresponding to the Bragg scattering minima due to the
destructive interference, the scattering from the MI state vanishes. In contrast,
the scattering pattern from the SF state is nonzero at all angles, also in the
directions where the classical component vanishes. We observe that a change of a
ratio $\lambda_{p}/\lambda_{L}$ affects the scattering pattern, in particular a
number and positions of the highest peaks resulting from the constructive interference.
\begin{figure}[b]
\centering
\subfigure[\:$\alpha=\pi/4$]{
\includegraphics[scale=0.39]{3a.eps}}
\subfigure[\:$\alpha=\pi/2$]{
\includegraphics[scale=0.39]{3b.eps}}
\caption{(Color online) Structure function $F(\mbf{q})$ for SF (blue top curve) and
MI (gray bottom curve) phases, in the deep lattice approximation, for a probing
laser set at different angles $\alpha$, and a detector set at angle $\beta$. Here,
$V_0 = 15 E_r$, $M=55$, $N=3 M$, $\lambda_{p}/ \lambda_{L}=1$. The black line
represents average distribution (\ref{eqn:FMI0rdAvg}).}
\label{fig:DiffA}
\centering
\subfigure[\:$\lambda_{p}/ \lambda_{L}=2$]{
\includegraphics[scale=0.39]{4a.eps}}
\subfigure[\:$\lambda_{p}/ \lambda_{L}=1/3$]{
\includegraphics[scale=0.39]{4b.eps}}
\caption{(Color online) Structure function $F(\mbf{q})$ for SF (blue top curve)
and MI (gray bottom curve) phases, in the deep lattice approximation, for
different ratios of $\lambda_{p} /\lambda_{L}$, and a detector set at angle $\beta$.
Here, $V_0 = 10 E_r$, $M=55$, $N=3 M$, $\alpha=\pi/8$. The black line represents the
average distribution (\ref{eqn:FMI0rdAvg}).}
\label{fig:DiffL}
\end{figure}
We note that for large $M$ the scattering pattern quickly oscillates and thus,
in the realistic measurement, one would detect photons scattered in some finite
solid angle $d \Omega$ which is characteristic for the detector and that
contains several interference fringes. Hence, we find it more appropriate
to calculate the angular distribution of photons that is averaged over few
neighboring maxima. The averaging does not affect the scattering pattern of
SF phase, which is rather smooth, but it is important for MI phase. In 1D
optical lattice, the angular distribution of photons scattered from MI state
is determined by
\begin{align}
F^{MI}(\mbf{q}) = \left| f_{0,0} (\mbf{q}) \right|^2 n^2 \frac{\sin^2
\left(\frac M 2 \mbf{q}\mbf{d} \right)}{\sin^2\left(\frac12 \mbf{q}\mbf{d} \right)}
\label{eqn:FMI0rd}
\end{align}
where $\mbf{d}$ denotes the translation vector of a 1D lattice. Averaging over
some finite solid angle around $\mbf{q}$ containing several maxima yields
\begin{align}
\overline{F^{MI}(\mbf{q})} = \left| f_{0,0} (\mbf{q}) \right|^2
\frac{n^2}{2} \frac{1}{\sin^2\left(\frac12 \mbf{q}\mbf{d} \right)}.
\label{eqn:FMI0rdAvg}
\end{align}
The above result is derived provided that the measurement is done not too close
to the main maxima determined by the directions of the constructive interference.
As can be observed in Fig.~\ref{fig:DiffA} and Fig.~\ref{fig:DiffL}, the
averaged distribution of the light scattered from MI phase still can be well
distinguished from the scattering from the SF phase, and result \eqref{eqn:FMI0rdAvg}
for the averaged distribution remains approximately valid even close to the points of
the destructive interference.
While performing the deep lattice approximation in Eq.~(\ref{eqn:FdeepLatt}) we
have dropped all the Franck-Condon factors corresponding to transitions between states
localized in different lattice sites. Obviously, with the decreasing lattice depth, the
Wannier states begin to overlap between neighboring sites and we expect the nonlocal
Franck-Condon factors to give larger contribution. In order to investigate this issue
in Fig.~\ref{fig:poprawki} we compare the deep lattice approximation \eqref{eqn:FdeepLatt}
with the result that includes summation over all pairs of the lattice sites in
Eq.~\eqref{eqn:Fq}. For the clarity of presentation we show the results for a relatively
small system of $M=11$ sites and a lattice depth $V=1 E_r$ and $V=5 E_r$ for SF and MI
phases, respectively, expressed in the units of the recoil energy $E_r = \hbar^2 k_L^2/(2 m)$.
We observe that even for the shallow lattice potential the nonlocal corrections give
negligible contribution for the scattering from the SF state. Moreover, for the MI state
the nonlocal corrections are even smaller because of the deeper lattice required to achieve
this phase. In Appendix~\ref{app:nonlocal} we show that corrections due to the nearest
neighbors in weak lattices are isotropic and scale as the total number of atoms $N$.
Therefore, nonlocal corrections give rise to a scattering at the angles of destructive
interference of the classical part. However, for typical lattice depths, the corresponding
contribution is small and can be totally neglected for both quantum phases.
\begin{figure}[t]
\centering
\subfigure[\hspace{0.07cm} SF scattering at $V_0=1E_r$]{{
\label{fig:poprawkiSF}}{{\includegraphics[scale=0.39]{5a.eps}}}}
\subfigure[\hspace{0.07cm} MI scattering at $V_0=5E_r$]{{
\label{fig:poprawkiMI}}{{\includegraphics[scale=0.39]{5b.eps}}}}
\caption{
(Color online) Distribution of light scattered from SF (left panel) and MI
(right panel) phases at zero temperature, versus the angle $\beta$. Here,
$M=11$, $N=3 M$, $\alpha = \pi/2$, and $\lambda_{p}/ \lambda_{L}=1$. The
red curves present $F(\beta)$ calculated in local approximation
\eqref{eqn:FdeepLatt}, whereas the blue curve (in the SF case) and the gray
curve (in the MI case) show results including also non-local Franck-Condon
factors as in Eq.~\eqref{eqn:Fq}. In the case of scattering from the SF
state at $V_0=1 E_r$ one notices slight differences between the two curves
in a vicinity of $\beta=0$ and $\beta= \pm \pi$. In the case of scattering
from the MI state at $V_0=5 E_{r}$ the two curves are indistinguishable.}
\label{fig:poprawki}
\end{figure}
\section{Statistical properties at finite temperatures}
\label{sec:statProperties}
\subsection{Bose-Hubbard model of an ultracold gas in a periodic potential}
As discussed in the previous section, the angular distribution of the
scattered light is determined by the occupation number statistics in
lattice sites. We investigate the occupation number statistics within
Bose-Hubbard (BH) model \cite{fisher:1989,jaksch:1998}, considering only
excitations within the lowest Bloch band. The BH Hamiltonian reads:
\begin{equation}
\mathcal{H} = -J \!\!\!\!\sum\limits_{\aver{\mbf{m},\mbf{m}'}} g^{\dagger}_{\mbf{m}} g_{\mbf{m}'}
+\frac{1}{2}U\!\sum\limits_{\mbf{m}} \hat{n}_{\mbf{m}} \left( \hat{n}_{\mbf{m}}-1 \right)
\label{eqn:BHHam}
\end{equation}
where the first sum on the right-hand side is restricted to nearest
neighbors only. The parameter
\begin{equation}
J \equiv -\int d^{3}r \:w^{\ast}_{\mbf{m}} (\mbf{r})\left[ -\frac{\hbar^2}{2m}\nabla^{2}+
V_{p} (\mbf{r}) \right] w_{\mbf{m}'}(\mbf{r})
\end{equation}
is the hopping matrix element between neighboring sites $\mbf{m}$
and $\mbf{m}'$, and parameter:
\begin{equation}
U \equiv \frac{4 \pi a_s \hbar^2}{m} \int d^{3}r \left| w_{\mbf{m}}(\mbf{r}) \right|^4
\end{equation}
corresponds to the strength of the on site repulsion of two atoms in
a lattice site $\mbf{m}$. As before, $w_{\mbf{m}}(\mbf{r})$ is the
single-particle Wannier's wavefunction of an atom occupying site $\mbf{m}$
in a lattice.
The BH Hamiltonian can be equivalently expressed in the momentum space,
which is convenient for the analysis of the SF phase at finite
temperatures and application of the Bogoliubov method. To this end
we introduce annihilation and creation operators in momentum space
\begin{align}
a_{\mbf{k}} = &\frac{1}{\sqrt{M}} \sum\limits_{\mbf{m}} g_{\mbf{m}}
e^{\imath \mbf{k} \mbf{r}_{\mbf{m}}} \label{eqn:anniFour}, \\
a^{\dagger}_{\mbf{k}} = &\frac{1}{\sqrt{M}} \sum\limits_{\mbf{m}}
g^{\dagger}_{\mbf{m}} e^{-\imath \mbf{k} \mbf{r}_{\mbf{m}}},
\label{eqn:creatFour}
\end{align}
respectively, in which index $\mbf{m}$ runs over all sites in a lattice.
A period of the cubic lattice is $d=\lambda_{p}/2$ and a size of the
system is equal to $L=M^{1/3} d$. The periodic boundary conditions imply
quantization of a wave vector: $\; \mbf{k} = \frac{2 \pi}{L} (n_x,n_y,n_z)$,
where $n_i$ are integer numbers ranging from $-\lfloor M/2 \rfloor$ to
$\lfloor M/2 \rfloor$ \cite{rem2}. By rewriting Eq.~(\ref{eqn:BHHam}) in terms of
$a_{\mbf{k}}$ and $a^{\dagger}_{\mbf{k}}$, we find:
\begin{align}
\mathcal{H} =& \sum\limits_{\mbf{k}} \epsilon_{\mbf{k}} a^{\dagger}_{\mbf{k}} a_{\mbf{k}}
+ \frac{U}{2 M} \sum\limits_{\mbf{k}, \mbf{k}', \mbf{k}''} a^{\dagger}_{\mbf{k} +
\mbf{k}''} a^{\dagger}_{\mbf{k} '- \mbf{k}''} a_{\mbf{k} '} a_{\mbf{k}}
\label{eqn:BHHamM}
\end{align}
where
\begin{equation}
\epsilon_{\mbf{k}} \equiv 6 J - 2 J \sum\limits_{i=1}^{3} \cos\left( k_i d \right).
\end{equation}
\subsection{Statistical properties of the superfluid phase}
The standard description of a weakly interacting Bose gas is
based on the Bogoliubov approximation \cite{bogoliubov:1947}
and can be also applied for a superfluid phase in periodic
potentials \cite{stoof:2001}. We perform the Bogoliubov
approximation to Hamiltonian (\ref{eqn:BHHamM}), replacing
the annihilation and creation operators in the zero
quasi-momentum modes ($\mbf{k}=0$) by $\mathbb{C}$-numbers
$a_0 \approx a_0^\dagger \approx \sqrt{N_0}$. By introducing
the quasi-particle annihilation and creation operators
$b_{\mbf{k}}$ and $b^{\dagger}_{\mbf{k}}$, respectively,
which fulfill the standard bosonic commutation rules
$\left[b_{\mbf{k}}, b^{\dagger}_{\mbf{k}'}\right] = \delta_{\mbf{k},\mbf{k}'}$,
and are related to $a_{\mbf{k}}$ and $a^{\dagger}_{\mbf{k}}$
by the canonical transformation
\begin{equation}
\begin{pmatrix} b_{\mbf{k}}\\b^{\dagger}_{-\mbf{k}} \end{pmatrix} =
\begin{pmatrix} u_{\mbf{k}} & v_{\mbf{k}}\\v_{\mbf{k}} & u_{\mbf{k}}\end{pmatrix}
\begin{pmatrix} a_{\mbf{k}}\\a^{\dagger}_{-\mbf{k}} \end{pmatrix},
\label{eqn:botranf}
\end{equation}
we diagonalize Hamiltonian (\ref{eqn:BHHamM}) obtaining
\begin{equation}
\mathcal{H} = E_0 + \sum\limits_{\mbf{k}} \hbar \omega_{\mbf{k}}
b^{\dagger}_{\mbf{k}} b_{\mbf{k}}.
\label{eqn:BogHamDiag}
\end{equation}
Here, $E_0$ represents constant, ground-state energy term,
$\hbar \omega_{\mbf{k}}$ are the energies of the quasi-particle
excitation spectrum
\begin{align}
\hbar \omega_{\mbf{k}} = \sqrt{\epsilon_{\mbf{k}}^{2} +
2 U \frac{N_0}{M} \epsilon_{\mbf{k}}},
\end{align}
and real-valued coefficients $u_{\mbf{k}}$ and
$v_{\mbf{k}}$ of the Bogoliubov transformation are given by
\begin{equation}
v_{\mbf{k}}^{2} = u_{\mbf{k}}^{2} - 1 = \frac{1}{2}
\left( \frac{\epsilon_{\mbf{k}} +
U \frac{N_0}{M}}{\hbar \omega_{\mbf{k}}} -1 \right).
\end{equation}
We note, that the excitation spectrum $\hbar \omega_{\mbf{k}}$
depends on the condensate population $N_0$, which is known
in the literature as the Bogoliubov-Popov spectrum, and
is well suited to describe the statistics of a BEC at finite
temperatures \cite{Svidzynsky,Idziaszek3}.
Below the critical temperature, when the condensate is
macroscopically occupied, the occupation statistics of the
quasi-particle modes is given by the Bose-Einstein distribution:
\begin{align}
\label{eqn:fk}
\aver{b^{\dagger}_{\mbf{k}} b_{\mbf{k}}} = & \frac{1}{e^{\beta \hbar
\omega_{\mbf{k}}}-1} \equiv f_\mbf{k},\quad\mbf{k}\neq 0,\\
\label{eqn:fk2}
\aver{b^{\dagger}_{\mbf{k}} b_{\mbf{k}} b^{\dagger}_{\mbf{k}'} b_{\mbf{k}'}} = &
f_\mbf{k} f_{\mbf{k}'} + \delta_{\mbf{k},\mbf{k}'} \left( f_\mbf{k}^2 +
f_{\mbf{k}} \right), \quad\mbf{k},\mbf{k}'\neq 0,
\end{align}
where the value of the chemical potential $\mu$ is set to zero.
This follows from the fact the condensate acts as a reservoir
of particles, and distributions of particles in excited modes
are not restricted by the particle number conservation, which is
consistent with the so-called Maxwell-Demon (MD) ensemble approximation
\cite{navez:1997,grossman:1997,wilkens:1997}. Applying
Bogoliubov transformation \eqref{eqn:botranf}, and
Eqs.~(\ref{eqn:fk}) and (\ref{eqn:fk2}) we can easily find
the mean occupation, fluctuations and correlations of the
number of atoms in the quantized quasi-momentum modes:
\begin{align}
\aver{n_{\mbf{k}}} =& (u_\mbf{k}^2 + v_\mbf{k}^2) f_{\mbf{k}} + v_\mbf{k}^2,
\label{eqn:nk} \\
\aver{\delta^{2} n_{\mbf{k}}} =& \left( u_{\mbf{k}}^{2} + v_{\mbf{k}}^{2} \right)^2
\left( f_{\mbf{k}}^{2} + f_{\mbf{k}} \right) + u_{\mbf{k}}^2 v_{\mbf{k}}^2, \\
\aver{n_{\mbf{k}} n_{\mbf{k}'}} = & u_{\mbf{k}}^{2} v_{\mbf{k}}^{2}
\left( 1 + 4 f_{\mbf{k}} + 4 f_{\mbf{k}}^{2} \right)
\delta_{\mbf{k},-\mbf{k}'} + \aver{n_{\mbf{k}}} \!\aver{n_{\mbf{k}'}}
\label{eqn:nknkp}
\end{align}
in which $\mbf{k} \neq \mbf{k}' \neq \mbf{0}$ and
$n_{\mbf{k}} \equiv a^{\dagger}_{\mbf{k}} a_{\mbf{k}}$
is a particle number operator for a quasi-momentum mode
$\mbf{k}$. Calculations of statistical quantities
\eqref{eqn:nk}-\eqref{eqn:nknkp} within the Bogoliubov-Popov
method require self-consistent determination of the mean
condensate population $N_0$. First, $N_0$ enters the
excitation spectrum as a parameter. Second, it is determined
by the statistics itself,
\begin{equation}
N_0 = N - \sum_{\mbf{k} \neq \mbf{0}} \aver{n_{\mbf{k}}},
\end{equation}
which yields
\begin{align}
N_0 = N - & \sum\limits_{\mbf{k} \neq 0} \left( \frac{\epsilon_{\mbf{k}}
+ U \frac{N_0}{M}}{\hbar \omega_{\mbf{k}}} f_{\mbf{k}} +
\frac{\epsilon_{\mbf{k}} + U \frac{N_0}{M} - \hbar \omega_{\mbf{k}}}
{2 \hbar \omega_{\mbf{k}}} \right).
\label{eqn:BOgN0}
\end{align}
\begin{figure}[t]
\centering
\subfigure[\:Number of condensate atoms in the lattice.]{{\label{fig:SFcomboStatisticsA}}{{
\includegraphics[scale=0.39]{6a.eps}}}}
\subfigure[\:Atoms number fluctuations in a single site of the lattice.]{{\label{fig:SFcomboStatisticsB}}{{
\includegraphics[scale=0.38]{6b.eps}}}}\\
\subfigure[\:Correlations between close neighbours. Here, $\mbf{m} = (0,0,0)$ and
$\mbf{m}' =(1,0,0)$.]{{\label{fig:SFcomboStatisticsC}}{{
\includegraphics[scale=0.38]{6c.eps}}}}
\subfigure[\:Correlations between distant neighbours. Here, $\mbf{m} = (0,0,0)$ and
$\mbf{m}' =(5,5,5)$.]{{\label{fig:SFcomboStatisticsD}}{{
\includegraphics[scale=0.39]{6d.eps}}}} \\
\subfigure[\:Momemtum modes correlations. Here, $\mbf{k}=(2,2,2).$]{{\label{fig:SFmomSpaceStatisticsE}}{{
\includegraphics[scale=0.39]{6e.eps}}}}
\subfigure[\:Momemtum modes correlations. Here, $\mbf{k}=(1,0,0)$ and
$\mbf{k}' = (1,1,1).$]{{\label{fig:SFmomSpaceStatisticsF}}{{
\includegraphics[scale=0.39]{6f.eps}}}}
\caption{(Color online) Statistical properties of the SF phase in an optical
lattice. Here, $M=11 \!\times\! 11 \!\times\! 11$ and $N= 3 M$.
Different colors of the curves refer to different values of the
parameter $U$: blue (solid) for $U=0$, green (dashed) for $U=1$,
yellow (dot-dashed) for $U=4$, red (dotted) for $U=6$. The selected
values of $U$ imply the following values of quantum depletion:
0\%, 1.6\%, 6.8\%, 9.7\%, respectively. The strength of interaction
$U$ and temperature $T$ are expressed in units of~$J$.}
\label{fig:SFcomboStatistics}
\end{figure}
Finally, by transforming to the position space with the help of
Eqs.~(\ref{eqn:anniFour}) and (\ref{eqn:creatFour}), we evaluate
single-site occupation number statistics, i.e. single-site
fluctuations, $\aver{\delta^2 n_{\mbf{m}}} = \aver{n_{\mbf{m}}^2} - n^2$,
and correlations, $\aver{n_{\mbf{m}} n_{\mbf{m}'}} - n^2$, between each pair
of sites in the lattice. At the final stage they are substituted
into Eq.~(\ref{eqn:FdeepLatt}) determining the angular
distribution of the scattered light.
In Fig. \ref{fig:SFcomboStatistics} we present results
for the statistics of the SF state realized in an optical
lattice. We have chosen four different values of the strength of
interactions $U$ that correspond to quantum depletion
ranging from $0$ to approximately $0.1$, which should be proper in
the regime of a weakly interacting gas where the Bogoliubov
method is applicable. An upper limit of temperatures we
consider is established by the conditions of validity of the MD
ensemble approximation that, for sufficiently large systems, works
well up to a temperature close to the critical temperature $T_C$.
As expected, we observe that the on-site fluctuations
increase monotonically up to $T_C$. In contrast, the correlations
between populations of different sites exhibit non-monotonic behavior
that is strongly dependent on a distance between the considered sites.
The phenomena can be understood by studying the behavior of
these statistical quantities at small and at large temperatures.
Readily, in the limit $T \rightarrow 0$ the on-site fluctuations and
correlations follow the behavior presented in Section
\ref{sec:oneDimTzeroScatt}:
$\aver{\delta^2 n_{\mbf{m}}} = n - \frac{n^2}{N}$ and
$\aver{n_{\mbf{m}} n_{\mbf{m}'}} - n^2 = -\frac{n^2}{N}$.
On the contrary, at large temperatures the fluctuations
and correlations can be described consistently within a model
of $N$ indistinguishable particles distributed over $M$
degenerate levels: $\aver{\delta^2 n_{\mbf{m}}} = n^2 + n$
and $\aver{n_{\mbf{m}} n_{\mbf{m}'}} - n^2 =0$.
For both of the limits, the analytical expressions are derived
under the assumption $U/J \rightarrow 0$, however the
approximations work reasonably well also for finite values
of the ratio $U/J$. The last two panels of Fig. \ref{fig:SFcomboStatistics}
present correlations between different modes in the momentum space.
We note that the correlation between an excited and the condensate
mode $\aver{n_{\mbf{k}} n_{\mbf{0}}}$, exhibits a maximum at some
moderate temperature. This follows simply from the competition
between the process of thermal depletion of the condensate
and a growth of the thermal fraction. Similarly, in case of
correlations between two excited modes, we observe that at
some temperature the initial growth of $\aver{n_{\mbf{k}} n_{\mbf{k}'}}$
is suppressed by decrease in population of these modes in
favor of population of modes of some higher quasi-momentum.
\subsection{Statistical properties of the Mott-insulator phase}
We introduce grand canonical Bose-Hubbard Hamiltonian ${\cal K}$:
\begin{equation}
{\cal K} = -J \!\!\!\!\sum\limits_{\aver{\mbf{m},\mbf{m}'}}
g^{\dagger}_{\mbf{m}} g_{\mbf{m}'}
+\frac{1}{2}U\!\sum\limits_{\mbf{m}} \hat{n}_{\mbf{m}}
\left( \hat{n}_{\mbf{m}}-1 \right) - \mu \sum_{\mbf{m}} \hat{n}_{\mbf{m}}.
\label{eqn:GCbhHam}
\end{equation}
In order to describe quantum statistics of the Mott insulator phase at
finite temperatures we adopt a mean-field decoupling approximation
\cite{sachdev:1999,stoof:2003}. In analogy to the Bogoliubov approach,
we introduce a complex mean-field parameter $\psi \equiv \aver{g_m}$
that can be physically interpreted as an order parameter that is
nonzero if the system is superfluid. Below the phase transition
point, the symmetry related to the gauge invariance of the phase
is spontaneously broken and without losing generality we can assume
that $\psi$ is real. The new parameter allows one to decouple
the hopping term occurring in Eq.~(\ref{eqn:GCbhHam})
\begin{equation}
g^{\dagger}_{\mbf{m}} g_{\mbf{m}'} = \psi \left( g^{\dagger}_{\mbf{m}}
+ g_{\mbf{m}'} \right) - \psi^{2}.
\label{eqn:midec}
\end{equation}
By performing this substitution we can decompose Hamiltonian~(\ref{eqn:GCbhHam})
into a sum of mean-field local Hamiltonians ${\cal K}^{\scriptscriptstyle{MF}}_{\mbf{m}}$:
$\displaystyle {\cal K} = \sum_{\mbf{m}} {\cal K}^{\scriptscriptstyle{MF}}_{\mbf{m}}$,
where
\begin{align}
{\cal K}^{\scriptscriptstyle{MF}}_{\mbf{m}} \equiv & -2 D J \psi \left( g_{\mbf{m}}
+ g^{\dagger}_{\mbf{m}} \right) + 2 D J\psi^2 - \mu \hat{n}_{\mbf{m}} \nonumber \\
& +\frac{1}{2}U \hat{n}_{\mbf{m}} \left( \hat{n}_{\mbf{m}}-1 \right).
\label{eqn:hamMF}
\end{align}
At zero temperature, calculation of the ground-state energy and its
minimization as a function of the superfluid order parameter $\psi$
yields the phase diagram analytically \cite{stoof:2001}. However,
for non-zero temperatures the model has no analytical solution,
and one has to resort to numerical calculations. Namely, by
diagonalization of Eq.~(\ref{eqn:hamMF}) we calculate grand
canonical partition function $\mathcal{Z}(\psi)$,
\begin{equation}
\mathcal{Z}(\psi) = \textrm{Tr} \left\{ e^{-\beta
{\cal K}^{\scriptscriptstyle{MF}}_{\mbf{m}}} \right\},
\end{equation}
and on its grounds we determine the grand thermodynamic
potential $\Omega (\psi)$,
\begin{equation}
\Omega (\psi)= -\frac{1}{\beta} \ln \mathcal{Z}(\psi).
\end{equation}
Subsequently, by minimizing $\Omega(\psi)$ with respect
to $\psi$, we obtain the equilibrium value of the order parameter
that we use to calculate all the relevant thermodynamic
quantities. In particular:
\begin{align}
\aver{n_{\mbf{m}}(\mu)} =& \frac{1}{\mathcal{Z}} \textrm{Tr}
\left\{ \hat{n}_{\mbf{m}} e^{-\beta {\cal K}^{\scriptscriptstyle{MF}}_{\mbf{m}}} \right\}
\label{eqn:nAverMI}, \\
\aver{\delta^2 n_{\mbf{m}}(\mu)} =& \frac{1}{\mathcal{Z}} \textrm{Tr}
\left\{ \hat{n}^{2}_{\mbf{m}} e^{-\beta {\cal K}^{\scriptscriptstyle{MF}}_{\mbf{m}}} \right\}
- n^{2}.
\end{align}
In the homogeneous lattice $\aver{n_{\mbf{m}}(\mu)} = n$, and this
identity is used to determine the value of the chemical potential,
for a given single site occupation $n$.
Our mean-field approach assumes decoupling of
different sites, $\aver{n_{\mbf{m}} n_{\mbf{m}'}} - n^2 =0$ ($\mbf{m}
\neq \mbf{m}'$), that agrees with the zero temperature statistics of the
MI assumed in section IV. This, in general, is not valid at higher
temperatures when the hopping between adjacent sites
in non-negligible. Nevertheless, it is satisfied at
smaller temperatures considered here.
In Fig. \ref{fig:statsMI} we present the mean single-site
occupation and fluctuations as a function of the chemical potential,
and for different system temperatures. The calculations have been performed
within the mean-field model for $U/J = 128$. One can see the existence of a
characteristic temperature above which, the flat steps in
$\aver{n_{\mbf{m}}(\mu)}$ disappear completely, and the curve
becomes monotonically increasing. This crossover is accompanied
by an appearance of nonzero fluctuations for
all values of $\mu$ presented in the plot. This corresponds to
the transition from the MI to the normal phase. In MI phase the
system is infinitely compressible: $\partial \aver{n}/\partial \mu = 0$,
while in the normal or SF phase the compressibility becomes finite:
$\partial \aver{n}/\partial \mu \neq 0$. Since the on-site fluctuations can be
expressed as $\langle \delta^2 n \rangle = \partial \aver{n}/\partial(\beta\mu)$,
therefore they can be nonzero only in the normal or SF phase. In order to
distinguish between the normal and SF phases one can resort to the
value of the order parameter $\psi$. Finally, we note that our
mean-field treatment neglects the effects of
correlations between different sites and the quantum fluctuations. In the more
accurate models that take these effects into account, the on-site fluctuations
become nonzero already in the MI regime, close to the boundaries with the SF or
normal phases.
\begin{figure}[t]
\centering
\subfigure[\:Average atoms number in a single site in the lattice.]{
\includegraphics[scale=0.376]{7a.eps}}
\subfigure[\:Atoms number fluctuations in a single site in the lattice.]{
\includegraphics[scale=0.39]{7b.eps}}
\caption{(Color online) Statistical properties of Mott insulator phase
for $U=128$. Colors of the curves refer to different values of
temperature being considered: blue (dotted) for $T=0$,
dark green (dot-dashed) for $T=4$, green (dashed) for $T=6$,
orange (long-dashed) for $T=8$ and red (solid) for $T=11$.
All the values of parameters are expressed in units of $J$.}
\label{fig:statsMI}
\end{figure}
\section{Finite temperature scattering in three dimensions}
\label{sec:3D}
\begin{figure}[b]
\centering
\includegraphics[scale=0.41]{8.eps}
\caption{(Color online) Setup. A three-dimensional optical cubic lattice
generated by lasers $\lambda_p$ (red arrows) is illuminated by a probing
laser $\lambda_{L}$ (yellow arrow) set at angles $({\phi}_L, {\theta}_L)$.
A detector of scattered photons (green arrow) is aligned in a direction
$({\phi}_d, {\theta}_d)$.}
\label{fig:3dsystem}
\end{figure}
We consider three-dimensional cubic lattice and assume a sufficiently
large value of the trapping potential depth $V_0$ to neglect
corrections from the nonlocal Franck-Condon factors. The setup
is depicted in Fig.~\ref{fig:3dsystem}. The lasers creating an optical
lattice ($\lambda_p$, red arrows) are set along $x,y$ and $z$ axes.
In general, the position of a probing laser ($\lambda_{L}$, yellow arrow,
characterized by angles $\phi_L$, $\theta_L$) and detector (green arrow,
characterized by angles $\phi_d$, $\theta_d$) can be optimized in order to
minimize the contribution from the classical component in the vicinity of the
direction of the measurement, cf. Appendix \ref{app:optimization}. This is of
particular importance in case of large lattices for which, due to a big number
of interference fringes, the detector would collect the photons from several
interference peaks. Here, though, we do not choose the optimal configuration,
we consider some example geometry which not only sufficiently reduces an
influence of the classical component but also offers relatively simple
experimental realization. Namely, we choose the direction of the probing
beam along one of the diagonals of the lattice cube,
$\mbf{k}_L = |\mbf{k}_L| (\frac{1}{\sqrt{3}},-\frac{1}{\sqrt{3}},
\frac{1}{\sqrt{3}})$, and a detector centered around $\mbf{k} =
|\mbf{k}_L| (\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0)$.
\begin{figure}[b]
\centering
\includegraphics[scale=0.78]{9.eps}
\caption{(Color online) Logarithm of the structure function $F(\mbf{q})$
for MI phase at zero temperature, as a function of spherical angles of
detection, $(\theta,\phi)$. The probing laser is set at $({\phi}_L, {\theta}_L)$. Bright regions
correspond to directions in which a large number of photons is scattered.
The yellow circle (pointed by the yellow arrow) indicates the direction of
the probing laser (global maximum of number of scattered photons). The green
circle (pointed by the green arrow) refers to the direction of a detection
$\mbf{k} = |\mbf{k}_L| (\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0)$ which
is discussed in details in the text. Here, $M=55$, $N=3 M$ and $V_0 = 20
E_{r}$.}
\label{fig:T0miPhaseSpace}
\end{figure}
In analogy to the one-dimensional case, we expect that sharp
differences in the intensity of light scattered from the SF and
MI phases can be observed at angles for which the classical
component $F^{clas}(\phi, \theta)$ is negligible. In Fig.~\ref{fig:T0miPhaseSpace}
we present the zero-temperature structure function $F(\phi,\theta)$ for MI
phase that is equivalent to the $F^{clas}(\phi, \theta)$. Keeping in mind
that the quantum component $F^{quant}(\phi, \theta)$ for the SF
state is slowly varying and of order of $N$, we observe that
$({\phi}_d, {\theta}_d)$ is indeed a promising direction for a
measurement that can distinguish the two quantum phases. In
Fig.~\ref{fig:T0compareMISF} we corroborate this observation by
presenting scattering patterns for the superfluid $F^{SF}(\phi,\theta)$
and Mott-insulator $F^{MI}(\phi,\theta)$ phases at $T=0$. The
plots show cross-sections of $F(\phi, \theta)$ along the planes
of constant $\theta$ and $\phi$, respectively. Evidently, for the
specific values of parameters we have chosen and for the assumed
directions of the probing laser and of the detector, the difference
of number of photons scattered from the SF and MI phases is of
order $N \sim 10^5$ and thus should be readily measurable
in experiment.
\begin{figure}[t]
\centering
\subfigure[\: $F(\phi, \theta = \theta_{d})$]
{\includegraphics[scale=0.40]{10a.eps}}
\subfigure[\: $F(\phi = \phi_{d}, \theta)$]
{\includegraphics[scale=0.40]{10b.eps}}
\caption{(Color online) Zero-temperature $F^{SF}(\theta, \phi)$ (top blue
curves) and $F^{MI}(\theta, \phi)$ (bottom gray curves) with domains
restricted to $\phi$ (left figure) and $\theta$ (right figure). The
position of the photons detector $({\phi}_d, {\theta}_d)$ is indicated
with vertical dashed lines. Here,
$M=55 \!\times\! 55 \!\times\! 55$, $N= 3 M$, $\lambda_{p} / \lambda_{L}=1$
and $V_0=20 E_{r}$.}
\label{fig:T0compareMISF}
\end{figure}
We turn now to the thermal effects and their influence on the angular
distribution of scattered photons. Analyzing Fig.~\ref{fig:FinTsf} and
Fig.~\ref{fig:FinTmi} we observe the isotropic and monotonic growth of
the intensity of scattered light with temperature, for both SF and MI
phases. In the case of SF phase, this behavior can be explained on the
grounds of the Eq.~(\ref{eqn:FdeepLatt}) rewritten in the momentum
representation by means of transformation (\ref{eqn:anniFour})-(\ref{eqn:creatFour}).
In particular, if we disregard anomalous averages while calculating
expectation values of the form
$\aver{a^{\dagger}_{\mbf{k}_1} a_{\mbf{k}_2} a^{\dagger}_{\mbf{k}_3} a_{\mbf{k}_4} }$,
i.e. if we perform the approximation
\begin{align}
\aver{a^{\dagger}_{\mbf{k}_1} a_{\mbf{k}_2} a^{\dagger}_{\mbf{k}_3} a_{\mbf{k}_4} } & \approx
\delta_{\mbf{k}_1, \mbf{k}_2} \delta_{\mbf{k}_3, \mbf{k}_4} \aver{a^{\dagger}_{\mbf{k}_1}
a_{\mbf{k}_1} a^{\dagger}_{\mbf{k}_2} a_{\mbf{k}_2}} \nonumber \\
& \phantom{\simeq} + \delta_{\mbf{k}_1, \mbf{k}_4} \delta_{\mbf{k}_2, \mbf{k}_3}
\aver{a^{\dagger}_{\mbf{k}_1} a_{\mbf{k}_2} a^{\dagger}_{\mbf{k}_2} a_{\mbf{k}_1}},
\end{align}
Eq.~(\ref{eqn:FdeepLatt}) can be rewritten as:
\begin{align}
F(\mbf{q}) =& \frac{1}{M^2} \left| f_{\textbf{0},\textbf{0}} (\mbf{q}) \right|^2 \!
\Biggl[ N(N-1) \left| \sum\limits_{\mbf{m}} e^{\imath \mbf{q} \mbf{r}_{\mbf{m}}} \right|^2
\nonumber \\
&+ \sum_{\mbf{k} \neq \mbf{k}'} \aver{n_{\mbf{k}} n_{\mbf{k}'}} \left|
\sum\limits_{\mbf{m}} e^{\imath \left(- \mbf{k} + \mbf{k}' + \mbf{q} \right)
\mbf{r}_{\mbf{m}}} \right|^2 \Biggr] \nonumber \\
&+ N \left| f_{\textbf{0},\textbf{0}} (\mbf{q}) \right|^2.
\label{eqn:FMomSpace}
\end{align}
When temperature increases, a number of particles occupying excited modes
grows (see Fig.~\ref{fig:SFcomboStatistics}), causing an increase in
correlations terms $\aver{n_{\mbf{k}} n_{\mbf{k}'}}$. In consequence, the
total intensity of the scattered light $F^{SF}(T)$ increases monotonically
with temperature in any direction of measurement. We note that some
correlations terms $\aver{n_{\mbf{k}} n_{\mbf{k}'}}$ start to decrease
above some characteristic temperature (see Fig.~\ref{fig:SFcomboStatistics}).
However this does not influence the total structure function $F^{SF}(T)$ that
grows monotonically with $T$.
\begin{figure}[t]
\centering
\includegraphics[scale=0.4]{11a.eps}
\includegraphics[scale=0.4]{11b.eps} \\
\includegraphics[scale=0.4]{11c.eps}
\includegraphics[scale=0.4]{11d.eps}
\caption{(Color online) Structure function $F(\phi, \theta)$ for SF phase
of bosons in a three-dimensional optical lattice. The preferred position of the
detector $({\phi}_d, {\theta}_d)$ is marked with vertical dashed lines. The plots
show the cross sections along the constant $\theta$ (left panels) and constant
$\phi$ (right panels). The calculations have been performed for $^{87}\textrm{Rb}$
atoms, $\lambda_{p} = \lambda_{L} = 850\textrm{nm}$, $M=11 \!\times\! 11 \!\times\! 11$
and $N= 3 M$. The upper panels show results for constant $V_0=6.80 E_r \hspace{0.07cm}
(U=4)$ and increasing value of temperature: $T=0, 3, 5, 7, 9, 11 $ (ordered from
the bottommost to the topmost curve) with $U$ and $T$ being expressed in units of
$J$. The bottom panels show results for constant temperature $T=10$ and increasing
value of trapping potential depth: $V_0 = 3.66 E_r \hspace{0.07cm} (U=1), V_0 = 5.15
E_r \hspace{0.07cm} (U=2), V_0 = 6.80 E_r \hspace{0.07cm} (U=4), V_0 = 7.82 E_r
\hspace{0.07cm} (U=6)$, ordered from the bottommost to the topmost curve. The insets
show the number of photons scattered in the direction of a detector
$({\phi}_d, {\theta}_d)$ versus temperature (top panel) and interaction strength
(bottom panel).}
\label{fig:FinTsf}
\end{figure}
Similarly, an increase in the interaction strength (an increase in the lattice
potential depth, equivalently) results in larger population of excited modes,
partially due to increase in the quantum depletion. This behavior leads to a
growth of the correlation terms in Eq.~(\ref{eqn:FMomSpace}) and again
to the monotonic increase in the full function $F^{SF}(U)$.
In the case of MI phase, presented in Fig.~\ref{fig:FinTmi}, the increase in
the number of scattered photons $F^{MI}(T)$ is fully determined by a
temperature-driven growth of single-site fluctuations that have been
presented in Fig.~\ref{fig:statsMI}.
\begin{figure}[t]
\centering
\includegraphics[scale=0.4]{12a.eps}
\includegraphics[scale=0.4]{12b.eps}
\caption{(Color online) Structure function $F(\phi, \theta)$ for MI phase of
bosons in a three-dimensional optical lattice. The preferred position of the
detector $({\phi}_d, {\theta}_d)$ is marked with vertical dashed lines. The
plots show the cross sections along the constant $\theta$ (left panels) and
constant $\phi$ (right panels). The calculations have been performed for
$^{87}\textrm{Rb}$ atoms, $\lambda_{p} = \lambda_{L} = 850\textrm{nm}$,
$M=55 \!\times\! 55 \!\times\! 55$, $N= 3 M$, $V_0 = 18.3 E_r \; (U=128)$,
$\mu = 320$ and temperatures: $T=0, 6, 8, 10, 11$ (ordered from the bottommost
to the topmost curve), with $U,T$ and $\mu$ being expressed in units of $J$.
The inset shows the number of photons scattered in the direction of a detector
$({\phi}_d, {\theta}_d)$ versus temperature.}
\label{fig:FinTmi}
\end{figure}
\section{Summary and conclusions}
\label{sec:conclusions}
We have investigated the scattering of a weak and far-detuned laser
light from a system of ultracold bosons in an optical lattice. We
have shown that the light scattering can be used as a probe of the
on-site quantum statistics, in particular fluctuations and correlations.
Calculating the statistics for the superfluid and Mott-insulator phases at
finite temperatures, we have determined the angular distributions of the
mean number of the scattered photons. The profiles of the scattered light
are fully determined by the on-site particle number fluctuations and
correlations and thus allow for an experimental verification of the
present theoretical models describing the statistics in ultracold gases.
For the 3D optical lattice we have determined the optimal geometry at
which the contribution from the Bragg scattering pattern is minimized.
We have shown that even at some non-optimal configurations, which can
be more accessible from the experimental point of view, this contribution
is sufficiently small and allows one to measure the effects of quantum
statistics. Our main conclusion is that by careful choice of the measurement
geometry one can distinguish between different phases, even at finite
temperatures, and observe the effects of the finite temperature statistics
of a Bose gas.
\begin{acknowledgments}
The authors acknowledge support of the Polish Government Research Grants
for years 2007-2009 (K. \L ., Z. I.) and for years 2007-2010 (M. T.).
\end{acknowledgments}
|
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{"url":"https:\/\/www.semanticscholar.org\/paper\/Duality-covariant-field-redefinitions-Baron\/2c197faaf9503b3d363a282264dcaf0054b831fa","text":"Duality covariant field redefinitions\n\n@article{Baron2022DualityCF,\ntitle={Duality covariant field redefinitions},\nauthor={Walter H. Baron},\njournal={Physical Review D},\nyear={2022}\n}\n\u2022 W. Baron\n\u2022 Published 31 December 2021\n\u2022 Physics\n\u2022 Physical Review D\nWe explore the role of the dilaton \ufb01eld on higher-derivative supergravity within the framework of Double Field Theory and use it to \ufb01x the Lorentz-noncovariant \ufb01eld rede\ufb01nitions connecting the metric and dilaton \ufb01elds with the duality multiplets.\n1 Citations\n\nThe $\\alpha'^2$ correction from double field theory\n\n\u2022 Physics\n\u2022 2022\nIt is known that the order \u03b1 \u2032 correction to the tree-level e\ufb00ective action for the bosonic and heterotic string can be described in the framework of Double Field Theory (DFT). Here we determine the\n\nReferences\n\nSHOWING 1-10 OF 39 REFERENCES\n\nT-duality and \u03b1\u2032-corrections\n\n\u2022 Physics\n\u2022 2015\nA bstractWe construct an O(d, d) invariant universal formulation of the first-order \u03b1\u2032-corrections of the string effective actions involving the dilaton, metric and two-form fields. Two free\n\nThe generalized Bergshoeff-de Roo identification. Part II\n\n\u2022 Mathematics\nJournal of High Energy Physics\n\u2022 2021\nWe recently introduced a T-duality covariant mechanism to compute all-order higher-derivative interactions in the heterotic string. Here we extend the formalism to account for a two-parameter family\n\nString theory at order \u03b1\u20322 and the generalized Bergshoeff-de Roo identification\n\n\u2022 Physics\nJournal of High Energy Physics\n\u2022 2021\nAbstract It has been shown by Marques and Nunez that the first \u03b1\u2032-correction to the bosonic and heterotic string can be captured in the O(D, D) covariant formalism of Double Field Theory via a\n\n\u03b1\u2032-corrections and their double formulation\n\n\u2022 Eric Lescano\n\u2022 Education\nJournal of Physics A: Mathematical and Theoretical\n\u2022 2021\nThe present notes are based on three lectures, each 90 min long, prepared for the school \u2018Integrability, Dualities and Deformations\u2019, that ran from 23 to 27 August 2021 in Santiago de Compostela and\n\nGeneral string cosmologies at order \u03b1\u20323\n\n\u2022 Physics, Mathematics\nPhysical Review D\n\u2022 2021\nWe compute the cosmological reduction of general string theories, including bosonic, heterotic and type II string theory to order \u03b1\u20323, i.e., with up to eight derivatives. To this end we refine\n\nNew non-perturbative de Sitter vacua in \u03b1\u2032-complete cosmology\n\n\u2022 Physics, Mathematics\nJournal of High Energy Physics\n\u2022 2021\nAbstract The \u03b1\u2032-complete cosmology developed by Hohm and Zwiebach classifies the O(d, d; \u211d) invariant theories involving metric, b-field and dilaton that only depend on time, to all orders in \u03b1\u2032.\n\nO(D,D)-covariant two-loop \u03b2-functions and Poisson-Lie T-duality\n\n\u2022 Mathematics, Physics\nJournal of High Energy Physics\n\u2022 2021\nAbstract We show that the one- and two-loop \u03b2-functions of the closed, bosonic string can be written in a manifestly O(D,D)-covariant form. Based on this result, we prove that1) Poisson-Lie\n\nQuantum Correction to Generalized T Dualities.\n\n\u2022 Mathematics\nPhysical review letters\n\u2022 2020\nIt is shown thatPoisson-Lie duality can be viewed as a map between solutions of the low-energy effective equations of string theory at the (super) gravity level, and this fact extends to the next order in \u03b1^{'} (two loops in \u03c3-model perturbation theory) provided that the map is corrected.\n\nSolution of the size and horizon problems from classical string geometry\n\n\u2022 Physics, Mathematics\n\u2022 2020\nIn a recent paper we developed a string cosmology background from classical string geometry. Here, we show that this background yields a solution to the size and horizon problems of Standard Big Bang\n\nGeneralized dualities and higher derivatives\n\n\u2022 Mathematics\n\u2022 2020\nGeneralized dualities had an intriguing incursion into Double Field Theory (DFT) in terms of local O(d, d) transformations. We review this idea and use the higher derivative formulation of DFT to","date":"2022-10-07 19:54:59","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\": 0.5665209889411926, \"perplexity\": 5241.08025330706}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-40\/segments\/1664030338244.64\/warc\/CC-MAIN-20221007175237-20221007205237-00255.warc.gz\"}"}
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\section{Introduction}
Moduli spaces of flat connections on punctured Riemann surfaces and their
quantisation are of interest to both mathematics and physics due to their rich
mathematical structure and their links with a variety of topics from geometry,
algebra and gauge theory. From the physics perspective, a major motivation for
their study is their role in Chern-Simons theory. Moduli spaces of flat
connections can be viewed as the gauge-invariant or reduced phase spaces of
Chern-Simons theories. Their quantisation is thus related to structures arising
in the quantisation of Chern-Simons theory such as quantum groups, aspects of
knot theory \cite{Witten:1989aa} and topological quantum field
theories. Another important feature of the theory is its relation to
three-dimensional gravity \cite{Achucarro:1986aa, Witten:1988aa,Witten:1989sx}.
The quantisation of moduli spaces of flat $G$-connections and their relation to
quantum Chern-Simons theory are well understood for the case of compact,
semisimple Lie groups $G$. In this setting, quantisation can be achieved via
many formalisms, and most of these formalisms involve the representation theory
of a quantum group, namely the $q$-deformed universal enveloping algebra
$U_q(\algebra{g})$ at a root of unity. In the case of non-compact non-semisimple Lie
groups, the quantisation proves more difficult. Although there are partial
results on the quantisation of these cases via analytic continuation
\cite{Witten:2011aa} and results for specific Lie groups \cite{Witten:1989ip,
Buffenoir:2002aa, Meusburger:2004aa, Meusburger:2010aa, Meusburger:2010bb},
there is currently no general framework to address this case.
From the viewpoint of Hamiltonian quantisation formalisms, these difficulties
are related to the fact that Chern-Simons theory and the associated moduli
spaces of flat connections can be viewed as constrained Hamiltonian systems. In
the non-compact setting, representation-theoretical complications lead to
difficulties in the implementation of the constraint operators in the quantum
theory. It therefore seems advisable to also consider other approaches to the
quantisation of moduli spaces of flat connections with non-compact gauge
groups. This includes in particular ``quantisation after constraint
implementation'' approaches, which proceed by first applying the Dirac gauge
fixing formalism to the classical theory and then quantising the resulting
Poisson structure. However, besides partial results
for $\SL(2,\mathbb{C})$-Chern-Simons theory \cite{Buffenoir:2003aa, Buffenoir:2005aa,
Noui:2005aa}, this avenue has not been pursued yet.
An independent mathematical motivation for investing gauge fixing procedures
related to moduli spaces of flat connections arises from Poisson geometry. Such
gauge fixing procedures can be interpreted as the Poisson counterpart of
symplectic reduction. Moduli spaces of flat connections played an important
role in many interesting developments in this subject such as Lie-group-valued
moment maps, see \cite{Alekseev:1998aa} and the references therein. Moreover,
the symplectic structure on the moduli space can be characterised in terms of
certain Poisson structures from the theory of Poisson-Lie groups. Gauge fixing
in this context has been shown to give rise to classical dynamical $r$-matrices
in some cases \cite{Feher:2001aa, Feher:2004aa}.
In this article, we undertake a systematic investigation of Dirac gauge fixing
for the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections on a Riemann surface $S_{g,n}$
of general genus $g$ and with $n\geq 2$ punctures. Our choice of the group
$\ensuremath{\ISO(2,1)}=\ensuremath{\SO_+(2,1)}\ltimes\mathbb R^3$ is motivated by the fact that it is an example
of a non-compact non-semisimple Lie group and that Chern-Simons theory with
gauge group $\ensuremath{\ISO(2,1)}$ is closely related to (2+1)-gravity
\cite{Achucarro:1986aa,Witten:1988aa}.
Our starting point is a description of the moduli space of flat
$\ensuremath{\ISO(2,1)}$-connections on $S_{g,n}$ in terms of a Poisson structure on the direct
product $\ensuremath{\mathcal{P}_{\mathrm{ext}}}=\ensuremath{\pogr^{n+2g}}$ due to Alekseev and Malkin
\cite{Alekseev:1995ab} and Fock and Rosly \cite{Fock:1998aa}. It is shown in
\cite{Alekseev:1995ab, Fock:1998aa} that this Poisson structure is given in
terms of certain Poisson structures related to Poisson-Lie groups and after
reduction induces the canonical Poisson structure on the moduli space. In the
case of the group $\ensuremath{\ISO(2,1)}$, this Poisson structure is given by the direct
product of $n$ copies of the dual Poisson-Lie structure on $\ensuremath{\ISO(2,1)}$ and $2g$
copies of the cotangent bundle Poisson structure for $\ensuremath{\SO_+(2,1)}$
\cite{Meusburger:2004aa}:
\begin{equation*}
\ensuremath{\mathcal{P}_{\mathrm{ext}}}=\underbrace{\ensuremath{\ISO(2,1)}^*\times\ldots\ensuremath{\ISO(2,1)}^*}_{n\;\times}\times \underbrace{T^*(\ensuremath{\SO_+(2,1)})\times\ldots\times T^*(\ensuremath{\SO_+(2,1)})}_{2g\;\times}.
\end{equation*}
From the physics perspective, the Poisson manifold $(\ensuremath{\mathcal{P}_{\mathrm{ext}}},
\{\;,\;\})$ can be viewed as a constrained system with a set of six first-class
constraints from which the moduli space and its symplectic structure are
obtained after constraint implementation. This allows one to choose appropriate
gauge fixing conditions and to apply the Dirac gauge fixing procedure
\cite{Dirac:1949aa,Dirac:1950aa} to this Poisson structure. We explicitly
compute the resulting Poisson bracket for a large class of gauge fixing
conditions and investigate the resulting Poisson structures. This yields our
first central result:
\begin{theorem*}
For suitable gauge fixing conditions, the Dirac gauge fixing procedure
applied to $(\ensuremath{\mathcal{P}_{\mathrm{ext}}},\{\;,\;\})$ gives rise to a Poisson structure on
$\mathbb{R}^2\times \ensuremath{\pogr^{n-2+2g}}$ which is determined uniquely by a solution of the
classical dynamical Yang-Baxter
equation.
\end{theorem*}
In particular, we find that this Poisson structure on $\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$ is
given by a formula directly analogous to the original Poisson structure on
$\ensuremath{\mathcal{P}_{\mathrm{ext}}}$. The only difference is that the classical $r$-matrix in the
original definition is replaced by a solution of the classical dynamical
Yang-Baxter equation for $\ensuremath{\iso(2,1)}$ whose two dynamical variables parametrise
$\mathbb{R}^2\subset \mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$.
We then investigate the relation between the Poisson structures and solutions
of the classical dynamical Yang-Baxter equation that result from different
choices of gauge fixing conditions. This leads to our second central result:
\begin{theorem*}
All solutions of the classical dynamical Yang-Baxter equation obtained from
gauge fixing are related by dynamical $\ensuremath{\ISO(2,1)}$-valued transformations that
generalise the gauge transformations of classical dynamical $r$-matrices from
\cite{Etingof:1998aa}. All such solutions are locally equivalent to one of
two standard solutions corresponding to Cartan subalgebras of $\ensuremath{\iso(2,1)}$.
\end{theorem*}
The second statement of the theorem refers to an interesting feature of our
solutions that does not appear to have been observed in the literature yet. The
solutions of the classical dynamical Yang-Baxter equation that arise from
generic gauge fixing conditions are not associated with a fixed Cartan
subalgebra of $\ensuremath{\iso(2,1)}$ but combine classical dynamical $r$-matrices for two
non-conjugate Cartan subalgebras of
$\ensuremath{\iso(2,1)}$.
Our paper is structured as follows. Section \ref{sec:notation} contains the
basic definitions, notation and conventions used in the remainder of the
paper. Section \ref{sec:physics} summarises the physics background and
motivation of this work. It can be skipped without loss by the reader
unfamiliar with this or interested mainly in the mathematical results. It
contains a brief discussion of Chern-Simons theory on manifolds of topology
$\mathbb{R} \times S_{g,n}$ and of the moduli space of flat connections on $S_{g,n}$
as a reduced or gauge-invariant phase space of Chern-Simons theory. It then
explains the Dirac gauge fixing formalism and its relation to symplectic
reduction and discusses the constraints and gauge fixing conditions imposed to
obtain the moduli space.
Section \ref{sec:gaugefixing} contains the first central result of this
article, namely the explicit description of the Poisson structure resulting
from Dirac gauge fixing for a general set of gauge fixing conditions. We show
that the resulting Poisson structures are associated with solutions of the
classical dynamical Yang-Baxter equation and discuss examples arising from
specific choices of gauge fixing conditions as well as two simple standard
solutions associated with Cartan subalgebras of $\ensuremath{\iso(2,1)}$.
In Section \ref{sec:dynamic-poincare-trafos} we introduce dynamical
$\ensuremath{\ISO(2,1)}$-transformations which can be viewed as transitions between different
gauge fixing conditions. We determine the associated transformations of the
Dirac bracket and show how they can be interpreted as transitions between
different solutions of the classical dynamical Yang-Baxter equation. In that
sense, the dynamical $\ensuremath{\ISO(2,1)}$-transformations generalise the gauge
transformations of classical dynamical $r$-matrices in
\cite{Etingof:1998aa}. We then apply these dynamical transformations to give a
complete classification of the classical dynamical $r$-matrices and Poisson
structures obtained from gauge fixing. Section \ref{sec:outlook} contains the
outlook and our conclusions.
\section{Notations and conventions}
\label{sec:notation}
We denote by $\boldify{e}_0=(1,0,0)$, $\boldify{e}_1=(0,1,0)$, $\boldify{e}_2=(0,0,1)$ the standard
basis of $\mathbb{R}^3$ and use Einstein's summation convention throughout this
paper. Unless stated otherwise, all indices run from 0 to 2 and are raised and
lowered with the three-dimensional Minkowski metric $\eta=\diag(1,-1,-1)$. By
$\ee_{abc}$ we denote the totally antisymmetric tensor in three dimensions with
the convention $\ee_{012}=1$. For vectors $\boldify{x},\boldify{y}\in\mathbb{R}^3$, we use the
notation $\eta(\boldify{x},\boldify{y})=\boldify{x}\cdot\boldify{y}=\eta_{ab}x^ay^b$ and $\boldify{x}^2=\boldify{x}\cdot\boldify{x}$,
and we write $\boldify{x}\wedge\boldify{y}$ for the vector with components
$(\boldify{x}\wedge\boldify{y})^a=\ee^{abc}x_by_c$. Note that this is a Lorentzian version of
the wedge product which does not coincide with the standard one.
We denote by $\ensuremath{\SO_+(2,1)} \cong \PSL(2,\mathbb{R})$ the proper orthochronous Lorentz group
in three dimensions and by $\ensuremath{\so(2,1)} \cong \algebra{sl}(2,\mathbb{R})$ its Lie algebra.
In the following, we fix a set of generators $\{J_a\}_{a = 0, 1, 2}$ of $\ensuremath{\so(2,1)}$
such that the Lie bracket takes the form $[J_a, J_b] = \tensor{\ee}{_a_b^c}
J_c$. As the representation of $\PSL(2,\mathbb{R})$ by $\ensuremath{\SO_+(2,1)}$ matrices coincides
with its adjoint representation, we denote both representations by $\Ad$ in the
following:
\begin{equation*}
g\cdot J_a\cdot g^{-1}=\tensor{\Ad(g)}{^b_a} J_b \qquad \forall g\in \ensuremath{\SO_+(2,1)}.
\end{equation*}
The Poincaré group in three dimensions is the semidirect product $\ensuremath{\ISO(2,1)} \equiv
\ensuremath{\SO_+(2,1)} \ltimes \mathbb{R}^3$ of the proper orthochronous Lorentz group $\ensuremath{\SO_+(2,1)}$ and the
translation group $\mathbb{R}^3$. We parametrise elements of $\ensuremath{\ISO(2,1)}$ as
\begin{equation*}
(u,\boldify{a})=(u,0)\cdot (\mathds{1},-\boldify{j})=(u, -\Ad(u)\boldify{j})\quad \text{with}\; u\in \ensuremath{\SO_+(2,1)},
\boldify{j}, \boldify{a}\in\mathbb{R}^3.
\end{equation*}
The group multiplication law then takes the form
\begin{equation*}
(u_1,\boldify{a}_1)\cdot(u_2,\boldify{a}_2)
=\bigl(u_1u_2,\boldify{a}_1+\Ad(u_1)\boldify{a}_2\bigr).
\end{equation*}
A basis of the Lie algebra $\ensuremath{\iso(2,1)}$ is given by the basis $\{J_a\}_{a=0,1,2}$ of
$\ensuremath{\so(2,1)}$ together with a basis $\{P_a\}_{a=0,1,2}$ of the abelian Lie algebra
$\mathbb{R}^3$. In this basis, the Lie bracket takes the form
\begin{equation}\label{eq:poincare-algebra-bracket}
[J_a,J_b]=\tensor{\ee}{_a_b^c} J_c,\qquad
[J_a,P_b]=\tensor{\ee}{_a_b^c} P_c,\qquad
[P_a,P_b]=0,
\end{equation}
and a non-degenerate $\Ad$-invariant symmetric bilinear form on $\ensuremath{\iso(2,1)}$ is
given by
\begin{equation}\label{eq:pairing}
\langle J_a,J_b\rangle=\langle P_a,P_b\rangle=0, \qquad
\langle J_a,P_b\rangle=\eta_{ab}.
\end{equation}
All Cartan subalgebras of $\ensuremath{\iso(2,1)}$ are abelian and can be parametrised in terms
of two vectors $\boldify{x},\boldify{y}\in\mathbb{R}^3$ with $\boldify{x}^2\in\{1,-1\}$ and $\boldify{x}\cdot \boldify{y}=0$
as
\begin{equation}\label{eq:csa}
\algebra{h} = \Span\{x^aP_a, \, x^aJ_a + y^aP_a\}
\end{equation}
If the vector $\boldify{x}$ is timelike ($\boldify{x}^2=1$), then the associated Cartan
subalgebra $\algebra{h}$ is conjugate under the adjoint action of $\ensuremath{\ISO(2,1)}$ to
$\Span\{P_0, J_0\}$. If $\boldify{x}$ is spacelike ($\boldify{x}^2=-1$), then $\algebra{h}$ is
conjugate to $\Span\{P_1, J_1\}$. Note that the set \eqref{eq:csa} with a
lightlike vector $\boldify{x}\in\mathbb{R}^3$ ($\boldify{x}^2=0$) does not form a Cartan subalgebra of
$\ensuremath{\iso(2,1)}$ because it is not self-normalising.
In the following, we will also need the right- and left-invariant vector fields
on $\ensuremath{\ISO(2,1)}$ associated with a basis $\{T_a\}_{a=0,\dots,5}$ of $\ensuremath{\iso(2,1)}$. They are
given by
\begin{equation}\label{eq:vector-fields}
L_a f(h)=\tdiffat{}{t}{t=0} f(e^{-tT_a} \cdot h),\quad
R_a f(h)=\tdiffat{}{t}{t=0} f(h\cdot e^{tT_a})\qquad
\forall f\inC^\infty(\ensuremath{\ISO(2,1)}),
\end{equation}
where $e: \ensuremath{\iso(2,1)}\to \ensuremath{\ISO(2,1)}$, $\boldify{x}\mapsto e^{\boldify{x}}$ is the exponential map for
$\ensuremath{\ISO(2,1)}$. For the basis $\{J_a,P_a\}_{a=0,1,2}$, we denote by $P_a^{L}, P_a^R$,
respectively, the right- and left-invariant vector fields associated with the
translations and by $J_a^L, J_a^R$ the ones associated with the Lorentz
transformations. The former act trivially on functions that depend only on the
Lorentzian component of $\ensuremath{\ISO(2,1)}$. For the latter, the action on such functions
coincides with the action of the right- and left-invariant vector fields of the
Lorentz group. The action of these vector fields on the coordinate functions
$j^a: \ensuremath{\ISO(2,1)}\to\mathbb{R}$, $(u,-\Ad(u)\boldify{q})\mapsto q^a$ is given by
\begin{equation}\label{eq:action-of-vector-fields}
\left.
\begin{aligned}
&P^L_a j^b (u,-\Ad(u)\boldify{j})=\tensor{\Ad(u)}{_a^b}, & \quad
&P^R_a j^b(u,-\Ad(u)\boldify{j})=-\tensor{\delta}{_a^b}, \\
&J^L_a j^b(u,-\Ad(u)\boldify{j})=0, &
&J^R_aj^b(u,-\Ad(u)\boldify{j})=-\tensor{\ee}{^a^b_c}\,j^c.
\end{aligned}
\qquad\right\}
\end{equation}
\section{Physics background and motivation}
\label{sec:physics}
\subsection{Chern-Simons theory with gauge group ISO(2, 1){} and the moduli space of flat ISO(2, 1)-connections}
\newcommand{\ensuremath{S_{g,n}}}{\ensuremath{S_{g,n}}}
\newcommand{\mathcal{D}_i}{\mathcal{D}_i}
\newcommand{\grporbit}[1][i]{\mathcal{C}_#1}
In the following, we consider Chern-Simons theory with gauge group $\ensuremath{\ISO(2,1)}$ on
manifolds of topology $M \approx \mathbb{R} \times \ensuremath{S_{g,n}}$, where $\ensuremath{S_{g,n}}$ is an
oriented surface of genus $g$ with $n$ punctures. In the absence of punctures,
the solutions of the theory are flat connections $A$ on an $\ensuremath{\ISO(2,1)}$-principal
bundle over $M$. The punctures of $\ensuremath{S_{g,n}}$ are incorporated
\cite{deSousaGerbert:1990aa} into the theory by assigning the coadjoint orbit
of an element $\mathcal{D}_i \in \ensuremath{\iso(2,1)}$ to the $i$-th puncture and coupling it
minimally to the connection $A$. Parametrising the coadjoint orbit of
$\mathcal{D}_i$ in terms of group-valued functions $h_i: \mathbb{R} \to \ensuremath{\ISO(2,1)}$, one
obtains the following expression for the Chern-Simons action:
\begin{equation}\label{eq:cs-action}
S(A)=\int_M \langle A \wedge \mathrm{d} A + \tfrac23 A \wedge A \wedge A \rangle
- 2 \sum_{i=1}^n \int_\mathbb{R} \langle \mathcal{D}_i, h_i^\inv A\big|_{l_i} h_i + h_i^\inv \mathrm{d} h_i \rangle \mathrm{d} t,
\end{equation}
where $\langle \;\;\rangle$ denotes the non-degenerate $\Ad$-invariant
symmetric bilinear form \eqref{eq:pairing} on $\ensuremath{\iso(2,1)}$ and $l_i: \mathbb{R} \to M$,
$i=1,\dots,n$, are the curves defined by the punctures. Up to a topological
term, the Chern-Simons action is invariant under gauge transformations $\gamma
\in C^\infty(M, \ensuremath{\ISO(2,1)})$ that are constant along $l_i$: $A \mapsto \gamma^\inv A
\gamma + \gamma^{-1} \mathrm{d}\gamma$, $h_i \mapsto \gamma(l_i)^\inv h_i$.
The connections that extremise the action \eqref{eq:cs-action} are those that
are flat everywhere on $M$ except at $l_i$ ($i=1,\dots,n$), where their
curvature $F \equiv \mathrm{d} A + A \wedge A$ develops $\delta$-singularities.
From the Hamiltonian formulation of the theory one then obtains that the
gauge-invariant phase space of the theory is the moduli space $\ensuremath{\mathcal{P}}$ of
flat $\ensuremath{\ISO(2,1)}$-connections on $\ensuremath{S_{g,n}}$ modulo gauge transformations
\cite{Witten:1988aa,deSousaGerbert:1990aa}.
A convenient parametrisation of the moduli space is given by group
homomorphisms $h:\pi_1(\ensuremath{S_{g,n}}) \to \ensuremath{\ISO(2,1)}$ that map the homotopy equivalence
class $m_i$ of a loop around the $i$-th puncture to the associated conjugacy
class
\begin{equation}\label{eq:particle-conjugacy-classes}
\grporbit=\{h\cdot\exp(\mathcal{D}_i)\cdot h^\inv \mid h\in \ensuremath{\ISO(2,1)}\}.
\end{equation}
Two such group homomorphisms describe gauge-equivalent connections if and only
if they are related by conjugation with an element of $\ensuremath{\ISO(2,1)}$. This implies
that the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections on $\ensuremath{S_{g,n}}$ is given by
\begin{align}\label{eq:phase-space}
\ensuremath{\mathcal{P}}
&=\Hom_{\grporbit[1], \dots, \grporbit[n]}\big(\pi_1(\ensuremath{S_{g,n}}), \ensuremath{\ISO(2,1)}\big)/\ensuremath{\ISO(2,1)}\\
&=\{h \in \Hom\bigl(\pi_1(\ensuremath{S_{g,n}}), \ensuremath{\ISO(2,1)}\bigr) \mid h(m_i)\in\grporbit[i]\}/\ensuremath{\ISO(2,1)}.\nonumber
\end{align}
\begin{figure}
\centering
\includegraphics{fundamental-group.pdf}
\caption{Generators of the fundamental group for an $n$-punctured genus $g$
surface $\ensuremath{S_{g,n}}$. The chosen generators of the fundamental group $\pi_1(\ensuremath{S_{g,n}})$
are the homotopy equivalence classes of the curves $m_1, \dots, m_n, a_1,
b_1, \dots, a_n, b_n$.
The short wavy line indicates the cilium that defines a linear ordering of
the incident edges at the basepoint.}
\label{fig:fundamental-group}
\end{figure}
The fundamental group $\pi_1(\ensuremath{S_{g,n}})$ of an oriented genus $g$ surface $\ensuremath{S_{g,n}}$
with $n$ punctures is generated by the homotopy equivalence classes of a loop
$m_i$ ($i=1,\dots,n$) around each puncture and the $a$- and $b$-cycles
$a_j,b_j$ ($j=1,\dots,g$) of each handle as shown in Figure
\ref{fig:fundamental-group}. It has a single defining relation, which states
that the curve $c$ in Figure \ref{fig:fundamental-group} is contractible:
\begin{equation*}
\pi_1(\ensuremath{S_{g,n}})=\langle m_1,\dots,m_n, a_1,b_1,\dots,a_g,b_g \mid b_g a_g^\inv b_g^\inv a_g\cdots b_1 a_1^\inv b_1^\inv a_1 m_n\cdots m_1=1 \rangle.
\end{equation*}
By characterising the group homomorphisms in \eqref{eq:phase-space} in terms of
the images of the generators of $\pi_1(\ensuremath{S_{g,n}})$, we can thus identify the moduli
space of flat connections with the set
\begin{multline}\label{eq:phase-space-with-holonomies}
\ensuremath{\mathcal{P}} = \{(M_1,\dots,M_n,A_1,B_1,\dots,A_g,B_g)\in\ensuremath{\pogr^{n+2g}} \mid \\
M_i\in\grporbit, \ [B_g,A_g^\inv] \cdot [B_1,A_1^\inv] \cdot M_n\cdots M_1=1\}/\ensuremath{\ISO(2,1)},
\end{multline}
where $[B_g, A_g^\inv]=B_g\cdot A_g^\inv\cdot B_g^\inv\cdot A_g$ denotes the
group commutator and the quotient is taken with respect to the diagonal action
of $\ensuremath{\ISO(2,1)}$ on $\ensuremath{\pogr^{n+2g}}$. In the gauge-theoretical description, the group
elements $M_1, \dots, B_g\in \ensuremath{\ISO(2,1)}$ correspond to the path-ordered exponentials
of the gauge field $A$ along the closed curves $m_1, \dots, b_g$ displayed in
Figure \ref{fig:fundamental-group}. In the following, we will sometimes refer
to these group elements as holonomies.
\subsection{Symplectic structure of the moduli space of flat connections}
The moduli space of flat $\ensuremath{\ISO(2,1)}$-connections carries a canonical symplectic
structure \cite{Atiyah:1983aa} that is obtained via symplectic reduction from
the canonical symplectic structure on the space of connections on $\ensuremath{S_{g,n}}$. A
convenient and explicit description of this symplectic structure is given in
the works of Alekseev and Malkin and Fock and Rosly \cite{Fock:1998aa,
Alekseev:1995ab}. They describe the canonical symplectic structure on the
moduli space $\ensuremath{\mathcal{P}}$ in terms of a (non-canonical) Poisson structure on
an enlarged ambient space $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$. Via symplectic reduction, this
Poisson structure then induces the canonical symplectic structure on
$\ensuremath{\mathcal{P}}$.
In the following, we will work with a specific form of the Poisson structure in
\cite{Fock:1998aa} which is associated with a choice of an ordered set of
generators of the fundamental group $\pi_1(\ensuremath{S_{g,n}})$. It plays an important role
as a starting point for the combinatorial quantisation of the theory
\cite{Alekseev:1995aa,Alekseev:1996aa,Alekseev:1996ab,Buffenoir:1995aa}. In
this description, the ambient space $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$ is given by $n+2g$ copies
of the Poincar\'e group, $\ensuremath{\mathcal{P}_{\mathrm{ext}}}=\ensuremath{\pogr^{n+2g}}$, each corresponding to the
holonomy along a generator of the fundamental group $\pi_1(\ensuremath{S_{g,n}})$.
Explicit expressions for this Poisson structure are given in Definition
\ref{def:fr} in the next section. Here, we only discuss its most important
structural features. Firstly, the definition of the bracket requires a
classical $r$-matrix for the Lie algebra $\ensuremath{\iso(2,1)}$, i.e.\ an element
$r\in\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ that is a solution of the classical Yang-Baxter equation
\begin{equation*}
[[r,r]] \coloneqq [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] = 0,
\end{equation*}
where $r_{12}=r^{\alpha\beta} T_\alpha\otimes T_\beta\otimes 1$,
$r_{13}=r^{\alpha\beta} T_\alpha\otimes 1\otimes T_\beta$,
$r_{23}=r^{\alpha\beta} 1\otimes T_\alpha\otimes T_\beta$. This property is
needed to ensure that the bracket satisfies the Jacobi identity. Moreover, it
is shown in \cite{Fock:1998aa} that this Poisson structure induces a symplectic
structure on the moduli space $\ensuremath{\mathcal{P}}$, which agrees with its canonical
symplectic structure if and only if the symmetric part of $r$ is dual to the
pairing \eqref{eq:pairing} in the Chern-Simons action:
\begin{equation}\label{eq:r-symm}
r_S \equiv r_{(s)}^{\alpha\beta} T_\alpha\otimes T_\beta
= \tfrac 1 2 (P_a\otimes J^a+J^a\otimes P_a).
\end{equation}
Note that Fock and Rosly's Poisson structure on the ambient space
$\ensuremath{\mathcal{P}_{\mathrm{ext}}}$ is therefore non-canonical in two ways. Firstly, it depends on
the choice of a set of generators of the fundamental group $\pi_1(\ensuremath{S_{g,n}})$ and
of a linear ordering of the incident edges at the basepoint. This ordering of
the edges is indicated in Figure \ref{fig:fundamental-group} by the short wavy
line (cilium) and gives rise to a partial ordering of the generators of
$\pi_1(\ensuremath{S_{g,n}})$: $m_1 < \dots < m_n < a_1, b_1 < \dots < a_g, b_g$.
Secondly, the definition of the Poisson structure on the ambient space
$\ensuremath{\mathcal{P}_{\mathrm{ext}}}$ requires the choice of a classical $r$-matrix, which is
generally not unique; for a classification of the classical $r$-matrices for
$\ensuremath{\iso(2,1)}$ see \cite{Stachura:1998aa}. However, it is shown in \cite{Fock:1998aa}
that the Poisson structures on $\ensuremath{\pogr^{n+2g}}$ associated with different choices of
generators and orderings of $\pi_1(\ensuremath{S_{g,n}})$ and different choices of classical
$r$-matrices that satisfy \eqref{eq:r-symm} induce the same symplectic
structure on the moduli space $\ensuremath{\mathcal{P}}$. This is apparent in formula
\eqref{eq:fr-bivector}, which shows that the Poisson bracket of two functions
on $\ensuremath{\pogr^{n+2g}}$ depends only on the symmetric component of $r$ if one of the two
functions is invariant under the diagonal action of $\ensuremath{\ISO(2,1)}$ on $\ensuremath{\pogr^{n+2g}}$.
In the following, we work with the classical $r$-matrix that corresponds to the
structure of $\ensuremath{\iso(2,1)}$ as a classical double of $\ensuremath{\so(2,1)}$. In terms of the basis
\eqref{eq:poincare-algebra-bracket} it is given by $ r = P_a \otimes J^a$. It
is shown in \cite{Meusburger:2004aa}, see also the discussion in Section
\ref{subsec:dirac-intro}, that Fock and Rosly's Poisson structure for this case
can be formulated in terms of functions and certain vector fields on $\ensuremath{\logr^{n+2g}}$
such that the Poisson bracket takes the form
\begin{equation*}
\{f,g\}=0,\quad
\{X, f\}=\mathcal L_Xf,\quad
\{X,Y\}=[X,Y],
\end{equation*}
where $f,g\inC^\infty(\ensuremath{\logr^{n+2g}})$, $X,Y\in\text{Vec}(\ensuremath{\logr^{n+2g}})$, $\mathcal L_Xf$
denotes the Lie derivative and $[X,Y]$ the Lie bracket of vector fields on
$\ensuremath{\logr^{n+2g}}$.
This implies that the associated Poisson algebra has a canonical $\mathbb{N}$-grading,
in which the subspaces of homogeneous degree are given by homogeneous
polynomials in the vector fields $X$ with $C^\infty(\ensuremath{\logr^{n+2g}})$-valued
coefficients. This $\mathbb{N}$-grading corresponds naturally to a physical dimension
of $\hbar$ and plays an important role in the quantisation of the theory.
\subsection{Quantisation}
The description of the moduli space of flat connections outlined in the
previous section serves as the starting point of the combinatorial quantisation
formalism \cite{Alekseev:1995aa, Alekseev:1996aa, Alekseev:1996ab,
Buffenoir:1995aa} for Chern-Simons theories with compact, semisimple gauge
groups. This formalism proceeds by quantising the auxiliary Poisson structure
on the ambient space $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$ and then imposing the constraints in the
quantum theory. As this auxiliary Poisson structure is closely related to
certain Poisson structures from the theory of Poisson-Lie groups, the
corresponding quantum algebra is given in terms of quantum groups. The
implementation of the constraints in the quantum theory then reduces to a
problem from the representation theory of the associated quantum group, namely
to determining the invariant subspace in the tensor product of certain
representations of this quantum group.
While this formalism is well-established for Chern-Simons theories with
compact, semisimple gauge groups, for which the corresponding quantum groups
are universal enveloping algebras at a root of unity, it cannot be extended
straightforwardly to Chern-Simons theories with non-compact gauge groups. This
is due to the fact that the representations of the corresponding quantum groups
no longer form a semisimple ribbon category and their characters become
distributions.
The generalisation of the combinatorial quantisation formalism to Chern-Simons
theory with gauge group $\SL(2,\mathbb{C})$ has been achieved in
\cite{Buffenoir:2002aa}, for partial results on semidirect product gauge groups
see \cite{Meusburger:2004aa, Meusburger:2010aa, Meusburger:2010bb}. However,
there is currently no general quantisation formalism for moduli spaces of flat
connections associated with non-compact or non-semisimple groups. Other
quantisation approaches such as Reshetikhin-Turaev invariants
\cite{Reshetikhin:1991aa} face similar problems. Although there is work that
investigates their extension by analytic continuation \cite{Bar-Natan:1991aa,
Witten:2011aa}, there is no general method that allows one to extend these
models to general non-compact or non-semisimple gauge groups.
For this reason, we pursue a different strategy, namely we implement the
constraints directly in the classical theory by means of the Dirac gauge fixing
procedure. This potentially avoids the issues with the implementation of the
constraints in the quantisation of the theory and leads to an explicit
description of the canonical Poisson structure on the moduli space of flat
connections in terms of holonomies. In the following, we apply this approach
to the description of the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections in terms of
the auxiliary Poisson structure on the ambient space $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$.
A specific example of such a gauge fixing procedure is investigated in
\cite{Meusburger:2011aa}. It is shown there that in the application to
(2+1)-gravity, this gauge fixing procedure has a direct physical
interpretation. The gauge fixing conditions can be viewed as a prescription
that specifies an observer with respect to the geometry of the spacetime. The
resulting gauge-fixed Poisson structure then depends on two variables that
correspond to the total mass and internal angular momentum of the spacetime as
measured by this observer.
\subsection{The Dirac gauge fixing procedure}
\label{subsec:dirac-constraints}
In general, the Dirac gauge fixing procedure applies to constrained dynamical
systems, i.e.\ to Poisson manifolds $(\ensuremath{\mathcal{P}_{\mathrm{ext}}}, \{,\})$ with a set of
constraint functions $\{\phi_i\}_{i=1,\dots,k}\subsetC^\infty(\ensuremath{\mathcal{P}_{\mathrm{ext}}})$. In
the following, we restrict attention to the case where the constraints are
first-class and such that $0$ is a regular value of $\Phi=(\phi_1, \dots
\phi_k): \ensuremath{\mathcal{P}_{\mathrm{ext}}} \to \mathbb{R}^k$. Gauge fixing then amounts to imposing an
additional set of constraints
$\{\chi_j\}_{j=1,\dots,k}\subsetC^\infty(\ensuremath{\mathcal{P}_{\mathrm{ext}}})$, the gauge fixing
conditions, which must satisfy the following requirements
\cite{Henneaux:1994aa}:
\begin{enumerate}
\item It is possible to map any point $q \in \{p\in \ensuremath{\mathcal{P}_{\mathrm{ext}}} \mid
\phi_i(p) = 0 \; \forall i=1,\dots,k\}$ to a point on the constraint surface
$\ensuremath{\Sigma} \coloneqq \{p\in \ensuremath{\mathcal{P}_{\mathrm{ext}}} \mid \phi_i(p) = 0 \text{ and }
\chi_i(p)=0 \; \forall i=1,\dots,k\}$ with the flows that the first-class
constraints $\phi_i$ generate via the Poisson bracket.
\item The matrix $C=(\{\chi_j, \phi_i\})_{i,j=1,\dots,k}$ is invertible
everywhere on the constraint surface $\ensuremath{\Sigma}$.
\end{enumerate}
The second condition implies that the gauge fixing conditions
$\{\chi_j\}_{j=1,\dots,k}$ together with the original constraints
$\{\phi_i\}_{i=1,\dots,k}$ can be collected in a single set
$\{C_i\}_{i=1,\dots,2k}$ of constraints such that $0$ is a regular value of the
function $C=(C_1,\dots,C_{2k}): M\to\mathbb{R}^{2k}$ and for which the Dirac matrix
$D=(D_{ij})_{i,j=1,\dots,2k}$, $D_{ij} \coloneqq \{C_i, C_j\}$, is invertible
anywhere on the constraint surface $\ensuremath{\Sigma}$. The Dirac bracket of two
functions $f,g\in C^\infty(\ensuremath{\Sigma})$ is then defined by
\begin{equation*}
\{f, g\}_D \coloneqq \{\tilde f, \tilde g\} - \smashoperator{\sum_{i,j=1}^{2k}} \{\tilde f, C_i\} (D^\inv)_{ij} \{C_j, \tilde g\},
\end{equation*}
where $\tilde f,\tilde g\in C^\infty(\ensuremath{\mathcal{P}_{\mathrm{ext}}})$ are arbitrary extensions
of $f,g\in C^\infty(\ensuremath{\Sigma})$. The Dirac bracket does not depend on the
choice of the extensions $\tilde f, \tilde g$ and defines a Poisson structure
on $\ensuremath{\Sigma}$ \cite{Dirac:1949aa,Dirac:1950aa,Henneaux:1994aa}.
The physical interpretation of this bracket can be summarised as follows: The
Poisson manifold $(\ensuremath{\mathcal{P}_{\mathrm{ext}}},\{\,,\,\})$ plays the role of an extended,
non-gauge-invariant phase space which contains redundant degrees of freedom
corresponding to different descriptions of a single physical state. The
gauge-invariant or physical phase space is given as the quotient
$\ensuremath{\mathcal{P}}=\equivalencequotientspace{\ensuremath{\mathcal{Q}}}$, where
$\ensuremath{\mathcal{Q}}\coloneqq\{p\in\ensuremath{\mathcal{P}_{\mathrm{ext}}}\,\vert\, \phi_i(p)=0\;\forall
i=1,\dots,k\}$ and two points on $\ensuremath{\mathcal{Q}}$ are identified if they
are mapped into each other by the flows the first-class constraints $\phi_i$
generate via the the Poisson bracket. The associated equivalence classes are
called gauge orbits.
Imposing gauge fixing conditions amounts to selecting a representative in each
gauge orbit. The first requirement on the gauge fixing conditions ensures that
the gauge fixing conditions select at least one representative in every gauge
orbit. The second requirement ensures that they select at most one
representative in each gauge orbit. Imposing gauge fixing conditions thus
amounts to constructing a diffeomorphism that identifies the quotient
$\ensuremath{\mathcal{P}}=\equivalencequotientspace{\ensuremath{\mathcal{Q}}}$ with the
submanifold $\ensuremath{\Sigma} \subset \ensuremath{\mathcal{Q}} \subset \ensuremath{\mathcal{P}_{\mathrm{ext}}}$.
From the viewpoint of symplectic reduction, this procedure can be interpreted
in the following way. If $(\ensuremath{\mathcal{P}_{\mathrm{ext}}},\{\,\,\})$ is symplectic, then the
submanifold $\ensuremath{\mathcal{Q}}\subset\ensuremath{\mathcal{P}_{\mathrm{ext}}}$ is coisotropic, and the
flows generated by the first-class constraints $\phi_i$ define a foliation of
$\ensuremath{\mathcal{Q}}$.
If we think of $\ensuremath{\mathcal{Q}}$ as a bundle over $\ensuremath{\mathcal{P}}$, then
choosing a representative for each equivalence class in $\ensuremath{\mathcal{P}}$ amounts
to specifying a global section on $\ensuremath{\mathcal{Q}}$. This is achieved via
the gauge fixing conditions, which define a diffeomorphism $\xi:
\ensuremath{\Sigma}\subset\ensuremath{\mathcal{P}_{\mathrm{ext}}}\to\ensuremath{\mathcal{P}}$, $p\mapsto [p]$. The manifold
$\ensuremath{\mathcal{P}}$ carries a canonical symplectic structure obtained via symplectic
reduction from the symplectic structure on $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$. The pull-back of
this symplectic structure to the submanifold $\ensuremath{\Sigma}\subset\ensuremath{\mathcal{P}_{\mathrm{ext}}}$
with $\xi$ is the Dirac bracket on $\ensuremath{\Sigma}$.
The situation is similar in the case where $(\ensuremath{\mathcal{P}_{\mathrm{ext}}},\{\,\,\})$ is not
symplectic but there is a function $\Psi=(\psi_1,\dots,\psi_l)\in
C^\infty(\ensuremath{\mathcal{P}_{\mathrm{ext}}},\mathcal \mathbb{R}^l)$ such that $0$ is a regular value of
$\Psi$, $\Psi^\inv(0)$ a symplectic submanifold of $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$ and
$\psi_1,\dots,\psi_l$ Poisson-commute with all functions on
$\ensuremath{\mathcal{P}_{\mathrm{ext}}}$. In this case, the reasoning above can be applied to the
submanifold $\Psi^\inv(0)$.
\subsection{Constraints and gauge fixing conditions for the moduli space}
\label{subsec:constraints-and-gauge-fixing}
The moduli space of flat $\ensuremath{\ISO(2,1)}$-connections can be viewed as a constrained
system in the sense of Dirac. From this viewpoint, the Poisson manifold
$(\ensuremath{\mathcal{P}_{\mathrm{ext}}}, \{\,,\})$ is identified with the ambient space
$\ensuremath{\mathcal{P}_{\mathrm{ext}}}=\ensuremath{\pogr^{n+2g}}$ equipped with Fock and Rosly's Poisson structure
\cite{Fock:1998aa}. From expression \eqref{eq:phase-space-with-holonomies} for
the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections, it is then apparent that the
moduli space is obtained from $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$ by imposing a group-valued
constraint that arises from the defining relation of the fundamental group
$\pi_1(\ensuremath{S_{g,n}})$, together with a set of constraints that restrict the holonomies
$M_1,\dots,M_n$ to the conjugacy classes \eqref{eq:particle-conjugacy-classes}.
The latter can be formulated as pairs of constraints of the form
\begin{equation*}
\Tr(u_{M_i})-c_i \approx 0, \qquad
\Tr(j_{M_i}^aJ_a\cdot u_{M_i})-d_i\approx 0,
\end{equation*}
with real parameters $c_i,d_i$ for each puncture. It turns out that these
constraints are Casimir functions of the Poisson bracket, i.e.\ Poisson-commute
with all functions on $\ensuremath{\mathcal{P}_{\mathrm{ext}}}$. For this reason, reducing the Poisson
structure to the relevant conjugacy classes presents no difficulties and does
not require any gauge fixing.
The group-valued constraint from the defining relation of the fundamental group
$\pi_1(\ensuremath{S_{g,n}})$ can be viewed as a set of six first-class constraints for the
Fock-Rosly bracket \cite{Meusburger:2011aa}. Parametrising the holonomy of the
curve $c$ in Figure \ref{fig:fundamental-group} as
\begin{equation*}
(u_C^\inv, \boldify{j}_C)
\coloneqq M_1^\inv\cdots M_n^\inv [A_1^\inv, B_1]\cdots[A_g^\inv, B_g],
\end{equation*}
one can express this group-valued constraint in the form of the six constraints
\begin{equation}\label{eq:constraints-in-jp}
\Tr(J_a\cdot u_C) \approx 0,\qquad j_C^a\approx 0\qquad\forall a\in\{0,1,2\}.
\end{equation}
The Poisson brackets of these six constraint functions are closely related to
the Lie bracket of $\ensuremath{\iso(2,1)}$. The associated gauge transformations which they
generate via the Poisson bracket are given by the diagonal action of $\ensuremath{\ISO(2,1)}$ on
$\ensuremath{\pogr^{n+2g}}.$
A specific choice of gauge fixing conditions for the constraints
\eqref{eq:constraints-in-jp} is investigated in \cite{Meusburger:2011aa}. It
imposes gauge fixing conditions on the holonomies associated with two punctures
on the surface $\ensuremath{S_{g,n}}$ and derives the associated Dirac bracket. In this paper
we consider general gauge fixing conditions, subject to certain structural
requirements, and investigate the resulting Dirac brackets. We require that the
gauge fixing conditions satisfy the conditions 1 and 2 in Section
\ref{subsec:dirac-constraints} and are subject to the following two additional
restrictions:
\begin{enumerate}[\quad a.]
\item The gauge fixing conditions depend only on the holonomies $M_i$, $M_j$
associated with two punctures on the surface $\ensuremath{S_{g,n}}$.
\item The gauge fixing conditions depend at most linearly on the variables
$j_{M_i}$ and $j_{M_j}$ associated with these holonomies.
\end{enumerate}
The first condition is motivated by convenience and by physics
considerations. Although it is feasible in principle to impose gauge fixing
conditions that involve the holonomies of more than two punctures, this would
complicate many details of the description without adding much on the
conceptual level. Moreover, in the application to (2+1)-gravity, gauge fixing
conditions based on the holonomies of two punctures have a direct physical
interpretation, while the interpretation of a complicated gauge fixing
condition involving more than two punctures is less obvious.
Note also that the first condition allows us to restrict attention to gauge
fixing conditions that depend only on the holonomies $M_1$, $M_2$ of the first
two punctures on $\ensuremath{S_{g,n}}$. This is due to the fact that different orderings of
the punctures are related by the action of the braid group on $\ensuremath{S_{g,n}}$ and the
braid group on the associated surface $\ensuremath{S_{g,n}}\setminus D$ with a disc removed
\cite{Birman:1974aa}. It is shown in \cite{Meusburger:2005aa} that the braid
group of the surface $\ensuremath{S_{g,n}}\setminus D$ acts by Poisson isomorphisms on the
Poisson manifold $(\ensuremath{\mathcal{P}_{\mathrm{ext}}}=\ensuremath{\pogr^{n+2g}},\{\,,\,\})$. The action of the braid
group thus allows one to permute the punctures and to suppose that the gauge
fixing conditions depend only on the holonomies of the first two punctures.
The second condition is motivated by structural considerations, namely the wish
to preserve the natural $\mathbb{N}$-grading of the Poisson structure. As we will see
in the following, gauge fixing conditions that are non-linear in the variables
$j_{M_1}$ or $j_{M_2}$ and, consequently, non-linear in the vector fields on
$\ensuremath{\logr^{n+2g}}$, would compromise this grading. However, the grading is an
important structural feature of the theory and plays a central role in its
quantisation \cite{Meusburger:2004aa,Meusburger:2010aa,Meusburger:2010bb}. For
this reason, it seems natural to impose that it is preserved by the gauge
fixing procedure.
Conditions 1 and 2 from Section \ref{subsec:dirac-constraints} together with
the additional assumptions a and b above imply that the gauge fixing conditions
can be brought into the form
\begin{equation*}
\sum_{i=1}^2 \ThetaFam{1}{M_i}{a}\,j^a_{M_i} \approx 0, \quad
\sum_{i=1}^2 \ThetaFam{2}{M_i}{a}\,j^a_{M_i} \approx 0, \quad
\sum_{i=1}^2 \ThetaFam{3}{M_i}{a}\,j^a_{M_i} \approx 0, \quad
\Delta_1 \approx 0, \quad
\Delta_2 \approx 0, \quad
\Delta_3 \approx 0,
\end{equation*}
where $\ThetaFam{j}{M_i}{a}, \Delta_j \in C^\infty(\ensuremath{\SO_+(2,1)}\times \ensuremath{\SO_+(2,1)})$ and the two
copies of the Lorentz group $\ensuremath{\SO_+(2,1)}$ are identified with the Lorentzian
components of the holonomies $M_1$ and $M_2$. These gauge fixing conditions
allow one to express the two constrained holonomies $M_1$ and $M_2$ as
functions of the four fixed parameters that characterise the conjugacy classes
$\grporbit[1],\grporbit[2]$ and of two conjugation-invariant dynamical
variables $\psi,\alpha$, which depend only on the product $M_2\cdot M_1$. As
there are many possible definitions of these variables, we will not adhere to
one of them, but impose that they are given in terms of the Lorentzian and
translational components of the product $M_2\cdot M_1=(u_{12},-\Ad(u_{12})
\boldify{j}_{12})$ as
\begin{equation*}
\psi=f(\Tr(u_{12})), \quad
\alpha=g(\Tr(u_{12})) \Tr(j_{12}^aJ_a\cdot u_{12})+h(\Tr(u_{12})),
\end{equation*}
with diffeomorphisms $f,g\inC^\infty(\mathbb{R})$ and a smooth function $h\inC^\infty(\mathbb{R})$.
As for the gauge fixing conditions investigated in \cite{Meusburger:2011aa},
the Dirac gauge fixing procedure with these gauge fixing conditions gives rise
to a Poisson structure $\{\,,\}_D$ on the constraint surface $\ensuremath{\Sigma} \subset
\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$, where $\mathbb{R}^2$ is parametrised by the variables
$\psi,\alpha$ and $\ensuremath{\pogr^{n-2+2g}}$ by the non-gauge-fixed holonomies
$M_3,\dots,B_g$. This Poisson structure is derived in the next section.
\section{Gauge fixing and solutions of the classical dynamical Yang-Baxter equation}
\label{sec:gaugefixing}
\subsection{General form of the Dirac bracket}
\label{subsec:dirac-intro}
In this section, we derive explicitly the Dirac bracket obtained by gauge
fixing the auxiliary Poisson structure from \cite{Fock:1998aa} on the ambient
space $\ensuremath{\pogr^{n+2g}}$. While the Poisson structure in \cite{Fock:1998aa} is
associated with general ciliated fat graphs on a genus $g$-surface $S_{g,n}$
with $n$ punctures, we restrict attention to the case where the graph is a set
of generators of the fundamental group $\pi_1(S_{g,n})$ as depicted in Figure
\ref{fig:fundamental-group}. In that case, the Poisson structure from
\cite{Fock:1998aa} takes the following form.
\begin{definition}[\cite{Fock:1998aa}]\label{def:fr}
Let $G$ be a Lie group with Lie algebra $\algebra{g}$,
$\{T_\alpha\}_{\alpha=1,\dots,\text{dim}(G)}$ a basis of $\algebra{g}$ and
$r=r^{\alpha\beta}T_\alpha\otimes T_\beta\in\algebra{g}\otimes\algebra{g}$. Then Fock and Rosly's
bivector $\ensuremath{B_r^{n,g}}$ is the antisymmetric section of the bundle
$TG^{n+2g}\otimes TG^{n+2g}$ defined by
\begin{align}\label{eq:fr-bivector}
&\ensuremath{B_r^{n,g}}
= \tfrac 1 2 r_{(a)}^{\alpha\beta} \Bigl(\smashoperator{\sum_{i=1}^{n+2g}} L^{i}_\alpha+R^{i}_\alpha\Bigr)\otimes\Bigl(\smashoperator{\sum_{j=1}^{n+2g}} L^{j}_\beta+R^{j}_\beta\Bigr)
+ \tfrac 1 2 r_{(s)}^{\alpha\beta} \smashoperator{\sum_{1\leq i<j\leq n+2g}} \left(L^{i}_\alpha+R^{i}_\alpha\right)\wedge\left(L^{j}_\beta+R^{j}_\beta\right) \\[-.5em]
&\!+\! \tfrac 1 2 r_{(s)}^{\alpha\beta}\Big(
\sum_{i=1}^n R^{i}_\alpha \!\wedge\! R^{i}_\beta \!+\!
\sum_{j=1}^{g} \Big[
L^{n+2j}_\alpha \!\wedge\! L^{n+2j}_\beta \!-\!
(R^{n+2j-1}_\alpha \!+\! L^{n+2j-1}_\alpha) \!\wedge\! R^{n+2j}_\beta \!-\!
L^{n+2j-1}_\alpha \!\wedge\! L^{n+2j}_\beta
\Big]
\Big),\nonumber
\end{align}
where $L_\alpha^i$ and $R_\alpha^i$ denote the right- and left-invariant vector
fields associated with the different components of $G^{n+2g}$ and the basis
elements $T_\alpha$:
\begin{align*}
L^i_\alpha f(g_1,\dots,g_{n+2g})&=\tdiffat{}{t}{t=0} f(u_1,\dots,u_{i-1}, e^{-t T_\alpha}\cdot u_i, u_{i+1},\dots,u_{n+2g}),\\
R^i_\alpha f(g_1,\dots,g_{n+2g})&=\tdiffat{}{t}{t=0} f(u_1,\dots,u_{i-1}, u_i\cdot e^{t T_\alpha}, u_{i+1},\dots,u_{n+2g}),
\end{align*}
and $r^{\alpha\beta}_{(a)}$, $r^{\alpha\beta}_{(s)}$ the coefficients of the
antisymmetric and symmetric component of $r$:
\begin{equation*}
r^{\alpha\beta}_{(a)}=\tfrac 1 2 (r^{\alpha\beta}-r^{\beta\alpha})\qquad r^{\alpha\beta}_{(s)}=\tfrac 1 2 (r^{\alpha\beta}+r^{\beta\alpha}).
\end{equation*}
The associated bracket on $G^{n+2g}$ is the antisymmetric bilinear map $\{\;\}:
C^\infty(G^{n+2g})\times C^\infty(G^{n+2g})\to C^\infty(G^{n+2g})$ given by
$$\{f,g\}=\ensuremath{B_r^{n,g}}(\mathrm{d} f \otimes \mathrm{d} g)\qquad\forall f,g\in C^\infty(G^{n+2g}).$$
\end{definition}
The main advantage of this description is that it defines an auxiliary Poisson
structure on $G^{n+2g}$ that is given in terms of a classical $r$-matrix for
$\algebra{g}$ and induces the canonical symplectic structure on the moduli space of
flat $G$-connections on $S_{g,n}$.
\begin{theorem}[\cite{Fock:1998aa}]
If $r$ is a solution of the classical Yang-Baxter equation
\begin{equation*}
[[r,r]]=[r_{12}, r_{13}]+[r_{12}, r_{23}]+[r_{13}, r_{23}]=0,
\end{equation*}
then $\ensuremath{B_r^{n,g}}$ defines a Poisson structure on $G^{n+2g}$. If additionally
$\langle\;,\;\rangle$ is a non-degenerate $\Ad$-invariant symmetric bilinear
form on $\algebra{g}$ and
\begin{equation*}
r^{\alpha\beta}_{(s)}=\tfrac12\kappa^{\alpha\beta}
\quad\text{ with }
\kappa^{\alpha\beta}\kappa_{\beta\gamma}=\tensor{\delta}{^\alpha_\gamma},\;
\kappa_{\alpha\beta}=\langle T_\alpha, T_\beta\rangle,
\end{equation*}
then the Poisson structure defined by $\ensuremath{B_r^{n,g}}$ induces the canonical
symplectic structure on the moduli space of flat $G$-connections on $S_{g,n}$.
\end{theorem}
It is shown in \cite{Alekseev:1995ab} that this Poisson structure can be
identified with the direct product of $n$ copies of the dual Poisson-Lie
structure on $G$ and $g$ copies of the Heisenberg double Poisson structure
associated with $G$ and $r$.
For the case at hand, where the Lie group is the Poincar\'e group in three
dimensions, $G=\ensuremath{\ISO(2,1)}$, a classical $r$-matrix is given by $r=P_a\otimes J^a$. The
corresponding Poisson structure on $\ensuremath{\pogr^{n+2g}}$ is computed explicitly in
\cite{Meusburger:2003aa}, and in \cite{Meusburger:2004aa} it is shown that this
Poisson structure can be identified with the direct product of $n$ copies of
the dual Poisson-Lie structure on $\ensuremath{\ISO(2,1)}$ and $2g$ copies of the cotangent
bundle Poisson structure:
\begin{equation*}
(\ensuremath{\pogr^{n+2g}},\{\})=\underbrace{\ensuremath{\ISO(2,1)}^*\times\ldots\times \ensuremath{\ISO(2,1)}^*}_{n\,\times}\times \underbrace{T^*(\ensuremath{\SO_+(2,1)})\times\ldots\times T^*(\ensuremath{\SO_+(2,1)})}_{2g\,\times}.
\end{equation*}
From this description, it is directly apparent that the symplectic leaves of
this Poisson structure are of the form
$\grporbit[1]\times\dots\times\grporbit[n]\times\ensuremath{\ISO(2,1)}^{2g}$, where
$\grporbit[i]\subset \ensuremath{\ISO(2,1)}$ are fixed conjugacy classes.
In the following, we will not use this identification, but we will work with a
description of the Poisson structure that is closer to the formula in
Definition \ref{def:fr}. This formulation has the advantage that it is more
adapted to physics applications, especially in the Chern-Simons formulation of
$(2+1)$-gravity, and that its geometrical interpretation is more apparent. To
emphasise the geometrical interpretation of the variables and their relation
with the set of generators of the fundamental group $\pi_1(S_{g,n})$ in Figure
\ref{fig:fundamental-group} we denote elements of $\ensuremath{\pogr^{n+2g}}$ and $\ensuremath{\logr^{n+2g}}$ as,
respectively,
\begin{equation*}
(M_1,\dots,M_n,A_1,B_1,\dots,A_g,B_g)\in \ensuremath{\pogr^{n+2g}},\qquad
(u_{M_1},\dots,u_{B_g})\in\ensuremath{\logr^{n+2g}},
\end{equation*}
and write $J_{X}^{L,a}$, $J_Y^{R,a}$ for the associated right- and
left-invariant vector fields on $\ensuremath{\logr^{n+2g}}$:
\begin{align*}
J_{X}^{L,a}f(u_{M_1},\dots,u_{B_g})&=\tdiffat{}{t}{t=0} f(u_{M_1},\dots, e^{-tJ_a}\cdot u_X,\dots,u_{B_g}),\\
J_{X}^{R,a}f(u_{M_1},\dots,u_{B_g})&=\tdiffat{}{t}{t=0} f(u_{M_1},\dots, u_X\cdot e^{tJ_a},\dots,u_{B_g}).
\end{align*}
It is shown in \cite{Meusburger:2004aa} that by using the identification
$\ensuremath{\ISO(2,1)}=T\ensuremath{\SO_+(2,1)}$ and the classical $r$-matrix $r=P_a\otimes J^a$, the Poisson
structure given by \eqref{eq:fr-bivector} can be expressed in terms of
functions $f\in C^\infty(\ensuremath{\logr^{n+2g}})$ and certain vector fields on
$\ensuremath{\logr^{n+2g}}$. For this, one identifies the coordinate functions $j^a_X:
(M_1,\dots,B_g)\mapsto q^a_X$, where we use the parametrisation
$X=(u_X,-\Ad(u_X)\boldify{q})$ for $X\in\{M_1,\dots,B_g\}$, with certain vector fields
on $\ensuremath{\logr^{n+2g}}$. The Poisson bracket of two variables $j^a_X,j^b_Y$ then
coincides with the Lie bracket of the associated vector fields, the Poisson
bracket of a variable $j^a_X$ with a function on $\ensuremath{\logr^{n+2g}}$ coincides with its
Lie derivative and any two functions on $\ensuremath{\logr^{n+2g}}$ Poisson-commute.
\begin{theorem}[\cite{Meusburger:2003aa, Meusburger:2004aa}]\label{thm:fr-simple}
For $G=\ensuremath{\ISO(2,1)}$ and $r=P_a\otimes J^a$, the Poisson structure \eqref{eq:fr-bivector}
on $\ensuremath{\pogr^{n+2g}}=T\ensuremath{\logr^{n+2g}}$ is characterised uniquely in terms of vector fields
$j^a_X$, $a\in\{0,1,2\}$, $X\in\{M_1,\dots,B_g\}$ on $\ensuremath{\logr^{n+2g}}$ and functions
on $\ensuremath{\logr^{n+2g}}$:
\begin{equation*}
\{f,g\}=0,\qquad
\{j^a_X, f\}=\mathcal L_{j^a_X} f,\qquad
\{j^a_X,j^b_Y\}=[j^a_{X}, j^b_{Y}]\qquad
\forall f\in C^\infty(\ensuremath{\logr^{n+2g}}),
\end{equation*}
where $\mathcal L$ denotes the Lie derivative and $[\;,\;]$ the Lie bracket on
$\ensuremath{\logr^{n+2g}}$. The vector fields $j^a_X$ on $\ensuremath{\logr^{n+2g}}$ are given by:
\begin{align*}
&j^a_{M_i}=-\left(J_{M_i}^{R,a}+J_{M_i}^{L,a}\right)-\tensor{(\mathds{1}\!-\!\Ad(u_{M_i}))}{^a_b} \smashoperator{\sum_{Y>M_i}} \left(J^{R,b}_Y\!+\!J^{L,b}_Y\right),\\
&j^a_{A_j}=-\left(J_{A_j}^{R,a}+J_{A_j}^{L,a}+J_{B_j}^{L,a}+\tensor{(\mathds{1}\!-\!\Ad(u_{A_j}^\inv u_{B_j}))}{^a_b} J_{B_j}^{R,b}\right)-\tensor{(\mathds{1}\!-\!\Ad(u_{A_j}))}{^a_b} \smashoperator{\sum_{Y>A_j}} \left(J^{R,b}_Y\!+\!J^{L,b}_Y\right),\\
&j^a_{B_j}=-\left(J_{B_j}^{R,a}+J_{B_j}^{L,a}+J_{A_j}^{L,b}\right)-\tensor{(\mathds{1}\!-\!\Ad(u_{B_j}))}{^a_b} \smashoperator{\sum_{Y>A_j}} \left(J^{R,b}_Y\!+\!J^{L,b}_Y\right),
\end{align*}
where $Y>X$ refers to the partial ordering of the generators of $\pi_1(\ensuremath{S_{g,n}})$:
$Y>X$ if $X=M_i$ and $Y=M_j$ with $i<j$ or if $X\in\{A_i,B_i\}$,
$Y\in\{A_j,B_j\}$ with $i<j$ or if $X\in\{M_1,\dots,M_n\}$,
$Y\in\{A_1,B_1,\dots,A_g,B_g\}$.
\end{theorem}
We will now determine the Dirac bracket associated with the Poisson structure
on $\ensuremath{\pogr^{n+2g}}$ and certain smooth constraint functions on $\ensuremath{\pogr^{n+2g}}$. The Dirac
bracket is a well-established formalism from the theory of constrained
Hamiltonian systems and, in a certain sense, can be viewed as the Poisson
counterpart or the physicist's version of symplectic reduction. A more detailed
discussion of this is given in Section \ref{subsec:dirac-constraints}, for the
general theory we refer the reader to
\cite{Dirac:1949aa,Dirac:1950aa,Henneaux:1994aa}.
\begin{definition}[\cite{Dirac:1949aa,Dirac:1950aa}]\label{def:dirac-bracket}
Let $(M,\{\,,\,\})$ be an $n$-dimensional Poisson manifold, $k<n$, and
$C=(C_1,\dots,C_k): M\to\mathbb{R}^k$ a smooth function such that $0$ is a regular
value of $C$ and the matrix $D(p)=(\{C_i,C_j\}(p))_{i,j=1,\dots,k}$ is
invertible for all $p\in \ensuremath{\Sigma}=C^\inv(0)$. The \definee{Dirac bracket} for
$C$ is the antisymmetric bilinear map $\{\,,\,\}_D: C^\infty(\ensuremath{\Sigma})\times
C^\infty(\ensuremath{\Sigma})\to C^\infty(\ensuremath{\Sigma})$ defined by
\begin{equation*}
\{f,g\}_D=\{\tilde f,\tilde g\}\vert_\csurface - \smashoperator{\sum_{i,j=1}^k} \{\tilde f, C_i\} \cdot (D\vert_\csurface)^\inv_{ij} \cdot \{C_j,\tilde g\}\vert_\csurface,
\end{equation*}
where $\tilde f,\tilde g\in C^\infty(M)$ are arbitrary extensions of $f, g\in
C^\infty(\ensuremath{\Sigma})$: $\tilde f\vert_\csurface=f$, $\tilde g\vert_\csurface=g$. The
Dirac bracket does not depend on the choice of these extensions and defines a
Poisson structure on $\ensuremath{\Sigma}$. The submanifold $\ensuremath{\Sigma}=C^\inv(0)\subset
M$ is called \definee{constraint surface}, the functions $C_i: M\to\mathbb{R}$ are
called \definee{constraint functions}.
\end{definition}
The aim is now to determine the Dirac bracket for the Poisson structure from
Definition~\ref{def:fr} for constraint functions that relate this Poisson
structure to the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections:
\begin{align}\label{eq:math-phase-space}
\ensuremath{\mathcal{P}}
&=\{h\in \Hom\bigl(\pi_1(S_{g,n}), \ensuremath{\ISO(2,1)}\bigr) \mid h(m_i)\in\grporbit[i]\}/\ensuremath{\ISO(2,1)}\\
&\cong\{(M_1, \dots, B_g)\in\grporbit[1]\times\dots\times\grporbit[n]\times\ensuremath{\ISO(2,1)}^{2g} \mid \nonumber\\
&\hspace{9em} [B_g,A_g^\inv]\cdots[B_1,A_1^\inv]\cdot M_n\cdots M_1=1\}/\ensuremath{\ISO(2,1)}.\nonumber
\end{align}
As discussed in Section \ref{subsec:constraints-and-gauge-fixing}, this leads
to constraint functions of the form
\begin{equation}\label{eq:generic-constraints}
\left.
\begin{aligned}
&C_1= j_C^0, &\quad &C_2= j_C^1, &\quad &C_3= j_C^2, \\
&C_4=\textstyle{\sum_{i=1}^2} \ThetaFam{1}{M_i}{a}\,j^a_{M_i}, &\quad
&C_5=\textstyle{\sum_{i=1}^2} \ThetaFam{2}{M_i}{a}\,j^a_{M_i}, &\quad
&C_6=\textstyle{\sum_{i=1}^2} \ThetaFam{3}{M_i}{a}\,j^a_{M_i}, \\
&C_7=\Tr(J_0\cdot u_C), &\quad &C_8=\Tr(J_1\cdot u_C), &\quad &C_9=\Tr(J_2\cdot u_C), \\
&C_{10}=\Delta_1, &\quad
&C_{11}=\Delta_2, &\quad
&C_{12}=\Delta_3,
\end{aligned}
\quad\right\}
\end{equation}
where $j^a_{M_i}$ is defined as in Theorem \ref{thm:fr-simple} and
$\ThetaFam{j}{M_i}{a}, \Delta_j \in C^\infty(\ensuremath{\SO_+(2,1)}\times \ensuremath{\SO_+(2,1)})$ are functions that
depend only on the $\ensuremath{\SO_+(2,1)}$-part of the first two copies of $\ensuremath{\ISO(2,1)}$ and
\begin{equation*}
(u_C^\inv, \boldify{j}_C)
\coloneqq M_1^\inv\cdots M_n^\inv \cdot [A_1^\inv, B_1]\cdots[A_g^\inv, B_g].
\end{equation*}
The functions $(C_i)_{i=1,2,3,7,8,9}$ have an interpretation as first-class
constraints in the Dirac gauge fixing formalism and the functions
$(C_i)_{i=4,5,6,10,11,12}$ play the role of gauge fixing conditions. While the
former are fixed and implement the condition
$[B_g,A_g^\inv]\cdots[B_1,A_1^\inv]\cdot M_n\cdots M_1=1$, the latter involve
functions $\ThetaFam{j}{M_i}{a}, \Delta_j $ which can be chosen arbitrarily as
long as the requirements from Definition \ref{def:dirac-bracket} and the
conditions a and b from Section~\ref{subsec:constraints-and-gauge-fixing} are
met. The latter ensure that the constraint functions are adapted to the tangent
bundle structure of $\ensuremath{\ISO(2,1)}=T\ensuremath{\SO_+(2,1)}$. Different choices of these functions
correspond to different gauge choices. They implement the quotient by $\ensuremath{\ISO(2,1)}$
in \eqref{eq:math-phase-space} and restrict the variables $M_1,M_2$ in such a
way that for all points $(M_1,\dots,B_g)\in \ensuremath{\Sigma}=C^\inv(0)$, the
components $M_1,M_2\in \ensuremath{\ISO(2,1)}$ are determined uniquely by two real parameters
\begin{align}\label{eq:psi-alpha-general}
\psi=f(\Tr(u_{12})), \quad
\alpha=g(\Tr(u_{12})) \Tr(j_{12}^aJ_a\cdot u_{12})+h(\Tr(u_{12})),
\end{align}
where $f,g\inC^\infty(\mathbb{R})$ are arbitrary diffeomorphisms and $h\inC^\infty(\mathbb{R})$.
This allows us to identify the constraint surface $\ensuremath{\Sigma}=C^\inv(0)$ with a
subset of $\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$, where the $\mathbb{R}^2$ is parametrised by
$(\psi,\alpha)$ and $\ensuremath{\pogr^{n-2+2g}}$ by $(M_3,\dots,B_g)$.
Given the expressions for the Poisson structure on $\ensuremath{\pogr^{n+2g}}$ and the
constraint functions \eqref{eq:generic-constraints}, we can explicitly compute
the associated Dirac bracket and obtain a Poisson structure on
$\ensuremath{\Sigma}\subset\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$.
\begin{theorem}\label{thm:generic-dirac-bracket}
For all constraint functions of the form \eqref{eq:generic-constraints} that
satisfy the requirements in Definition \ref{def:dirac-bracket} and conditions
a and b from Section \ref{subsec:constraints-and-gauge-fixing}, the
associated Dirac bracket defines a Poisson structure $\{\,,\,\}_D$ on
$\ensuremath{\Sigma} \subset \mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$, which takes the following form:
\begin{enumerate}
\item The Dirac bracket of $\psi$ and $\alpha$ vanishes: $\{\psi, \alpha\}_D
= 0$.
\item For all $X \in \{M_3, \dots, B_g\}$ and $f\inC^\infty(\ensuremath{\logr^{n-2+2g}})$:
\begin{equation*}
\begin{aligned}
\{\psi, f \}_D &= 0, &
\{\psi, \boldify{j}_X \}_D &=
-\idadi{X} \, \boldify{q}_\psi, \\
\{\alpha, f\}_D &=
\smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} q_\alpha^a(J_a^{R,Y}+J_a^{L,Y})f, &
\{\alpha, \boldify{j}_X\}_D &=
-\idadi{X} \boldify{q}_\theta - \boldify{q}_\alpha \wedge \boldify{j}_X,
\end{aligned}
\end{equation*}
with $\boldify{q}_\psi, \boldify{q}_\alpha,\boldify{q}_\theta: \mathbb{R}^2 \to \mathbb{R}^3$ satisfying
$\boldify{q}_\psi \wedge \boldify{q}_\alpha = 0$ and
$\partial_\alpha\boldify{q}_\psi=\partial_\alpha\boldify{q}_\alpha=\partial_\alpha^2
\boldify{q}_\theta= 0$.
\item For $F, G \in C^\infty(\ensuremath{\pogr^{n-2+2g}})$: $$\{F, G\}_D = \frrestbivector(\mathrm{d} F
\otimes \mathrm{d} G),$$ where $\frrestbivector$ is the Poisson bivector
\eqref{eq:fr-bivector} and $r: \mathbb{R}^2 \to \ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ is given by
\begin{equation*}
r (\psi,\alpha)= P_a \otimes J^a - V^{bc}(\psi)(P_b \otimes J_c - J_c \otimes P_b) + \ee^{bcd}m_d(\psi,\alpha) P_b \otimes P_c,
\end{equation*}
where $V:\mathbb{R} \to \Mat(3, \mathbb{R})$ and $\boldify{m}:\mathbb{R}^2 \to \mathbb{R}^3$ satisfies
$\partial_\alpha^2 \boldify{m}=0$.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof is a direct generalisation of the proof of Theorem 5.1 in
\cite{Meusburger:2011aa}.
\begin{enumerate}
\item The Dirac matrix associated to the constraints
\eqref{eq:generic-constraints} takes the form
\begin{equation*}
D = \begin{pmatrix}J & P \\ -P^T & 0\end{pmatrix} \quad\text{with}\quad
J \coloneqq (\{C_i, C_j\})_{i,j=1,\ldots,6}, \;
P \coloneqq (\{C_i,C_{j+6}\})_{i,j=1,\ldots,6}.
\end{equation*}
On the constraint surface, the $(6 \times 6)$-matrices $J$ and $P$ can be
expressed as
\begin{equation*}
J\vert_\csurface=\begin{pmatrix}0 & H \\ -H^T & G\end{pmatrix}, \qquad
P\vert_\csurface=\begin{pmatrix}0 & A \\ B & C\end{pmatrix},
\end{equation*}
with $(3\times 3)$-matrices $A, B, C, G, H$ given by
\begin{equation*}
\left.
\begin{gathered}
A_{ij} \coloneqq \{C_i, C_{j+9}\}\vert_\csurface, \quad
B_{ij} \coloneqq \{C_{i+3}, C_{j+6}\}\vert_\csurface, \quad
C_{ij} \coloneqq \{C_{i+3}, C_{j+9}\}\vert_\csurface, \\
G_{ij} \coloneqq \{C_{i+3}, C_{j+3}\}\vert_\csurface, \quad
H_{ij} \coloneqq \{C_i, C_{j+3}\}\vert_\csurface,
\end{gathered}
\;\right\}\; i,j=1,2,3.
\end{equation*}
This implies that the inverse of the Dirac matrix $D$ on the constraint
surface is given by
\begin{align*}
&(D\vert_\csurface)^\inv = \begin{pmatrix}0 & -(P^\inv)^T \\ (P\vert_\csurface)^\inv & (P\vert_\csurface)^\inv (J\vert_\csurface) (P\vert_\csurface^\inv)^T\end{pmatrix}
\;\;\text{with}\;\;
(P\vert_\csurface)^\inv = \begin{pmatrix}-B^\inv C A^\inv & B^\inv \\ A^\inv & 0\end{pmatrix}, \\
&(P\vert_\csurface)^\inv (J\vert_\csurface) (P\vert_\csurface^\inv)^T =
\begin{pmatrix}
B^\inv\bigl[G-CA^\inv H+(CA^\inv H)^T\bigr](B^\inv)^T & -B^\inv H^T(A^\inv)^T \\
A^\inv H (B^\inv)^T & 0
\end{pmatrix}.
\end{align*}
Inserting these expression into the general formula in Definition
\ref{def:dirac-bracket}, one finds that for all $X, Y \in \{M_1, \dots,
B_g\}$ and $f,g\inC^\infty(\ensuremath{\logr^{n+2g}})$, the Dirac bracket takes the form
\begin{subequations}\label{eq:generic-dirac-general}
\begin{align}
\{f, g\}_D &= 0, \\
\begin{split}
\{j_X^a, f\}_D &= \{j_X^a, f\}\vert_\csurface + \smashoperator{\sum_{i,j=1}^3} \Bigl[
\{j_X^a, C_{i+6}\} (B^\inv)_{ij} \{f, C_{j+3}\}
+\{j_X^a, C_{i+9}\} (A^\inv)_{ij} \{f, C_j\} \\[-0.6em] &\hspace{8em}
-\{j_X^a, C_{i+6}\} (B^\inv C A^\inv)_{ij} \{f, C_j\}
\Bigr]\vert_\csurface,
\end{split} \raisetag{3.8em}\\
\begin{split}
\{j^a_X, j^b_Y\}_D &= \{j^a_X,j^b_Y\}\vert_\csurface
+ \sum_{i=1}^6\sum_{j=7}^{12} \{j^a_X, C_i\}\vert_\csurface (D\vert_\csurface^\inv)_{ij} \{j^b_Y, C_j\}\vert_\csurface \\[-0.8em]
+ &\sum_{i=7}^{12}\sum_{j=1}^6 \{j^a_X, C_i\}\vert_\csurface (D\vert_\csurface^\inv)_{ij} \{j^b_Y, C_j\}\vert_\csurface
+ \sum_{i=7}^{12}\sum_{j=7}^{12} \{j^a_X, C_i\}\vert_\csurface (D\vert_\csurface^\inv)_{ij} \{j^b_Y, C_j\}\vert_\csurface.
\end{split}\raisetag{5.0em}
\end{align}
\end{subequations}
\item To prove the relations for the brackets involving $\psi$ and $\alpha$,
we use \eqref{eq:generic-dirac-general} to compute the Dirac brackets of
$j_{M_1}$, $j_{M_2}$ and functions $g\inC^\infty(\ensuremath{\SO_+(2,1)}\times \ensuremath{\SO_+(2,1)})$ of the
variables $u_{M_1}$, $u_{M_2}$ with functions $f\inC^\infty(\ensuremath{\logr^{n-2+2g}})$ of the
variables $M_3,\dots,B_g$. It follows directly from the block form of
$(D\vert_\csurface)^\inv$ that $\{g, f\}_D = 0$ and hence $\{\psi, f\}_D = 0$
by \eqref{eq:psi-alpha-general}. For $X\in\{M_3,\dots,B_g\}$, we have
$\{j^a_X, C_{i+9}\} = 0$ for all $i\in\{1,2,3\}$ and thus
\begin{equation*}
\{j_{X}^a, g\}_D = -\tidadi{X}{^a_c} \smashoperator{\sum_{j,k=1}^3} \tensor{W}{^c_{k-1}} \bigl[
(B^{-1})_{kj} \{g, C_{j+3}\} - (B^{-1} C A^{-1})_{kj} \{g, C_j\}
\bigr]\vert_\csurface,
\end{equation*}
where $W: \mathbb{R} \to \Mat(3, \mathbb{R})$ is a function of the variable $\psi$ from
\eqref{eq:psi-alpha-general} defined by the condition $\{j_X^a, C_{i+6}\} =
-\tidadi{X}{^a_b} \tensor{W}{^b_{i-1}}$ for all $i = 1, 2, 3$,
$X\in\{M_3,\dots,B_g\}$. From equations \eqref{eq:generic-constraints} for
the constraints and the definition of the matrices $A,B,C$ it follows that
the right-hand side of this equation can be expressed as a function of
$\psi$ and the fixed parameters that characterise the conjugacy classes
$\grporbit[1], \grporbit[2]$. This implies that there is a map $\boldify{q}_\psi:
\mathbb{R}^2 \to \mathbb{R}^3$ with $\partial_\alpha\boldify{q}_\psi=0$ such that
\begin{equation}\label{eq:psijX}
\{\psi, \boldify{j}_X\}_D = -\idadi{X} \boldify{q}_\psi.
\end{equation}
Similarly, we obtain for $i\in\{1,2\}$ and functions
$f\inC^\infty(\ensuremath{\logr^{n-2+2g}})$:
\begin{align*}
&\{j_{M_i}^a, f\}_D= \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} (J^{R,c}_Yf+J^{L,c}_Yf) \Bigl\{
-\tidadi{M_i}{^a_c}
+\smashoperator{\sum_{k,j=1}^3} \Bigl[\{j_{M_i}^a, C_{k+9}\} (A^\inv)_{kj} \delta^{j-1}_c \\[-1em]
&+\{j_{M_i}^a, C_{k+6}\} (B^\inv)_{kj} \sum_{l=1}^2 \ThetaFam{j}{M_l}{d} \tidadi{M_l}{^d_c}
- \{j_{M_i}^a, C_{k+6}\} (B^\inv C A^\inv)_{kj} \delta^{j-1}_c
\Bigr]
\Bigr\}\vert_\csurface.
\end{align*}
The term inside the curly brackets on the right-hand side again depends on
$\psi$ only, which shows that there is a map $\boldify{q}_\alpha: \mathbb{R}^2 \to \mathbb{R}^3$
with $\partial_\alpha\boldify{q}_\alpha=0$ such that
\begin{equation}\label{eq:alphapX}
\{\alpha, f\}_D = \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} q_\alpha^a(J^Y_{R,a}+J^Y_{L,a}) f\vert_\csurface
\end{equation}
The remaining brackets which involve $\psi,\alpha$ and the variables
$j^a_X$, $X\in\{M_3,\dots,B_g\}$, are obtained from the Dirac brackets of
$j^a_{M_i}$, $i=1,2$, with $j^a_X$:
\begin{align*}
& \{j^a_{M_i}, j^b_X\}_D =
- \tensor{\ee}{^b^c_d} j_X^d \Bigl[
-\tidadi{M_i}{^a_c}
+ \sum_{k=7}^{12}\sum_{j=1}^3 \{j^a_{M_i}, C_k\} (D^\inv)_{kj} \delta^{j-1}_c \\[-1em]
&\hspace{10em} + \sum_{k=7}^{12}\sum_{j=4}^6 \{j^a_{M_i}, C_k\} (D^\inv)_{kj} \sum_{l=1}^2 \ThetaFam{j-4}{M_l}{e} \tidadi{M_l}{^e_c}
\Bigr]\vert_\csurface\\[-.5em]
& - \tidadi{X}{^b_c} \Bigl[
\sum_{k=1}^6\sum_{j=7}^{9} \{j^a_{M_i}, C_k\} (D^\inv)_{kj} \tensor{W}{^c_{j-7}}
- \sum_{k=7}^{12}\sum_{j=7}^{9} \{j^a_{M_i}, C_k\} (D^\inv)_{kj} \tensor{W}{^c_{j-7}}
\Bigr]\vert_\csurface.
\end{align*}
The term in the second set of square brackets depends on $\psi$ and
$\alpha$ while the term in the first set of square brackets coincides with
the term in the curly brackets in the expression for $\{j^a_{M_i},f\}_D$.
This implies that there is a map $\boldify{q}_\theta: \mathbb{R}^2 \to \mathbb{R}^3$,
$\partial_\alpha^2\boldify{q}_\theta=0$ such that for all $X\in\{M_3,\dots,B_g\}$:
\begin{equation*}
\{\alpha, \boldify{j}_X\}_D = -\idadi{X}\boldify{q}_\theta - \boldify{q}_\alpha \wedge \boldify{j}_X.
\end{equation*}
It remains to show that $\{\alpha, \psi\}_D = 0$ and that $\boldify{q}_\psi\wedge
\boldify{q}_\alpha = 0$. With the definitions
\begin{equation*}
\begin{aligned}
(u_{12}^\inv, \boldify{j}_{12}) &\coloneqq M_1^\inv \cdot M_2^\inv, \\
(u_R^\inv, \boldify{j}_R) &\coloneqq M_3^\inv\cdots M_n^\inv[A_1^\inv,B_1]\cdots [A_g^\inv,B_g],
\end{aligned}
\end{equation*}
the constraints $C_7,C_8,C_9$ imply $\Tr(u_{12})=\Tr(u_R)$. From the Dirac
brackets \eqref{eq:alphapX} of $\alpha$ with functions
$f\inC^\infty(\ensuremath{\logr^{n-2+2g}})$ of the holonomies $M_3,\dots,B_g$, it follows that
the Dirac bracket of $\alpha$ and $\psi$ vanishes. Moreover, the
constraint functions $C_1,C_2,C_3$ imply $\boldify{j}_{12} = -\Ad(u_{12}^\inv)
\boldify{j}_R$ on $\ensuremath{\Sigma}$ and it follows from \eqref{eq:psijX} that
\begin{equation}\label{eq:psi-j12}
0 = \{\psi, \boldify{j}_{12}\}_D = -\Ad(u_{12}^\inv) \{\psi, \boldify{j}_R\}_D = -\idadi{12} \boldify{q}_\psi.
\end{equation}
For each function $g\inC^\infty(\ensuremath{\SO_+(2,1)})$, we have two associated functions
$g_{\mathbb{R}^2}\in C^\infty(\mathbb{R}^2)$, $g_{\mathbb{R}^2}(\psi,\alpha)\coloneqq g(u_{12}^\inv)$
and $\bar g\inC^\infty(\ensuremath{\logr^{n-2+2g}})$, $\bar g(u_{M_3},\dots,u_{B_g})\coloneqq
g(u_R)$. With the identity $\{\psi,\alpha\}_D= 0$, we obtain
\begin{equation*}
0 = \{\alpha, g_{\mathbb{R}^2}\}_D= \{\alpha, \bar g\}_D= \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} q^a_\alpha (J_{R,a}^Y+J_{L,a}^Y)\bar g.
\end{equation*}
Together with \eqref{eq:psi-j12}, this implies that both,
$\exp(q_\alpha^aJ_a)$ and $\exp(q_\psi^a J_a)$, stabilise $u_{12}$ and
hence $\boldify{q}_\psi \wedge \boldify{q}_\alpha = 0$.
\item To prove the second part of the theorem, we explicitly compute the
Dirac brackets of the variables $\boldify{j}_X$ for $X \in \{M_3, \dots, B_g\}$ and
functions $f\inC^\infty(\ensuremath{\logr^{n-2+2g}})$ from expressions
\eqref{eq:generic-dirac-general}. To determine the brackets $\{\boldify{j}_X,
f\}_D$, we note that $\{j^a_X, C_{i+9}\} = 0$ for all $i\in\{1,2,3\}$,
which implies
\begin{align}\label{eq:j-f-bracket}
&\{j_X^a, f\}_D = \{j_X^a, f\}\vert_\csurface - \tidadi{X}{^a^e} \, V_{ed} \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} (J^{R,d}_Y+J^{L,d}_Y) f\vert_\csurface
\quad\text{with} \\[-.3em]
&V_{ed} \coloneqq
\tensor{W}{_e^f}(B^\inv C A^\inv)_{f+1,d+1}\vert_\csurface
- \tensor{W}{_e^f} \sum_{j=1}^3 (B^\inv)_{f+1,j} \smashoperator{\sum_{i =1}^2}
\ThetaFam{j}{M_i}{a} \tidad{M_i}{_d^a}\vert_\csurface.\nonumber
\end{align}
As none of the terms in the expression for $V$ depend on $\alpha$, it gives
rise to a map $V:\mathbb{R}^2\to \Mat(3, \mathbb{R})$ that satisfies $\partial_\alpha
V=0$. Similarly, we obtain
\begin{align}\label{eq:j-j-bracket}
\{j^a_X, j^b_Y\}_D &= \{j^a_X,j^b_Y\}\vert_\csurface
+ \idadi{X}^{ad} \, V_{dg} \, \tensor{\ee}{^g^b_f} j_Y^f \vert_\csurface\nonumber
- \idadi{Y}^{bd} \, V_{dg} \, \tensor{\ee}{^g^a_f} j_X^f\vert_\csurface \\ &\hspace{5em}
+ \idadi{X}^{ac} \idadi{Y}^{bd} U_{cd}\vert_\csurface
\end{align}
with $U_{cd} \coloneqq \tensor{W}{_c^e} \tensor{W}{_d^f} (D^\inv)_{e+7,f+7}$
for all $c,d\in\{0,1,2\}$. The matrix $U$ depends only on the parameters
$\psi$ and $\alpha$, and its dependence on $\alpha$ is at most
linear. Moreover, it follows directly from the definition of the matrix $D$
that $U$ is antisymmetric. This allows us to expand $U$ in a basis:
$U^{ab}=\ee^{abc}m_c$ with $\boldify{m}:\mathbb{R}^2\to \mathbb{R}^3$, $\partial_\alpha^2\boldify{m}=0$.
\item By inserting the expressions \eqref{eq:vector-fields},
\eqref{eq:action-of-vector-fields} for the left- and right-invariant vector
fields on $\ensuremath{\ISO(2,1)}$ into the Poisson bivector \eqref{eq:fr-bivector} together
with the expression for $r(\psi,\alpha)$, one obtains after some
computations expressions \eqref{eq:j-f-bracket},
\eqref{eq:j-j-bracket}. This proves the claim.
\end{enumerate}
\end{proof}
Theorem \ref{thm:generic-dirac-bracket} gives explicit expressions for the
Dirac bracket for a rather general set of gauge fixing conditions. This
generalises the results from \cite{Meusburger:2011aa}, which investigates
specific gauge fixing conditions of this type. Given the fact that the Dirac
bracket is obtained from six first-class constraints with six associated gauge
fixing conditions and hence involves inverting a $(12\times 12)$-Dirac matrix,
its structure is surprisingly simple. This is partly due to the restriction
that the gauge fixing conditions are adapted to the tangent bundle structure of
$\ensuremath{\ISO(2,1)}=T\ensuremath{\SO_+(2,1)}$.
\subsection{The Dirac bracket and the classical dynamical Yang-Baxter equation}
The Dirac bracket in Theorem \ref{thm:generic-dirac-bracket} defines a Poisson
structure on the constraint surface $\ensuremath{\Sigma}= C^\inv(0)$ which can be
identified with a subset of $\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$. However, this
identification is implicit, and it is cumbersome to give an explicit
parametrisation of this subset for general gauge fixing conditions. For this
reason, we consider in the following the bracket on $\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$
defined by the expressions in Theorem \ref{thm:generic-dirac-bracket}.
\begin{definition}\label{def:extended-dirac-bracket}
We denote by $\{\,,\,\}_D$ the antisymmetric bilinear function $\{\,,\,\}_D:
C^\infty(\mathbb{R}^2\times \ensuremath{\pogr^{n-2+2g}})\times C^\infty(\mathbb{R}^2\times \ensuremath{\pogr^{n-2+2g}})\to
C^\infty (\mathbb{R}^2\times \ensuremath{\pogr^{n-2+2g}})$ that takes the form described in Theorem
\ref{thm:generic-dirac-bracket}. With $\mathbb{R}^2$ parametrised by $\psi,\alpha$ and
the different copies of $\ensuremath{\ISO(2,1)}$ labelled by
$\{M_3,\dots,M_n,A_1,B_1,\dots,A_g,B_g\}$, this bracket is given by:
\begin{enumerate}
\item $\{\psi,\alpha\}_D=0$, and for all $f\inC^\infty(\ensuremath{\logr^{n-2+2g}})$,
$X,Y\in\{M_3,\dots,B_g\}$:
\begin{equation}\label{eq:extended-dirac-bracket-for-gauge-fixed}
\begin{aligned}
\{\psi, f \}_D &= 0, &
\{\psi, \boldify{j}_X \}_D &= -\idadi{X} \, \boldify{q}_\psi, \\
\{\alpha, f\}_D &= \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} q_\alpha^a(J_{R,a}^Y+J_{L,a}^Y)f, &
\{\alpha, \boldify{j}_X\}_D &= -\idadi{X} \boldify{q}_\theta - \boldify{q}_\alpha \wedge \boldify{j}_X,
\end{aligned}
\end{equation}
with $\boldify{q}_\psi, \boldify{q}_\alpha,\boldify{q}_\theta: \mathbb{R}^2 \to \mathbb{R}^3$ satisfying $\boldify{q}_\psi
\wedge \boldify{q}_\alpha = 0$ and
$\partial_\alpha\boldify{q}_\psi=\partial_\alpha\boldify{q}_\alpha=\partial_\alpha^2
\boldify{q}_\theta= 0$.
\item For all functions $F, G \in C^\infty(\ensuremath{\pogr^{n-2+2g}})$: $\{F, G\}_D =
\frrestbivector(\mathrm{d} F \otimes \mathrm{d} G)$, where $\frrestbivector$ is the
Poisson bivector \eqref{eq:fr-bivector} and $r: \mathbb{R}^2 \to \ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ is
given by
\begin{equation}\label{eq:extended-dirac-r}
r (\psi,\alpha)= P_a \otimes J^a - V^{bc}(\psi)(P_b \otimes J_c - J_c \otimes P_b) + \ee^{bcd}m_d(\psi,\alpha) P_b \otimes P_c,
\end{equation}
with a map $V:\mathbb{R} \to \Mat(3, \mathbb{R})$ that does not depend on $\alpha$ and a
vector-valued function $\boldify{m}:\mathbb{R}^2\to \mathbb{R}^3$ satisfying
$\partial_\alpha^2\boldify{m}=0$.
\end{enumerate}
\end{definition}
This bracket has a particularly simple structure. The two variables $\psi$ and
$\alpha$ Poisson-commute, and their Dirac brackets with functions on
$\ensuremath{\pogr^{n-2+2g}}$ are given by three functions
$\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta:\mathbb{R}^2\to\mathbb{R}^3$. The Dirac bracket of two
functions on $\ensuremath{\pogr^{n-2+2g}}$ is again given by the Poisson bivector
\eqref{eq:fr-bivector}. The only difference is that the classical $r$-matrix
$r=P_a\otimes J^a$ in the Poisson bivector $\frrestbivector$ is now replaced by the
map $r:\mathbb{R}^2 \to \ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ that depends on the variables $\psi$ and
$\alpha$.
Note that it is a priori not guaranteed that the bracket $\{\,,\,\}_D$ on
$\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$ satisfies the Jacobi identity. The Dirac gauge fixing
formalism only guarantees that this is the case on the constraint surface
$\ensuremath{\Sigma}=C^\inv(0)\subset \mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$. Moreover, it is natural to
ask how the Jacobi identity is encoded in the structures that characterise the
bracket in Definition \ref{def:extended-dirac-bracket}: the map $r:\mathbb{R}^2\to
\ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ and the vector-valued functions $\boldify{q}_\psi,
\boldify{q}_\alpha,\boldify{q}_\theta: \mathbb{R}^2 \to \mathbb{R}^3$. As the classical Yang-Baxter equation
for the $r$-matrix in the Poisson bivector \eqref{eq:fr-bivector} ensures that
the associated bracket satisfies the Jacobi identity, it is natural to expect
that the Jacobi identity for the Dirac bracket follows from an analogous
property of the map $r$. This suggests that $r$ should be related to solutions
of the classical {\em dynamical} Yang-Baxter equation and hence to classical
dynamical $r$-matrices for the Lie algebra $\ensuremath{\iso(2,1)}$.
This intuition is also supported by the fact that Fock and Rosly's Poisson
structure is related to certain Poisson structures from the theory of
Poisson-Lie groups \cite{Fock:1998aa, Alekseev:1995ab}. It is shown in
\cite{Feher:2001aa, Feher:2004aa} that Dirac gauge fixing in the context of
Poisson-Lie groupoids is linked to classical dynamical $r$-matrices. Note,
however, that our case is more involved. While the references
\cite{Feher:2001aa, Feher:2004aa} consider a gauge fixing procedure for a
generalisation of the Sklyanin bracket in the context of Poisson-Lie groupoids,
our Poisson structure involves several copies of the dual Poisson-Lie structure
and the Heisenberg double Poisson structure which interact in a non-trivial
way. Moreover, the gauge fixing conditions we consider are associated with two
punctures and hence with two non-Poisson-commuting dual Poisson-Lie structures
whose Poisson brackets with the remaining punctures and handles do not
vanish. Nevertheless, it is natural to expect that our gauge fixing procedure
should be related to solutions of the classical dynamical Yang-Baxter equation.
The concept of a classical dynamical $r$-matrix generalises the notion of
classical $r$-matrices $r\in\algebra{g}\otimes\algebra{g}$ for a Lie algebra $\algebra{g}$ to maps $r: U\to
\algebra{g}\otimes\algebra{g}$ that depend non-trivially on variables in $U$. The domain $U$ is an
open subset of the dual $\algebra{h}^*$ of an abelian Lie subalgebra $\algebra{h}\subset\algebra{g}$,
and the map $r$ is required to be invariant under the action of $\algebra{h}$. Instead
of the classical Yang-Baxter equation (CYBE), the classical dynamical
$r$-matrix is required to satisfy the classical dynamical Yang-Baxter equation
(CDYBE). The latter is obtained by replacing the right-hand side of the CYBE
by a term that contains the derivatives of $r$ with respect to the coordinates
on $U$.
\begin{definition}[\cite{Etingof:1998aa}]\label{def:dyn-r-matrix}
Let $\algebra{g}$ be a finite-dimensional Lie algebra, $\algebra{h} \subset \algebra{g}$ an abelian
Lie subalgebra, and $U \subset \algebra{h}^*$ an open subset. A \definee{classical
dynamical $r$-matrix} for $(\algebra{g}, \algebra{h}, U)$ is an $\algebra{h}$-invariant,
meromorphic function $r: U \to \algebra{g} \otimes \algebra{g}$ that satisfies the
\definee{classical dynamical Yang-Baxter equation} (CDYBE):
\begin{equation}\label{eq:dcybe}
[[r,r]] \coloneqq [r_{12}, r_{13}] + [r_{12}, r_{23}] + [r_{13}, r_{23}] =
\sum_{i=1}^{\text{dim}\,\algebra{h}} \left(
x_i^{(1)}\partial_{x^i}\,r_{23} - x_i^{(2)}\partial_{x^i}\,r_{13} + x_i^{(3)}\partial_{x^i}\,r_{12}
\right),
\end{equation}
where $\{x_i\}_{i=1,\dots,\text{dim}\, \algebra{h}}$ is a basis of $\algebra{h}$ and
$\{x^i\}_{i=1,\dots,\text{dim}\, \algebra{h}}$ the associated dual basis of $\algebra{h}^*$.
\end{definition}
In the following, we only require the case where $\algebra{g} = \ensuremath{\iso(2,1)}$ and $\algebra{h}$ is a
two-dimensional abelian Lie subalgebra of $\algebra{g}$. We thus identify $\algebra{h}^*$ with
$\mathbb{R}^2$ and parametrise it by two variables $x^1=\psi$, $x^2=\alpha$.
Moreover, we temporarily drop the requirements that the elements $x_1,x_2$ in
the CDYBE form a fixed basis of $\algebra{h}\subset\ensuremath{\iso(2,1)}$ and that $r$ is invariant
under the action of $\algebra{h}$.
Instead, we investigate solutions of the CDYBE \eqref{eq:dcybe} associated with
maps $x_1,x_2:\mathbb{R}^2\to\ensuremath{\iso(2,1)}$ of the form $x_1=q_\psi^aP_a$ and
$x_2=q_\alpha^aJ_a+q_\theta^aP_a$ with $\boldify{q}_\psi,\boldify{q}_\alpha:\mathbb{R}\to\mathbb{R}^3$,
$\boldify{q}_\theta:\mathbb{R}^2\to\mathbb{R}^3$ satisfying $\boldify{q}_\psi\wedge\boldify{q}_\alpha=0$. Note that
this implies that $\algebra{h}(\psi,\alpha)=\Span\{x_1(\psi,\alpha),
x_2(\psi,\alpha)\}$ is a two-dimensional abelian Lie subalgebra of $\ensuremath{\iso(2,1)}$ for
all values of $\psi$ and $\alpha$. It is a Cartan subalgebra if and only if
$\boldify{q}_\psi, \boldify{q}_\alpha$ satisfy the additional requirement $\boldify{q}_\alpha^2,
\boldify{q}_\psi^2\neq 0$. We do not assume that $r(\psi,\alpha)$ is invariant under
the subalgebra $\algebra{h}(\psi,\alpha)$.
Although such solutions of the CDYBE \eqref{eq:dcybe} do not correspond to
classical dynamical $r$-matrices in the sense of Definition
\ref{def:dyn-r-matrix}, admitting such generalised solutions allows us to apply
the CDYBE to the maps $r$ and $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta$ in Definition
\ref{def:extended-dirac-bracket} and to determine under which conditions they
give rise to a solution of the CDYBE. By comparing these conditions to the
requirement that the bracket $\{\,,\}_D$ in Definition
\ref{def:extended-dirac-bracket} satisfies the Jacobi identity, we obtain the
following theorem.
\begin{theorem}\label{thm:jacobi-for-extended-dirac}
The bracket $\{\,,\,\}_D$ in Definition \ref{def:extended-dirac-bracket}
satisfies the Jacobi identity and hence defines a Poisson structure on
$\ensuremath{\pogr^{n-2+2g}}$ if and only if:
\begin{compactenum}
\item The map $r:\mathbb{R}^2\to \ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ in \eqref{eq:extended-dirac-r}
satisfies the CDYBE with $x^1=\psi$, $x^2=\alpha$ and $x_1=q_\psi^aP_a$,
$x_2=q_\alpha^aJ_a+q_\theta^a P_a$.
\item The following additional conditions hold:
\begin{equation}\label{eq:q-relations}
\left.
\begin{aligned}
0 &= q_\psi^a + \tensor{\ee}{^a_b_c} q_\psi^b \partial_\psi q_\psi^c + q_\psi^b \tensor{V}{_b^a} - q_\psi^a \tensor{V}{^b_b}, \\
0 &= \tensor{\ee}{^a_d_h} q_\alpha^d V^{bh} + \tensor{\ee}{^b_d_h} q_\alpha^d V^{ah} + \tensor{\ee}{_c_d_e} q_\alpha^c V^{de} \eta^{ab} - \tensor{\ee}{^b_d_e} q_\alpha^a V^{de} + q_\alpha^a \partial_\alpha q_\theta^b - q_\psi^b \partial_\psi q_\alpha^a, \\
0 &= q_\theta^a + \tensor{\ee}{^a_b_c} q_\theta^b \partial_\alpha q_\theta^c + \tensor{\ee}{^a_b_c} q_\psi^b \partial_\psi q_\theta^c - \tensor{\ee}{^a_b_c} m^b q_\alpha^c + q_\theta^d \tensor{V}{_d^a} - q_\theta^a \tensor{V}{_d^d}.
\end{aligned}
\quad\right\}
\end{equation}
\end{compactenum}
\end{theorem}
\begin{proof} $\quad$
\begin{enumerate}
\item As a first step, we show that a map $r: \mathbb{R}^2 \to \ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ of
the form \eqref{eq:extended-dirac-r} is a solution of the CDYBE with
$x^1=\psi$, $x^2=\alpha$, $x_1=q_\psi^a P_a:\mathbb{R}\to\mathbb{R}^3$, $x_2=q_\alpha^a
J_a+q_\theta^a P_a:\mathbb{R}^2\to\mathbb{R}^3$ if and only if it satisfies the equations
\begin{equation}\label{eq:dcybe-with-UV}
\left.
\begin{aligned}
0 &= \Upsilon^{abc} \coloneqq q_\alpha^a \ee ^{bcd} \partial_\alpha m_d - q_\psi^b \partial_\psi V^{ca} + q_\psi^c \partial_\psi V^{ba} \\
&\qquad\qquad\quad - V^{bd}V^{cg}\tensor{\ee}{_d_g^a} - V^{da}V^{cg}\tensor{\ee}{_d_g^b} + V^{da}V^{bg}\tensor{\ee}{_d_g^c} - V^{da} \tensor{\ee}{^b^c_d}, \\
0 &= \Omega\coloneqq
\boldify{q}_\psi \cdot \partial_\psi \boldify{m} + \boldify{q}_\theta \cdot \partial_\alpha \boldify{m} +\boldify{w} \cdot \boldify{m}
\qquad\text{with}\quad \ee^{abc}w_c=V^{ab}-V^{ba}.
\end{aligned}
\qquad\right\}
\end{equation}
Inserting expression \eqref{eq:extended-dirac-r} for $r$ into the left-hand
side of the CDYBE \eqref{eq:dcybe} and using expressions
\eqref{eq:poincare-algebra-bracket} for the Lie bracket of $\ensuremath{\iso(2,1)}$, we obtain
\begin{equation*}
\begin{split}
&[[r, r]] = -\boldify{w} \cdot \boldify{m} \, \ee^{abc} P_a \otimes P_b \otimes P_c + {} \\
&\bigl[V^{bd}V^{cg}\tensor{\ee}{_d_g^a} \!+\! V^{da}V^{cg}\tensor{\ee}{_d_g^b}\! -\! V^{da}V^{bg}\tensor{\ee}{_d_g^c} \!+ \!V^{da} \tensor{\ee}{^b^c_d}\bigr]
(J_a\!\!\otimes\!\!P_b\!\!\otimes\!\!P_c \!-\! P_b\!\!\otimes\!\!J_a\!\!\otimes\!\!P_c \!+\! P_b\!\!\otimes\!\!P_c\!\!\otimes\!\!J_a).
\end{split}
\end{equation*}
Setting $x^1 = \psi$, $x^2 = \alpha$, $x_1 = q_\psi^a P_a$, $x_2 = q_\alpha^a
J_a + q_\theta^a P_a$ and using $\partial_\alpha V = 0$, we find that the
right-hand side of the CDYBE is given by:
\begin{equation*}
\begin{split}
&\sum_{i=1}^2 x_i^{(1)} \partial_{x_i} r_{23} -
x_i^{(2)} \partial_{x_i} r_{13} +
x_i^{(3)} \partial_{x_i} r_{12} =
(\boldify{q}_\psi\cdot\partial_\psi\boldify{m} + \boldify{q}_\theta\cdot\partial_\alpha\boldify{m})\ee^{abc} P_a \!\otimes\! P_b \!\otimes\! P_c + {} \\
& \left(q_\alpha^a \ee^{bcd}\partial_\alpha m_d-
q_\psi^b \partial_\psi V^{ca} +
q_\psi^c \partial_\psi V^{ba}\right)
(J_a \otimes P_b \otimes P_c - P_b \otimes J_a \otimes P_c + P_b \otimes P_c \otimes J_a).
\end{split}
\end{equation*}
A comparison of the coefficients in these two expressions then yields
equations \eqref{eq:dcybe-with-UV}.
\item To determine under which conditions the bracket in Definition
\ref{def:extended-dirac-bracket} satisfies the Jacobi identity, we consider
the variables $\psi, \alpha$ , functions $h\inC^\infty(\ensuremath{\logr^{n-2+2g}})$ and the
variables $ j^a_X$ for $X \in \{M_3, \dots, B_g\}$. The structure of the
Poisson algebra in Theorem \ref{thm:generic-dirac-bracket} allows us to
reduce the proof to six cases which are distinguished by the number of
variables $j_X^a$, $\psi$, $\alpha$ in the brackets.
\begin{enumerate}
\item For cyclic sums over brackets of the form $\{h, \{j^b_Y,j^c_Z\}_D\}_D$
with $Y,Z\in\{M_3,\dots,B_g\}$, we obtain
\begin{multline*}
\{h, \{j_Y^b, j_Z^c\}_D\}_D + \{j_Y^b, \{j_Z^c, h\}_D\}_D + \{j_Z^c, \{h, j_Y^b\}_D\}_D \\
= \tidadi{Y}{^b_d} \tidadi{Z}{^c_e} \Upsilon^{deg} \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} (R^g_Y+L^g_Y)h,
\end{multline*}
where $\Upsilon^{deg}$ is the term in the first equation of
\eqref{eq:dcybe-with-UV}. Consequently, it vanishes if $r$ satisfies the
CDYBE.
\item For cyclic sums over brackets of the form $\{j_X^a,
\{j^b_Y,j^c_Z\}_D\}_D$ with $X,Y,Z\in\{M_3,\dots,\linebreak[0]
B_g\}$, we have
\begin{equation*}
\begin{split}
&\{j_X^a, \{j_Y^b, j_Z^c\}_D\}_D + \{j_Y^b, \{j_Z^c, j_X^a\}_D\}_D + \{j_Z^c, \{j_X^a, j_Y^b\}_D\}_D \\
&\qquad = \tidadi{Y}{^b_f} \tidadi{Z}{^c_e} \, \tensor{\ee}{^a_d_g} \, j_X^d \Upsilon^{efg} \\
&\qquad + \tidadi{X}{^a_e} \tidadi{Z}{^c_f} \, \tensor{\ee}{^b_d_g} \, j_Y^d \Upsilon^{efg} \\
&\qquad + \tidadi{X}{^a_f} \tidadi{Y}{^b_e} \, \tensor{\ee}{^c_d_g} \, j_Z^d \Upsilon^{efg} \\
&\qquad + \tidadi{X}{^a_d} \tidadi{Y}{^b_e} \tidadi{Z}{^c_f} \ee^{def} \Omega,
\end{split}
\end{equation*}
where $\Upsilon^{efg}$ and $\Omega$ are, respectively, the terms in the
first and second lines of \eqref{eq:dcybe-with-UV}. This shows that the
Jacobi identity for brackets of this type is satisfied if and only if $r$
is a solution of the CDYBE.
\item The remaining cases involve cyclic sums over brackets of the form
$\{\psi, \{j^a_X. j^b_Y\}_D\}_D$, $\{\psi,\{\alpha,j^a_X\}_D\}_D$,
$\{\alpha,\{h,j^b_Y\}_D\}_D$ and $\{\alpha,\{j^a_X,j^b_Y\}_D\}_D$ with
$X,Y\in\{M_3,\dots,B_g\}$. A direct calculation along the same lines as in
cases (a) and (b) shows that the Jacobi identity is satisfied for brackets
of this type if and only if the identities in \eqref{eq:q-relations} hold.
\end{enumerate}
\end{enumerate}
\end{proof}
Theorem \ref{thm:jacobi-for-extended-dirac} gives a direct link between Poisson
structures on $\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$ of the form in Definition
\ref{def:extended-dirac-bracket} and solutions of the CDYBE. As is apparent in
the proof, the CDYBE is a necessary and sufficient condition which ensures that
the Poisson brackets of functions $F, G\inC^\infty(\ensuremath{\pogr^{n-2+2g}})$ satisfy the Jacobi
identity for all values of $\psi$ and $\alpha$. The additional conditions
\eqref{eq:q-relations} ensure that the Jacobi identity also holds for mixed
brackets involving the variables $\psi,\alpha$ as well as functions
$F\inC^\infty(\ensuremath{\pogr^{n-2+2g}})$. We will show in the next section that these conditions
have a direct geometrical interpretation. They allow one to locally transform a
solution $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes \ensuremath{\iso(2,1)}$ of the CDYBE into a classical dynamical
$r$-matrix in the sense of Definition \ref{def:dyn-r-matrix} that is invariant
under a fixed Cartan subalgebra $\algebra{h}\subset\ensuremath{\iso(2,1)}$.
\subsection{Examples of solutions}
The conditions \eqref{eq:dcybe-with-UV} that characterise the classical
dynamical Yang-Baxter equation and the supplementary conditions
\eqref{eq:q-relations} in Theorem \ref{thm:jacobi-for-extended-dirac} are quite
complicated. It is therefore not obvious to determine solutions of these
equations. In the following, we show that the specific gauge fixing conditions
investigated in \cite{Meusburger:2011aa} give rise to a solution of the CDYBE
that also satisfies the additional conditions \eqref{eq:q-relations} in Theorem
\ref{thm:jacobi-for-extended-dirac}. We also determine a simplified standard
set of solutions of these equations that are classical dynamical $r$-matrices
in the sense of Definition \ref{def:dyn-r-matrix}.
The publication \cite{Meusburger:2011aa} investigates a specific set of gauge
fixing conditions of the type discussed in Section
\ref{subsec:constraints-and-gauge-fixing}, which is motivated by its direct
physical interpretation in the application to the Chern-Simons formulation of
(2+1)-gravity. These gauge fixing conditions consider the case where the two
gauge-fixed holonomies $M_1,M_2$ are restricted to conjugacy classes
\begin{equation*}
\grporbit[j]=\{h\cdot \exp(-\mu_j J_0-s_j P_0)\cdot h^\inv \mid h\in \ensuremath{\ISO(2,1)}\} \qquad
\forall j=1,2,
\end{equation*}
with $\mu_1,\mu_2\in(0,2\pi)$, $s_1,s_2\in\mathbb{R}$. The resulting Dirac bracket is
determined in \cite{Meusburger:2011aa}. It takes the form of Theorem
\ref{thm:generic-dirac-bracket} and Definition \ref{def:extended-dirac-bracket}
with $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ given by
\begin{equation}\label{eq:special-w-m}
\left.
\begin{aligned}
&r(\psi,\alpha)= \tfrac 1 2 \left(P_a \!\otimes\! J^a\!+\!J^a\!\otimes\! P_a\right) \!-\! \tfrac 1 2 \ee^{abc} w_c(\psi)(P_a \!\otimes\! J_b \!-\! J_b \!\otimes\! P_a) \!+\! \ee^{abc}m_c(\psi,\alpha) P_a \!\otimes\! P_b,\\
&\boldify{w}(\psi)=\cot\tfrac{\mu_1} 2\, \boldify{e}_0+\cot\tfrac{\mu_1} 2 \coth\psi \, \boldify{e}_1-\coth\psi \, \boldify{e}_2,\\
&\boldify{m}(\psi,\alpha)=s_1/(4\sin^2\tfrac{\mu_1} 2)\,\boldify{e}_0 + s_1\coth\psi/(4\sin^2\tfrac{\mu_1} 2)\,\boldify{e}_1 + \tfrac12 \alpha \, \partial_\psi \boldify{w}(\psi).
\end{aligned}
\;\;\right\}
\end{equation}
The associated maps $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta:\mathbb{R}^2\to\mathbb{R}^3$ take the
form
\begin{equation}\label{eq:special-q-psi-alpha}
\left.
\begin{aligned}
&\boldify{q}_\psi(\psi,\alpha)=-\boldify{q}_\alpha(\psi,\alpha)=\tfrac 1 2 ({\coth\psi \cot\tfrac{\mu_1}{2}+ \cot\tfrac{\mu_2}{2}/\sinh\psi})\,\boldify{e}_0+\tfrac 1 2 \cot\tfrac{\mu_1} 2\,\boldify{e}_1-\tfrac 1 2 \, \boldify{e}_2,\\
&\boldify{q}_\theta (\psi,\alpha)\!=\!\left[
\frac{s_1\coth\psi}{4\sin^2\tfrac{\mu_1}{2}} \!+\!
\frac{s_2}{4\sin^2\tfrac{\mu_2}{2}\sinh\psi} \!-\!
\frac{\alpha (\cot\frac{\mu_1}{2} \!+\! \cosh\psi\cot\frac{\mu_2}{2})}{2\sinh^2\psi}\!
\right]\boldify{e}_0 +
\frac{s_1}{4\sin^2\tfrac{\mu_1}{2}}\,\boldify{e}_1.
\end{aligned}
\;\right\}
\end{equation}
We will now show that this defines a solution of the CDYBE which also satisfies
the additional conditions \eqref{eq:q-relations}.
\begin{lemma}\label{lem:dcybe-for-special-r}
The map $r: \mathbb{R}^2 \to \ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ in \eqref{eq:special-w-m} is a solution
of the CDYBE with $x^1=\psi$, $x^2=\alpha$, $x_1=q_\psi^a P_a$,
$x_2=q_\alpha^a J_a+q_\theta^a P_a$, $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta$ as in
\eqref{eq:special-q-psi-alpha}, and satisfies the additional conditions
\eqref{eq:q-relations}. The maps $x_1,x_2:\mathbb{R}^2\to\ensuremath{\iso(2,1)}$ define a Cartan
subalgebra $\algebra{h}(\psi,\alpha)$ for all values of $\psi$ for which
$\boldify{q}_\psi^2(\psi), \boldify{q}_\alpha^2(\psi) \neq 0$.
\end{lemma}
\begin{proof} $\quad$
\begin{enumerate}
\item That the maps $x_1,x_2:\mathbb{R}^2\to\ensuremath{\iso(2,1)}$ define a two-dimensional abelian
Lie subalgebra of $\ensuremath{\iso(2,1)}$ follows directly from the condition
$\boldify{q}_\psi\wedge\boldify{q}_\alpha=0$. One finds that this Lie subalgebra is a Cartan
subalgebra for all values of $\psi$ for which $\boldify{q}_\psi^2(\psi) \neq 0$.
\item That $r$ solves the CDYBE can be shown with a direct calculation.
Inserting the expressions \eqref{eq:special-w-m} for $\boldify{w},\boldify{m}$ and
expressions \eqref{eq:special-q-psi-alpha} for $\boldify{q}_\psi,\boldify{q}_\theta$ into the
left-hand-side of the second equation in \eqref{eq:dcybe-with-UV}, one finds
after some computations that this expression vanishes. To verify that the
first set of equations in \eqref{eq:dcybe-with-UV} is satisfied, we note that
for maps $V:\mathbb{R}\rightarrow\Mat(3, \mathbb{R})$ of the form $V^{ab}(\psi) = \frac12
\eta^{ab} + \frac12 \tensor{\ee}{^a^b_c} w^c(\psi)$ with $\boldify{w}: \mathbb{R} \to
\mathbb{R}^3$, the first line of \eqref{eq:dcybe-with-UV} is equivalent to the
following conditions
\begin{equation}\label{eq:dcybe-with-simple-UV}
1+\boldify{w}^2+2\boldify{q}_\psi\cdot \partial_\psi \boldify{w}=0,\qquad
\partial_\psi\boldify{w}\wedge \partial_\alpha\boldify{m}=0,\qquad
\partial_\alpha m^a q_\alpha^b=-\tfrac 1 2 \partial_\psi w^a q_\psi^b.
\end{equation}
Setting $\boldify{q}_\alpha=-\boldify{q}_\psi$ and inserting expressions
\eqref{eq:special-w-m} for $\boldify{w},\boldify{m}$ together with expression
\eqref{eq:special-q-psi-alpha} for $\boldify{q}_\psi$ into
\eqref{eq:dcybe-with-simple-UV}, one verifies the first condition in
\eqref{eq:dcybe-with-UV}.
\item To show that $r$ and $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta$ satisfy the
additional conditions \eqref{eq:q-relations}, we note that for matrices $V$
of the form $V^{ab}(\psi)=\tfrac 1 2 \eta^{ab}+\tfrac 1 2 \ee^{abc}
w^c(\psi)$, these conditions reduce to the following set of equations
\begin{equation}
\left.
\begin{aligned}\label{eq:q-relations-simple}
0&=\boldify{q}_\psi\wedge (\partial_\psi\boldify{q}_\psi-\tfrac 1 2 \boldify{w}),\\
0&=\tfrac 1 2 (q_\alpha^bw^a-q_\alpha^aw^b)+q^a_\alpha\partial_\alpha q^b_\theta-q^b_\psi\partial_\psi q^a_\alpha\quad\forall a,b\in\{0,1,2\},\\
0&=\boldify{q}_\theta\wedge(\partial_\alpha\boldify{q}_\theta-\tfrac 1 2 \boldify{w})+\boldify{q}_\psi\wedge \partial_\psi \boldify{q}_\theta+\boldify{q}_\alpha\wedge\boldify{m}.
\end{aligned}
\qquad\right\}
\end{equation}
Inserting expressions \eqref{eq:special-w-m} for $\boldify{w},\boldify{m}$ and expressions
\eqref{eq:special-q-psi-alpha} for $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta$ into
these equations, one finds that they are indeed satisfied.
\end{enumerate}
\end{proof}
Note that the resulting solution of the CDYBE in \cite{Meusburger:2011aa} is
{\em not} a classical dynamical $r$-matrix in the sense of Definition
\ref{def:dyn-r-matrix}. While Definition \ref{def:dyn-r-matrix} requires the
choice of an abelian Lie subalgebra $\algebra{h}\subset \ensuremath{\iso(2,1)}$ and an identification of
the two variables in the solution with its dual, the abelian Lie subalgebra
$\algebra{h}(\psi,\alpha)=\Span\{ q_\psi^aP_a, q_\alpha^aJ_a+q_\theta^aP_a\}$
associated with the above solution varies with $\psi$ and $\alpha$. A direct
calculation shows that depending on the value of $\psi$, the Lie subalgebra
$\algebra{h}(\psi,\alpha)$ is conjugate either to the Cartan subalgebra
$\algebra{h}_a=\Span\{J_0,P_0\}$ for $\boldify{q}_\psi^2(\psi)>0$, to the Cartan subalgebra
$\algebra{h}_b=\Span\{J_1,P_1\}$ for $\boldify{q}_\psi^2(\psi)<0$ or to the two-dimensional Lie
subalgebra $\algebra{h}_c=\Span\{J_0+J_1,P_0+P_1\}$ for $\boldify{q}_\psi^2(\psi)=0$.
The solution therefore combines solutions of the CDYBE that are associated with
different, non-conjugate two-dimensional Lie subalgebras of $\ensuremath{\iso(2,1)}$. To show
that the existence of solutions associated with different Lie subalgebras is a
generic phenomenon and not a consequence of the specific gauge fixing
conditions in \cite{Meusburger:2011aa}, we determine a simple set of solutions
of a similar form.
\begin{lemma}\label{lem:simple-gen-solutions}
For all $c\in\mathbb{R}$, $\gamma\inC^\infty(\mathbb{R})$, the map $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes
\ensuremath{\iso(2,1)}$ given by
\begin{equation*}
r(\psi,\alpha)=\tfrac 1 2 (P_a\!\otimes\!J^a + J^a\!\otimes\!P_a) - \tensor{\ee}{^a^b_c}\,\partial_\psi q_\psi^c(\psi)(P_a{\otimes} J_b - J_b {\otimes} P_a) - \alpha\, \tensor{\ee}{^a^b_c}\,\partial_\psi^2 q_\psi^c (\psi)\, P_a {\otimes} P_b
\end{equation*}
is a solution of the CDYBE with $x_1=\psi$, $x_2=\alpha$, $x^1=q_\psi^aP_a$,
$x^2=q_\alpha^aJ_a+q_\theta^aP_a$ and
\begin{equation*}
\boldify{q}_\psi(\psi)=\boldify{q}_\alpha(\psi)=\gamma(\psi) \boldify{e}_0+\sqrt{\gamma^2(\psi)+\tfrac 1 4 (\psi-c)^2} \,\boldify{e}_1,\qquad
\boldify{q}_\theta(\psi,\alpha)=\alpha\,\partial_\psi\boldify{q}_\psi(\psi).
\end{equation*}
\end{lemma}
\begin{proof}
This follows by a direct calculation. As $r$ is of a form similar to the
solution in Lemma~\ref{lem:dcybe-for-special-r} with
$\boldify{w}=2 \partial_\psi\boldify{q}_\psi$, $\boldify{m}=-\alpha\, \partial_\psi^2 \boldify{q}_\psi$,
inserting these expressions into \eqref{eq:q-relations-simple} shows directly
that the conditions \eqref{eq:q-relations} are satisfied. The CDYBE then
reduces to the requirement $1+2\partial_\psi^2(\boldify{q}_\psi^2)=0$, which is
verified by a simple computation.
\end{proof}
Note, however, that the CDYBE \eqref{eq:dcybe-with-UV} and the additional
requirements \eqref{eq:q-relations} also admit solutions which are associated
with fixed Cartan subalgebras of $\ensuremath{\iso(2,1)}$ and define classical dynamical
$r$-matrices in the sense of Definition \ref{def:dyn-r-matrix}. To obtain such
solutions, we set $\boldify{q}_\theta(\psi,\alpha)=0$ and either
$\boldify{q}_\psi(\psi)=\boldify{q}_\alpha(\psi)=\boldify{e}_0$ or
$\boldify{q}_\psi(\psi)=\boldify{q}_\alpha(\psi)=\boldify{e}_1$ for all admissible values of
$\psi$. The conditions \eqref{eq:q-relations-simple} then reduce to the
requirements $\boldify{w},\boldify{m}\in\Span\{\boldify{q}_\psi\}$, and the expressions
\eqref{eq:dcybe-with-simple-UV} to $\partial_\alpha\boldify{m}=-\tfrac 1
2 \partial_\psi\boldify{w}$, $1+\boldify{w}^2+2\boldify{q}_\psi\cdot\partial_\psi\boldify{w}=0$. From this,
one then obtains two solutions associated with the Cartan subalgebras
$\algebra{h}_a=\Span\{J_0,P_0\}$ and $\algebra{h}_b=\Span\{J_1,P_1\}$ in $\ensuremath{\iso(2,1)}$.
\begin{lemma}\label{lem:dcybe-solutions}
Two solutions of the CDYBE with $x^1=\psi$, $x^2=\alpha$, $x_1=q_\psi^aP_a$ and
$x_2=q_\alpha^aJ_a+q^a_\theta P_a$ that also satisfy the additional conditions
\eqref{eq:q-relations} are given by
\begin{compactenum}[\quad a)]
\item $\boldify{q}_\psi=\boldify{q}_\alpha=\boldify{e}_0$, $\boldify{q}_\theta=0$ and $r:{(-\tfrac \pi 2,\tfrac \pi 2)}\times \mathbb{R}\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$,
\begin{equation*}
r_a(\psi,\alpha)=\tfrac 1 2 (P_a\!\otimes\!J^a+J^a\!\otimes\!P_a) + \tfrac12 \tan\tfrac\psi 2\left(P_1{\wedge} J_2 - P_2{\wedge} J_1\right) + \frac{\alpha}{4\cos^2\tfrac \psi 2} P_1{\wedge} P_2,
\end{equation*}
\item $\boldify{q}_\psi=\boldify{q}_\alpha=\boldify{e}_1$, $\boldify{q}_\theta=0$ and $r:
\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$,
\begin{equation*}
r_b(\psi,\alpha)=\tfrac 1 2 (P_a\!\otimes\!J^a+J^a\!\otimes\!P_a) + \tfrac 1 2 \tanh\tfrac\psi 2 \left(P_2{\wedge} J_0 - P_0{\wedge} J_2\right) + \frac{\alpha}{4\cosh^2\tfrac \psi 2} P_2{\wedge} P_0,
\end{equation*}
\end{compactenum}
where $X\wedge Y:=X\otimes Y-Y\otimes X$. They are classical dynamical $r$-matrices as
in Definition \ref{def:dyn-r-matrix} for, respectively, the Cartan subalgebras
$\algebra{h}_a=\Span\{J_0,P_0\}$ and $\algebra{h}_b=\Span\{J_1,P_1\}$.
\end{lemma}
Lemma \ref{lem:dcybe-solutions} provides us with two particularly simple
classical dynamical $r$-matrices for $\ensuremath{\iso(2,1)}$. We will show in the next section
that every solution of the CDYBE of the form \eqref{eq:extended-dirac-r} which
satisfies the additional conditions \eqref{eq:q-relations} can be transformed
into one of these two solutions for all values of $\psi$ for which either
$\boldify{q}_\psi^2(\psi),\boldify{q}_\alpha^2(\psi)>0$ (case a) or
$\boldify{q}_\psi^2(\psi),\boldify{q}_\alpha^2(\psi)<0$ (case b).
One might wonder if there are similar solutions of the CDYBE and conditions
\eqref{eq:q-relations} for which $\boldify{q}_\psi, \boldify{q}_\alpha$ are fixed lightlike
vectors that do not depend on $\psi$ and $\alpha$. However, it turns out that
such solutions do not exist. This appears to be linked to the fact that the
vectors $\boldify{q}_\psi, \boldify{q}_\alpha$ associated with the solutions in Lemma
\ref{lem:dcybe-for-special-r} and Lemma \ref{lem:simple-gen-solutions} are
spacelike or timelike for values of $\psi$ in certain open intervals of $\mathbb{R}$,
but can become lightlike only for a very specific discrete set of values of
$\psi$. This again suggests that the variation of $\boldify{q}_\psi, \boldify{q}_\alpha$ with
$\psi$ is a generic feature of the gauge fixing procedure, and that there are
no gauge fixing conditions that allow one to obtain a Poisson structure
determined by vectors $\boldify{q}_\psi, \boldify{q}_\alpha$ that are lightlike for all $\psi$.
We have the following lemma:
\begin{lemma}
There are no simultaneous solutions of the CDYBE \eqref{eq:dcybe} and
conditions \eqref{eq:q-relations} for which $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta$
are constant vectors with $\boldify{q}_\psi\wedge\boldify{q}_\alpha=0$ and
$\boldify{q}_\psi^2=\boldify{q}_\alpha^2=0$.
\end{lemma}
\begin{proof}
Suppose that $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ is of the form in
Definition~\ref{def:extended-dirac-bracket} with $V^{ab}(\psi)=\tfrac 1 2
\eta^{ab}+Q^{ab}(\psi)+\tfrac12\ee^{abc} w_c(\psi)$, where $Q:\mathbb{R}\to\Mat(3,
\mathbb{R})$ is symmetric. Then conditions~\eqref{eq:q-relations} imply
$Q^{02}=Q^{12}=Q^{22}=0$, $Q^{00}=Q^{01}=Q^{11}$ and $\boldify{w} \in
\Span\{\boldify{e}_0+\boldify{e}_1\}$. Inserting this into the first equation of
\eqref{eq:dcybe-with-UV} yields a contradiction and thus proves the claim.
\end{proof}
\section{Transformations between Dirac brackets}
\label{sec:dynamic-poincare-trafos}
\subsection{Dynamical Poincar\'e transformations}
The results of the previous section show how constraint functions that satisfy
the conditions in Definition \ref{def:dirac-bracket} and the requirements a and
b from Section~\ref{subsec:constraints-and-gauge-fixing} give rise to a Dirac
bracket that is given by solutions of the CDYBE. In this section, we
investigate the transformation of the Dirac bracket under a change of
constraint functions.
For this, recall from the discussion before Theorem
\ref{thm:generic-dirac-bracket} that any set of admissible constraint functions
restricts the variables $M_1,M_2$ which parametrise the first two copies of
$\ensuremath{\ISO(2,1)}$ in $\ensuremath{\pogr^{n+2g}}$ in such a way that they are determined uniquely by the
two conjugation invariant quantities $\psi,\alpha$ which depend only on the
product $M_2\cdot M_1$. This suggests that for any two sets of variables
$M_1,M_2$ and $M_1',M_2'$ obtained in this way, there should be a Poincar\'e
transformation $p(\psi,\alpha)\in \ensuremath{\ISO(2,1)}$ such that $M_1'=p(\psi,\alpha)\cdot
M_1\cdot p(\psi,\alpha)^\inv$ and $M_2'=p(\psi,\alpha)\cdot M_2\cdot
p(\psi,\alpha)^\inv$. If the associated gauge fixing conditions satisfy
conditions a and b from Section \ref{subsec:constraints-and-gauge-fixing}, it
follows from \eqref{eq:psi-alpha-general} that one can restrict attention to
Poincar\'e transformations $p(\psi,\alpha)$ whose Lorentzian components do not
depend on $\alpha$ and whose translational components depend on $\alpha$ at
most linearly.
This is a strong motivation to investigate the transformation of the bracket in
Definition \ref{def:extended-dirac-bracket} under such Poincar\'e
transformations. We therefore consider smooth maps
\begin{align}\label{eq:poincare-diffeo}
\Phi^p:\; &\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}\to \mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}},\\
&(\psi,\alpha,M_3,\dots,B_g)\mapsto (\psi,\alpha,\; p(\psi,\alpha)\cdot M_3\cdot p(\psi,\alpha)^\inv,\dots,\;p(\psi,\alpha)\cdot B_g\cdot p(\psi,\alpha)^\inv),\nonumber
\end{align}
where $p=(g,-\Ad(g)\boldify{t})\inC^\infty(\mathbb{R}^2,\ensuremath{\ISO(2,1)})$ with $\partial_\alpha
g=\partial_\alpha^2\boldify{t}=0$. We find that the transformation of the bracket
$\{\,,\}_D$ under such a dynamical Poincar\'e transformation corresponds to a
simultaneous transformation of the maps $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ and
$\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta:\mathbb{R}^2\to\mathbb{R}^3$.
\begin{lemma}\label{lem:poinc-trafo}
Let $\{,\}_D$ and $r: \mathbb{R}^2 \to \ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ be given as in Definition
\ref{def:extended-dirac-bracket} and consider a dynamical Poincar\'e
transformation $\Phi^p:\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}\to \mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$ as
above. Then for all $F,G\inC^\infty(\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}})$:
\begin{equation*}
\{ F\circ \Phi^p, G\circ \Phi^p\}_D=\{F,G\}^p_D\circ \Phi^p,
\end{equation*}
where $\{\,,\,\}_D^p$ is the bracket from Definition
\ref{def:extended-dirac-bracket} associated with
\begin{equation}\label{eq:trafo-quant}
\left.
\begin{aligned}
&\boldify{q}_\psi^p=\Ad(g)\boldify{q}_\psi,\qquad \boldify{q}_\alpha^p=\Ad(g)\boldify{q}_\alpha,\qquad \boldify{q}_\theta^p=\Ad(g)(\boldify{q}_\theta-\boldify{q}_\alpha\wedge\boldify{t}),\\
&r^p=\left(\Ad(p)\otimes\Ad(p)\right)\left[r+\bar\eta^p-\bar\eta^p_{21}\right],
\end{aligned}
\qquad\right\}
\end{equation}
and $\bar\eta^p:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes \ensuremath{\iso(2,1)}$ is given by
\begin{equation}\label{eq:trafo-eta}
\bar\eta^p = q_\psi^a P_a \otimes p^\inv\partial_\psi p + (q^a_\alpha J_a+q^a_\theta P_a) \otimes p^\inv\partial_\alpha p.
\end{equation}
\end{lemma}
\begin{proof} $\quad$
\begin{enumerate}
\item To derive explicit expressions for the transformed bracket $\{,\}_D^p$,
it is convenient to consider two cases separately, namely Lorentz
transformations $g: \mathbb{R}^2 \to \ensuremath{\SO_+(2,1)}$, which do not depend on $\alpha$, and
translations $t: \mathbb{R}^2 \to \mathbb{R}^3$, which depend on $\alpha$ at most linearly.
We start by determining concrete expressions for $r^p$ from formula
\eqref{eq:trafo-quant}. For a Lorentz transformation $p=(g,0):\mathbb{R}\to \ensuremath{\SO_+(2,1)}$
with $\partial_\alpha p=0$, we have $\bar\eta^p(\psi)=q^a_\psi(\psi) P_a \otimes
g^\inv\partial_\psi g(\psi)$. Expanding this terms of a basis as
$g^\inv\partial_\psi g(\psi)=n^a(\psi)J_a$ then yields
\begin{equation}\label{eq:r-gauged-classification-Ltrafo}
r^p = \bigl(\Ad(g) \otimes \Ad(g)\big) [r + q_\psi^a n^b (P_a \otimes J_b - J_b \otimes P_a)].
\end{equation}
In the case of a translation $p=(1,\boldify{t}):\mathbb{R}^2\to\mathbb{R}$ with
$\partial_\alpha^2\boldify{t}=0$, we have $\bar\eta^p=q_\psi^a\partial_\psi t^b P_a
\otimes P_b + q_\alpha^a\partial_\alpha t^b J_a \otimes P_b +
q_\theta^a\partial_\alpha t^b P_a \otimes P_b$. Inserting this with expression
\eqref{eq:extended-dirac-r} for $r$ into \eqref{eq:trafo-quant} and using the
identities $\Ad(\boldify{t}) J_a=J_a+\tensor{\ee}{_a_b^c} t^b P_c$,
$\Ad(\boldify{t})P_a=P_a$, we obtain after some computations
\begin{equation}\label{eq:r-gauged-classification-translation}
\begin{split}
r^p = r &- \partial_\alpha t^a q_\alpha^b (P_a \otimes J_b - J_b \otimes P_a) \\
&+ \tensor{\ee}{^a^b_c}\bigl[
(1-V^d_{\;\;d})\boldify{t} + V^T\boldify{t}
+ \boldify{q}_\psi\wedge\partial_\psi\boldify{t}
+ [\boldify{q}_\theta - \boldify{q}_\alpha\wedge\boldify{t}]\wedge\partial_\alpha\boldify{t}
\bigr]^c P_a \otimes P_b.
\end{split}
\end{equation}
\item We now derive explicit expressions for the transformed Poisson brackets
$\{\,,\,\}_D^p$. For a Lorentz transformation $p=(g,0):\mathbb{R}\to \ensuremath{\SO_+(2,1)}$ with
$\partial_\alpha p=0$, it follows directly from the identity
$\{\psi,\alpha\}_D=0$ that the Poisson brackets involving the variables
$\psi$ and $\alpha$ with functions on $\ensuremath{\pogr^{n-2+2g}}$ are given by:
\begin{equation*}
\begin{aligned}
&\{\psi\circ \Phi^p, h\circ \Phi^p\}_D = 0, \\
&\{\psi\circ \Phi^p, \boldify{j}_X\circ \Phi^p\}_D = \big[-\idadi{X} \Ad(g) \, \boldify{q}_\psi\big] \circ \Phi^p, \\
&\{\alpha\circ \Phi^p, h\circ \Phi^p\}_D = \big[\smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} \tensor{\Ad(g)}{^a_b}q_\alpha^b(J_{R,a}^Y+J_{L,a}^Y)h\big] \circ \Phi^p, \\
&\{\alpha\circ \Phi^p, \boldify{j}_X\circ \Phi^p\}_D = \big[-\idadi{X} \Ad(g)\boldify{q}_\theta - (\Ad(g)\boldify{q}_\alpha) \wedge \boldify{j}_X\big] \circ \Phi^p,
\end{aligned}
\end{equation*}
for all $h\inC^\infty(\ensuremath{\logr^{n-2+2g}})$ and $X\in\{M_3,\dots,B_g\}$. To determine the
brackets of the type $\{F, G\}_D^p$ with $F,G\inC^\infty(\ensuremath{\pogr^{n-2+2g}})$, we again
consider functions $h\inC^\infty(\ensuremath{\logr^{n-2+2g}})$ and the variables $j_X$,
$X\in\{M_3,\dots,B_g\}$. After some straightforward computations, we obtain
\begin{multline*}
\{j_X^a\circ\Phi^p, h\circ \Phi^p\}_D = \\
\{j_X^a, h\} \circ \Phi^p
- \tensor{\Ad(g)}{^a_c} \tidadi{X}{^c_h} \bigl(\tensor{V}{^h_e} - q_\psi^h n_e\bigr) \Bigl[\smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} (J^{R,e}_Y+J^{L,e}_Y)h\Bigr] \circ \Phi^p,
\end{multline*}
which allows us to express the transformed bracket as
\begin{align*}
&\{j_X^a\circ \Phi^p, h\circ \Phi^p\}_D
= \Bigl[
\{j_X^a, h\} -
\tidadi{X}{^a_h} \tensor{({V}^p)}{^h_e} \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} (J^{R,e}_Y+J^{L,e}_Y)h
\Bigr] \circ \Phi^p,\\[-.4em]
&\text{with}\quad \tensor{({V}^p)}{^h_e} = \tensor{\Ad(g)}{^h_m}\tensor{\Ad(g)}{_e^p}\big[\tensor{V}{^m_p} - q_\psi^m n_p\big].
\end{align*}
An analogous calculation for the brackets of the form $\{j_X^a\circ \Phi^p,
j_Y^b\circ \Phi^p\}_D$ shows that it is obtained by transforming $V\to V^p$
as above and replacing $\boldify{m}:\mathbb{R}^2\to\mathbb{R}^3$ by
$\boldify{m}^p=\Ad(g)\boldify{m}:\mathbb{R}^2\to\mathbb{R}^3$. This implies that the transformed bracket
takes the form $\{F\circ\Phi^p, G\circ\Phi^p\}_D = [\frrestbivector[r^p]
(\mathrm{d} F \otimes \mathrm{d} G)]\circ\Phi^p$ with $r^p$ given by
\eqref{eq:r-gauged-classification-Ltrafo} and proves the claim for the
Lorentz transformations.
\item To determine the transformation of the bracket $\{\,,\,\}_D$ under
translations, we again use the parametrisation in terms of the variables
$j_X$, $X\in\{M_3,\dots,B_g\}$ and functions $h\inC^\infty(\ensuremath{\logr^{n-2+2g}})$. The
transformation of these variables under a translation
$p=(0,\boldify{t}):\mathbb{R}^2\to\mathbb{R}^3$ is given by $h\circ \Phi^p= h$ and $\boldify{j}_X \circ
\Phi^p=\boldify{j}_X + \idadi{X}\boldify{t}$. This implies directly that the brackets
$\{\psi,h\}_D$ and $\{\alpha,h\}_D$ are preserved, and with the relations
$\{\alpha,\psi\}_D=0$, $\{\psi, h\}_D=0$, one obtains the same for the
brackets $\{\psi,\boldify{j}_X\}_D$. The formula for the brackets
$\{\alpha,\boldify{j}_X\}_D^p$ follows directly from the relation
\begin{equation*}
\Big\{\alpha, \tidadi{X}{^a_b}\Big\}_D =
\Big[\tensor{\ee}{_d^a_m} \tidadi{X}{^m_b} +
\tensor{\ee}{_d_b^m} \tidadi{X}{^a_m}\Big] q_\alpha^d.
\end{equation*}
We thus find that the transformed brackets involving the variables
$\psi,\alpha$ are given by
\begin{equation*}
\begin{aligned}
&\{\psi \circ \Phi^p, h\circ \Phi^p\}_D = 0, \\
&\{\psi\circ \Phi^p, \boldify{j}_X\circ \Phi^p\}_D = \bigl[-\idadi{X} \, \boldify{q}_\psi\bigr] \circ \Phi^p, \\
&\{\alpha\circ \Phi^p, h\circ \Phi^p\}_D = \bigl[\smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} q_\alpha^a(J_{R,a}^Y+J_{L,a}^Y)h\bigr] \circ \Phi^p, \\
&\{\alpha\circ \Phi^p, \boldify{j}_X\circ \Phi^p\}_D = \bigl[-\idadi{X} (\boldify{q}_\theta - \boldify{q}_\alpha \wedge \boldify{t}) - \boldify{q}_\alpha \wedge \boldify{j}_X\bigr] \circ \Phi^p.
\end{aligned}
\end{equation*}
To determine the transformed brackets $\{F\circ \Phi^p, G\circ \Phi^p\}_D$
for $F,G\inC^\infty(\ensuremath{\pogr^{n-2+2g}})$, we calculate the brackets of functions
$h\inC^\infty(\ensuremath{\logr^{n-2+2g}})$ and variables $\boldify{j}_X, \boldify{j}_Y$ for $X, Y\in
\{M_3,\dots,B_g\}$. A direct computation yields
\begin{align*}
&\{j_X^a\circ \Phi^p, h\circ \Phi^p\}_D
= \Bigl[\{j_X^a, h\} - \tidadi{X}{^a_g} \tensor{(V^p)}{^g_d} \smashoperator{\sum_{Y\in\{M_3,\dots,B_g\}}} (J^{R,d}_Y+J^{L,d}_Y)h\Bigr]\circ \Phi^p, \\
\begin{split}
&\{j_X^a\circ \Phi^p, j_Y^b\circ \Phi^p\}_D
= \Bigl[\{j_X^a, j_Y^b\}
+ \tidadi{X}{^a_g} \tensor{({V^p})}{^g_d}\,\tensor{\ee}{^d^b_f} j_Y^f\\[-.5em]
&\quad - \tidadi{Y}{^b_g} \tensor{({V}^p)}{^g_d}\,\tensor{\ee}{^d^a_f} j_X^f
+ \tidadi{X}{^a_c}\tidadi{Y}{^b_d} \ee^{cdf} m^p_f\Bigr] \circ \Phi^p,
\end{split}\\
&\text{with } \tensor{({V}^p)}{^b^c} = \tensor{V}{^b^c} + \partial_\alpha t^b q_\alpha^c,\\
&\text{and } \boldify{m}^p= \boldify{m}+(1-\tensor{V}{^d_d}) \boldify{t}
+ V^T \boldify{t}
+ \boldify{q}_\psi \wedge \partial_\psi\boldify{t}
+ (\boldify{q}_\theta - \boldify{q}_\alpha\wedge\boldify{t})\wedge\partial_\alpha\boldify{t}.
\end{align*}
For all $F,G\inC^\infty(\ensuremath{\pogr^{n-2+2g}})$ the transformed bracket therefore takes
the form $\{F\circ \Phi^p, G\circ \Phi^p\}_D= [\frrestbivector[r^p] (\mathrm{d} F
\otimes \mathrm{d} G)] \circ \Phi^p$ with $r^p$ given by
\eqref{eq:r-gauged-classification-translation}. This proves the claim.
\end{enumerate}
\end{proof}
Lemma \ref{lem:poinc-trafo} gives explicit expressions for the transformation
of the bracket in Definition~\ref{def:extended-dirac-bracket} under dynamical
Poincar\'e transformations which depend on the variables $\psi, \alpha$ and act
diagonally on $\ensuremath{\pogr^{n-2+2g}}$. It allows one to identify the transformed bracket
with another bracket of the form in Definition \ref{def:extended-dirac-bracket}
associated with transformed maps $r^p:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ and
$\boldify{q}_\psi^p,\boldify{q}_\alpha^p,\boldify{q}_\theta^p:\mathbb{R}^2\to\mathbb{R}^3$.
As the bracket $\{\,,\,\}_D$ in Definition \ref{def:extended-dirac-bracket} is
modelled after the Dirac bracket in Theorem \ref{thm:generic-dirac-bracket} and
Poincar\'e transformations of this type can be viewed as transitions between
different gauge fixing conditions, it is natural to ask whether these
Poincar\'e transformations preserve the Jacobi identity. For this, note that,
given any Poisson manifold $(M, \{\,,\,\})$ and a diffeomorphism $\Phi: M \to
M$, one obtains a new Poisson bracket $\{\,,\,\}^\Phi$ on $M$ by setting $\{f,
g\}^\Phi \coloneqq \{f \circ \Phi, g \circ \Phi\} \circ \Phi^\inv$. As shown in
Lemma \ref{lem:poinc-trafo}, the bracket $\{\,,\,\}^p_D = \{\,,\,\}^{\Phi^p}_D$
obtained from $\{\,,\,\}_D$ by applying the diffeomorphism $\Phi^p$ from
\eqref{eq:poincare-diffeo} is again of the form in Definition
\ref{def:extended-dirac-bracket}, but with the transformed maps $r^p,
\boldify{q}_\psi^p, \boldify{q}_\alpha^p, \boldify{q}_\theta^p$. From Theorem
\ref{thm:jacobi-for-extended-dirac} we thus deduce that these transformed maps
are solutions of the CDYBE \eqref{eq:dcybe} and the additional conditions
\eqref{eq:q-relations} if and only if the original maps $r, \boldify{q}_\psi,
\boldify{q}_\alpha, \boldify{q}_\theta$ are.
\begin{corollary}
Let $r: \mathbb{R}^2\to \ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ as in \eqref{eq:extended-dirac-r} be a
solution of the CDYBE with $x^1=\psi$, $x^2=\alpha$, $x_1=q_\psi^a P_a$,
$x_2=q_\alpha^a J_a+q_\theta^a P_a$ such that the conditions in
\eqref{eq:q-relations} are satisfied and let $p:\mathbb{R}^2\to \ensuremath{\ISO(2,1)}$ be a
dynamical Poincaré transformation as in Lemma \ref{lem:poinc-trafo}. Then
$r^p:\mathbb{R}^2\to \ensuremath{\iso(2,1)} \otimes \ensuremath{\iso(2,1)}$ is a solution of the CDYBE with $x^1=\psi$,
$x^2=\alpha$ and $x_1=q_\psi^{p,a}P_a$,
$x_2=q_\alpha^{p,a}J_a+q_\theta^{p,a}P_a$ and satisfies
\eqref{eq:q-relations}. The map $\Phi^p$ is a Poisson isomorphism between
the Poisson structures $\{\,,\,\}_D$ and $\{\,,\,\}_D^p$.
\end{corollary}
As discussed in the previous section, the equivalence of the CDYBE and
conditions \eqref{eq:q-relations} with the Jacobi identity for the bracket in
Definition \ref{def:extended-dirac-bracket} suggests that the solutions should
be viewed as a generalisation of the classical dynamical $r$-matrices in
Definition \ref{def:dyn-r-matrix} for which the associated abelian subalgebra
of $\ensuremath{\iso(2,1)}$ is allowed to vary with the variables $\psi$ and $\alpha$. The
transformation formula \eqref{eq:trafo-quant} for the maps
$r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ under dynamical Poincar\'e transformations and the
fact that these Poincar\'e transformations preserve the Jacobi identity
suggests that these Poincar\'e transformations should be interpreted as a
generalised version of the gauge transformations of classical dynamical
$r$-matrices introduced by Etingof and Varchenko in their work on the
classification of classical dynamical $r$-matrices \cite{Etingof:1998aa} (see
also \cite{Schiffmann:1998aa, Etingof:1999aa, Etingof:2000aa, Xu:2002aa} for
further work on the classification). We summarise the relevant definitions and
results from \cite{Etingof:1998aa}.
\begin{definition}[\cite{Etingof:1998aa}]\label{def:gauge-trafo}
Let $G$ be a Lie group, $H\subset G$ an abelian subgroup and
$r:\algebra{h}^*\to\algebra{g}\otimes\algebra{g}$ a classical dynamical $r$-matrix for $(\algebra{h},\algebra{g})$. A
\definee{gauge transformation} of $r$ is a smooth function $\Pi: \algebra{h}^*\to G^H$
into the centraliser $G^H$ of $H$ in $G$ which acts on $r$ according
to\footnote{The sign difference between this formula and the one in
\cite{Etingof:1998aa} is due to a different sign convention for the
CDYBE~\eqref{eq:dcybe}.}
\begin{equation}\label{eq:g-trafo-r}
r^{\Pi} = \bigl(\Ad(\Pi) \otimes \Ad(\Pi)\bigr) [r + \bar{\eta}^\Pi - \bar{\eta}^\Pi_{21}],
\end{equation}
where $\bar \eta_\Pi:\algebra{h}^*\to \algebra{h}\otimes \algebra{g}^{H}$ is the map dual to the
$\algebra{g}^\algebra{h}$-valued one-form $\eta_\Pi=\Pi^\inv\mathrm{d}\Pi$ on $\algebra{h}^*$ and
$\bar\eta^{21}_\Pi$ denotes its flip with values in $\algebra{g}^\algebra{h}\otimes\algebra{h}$.
\end{definition}
The name gauge transformation is motivated by the fact that it maps classical
dynamical $r$-matrices for $(\algebra{h},\algebra{g})$ to classical dynamical $r$-matrices for
$(\algebra{h},\algebra{g})$. It is shown in \cite{Etingof:1998aa} that if $r$ is an
$\algebra{h}$-invariant solution of the CDYBE, then this also holds for the transformed
$r$-matrix $r^{\Pi}$.
\begin{theorem}[\cite{Etingof:1998aa}]
Let $G$ be a Lie group, $H\subset G$ an abelian subgroup and
$r:\algebra{h}^*\to\algebra{g}\otimes\algebra{g}$ a classical dynamical $r$-matrix for $(\algebra{h},\algebra{g})$. Then
for every gauge transformation $\Pi:\algebra{h}^*\to G^H$, $r^\Pi$ is a classical
dynamical $r$-matrix for $(\algebra{h},\algebra{g})$.
\end{theorem}
By comparing formula \eqref{eq:trafo-quant} for the action of dynamical
Poincar\'e transformations on the solutions of the CDYBE with the one in
Definition \ref{def:gauge-trafo}, it becomes apparent that the two expressions
agree. The only difference is that in Definition \ref{def:gauge-trafo} the
gauge transformations are restricted to take values in the centraliser of the
subgroup $H\subset G$, while no such condition is imposed in our
case. Consequently, the dynamical Poincar\'e transformations also act on the
maps $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta:\mathbb{R}^2\to \mathbb{R}^3$ and hence on the
associated two-dimensional Lie subalgebra $\algebra{h}(\psi,\alpha)=\Span\{q_\psi^aP_a,
q_\alpha^aJ_a+q_\theta^aP_a\}$. This is also apparent in the formula for the
map $\bar\eta^p: \mathbb{R}^2\to \ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ in \eqref{eq:trafo-eta}, which depends
on the chosen basis of the subalgebra $\algebra{h}(\psi,\alpha)$.
It is therefore instructive to consider the gauge transformations
\eqref{eq:g-trafo-r} for the classical dynamical $r$-matrices from Lemma
\ref{lem:dcybe-solutions} which are associated with fixed Cartan subalgebras
$\algebra{h}_a=\Span\{P_0,J_0\}\subset \ensuremath{\iso(2,1)}$ and $\algebra{h}_b=\Span\{P_1,J_1\}\subset\ensuremath{\iso(2,1)}$.
In that case, the abelian subgroup $H$ in Definition \ref{def:gauge-trafo} is
obtained by exponentiating, respectively, the Cartan subalgebras $\algebra{h}_a$ and
$\algebra{h}_b$, and the associated centraliser $G^H$ coincides with $H$. With our
additional restriction that the Lorentzian component of $\Pi$ does not depend
on $\alpha$ and its translational component depends on $\alpha$ at most
linearly, the map $\Pi:\algebra{h}^*\to G^H$ in Definition \ref{def:gauge-trafo}
therefore takes the form $\Pi(\psi,\alpha)=(\exp(-\beta(\psi)J_j),
-[\gamma(\psi)+\alpha\delta(\psi)]\boldify{e}_j)$ with $\beta,\gamma,\delta: \mathbb{R}\to
\mathbb{R}$ and $j=0$ in case $a$ and $j=1$ in case b). The transformation of the
classical dynamical $r$-matrices $r_{a,b}:\mathbb{R}\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ in Lemma
\ref{lem:dcybe-solutions} under $\Pi$ is thus given by
\begin{equation*}
r^\Pi(\psi,\alpha)=r(\psi,\alpha)-[\beta'(\psi)-\delta(\psi)] P_j\wedge J_j,
\end{equation*}
with $j=0$ in case a) and $j=1$ in case b). As $[\beta'(\psi)-\delta(\psi)]
P_j\wedge J_j$ satisfies the classical dynamical Yang-Baxter equation and
because $[[r(\psi,\alpha), P_j\wedge J_j]]+[[P_j\wedge J_j,
r(\psi,\alpha)]]=0$, it is directly apparent that this yields another classical
dynamical $r$-matrix for $(\algebra{h},\ensuremath{\iso(2,1)})$ and only modifies $r$ by adding a twist.
Note in particular that $r_{a,b}$ are invariant under gauge transformations
$\Pi:\algebra{h}_{a,b}\to H$ of the form $\Pi(\psi,\alpha)=(\exp(-\beta(\psi)J_j),
-\alpha\beta'(\psi) P_j)$ and $\Pi(\psi,\alpha)=(\mathds{1},
-\gamma(\psi)P_j)$. The former correspond to combinations of a Lorentz
transformation that preserves $\algebra{h}_{a,b}$ and a translation in the direction of
its axis. The latter correspond to translations which do not depend on the
parameter $\alpha$. By specialising formula
\eqref{eq:extended-dirac-bracket-for-gauge-fixed} to the case at hand in which
$\boldify{q}_\psi=\boldify{q}_\alpha=e_j$, $\boldify{q}_\theta=0$ one finds that these are precisely
the flows that the variables $\psi\cdot \alpha$ and $\psi$ generate via the
bracket $\{\,,\,\}_D$ in Definition \ref{def:extended-dirac-bracket}.
\subsection{Standard solutions and classical dynamical r-matrices}
Although Theorem \ref{thm:jacobi-for-extended-dirac} provides a direct link
between the Jacobi identity for the bracket in Definition \ref{def:extended-dirac-bracket}
and solutions of the CDYBE that are subject to the additional conditions
\eqref{eq:q-relations}, the disadvantage of this description is that the
associated solutions $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ are in general quite complicated
and the additional conditions \eqref{eq:q-relations} do not have an immediate
geometrical interpretation.
It is therefore natural to ask if they can be related to a simple set of
standard solutions which define classical dynamical $r$-matrices in the sense
of Definition \ref{def:dyn-r-matrix}. The results of the previous subsection
suggest that this can be achieved by applying dynamical Poincar\'e
transformations. As we will see in the following, this is possible for those
values of the parameters $\psi$ for which $\boldify{q}_\psi$ and $\boldify{q}_\alpha$ are
timelike or spacelike. A necessary and sufficient condition then is that the
maps $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ and $\boldify{q}_\psi,
\boldify{q}_\alpha,\boldify{q}_\theta:\mathbb{R}^2\to\mathbb{R}^3$ satisfy the equations
\eqref{eq:q-relations} in Theorem~\ref{thm:jacobi-for-extended-dirac}. We have
the following lemma:
\begin{lemma}\label{lem:standard-form}
Consider $r:I\times \mathbb{R}\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ as in \eqref{eq:extended-dirac-r} and
$\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta: I\times \mathbb{R}\to\mathbb{R}^3$, where $I\subset\mathbb{R}$ is
an open interval with $\boldify{q}_\psi^2(\psi),\boldify{q}_\alpha^2(\psi)\neq 0$,
$\boldify{q}_\psi(\psi)\wedge\boldify{q}_\alpha(\psi)=0$ and
$\partial_\alpha\boldify{q}_\psi(\psi)=\partial_\alpha\boldify{q}_\alpha(\psi)=\partial_\alpha^2\boldify{q}_\theta(\psi)=0$
for all $\psi\in I$. If $r$ and $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta$ satisfy the
conditions \eqref{eq:q-relations}, then there exists a Poincar\'e
transformation as in Lemma \ref{lem:poinc-trafo} such that the transformed
quantities $r^p: I \times \mathbb{R}\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$,
$\boldify{q}_\psi^p,\boldify{q}_\alpha^p,\boldify{q}_\theta^p:I\times \mathbb{R}\to \mathbb{R}^3$ defined by
\eqref{eq:trafo-quant} are of the form
\begin{equation}\label{eq:r-simple}
r^p=\tfrac 1 2 (P_a\otimes J^a+J^a\otimes P_a)-\tfrac 1 2 \ee^{abc} w^p_c P_a\wedge J_b+\tfrac 1 2 \ee^{abc} m^p_c P_a\wedge P_b,
\end{equation}
with one of the following:
\begin{compactenum}[\quad a)]
\item $\boldify{q}_\psi^p,\boldify{q}^p_\alpha,\boldify{q}^p_\theta,\boldify{w}^p,\boldify{m}^p\in\Span\{\boldify{e}_0\}$ and
$q_\alpha^{p,0}\partial_\alpha q_\theta^{p,0}=q_\psi^{p,0}\partial_\psi
q_\alpha^{p,0}$,
\item $\boldify{q}_\psi^p,\boldify{q}_\alpha^p,\boldify{q}_\theta^p,\boldify{w}^p,\boldify{m}^p\in\Span\{\boldify{e}_1\}$ and
$q_\alpha^{p,1}\partial_\alpha q_\theta^{p,1}=q_\psi^{p,1}\partial_\psi
q_\alpha^{p,1}$.
\end{compactenum}
\end{lemma}
\begin{proof} $\quad$
\begin{enumerate}
\item Let $r$ be of the form \eqref{eq:extended-dirac-r} and
$\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta:I\times\mathbb{R}\to\mathbb{R}^3$ with
$\boldify{q}_\psi\wedge\boldify{q}_\alpha=0$ and either $\boldify{q}_\psi^2,\boldify{q}_\alpha^2>0$ (case a)
or $\boldify{q}_\psi^2,\boldify{q}_\alpha^2<0$ (case b). Then the formulas for the action of
dynamical Lorentz transformations in Lemma \ref{lem:poinc-trafo} imply that
via a suitable Lorentz transformation $g:\mathbb{R}^2\to \ensuremath{\SO_+(2,1)}$, $\partial_\alpha
g=0$, we can achieve one of the following: a)
$\boldify{q}_\psi,\boldify{q}_\alpha\in\Span\{\boldify{e}_0\}$ or b)
$\boldify{q}_\psi,\boldify{q}_\alpha\in\Span\{\boldify{e}_1\}$.
The resulting matrix $V: \mathbb{R}^2 \to \Mat(3, \mathbb{R})$ in
\eqref{eq:extended-dirac-r} can be decomposed into a symmetric and an
antisymmetric component according to
\begin{equation*}
V^{ab}(\psi)=\tfrac 1 2 \eta^{ab}+ Q^{ab}(\psi)+\tfrac 1 2 \ee^{abc} w_c(\psi)
\end{equation*}
with $\boldify{w}:I\to\mathbb{R}^3$ and $Q^{ab}:I\to\Mat(3, \mathbb{R})$ symmetric. By applying a
suitable translation $\boldify{t}: I\times \mathbb{R}\to\mathbb{R}^3$, $\partial_\alpha^2 \boldify{t}=0$,
which does not affect $\boldify{q}_\psi,\boldify{q}_\alpha$, one can then achieve that the
symmetric matrix $Q^{ab}:I\times \mathbb{R}\to\Mat(3, \mathbb{R})$ satisfies
$Q^{0a}=Q^{a0}=0$ $\forall a\in\{0,1,2\}$ in case a) and $Q^{1a}=Q^{a1}=0$
$\forall a\in\{0,1,2\}$ in case b). With a further translation $\boldify{t}':\mathbb{R}^2\to
\mathbb{R}^3$ which satisfies $\partial_\alpha\boldify{t}'=0$ and hence does not affect $V$,
one can achieve that $\boldify{q}_\theta$ takes the form
$\boldify{q}_\theta=\alpha\cdot \partial_\alpha\boldify{q}_\theta+\tilde\boldify{q}_\theta$ with
$\partial_\alpha\tilde\boldify{q}_\theta=0$ and $\tilde\boldify{q}_\theta\in\Span\{\boldify{e}_0\}$
in case a) or $\tilde\boldify{q}_\theta\in\Span\{\boldify{e}_1\}$ in case b).
\item After these transformations, the first condition in
\eqref{eq:q-relations} is satisfied if and only if $\boldify{w}\in\Span\{\boldify{e}_0\}$ and
$Q^{11}+Q^{22}=0$ in case a) and $\boldify{w}\in\Span\{\boldify{e}_1\}$ and $Q^{00}-Q^{22}=0$
in case b). Under these conditions, the second equation in
\eqref{eq:q-relations} simplifies to
\begin{equation*}
q^a_\alpha\partial_\alpha q_\theta^b-q_\psi^b\partial_\psi q^a_\alpha
+q_\alpha^d(\ee^{a}_{\;\;dh} Q^{bh}+\ee^{b}_{\;\;dh} Q^{ah})=0\qquad\forall a,b\in\{0,1,2\}.
\end{equation*}
This implies $\partial_\alpha\boldify{q}_\theta\in\Span\{\boldify{e}_0\}$, $Q=0$ in case a)
and $\partial_\alpha\boldify{q}_\theta\in\Span\{\boldify{e}_1\}$, $Q=0$ in case b).
Combining this with the previous results, we obtain
$\boldify{q}_\theta\in\Span\{\boldify{e}_0\}$ and $q_\alpha^0 \partial_\alpha
q_\theta^0=q_\psi^0\partial_\psi q_\alpha^0$ in case a) and
$\boldify{q}_\theta\in\Span\{\boldify{e}_1\}$ and $q_\alpha^1 \partial_\alpha
q_\theta^1=q_\psi^1\partial_\psi q_\alpha^1$ in case b). Inserting these
conditions into the third equation in \eqref{eq:q-relations}, one finds that
this equation simplifies to the condition $\boldify{m} \wedge\boldify{q}_\alpha=0$, which
implies $\boldify{m}\in\Span\{\boldify{e}_0\}$ in case a) and $\boldify{m}\in\Span\{\boldify{e}_1\}$ in case
b). This proves the claim.
\end{enumerate}
\end{proof}
If $r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ and $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta:
\mathbb{R}^2\to\mathbb{R}^3$ are of the form in Lemma \ref{lem:standard-form}, then
$\algebra{h}(\psi,\alpha)=\Span\{q_\psi^a P_a, q^a_\alpha J_a+q_\theta^aP_a\}$ is a
fixed Cartan subalgebra of $\ensuremath{\iso(2,1)}$ which no longer varies with $\psi$ and
$\alpha$. Moreover, a direct calculation shows that $r(\psi,\alpha)$ is then
invariant under the action of the Cartan subalgebra $\algebra{h}(\psi,\alpha)$: $
[\boldify{y}\otimes 1+1\otimes \boldify{y}, r(\psi,\alpha)]=0$ for all $\boldify{y}\in \algebra{h}(\psi,\alpha)$.
This provides us with a natural geometrical interpretation of the conditions
\eqref{eq:q-relations} in Theorem~\ref{thm:jacobi-for-extended-dirac}. These
conditions ensure that for all values of $\psi$ for which
$\boldify{q}_\psi^2(\psi),\boldify{q}_\alpha^2(\psi)\neq 0$, the maps
$r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$, $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta:\mathbb{R}^2\to\mathbb{R}^3$
(which are {not} required to satisfy the CDYBE in Lemma
\ref{lem:standard-form}) can be brought into a standard form via a suitable
dynamical Poincar\'e transformation. The resulting map
$r:\mathbb{R}^2\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ is then invariant under the fixed Cartan subalgebra
spanned by $\boldify{q}_\psi,\boldify{q}_\theta,\boldify{q}_\alpha$. The conditions
\eqref{eq:q-relations} can therefore be viewed as a generalised or
Poincar\'e-transformed version of the restriction to a fixed Cartan subalgebra
in Definition \ref{def:dyn-r-matrix}. As shown in the proof of Theorem
\ref{thm:jacobi-for-extended-dirac}, they ensure that the Jacobi identity holds
for mixed brackets involving functions of the variables $\psi,\alpha$ and
functions on $\ensuremath{\pogr^{n-2+2g}}$.
Lemma \ref{lem:standard-form} applies in particular to solutions of the CDYBE
that satisfy the conditions \eqref{eq:q-relations} and hence give rise to
Poisson structures $\{\,,\,\}_D$ on $\mathbb{R}^2\times\ensuremath{\pogr^{n-2+2g}}$. This allows one to
(locally) classify all possible solutions of the CDYBE that arise from gauge
fixing conditions satisfying the conditions in Section
\ref{subsec:constraints-and-gauge-fixing} and hence all associated Dirac
brackets.
\begin{theorem}\label{thm:dcybe-standard-transformation}
Let $r:I\times \mathbb{R}\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ be a solution of the CDYBE with $x^1=\psi$,
$x^2=\alpha$, $x_1=q_\psi^aP_a$, $x_2=q_\alpha^aJ_a+q_\theta^aP_a$ that
satisfies conditions \eqref{eq:q-relations} in Theorem
\ref{thm:jacobi-for-extended-dirac} and for which $\boldify{q}_\psi^2,\boldify{q}_\alpha^2\neq
0$, $\boldify{q}_\psi\wedge\boldify{q}_\alpha=0$ and
$\partial_\alpha\boldify{q}_\psi=\partial_\alpha\boldify{q}_\alpha=\partial_\alpha^2\boldify{q}_\theta=0$
on $I\times\mathbb{R}$. Then there exists a Poincar\'e transformation $p:I\times\mathbb{R}\to
\ensuremath{\ISO(2,1)}$ as in Lemma \ref{lem:standard-form} and a diffeomorphism $\boldify{y}=(y^1,
y^2): I\times\mathbb{R}\to I'\times\mathbb{R}$ with $\partial_\alpha y_1 = 0$ and
$\partial_\alpha^2 y_2=0$ such that one of the following holds:
\begin{compactenum}[\quad a)]
\item $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta \in \Span\{\boldify{e}_0\}$ and
\vspace{-0.3cm}
\begin{equation*}
r^p(\psi,\alpha)=\tfrac 1 2 (P_a\!\otimes\!J^a \!+\! J^a\!\otimes\!P_a) + \tfrac12 \tan\tfrac{y^1(\psi)}{2} \left(P_1\!\wedge\!J_2 \!-\! P_2\!\wedge\!J_1\right) + \frac{y^2(\psi,\alpha)}{4\cos^2\tfrac{y^1(\psi)}{2}}P_1\!\wedge\! P_2,
\end{equation*}
\vspace{-0.4cm}
\item $\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta \in \Span\{\boldify{e}_1\}$ and
\vspace{-0.3cm}
\begin{equation*}
r^p(\psi,\alpha)=\tfrac 1 2 (P_a\!\otimes\!J^a \!+\! J^a\!\otimes\!P_a) + \tfrac12 \tanh\tfrac{y^1(\psi)}{2} \left(P_2\!\wedge\!J_0 \!-\! P_0\!\wedge\!J_2\right) + \frac{y^2(\psi,\alpha)}{4\cosh^2\tfrac{y^1(\psi)}{2}}P_2\!\wedge\! P_0.
\end{equation*}
\end{compactenum}
\end{theorem}
\begin{proof} $\quad$
\begin{enumerate}
\item By Lemma \ref{lem:standard-form} there exists a Poincar\'e-valued
function $p:I\times \mathbb{R}\to \ensuremath{\ISO(2,1)}$ whose Lorentzian component does not depend
on $\alpha$ and whose translation component depends on $\alpha$ at most
linearly such that $r^p:I\times \mathbb{R}\to\ensuremath{\iso(2,1)}\otimes\ensuremath{\iso(2,1)}$ and
$\boldify{q}_\psi^p,\boldify{q}_\alpha^p,\boldify{q}_\theta^p:I\times \mathbb{R}\to\mathbb{R}^3$ satisfy again the
CDYBE as well as conditions \eqref{eq:q-relations} and are of the form given
in Lemma~\ref{lem:standard-form}. This implies that $r$ is of the form
\eqref{eq:r-simple}:
\begin{align*}
r^p=\tfrac 1 2 (P_a\otimes J^a+J^a\otimes P_a)-\tfrac 1 2 \ee^{abc} w^p_c P_a\wedge J_b+\tfrac 1 2 \ee^{abc} m^p_c P_a\wedge P_b
\end{align*}
with $\boldify{q}_\psi^p,\boldify{q}_\alpha^p,\boldify{q}_\theta^p,\boldify{w}^p,\boldify{m}^p$ subject to conditions
a) or b) in Lemma \ref{lem:standard-form}. It follows that there are
smooth functions $\beta,\gamma,\delta,\epsilon, \varphi_0,\varphi_1: I \to
\mathbb{R}$ with $\beta(\psi),\gamma(\psi)\neq 0$ for all $\psi\in I$ such that
\begin{equation*}
\boldify{q}^p_\psi=\beta \boldify{e}_j,\quad
\boldify{q}^p_\alpha=\gamma \boldify{e}_j,\quad
\boldify{q}^p_\theta=\left(\delta+{\alpha\, \gamma'\beta}/{\gamma}\right)\boldify{e}_j,\quad
\boldify{w}^p=\epsilon\,\boldify{e}_j,\quad
\boldify{m}^p=(\varphi_0+\alpha\varphi_1)\boldify{e}_j,
\end{equation*}
where $j=0$ in case a) and $j=1$ in case b). Inserting these expressions
into \eqref{eq:dcybe-with-simple-UV}, one finds that $r^p$ satisfies the
CDYBE with $x^1=\psi$, $x^2=\alpha$, $x_1=q_\psi^{p,a}P_a$ and
$x_2=q_\alpha^{p,a}J_a+q_\theta^{p,a}P_a$ if and only if the coefficient
functions $\beta,\gamma,\delta,\epsilon, \varphi_0,\varphi_1: I \to \mathbb{R}$
satisfy the following set of differential equations:
\begin{align}
&\gamma \varphi_1+\tfrac 1 2 \beta\epsilon'=0,\qquad
1\pm\epsilon^2\pm 2\beta\epsilon'=0,\qquad
\beta\varphi_0'+\delta\varphi_1+\epsilon\varphi_0=0,
\end{align}
where the sign $+$ in the second equation refers to case a), the sign $-$ to
case b). Set $g=1/\gamma$ and let $f:\mathbb{R}\to\mathbb{R}$ be a function with
$f'=1/\beta$. Then the second equation can be integrated to $\epsilon(\psi)=-
\tan(f(\psi)/2)$ in case a) and $\epsilon(\psi)=\tanh(f(\psi)/2)$ in case
b). Inserting this result into the remaining two equations, we obtain
\begin{equation*}
\varphi_1(\psi)=-\tfrac 1 2 g(\psi)\epsilon'(\psi),\qquad
\delta(\psi)=-\frac{h'(\psi)}{f'(\psi) g(\psi)},
\end{equation*}
with $h(\psi)=4\varphi_0(\psi)\cos^2(f(\psi)/2)$ in case a) and
$h(\psi)=4\varphi_0(\psi)\cosh^2(f(\psi)/2)$ in case b). It follows that
$\boldify{q}_\psi,\boldify{q}_\alpha,\boldify{q}_\theta,\boldify{w},\boldify{m}$ are given by:
\begin{equation}\label{eq:transf-qs}
\left.
\begin{aligned}
&\boldify{q}_\psi^p(\psi)=\frac {\boldify{e}_0}{f'(\psi)},\quad
\boldify{q}_\alpha^p(\psi)=\frac {\boldify{e}_0}{g(\psi)},\quad
\boldify{q}_\theta^p(\psi,\alpha)=-\frac{h'(\psi)+\alpha g'(\psi)}{f'(\psi) g(\psi)}\,\boldify{e}_0,\\
&\boldify{w}^p(\psi)=-\tan(f(\psi)/2)\,\boldify{e}_0,\quad\;\boldify{m}^p(\psi,\alpha)=\frac{h(\psi)+\alpha g(\psi)}{4\cos^2(f(\psi)/2)}\,\boldify{e}_0\quad\;\;\text{in case a)},\\
&\boldify{w}^p(\psi)=\tanh(f(\psi)/2)\,\boldify{e}_0,\quad\boldify{m}^p(\psi,\alpha)=-\frac{h(\psi)+\alpha g(\psi)}{4\cosh^2(f(\psi)/2)}\,\boldify{e}_0\quad\text{in case b)}.
\end{aligned}
\quad\right\}
\end{equation}
\item With the definitions $y^1(\psi)=f(\psi)$,
$y^2(\psi,\alpha)=h(\psi)+\alpha g(\psi)$ this yields expressions a) and b)
for $r$. For any $y_1,y_2\in \ensuremath{\iso(2,1)}$ the right-hand side of the CDYBE then
takes the form
\begin{align*}
&y_1^{(1)}\partial_{y^1}\,r_{23}- y_1^{(2)}\partial_{y^1}\, r_{13}+y_1^{(3)}\partial_{y^1}\, r_{12}+y_2^{(1)}\partial_{y^2}\,r_{23}- y_2^{(2)}\partial_{y^2}\, r_{13}+y_2^{(3)}\partial_{y^2}\, r_{12}\\
&=x_1^{(1)}\partial_{\psi}\,r_{23}- x_1^{(2)}\partial_{\psi}\, r_{13}+x_1^{(3)}\partial_{\psi}\, r_{12}+x_2^{(1)}\partial_{\alpha}\,r_{23}- x_2^{(2)}\partial_{\alpha}\, r_{13}+x_2^{(3)}\partial_{\alpha}\, r_{12},
\end{align*}
where
\begin{equation*}
x_1(\psi)=\frac{y_1(\psi)}{f'(\psi)},\qquad
x_2(\psi, \alpha)=\frac{y_2(\psi, \alpha)}{g(\psi)}-\frac{(h'(\psi)+\alpha g'(\psi))\,y_1(\psi)}{f'(\psi)g(\psi)}.
\end{equation*}
Setting $y_1=P_0$, $y_2=J_0$ in case a) and $y_1=P_1$, $y_2=J_1$ in case b),
we obtain $x_1=q_\psi^{p,a}P_a$, $x_2=q_\alpha^{p,a}J_a+q_\theta^{p,a}P_a$
with $\boldify{q}_\psi^p,\boldify{q}_\alpha^p,\boldify{q}_\theta^p$ given by
\eqref{eq:transf-qs}. This proves the claim.
\end{enumerate}
\end{proof}
Theorem \ref{thm:dcybe-standard-transformation} amounts to a classification of
all possible solutions of the CDYBE \eqref{eq:dcybe} of the form in Definition
\ref{def:extended-dirac-bracket} that satisfy the additional conditions
\eqref{eq:q-relations} and hence of all Poisson structures of the type in
Definition \ref{def:extended-dirac-bracket}. It states that locally all such
Poisson structures are obtained from one of the classical dynamical
$r$-matrices in Lemma \ref{lem:dcybe-solutions} by applying a suitable
Poincar\'e transformation together with a rescaling of the variables $\psi$,
$\alpha$. Note, however, that this classification is only local in the
following sense: for any given value $\psi_0$ of the variable $\psi$ for which
$\boldify{q}_\psi^2,\boldify{q}_\alpha^2\neq 0$, there is an open interval $I\subset\mathbb{R}$,
$\psi_0\in I$, such that $r(\psi,\alpha)$ can be transformed into one of the
classical dynamical $r$-matrices in Theorem
\ref{thm:dcybe-standard-transformation} for all $(\psi,\alpha)\in I\times\mathbb{R}$.
In particular, this locally determines all possible Dirac brackets that arise
from gauge fixing procedures that satisfy the well-motivated conditions in
Section \ref{subsec:constraints-and-gauge-fixing}. The Dirac bracket of such a
gauge fixing procedure is always determined by a solution of the CDYBE that
satisfies the additional conditions \eqref{eq:q-relations} and
$\boldify{q}_\psi\wedge\boldify{q}_\alpha=0$. For those values of the variable $\psi$ for which
$\boldify{q}_\psi^2(\psi),\boldify{q}_\alpha^2(\psi)\neq 0$ , the resulting Dirac bracket can
be transformed into the bracket defined by one of the two classical dynamical
$r$-matrices in Lemma \ref{lem:dcybe-solutions}.
However, the Dirac bracket resulting from a generic gauge fixing condition is
associated with maps $\boldify{q}_\psi,\boldify{q}_\alpha:\mathbb{R}\to\mathbb{R}^3$ for which the signature
of $\boldify{q}_\psi^2,\boldify{q}_\alpha^2$ changes as a function of the variable $\psi$. In
particular, this is the case for the specific gauge fixing conditions
investigated in \cite{Meusburger:2011aa}. It is shown there that the map
$\boldify{q}_\psi:\mathbb{R}^2\to\mathbb{R}^3$ is timelike, lightlike or spacelike for those values
of $\psi$ for which, respectively, the product $u_{M_2}\cdot u_{M_1}$ of the
Lorentzian components of the gauge-fixed holonomies is elliptic, parabolic or
hyperbolic. It is well-known that the product of two elliptic elements of
$\ensuremath{\SO_+(2,1)}$ can be either elliptic, parabolic or hyperbolic. The first requirement
on the gauge fixing conditions in Section \ref{subsec:dirac-constraints}
implies that all of these cases must arise in the Dirac bracket. This suggests
that such transitions between timelike, spacelike and lightlike solutions are a
generic outcome of the gauge fixing procedure when the two holonomies $M_1,M_2$
are elliptic.
It is therefore not possible to reduce the investigation of gauge fixing
procedures and the resulting Poisson structures to classical dynamical
$r$-matrices in the sense of Definition \ref{def:dyn-r-matrix}. Instead, one
needs to admit more general solutions of the CDYBE and to allow the associated
Lie subalgebras $\algebra{h}(\psi,\alpha)\subset \ensuremath{\iso(2,1)}$ to vary non-trivially with the
variables $\psi$ and $\alpha$. Such solutions are no longer invariant under the
action of the subalgebra $\algebra{h}(\psi,\alpha)$. Instead, they satisfy the
generalised consistency conditions \eqref{eq:q-relations} which together with
the CDYBE ensure the Jacobi identity for the associated Poisson bracket.
\section{Outlook and conclusions}
\label{sec:outlook}
In this paper we applied the Dirac gauge fixing procedure to the description of
the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections in terms of an ambient space with
an auxiliary Poisson structure \cite{Fock:1998aa}. We investigated a large
class of gauge fixing conditions subject to two well-motivated structural
requirements, namely that the gauge fixing is based on the choice of two
punctures on the underlying Riemann surface and that it preserves the natural
$\mathbb{N}$-grading of the auxiliary Poisson structure.
We showed that the Poisson algebras obtained from gauge fixing are in
one-to-one correspondence with solutions of the classical dynamical Yang-Baxter
equation (CDYBE) and of a set of additional equations. The latter can be viewed
as the counterpart of the usual requirement of invariance of the classical
dynamical $r$-matrices under the action of a fixed Cartan subalgebra. These
solutions of the CDYBE are a generalisation of classical dynamical $r$-matrices
in which the associated two-dimensional Lie subalgebras of $\ensuremath{\iso(2,1)}$ vary with
the dynamical variables.
We also demonstrated how a change of gauge fixing conditions affects the
associated solutions of the CDYBE and showed that this change corresponds to
the action of dynamical Poincar\'e transformations. These dynamical Poincar\'e
transformations generalise the gauge transformations of classical dynamical
$r$-matrices in \cite{Etingof:1998aa}. We found that every solution obtained
via gauge fixing can be transformed into one of two classical dynamical
$r$-matrices via these transformations and a rescaling for almost all values of
the dynamical variables. This gives rise to a complete (local) classification
of all possible outcomes of gauge fixing in the context of the moduli space of
flat $\ensuremath{\ISO(2,1)}$-connections.
However, for generic solutions there are also certain values of the dynamical
variables for which the solutions cannot be transformed into standard classical
dynamical $r$-matrices. These singular points appear at the transition between
classical dynamical $r$-matrices for two non-conjugate Cartan subalgebras of
$\ensuremath{\iso(2,1)}$ and are associated with two-dimensional Lie subalgebras of $\ensuremath{\iso(2,1)}$
which contain parabolic elements of $\ensuremath{\so(2,1)}$. This occurs for instance in the
solutions in Lemma \ref{lem:dcybe-for-special-r} and cannot be excluded by
suitable gauge fixing conditions.
To our knowledge this phenomenon does not appear in earlier references on the
topic. We expect that this is due to the fact that most of these references
investigate classical dynamical $r$-matrices for complex (semi)simple Lie
groups, for which all Cartan subalgebras are conjugate. Based on our results,
we would predict that such transitions between classical dynamical $r$-matrices
for non-conjugate Cartan subalgebras arise for real Lie groups for which the
underlying symmetric, non-degenerate $\Ad$-invariant symmetric form
$\langle\,,\,\rangle$ is indefinite.
Another aspect which merits further investigation is the physical
interpretation of these classical dynamical $r$-matrices in the context of
(2+1)-dimensional (quantum) gravity. It was shown in \cite{Meusburger:2011aa}
that gauge fixing procedures of the type investigated in this paper can be
viewed as the specification of an observer in a (2+1)-dimensional
spacetime. Moreover, the results in \cite{Meusburger:2011aa} suggest a direct
geometrical interpretation for the two dynamical variables in the classical
dynamical $r$-matrices: they correspond to the total mass and angular momentum
of the spacetime as measured by this observer. In this interpretation, the two
standard classical dynamical $r$-matrices would be associated with the
centre-of-mass frame of the universe and the transition points between
different Cartan subalgebras would correspond to the formation of Gott pairs
\cite{Gott:1991aa}.
We expect that our results could be generalised to the other moduli spaces of
flat connections that arise in the description of (2+1)-gravity for different
signatures and different values of the cosmological constant. For Lorentzian
signature, the relevant Lie groups are $\ensuremath{\ISO(2,1)}$, $\PSL(2,\mathbb{R})\times \PSL(2,\mathbb{R})$
and $\PSL(2,\mathbb{C})$, respectively, for vanishing, negative and positive
cosmological constant. For Euclidean signature, the corresponding Lie groups
are $\ISO(3)$, $\SO(3)\times\SO(3)$ and $\PSL(2,\mathbb{C})$. It seems plausible that
analogous gauge fixing procedures applied to these groups would yield similar
outcomes, with the exception of the transition between different Cartan
subalgebras, which should not occur for Euclidean signature.
In this context, it would also be desirable to understand in more detail how
our results are related to the classical dynamical $r$-matrix symmetries
obtained by Buffenoir, Noui and Roche via a regularisation procedure for point
particles coupled to Chern-Simons theory with gauge group $\SL(2,\mathbb{C})$
\cite{Buffenoir:2003aa, Buffenoir:2005aa, Noui:2005aa}. Although the approach
and setting in this work are very different, there should be an underlying
reason that forces the appearance of classical dynamical $r$-matrices in both
cases.
It would also be interesting to investigate in more detail the relation between
gauge-fixed Poisson structures on the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections
and mathematical structures associated with classical dynamical $r$-matrices
such as Poisson-Lie groupoids. As the auxiliary Poisson algebra which is
gauge-fixed is closely related to certain structures from the theory of
Poisson-Lie groups, namely the dual Poisson-Lie structure and the Heisenberg
double, it could be expected that gauge fixing should be related to the
construction of dynamical versions of these structures.
Finally, we expect our results to have useful application in the
quantisation of the moduli space of flat $\ensuremath{\ISO(2,1)}$-connections. This is due to
the fact that the resulting Poisson structure is very closely related to Fock
and Rosly's Poisson structure on the ambient space. The only difference is that
the classical $r$-matrix is replaced by a classical dynamical $r$-matrix
associated with two-dimensional Lie subalgebras of $\ensuremath{\iso(2,1)}$. As Fock and Rosly's
Poisson structure serves as the starting point for the combinatorial
quantisation formalism, this suggests that this formalism could be extended
straightforwardly to include the gauge-fixed Poisson structure. This would
reduce the task of quantising the theory to the construction of the associated
dynamical quantum group.
\section*{Acknowledgements}
This work was supported by the Emmy Noether research grant ME 3425/1-1 of the
German Research Foundation (DFG). We thank Karim Noui for
useful remarks and discussions and Stefan Waldmann and Winston Fairbairn for comments on a draft of
this paper. The \textsc{xAct} tensor calculus package
\cite{Martin-Garcia:2008aa} proved useful in some of the computations.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,348
|
#ifndef __itktubeVectorImageToListGenerator_h
#define __itktubeVectorImageToListGenerator_h
#include <itkDataObject.h>
#include <itkDataObjectDecorator.h>
#include <itkFixedArray.h>
#include <itkListSample.h>
#include <itkPixelTraits.h>
#include <itkProcessObject.h>
namespace itk
{
namespace tube
{
namespace Statistics
{
/** \class VectorImageToListGenerator
* \brief The class takes an image as input and generates a list sample as
* output.
*
* There are differences between this class and VectorImageToListAdaptor.
* This class is not an adaptor. It creates a new list sample and does not
* provide a pseudo interface to the actual image to make it look like a
* list sample.
*
* The class optionally allows you to specify a mask image as an input. The
* list sample ( if a mask is specified ) is constructed from pixels that are
* within the mask
*
* \todo
* In future allow the filter to take a Spatial object as input so a
* generic spatial object like an ellipse etc can be used as a mask.
* Sure the ImageMaskSpatialObject
* can represent image masks too, so why not make SpatialObjects the
* default. I think the ImageMaskSpatialObject is slow in terms of
* inefficient iteration through the image.
*
* \sa VectorImageToListAdaptor
*/
template< class TImage, class TMaskImage >
class VectorImageToListGenerator : public ProcessObject
{
public:
/** Standard class typedefs */
typedef VectorImageToListGenerator Self;
typedef ProcessObject Superclass;
typedef SmartPointer< Self > Pointer;
typedef SmartPointer< const Self > ConstPointer;
/** Run-time type information ( and related methods ). */
itkTypeMacro( VectorImageToListGenerator, ProcessObject );
/** Method for creation through the object factory. */
itkNewMacro( Self );
/** Image typedefs */
typedef TImage ImageType;
typedef typename ImageType::Pointer ImagePointer;
typedef typename ImageType::ConstPointer ImageConstPointer;
typedef typename ImageType::PixelType PixelType;
typedef PixelType MeasurementVectorType;
/** Mask Image typedefs */
typedef TMaskImage MaskImageType;
typedef typename MaskImageType::Pointer MaskImagePointer;
typedef typename MaskImageType::ConstPointer MaskImageConstPointer;
typedef typename MaskImageType::PixelType MaskPixelType;
/** Type of the output list sample */
typedef itk::Statistics::ListSample< MeasurementVectorType > ListSampleType;
/** Superclass typedefs for Measurement vector, measurement,
* Instance Identifier, frequency, size, size element value */
typedef PixelTraits< typename ImageType::PixelType > PixelTraitsType;
typedef typename ListSampleType::MeasurementVectorSizeType
MeasurementVectorSizeType;
typedef DataObject::Pointer DataObjectPointer;
/** ListSample is not a DataObject, we need to decorate it to push it down
* a ProcessObject's pipeline */
typedef DataObjectDecorator< ListSampleType > ListSampleOutputType;
/** the number of components in a measurement vector */
itkStaticConstMacro( MeasurementVectorSize, unsigned int,
PixelTraitsType::Dimension );
/** Standard itk::ProcessObject subclass method. */
using Superclass::MakeOutput;
virtual DataObjectPointer MakeOutput( DataObjectPointerArraySizeType idx )
override;
virtual void SetMeasurementVectorSize( const MeasurementVectorSizeType s )
{
// Measurement vector size for this class is fixed as the pixel's
// dimension. This method should throw an exception if the user tries to
// set the dimension to a different value.
if( s != MeasurementVectorSize )
{
itkExceptionMacro(
<< "Measurement vector size for the image adaptor obtained"
<< " from the pixel dimension is: "
<< MeasurementVectorSize << " but you "
<< "are setting it to " << s );
}
}
unsigned int GetMeasurementVectorSize( void ) const
{
return MeasurementVectorSize;
}
/** Method to set/get the image */
void SetInput( const ImageType* image );
const ImageType* GetInput( void ) const;
/** Method to set/get the mask */
void SetMaskImage( const MaskImageType* image );
const MaskImageType* GetMaskImage( void ) const;
/** Method to get the list sample, the generated output. Note that this does
* not invoke Update(). You should have called update on this class to get
* any meaningful output. */
const ListSampleType * GetListSample( void ) const;
/** Set the pixel value treated as on in the mask. If a mask has been
* specified, only pixels with this value will be added to the list sample, if
* no mask has been specified all pixels will be added as measurement vectors
* to the list sample. */
void SetMaskValue( const MaskPixelType maskValue );
itkGetMacro( MaskValue, MaskPixelType );
itkSetMacro( UseSingleMaskValue, bool );
itkGetMacro( UseSingleMaskValue, bool );
/** This method causes the filter to generate its output. */
virtual void GenerateData( void ) override;
/** This method ensures that a mask image if specified has requested regions
* that at least contain the input image's buffered region. */
virtual void GenerateInputRequestedRegion( void ) override;
virtual void GenerateOutputInformation( void ) override;
protected:
VectorImageToListGenerator( void );
virtual ~VectorImageToListGenerator( void ) {}
void PrintSelf( std::ostream& os, Indent indent ) const override;
private:
VectorImageToListGenerator( const Self& ); //purposely not implemented
void operator=( const Self& ); //purposely not implemented
// To remove warning "was hidden [-Woverloaded-virtual]"
void SetInput( const DataObjectIdentifierType &, itk::DataObject * )
override {};
MaskPixelType m_MaskValue;
bool m_UseSingleMaskValue;
}; // End class VectorImageToListGenerator
} // End namespace Statistics
} // End namespace tube
} // End namespace itk
#ifndef ITK_MANUAL_INSTANTIATION
#include "itktubeVectorImageToListGenerator.hxx"
#endif
#endif // End !defined( VectorImageToListGenerator )
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,719
|
require "net/http"
require "time"
RSpec.describe HttpSignatures::Verifier do
DATE = "Fri, 01 Aug 2014 13:44:32 -0700"
DATE_DIFFERENT = "Fri, 01 Aug 2014 13:44:33 -0700"
subject(:verifier) { HttpSignatures::Verifier.new(key_store: key_store) }
let(:key_store) { HttpSignatures::KeyStore.new("pda" => "secret") }
let(:message) { Net::HTTP::Get.new("/path?query=123", headers) }
let(:headers) { {"Date" => DATE, "Signature" => signature_header} }
let(:signature_header) do
'keyId="%s",algorithm="%s",headers="%s",signature="%s"' % [
"pda",
"hmac-sha256",
"(request-target) date",
"cS2VvndvReuTLy52Ggi4j6UaDqGm9hMb4z0xJZ6adqU=",
]
end
it "verifies a valid message" do
expect(verifier.valid?(message)).to eq(true)
end
it "rejects message with missing headers" do
headers.clear
expect(verifier.valid?(message)).to eq(false)
end
it "rejects message with tampered path" do
message.path << "x"
expect(verifier.valid?(message)).to eq(false)
end
it "rejects message with tampered date" do
message["Date"] = DATE_DIFFERENT
expect(verifier.valid?(message)).to eq(false)
end
it "rejects message with tampered signature" do
message["Signature"] = message["Signature"].sub('signature="', 'signature="x')
expect(verifier.valid?(message)).to eq(false)
end
it "rejects message with malformed signature" do
message["Signature"] = "foo=bar,baz=bla,yadda=yadda"
expect(verifier.valid?(message)).to eq(false)
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 952
|
\subsection{Introduction}
\label{sec:A}
The multiplicity of produced particles is an important characteristic of the final state
in high energy proton-proton collisions.
The shapes of the charged-particle multiplicity distributions (MDs) measured with high accuracy
can provide valuable information on various processes in these interactions.
The distributions reflect correlations in the system encoded in an integrated form.
At lower collision energies, the two-particle correlations are dominant and a single negative
binomial distribution (NBD) \cite{Giov1} gives a satisfactory description of the MD up to the ISR energies.
At higher collision energies, a shoulder structure of the MD at high multiplicities \cite{UA5}
and oscillation
of the ratio $H_q=K_q/F_q$ of the cumulant to factorial moments \cite{Dremin0} have been found.
No pattern of this kind appears in a phenomenological fit by a single NBD \cite{Dremin1}.
It has been proposed \cite{GiovUgo1} to describe the observed shoulder structure as a weighted
superposition of soft events (without mini-jets) and semi-hard events (with mini-jets),
each of the NBD type. The suggestion was emphasised by the interesting and remarkable observation
that the oscillations of $H_q$ vs. rank $q$ obtained from data is reproduced
by the $K_q$ over $F_q$ ratio calculated from the superposition of two NBDs.
The appearance of the second multiplicity component under the tail of the MD is
connected with cumulants each of
which involves an infinite ``cumulative" sum over ${\it all}$ multiplicity probabilities~$P(n)$.
The extrapolation of the weighted superposition of the two defined classes
of events from the TeV to the multi-TeV energy domain
and a study of the properties of particle clans raised intriguing questions concerning the onset
of a third class of events in the tail of the distribution, both
in the full phase space \cite{GiovUgo2} and in the limited windows in pseudorapidity \cite{GiovUgo3}.
Later study \cite{IZ} of the MDs measured
at $\sqrt{s}= 7$~TeV indicated
existence of the third multiplicity component at low $n$.
In this region, near the maximum of $P(n)$,
there is a suitable tool, the method of combinants \cite{Gyulassy}, to be used
in searching for new phenomena in the behavior of the MDs.
The combinants ${\cal C}(i)$ have similar additive properties as cumulants
which are expressible as an infinite sum over all probabilities.
The latter are therefore convenient for an analysis of the total shape of the MDs.
On the other hand, the combinants can be expressed as a finite combination of the ratios $P(n)/P(0)$.
The combinants are thus extremely suitable
for study of the final set of the multiplicity frequencies which make up the
experimental data at low multiplicities.
In this paper we study in terms of a three-component description the data on MD of charged particles produced
in $pp$ collisions at the LHC at new high energies, $\sqrt{s}=8$ and 13~TeV.
Each component is represented by a single NBD and parametrized by two parameters, $\bar{n}$ and $k$.
The total distribution is weighted sum of three NBDs.
The paper extends an earlier analysis \cite{IZ} performed with data on MD measured
at $\sqrt{s}=0.9$ and 7~TeV.
The study of the ATLAS measurements \cite{ATLAS1,ATLAS2,ATLAS3,ATLAS4}
show that the two-component description of the MD is unsatisfactory. Within the
multi-component NBD parametrization, the structure of the data indicates the necessity
of a third component in the region of low multiplicities.
We demonstrate that, unlike the two-component fits to the MDs measured at the LHC,
the phenomenological fits with the third NBD component at low $n$
can give oscillations of the combinants multiplied by their rank discovered lately \cite{WW1}
in multiplicity data measured by the CMS \cite{CMS} and ALICE \cite{ALICE8} Collaborations.
Although the sensitivity of the oscillations to the systematic uncertainties
of the measurements is large, the natural applicability and power of the method of combinants
in the analysis of the fine structure of the MDs observed in $pp$ collisions at the LHC
is of interest in the search for and study of new phenomena emerging in the global characteristics
of particle production.
The second part of the paper is dedicated to a study of the obtained results within
the clan structure analysis of data on MD.
The qualifying assumption is that each of the components of the weighted superposition
used to parametrize the experimental data has the NBD form.
The emergence of the peak in the MD at low $n$, described by the third NBD, has a serious impact
on the clan parameters of the first and the second component
of the total distribution.
The parameters changed dramatically in comparison with the two-NBD analyses
considered usually in different scenarios of soft and semi-hard events.
The most appealing observation is the increase of the average number of clans
in the high-$n$ (semi-hard) component with the center-of-mass
energy $ \sqrt{s} $. This is in sharp contrast with two-NBD parametrizations of the multiplicity data.
The contribution of the third NBD to the low multiplicity part of the charged particle distribution
has a serious consequence
for the average number of particles per clan in the first and the second component.
It turns out that the dominant component with largest probability is characterized with few clans
containing many particles. We illustrate that the second, high-$n$ component consists
of large number of clans containing less particles.
The properties of clans of the three-NBD parametrization of the MDs measured at the LHC and some
corresponding mechanisms of particle production are discussed.
\subsection{Weighted superposition of NBDs}
\label{sec:B}
The data on MD provide information on various processes underlying the production
mechanism contain information as regards correlations in the system and can serve as a test for
probing the dynamics of the interaction.
The particle production
is based on quantum chromodynamics (QCD).
The multi-particle processes responsible for the final multiplicity
involve also soft scales,
including hadronization,
which remain beyond the reach of perturbative QCD.
The complex phenomena that influence MD are therefore hard to describe in detail and one has
to rely on phenomenology.
One of the most successful distribution functions used in describing the probability distributions
of produced particles in $pp/\bar{p}p$ collisions is the two-parameter NBD,
\begin{equation}
P(n,\bar{n},k)=\frac{\Gamma(n+k)}{\Gamma(k)\Gamma(n+1)}
\left[\frac{\bar{n}}{k+\bar{n}}\right]^n
\left[\frac{k}{k+\bar{n}}\right]^k,
\label{eq:r1}
\end{equation}
where $\bar{n}$ is the average multiplicity and $k$ characterizes the width of
the distribution.
The history and genesis of NBD can be found e.g. in \cite{NBDhistory}.
An important application of this distribution in the particle production is
connected with cascading mechanisms.
A frequently used and popular interpretation of NBD comes
from the clan model \cite{GioVanHove} where a particle emits additional particles in a
self-similar branching pattern. The clans are produced independently and contain particles of the same ancestry.
The Poisson distribution is the distribution of clans consisting of single particles. It is obtained for $k=\infty$.
A superposition of NBDs was exploited by different approaches
to multiple production in hadron collisions.
It is based on the decomposition
\begin{equation}
P(n)=
\sum_{i=1}^N \alpha_i P(n,\bar{n}_i,k_i), \ \ \ \
\sum_{i=1}^N \alpha_i = 1,
\label{eq:r2}
\end{equation}
where $P(n,\bar{n}_i,k_i)$ is given by (\ref{eq:r1}).
In this paper we study high statistic multiplicity data \cite{ATLAS2,ATLAS3,ATLAS4}
obtained by the ATLAS Collaboration at the LHC
in the framework of a weighted superposition of three NBDs.
The aim is to achieve a detailed description of the data including the shape
of the distributions in the region of maximal values of $P_n$.
Specifically, we consider the function (\ref{eq:r2}) for $N=3$
and extract values of the eight independent parameters ($\bar{n}_i$, $k_i$, $\alpha_2$ and $\alpha_3$)
corresponding to the ATLAS data
with different transverse momentum cuts
at the energies $\sqrt{s}=8$ and 13~TeV.
The results are compared with our previous analysis \cite{IZ}
at $\sqrt{s}=0.9$ and 7~TeV and with a two-NBD parametrization of the same data.
\subsection{Analysis of new ATLAS data at midrapidity}
\label{sec:2}
The multiplicity distributions of charged particles produced in $pp$ collisions
have been measured at the LHC by different experiments, in different kinematic regions,
at different energies and for different classes of events.
The experimental data accumulated by the ATLAS Collaboration are exceptional, for they have small systematic errors
and are based on the analysis of an extremely large number of events.
The ATLAS data at $\sqrt{s}=8$~TeV \cite{ATLAS2} and $\sqrt{s}=13$~TeV \cite{ATLAS3,ATLAS4}
exploit a similar methodology to that used at lower centre-of-mass energies \cite{ATLAS1}.
The analyzed charged particle MDs were obtained in the same
phase-space regions with application of the same multiplicity cuts
in the same pseudorapidity window $|\eta|<2.5$.
The recorded number of events for the conditions $p_T > 100$~MeV/c and $n_{ch}>1$
exceeds 9 million for both energies, $\sqrt{s}=8$ and 13~TeV.
A similar large number of events was considered in the data sample with the
higher transverse momentum cut $p_T > 500$~MeV/c and $n_{ch}>0$.
The measurements in both regions give a severe restriction on models
of multiparticle production in $pp$ collisions at high energies.
This concerns in particular the weighted superposition of two NBDs, which
is usually attributed to the classification of events into soft and semi-hard events
with respect to the momentum
transfer in parton-parton scatterings.
\subsubsection{MDs at low transverse momentum}
\label{sec:C.1}
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure1a.eps}
\vskip -0.6cm
\includegraphics[width=78mm,height=78mm]{Figure1b.eps}
\vskip -0.7cm
\caption{
MD of charged particles
measured by the ATLAS Collaboration \protect\cite{ATLAS2,ATLAS4}
in the pseudorapidity interval $|\eta|<2.5$ for $p_T > 100$~MeV/c, $n>1$
at {\bf a} $\sqrt s=8$~TeV and {\bf b} $\sqrt s=13$~TeV.
The error bars include both the statistical and the systematic uncertainties summed in quadrature.
The solid lines represent a superposition of three NBDs
with the parameters from Tables \ref{tab:1} and \ref{tab:2}, respectively.
The dash-dot, dash and dash-dot-dot lines show
single components of the total MD
}
\label{fig:1}
\end{figure}
The most inclusive phase-space region covered by the ATLAS measurements corresponds to
the conditions $p_T > 100$~MeV/c and $n_{ch}>1$.
We have fitted the data \cite{ATLAS2,ATLAS4} on MD measured
in the interval $ |\eta|<2.5$ in this region with a weighted superposition of three NBDs.
The decomposition of the total MD at the energies $\sqrt s=8$~TeV and 13~TeV
is shown in Fig. \ref{fig:1}a,~b, respectively.
The experimental data are indicated by symbols and the fitted three-component
function (\ref{eq:r2}) is depicted by the solid line.
The dash-dot, dash and dash-dot-dot lines represent
single NBD components of the total distribution.
The corresponding parameters and values of $\chi^2$ are quoted in Tables~\ref{tab:1} and \ref{tab:2}.
Terms of the fitting procedure are explained in Appendix A.
\begin{table}
\caption{
The parameters of the superposition of three and two NBDs
obtained from fits to data \protect\cite{ATLAS2} on MD measured by the ATLAS
Collaboration in the pseudorapidity window $|\eta|<2.5$ with the cut $p_T>100$~MeV/c, $n>1$ at
$ \sqrt{s}=8$~TeV.
The parameter values were obtained by minimization of Eq.~(\ref{eq:a1})
}
\label{tab:1}
\begin{tabular*}{\columnwidth}{@{}l@{\extracolsep{\fill}}lll@{}}
\hline\noalign{\smallskip}
{\it i} & $\ \ \ \alpha_i$ & \ \ \ $\bar{n}_i$ & \ \ \ $k_i$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.763$^{+0.041}_{-0.055}$ & 24.2$^{+1.4 }_{-1.9 }$ &\ \,1.44$^{+0.06 }_{-0.04 }$ \\ \noalign{\smallskip}
2 & 0.149$^{+0.046}_{-0.032}$ & 62.7$^{+1.7 }_{-2.3 }$ &\ \,6.0$^{+0.8 }_{-0.6 }$ \\ \noalign{\smallskip}
3 & 0.088$^{+0.009}_{-0.009}$ & 10.8$^{+0.2 }_{-0.2 }$ & 18.9$^{+7.8 }_{-4.5 }$ \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 10.2/(85-8)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.440\,$\pm$\,0.019 & 11.90\,$\pm$\,0.25 & \ 2.58\,$\pm$\,0.10 \\
2 & 0.560\,$\pm$\,0.019 & 42.78\,$\pm$\,0.76 & \ 2.72\,$\pm$\,0.12 \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 119/(85-5)} \\
\noalign{\smallskip}\hline
\end{tabular*}
\end{table}
\begin{table}[h!]
\caption{
The parameters of the superposition of three and two NBDs
obtained from fits to data \protect\cite{ATLAS4} on MD measured by the ATLAS
Collaboration in the pseudorapidity window $|\eta|<2.5$ with the cut $p_T\!>\!100$~MeV/c, $n\!>\!1$ at
$ \sqrt{s}\!~=\!~13$~TeV.
The parameter values were obtained by minimization of Eq.~(\ref{eq:a1})
}
\label{tab:2}
\begin{tabular*}{\columnwidth}{@{}l@{\extracolsep{\fill}}lll@{}}
\hline\noalign{\smallskip}
{\it i} & $\ \ \ \alpha_i$ & \ \ \ $\bar{n}_i$ & \ \ \ $k_i$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.771$^{+0.036}_{-0.048}$ & 26.9$^{+1.5 }_{-2.0 }$ &\ \,1.28$^{+0.04 }_{-0.03 }$ \\ \noalign{\smallskip}
2 & 0.145$^{+0.041}_{-0.029}$ & 71.2$^{+1.4 }_{-2.2 }$ &\ \,5.8$^{+0.8 }_{-0.6 }$ \\ \noalign{\smallskip}
3 & 0.084$^{+0.007}_{-0.007}$ & 11.1$^{+0.1 }_{-0.2 }$ & 16.3$^{+3.5 }_{-2.5 }$ \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 18.9/(86-8)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.337\,$\pm$\,0.013 & 10.78\,$\pm$\,0.10 & \ 2.55\,$\pm$\,0.11 \\
2 & 0.663\,$\pm$\,0.013 & 43.62\,$\pm$\,0.51 & \ 1.96\,$\pm$\,0.06 \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 265/(86-5)} \\
\noalign{\smallskip}\hline
\end{tabular*}
\end{table}
One can see from Fig.~\ref{fig:1}
that the decomposition into three components is similar at both energies and reveals identical
features to those found \cite{IZ} in the same class of events at lower $\sqrt s$.
The dominant component with the largest probability gives the main contribution $ \alpha_1\bar{n}_1 $
to the total average multiplicity.
The other two components contribute to the high- and low-multiplicity region, respectively.
The average multiplicities of the first and the second component, $\bar{n}_1$ and $\bar{n}_2$,
increase with energy. This results in broadening of the total distribution.
The average multiplicity $\bar{n}_3 \simeq 11$ of the third component
is nearly energy independent.
Within the errors quoted in Table~\ref{tab:1},
the probabilities $ \alpha_i $ show a weak energy dependency as well.
The values of the parameters $k_i$ increase with decreasing probabilities $\alpha_i$.
The first NBD with the largest probability $\alpha_1$ is characterized by the smallest parameter $k_1$.
The NBD component under the peak of the distribution at low multiplicities
is narrow with large value of $k_3$.
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure2a.eps}
\vskip -0.7cm
\includegraphics[width=78mm,height=78mm]{Figure2b.eps}
\vskip -0.7cm
\caption{
Normalized residues of the MDs relative to superposition of two NBDs $(\mathrm{P_n^{2NBD}})$
with the parameters listed in Tables~\ref{tab:1} and \ref{tab:2}, respectively.
The symbols correspond to data on MDs measured by the ATLAS Collaboration \protect\cite{ATLAS2,ATLAS4}
in the interval $|\eta|<2.5$ with the cut $p_T > 100$~MeV/c, $n>1$
at {\bf a} $\sqrt s=8$~TeV and {\bf b} $\sqrt s=13$~TeV.
The error bars include both the statistical and the systematic uncertainties summed in quadrature.
The solid lines represent a three-NBD
superposition with the parameters from Tables \ref{tab:1} and \ref{tab:2}, respectively.
The inserts show the detailed behaviour of the residues at
low~$n$
}
\label{fig:2}
\end{figure}
We have fitted the ATLAS data with a weighted superposition of two NBDs.
The obtained values of the parameters for the two-component hypothesis
are given in Tables~\ref{tab:1} and \ref{tab:2}.
Figure \ref{fig:2} shows the relative residues of data with respect to the two-NBD fits.
The points correspond to the measurements of the ATLAS Collaboration
in the pseudorapidity interval $|\eta|<2.5$ for $p_T > 100$~MeV/c, $n>1$.
The inserts show the detailed structure of the residues at low multiplicities.
The solid lines are given by the three-component description of the ATLAS data
with the parameters quoted in Tables \ref{tab:1} and \ref{tab:2}.
One can see that the two-NBD approximation of the measured MDs is unsatisfactory at
both energies.
The corresponding values of $ \chi^2/dof $ are too large,
especially at the energy $\sqrt{s}=13$~TeV.
The high statistic ATLAS data measured with relatively small systematic uncertainties
manifest a distinct peak around $n\sim 11$.
The description of the peak clearly seen in the residues in Fig.~\ref{fig:2}
was obtained by the third negative binomial component with $\bar{n}_3 \simeq 11$,
similar to the demonstration in \cite{IZ} at lower $\sqrt s$.
The third component is depicted by the dash-dot-dot lines in Fig.~\ref{fig:1}.
\subsubsection{MDs with the cut $p_T>$500 MeV/c}
\label{sec:C.2}
The ATLAS Collaboration presented data on MD produced in $pp$ collisions
in the separate phase-space region defined by the conditions $p_T > 500$~MeV/c and $n_{ch}>0$
at the energies $\sqrt s=8$~TeV \cite{ATLAS2} and 13~TeV \cite{ATLAS3}.
The distributions were measured in the window $|\eta|<2.5$.
They differ from earlier analyses of primary charged particles
in that charged particles with a life time 30~ps $<\tau <300$~ps were considered as secondary
particles and thus excluded.
The ATLAS data at $\sqrt s=8$ and 13~TeV together with three-NBD fits are shown
in Fig.~\ref{fig:3}a, b, respectively.
The symbols and the lines have the same meaning as in Fig.~\ref{fig:1}.
The corresponding parameters and values of $\chi^2/dof$ are given
in Tables \ref{tab:3} and \ref{tab:4}.
One can see some similarities when comparing the description of the most inclusive and
the $p_T$-cut data shown in Figs. \ref{fig:1} and \ref{fig:3}, respectively.
The component with the largest probability $\alpha_1$ has the smallest parameter $k_1$.
The parameters $k_i$ increase as the probabilities $\alpha_i$ decrease.
The average multiplicity $\bar{n}_3 \simeq 3$ of the third component
is nearly energy independent.
The probability $\alpha_3$ of the third component is non-zero also in the $p_T$-cut data sample.
On the other hand, there are differences between the description of data shown
in Figs. \ref{fig:1} and \ref{fig:3}, respectively.
With the imposed $p_T$ cut, the value of $\bar{n}_3$ becomes significantly smaller
relative to the average multiplicities of the first and the second NBD component.
The multiplicity $\bar{n}_1$ does not increase with energy but shows signs of saturation.
The contribution $\alpha_2\bar{n}_2$ of the second component to the total average multiplicity
increases rapidly with $\sqrt{s}$ and becomes dominant at $\sqrt s=13$~TeV.
A decreasing tendency of the probabilities $\alpha_i$ with the index $i$
(seen in minimal $p_T$-biased data)
is well visible for $p_T>500$~MeV/c at $\sqrt s=13$~TeV only.
A similar description applies to the hierarchy of the parameters $k^{-1}_i$ characterizing the widths
of the single NBD components.
\begin{table}
\caption{
The parameters of the superposition of three and two NBDs
obtained from fits to MDs \protect\cite{ATLAS2} measured by the ATLAS
Collaboration in the pseudorapidity window $|\eta|<2.5$ with the cut $p_T>500$~MeV/c, $n>0$ at
$ \sqrt{s}=8$ TeV.
The parameter values were obtained by minimization of Eq. (\ref{eq:a1})
}
\label{tab:3}
\begin{tabular*}{\columnwidth}{@{}l@{\extracolsep{\fill}}lll@{}}
\hline\noalign{\smallskip}
{\it i} & $\ \ \ \alpha_i$ & \ \ \ $\bar{n}_i$ & \ \ \ $k_i$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.68$^{+0.15 }_{-0.36 }$ & 10.5$^{+2.0 }_{-2.4 }$ & 1.32$^{+1.64 }_{-0.25 }$ \\ \noalign{\smallskip}
2 & 0.14$^{+0.17 }_{-0.06 }$ & 29.7$^{+3.1 }_{-5.4 }$ & 4.7$^{+1.3 }_{-1.3 }$ \\ \noalign{\smallskip}
3 & 0.18$^{+0.19 }_{-0.09 }$ &\ \,2.8$^{+0.2 }_{-0.2 }$ & 3.8$^{+10.9 }_{-1.5}$ \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 4.5/(39-8)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.548\,$\pm$\,0.030 &\ \,4.47\,$\pm$\,0.24 & 1.38\,$\pm$\,0.08 \\
2 & 0.452\,$\pm$\,0.030 & 20.05\,$\pm$\,0.71 & 2.47\,$\pm$\,0.16 \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 43.8/(39-5)} \\
\noalign{\smallskip}\hline
\end{tabular*}
\end{table}
\begin{table}
\caption{
The parameters of the superposition of three and two NBDs
obtained from fits to MDs \protect\cite{ATLAS3} measured by the ATLAS
Collaboration in the pseudorapidity window $|\eta|<2.5$ with the cut $p_T>500$~MeV/c, $n>0$ at
$ \sqrt{s}=13$ TeV.
The parameter values were obtained by minimization of Eq. (\ref{eq:a1})
}
\label{tab:4}
\begin{tabular*}{\columnwidth}{@{}l@{\extracolsep{\fill}}lll@{}}
\hline\noalign{\smallskip}
{\it i} & $\ \ \ \alpha_i$ & \ \ \ $\bar{n}_i$ & \ \ \ $k_i$ \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.679$^{+0.059}_{-0.073}$ &\ \,9.4$^{+0.7 }_{-0.7 }$ & 1.06$^{+0.15 }_{-0.09 }$ \\ \noalign{\smallskip}
2 & 0.213$^{+0.042}_{-0.035}$ & 31.4$^{+1.2 }_{-1.4 }$ & 3.7$^{+0.3 }_{-0.3 }$ \\ \noalign{\smallskip}
3 & 0.108$^{+0.031}_{-0.024}$ &\ \,2.9$^{+0.1 }_{-0.1 }$ & 6.8$^{+4.6 }_{-2.2 }$ \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 46.5/(81-8)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
1 & 0.532\,$\pm$\,0.013 &\ \,4.37\,$\pm$\,0.11 & 1.01\,$\pm$\,0.02 \\
2 & 0.468\,$\pm$\,0.013 & 22.38\,$\pm$\,0.33 & 2.11\,$\pm$\,0.06 \\ \noalign{\smallskip}
& \multicolumn{3}{c}{$\chi^2/dof$ = 548/(81-5)} \\
\noalign{\smallskip}\hline
\end{tabular*}
\end{table}
The normalized residues relative
to the two-NBD parametrization (see Tables~\ref{tab:3} and \ref{tab:4})
of the ATLAS data measured in the pseudorapidity window $|\eta|<2.5$
with $p_T > 500$~MeV/c,
$n>0$ at the energies $\sqrt s=8$~TeV \cite{ATLAS2} and 13~TeV \cite{ATLAS3}
are depicted in Fig. \ref{fig:3}c,~d, respectively.
Both data show sizeable discrepancies with respect to the weighted superposition
of two~NBDs.
The values of $\chi^2/dof=43.8/34$ at $\sqrt s=8$~TeV and $\chi^2/dof=548/76$
at $\sqrt s=13$~TeV for the two-NBD fits are too large, especially at the higher energy.
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure3a.eps}
\includegraphics[width=78mm,height=78mm]{Figure3b.eps}
\vskip -1.0cm
\includegraphics[width=78mm,height=78mm]{Figure3c.eps}
\includegraphics[width=78mm,height=78mm]{Figure3d.eps}
\end{center}
\vskip -1.2cm
\caption{
MD of charged particles
measured by the ATLAS Collaboration \protect\cite{ATLAS2,ATLAS3}
in the pseudorapidity interval $|\eta|<2.5$ for $p_T > 500$~MeV/c, $n>0$
at {\bf a} $\sqrt s=8$~TeV and {\bf b} $\sqrt s=13$~TeV.
Normalized residues of the data on MDs relative to weighted superposition
of two NBDs $(\mathrm{P_n^{2NBD}})$
with the parameters listed in Table~\ref{tab:3} and Table~\ref{tab:4} at {\bf c} $\sqrt s=8$~TeV and
{\bf d} $\sqrt s=13$~TeV, respectively.
The error bars include both the statistical and the systematic uncertainties summed in quadrature.
The lines have the same meaning as in Figs. \ref{fig:1} and \ref{fig:2}
}
\label{fig:3}
\end{figure*}
The residues manifest
a clear peak visible at low multiplicities around $n\!\simeq\!3$.
The description of the peak is obtained by the third negative binomial component
with $\bar{n}_3\!\simeq\!3$ shown by the dash-dot-dot lines in Fig.~\ref{fig:3}a, b.
The analysis of the new ATLAS data at $ \sqrt{s}=8$ and 13~TeV
confirms the existence of the peak \cite{IZ} emerging near $n\simeq 3$ at $ \sqrt{s}=7$~TeV.
Such an effect is negligible in the ATLAS measurements
with $p_T > 500$~MeV/c at $\sqrt{s}=0.9$~TeV (see Fig. 6(b) in \cite{IZ}).
\subsection{Energy dependence of three-NBD description}
\label{sec:D}
The ATLAS data on MDs in two $p_T$-cut regions
show a distinct peak at multiplicities where the soft production processes dominate.
The peaky structure can be well described by a separate component
within the three-component
parametrization of the experimental data.
A description of the MDs by superposition of
three NBDs reveals some properties which are compared below for both analyzed kinematic regions.
It concerns the energy dependence of the corresponding parameters obtained
at lower energies ($\sqrt{s}=0.9$ and 7~TeV) \cite{IZ}
and in this analysis ($\sqrt{s}=8$ and 13~TeV).
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure4a.eps}
\vskip -0.2cm
\includegraphics[width=78mm,height=78mm]{Figure4b.eps}
\vskip 0.5cm
\caption{
Energy dependence of the probabilities $\alpha_i$ of single components of the
weighted superposition of three NBDs fitted to the charged particle MDs
\protect\cite{ATLAS1,ATLAS2,ATLAS3,ATLAS4}
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
for {\bf a} $p_T>100$~MeV/c and {\bf b} $p_T>500$~MeV/c.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:4}
\vskip -0.5cm
\end{figure}
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure5a.eps}
\includegraphics[width=78mm,height=78mm]{Figure5b.eps}
\end{center}
\vskip -1.0cm
\caption{
Energy dependence of the average multiplicities $\bar{n}_i$
of single components of the weighted superposition of three NBDs
fitted to the charged-particle MDs
\protect\cite{ATLAS1,ATLAS2,ATLAS3,ATLAS4}
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
for {\bf a} $p_T>100$~MeV/c and {\bf b} $p_T>500$~MeV/c.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively.
The lines correspond to the quoted formulas where $m_p$ and $m_{\pi}$
are the proton and pion mass, respectively
}
\label{fig:5}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure6a.eps}
\includegraphics[width=78mm,height=78mm]{Figure6b.eps}
\end{center}
\vskip -0.4cm
\caption{
Energy dependence of the inverse values of the parameters $k_i$
of single components of the weighted superposition of three NBDs
fitted to the charged-particle MDs
\protect\cite{ATLAS1,ATLAS2,ATLAS3,ATLAS4}
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
for {\bf a} $p_T>100$~MeV/c and {\bf b} $p_T>500$~MeV/c.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:6}
\vskip -0.3cm
\end{figure*}
Figure \ref{fig:4} shows the probabilities $\alpha_i$ of single NBD components
in dependence on $\sqrt{s}$.
The parameters $\alpha_i$ were obtained from fits to the ATLAS data on multiplicities
in the window $|\eta|<2.5$
for (a) an almost inclusive ($p_T>100$~MeV/c) and (b) a transverse momentum cut ($p_T>500$~MeV/c)
data sample.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs.~(\ref{eq:a1})~and~(\ref{eq:a4}), respectively.
Despite considerable errors,
there are some trends visible in the behavior of the probabilities.
The values of $\alpha_i$ in Fig.~\ref{fig:4}a show a weak or nearly no energy dependence
in the multi-TeV energy
region.~Errors of their determination allow to trace a hierarchy in relation to the index $i$
which is rather stable against $\sqrt{s}$.
The same ordering of $\alpha_i$
is obvious in Fig.~\ref{fig:4}b at $\sqrt{s}=13$~TeV only.
The ordering of
$\alpha_2$ and $\alpha_3$ for the data with $p_T>500$~MeV/c is due to large errors
unclear at $\sqrt{s}=$7 and 8~TeV.
The probability of the third NBD component is non-zero
in all analyzed cases.
Figure \ref{fig:5} shows the $\sqrt{s}$ dependence of the average multiplicities $\bar{n}_i$ of
single NBDs in the window $|\eta|<2.5$
for (a) $p_T>100$~MeV/c and (b) $p_T>500$~MeV/c cuts.
The values of $\bar{n}_i$ extracted from the ATLAS data \cite{ATLAS1,ATLAS2,ATLAS3,ATLAS4}
are depicted in the log-log plot to check their power behavior.
As seen from Fig.~\ref{fig:5}a, the average multiplicities $\bar{n}_1$ and $\bar{n}_2$
demonstrate a power increase with the energy $\sqrt s$.
Both dependences are well parametrized in terms of
$Y_{max}\!=\!\ln[(\sqrt{s}\!-\!2m_p)/m_{\pi}]$, which is
the maximal rapidity of pions in the $pp$ c.m. system.
A similar description seems to be valid for the $\sqrt{s}$ dependence of $\bar{n}_2$
shown in Fig.~\ref{fig:5}b, thought
for the multiplicity $\bar{n}_1$ with $p_T>500$~MeV/c it is problematic to draw such a conclusion.
The third NBD component at low $n$ reveals remarkable properties.
Its average multiplicity $ \bar{n}_3$ is nearly energy independent
for both $p_T$-cut data samples.
The inverse values of the parameters $k_i$
of the weighted superposition of three NBDs are depicted
as a function of $\sqrt s$ in Fig.~\ref{fig:6}.
They were found from an analysis of the same data as the parameters shown in
Figs.~\ref{fig:4} and \ref{fig:5}.
As seen from Fig.~\ref{fig:6}a, the parameters demonstrate a clear hierarchy
with the index $i$ for the $p_T>100$~MeV/c data sample.
The width of the dominant multiplicity component characterized by $k^{-1}_1$ shows
a slight growth with energy.
The parameter $k^{-1}_2$ of the component under the tail of the distributions reveals
approximate constancy with respect to $\sqrt{s}$ in the multi-TeV energy region.
The width of the peak at low multiplicities ($k^{-1}_3$) indicates a similar tendency, though
such a statement is not too conclusive due to errors.
The same hierarchy of the parameters $k^{-1}_i$ with respect to the index $i$
as in Fig.~\ref{fig:6}a
is observed for the data sample with $p_T>500$~MeV/c
at $\sqrt{s}=$ 13~TeV only. The ordering of the parameters at lower energies is hard to tell
and their energy dependence is unclear from Fig.~\ref{fig:6}b.
\subsection{Combinants of the MDs}
\label{sec:E}
A general form of the MD is useful to make a characterization
in terms of its deviations from the Poisson distribution.
The natural logarithm of the Poisson generating functional is given by a linear
dependence on its argument.
The higher order terms of the power series expansion of the logarithm of
a general generating functional denote the deviation from the Poisson distribution.
The expansion coefficients ${\cal C}(i)$, named combinants \cite{Gyulassy},
characterize the MD in terms of the generating function
\begin{equation}
G(z)=\exp\left(\sum_{i=1}^{\infty} {\cal C}(i)(z^i-1)\right).
\label{eq:r3}
\end{equation}
The combinants possess the additivity property and are expressible
as a finite combination of the ratios $P(n)/P(0)$.
Some of them are permitted to have negative values \cite{Gyulassy}.
Individual interpretations of the coefficients
depend on different models of multiparticle production.
The quantity of interest, named here ``cumulative combinant", is $i{\cal C}(i)$.
In the model of independent boson production with geometric distributions of particles
in each production mode (e.g. in the mode with an average number of bosons $\overline{n}_i$
with momenta in the interval $(p_i, p_i+dp_i)$),
the cumulative combinant $(i\!+\!1){\cal C}(i\!+\!1)$
is the mean number of modes which have an occupation number greater than $i$ \cite{Gyulassy78}.
The cumulative combinants can be rewritten in the form
$(i\!+\!1){\cal C}(i\!+\!1) = \langle N\rangle C_i$ where
$\langle N\rangle=G'(z=1)$ is the average multiplicity and
the coefficients $C_i$ are commonly
used in the relation
\begin{equation}
(n+1)P(n+1)=\langle N\rangle \sum_{i=0}^{n} C_i P(n-i).
\label{eq:r4}
\end{equation}
\vskip -0.1cm
Such a kind of recurrence formula occurs in cascade-stochastic processes \cite{SalehTeich,SchmittMarsan}.
The relation was applied to the parametrization of the MD of charged particles in hadron-hadron collisions
at high energies \cite{Rusov1,Rusov2}.
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure7a.eps}
\vskip -1.0cm
\includegraphics[width=78mm,height=78mm]{Figure7b.eps}
\vskip -0.7cm
\caption{
Energy dependence of the cumulative combinants $\langle N\rangle C_i$
calculated from a weighted superposition of {\bf a} three and {\bf b} two NBDs
fitted to the charged-particle MDs
\protect\cite{ATLAS1,ATLAS2,ATLAS4}
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
for $p_T>100$~MeV/c
}
\label{fig:7}
\vskip -0.5cm
\end{figure}
The quantities $\langle N\rangle C_i$
calculated by Eq. (\ref{eq:r4}) from MDs measured
by the CMS \cite{CMS} and ALICE \cite{ALICE8} Collaborations in $pp$ collisions
possess remarkable oscillating properties \cite{WW1}.
Unlike this,
the two-NBD superposition used to fit the data is not able to account
for the oscillating structure of $\langle N\rangle C_i$.
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure8a.eps}
\includegraphics[width=78mm,height=78mm]{Figure8b.eps}
\end{center}
\vskip -0.8cm
\caption{
Energy dependence of the cumulative combinants $\langle N\rangle C_i$
calculated from a weighted superposition of {\bf a} three and {\bf b} two NBDs
fitted to the charged-particle MDs
\protect\cite{ATLAS1,ATLAS2,ATLAS3}
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
for $p_T>500$~MeV/c
}
\label{fig:8}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure9a.eps}
\includegraphics[width=78mm,height=78mm]{Figure9b.eps}
\end{center}
\vskip -0.8cm
\caption{
The cumulative combinants $\langle N\rangle C_i$
calculated from a weighted superposition of {\bf a} three and {\bf b} two NBDs
fitted to the charged-particle MDs
\protect\cite{CMS}
measured by the CMS Collaboration at $\sqrt s=$7~TeV for $p_T>0$
in different pseudorapidity intervals
}
\label{fig:9}
\end{figure*}
Next we show that oscillations of the cumulative combinants
$\langle N\rangle C_i=(i\!+\!1){\cal C}(i\!+\!1)$
can arise from the three-NBD superposition fitted to the data with the third
component, which accounts for the peak at low~$n$.
We calculate the coefficients $\langle N\rangle C_i$
from Eq. (\ref{eq:r4}) using
the values of $P(n)$ obtained from the corresponding fits.
Figure \ref{fig:7} shows the quantities
$\langle N\rangle C_i$ in dependence on the rank $i$
calculated from a weighted superposition of (a) three
and (b) two NBDs used to fit the data on MD
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
with the transverse momentum cut $p_T>100$~MeV/c.
The parameters of the fitted functions
are taken from Table 1 of \cite{IZ} (for $\sqrt s=$0.9 and 7~TeV)
and from Tables \ref{tab:1} and \ref{tab:2} of the present paper (for $\sqrt s=$8 and 13~TeV).
One can see that two-NBD fits to the ATLAS data give a smooth behavior of the cumulative combinants
in dependence on their rank.
In contrast to this, the three-NBD fits to the ATLAS data are characterized by the oscillations
of $\langle N\rangle C_i$ at low $i$.
The shape of the first oscillation is nearly energy independent.
The position of the first minimum corresponds to the energy independent
value of the average multiplicity $\bar{n}_3\sim 11$
of the third NBD component of the total distribution.
As discussed in \cite{WW1}, the oscillations can be connected with memory
(or correlations) in the produced multi-particle system.
The correlation of particles in the minima is weaker.
The first minimum
corresponds to the production of particles within the third independent NBD component which
makes the average correlation of the total system weaker. The consecutive oscillations
represent non-monotonic loss of memory (or correlations) away from a considered multiplicity $n$.
The cumulative combinants obtained from
a weighted sum of three and two NBDs used to fit the ATLAS data
in the interval $|\eta|<2.5$
with the cut $p_T>500$~MeV/c are shown in Fig.~\ref{fig:8}a, b, respectively.
One can see that the oscillations have disappeared for the data sample with
$p_T>500$~MeV/c even for the three-NBD description.
This is despite the fact that a three-NBD fit to the data is still needed to account
for the peak at low multiplicities (see Fig.~\ref{fig:3}c, d).
The disappearance of the oscillations is caused by the small value of $\bar{n}_3\sim 3$
(cf. the diminishing of the oscillations for small windows in Fig.~\ref{fig:9}a where
$\bar{n}_3=2.7$ for $ |\eta|<$0.5).
The data on MD of charged particles measured by the CMS Collaboration
in the central interaction region \cite{CMS} allow one to study
the cumulative combinants in dependence on the pseudorapidity window.
As shown in \cite{WW1}, the amplitude and periodicity of the oscillations increase with
the window size. For small windows they practically vanish.
Figure \ref{fig:9}a demonstrates that similar properties of the cumulative combinants
can be obtained from three-NBD fits \cite{IZ} to the CMS data.
As seen from Fig.~\ref{fig:9}b,
the two-NBD fits \cite{IZ} to the CMS data do not give any oscillations.
The cumulative combinants calculated from weighted superposition of two NBDs used to fit the data
are decreasing functions of the rank $i$.
In contrast, the combinants calculated from three-NBD fits to the data
reveal oscillations at low $i$.
The magnitude of the oscillations decreases with the decreasing window size.
The larger periodicity is in the larger windows.
The first minimum
corresponds to production of particles within the third-NBD component, which
makes the average correlation of the total system weaker.
Its position (for larger windows) corresponds to the pseudorapidity dependence
of the average multiplicity $\bar{n}_3$ (see Table 4 of \cite{IZ})
of the third-NBD component of the total distribution.
Let us note that, for the CMS data, the multiplicity $n=0$ was excluded from the fits.
Therefore the cumulative combinants shown in Fig.~\ref{fig:9} are calculated
with the parameters quoted in Tables 4 and 5 of \cite{IZ}
without any correction to the value of $P(0)$ obtained from the fits.
We have studied the sensitivity of the oscillations of the cumulative combinants
to changes of the
probabilities $\alpha_i$ of single NBD components of the total distribution.
For this purpose, we used auxiliary parameters $\epsilon_i$ by which
the corresponding probabilities $\alpha_i$ are gradually turned down.
In order to keep the total probability equal to unity,
the attenuation of $\alpha_1$ with $\epsilon_{1} < 1$ is compensated
by the proportional amplification of $\alpha_2$ and $\alpha_3$, which
is realised as follows:
\begin{equation}
\alpha^{'}_{1}(\epsilon_1)=\alpha_1\epsilon_1,\ \ \
\alpha^{'}_{i}(\epsilon_1)=\alpha_i \frac{(1-\alpha^{'}_{1})}{(1-\alpha_1)},\ \ \
i=2,3.
\label{eq:r5}
\end{equation}
Similar equations were used for weakening of the second and
the third component by $\epsilon_2$ and $\epsilon_3$, respectively.
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure10a.eps}
\vskip -0.7cm
\includegraphics[width=78mm,height=78mm]{Figure10b.eps}
\vskip -0.7cm
\includegraphics[width=78mm,height=78mm]{Figure10c.eps}
\vskip -0.6cm
\caption{
The cumulative combinants $\langle N\rangle C_i$
calculated from three-NBD fit to the ATLAS data \protect\cite{ATLAS4} at $\sqrt s=$13~TeV
(black circles from Fig.~\ref{fig:7}a).
The modifications of $\langle N\rangle C_i$ (lines) with reduction
of the {\bf a} first, {\bf b} second, and {\bf c} third NBD component (see text)
}
\label{fig:10}
\end{figure}
Figure \ref{fig:10}a demonstrates the increase of the amplitude of oscillations when
the probability $\alpha_1$ of the main NBD component is reduced by 5 or 10\%. The locations
of the minima and maxima of the oscillations remain intact as the other parameters were unchanged.
Figure \ref{fig:10}b depicts the sensitivity of the cumulative combinants to the probability
$\alpha_2$ of the NBD component under the tail of the total distribution.
One can see that the first wave up to $n\sim 24$ is not affected, even for $\epsilon_2=0$.
The behavior of the cumulative combinants in dependence on $\epsilon_3$ is shown
in Fig.~\ref{fig:10}c. One can see that the amplitudes of the oscillations diminish
with decreasing probability $\alpha^{'}_{3}$ of the third NBD component at low multiplicities.
The oscillations disappear completely in the case of $\alpha^{'}_3=0$.
The uncertainties of the quantities $\langle N\rangle C_i$
and, more specifically, correlation of their size with the pattern of their oscillations are sensitive to statistical fluctuations and systematic uncertainties of the raw data to such an
extent that the wavy structure in $\langle N\rangle C_i$ may be rendered
insignificant given sufficiently large experimental uncertainties,
as demonstrated above by varying the fitted three-NBD parameters.
This will be further confirmed by direct extraction of
modified versions of the cumulative combinants from experimental data on MDs in Sect.~\ref{sec:G}.
The natural applicability of the method of combinants
to the study of fine structure of the MDs appears to be useful.
This together with the relation of the combinants to the
finite set of multiplicities which make up the data seems to be an appropriate tool
in justification of the existence of the third multiplicity component emerging
in $pp$ collisions at the LHC at low
multiplicities.
\subsection{Clan structure of three-NBD description}
\label{sec:F}
The correlation structures of multi-particle states created in high energy collisions are often studied
in the framework of the clan concept introduced to interpret NBD occurring in
different reactions over a wide range of energies.
In hadron-hadron collisions, the properties of clans have been investigated and discussed
within a two-component model used to describe the MD of the produced particles
by a superposition of two NBDs \cite{GiovUgo1,GiovUgo4}.
The first component, connected with soft events, reveals invariant properties
as a function of the c.m. energy \cite{CDF}.
The second component, under the shoulder of the MD at high $n$, is associated with semi-hard processes
including the production of mini-jets.
It is believed that mini-jets with low $p_T$ are an important part of
particle production at high energies~\cite{Minijets}.
The clan model is based on the decomposition of NBD to Poisson distributed clusters
or independent bunches of produced particles \cite{GioVanHove,SimakSumbera}.
According to the common definition, a single cluster (clan) contains at least one particle.
The number of particles inside the clan follows a logarithmic distribution.
Among possible interpretations of the logarithmic distribution
let us mention the time average of time-dependent cascading \cite{GioVanHove}
and a model of cascade processes with collapsing of clans \cite{IZ_clans}.
The average number of clans, $\bar{N}$,
and the average number of particles per clan, $\bar{n}_c$, are expressed in terms of the
NBD parameters $\bar{n}$ and $k$ as follows:
\begin{equation}
\bar{N}=k\ln\left(1+\frac{\bar{n}}{k}\right), \ \ \ \ \ \bar{n}_c=\frac{\bar{n}}{\bar{N}}.
\label{eq:r6}
\end{equation}
For a weighted superposition of NBDs, the clan characteristics of a single NBD depend
on the number of components used for a description of the experimental data.
The analysis of the data in terms of
two and three NBDs is instructive,
because it allows one to study changes of clan parameters
with the emergence of a distinct peak observed in MD at the LHC at low $n$.
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure11a.eps}
\vskip -1.8cm
\includegraphics[width=78mm,height=78mm]{Figure11b.eps}
\caption{
Energy dependence of the average number of clans $\bar{N}_i$
calculated from a weighted superposition of {\bf a} two and {\bf b} three NBDs
fitted to the charged-particle MDs
\protect\cite{ATLAS1,ATLAS2,ATLAS4}
measured by the ATLAS Collaboration in the interval $|\eta|<~2.5$
for $p_T>100$~MeV/c.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:11}
\end{figure}
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure12a.eps}
\vskip -1.8cm
\includegraphics[width=78mm,height=78mm]{Figure12b.eps}
\caption{
Energy dependence of the average number of particles per clan $\bar{n}_{ci}$
calculated from weighted superposition of {\bf a} two and {\bf b} three NBDs
fitted to the charged-particle MDs
\protect\cite{ATLAS1,ATLAS2,ATLAS4}
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
for $p_T>100$~MeV/c.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:12}
\end{figure}
Figure \ref{fig:11}a shows the $\sqrt s$ dependence of the average number of clans calculated
from a weighted superposition of two NBDs. The parameters were calculated from fits
to the ATLAS data
measured in the interval $|\eta|<2.5$ for $p_T>100$~MeV/c.
The symbols with error bars and shaded rectangles are given by parameter values
using Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively.
As one can see, the average number of clans of the first component, $\bar{N}_1$,
is approximately constant with energy (except the last shaded rectangle corresponding
to large value of $\widetilde{\chi}^2/dof$ = 776/81).
A similar observation \cite{Ghosh} follows from the measurements of the CMS Collaboration \cite{CMS}
in the interval $|\eta|<2.4$.
The most peculiar feature of the two-NBD description is the decrease of the average number of
clans $\bar{N}_2$ of the semi-hard component with energy (upper panel in Fig. \ref{fig:11}a).
This property, confirmed by the analysis \cite{Ghosh} of CMS data,
was discussed \cite{GiovUgo4} within
different scenarios for soft and semi-hard events.
The $\sqrt s$ dependence of the average number of clans
is quite different in the three-NBD model.
As seen from Fig. \ref{fig:11}b,
the values of $\bar{N}_1$ for the dominant component with largest probability $\alpha_1$
are nearly constant at high energies.
The average number of clans
in the second component, $\bar{N}_2$, increases with energy
and becomes much larger than $\bar{N}_1$ at high $\sqrt s$.
This is a new feature, substantially different from the parametrization of the data by two NBDs
(cf. the upper panel in Fig. \ref{fig:11}a).
Another observation is that the values of $\bar{N}_3$ corresponding
to the third NBD are relatively high and nearly constant.
In a certain sense, the energy independence of $\bar{N}_1$ and $\bar{N}_3$
substitutes the approximate constancy of the average number of clans
of the soft component in the two-NBD scenario
(cf. the lower panel in Fig. \ref{fig:11}a).
Figure \ref{fig:12} shows the average number of particles per clan, $\bar{n}_{ci}$,
calculated from a weighted superposition of (a) two and (b) three NBDs
fitted to the ATLAS data in the interval $|\eta|<2.5$
for $p_T>100$~MeV/c.
In the two-NBD scenario (Fig. \ref{fig:12}a),
$\bar{n}_{c1}$ for the soft component does not change much with energy.
The value of $\bar{n}_{c2}$ grows with energy and becomes larger than $\bar{n}_{c1}$.
Such a trend
is compensated with the decrease of the average number of the semi-hard clans with $\sqrt s$,
as is illustrated in the upper panel in Fig. \ref{fig:11}a.
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure13a.eps}
\includegraphics[width=78mm,height=78mm]{Figure13b.eps}
\end{center}
\vskip -2.2cm
\caption{
The $\eta_c$ dependence of {\bf a} the average number of clans $\bar{N}_{i}$ and {\bf b}
the average number of charged particles per clan, $\bar{n}_{ci}$, for a two-NBD description
of the CMS data \protect\cite{CMS}
on MD measured in different pseudorapidity intervals $|\eta|<\eta_c$ at $\sqrt s =$ 7 TeV.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:13}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure14a.eps}
\includegraphics[width=78mm,height=78mm]{Figure14b.eps}
\end{center}
\vskip -0.5cm
\caption{
The $\eta_c$ dependence of {\bf a} the average number of clans $\bar{N}_{i}$ and {\bf b}
the average number of charged particles per clan,~$\bar{n}_{ci}$, for a three-NBD description
of the CMS data \protect\cite{CMS}
on MD measured in different pseudorapidity intervals $|\eta|<\eta_c$ at $\sqrt s =$~7~TeV.
The symbols with error bars and shaded rectangles correspond to the parameter values
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:14}
\end{figure*}
The situation becomes completely different for the three-NBD description.
As demonstrated in Fig. \ref{fig:12}b,
the average number of particles inside clans of the first (dominant) NBD component, $\bar{n}_{c1}$,
increases rapidly with~$\sqrt s$.
The values of $\bar{n}_{c2}$ of the second
component under the tail of MD reveal growth with energy as well, but not as rapid as in Fig. \ref{fig:12}a.
The substantial difference is that, contrary to the two-NBD description, $\bar{n}_{c1} > \bar{n}_{c2}$
in the multi-TeV energy domain.
The average number of particles inside clans of the third NBD component, $\bar{n}_{c3}$, is only slightly
larger than unity and depends weakly on~$\sqrt s$.
Based on the performed analysis of the ATLAS data with
weighted superposition of three NBDs,
we draw the following conclusions concerning the clan properties of the first
and the second NBD component of the total distribution.
The first component, corresponding to the dominant class of events,
contains large clans consisting of many particles.
The number of particles inside the clans grows strongly with energy.
The average number of such clans is small ($ \bar{N}_1\simeq~4$) and independent of $\sqrt s$.
The small and constant number of clans suggests
that there are few (or no) ``mini-jet clans" in these events.
The multiplicity component under the tail of the distribution corresponds to
a class of events in which much more clans are produced than in the first, dominant one.
The average number of clans of the second component increases with energy
and reaches large values ($ \bar{N}_2\sim~15$) at high $\sqrt s$.
The corresponding number of particles per clan $\bar{n}_{c2}$ is somewhat smaller
than $\bar{n}_{c1}$ in the high energy region.
It grows with $\sqrt s$ as well.
The large, increasing number of clans and the growing number of particles per
clan suggest that there are many clans ``mini-jets" in this class of events.
The properties of the second multiplicity component under the tail of the MD are
characteristic for semi-hard processes with abundant production of mini-jets.
We have studied the pseudorapidity dependence of the clan parameters
using the data \cite{CMS}
measured by the CMS Collaboration in the intervals $ |\eta|< \eta_c =$ 0.5, 1.0, 1.5, 2.0, and 2.4
at $\sqrt s =$ 7~TeV.
Figure \ref{fig:13} shows (a) the average number of clans
$\bar{N}_{i}$ and (b) the average number of particles per clan $\bar{n}_{ci}$
calculated from two-NBD description \cite{IZ} of the CMS data in
dependence on the width of the pseudorapidity window.
The behavior of the parameters calculated from three-NBD fits \cite{IZ} to the same data
is depicted in Fig. \ref{fig:14}.
As one can see from Figs. \ref{fig:13}a and \ref{fig:14}a,
the $\eta_c$ dependence of the average number of clans of the first component, $\bar{N}_{1}$,
is nearly the same for two- and three-NBD parametrization.
Despite large errors, a difference is visible in the behavior of $\bar{N}_{2}$
for the semi-hard events with mini-jets,
especially in the large pseudorapidity windows.
The values of $\bar{N}_{2}$ increased slightly
for the three-NBD description
and do not show a sign of saturation for larger values of $\eta_c$.
The average number of clans grows with $\eta_c$ for all three components.
Another difference between the parametrization of the CMS data with two and three NBDs
is seen in the $\eta_c$ dependence of the average number of particles per clan.
There is some indication that
clans of the first component contain more particles
than the clans of the second one ($\bar{n}_{c1}>\bar{n}_{c2}$)
when fitting the data with weighted superposition of three NBDs in larger windows.
This is in accord with the clan analysis of the ATLAS data shown in Fig. \ref{fig:12}.
The above observations are consistent with the interpretation of $1/k_i$ as an aggregation parameter \cite{GiovUgo4}
\begin{equation}
\frac{1}{k_i} = \frac{P_i(1,2)}{P_i(2,2)},
\label{eq:r7}
\end{equation}
where $P_i(N,m)$ is the probability that $m$ particles belong to $N$ clans.
As seen from Figs. \ref{fig:11}b and \ref{fig:12}b,
clans of the second component are more numerous and contain
fewer particles in comparison with the clans of the first component.
This means that there is
much less aggregation of particles in the semi-hard events with mini-jets than in the
first, dominant class of events.
Such a property is reflected
by the relation $1/k_2<1/k_1$ observed in central pseudorapidity intervals at $\sqrt s =$ 7~TeV
(Fig. 18(a) in \cite{IZ})
and at different energies in the window $|\eta|<$ 2.5 (Fig.~\ref{fig:6}a).
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure15a.eps}
\vskip -0.5cm
\includegraphics[width=78mm,height=78mm]{Figure15b.eps}
\vskip -0.5cm
\caption{
Ratios of {\bf a} the average numbers of clans
and {\bf b} the average numbers of particles per clan
extracted for single NBD components from MDs with the cuts $p_T>$ 500 MeV/c and $p_T>$ 100 MeV/c.
The upper and lower panels correspond to the respective two- and three-NBD parametrizations
of the ATLAS data \protect\cite{ATLAS3,ATLAS4}
measured in the interval $|\eta|<$ 2.5 at $\sqrt s =$ 13~TeV.
The symbols with error bars and shaded rectangles depict the ratios corresponding to fitted parameters
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:15}
\vskip -0.6cm
\end{figure}
We have studied the influence of transverse momentum on clan structure analysis
using the ATLAS data on MDs \cite{ATLAS1,ATLAS2,ATLAS3,ATLAS4} with different $p_T$ cuts.
Figure \ref{fig:15} shows the ratios of the clan parameters extracted from fits to the data
with $p_T>$ 500 MeV/c and $p_T>$ 100~ MeV/c at $\sqrt s =$ 13~TeV.
The ratios for the two- and three-NBD superposition are depicted in the upper and lower panels, respectively.
As one can see from Fig. \ref{fig:15}a,
the average number of clans in the third component shows the greatest suppression with the $p_T$ cut.
It means that these clans, consisting of very few particles,
are produced mostly with low transverse momenta.
For the three-NBD description, the average number of clans of the first component
$\bar{N}_1$ is reduced the least,
slightly less than $\bar{N}_2$.
This trend is quite opposite in the two-NBD model, as depicted in the upper panel in Fig. \ref{fig:15}a.
The suppression of the average number of particles per clan with $p_T$ is illustrated in
Fig. \ref{fig:15}b.
The value of $\bar{n}_{c3}$ is reduced minimally.
It is natural because, by definition, a clan contains a minimum of one particle and because
of the very small number of particles in these clans ($\bar{n}_{c3}\!\sim\!1$).
At the same time $\bar{n}_{c1}$ is diminished considerably with the harder $p_T$ cut for the three-NBD description.
The opposite follows from the two-NBD scenario.
There is a very small suppression of particles inside clans
of the soft multiplicity component, as depicted for the first NBD in the upper panel in Fig. \ref{fig:15}b.
Let us interpret the results obtained by the description of the ATLAS data on MD with different $p_T$ cuts
when using weighted superposition of three NBDs.
The relatively small suppression of the average number of clans $\bar{N}_{1}$
and considerable reduction of the average number of particles per clan $\bar{n}_{c1}$
suggest that clans of the first, dominant component contain particles with a wide spectrum of transverse momenta.
There are many particles in these clans (see Fig. \ref{fig:12}b),
some of them with larger, some of them with smaller $p_T$.
The imposed cut
$p_T>500$ MeV/c reduces the number of particles $\bar{n}_{c1}$ inside the clans by almost half,
as those with small $p_T$ are lost.
At the same time, $\bar{N}_{1}$ is least suppressed with the $p_T$ cut.
This observation suggests that the large clans of the first multiplicity component
consisting mostly of soft particles
must contain also some particles with considerably high $p_T$.
It is therefore reasonable to assume that there are semi-hard processes with relatively
large momentum transfer in the dominant class of events.
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure16a.eps}
\vskip -0.6cm
\includegraphics[width=78mm,height=78mm]{Figure16b.eps}
\vskip -0.6cm
\caption{
Ratios of {\bf a} the average numbers of particles $\bar{n}_i$
and {\bf b} the aggregation parameters $k^{-1}_i$
extracted for single NBD components from MDs with the cuts
$p_T>$ 500 MeV/c and $p_T>$ 100 MeV/c.
The upper and lower panels correspond to the respective two- and three-NBD parametrizations
of data \protect\cite{ATLAS3,ATLAS4}
measured by the ATLAS Collaboration in the interval $|\eta|<$ 2.5 at $\sqrt s =$ 13~TeV.
The symbols with error bars and shaded rectangles depict the ratios corresponding to fitted parameters
obtained by minimization of
Eqs. (\ref{eq:a1}) and (\ref{eq:a4}), respectively
}
\label{fig:16}
\vskip -0.6cm
\end{figure}
The properties of the clan structure of the second multiplicity component are different.
As depicted in the lower panel in Fig. \ref{fig:15}b,
the number of particles per clan
$\bar{n}_{c2}$ is less suppressed with the cut $p_T>500$ MeV/c than $\bar{n}_{c1}$.
The relatively small reduction of $\bar{n}_{c2}$ means that fewer particles
are affected by the $p_T$ cut.
It suggests that there are fewer particles with low $p_T$ in the clans of the second component
in comparison with the clans of the first, dominant one.
At the same time
(see lower panel in Fig. \ref{fig:15}a),
the average number of clans, $\bar{N}_2$, is reduced
similarly and even slightly more than $\bar{N}_1$.
The considerable suppression of $\bar{N}_2$ and small reduction of $\bar{n}_{c2}$ indicate that
the transverse momenta of particles inside clans of the second component
have a quite narrow distribution concentrated around some $\tilde{p}_T$.
This $\tilde{p}_T$ is higher than the average $<\!\!p_T\!\!>$ of particles of the
first component, most of which carry low transverse momenta.
Moreover, considering that $\bar{n}_{c2}$ is obviously smaller than
$\bar{n}_{c1}$ for $p_T>100$ MeV/c at $\sqrt s =$ 13~TeV (see Fig. \ref{fig:12}b),
one can make the following conclusions.
The clans of the second NBD component contain fewer particles
than the large clans of the first component.
The particles of the smaller clans carry on average higher $p_T$
than most of the particles in clans of the dominant component.
Their transverse momenta are concentrated around some $\tilde{p}_T$.
The average number of these clans, $\bar{N}_{2}$, increases with energy
and becomes much larger than $\bar{N}_{1}$ at high $\sqrt s$.
The mentioned properties of the clan structure of the second semi-hard component resemble
some basic features of mini-jets: clustering of particles of close
transverse momenta with increasing frequency at high energies.
The application of the clan structure analysis to MDs measured by the ATLAS Collaboration
with different limitations on transverse momenta
reveals properties which differentiate between two- and three-NBD description of the data.
It concerns both the average number of clans and the average number of particles per clan, which are
reduced with varying intensity when applying the higher $p_T$ cut.
As seen from Fig. \ref{fig:15}, there are opposite trends of the suppression for both models
in the first and the second multiplicity component.
In order to clarify what is the cause of such a difference, one can look at the $p_T$ dependence of
the corresponding NBD parameters.
Figure \ref{fig:16}a shows the ratios of the average number of particles, $\bar{n}_{i}$,
obtained from the fits to the ATLAS data measured in the interval $|\eta|<2.5$
for $p_T>500$~MeV/c and $p_T>100$~MeV/c.
The ratios of the NBD parameters $1/k_i$ extracted from the data are depicted in
Fig. \ref{fig:16}b.
As concerns the first and the second multiplicity component, the ratios of $\bar{n}_{i}$ follow
a similar trend for two- and three-NBD parametrization of the data.
The average multiplicities $\bar{n}_{1}$ and $\bar{n}_{2}$ are suppressed almost proportionally for both models.
Contrary to this, the corresponding ratios of the aggregation parameters, $1/k_i$,
behave completely opposite for parametrization of the ATLAS data with two and three NBDs.
The aggregation of particles into clans increases strongly with the cut $p_T>500$ MeV/c
for the soft component in the two-NBD model.
At the same time, the aggregation parameter $1/k_2$
changes very little for the semi-hard component.
In the three-NBD model, the aggregation of particles into clans increases slightly with $p_T$ for the first,
dominant component. The aggregation is slightly stronger for the semi-hard component with mini-jets.
As concerns the third NBD, it is hard to tell because of the large error.
The observations indicate that it is just another aggregation of particles into clans, which
results in different properties of clans in two- and three-NBD models
used for the description of the ATLAS~data.
\subsection{Discussion}
\label{sec:G}
A detailed examination of the sensitivity of cumulative combinants to the systematic uncertainties of measurement and to the unfolding procedures applied to raw data requires information on the response matrix for the considered experiments as well as knowledge of the experimental methods used
by obtaining the final MD.
Though a study of the raw data is not the subject of the present paper, we discuss here an estimation
of errors of the cumulative combinants
based on the published data.
The combinants are expressible in terms of the first $i$ probability ratios,
$P_n/P_0, n=1,..,i$, which depend on $P_0$.
For the ATLAS data, the analysed events are required to satisfy special criteria in order to minimize the systematic uncertainties. One of them is the minimal number of charged tracks that depends
on the particular phase space region.
The most inclusive phase space region covered by the measurements
corresponds to the conditions $n_{ch}\ge 2$ and $p_T > 100$~MeV/c.
The experimental information on $P_0$ and $P_1$ is missing in these data.
For the CMS data at $\sqrt s =$ 7~TeV,
the measured value of $P_0$ is very large.
Due to the experimental difficulties connected with the large error of this probability
and because of the rise of the MDs in the zeroth bin, $P_0$ is usually
omitted in the NBD fits to the data.
In order to avoid inaccuracies as to the assumptions with regard to
the multiplicity probabilities and their errors in the first two bins,
let us consider a modification of combinants based on the relation \ref{eq:r4}.
Instead of the full MD $\{P_0, P_1, P_2,...\}$ one can apply
this recurrence relation to the truncated sets
$\{P_2, P_3, P_4,...\}$ or $\{P_1, P_2, P_3,...\}$.
The corresponding modified quantities $(i\!+\!1){\cal C}_2(i\!+\!1)$
and $(i\!+\!1){\cal C}_1(i\!+\!1)$ are defined by the formulas
\begin{equation}
{\cal C}_2(1)=\frac{P_3}{P_2}, \ \ \ \
2{\cal C}_2(2)=2\frac{P_4}{P_2}-{\cal C}_2(1)\frac{P_3}{P_2},...
\label{eq:r8}
\end{equation}
and
\begin{equation}
{\cal C}_1(1)=\frac{P_2}{P_1}, \ \ \ \
2{\cal C}_1(2)=2\frac{P_3}{P_1}-{\cal C}_1(1)\frac{P_2}{P_1},...,
\label{eq:r9}
\end{equation}
respectively.
The lower index $2$ ($1$) at ${\cal C}$ means that
the truncated probability set begins with $P_2$ ($P_1$).
As illustrated below, the quantities $(i\!+\!1){\cal C}_2(i\!+\!1)$ ($(i\!+\!1){\cal C}_1(i\!+\!1)$)
defined by Eqs. (\ref{eq:r8}) ((\ref{eq:r9})) reveal
similar oscillating properties to the cumulative combinants calculated by Eq. (\ref{eq:r4})
from a superposition of NBDs with parameters obtained from the corresponding fits.
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure17a.eps}
\vskip -0.6cm
\includegraphics[width=78mm,height=78mm]{Figure17b.eps}
\vskip -0.6cm
\caption{
The coefficients $(i+1) {\cal C}_2(i+1)$ calculated from
MDs measured by the ATLAS Collaboration \protect\cite{ATLAS1,ATLAS2,ATLAS4}
in the interval $|\eta|<~2.5$ with $p_T > 100$~MeV/c, $n>1$
at {\bf a} $\sqrt s=$13, 8~TeV
and {\bf b} $\sqrt s=$ 7, 0.9~TeV.
The error bars correspond to the statistical and the systematic uncertainties summed in quadrature.
The open symbols are shifted by the factor -1.
The full (dashed) lines are obtained by Eq. (\ref{eq:r8}) from three- (two-) NBD
fits to the data on MDs using Eq. (\ref{eq:a4})
}
\label{fig:17}
\vskip -0.6cm
\end{figure}
Figure \ref{fig:17} shows the coefficients $(i\!+\!1){\cal C}_2(i\!+\!1)$
in dependence on the rank $i$.
Their values were calculated from the data on MD ($n_{ch}\ge 2$)
measured by the ATLAS Collaboration in the interval $|\eta|<2.5$
with the transverse momentum cut $p_T>100$~MeV/c
at $\sqrt s=$13, 8, 7 and 0.9~TeV.
The error bars shown in the figure were obtained from experimental errors of the data on
$P_n$. The latter include both the statistical and the systematic uncertainties summed in quadrature.
In determination of the errors of $(i+1) {\cal C}_2(i+1)$ we have proceeded in the following way
(see Appendix A).
Taking into account the positive correlation between adjacent multiplicity bins and the anti-correlation
between the opposite sides of the MD distribution maximum, we have considered the distributions shifted
to the right and left with respect to the error variation in single multiplicity bins.
The right shifted distribution is constrained by the upper values of the errors of $P_n$ for $ n > m $
and by their lower values for $n < m$ where $m$ is the multiplicity with the maximal value of $P_m$.
The left shifted distribution was considered in the analogous way, exchanging mutually
upper and lower errors at both sides of the maximum.
The right shifted distribution of $P_n$ is responsible for
the upper error bars of $(i+1) {\cal C}_2(i+1)$ at maxima and the lower error bars near the minimum
visible in Fig. \ref{fig:17}. The left shifted $P_n$ governs the error bars in the opposite direction.
For higher values of the rank $i$,
one can see a scattering of points around the curves calculated from three- and two-NBD fits
to the experimental MDs.
This corresponds to a statistical spread of the mean values of $P_n$
with respect to the fits. As the statistical uncertainties of the data are much smaller than the spread
of the mean values in the ATLAS data,
the error bars shown in Fig. \ref{fig:17} are practically hidden within the symbols
for increasing $i$.
\begin{figure}
\includegraphics[width=78mm,height=78mm]{Figure18a.eps}
\vskip -0.6cm
\includegraphics[width=78mm,height=78mm]{Figure18b.eps}
\vskip -0.6cm
\caption{
{\bf a} The coefficients $(i+1) {\cal C}_1(i+1)$ calculated from
data on MD measured by the ATLAS Collaboration
\protect\cite{ATLAS1,ATLAS2,ATLAS3}
in the interval $|\eta|<~2.5$ with $p_T > 500$~MeV/c, $n>0$
at {\bf a} $\sqrt s=$13, 8~TeV
and {\bf b} $\sqrt s=$ 7, 0.9~TeV.
The error bars correspond to the statistical and the systematic uncertainties summed in quadrature.
The open symbols are shifted by the factor -1.
The full (dashed) lines are obtained by Eq. (\ref{eq:r9}) from three- (two-) NBD
fits to the data using Eq. (\ref{eq:a4})
}
\label{fig:18}
\vskip -0.6cm
\end{figure}
Because of relatively small systematic uncertainties of the measurements,
one can see a sizeable wavy structure of
$(i+1) {\cal C}_2(i+1)$ in Fig. \ref{fig:17},
which clearly discriminates between
the two- and three-NBD parametrization of the ATLAS data with $p_T>100$ MeV/c.
The difference is seen in the region of low $i$ and corresponds to a distinct peak near
the maximum of the measured MD.
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure19a.eps}
\includegraphics[width=78mm,height=78mm]{Figure19b.eps}
\end{center}
\vskip -0.8cm
\caption{
{\bf a} The coefficients $(i+1) {\cal C}_1(i+1)$ calculated from
data on MD measured by the CMS Collaboration \protect\cite{CMS}
in the interval $|\eta|<2.4$ at $\sqrt s=7$~TeV for $p_T>0$.
The error bars correspond to the statistical and the systematic uncertainties of $P_n$
summed in quadrature.
{\bf b} The coefficients $(i+1) {\cal C}_1(i+1)$ calculated from the same data
with the systematic errors reduced by 50$\%$.
The solid (dashed) lines were calculated from three- (two-) NBD fits
to the measured MD using Eq. (\ref{eq:r9})
}
\label{fig:19}
\end{figure*}
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure20a.eps}
\includegraphics[width=78mm,height=78mm]{Figure20b.eps}
\end{center}
\vskip -0.8cm
\caption{
The coefficients $(i+1) {\cal C}_1(i+1)$ calculated from
a weighted superposition of {\bf a} two and {\bf b} three NBDs
fitted to the charged-particle MDs
\protect\cite{CMS}
measured by the CMS Collaboration at $\sqrt s=$7~TeV in the pseudorapidity interval $|\eta|<2.4$ for $p_T>0$.
The error bars correspond to the systematic and statistical uncertainties
of $P_n$ summed in quadrature
}
\label{fig:20}
\end{figure*}
Figure \ref{fig:18} shows
the coefficients $(i+1) {\cal C}_1(i+1)$ calculated from data
on MD ($n_{ch}\ge 1$)
measured by the ATLAS Collaboration in the window $|\eta|<2.5$ with the cut
$p_T > 500$~MeV/c at $\sqrt s=$13, 8, 7 and 0.9~TeV.
The errors of the coefficients were obtained from experimental uncertainties of
$P_n$ in the same way as in Fig.~\ref{fig:17}.
Except for a few points at low $i$ in Fig. \ref{fig:18}a,
the errors are smaller than the size of the displayed symbols.
One can see that there are no oscillations of $(i+1) {\cal C}_1(i+1)$
for the ATLAS data
with $p_T > 500$~MeV/c at all considered energies.
The quantities calculated from the data on MD reveal
approximately the same behavior as the cumulative
combinants obtained from weighted sum of three and two NBDs used to fit the data
(see Fig. \ref{eq:r8}).
The disappearance of oscillations is connected with
the small average multiplicity $\bar{n}_3 \sim 3$
of the third NBD component for the data with higher $p_T$ cut
(cf. the diminishing of the oscillations in
small windows in Fig. 9a where $\bar{n}_3 = 2.7$ for $|\eta| <0.5$).
The situation is somewhat different for the CMS measurements with $p_T>0$ \cite{CMS}
in sufficiently large pseudorapidity intervals.
Figure \ref{fig:19}a shows the quantities $(i\!+\!1){\cal C}_1(i\!+\!1)$
in dependence on the rank $i$ calculated from the $n_{ch}$ distributions
in the window $|\eta|<2.4$ at $\sqrt s=$7~TeV.
The black symbols correspond to the mean values of $P_n$.
The error bars were obtained in the same way as
in the case of the ATLAS data.
\begin{figure*}
\begin{center}
\includegraphics[width=78mm,height=78mm]{Figure21a.eps}
\includegraphics[width=78mm,height=78mm]{Figure21b.eps}
\end{center}
\vskip -0.8cm
\caption{
{\bf a} The coefficients $(i+1) {\cal C}_2(i+1)$ calculated from
data on MD measured by the CMS Collaboration \protect\cite{CMS}
in the interval $|\eta|<2.4$ at $\sqrt s=7$~TeV for $p_T>0$.
The error bars correspond to the statistical and the systematic uncertainties of $P_n$
summed in quadrature.
{\bf b} The coefficients $(i+1) {\cal C}_2(i+1)$ calculated from the same data
with the systematic errors reduced by 50$\%$.
The solid (dashed) lines were calculated from three- (two-) NBD fits
to the measured MD using Eq. (\ref{eq:r8})
}
\label{fig:21}
\end{figure*}
The right shifted distribution of $P_n$ results in a large oscillation of errors
that increases strongly in the region of higher rank $i$.
The right shift determines the upper errors at maxima
and the lower errors at minima of the first oscillations and creates a repeating pattern.
The left shift of the distribution is mostly responsible for errors in the opposite direction.
One can see from Fig. \ref{fig:19}a that the structure of the error bars agrees with
the initial wavy character of the black symbols quite well.
The errors grow very rapidly at extrema
with the increasing number of oscillations.
Such a behavior is caused by the systematic uncertainties of the CMS data, which are
much larger than in the ATLAS measurements.
The systematic uncertainties govern the errors of residues of the MD with respect to the two-NBD fits \cite{IZ}.
They form a characteristic envelope around the mean values of $P_n$
that follows the fine structure of the residual
mean values quite accurately.
We have examined how a reduction of the systematic uncertainties of MD can affect the information
on the oscillatory behavior of the coefficients $(i\!+\!1){\cal C}_1(i\!+\!1)$. Assuming that
the systematic errors of $P_n$ could be reduced by 50$\%$, the oscillating structure
of cumulative combinants including errors is expected
to be similar to Fig.~\ref{fig:19}b.
Such an increase in accuracy would help in discriminating between
two- and three-component description of the data considerably.
The oscillating pattern depicted in Fig. \ref{fig:19}a can help to discriminate
between the two- and three-NBD hypotheses also on the basis of the following considerations.
Let us assume that the distribution of mean values of $P_n$ is described exactly
by two NBDs, e.g. with parameters quoted in~\cite{IZ}.
In this model situation we take the
statistical and systematic uncertainties provided by the CMS Collaboration
and construct the quantities $(i\!+\!1){\cal C}_1(i\!+\!1)$.
The result for the interval $|\eta|<2.4$ at $\sqrt s=7$~TeV and $p_T>0$ is shown in Fig.~\ref{fig:20}a.
No characteristic pattern of oscillation can be seen there.
If we assume, however, that the mean values of $P_n$ follow exactly a three-NBD curve,
with parameters taken e.g. from \cite{IZ}, and take the experimental errors from
the corresponding measurement, the uncertainties of $(i\!+\!1){\cal C}_1(i\!+\!1)$ constructed as above
look like Fig.~\ref{fig:20}b. One can see that the oscillating pattern based on the three-NBD
hypothesis shows a very similar structure, as depicted in Fig.~\ref{fig:19}a. The later corresponds to the modified cumulative combinants and their errors calculated from data published by the CMS Collaboration.
We examined the sensitivity of the oscillation pattern of errors to the change of
the truncation condition from $n\ge 1$ to $n\ge 2$.
For that purpose we used data on MD measured by the CMS Collaboration \cite{CMS}
for $p_T>0$ in the interval $|\eta|<2.4$ at $\sqrt s=7$~TeV and calculated
the coefficients $(i\!+\!1){\cal C}_2(i\!+\!1)$.
The result depicted in Fig.~\ref{fig:21} demonstrates
that the error pattern seen in Fig.~\ref{fig:19}
is not destroyed with the increase of the truncation point.
A comparison of both figures shows that omitting of $P_1$ from
the construction of the modified combinants results in a reduction of the amplitude of their errors
and is accompanied by a slight decrease of their periodicity.
Let us note that the structure illustrated in Fig.~\ref{fig:21}b is similar to that
depicted by full symbols in Fig.~\ref{fig:17}b.
A detailed analysis of the cumulative combinants requires study of raw data and
knowledge of proper response of detectors. The discussed estimation of their uncertainties
indicates that oscillations of the cumulative combinants seem to be a true
property of $n_{ch}$ distributions. The ATLAS data are most conclusive in that respect.
The CMS data are less decisive as to the oscillations, although some supplementary support
for such a hypothesis is seen in larger pseudorapidity intervals.
In the following part we discuss some ideas concerning the development of the clan structure
in the context of the present study of the clan parameters obtained from
the three-NBD description of the analyzed data.
The emerging picture
is consistent with a stochastic-physical scenario \cite{IZ_clans} of clans' evolution
involving the ingredients of QCD branching processes.
In this scenario, clans, during their (QCD) evolution,
can perform collapses onto single particle states. The states evolve further by
subsequent branching.
The collapses of clans are most probable in the early stage of their development.
We consider that such collective behavior reflects early
neutralization of color configurations and simulates rapid freeze-out of QCD
degrees of freedom inside clans.
The processes of a clan's collapsing define directions along which the final particles originated from
their ancestors are spreaded.
Those clans that experienced (one or more) collapses can be
identified (under additional conditions) with mini-jets.
The ``mini-jet clans" are expected to have different properties
in comparison with clans that evolved without collapses.
It concerns the transverse momentum distribution of particles which,
due to collapses, should be concentrated around some $\tilde{p}_T$.
Moreover, it is natural to expect that the number of clans mini-jets should increase with energy
and should become larger than the number of clans of which the evolution happened without collapses.
Due to collapses, clans identified with mini-jets should contain less particles than
clans that did not suffer any collapse.
As can be seen from Figs. \ref{fig:11}b and \ref{fig:12}b,
this picture is in accord with the values of the clan parameters
extracted from three-NBD description of the ATLAS data.
The first multiplicity component consists of large clans containing many particles.
The average number of these clans is low and constant with energy.
These properties are in line with the conjecture that the
occurrence of the mini-jets in the dominant component is small.
The second NBD component describes a high-multiplicity tail of the total distribution.
As usually considered, the component is enriched with mini-jets.
Their symptoms concern clustering of particles into smaller clans with increasing intensity
at high energies.
This assumption complies with the clan properties extracted from the experimental data.
First, the clans of the second component contain fewer particles
in comparison with the clans of the first one ($\bar{n}_{c2}<\bar{n}_{c1}$).
The value of $\bar{n}_{c2}$ increases with energy.
Second, the average number of the smaller clans, $\bar{N}_2$,
increases with energy and becomes much larger than $\bar{N}_1$ at high $\sqrt{s}$.
The properties of clans in the third component are different.
There is only one or very few particles in these clans,
as illustrated by empty squares in Fig. \ref{fig:12}b.
It points to little or nearly no branching in such a case.
The parameter $1/k_3$ is small (see Fig. \ref{fig:6}a),
reflecting a minimal aggregation of particles in comparison with the first and the second component.
A characteristic feature of the three-NBD fits to the ATLAS data is the approximate constancy
of the average multiplicity $\bar{n}_{3}$ with energy.
Moreover, the average number of clans, $\bar{N}_{3}\simeq 9$,
depends very weakly on $\sqrt s$ as well (see Fig. \ref{fig:11}b).
These observations suggest that there are different mechanisms responsible
for particle production in the first two components of the MD and the third one
emerging at low~$n$.
The performed study of the clan parameters indicates that
the peak around maximum of the MD is influenced by a mechanism which depends weakly on energy and
is characterized by minimum branching and small transverse momenta.
This is typical for soft particle production.
It is unlikely that the peak is a remnant of diffractive events.
In the central rapidity intervals studied at the LHC, the single-diffractive events
influence 0- and 1-bin in multiplicity.
The double-diffractive events influence 1- and 2-bin.
It is therefore hard to assume that the peak at $ \bar{n}_3\simeq 11$
in the central region ($ |\eta|<2.5$) is caused by diffraction.
The relevant mechanism could be the production of particles from fragmentation
of (color) strings stretched between the leading partons of the colliding protons.
The fragmentation of the longitudinally stretched strings is a soft process and results
in narrow (nearly Poissonian) MD of particles with a limited range of transverse momenta.
The properties of the clan parameters in the first and the second NBD component
point to a quite different kind of mechanism
responsible for particle production.
The large amount of branching in both of them can be understood by a two-stage mechanism;
via collisions of constituents of the colliding protons with
production of massive secondary objects (fireballs, clans' ancestors, etc.)
in the first stage, and the successive branching of these objects into observable particles
in the second one.
The events corresponding to the first, dominant multiplicity component
are characterized by the production of a small number of clans containing many particles.
The independence of their number, $\bar{N}_1$, from energy
corresponds to the occurrence of a small number of mini-jets.
Most particles in the clans carry relatively low transverse momenta.
However, the weak suppression of $\bar{N}_1$ with $p_T$ indicates
that the clans contain also particles with relatively high $p_T$.
The above-mentioned properties suggest
that there are semi-hard processes with a small number of mini-jets followed by
intensive branching and successive production of many low-$p_T$ particles in
the dominant class of events.
The second multiplicity component, under the tail of the MD, describes production
of particles in semi-hard processes with extensive production of mini-jets.
The corresponding NBD consists of many clans containing
less particles on average than in the first component.
We consider these clans to represent to a certain extent mini-jets,
i.e. clusters of directed particles of a common ancestor.
Such assumption is supported by the simultaneous increase of the average number of clans $\bar{N}_2$
and the average number of particles per clan $\bar{n}_{c2}$ with energy.
\subsection{Summary}
\label{sec:H}
We have analysed charged-particle MDs
using new high-statistics data \cite{ATLAS2,ATLAS3,ATLAS4} measured by the ATLAS Collaboration
in $pp$ collisions at energies $\sqrt s=$8 and 13~TeV.
The data include measurements in the low-$p_T$ regime ($p_T>100$~MeV/c) and with the
cut $p_T>500$~MeV/c.
The analysis extends our previous study \cite{IZ} of the MDs
obtained at the LHC at lower energies.
The results of our investigations show that the new ATLAS data confirm the existence of a
distinct peak in MD at low multiplicities up to the energy
$\sqrt s=$13~TeV.
The description of the data within a two-component superposition of NBDs is unsatisfactory.
The ATLAS data can be well parametrized by a weighted sum of three NBDs.
The probabilities of the single NBD components show a weak or nearly no energy dependence
in the most inclusive phase-space region ($p_T>100$~MeV/c).
The third NBD component accounts for the peak-like structure of the experimental MD at low $n$.
Its average multiplicity $\bar{n}_3$ is approximately energy independent and decreases
with the imposed transverse momentum cut.
The parameters $k^{-1}_i$, characterizing widths of the single NBDs,
demonstrate a clear hierarchy with the index i for the $p_T >$~100~MeV/c data sample.
The third component of the total distribution is
narrow, well described by the NBD with a large value of $k_3$.
The probability $\alpha_3$ of the low-multiplicity component does not diminish
when a cut on the transverse momentum is imposed.
We have studied the properties of the cumulative combinants $\langle N\rangle C_i$ calculated
from a weighted superposition of three- and two-NBD parametrization of the ATLAS and CMS
data on MDs.
It was demonstrated that the third NBD component at low multiplicities with sufficiently
large $\bar{n}_3$ can be responsible for the oscillating structure of the combinants.
The oscillations are clearly visible when they are calculated from the three-NBD fits to
the ATLAS data measured
in the pseudorapidity interval $ |\eta|<$2.5 with $p_T >$ 100~MeV/c.
It was demonstrated that the oscillatory behaviour of $\langle N\rangle C_i$ obtained
from the three-NBD fits \cite{IZ} to the CMS data on MDs
at $\sqrt s=$7~TeV in different pseudorapidity windows reveal similar properties as those found in \cite{WW1}.
They are the decrease of the magnitude and periodicity of the oscillations
with the decreasing window size. The position of the first minimum
of the oscillations corresponds to the average multiplicity $\bar{n}_3$ of the third NBD component.
The result suggests that
the average correlation of the system is weaker if just this multiplicity contributes
to the build-up of the probability $P(n)$ for a higher multiplicity $n$.
To the contrary, the cumulative combinants $\langle N\rangle C_i$ calculated from a
weighted superposition of two NBDs used to fit the data give no oscillations and
reveal a monotonic decrease only.
We conclude that a detailed parametrization of the LHC data on MD which would give oscillations
of the cumulative combinants $\langle N\rangle C_i=(i\!+\!1){\cal C}(i\!+\!1)$
requires third component within weighted sum of NBDs.
This is analogous to the oscillations of the ratio $H_q=K_q/F_q$
of the cumulant to the factorial moments which were described by two-NBD fits
used to account for the shoulder of MDs observed already at lower energies.
The oscillating structures corresponding
to the low- and high-multiplicity regions
cannot be obtained from the two- and one-NBD fits to the experimental data, respectively.
A modification of cumulative combinants based on a truncation of the MD in the first two multiplicity
bins was examined. The modified combinants, $(i\!+\!1){\cal C}_2(i\!+\!1)$ and
$(i\!+\!1){\cal C}_1(i\!+\!1)$, were obtained from experimental data on MD using
Eqs. (\ref{eq:r8}) and (\ref{eq:r9}), respectively.
It was illustrated that they reveal similar
properties as the cumulative combinants
calculated by Eq. (\ref{eq:r4}) from a superposition of three NBDs with
the parameters obtained from the corresponding fits.
The results of the analysis of the ATLAS data measured
in the interval $|\eta| < $2.5 with the transverse
momentum cut $p_T > $100 MeV/c at $\sqrt{s} = $ 13, 8, 7 and 0.9 TeV show that
the quantity $(i\!+\!1){\cal C}_2(i\!+\!1)$ reveals an oscillating structure in the
region of low $i$.
The wavy structure is clearly visible
because of relatively small systematic uncertainties of the ATLAS measurements.
The oscillation is a consequence of
the distinct peak at the maximum of the MD.
The oscillating wave at low $i$ allows to discriminate
between the two- and three-NBD parametrization of the ATLAS data.
We have computed the quantities $(i\!+\!1){\cal C}_1(i\!+\!1)$ and
$(i\!+\!1){\cal C}_2(i\!+\!1)$ using data on MD measured by the CMS Collaboration in the
pseudorapidity interval $|\eta|<$ 2.4 at $\sqrt{s}$=7 TeV.
The results of the analysis show that
the CMS data is inconclusive as regards the discussed oscillations
(see Figs. \ref{fig:19}a and \ref{fig:21}a).
We have analyzed the clan structure of the MD of charged particles produced in $pp$ collisions.
The clan parameters were studied in the framework of the
weighted superposition of three NBDs used to describe
the ATLAS data on MD in the interval $ |\eta|<$2.5 and $p_T >$ 100~MeV/c.
The examination of the experimental data shows considerable differences
in the behavior of the average number of clans $\bar{N}_{i}$ and
the average number of particles per clan $\bar{n}_{ci}$
in comparison with two-component parametrization of the same data.
The differences concerning the average values of the parameters are summarized as follows.
The three-NBD model gives:
\begin{itemize}
\item[1.]
increasing number of particles per clan $\bar{n}_{c1}$ with $\sqrt s$
in the first, dominant component,
\item[2.]
increasing number of clans $\bar{N}_{2}$ with $\sqrt s$
in the semi-hard component with mini-jets,
\item[3.]
more particles per clan in the first component
than in the semi-hard one with mini-jets ($\bar{n}_{c1}>\bar{n}_{c2}$).
\end{itemize}
The two-NBD model gives:
\begin{itemize}
\item[1.]
constant number of particles per clan $\bar{n}_{c1}$ with $\sqrt s$ in the soft component,
\item[2.]
decreasing number of clans $\bar{N}_{2}$ with $\sqrt s$ in the semi-hard component with mini-jets,
\item[3.]
fewer particles per clan in the soft component
than in the semi-hard one ($\bar{n}_{c1}<\bar{n}_{c2}$).
\end{itemize}
It was shown that the above properties of clans are consistent with the results of the analysis performed
with data on MD measured by the CMS Collaboration in different pseudorapidity intervals at $\sqrt s=$ 7 TeV.
As concerns the third multiplicity component,
the three-NBD model gives approximately constant value of the average number of clans $\bar{N}_{3}$
with energy.
These clans contain one or very few particles on average.
We have studied the influence of the transverse momentum cut on the clan parameters using the
ATLAS data with $p_T >$ 500~MeV/c.
The average number of clans and the average number of particles per clan are reduced
in all three multiplicity components with the $p_T $ cut.
It was shown that the reduction reveals a different behavior from the two-NBD model.
The size of the changes
suggests that there are semi-hard processes with simultaneous
production of many low-$p_T$ particles in the first, dominant class of events.
The values of the clans' parameters point to intensive branching and
a small number of clans mini-jets in these events.
On the other hand, the semi-hard component under the tail of the MD reveals
the existence of many particle clusters
with properties expected for mini-jets.
The transverse momenta of particles in these clusters are on average higher than for
the most particles belonging to the clans of the first component.
The suppression of the clan parameters with the cut $p_T>500$ MeV/c indicates that
the transverse momenta are concentrated around some $\tilde{p}_T$.
A novel feature is growth of the average number of the clans, $\bar{N}_{2}$, with $\sqrt{s}$.
As concerns the third multiplicity component, the present study shows that
it consists mostly of particles with low transverse momenta.
The biggest suppression of
$\bar{n}_{3}$ with $p_T $
suggests that there is a soft mechanism responsible for
particle production under the peak of the MD
at low $n$.
Based on the performed analysis we argue that there is
strong evidence of a new component in the MD
of charged particles measured by the ATLAS Collaboration in
$pp$ collisions up to the energy $\sqrt s=$13~TeV.
The data are well described by a superposition of three NBD functions in a wide range of $\sqrt s$.
The third component at low $n$ gives oscillations of the cumulative combinants
$\langle N\rangle C_i=(i\!+\!1){\cal C}(i\!+\!1)$
observed for the first time in~\cite{WW1}.
The analysis of the clan properties applied to three-NBD fits to the ATLAS data
on multiplicities reveals new features in the LHC energy domain.
Further study of the clan characteristics and the fine structure of the MD with future high quality data
is a challenging problem, which can give important information on
production mechanism and correlation structures
of the multi-particle states created in high energy proton-proton collisions.
\begin{acknowledgements}
The investigations were supported by the institutional support
RVO61389005 and
by the grant LG 15052 of the Ministry of Education of the Czech Republic.
\end{acknowledgements}
\newpage
\setcounter{equation}{0}
\begin{Aeqno}
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War on I-4 Set For Black Friday, November 24 at 3:30 p.m. on ABC
The War on I-4 will be televised in front of a national audience on Black Friday with a 3:30 p.m. kickoff time. Start making your plans now.
By Nick Simon@Nick_JSimon Nov 13, 2017, 1:48pm EST
Share All sharing options for: War on I-4 Set For Black Friday, November 24 at 3:30 p.m. on ABC
Photo by Jason Behnken / Getty Images
The War on I-4 will be broadcasted in front of a national audience.
It was announced Monday morning that USF's pivotal November 24 rivalry showdown at C. will kickoff at 3:30 p.m. on ABC.
The #WarOnI4 is going coast-to-coast.
Full national broadcast on ABC.#BullStrong pic.twitter.com/rmZuubjOsK
— USF Football (@USFFootball) November 13, 2017
With the AAC East division title on the line (as long as USF doesn't trip up against Tulsa this Thursday), it makes sense that this high-stakes rivalry would clash garner national attention on an afternoon lacking in marquee matchups.
This is certainly the type of exposure Commissioner Mike Aresco and others were looking for to showcase the conference this late in the year.
Now that the time of #BlackFridayArmageddon has officially been set, spread the word and plan accordingly. With the usual Orlando traffic combined with Black Friday, you may want to leave super early to get there on time... or find a way around the Disney/outlet mall hellscape between here and there.
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\section{Experimental Results}
Recent high precision measurements of $Z$ pole observables by the
ALEPH, DELPHI, L3, and OPAL \cite{v1,4lep,ds} collaborations
at LEP and SLD at the SLC \cite{I2A},
the $W$ mass by
CDF \cite{v2} and UA2 \cite{v3}, atomic parity violation in cesium
\cite{v4,v5}, neutrino-electron scattering by CHARM II \cite{i1},
and other weak
neutral current observables \cite{iccfr,v6}, as well as the direct lower bounds
$m_t > 91$ GeV (CDF \cite{v7}) and $M_H > 60$ GeV (LEP average \cite{v8})
and the determination $\mbox{$\alpha_s(M_Z)$} = 0.12 \pm 0.01$ from $Z$-pole and
low-energy observables \cite{v9} allow precise tests of
the standard electroweak model and searches for certain types of new physics.
In
this talk, which is an update of previously presented analyses
\cite{ival}-\cite{ipol}, I review the status of the standard model tests
and parameters, and
the coupling constant predictions in ordinary and supersymmetric
grand unified theories (GUTs).
Many of the recent results
are summarized in Table \ref{data}. The LEP results are averages by D.
Schaile of the four LEP experiments as of March, 1993 \cite{ds}, which
includes nearly final results for the 1991 LEP run and contains a proper
treatment of common systematic errors \cite{4lep}.
$M_Z$ is now known to the incredible precision of better than 0.01\%. This was
achieved by the method of resonant depolarization, in which the
(calculable) energies at which the small ($\sim 10 \%$) transverse polarization
of the leptons is destroyed by an oscillating $B$ field was used to
calibrate the energy of the LEP beams. The method is so precise that the tidal
effects of the moon, which cause the size of the LEP ring to change by a few
parts
in $10^8$ and thus change the energy by $\sim 8\ MeV$, had to be measured and
corrected for\footnote{This is the first experiment in which all four
interactions were important simultaneously!}.
\begin{table}
\centering
\begin{tabular}{|l|l|l|} \hline
Quantity & Value & standard model \\ \hline
$M_Z$ (GeV) & $91.187 \pm 0.007$ & input \\
$\Gamma _Z$ (GeV) & $2.491 \pm 0.007$ & $2.490 \pm 0.001 \pm 0.005
\pm [0.006]$\\
$ R = \Gamma_{had}/ \Gamma_{l \bar{l}}$ & $20.87 \pm 0.07$ & $20.78
\pm 0.01 \pm 0.01 \pm [0.07]$ \\
$\sigma^h_p (nb)$ & $41.33 \pm 0.18$ & $41.42 \pm 0.01 \pm 0.01 \pm
[0.06]$\\
$\Gamma_{b \bar{b}}$ (MeV) & $373 \pm 9$ & $375.9 \pm 0.2 \pm 0.5 \pm [1.3]$ \\
$A_{FB} (\mu)$ & $0.0152\pm 0.0027$ & $0.0141 \pm 0.0005 \pm 0.0010$
\\
$A_{pol} (\tau)$ & $0.140\pm 0.018$ & $0.137 \pm 0.002 \pm 0.005$ \\
$A_{e} (P_\tau)$ & $0.134\pm 0.030$ & $0.137 \pm 0.002 \pm 0.005$ \\
$A_{FB} (b)$ & $0.093\pm 0.012$ & $0.096 \pm 0.002 \pm 0.003$ \\
$A_{FB} (c)$ & $0.072\pm 0.027$ & $0.068 \pm 0.001 \pm 0.003$ \\
$A_{LR} $ & $0.100\pm 0.044$ & $0.137 \pm 0.002 \pm 0.005$ \\
\hline
$\Gamma_{l \bar{l}}$ (MeV) & $83.43 \pm 0.29$ & $83.66 \pm 0.02 \pm 0.13$ \\
$\Gamma_{had}$ (MeV) & $1741.2 \pm 6.6$ & $1739 \pm 1 \pm 4 \pm [6]$ \\
$\Gamma_{inv}$ (MeV) & $499.5 \pm 5.6$ &
$500.4 \pm 0.1 \pm 0.9$ \\
$N_{\nu}$ & $3.004 \pm 0.035 $ & $3$ \\
$\bar{g}_A$ & $-0.4999 \pm 0.0009$ & $-0.5$\\
$\bar{g}_V$ & $-0.0351 \pm 0.0025$ & $-0.0344\pm 0.0006 \pm 0.0013$
\\
$\bar{s}^2_W \ (A_{FB}(q))$& $0.2329\pm 0.0031$ & $0.2328 \pm 0.0003
\pm 0.0007 \pm$ ? \\
\hline
$M_W$ (GeV) & $79.91 \pm 0.39$ & $80.18 \pm 0.02 \pm 0.13$ \\
$M_W/M_Z$ & $0.8813 \pm 0.0041$ & $0.8793 \pm 0.0002 \pm 0.0014$ \\
$Q_W (Cs)$ & $-71.04 \pm 1.58 \pm [0.88]$ & $-73.20 \pm 0.07
\pm 0.02$ \\
$g_A^e (\nu e \mbox{$\rightarrow$} \nu e)$ & $-0.503 \pm 0.017$ & $-0.505 \pm 0 \pm 0.001$ \\
$g_V^e (\nu e \mbox{$\rightarrow$} \nu e)$ & $-0.025 \pm 0.020$ & $-0.036 \pm 0.001
\pm 0.001$ \\
\mbox{$\sin^2\theta_W\,$} & $ 0.2242 \pm 0.0042 \pm [0.0047]$ & $0.2269 \pm 0.0003
\pm 0.0025$ \\ \hline
\end{tabular}
\caption[]{Experimental values for LEP \cite{v1,4lep,ds}
and SLC \cite{I2A} observables,
$M_W$ \cite{v2}, $M_W / M_Z$
\cite{v3}, the weak charge in cesium $Q_W$ \cite{v4,v5},
the parameters $g_{V,A}^e$ relevant to $\nu_\mu e$ scattering from CHARM II
\cite{i1}, and $\mbox{$\sin^2\theta_W\,$} \equiv 1 - M_W^2/M_Z^2$ from CCFR \cite{iccfr},
compared with the standard model predictions
for $M_Z = 91.187 \pm 0.007$ GeV, $m_t = 150 ^{+19}_{-24}$ GeV,
and 60 GeV $< M_H < 1$ TeV. Only the first eleven $Z$-pole observables are
independent. The ? for the $\bar{s}^2_W \ (A_{FB}(q))$
prediction refers to the scheme dependence.
The two errors for $Q_W (Cs)$ and \mbox{$\sin^2\theta_W\,$} are experimental and
theoretical (in brackets). The first error in the predictions is
from the uncertainties in $M_Z$ and $\Delta r$, the second is from
$m_t$ and $M_H$, and the third (in brackets) is the theoretical
QCD uncertainty for $\mbox{$\alpha_s(M_Z)$} = 0.12 \pm 0.01$ \cite{v9} . The older
neutral current quantities described
in \cite{v6} are also used in the analysis.}
\label{data}
\end{table}
$\Gamma_Z, \Gamma_{l \bar{l}}, \Gamma_{had}$, $\Gamma_{b \bar{b}}$,
and
$\Gamma_{inv}$ refer respectively to the total, leptonic (average of $e, \mu,
\tau$), hadronic, $b \bar{b}$,
and invisible $Z$ widths; $ R \equiv \Gamma_{had}/
\Gamma_{l \bar{l}}$; $\sigma^h_p
= 12 \pi \Gamma _{e \bar{e}} \Gamma _{had} / M ^2_Z \Gamma^2_Z$ is the
hadronic cross section on the pole; and $N_\nu \equiv \Gamma _{inv} /
\Gamma _{\nu \bar{\nu}}$ is the number of light neutrino flavors.
A number of asymmetries have also been measured.
$A_{FB} (f)$ is the forward-backward asymmetry for $e^+ e^- \mbox{$\rightarrow$} f \bar{f}$;
$A_{pol} (\tau)$ is the polarization of a
final $\tau$ ($L$ is positive), while $A_{e} (P_\tau)$ is essentially the
forward-backward asymmetry in the polarization; $A_{LR}$ is the
polarization asymmetry, which has recently been measured for
the first time by the SLD collaboration at the SLC \cite{I2A}.
All of the asymmetries are Born contributions, from which various
QED, QCD, interference, and box contributions have been removed by
the experimenters.
Finally, $\bar{g}_A, \bar{g}_V$ are
effective Born couplings, related, for example,
to $\Gamma_{l \bar{l}}$ and $A_{FB} (\mu)$
by\footnote{I assume lepton universality, throughout. This is
strongly supported by the LEP data.}
\begin{equation}
\Gamma_{l \bar{l}} = \frac{G_F M^3_Z}{6\sqrt{2} \pi} (\bar{g}^2_A
+ \bar{g}^2_V) \ \ \ \ \ \ \
A_{FB}(\mu) = \frac{3 \ \bar{g}^2_V \ \bar{g}^2_A}
{(\bar{g}^2_V + \bar{g} ^2 _A)^2}.
\end{equation}
Similarly,
\begin{equation} A_{LR} = \frac{2 \ \bar{g}_V \ \bar{g}_A}
{\bar{g}^2_V + \bar{g} ^2 _A}, \end{equation}
with the same expression for
$A_{pol} (\tau)$ and $A_{e} (P_\tau)$.
Of the $Z$-pole observables only $M_Z$, $\Gamma_Z, R,$ $\sigma^h_p$,
$\Gamma_{b \bar{b}}$, $A_{FB} (\mu)$, $A_{pol} (\tau)$, $A_{e} (P_\tau)$,
$A_{FB}(b)$ (which is corrected for $b\bar{b}$ oscillations),
$A_{FB}(c),$ and $A_{LR}$
are used in the analysis. $ \Gamma_Z,\ R,$ and $\sigma^h_p$ are used
rather than the more physically transparent $ \Gamma_Z,\ \Gamma_{l \bar{l}},$
and $ \Gamma_{had}$ because the former are closer to what is actually
measured and are relatively weakly correlated. (The combined LEP
values \cite{ds} for the correlations are used.)
$\bar{s}^2_W \ (A_{FB}(q))$,
which is the effective weak angle obtained from the
charge asymmetry in hadronic decays, is not used because the results have
only been presented assuming the validity of the standard model. The other
LEP observables are not independent but are displayed
for completeness.
Recent measurements of the $W$ mass and weak neutral
current data are also displayed in Table \ref{data}. $Q_W (Cs)$ is the
effective charge of the parity-violating interaction in cesium \cite{v4},
while $g_{V,A}^e$ are the coefficients of the vector and axial electron
currents in the effective four-fermi interaction for $\mbox{$\stackrel{(-)}{\nu}$}_\mu e
\mbox{$\rightarrow$} \mbox{$\stackrel{(-)}{\nu}$}_\mu e$ as obtained by CHARM II \cite{i1}.
The preliminary value of the on-shell weak angle
$\mbox{$\sin^2\theta_W\,$} \equiv 1 - M_W^2/M_Z^2 = 0.2242 \pm 0.0042 \pm [0.0047]$
obtained from deep inelastic neutrino scattering
from CCFR \cite{iccfr} at Fermilab is in reasonable
agreement with the earlier
CERN values $0.228 \pm 0.005 \pm [0.005]$ \cite{cdhs}, and
$0.236 \pm 0.005 \pm [0.005]$ \cite{charm}, though the central
value is somewhat lower. The errors in brackets are theoretical.
They are dominated by the $c$-quark threshold in the charged current
scattering
used to normalize the neutral current process, and are
strongly correlated between the experiments. Older neutral current
results, included in the analysis, are described in~\cite{v6}.
The
standard model predictions for each quantity other than $M_Z$ are also
shown.
These are computed using $M_Z = 91.187 \pm 0.007$ GeV
as input, using the range of $m_t$ determined from the global fit and
60 GeV $< M_H < 1$ TeV. The agreement is excellent.
The $b$ observables $\Gamma_{b\bar{b}}$ and $ A_{FB} (b)$ are especially
important because the predictions depend on
the $SU_2$ assignments of the $b$. In Table~\ref{b} the
experimental values are compared with topless models and
other
alternatives with $V+A$ currents. It is seen that the data uniquely
picks out the standard model from these alternatives~\cite{v15}. This
conclusion is strenghtened by a recent detailed analysis by Schaile and
Zerwas \cite{v16} of LEP and lower energy data, which yields
\begin{equation} t_{3L}(b) = -0.490^{+0.015}_{-0.012}\ \ \ \ \ \ \ \
t_{3R}(b) = -0.028 \pm 0.056 \end{equation}
for the third component of the weak isospin of the $b_{L,R}$, respectively,
in agreement with the standard model expectations of $-1/2$ and $0$ --
{\em i.e., } topless models are excluded and the $b_L$ must be
in a weak doublet with
the $t_L$.
\begin{table} \centering
\small \def\baselinestretch{1} \normalsize
\begin{tabular}{|c|c|c|c|c|c|} \hline
Quantity & Experiment & SM & Topless & Mirror & Vector \\
\hline
$\Gamma_{b\bar{b}}$ (MeV)&
373 \mbox{$\pm$} 9 & 376 & 24 & 376 & 728 \\
$A_{FB} {(b)}$ & 0.093 \mbox{$\pm$} 0.012 & 0.096 & 0 & $-0.096$ & 0 \\ \hline
\end{tabular}
\caption[]{Predictions of the standard model (SM),
topless models, a mirror
model with $(t \; b)_R$ in a doublet, and a vector model with left and
right-handed doublets, for $\Gamma_{b\bar{b}}$
and $A_{FB} (b)$, compared with the experimental values.}
\label{b}
\end{table}
\section{Standard Model Tests and $m_t$}
Results will be presented in the $\overline{\rm{MS}}$ \cite{v17} and
on-shell \cite{v18} schemes. I use the radiative corrections calculated by
Degrassi {\it et al.}
\cite{v19} for the $W$ and $Z$ masses, those of Hollik
\cite{v20} for the $Z$ widths, and generalized Born expressions for
the Born contributions to the asymmetries. The latter are obtained
from the data, {\it e.g., } by using the program
ZFITTER \cite{v22}. The calculations in \cite{v19}-\cite{v22} are in
excellent agreement with each other and with those in \cite{v21}.
Radiative corrections to low energy
neutral current processes are described in \cite{v6}.
In the standard model
\begin{eqnarray}
M^2_Z & = & \frac{A^2_0}
{\hat{\rho} \hat{c}^2 \hat{s}^2 (1 - \Delta \hat{r}_W)}
= \frac{A^2_0}{c^2 s^2 (1-\Delta r)} \nonumber \\
M^2_W & = & \hat{\rho} \hat{c}^2 M^2_Z = c^2 M^2_Z \label{mwz}
\end{eqnarray}
where $A^2_0 = \pi \alpha/ \sqrt{2} G_F = (37.2803 \ \rm{GeV})^2,
\hat{s}^2 \equiv$ \mbox{$\sin^2\hat{\theta}_W(M_Z)\ $} refers to the weak angle in the
$\overline{\rm{MS}}$
scheme \cite{v17}, $s^2 \equiv$ \mbox{$\sin^2\theta_W\,$} $= 1 - M^2_W / M^2_Z$ refers to
the on-shell scheme \cite{v18}, $\hat{c}^2 \equiv 1 - \hat{s}^2$, and
$c^2 \equiv 1 -s^2$; $\Delta
\hat{r} _W, \hat{\rho} - 1$, and $\Delta r$ are radiative correction
parameters. As is well known \cite{v23}, $\hat{\rho}
\sim 1 + \Delta \rho_t$, where
\begin{equation}
\Delta \rho_t = \frac{3G_F m_t^2}{8 \sqrt{2} \pi^2} \simeq 0.0031
\left( \frac{m_t}{100 \rm{GeV}}\right)^2 \ , \label{rhat}
\end{equation}
has a strong $m_t$ dependence, while
$\Delta r \simeq \Delta r_0 - \Delta \rho_t / \tan^2 \theta_W$ is
even more sensitive.
$\Delta \hat{r}_W \sim \Delta r_0 \sim 1 - \alpha/\alpha(M_Z) \sim
0.07$ has no quadratic
$m_t$ dependence. There is additional logarithmic dependence on $m_t$
and $M_H$ in $\hat{\rho}, \Delta \hat{r}_W$, and $\Delta r$, as
well as $O (\alpha)$ effects associated with low energy physics. These
effects are important and are fully incorporated in the analysis, but
will not be displayed here.
Gluonic corrections
to $ \Delta \rho_t$ of order $-\alpha \alpha_s m_t^2/M_Z^2$ can be
important for large $m_t$ \cite{v24,v25}. The leading perturbative term is
\cite{v24} $-2
\alpha_s(m_t) (\pi^2 + 3)/9 \pi \sim -0.10$ times the expression in
(\ref{rhat}). These corrections, which increase the
predicted value of $m_t$ by about 5\%, are included in
the analysis\footnote{These
terms were omitted from previous analyses (e.g., \cite{ival}), because of
uncertainties in both the magnitude and sign of important nonperturbative
effects \cite{v25}. However, a careful new analysis \cite{v25a} indicates
that the perturbative esimate is an excellent approximation. A future
global analysis will include small additional corrections.}.
The most accurate determination of \mbox{$\sin^2\hat{\theta}_W(M_Z)\ $} and \mbox{$\sin^2\theta_W\,$} are from $M_Z =
91.187 \pm 0.007$~GeV.
The values are shown in
Table~\ref{tmz}
for $m_t = 100$ and 200~GeV and $M_H = 300$~GeV, and also for the
global best fit range for $m_t$ and $M_H$. It is apparent that
the extracted value of $\sin^2\theta_W$ depends strongly on $m_t$,
while $\sin^2\hat{\theta}_W (M_Z)$ is considerably less
sensitive due to the smaller coefficient of the quadratic $m_t$ term in $
\hat{\rho}$ than in $\Delta r$.
For fixed $m_t$
and $M_H$ the
uncertainty of $\pm 0.0003$ in $\sin^2\theta_W$ has two components:
the experimental error from $\Delta M_Z$ is only $\pm 0.0001$, while
the theoretical error (from the uncertainty of $\pm 0.0009$ in $\Delta
r$ from low energy hadronic contributions) is larger, $\pm 0.0003$.
The $\pm 1 \sigma$ limits on $\sin^2 \hat{\theta}_W(M_Z)$ as a
function of $m_t$ are shown in Figure~\ref{mtop}.
\begin{table}\centering
\small \def\baselinestretch{1} \normalsize
\begin{tabular}{|ccc|} \hline
$m_t$ (GeV) & $\sin^2\theta_W$ & $\sin^2\hat{\theta}_W(M_Z)$ \\
\hline
100 & $0.2322 \pm 0.0003$ & $ 0.2340 \pm 0.0003$ \\
200 & $0.2204 \pm 0.0003$ & $0.2311 \pm 0.0003$ \\
$150^{+19}_{-24}$ & $0.2269 \pm 0.0025$ & $0.2328 \pm 0.0007$ \\
\hline
\end{tabular}
\caption[]{Values of $\sin^2\theta_W = 1 - M^2_W/M^2_Z$ and
$\sin^2\hat{\theta}_W(M_Z)$ obtained from $M_Z = 91.187 \pm
0.007$~GeV, assuming $(m_t, M_H) =$~(100, 300) and (200, 300) GeV. In
the last row $m_t = 150^{+19}_{-24}$~GeV (obtained from the global fit
to all data) and 60~GeV$< M_H < 1000$~GeV.}
\label{tmz}
\end{table}
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxmt.ps}{0.8}
\caption[]{One $\sigma$ uncertainty in $\sin^2\hat{\theta}_W(M_Z)$ as a
function of $m_t$ for $\sin^2\hat{\theta}_W(M_Z)$ determined from
various inputs for $M_H = 300$~GeV. The direct lower limit $m_t >
91$~GeV and the 90\% CL fit to all data are also shown.}
\label{mtop}
\end{figure}
The ratio $M_W/M_Z = 0.8813 \pm 0.0041$ determined by UA2 \cite{v3}
and $M_W =
79.91 \pm 0.39$~GeV from CDF \cite{v2} determine the values of $\sin^2
\hat{\theta}_W (M_Z)$ shown in Table~\ref{stab}. From
Figure~\ref{mtop} it is apparent that
$M_Z$, $M_W$, and $M_W/M_Z$ together imply an upper limit of
$O$(200~GeV) on $m_t$. A simultaneous fit of $M_Z$, $M_W$, and
$M_W/M_Z$ to $\sin^2\hat{\theta}_W(M_Z)$, and $m_t$ yields $m_t
= 145^{+42}_{-49} \pm 16$~GeV, where the second uncertainty is from $M_H$ in
the range 60-1000~GeV, with a central value of 300~GeV. The
90(95)\%~CL upper limits on $m_t$ are 211~(223)~GeV
(Table~\ref{ttab}). The upper limits are for $M_H = 1000$~GeV,
which gives the weakest constraint. The value of
$\sin^2\hat{\theta}_W(M_Z)$, including the uncertainties from $m_t$ and
$M_H$, is also given in Table~\ref{ttab}.
\begin{table}\centering \small \def\baselinestretch{1} \normalsize
\begin{tabular}{|c|c|c|c|} \hline
Data & $\sin^2\hat{\theta}_W(M_Z)$ & \ \ & \ \ \\
\ \ & $m_t = 100$ & $m_t=200$& $m_t=150^{+19}_{-24}$ \\ \hline
$M_Z$ & 0.2340 \mbox{$\pm$} 0.0003 & 0.2311 \mbox{$\pm$} 0.0003 & 0.2328 \mbox{$\pm$} 0.0007
\\
$M_W, \frac{M_W}{M_Z}$ & 0.2331 \mbox{$\pm$} 0.0022 & 0.2345 \mbox{$\pm$} 0.0022 &
0.2339 \mbox{$\pm$} 0.0022 \\
$ \Gamma_{Z},R,\sigma^h_p $ & 0.2333 \mbox{$\pm$} 0.0006 &
0.2319 \mbox{$\pm$} 0.0006 & 0.2327 \mbox{$\pm$}
0.0006 \\
$\Gamma_Z$ & 0.2332 \mbox{$\pm$} 0.0006 & 0.2320 \mbox{$\pm$} 0.0006 & 0.2327 \mbox{$\pm$}
0.0007 \\
$\Gamma_{l\bar{l}}$ & 0.2340 \mbox{$\pm$} 0.0007 & 0.2326 \mbox{$\pm$} 0.0007 & 0.2333
\mbox{$\pm$}
0.0008 \\
$\Gamma_{b\bar{b}}$ & 0.236 \mbox{$\pm$} 0.004 & 0.232 \mbox{$\pm$} 0.004 & 0.234
\mbox{$\pm$}
0.004 \\
$\Gamma_Z/M_Z$ & 0.2329 \mbox{$\pm$} 0.0009 & 0.2323 \mbox{$\pm$} 0.0009 & 0.2326
\mbox{$\pm$}
0.0009 \\
$\Gamma_{l\bar{l}}/M_Z$ & 0.2341 \mbox{$\pm$} 0.0010 & 0.2332 \mbox{$\pm$} 0.0010 & 0.2336
\mbox{$\pm$} 0.0010 \\
$\Gamma_Z/M^3_Z$ & 0.230 \mbox{$\pm$} 0.003 & 0.236 \mbox{$\pm$} 0.003 & 0.232 \mbox{$\pm$}
0.004 \\
$\Gamma_{l\bar{l}}/M_Z^3,R$ & 0.231 \mbox{$\pm$} 0.004 & 0.234 \mbox{$\pm$} 0.003 & 0.232
\mbox{$\pm$} 0.004 \\
$A_{FB} (\mu)$ & 0.232 \mbox{$\pm$} 0.002 & 0.232 \mbox{$\pm$} 0.002 & 0.232
\mbox{$\pm$}
0.002 \\
$A_{pol} (\tau)$ & 0.232 \mbox{$\pm$} 0.002 & 0.232 \mbox{$\pm$} 0.002 & 0.232 \mbox{$\pm$}
0.002 \\
$A_{FB} (b)$ & 0.233 \mbox{$\pm$} 0.002 & 0.233 \mbox{$\pm$} 0.002 & 0.233 \mbox{$\pm$}
0.002 \\
$A_{LR}$ & 0.238 \mbox{$\pm$} 0.006 & 0.238 \mbox{$\pm$} 0.006 & 0.238 \mbox{$\pm$}
0.006 \\
$\nu N \rightarrow \nu X$ & 0.233 \mbox{$\pm$} 0.005 & 0.238 \mbox{$\pm$} 0.005 & 0.235 \mbox{$\pm$}
0.005 \\
$\nu p \rightarrow \nu p$ & 0.212 \mbox{$\pm$} 0.032 & 0.212 \mbox{$\pm$} 0.031 & 0.212 \mbox{$\pm$}
0.032 \\
$\nu_\mu e \rightarrow \nu_\mu e$ & 0.232 \mbox{$\pm$} 0.009 & 0.231 \mbox{$\pm$} 0.009 &
0.232 \mbox{$\pm$} 0.009 \\
$e^{\uparrow \!\downarrow} D \rightarrow eX$ & 0.222 \mbox{$\pm$} 0.018 & 0.223 \mbox{$\pm$}
0.018 & 0.222 \mbox{$\pm$} 0.018 \\
atomic parity & 0.224 \mbox{$\pm$} 0.008 & 0.221 \mbox{$\pm$} 0.008 & 0.223 \mbox{$\pm$} 0.008
\\
All & 0.2337 \mbox{$\pm$} 0.0003 & 0.2314 \mbox{$\pm$} 0.0003 & 0.2328 \mbox{$\pm$} 0.0007 \\
\hline
\end{tabular}
\caption[]{Values of $\sin^2\hat{\theta}_W (M_Z)$ obtained from various
inputs, for $(m_t, \; M_H) =$~(100, 300) and (200, 300)~GeV. In the
last column $m_t = 150^{+19 +15}_{-24 -20}$~GeV, from the global fit,
correlated with
60~GeV $< M_H < 1000$~GeV. For $\nu N \rightarrow \nu X$ the uncertainty includes
0.003
(experiment) and 0.005 (theory). For atomic parity, the experimental and
theoretical components of the error are 0.007 and 0.004 respectively.}
\label{stab}
\end{table}
\begin{table} \centering
\small \def\baselinestretch{1} \normalsize
\begin{tabular}{|l|l|l|l|} \hline
data & \mbox{$\sin^2\hat{\theta}_W(M_Z)\ $} & $m_t$ (GeV) & $m^{max}_t$ (GeV) \\ \hline
$M_Z, M_W,M_W/M_Z$ & $0.2329 \pm 0.0014$ & $145^{+42}_{-49} \pm 16$ &
$211 (223)$\\
$Z$-POLE & $0.2328 \pm 0.0008$ &
$150 \pm 27 \pm 18$ & $198 \ (206)$\\
$Z$-POLE $ + M_W, M_W/M_Z$ & $0.2328 \pm 0.0007$
& $150^{+21}_{-26}
\pm 18$ & $193 (200)$\\
$M_Z, \nu N$ & $0.2333 ^{+0.0011}_{-0.0016}$ &
$132^{+54}_{-40} \pm 19$
& $210 \ (223)$\\
All & $0.2328 \pm 0.0007$ & $150 ^{+19 +15}_{-24 -20}$ & $190 \ (197)$
\\ \hline
\end{tabular}
\caption[]{
Values of \mbox{$\sin^2\hat{\theta}_W(M_Z)\ $} and $m_t$ obtained for various data sets.
$Z$-POLE refers to the first 11 constraints in Table \ref{data}
(with correlations).
The \mbox{$\sin^2\hat{\theta}_W(M_Z)\ $} error includes $m_t$ and $M_H$. The first error for $m_t$
includes experimental and theoretical uncertainties for $M_H = $ 300 GeV.
The second error is the variation for $M_H \rightarrow 60$ GeV $(-)$ and
$M_H \rightarrow 1000$ GeV $(+)$. The last column lists the upper limits
on $m_t$ at 90 (95)\% CL for $M_H = 1000$ GeV, which gives the weakest
upper limit. The direct CDF constraint $m_t > 91$ GeV is included.}
\label{ttab}
\end{table}
The partial width for $\Gamma \rightarrow f \bar{f}$ is given
by~\cite{v20,v21},
\begin{equation} \Gamma_{f\bar{f}} = \frac{C_f \hat{\rho} G_F M_Z^3}{6
\sqrt{2} \pi} \left( |a_f|^2 + |v_f|^2 \right). \label{eq27} \end{equation}
The axial and vector couplings are
\begin{eqnarray} a_f &=& t_{3L} (f) = \pm \frac{1}{2} \nonumber \\
v_f &=& t_{3L} (f) - 2 \sin^2\hat{\theta}_W (M_Z) q_f, \label{eq28}
\end{eqnarray}
where $t_{3L}(f)$ and $q_f$ are respectively the third component
of weak isospin and electric charge of fermion $f$;
$\hat{\rho}$ is dominated by the
$m_t$ term (cf (\ref{rhat}). The coefficient comes about by
rewriting the tree-level formula
\begin{equation} \frac{g^2(M_Z) M_Z}{8 \cos^2 \theta_W} = \frac{G_F
M_Z^3}{\sqrt{2}}. \label{eq29} \end{equation}
Expressing the width in this way incorporates the bulk of the
radiative
corrections, except for the large $m_t$ dependence in $\hat{\rho}$.
Additional small radiative corrections are included but not displayed
here.
The factor in front incorporates the color factor and QED
and QCD corrections:
\begin{equation} C_f = \left\{ \begin{array}{lll}
1 + \frac{3\alpha}{4\pi} q_f^2 & ,& {\rm leptons} \\
3 \left( 1 + \frac{3\alpha}{4\pi} q^2_f \right) \left( 1 +
\frac{\alpha_s}{\pi} + 1.405 \frac{\alpha_s^2}{\pi^2} \right)
\ &,& {\rm quarks} \end{array} \right. \label{eq30} \end{equation}
where the range
$\alpha_s (M_Z) \simeq 0.12 \pm 0.01$ from $Z$-decay event
topologies and other data \cite{v9} is used.
(\ref{eq27}) is written neglecting the fermion masses. In practice,
fermion
mass corrections~\cite{v20} must be applied for $\Gamma_{b\bar{b}}$.
They are also included in the following for
$\Gamma_{c\bar{c}}$ and $\Gamma_{\tau \bar{\tau}}$, though
the effects are small. There are significant correlations between
the experimental values of the various total and partial $Z$ widths,
which must be included in a global analysis.
The vertex corrections for $\Gamma_{b\bar{b}}$
depend strongly on $m_t$ and must be included as an
extra correction~\cite{v26}. For fixed $M_Z$ the $b\bar{b}$ width
actually decreases
with $m_t$, while the other modes all increase (because of the
$\hat{\rho}$ factor). This gives a means of separating
$\hat{\rho} (m_t)$ from such new physics as nonstandard Higgs
representations by comparing $\Gamma_{b\bar{b}}$ or
$\Gamma_Z$ with the other data~\cite{v10}.
The standard model predictions for $\Gamma_Z$, $\Gamma_{\ell^+\ell^-}
(\ell = e, \mu,$ or $\tau), \; R \equiv \Gamma_{had}
/\Gamma_{\ell^+\ell^-}$, and the invisible width $\Gamma_{inv}$
as a function of $m_t$ are compared with the experimental
results in Figures~\ref{gam1} and \ref{gam2}. ($\sin^2
\hat{\theta}_W(M_Z)$ in $v_f$ is obtained from $M_Z$). One sees that
the agreement is excellent for $m_t$ in the 100 -- 200 GeV range. The
results of fits to the $Z$ widths are listed in Tables \ref{stab} and
\ref{ttab}. The $R$ ratio, which is insensitive to $m_t$, is slightly above
the standard model prediction, though only at the 1$\sigma$ level. As
will be discussed, $R$ favors a slightly higher value of \mbox{$\alpha_s(M_Z)$} than the
value obtained from event topologies and low energy data.
The invisible width in Figure~\ref{gam2} is
clearly in agreement with $N_\nu = 3$ but not $N_\nu = 4$. In fact, the
result \cite{ds} $N_\nu = 3.004 \pm 0.035$ not only eliminates extra
fermion families with $m_\nu \ll M_Z/2$, but also supersymmetric
models with light
sneutrinos ($\Delta N_\nu = 0.5$) and models with triplet ($\Delta N_\nu =
2$) or doublet ($\Delta N_\nu = 0.5$) Majorons \cite{numass}.
$N_\nu$ does not include
sterile ($SU_2$-singlet) neutrinos. However, the complementary bound
$N_\nu' < 3.3$ (95\% CL) from nucleosynthesis \cite{v27} {\it does}
include sterile neutrinos for a wide range of masses and mixings, provided
their mass is less than $\sim 20$ MeV.
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxgamf.ps}{0.8}
\caption[]{Theoretical predictions for $\Gamma_Z,\Gamma_{\ell^+\ell^-}$,
and $R = \Gamma_{had}/\Gamma_{\ell^+\ell^-}$ in the standard model
as a function of $m_t$, compared with the experimental
results. The $M_H$ dependence is too small to see on the scale of the
graph.
The QCD uncertainties in $\Gamma_Z$ and
$R$ are indicated.}
\label{gam1}
\end{figure}
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxginv.ps}{0.55}
\caption[]{Theoretical prediction for $\Gamma_{inv}$ in the standard
model with $N_\nu = 3$ and 4, compared with the experimental
value.}
\label{gam2}
\end{figure}
One can obtain precise $(\Delta = O (\pm 0.0007))$ values of
$\sin^2\hat{\theta}_W(M_Z)$ from $\Gamma_Z$ and $\Gamma_{\ell^+
\ell^-}$
(Table~\ref{stab}). The major sensitivity is through the $M^3_Z$
factor in (\ref{eq27}) rather than from the vertices ({\em i.e., } the $v_f$).
It is useful to also obtain the $\sin^2\hat{\theta}_W(M_Z)$ from the
vertices.
Values can be obtained from the ``reduced widths''
$\Gamma_Z/M^3_Z, \; \Gamma_{\ell^+\ell^-}/M_Z^3$, and $R$. As can be
seen in Table~\ref{stab}, the $\sin^2\hat{\theta}_W(M_Z)$
sensitivity from
$\Gamma_Z/M_Z^3$ and the combination $(\Gamma_{\ell^+\ell^-}/M_Z^3,
R)$ is around $\pm 0.004\; (\Gamma_{\ell^+\ell^-}/M_Z^3$ and $R$
individually give large asymmetric errors). Yet another determination
of $\sin^2\hat{\theta}_W(M_Z)$ comes from $\Gamma_Z/M_Z$ and
$\Gamma_{l^+l^-}/M_Z$. As can be seen in Table~\ref{stab} the
values obtained are insensitive to $m_t$. This can be understood from
(\ref{mwz}) and (\ref{eq27}), from which one sees that
$\Gamma_{f\bar{f}}/M_Z$ has no quadratic $m_t$ dependence (except
$f=b)$. Of course, the various values of $\sin^2\hat{\theta}_W$
obtained from the $\Gamma$'s are not all independent.
At tree level the asymmetries can be written
\begin{equation} A_{FB}(f) \simeq 3 \eta_e \eta_f, \end{equation}
and \begin{equation} A_{pol}(\tau) \simeq 2 \eta_\tau, \end{equation}
where
\begin{equation} \eta_f \equiv \frac{v_f a_f}{v_f^2 + a_f^2}, \end{equation}
and $v_f$ and $a_f$ are the tree-level vector and axial couplings in
(\ref{eq28}).
These expressions are an excellent first approximation even in
the presence of higher-order corrections, provided that $v_f$ is expressed
in terms of \mbox{$\sin^2\hat{\theta}_W(M_Z)\ $}, {\it i.e.,} one identifies $v_f$ and $a_f$ with the
effective
Born couplings $\bar{g}_V$ and $\bar{g}_A$.
$ A_{FB}(b) = 0.093 \pm 0.012$ has been corrected for $b
\bar{b}$ oscillations \cite{v1}, using
\begin{equation} A_{FB}(b) = \frac{ A_{FB}^{obs}(f)}{1-2 \chi}, \end{equation}
where $\chi = 0.126 \pm 0.012$ is the oscillation probability at the
$Z$-pole. $Z b \bar{b}$ vertex corrections can be added to $ A_{FB}(b)$ but
are negligible numerically. The predictions for $A_{FB} (\mu)$,
$A_{pol} (\tau)$, and $A_{FB} (b)$ are compared with the experimental
data in Figure~\ref{asym}. Again, the agreement is excellent.
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxafb.ps}{0.8}
\caption[]{Theoretical prediction for $A_{FB} (\mu)$, $A_{pol} (\tau)$, and
$A_{FB} (b)$ in the standard
model as a function of $m_t$ for $M_H$ = 60 (dotted line), 300 (solid), and
1000 (dashed) GeV, compared with the experimental values. The
theoretical uncertainties from $\Delta \Delta
r = \pm 0.0009$ are also indicated.}
\label{asym}
\end{figure}
The results for $\sin^2\hat{\theta}_W$ obtained from a variety of low
energy neutral current processes are listed in Table~\ref{stab}.
The values obtained from atomic parity violation, $e^{\uparrow
\downarrow}D$, and $\nu e$ and $\nu p$ elastic scattering are
consistent with the value obtained from $M_Z$. They all have a
similar dependence on $m_t$ as the $M_Z$ value and therefore do not
significantly constrain $m_t$. They are, however, quite important in
searches for new physics.
On the other hand, the value of the on-shell $\sin^2\theta_W$ obtained
from deep inelastic $\nu N$ scattering
\cite{v28} is insensitive to $m_t$.
As can be seen in Table~\ref{stab} and
Figure~\ref{mtop} the corresponding $\sin^2\hat{\theta}_W (M_Z)$
increases rapidly with $m_t$. From Table~\ref{ttab} deep inelastic
$\nu N$ scattering (combined with $M_Z$) gives $ m_t < 210(223)$~GeV
at 90(95)\% CL. These are somewhat weaker than previous limits (193 (207)
GeV) \cite{ival} due to the inclusion of the new CCFR result \cite{iccfr},
with its slightly lower value for \mbox{$\sin^2\theta_W\,$} (+6 GeV) and due to the
inclusion of $O(\alpha \alpha_s m_t^2)$ radiative corrections (+11 GeV).
The results of global fits to all data
are shown in
Tables~\ref{stab} and \ref{ttab}. All results include full statistical and
systematic uncertainties in the experimental data as well as all of the
important correlations.
In particular, one
obtains the prediction\footnote{This is in excellent agreement with the result
$148^{+18 +17}_{-20 -19}$ of Schaile \cite{ds}.}
\begin{equation} m_t = 150^{+19 +15}_{-24 -20} \ {\rm GeV}, \label{eq52} \end{equation}
where the central value assumes \mbox{$M_H$} \ = 300 GeV.
The second error is from the Higgs mass, assuming $60$~GeV
$<M_H< 1000$~GeV. The \mbox{$m_t$} \ and \mbox{$M_H$} \ dependences are strongly correlated.
The relation between the two in the radiative corrections is not universal,
but a reasonable interpolation of the \mbox{$M_H$} dependence is
\begin{equation} m_t {\rm (GeV)} = 150^{+19}_{-24} + 12.5 \ln (\mbox{$M_H$}/300 {\rm GeV}).
\label{eq52a} \end{equation}
Alternately, we can allow \mbox{$M_H$} \ to be a free parameter in the
range 60 -1000 GeV, with the result that $m_t = 131^{+47}_{-28}$
GeV, with the lower central value occurring because the best fit
is for \mbox{$M_H$} \ = 60 GeV.
The upper limit on \mbox{$m_t$} \ is
\begin{equation} m_t < \left\{ \begin{array}{clc} 190 \ {\rm GeV},
& 90\%& \ CL \\
197\ {\rm GeV}, & 95\%& \ CL \\ 208\ {\rm GeV}, & 99\%& \ CL
\end{array}
\right.\ \ \ \ \ , \label{eq53} \end{equation}
which occurs for $M_H = 1000$~GeV. For $M_H = 60 (300)$~GeV, the 90\%
CL limit is 158 (175)~GeV and the 95\%
CL limit is 165 (182)~GeV. The upper and lower limits on $m_t$ are
shown as a function of $M_H$ in Figure \ref{ht}.
The values of $\sin^2\hat{\theta}_W$ and
$m_t$ and the $m_t$ limits for various subsets of the data are given
in Table~\ref{ttab}. The $\chi^2$ distribution as a function of
$m_t$ is shown in Figure~\ref{chis} for $M_H = $~60, \mbox{$M_Z$}, 300, and
1000~GeV. The fit is excellent\footnote{In fact, the fit is {\em too}
good. This has always been the case for precision neutral current
and $Z$-pole experiments \cite{v6}. The most likely explanation is a tendency
for experimenters to overestimate systematic errors.}, with a $\chi^2/df$ of
168/206 $\sim$ 0.82 for $m_t = 150,\ M_H = 300$ GeV.
The result in (\ref{eq52}) is very close to the value
$149^{+21}_{-27} \pm 16 $ obtained about 1 year ago. The agreement
is somewhat fortuitous: the new 1991 LEP and other data lower
the prediction by $\sim 9$ GeV, but this is compensated by
the inclusion of $O(\alpha \alpha_s m_t^2)$ radiative corrections
(+8 GeV) and the use of 300 (rather than 250) GeV as the central
\mbox{$M_H$} \ value (+2 GeV).
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxmtmh.ps}{0.55}
\caption[]{Best fit value for $m_t$ and upper and lower limits as a
function of $M_H$. The direct lower limit $M_H >$ 60 GeV \cite{v8} and
the approximate triviality limit \cite{v29} $M_H < $ 600 GeV are also
indicated. The latter becomes $M_H < $ 200 GeV if one requires that the
standard model holds up to the Planck scale.}
\label{ht}
\end{figure}
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxchis.ps}{0.55}
\caption[]{$\chi^2$ distribution
for all data (207 df) in the standard model
as a function of $m_t$, for $M_H = 60$, \mbox{$M_Z$}, 300, and
1000~GeV. The direct constraint \mbox{$m_t$} $> 91$ GeV is {\em not}
included.} \label{chis}
\end{figure}
The prediction in (\ref{eq52}) is for the minimal standard model.
In the minimal supersymmetric extension (MSSM),
for almost all of the allowed parameter range for the superpartner
spectrum the only significant effect on the analysis is in the Higgs
sector \cite{susyrad}. There is a light ($M < 150$ GeV) scalar which
acts like
a light standard model Higgs
(as far as radiative corrections are concerned), and
(typically) the other Higgs particles and superpartners do not contribute
significantly. Thus, for the MSSM we will take 60 GeV $< \mbox{$M_H$} <$ 150 GeV with a
central of \mbox{$M_Z$} , yielding:
\begin{equation}
{\rm MSSM:} \ \ \ \ \
m_t = 134^{+23}_{-28} \pm 5 \ {\rm GeV}. \label{eq52b}
\end{equation}
For \mbox{$M_H$} \ a free parameter in the
range 60 -150 GeV, one obtains $m_t = 131^{+31}_{-28}$
GeV, with the best fit
for \mbox{$M_H$} \ = 60 GeV.
The data also yield an indirect {\it lower} limit on $m_t$ (Figure
\ref{mtop}). For $M_H$ = 60 GeV
one obtains $m_t > $ 95(83) GeV at 90(95)\% CL.
The corresponding limits are 118(108) GeV for $M_H = 300$ GeV and
138(129) GeV for $M_H = 1000$ GeV. The lower bound is comparable to the
direct CDF limit $m_t > 91$ GeV (95\% CL) \cite{v7}. However, it is more
general in that it applies even for nonstandard $t$ decay modes, for which
the direct lower limit is $\sim 60$ GeV.
The data will not significantly constrain $M_H$ until $m_t$ is
known separately. At present the best fit occurs
for lower values of
$M_H$, but the change in $\chi^2$ between $M_H$ = 60 and 1000 GeV is
only 0.6. From Figure \ref{chis} is it obvious that if $m_t$ is measured
directly to within 5-10 GeV it may be possible to constrain $M_H$,
particularly if $m_t$ is in the lower part of the allowed range. This
is further illustrated in Figure \ref{mtdirect}, in which are displayed
the 68 and 90\% CL \mbox{$M_H$} \ ranges that could be obtained from present data
if \mbox{$m_t$} \ were known to 10 GeV.
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxhiggs.ps}{0.55}
\caption[]{68 and 90\% CL \mbox{$M_H$} \ ranges that could be obtained from present data
if \mbox{$m_t$} \ were known by direct measurement to $\pm$ 10 GeV as a function of the
central value of \mbox{$m_t$}.} \label{mtdirect} \end{figure}
Assuming the standard model, one therefore concludes 91~GeV~$< m_t <
197$~GeV at 95\% CL. In
most cases, the effect of new physics is to
{\em strengthen} the upper bound rather than weaken it. The obvious
question is, why is $m_t$ so large (or why are the other fermion
masses so small)?
Note that the value of \mbox{$m_t$} \ considered here is the position of
the pole in the $t$ propagator (not the running mass). It should
coincide (with a theoretical ambiguity of
$\sim $ 5 GeV) with the kinematic mass
relevant for the production of the $t$ quark at hadron colliders.
For the weak angle one obtains (in the standard model)
\begin{eqnarray} \sin^2\theta_W & = & 0.2267 \pm 0.0024 \nonumber \\
\sin^2\hat{\theta}_W (M_Z) & = & 0.2328\pm 0.0007, \label{eq54} \end{eqnarray}
where the uncertainty is mainly from $m_t$.
The small uncertainty from $M_H$ in the range 60 - 1000~GeV is
included in the errors in (\ref{eq54}).
The corresponding value in the MSSM is
$\sin^2\hat{\theta}_W (M_Z)
= 0.2326\pm 0.0006$.
Of course, $\sin^2
\hat{\theta}_W (M_Z)$ is much less sensitive to $m_t$ and $M_H$ than
$\sin^2\theta_W$. All of the values obtained from individual observables
are in excellent agreement with (\ref{eq54}). In particular, the
$\sin^2\hat{\theta}_W$ values obtained assuming $m_t =
150^{+19}_{-24}$~GeV and 60~GeV~$<M_H < 1000$~GeV are shown in
Table~\ref{stab} and in Figure~\ref{sq}. The agreement is
remarkable.
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxxq.ps}{0.7}
\caption[]{$\sin^2\hat{\theta}_W(M_Z)$ obtained from various observables
assuming $m_t = 150^{+19}_{-24}$~GeV, 60~$< M_H < 1000$~GeV.}
\label{sq}
\end{figure}
One can also extract the radiative correction parameter
$\Delta r$ (eqn. (\ref{mwz})). One finds
\begin{equation} \Delta r = 0.049 \pm 0.008 \label{eq55} \end{equation}
compared to the expectation $0.0626 \pm 0.0009 (0.0273)$ for $m_t =
100(200)$, $M_H = 300$. Similarly, in the
$\overline{\rm MS}$ scheme, one finds
\begin{equation} \Delta \hat{r}_W = 0.069 \pm 0.006, \end{equation}
compared with the expectation $0.0696 \pm 0.0009 (0.0723)$.
The hadronic $Z$ width depends on the value of \mbox{$\alpha_s(M_Z)$}. The quoted results
use the value $0.12 \pm 0.01$ obtained from $Z$-decay event topologies
and low energy data \cite{v9}. One can also obtain a value of \mbox{$\alpha_s(M_Z)$} from the
hadronic widths, and in particular from $R$, which is insensitive to $m_t$. A
fit to all $Z$-pole and other data (but not including the constraint \mbox{$\alpha_s(M_Z)$} =
$0.12 \pm 0.01$) to
\mbox{$\alpha_s(M_Z)$}, \mbox{$\sin^2\hat{\theta}_W(M_Z)\ $}, and $m_t$ yields \mbox{$\alpha_s(M_Z)$} = $0.130 \pm 0.009$, which is consistent
but slightly above the other determinations. From
$M_Z, \Gamma_Z, R, \sigma^h_p, $ and $\Gamma_{b \bar{b}}$ only,
one finds the higher value $0.135 \pm 0.011$.
When \mbox{$\alpha_s(M_Z)$} = $0.12 \pm 0.01$ is
included as a separate constraint in the fit to all data
one obtains the average $\mbox{$\alpha_s(M_Z)$} = 0.126 \pm 0.007$.
These values are listed in Table \ref{als}, along with
the most important low energy determinations \cite{v9}. There is a
slight tendency for higher values from the $Z$-pole data, but given
the uncertainties (which are usually dominated by
theoretical errors) there is no real discrepancy.
\begin{table}\centering \small \def\baselinestretch{1} \normalsize
\begin{tabular}{|cc|} \hline
\mbox{$\alpha_s(M_Z)$} & source \\
\hline
$0.130 \pm 0.009$ & precision $Z$-pole and low energy \\
$0.135 \pm 0.011$ & $M_Z, \Gamma_Z, R, \sigma^h_p, \Gamma_{b \bar{b}}$
\\ \hline \hline
$0.123 \pm 0.005$ & event topologies \cite{v9} \\
$0.118 \pm 0.005$ & $\tau$ decays \cite{v9} \\
$0.112 \pm 0.005$ & deep inelastic scattering (DIS) \cite{v9} \\
$0.113 \pm 0.006$ & $\Upsilon,\ J/\psi$ \cite{v9} \\ \hline
$0.12 \pm 0.01$ & event topologies, $\tau$, DIS,
$\Upsilon,\ J/\psi$ \\ \hline \hline
$ 0.126 \pm 0.007$ & combined \\
\hline
\end{tabular}
\caption[]{Values of \mbox{$\alpha_s(M_Z)$} from indirect precision data,
event topologies, low energy data, and all data.}
\label{als}
\end{table}
The value of \mbox{$\alpha_s(M_Z)$} \ from the precision experiments is strongly
anticorrelated with \mbox{$m_t$}, as can be seen in Figure \ref{alsmt}.
In particular, larger \mbox{$m_t$} \ corresponds to smaller \mbox{$\alpha_s(M_Z)$}, in
better agreement with the low energy data.
\begin{figure}
\small \def\baselinestretch{1} \normalsize
\postscript{xxalsmt.ps}{0.55}
\caption[]{90\% CL allowed region in \mbox{$\alpha_s(M_Z)$} and \mbox{$m_t$} \ from a combined
fit to precision $Z$-pole and other data (but not including event
topology and low energy determinations of \mbox{$\alpha_s(M_Z)$}).}
\label{alsmt}
\end{figure}
\section{Implications for Grand Unification}
These results are in excellent agreement with the predictions
of grand unification in the minimal supersymmetric extension of the
(MSSM), but not in the simplest (and most predictive)
non-supersymmetric GUT (SM) \cite{recent}.
In particular, using
$\alpha^{-1}(M_Z) = 127.9 \pm 0.2$ and \mbox{$\alpha_s(M_Z)$} = 0.12 $\pm$ 0.01
one predicts
\begin{eqnarray} \sin^2\hat{\theta}_W (M_Z) & = & 0.2334\pm
0.0025 \pm 0.0025\ {\rm (MSSM)}, \nonumber \\
\sin^2\hat{\theta}_W (M_Z) & = & 0.2100\pm 0.0025 \pm
0.0007 \ {\rm (SM)}, \label{eqgut} \end{eqnarray}
where the first uncertaintly is from $\alpha_s$ and $\alpha^{-1}$,
and the second is an estimate of theoretical uncertainties
from the superspectrum, high-scale thresholds, and possible
non-renormalizable operators \cite{ipol}.
The MSSM prediction is in excellent agreement with the
experimental value $0.2326 \pm 0.0006$, while the SM prediction is
in conflict with the data.
Because of the large uncertainty in \mbox{$\alpha_s(M_Z)$}, it is convenient
to invert the logic and use the precisely known
$\alpha^{-1}$ and $\sin^2\hat{\theta}_W (M_Z)$
to predict \mbox{$\alpha_s(M_Z)$}:
\begin{eqnarray} \mbox{$\alpha_s(M_Z)$} & = & 0.125 \pm 0.002 \pm 0.009 \ {\rm (MSSM)}, \nonumber \\
\mbox{$\alpha_s(M_Z)$} & = & 0.072 \pm 0.001 \pm 0.001 \ {\rm (SM)}, \label{eqgut1} \end{eqnarray}
where again the second error is theoretical.
It is seen that the
SUSY case is in excellent agreement with the experimental
\mbox{$\alpha_s(M_Z)$} = 0.12 $\pm$ 0.01, while the
simplest ordinary GUTs are excluded
(this is completely independent of proton decay).
The unification slightly prefers larger values of \mbox{$\alpha_s(M_Z)$}, as suggested
by the $Z$-pole data, but the theoretical uncertainties are
comparable to the error on the observed \mbox{$\alpha_s(M_Z)$} (which is also
dominated by theory). Proton decay is strongly
suppressed in SUSY-GUTs. Perhaps, the coupling constants will
indeed prove to be the ``first harbinger of supersymmetry''
\cite{amaldi}.
\section{Conclusions}
\begin{itemize}
\item There is no evidence for any deviation from the standard model.
\item \mbox{$\overline{\rm{MS}}\ $}: $\mbox{$\sin^2\hat{\theta}_W(M_Z)\ $} = 0.2328 \pm 0.0007$
\item On-shell: $\mbox{$\sin^2\theta_W\,$} \equiv 1 - \mbox{$M_W$}^2/\mbox{$M_Z$}^2 = 0.2267 \pm 0.0024$,
where the uncertainties are mainly from \mbox{$m_t$}.
\item In the standard model one predicts:
$\mbox{$m_t$} = 150^{+19 + 15}_{-24 - 20}$ GeV, where the central value assumes
\mbox{$M_H$} = 300 GeV and the second uncertainty is for \mbox{$M_H$} $\rightarrow$ 60 GeV ($-$) or
1 TeV (+).
\item In the
MSSM $\mbox{$m_t$} = 134^{+23}_{-28} \pm 5$ GeV, where the difference
is due the light Higgs scalar expected in the MSSM.
\item Precision data yield the 95\% CL constraints
\begin{equation} 83 \ {\rm GeV} < \mbox{$m_t$} < 197 \ {\rm GeV}, \end{equation}
where the lower (upper) limits are for \mbox{$M_H$} \ = 60 (1000) GeV.
The lower limit is valid for any decay mode, and is to be compared with the
direct CDF limit $\mbox{$m_t$} > 91$ GeV, which assumes canonical decays.
\item There is no significant
constraint on \mbox{$M_H$} \ until \mbox{$m_t$} \ is known independently.
\item Precision $Z$-pole and low-energy data yield the indirect
result $\mbox{$\alpha_s(M_Z)$} = 0.130 \pm 0.009$, in reasonable agreement with the
value $0.12 \pm 0.01$ obtained from jet event topologies and low
energy direct determinations.
\item
The low energy couplings are in excellent agreement with
the predictions of supersymmetric grand unification, but not with
the simplest (and most predictive) non-supersymmetric grand unified
theories.
\item The precision data place stringent limits on many types of new
physics into the TeV range.
\item In the future, precision electroweak experiments will be a useful
complement to high energy colliders.
\end{itemize}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Q: Pullparser doesn't get inputStream I am trying to get the data from an xml response, the data is there, I have printed it in the log converting it to a String, but when I try to parse it with the xml parser, it doesn't work.
Relevant code:
URL myUrl = new URL(_url);
URLConnection connection = myUrl.openConnection();
HttpURLConnection httpConnection = (HttpURLConnection)connection;
httpConnection.setReadTimeout(10000);
httpConnection.setConnectTimeout(15000);
//>>
httpConnection.setRequestMethod("POST");
httpConnection.setDoOutput(true);
httpConnection.setDoInput(true);
httpConnection.connect();
int responseCode = httpConnection.getResponseCode();
Log.d("OpenDataStream", "response: " + responseCode);
if(responseCode== HttpURLConnection.HTTP_OK)
{
InputStream inStream = httpConnection.getInputStream();
//Log.v("OpenDataStream", readIt(inStream, 500)); << This method shows me that data is getting retrieved
switch (process)
{
case LOGIN:
{
ProcessLogin(inStream, responseCode);
}
}
}
...
public boolean ProcessLogin(InputStream myInStream, int responseCode)
{
Player myPlayer = new Player();
XmlPullParserFactory myXMLFactory;
try
{
myXMLFactory = XmlPullParserFactory.newInstance();
myXMLFactory.setNamespaceAware(true);
XmlPullParser pullParser = myXMLFactory.newPullParser();
pullParser.setInput(myInStream, null);
int eventType = pullParser.getEventType();
while(eventType!= XmlPullParser.END_DOCUMENT)
{
Log.w("PULLPARSER", "eventType: " + eventType);
Log.w("PULLPARSER", "pullParser: " + pullParser.getName());
if(eventType==XmlPullParser.START_TAG && pullParser.getName().equals("response"))
{
Log.w("PULLPARSER", "Starting doc");
eventType= pullParser.next();
while(!(eventType==XmlPullParser.END_TAG && pullParser.getName().equals("response")))
{
Log.w("PULLPARSER", "Starting response");
if(eventType==XmlPullParser.START_TAG)
{
if(pullParser.getName().equals("user_id"))
{
myPlayer.setIdUser(pullParser.nextText());
}
}
eventType=pullParser.next();
}
}
eventType=pullParser.next();
}
Log.v("PullParser" , "Finished");
return new SharedPrefController(myContext).savePlayer(myPlayer);
}
catch (XmlPullParserException ex)
{
Log.w("PULLPARSER","XML PUll Parser Exception",ex);
}
catch (IOException ex)
{
Log.w("PULLPARSER","IO Exception",ex);
}
return false;
}
And what I get in the log is this:
D/OpenDataStream﹕ response: 200
V/OpenDataStream﹕ <?xml version="1.0" encoding="utf-8"?>
<response><user_id>6386</user_id><user_email>test@test.com</user_email><user_name>Test</user_name><user_lastname>Tests</user_lastname><user_status>2</user_status></response>
W/PULLPARSER﹕ pullParser: null
W/PULLPARSER﹕ eventType: 1
V/PullParser﹕ Finished
Any idea what is wrong?
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,771
|
Q: sleep() causes program to hang indefinitely I'm working on a mail sending python script in Linux.
I can send 10 messages, then need to wait 5 - 10 minutes before sending again.
I am executing the following code inside of a while() loop.
#Pausing so the MailServer doesn't complain
delay = randint(300,600)
now = datetime.now()
current_time = now.strftime("%H:%M:%S")
print(current_time + " sleeping " + str(timedelta(seconds=delay)))
sleep(delay)
This causes the program to hang whether it is at the beginning of the loop or end.
While stepping through with pdb, I can't detect the issue.
I am not using threading (just because I haven't learned how yet).
I'm open to ideas.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 7,818
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The total county population amounted to ,, or In counties are included all towns not constituted municipal boroughs; but, on the other hand, the people living in many of the boroughs can hardly be called town population. The population in boroughs was , persons, or For every persons resident in counties in there were In the counties had , persons, and the boroughs ,, or, in other words, for every persons in counties, 84 were residents of the boroughs. Thus it will be seen that the proportion of the town to the county population was greater in than in The Cities of Auckland, Christchurch, and Dunedin have considerable suburbs.
The suburban population of Wellington is comparatively small. The following gives-the names and populations of the several localities, as at the date of the census , which might fairly be termed suburbs of the four principal cities at that time :—. The increase of population for ten years prior to the census of at the four chief centres, with their suburbs, was:—.
Thus the two principal cities of the North Island are found to have progressed between and at a greater rate than those of the South Island, and Wellington in particular to have developed at nearly three times the rate of Dunedin, and considerably faster than Christchurch. Besides the boroughs, there were 40 town districts including the special town district of Rotorua. Two only of these, Rotorua and Hampstead, have more than 1, inhabitants. A list of these town districts is subjoined, with populations, as in :—. In addition to the boroughs and town districts above referred to, the census results showed for throughout the colony no less than places of the nature of townships, villages, or small centres without boundaries.
The populations so brought out may not in all cases be locally considered strictly accurate, even for the census-date, or. The question of including, with the nucleus, more or less of the surrounding country, is dealt with in different ways by the Sub-Enumerators. But even if objections are raised in a few cases, a great deal of the information now given is held to be valuable, and there is in for every place some kind of centre.
The names and populations of the colony were, in April, :—. Only three of these islands had a population over persons at last census. Since the boundaries of the colony have been extended to include the Cook and certain other Pacific islands, the population of which is shown elsewhere.
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The numbers of persons on shipboard at the various ports of the colony were 4, persons 4, males and females. This number does not include persons—officers and crews of two British men-of-war in ports of New Zealand on the census night. The gradual equalisation of the numbers of the sexes and growing density of population and dwellings in the colony are exhibited below.
The proportion of persons to a square mile increased from 7. In there were 6. Since the proportions at the different census years were:—. Of the different provincial districts, the most thickly populated is Wellington, and the one with the fewest people in proportion to size is Marlborough.
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The table below shows the area of the provincial districts, and the average number of persons to a square mile:—. The population in the boroughs, amounting to ,, gives an average of 1, persons to every square mile in these towns. The people lay closest in the City of Auckland, where they are Since the census of the area of the City of Wellington has been increased by the inclusion within its boundaries of the Borough of Melrose, containing a large extant of unoccupied land.
This circumstance reduced the average density of the population of the city to 8. Outside the boroughs and excluding persons on shipboard the population shows an average of 4. At the census of the number of females to males was found to be From this last year the proportion of females steadily increased to at the census of , but owing to a larger immigration between and , mainly of males, the proportion of females was reduced in the latter year to The numbers of the sexes are shown to be gradually becoming equal as time advances, with the exception above indicated.
The proportion of females to males was. The centesimal increase of the population is found to be greater in respect of the males than the females in all the provincial districts except Otago and Taranaki. The dwellings in the colony on the census night numbered ,, of which , were occupied houses, 11, unoccupied, and. Besides these there were 5, tents or dwellings with canvas roofs. The average number of persons to an inhabited dwelling has increased from 4. The average number of inhabited dwellings to a square mile was only 0. Of , dwellings, , were built of wood, iron, or lath and plaster, and 8, of brick, stone, or concrete.
There were also 1, cob or sod houses, 42 of raupo, besides 5, tents and dwellings with canvas roofs, and 3, houses and huts of miscellaneous materials. The inhabitants of the several classes of dwellings were distributed as under at the last two censuses:—.
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With an increase of population amounting to The following are the proportions of the population excluding Chinese and Maoris residing in the different classes of dwelling at the last five census periods:—. The number of brick, stone, or concrete houses increased between and from 7, to 8,, or at the rate of The accommodation in the dwellings of the people has improved greatly in the time.
This is exhibited by the following comparative table:—.
It will be noticed that the increase lies mainly in the houses of five to six and more than six rooms, which are more numerous by 24, than in ; whereas the dwellings of one to four rooms including tents and houses of which the rooms were not stated only increased by 1, in five years. The actual number of houses was greatest in the group of those having five and six rooms 68, , while the houses of three to four rooms numbered 47, Of houses of more than six rooms, the number was 45, Of the four chief cities, Wellington shows the greatest number of persons to a house since Auckland shows an equal average for The proportion in Wellington for is higher than that which obtained in , , and in the same city, but lower than in , when the average was 5.
At Christchurch and Dunedin the proportions fall regularly from At Auckland the proportion is highest for and lowest for For the whole colony the average number of persons to each inhabited dwelling was 4. The succeeding statement gives the number of inhabited and uninhabited dwellings at each of the five past census dates:. The number of uninhabited dwellinghouses in was 11, being in the proportion of 1—27 to each of population , as against 10, in , and 8, in In the counties excluding the boroughs contained 7, uninhabited houses, or 1.
The number of houses in course of erection at the census of was 1,, an increase of on that of the census of The numbers of houses being built and uninhabited at the three last census periods are shown. O F the various religious denominations, the Church of England has most adherents in the colony.
They numbered , at the date of the census; or, including 1, Protestants not more specifically described, , persons, being The Presbyterians numbered , persons or The Methodists were 89,, or Of other denominations, the Baptists, of whom there were 17, persons, returned 2 per cent, of the total population. The numbers and percentages for five censuses are given in tabular form, so as to allow of the degree relatively to the population being observed —.
Here the proportion belonging to the Church of England is shown to have increased from 40 per cent, in to Baptists declined from 2. Lutherans are fewer in proportion to the total at each succeeding census, while the Salvation Army increased from 0. Roman Catholics and Catholics undefined formed practically 14 per cent. The proportion of Buddhists and Confucians diminishes with the number of Chinese in the colony. In the percentage of persons objecting to state their religion was 3. A full statement of the particulars of all denominations as at the censuses of and is given, with the numerical and centesimal increase or decrease in each case.
The Complete descriptions will be published in the census volume. It will be seen by the table that, of the larger Protestant denominations, the Church of England increased from , to ,, or Roman Catholics added 17, to their number, being an increase of Hebrews were 1, in , and 1, in , a difference of Spiritualists progressed, the numbers being 1, and , an increase of more than per cent. Freethinkers also increased by 9. While the number of males is found to be greater than that of females in the Church of England, Presbyterian, Roman Catholics, and sundry other religious denominations, the contrary result is found in the following cases, the proportions per cent, being—.
O F the population exclusive of Maoris , persons , all but were described as to birthplace on the census schedules. The number of the New-Zealand-born was ,, and of those born in Australia, Tasmania, and Fiji, 47,; making , born in Australasia. The New-Zealand-born increase in proportion to the whole with every successive census.
In , Besides these there were 4, persons born in other British possessions. Summarising these results, it is found that , of the population, or There remained 19, persons born in foreign countries or 2. Helena, and in British West Indies. Out of 19, persons born abroad 14, were born in Europe, 4, of these were born-in Germany and possessions, 2, in Denmark and possessions, 2, in Austria-Hungary, 1, in Sweden, 1, in Norway, in France, in Italy, in Russia, in Switzerland.
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Outside of Europe 2, were found to have been born in the United States, and North America not more specifically defined, also in South America. Those born in foreign parts of Asia numbered 3,, in which are included 2, born in China 53 of European blood , and in Syria all Asiatics.
Only persons were returned as born in Africa, outside of the British possessions in that continent. The foreign-born decrease at successive censuses having been 2. The New-Zealand-born increased from , in to ,, or at the rate of The numbers born in the United Kingdom increased altogether by 3, in the quinquennium. The numbers of the Australian-born are found to have increased for each State.
The number born in New South Wales, living in New Zealand, was 6, in the year , but 13, in , an increase of There were 12, persons in this colony in born in Victoria, but 19, at last census, or an increase of New Zealand also gained on the number born in Queensland, there being 3, in , against 1, in , or The numbers of those born in Germany, the Netherlands, Portugal, and China all decreased since The following table gives full details, and exhibits under the head of allegiance the number of British and foreign subjects in New Zealand:—.
I N connection with this subject it is desirable to consider first the numbers for eight groups of important age-periods which are given below, and compared with those of three previous censuses.
The table below is worthy of notice as exhibiting a much stronger position than obtained in as to component parts of the population. In the population under five years had fallen from 86, persons in to 83,, a loss of 3,, or 4. In those under five had increased to the number of 83,, being more than in ,. By the children under five had increased to 86,, being 3, more, or 3. The effect of the deficiencies had been overcome by , when the census showed , children under five, being an increase of 15,, or The number at the group five to ten was 90,, or an increase of 4, in the quinquennium, and at the ten to fifteen the number was 86,, an increase of 1,, or 1.
Thus the minus sign has been eliminated in regard to the three groups comprising the population under fifteen years of age. The group 15 to 21 years still suffers from losses in the previous periods, but an increase of 2, over the number for is nevertheless exhibited, or 2. Again at 40 to 55 there are , persons, an increase of There is nothing remarkable about the numbers at the groups 55—65 and 65— The number of children under one year, and the total population at all ages, according to the results of three censuses, was:—.
Thus, in , with a population of , persons, there were 17, children under one year, against 22, children of that age in to a population of , persons. The births registered in were 18,, against 23, in , and the birth-rate, which was Deducting 1,, the number of deaths of children under one year registered in , from 23,, the number of births for that year, leaves 22,, or within of the living children under one year at the time of the last census.
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|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,465
|
Spotlight on… Non Fiction
June 17, 2020 By IT 2 Comments
Hello booklovers,
Our 2020 online programme is packed this year! So far in our spotlight series, we've brought you the delights of fiction, thrillers, crime fiction and local writing. Today we are celebrating non-fiction. It is often an extremely underrated genre, pushed aside to the memories of dusty textbooks in the corners of classrooms.
Communicating facts and information in a interesting and engaging way is a tricky business but that is where the genre of non fiction scores highly. The variety included within this umbrella term is endless: the 'non fiction' section of any bookshop or library will bring you history, philosophy, science, gardening, cooking, biography, crafts, medicine and art, to name a few. Indeed, Groucho Marx was such a fan that he said "I find television very educating. Every time somebody turns on the set, I go into the other room and read a book."
This year we a re holding virtual host to the authors of two excellent non-fiction works, including Nick Holland, author of the biography of Anne Brontë Crave the Rose: Anne Brontë at 200.
'Crave the Rose takes a fresh look at Anne, revealing a woman whose work was more radical than that of her sisters, and which is therefore as relevant today as it has ever been. Alongside a biography of Anne's remarkable, but tragically short, life, this book contains a comprehensive selection of first-person encounters with the Brontës from 19th-century newspapers and archives, giving a fresh insight into the real character of Anne and her family. Also contained exclusively within this landmark book is a newly-discovered essay by Anne Brontë – what may well be the last words that she ever wrote, in print here for the first time.'
Nick will be joining the Felixstowe Book Festival book club on Friday the 26th of June in order to celebrate Anne's 200th birthday. Along with discussion of Crave the Rose, our literary thoughts will also be drawn towards Nick's other work, In Search of Anne Brontë. We are very much looking forward to what promises to be a fascinating evening.
We will also be joined by Martin Bell in a pre-recorded interview with Richard Walker. A familiar face as a former MP and BBC War correspondent, Martin will be chatting about his book War and the Death of News: From Battlefield to Newsroom – My Fifty Years in Journalism.
'A smoke bomb went off. Then shots were fired from buildings overlooking the square… The camera had a BBC News sign on it. Someone cried out from the crowd: 'You are the world, you are the world, you have to tell what they are doing to our people.'
From Vietnam to Iraq, Martin Bell has seen how war has changed over the last fifty years, neither fought nor reported the way it used to be. Truth is degraded in the name of balance and good taste, reports are delivered from the sidelines, and social media, with rumours and unverifiable videos, has ushered in a post-truth world.
As modern news increasingly seeks to entertain first and inform second, the man in the white suit provides a moving account of all he has witnessed throughout his career and issues an impassioned call to put the substance back into reporting.'
Martin will bring us an insight into his varied and, at times, dramatic career. From the junior reaches of the BBC Norwich newsroom to the battlefields of Vietnam, El Salvador, Kuwait, Iraq and Bosnia, Martin Bell became the voice of foreign news, and eventually became the news himself when he was wounded by mortar fire on camera in Bosnia.
Make sure you tune in for an informative and sobering event.
We hope you have enjoyed our celebration of non fiction – our varied programme just shows how far the genre can reach.
Bookish best,
The Felixstowe Book Festival Team
Jacqueline Collier says
Can you please tell me how and when we can watch these especially Martin Bell who we have heard many times as members of the Adrian Bell Society, sadly now disbanded? Many thanks. Jacqueline Collier
Hello – thank you for getting in touch.
Our full programme contains a mix of live events and pre-recorded events which you can check out here:
https://felixstowebookfestival.co.uk/events/categories/adult-programme
We have created a guide to how to view our live events on our website – you will find it under the Programme button in the top right corner of your screen.
All other videos can be viewed via our Facebook page or via the Online section of our website which you can find a little further along from the programme button.
All the best,the FbF Team
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 2,886
|
Q: How is memory allocated in c , and why diffrence between two contiguous is always 4? #include <stdio.h>
#include <cs50.h>
int main (void)
{
int *x;
x = malloc(sizeof(long long)*3);
scanf("%i %i %i",x, (x+1), (x+2));
printf("%i\t %i\t %i\n",(int)x, (int)(x+1), (int)(x+2));
printf("%i\t %i\t %i\n",*x, *(x+1), *(x+2));
free(x);
}
The output of this program for input 12,2,3 is :
43171856 43171860 43171864
12 2 3
so, my question is why difference between address is 4 in each case ,
and if *x points to 43171856 then *(x+1) should point to 4317185 not 43171860? sizeof(long long) is also 8 bytes , so how allocated memory allocates 8 bytes between those 4 bytes between 43171856 and 43171860.
A: First of all, in your code
printf("%i\t %i\t %i\n",(int)x, (int)(x+1), (int)(x+2));
invokes implementation defined behaviour, as you're trying to cast a pointer to integer.
If you want to print pointers
*
*use %p format specifier
*cast the argument to void *.
That said, pointer arithmetic honors the data type. You had declared x to be a pointer to int, so any pointer arithmetic will be based on sizeof(int), whatever that evaluates to in your platform.
Quoting C11, chapter §6.5.6/P8, (emphasis mine)
When an expression that has integer type is added to or subtracted from a pointer, the
result has the type of the pointer operand. If the pointer operand points to an element of
an array object, and the array is large enough, the result points to an element offset from
the original element such that the difference of the subscripts of the resulting and original
array elements equals the integer expression. In other words, if the expression P points to
the i-th element of an array object, the expressions (P)+N (equivalently, N+(P)) and
(P)-N (where N has the value n) point to, respectively, the i+n-th and i−n-th elements of
the array object, provided they exist. [....]
In your code, you wrote
x = malloc(sizeof(long long)*3);
which is erroneous. In this case, you may be on the safer side, as sizeof(long long) is >= sizeof(int), but that is not true for any arbitary type.
*
*Best case: You'll end up wasting memory.
*Worst case: You'll end up accessing out of bound (invalid) memory.
A better and preferred way to write this would be
x = malloc(sizeof*x * 3); //sizeof is not a function :)
and then, check for malloc() success. This allocates the exact amount of memory required, no more, no less.
A: This is one of the really confusing bits of C: x+1, when x has a pointer type, increments the numeric value of x by sizeof(*x), not by 1.
It has to be that way, because, for any pointer type T *x, x+1 is the same as &x[1]. &x[1] is the address of the second T in the pseudo-array pointed to by x. Therefore the numeric value of x+1 must be equal to the numeric value of x plus sizeof(T), which in your case is 4.
malloc, meanwhile, doesn't know that you passed it 3*sizeof(long long). It sees malloc(24) and it gives you 24 bytes, which (on your platform) is six ints. You are using only the first three, which is fine, it just wastes a little memory. You probably meant to write 3*sizeof(int).
A: You are using an int* which usually is 32 bits and thus 4 bytes.
Try using a long long *x instead?
You have as you stated allocated 8*3 bytes byt are only using 4*3 bytes of them..
Each offeset i.e (x+1) is only offseted the size of an int.
A: C 2011 Online Draft
6.3.2.1 Lvalues, arrays, and function designators
...
3 Except when it is the operand of the sizeof operator, the _Alignof operator, or the
unary & operator, or is a string literal used to initialize an array, an expression that has
type ''array of type'' is converted to an expression with type ''pointer to type'' that points
to the initial element of the array object and is not an lvalue. If the array object has
register storage class, the behavior is undefined.
Under most circumstances, an expression of type "array of T" will be converted ("decay") to an expression of "pointer to T", and the value of the expression will be the address of the first element of the array.
AFAIK, the _Alignof operator clause is a mistake in the online draft that was corrected in the official™, not freely-available standard, which is why it's struck out in the quote above.
6.5.2 Array subscripting
...
2 A postfix expression followed by an expression in square brackets [] is a subscripted
designation of an element of an array object. The definition of the subscript operator []
is that E1[E2] is identical to (*((E1)+(E2))). Because of the conversion rules that
apply to the binary + operator, if E1 is an array object (equivalently, a pointer to the
initial element of an array object) and E2 is an integer, E1[E2] designates the E2-th
element of E1 (counting from zero).
Given an array a of type T and an integer i, the expression a[i] is equivalent to (defined as) *(a + i) - given an address a, offset i elements of type T (not bytes) from that address and dereference the result.
If a is an array or pointer expression and i is an integral expression, then a[i] and i[a] will yield the same result.
6.5.6 Additive Operators
...
8 When an expression that has integer type is added to or subtracted from a pointer, the result has the type of the pointer operand. If the pointer operand points to an element of an array object, and the array is large enough, the result points to an element offset from the original element such that the difference of the subscripts of the resulting and original array elements equals the integer expression. In other words, if the expression P points to
the i-th element of an array object, the expressions (P)+N (equivalently, N+(P)) and
(P)-N (where N has the value n) point to, respectively, the i+n-th and i−n-th elements of the array object, provided they exist. Moreover, if the expression P points to the last
element of an array object, the expression (P)+1 points one past the last element of the array object, and if the expression Q points one past the last element of an array object, the expression (Q)-1 points to the last element of the array object. If both the pointer operand and the result point to elements of the same array object, or one past the last element of the array object, the evaluation shall not produce an overflow; otherwise, the
behavior is undefined. If the result points one past the last element of the array object, it shall not be used as the operand of a unary * operator that is evaluated.
If p is a pointer to an object of type T, then the expression p + 1 yields the address of the next object of that type. If sizeof (T) is 1, then p + 1 adds 1 to the address. If sizeof (T) is 4, then p + 1 adds 4 to the address.
Similarly, the expression ++p and p++ advance p to point to the next object of type T.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,981
|
package com.planet_ink.coffee_mud.core.intermud;
import com.planet_ink.coffee_mud.core.intermud.imc2.*;
import com.planet_ink.coffee_mud.core.intermud.i3.packets.*;
import com.planet_ink.coffee_mud.core.intermud.i3.persist.*;
import com.planet_ink.coffee_mud.core.intermud.i3.server.*;
import com.planet_ink.coffee_mud.core.intermud.i3.net.*;
import com.planet_ink.coffee_mud.core.intermud.*;
import com.planet_ink.coffee_mud.core.*;
import com.planet_ink.coffee_mud.core.CMSecurity.DisFlag;
import com.planet_ink.coffee_mud.core.collections.*;
import com.planet_ink.coffee_mud.Libraries.interfaces.*;
import com.planet_ink.coffee_mud.Abilities.interfaces.*;
import com.planet_ink.coffee_mud.Areas.interfaces.*;
import com.planet_ink.coffee_mud.Behaviors.interfaces.*;
import com.planet_ink.coffee_mud.CharClasses.interfaces.*;
import com.planet_ink.coffee_mud.Commands.interfaces.*;
import com.planet_ink.coffee_mud.Common.interfaces.*;
import com.planet_ink.coffee_mud.core.interfaces.*;
import com.planet_ink.coffee_mud.Items.interfaces.*;
import com.planet_ink.coffee_mud.Locales.interfaces.*;
import com.planet_ink.coffee_mud.MOBS.interfaces.*;
import com.planet_ink.coffee_mud.Races.interfaces.*;
import java.util.*;
import java.net.*;
@SuppressWarnings({"unchecked","rawtypes"})
public class IMudClient implements I3Interface
{
@Override public String ID(){return "IMudClient";}
@Override public String name() { return ID();}
@Override public CMObject newInstance(){try{return getClass().newInstance();}catch(final Exception e){return new IMudClient();}}
@Override public void initializeClass(){}
@Override public CMObject copyOf(){try{return (CMObject)this.clone();}catch(final Exception e){return newInstance();}}
@Override public String L(final String str, final String ... xs) { return CMLib.lang().fullSessionTranslation(str, xs); }
@Override public int compareTo(CMObject o){ return CMClass.classID(this).compareToIgnoreCase(CMClass.classID(o));}
@Override public boolean activate(){ return true;}
@Override public boolean shutdown(){ return true;}
@Override public void propertiesLoaded(){}
@Override public TickClient getServiceClient() { return null;}
public IMC2Driver imc2=null;
@Override
public void registerIMC2(Object O)
{
if(O instanceof IMC2Driver)
imc2=(IMC2Driver)O;
}
@Override
public void i3who(MOB mob, String mudName)
{
if(mob==null)
return;
if((!i3online())&&(!imc2online()))
return;
if((mudName==null)||(mudName.length()==0))
{
mob.tell(L("You must specify a mud name."));
return;
}
if(i3online()&&Intermud.isAPossibleMUDName(mudName))
{
mudName=Intermud.translateName(mudName);
if(!Intermud.isUp(mudName))
{
mob.tell(L("@x1 is not available.",mudName));
return;
}
final WhoPacket wk=new WhoPacket();
wk.type=Packet.WHO_REQUEST;
wk.sender_name=mob.Name();
wk.target_mud=mudName;
wk.who=new Vector();
try
{
wk.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
else
if(imc2online()&&(imc2.getIMC2Mud(mudName)!=null))
imc2.imc_send_who(mob.name(),imc2.getIMC2Mud(mudName).name,"who",mob.phyStats().level(),0);
else
{
mob.tell(L("'@x1' is not a mud name.",mudName));
return;
}
}
@Override
public boolean i3online()
{
return Intermud.isConnected() && (!CMSecurity.isDisabled(DisFlag.I3));
}
@Override
public boolean imc2online()
{
if((imc2==null)||(CMSecurity.isDisabled(DisFlag.IMC2)))
return false;
return imc2.imc_active==IMC2Driver.IA_UP;
}
@Override
public void imc2mudInfo(MOB mob, String parms)
{
if((mob==null)||(!imc2online()))
return;
if((parms==null)||(parms.length()==0)||(imc2.getIMC2Mud(parms)==null))
{
mob.tell(L("You must specify a mud name."));
return;
}
imc2.imc_send_who(mob.name(),imc2.getIMC2Mud(parms).name,"info",mob.phyStats().level(),0);
}
@Override
public void i3chanwho(MOB mob, String channel, String mudName)
{
if((mob==null)||(!i3online()))
return;
if((mudName==null)||(mudName.length()==0))
{
mob.tell(L("You must specify a mud name."));
return;
}
if((channel==null)||(channel.length()==0)||(Intermud.getRemoteChannel(channel).length()==0))
{
mob.tell(L("You must specify an InterMud 3 channel name."));
return;
}
if(!Intermud.isAPossibleMUDName(mudName))
{
mob.tell(L("'@x1' is an unknown mud.",mudName));
return;
}
mudName=Intermud.translateName(mudName);
if(!Intermud.isUp(mudName))
{
mob.tell(L("@x1 is not available.",mudName));
return;
}
final ChannelWhoRequest ck=new ChannelWhoRequest();
ck.sender_name=mob.Name();
ck.target_mud=mudName;
ck.channel=channel;
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
@Override
public void i3channelAdd(MOB mob, String channel)
{
if((mob==null)||(!i3online()))
return;
if((channel==null)||(channel.length()==0)||(Intermud.getLocalChannel(channel).length()==0))
{
mob.tell(L("You must specify an existing channel to add it to the i3 network."));
return;
}
final ChannelAdd ck=new ChannelAdd();
ck.sender_name=mob.Name();
ck.channel=channel;
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
@Override
public void i3channelListen(MOB mob, String channel)
{
if((mob==null)||(!i3online()))
return;
if((channel==null)||(channel.length()==0))
{
mob.tell(L("You must specify a channel name listed in your INI file."));
return;
}
if(Intermud.getLocalChannel(channel).length()==0)
{
if(Intermud.registerFakeChannel(channel).length()>0)
mob.tell(L("Channel was not officially registered."));
else
mob.tell(L("Channel listen failed."));
}
final ChannelListen ck=new ChannelListen();
ck.sender_name=mob.Name();
ck.channel=channel;
ck.onoff="1";
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
@Override
public void i3channelSilence(MOB mob, String channel)
{
if((mob==null)||(!i3online()))
return;
if((channel==null)
||(channel.length()==0)
||(Intermud.getLocalChannel(channel).length()==0))
{
mob.tell(L("You must specify an actual channel name."));
return;
}
if(Intermud.removeFakeChannel(channel).length()>0)
mob.tell(L("Unofficial channel closed."));
final ChannelListen ck=new ChannelListen();
ck.sender_name=mob.Name();
ck.channel=channel;
ck.onoff="0";
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
@Override
public void i3channelRemove(MOB mob, String channel)
{
if((mob==null)||(!i3online()))
return;
if((channel==null)||(channel.length()==0)||(Intermud.getRemoteChannel(channel).length()==0))
{
mob.tell(L("You must specify a valid InterMud 3 channel name."));
return;
}
final ChannelDelete ck=new ChannelDelete();
ck.sender_name=mob.Name();
ck.channel=channel;
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
@Override
public void i3tell(MOB mob, String tellName, String mudName, String message)
{
if(mob==null)
return;
if((!i3online())&&(!imc2online()))
return;
if((mudName==null)||(mudName.length()==0))
{
mob.tell(L("You must specify a mud name."));
return;
}
if((tellName==null)||(tellName.length()<1))
{
mob.tell(L("You must specify someone to talk to."));
return;
}
if((message==null)||(message.length()<1))
{
mob.tell(L("You must enter a message!"));
return;
}
if(i3online()&&Intermud.isAPossibleMUDName(mudName))
{
mudName=Intermud.translateName(mudName);
if(!Intermud.isUp(mudName))
{
mob.tell(L("@x1 is not available.",mudName));
return;
}
mob.tell(L("You tell @x1 '@x2'",tellName,message));
final TellPacket tk=new TellPacket();
tk.sender_name=mob.Name();
tk.sender_visible_name=mob.Name();
tk.target_mud=mudName;
tk.target_name=tellName;
tk.message=message;
if(mob.playerStats()!=null)
mob.playerStats().addTellStack("You tell "+tellName+" '"+message+"'");
try
{
tk.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
else
if(imc2online()&&(imc2.getIMC2Mud(mudName)!=null))
{
tellName=CMStrings.capitalizeAndLower(tellName)+"@"+imc2.getIMC2Mud(mudName).name;
mob.tell(L("^CYou tell @x1 '@x2'^?",tellName,message));
if(mob.playerStats()!=null)
mob.playerStats().addTellStack("You tell "+tellName+" '"+message+"'");
imc2.imc_send_tell(mob.name(),tellName,message,0,CMLib.flags().isInvisible(mob)?1:0);
}
else
{
mob.tell(L("@x1 is an unknown mud.",mudName));
return;
}
}
public void destroymob(MOB mob)
{
if(mob==null)
return;
final Room R=mob.location();
mob.destroy();
if(R!=null)
R.destroy();
}
@Override
public void i3channel(MOB mob, String channelName, String message)
{
if(mob==null)
return;
if((!i3online())&&(!imc2online()))
return;
if((channelName==null)||(channelName.length()==0))
{
mob.tell(L("You must specify a channel name."));
return;
}
if((message==null)||(message.length()<1))
{
mob.tell(L("You must enter a message!"));
return;
}
if(i3online()&&Intermud.getRemoteChannel(channelName).length()>0)
{
final ChannelPacket ck=new ChannelPacket();
ck.channel=channelName; // ck will translate it for us
ck.sender_name=mob.Name();
ck.sender_visible_name=mob.Name();
if((message.startsWith(":")||message.startsWith(","))&&(message.trim().length()>1))
{
String msgstr=message.substring(1);
final Vector<String> V=CMParms.parse(msgstr);
Social S=CMLib.socials().fetchSocial(V,true,false);
if(S==null)
S=CMLib.socials().fetchSocial(V,false,false);
CMMsg msg=null;
if(S!=null)
{
msg=S.makeChannelMsg(mob,0,channelName,V,true);
if((msg.target()!=null)&&(msg.target().name().indexOf('@')>=0))
{
final int x=msg.target().name().indexOf('@');
String mudName=msg.target().name().substring(x+1);
final String tellName=msg.target().name().substring(0,x);
if((mudName==null)||(mudName.length()==0))
{
mob.tell(L("You must specify a mud name."));
return;
}
if((tellName==null)||(tellName.length()<1))
{
mob.tell(L("You must specify someone to emote to."));
return;
}
if(!Intermud.isAPossibleMUDName(mudName))
{
mob.tell(L("'@x1' is an unknown mud.",mudName));
return;
}
mudName=Intermud.translateName(mudName);
if(!Intermud.isUp(mudName))
{
mob.tell(L("@x1 is not available.",mudName));
return;
}
ck.target_mud=mudName;
ck.target_name=tellName;
ck.target_visible_name=tellName;
}
else
if(msg.target()!=null)
{
ck.target_name=msg.target().name();
ck.target_visible_name=msg.target().name();
}
if((msg.target()!=null)&&(msg.targetMessage()!=null)&&(msg.targetMessage().length()>0))
ck.message_target=socialFixOut(CMStrings.removeColors(msg.targetMessage()));
if((msg.othersMessage()!=null)&&(msg.othersMessage().length()>0))
ck.message=socialFixOut(CMStrings.removeColors(msg.othersMessage()));
else
ck.message=socialFixOut(CMStrings.removeColors(msg.sourceMessage()));
}
else
{
if(msgstr.trim().startsWith("'")||msgstr.trim().startsWith("`"))
msgstr=msgstr.trim();
else
msgstr=" "+msgstr.trim();
ck.message=socialFixOut("<S-NAME>"+msgstr);
}
if((ck.target_name!=null)&&(ck.target_name.length()>0))
ck.type=Packet.CHAN_TARGET;
else
ck.type=Packet.CHAN_EMOTE;
}
else
ck.message=message;
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
else
if(imc2online()&&(imc2.getAnIMC2Channel(channelName)!=null))
{
int emote=0;
if((message.startsWith(":")||message.startsWith(","))&&(message.trim().length()>1))
{
message=message.substring(1);
final MOB mob2=CMClass.getFactoryMOB();
mob2.setName(mob.Name()+"@"+imc2.imc_name);
mob2.setLocation(CMClass.getLocale("StdRoom"));
final Vector<String> V=CMParms.parse(message);
Social S=CMLib.socials().fetchSocial(V,true,false);
if(S==null)
S=CMLib.socials().fetchSocial(V,false,false);
CMMsg msg=null;
if(S!=null)
{
msg=S.makeChannelMsg(mob,0,channelName,V,true);
if((msg.target()!=null)&&(msg.target().name().indexOf('@')>=0))
{
final int x=msg.target().name().indexOf('@');
final String mudName=msg.target().name().substring(x+1);
final String tellName=msg.target().name().substring(0,x);
if((mudName==null)||(mudName.length()==0))
{
mob.tell(L("You must specify a mud name."));
destroymob(mob2);
return;
}
if((tellName==null)||(tellName.length()<1))
{
mob.tell(L("You must specify someone to emote to."));
destroymob(mob2);
return;
}
if(imc2.getIMC2Mud(mudName)==null)
{
mob.tell(L("@x1 is not available.",mudName));
destroymob(mob2);
return;
}
}
if((msg.othersMessage()!=null)&&(msg.othersMessage().length()>0))
message=CMLib.coffeeFilter().fullOutFilter(null,CMClass.sampleMOB(),mob2,msg.target(),null,CMStrings.removeColors(msg.othersMessage()),false);
else
message=CMLib.coffeeFilter().fullOutFilter(null,CMClass.sampleMOB(),mob2,msg.target(),null,CMStrings.removeColors(msg.sourceMessage()),false);
if(message.toUpperCase().startsWith((mob.Name()+"@"+imc2.imc_name).toUpperCase()))
message=message.substring((mob.Name()+"@"+imc2.imc_name).length()).trim();
emote=2;
}
emote=1;
destroymob(mob2);
}
final IMC_CHANNEL c=imc2.getAnIMC2Channel(channelName);
imc2.imc_send_chat(mob.name(),c.name,message,c.level,emote);
}
else
{
mob.tell(L("You must specify a channel name."));
return;
}
}
@Override
public void i3locate(MOB mob, String mobName)
{
if(mob==null)
return;
if((!i3online())&&(!imc2online()))
return;
if((mobName==null)||(mobName.length()==0))
{
mob.tell(L("You must specify a name."));
return;
}
if(i3online())
{
final LocateQueryPacket ck=new LocateQueryPacket();
ck.sender_name=mob.Name();
ck.user_name=mobName;
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
if(imc2online())
imc2.imc_send_whois(mob.Name(),mobName,mob.phyStats().level());
}
@Override
public void i3pingRouter(MOB mob)
{
if(mob==null)
return;
if((!i3online())&&(!imc2online()))
return;
if(i3online())
{
final PingPacket ck=new PingPacket(I3Server.getMudName());
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
}
@Override
public void i3finger(MOB mob, String mobName, String mudName)
{
if(mob==null)
return;
if((!i3online())&&(!imc2online()))
return;
if((mobName==null)||(mobName.length()==0))
{
mob.tell(L("You must specify a name."));
return;
}
if(i3online())
{
final FingerRequest ck=new FingerRequest();
ck.sender_name=mob.Name();
ck.target_name=mobName;
ck.target_mud=mudName;
try
{
ck.send();
}catch(final Exception e){Log.errOut("IMudClient",e);}
}
if(imc2online())
imc2.imc_send_whois(mob.Name(),mobName,mob.phyStats().level());
}
public String getMudInfo(I3Mud mudToShow)
{
final StringBuilder buf=new StringBuilder("");
buf.append(CMStrings.padRight(L("Name"),10)+": "+mudToShow.mud_name+"\n\r");
buf.append(CMStrings.padRight(L("Address"),10)+": "+mudToShow.address+"\n\r");
buf.append(CMStrings.padRight(L("Port"),10)+": "+mudToShow.player_port+"\n\r");
buf.append(CMStrings.padRight(L("Admin@"),10)+": "+mudToShow.admin_email+"\n\r");
buf.append(CMStrings.padRight(L("Base"),10)+": "+mudToShow.base_mudlib+"\n\r");
buf.append(CMStrings.padRight(L("MudLib"),10)+": "+mudToShow.mudlib+"\n\r");
buf.append(CMStrings.padRight(L("Type"),10)+": "+mudToShow.mud_type+"\n\r");
buf.append(CMStrings.padRight(L("Driver"),10)+": "+mudToShow.driver+"\n\r");
buf.append(CMStrings.padRight(L("Status"),10)+": "+mudToShow.status+"\n\r");
return buf.toString();
}
public List<I3Mud> mudFinder(String parms)
{
final MudList list=Intermud.getAllMudsList();
if(list==null)
return null;
final Map<String,I3Mud> l=list.getMuds();
for(final I3Mud m : l.values())
{
if(m.mud_name.equals(parms))
return new XVector<I3Mud>(m);
}
for(final I3Mud m : l.values())
{
if(m.mud_name.equalsIgnoreCase(parms))
return new XVector<I3Mud>(m);
}
if(parms.startsWith("*")&&(!parms.endsWith("*")))
{
final List<I3Mud> muds=new XVector<I3Mud>();
for(final I3Mud m : l.values())
{
if(m.mud_name.toLowerCase().endsWith(parms.toLowerCase()))
muds.add(m);
}
return muds;
}
if(parms.endsWith("*")&&(!parms.startsWith("*")))
{
final List<I3Mud> muds=new XVector<I3Mud>();
for(final I3Mud m : l.values())
{
if(m.mud_name.toLowerCase().startsWith(parms.toLowerCase()))
muds.add(m);
}
return muds;
}
if(parms.endsWith("*")&&(parms.startsWith("*")))
{
final List<I3Mud> muds=new XVector<I3Mud>();
for(final I3Mud m : l.values())
{
if(m.mud_name.toLowerCase().indexOf(parms.toLowerCase())>=0)
muds.add(m);
}
return muds;
}
final List<I3Mud> muds=new XVector<I3Mud>();
for(final I3Mud m : l.values())
{
if((m.state<0)&&(CMLib.english().containsString(m.mud_name,parms)))
muds.add(m);
}
return muds;
}
@Override
public void i3mudInfo(MOB mob, String parms)
{
if((mob==null)||(!i3online()))
return;
if(mob.isMonster())
return;
final StringBuffer buf=new StringBuffer("\n\r");
final List<I3Mud> muds=this.mudFinder(parms);
if(muds.size()==0)
buf.append("Not found!");
else
for(final I3Mud mudToShow : muds)
{
buf.append(CMStrings.padRight(L("Name"),10)+": "+mudToShow.mud_name+"\n\r");
buf.append(CMStrings.padRight(L("Address"),10)+": "+mudToShow.address+"\n\r");
buf.append(CMStrings.padRight(L("Port"),10)+": "+mudToShow.player_port+"\n\r");
buf.append(CMStrings.padRight(L("Admin@"),10)+": "+mudToShow.admin_email+"\n\r");
buf.append(CMStrings.padRight(L("Base"),10)+": "+mudToShow.base_mudlib+"\n\r");
buf.append(CMStrings.padRight(L("MudLib"),10)+": "+mudToShow.mudlib+"\n\r");
buf.append(CMStrings.padRight(L("Type"),10)+": "+mudToShow.mud_type+"\n\r");
buf.append(CMStrings.padRight(L("Driver"),10)+": "+mudToShow.driver+"\n\r");
buf.append(CMStrings.padRight(L("Status"),10)+": "+mudToShow.status+"\n\r");
}
mob.session().wraplessPrintln(buf.toString());
}
@Override
public void giveIMC2MudList(MOB mob)
{
if((mob==null)||(!imc2online()))
return;
if(mob.isMonster())
return;
final Hashtable l=imc2.query_muds();
final Vector V=new Vector();
for(final Enumeration e=l.elements();e.hasMoreElements();)
{
final REMOTEINFO m=(REMOTEINFO)e.nextElement();
boolean done=false;
for(int v=0;v<V.size();v++)
{
final REMOTEINFO m2=(REMOTEINFO)V.elementAt(v);
if(m2.name.toUpperCase().compareTo(m.name.toUpperCase())>0)
{
V.insertElementAt(m,v);
done=true;
break;
}
}
if(!done)
V.addElement(m);
}
final StringBuffer buf=new StringBuffer("\n\rIMC2 Mud List:\n\r");
for(int v=0;v<V.size();v++)
{
final REMOTEINFO m=(REMOTEINFO)V.elementAt(v);
buf.append("["+CMStrings.padRight(m.name,15)+"]["+CMStrings.padRight(m.version,30)+"] "+CMStrings.padRight(m.network,13)+" ("+CMStrings.padRight(m.hub,10)+")\n\r");
}
mob.session().wraplessPrintln(buf.toString());
}
@Override
public void giveI3MudList(MOB mob)
{
if((mob==null)||(!i3online()))
return;
if(mob.isMonster())
return;
final StringBuffer buf=new StringBuffer("\n\rI3 Mud List:\n\r");
final MudList list=Intermud.getAllMudsList();
final Vector V=new Vector();
int col1Width=ListingLibrary.ColFixer.fixColWidth(25, mob);
int col2Width=ListingLibrary.ColFixer.fixColWidth(25, mob);
if(list!=null)
{
for(final I3Mud m : list.getMuds().values())
{
if(m.state<0)
{
boolean done=false;
for(int v=0;v<V.size();v++)
{
final I3Mud m2=(I3Mud)V.elementAt(v);
if(m2.mud_name.toUpperCase().compareTo(m.mud_name.toUpperCase())>0)
{
V.insertElementAt(m,v);
done=true;
break;
}
}
if(!done)
V.addElement(m);
}
}
for(int v=0;v<V.size();v++)
{
final I3Mud m=(I3Mud)V.elementAt(v);
if((m!=null)&&(m.base_mudlib!=null))
{
final String mudlib = m.base_mudlib.startsWith("CoffeeMud") ? "^H"+m.base_mudlib+"^?" : m.base_mudlib;
buf.append("["+CMStrings.padRight(m.mud_name,col1Width)+"]["+CMStrings.padRight(mudlib,col2Width)+"] "+m.address+" ("+m.player_port+")\n\r");
}
}
}
mob.session().wraplessPrintln(buf.toString());
}
@Override
public void giveI3ChannelsList(MOB mob)
{
if((mob==null)||(!i3online()))
return;
if(mob.isMonster())
return;
final StringBuffer buf=new StringBuffer("\n\rI3 Channels List:\n\r");
final ChannelList list=Intermud.getAllChannelList();
if(list!=null)
{
final Hashtable l=list.getChannels();
for(final Enumeration e=l.elements();e.hasMoreElements();)
{
final Channel c=(Channel)e.nextElement();
if(c.type==0)
buf.append("["+CMStrings.padRight(c.channel,20)+"] "+c.owner+"\n\r");
}
}
mob.session().wraplessPrintln(buf.toString());
}
@Override
public void giveIMC2ChannelsList(MOB mob)
{
if((mob==null)||(!imc2online()))
return;
if(mob.isMonster())
return;
final StringBuffer buf=new StringBuffer("\n\rIMC2 Channels List:\n\r");
final Hashtable channels=imc2.query_channels();
buf.append(CMStrings.padRight(L("Name"), 22)+CMStrings.padRight(L("Policy"),25)+CMStrings.padRight(L("Owner"),20)+"\n\r");
final Enumeration e = channels.keys();
while (e.hasMoreElements())
{
final String key = (String) e.nextElement();
final IMC_CHANNEL r = (IMC_CHANNEL) channels.get(key);
if (r != null)
{
String policy = "final public";
if (r.policy == IMC2Driver.CHAN_PRIVATE)
policy = "(private)";
else
if (r.policy == IMC2Driver.CHAN_COPEN)
policy = "open";
else
if (r.policy == IMC2Driver.CHAN_CPRIVATE)
policy = "(cprivate)";
buf.append(CMStrings.padRight(key, 22)+
CMStrings.padRight(policy+"("+r.level+")",25)+
r.owner+"\n\r");
}
}
mob.session().wraplessPrintln(buf.toString());
}
@Override
public boolean isIMC2channel(String channelName)
{
if(!imc2online())
return false;
final Object remote=imc2.getAnIMC2Channel(channelName);
if(remote==null)
return false;
return true;
}
@Override
public boolean isI3channel(String channelName)
{
if(!i3online())
return false;
final String remote=Intermud.getRemoteChannel(channelName);
if(remote.length()==0)
return false;
return true;
}
public String socialFixOut(String str)
{
str=CMStrings.replaceAll(str,"<S-NAME>","$N");
str=CMStrings.replaceAll(str,"<T-NAME>","$O");
str=CMStrings.replaceAll(str,"<T-NAMESELF>","$O");
str=CMStrings.replaceAll(str,"<S-HIM-HER>","$m");
str=CMStrings.replaceAll(str,"<T-HIM-HER>","$M");
str=CMStrings.replaceAll(str,"<S-HIS-HER>","$s");
str=CMStrings.replaceAll(str,"<T-HIS-HER>","$S");
str=CMStrings.replaceAll(str,"<S-HE-SHE>","$e");
str=CMStrings.replaceAll(str,"<T-HE-SHE>","$E");
str=CMStrings.replaceAll(str,"\'","`");
if(str.equals(""))
return "$";
return str.trim();
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 2,453
|
Mycodrosophila matilei är en tvåvingeart som beskrevs av Chassagnard och Lachaise 2000. Mycodrosophila matilei ingår i släktet Mycodrosophila och familjen daggflugor. Inga underarter finns listade i Catalogue of Life.
Källor
Externa länkar
Daggflugor
matilei
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,793
|
Q: States in React, how to preserve the original`? I am working in a SP framework application that read, group and filter data from a list.
To do that I have a method in the app that get called from an onclick eventhandler.
This method contains just a switch to check what other method needs to be called. It looks like this:
private _handleRequest(request: string): void {
switch (request) {
case 'Customer':
case 'Sales Manager':
this.groupHandler(request);
break;
case 'Agreement Ended':
this.getEnded();
break;
case 'Last Price Adjustment':
this.getPassed();
break;
default:
break;
}
}
The groupHandler method send a request to Sharepoint that fetchs all the data and then send the data to a ListView component which render it on the screen:
private groupHandler(group: string): void {
group = group.replace(/ +/g, "");
this.setState({
groupByFields: [{
name: group,
order: GroupOrder.ascending
}]
});
}
}
The getEnded and getPassed are almost similar, fetch data from sharepoint and filter the result before it is sended to the ListView component.
Each method update the state:
private getEnded(): void {
this.props.provider.getEnded().then((listItems: IList[]) => {
this.setState({
listItems: listItems
});
});
}
private getPassed(): void {
this.props.provider.getPassed().then((listItems: IList[]) => {
this.setState({
listItems: listItems
});
});
}
and it is initialize like this:
export interface IListState {
listItems: IList[];
}
export default class AgreementDatabase extends React.Component {
constructor(props: IAgreementDatabaseProps) {
super(props);
this.state = {
listItems: []
};
this.groupHandler = this.groupHandler.bind(this);
}
This is the componentDidMount where I populate the state with data fetched from SharePoin List:
public componentDidMount(): void {
this.props.provider.getContent().then((listItems: IList[]) => {
this.setState({
listItems: listItems
});
});
}
Now the problem is, when I run the app and click on a link that call the groupHandler method the list get grouped without problem. The problem arise when, with the list grouped, I click on a link that call getEnded or getPassed method. In that case the list doesn't reload before filtering the result. What I get is a grouped list with filtered results.
I post some images to better understanding.
How can I reset the state so every time I click on a button the app fetch the original state / values before applying filtering or grouping?
Best regards
Americo
A: I think that I found the real problem and it has nothing to do with the state.
In the render method I am using ListView to render the list and in this component I am using the groupByFields property which automatically groups the results:
<ListView
items={this.state.listItems}
viewFields={this._viewFields}
groupByFields={this.state.groupByFields}
/>
The groups comes from a groupByFields state.
What I really need to do is find the way to deactivate this groupByfield property when I want to filter otherwise it doesn't matter how I filter or manipulate the state, the result will be always grouped.
Do you know a way to do that? or a way to add this property only when the groupHandler method gets called?
EDIT: I found this solution:
I call a method called renderList:
private _renderList(): any {
if (this.state.filter) {
return ( <ListView
items={this.state.filteredListItems}
viewFields={this._viewFields}
/>)
} else {
return (<ListView
items={this.state.originalitems}
viewFields={this._viewFields}
groupByFields={this.state.groupByFields}
/>)
}
}
And I use state to see if is the filter or the grouping list that should be rendered
Thanks for your time, it helped med to think a little out of the box
Best regards
Americo
A: So, what I was thinking it's to keep a property filter in the state, which is set in getPassed() and getEnded() methods.
For example, in getPassed(), you set filter = "passed"; instead, in getEnded() you set filter = "ended". Finally, by default, filter property could be set to null.
Then, in write something like this:
/* Inside your Component */
renderPassed = () => {
const itemToRender = [];
this.state.listItems.forEach(item => {
if (item.status === "passed") {
itemToRender.push(item);
}
}
return itemToRender;
}
renderEnded = () => {
const itemToRender = [];
this.state.listItems.forEach(item => {
if (item.status === "ended") {
itemToRender.push(item);
}
}
return itemToRender;
}
render() {
let itemToRender;
switch (this.state.filter) {
case "passed": itemToRender = this.renderPassed(); break;
case "ended": itemToRender = this.renderEnded(); break;
default: itemToRender = this.state.listItems;
}
return <div>{itemToRender}</div>;
}
It's really a simplification, but I think you can get my idea.
I don't know you know if one item is passed and/or ended, I just supposed a status property in each item.
Does it help? Can it be implemented in your App?
EDIT: Notice that I don't "reset" the state. It's not useful since you would have two option to do that:
*
*Executing again the fetch;
*Keep your original data into a property of the class, but then you would have some items that are repeated in the original data and in the filtered data;
A: I think that I found the real problem and it has nothing to do with the state.
In the render method I am using ListView to render the list and in this component I am using the groupByFields property which automatically groups the results:
<ListView
items={this.state.listItems}
viewFields={this._viewFields}
groupByFields={this.state.groupByFields}
/>
The groups comes from a groupByFields state.
What I really need to do is find the way to deactivate this groupByfield property when I want to filter otherwise it doesn't matter how I filter or manipulate the state, the result will be always grouped.
Do you know a way to do that? or a way to add this property only when the
groupHandler method gets called?
Best regards
Americo
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,221
|
Radiohead guitarist, Jonny Greenwood, has been nominated for Best Original Score at this year's Oscars. Greenwood scored Paul Thomas Anderson's Phantom Thread, which stars Daniel Day-Lewis and Vicky Krieps. It's his first Oscar nomination.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,866
|
{"url":"https:\/\/rstudio-pubs-static.s3.amazonaws.com\/188117_29a7a382c1884bd98ff54b9c7805613f.html","text":"This tutorial provides a brief overview of how to create Sweave (LaTeX + R) documents in APA style.\n\n# 0: Setup\n\n### Install LaTeX packages\n\nIn order to create LaTeX documents, you need to install LaTeX libraries on your computer. Here are links to Mac and Windows LaTeX libraries\n\n### Install R packages\n\nNext you should install some R packages that will help you create Sweave documents. Open RStudio and run the following two lines of code:\n\n# These packages help with Sweave\ninstall.packages(\"knitr\")\ninstall.packages(\"xtable\")\n\n### Update RStudio preferences\n\nNext, you\u2019ll need to change some of the the RStudio Sweave preferences.\n\nGo to RStudio \u2013 Preferences \u2013 Sweave and make the following two changes:\n\n\u2022 Select \u201cSweave Rnw files using Knitr\n\u2022 Set \u201cPreview PDF\u201d with \u201cSystem Viewer\u201d\n\n# 1: A simple APA Sweave Template\n\n1. Open a new Sweave file in RSTudio: File \u2013 New File \u2013 R Sweave.\n\nWhen you do this, you\u2019ll see a bare-bones file with a three commands.\n\n1. Save the file under a new name with the .Rnw suffix (e.g.; myfirstsweave.Rnw)\n2. Now open a web-browser and go to the following address to find an APA style Sweave template (Simple APA Sweave File)\n3. Copy and paste all of the text from the template and paste it into your Sweave file.\n4. Click the \u201cCompile PDF\u201d button at the top of your screen. When you do this, you should see some processing output, followed by a PDF!\n\n# 2: Basic LaTeX commands\n\nImportant aspects of the document, such as the title, author, and affiliation are specified with commands in the form \\command{}.\n\n\u2022 The main aspects of your document such as the title, header, and author are specified with commands like \\title{}, \\shorttitle{}, \\author{}.\n\n\u2022 Sections are defined using the \\section{} and \\subsection{} commands. For example, to start the Method section, you write \\section{Method}. To include the Participants subsection, you write \\subsection{Participants}\n\n\u2022 Bold and italic text are created with the \\bf{} and \\emph{} commands:\n\n\u2022 SWEAVE: Here is how you write \\bf{bold text}, here is how you write \\emph{italic text}.\n\u2022 PDF: Here is how you write bold text, here is how you write italic text.\n\n# 3: R code chunks\n\nYou can incorporate R code into your Sweave document in one of two ways: with regular chunks, or with mini-chunks. Everything you put in an R chunk will be evaluated as R code.\n\n## Regular R chunks\n\nRegular R chunks start with a start \u2018tag\u2019 <<>>= and end with an end \u2018tag\u2019 @. They are indicated with gray backgrounds and look like this:\n\nYou can create a new R chunk either by manually typing the start <<>>= and end @ tags, or by clicking the green \u201cInsert a new code chunk\u201d button on the top right of your editor window in RStudio.\n\nYou can specify many arguments at the top of the chunk (in between <<>>=) that will change how the code is evaluated. Here are some key arguments:\n\n\u2022 eval: Should to code in the chunk be evaluated (eval = T) or completely ignored (eval = F)? If you have code in a chunk that is not complete, but you still want to create the document without that chunk, set eval = F.\n\n\u2022 echo: Should the raw code be repeated in the final document (echo = T) or not? (echo = F). In most APA style papers you\u2019ll never repeat the code so you\u2019ll usually set echo = F.\n\n\u2022 fig.width, fig.height, fig.align: If your chunk creates a figure, these arguments will change its size and location. For example, fig.width = 4, fig.height = 10, fig.align = \u2018center\u2019 will create a centered figure that is 4 inches wide and 10 inches tall.\n\nIf you want to change some options for all of your R chunks, you can do this using the ops_chunk$set() function in an R chunk at the beginning of your document. For example, in virtually all of your R chunks you will not want to echo (aka repeat) the source code in the document. For this reason, it\u2019s good to set the global chunk option to echo = F. ## Mini-code chunks If you want to include the output of R code (e.g.; the p-value of a t.test) directly into your text, you should use a mini-chunk. Mini-chunks are specified with the \\Sexpr{} command. Everything in the curly braces will be interpreted as R code. For example, the following sentence in Sweave will be printed in the PDF as follows \u2022 SWEAVE: The mean age of participants was \\Sexpr{mean(data$age)} and there were \\Sexpr{sum(data\\$sex == \u201cf\u201d)} females.\n\u2022 PDF: The mean age of participants was 21 and there were 100 females.\n\n# 4: Including figures\n\nTo include figures, use the \\begin{figure} and \\end{figure} commands. In between these commands, you\u2019ll specify how to create, or find, the figure.\n\n## Creating figures in R Chunks\n\nTo create the figure directly using R code, create an R chunk that creates the figure. For example, the following code chunk creates a histogram\n\n## Importing figures from external files\n\nIf your figure is in an external file (like a pdf or jpg), you don\u2019t need to use an R chunk. Instead, you include the figure using the \\includegraphics{} command. Just put the name (and possibily the file path) of the figure in the braces.","date":"2021-11-30 03:40:42","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\": 0.5740199089050293, \"perplexity\": 4163.58766270013}, \"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-2021-49\/segments\/1637964358903.73\/warc\/CC-MAIN-20211130015517-20211130045517-00560.warc.gz\"}"}
| null | null |
Emily Kinkead is an American politician. She is a Democrat in the Pennsylvania House of Representatives. In 2020, Kinkead was elected to represent District 20, which encompasses several communities in Pittsburgh and Allegheny County.
Education
In 2009, Kinkead earned a Bachelor of Science degree in Biology and a Bachelor of Arts degree in Political Science from Bloomsburg University of Pennsylvania. In 2016, she earned her Juris Doctor degree from the University of Pittsburgh School of Law.
Background
Kinkead is a private-practice lawyer from the Brighton Heights neighborhood of Pittsburgh. Previously, Kinkead has worked in Washington D.C. for Common Cause on government ethics reform, for the National Institutes of Health and as a Judicial Law Clerk for Allegheny County Commonwealth Court Judge, Michael Wojcik.
Political career
In 2019, Kinkead and Emily Marburger, mayor of Bellevue, Pennsylvania, entered the Democratic primary against Adam Ravenstahl, the incumbent in their district. In November 2019, Marburger dropped out of the race. Kinkead defeated Ravenstahl in the primary on June 2, 2020.
She was unopposed in the general election on November 3, 2020, and took office on December 1, 2020.
Committee assignments
Agriculture & Rural Affairs
Appropriations
Committee On Committees
Human Services
Judiciary
References
External links
Official Pennsylvania House profile
Democratic Party members of the Pennsylvania House of Representatives
Women state legislators in Pennsylvania
Politicians from Pittsburgh
Living people
Year of birth missing (living people)
21st-century American women
Bloomsburg University of Pennsylvania alumni
University of Pittsburgh School of Law alumni
1987 births
Place of birth missing (living people)
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 4,505
|
Q: Notation on fibre bundles I came up this morning with the following question and after looking for it for a while on the internet i found this old question on math.stackexchange with no answers. Could anyone please give some clue? Thanks a lot in advance
Question on notation (topology & fiber bundles)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 9,085
|
\section{Introduction}
In 2015, the direct detection of gravitational waves (GW) by the LIGO-Virgo Collaboration~\cite{Abbott:2016blz} emitted by inspiralling compact binary, opened new avenues for probing the dynamics in strong gravity regime~\cite{Arun:2006yw,Mishra:2010tp,Li:2011cg,Agathos:2013upa,Cornish:2011ys,Berti:2015itd,Yunes:2016jcc}. It is expected that future GW detectors, like the Einstein Telescope~\cite{Maggiore:2019uih} and Cosmic Explorer~\cite{Evans:2021gyd} will shed more light on alternative theories of gravity by constraining the parameters of such theories.
The simplest theory amongst the alternative theories of gravity is the addition of a massless scalar field to GR, scalar-tensor (ST) theories, which are
are extensively studied~\cite{Damour:1992we,Damour:1993hw,Damour:1995kt,Freire:2012mg,Khalil:2022sii,Gautam:2022cpb}.
The motivation for ST theories is to explain both the accelerated expansion of the universe as $f(R)$-theories \cite{DeFelice:2010aj} as well as UV complete alternate theories of GR. The two-body PN formalism for ST theories has been extensively studied~\cite{Lang:2013fna,Lang:2014osa,Bernard:2018hta,Bernard:2018ivi,Bernard:2019yfz,Schon:2021pcv,Brax:2021qqo,Sennett:2016klh,Bernard:2022noq}.
The important violations for ST theories arise through the non-perturbative strong field effects in neutron-stars such as spontaneous scalarisation \cite{Damour:1995kt}. Although the current constraints come from the binary pulsar observations, the future GW detections can better constraint the parameters using strong-field information and additional terms in radiation,\ i.e. dipolar radiation which is not present in GR, due to the scalar extension of GR~\cite{Palenzuela:2013hsa,Sennett:2016rwa,Khalil:2022sii}.
The EOB formalism was introduced to construct analytical waveform templates for GR~\cite{Buonanno:1998gg,Buonanno:2000ef,Damour:2000we,Damour:2008qf,Damour:2014jta,Damour:2015isa,Damour:2016abl}. Recently, the two-body PN dynamics has also been mapped within the EOB formalism to construct waveform templates for ST theories~\cite{Julie:2017ucp,Julie:2017pkb, Jain:2022nxs}. In our previous work~\cite{Jain:2022nxs}, we determined the EOB potentials for the local part of dynamics at 3PN order.
The aim of this paper is to determine the complete nonlocal-in-time EOB potentials following our results of local part in Ref. ~\cite{Jain:2022nxs}
starting from the 3PN nonlocal-in-time Lagrangian of Ref.~\cite{Bernard:2018hta,Bernard:2018ivi}. Hereafter, the companion paper \cite{Jain:2022nxs} will be referred as Paper I.
The paper is organised as follows. In Sec.~\ref{Sec-reminder},
we give a summary of results obtained in Paper I.
Then, in Sec.~\ref{Sec-OrdinaryHamiltonian} we derive the conserved energy for nonlocal-in-time part using two methods, (i) non-order-reduced nonlocal Hamiltonian using nonlocal phase shift, and (ii) order-reduction of nonlocal dynamics to local ordinary action-angle Hamiltonian. Finally, in Sec.~\ref{Sec-EOB} we map the nonlocal-in-time ordinary Hamiltonian
into an EOB Hamiltonian at 3PN order.
\section{Summary of Previous results}
\label{Sec-reminder}
We consider mono-scalar massless ST theories described by the following action in the Einstein Frame (the scalar field minimally couples to the metric),
\begin{align}
S&=\frac{c^4}{16 \pi G}\int d^4x \sqrt{-g}(R-2g^{\mu \nu} \partial_\mu \varphi \partial_\nu \varphi)\nonumber\\
&\qquad\qquad\qquad\qquad+S_m[\Psi, {\mathcal{A}(\varphi)}^2g_{\mu\nu}]~,
\end{align}
where $g_{\mu\nu}$ is the Einstein metric, $R$ is the Ricci scalar, $\varphi$ is the scalar field, $\Psi$ collectively denotes the matter fields, $g \equiv \det(g_{\mu\nu})$ and $G$ is the bare Newton's constant~\cite{Jain:2022nxs}.
As Paper I (see, Table I), we adopt the conventions and notations of Refs.~\cite{Damour:1992we,Damour:1995kt}.
In Einstein Frame, the dynamics of the scalar field arises from its coupling to the matter fields $\Psi$, and the field equations can be found in Ref.~\cite{Damour:1992we} where the parameter
\begin{equation}
\alpha(\varphi)=\frac{\partial \ln \mathcal{A}}{\partial \varphi} \ ,
\end{equation}
measures the coupling between the matter and the scalar field.
The scalar field is non-minimally coupled to the metric in Jordan Frame (physical frame)
\begin{equation}
\tilde{g}_{\mu\nu}={\mathcal{A}(\varphi)}^2 g_{\mu\nu} \ ,
\end{equation}
where $\tilde{g}_{{\mu\nu}}$ is the metric in Jordan frame.
We follow the approach suggested by~\cite{1975ApJ...196L..59E} to ``skeletonize'' the compact, self-gravitating objects in ST theories as point particles, i.e. the total mass of each body is dependent on the local value of the scalar field.
The skeletonized matter action with the scalar field dependent mass $\tilde{m}_I(\varphi)$ is then given by
\begin{equation}
S_{m}=-\sum_{J=A,B}\int \sqrt{-\tilde{g}_{\mu \nu}\frac{dx^{\mu}}{d\lambda}\frac{d x^{\nu}}{d\lambda}} \tilde{m}_{J}(\varphi)~,
\end{equation}
where $\lambda$ is the affine parameter.
Since $ \tilde{g}_{\mu\nu}={\mathcal{A}(\varphi)}^2 g_{\mu\nu}$, the Einstein-frame mass is defined as
\begin{equation}
m(\varphi)=\mathcal{A}(\varphi)\tilde{m}(\varphi) \ .
\end{equation}
In Paper I, we first derive the ordinary Hamiltonian (dependent only on the positions and momenta) using the contact transformation at 3PN order starting from the Lagrangian of Ref.~\cite{Bernard:2018ivi} only for the local-in-time part of the dynamics. The Jordan-Frame parameters of Ref.~\cite{Bernard:2018ivi} that encompass the scalar field effect are converted to the dimensionless Einstein-Frame parameters (see, Table I).
The mass function $m(\varphi)$ is used to define these dimensionless body-dependent parameters following Refs.~\cite{Damour:1992we,Damour:1995kt,Julie:2017pkb} \ i.e.
\begin{align}
\alpha_I&=\frac{d\ln m(\varphi)_I}{d\varphi},\\
\beta_I&=\frac{d\alpha_I}{d\varphi},\\
\beta'_I&=\frac{d\beta_I}{d\varphi},\\
\beta''_I&=\frac{d\beta'_I}{d\varphi}~.
\end{align}
Here, we follow the notations of Paper I for the binary parameters and use the same notation as \cite{Bernard:2018hta,Bernard:2018ivi} to denote weak-field and strong-field parameters.
Finally, we then determine the ST corrections to the EOB metric potential $(A, B, Q_e)$ at 3PN order for the local in time (instantaneous) part of the dynamics by mapping the EOB Hamiltonian in DJS gauge \cite{Damour:2000we}
\begin{align}
\label{heff-start}
\hat{H}_{\text{eff}}=\frac{H_{\text{eff}}}{\mu}=\sqrt{A(\hat{r})\left(1+\frac{\hat{p}_r^2}{B(\hat{r})}+\frac{\hat{p}_{\phi}^2}{\hat{r}^2}+q_3\frac{\hat{p}_r^4}{\hat{r}^2}\right)}~,
\end{align}
where $\hat{p}_r, \hat{p_{\phi}}$ are the dimensionless radial and angular momenta, and $\hat{r}(=r/(G_{AB}M)$ is the dimensionless radial separation, to the ordinary two-body Hamiltonian (hereafter the superscript \textit{hat} is used to denote the dimensionless variables).
The three EOB potentials at 3PN are
\begin{align}
A(\hat{r})&=1-\frac{2}{\hat{r}}+\frac{a_2}{\hat{r}^2}+\frac{a_3}{\hat{r}^3}+\frac{a_4}{\hat{r}^4}~,\\
B(r)&=1+\frac{b_1}{\hat{r}}+\frac{b_2}{\hat{r}^2}+\frac{b_3}{\hat{r}^3}~,\\
Q_e(\hat{r})&= q_3\frac{\hat{p}_r^4}{\hat{r}^2 }~.
\end{align}
The GR and ST corrections in coefficients ($a_i$, $b_i$) are separated as
\begin{align}
a_{i}&=a_{i}^{\rm GR}+\delta a_{i}^{\rm ST}~,\\
b_{i}&=b_{i}^{\rm GR}+\delta b_{i}^{\rm ST},\\
q_{3}&=q_3^{\rm GR}+\delta q_3^{\rm ST}~.
\end{align}
Since there are also nonlocal-in-time and tidal contributions at 3PN order in ST theory, all the 3PN ST coefficients can thus be decomposed as Eq.~(5.23) of Paper I.
The complete expressions of local-in-time ST corrections at 3PN can be found in Eqs.(5.14)-(5.16) of Paper I.
In Paper I, we also derive the nonlocal-in-time (tail) and tidal corrections only for the circular orbits using the gauge invariant energy for circular orbits given in Ref.~\cite{Bernard:2018ivi, Bernard:2019yfz}. The complete expression for these coefficients can be found in Eqs. (5.25)-(5.27) of Paper I.
\section{Tail contribution to the 3PN dynamics}
\label{Sec-OrdinaryHamiltonian}
The nonlocal-in-time two-body 3PN Lagrangian for massless ST theory obtained in Ref.~\cite{Bernard:2018ivi} is in harmonic coordinates, i.e. it depends (linearly) on the acceleration of the two bodies. In this section, we will use two different methods to derive the \textit{Noetherian} conserved energy for the tail contributions. First, we will remove the acceleration dependence from the Lagrangian (hence, the Hamiltonian) and stay within the non-order-reduced nonlocal framework (as done in Refs.~\cite{Bernard:2015njp,Bernard:2016wrg} for GR). Second, we will derive the order-reduced, local Hamiltonian using the action-angle variables (see, Ref.~\cite{Damour:2015isa} for GR).
\subsection{Non-order-reduced Ordinary Hamiltonian}
In Paper I, we derived the ordinary (dependent only on positions and momenta) Hamiltonian for local-in-time contribution using contact transformation (see, Appendix A of Paper I for the contact transformation). Now, concerning the nonlocal-in-time part we need to find the nonlocal shift that removes the acceleration dependence from the tail part of the Lagrangian of Ref.~\cite{Bernard:2018ivi} (see, Refs.~\cite{Bernard:2015njp,Bernard:2016wrg} for GR). Corresponding to this ordinary Lagrangian, we can then derive the ordinary Hamiltonian.
The tail part of the Lagrangian at 3PN order reads \cite{Bernard:2018ivi},
\begin{align}
\label{tail-Bernard}
L^{\rm tail}&=\frac{2 G^2 M}{3 c^6}(3+2\omega_0)~\mathrm{Pf}_{2r_{AB}/c}\int_{-\infty}^{\infty}\frac{d\tau}{\abs{\tau}}I_{s,i}^{(2)}(t)I_{s,i}^{(2)}(t+\tau),
\end{align}
where $\mathrm{Pf}$ is the Hadamard partie finie function, Hadamard scale $r_{AB}(=r)$ is the relative separation of two bodies, and $I_{s,i}^{(2)}$ is the second time derivative of the dipole moment. Here, we find the shift that transforms this Lagrangian into the same expression but with the derivatives of the dipole moment evaluated using the Newtonian equations of motion. In the centre of mass (COM) frame in notations of Ref.~\cite{Bernard:2018hta,Bernard:2018ivi} it is,
\begin{align}
\label{ord-dipole}
\acute{I}_{s,i}^{(2)}=\frac{2 M\nu(s_A-s_B)}{\phi_0(3+2w_0)} \left(-\frac{G_{AB} M}{r^2}n_{AB}^i\right)~,
\end{align}
where $s_A$, $s_B$ are the sensitivity of two bodies.
As the nonlocal contribution starts at 3PN order, the ordinary Lagrangian is
\begin{align}
L_{\text{ord}}^{\text{tail}}=L^{\text{tail}}+\sum_{J=A,B}m_J\left(-a_J^i-\sum_{J\neq K} \frac{G_{AB}~m_K}{r^2} n_{JK}^i\right)\xi_{J,i}~,
\end{align}
where $L_{\text{ord}}^{\rm tail}$ is given by the same expression as Eq.~\eqref{tail-Bernard} but with second time derivative of the dipole moment replaced by its on-shell value given in Eq.~\eqref{ord-dipole}, and the nonlocal shift, $\xi_{J,j}$,
\begin{align}
\xi_{J,j}=\frac{1}{m_J} \frac{2 G^2 M}{3c^6 }&(3+2w_0)\left[-\frac{m_J(1-2s_J)}{\phi_0(3+2w_0)}\right] \delta_j^i \nonumber \\
&\mathrm{Pf}_{2r/c} \int_{-\infty}^{\infty} \frac{d\tau}{\abs{\tau}}\acute{I}_{s,i}^{(2)}(t+\tau)~.
\end{align}
The ordinary Hamiltonian is then derived using the ordinary Legendre transformation, $H_{\text{ord}} = \sum_A p_A v_A - L_{\text{ord}}$ which reads
$H_{\rm ord}=H_{\rm ord}^{\text{loc}}+H_{\rm ord}^{\text{tail}}$, where the local contribution $H_{\rm ord}^{\text{loc}}$ is derived in Paper I (see, Appendix C) and the tail contribution is
\begin{align}
\label{nonlocal-Ham}
H_{\rm ord}^{\text{tail}}&=-\frac{2 G^2 M}{3c^6}(3+2w_0) ~\mathrm{Pf}_{2 r/c}\int_{-\infty}^{\infty}\frac{d\tau}{\abs{\tau}}\acute{I}_{s,i}^{(2)}(t)\acute{I}^{(2)}_{s,i}(t+\tau).
\end{align}
The tail part of the Hamiltonian is just opposite to tail part of Lagrangian.
As shown in Ref.~\cite{Damour:2016abl, Bernard:2016wrg} for the non-order-reduced, nonlocal framework the \textit{Noetherian} conserved energy ($E_{\rm cons}$) is not given by the Hamiltonian but is given by, $E_{\rm cons}=H_{\rm ord}^{\text{tail}}+\delta H$. This additional term $\delta H$ consists of purely a constant term (DC type) and time oscillating term with zero average value (AC type) and is same as given in Eq. (4.10) of Ref.~\cite{Bernard:2018ivi}.
\subsection{Order-reduced Ordinary Hamiltonian}
\label{Action-angleH}
The second method to derive the conserved energy for tail part is to work in the order-reduced, local framework as given in Ref.~\cite{Damour:2015isa,Damour:2016abl} for GR.
The tail part of the Hamiltonian in ST theory is,
\begin{align}
\label{tail-BernardH}
H^{\rm tail}=-\frac{2 G^2 M}{3 c^6}(3+2\omega_0)&\left[\mathrm{Pf}_{2r/c}\int_{-\infty}^{\infty}\frac{d\tau}{\abs{\tau}}I_{s,i}^{(2)}(t)I_{s,i}^{(2)}(t+\tau)\right.\nonumber \\
&\left.-2\ln\left(\frac{\hat{r}}{a}\right)I_{s,i}^{(2)}(t)^2\right] .
\end{align}
As mentioned in Ref.~\cite{Bernard:2016wrg}, in the action-angle form there should be an additional term (second term in Eq.~\eqref{tail-BernardH}) which is local and accounts for dependence of Hadamard Partie finie function on the radial separation ($r$) at time $t$ i.e., $\hat{r} = a(1-e\cos(u))$ in action-angle variables.
The basic methodology we use to order-reduce the nonlocal dynamics of the above form is based on Refs.~\cite{Damour:2015isa,Damour:2016abl} for GR, and consists of four main steps: (i) Re-express the Hamiltonian in terms of action angle variables, (ii)``order-reduce" the nonolocal dependence on action angle variable, (iii) expand it in powers of eccentricity, and (iv) eliminate the periodic terms in order-reduced Hamiltonian by a canonical transformation. All of these steps lead to the order-reduced ordinary local Hamiltonian for the tail part in terms of action-angle variables.
Let us consider the expression of nonlocal-in-time piece of Eq.~\eqref{tail-BernardH}, \ i.e.
\begin{align}
\label{nonloc-expr}
\mathcal{K}(t,\tau)=\ddot{I}_{s,i}(t)\ddot{I}_{s,i}(t+\tau)~.
\end{align}
To order reduce the nonlocal piece, we use the equations of motion to express the phase-space variables at shifted time $t+\tau$ in terms of the phase-space variables at time $t$. As the zeroth order equations are Newtonian equations, it will be convenient to use the action-angle form of the Newtonian equations of motion,
\begin{align}
\label{actioneq}
\frac{\partial l}{\partial \hat{t}}&=\frac{\partial H_0}{\partial \mathcal{L}}=\frac{1}{\mathcal{L}^3}=\Omega(\mathcal{L})~,~~\frac{\partial \mathcal{L}}{\partial \hat{t}}=\frac{\partial H_0}{\partial l}=0~,\nonumber \\
\frac{\partial \mathcal{G}}{\partial \hat{t}}&=\frac{\partial H_0}{\partial g}=0~,\hspace{1.7cm}\frac{\partial g}{\partial \hat{t}}=\frac{\partial H_0}{\partial \mathcal{G}}=0~,
\end{align}
where $\hat{t}={t}/({G_{AB}M})$ is the dimensionless time variable, $(\mathcal{L},l,\mathcal{G},g)$ are the action-angle variables. The zeroth-order (Newtonian) Hamiltonian in action-angle variable is $H_0=-1/(2\mathcal{L}^2)$.
Here, the variable $\mathcal{L}$ is conjugate to the ``mean anamoly" $l$ and $\mathcal{G}$ is conjugate to argument of periastron $g$. In terms of the Keplerian variables, semi-major axis $a$, and eccentricity $e$, these are
\begin{align}
\mathcal{L}=\sqrt{a},\hspace{0.4cm}\mathcal{G}=\sqrt{a(1-e^2)}~.
\end{align}
From Eq.~\eqref{actioneq}, the variables $\mathcal{L}$, $\mathcal{G}$ and $g$ are independent of time, and $l$ varies linearly with time, hence it will be sufficient to use
\begin{align}
l(t+\tau)=l(t)+\Omega~ \hat{\tau}~,
\end{align}
where $\hat{\tau}=\tau/(G_{AB}M)$. The order-reduced non-local in time expression of Eq.~\eqref{nonloc-expr} becomes
\begin{align}
\mathcal{K}(t,\tau)&=\left(\frac{1}{G_{AB}M}\right)^4\mathcal{K}(\hat{t},\hat{\tau})\nonumber\\
&=\left(\frac{\Omega}{G_{AB}M}\right)^4\frac{d^2}{dl^2} {I}_{s,i}(l)\frac{d^2}{dl^2}{I}_{s,i}(l+\Omega \hat{\tau})~.
\end{align}
Using the Fourier decomposition of dipole moment given in Eq.~\eqref{Fouier-exp}, we find the structure of nonlocal-in-time expression $\mathcal{K}(t,\hat{\tau})$ and hence the Hamiltonian. As shown in \cite{Damour:2015isa} for GR, all the periodically varying terms can be eliminated by a suitable canonical transformation. Hence, the order-reduced Hamiltonian can be further simplified by replacing $H^{\rm tail}$ with its $l$-average value
\begin{align}
\label{order-red_ham}
\bar{H}_{\text{tail}}=\int_0^{2 \pi} dl H^{\rm tail}~.
\end{align}
Using the result
\begin{align}
\mathrm{Pf}_T\int_0^{\infty}\frac{dv}{v}\cos(\omega v)=-(\gamma_{\rm E}+\ln(\omega~T))~\hspace{1.4cm}\forall~(\omega>0)~,
\end{align}
where $\gamma_{\rm E}$ is the Euler's constant, and inserting the expression of $r$ from Eq.~\eqref{action-angleformr}, the Hamiltonian, Eq.~\eqref{order-red_ham}, reads
\begin{align}
\label{nonloc-f}
\bar{H}_{\text{tail}}=\frac{8 G^2 M}{3}\left(\frac{\Omega}{G_{AB}M}\right)^4&(3+2w_0)\sum_{p=1}^{\infty}p^4\abs{I_{s,i}(p)}^2 \nonumber \\
&\ln\left(\mathrm{e}^{\rm \gamma_{\rm E}}\frac{2p~a~\Omega }{c}\right).
\end{align}
Now, inserting the Fourier-Bessel expansion of scalar dipole moment from Eqs.~\eqref{BesselIx}-\eqref{BesselIy} (see, Appendix \ref{Fourier-Bderived} for derivation) in Eq.~\eqref{nonloc-f}, the real two-body nonlocal-in-time Hamiltonian in order-reduced, local framework is (in notations of Paper I)
\begin{widetext}
\begin{align}
\hat{\bar{H}}_{\text{tail}}\equiv\frac{\bar{H}_{\text{tail}}}{\mu}=\frac{2\nu}{3a^4}\left(2 \delta_+ +\frac{\bar{\gamma}_{AB}(\bar{\gamma}_{AB}+2)}{2}\right)\sum_{p=1}^{\infty}\frac{p^2}{e^2}&\left\{4e^2J^2_{p-1}(pe)+(8-4e^2) J^2_p(pe)-8eJ_{p-1}(pe)J_p(pe)\right\}\nonumber\\
& \left[\gamma_{\rm E}+\ln\left(\frac{2p~a^{-1/2}}{c}\right)\right]~.
\end{align}
\end{widetext}
Expanding the result in powers of eccentricity, the Hamiltonian as an expansion in eccentricity upto order of $e^4$ reads
\begin{widetext}
\begin{align}
\hat{\bar{H}}_{\text{tail}}&=\frac{2\nu}{3a^4}\left(2 \delta_+ +\frac{\bar{\gamma}_{AB}(\bar{\gamma}_{AB}+2)}{2}\right)\left\{2\ln(2)-\ln(a)+2\gamma_{\rm E}+e^2\left(14\ln(2)+6\gamma_{\rm E}-3\ln(a)\right)\right.\nonumber\\
&\left.+e^4\left(\frac{45}{4}\gamma_{\rm E}-\frac{3}{4}\ln(2)+\frac{729}{32}\ln(3)-\frac{45}{8}\ln(a)\right)+\mathcal{O}(e^6)\right\}~.
\end{align}
\end{widetext}
\section{Scalar Tensor corrections to Effective One Body at 3PN:Tail}
\label{Sec-EOB}
In this section, we will derive the complete tail corrections to the EOB metric
potentials $(A,B,Q_e)$ for ST theories at 3PN order.
Similar to the decomposition of complete 3PN coefficient $\delta a_4^{\text{ST}}$
in Eq. (5.23) of Paper I, we decompose the complete
3PN ST coefficients $\delta b_3^{\text{ST}},\delta q_3^{\text{ST}}$ as
\begin{align}
\label{b-decompose}
\delta b_3^{ST}&=\delta b_{3,\rm loc}^{\rm ST}+\delta b_{3,\rm nonloc}^{\rm ST}+\delta b_{3,\rm tidal}^{\rm ST}~,\\
\label{q-decompose}
\delta q_3^{ST}&=\delta q_{3,\rm loc}^{\rm ST}+\delta q_{3,\rm nonloc}^{\rm ST}+\delta q_{3,\rm tidal}^{\rm ST}~,
\end{align}
where the local contributions $(\delta b_{3,\rm loc}^{\rm ST},\delta q_{3,\rm loc}^{\rm ST})$ are derived in Paper I (see, Eqs. (5.14)-(5.15)),
$(\delta b_{3,\rm nonloc}^{\rm ST},\delta q_{3,\rm nonloc}^{\rm ST})$ are the nonlocal contributions,
and $(\delta b_{3,\rm tidal}^{\rm ST},\delta q_{3,\rm tidal}^{\rm ST})$ are the tidal contributions.
The nonlocal contributions can be further decomposed similar to Eq. (5.24) of Paper as
\begin{align}
\label{tail-decompose}
\delta a_{4,\rm nonloc}^{\rm ST}&=\delta a_{4,\rm nonloc,0}^{\rm ST}+\delta a_{4,\rm nonloc,\rm log}^{\rm ST}\ln (\hat{r})~,\\
\label{b-tail}
\delta b_{3,\rm nonloc}^{\rm ST}&=\delta b_{3,\rm nonloc,0}^{\rm ST}+\delta b_{3,\rm nonloc,\rm log}^{\rm ST}\ln(\hat{ r})~,\\
\label{q-tail}
\delta q_{3,\rm nonloc}^{\rm ST}&=\delta q_{3,\rm nonloc,0}^{\rm ST}+\delta q_{3,\rm nonloc,\rm log}^{\rm ST}\ln(\hat{r})~.
\end{align}
Inserting the split of the EOB functions ($A$, $B$, $q_3$) using
Eqs.~\eqref{b-decompose}-\eqref{q-decompose} and Eq. (5.23) of Paper I
in the effective Hamiltonian of Eq. \eqref{heff-start},
and then after expanding the right-side into a Taylor series of $1/c^2$, we obtain
\begin{align}
\hat{H}_{\rm eff}=\hat{H}^{\rm loc}_{\rm eff}+\hat{H}^{\rm nonloc}_{\rm eff}~,
\end{align}
where $\hat{H}^{\rm loc}_{\rm eff}$ is computed only by the local contributions
$(\delta a_{4,\rm loc}^{\rm ST}, \delta b_{3,\rm loc}^{\rm ST},\delta q_{3,\rm loc}^{\rm ST})$ and
$\hat{H}^{\rm nonloc}_{\rm eff}$ is the nonlocal contribution of Hamiltonian computed by $(\delta a_{4,\rm nonloc}^{\rm ST}, \delta b_{3,\rm nonloc}^{\rm ST},\delta q_{3,\rm nonloc}^{\rm ST})$.
The nonlocal contribution $\hat{H}^{\rm nonloc}_{\rm eff}$ reads
\begin{align}
\label{heff}
\hat{H}^{\rm nonloc}_{\rm eff}=\frac{1}{2}\left(\delta a_{4,\rm nonloc}^{\rm ST}\frac{1}{\hat{r}^4}-\delta b_{3,\rm nonloc}^{\rm ST}\frac{\hat{p}_r^2}{\hat{r}^3}+\delta q_{3,\rm nonloc}^{\rm ST}\frac{\hat{p}_r^4}{\hat{r}^2}\right)~.
\end{align}
To map the real two-body dynamics to EOB, we express the nonlocal effective Hamiltonian, $\hat{H}^{\rm nonloc}_{\rm eff}$,
in action-angle variables $\mathcal{L}$, $l$, $\mathcal{G}$, and $g$ (hence the Keplerian variables $a$ and $e$)
and compute its $l$-averaged value,
\begin{align}
\hat{\bar{H}}^{\rm nonloc}_{\rm eff}=\frac{1}{2\pi}\int_0^{2\pi}d l \hat{H}^{\rm nonloc}_{\rm eff}~.
\end{align}
The explicit expression of $\hat{H}^{\rm nonloc}_{\rm eff}$ depends on $l$-average monomials involving powers of
$1/\hat{r}$ and $\hat{p}_r$ (and also $\ln(\hat{r})$ from Eqs.~\eqref{tail-decompose}, \eqref{b-tail}, and \eqref{q-tail}).
These computations can be performed by expanding Eq.~\eqref{heff} in terms of eccentricity upto $e^5$
using the Newtonian equations of motion in action-angle form
recalled in Sec.~\ref{Action-angleH}.
The $l$-averaged value we obtain is
\begin{widetext}
\begin{align}
\label{last}
\hat{\bar{H}}^{\rm II}_{\rm eff}&=\frac{1}{2a^4}\left\{\delta a_{4,\rm nonloc,0}^{\rm ST}+\delta a_{4,\rm nonloc,\rm log}^{\rm ST}\ln(a)\right.\nonumber\\
&\left.+\left(3\delta a_{4,\rm nonloc,0}^{\rm ST}-\frac{7}{4}\delta a_{4,\rm nonloc,\rm log}^{\rm ST}-\frac{1}{2}\delta b_{3,\rm nonloc,0}^{\rm ST}+3\delta a_{4,\rm nonloc,\rm log}^{\rm ST}\ln(a)-\frac{1}{2}\delta b_{3,\rm nonloc,\rm log}^{\rm ST}\ln(a)\right)e^2\right.\nonumber\\
&\left.+\left( \frac{45}{8}\left[\delta a_{4,\rm nonloc,0}^{\rm ST}+\delta a_{4,\rm nonloc,\rm log}^{\rm ST}\ln(a)\right]-\frac{5}{4}\left[\delta b_{3,\rm nonloc,0}^{\rm ST}+\delta b_{3,\rm nonloc,\rm log}^{\rm ST}\ln(a)\right]+\frac{3}{8}\left[\delta q_{3,\rm nonloc,0}^{\rm ST}+\delta q_{3,\rm nonloc,\rm log}^{\rm ST}\ln(a)\right]\right.\right.\nonumber\\
&\left.\left.-\frac{171}{32}\delta a_{4,\rm nonloc,\rm log}^{\rm ST}+\frac{9}{16}\delta b_{3,\rm nonloc,\rm log}^{\rm ST}\right)e^4+\mathcal{O}(e^6)\right\}~.
\end{align}
\end{widetext}
The final step is then to map the real two-body dynamics to EOB metric by the \textit{nontrivial} map,
\begin{align}
\hat{H}_{\rm real}=\frac{H_{\rm real}}{\mu}=\frac{1}{\nu}\sqrt{1+2\nu(\hat{H}_{\rm eff}-1)}~,
\end{align}
between the EOB Hamiltonian $(\hat{H}_{\rm eff})$ and real two-body Hamiltonian $(\hat{H}_{\rm real})$.
The quadratic map relating the two Hamiltonians is proven at \textit{all} PN orders in GR and ST within the Post-Minkowskian scheme in Ref.~\cite{Damour:2016gwp}. However, it can be seen that only for the nonlocal contributions at 3PN order,
the map relating the two nonlocal Hamiltonians is
\begin{align}
\hat{\bar{H}}^{\rm nonloc}_{\rm eff}=\hat{\bar{H}}^{\rm II}_{\rm real, nonloc}~.
\end{align}
The \textit{unique} nonlocal ST contributions at 3PN from this matching are
\begin{widetext}
\begin{align}
\label{circ-0}
\delta a^{\rm ST}_{4,\rm nonloc,0}&=\frac{4}{3}\nu\left[2\delta_++\frac{\bar{\gamma}_{AB}(\bar{\gamma}_{AB}+2)}{2}\right](2\ln 2+2\gamma_{\rm E}),\\
\label{circ-log}
\delta a^{\rm ST}_{4,\rm nonloc,\rm log}&=-\frac{4}{3}\nu\left[2\delta_++\frac{\bar{\gamma}_{AB}(\bar{\gamma}_{AB}+2)}{2}\right],\\
\delta b_{3,\rm nonloc,0}^{\rm ST}&=\frac{4}{3}\nu\left[2\delta_++\frac{\bar{\gamma}_{AB}(\bar{\gamma}_{AB}+2)}{2}\right]\left(\frac{21}{2}-16\ln 2\right),\\
\delta b_{3,\rm nonloc,\rm log}^{\rm ST}&=0,\\
\delta q_{3,\rm nonloc,0}^{\rm ST}&=\frac{4}{3}\nu\left[2\delta_++\frac{\bar{\gamma}_{AB}(\bar{\gamma}_{AB}+2)}{2}\right]\left(-\frac{31}{4}-\frac{256}{3}\ln 2+\frac{243}{4}\ln 3\right),\\
\delta q_{3,\rm nonloc,\rm log}^{\rm ST}&=0~.
\end{align}
\end{widetext}
The ST tensor correction $\delta a^{\rm ST}_{4,\rm nonloc}$ for the circular orbit case, Eqs.~\eqref{circ-0}-\eqref{circ-log}, matches with the results obtained in Paper I (see, Eqs.(5.25)-(5.26)) except a negative sign in Eq.~\eqref{circ-log}. The negative sign is due to the difference in the definition of $\delta a^{\rm ST}_{4,\rm nonloc}$ in Eq.~\eqref{tail-decompose} used in this work with the Eq.~(5.24) of Paper I.
\section{Conclusions}
In Paper I, building upon the results of \cite{Bernard:2018ivi} for massless scalar-tensor theory, we determined the EOB coefficients at 3PN order though restricting ourselves to local-in-time part of the dynamics and nonlocal-in-time and tail contributions only for the circular case. In the present paper, we derived the complete nonlocal-in-time EOB coefficients starting from the nonlocal-in-time Lagrangian of Ref.~\cite{Bernard:2018ivi}. First, we derived the two-body \textit{conserved} ordinary Hamiltonian (dependent only on positions and momenta) for nonlocal-in-time part by two methods: (i) non-order-reduced nonlocal Hamiltonian using nonlocal phase shift (see, Ref.~\cite{Bernard:2015njp,Bernard:2016wrg} for GR), and (ii) order-reduction of nonlocal dynamics to local ordinary action-angle Hamiltonian \cite{Damour:2015isa}. We then expressed the effective Hamiltonian in Delaunay variables to recast the order-reduced ordinary action-angle Hamiltonian into equivalent, 3PN-accurate, nonlocal part of EOB potentials $(A, B, Q_e)$, see Eqs. (4.12)-(4.17).
By combining the results of Paper I and the present work, we could transcribe the two-body Hamiltonian into equivalent 3PN-accurate EOB potentials $(A, B, Q_e)$ for both local-in-time and nonlocal-in-time part of dynamics.
\\
\\
\textbf{Note:} During the preparation of the final manuscript of this work, the author became aware of the independent effort which recently arrived on arXiv \cite{Julie:2022qux}.
\begin{acknowledgements}
The author is grateful to P. Rettegno, M. Agathos and A. Nagar for useful discussions and suggestions during the preparation of this work. The author is jointly funded by the University of Cambridge Trust, Department of Applied Mathematics and Theoretical Physics (DAMTP), and Centre for Doctoral Training, University of Cambridge.
\end{acknowledgements}
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Home » Newsletter sign up » Permaculture Publishers Now Reach the Entire English Speaking World
Permaculture Publishers Now Reach the Entire English Speaking World
On the 1st January 2013, Permanent Publications will be joining forces with The GMC Group.
The GMC Group will provide sales representation as well as the marketing and distribution of all Permanent Publications' book titles, which will help its publications reach a much broader and much more global market.
Permanent Publications is a growing business, established in 1990 with the aim to publish and produce books, eBooks, DVDs and websites that provide information to encourage people to live more healthily, self-reliant and ecological sound lives. Permanent Publications specialises in permaculture, a practical toolkit for creating low carbon ways of living. Based on natural principles found in ecosystems, these can be used by anyone, anywhere – in our homes, gardens or the wider community.
Permanent Publications also publishes Permaculture magazine, launched in 1992, available quarterly in print and as a digital edition and an App. The magazine has grown from tiny beginnings and now has over 100,000 readers in readers in 77 countries.
"Permaculture magazine has always been our flagship publication, with readers all over the world. We have always wanted to make our books similarly available. The GMC group, with its representation all over the English speaking world, will make that possible," says Maddy Harland, Permanent Publications' CEO.
Based in East Sussex, The GMC Group publishes and distributes over 4,500 books and magazines nationally and internationally, with a very diverse range of titles, from photography to woodworking. They have a network of dedicated employees, professionals and individuals, all focusing on building lasting relationships with their customers, while providing the best products available.
The new agreement will enable distribution into stores all over the UK, Ireland, Western, Eastern and Southern Europe, Scandinavia, Middle East, South America, Asia, Africa and The Caribbean. Permanent Publications will retain Chelsea Green as their North Amercian distributors, a longstanding relationship between two pioneering green publishers.
Tony Rollinson, Sales Director of Permanent Publications, comments on this exciting new development:
"Permanent Publications is a growing business that is rising to the challenge of making books available to the increasing number of people who wish to enjoy growing their own food, saving themselves money and vitally to tread more lightly upon our planet. To be found by this ever-increasing number of people, we needed a distributor who could help us be in the right bookshops, garden centres, libraries and specialist retailers. But, as well as this, we also wanted to grow with a company that holds a similar ethos and with staff who understand and empathise with our vision – The GMC Group fulfil this criteria admirably.
"GMC's marketing, distribution and sales team are already putting our titles in front of a new audience and 2013 looks like being a breakthrough year, both for our titles and the permaculture movement that we are a part of."
For more details about our distribution network and The GMC Group see Trade Contacts & Distribution.
By hayley Posted in News on Dec 21, 2012
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Q: PKI with openSSL and Windows 2008 Could anyone tell me about advantages and disadvantages of PKI with openSSL and Windows Certificate authorities ?
A: OpenSSL-based systems have no problems using Microsoft PKI certificates. They're just certificates, same as you'd get from the pay vendors.
Where the issues creep in is the Certificate Signing Requests. MS-PKI requires extended attributes on the CSRs to correlate with Certificate Templates it track internal to the system. Creating a correct CSR using OpenSSL tools can take some work to get right.
However, if you have the ability to just create certificates and CSRs aren't required, doing that and exporting the certificate will make things go a lot easier.
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"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
The notion of entanglement entropy (and more generally quantum entanglement) looms large in theoretical physics today.
Entanglement entropy may be a good order parameter for topological phase transitions in condensed matter systems. For conformal field theories in 1+1 dimensions, numerical computation of the entanglement entropy provides a rapid way to calculate the central charge $c$. In relativistic field theories more generally,
certain special kinds of entanglement entropy show monotonicity properties under renormalization group flow
\cite{Casini:2006es,Casini:2012ei}. See \cite{Calabrese:2009qy} and \cite{EislerPeschel} for reviews.
To compute the entanglement entropy for a quantum mechanical system, we must first divide the associated Hilbert space up into two pieces. Usually, the division is made with respect to spatial regions $A$ and complement $\bar A = B$.
We find the reduced density matrix $\rho_A \equiv \operatorname{tr}_{B} \rho$ by tracing over the degrees of freedom in $B$. Finally, the entanglement entropy is defined to be
\begin{equation}
S \equiv - \operatorname{tr} \rho_A \log \rho_A \ .
\end{equation}
It is surprising that even for what many consider to be the simplest field theoretic system -- a massive scalar field in 1+1 dimensions -- the entanglement entropy has thus far been computed analytically only in certain limits.
In the limit $m=0$, one can use results from conformal field theory \cite{Calabrese:2004eu,Korepin:2004zz}.
In particular, for the massless scalar field on the cylinder $\mathbb{R} \times S^1$ where ${\mathbb R}$ is interpreted as the time direction, one has
\begin{equation}
\label{circleSE}
S = \frac{1}{3} \log \left( \frac{L}{\pi \epsilon} \sin \frac{\pi \ell}{L} \right) + c_0 \ ,
\end{equation}
where $L$ is the circumference of the $S^1$, $\ell$ is the length of the interval, $\epsilon$ is a UV regulator and $c_0$ is a constant that depends on the regulation scheme. (In fact, for the massless scalar, there is an additional IR divergence, and $c_0$ depends also on an IR cutoff.)
Reinterpreting $S^1$ as a Euclidean time direction, one obtains a result at nonzero temperature $T=1/\beta$ for the scalar on $\mathbb{R}$.
\begin{equation}
\label{thermalSE}
S = \frac{1}{3} \log \left( \frac{\beta}{\pi \epsilon} \sinh \frac{\pi \ell}{\beta} \right) + c_0 \ .
\end{equation}
When $m \neq 0$ for the scalar field on $\mathbb{R}^2$, Huerta and Casini \cite{Casini:2005zv}
have shown that the entanglement entropy can be computed from the solution to a certain Painlev\'{e} equation. Their work allows analytic access to the small and large mass limits.
For $m\ell \ll 1$, one obtains
\begin{equation}
S \sim \frac{1}{3} \log \frac{\ell}{\epsilon} + \frac{1}{2} \log\left( \frac{\log(m \epsilon)}{\log (m \ell)} \right) \ ,
\end{equation}
while for $m\ell \gg 1$, one finds instead exponential suppression\footnote{%
A generalization was obtained by Doyon and collaborators \cite{Doyon:2008} and \cite{Doyon2} allowing for multiple masses.}
\begin{equation}
S \sim \frac{1}{16} \sqrt{\frac{\pi}{m \ell}} \, e^{-2 m \ell} \ .
\end{equation}
Ideally, one would like to understand the case where $m$, $T$, and $1/L$ are all nonzero.
Numerically, the entanglement entropy can be computed with ease using a generalization \cite{Peschel} of the procedure introduced by Srednicki \cite{Srednicki}. One realizes the scalar field as the continuum limit of an $N$-site harmonic chain.
For such a chain, one introduces two point functions $\langle \phi_i \phi_j \rangle$ and $\langle \pi_i \pi_j \rangle$ of the oscillator positions and conjugate momenta respectively. Restricting now to an interval $n \epsilon = \ell<L$ where $1 \leq i,j, k \leq n$, one constructs the $n \times n$ matrix
\begin{equation}
\label{defC2}
(C^2)_{ij} \equiv \sum_{k=1}^n \langle \phi_i \phi_k \rangle \langle \pi_k \pi_j \rangle \ .
\end{equation}
The entanglement entropy is then
\begin{equation}
\label{SEfromC2}
S = \operatorname{tr} \left[ (C +1/2) \ln (C+1/2) - (C-1/2) \ln (C-1/2) \right] \ .
\end{equation}
To our knowledge, this quantity has not been computed analytically for the real scalar field with two or more of the quantities $m$, $T$, and $1/L$ nonzero. Happily, with today's desktop computers, it is relatively quick to diagonalize $C$ numerically for $N \sim 10^3$. Ref.\ \cite{BoteroReznik} provides a numerical analysis of the harmonic chain using this approach.
In this paper, we take some steps toward an analytic understanding of the eigenvalues of $C$.
As noted in \cite{BoteroReznik}, the parity operator $P$ commutes with $C$ where parity here is a reflection of the circle $S^1$ with respect to the midpoint of the interval. Thus, one may divide $C$ into even and odd parity blocks $C_e$ and $C_o$.
We compute the two partial traces $\operatorname{tr} C_e^2$ and $\operatorname{tr} C_o^2$ in the limit $m, T \ll 1/L$. As the spectrum of $C^2$ is bounded below by 1/4, these traces give us upper bounds on the two largest eigenvalues of $C$. A variational approach gives a lower bound to the largest (parity even) eigenvalue. These bounds in turn give us some intuition for the $m$, $T$, and $L$ dependence of the entanglement entropy in the limit $m,T \ll 1/L$.
The original motivation for this project came from our interest in the Ryu-Takayanagi proposal \cite{Ryu:2006bv} for computing the entanglement entropy of field theories with dual holographic classical gravity descriptions.
Given two complementary regions $A$ and $B$ in the field theory, the Ryu-Takayanagi proposal associates a nonzero
$S_A - S_B$ to gravity descriptions with black holes, while in the absence of such defects $S_A = S_B$.
In the dual field theory, the existence of a black hole typically implies deconfined gauge theory degrees of freedom \cite{Witten:1998zw,Herzog:2006ra}.
We may contrast this result with the quantum mechanical point of view where at $T=0$, the density matrix is constructed from a pure state. (We are assuming the existence of a unique ground state.)
It follows from a Schmidt decomposition of the Hilbert space that for pure states $S_A = S_B$ (see for example \cite{EislerPeschel}).
However, at any nonzero temperature, regardless of the presence of deconfined degrees of freedom, the density matrix is not constructed from a pure state and one would generically expect $S_A \neq S_B$. As gauge theories are more difficult to study than the free scalar field and as the entanglement entropy of the free scalar field has not yet been completely understood, our toy model of confinement in this paper is a 1+1 dimensional massive scalar field on a circle at $T>0$. Morally, the regime $T \ll m$ can be thought of as ``confining''.\footnote{%
Klebanov et.~al.\ \cite{Klebanov:2007} were the first to consider the entanglement entropy of confining theories from a holographic perspective. Their work at zero temperature was later followed up by lattice computations \cite{Buividovich, Velytsky, Nakagawa:2011su}.} One of our results is that in this regime, the entanglement entropy difference does not vanish but rather scales
as\footnote{%
After finishing this work, we became aware of ref.\ \cite{Cacciatori:2008qs} where the same exponential behavior was found for a ``renormalized thermal entropy'' similar in some respects to the entanglement entropy we study here.}
\[
S_A - S_B \sim e^{-m/T} \ .
\]
\section{From the Harmonic Chain to the Scalar Field}
Consider the Hamiltonian for a real free massive scalar field on a circle of circumference $L$ at $T>0$:
\begin{equation}
H = \frac{1}{2} \int dx \left[ \pi(x)^2 + (\partial_x \phi(x))^2 + m^2 \phi(x)^2 \right] \ .
\end{equation}
We discretize the circle into $N$ points where $L = N \epsilon$:
\begin{equation}
H = \frac{1}{2 \epsilon} \sum_{j=1}^N \left[ \pi_j^2 + (\phi_{j+1} - \phi_{j})^2 + m^2 \epsilon^2 \phi_j^2 \right] \ ,
\end{equation}
where $\pi(j \epsilon) = \pi_j/ \epsilon$ but $\phi(j \epsilon) = \phi_j$.
The thermal density matrix can be written in terms of $H$ in the standard way:
\begin{equation}
\rho = \frac{e^{-H/T}}{ \operatorname{tr} (e^{-H/T})} \ ,
\end{equation}
and expectation values are defined via $\langle X \rangle \equiv \operatorname{tr} (\rho X)$.
A short calculation yields the two point functions of the oscillator positions $\phi_j$ and their conjugate momenta $\pi_j$:
\begin{eqnarray}
\label{phiphi}
\langle \phi_j \phi_k \rangle &=&
\frac{1}{2N} \sum_{a=0}^{N-1} \frac{1}{\epsilon \omega_a} \coth \left( \frac{ \omega_a }{2 T} \right) \cos \left( \frac{2 \pi (j-k)a}{N} \right) \ , \\
\label{pipi}
\langle \pi_j \pi_k \rangle &=&
\frac{1}{2N} \sum_{a=0}^{N-1} \epsilon \omega_a \coth \left( \frac{ \omega_a }{2 T} \right) \cos \left( \frac{2 \pi (j-k)a}{N} \right) \ ,
\end{eqnarray}
where
\[
\omega_a^2 = m^2 + \frac{4}{\epsilon^2} \sin^2 \frac{\pi a}{ N} \ .
\]
From eqs.\ (\ref{defC2}) and (\ref{SEfromC2}), we may compute the entanglement entropy from the matrix $C^2 = \langle \pi \pi \rangle \cdot \langle \phi \phi \rangle$ where the two point functions are now restricted to the interval $A$: $-s \leq j,k \leq s$. In terms of $n$, we have $2s+1=n$. For simplicity, we choose $n$ to be an odd number. Any dependence on the parity of $n$ should disappear in the large $N$ limit.
The Hamiltonian $H$ is a set of $N$ coupled harmonic oscillators. Diagonalizing the Hamiltonian, one finds $H = \sum_a \omega_a b_a^\dagger b_a$ where $[b_a, b_b^\dagger] = \delta_{ab}$. Surprisingly for a free scalar field, the reduced density matrix $\rho_A \sim e^{-H_A}$ can be written in terms of a similar entanglement Hamiltonian $H_A = \sum_k \epsilon_k b_k^\dagger b_k$ (see for example \cite{EislerPeschel}). Moreover, there is a one-to-one correspondence between eigenvalues $\lambda_k$ of $C^2$ and the energies $\epsilon_k$:
\begin{equation}
\lambda_k = \frac{1}{4} \coth^2 \frac{\epsilon_k}{2} \ .
\end{equation}
As the $\epsilon_k$ are real, we conclude that $\lambda_k \geq 1/4$.
\section{Taking Traces}
For a region $-s \leq k \leq s$, the matrix $C^2$ commutes with the parity operator\footnote{%
Note that $C^2$ commutes with the parity operator for both odd and even $n$.
For example, if we indexed $C^2$ from $1\leq k \leq n$, parity would send $k \to n-k+1$.
}
which sends $k \to -k$.
Thus, we can decompose $C^2$ into even and odd parity pieces, $C^2 = C_e^2 + C_o^2$.
The matrices $C_o^2$ and $C_e^2$ are then given by
\begin{eqnarray}
C_e^2 &=&
\frac{1}{4N^2} \sum_{a,b} \frac{\omega_a}{\omega_b} \coth \left( \frac{\omega_a}{2 T} \right)
\coth \left(\frac{\omega_b}{2 T} \right) \frac{\sin \frac{\pi n(a-b) }{N}}{\sin \frac{\pi (a-b)}{N}}
\cos \frac{2\pi j a}{N} \cos \frac{2 \pi k b}{N} \ , \\
C_o^2 &=&
\frac{1}{4N^2} \sum_{a,b} \frac{\omega_a}{\omega_b} \coth \left( \frac{\omega_a}{2 T} \right)
\coth \left(\frac{\omega_b}{2 T} \right) \frac{ \sin \frac{\pi n(a-b) }{N}}{\sin \frac{\pi (a-b)}{N}}
\sin \frac{2\pi j a}{N} \sin \frac{2 \pi k b}{N} \ ,
\end{eqnarray}
While our main interest is a circle with periodic boundary conditions, the eigenvalues of $C_e$ and $C_o$ also allow us to compute the entanglement entropy for an interval of length $s$ sitting at one end of a strip of length $N/2$. The matrix $C_o$ gives the two point function of a strip with Dirichlet boundary conditions, while $C_e$ corresponds to Neumann boundary conditions.
The numerics suggest that for small masses ($m L \ll 1$) and low temperatures ($T L \ll 1$), the matrix $C^2$ has only a handful of eigenvalues which are significantly different from $1/4$.
The largest of these eigenvalues corresponds to an eigenvector with even parity, while the second largest has odd parity. We approximate these eigenvalues by computing $\operatorname{tr} C_e^2$ and $\operatorname{tr} C_o^2$. We find in the even sector that
\begin{eqnarray}
\label{trCe2}
\operatorname{tr} C_e^2 &=&
\frac{1}{2\pi m L} \coth \left(\frac{m}{2T} \right) \left[ \gamma + \ln \left( \frac{4N \sin ( \pi r)}{\pi} \right) \right]
+ \frac{r^2}{4} \operatorname{csch}^2 \left(\frac{m}{2T}\right)
\nonumber \\
&&
+\frac{1}{4} \Biggl[ s +
\frac{11}{12} - \frac{1}{\pi^2} \nonumber \\
&& + \frac{1}{2 \pi^2} \left( -2 + \gamma + 4 \ln \frac{2N}{\pi} - 3 \ln \frac{4 N \sin ( \pi r)}{\pi} \right)
\left( \gamma + \ln \frac{4N \sin ( \pi r)}{\pi} \right) \Biggr] \nonumber
\\
&&
- \frac{3m L}{32 \pi^3} \coth \left( \frac{m}{2 T} \right)
\left[ \operatorname{Li}_3(e^{2 \pi i r})+ \operatorname{Li}_3(e^{-2 \pi i r}) - 2 \zeta(3) \right]
\nonumber \\
&&
+ O((m L)^2, e^{-2\pi/ T L}, \log N /N)
\ ,
\end{eqnarray}
and that in the odd sector
\begin{eqnarray}
\label{trCo2}
\operatorname{tr} C_o^2 &=&
\frac{1}{4} \Bigl[
s + \frac{1}{12} - \frac{3}{2 \pi^2} +
\frac{1}{2\pi^2}
\left( \gamma-1+ \ln \frac{4N \sin ( \pi r)}{\pi}\right)^2
\Bigl]
\nonumber \\
&&
+ O((m L)^2, e^{-2\pi/ T L}, \log N /N)\ ,
\end{eqnarray}
where $r = \ell/L$ and $2s+1$ is the number of lattice sites. We make some brief remarks about how these traces were computed below.
Because of the relation $\lambda_k = \frac{1}{4} \coth^2(\epsilon_k/2)$ between the entanglement spectrum and the eigenvalues of $C^2$, we know that the eigenvalues of $C^2$ are bounded below by $1/4$. The largest even eigenvalue $\lambda_e$ and odd eigenvalue $\lambda_o$ are thus bounded above by
\begin{eqnarray}
\label{evenupper}
\lambda_e &\leq& \operatorname{tr} C_e^2 - \frac{s}{4} \ , \\
\label{oddupper}
\lambda_o &\leq& \operatorname{tr} C_o^2 - \frac{s-1}{4} \ .
\end{eqnarray}
We can also put a lower bound on $\lambda_e$ by using the variational principle and a ``trial wave function''. In this case, we use a constant trial wave function, $\psi_e = (1, 1, \ldots, 1) / \sqrt{n}$. The expectation value then provides a lower bound:
\begin{eqnarray}
\label{lowerbound}
\lambda_e &\geq& \langle \psi_e | C_e^2 | \psi_e \rangle \\
&=& \frac{1}{2\pi mL} \coth \left(\frac{m}{2T} \right) \left[ \gamma + \ln \left( \frac{4N \sin ( \pi r)}{\pi} \right) \right]
+ \frac{1}{12}
\nonumber
\\
&&
- \frac{i}{8 \pi^3 r} \left[ \gamma + \ln \left( \frac{4 N \sin(\pi r)}{\pi} \right) \right]
\left[ \operatorname{Li}_2(e^{2 \pi i r}) - \operatorname{Li}_2(e^{-2 \pi i r}) \right]
\nonumber \\
&&
-\frac{r^2}{4} \left[ \frac{1}{3} - \coth^2 \left( \frac{m}{2T} \right) \right] + O(m L,e^{-2\pi/ TL}, \log N /N)
\nonumber
\ .
\end{eqnarray}
Figure \ref{boundplot} demonstrates that our upper and lower bounds provide relatively good estimates of the two largest eigenvalues at $T=0$.
We could
try to produce an analytic lower bound on $\lambda_o$ by similar methods.
However,
simple trial wave functions such as $(\psi_o)_j \sim \sin (\pi j / N)$ or $(\psi_o)_j \sim j$
do not seem to give strong lower bounds numerically and are harder to work with analytically than the constant trial wave function used above in the even case.
\begin{figure}
\begin{center}
a) \includegraphics[width=2.5in]{boundplot}
b)\includegraphics[width=2.5in]{bound2plot}
\end{center}
\caption{
The largest (a) and second largest (b) eigenvalue of $C^2$ plotted against the interval length for $mL=1/10$, $T=0$, and $N=1000$. The points are numerically computed. The curves above the points are the analytic upper bounds (\ref{evenupper}) and (\ref{oddupper}) computed from the traces. The solid curve below the points on the left is the lower bound (\ref{lowerbound}) computed from the variational principle.
\label{boundplot}
}
\end{figure}
The zero mode $a=0$ terms in $\langle \phi \phi \rangle$ and $\langle \pi \pi \rangle$ have a large influence on the structure of these traces in our $m, T \ll 1/L$ limit.
As these zero modes have even parity, they do not contribute to $C_o^2$.
For example, note that $\operatorname{tr} C_e^2 = O(1/m L)$ is much larger than $\operatorname{tr} C_o^2 = O(1)$ because
the zero mode $a=0$ term in $\langle \phi \phi \rangle$
is $O(1/m L)$ but only contributes to the even sector of $C^2$.
Also note that only $\operatorname{tr} C_e^2$ depends on $T$. The reason is that $\coth (\omega_a / 2T )\approx 1$ up to exponentially suppressed terms except when $a=0$.
Another interesting feature of these traces is their behavior under the exchange of the interval $A$ with its complement $B$. By translation invariance, this exchange can be implemented by sending $r \to 1 -r$. At $T=0$, both $\operatorname{tr} C_o^2$ and $\operatorname{tr} C_e^2$ are invariant under this transformation. This invariance is expected in order to guarantee that $S_A = S_B$.
For $T\neq 0$, the breaking of this symmetry is due entirely to the
$r^2 \operatorname{csch}^2 (m/2T)$ term in $\operatorname{tr} C_e^2$.\
This symmetry breaking term comes from multiplying the $a=0$ zero modes in $\langle \phi \phi \rangle$ and $\langle \pi \pi \rangle$ together.
Figure \ref{bound2plot} demonstrates that $\operatorname{tr} C_e^2$ gives a remarkably good estimate of the temperature dependence of the largest eigenvalues for regions $A$ and $B$, and also for their difference.
\begin{figure}
\begin{center}
a) \includegraphics[width=2.4in]{largestTplot}
b)\includegraphics[width=2.4in]{ilargestTplot}
\end{center}
\begin{center}
c)\includegraphics[width=2.4in]{difflargestplot}
\end{center}
\caption{
The largest eigenvalue of $C^2$ as a function of temperature for $mL = 1/50$: a) $\ell/L=1/5$; b) $\ell/L=4/5$; c) the difference between the two for a lattice with $N=200$.
The points are numerical while the curve is the upper bound computed from $ \operatorname{tr} C_e^2$.
\label{bound2plot}
}
\end{figure}
We should say a few words about the lengthy computation performed to obtain (\ref{trCe2}), (\ref{trCo2}), and (\ref{lowerbound}).
Consider first the $O(1/mL)$ contribution to $\operatorname{tr} C_e^2$:
\begin{equation}
\label{trCe2initial}
\operatorname{tr} C_e^2 = \frac{1}{2 m L} \coth \left(\frac{m}{2T} \right) f(n,N) + O(m L)^0 + O(e^{-2 \pi / TL}) \ ,
\end{equation}
where
\begin{equation}
f(n, N) \equiv \frac{1}{N} \sum_{a=1}^N \frac{ \sin^2 (\pi a n / N) }{\sin (\pi a / N)} = \frac{1}{N} \sum_{j=1}^{n} \cot \frac{\pi}{N} (j-1/2 ) \ .
\end{equation}
We want to evaluate this sum in the continuum limit where $n$ and $N$ are both large but $r = n/N$ is held fixed between zero and one.
Replacing the sum over $j$ by an integral introduces unacceptably large errors because of the divergence at $j=1/2$. Instead, we compute a related integral that does not have this divergence:
\begin{eqnarray}
\label{fnN}
f(n,N)
&\approx& \int_{1/N}^r \left( \cot \pi (x - 1/2N) - \frac{1}{\pi(x-1/2N)} \right) dx \nonumber \\
&& \; \; \; \; + \frac{1}{N}\sum_{j=1}^n \frac{N}{\pi (j-1/2)} \\
&=& \frac{1}{\pi} \left[ \ln \left( \frac{4 N \sin (\pi r)}{\pi} \right) + \gamma \right]
+ O(1 /N^2) \ .
\end{eqnarray}
Calculating the $O(m L)^0$ and $O(mL)$ terms is a more complicated enterprise.
As mentioned already above, one contribution to $C_e^2$ comes from multiplying the zero modes in $\langle \phi \phi \rangle$ and $\langle \pi \pi \rangle$ together and yields $r^2 \operatorname{csch}^2(m/2T)$.
The remaining order one pieces can be computed from the matrix $C_e^2$ with the zero modes removed in the limit $m=0=T$:
\begin{eqnarray}
\label{CeOzero}
(\tilde C_e^2)_{jk} &=& \frac{1}{4N^2} \sum_{a,b=1}^{N-1} \sum_{l=-s}^s
\frac{\sin \frac{\pi a}{ N}}{\sin \frac{\pi b}{ N}} \cos \frac{2\pi l a}{N} \cos \frac{2 \pi l b}{N} \cos \frac{2 \pi j a}{N} \cos \frac{2 \pi k b}{N} \ .
\end{eqnarray}
Similarly, the $O(m L)^0$ contribution to $\operatorname{tr} C_o^2$ can be calculated from the
$m=T=0$ limit of the matrix $C_o$:
\begin{eqnarray}
\label{CoOzero}
(\tilde C_o^2)_{jk} &=& \frac{1}{N^2} \sum_{a,b=1}^{N-1} \sum_{l=1}^s
\frac{\sin \frac{\pi a}{ N}}{\sin \frac{\pi b}{ N}} \sin \frac{2\pi l a}{N} \sin \frac{2 \pi l b}{N} \sin \frac{2 \pi j a}{N} \sin \frac{2 \pi k b}{N} \ .
\end{eqnarray}
The $O(mL)$ term of $(C_e^2)_{jk}$ comes from zero modes pieces of $C^2$ where either $a=0$ in the $\langle \phi \phi \rangle$ sum or $a=0$ in the $\langle \pi \pi \rangle$ sum:
\begin{equation}
\label{CeOone}
\frac{3 m L}{16 N^3} \coth \left( \frac{m}{2T} \right) \sum_{b=1}^{N-1} \sum_{l=-s}^s \frac{\cos \frac{2 \pi l b}{N} \cos \frac{2 \pi k b}{N}}{\sin \frac{\pi b}{N}} \ .
\end{equation}
(For $(C_e)_{jk}$, the indices have the range $-s \leq j,k \leq s$, while for $(C_o)_{jk}$, we restrict to $1 \leq j,k \leq s$.)
In the appendix,
we describe how to perform the sums (\ref{CeOzero}), (\ref{CoOzero}), and (\ref{CeOone}) along with (\ref{lowerbound}) in the the large $N$ limit with $s/N$ held fixed.
\section{Raising the Temperature}
We present three arguments that the entanglement entropy depends exponentially on the ratio $m/T$ in the limit $T \ll m$. The first argument is heuristic and relies on the structure of the matrix $C^2$. The second argument is based on our earlier calculation of $ \operatorname{tr} C_e^2$.
The third argument is based on numerical evidence.
We would like to show two things. The first is that for a fixed interval $A$,
\begin{equation}
S(T) - S(0) \sim e^{-m/T} \ .
\label{STone}
\end{equation}
The second is that for two complementary intervals $A$ and $\bar A = B$,
\begin{equation}
S_A - S_B \sim e^{-m/T} \ .
\label{STtwo}
\end{equation}
The first argument relies on the fact that the temperature dependence of $C^2$ comes entirely from the factors of
$\coth(\omega_a / 2 T)$ in $\langle \phi \phi \rangle$ and $\langle \pi \pi \rangle$. The frequency $\omega_a$ is bounded below by $m$. Thus we conclude that
\begin{equation}
\coth\left(\frac{\omega_a}{2 T} \right) \leq \coth \left( \frac{m}{2T} \right ) = 1 + 2 e^{-m/T} + O(e^{-2m/T}) \ .
\end{equation}
In other words, the matrix $C$ has a low temperature expansion of the form
\begin{equation}
C(T) = C(0) + e^{-m/T} \delta C + \ldots
\end{equation}
where the ellipsis denotes terms that are more exponentially suppressed.
Now if $C(T)$ has such an expansion, then the eigenvalues $\nu_k(T) = \nu_k(0) + e^{-m/T} \delta \nu_k +\ldots$ will as well.
Assuming $\nu_k(0) -1/2 \gg e^{-m/T}$, expanding eq.\ (\ref{SEfromC2}) in the small $T$ limit, one concludes that the entanglement entropy for a single interval shifts by an amount
\begin{equation}
\delta S = 2 \sum_k [ \ln (\nu_k(0) + 1/2) - \ln (\nu_k(0) - 1/2) ] \delta \nu_k e^{-m/T} + \ldots\ ,
\end{equation}
implying the scaling (\ref{STone}).
Assuming $\delta S$ is different for an interval and its complement, one also concludes the scaling (\ref{STtwo}).
While, the numerical evidence we present below suggests both scalings (\ref{STone}) and (\ref{STtwo}) are correct, there are some loop holes in our argument. An obvious problem is that the $e^{-m/T}$ term in the small $T$ expansion may vanish; the temperature dependence may be of the form $e^{M/T}$ for some $M>m$.
A more subtle loop hole involves the fact that many of the $\nu_k(0)$ are close to $1/2$. In this case, the correction to the entanglement entropy $\delta S$ can scale as $(m/T) e^{-m/T}$ instead of just $e^{-m/T}$. Numerically, we see no evidence for this behavior.
Instead, in these cases we find that the logarithmic enhancement is not enough to make up for the smallness of $\delta \nu_k$; these eigenvalues contribute negligibly to the entanglement entropy.
The second argument for the scalings (\ref{STone}) and (\ref{STtwo}) is based on using
$\operatorname{tr} C_e^2$ as an estimate of the largest eigenvalue $\lambda_e$.
Using $\operatorname{tr} C_e^2$, we estimate the contribution of $\lambda_e$ to $S$ and infer the scalings from this contribution.
The temperature dependence of a single interval
comes principally from the leading $\coth(m/2T) /m L$ term in (\ref{trCe2}). One finds agreement with (\ref{STone}):
\begin{equation}
\left. [S(T) - S(0)]\right|_{\lambda_e} \sim e^{-m/T} \ .
\end{equation}
Next we consider the entanglement difference $S_B - S_A$.
This type of temperature dependence comes from the $r^2 \operatorname{csch}^2(m/2T)$ piece of (\ref{trCe2}). One finds agreement with (\ref{STtwo}):
\begin{equation}
\left. [S_B(T) - S_A(T)] \right|_{\lambda_{e,A}, \lambda_{e,B}} \sim \frac{\pi}{2} \frac{m L}{\log N}\left(1 - 2r \right)e^{-m/T} \ .
\label{dSapprox}
\end{equation}
We should emphasize that using $\operatorname{tr} C_e^2$ and the largest eigenvalue $\lambda_e$ to estimate the temperature scalings is flawed.
An obvious limitation is that we only have a result for $\operatorname{tr} C_e^2$ in the limit $mL \ll 1$ while we expect the temperature scalings to hold more generally.
A less obvious limitation is that
despite the fact that $\lambda_e$ is much larger than the other eigenvalues in the small mass limit, the logarithms in (\ref{SEfromC2}) play a democratizing role and let smaller eigenvalues contribute substantially to the entanglement entropy.
For example, in this small mass limit numerical analysis shows that the dominant contribution to $S_A-S_B$ comes from the second largest even eigenvalue (see figure \ref{evalplot}).
\begin{figure}
\begin{center}
\includegraphics[width=3.2in]{evalplot}
\end{center}
\caption{
The contribution to $\delta S = S_A - S_B$ for the ten largest pairs of eigenvalues $\lambda_{j,A}$ and $\lambda_{j,B}$, arranged from largest to smallest. In this plot $mL = 0.02$, $m/T = 1$ and $\ell/L = 1/5$ for region $B$. Note that the odd parity eigenvalues do not contribute. ($N=200$ was used for this plot.)
}
\label{evalplot}
\end{figure}
Our most convincing evidence for the scalings (\ref{STone}) and (\ref{STtwo}) is numerical and is presented in figures \ref{SEAinterval} and \ref{Sdiffplot}.
Figure \ref{SEAinterval} demonstrates unambiguous evidence for (\ref{STone}), not only for $mL \ll 1$ but also for $mL > 1$.
Figure \ref{Sdiffplot}a displays unambiguous evidence for (\ref{STtwo}), again both for small and large values of $mL$.
More ambitiously, we can try to investigate numerically whether the $m L (1-2r) / \log N$ behavior of eq.\ (\ref{dSapprox}) is correct as well.
Figure \ref{Sdiffplot}a provides evidence for the $mL$ scaling. Figure \ref{Sdiffplot}b provides some limited evidence for the $1-2r$ behavior for large values of $mL$ and for intervals with $r \sim 1/2$. However, we find no evidence for the $\log N$ behavior of (\ref{dSapprox}).
\begin{figure}
\begin{center}
a) \includegraphics[width=2.7in]{SEAintervalsmall}
b) \includegraphics[width=2.7in]{SEAintervallarge}
\end{center}
\caption{
A log plot of the entanglement entropy $\delta S = S(T) - S(0)$ vs.\ $m/T$ with an interval size $\ell/L = 3/10$. The points are numerically computed, and the line $\log(\delta S) = -m/T$ is a guide to the eye:
a) $m L = 5 \times 10^{-3}$; b) $m L = 5$.
(For both plots, the points were computed with $N=50$, 100, 200, and 400.
The data points for different values of $N$ all lie roughly on top of each other.)}
\label{SEAinterval}
\end{figure}
\begin{figure}
\begin{center}
a) \includegraphics[width=2.8in]{Sdiffplot}
b)
\includegraphics[width=2.6in]{rplot}
\end{center}
\caption{
a)
A log plot of the entanglement entropy difference $\delta S = S_A - S_B$ vs.\ $m/T$ for $m L = 5$ and $5 \times 10^{-3}$, and an interval B of size $\ell/L= 1/5$. At fixed $m/T$, the larger mass points lie below the smaller ones. The line $\log(\delta S / m L) = -m/T$ is a guide to the eye. (The lattice was taken to have size $N=200$, but there is no noticeable difference between this graph and a graph with $N=100$.)
b)
The entanglement entropy difference $\delta S$ vs.\ $\ell/L$ for (from bottom to top) $mL = 5 \times 10^{-3}$, $2$, and $5$. The mass to temperature ratio is $m/T = 10$. The line $e^{m/T} \delta S / m L = 3m / T - 3/2$ is a guide to the eye.
(The lattice was taken to have $N=400$, but there is no difference between this graph and a graph with $N=200$.)
}
\label{Sdiffplot}
\end{figure}
\section{Discussion}
As mentioned in the introduction, the original motivation for this paper came from the Ryu-Takayanagi proposal \cite{Ryu:2006bv} for computing the entanglement entropy of field theories with holographic dual classical gravity descriptions.
In their proposal, the field theory lives on the boundary of the space-time in the dual description. Let $C$ be the curve that separates region $A$ from region $B$ in the field theory. Let $C$ also be the boundary of a minimal surface $M$ that falls into the space-time. The proposal is that the entanglement entropy is proportional to the area of $M$:
\begin{equation}
S_A = \frac{\operatorname{Area}(M)}{4 G_N} \ ,
\end{equation}
where $G_N$ is Newton's constant. Assuming a unique such $M$, the entanglement entropy of a region and its complement are always equal, $S_A = S_B$.
When the space-time contains a black hole, Ryu-Takayanagi modified their proposal to account for the existence of two minimal surfaces $M_A$ and $M_B$.
The entanglement entropy for $A$ must be computed from the surface $M_A$ that is deformable into $A$. Correspondingly, for region $B$, we must use $M_B$. For
large black holes, $\operatorname{Area}(M_A) - \operatorname{Area}(M_B)$ will come
mostly from the differing amount of black hole horizon area that the two surfaces wrap (see figure \ref{RT}).
The Hawking temperature of the black hole corresponds to the temperature of the field theory, and thus this modification of the proposal provides a way for $S_A - S_B$ to be nonzero for certain thermal field theories.
\begin{figure}
\begin{center}
\includegraphics[width=4in]{blackhole}
\end{center}
\caption{
The two minimal surfaces $M_A$ and $M_B$ corresponding to a region $A$ and its complement $B$ when the dual space time contains a black hole (BH).
\label{RT}
}
\end{figure}
However, there are instances where field theories at $T>0$ have dual gravity descriptions without a black hole. A classic example is the large $N$, strong coupling limit of maximally supersymmetric $SU(N)$ Yang-Mills theory on $S^3 \times S^1$ \cite{Witten:1998zw}. At temperatures small compared to the inverse radius of the $S^3$, the dual description is thermal $AdS_5 \times S^5$. At a critical temperature $T_c$, the gravity description undergoes a first order Hawking-Page phase transition to a state with a large black hole. For the field theory, this transition is understood as a deconfinement phase transition.
On the one hand, their proposal implies that the entanglement entropy will serve as an order parameter for the phase transition:
for $T< T_c$, $S_A - S_B =0$, while for $T>T_c$, $S_A - S_B \neq 0$.
On the other, at any finite $N$, we have a system at finite volume for which there can be no phase transitions. The transition from
$S_A = S_B$ at $T=0$ to $S_A \neq S_B$ at $T>T_c$ must be smooth.
We conclude that the Ryu-Takayanagi formula is only valid in the strict large $N$ limit,
but it would be nice to understand the form of the $1/N$ corrections. In principle, one should be able to compute the entanglement entropy for maximally supersymmetric Yang-Mills at weak coupling. In practice, such a computation
is substantially more difficult, and
we instead considered a 1+1 dimensional massive scalar field on a circle at $T>0$.
Morally, the regime $T<m$ should correspond to the confining regime of the Yang-Mills theory where the fields get a mass through their coupling to the curvature of the $S^3$.
For our scalar field, we argued that in the regime $T \ll m$, the entanglement entropy difference scales as
\[
S_B - S_A \sim e^{-m/T} \ .
\]
We conjecture that this type of scaling should be a generic feature of all gapped systems.
\subsection*{Acknowledgements}
We would like to thank A.~Abanov, K.~Balasubramanian, P.~Gao, G.~Giecold, D.~Gulotta, T.~Nishioka, and T.-C. Wei for discussion. This work was supported in part by the National Science Foundation under Grants No. PHY-0844827 and PHY-0756966. C.~H. also thanks the Sloan Foundation for partial support.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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Click here to view or download autism cluster maps.
Researchers at UC Davis have identified 10 locations in California where the incidence of autism is higher than surrounding areas in the same region. Most of the areas, or clusters, are in locations where parents have higher-than-average levels of educational attainment. The other clusters are located close to major autism treatment centers.
The clusters are primarily in the high-population areas of Southern California and, to a lesser extent, in the San Francisco Bay Area. In order to qualify as a cluster for the study, the rate of autism had to be at least 70 percent higher than in surrounding areas. The majority of children with autism in California, however, live outside of the clusters.
For the rigorous study, which was published online today by Autism Research and will appear in the February issue of the journal, scientists examined nearly all of the approximately 2-1/2 million births recorded in the state of California from 1996 through 2000. About 10,000 children born during that five-year period were later diagnosed with an autism spectrum disorder, according to the California Department of Developmental Services (DDS).
"Because of the strong link between demographics, particularly parental education, and the locations of clusters, other explanations for these pockets of high autism incidence, such as localized sources of exposure, are not likely," Van Meter explained.
"The risk for a child with highly educated parents to be diagnosed with autism is probably not caused by the location of the mother's residence or any local shared environmental exposures," she said. "Our result indicates that the most likely sources of environmental hazards for autism in California are in or around the home or else are widespread."
"The strong link between demographics, particularly parental education, and the locations of the clusters validated the effectiveness of the statistical method that we employed because it successfully identified areas where a known risk factor was concentrated," she added.
The UC Davis MIND Institute in Sacramento, Calif., was founded in 1998 as a unique interdisciplinary research center where parents, community leaders, researchers, clinicians and volunteers collaborate to study and treat autism and other neurodevelopmental disorders. More information about the institute is available on the Web at https://www.ucdmc.ucdavis.edu/mindinstitute/.
|
{
"redpajama_set_name": "RedPajamaC4"
}
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\section{Introduction}
There has been significant recent interest in the topology and the geometry of networks \cite{Perspectives,Lambiotte,Leskovec_science,giusti}. Network topology and geometry allow us to tackle fundamental theoretical questions concerning the principles determining the emergence of network geometry \cite{Emergent, Hyperbolic} or the proper definition of curvature in the discrete setting of networks and simplicial complexes \cite{Jost1,Jost2,Loll}.
Moreover, network topology has been shown to be an important tool for network inference, and has been extensively used to explore network structure \cite{Vaccarino1,petri2014homological,maletic,scolamiero,sporns,expert} and dynamics \cite{Ziff,Petri1,Petri2}.
In this paper, we propose a new network topology framework to investigate the local structure of real networks determined by the neighbourhoods of their nodes, which we often refer to as \textit{node neighbourhoods}. This approach will allow us to detect non-random statistical properties (with respect to a specific random model) in the local organization of these structures.
Node neighbourhoods, also called \textit{ego-centered networks}, have been studied extensively in Network Science \cite{SW,Newman}. In particular the clustering coefficient of a node is the most famous measure that quantifies the local density of triangles, or the so called \textit{transitivity}, of connections \cite{SW}.
In particular the clustering coefficient has been used extensively to quantify to what extent a network satisfies the principle of \textit{triadic closure}. This principle was originally formulated in the context of social networks, where it is observed that two friends of a common person are more likely to be friends of each other than in a set of random relations. However this tendency of real networks of displaying a large number of triangles has also been observed in other complex networks. Interestingly it is to mention that triadic closure is a basic mechanism for generating self-organised communities, as demonstrated by models enforcing this mechanism of network evolution \cite{Santo}.
Recently, higher-order clustering coefficients have been formulated in order to measure the density of cliques larger than triangles in a given node neighbourhood. The closure of higher dimensional cliques has also been used for link prediction \cite{Cannistraci_common,Kleinberg}.
However, the clustering coefficient and its generalisations are insufficient to fully characterise the topology of a node neighbourhood, and therefore it is important to develop new Topological Data Analysis tools \cite{Nanda,Skraba} that allow us to go beyond these simple metrics.
To perform a topological analysis of real networks, the first step is to construct a \textit{simplicial complex} starting from the network. A simplicial complex is a higher-order network structure that is not just formed by nodes ($0$-dimensional simplices) and links ($1$-dimensional simplices) like a network but it is also formed by higher dimensional simplices such as triangles ($2$-dimensional simplices), tetrahedra ($3$-dimensional simplices), and so on. Starting from a real network, we can extract a simplicial complex (called clique complex) by associating to each $c$-clique of the network a $(c-1)$-dimensional simplex. The topology of the clique complex can be investigated by calculating the Betti numbers, therein measuring the number of connected components (Betti number $\beta_0$), the number of cycles that form $1$-dimensional holes (Betti number $\beta_1$), the number of $2$-dimensional cavities (Betti number $\beta_2$), and their higher dimensional generalisations.
Here we propose a novel topological approach for analysing node neighbourhoods which goes beyond the traditional measure of the clustering coefficient, performing a large scale statistical analysis of the topology of the node neighbourhoods of real network datasets, with size ranging from 82,168 to 12,394,385 nodes. The results we obtain are compared to random simplicial complexes \cite{kahle2009} or random Vietoris-Rips complexes \cite{kahle2}, which take on the role of null models. We show how the topology the node neighbourhoods of real datasets significantly differ from these null models, showing that they obey organisation principles. Moreover, we show how the proposed topological analysis of node neighbourhoods reveals significant statistical differences between scale-free networks, and planar road networks.
The paper is organised as follows. In Sec. \ref{sec:2} we define networks, simplicial complexes and clique complexes, and discuss how the clique complexes can be extracted from a network dataset. In Sec. \ref{sec:3} we define node neighbourhoods, and we describe their topology in terms of the total number of nodes, link density, and Betti numbers. In Sec. \ref{sec:4} we show evidence of the relevant diversity found in the topology of real network neighbourhoods with comparable node and link density. In Sec. \ref{sec:5} we study the topology of null model of node neighbourhoods, and we discuss general differences observed between real network neighbourhoods and the null models. In Sec. \ref{sec:6} we provide the results of a large scale statistical comparison between the topology of neighbourhoods of scale-free hierarchical networks and neighbourhoods of road networks, and we compare the statistical properties of these neighbourhoods with the statistical properties of the considered null models. Finally in Sec. \ref{sec:6} we provide the conclusions.
\section{Networks, simplicial complexes and clique complexes}\label{sec:2}
A network is a graph $G=(V,E)$ formed by a set of nodes $V$ and a set of links $E$ that represent the elements of a complex system and their interactions, respectively.
Networks are ubiquitous and include systems as different as the WWW (web graphs), infrastructures (as airport networks or road networks) and biological networks (as the brain of the protein interaction network in the cell).
Simplicial complexes represent higher-order networks, which include interactions between two or more nodes, described by simplices. A simplex $\mu$ of dimension $c-1$ is formed by a subset of $c$ nodes. for instance a node is a $0$ dimensional simplex, a link is a $1$-dimensional simplex, a triangle is a $2$-dimensional simplex and a tetrahedra is a $4$-dimensional simplex and so on. A simplicial complex ${\mathcal K}$ is formed a by a set of simplices that satisfy the following two conditions:
\begin{itemize}
\item[(a)] if a simplex $\mu$ belongs to the simplicial complex, i.e. $\mu\in {\mathcal K}$ then any simplex $\mu^{\prime}$ formed by a subset of its nodes is also included in the simplicial complex, i.e. if $\mu^{\prime}\subset \mu$ then $\mu^{\prime}\in \mathcal K$;
\item[(b)] given two simplices of the simplicial complex $\mu\in {\mathcal K}$ and $\mu^{\prime}\in {\mathcal K}$ then either their intersection belongs to the simplicial complex, i.e. $\mu\cap \mu^{\prime}\in {\mathcal K}$ or their intersection is a null set, i.e. $\mu\cap \mu^{\prime}=\emptyset$.
\end{itemize}
Given a simplicial complex it is always possible to extract a network known as the $1$-skeleton of the simplicial complex by considering exclusively the nodes and links belonging to the simplicial complex.
Conversely, given a network, it is possible to derive deterministically a simplicial complex defining its \textit{clique complex}, obtained by taking a $(c-1)$ dimensional simplex for every $c$-clique in the network. The clique complex is a simplicial complex. In fact, if a simplex is included in a clique complex, then all its sub-simplicies are also included. Moreover any two simplices of the clique complex have an intersection that is either the null set or it is a simplex of the clique complex.
\section{Node neighbourhoods}\label{sec:3}
\subsection{Definition}
A complex network can be described locally in terms of $d$-hop \textit{neighbourhoods}, or \textit{ego-centered} networks.
Starting from a given node $i$, we consider the subgraph induced by the set of the nodes at hopping distance $\delta$ (with $0<\delta\leq d$). The neighbourhood of node $i$ is the clique complex of this induced subgraph. For example, if node $i$ has degree 12, of which all nodes apart from one pair are disconnected, the corresponding clique complex contains 11 disconnected components, of which 10 are $0$-simplices, and the other is a $1$-simplex. Fig. \ref{fig:simplicial_complexes} depicts an example of a $1$-hop neighbourhood. On panel (a), the induced subgraph on the neighbours is shown, with the corresponding clique complex on the right in panel (b). The simplices, show up to dimension two, are randomly coloured.
\subsection{Number of nodes and link density of the neighbourhoods}
The neighbourhood of node $i$ is characterised by its number of nodes $n_i$, and the density of links $\rho_i$, given by
\begin{eqnarray}
\rho_i=\frac{\mbox{\#\ of\ links\ in\ the\ neighbourhood}}{n_i(n_i-1)/2}.
\end{eqnarray}
When the neighbourhoods are formed by nodes at distance $\delta=1$ (i.e. when we consider $d=1$ and $1$-hop neighbourhoods), the number of nodes $n_i$ is given by the degree $k_i$ of node $i$ in the original network, i.e.
\begin{eqnarray}
n_i=k_i.
\end{eqnarray} Additionally in this case the density of links $\rho_i$ in the neighbourhood is given by the local clustering coefficient $C_i$ of node $i$, \cite{SW} i.e.
\begin{eqnarray}
\rho_i=C_i.
\end{eqnarray}
\subsection{Betti numbers}
The neighbourhood topology can be studied by computing the \textit{Betti numbers} of the resulting simplicial complex. These are topological invariants derived from the simplicial complex, and correspond, for each $i \geq 0$, to the number of linearly independent $i$-dimensional holes in the space. Specifically the Betti number $\beta_0$ provides the number of connected components of the neighbourhood, the Betti number $\beta_1$ measures the number of $1$-dimensional holes, i.e. cycles that are not boundaries of $2$-dimensional subsets of the simplicial complex, and so on for spheres, hyperspheres, etc.
\begin{figure*}[!t]
\begin{centering}
\begin{adjustbox}{width=\columnwidth}
\centering
\includegraphics[scale=0.56]{figure1}
\end{adjustbox}
\caption{The construction of the neighbourhoods of a node include two steps. First the subgraph induced by the neighbours of a node found at distance $0<\delta\leq d$ is considered (panel (a)). Subsequently the clique complex of this subgraph is constructed, by adding $(c-1)$-dimensional simplices for any $c$-cliques (panel (b)). }
\end{centering}
\label{fig:simplicial_complexes}
\end{figure*}
\section{Topology of neighbourhoods in real complex networks}\label{sec:4}
We have considered a number of large real complex networks, including WWW graphs, social networks and road networks with the total number of nodes $N$, total number of links $L$, average local coefficient $C$, and diameter $D$ indicated in Table \ref{table:tableofvals}.
All these networks are small-world \cite{SW}, hierarchical \cite{hierarchical} and scale-free \cite{BA}, with the exception of the road networks that are both spatial \cite{Spatial} and planar. For the road networks alone, we have considered neighbourhoods formed by nodes at distance less or equal than $d=5$ from the central nodes. In fact, due to their planarity, the $1$-hop neighbourhoods ($d=1$) are typically formed by isolated nodes. However, for all other small-world networks, considering $5$-hop neighbourhoods would capture non-local properties, since these neighbourhoods would include a large fraction of the nodes of the network. Therefore, for the scale-free, small-world networks (i.e. all the networks considered, with the exception of road networks) we have considered only neighbourhoods formed by nodes at hopping distance $d=1$.
For each studied dataset we have performed statistical analysis of the topology of its neighbourhoods by computing the Betti numbers, using the computational homology software CHomP \cite{chomp}. Specifically, in order to define an efficient computational framework, we have restricted our attention to neighbourhoods formed by clique complexes of dimension equal to $3$, which leads to an accurate information about the Betti numbers $\beta_0$ and $\beta_1$.
In fact, this procedure guarantees that the value of the measured Betti numbers remain unchanged if clique complexes including simplicies of higher dimensions are also included.
The Betti number $\beta_0$ indicates the number of components of the local neighbourhood. Therefore, a high Betti number $\beta_0$ of a neighbourhood around node $i$ indicates that the node $i$ determining the neighbourhood acts as broker between different otherwise disconnected components of its neighbourhood.
The Betti number $\beta_1$ indicates the number of cycles forming $1$-dimensional holes. Therefore a high ratio $\beta_1/\beta_0$ indicates that the neighbourhood is not maximally dense.
In Figure \ref{fig:2}, we plot several examples of node neighbourhoods found in the analysed datasets.
The first observation that we can draw from the statistical analysis is that if we compare the homology of neighbourhoods with comparable number of nodes $n$ and density of links $\rho$, but coming from different network datasets, there are significant fluctuations.
The large variability of the network topology of neighbourhoods with the same density of links $\rho_i$ indicates that the density of links in the neighbourhood (equivalent to the local clustering coefficient $C_i$ for neighbourhood with $d=1$) cannot fully capture the variability observed in the topology of the node neighbourhoods. In fact for fixed value of the density of links $\rho$ that can have very different Betti numbers $\beta_0$ and $\beta_1$ across different network datasets.
In particular while the small-world and scale-free datasets are characterised often by neighbourhood with high value of the Betti number $\beta_0$, the planar nature of the road network is revealed by the high value of the Betti number $\beta_1$ of its neighbourhoods with respect to the neighbourhoods of the other not planar networks.
\begin{figure}
\centering
\centering
\includegraphics[width=0.9\columnwidth]{figure2}
\caption{We show several examples of neighbourhoods with number of nodes $n\simeq 20$ and different density of links $\rho$ for different real network datasets and for the random clique complexes. The topology of network neighbourhoods evaluated by the Betti numbers $\beta_0$ and $\beta_1$ can have significant fluctuations even for neighbourhoods with comparable values of the local parameters $n$ and $\rho$.}
\label{fig:2}
\end{figure}
\begin{figure}
\begin{adjustbox}{width=\columnwidth}
\centering
\includegraphics[scale=1]{figure3}
\end{adjustbox}
\caption{We show several examples of neighbourhoods with number of nodes $n\simeq 100$ and different density of links $\rho$ for different real network datasets and for the random clique complexes. The topology of network neighbourhoods evaluated by the Betti numbers $\beta_0$ and $\beta_1$ can have significant fluctuations even for neighbourhoods with comparable values of the local parameters $n$ and $\rho$.}
\label{fig:2part2}
\end{figure}
\begin{table}[tb!]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
\textbf{Network} & $N$ & $L$ & $C$ & $D$ \\
\hline
Notre Dame Web Graph & 325,729 & 1,497,134 & 0.023 & 46 \\
\hline
Google Web Graph & 875,713 & 5,105,039 & 0.049 & 21 \\
\hline
Slashdot Social Network & 82,168 & 948,464 & 0.060 & 11 \\
\hline
Pokec Social Network & 1,632,803 & 30,622,564 & 0.010 & 11 \\
\hline
WikiTalk Social Network & 2,394,385 & 5,021,410 & 0.010 & 9 \\
\hline
Texas Road Network & 1,379,917 & 1,921,660 & 0.0015 & 1054 \\
\hline
Pensylvania Road Network & 1,088,092 & 1,541,898 & 0.0015 & 786 \\
\hline
California Road Network & 1,965,206 & 2,766,607 & 0.0015 & 849 \\
\hline
\end{tabular}
\caption{Table of used network datasets including number of nodes $N$, number of links $L$, average clustering coefficient $C$, and Diameter $D$. }
\label{table:1}
\label{table:tableofvals}
\end{table}
\section{Null models of random complexes}\label{sec:5}
\subsection{Why null models?}
In order to characterize real datasets it is always important to refer to suitable null models. A null model allows us to compare the results obtained in real datasets with the results obtained when minimal assumptions are made on the underlying network topology. To this end, here we consider two popular models of simplicial complexes used in the Applied Topology literature, the random clique complex \cite{kahle2009} and the random Vietoris-Rips complex \cite{kahle2}.
It is to be mentioned that this is not the only possible choice of null models. Indeed several more complex models of stochastic random simplicial complexes \cite{Farber1,Owen1} and evolving simplicial complexes \cite{NGF,Weighted} have been proposed in the literature. However, here the choice of the random clique complex and the random Vietoris-Rips complex is driven by the need to make minimal assumptions on the topology of the node neighbourhoods.
\subsection{Random Clique Complex}
The random clique complex \cite{kahle2009} also known as flag complex is the most fundamental null model for simplicial complexes. The random clique complex is the clique complex of the Erd\"os-R\'enyi random graph $G(n,p)$ of $n$ nodes and density of links $p$.
\subsection{Random Vietoris-Rips Complex}
The random Vietoris-Rips complex is the clique complex derived from the random geometric graph $\mathcal{G}(n, \pi r_0^2)$ \cite{kahle2}. This ensemble is formed by $n$ nodes distributed in the unit square $[0,1]^{2}$ with periodic boundary conditions (i.e. $\mathbb{R}^2/\mathbb{Z}^2$)
according to a Poisson point process and connected when at distance less or equal to $r_0$. Therefore the expected average degree of the random Vietoris-Rips $1$-skeleton is given by $n \pi r_0^2$.
\subsection{Betti numbers of random clique complexes and random Vietoris-Rips complexes}
For random clique complexes and random Vietoris-Rips complexes, there are analytical results indicating
the expected value of the Betti numbers and sometimes also their distribution in the limit of large $n$, i.e. for $n\to \infty$ \cite{kahle2009,kahle2}.
In particular, since the Betti number $\beta_0$ of a simplicial complex is equal to the number of connected components, its value is predicted by the theory of percolation.
For the random clique complex $G(n,p)$ it is well known that in the limit $n\to \infty$ the Betti number $\beta_0$ is a monotone graph property and is always decreasing with $p$. In particular for $n\to \infty $ and $np = a\log{n}$ as long as the constant $a$ is greater than one, the network has almost surely a Betti number $\beta_0=1$, i.e. it is formed by a single connected component.
The higher Betti numbers $\beta_j$, with $j>0$ however, display a non-monotone monotonic behaviour with the density of the network $p$. In a random clique complex this implies for instance that the Betti number $\beta_1$ which is zero for $p \ll 1$ initially grows with increasing values of $p$. As the density of links $p$ increases further, $\beta_1$ decreases. This result has an intuitive interpretation. Take for example a 4-cycle, which contributes one to the Betti number $\beta_1$. As the density of links increases, the four nodes on the cycle connect into a four clique, which then contributes zero to the Betti number $\beta_1$.
In Ref. \cite{kahle2009}, Kahle shows how this \textit{uni-modal} transition of the Betti number $\beta_j,j> 0$, which goes from vanishing, to non-vanishing, and back to vanishing again, occurs in the limit $n\to \infty$ in a almost sequential way. For $\alpha,j>0$, if $p = 1/n^{\alpha}$ and $\alpha > 1/j$ or $\alpha < 1/(2j+1)$, then in the limit $n\to \infty$ almost surely we have that the Betti number $\beta_j$ is vanishing, i.e.
\begin{equation}
\beta_j= 0.
\end{equation}
For example, for $j=1$ we have the threshold for percolation, and the theorem states that all components of $G(n,p)$ with $p=1/n^{\alpha}$ and $\alpha>1$ are trees. This is a classic topological property of the component structure in the sub-critical phase of the random graph. The above result implies that for $\alpha < 1/3$, all the cycles belong to the boundary of a higher dimensional clique.
\begin{figure}[!htb]
\begin{adjustbox}{width=\columnwidth}
\centering
\includegraphics[scale=1]{figure4}
\end{adjustbox}
\caption{ In the left panel the average Betti numbers $\beta_0$ and $\beta_1$ of the random clique complex of $n=20,50$ nodes are plotted as a function of the density of links $p$. In the right panel the the average Betti numbers $\beta_0$ and $\beta_1$ of the random Vietoris-Rips complex of $n=20,50$ nodes are plotted as a function of the connection range $r_0$. }
\label{fig:model}
\end{figure}
Similarly it has been shown \cite{kahle2} that for the random Vietoris-Rips complex, the Betti number $\beta_0$ is monotonically decreasing with the increasing connection range, while the Betti number $\beta_j$ with $j>0$ displays a uni-modal transition with $r_0$.
While these results are clearly fundamental to shed light on the topology of random simplicial complexes, since in this work we consider node neighbourhoods, we need to study how these asymptotic results are reflected in the small or middle sized networks which are available to us.
In Figure \ref{fig:model}, we show evidence that the discussed asymptotic behaviour of the two chosen null models is also reflected on random clique complexes and random Vietoris-Rips complexes of relatively small size ($n=20$ and $n=50)$. From this figure, it is apparent that as a consequence of the uni-modality of the Betti number $\beta_1$, typically in a random clique complex with given value of density of links $p$, only one of the two Betti numbers $\beta_0$ and $\beta_1$ are significant. A similar behaviour is observed also for the random Vietoris-Rips complex with given connection range $r_0$.
\subsection{Comparison between Betti numbers of null models and Betti number of real complex network neighbourhoods}
\begin{figure*}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.3]{figure5}
\end{adjustbox}
\caption{The typical neighbouroods of the random clique complex are shown for $\rho\simeq 0.05$, and increasing values of the number of nodes $n=10,20,30,50,75,125$.} \label{fig:a1}
\vspace{2mm}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.3]{figure6}
\end{adjustbox}
\caption{The typical neighbourhoods of the random Vietoris-Rips complex are shown for $\rho\simeq 0.05$ and increasing values of the number of nodes $n=10,20,30,50,75,125$.} \label{fig:a2}
\end{figure*}
The neighbourhoods of real complex networks datasets can be compared with the two considered null models (random clique complexes and random Vietoris-Rips complexes), whose underlying network skeleton has the same average degree.
When comparing the neighbourhood topology with an ensemble of random complexes (whose members act as an hypothesis of neighbourhood graphs), the neighbourhoods are constructed according to the null model, rather than obtained by searching large random graphs, as it is done for the real datasets rather random neighbourhoods are directly sampled from the corresponding ensemble of the null models.
One of the first observation that we draw from this comparison is that the neighbourhoods of the real network datasets can display at the same time large Betti number $\beta_0$ and large Betti number $\beta_1$ while as we previously discussed, this is in sharp contrast with what it is typically observed in the two considered null models.
In particular in Figures $\ref{fig:a1}$ and $\ref{fig:a2}$ we show typical realizations of the random clique complexes and the random Vietoris-Rips complexes for given density of links $\rho$ but tuneable size $n$. As the average degree $n\rho$ increases a single connected component quickly forms. We also observe that due to the uni-modal transitions of the Betti number $\beta_1$ the Betti number $\beta_1$ is typically maximal when the Betti number $\beta_0$ is low.
This behaviour is in sharp contrast to what is observed in real scale-free social network datasets (see Figures $\ref{fig:a3},\ref{fig:a4},\ref{fig:a5}$) tend to have dense neighbourhood formed by several components formed by isolated nodes and a large well clustered component. The observed topology of the node neighbourhoods suggest a local topology of the neighbourhoods similar to the one expected in the condensed phase of the Strauss model \cite{Strauss,Doro}. This is a model that enforce formation of many triangles leading to a separation between many disconnected isolated nodes and a large, highly clustered component in its condensed phase.
Additionally in these networks is not rare to observe neighbourhoods displaying at the same time large Betti number $\beta_0$ and large Betti number $\beta_1$, leading to a topology of network neighbourhoods significantly different from the ones of the considered null models.
In Figure $\ref{fig:a7}$ we show typical instance of neighbourhoods of the California road network which has instead rather different topology. In fact denser neighbourhoods are typically formed by a single component but display a large Betti number $\beta_1$ reflecting the planar nature of the underlying network.
\begin{figure*}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.3]{figure7}
\end{adjustbox}
\caption{A set of typical neighbouroods of the Slashdot social networks are shown for $\rho\simeq 0.05$ and increasing values of the number of nodes $n=10,20,30,50,75,125$.}\label{fig:a3}
\vspace{2mm}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.3]{figure8}
\end{adjustbox}
\caption{A set of typical neighbouroods of the Pokec social network are shown for $\rho\simeq 0.05$ and increasing values of the number of nodes $n=10,20,30,50,75,125$.}\label{fig:a4}
\vspace{2mm}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.3]{figure9}
\end{adjustbox}
\caption{A set of typical neighbourhoods of the WikiTalk social network are shown for $\rho\simeq 0.05$ and increasing values of the number of nodes $n=10,20,30,50,75,125$.}\label{fig:a5}
\vspace{2mm}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.3]{figure10}
\end{adjustbox}
\caption{A set of typical neighbourhoods of the California road networks are shown for $\rho\simeq 0.05$ and increasing values of the number of nodes $n=10,20,30,50,75,125$.}\label{fig:a7}
\end{figure*}
\section{Statistical topological analysis of complex networks neighbourhoods}\label{sec:6}
\subsection{Homology of hierarchical scale-free networks versus homology of road networks}
The datasets that we have considered include among them several hierarchical scale-free networks \cite{hierarchical} (Notre Dame and Google Web Graphs, the Pokec, Slashdot and WikiTalk social networks)
characterized by an average clustering coefficient of nodes of degree $k$ scaling like
\begin{eqnarray}
C(k)\simeq k^{-\theta}
\label{theta}
\end{eqnarray}
and several road networks (Texas, Pensylvania, California road networks).
\begin{figure*}
\noindent \begin{centering}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.8]{figure11}
\end{adjustbox}
\caption{The average density of links $\rho$ of node neighbourhoods of size $n$ is plotted as a function of $n$ different network datasets.}
\label{fig:rcc0}
\par\end{centering}
\noindent \begin{centering}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.8]{figure12}
\end{adjustbox}
\caption{The average Betti number $\beta_0$ of node neighbourhodds of size $n$ is plotted as a function of $n$ for different network datasets.}
\label{fig:rcc}
\par\end{centering}
\end{figure*}
\begin{figure*}
\noindent \begin{centering}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.8]{figure13}
\end{adjustbox}
\caption{The average Betti numbers $\beta_0$ and $\beta_1$ of node neighbourhoods with given density of links $\rho$ are plotted as a function of $\rho$ for different network datasets.}\label{fig:b0C1}
\par\end{centering}
\end{figure*}
Clearly we expect to see significant differences in the topology of the neighbourhoods of hierarchical networks and road networks. The planarity of road networks clearly constraint all the Betti numbers $\beta_j$ with $j>1$ to be zero. However as we have seen in the typical neighbourhood of road networks shown in Figure $\ref{fig:2}$ and in Figure $\ref{fig:a7}$, the road networks neighbourhood tends also to have a lower $\beta_0$ and a higher $\beta_1$ with respect to neighbourhoods in the other datasets having the same number of nodes $n$ and density of links $\rho$.
To investigate further the statistical differences between road network neighbourhoods and scale-free hierarchical networks neighbourhoods we have evaluated the average value of the Betti numbers $\beta_0$ and $\beta_1$ over neighbourhoods with fixed number of nodes $n$ or density of links $\rho$.
In the hierarchical scale-free networks a node with degree $k$ will give rise to a node neighbourhood of $n=k$ nodes as long as all the nodes of the neighbourhood are at distance $\delta=1$ from the original node. Correspondingly the density of links $\rho$ of the node neighbourhood can be approximated with the average clustering coefficient $C(k)$ of nodes of degree $k=n$, i.e. $\rho \simeq C(k)$ that obeys Eq. ($\ref{theta}$) providing the following power-law scaling of $\rho$ as a function of $n$ (see Figure $\ref{fig:rcc0}$)
\begin{eqnarray}
\rho \propto n^{-\theta}.
\label{rhon}
\end{eqnarray}
From the statistical topological analysis of the average Betti number $\beta_0$ of the network neighbourhood we find that the average Betti number $\beta_0$ increases as a power law with $n$, i.e.
\begin{equation}
\beta_0 \propto n^{\nu},
\end{equation}
with $\nu>0$ (see Figure $\ref{fig:rcc}$).
This scaling indicates a monotonic power-law increases of the number of components of the local neighbourhoods as a function of the number of nodes $n$ of the neighbourhoods. Therefore hubs tend to be broker between different otherwise unconnected communities.
In Figure $\ref{fig:b0C1}$ we report the average Betti numbers $\beta_0$ and $\beta_1$ for node neighbourhoods with given density of links $\rho$.
For hierarchical scale-free networks, using the scaling indicated in Eq.$(\ref{rhon})$, it is easy to predict that the average Betti number $\beta_0$ should decay as an inverse power of the density of links $\rho$, i.e.
\begin{equation}
\beta_0\propto \rho^{-\alpha}
\label{alpha}
\end{equation}
with $\alpha=\nu/\theta$.
These scaling relations imply that more densely connected neighbourhoods are typically smaller, and characterised by a smaller Betti number $\beta_0$.
The road networks, that are not hierarchical are characterised by a significant different trend of the average Betti number $\beta_0$ as a function of $\rho$. In fact for the road networks the Betti number $\beta_0$ is a non-decreasing function of $\rho$.
This result reveal that the number of connected components of the road neighbourhoods decreases for the neighbourhood of larger road junctions.
Therefore the Betti number $\beta_0$ of road networks neighbourhoods display very relevant statistical differences with respect to the Betti number $\beta_0$ of scale-free hierarchical networks.
\begin{figure*}
\noindent \begin{centering}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.8]{figure14}
\end{adjustbox}
\caption{The average Betti numbers $\beta_0$ and $\beta_1$ of the neighbourhoods to the Slashdot social network as plotted as a function of the number of nodes $n$ and the density of links $\rho$ of the neighbouroods. The real data are then compared with the results obtained on the random clique complex and the random Vietoris-Rips complex observing significant difference highlighting the non-random character of the real dataset.}\label{fig:3dslash}
\par\end{centering}
\end{figure*}
\begin{figure*}
\noindent \begin{centering}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[scale=0.8]{figure15}
\end{adjustbox}
\caption{The average Betti numbers $\beta_0$ and $\beta_1$ of the neighbourhoods to the Pokec social network as plotted as a function of the number of nodes $n$ and the density of links $\rho$ of the neighbourhoods. The real data are then compared with the results obtained on the random clique complex and the random Vietoris-Rips complex observing significant difference highlighting the non-random character of the real dataset.}\label{fig:3dpokec}
\par\end{centering}
\end{figure*}
\subsection{Topological data analysis as a function of the number of nodes $n$ and the density of links $\rho$}
The non-random behaviour in the datasets becomes apparent when we characterise the average topology of the neighbourhoods of real datasets with given number of nodes $n$ and density of links $\rho$. To this end we plot the average Betti numbers $\beta_0$ and $\beta_1$ as a function of $n$ and $\rho$ and we compare the results with the average Betti numbers of the random clique complexes and the random Vietoris-Rips complexes with the same number of nodes and density of links.
The Figures \ref{fig:3dslash} and \ref{fig:3dpokec} display the numerical results for the neighbourhoods of the Slashdot social network and of the Pokec social network, respectively.
\begin{figure*}
\noindent \begin{centering}
\begin{adjustbox}{width=\columnwidth}
\includegraphics[width=0.9\columnwidth]{figure16}
\end{adjustbox}
\caption{The average Betti number $\beta_0$ of neighbourhoods of the Slashdot and Pokec social networks, of the Notre Dame Web graph and of the Texas Roads Networks, are plotted as a function of the density of links $\rho$ by distinguishing between neighbouroods of different size $n$ ($n<20$, $20\leq n\leq 70$ and $n>70$). The results obtained with the real datasets are compared with the results obtained for the random clique complex and the random Vietoris-Rips complex.}
\label{fig:ndep}
\par\end{centering}
\end{figure*}
In order to emphasise the quantitative differences that we observe in the real datasets with respect to the null models in Figure \ref{fig:ndep} we plot the average Betti number $\beta_0$ of neighbourhoods of hubs nodes ($n>70$), peripheral nodes $(n<20)$ and of bridge nodes $n\in [20,70]$ as a function of $\rho$.
All the considered real scale-free networks display a power law dependence of the Betti number $\beta_0$ with respect to the density of links $\rho$ that persists throughout all instances, independently of $n$. However, this behaviour is not captured by the null models. This is the clearest distinction between the stochastic topology of the null models, and the neighbourhoods in the real datasets.
This apparently follows from the scale-free, hierarchical structure of the datasets, which acts to ensure that large neighbourhoods display multiple disconnected components, unlike random cliques or random Vietoris Rips complexes.
The Texas road network datasets, also, display a topology that is significantly different from the null models, where the Betti number $\beta_0$ is not decreasing with the density of links $\rho$ while it is decreasing for the random clique complex and for the random Vietoris-Rips complex.
\section{Conclusions}\label{sec:7}
In this paper, we have analysed the topology of node neighbourhoods in large network datasets. The node neighbourhood of a generic node $i$ is the clique complex of the network induced by the nodes up to distance $d$ from $i$. The topology of node neighbourhoods is then investigated by calculating their Betti numbers $\beta_0$ and $\beta_1$. A node neighbourhood with a high $\beta_0$ indicates that the central node is connected to many nodes that are not directly connected to any other node in the neighbourhood. A high $\beta_1$ indicates instead that among connected nodes in the neighbourhood, there are several open cycles, implying that the corresponding cluster is not densely connected.
Our large scale statistical analysis reveals that the topology of these neighbourhoods is not only determined by their size and their link density. In fact, for a given size and link density of a neighbourhood, different topologies can be observed. Specifically we show that the topological study of node neighbourhoods is able to distinguish between the neighbourhoods of scale-free hierarchical networks and the neighbourhoods of spatial road networks. Moreover both types of real datasets obey significant organisation principles that impose a local topology of the node neighbourhoods that is significantly different from the random clique complex and the random Vietoris-Rips complex.
In the future an interesting question that we would like to investigate is to what extent the recently proposed local curvatures of discrete networks and simplicial complexes \cite{Jost1,Jost2,Loll} capture the local topology of node neighbourhoods.
\section*{References}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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La ville de Brno est officiellement fondée en 1243 par Venceslas de Bohême, mais le site est habité depuis le .
Histoire de la ville
Préhistoire
Très boisé, le bassin de Brno était, à l'époque préhistorique, parcouru de groupes épars de chasseurs-cueilleurs, et, à l'emplacement de la motte castrale morave de Staré Zámky, les fouilles ont révélé un campement permanent établi au néolithique, qui a perduré jusqu'au début du .
Antiquité
La ville est mentionnée dans l'atlas de Ptolémée sous le nom celte dEburodunum soit « gué des sangliers ». C'est une des étapes de la route de l'ambre, qui relie la Baltique à l'Adriatique par Wrocław (Vratislavie, Breslau), Olomouc (Olmuntium, Olmütz), Brno (Eburodunum, Bruna, Brünn, Brin), Bratislava (Prešporok, Prešpurk, Pressbourg, Pozsony), Sopron (Scarbantia, Ödenbourg) et Ljubljana (Labacum, Loubiana, Laubach, Laibach).
Moyen Âge
La Moravie est décrite comme un pays initialement forestier et marécageux, successivement peuplé de celtes, de germaniques et de slaves, parfois envahi par les magyars, et de ce fait l'étymologie de Brno est discutée. On l'a reliée le plus souvent aux mots vieux-slaves brneno, brno « marais », brŭvĩno « poutre, pieu » ou brniti « fortifier ». Dans les langues celtiques bren, bryn signifie « colline », en allemand Brünn signifie « source » et en hongrois bárna signifie « sombre ». Au la dynastie des Přemyslides remplace la motte castrale par un château-fort et s'y établit, faisant de Brno un des centres de la Moravie avec Olomouc et Znojmo. À la même époque, une chapelle est fondée sur la colline Petrov. Après beaucoup de remaniements, elle devient plus tard la Cathédrale Saint-Pierre-et-Saint-Paul de Brno.
La forteresse du Spielberg est bâtie au début du . Elle devient le principal château-fort de la Moravie et en 1243, Brno reçoit les privilèges de Ville Royale (Königsstadt) par le roi de Bohême Wenzel Ier, qui encourage l'installation des Allemands dans son royaume. En 1324, la reine Élizabeth Ryksa fonde la basilique Notre-Dame de l'Assomption, toujours visible aujourd'hui.
En 1349, elle devient siège permanent du Margraviat de Moravie, mais ne remplacera Olomuc (Olmütz) comme capitale du margraviat qu'en 1641. Auparavant, au cours du , Brno accueille, en alternance avec Olomouc, les assemblés régionales de la Moravie. Ces assemblées contrôlaient la vie politique et financière de la région. La ville abritait aussi les (les tables de la loi tchèque) et la cour suprême de Moravie.
L'un des Margraves fut élu plus tard « roi des Romains » c'est-à-dire Empereur du Saint-Empire « romain » germanique.
Lors de la guerre des hussites, la ville reste fidèle à Sigismond . Les hussites assiègent par deux fois en vain la ville, en 1428 et 1430.
Époque moderne
Durant la Guerre de Trente Ans, entre 1643 et 1645, la ville est un obstacle sur la route de Vienne. Du au elle résiste au siège des Suédois commandées par le général Lennart Torstensson. Les défenseurs de Brno sont menés par le colonel Jean-Louis Raduit de Souches. Le siège donne largement le temps à l'empire autrichien d'organiser sa défense. La ville fut par la suite récompensée généreusement, avec un renouvellement de ses privilèges. Dans les années suivant le siège, le château Spielberg est remanié en citadelle baroque.Cette victoire figure parmi les chapitres les plus glorieux de l'histoire de la ville, et en 1995, le de cet événement a été célébré par d'importantes festivités.
En 1742, les Prussiens, dirigés par Frédéric II de Prusse, échouent à conquérir la ville.
Le statut important de Brno est confirmé par la construction d'un évêché en 1777.
Epoque contemporaine
La ville est connue pour sa proximité avec la ville de Slavkov, dont le nom allemand est Austerlitz, où s'est tenue le la bataille du même nom. La ville a ainsi successivement accueilli les principaux protagonistes de la bataille à commencer par Koutouzov, le général russe et Napoléon . Des plaques marquent encore dans la ville le souvenir de leur passage.
Révolution industrielle
Au commence le développement de l'industrie et du commerce. Peu de temps après la révolution industrielle, la ville devient le centre industriel de la Moravie et de l'empire austro-hongrois. Elle est quelquefois surnommée « la Manchester Tchèque ». Le premier train arrive en 1839 et marque le début de l'histoire du rail en république tchèque. La ville est reliée au gaz en 1847 et le tramway hippomobile, le premier du pays, est mis en circulation en 1869. Le Théâtre Mahen est un des premiers en Europe à utiliser les lampes électriques de Thomas Edison. Ce dernier visitera d'ailleurs la ville en 1911.
Avec le développement de la banlieue, la ville perd ses fortifications et la forteresse Spielberg devient une prison, accueillant surtout les opposants politiques à l'empire d'Autriche.
Durant la Première République tchécoslovaque (1918-1938), Brno continue son développement. L'Université Masaryk est construite en 1919, tout comme la manufacture d'armes Ceská Zbrojovka Brno. La foire de Brno est inaugurée en 1928, avec une exposition de culture contemporaine. La ville n'est pas seulement un centre industriel et commercial mais aussi un centre culturel et éducatif.
En 1919, deux villes voisines, Královo Pole et Husovice, et municipalités sont annexées à Brno pour créer « la grande Brno » (Velké Brno). La taille de la ville est multipliée par sept et passe de à .
En 1921, Brno devient la capitale de la région de Moravie (země Moravská), puis sept ans après, de la région de Moravie-Silésie (země Moravskoslezská).
Quand la Première Guerre mondiale se termine en 1919, la ville compte environ .
En 1938, Brno est annexée par l'Allemagne nazie comme toute la Moravie et la Bohême. Tous les centres éducatifs sont fermés le (notamment les quatre universités). sont envoyés au camp de Sachsenhausen et la résidence étudiante de Kounic est transformée en prison et quartier général pour la Gestapo.
La plupart de la population juive de Brno () est déportée et tuée durant l'occupation nazie entre 1939 et 1945. Environ tchèques, américains et anglais sont arrêtés et torturés à Brno et tués.
Brno est libérée par le troisième front biélorusse de l'armée rouge, commandée par le général Hovhannes Bagramian, le après deux semaines de combats.
Après la fin de la guerre et le rétablissement de la Tchécoslovaquie, , la majorité de la population allemande de Brno (excepté les antifascistes, membres de la résistance et couples mixtes) sont expulsés vers l'Allemagne et l'Autriche, comme dans toute la Tchécoslovaquie à cette époque.
La « marche de la mort de Brno » commence le : de Brno parcourent jusqu'à la frontière autrichienne. D'après les témoignages allemands, trouvent la mort dans cette marche forcée. Les estimations tchèques donnent environ , due en grande partie à une épidémie de shigellose.
Statut de capitale de la Moravie
Au milieu du , la Moravie est divisée en 3 territoires. Chacune a son propre dirigeant, indépendant des deux autres. Les capitales de ces territoires étaient les villes fortifiées de Brno, Olomouc et Znojmo. À la fin du la Moravie entame sa réunification et devient le Margraviat de Moravie. Dès lors et jusqu'en 1641, le statut de la ville n'est pas complètement défini. Le pouvoir est divisé entre Brno et Olomouc mais Znojmo garda un rôle important. La diète Morave (zemský sněm), les tables de la lois et la cour suprême de Moravie ont siégé en alternance à Brno et Olomouc. Brno fut le siège des margraves moraves et plus tard très proche de Vienne. Olomouc, ville plus grande par la taille, fut jusqu'en 1777 le seul siège de l'évêché de Moravie.
En 1641, au milieu de la Guerre de , le roi Ferdinand III ordonne le transfert de la Diète, des tables des tables de la loi et de la cour d'Olomouc à Brno, le Collegium Nordicum d'Olomouc étant l'une des principales cibles de l'armée suédoise. En 1642, Olomouc se rend à l'armée suédoise qui y restera huit ans. La même année Brno devient l'unique capitale de la Moravie.
Après la guerre (1648), Brno conserve son statut d'unique capitale de la Moravie. Cet état confirmé en 1784 par Joseph II du Saint-Empire puis en 1849 par la constitution Morave.
En 1948, le gouvernement communiste de Tchécoslovaquie met fin à l'autonomie morave, enlève à Brno son titre de capitale de la Moravie et transfère tous les pouvoirs politiques du pays à Prague. La région est divisée en plusieurs administrations dirigées par Prague.
Brno devient par la suite la capitale administrative de la Moravie-du-Sud, appelée à l'origine « région de Brno ». En 1968, elle obtient le statut de .
Notes et références
Notes
Références
Articles connexes
Histoire de la République tchèque
Brno
Brno
|
{
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Q: Loading HashMap to a nested configuration bean throws Binding Exception I was attempting this example given here :
https://docs.spring.io/spring-boot/docs/current/reference/htmlsingle/#boot-features-external-config-typesafe-configuration-properties
All worked well except when attempted added another properties to load hashmap values
property added as:
demoapp.security.policies={'KEY1': 'value1', 'KEY2': 'value3', 'KEY3': 'value5'}
And inside Secutiry inner class, added another variable as below:
private Map<String, String> policies;
public Map<String, String> getPolicies() {
return policies;
}
public void setPolicies(Map<String, String> policies) {
this.policies = policies;
}
But this throws error as :
Caused by: org.springframework.core.convert.ConverterNotFoundException: No converter found capable of converting from type [java.lang.String] to type [java.util.Map<java.lang.String, java.lang.String>]
Interestingly if I put this in a normal(non-nested) configuration class it works fine for me.
What is going wrong here, any suggestions please
A: When binding to a map, you're binding nested properties, so you need to specify the properties separately.
Properties file:
demoapp.security.policies.KEY1=value1
demoapp.security.policies.KEY2=value3
demoapp.security.policies.KEY3=value5
YAML file:
demoapp.security.policies:
"[KEY1]": value1
"[KEY2]": value3
"[KEY3]": value5
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 1,741
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Q: How to automatically add hyphen on the state textbox when onchange handler detect on length 4 using javascript I have module where I need to format the value of textbox to xxxx-xxxxxxx. Now I already detect the length of my textbox via using e.target.value.length my goal is when the handler detect the length of 4 the hyphen will automatically join after the 3rd length value of textbox. Ex. xxxx- I currently using react js.
Problem: the hyphen is inserting in the 4 length.
Goal: xxxx-xxxxxxx.
Here is my handler:
handleChange = (e) => {
let val = e.target.id ? e.target.id : e.target.name
var value = { [val]: e.target.value };
if(val == 'mobile_number') {
if(e.target.value.length === 4) {
value = this.state.formData.mobile_number += "-"
}
}
this.setState((prevState) => ({
formData: {
...prevState.formData,
...value,
},
}));
};
Here is my mobile_number textbox:
<input
type="text"
className="form-control r-5"
id="mobile_number"
value={mobile_number || ''}
required
onChange={this.handleChange}
maxLength='12'
/>
My Work:
A: You need to use value from event.target instead of using your state, while appending the hyphen. That is because your state has the 'not-still-updated' value which arrives from the handler via event.
Instead
if(e.target.value.length === 4) {
value = this.state.formData.mobile_number += "-"
}
You should do something like, to support both cases when adding inputs and when deleting inputs:
let value;
if (
event.target.value.length === 5 &&
event.target.value.includes("-")
) {
value = event.target.value.replace("-", "");
}
if (event.target.value.length === 4) {
value = event.target.value + "-";
}
}}
Edit: link to sandbox -> https://codesandbox.io/s/add-hyphen-after-4-input-pxohf?file=/src/App.js
A: You can do something like this
let input = "78945612378584"
let firstFour = input.substring(0,4)
let last = input.substring(4,input.length)
let d = firstFour+'-'+last
console.log(d) //"7894-5612378584"
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,963
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Authorities identify Suwannee County hit-and-run victim
J.C. Derrick, Staff Reports
Update (1/15/21): Florida Highway Patrol (FHP) announced it has identified the hit-and-run victim and notified the next of kin, but it will not release the name of the deceased man.
In a press release FHP reported the man was 34 years old and—for reasons that are still under investigation—was apparently lying in the road when a 2007 Dodge Caliber struck and killed him. Investigators have located a person of interest in connection with the crash.
Our original story:
Investigators for the Florida Highway Patrol (FHP) are asking for the public's help identifying a hit-and-run victim in Suwannee County.
According to the accident report, the crash occurred at 1:22 a.m. just north of Live Oak. An unidentified vehicle traveling northbound on US-129 struck and killed a middle-aged white man before fleeing the scene, the report says. The collision also killed a mid-sized tan dog.
Investigators say the man had long dark hair, a beard, and was wearing blue pajama pants and a red shirt at the time of the accident. They say he also had numerous tattoos, including "Live, Love" on his right hand; "Free, Life" on his left hand; and "Fido" on his right bicep.
FHP asks that anyone with information about the accident or the deceased victim contact the FHP communications center at 800-387-1290. Members of the public can also call Crime Stoppers of Suwannee County at 386-208-8477.
Tags:none
J.C. Derrick
J.C. Derrick is publisher for Mainstreet Daily News. He spent 18 years covering sports, education, and politics in Texas and Washington, D.C., before joining Mainstreet in 2020.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 7,844
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El Palazzo Platoni Farnese es una construcción de cuatro pisos que se asoma sobre un eje vial con una fuerte pendiente ubicado en la Avenida Vittorio Emanuele III, en la comuna de Farnese provincia de Viterbo en la región del Lacio en Italia.
La propiedad anteriormente había sido un hospital de la Casa de Farnesio. El ilustre Muzio Platoni quien pertenecía a una de las familias más influyente y de la aristocracia de la comuna de Farnese compró el hospital en el año 1570 a los Farnesio por la suma de 100 coronas.
El noble Muzio era originario de Borgo Val di Taro y era reconocido por contar con la representación de los habitantes de su ciudad natal. Provenía de la antiquísima dinastía de Platoni de origen lombardos. En 1571 junto a su hermano Scipione contrataron al arquitecto Sallustio de Bernardino para la construcción del palacio, posteriormente se sumó a la obra el arquitecto Gerolamo de Valentano.
Posee una fachada simple, subdividida por las cornisa de teclas de piano (marcapiano en italiano) en toba volcánica y ventanas rectangulares en los pisos inferiores. Bajo el techo las ventas son ovaladas, también se encuentran enmarcadas en peperino. Parte de sus habitaciones y techos se encuentran decorados con frescos.
La propiedad es llamada Palazzo Platoni Farnese porque la realización del portal y parte de la fachada se inspiró en el palacio de la Villa Farnesio (Caprarola), evidentemente a una escala más reducida y simple. Ambas estructuras son similares, con la distinción modular del registro inferior y del ático, en donde se aprecia las metopa decoradas con la flor de lis en relieve propia de la Dinastía Farnesio, además se aprecia el empleo del almohadillado y sillar lisos.
Muzio al igual que otros miembros de la familia Platoni trabajaron directamente con el Ducado Farnesio, en este caso con el Duque Octavio Farnesio. Pero más tarde siguió la conexión con los Duques posteriores como en el caso del gobernador de Parma Giulio Platoni por todo lo anterior se le reconoce hasta el día de hoy con el nombre de Palazzo Platoni Farnese.
Sobre el portal principal se encuentra el escudo de armas de la Familia Platoni o de Platis (en latín).
Descripción del Escudo de Armas original: Dividido en dos partes. La primera en color azul está el castillo de plata con almena y contornos negros sostenido sobre un monte de oro de tres cumbres. La segunda parte, de azul con bandas de oro.
Existe además un segundo portal con la reja original de hierro, decorada con la flor de lis Farnesio
Acontecimientos
Los Platoni que habitaron el palacio fueron familiares del emperador Francisco de Habsburgo-Lorena y Borbón Francisco I de Austria y II del Sacro Imperio Romano Germánico
También hubo uniones en matrimonio con varias de las familias nobles incluyendo Humani y Castiglioni.
Destaca entre habitantes del palacio, el Padre Samuel de Farnese Andrea Platoni, quien se fue muy joven a vivir en conventos optando por una vida austera para morir en santidad.
Referencias
Platoni Farnese
Casa de Platoni
|
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| 8,574
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{"url":"http:\/\/openstudy.com\/updates\/51843b63e4b0bc9e008a0764","text":"## anonymous 3 years ago 7\/8 times 39\/180 and simplify. I keep getting 273\/1440 but it's telling me I'm wrong.\n\n1. jdoe0001\n\nit's correct :), you just need to SIMPLIFY it :)\n\n2. anonymous\n\nHow?\n\n3. jdoe0001\n\ndo you know what simplifying means?\n\n4. anonymous\n\nSometimes. (-:. I guess I should divide 273 by 1440. Is that right?\n\n5. jdoe0001\n\nwell, yes and no, you should to the extent you're still left with a fraction, so, $$\\cfrac{2}{4} \\ simplifies\\ to\\ \\cfrac{1}{2}$$\n\n6. naveen\n\nOk, so that was easy. But I cant think of where to start. 273 is a large number as is 1440. Is there a trick?\n\n7. jdoe0001\n\nso $$\\cfrac{1}{2}\\times 2 = \\cfrac{2}{4}$$, same proportions, with just a multiplier, in this case \"2\"\n\n8. jdoe0001\n\nwell, sorta, 273, 2+7+3 =12, 12 is divisible by 3, start from there, divide both by 3 :), see if that works\n\n9. naveen\n\nSuppose it was 123\/457?\n\n10. jdoe0001\n\nwell, then I'd check what I can get from the numbers, usually you start small, 2, or 3, or 4 and so on, the smallest integer you can divide, NOT 1 of course, so 2 and so on, yo have 273, is not divisible by 2, so I checked with 3 :)\n\n11. jdoe0001\n\nlet me see if I can make a quick table of both of those numbers, simplifying\n\n12. jdoe0001","date":"2016-12-11 02:20:54","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\": 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\": 0.7728895545005798, \"perplexity\": 1600.9030270864885}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-50\/segments\/1480698543782.28\/warc\/CC-MAIN-20161202170903-00463-ip-10-31-129-80.ec2.internal.warc.gz\"}"}
| null | null |
Gheorghe Lichiardopol (2 August 1913 – 1991) was a Romanian sport shooter who competed in the 1952 Summer Olympics and at the 1956 Summer Olympics, winning two bronze medals.
References
1913 births
1991 deaths
Romanian male sport shooters
Romanian people of Greek descent
ISSF pistol shooters
Olympic shooters of Romania
Shooters at the 1952 Summer Olympics
Shooters at the 1956 Summer Olympics
Olympic bronze medalists for Romania
Olympic medalists in shooting
Medalists at the 1952 Summer Olympics
Medalists at the 1956 Summer Olympics
Sportspeople from Bucharest
20th-century Romanian people
|
{
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| 5,716
|
{"url":"https:\/\/ng.mathematicstip.com\/8062-75-solve-rational-equations.html","text":"# 7.5: Solve Rational Equations\n\nAfter defining the terms \u2018expression\u2019 and \u2018equation\u2019 earlier, we have used them throughout this book. Now we will solve a rational equation.\n\nYou must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.\n\n[dfrac{1}{8} x+dfrac{1}{2} quad quad dfrac{1}{8} x+dfrac{1}{2}=dfrac{1}{4} onumber ]\n\n## Solve Rational Equations\n\nWe have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to \u201cclear\u201d the fractions.\n\nWe will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then, we will have an equation that does not contain rational expressions and thus is much easier for us to solve. But because the original equation may have a variable in a denominator, we must be careful that we don\u2019t end up with a solution that would make a denominator equal to zero.\n\nSo before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.\n\nAn algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution to a rational equation.\n\nExtraneous Solution to a Rational Equation\n\nAn extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.\n\nWe note any possible extraneous solutions, (c), by writing (x eq c) next to the equation.\n\nExample (PageIndex{1}): How to Solve a Rational Equation\n\nSolve: [dfrac{1}{x}+dfrac{1}{3}=dfrac{5}{6} onumber ]\n\nSolution\n\nStep 1. Note any value of the variable that would make any denominator zero.\n\nIf (x=0), then (dfrac{1}{x}) is undefined. So we'll write (x eq 0) next to the equation.\n\n[dfrac{1}{x}+dfrac{1}{3}=dfrac{5}{6}, x eq 0 onumber ]\n\nStep 2. Find the least common denominator of all denominators in the equation.\n\nFind the LCD of (dfrac{1}{x}), (dfrac{1}{3}), and (dfrac{5}{6})\n\nThe LCD is (6x).\n\nStep 3. Clear the fractions by multiplying both sides of the equation by the LCD.\n\nMultiply both sides of the equation by the LCD, (6x).\n\n[{color{red}6 x} cdotleft(dfrac{1}{x}+dfrac{1}{3} ight)={color{red}6 x} cdotleft(dfrac{5}{6} ight) onumber ]\n\nUse the Distributive Property.\n\n[{color{red}6 x} cdot dfrac{1}{x}+{color{red}6 x} cdot dfrac{1}{3}={color{red}6 x} cdotleft(dfrac{5}{6} ight) onumber ]\n\nSimplify - and notice, no more fractions!\n\n[6+2 x=5 x onumber ]\n\nStep 4. Solve the resulting equation.\n\nSimplify.\n\n[\begin{aligned} &6=3 x &2=x end{aligned} onumber ]\n\nStep 5. Check.\n\nIf any values found in Step 1 are algebraic solutions, discard them. Check any remaining solutions in the original equation.\n\nWe did not get 0 as an algebraic solution.\n\n[dfrac{1}{x}+dfrac{1}{3}=dfrac{5}{6} onumber ]\n\nWe substitute (x=2) into the original equation.\n\n[\begin{aligned} frac{1}{2}+frac{1}{3}&overset{?}{=}frac{5}{6} frac{3}{6}+frac{2}{6}&overset{?}{=}frac{5}{6} frac{5}{6}&=frac{5}{6} surd end{aligned} onumber ]\n\nThe solution is (x=2)\n\nExercise (PageIndex{1})\n\nSolve: [dfrac{1}{y}+dfrac{2}{3}=dfrac{1}{5} onumber ]\n\n(y=-dfrac{7}{15})\n\nExercise (PageIndex{2})\n\nSolve: [dfrac{2}{3}+dfrac{1}{5}=dfrac{1}{x} onumber ]\n\n(x=dfrac{13}{15})\n\nThe steps of this method are shown.\n\nhow to Solve equations with rational expressions.\n\n\u2022 Step 1. Note any value of the variable that would make any denominator zero.\n\u2022 Step 2. Find the least common denominator of all denominators in the equation.\n\u2022 Step 3. Clear the fractions by multiplying both sides of the equation by the LCD.\n\u2022 Step 4. Solve the resulting equation.\n\u2022 Step 5. Check:\n\u2022 If any values found in Step 1 are algebraic solutions, discard them.\n\u2022 Check any remaining solutions in the original equation.\n\nWe always start by noting the values that would cause any denominators to be zero.\n\nExample (PageIndex{2}): How to Solve a Rational Equation using the Zero Product Property\n\nSolve: [1-dfrac{5}{y}=-dfrac{6}{y^{2}} onumber ]\n\nSolution\n\nNote any value of the variable that would make any denominator zero.\n\n[1-dfrac{5}{y}=-dfrac{6}{y^{2}}, y eq 0 onumber ]\n\nFind the least common denominator of all denominators in the equation. The LCD is (y^2).\n\nClear the fractions by multiplying both sides of the equation by the LCD.\n\n[y^{2}left(1-dfrac{5}{y} ight)=y^{2}left(-dfrac{6}{y^{2}} ight) onumber ]\n\nDistribute.\n\n[y^{2} cdot 1-y^{2}left(dfrac{5}{y} ight)=y^{2}left(-dfrac{6}{y^{2}} ight) onumber ]\n\nMultiply.\n\n[y^{2}-5 y=-6 onumber ]\n\nSolve the resulting equation. First write the quadratic equation in standard form.\n\n[y^{2}-5 y+6=0 onumber ]\n\nFactor.\n\n[(y-2)(y-3)=0 onumber ]\n\nUse the Zero Product Property.\n\n[y-2=0 ext { or } y-3=0 onumber ]\n\nSolve.\n\n[y=2 ext { or } y=3 onumber ]\n\nCheck. We did not get (0) as an algebraic solution.\n\nCheck (y=2) and (y=3)in the original equation.\n\n[1-dfrac{5}{2} overset{?}{=} -dfrac{6}{2^{2}} quad quad quad 1-dfrac{5}{3} overset{?}{=} -dfrac{6}{3^{2}} onumber ]\n\n[1-dfrac{5}{2} overset{?}{=}-dfrac{6}{4} quad quad quad 1-dfrac{5}{3} overset{?}{=} -dfrac{6}{9} onumber ]\n\n[dfrac{2}{2}-dfrac{5}{2} overset{?}{=} -dfrac{6}{4} quad quad quad dfrac{3}{3}-dfrac{5}{3} overset{?}{=} -dfrac{6}{9} onumber ]\n\n[-dfrac{3}{2} overset{?}{=} -dfrac{6}{4} quad quad quad -dfrac{2}{3} overset{?}{=} -dfrac{6}{9} onumber ]\n\nThe solution is (y=2,y=3)\n\nExercise (PageIndex{3})\n\nSolve: [1-dfrac{2}{x}=dfrac{15}{x^{2}} onumber ]\n\n(x=-3, x=5)\n\nExercise (PageIndex{4})\n\nSolve: [1-dfrac{4}{y}=dfrac{12}{y^{2}} onumber ]\n\n(y=-2, y=6)\n\nIn the next example, the last denominators is a difference of squares. Remember to factor it first to find the LCD.\n\nExample (PageIndex{3})\n\nSolve: [dfrac{2}{x+2}+dfrac{4}{x-2}=dfrac{x-1}{x^{2}-4} onumber ]\n\nSolution\n\nNote any value of the variable that would make any denominator zero.\n\n[dfrac{2}{x+2}+dfrac{4}{x-2}=dfrac{x-1}{(x+2)(x-2)}, x eq-2, x eq 2 onumber ]\n\nFind the least common denominator of all denominators in the equation. The LCD is ((x+2)(x-2)).\n\nClear the fractions by multiplying both sides of the equation by the LCD.\n\n[(x+2)(x-2)left(dfrac{2}{x+2}+dfrac{4}{x-2} ight)=(x+2)(x-2)left(dfrac{x-1}{x^{2}-4} ight) onumber ]\n\nDistribute.\n\n[(x+2)(x-2) dfrac{2}{x+2}+(x+2)(x-2) dfrac{4}{x-2}=(x+2)(x-2)left(dfrac{x-1}{x^{2}-4} ight) onumber ]\n\nRemove common factors.\n\n[cancel {(x+2)}(x-2) dfrac{2}{cancel {x+2}}+(x+2){cancel {(x-2)}} dfrac{4}{cancel {x-2}}=cancel {(x+2)(x-2)}left(dfrac{x-1}{cancel {x^{2}-4}} ight) onumber ]\n\nSimplify.\n\n[2(x-2)+4(x+2)=x-1 onumber ]\n\nDistribute.\n\n[2 x-4+4 x+8=x-1 onumber ]\n\nSolve.\n\n[\begin{aligned} 6 x+4&=x-1 5 x&=-5 x&=-1 end{aligned}]\n\nCheck: We did not get 2 or \u22122 as algebraic solutions.\n\nCheck (x=-1) in the original equation.\n\n[\begin{aligned} dfrac{2}{x+2}+dfrac{4}{x-2} &=dfrac{x-1}{x^{2}-4} dfrac{2}{(-1)+2}+dfrac{4}{(-1)-2} &overset{?}{=} dfrac{(-1)-1}{(-1)^{2}-4} dfrac{2}{1}+dfrac{4}{-3} &overset{?}{=} dfrac{-2}{-3} dfrac{6}{3}-dfrac{4}{3} &overset{?}{=} dfrac{2}{3} dfrac{2}{3} &=dfrac{2}{3} surd end{aligned} onumber ]\n\nThe solution is (x=-1).\n\nExercise (PageIndex{5})\n\nSolve: [dfrac{2}{x+1}+dfrac{1}{x-1}=dfrac{1}{x^{2}-1} onumber ]\n\n(x=dfrac{2}{3})\n\nExercise (PageIndex{6})\n\nSolve: [dfrac{5}{y+3}+dfrac{2}{y-3}=dfrac{5}{y^{2}-9} onumber ]\n\n(y=2)\n\nIn the next example, the first denominator is a trinomial. Remember to factor it first to find the LCD.\n\nExample (PageIndex{4})\n\nSolve: [dfrac{m+11}{m^{2}-5 m+4}=dfrac{5}{m-4}-dfrac{3}{m-1} onumber ]\n\nSolution\n\nNote any value of the variable that would make any denominator zero. Use the factored form of the quadratic denominator.\n\n[dfrac{m+11}{(m-4)(m-1)}=dfrac{5}{m-4}-dfrac{3}{m-1}, m eq 4, m eq 1 onumber ]\n\nFind the least common denominator of all denominators in the equation. The LCD is ((m-4)(m-1))\n\nClear the fractions by multiplying both sides of the equation by the LCD.\n\n[(m-4)(m-1)left(dfrac{m+11}{(m-4)(m-1)} ight)=(m-4)(m-1)left(dfrac{5}{m-4}-dfrac{3}{m-1} ight) onumber ]\n\nDistribute.\n\n[(m-4)(m-1)left(dfrac{m+11}{(m-4)(m-1)} ight)=(m-4)(m-1) dfrac{5}{m-4}-(m-4)(m-1) dfrac{3}{m-1} onumber ]\n\nRemove common factors.\n\n[cancel {(m-4)(m-1)}left(dfrac{m+11}{cancel {(m-4)(m-1)}} ight)=cancel {(m-4)}(m-1) dfrac{5}{cancel{m-4}}-(m-4)cancel {(m-1)} dfrac{3}{cancel {m-1}} onumber ]\n\nSimplify.\n\n[m+11=5(m-1)-3(m-4) onumber ]\n\nSolve the resulting equation.\n\n[\begin{aligned} m+11&=5 m-5-3 m+12 4&=m end{aligned} onumber ]\n\nCheck. The only algebraic solution was 4, but we said that 4 would make a denominator equal to zero. The algebraic solution is an extraneous solution.\n\nThere is no solution to this equation.\n\nExercise (PageIndex{7})\n\nSolve: [dfrac{x+13}{x^{2}-7 x+10}=dfrac{6}{x-5}-dfrac{4}{x-2} onumber ]\n\nThere is no solution.\n\nExercise (PageIndex{8})\n\nSolve: [dfrac{y-6}{y^{2}+3 y-4}=dfrac{2}{y+4}+dfrac{7}{y-1} onumber ]\n\nThere is no solution.\n\nThe equation we solved in the previous example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. In the next example we get two algebraic solutions. Here one or both could be extraneous solutions.\n\nExample (PageIndex{5})\n\nSolve: [dfrac{y}{y+6}=dfrac{72}{y^{2}-36}+4 onumber ]\n\nSolution\n\nFactor all the denominators, so we can note any value of the variable that would make any denominator zero.\n\n[dfrac{y}{y+6}=dfrac{72}{(y-6)(y+6)}+4, y eq 6, y eq-6 onumber ]\n\nFind the least common denominator. The LCD is ((y-6)(y+6))\n\nClear the fractions.\n\n[(y-6)(y+6)left(dfrac{y}{y+6} ight)=(y-6)(y+6)left(dfrac{72}{(y-6)(y+6)}+4 ight) onumber ]\n\nSimplify.\n\n[(y-6) cdot y=72+(y-6)(y+6) cdot 4 onumber ]\n\nSimplify.\n\n[y(y-6)=72+4left(y^{2}-36 ight) onumber ]\n\nSolve the resulting equation.\n\n[\begin{aligned} y^{2}-6 y&=72+4 y^{2}-144 0&=3 y^{2}+6 y-72 0&=3left(y^{2}+2 y-24 ight) 0&=3(y+6)(y-4) y&=-6, y=4 end{aligned} onumber ]\n\nCheck.\n\n(y=-6) is an extraneous solution. Check (y=4) in the original equation.\n\n[\begin{aligned} dfrac{y}{y+6} &=dfrac{72}{y^{2}-36}+4 dfrac{4}{4+6} &overset{?}{=}dfrac{72}{4^{2}-36}+4 dfrac{4}{10} &overset{?}{=} dfrac{72}{-20}+4 dfrac{4}{10} &overset{?}{=} -dfrac{36}{10}+dfrac{40}{10} dfrac{4}{10} &=dfrac{4}{10} surd end{aligned} onumber ]\n\nThe solution is (y=4).\n\nExercise (PageIndex{9})\n\nSolve: [dfrac{x}{x+4}=dfrac{32}{x^{2}-16}+5 onumber ]\n\n(x=3)\n\nExercise (PageIndex{10})\n\nSolve: [dfrac{y}{y+8}=dfrac{128}{y^{2}-64}+9 onumber ]\n\n(y=7)\n\nIn some cases, all the algebraic solutions are extraneous.\n\nExample (PageIndex{6})\n\nSolve: [dfrac{x}{2 x-2}-dfrac{2}{3 x+3}=dfrac{5 x^{2}-2 x+9}{12 x^{2}-12} onumber ]\n\nSolution\n\nWe will start by factoring all denominators, to make it easier to identify extraneous solutions and the LCD.\n\n[dfrac{x}{2(x-1)}-dfrac{2}{3(x+1)}=dfrac{5 x^{2}-2 x+9}{12(x-1)(x+1)} onumber ]\n\nNote any value of the variable that would make any denominator zero.\n\n[dfrac{x}{2(x-1)}-dfrac{2}{3(x+1)}=dfrac{5 x^{2}-2 x+9}{12(x-1)(x+1)}, x eq 1, x eq-1 onumber ]\n\nFind the least common denominator. The LCD is (12(x-1)(x+1)).\n\nClear the fractions.\n\n[12(x-1)(x+1)left(dfrac{x}{2(x-1)}-dfrac{2}{3(x+1)} ight)=12(x-1)(x+1)left(dfrac{5 x^{2}-2 x+9}{12(x-1)(x+1)} ight) onumber ]\n\nSimplify.\n\n[6(x+1) cdot x-4(x-1) cdot 2=5 x^{2}-2 x+9 onumber ]\n\nSimplify.\n\n[6 x(x+1)-4 cdot 2(x-1)=5 x^{2}-2 x+9 onumber ]\n\nSolve the resulting equation.\n\n[\begin{aligned} 6 x^{2}+6 x-8 x+8&=5 x^{2}-2 x+9 x^{2}-1&=0 (x-1)(x+1)&=0 x&=1 ext { or } x=-1 end{aligned} onumber ]\n\nCheck.\n\n(x=1) and (x=-1) are extraneous solutions.\n\nThe equation has no solution.\n\nExercise (PageIndex{11})\n\nSolve: [dfrac{y}{5 y-10}-dfrac{5}{3 y+6}=dfrac{2 y^{2}-19 y+54}{15 y^{2}-60} onumber ]\n\nThere is no solution.\n\nExercise (PageIndex{12})\n\nSolve: [dfrac{z}{2 z+8}-dfrac{3}{4 z-8}=dfrac{3 z^{2}-16 z-16}{8 z^{2}+2 z-64} onumber ]\n\nThere is no solution.\n\nExample (PageIndex{7})\n\nSolve: [dfrac{4}{3 x^{2}-10 x+3}+dfrac{3}{3 x^{2}+2 x-1}=dfrac{2}{x^{2}-2 x-3} onumber ]\n\nSolution\n\nFactor all the denominators, so we can note any value of the variable that would make any denominator zero.\n\n[dfrac{4}{(3 x-1)(x-3)}+dfrac{3}{(3 x-1)(x+1)}=dfrac{2}{(x-3)(x+1)}, x eq-1, x eq dfrac{1}{3}, x eq 3 onumber ]\n\nFind the least common denominator. The LCD is ((3 x-1)(x+1)(x-3)).\n\nClear the fractions.\n\n[(3 x-1)(x+1)(x-3)left(dfrac{4}{(3 x-1)(x-3)}+dfrac{3}{(3 x-1)(x+1)} ight)=(3 x-1)(x+1)(x-3)left(dfrac{2}{(x-3)(x+1)} ight) onumber ]\n\nSimplify.\n\n[4(x+1)+3(x-3)=2(3 x-1) onumber ]\n\nDistribute.\n\n[4 x+4+3 x-9=6 x-2 onumber ]\n\nSimplify.\n\n[7 x-5=6 x-2 onumber ]\n\n[x=3 onumber ]\n\nThe only algebraic solution was (x=3)$,$ but we said that (x=3) would make a denominator equal to zero. The algebraic solution is an extraneous solution.\n\nThere is no solution to this equation.\n\nExercise (PageIndex{13})\n\nSolve: [dfrac{15}{x^{2}+x-6}-dfrac{3}{x-2}=dfrac{2}{x+3} onumber ]\n\nThere is no solution.\n\nExercise (PageIndex{14})\n\nSolve: [dfrac{5}{x^{2}+2 x-3}-dfrac{3}{x^{2}+x-2}=dfrac{1}{x^{2}+5 x+6} onumber ]\n\nThere is no solution.\n\n## Use Rational Functions\n\nWorking with functions that are defined by rational expressions often lead to rational equations. Again, we use the same techniques to solve them.\n\nExample (PageIndex{8})\n\nFor rational function, (f(x)=dfrac{2 x-6}{x^{2}-8 x+15}):\n\n1. Find the domain of the function\n2. Solve (f(x)=1)\n3. Find the points on the graph at this function value.\n\nSolution\n\n1. The domain of a rational function is all real numbers except those that make the rational expression undefined. So to find them, we will set the denominator equal to zero and solve.\n\n[\begin{aligned} x^{2}-8 x+15&=0 (x-3)(x-5)&=0 quad ext{Factor the trinomial.} x-3&=0 quad ext {Use the Zero Product Property.} x-5&=0 quad ext {Use the Zero Product Property.} x=3 &; x=5 ext{ Solve.} end{aligned} onumber ]\n\nThe domain is all real numbers except (x eq 3, x eq 5)\n\n1. [f(x)=1 onumber ]\n\nSubstitute in the rational expression.\n\n[dfrac{2 x-6}{x^{2}-8 x+15}=1 onumber ]\n\nFactor the denominator.\n\n[dfrac{2 x-6}{(x-3)(x-5)}=1 onumber ]\n\nMultiply both sides by the LCD, ((x-3)(x-5))\n\n[(x-3)(x-5)left(dfrac{2 x-6}{(x-3)(x-5)} ight)=(x-3)(x-5)(1) onumber ]\n\nSimplify.\n\n[2 x-6=x^{2}-8 x+15 onumber ]\n\nSolve.\n\n[0=x^{2}-10 x+21 onumber ]\n\nFactor.\n\n[0=(x-7)(x-3) onumber ]\n\nUse the Zero Product Property.\n\n[x-7=0 quad x-3=0 onumber ]\n\nSolve.\n\n[x=7 quad x=3 onumber ]\n\n1. The value of the function is 1 when (x=7, x=3)$.$ So the points on the graph of this function when (f(x)=1)$,$ will be ((7,1),(3,1)).\n\nExercise (PageIndex{15})\n\nFor rational function, (f(x)=dfrac{8-x}{x^{2}-7 x+12})\n\n1. Find the domain of the function.\n2. Solve (f(x)=3).\n3. Find the points on the graph at this function value.\n1. The domain is all real numbers except (x eq 3) and (x eq 4)\n2. (x=2, x=dfrac{14}{3})\n3. ((2,3),left(dfrac{14}{3}, 3 ight))\n\nExercise (PageIndex{16})\n\nFor rational function, (f(x)=dfrac{x-1}{x^{2}-6 x+5})\n\n1. Solve (f(x)=4).\n2. Find the points on the graph at this function value.\n1. The domain is all real numbers except (x eq 1) and (x eq 5)\n2. (x=dfrac{21}{4})\n3. (left(dfrac{21}{4}, 4 ight))\n\n## Solve a Rational Equation for a Specific Variable\n\nWhen we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.\n\nWhen we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD.\n\n[\begin{aligned} m &=frac{y-y_{1}}{x-x_{1}} mleft(x-x_{1} ight) &=left(frac{y-y_{1}}{x-x_{1}} ight)left(x-x_{1} ight) quad ext{Multiply both sides of the equation by } x-x_1. mleft(x-x_{1} ight) &=y-y_{1} quad ext {Simplify.} y-y_{1} &=mleft(x-x_{1} ight) quad ext {Rewrite the equation with the y terms on the left.} end{aligned} onumber ]\n\nIn the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through the point ((2,3)). We will add one more step to solve for (y).\n\nExample (PageIndex{9})\n\nSolve: (m=dfrac{y-2}{x-3}) for (y).\n\nSolution\n\n[m=dfrac{y-2}{x-3} onumber ]\n\nNote any value of the variable that would make any denominator zero.\n\n[m=dfrac{y-2}{x-3}, x eq 3 onumber ]\n\nClear the fractions by multiplying both sides of the equation by the LCD, (x-3).\n\n[(x-3) m=(x-3)left(dfrac{y-2}{x-3} ight) onumber ]\n\nSimplify.\n\n[x m-3 m=y-2 onumber ]\n\nIsolate the term with (y).\n\n[x m-3 m+2=y onumber ]\n\nExercise (PageIndex{17})\n\nSolve: (m=dfrac{y-5}{x-4}) for (y).\n\n(y=m x-4 m+5)\n\nExercise (PageIndex{18})\n\nSolve: (m=dfrac{y-1}{x+5}) for (y).\n\n(y=m x+5 m+1)\n\nRemember to multiply both sides by the LCD in the next example.\n\nExample (PageIndex{10})\n\nSolve: (dfrac{1}{c}+dfrac{1}{m}=1) for (c)\n\nSolution\n\n[dfrac{1}{c}+dfrac{1}{m}=1 ext { for } c onumber ]\n\nNote any value of the variable that would make any denominator zero.\n\n[dfrac{1}{c}+dfrac{1}{m}=1, c eq 0, m eq 0 onumber ]\n\nClear the fractions by multiplying both sides of the equations by the LCD, (cm).\n\n[cmleft(dfrac{1}{c}+dfrac{1}{m} ight)=cm(1) onumber ]\n\nDistribute.\n\n[cmleft(frac{1}{c} ight)+cm frac{1}{m}=cm(1) onumber ]\n\nSimplify.\n\n[m+c=cm onumber ]\n\nCollect the terms with (c) to the right.\n\n[m=cm-c onumber ]\n\nFactor the expression on the right.\n\n[m=c(m-1) onumber ]\n\nTo isolate (c), divide both sides by (m-1).\n\n[dfrac{m}{m-1}=dfrac{c(m-1)}{m-1} onumber ]\n\nSimplify by removing common factors.\n\n[dfrac{m}{m-1}=c onumber ]\n\nNotice that even though we excluded (c=0) and (m=0) from the original equation, we must also now state that (m eq 1).\n\nExercise (PageIndex{19})\n\nSolve: (dfrac{1}{a}+dfrac{1}{b}=c) for (a).\n\n(a=dfrac{b}{c b-1})\n\nExercise (PageIndex{20})\n\nSolve: (dfrac{2}{x}+dfrac{1}{3}=dfrac{1}{y}) for (y)\n\n(y=dfrac{3 x}{x+6})\n\n## 67 Solve Rational Equations\n\nIf you miss a problem, go back to the section listed and review the material.\n\n1. Solve:\nIf you missed this problem, review (Figure).\n2. Solve:\nIf you missed this problem, review (Figure).\n3. Solve for in terms of : for\nIf you missed this problem, review (Figure).\n\nAfter defining the terms expression and equation early in Foundations, we have used them throughout this book. We have simplified many kinds of expressions and solved many kinds of equations. We have simplified many rational expressions so far in this chapter. Now we will solve rational equations.\n\nThe definition of a rational equation is similar to the definition of equation we used in Foundations.\n\nA rational equation is two rational expressions connected by an equal sign.\n\nYou must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.\n\n### Solve Rational Equations\n\nWe have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to \u201cclear\u201d the fractions.\n\nHere is an example we did when we worked with linear equations:\n\n We multiplied both sides by the LCD. Then we distributed. We simplified\u2014and then we had an equation with no fractions. Finally, we solved that equation.\n\nWe will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then we will have an equation that does not contain rational expressions and thus is much easier for us to solve.\n\nBut because the original equation may have a variable in a denominator we must be careful that we don\u2019t end up with a solution that would make a denominator equal to zero.\n\nSo before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.\n\nAn algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution.\n\nAn extraneous solution to a rational equation is an algebraic solution that would cause any of the expressions in the original equation to be undefined.\n\nWe note any possible extraneous solutions, c, by writing next to the equation.\n\nSolve:\n\nSolve:\n\nSolve:\n\nThe steps of this method are shown below.\n\n1. Note any value of the variable that would make any denominator zero.\n2. Find the least common denominator of all denominators in the equation.\n3. Clear the fractions by multiplying both sides of the equation by the LCD.\n4. Solve the resulting equation.\n5. Check.\n\u2022 If any values found in Step 1 are algebraic solutions, discard them.\n\u2022 Check any remaining solutions in the original equation.\n\nWe always start by noting the values that would cause any denominators to be zero.\n\nSolve:\n\n Note any value of the variable that would make any denominator zero. Find the least common denominator of all denominators in the equation. The LCD is. Clear the fractions by multiplying both sides of the equation by the LCD. Distribute. Multiply. Solve the resulting equation. First write the quadratic equation in standard form. Factor. Use the Zero Product Property. Solve. Check. We did not get 0 as an algebraic solution.\n\nSolve:\n\nSolve:\n\nSolve:\n\n Note any value of the variable that would make any denominator zero. Find the least common denominator of all denominators in the equation. The LCD is. Clear the fractions by multiplying both sides of the equation by the LCD. Remove common factors. Simplify. Multiply. Solve the resulting equation. We did not get 0 or as algebraic solutions.\n\nSolve:\n\nSolve:\n\nWhen one of the denominators is a quadratic, remember to factor it first to find the LCD.\n\nSolve:\n\n Note any value of the variable that would make any denominator zero. Find the least common denominator of all denominators in the equation. The LCD is. Clear the fractions by multiplying both sides of the equation by the LCD. Distribute. Remove common factors. Simplify. Distribute. Solve. We did not get as algebraic solutions.\n\nSolve:\n\nSolve:\n\nSolve:\n\n Note any value of the variable that would make any denominator zero. Find the least common denominator of all denominators in the equation. The LCD is. Clear the fractions by multiplying both sides of the equation by the LCD. Distribute. Remove common factors. Simplify. Simplify. Combine like terms. Solve. First write in standard form. Factor. Use the Zero Product Property. We did not get 4 or 3 as algebraic solutions.\n\nSolve:\n\nSolve:\n\nSolve:\n\n Factor all the denominators, so we can note any value of the variable the would make any denominator zero. Find the least common denominator of all denominators in the equation. The LCD is. Clear the fractions. Distribute. Remove common factors. Simplify. Solve the resulting equation. Check. The only algebraic solution was 4, but we said that 4 would make a denominator equal to zero. The algebraic solution is an extraneous solution. There is no solution to this equation.\n\nSolve:\n\nSolve:\n\nThe equation we solved in (Figure) had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. Some equations have no solution.\n\nSolve:\n\n Note any value of the variable that would make any denominator zero. Find the least common denominator of all denominators in the equation. The LCD is. Clear the fractions by multiplying both sides of the equation by the LCD. Distribute. Remove common factors. Simplify. Solve the resulting equation. Check. is an extraneous solution.\n\nSolve:\n\nSolve:\n\nSolve:\n\n Factor all the denominators, so we can note any value of the variable that would make any denominator zero. Find the least common denominator. The LCD is. Clear the fractions. Simplify. Simplify. Solve the resulting equation. Check. is an extraneous solution.\n\nSolve:\n\nSolve:\n\nSolve:\n\n We will start by factoring all denominators, to make it easier to identify extraneous solutions and the LCD. Note any value of the variable that would make any denominator zero. Find the least common denominator.The LCD is Clear the fractions. Simplify. Simplify. Solve the resulting equation. Check. and are extraneous solutions. The equation has no solution.\n\nSolve:\n\nSolve:\n\n### Solve a Rational Equation for a Specific Variable\n\nWhen we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.\n\nWe\u2019ll start with a formula relating distance, rate, and time. We have used it many times before, but not usually in this form.\n\nSolve:\n\n Note any value of the variable that would make any denominator zero. Clear the fractions by multiplying both sides of the equations by the LCD, T. Simplify. Divide both sides by R to isolate T. Simplify.\n\nSolve: for\n\nSolve: for\n\n(Figure) uses the formula for slope that we used to get the point-slope form of an equation of a line.\n\nSolve:\n\n Note any value of the variable that would make any denominator zero. Clear the fractions by multiplying both sides of the equations by the LCD, . Simplify. Isolate the term with y. Divide both sides by m to isolate y. Simplify.\n\nSolve: for\n\nSolve: for\n\nBe sure to follow all the steps in (Figure). It may look like a very simple formula, but we cannot solve it instantly for either denominator.\n\nSolve\n\n Note any value of the variable that would make any denominator zero. Clear the fractions by multiplying both sides of the equations by the LCD, . Distribute. Simplify. Collect the terms with c to the right. Factor the expression on the right. To isolate c, divide both sides by . Simplify by removing common factors.\n\nNotice that even though we excluded from the original equation, we must also now state that .\n\nSolve: for\n\nSolve: for\n\n### Key Concepts\n\n\u2022 Strategy to Solve Equations with Rational Expressions\n1. Note any value of the variable that would make any denominator zero.\n2. Find the least common denominator of all denominators in the equation.\n3. Clear the fractions by multiplying both sides of the equation by the LCD.\n4. Solve the resulting equation.\n5. Check.\n\u2022 If any values found in Step 1 are algebraic solutions, discard them.\n\u2022 Check any remaining solutions in the original equation.\n\n#### Practice Makes Perfect\n\nSolve Rational Equations\n\nIn the following exercises, solve.\n\nSolve a Rational Equation for a Specific Variable\n\nIn the following exercises, solve.\n\n#### Everyday Math\n\nHouse Painting Alain can paint a house in 4 days. Spiro would take 7 days to paint the same house. Solve the equation for t to find the number of days it would take them to paint the house if they worked together.\n\ndays\n\nBoating Ari can drive his boat 18 miles with the current in the same amount of time it takes to drive 10 miles against the current. If the speed of the boat is 7 knots, solve the equation for c to find the speed of the current.\n\n#### Writing Exercises\n\nWhy is there no solution to the equation ?\n\nPete thinks the equation has two solutions, . Explain why Pete is wrong.\n\n#### Self Check\n\n\u24d0 After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.\n\n\u24d1 After reviewing this checklist, what will you do to become confident for all objectives?\n\n## 7.5: Solve Rational Equations\n\nApplying Rational Equations\n\n\u00b7 Solve real world problems using rational functions.\n\nRational expressions and rational equations can be useful tools for representing real life situations and for finding answers to real problems. In particular, they are quite good for describing distance-speed-time questions, and modeling multi-person work problems.\n\nSolving Work Problems\n\nWork problems often ask us to calculate how long it will take different people working at different speeds to finish a task. The algebraic models of such situations often involve rational equations derived from the work formula, W = rt. The amount of work done (W) is the product of the rate of work (r) and the time spent working (t). The work formula has 3 versions:\n\nSome work problems have multiple machines or people working on a project together for the same amount of time but at different rates. In that case, we can add their individual work rates together to get a total work rate. Let\u2019s look at an example:\n\nMyra takes 2 hours to plant 500 flower bulbs. Francis takes 3 hours to plant 450 flower bulbs. Working together, how long should it take them to plant 1500 bulbs?\n\nMyra : 500 bulbs\/2 hours = 250 bulbs\/hour\n\nThink about how many bulbs each person can plant in one hour. This is their planting rate.\n\n250 + 150 bulbs\/hour = 400 bulbs\/hour\n\nCombine their hourly rates to determine the rate they work together.\n\nUse one of the work formulas to write a rational equation, for example . We know r, the combined work rate, and we know W, the amount of work that must be done. What we don't know is how much time it will take to do the required work at the designated rate.\n\nSolve the equation by multiplying both sides by the common denominator, then isolating t.\n\nIt takes 3 hours 45 minutes for Myra and Francis to plant 1500 bulbs together.\n\nOther work problems go the other way. We'll calculate how long it will take one person to do a job alone when we know how long it takes a group to get it done:\n\nJamie, Pria and Paul can paint a room together in 2 hours. If Pria does the job alone she can paint the room in 5 hours. If Paul works alone, he can paint the room in 6 hours. If Jamie works alone, how long would it take her to paint the room?\n\nPria + Paul + Jamie = room\/ hour\n\nDetermine the hourly rates for each person and for the whole group using the formula .\n\nWork is painting 1 room, so W = 1.\n\nWe don\u2019t know how long Jamie will take, so we need to keep the variable t.\n\nWrite an equation to show that the sum of their individual rates equals the group rate.\n\n(Think of it this way: Pria works for one hour and paints of the room. Paul works for an hour and paints of the room. Jamie works for an hour and paints of the room. Together they have painted half the room in one hour.\n\nSolve the rational equation.\n\nFirst find the least common denominator of the individual rates. It is 5 \u2022 6 \u2022 t = 30t.\n\nThen multiply each term on the left by a fractional form of 1 so that all rates have the same denominator and can be added. (Note: we could also have found the common denominator of the entire equation, which is also 30t, and multiplied both sides by it.)\n\n11t - 11t + 30 = 15t \u2013 11t\n\nNow multiply both sides of the equation by the common denominator, then simplify.\n\nIt would take Jamie 7.5 hours to paint the room by herself.\n\nTanya and Cam can each wash a car and vacuum its interior in 2 hours. Pat needs 3 hours to do this same job by himself. If Pat, Cam and Tanya work together, how long will it take them to clean a car?\n\nA). Incorrect. If all three worked at the same rate, then the time for one car could be calculated by dividing the time it takes them working alone to clean 3 cars by 3. But that doesn't work in this case because Pat has a different cleaning rate than the others. The correct answer is 45 minutes.\n\nB) Correct. According to the formula, . Tanya and Cam each have a rate of car in one hour, and Pat\u2019s rate is car in one hour. Working together, they have a rate of , or . W is one car, so the formula becomes = , and so t = . It takes three-quarters of an hour, or 45 minutes, to clean one car.\n\nC) Incorrect. Tanya and Cam clean a car in two hours each, not as a team. They each have a rate of car in one hour, and Pat\u2019s rate is car in one hour. Working together, they have a rate of . The correct answer is 45 minutes.\n\nD) Incorrect. This is the time it would take for Cam and Tanya to clean one car together. Since Pat is helping, it will take less time than that. The correct answer is 45 minutes.\n\nSometimes work problems describe rates in a relative way: someone works 3 times as fast as someone else or a machine takes 2 fewer hours to finish a job than another model of machine. In these instances, we express one rate using information about another rate. Let\u2019s look at an example:\n\nOne pipe can fill a pool 1.5 times faster than a second pipe. If both pipes are open, the pool can be filled in 6 hours. If only the slower pipe is open, how long would it take to fill the pool?\n\nFind the rates of each pipe alone and the two working together.\n\nHours needed for fast pipe to fill pool: p\n\nHours needed for slow pipe to fill pool alone: 1.5 p\n\nHours needed for both pipes together: 6\n\nWrite an equation that shows that the amount of work completed by both pipes in one hour is equal to the sum of the work of each pipe.\n\nSolve for p. One way to do this is to rewrite the rational expressions using a common denominator.\n\nCommon denominator of p, 1.5p and 6 is 6p.\n\nSlow pipe takes 1.5p hours to do the work alone.\n\nThe slower pipe will take 15 hours to fill the pool alone.\n\nRational equations can be used to solve a variety of problems that involve rates, times and work. Using rational expressions and equations can help us answer questions about how to combine workers or machines to complete a job on schedule.\n\n## Open Resources for Community College Algebra\n\nWe open this section looking back on Example 12.3.2. Julia is taking her family on a boat trip (12) miles down the river and back. The river flows at a speed of (2) miles per hour and she wants to drive the boat at a constant speed, (v) miles per hour downstream and back upstream. Due to the current of the river, the actual speed of travel is (v+2) miles per hour going downstream, and (v-2) miles per hour going upstream. If Julia plans to spend (8) hours for the whole trip, how fast should she drive the boat?\n\nThe time it takes Julia to drive the boat downstream is (frac<12>) hours, and upstream is (frac<12>) hours. The function to model the whole trip's time is\n\nwhere (t) stands for time in hours. The trip will take (8) hours, so we want (t(v)) to equal (8 ext<,>) and we have:\n\nInstead of using the function's graph, we will solve this equation algebraically. You may wish to review the technique of eliminating denominators discussed in Subsection 2.3.2. We can use the same technique with variable expressions in the denominators. To remove the fractions in this equation, we will multiply both sides of the equation by the least common denominator ((v-2)(v+2) ext<,>) and we have:\n\n###### Remark 12.5.2 .\n\nAt this point, logically all that we know is that the only possible solutions are (-1) and (4 ext<.>) Because of the step where factors were canceled, it's possible that these might not actually be solutions to the original equation. They each might be what is called an . An extraneous solution is a number that would appear to be a solution based on the solving process, but actually does not make the original equation true. Because of this, it is important that these proposed solutions be checked. Note that we're not checking to see if we made a calculation error, but are instead checking to see if the proposed solutions actually solve the original equation.\n\nAlgebraically, both values do check out to be solutions. In the context of this scenario, the boat's speed can't be negative, so we only take the solution (4 ext<.>) If Julia drives at (4) miles per hour, the whole trip would take (8) hours. This result matches the solution in Example 12.3.2.\n\n###### Definition 12.5.3 . Rational Equation.\n\nA rational equation is an equation involving one or more rational expressions. Usually, we consider these to be equations that have the variable in the denominator of at least one term.\n\nLet's look at another application problem.\n\n###### Example 12.5.4 .\n\nIt takes Ku (3) hours to paint a room and it takes Jacob (6) hours to paint the same room. If they work together, how long would it take them to paint the room?\n\nSince it takes Ku (3) hours to paint the room, he paints (frac<1><3>) of the room each hour. Similarly, Jacob paints (frac<1><6>) of the room each hour. If they work together, they paint (frac<1><3>+frac<1><6>) of the room each hour.\n\nAssume it takes (x) hours to paint the room if Ku and Jacob work together. This implies they paint (frac<1>) of the room together each hour. Now we can write this equation:\n\nTo clear away denominators, we multiply both sides of the equation by the common denominator of (3 ext<,>) (6) and (x ext<,>) which is (6x ext<:>)\n\nDoes the possible solution (x=2) check as an actual solution?\n\nIt does, so it is a solution. If Ku and Jacob work together, it would take them (2) hours to paint the room.\n\nWe are ready to outline a general process for solving a rational equation.\n\n###### Process 12.5.5 . Solving Rational Equations.\n\nTo solve a rational equation,\n\nFind the least common denominator for all terms in the equation.\n\nMultiply every term in the equation by the least common denominator\n\nEvery denominator should cancel leaving a simpler kind of equation to solve. Use previous method to solve that equation.\n\nLet's look at a few more examples of solving rational equations.\n\n###### Example 12.5.6 .\n\nThe common denominator is (y(y+1) ext<.>) We will multiply both sides of the equation by (y(y+1) ext<:>)\n\nDoes the possible solution (y=-3) check as an actual solution?\n\nIt checks, so (-3) is a solution. We write the solution set as (<-3> ext<.>)\n\n###### Example 12.5.7 .\n\nThe common denominator is (z-4 ext<.>) We will multiply both sides of the equation by (z-4 ext<:>)\n\nDo the possible solutions (z=1) and (z=4) check as actual solutions?\n\nThe possible solution (z=4) does not actually work, since it leads to division by (0) in the equation. It is an extraneous solution. However, (z=1) is a valid solution. The only solution to the equation is (1 ext<,>) and thus we can write the solution set as (<1> ext<.>)\n\n###### Example 12.5.8 .\n\nTo find the common denominator, we need to factor all denominators if possible:\n\nNow we can see the common denominator is ((p+2)(p-2) ext<.>) We will multiply both sides of the equation by ((p+2)(p-2) ext<:>)\n\nDoes the possible solution (p=2) check as an actual solution?\n\nThe possible solution (p=2) does not actually work, since it leads to division by (0) in the equation. So this is an extraneous solution, and the equation actually has no solution. We could say that its solution set is the empty set, (emptyset ext<.>)\n\n###### Example 12.5.9 .\n\nSolve (C(t)=0.35 ext<,>) where (C(t)=frac<3t>) gives a drug's concentration in milligrams per liter (t) hours since an injection. (This function was explored in the introduction of Section 12.1.)\n\nTo solve (C(t)=0.35 ext<,>) we'll begin by setting up (frac<3t>=0.35 ext<.>) We'll begin by identifying that the LCD is (t^2+8 ext<,>) and multiply each side of the equation by this:\n\nThis results in a quadratic equation so we will put it in standard form and use the quadratic formula:\n\nEach of these answers should be checked in the original equation they both work. In context, this means that the drug concentration will reach (0.35) milligrams per liter about (1.066) hours after the injection was given, and again (7.506) hours after the injection was given.\n\n### Subsection 12.5.2 Solving Rational Equations for a Specific Variable\n\nRational equations can contain many variables and constants and we can solve for any one of them. The process for solving still involves multiplying each side of the equation by the LCD. Instead of having a numerical answer though, our final result will contain other variables and constants.\n\n###### Example 12.5.10 .\n\nIn physics, when two resistances, (R_1) and (R_2 ext<,>) are connected in a parallel circuit, the combined resistance, (R ext<,>) can be calculated by the formula\n\nSolve for (R) in this formula.\n\nThe common denominator is (R R_1 R_2 ext<.>) We will multiply both sides of the equation by (R R_1 R_2 ext<:>)\n\n###### Example 12.5.11 .\n\nHere is the slope formula\n\nSolve for (x_1) in this formula.\n\nThe common denominator is (x_2-x_1 ext<.>) We will multiply both sides of the equation by (x_2-x_1 ext<:>)\n\n###### Example 12.5.12 .\n\nSolve the rational equation (x=frac<4y-1><2y-3>) for (y ext<.>)\n\nOur first step will be to multiply each side by the LCD, which is simply (2y-3 ext<.>) After that, we'll isolate all terms containing (y ext<,>) factor out (y ext<,>) and then finish solving for that variable.\n\n### Subsection 12.5.3 Solving Rational Equations Using Technology\n\nIn some instances, it may be difficult to solve rational equations algebraically. We can instead use graphing technology to obtain approximate solutions. Let's look at one such example.\n\n###### Example 12.5.13 .\n\nSolve the equation (frac<2>=frac<8>) using graphing technology.\n\nWe will define (f(x)=frac<2>) and (g(x)=frac<8> ext<,>) and then look for the points of intersection.\n\nSince the two functions intersect at approximately ((-1.524,-0.442)) and ((3.405,4.936) ext<,>) the solutions to (frac<2>=frac<8>) are approximately (-1.524) and (3.405 ext<.>) We can write the solution set as (<-1.524ldots, 3.405ldots>) or in several other forms. It may be important to do something to communicate that these solutions are approximations. Here we used \u201c(ldots)\u201d, but you could also just say in words that the solutions are approximate.\n\nDescribe what an \u201cextraneous solution\u201d to a rational equation is.\n\nIn general, when solving a rational equation, multiplying through by the will leave you with a simpler equation to solve.\n\nWhen you believe you have the solutions to a rational equation, what is more important than usual (compared to other kinds of equations) for you to do?\n\n### Exercises 12.5.5 Exercises\n\n###### Review and Warmup\n\nRecall the time that Filip traveled with his kids to kick a soccer ball on Mars? We should examine one more angle to our soccer kick question. The formula (H(t)=-6.07t^2+27.1t) finds the height of the soccer ball in feet above the ground at a time (t) seconds after being kicked.\n\nUsing technology, find out what the maximum height of the ball was and when it reached that height.\n\nUsing technology, solve for when (H(t)=20) and interpret the meaning of this in a complete sentence.\n\nUsing technology, solve for when (H(t)=0) and interpret the meaning of this in a complete sentence.\n\n###### Solving Rational Equations for a Specific Variable\n\nSolve this equation for (a ext<:>)\n\nSolve this equation for (m ext<:>)\n\nSolve this equation for (x ext<:>)\n\nSolve this equation for (C ext<:>)\n\nSolve this equation for (a ext<:>)\n\nSolve this equation for (c ext<:>)\n\nSolve this equation for (B ext<:>)\n\nSolve this equation for (a ext<:>)\n\n###### Solving Rational Equations Using Technology\n\nUse technology to solve the equation.\n\nUse technology to solve the equation.\n\nUse technology to solve the equation.\n\nUse technology to solve the equation.\n\nUse technology to solve the equation.\n\nUse technology to solve the equation.\n\n###### Application Problems\n\nScot and Jay are working together to paint a room. If Scot paints the room alone, it would take him (18) hours to complete the job. If Jay paints the room alone, it would take him (12) hours to complete the job. Answer the following question:\n\nIf they work together, it would take them hours to complete the job. Use a decimal in your answer if needed.\n\nThere are three pipes at a tank. To fill the tank, it would take Pipe A (3) hours, Pipe B (12) hours, and Pipe C (4) hours. Answer the following question:\n\nIf all three pipes are turned on, it would take hours to fill the tank.\n\nCasandra and Tien are working together to paint a room. Casandra works (1.5) times as fast as Tien does. If they work together, it took them (9) hours to complete the job. Answer the following questions:\n\nIf Casandra paints the room alone, it would take her hours to complete the job.\n\nIf Tien paints the room alone, it would take him hours to complete the job.\n\nTwo pipes are being used to fill a tank. Pipe A can fill the tank (4.5) times as fast as Pipe B does. When both pipes are turned on, it takes (18) hours to fill the tank. Answer the following questions:\n\nIf only Pipe A is turned on, it would take hours to fill the tank.\n\nIf only Pipe B is turned on, it would take hours to fill the tank.\n\nKandace and Jenny worked together to paint a room, and it took them (2) hours to complete the job. If they work alone, it would take Jenny (3) more hours than Kandace to complete the job. Answer the following questions:\n\nIf Kandace paints the room alone, it would take her hours to complete the job.\n\nIf Jenny paints the room alone, it would take her hours to complete the job.\n\nIf both Pipe A and Pipe B are turned on, it would take (2) hours to fill a tank. If each pipe is turned on alone, it takes Pipe B (3) fewer hours than Pipe A to fill the tank. Answer the following questions:\n\nIf only Pipe A is turned on, it would take hours to fill the tank.\n\nIf only Pipe B is turned on, it would take hours to fill the tank.\n\nTown A and Town B are (570) miles apart. A boat traveled from Town A to Town B, and then back to Town A. Since the river flows from Town B to Town A, the boat\u2019s speed was (25) miles per hour faster when it traveled from Town B to Town A. The whole trip took (19) hours. Answer the following questions:\n\nThe boat traveled from Town A to Town B at the speed of miles per hour.\n\nThe boat traveled from Town B back to Town A at the speed of miles per hour.\n\nA river flows at (7) miles per hour. A boat traveled with the current from Town A to Town B, which are (260) miles apart. Then, the boat turned around, and traveled against the current to reach Town C, which is (160) miles away from Town B. The second leg of the trip (Town B to Town C) took the same time as the first leg (Town A to Town B). During this whole trip, the boat was driving at a constant still-water speed. Answer the following question:\n\nDuring this trip, the boat\u2019s speed on still water was miles per hour.\n\nA river flows at (5) miles per hour. A boat traveled with the current from Town A to Town B, which are (100) miles apart. The boat stayed overnight at Town B. The next day, the water\u2019s current stopped, and boat traveled on still water to reach Town C, which is (190) miles away from Town B. The second leg of the trip (Town B to Town C) took (9) hours longer than the first leg (Town A to Town B). During this whole trip, the boat was driving at a constant still-water speed. Find this speed.\n\nNote that you should not consider the unreasonable answer.\n\nDuring this trip, the boat\u2019s speed on still water was miles per hour.\n\nTown A and Town B are (600) miles apart. With a constant still-water speed, a boat traveled from Town A to Town B, and then back to Town A. During this whole trip, the river flew from Town A to Town B at (20) miles per hour. The whole trip took (16) hours. Answer the following question:\n\nDuring this trip, the boat\u2019s speed on still water was miles per hour.\n\nTown A and Town B are (350) miles apart. With a constant still-water speed of (24) miles per hour, a boat traveled from Town A to Town B, and then back to Town A. During this whole trip, the river flew from Town B to Town A at a constant speed. The whole trip took (30) hours. Answer the following question:\n\nDuring this trip, the river\u2019s speed was miles per hour.\n\nSuppose that a large pump can empty a swimming pool in (43 < m hr>) and that a small pump can empty the same pool in (53 < m hr> ext<.>) If both pumps are used at the same time, how long will it take to empty the pool?\n\nIf both pumps are used at the same time, it will take to empty the pool.\n\nThe winner of a (9 < m mi>) race finishes (14.73 < m min>) ahead of the second-place runner. On average, the winner ran (0.6 < extstylefrac< mmathstrut mi>< mmathstrut hr>>) faster than the second place runner. Find the average running speed for each runner.\n\nThe winner's average speed was and the second-place runner's average speed was .\n\nIn still water a tugboat can travel (15 < extstylefrac< mmathstrut mi>< mmathstrut hr>> ext<.>) It travels (42 < m mi>) upstream and then (42 < m mi>) downstream in a total time of (5.96 < m hr> ext<.>) Find the speed of the current.\n\nWithout any wind an airplane flies at (300 < extstylefrac< mmathstrut mi>< mmathstrut hr>> ext<.>) The plane travels (600 < m mi>) into the wind and then returns with the wind in a total time of (4.04 < m hr> ext<.>) Find the average speed of the wind.\n\nWhen there is a (11.8 < extstylefrac< mmathstrut mi>< mmathstrut hr>>) wind, an airplane can fly (770 < m mi>) with the wind in the same time that it can fly (702 < m mi>) against the wind. Find the speed of the plane when there is no wind.\n\nIt takes one employee (2.5 < m hr>) longer to mow a football field than it does a more experienced employee. Together they can mow the grass in (1.9 < m hr> ext<.>) How long does it take each person to mow the football field working alone?\n\nThe more experienced worker takes to mow the field alone, and the less experienced worker takes .\n\nIt takes one painter (13 < m hr>) longer to paint a house than it does a more experienced painter. Together they can paint the house in (30 < m hr> ext<.>) How long does it take for each painter to paint the house working alone?\n\nThe more experienced painter takes to paint the house alone, and the less experienced painter takes .\n\n## 7.5: Solve Rational Equations\n\nSolving Rational Equations\n\n\u00b7 Solve rational equations using the techniques for simplifying and manipulating rational expressions.\n\nEquations that contain rational expressions are called rational equations. We can solve these equations using the techniques for performing operations with rational expressions and for solving algebraic equations.\n\nSolving Rational Equations Using Common Denominators\n\nOne method for solving rational equations is to rewrite the rational expressions in terms of a common denominator. Then, since we know the numerators are equal, we can solve for the variable. To illustrate this, let\u2019s look at a very simple equation:\n\nSince the denominator of each expression is the same, the numerators must be equivalent as well. This means that x = 3.\n\nThis is true for rational equations with polynomials too:\n\nAgain, since the denominators are the same, we know the numerators must also be equal. So we can set them equal to one another and solve for x.\n\nWe should check our solution in the original rational expression:\n\nThe solution checks, and since x = 8 does not result in division by 0, the solution is valid.\n\nWhen the terms in a rational equation have unlike denominators, solving the equation will involve some extra work. Here\u2019s an example:\n\nThere are no excluded values because the denominators are both constants.\n\nFind a common denominator and rewrite each expression with that denominator.\n\nThe common denominator is 8.\n\nSince the denominators are the same, the numerators must be equal for the equation to be true. Solve for x.\n\nCheck the solution by substituting 4 for x in the original equation.\n\nAnother way of solving rational equations is to multiply both sides of the equation by the common denominator. This eliminates the denominators and turns the rational equation into a polynomial equation. Here is the same equation we just solved:\n\nThere are no excluded values because the denominators are both constants.\n\nMultiply both sides by the least common denominator\n\nNow that we understand the techniques, let\u2019s look at an example that has variables in the denominator too. Remember that whenever there are variables in the denominator, we need to find any values that are excluded from the domain because they'd make the denominator zero.\n\nTo solve this equation, we can multiply both sides by the least common denominator:\n\n(x + 2)(x \u2013 2) = 0\n\nFirst determine the excluded values. These are the values of x that result in a 0 denominator.\n\nx 2 \u2013 4 = (x \u2013 2)(x + 2)\n\nFind the common denominator of x \u2013 2, x + 2, and x 2 \u2013 4\n\nSince (x \u2013 2) and (x + 2) are both factors of x 2 \u2013 4, the least common denominator is (x \u2013 2)(x + 2) or x 2 \u2013 4\n\nMultiply both sides of the equation by the common denominator.\n\n7x \u2013 14 + 5x + 10 =10x \u2013 2\n\n12x \u2013 4 =10x \u2013 2\n\n12x \u2013 10x \u2013 4 = 10x \u2013 10x \u2013 2\n\nCheck to be sure that the solution is not an excluded value. (It is not.)\n\nCheck the solution in the original equation.\n\nSolve the equation , m 0 or 2\n\nA) Incorrect. You probably found the common denominator correctly, but forgot to distribute when you were simplifying. You also forgot to check your solution or note the excluded values m \u2260 2 because it makes the expression on the right side undefined. Multiplying both sides by the common denominator gives , so . The correct answer is m = 8.\n\nB) Incorrect. , so . The solution, 8, is not an excluded value. The correct answer is m = 8.\n\nC) Correct. Multiplying both sides of the equation by the common denominator gives , so . . The correct answer is m = 8.\n\nWe've seen that there is more than one way to solve rational equations. Because both of these techniques manipulate and rewrite terms, sometimes they can produce solutions that don't work in the original form of the equation. These types of answers are called extraneous solutions. These solutions are artifacts of the solving process and not real answers at all. That's why we should always check solutions in the original equations\u2014we may find that they yield untrue statements or produce undefined expressions.\n\nA) Correct. x \u2013 2 + x 2 \u2013 6x = 4 (x \u2013 6 )(x + 1) = 0. Since 6 is an excluded value, it is an extraneous solution. Only -1 is a real solution.\n\nB) Incorrect. 6 is an excluded value because it makes the denominator of the first rational expression equal to 0. Since 6 is an extraneous solution, it can't be included in the solution. The correct answer is -1.\n\nC) Incorrect. The common denominator is (x \u2013 6 )(x -2). Each term on the left side must be multiplied by a fraction equivalent to 1 that will produce that denominator: = . The correct answer is -1.\n\nD) Incorrect. When the equation is solved by finding the common denominator, the answers are -1 and 6. It is true that 6 is an excluded value and thus an extraneous solution that must be discarded. But -1 works in the original equation and it is a valid solution. The correct answer is -1.\n\nWe solve rational equations by finding a common denominator. We can then follow either of two methods. We can rewrite the equation so that all terms have the common denominator and we can solve for the variable with just the numerators. Or we can multiply both sides of the equation by the common denominator so that all terms become polynomials instead of rational expressions.\n\nAn important step in solving rational equations is to reject any extraneous solutions from the final answer. Extraneous solutions are solutions that don't satisfy the original form of the equation because they produce untrue statements or are excluded values that make a denominator equal to 0.\n\n## University of Transnational Business Law\n\nThis is \u201cSolving Rational Equations\u201d, section 7.5 from the book Beginning Algebra (v. 1.0). For details on it (including licensing), click here.\n\nHas this book helped you? Consider passing it on:\n\nCreative Commons supports free culture from music to education. Their licenses helped make this book available to you.\n\nDonorsChoose.org helps people like you help teachers fund their classroom projects, from art supplies to books to calculators.\n\n7.5 Solving Rational Equations\n\nLearning Objectives\n1.Solve rational equations.\n2.Solve literal equations, or formulas, involving rational expressions.\n\nSolving Rational Equations\n\nA rational equationAn equation containing at least one rational expression. is an equation containing at least one rational expression. Rational expressions typically contain a variable in the denominator. For this reason, we will take care to ensure that the denominator is not 0 by making note of restrictions and checking our solutions.\n\nSolve rational equations by clearing the fractions by multiplying both sides of the equation by the least common denominator (LCD).\n\nSolution: We first make a note that x\u22600 and then multiply both sides by the LCD, 3x:\n\nCheck your answer by substituting 12 for x to see if you obtain a true statement.\n\nAnswer: The solution is 12.\n\nAfter multiplying both sides of the previous example by the LCD, we were left with a linear equation to solve. This is not always the case sometimes we will be left with a quadratic equation.\n\nSolution: In this example, there are two restrictions, x\u22600 and x\u2260\u22121. Begin by multiplying both sides by the LCD, x(x+1).\n\nAfter distributing and dividing out the common factors, a quadratic equation remains. To solve it, rewrite it in standard form, factor, and then set each factor equal to 0.\n\nCheck to see if these values solve the original equation.\n\nAnswer: The solutions are \u22121\/2 and 1.\n\nUp to this point, all of the possible solutions have solved the original equation. However, this may not always be the case. Multiplying both sides of an equation by variable factors may lead to extraneous solutionsA solution that does not solve the original equation., which are solutions that do not solve the original equation. A complete list of steps for solving a rational equation is outlined in the following example.\n\nExample 3: Solve: xx+2+2x2+5x+6=5x+3.\n\nStep 1: Factor all denominators and determine the LCD.\n\nStep 2: Identify the restrictions. In this case, they are x\u2260\u22122 and x\u2260\u22123.\n\nStep 3: Multiply both sides of the equation by the LCD. Distribute carefully and then simplify.\n\nStep 4: Solve the resulting equation. Here the result is a quadratic equation. Rewrite it in standard form, factor, and then set each factor equal to 0.\n\nStep 5: Check for extraneous solutions. Always substitute into the original equation, or the factored equivalent. In this case, choose the factored equivalent to check:\n\nHere \u22122 is an extraneous solution and is not included in the solution set. It is important to note that \u22122 is a restriction.\n\nIf this process produces a solution that happens to be a restriction, then disregard it as an extraneous solution.\n\nTry this! Solve: xx\u22125+3x+2=7xx2\u22123x\u221210.\n\nVideo Solution\n(click to see video)\nSometimes all potential solutions are extraneous, in which case we say that there is no solution to the original equation. In the next two examples, we demonstrate two ways in which a rational equation can have no solutions.\n\nExample 4: Solve: 3xx2\u22124\u22122x+2=1x+2.\n\nSolution: To identify the LCD, first factor the denominators.\n\nMultiply both sides by the least common denonominator (LCD), (x+2)(x\u22122), distributing carefully.\n\nThe equation is a contradiction and thus has no solution.\n\nExample 5: Solve: xx\u22124\u22124x+5=36x2+x\u221220.\n\nSolution: First, factor the denominators.\n\nTake note that the restrictions are x\u22604 and x\u2260\u22125. To clear the fractions, multiply by the LCD, (x\u22124)(x+5).\n\nBoth of these values are restrictions of the original equation hence both are extraneous.\n\nTry this! Solve: 1x+1+xx\u22123=4xx2\u22122x\u22123.\n\nVideo Solution\n(click to see video)\nIt is important to point out that this technique for clearing algebraic fractions only works for equations. Do not try to clear algebraic fractions when simplifying expressions. As a reminder, we have\n\nExpressions are to be simplified and equations are to be solved. If we multiply the expression by the LCD, x(2x+1), we obtain another expression that is not equivalent.\n\nLiteral equations, or formulas, are often rational equations. Hence the techniques described in this section can be used to solve for particular variables. Assume that all variable expressions in the denominator are nonzero.\n\nExample 6: Solve for x: z=x\u22125y.\n\nSolution: The goal is to isolate x. Assuming that y is nonzero, multiply both sides by y and then add 5 to both sides.\n\nExample 7: Solve for c: 1c=1a+1b.\n\nSolution: In this example, the goal is to isolate c. We begin by multiplying both sides by the LCD, a\u22c5b\u22c5c, distributing carefully.\n\nOn the right side of the equation, factor out c.\n\nNext, divide both sides of the equation by the quantity (b+a).\n\nTry this! Solve for y: x=y+1y\u22121.\n\nVideo Solution\n(click to see video)\n\nKey Takeaways\n\u2022Begin solving rational equations by multiplying both sides by the LCD. The resulting equivalent equation can be solved using the techniques learned up to this point.\n\u2022Multiplying both sides of a rational equation by a variable expression introduces the possibility of extraneous solutions. Therefore, we must check the solutions against the set of restrictions. If a solution is a restriction, then it is not part of the domain and is extraneous.\n\u2022When multiplying both sides of an equation by an expression, distribute carefully and multiply each term by that expression.\n\u2022If all of the resulting solutions are extraneous, then the original equation has no solutions.\n\n## Solving Literal Equations and Applications Involving Reciprocals\n\nLiteral equations, or formulas, are often rational equations. Hence the techniques described in this section can be used to solve for particular variables. Assume that all variable expressions in the denominator are nonzero.\n\n### Example 9\n\nThe reciprocal of the combined resistance R of two resistors R 1 and R 2 in parallel is given by the formula 1 R = 1 R 1 + 1 R 2 . Solve for R in terms of R 1 and R 2 .\n\nThe goal is to isolate R on one side of the equation. Begin by multiplying both sides of the equation by the LCD, R R 1 R 2 .\n\nR R 1 R 2 \u22c5 1 R = R R 1 R 2 \u22c5 1 R 1 + R R 1 R 2 \u22c5 1 R 2 R 1 R 2 = R R 2 + R R 1 R 1 R 2 = R ( R 2 + R 1 ) R 1 R 2 R 2 + R 1 = R\n\nAnswer: R = R 1 R 2 R 1 + R 2\n\nTry this! Solve for y: x = 2 y + 5 y \u2212 3 .\n\nRecall that the reciprocal of a nonzero number n is 1 n . For example, the reciprocal of 5 is 1 5 and 5 \u22c5 1 5 = 1 . In this section, the applications will often involve the key word \u201creciprocal.\u201d When this is the case, we will see that the algebraic setup results in a rational equation.\n\n### Example 10\n\nA positive integer is 3 less than another. If the reciprocal of the smaller integer is subtracted from twice the reciprocal of the larger, then the result is 1 20 . Find the two integers.\n\nLet n represent the larger positive integer.\n\nLet n \u2212 3 represent the smaller positive integer.\n\nSet up an algebraic equation.\n\nSolve this rational expression by multiplying both sides by the LCD. The LCD is 20 n ( n \u2212 3 ) .\n\n2 n \u2212 1 n \u2212 3 = 1 20 20 n ( n \u2212 3 ) \u22c5 ( 2 n \u2212 1 n \u2212 3 ) = 20 n ( n \u2212 3 ) \u22c5 ( 1 20 ) 20 n ( n \u2212 3 ) \u22c5 2 n \u2212 20 n ( n \u2212 3 ) \u22c5 1 n \u2212 3 = 20 n ( n \u2212 3 ) \u22c5 ( 1 20 )\n\n40 ( n \u2212 3 ) \u2212 20 n = n ( n \u2212 3 ) 40 n \u2212 120 \u2212 20 n = n 2 \u2212 3 n 20 n \u2212 120 = n 2 \u2212 3 n 0 = n 2 \u2212 23 n + 120 0 = ( n \u2212 8 ) ( n \u2212 15 ) n \u2212 8 = 0 or n \u2212 15 = 0 n = 8 n = 15\n\nHere we have two viable possibilities for the larger integer n. For this reason, we will we have two solutions to this problem.\n\nIf n = 8 , then n \u2212 3 = 8 \u2212 3 = 5 .\n\nIf n = 15 , then n \u2212 3 = 15 \u2212 3 = 12 .\n\nAs a check, perform the operations indicated in the problem.\n\n2 ( 1 8 ) \u2212 1 5 = 1 4 \u2212 1 5 = 5 20 \u2212 4 20 = 1 20 \u2713\n\n2 ( 1 15 ) \u2212 1 12 = 2 15 \u2212 1 12 = 8 60 \u2212 5 60 = 3 60 = 1 20 \u2713\n\nAnswer: Two sets of positive integers solve this problem: <5, 8>and <12, 15>.\n\nTry this! When the reciprocal of the larger of two consecutive even integers is subtracted from 4 times the reciprocal of the smaller, the result is 5 6 . Find the integers.\n\n## Illustrative Mathematics Grade 7, Unit 5, Lesson 15: Solving Equations with Rational Numbers\n\nThe following diagram shows how to solve equations that include rational numbers and have rational solutions.\n\n#### Lesson 15.1 Number Talk: Opposites and Reciprocals\n\nThe variables a through h all represent different numbers. Mentally find numbers that make each equation true.\n3\/5 \u00b7 5\/3 = a\n7 \u00b7 b = 1\nc \u00b7 d = 1\n-6 + 6 = e\n11 + f = 0\ng + h = 0\n\n#### Lesson 15.2 Match Solutions\n\nMatch each equation to a value that makes it true by dragging the answer to the corresponding equation. Be prepared to explain your reasoning.\nOpen Applet\n\n#### Lesson 15.3 Trip to the Mountains\n\nThe Hiking Club is on a trip to hike up a mountain.\n\n1. The members increased their elevation 290 feet during their hike this morning. Now they are at an elevation of 450 feet.\na. Explain how to find their elevation before the hike.\nb. Han says the equation describes the situation. What does the variable represent?\nc. Han says that he can rewrite his equation as to solve for . Compare Han&rsquos strategy to your strategy for finding the beginning elevation.\n2. The temperature fell 4 degrees in the last hour. Now it is 21 degrees. Write and solve an equation to find the temperature it was 1 hour ago.\n3. There are 3 times as many students participating in the hiking trip this year than last year. There are 42 students on the trip this year.\na. Explain how to find the number of students that came on the hiking trip last year.\nb. Mai says the equation 3s = 42 describes the situation. What does the variable represent?\nc. Mai says that she can rewrite her equation as to solve for s = 1\/3 \u00b7 42. Compare Mai&rsquos strategy to your strategy for finding the number of students on last year\u2019s trip.\n4. The cost of the hiking trip this year is 2\/3 of the cost of last year&rsquos trip. This year&rsquos trip cost \\$32. Write and solve an equation to find the cost of last year&rsquos trip.\n\n#### Lesson 15.4 Card Sort: Matching Inverses\n\nYour teacher will give you a set of cards with numbers on them.\n\n1. Match numbers with their additive inverses.\n2. Next, match numbers with their multiplicative inverses.\n3. What do you notice about the numbers and their inverses?\n\n#### Lesson 15 Practice Problems\n\n1. Solve.\na. 2\/5 t = 6\nb. -4.5 = a - 8\nc. 1\/2 + p = -3\nd. 12 = x \u00b7 3\ne. -12 = -3y\n2. Evaluate each expression if x is 2\/5, y is -4, and z is -0.2.\na. x + y\nb. 2x - z\nc. x + y + z\nd. y \u00b7 x\n3. Match each equation to a step that will help solve the equation.\n4. a. Write an equation where a number is added to a variable, and a solution is -8.\nb. Write an equation where a number is multiplied by a variable, and a solution is\n5. The markings on the number line are evenly spaced. Label the other markings on the number line.\n6. In 2012, James Cameron descended to the bottom of Challenger Deep in the Marianas Trench the deepest point in the ocean. The vessel he rode in was called DeepSea Challenger.\nChallenger Deep is 35,814 feet deep at its lowest point\na. DeepSea Challenger\u2019s descent was a change in depth of (-4) feet per second. We can use the equation y = -4x to model this relationship, where y is the depth and x is the time in seconds that have passed. How many seconds does this model suggest it would take for DeepSea Challenger to reach the bottom?\nb. To end the mission DeepSea Challenger made a one-hour ascent to the surface. How many seconds is this?\nc. The ascent can be modeled by a different proportional relationship y = kx. What is the value of k in this case?\n\nThe Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.\n\nTry the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.","date":"2021-12-06 17:45:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7321842312812805, \"perplexity\": 1204.252760119165}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"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-2021-49\/segments\/1637964363309.86\/warc\/CC-MAIN-20211206163944-20211206193944-00475.warc.gz\"}"}
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Q: Beamer: compatibility issue natbib I'm trying to create a beamer presentation that shows, at the start of each new section, a slide with the table of contents, with the current section highlighted. I know how to do that (lines 4-8 in the MWE below). However, this seems to be incompatible with the natbib package.
If I try to typeset the MWE below, I get the following error: ./mwe.vrb:2: Extra }, or forgotten \endgroup.
\endframe ->\egroup
\begingroup \def @currenvir {frame}
l.2 \begin{thebibliography}{1}
Commenting out the natbib package removes the error, as does commenting out lines 4-8 of the MWE.
\documentclass{beamer}
\usepackage[authoryear]{natbib}
\AtBeginSection{%
\begin{frame}[c]{Outline}
\tableofcontents[currentsection, subsectionstyle=show/hide/hide]
\end{frame}
}
\title{Title}
\author{Author}
\begin{document}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}[c]{Outline}
\tableofcontents[hideallsubsections]
\end{frame}
\section{Section 1}
\begin{frame}
\frametitle{\insertsectionhead}
Some text.
\end{frame}
\section{Section 2}
\begin{frame}{\insertsectionhead}
Some more text; see in particular \cite{Chomsky1965}.
\end{frame}
\begin{frame}[fragile]{References}
\begin{thebibliography}{1}
\providecommand{\natexlab}[1]{#1}
\providecommand{\url}[1]{\texttt{#1}}
\providecommand{\urlprefix}{}
\expandafter\ifx\csname urlstyle\endcsname\relax
\providecommand{\doi}[1]{doi:\discretionary{}{}{}#1}\else
\providecommand{\doi}{doi:\discretionary{}{}{}\begingroup
\urlstyle{rm}\Url}\fi
\bibitem[{Chomsky(1965)}]{Chomsky1965}
Chomsky, Noam. 1965.
\newblock \emph{Aspects of the theory of syntax}.
\newblock Cambridge, MA: MIT Press.
\end{thebibliography}
\end{frame}
\end{document}
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"redpajama_set_name": "RedPajamaStackExchange"
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Inspiring website giving unparalled Buddhist insight into death & dying, helping people develop an awareness of their mortality in a way that will totally enrich and transform their lives.
Meetings where people gather to eat cake, drink tea and discuss death. The objective is 'to increase awareness of death with a view to helping people make the most of their finite lives'.
A global network of online obituaries that provides timely news of death and allows users to pay respect and celebrate life.
A collection of portraits and stories of suicide attempt survivors, as told by those survivors.
Official home of the International Association for Near-Death Studies, a Christian website which studies NDEs and the afterlife.
Online utility which is a friendly reminder that life is slipping away, estimating roughly when the average person is expected to die based on their date of birth, smoking status and BMI.
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{
"redpajama_set_name": "RedPajamaC4"
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\section{Introduction}
An explosive (discontinuous) transition
occurs when an infinitesimal increase of the
control parameter produces an abrupt change in macroscopic
quantities. This kind of transition has attracted
a lot of interest in the recent years, inspired by
the discovery of a procedure (the ``Achlioptas process") that gives rise to an abrupt percolation transition in complex networks \cite{explosive1,explosive2,explosive3,explosive4}.
While subsequent works have shown the Achlioptas process transition was, in fact, a continuous phase transition with unusual finite-size scaling \cite{continuous1,continuous2,continuous3}, many related models with alternative mechanisms showing genuinely discontinuous and anomalous transitions have now been discovered (see Ref. \cite{expreview} and references therein).
One of these main examples appear in the context of coupled
oscillators in which the Kuramoto Model (KM)
\cite{kuramoto} plays a central role.
The original KM describes self-sustained coupled phase
oscillators and exhibits a continuous phase transition at a critical
coupling, beyond which a collective behavior is achieved.
A few years ago, in a pioneering work, Garde\~nes et al. \cite{prl2011},
discovered
that a discontinuous phase transition to synchronization emerges as a consequence of the correlation
between structure and local dynamics when a scale-free network is considered.
Subsequent studies have confirmed the transition robustness
under changing ingredients, such as lattice topology \cite{prl2011}, time delay \cite{peron}, disorder
\cite{skardal} and inertia \cite{inertia}. Analysis of the explosive transition in simpler structures, such
as star graphs, for which exact treatment is possible \cite{tiago}, also confirmed that the transition to collective behavior is discontinuous.
Investigating the explosive synchronization in a generic complex network, Zhang et al. \cite{kurths} have found that a positive correlation between
the oscillators frequency and the degree of
their corresponding vertices is the required condition
for its appearance.
More recently, Pinto et al. \cite{saa} have verified that
it suffices to fulfill above minimal requirement for
the hubs (e.g. the vertices with higher degrees)
for promoting an abrupt transition.
Besides dynamical systems, they
manifest in markovian nonequilibrium
reaction-diffusion processes. Two groups are important
in this context: those presenting absorbing states and
the up-down symmetric systems. In the former, distinct mechanisms, such as the inclusion of a quadratic term in the particle
creation rates \cite{quadratic1,quadratic2}, the need of a minimal
neighborhood for generating subsequent offsprings \cite{fiore14}, synergetic effects in multi species models \cite{scp1,scp2} or cooperative coinfection in multiple diseases epidemic models \cite{coinfect1,coinfect2,coinfect3}
can be taken into account for shifting,
from a continuous transition (belonging generically to the directed
percolation (DP) universality class \cite{marr99,odor07,henkel})
to a discontinuous one.
The Majority Vote (MV) model is one of
the simplest nonequilibrium up-down symmetric systems exhibiting
an order-disorder phase transition \cite{mario92}.
Extensive studies of this model in distinct lattice topologies
(besides the usual regular ones)
showed that the symmetry-breaking phase transition
is not affected by the kind of the underlying networks \cite{chen1},
although the critical behavior results in
set of critical exponents entirely different \cite{pereira}.
However, very recently, Chen et al. \cite{chen2} verified that the usual
second-order phase transition in the majority vote (MV)
becomes first-order when
a term depending on the local density is included in the dynamics
(an {\em inertial} effect).
Aimed at investigating how the network topology and inertial effects contribute
to the emergence of the explosive transition in the MV model,
in this work we include the inertia only in a given fraction of sites
with degree $k$ larger than a threshold $\langle k \rangle k^*$.
We observe that the MV transition remains
explosive only for a low/intermediate fraction
of restriction $k^*$ in homogeneous
structures. On the other hand, in heterogeneous networks, it is
sufficient to include inertia only in the hubs for promoting an abrupt
behavior.
Remarkably, a new feature induced by the partial (but large)
inertia is the emergence
of an extra phase, in which the system is partially synchronized, whose
phase transition can be continuous or discontinuous, according to the
inertia magnitude. In this region
of the phase diagram, the system presents two phase transitions.
This paper is organized as follows: In Sec. II, we derive
the mean field theory for the model. Next, numerical results are shown in Sec. III.
Conclusions are drawn in Sec. IV.
\section{Model and mean field analysis}
The MV model is defined in
an arbitrary lattice topology, in which
each node $i$ of degree $k_i$ is attached to a
spin variable, $\sigma_i$, that can take the values
$\sigma_i=\pm 1$. In the original case, with probability $1-f$
each node $i$ tends to align itself with its local neighborhood majority,
and with complementary probability $f$, the majority rule is not
followed. The quantity $f$ is a misalignment term whose increasing
gives rise to an order-disorder (continuous) phase transition.
Chen et al. \cite{chen2} added to the transition rate a term
depending only on the
local state $\sigma_i$, {\em irrespectively}
the majority nearest neighbor spins.
Mathematically, one has the following transition rate
\begin{equation}
w(\sigma_i)=\frac{1}{2}\left\{1-(1-2f)\sigma_{i}
S\left[(1-\theta)\sum_{j=1}^{k_i}\sigma_j/k_i+\theta \sigma_i\right]\right\},
\label{eq2}
\end{equation}
where $\theta$ denotes the inertia strength and
$S(X)$ is defined by $S(X)={\rm sign(X)}$
if $X \neq 0$ and $S(0)=0$.
Note that for $\theta=0$ one recovers the original MV model,
whose critical transition depends on the nodes distribution.
In particular, for $\theta > 0.5$,
the dynamics is fully dominated by the inertia and no
phase transition is observed \cite{chen2}.
The time evolution of the density $\rho_k$ of
``up'' spins ($\sigma_i= 1$) of a node with degree $k$ given by
\begin{equation}
\frac{d }{dt}\rho_k=w_{-1 \rightarrow 1}(1-\rho_k)-\rho_k w_{1 \rightarrow -1},
\label{eq3}
\end{equation}
where $w_{-1 \rightarrow 1}$ and $w_{1 \rightarrow -1}$ denote the transition
rates to states with opposite spin. In the steady state one has that
\begin{equation}
\rho_k=\frac{w_{-1 \rightarrow 1}}{w_{-1 \rightarrow 1}+
w_{1 \rightarrow -1}}.
\label{eq4}
\end{equation}
The average magnetization of a node of degree $k$
is related to $\rho_k$ through the relation
$m_k=2\rho_k-1$.
Our first inspection of the inertia effect is carried out
through a mean-field treatment.
Here, we follow the ideas from Ref. \cite{romualdo,chen2}, in which
the transition rates in Eq. (\ref{eq3}) are rewritten in terms of
the majority and minority rules, given by
\begin{equation}
w_{-1 \rightarrow 1}=(1-2f){\bar P_{-}}+f,
\end{equation}
where ${\bar P_{-}}$ is the probability that the node $i$ of degree $k$, with spin
$\sigma_i=-1$, changes its state according to the majority rule. In
particular, ${\bar P_{-}}$ depends on the number $n_{k}^{-}$
of nearest neighbor $+$ spins
in such a way that
${\bar P_{-}}=\sum_{n=\lceil n_k^{-} \rceil}^{k}(1-\frac{1}{2}\delta_{n,n_{k}^{-}})C_{n}^{k}
p_{+1}^{n}p_{-1}^{k-n}$, where $p_{\pm 1}$ is
the probability that one of its nearest-neighbors is in
the spin state $\pm 1$. Since the quantity
$n_{k}^{-}$ corresponds to the lower limit, it is evaluated
from the condition $S(X)=0$, reading $n_{k}^{-}=\frac{k}{2(1-\theta)}$.
A similar expression is obtained by writing down
the transition rate $w_{1 \rightarrow -1}$ in terms of the minority rule
(instead of the majority one)
\begin{equation}
w_{1 \rightarrow -1}=(1-f)-(1-2f){\bar P_{+}},
\end{equation}
where ${\bar P_+}$ also reads
${\bar P_{+}}=\sum_{n=\lceil n_k^{+} \rceil}^{k}(1-\frac{1}{2}\delta_{n,n_{k}^{+}})C_{n}^{k}
p_{+1}^{n}p_{-1}^{k-n}$, but the bottom limit reads
$n_{k}^{+}=\frac{k(1-2\theta)}{2(1-\theta)}$.
For large $k$, each term of the above binomial distributions
${\bar P_{\pm}}$ approach to gaussian ones
with mean $m=kp_{+}$ and variance $\sigma^{2}=kp_{+}p_{-}$.
So that
\begin{equation}
{\bar P_{\pm}} \rightarrow \frac{1}{2}\left\{1 \pm {\rm erf}\left[
\sqrt {2k}\left(\frac{\theta}{2(1-\theta)} \pm y \right) \right]\right\},
\label{eq7}
\end{equation}
where ${\rm erf(x)}$ denotes the error function and
the nearest neighbor probability $p_+$ has been rewritten in terms of the
quantity $y$ through the relation $p_+\equiv 1/2+y$.
For any node without degree correlation, the probability
that a randomly nearest neighbor has degree $k$
is $kP(k)/\langle k \rangle$. Thus, $p_+$ and $\rho_k$
are related by $p_+=\sum_{k}kP(k)/\langle k \rangle \rho_k$
and finally we arrive at the following expression
\begin{equation}
y+1/2=\sum_{k}\frac{k P(k)}{\langle k \rangle}
\frac{(1-2f) {\bar P_-}+f}{1+(1-2f)({\bar P_-}-{\bar P_+})},
\label{eq8}
\end{equation}
with ${\bar P_\pm}$ evaluated from Eq. (\ref{eq7}). By splitting
Eq. (\ref{eq8}) in two parts, the first and
second terms get restricted to the nodes {\em in the absence of} and {\em with} inertia
respectively, in such a way that
\begin{widetext}
\begin{equation}
y+1/2=\sum_{k=k_{0}}^{\langle k \rangle k^{*}-1}
\frac{kP(k)}{2\langle k \rangle}[1+(1-2f){\rm erf}(\sqrt{2k}y)]+
\sum_{k=\langle k \rangle k^{*}}^{\infty}
\frac{kP(k)}{\langle k \rangle}\frac{1-(1-2f){\rm erf}\left[
\sqrt {2k}\left(\frac{\theta}{2(1-\theta)} - y \right) \right]}
{2-(1-2f)\{{\rm erf}\left[
\sqrt {2k}\left(\frac{\theta}{2(1-\theta)} - y \right) \right]+{\rm erf}\left[
\sqrt {2k}\left(\frac{\theta}{2(1-\theta)} + y \right) \right]\}},
\label{eq9}
\end{equation}
\end{widetext}
where $k_0$ denotes the minimum degree.
In particular,
for $\theta=0$, Eq. (\ref{eq9}) reduces to $y=\frac{1-2f}{2\langle k \rangle}
\left\{\sum_{k}k[{\rm erf}(\sqrt{2k}y)]P(k)\right\}$,
in consistency with results from Refs. \cite{chen1,chen2}.
Thus, the solution(s) of Eq. (\ref{eq9}) give
us the values of $y$, whose corresponding $\rho_k$'s
are obtained from Eq. (\ref{eq4}).
The mean magnetization $|m|$ is achieved by
summing over all values of $k$
with their correspondent weights $P(k)$, so
that $|m|=2\left[\sum_{k}\rho_kP(k)\right]-1$.
Figs. \ref{fig1} and \ref{fig1-2} show (for $\langle k\rangle=20$), the behavior
of $|m|$ versus $f$, for two distinct network topologies.
The first is an Erdos-Renyi (ER) graph, a prototypical model of a homogeneous random network,
with the degree distribution given by
$P(k)=\langle k \rangle^{k}e^{-\langle k \rangle}/k!$.
The second case is a
representative description of heterogeneous networks, in which
nodes are distributed according to the probability distribution
$P(k) \sim k^{-\gamma}$. From now on, such case will be referred
as power law (PL) graph. Here, we take $k_0=0$ for the ER and,
for avoiding divergences when $k \rightarrow 0$ in the PL,
we have imposed $k_0$ constrained to the mean
degree $\langle k \rangle$ through the relation
$k_0=\frac{\gamma-2}{\gamma-1}\langle k \rangle$.
\begin{figure}[h!]
\epsfig{file=fig1.eps,width=9cm,height=8cm}
\caption{For the ER and $\langle k\rangle=20$,
$|m|$ versus $f$ for distinct inertia rates.
Panels $(a)$ and $(b)$ show
the full inertia cases for $\theta=0.2$ and $0.4$, respectively.
In panels $(c)$ and $(d)$, the restrictive case
for $\theta=0.3$ and
$\theta=0.45$ and distinct restrictions $k^*$'s. In all cases,
red lines correspond the unstable branches.}
\label{fig1}
\end{figure}
\begin{figure}[h!]
\epsfig{file=fig2.eps,width=9cm,height=8cm}
\caption{For the PL and $\langle k\rangle=20$,
$|m|$ versus $f$ for distinct inertia rates.
Panels $(a)$ and $(b)$ show
the full inertia cases for $\theta=0.2$ and $0.4$, respectively.
In panels $(c)$ and $(d)$, the restrictive case
for $\theta=0.3$ and
$\theta=0.45$ and distinct restrictions $k^*$'s. In all cases,
red lines correspond the unstable branches.}
\label{fig1-2}
\end{figure}
As for ER and PL cases, top panels $(a)-(b)$ correspond to the full
inertia cases whose phase transition is continuous
for low $\theta$ and its increasing gives rise to
a discontinuous one. Its emergence is signed by
the appearance of
an unstable branch (red lines),
ending at to lower $f$'s when $\theta$ goes up.
Despite the similarity between both cases, note
the transition and the crossover (from continuous to discontinuous)
points depend on $P(k)$.
For example, for $\theta=0.2$ the transition is continuous
for the ER and discontinuous for the PL.
All these results are consistent with those
obtained in Ref. \cite{chen2}.
Next, we examine the partial inertia case,
in which the inertia appears only in a specific
fraction of nodes (the ones with larger degrees).
This analysis is inspired by the work by Pinto et al. \cite{saa}
for the KM, in which
a positive correlation between frequency-degree taken only for the hubs
is enough for promoting an explosive transition.
For instance, we introduce
the fraction parameter $k^{*}$, in such a way that
$\theta \neq 0$ only if $k \ge \langle k \rangle k^{*}$
and $\theta=0$ otherwise. Extremely large $k^{*}$ implies that most of the
nodes will be absent of inertia and thus the phase transition is
expected to be similar to the $\theta=0$ case (continuous).
In the opposite case, low $k^{*}$ makes the majority
of sites to have inertia and one expects a scenario around the
panels $(a)$ and $(b)$.
Panels $(c)$ and $(d)$ in Figs. \ref{fig1} and \ref{fig1-2}
show the results of the mean field theory (MFT) for
the ER and PL [$\langle k\rangle=20$] and distinct sorts
of $k^{*}$.
As expected, for low $k^{*}$'s MFT predicts similar behaviors
than the full inertia cases (see, e.g., panels $(c)$ and $(d)$
for $k^*=0.8$).
However, the increase of restriction leads to opposite scenarios.
Whenever the discontinuous phase transition is suppressed for the ER
when $k^*=1.2$, it is maintained for the PL. Despite a similar fraction of
nodes with inertia (about $21\%$ and $17\%$
for the ER and PL, respectively), the
presence of hubs in the PL
sustains the discontinuous transition.
Surprising, an additional phase transition
emerges for intermediate $k^{*}$'s and large $\theta$.
This is clearly exemplified for $\theta=0.45$ and $k^{*}=1.05$ [panels $(d)$] in
which the presence
of a jump and an unstable envelope signals a discontinuous
transition between two synchronized phases for low $f$
($|m| \neq 0$ in both cases) followed by a smooth vanishing of
$|m|$ for large $f$. In all cases, the transition between partially-ordered
and disordered phase is continuous.
To examine such a new feature in more details, we plot in
Fig. \ref{fig1-5} the contribution of each part
on the right side of Eq. (\ref{eq9}) separately.
That reveals the former transition comes from
the subsystem with inertia (which becomes disordered),
whereas the other gets ordered. By keeping the increase
of $f$, the remaining subset (without inertia) also loses the ordering
at $f_c$.
\begin{figure}[h!]
\epsfig{file=fige.eps,width=7cm,height=6cm}
\caption{For the ER, $\langle k\rangle=20$
and $k^*=1.05$, the plot of the
right side of Eq. (\ref{eq9}) versus $f$ restricted to the inertial
part ($\theta=0.45$ in the present case) and $\theta=0$.
Dashed lines correspond to the unstable solutions.}
\label{fig1-5}
\end{figure}
In the next section, we continue the study of the effects
of partial inertia through numerical simulations,
in order to compare with our MFT predictions.
\section{Numerical results}
We performed extensive numerical simulations of the MV model on random graphs
with ER and PL topologies
and system sizes $N$ ranging from $N=3600$ to $20000$.
To construct an ER graph, we connect each pair
of nodes with probability $\langle k \rangle/N$.
When the size of the graph tends to infinity $N\to\infty$,
its degree distribution is Poissonian, with mean $\langle k \rangle$.
The PL graphs were generated using the uncorrelated
configuration model (UCM) \cite{ucm}, so the degrees are
uncorrelated. As in the MFT, we use $\gamma=3$ and distinct sets of inertia values.
In order to classify the phase transition, we begin by analyzing the absolute value of the mean
magnetization per site $|m|$ as a function of $f$, starting from a
full ordered phase ($|m|= 1$) and increasing $f$ towards the completely
disordered phase ($|m|=0$).
In sequence, we take the opposite case,
wherein the system is in the disordered phase and the
parameter $f$ is gradually decreased approaching to the ordered phase.
Both increasing (``forward'') and decreasing (``backward'')
curves are expected to coincide
when the phase transition is critical, but
they are different at the phase coexistence (a trademark of a
discontinuous transition). The presence of hysteresis
indicates the system bistability with respect to the
ordered/disordered phase according to its initial condition.
They are also signed by the presence of
a bimodal probability distribution of the order-parameter $P(m)$.
On the other hand, if the phase transition is continuous, $P(m)$
exhibits a single peak, whose position depends on $f$.
Another feature distinguishing them relies that
the continuous case presents an algebraic divergence
of its order parameter variance
$\chi=N[\langle m^{2} \rangle-\langle m \rangle^{2}]$
at the critical point $f_c$ \cite{martins-fiore}.
(In simulations of finite systems, we observe a maximum
that increases with the system size $N$).
The transition point $f_c$ can also be identified
through the reduced cumulant
$U_4=1-\frac{\langle m^{4} \rangle}{3\langle m^{2} \rangle^{2}}$,
since curves for distinct $N$'s cross at $f_c$. Off the critical
point, $U_4 \rightarrow 2/3$ and $0$
for the ordered and disordered phases, respectively when $N \rightarrow\infty$.
In order to compare with the MFT, Fig. \ref{fig2} shows results for
the ER and PL topologies and lower inertia values
(exemplified here for $\theta=0.3$).
In both cases, the presence of hysteresis for
$k^*=0.8$ and $k^*=1$ reveal, in similarity
with MFT, discontinuous transitions
with hysteretic loop decreasing by
elevating $k^*$.
Also, the network
structure leads to opposite features for $k^*=1.2$ with
a continuous phase transition for the ER (panel $(b)$).
The behavior is different for
the PL (panel $(d)$), showing a small
jump of $|m|$ at $f \sim 0.22$ for a partially ordered phase
(see panel (d) in Fig. \ref{fig2}). This also contrasts with MFT results,
in which a discontinuous order-disordered transition is predicted.
Despite the evidence of a discontinuous transition for the PL,
we believe that sufficient larger $N$'s are
required for observing a hysteretic loop for such case.
\begin{figure}
\epsfig{file=fig3-1.eps,width=9.cm,height=4.5cm}
\epsfig{file=fig3-2.eps,width=9.cm,height=4.5cm}
\caption{For ER [$(a)-(b)$] and PL [$(c)-(d)$]
networks with $\theta=0.3$, the order parameter
$|m|$ versus the control parameter $f$
for distinct $k^*$. Circles (stars) correspond the increase
(decrease) of $f$
starting from an(a) ordered (disordered) phase.
Except in panel $(d)$ ($N=40000$), results are for $N=20000$.}
\label{fig2}
\end{figure}
As in the MFT,
numerical simulations also exhibit an additional phase transition for
large $\theta$ and intermediate sets of $k^*$ (exemplified
in Fig. \ref{fig3} for $\theta=0.45$).
Our results for low $f$ show a hysteretic loop signaling
a phase coexistence between two synchronized phases
(see panels $(a)$ and $(c)$ for $k^{*}=1.05$
and $1.1$, respectively),
whereas by further increasing $f$, $|m|$
vanishes continuously.
To achieve complementary
information about the hysteretic loop, we evaluate the difference
of magnetization restricted to the
subsets of nodes with and without inertia ($m_\theta$ and $m_0$, respectively)
given by $\phi=m_{\theta}-m_{0}$. For sufficient low $f$,
$m_\theta \approx m_0 \approx 1$, consistent with a full ordered phase.
The jump of $\phi$ to a moderate value indicates
that the nodes with inertia become
unsynchronized, but the vertices absent of inertia
remains ordered. Thus, in similarity
with the MFT, we observe that the system, in fact,
exhibits a {\em partial} synchronization.
Although for $k^{*}=1.05$ the decrease of
$f$ does not lead the system to the full synchronization
(in similarity with the original
order-disorder transition for large $\theta$), a closed
hysteretic loop is observed for $k^{*}=1.1$. A bimodal distribution
in the hysteretic region (inset) reinforces a discontinuous transition
between partially-ordered and ordered phases. As expected for the ER,
the additional transition is absent for $k^{*}=1.3$ (inset).
\begin{figure}[h!]
\epsfig{file=fig4.eps,width=9cm,height=7cm}
\caption{For the ER network of size $N=20000$ and $\theta=0.45$,
panels $(a)$ and $(b)$ show
$|m|$ and $\phi$ vs $f$ for $k^{*}=1.05$, respectively.
Circles (stars) correspond the increase
(decrease) of $f$
starting from an(a) ordered (disordered) phase. The same
in panels
$(c)$ and $(d)$, but for $k^{*}=1.1$. Insets: Top and bottom
show $|m|$ vs $f$ for
$k^{*}=1.3$ and the probability distribution $P(m)$
vs $m$ for distinct $f$'s and $N=3600$, respectively.}
\label{fig3}
\end{figure}
Now, we employ a finite size scaling analysis to characterize the order-disorder
phase transition.
In Fig. \ref{fig4}, we plot $\chi$ and $U_4$ for distinct network sizes $N$. We observe that they exhibit the typical
behaviors expected for continuous phase
transitions: the variance $\chi$ presents a maximum increasing with
$N$ and their positions also systematically deviates on $N$.
Analysis of $U_4$ (Fig. \ref{fig4}: panels $(b)$, $(d)$ and its inset) show crossings
at $f_c=0.179(2)$, $f_c=0.236(3)$ and $f_c=0.330(5)$ with $U_4=0.27(2)$,
close to the values found in Ref. \cite{pereira}. Note that all
critical points can be clearly distinguished from their previous
hysteric loops.
\begin{figure}[h!]
\epsfig{file=fig5.eps,width=9cm,height=7.5cm}
\caption{For the ER network and $\theta=0.45$,
panels $(a)$ and $(b)$ show
$\chi$ and $U_4$ vs $f$ for $k^{*}=1.05$.
Panels $(c)$ and $(d)$, the same but
for $k^{*}=1.1$. Insets: Results for $k^{*}=1.3$.}
\label{fig4}
\end{figure}
Similar trends are visualized in Fig. \ref{fig5} where we show
the results for PL networks
with $k^{*}=1.05$ and $1.3$, respectively. However, in this case,
the existence of hubs prolongs the partially synchronized
discontinuous transition for a larger set of restrictions than
the observed in ER networks.
\begin{figure}[h!]
\epsfig{file=fig6.eps,width=9cm,height=6.5cm}
\caption{For the PL network and $\theta=0.45$,
$|m|$ versus $f$ for $k^{*}=1.05$ [panel $(a)$]
and $1.3$ [panel $(c)$].
Circles (stars) correspond the increase
(decrease) of $f$ starting from an(a) ordered (disordered) phase.
Insets show the order parameter $\phi$ vs
$f$. Panels $(b)$ and $(d)$ shows the variance $\chi$
vs $f$ for distinct $N$'s.}
\label{fig5}
\end{figure}
\begin{figure}[h!]
\epsfig{file=fig7.eps,width=7cm,height=5cm}
\caption{Phase diagram $f \times \theta$ for partial
inertia $k^*=1.05$ and ER network with $\langle k \rangle=20$.
ORD, DIS and PO denote the ordered, disordered
and partially-ordered phases, respectively. Full lines correspond
to the continuous transitions, whereas dashed (dotted)
lines correspond to the increase (decrease) of $f$
starting from an(a) ordered (disordered) phase.}
\label{fig6-1}
\end{figure}
\begin{figure}[h!]
\epsfig{file=fig8.eps,width=8cm,height=5cm}
\caption{Phase diagram $f \times \theta$ for partial
inertia $k^*=1.05$ and PL network with $\langle k \rangle=20$.
In the left and right panels, numerical simulations
for $N=20000$ and MFT, respectively.
ORD, DIS and PO denote the ordered, disordered
and partially-ordered phases, respectively.
Full lines correspond
to the continuous transitions, whereas dashed (dotted)
lines correspond to the increase (decrease) of $f$
starting from an(a) ordered (disordered) phase.}
\label{fig7-1}
\end{figure}
In Fig. \ref{fig6-1} we plot the phase diagram for the ER and
$k^*=1.05$. As expected, both MFT and
numerical simulations predict continuous transition between
ordered (ORD) and disordered (DIS) phases for small $\theta$.
MFT predicts the appearance of the partially ordered (PO)
for $\theta\ge\theta_c=0.330(1)$, whose transition is
continuous in the interval
$\theta_c \le \theta \le 0.41$
and discontinuous for $\theta > 0.41$. Numerical
simulations exhibit an additional peak of $\chi$ (absent of hysteresis),
consistent to the emergence of the PO for
$0.36 \le \theta \le 0.39$ and a clear hysteretic loop for $\theta > 0.39$.
Despite the excellent qualitative agreement between approaches,
it is worth mentioning the difficulty of locating
(and classifying) the PO phase transition for $\theta_c \le \theta \le 0.36$
under numerical simulations.
Another point to mention concerns that the PO-DIS transition is
always continuous and practically independent on the inertia
for large $\theta$. A qualitative similar phase diagram is
shown in Fig. \ref{fig7-1} for the PL case. However, the ordered-PO
transition line is always discontinuous, in qualitative agreement
with MFT predictions.
\section{Conclusions}
Recently, Chen et al. \cite{chen2} have found that inertia is responsible
for the appearance of an abrupt transition
in the majority vote model in complex networks.
In the present work, we advance by scrutinizing the inertia
acting only in the most connected nodes.
We show, through mean field analysis and numerical simulations
for homogeneous and heterogeneous networks,
that $partial$ inertia can change
the system behavior depending on the inertia strength.
Our results also reveal that although relevant inertia rates
are required for preserving the discontinuous transitions
for homogeneous networks, this is not the
case of heterogeneous structures, in which a rather small fraction a
($17 \%$) already promotes an abrupt behavior. In other
words, by including only such above fraction in the sites
with larger degrees, the phase transition is discontinuous. This shares
some similarities
with the KM model, in which a positive frequency-degree
correlation included only in the hubs
is sufficient for sustaining an explosive synchronization
\cite{saa}.
A second remarkable effect of partial inertia concerns
in the appearance of an additional phase characterized
by a partial ordering of the system. The nature of the phase
transition from the disordered phase to this partially ordered
phase depends on the inertia strength. Therefore, there is a region
in the phase diagram in which we observe two phase transitions: a
continuous transition from the disordered to the partially ordered
phase, and a discontinuous transition from the latter to the full
ordered phase.
As pointed in \cite{chen2}, behavioral inertia is an essential
characteristic of human being and animal groups. Therefore,
inertia can be a significant ingredient in transitions
that arise in social systems \cite{social}, such as the
emergence of a common culture \cite{social2} or the appearance of
consensus \cite{sood} and decision-making systems \cite{couzin}. Our results suggest that inertia only in a small fraction of the population can
produce dramatic effects if it is concentrated in the most connected individuals.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 6,620
|
package me.zhengjie.utils;
import cn.hutool.core.collection.CollUtil;
import cn.hutool.core.collection.CollectionUtil;
import cn.hutool.core.util.ObjectUtil;
import lombok.extern.slf4j.Slf4j;
import me.zhengjie.annotation.DataPermission;
import me.zhengjie.annotation.Query;
import javax.persistence.criteria.*;
import java.lang.reflect.Field;
import java.util.*;
/**
* @author Zheng Jie
* @date 2019-6-4 14:59:48
*/
@Slf4j
@SuppressWarnings({"unchecked","all"})
public class QueryHelp {
public static <R, Q> Predicate getPredicate(Root<R> root, Q query, CriteriaBuilder cb) {
List<Predicate> list = new ArrayList<>();
if(query == null){
return cb.and(list.toArray(new Predicate[0]));
}
// 数据权限验证
DataPermission permission = query.getClass().getAnnotation(DataPermission.class);
if(permission != null){
// 获取数据权限
List<Long> dataScopes = SecurityUtils.getCurrentUserDataScope();
if(CollectionUtil.isNotEmpty(dataScopes)){
if(StringUtils.isNotBlank(permission.joinName()) && StringUtils.isNotBlank(permission.fieldName())) {
Join join = root.join(permission.joinName(), JoinType.LEFT);
list.add(getExpression(permission.fieldName(),join, root).in(dataScopes));
} else if (StringUtils.isBlank(permission.joinName()) && StringUtils.isNotBlank(permission.fieldName())) {
list.add(getExpression(permission.fieldName(),null, root).in(dataScopes));
}
}
}
try {
List<Field> fields = getAllFields(query.getClass(), new ArrayList<>());
for (Field field : fields) {
boolean accessible = field.isAccessible();
// 设置对象的访问权限,保证对private的属性的访
field.setAccessible(true);
Query q = field.getAnnotation(Query.class);
if (q != null) {
String propName = q.propName();
String joinName = q.joinName();
String blurry = q.blurry();
String attributeName = isBlank(propName) ? field.getName() : propName;
Class<?> fieldType = field.getType();
Object val = field.get(query);
if (ObjectUtil.isNull(val) || "".equals(val)) {
continue;
}
Join join = null;
// 模糊多字段
if (ObjectUtil.isNotEmpty(blurry)) {
String[] blurrys = blurry.split(",");
List<Predicate> orPredicate = new ArrayList<>();
for (String s : blurrys) {
orPredicate.add(cb.like(root.get(s)
.as(String.class), "%" + val.toString() + "%"));
}
Predicate[] p = new Predicate[orPredicate.size()];
list.add(cb.or(orPredicate.toArray(p)));
continue;
}
if (ObjectUtil.isNotEmpty(joinName)) {
String[] joinNames = joinName.split(">");
for (String name : joinNames) {
switch (q.join()) {
case LEFT:
if(ObjectUtil.isNotNull(join) && ObjectUtil.isNotNull(val)){
join = join.join(name, JoinType.LEFT);
} else {
join = root.join(name, JoinType.LEFT);
}
break;
case RIGHT:
if(ObjectUtil.isNotNull(join) && ObjectUtil.isNotNull(val)){
join = join.join(name, JoinType.RIGHT);
} else {
join = root.join(name, JoinType.RIGHT);
}
break;
case INNER:
if(ObjectUtil.isNotNull(join) && ObjectUtil.isNotNull(val)){
join = join.join(name, JoinType.INNER);
} else {
join = root.join(name, JoinType.INNER);
}
break;
default: break;
}
}
}
switch (q.type()) {
case EQUAL:
list.add(cb.equal(getExpression(attributeName,join,root)
.as((Class<? extends Comparable>) fieldType),val));
break;
case GREATER_THAN:
list.add(cb.greaterThanOrEqualTo(getExpression(attributeName,join,root)
.as((Class<? extends Comparable>) fieldType), (Comparable) val));
break;
case LESS_THAN:
list.add(cb.lessThanOrEqualTo(getExpression(attributeName,join,root)
.as((Class<? extends Comparable>) fieldType), (Comparable) val));
break;
case LESS_THAN_NQ:
list.add(cb.lessThan(getExpression(attributeName,join,root)
.as((Class<? extends Comparable>) fieldType), (Comparable) val));
break;
case INNER_LIKE:
list.add(cb.like(getExpression(attributeName,join,root)
.as(String.class), "%" + val.toString() + "%"));
break;
case LEFT_LIKE:
list.add(cb.like(getExpression(attributeName,join,root)
.as(String.class), "%" + val.toString()));
break;
case RIGHT_LIKE:
list.add(cb.like(getExpression(attributeName,join,root)
.as(String.class), val.toString() + "%"));
break;
case IN:
if (CollUtil.isNotEmpty((Collection<Object>)val)) {
list.add(getExpression(attributeName,join,root).in((Collection<Object>) val));
}
break;
case NOT_IN:
if (CollUtil.isNotEmpty((Collection<Object>)val)) {
list.add(getExpression(attributeName,join,root).in((Collection<Object>) val).not());
}
break;
case NOT_EQUAL:
list.add(cb.notEqual(getExpression(attributeName,join,root), val));
break;
case NOT_NULL:
list.add(cb.isNotNull(getExpression(attributeName,join,root)));
break;
case IS_NULL:
list.add(cb.isNull(getExpression(attributeName,join,root)));
break;
case BETWEEN:
List<Object> between = new ArrayList<>((List<Object>)val);
list.add(cb.between(getExpression(attributeName, join, root).as((Class<? extends Comparable>) between.get(0).getClass()),
(Comparable) between.get(0), (Comparable) between.get(1)));
break;
default: break;
}
}
field.setAccessible(accessible);
}
} catch (Exception e) {
log.error(e.getMessage(), e);
}
int size = list.size();
return cb.and(list.toArray(new Predicate[size]));
}
@SuppressWarnings("unchecked")
private static <T, R> Expression<T> getExpression(String attributeName, Join join, Root<R> root) {
if (ObjectUtil.isNotEmpty(join)) {
return join.get(attributeName);
} else {
return root.get(attributeName);
}
}
private static boolean isBlank(final CharSequence cs) {
int strLen;
if (cs == null || (strLen = cs.length()) == 0) {
return true;
}
for (int i = 0; i < strLen; i++) {
if (!Character.isWhitespace(cs.charAt(i))) {
return false;
}
}
return true;
}
public static List<Field> getAllFields(Class clazz, List<Field> fields) {
if (clazz != null) {
fields.addAll(Arrays.asList(clazz.getDeclaredFields()));
getAllFields(clazz.getSuperclass(), fields);
}
return fields;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 6,436
|
The 1975–76 Quebec Nordiques season was the Nordiques fourth season, as they were coming off their best season to date in 1974–75, earning 92 points and finishing on top of the Canadian Division in the regular season, and making it to the Avco Cup finals in the playoffs, where they were swept by the Houston Aeros.
Quebec would have a very strong start to the season, and would battle with the Winnipeg Jets all season long on top of the Canadian Division. Quebec would finish the season with a franchise record 50 wins and 104 points, but would finish behind the Jets, who tied the Houston Aeros with the most points in the league at 106. The Nordiques scored a league high 371 goals, and finished with a very impressive record of 33–7–0 at home, tying the Aeros for the best home record in the league.
Offensively, Quebec was led by Marc Tardif, who had the most goals and points in the league with 71 and 148 respectively, while his 77 assists tied teammate J. C. Tremblay for the most in the WHA. Tremblay would finish the year with 89 points, leading the Nordiques blueline. Real Cloutier had a breakout season, scoring 60 goals and earning 114 points, as did Chris Bordeleau, who had 37 goals and 109 points. Rejean Houle and Serge Bernier also finished with over 100 points, as they earned 103 and 102 respectively. Gord Gallant had a team high 297 penalty minutes, while Pierre Roy was not too far behind with 259.
In goal, Richard Brodeur shattered the Nordiques record for wins, earning 44, while he posted a 3.69 GAA and earned 2 shutouts in 69 games. Michel DeGuise backed him up, winning 6 games.
In the opening round of the playoffs, Quebec would face the Calgary Cowboys, who finished the season 3rd in the Canadian Division with 86 points, which was 18 fewer than the Nordiques. Calgary would quiet the Nordiques home crowd in the first game, beating Quebec 3–1, then the Cowboys would win the second game by a score of 8–4 to take a 2–0 series lead. Quebec would fall behind 3–0 in the series after Calgary took the third game by a 3–2 score. The Nordiques managed to squeak out a 4–3 win in the fourth game, however, Calgary would end the series with a 6–4 win at Le Colisée in the fifth game, ending the Nordiques season much sooner than anyone expected.
Season standings
Schedule and results
Playoffs
Calgary Cowboys 4, Quebec Nordiques 1
Season stats
Scoring leaders
Goaltending
Playoff stats
Scoring leaders
Goaltending
Draft picks
Quebec's draft picks at the 1975 WHA Amateur Draft.
References
SHRP Sports
The Internet Hockey Database
Quebec Nordiques season, 1975-76
Quebec Nordiques seasons
Quebec
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,607
|
Q: Customize Order received page based on shipping method in WooCommerce How could I customise the order thank you page based on the order's shipping method? So for example if a customer used 'Delivery on request' option, the thank you page would display a different title.
add_filter( 'the_title', 'woo_personalize_order_received_title', 10, 2 );
function woo_personalize_order_received_title( $title, $id ) {
if ( is_order_received_page() && get_the_ID() === $id ) {
global $wp;
// Get the order. Line 9 to 17 are present in order_received() in includes/shortcodes/class-wc-shortcode-checkout.php file
$order_id = apply_filters( 'woocommerce_thankyou_order_id', absint( $wp->query_vars['order-received'] ) );
$order_key = apply_filters( 'woocommerce_thankyou_order_key', empty( $_GET['key'] ) ? '' : wc_clean( $_GET['key'] ) );
if ( $order_id > 0 ) {
$order = wc_get_order( $order_id );
if ( $order->get_order_key() != $order_key ) {
$order = false;
}
}
if ( isset ( $order ) ) {
$chosen_titles = array();
$available_methods = $wp->shipping->get_packages();
$chosen_rates = ( isset( $wp->session ) ) ? $wp->session->get( 'chosen_shipping_methods' ) : array();
foreach ($available_methods as $method)
foreach ($chosen_rates as $chosen) {
if( isset( $method['rates'][$chosen] ) ) $chosen_titles[] = $method['rates'][ $chosen ]->label;
}
if( in_array( 'Delivery price on request', $chosen_titles ) ) {
//$title = sprintf( "You are awesome, %s!", esc_html( $order->billing_first_name ) ); // use this for WooCommerce versions older then v2.7
$title = sprintf( "You are awesome, %s!", esc_html( $order->get_billing_first_name() ) );
}
}
}
return $title;
}
A: is_order_received_page() doesn't exist. Instead use is_wc_endpoint_url( 'order-received' )…
Also $wp->session or $wp->shipping will not work. Instead you can find the chosen shipping method data in the order item "shipping".
Try this instead:
add_filter( 'the_title', 'customizing_order_received_title', 10, 2 );
function customizing_order_received_title( $title, $post_id ) {
if ( is_wc_endpoint_url( 'order-received' ) && get_the_ID() === $post_id ) {
global $wp;
$order_id = absint( $wp->query_vars['order-received'] );
$order_key = isset( $_GET['key'] ) ? wc_clean( $_GET['key'] ) : '';
if ( empty($order_id) || $order_id == 0 )
return $title; // Exit
$order = wc_get_order( $order_id );
if ( $order->get_order_key() != $order_key )
return $title; // Exit
$method_title_names = array();
// Loop through Order shipping items data and get the method title
foreach ($order->get_items('shipping') as $shipping_method )
$method_title_names[] = $shipping_method->get_name();
if( in_array( 'Delivery price on request', $method_title_names ) ) {
$title = sprintf( "You are awesome, %s!", esc_html( $order->get_billing_first_name() ) );
}
}
return $title;
}
Code goes on function.php file of your active child theme (or active theme). Tested and works.
Similar: Adding custom message on Thank You page by shipping method in Woocommerce
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,399
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\section{Introduction}
In this work, we introduce LDRANK (see section~\ref{ldrank}), an efficient query-biased and context-aware ranking algorithm that applies to the resources of a LOD graph. When combined with the automatic annotation of resources in Web pages (e.g. through DBpedia Spotlight~\cite{mendes2011dbpedia}), LDRANK offers the opportunity to build useful semantic snippet that can apply to any Web page regardless of its provenance (see section~\ref{ensen}). In this introduction, we provide the background information from which the necessity for this new algorithm will appear.
On the web of documents links are indications of a relationship between information carried by the documents. Although these indications are coarse-grained, they revealed themselves as essential for the most-effective ranking algorithms (PageRank~\cite{page1999pagerank}, HITS~\cite{kleinberg1999authoritative}, SALSA~\cite{lempel2001salsa}).
On the web of data, links are fine-grained explicit relationships between resources (i.e., URI for things of the phenomenal world, be they mental or physical). The vast majority of the existing ranking strategies for the web of data (see~\cite{roasurvey} and~\cite{jindal2014review} for recent surveys) are relying on adaptations of PageRank. The modifications made to adapt the PageRank algorithm to the web of data are necessary due to the high heterogeneity of both the provenance of the datasets and the types of the relationships. Otherwise, there are also a few experiments with learning-to-rank approaches applied to the web of data (e.g.,~\cite{dali2012query}). These techniques depend on the availability of relevance judgments for training (although indirect measures of correlated quantities can sometimes be used, e.g. the number of visits agents made to a resource).
In order to manage the aforementioned intrinsic heterogeneity of the web of data, the Linked Open Data (LOD) initiative promotes simple principles for publishing resources in a way conducive to a web of linked data with shared knowledge expressed in a common formalism (RDF) and accessible through a common interface (HTTP). As a key use-case, DBpedia has been used in conjunction with NLP strategies in order to associate resources with their surface forms in a text document. The main current applications for this use-case are: DBpedia Spotlight~\cite{mendes2011dbpedia}, AlchemyAPI\footnote{{\scriptsize\url{www.alchemyapi.com}~; \url{www.opencalais.com}~; \url{www.ontos.com}}} (similar to DBpedia Spotlight, but finds resources in various LOD datasets and thus includes a coreference resolution step), OpenCalais\footnotemark[\value{footnote}], SemanticAPI from Ontos\footnotemark[\value{footnote}], ZenCrowd~\cite{demartini2012zencrowd}\dots
In this context, we address the problem of ranking resources that come from the automatic annotation of a Web page selected by a web search engine in response to a user query. The main challenge is to make good use of the knowledge given by the query and the Web page's text in order to palliate the sparsity and heterogeneity of the graph of resources.
We propose an algorithm, LDRANK, and we compare it to other modified PageRank algorithms. Moreover, we apply it to the construction of semantic snippets\footnote{{\scriptsize\url{http://liris.cnrs.fr/drim/projects/ensen/}: live demo, source code, technical report, datasets}}. A snippet is an excerpt from a Web page determined at query-time and used to express how a Web page may be relevant to the query. A semantic snippet is meant to improve the process of matching the ranked Web pages presented within a Search Engine Result Page (SERP) with the user's mental model of her information need. It achieves this objective by making apparent the relationships existing between the information need and the more relevant resources present in the Web page.
In section~\ref{related-works} we introduce the related works about enhanced snippets for the web of documents and for the web of data. In section~\ref{dataset}, we describe the construction of a dataset for the evaluation query-biased entity ranking algorithms. In section~\ref{ldrank} we present the LDRANK algorithm and its evaluation. In section~\ref{ensen}, we introduce ENsEN, the software system we developed to provide semantic snippets. In section~\ref{user} we present the results of an evaluation of the usefulness of ENsEN.
\section{Related Works}\label{related-works}
We first mention works that generate snippets for native RDF documents. Ge~{\it et al.}~\cite{ge2012incorporating}, and Penin~{\it et al.}~\cite{penin2008snippet} focus on the generation of snippets for ontology search. Bai~{\it et al.}~\cite{bai2008rdf} generate snippets for a semantic web search engine.
In~\cite{penin2008snippet}, the authors first identify a topic thanks to an off-line hierarchical clustering algorithm. Next, they compute a list of RDF sentences (i.e. sets of connected RDF statements) semantically close to the topic. Finally, they rank the selected RDF statements by considering both structural properties of the RDF graph and lexical features of the terms present in the ontology (by way of a Wordnet-based similarity measure).
In~\cite{ge2012incorporating}, the authors first transform the RDF graph into a term association graph in which each edge is associated with a set of RDF sentences. Their objective is to produce a compact representation of the relationships existing between the terms of the query. These relationships are to be found in the RDF graph. To do this, they decompose the term association graph into maximum r-radius components in order to avoid long distance relations between query terms. Next, they search sub-snippets in these components (i.e. connected subgraphs that link some of the query-terms). Finally, they select some of the sub-snippets to form the final snippet.
In~\cite{bai2008rdf}, the authors first assign a topic to the RDF document (they use a property such as {\it p:primaryTopic} if it exists, otherwise they rely on a heuristic based on the comparison of the URI of the candidates topic-nodes with the text of the URL of the RDF document). Next they design a ranking algorithm for RDF statements. Particularly, they introduce the notions of {\it correlative} (e.g. \texttt{foaf:surname} and \texttt{foaf:family\_name}) and {\it exclusive} (e.g. \texttt{foaf:name} and \texttt{foaf:surname}) properties. Finally, they use this ranking algorithm to give the user a set of relationships between the query-related statements and the topic-related statements.
To sum up, we agree with Ge~{\it et al.}~\cite{ge2012incorporating} that the main benefit of possessing highly structured data from an RDF graph is the possibility to find non-trivial relationships among the query terms themselves, and also between the query terms and the main concepts of the document. Moreover, we agree with Penin~{\it et al.}~\cite{penin2008snippet} and Bai~{\it et al.}~\cite{bai2008rdf} about the necessity to design a ranking algorithm for RDF statements that considers both the structure of the RDF graph and lexical properties of the textual data. However, we find ourselves in an inverted situation with genuine text extracted from classical Web pages, and RDF graphs automatically generated from these Web pages.
Indeed, LOD resources can either come from: (i) a LOD dataset (e.g. by way of SPARQL queries), (ii) semantic annotations embedded in a Web page (i.e., by using RDFa, Microdata, or Microformats\footnote{{\scriptsize\url{www.w3.org/TR/xhtml-rdfa-primer/}~;
\url{microformats.org/}~; \url{www.w3.org/TR/microdata/}}}), or (iii) automatic association of resources with surface forms of the Web page by way of NLP strategies (e.g. DBpedia Spotlight~\cite{mendes2011dbpedia}, ZenCrowd~\cite{demartini2012zencrowd},\dots). Among the approaches that offer to enhance the snippets of a SERP by using the web of data~\cite{haas2011enhanced}~\cite{steiner2010google}, none rely on automatic annotation: they use only embedded annotations. Haas~{\it et al.}~\cite{haas2011enhanced} employed structured metadata (i.e. RDFa and several microformats) and information extraction techniques (i.e. handwritten or machine-learned wrappers designed for the top host names e.g., \url{en.wikipedia.org}, \url{youtube.com},\dots) to enhance the SERP with multimedia elements, key-value pairs and interactive features. By combining metadata authored by the documents' publishers with structured data extracted by ad-hoc wrappers designed for a few top host names, they are able to build enhanced snippets for many results of a SERP. They chose not to use the LOD graph to avoid the problem of the transfer of trust between the Web of documents and the Web of Data. Indeed, they argue that the quality of the editorial processes that produce the Web of Data from the Web of documents (e.g. the transformation from Wikipedia to DBPedia) cannot be controlled. Therefore, from their point of view, making use of the LOD graph for enhancing snippets would introduce too much noise. Also, Google Rich Snippet (GRS)~\cite{steiner2010google} is a similar initiative that relies exclusively on structured metadata authored by the Web pages' publishers.
Moreover, a study made in 2012\cite{bizer2013deployment} on the over 40 million websites of the Common Crawl corpus\footnote{\url{http://commoncrawl.org}} shows that 5.64\% of the websites contained embedded structured data. However, nearly 50\% of the top 10,000 websites of the Alexa list of popular websites\footnote{\url{http://www.alexa.com/topsites}} had structured data. Moreover, the authors of the study say that: ``The topics of the data [\ldots] seem to be largely determined by the major consumers the data is targeted at: Google, Facebook, Yahoo!, and Bing''. Therefore, there is still a clear need for a high quality process that, given a document relevant to a Web search query, can select the most relevant resources among those automatically discovered within the document (e.g., through state of the art NLP algorithms), and this, whatever the document's provenance may be. An efficient algorithm for ranking the resources of a LOD graph while taking into account their textual context could serve this purpose.
However, most of the existing approaches that can be used to rank the resources of graphs coming from the Web of data are not well adapted to this task. Thus, OntologyRank~\cite{ding2004swoogle} (used by Swoogle) introduces a modified version of PageRank with a teleportation matrix that takes into account the types of the links between ontologies. Similarly, PopRank~\cite{nie2005object} offers a modified PageRank that considers the different types of predicates between resources. RareRank~\cite{wei2011rational} introduces a modified PageRank with a teleportation matrix that takes into account topical relationships between resources as available from ontologies. The approach introduced in~\cite{fafalios2014post} modifies the teleportation matrix by taking into account the ranking of the Web pages within which the resources were discovered. Since this approach can be applied to our context, we include it to our evaluations (see section~\ref{eval}). Finally, TRank~\cite{tonon2013trank} addresses the task of ranking entity types given an initial entity and its textual context.
Given this context, we introduce LDRANK, a query-biased and context-aware ranking algorithm for LOD resources. Moreover, we apply LDRANK to the construction of generic semantic snippets that can apply to any Web page. In the next section, we introduce how we built a dataset through crowdsourced relevance judgments to evaluate our algorithm, LDRANK.
\section{Dataset for Evaluating Query-biased Ranking of LOD resources}\label{dataset}
We are interested in query-biased algorithms for the ranking of resources in sparse and heterogeneous LOD graphs associated with a textual context. To our knowledge, there is no evaluation dataset suited to this context (this can be verified for example through a recent survey~\cite{roasurvey}). Therefore, we used a crowdsourcing approach for making our evaluation dataset (freely available online\footnote{\url{http://liris.cnrs.fr/drim/projects/ensen/}}). We now describe how this dataset was obtained.
\subsection{Data Collection}
We took randomly 30 queries from the ``Yahoo! Search Query Tiny Sample'' offered by Yahoo! Webscope\footnote{\url{http://webscope.sandbox.yahoo.com/catalog.php?datatype=l}}. We submitted the queries to the Google search engine and we kept the top-5 Web pages for each query. For each one of the 150 HTML Web pages, we extracted its main raw textual content by applying the algorithm proposed by Kohlschütter, Fankhauser, and Nejdl~\cite{kohlschutter2010boilerplate}. On average, the text we kept for each Web page is made of 467 words. We applied DBpedia Spotlight~\cite{mendes2011dbpedia} on these texts to detect resources. There are on average 81 detected resources by Web page.
\subsection{Microtasks Generation}
Considering the length of our texts, the task of evaluating all the annotations of a Web page would be too demanding. Therefore, we divide this task into smaller ``microtasks''. A microtask will consist in scoring the relevance of the annotations of a single sentence. We split the text of a Web page into sentences with the ICU BreakIterator algorithm\footnote{\url{http://icu-project.org/apiref/icu4c/classicu\_1\_1BreakIterator.html}}. There are on average 22 sentences by document. Moreover, if a sentence contains more than 10 annotated resources, the work will be split over multiple microtasks. We used the CrowdFlower\footnote{\url{http://www.crowdflower.com/}} crowdsourcing platform. It distributes work to contributors in the U.S. and 153 other countries while maintaining quality and controlling costs. It has a global pool of 5 million contributors. A microtask is called a job by CrowdFlower. The design of a job is specified in CML, a markup language provided by CrowdFlower. For each job, we give the worker a short list of instructions about how to complete the job (we tested many formulations until finding a suitable one understood by all workers). We provide the worker with a topic made of a title (the query) and a short text (the sentence). For each resource in the sentence, there is a question asking the worker to evaluate the correctness and the relevance of the annotation. We used the ordinal scale proposed by J{\"a}rvelin and Kek{\"a}l{\"a}inen when they introduced the DCG graded relevance\cite{jarvelin2000ir}: irrelevant (0), marginally relevant (1), fairly relevant (2), and highly relevant (3). Each question is associated with a small text that describe the resource (viz. the beginning of its DBpedia abstract). Each job was given to 10 workers. Therefore, for each job we have 10 judgments. Each job was paid \$.01.
\subsection{Quality Control}
We only accepted workers that had completed over a hundred questions across a variety of job types and had an high overall accuracy. Workers had a maximum of 30 minutes to provide an answer. Workers had to spend at least 10 seconds on the job before giving an answer. We measured the agreement between workers with the Krippendorff's alpha coefficient~\cite{krippendorff2012content}. This coefficient uses by default a binary distance to compare answers, but other distances can be used. To take into account the fact that we used an ordinal scale encoding both correctness and relevance, we used the following symmetric distance: $d(0, 1)=0.5$ ; $d(0, 2)=0.75$ ; $d(0, 3)=1$ ; $d(1, 2)=0.25$ ; $d(1 ; 3)=0.5$ ; $d(2 ; 3)=0.25$ ; $d(x,x)=0$. With these parameters, we obtained an alpha of $0.22$. According to Landis and Koch's scale~\cite{landis1977measurement}, this can be considered a fair agreement (the scale was designed for Fleiss' kappa, but the Krippendorff's alpha is in most ways compatible with the kappa). However, by comparison with existing works that applied crowdsourcing to an information retrieval context, we cannot be satisfied with an alpha of $0.22$. For example, Jeong~{\it et al.}~\cite{jeong2013crowd} obtained a Fleiss' kappa of $0.41$ (i.e. moderate agreement) for a crowd-powered socially embedded search engine. However, Alonso, Marshall, and Najork~\cite{alonsocrowdsourcing} obtained a Krippendorff's alpha between $0.03$ and $0.19$ for a more subjective task: deciding if a tweet is or is not interesting. To improve the quality of our dataset, we found the workers that often disagreed with the majority. In fact, by removing the workers that disagree with the majority in more than $41.2\%$ of the cases, we obtained a Krippendorff's alpha of $0.46$. Then, $96.5\%$ of the jobs are done by at least 3 workers, $66\%$ of the jobs are done by at least 5 workers, and we have only $0.7 \%$ of the jobs done by only 1 worker.
\subsection{Aggregation of the Results}
We used majority voting for aggregating the results within each sentence. We used two different methods to break ties : (i) the maximum of the mean of the workers' trust (a metric provided by CrowdFlower), or (ii) the highest value. We discovered later that these two choices result in very similar outcomes when the dataset is used to compare ranking algorithms. We used the same majority voting strategy to aggregate the results at the level of a Web page.
In the next section, we introduce LDRANK, a query-biased ranking algorithm for LOD resources. The dataset we just described will be used in section~\ref{eval} to evaluate LDRANK and to compare it to the state of the art.
\section{LDRANK, a Query-biased Ranking Algorithm for LOD Resources}\label{ldrank}
\subsection{Context}
We introduce LDRANK (Linked Data Ranking Algorithm), a quey-biased algorithm for ranking the resources of a RDF graph. We suppose that the resources were discovered in a Web page found by a Web search engine in answer to a user's query.
In our experiments, the resources are detected in the Web page by DBpedia Spotlight~\cite{mendes2011dbpedia}. From this set of resources and through queries to a DBpedia SPARQL endpoint, we obtain a graph by finding all the relationships between the resources. To each resource, we associate a text obtained by merging its DBpedia abstract and windows of text (300 characters) from the Web page centered on the surface forms associated with the resource. We remove the empty words and we apply stemming\footnote{\url{http://snowball.tartarus.org}} to this text.
LDRANK is adapted by design to such sparse graphs of LOD resources detected in a Web page.
First, LDRANK uses the explicit structure of the graph through a PageRank-like algorithm; second, it uses the implicit relationships that can be inferred from the text associated with the resources through an original variation of the Singular Value Decomposition (SVD); and third, it takes into account the ranking of the Web pages where the resources were found thanks to a scoring function first introduced by Fafalios and Tzitzikas~\cite{fafalios2014post}.
More precisely, the SVD-based textual analysis and the exploitation of the ranking obtained from a Web search engine result page, each produce a different probability vector expressing some prior knowledge (or belief) about the importance of the resources (see sections~\ref{srp} and~\ref{svd}). Next, these probability vectors are combined through a consensual linear opinion aggregation strategy first introduced by Carvalho and Larson~\cite{carvalho2013consensual} (see section~\ref{belief}). Finally, we use this combined prior knowledge to influence the convergence of a PageRank-like algorithm towards a stable probability distribution corresponding to the final ranking of the resources (see section~\ref{mainalgo}).
\subsection{Prior Knowledge Based on a Web Search Engine Result Page}\label{srp}
\algbegin Algorithm H (Hit Score). This algorithm computes a probability vector ($hitdistrib$) that represents prior knowledge about the importance of the resources based on the rank of the Web pages in which they were detected. This strategy was first introduced by Fafalios and Tzitzikas~\cite{fafalios2014post}.
\algstep H1. $A \leftarrow $ the list of the top Web pages ranked by a Web search engine.
\algstep H2. $E \leftarrow $ the set of detected resources.
\algstep H3. $docs(e) \equiv $ the documents of $A$ containing the detected resources $e$.
\algstep H4. $rank(a) \equiv $ the rank of document $a$ in $A$.
\algstep H5. $hitscore(e) \equiv \sum_{a \in docs(e)} (size(A) + 1) - rank(a)$
\algstep H6. $hitdistrib[e] \leftarrow hitscore(e) / \sum_{e' \in E} hitscore(e')$
\algstep H7. [{\it End.}] \quad\hbox{\kern1.5pt\vrule width2.5pt height6pt depth1pt\kern1.5pt}
\subsection{Prior Knowledge Based on a Latent Analysis of Textual Data}\label{svd}
\aalgbegin Algorithm S (Linked Data Iterative SVD). This algorithm computes a probability vector ($svddistrib$) that represents prior knowledge about the importance of the resources based on the textual data associated to them.
\algstep S1. [{\it Initial matrix.}] $R \leftarrow $ the sparse resource-stem matrix (i.e., resources in rows, stems in columns) in Compressed Column Storage (CCS) format\footnote{\url{http://netlib.org/linalg/html\_templates/node92.html}}.
\algstep S2. [{\it Initial important resources.}] $info\_need \leftarrow $ a set of resources made of the union of the resources detected in the text of the query and the one resource with the best hitscore (for the case when no resources were detected in the query). We assume that these resources are likely to be close to the information need of the user.
\algstep S3. [{\it First SVD.}] $(U,S,V^T) \leftarrow svdLAS2A(R, nb\_dim)$ Compute the singular value decomposition (SVD) of $R$ at rank $k = nb\_dim$. Since $R$ is very sparse, we use the {\it las2} algorithm developed by Michael W. Berry~\cite{berry1992large} to compute the decomposition: $R_k = U_k S_k V_k^T$ with $U_k$ and $V_k$ orthogonal, $S_k$ diagonal, such that $\|R-R_k\|_F$ is minimized (i.e. from the perspective of the Frobenius norm, $R_k$ is the best rank-$k$ approximation of $R$).
\algstep S4. [{\it Resources' coordinates in the reduced space.}] $SUT \leftarrow S U^T$ In the new $k$-dimensional space, this operation scales the coordinates of the resources (i.e. the rows of $U$) by their corresponding factor in $S$. This is done by the matrix product: $S U^T$. Thus, we obtain the coordinates of the resources in the reduced space (i.e. the columns of $SUT$).
\algstep S5. $prev\_norms \leftarrow $ euclidean norms of the resources in the reduced space.
\algstep S6. [{\it Updated matrix.}] $R' \leftarrow $ $R$ where the rows corresponding to the resources of $info\_need$ have been multiplied by the parameter $stress$ (since $R$ is in CCS format, it is more convenient to do this operation on the transpose of $R$).
\algstep S7. [{\it Second SVD.}] $(U',S',V'^T) \leftarrow svdLAS2A(R', nb\_dim)$
\algstep S8. [{\it Updated resources' coordinates in the reduced space.}] $SUT' \leftarrow S' U'^T$
\algstep S9. $norms \leftarrow $ updated euclidean norm of the resources in the reduced space.
\algstep S10. [{\it Drift of the resources away from the origin of the reduced space}.] \\$svdscore(e) \equiv norms[e] - prev\_norms[e]$.
\algstep S11. $svddistrib[e] \leftarrow svdscore(e) / \sum_{e'} svdscore(e')$
\algstep S12. [{\it End.}] \quad\hbox{\kern1.5pt\vrule width2.5pt height6pt depth1pt\kern1.5pt}
We shall now introduce the essential property of the SVD on which relies Algorithm~S. For a strong dimensional reduction (i.e. for small values of $k$), the transformation $S_k U^T$ tends to place resources that were orthogonal to many other resources in the row space of $R$ near the origin of the $k$-dimensional resulting space. Indeed, as we said above, the SVD can be seen as an
optimization algorithm, and to minimize the error due to the impossibility for a resource to be orthogonal to more than $k$ non co-linear resources, this resource should be placed as close to the origin as possible for its dot product with other resources to remain small. A similar argument can be used to show that resources co-linear to many other resources in the row space of $R$ will also tend to be near the origin of the $k$-dimensional space.
Algorithm~S uses this property for ranking the resources by their importance relatively to the user's information need. In $R'$ the resources that are believed to be close to the information need are given artificially more importance. Therefore, resources having interesting relationships with the resources artificially pushed away from the origin will also move away from the origin. By ``interesting'', we mean different from the relationships they maintain with much of the other resources (cf. the geometric argument developed above about the SVD seen as an optimization algorithm).
We obtained the best experimental results with a reduction to the 1 dimensional line (i.e. with $nb\_dim = 1$ in steps S3 and S7 of Algorithm~S), and with a stress factor (step S6 of Algorithm~S) of $1000.0$.
\subsection{Belief Aggregation Strategy}\label{belief}
We consider $hitdistrib$ (from Algorithm~H), $svddistrib$ (from Algorithm~S), and the equiprobable distribution ($equidistrib$) as three experts' beliefs (or prior knowledge) about the importance of the resources. To aggregate these beliefs, we apply Carvalho and Larson~\cite{carvalho2013consensual} consensual linear opinion pool algorithm. It is an iterative algorithm where at each step expert $i$ re-evaluates its distribution as a linear combination of the distributions of all the experts. The weight associated by expert $i$ to the distribution of expert $j$ is proportional to the distance between the two distributions. The authors define this distance such that the process converges towards a consensus. We will refer to this resulting consensual probability vector by the name $finaldistrib$.
\subsection{LDRANK}\label{mainalgo}
The PageRank~\cite{page1999pagerank} algorithm transforms the adjacency matrix ($M$) of a network of Web pages into a matrix $H$ which is both stochastic (i.e., each row of $H$ sums to 1) and primitive (i.e., $\exists k$ s.t. $H^k > 0$), thus assuring the existence of a stationary vector (i.e., the positive eigenvector corresponding to the eigenvalue $1$). This stationary vector is a probability vector that can been interpreted as representing the importance of each Web page. Moreover, it can be computed efficiently with the power iteration algorithm by taking into account the sparsity of the stochastic matrix.
In the original version of the PageRank algorithm, no assumption is made about the probability of importance of the Web pages before the link analysis takes place. In other words: first, the matrix $M$ is transformed into a stochastic matrix $S$ by replacing each null row by the equiprobable distribution ($equidistrib$); second, the matrix $S$ is transformed into a primitive matrix $H$ by a linear combination with the so-called teleportation matrix ($T$): $H = \alpha S + (1-\alpha)T$ where each row of $T$ is the equiprobable distribution ($equidistrib$).
In algorithm~LDRANK, instead of using the equiprobable distribution, we use the consensual distribution ($finaldistrib$) introduced above in section~\ref{belief}. We obtained the best experimental results for $0.6 \leq \alpha \leq 0.8$. Moreover, we set at $1E-10$ the value of the convergence threshold controlling the termination of the power iteration method that computes the stationary vector.
LDRANK is available online under an open-source license\footnote{Source code available online under an opensource license \url{http://liris.cnrs.fr/drim/projects/ensen/}}.
\subsection{LDRANK evaluation}\label{eval}
We compared four ranking strategies, each one of them is based on a different source of prior knowledge used to inform a PageRank-like algorithm: unmodified PageRank i.e., prior knowledge about the importance of the resources is modeled by an equiprobable distribution (we name this strategy EQUI); PageRank modified with the hitscore prior knowledge introduced in section~\ref{srp} and due to Fafalios and Tzitzikas~\cite{fafalios2014post} (named HIT); PageRank modified with our new SVD-based prior knowledge introduced in section~\ref{svd} (named SVD); and PageRank modified with a consensual mixture of the three previous sources of prior knowledge (named LDRANK).
In order to compare the four strategies (EQUI, HIT, SVD and LDRANK), we used the NDCG (Normalized Discounted Cumulative Gain) metric. The DCG (Discounted Cumulative Gain) at rank $r$ is defined as: $DCG_r = rel_1 + \sum_{i=1}^r \frac{rel_i}{log_2 i}$. NDCG at rank $r$ is DCG at rank $r$ normalized by the ideal ranking at rank $r$. The construction of the dataset used for the evaluation was introduced in section~\ref{dataset}.
The results are presented in Figure~\ref{fig:ndcg}. We can see that the SVD and HIT strategies obtain similar performances. However, they are clearly outperformed by their consensual combination. Moreover, since we systematically took into account the sparsity of the data, we obtain good execution time performances (see Figure~\ref{fig:perfs}). The SVD strategy takes more time than the HIT strategy since it needs to compute the SVD. The additional time spent by the combined strategy is due to the time necessary to converge towards a consensus. Finally, we did similar experiments by considering the edges of the graph bidirectional. The relative performance and accuracy of the algorithms were similar, but the absolute NDCG scores were slightly better.
It should be noted that through these experiments, beside introducing a new efficient ranking strategy based on an original use of the SVD dimensionality reduction, we are also offering evidence that different strategies based on a modification of the teleportation matrix of the PageRank algorithm can profitably be combined when considered as concurrent sources of prior knowledge about the importance of the resources.
\begin{figure}
\centering
\includegraphics[width=4in, keepaspectratio]{ndcg_nosym.png}
\caption{Comparison of the NDCG scores for the four different strategies}
\label{fig:ndcg}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=4in, keepaspectratio]{clock_nosym.png}
\caption{Comparison of the execution time for the four different strategies (with processor: 2.9~GHz Intel Core i7, and memory: 8~GB 1600~MHz DDR3)}
\label{fig:perfs}
\end{figure}
\section{Overview of ENsEN}\label{ensen}
In order to better convince the reader of the usefulness and efficiency of LDRANK, we used it at the core of ENsEN (Enhanced Search Engine): a software system that enhances a SERP with semantic snippets (a live demonstration is available online, see a previous footnote for the URL).
Given the query, we obtain the SERP (we used Google for our experiments). For each result of the SERP, we use DBpedia Spotlight to obtain a set of DBpedia resources. In the same way, we find resources from the terms of the query. From this set of resources and through queries to a DBpedia SPARQL endpoint, we obtain a graph by finding all the relationships between the resources. To each resource, we associate a text obtained by merging its DBpedia's abstract and windows of text from the Web page centered on the surface forms associated with the resource. With as input the graph, its associated text, and the resources extracted from the query, we execute LDRANK and we obtain a ranking of the resources. The top-ranked resources (viz. ``main-resources'') are displayed on the snippet. From a DBpedia SPARQL endpoint, we do a 1-hop extension of the main-resources in order to increase the number of triples among which we will then search for the more important ones. To do this, we build a 3-way tensor from the extended graph: each predicate corresponds to an horizontal slice that represents the adjacency matrix for the restriction of the graph to this predicate. We compute the PARAFAC decomposition of the tensor into a sum of factors (rank-one three-way tensors) and interpret it in manner similar to~\cite{franz2009triplerank}: for each main-resource, we select the factors to which it contributes the most (as a subject or as an object), and for each one of these factors we select the triples with the best ranked predicates. Thus, we associate to each main-resource a set of triples that will appear within its description. Finally, we used a machine learning approach to select short excerpts of the Web page to be part of the description of each main-resource. In the context of this paper, for lack of space, we cannot describe this process but full details are available in an online technical report (see a previous footnote for the URL).
\section{Crowdsourcing-based User Evaluation}\label{user}
We selected randomly 10 tasks from the ``Yahoo! Answers Query To Questions'' dataset\footnote{\url{http://webscope.sandbox.yahoo.com/catalog.php?datatype=l}}. Each task was made of three questions on a common topic. To each task corresponds a job on the CrowdFlower platform. Each job was priced \$0.20. We collected 20 judgments for each task. Half of the workers was asked to use our system, and the other half used Google. In order to control that a worker answered the task by using our system, we generated a code that the worker had to copy and paste into her answer. The correctness results are shown on Figure~\ref{fig:Correctness}. Only complete answers were considered correct. We also monitored the time spent to answer the tasks (see Figure~\ref{fig:Timespent}). Thus, ENsEN is clearly beneficial to its users in terms of usefulness.
\begin{figure}
\centering
\includegraphics[width=4.9in, keepaspectratio]{correct.jpg}
\caption{Average Number of Correct Answers}
\label{fig:Correctness}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=4.9in, keepaspectratio]{time_spent.png}
\caption{Time Spent for Answering the Tasks}
\label{fig:Timespent}
\end{figure}
\section{Conclusion}\label{conclusion}
We proposed a new algorithm, LDRANK, for ranking the resources of a sparse LOD RDF graph given the knowledge of a user's information need expressed as a query made of keywords. These kind of graphs appear in particular as the result of the automatic detection of resources in a Web page. LDRANK takes advantage of both the explicit structure given by the Web of data and the implicit relationships that can be found by text analysis of a Web page. We applied LDRANK in the context of semantic snippets where its high accuracy allowed for the construction of useful and usable enhanced snippets that integrate resources obtained from the automatic annotation of a Web page. Future work could evaluate the potential of this approach for exploratory search.
\bibliographystyle{splncs03}
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Q: AWS Server-less Architecture using Lambda and SQS I've been learning more and more about AWS lately. I've been reading through the white papers and working my way through the various services. I've been working on PHP applications and front-end dev for a while now. Two things really stuck out to me. Those two things are server-less architecture using Lambdas with event-triggers and SQS (queues). The last three years I have been working with REST over HTTP with frameworks like Angular.
It occurred to me though that one could create an entire back-end/service layer through Lambda's and message queues alone. Perhaps I'm naive as I have never used that type of architecture for a real world project but it seems like a very simple means to build a service layer.
Has anyone built a web application back-end consisting of only Lambdas and message queues as opposed to "traditional" http request with REST. If so what types of drawbacks are there to this type of architecture besides relying so heavily on a vendor like AWS?
For example, wouldn't it be entirely possible to build a CMS using these technologies where the scripts create the AWS assets programmatically given a key with full admin rights to an account?
A: Yes, you can practically create the entire backend service using serverless architecture.
There are a lot of AWS services that usually play into the serverless gambit of things.
DynamoDB, SNS, SQS, S3 to name a few.
AWS Lambda is the backbone and sort of acts as a glue to bind these services.
Serverless doesn't mean you move away from "traditional" http request to message queues. If you need the web interface you would still need to use HTTP. You would primarily use message queues to decouple your services.
So, if you want the service to be accessible over HTTP just like your REST services and still be serverless then you can do that as well. And for that you will need to use AWS API Gateway in conjunction with AWS Lambda
One primary drawback/limitation is that debugging is not very straightforward. You cannot login to the system and cannot attach remote debuggers. And then obviously you get tied into the vendor.
Then there are limitations on the resources. E.g. Lambda can offer you a maximum memory footprint of 5GB, so if you need to do some compute intensive job that needs more memory and can't be broken down into sub tasks then serverless (AWS Lambda) is not an option for you.
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import { goRight, keydown, state } from "../../src/base/internal.js";
import KeyboardDirectionMixin from "../../src/base/KeyboardDirectionMixin.js";
import { assert, sinon } from "../testHelpers.js";
class KeyboardDirectionMixinTest extends KeyboardDirectionMixin(HTMLElement) {
constructor() {
super();
this[state] = { orientation: "" };
}
[goRight]() {
if (super[goRight]) {
super[goRight]();
}
return true;
}
}
customElements.define("keyboard-direction-test", KeyboardDirectionMixinTest);
describe("KeyboardDirectionMixin", () => {
it("maps a Right arrow key to a goRight action", () => {
const fixture = new KeyboardDirectionMixinTest();
const spy = sinon.spy(fixture, goRight);
const result = fixture[keydown]({
key: "ArrowRight",
target: fixture,
});
assert(spy.calledOnce);
assert(result);
});
it("ignores a Right arrow key when orientation is vertical", () => {
const fixture = new KeyboardDirectionMixinTest();
Object.assign(fixture[state], {
orientation: "vertical",
});
const spy = sinon.spy(fixture, goRight);
const result = fixture[keydown]({
key: "ArrowRight",
});
assert(!spy.calledOnce);
assert(!result);
});
it("ignores a Right arrow key if the meta (command) key was pressed", () => {
const fixture = new KeyboardDirectionMixinTest();
const spy = sinon.spy(fixture, goRight);
const result = fixture[keydown]({
altKey: true,
key: "ArrowRight",
});
assert(!spy.calledOnce);
assert(!result);
});
});
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\section{INTRODUCTION}
The {\it Kepler} satellite provides the highly precise and nearly-continuous photometric data for $\sim$ 200,000 objects that
has helped to revolutionize the study of stars themselves, as well as extrasolar planets (Borucki et al. 2010; Koch et al. 2010).
There are 2878 eclipsing and ellipsoidal binaries in the {\it Kepler} main field of view (Kirk et al. 2016), corresponding
to about 1.3 \% of all observed targets. Eclipsing binaries (EBs) serve as critical tools that provide an accurate and
direct determination of fundamental stellar parameters such as the mass and radius for each component. These data allow us
to test stellar evolution models and to determine distances of binary systems (Torres et al. 2010). Furthermore, it is
possible to measure precisely mid-eclipse times from binary light curves. The timing measurements are used to investigate
a variety of physical phenomena causing the orbital period changes of EBs (Hilditch 2001; Kreiner et al. 2001).
Such examples are mass transfer, angular momentum loss, apsidal motion in an elliptical orbit, third-body effect, and
magnetic activity cycle.
EBs with pulsating components are very promising objects for the study of stellar structure and evolution, because binarity
provides useful information about the components and asteroseismology assists in probing the interiors of stars. Most of
them have been found to be $\delta$ Sct-type pulsators of classical semi-detached Algols (Mkrtichian et al. 2004;
Liakos \& Niarchos 2016b). The $\delta$ Sct stars are dwarfs and subgiants with spectral types A and F located in the lower
portion of the Cepheid instability strip. They pulsate in low-order pressure ($p$) modes with typical periods of 0.02$-$0.25 days
and amplitudes of less than 0.1 mag (Breger 2000; Rodr\'iguez \& Breger 2001). The pulsations are driven by the $\kappa$ mechanism
mostly due to partial ionization of He II. The $\delta$ Sct variables in binaries have pulsation features similar to single
$\delta$ Sct stars, but their pulsations can be affected by mass transfer between both components and gravitation forces from
companions. Recently, Liakos \& Niarchos (2015, 2016a) showed that there is a threshold in the orbital period of $\sim$ 13 days
below which the pulsations are influenced by the binarity. In eccentric-orbit binaries, some pulsations can be excited by
tidal interaction. The signature of the tidally excited modes is the frequencies at multiples of the orbital frequency
(Welsh et al. 2011; Hambleton et al. 2013).
This paper is the fifth contribution in a series of detailed studies for pulsating stars in the {\it Kepler} EBs
(Lee et al. 2014, 2016a,b; Lee 2016). We present that KIC 11401845 (R.A.$_{2000}$ = 19$^{\rm h}$25$^{\rm m}$11$\fs275$;
decl.$_{2000}$ = +49$^{\circ}$14${\rm '}$40$\farcs$09; $K_{\rm p}$ = $+$14.355; $g$ = $+$14.443; $g-r$ = $+$0.118) is
a detached EB exhibiting multiperiodic pulsations and light-travel-time (LTT) delay. The system was announced to be an EB
pulsating at frequencies of 13$-$25 d$^{-1}$ by Gaulme \& Guzik (2014).
\section{{\it KEPLER} PHOTOMETRY AND LIGHT-CURVE SYNTHESIS}
The {\it Kepler} data of KIC 11401845 were obtained in a long cadence mode of 29.42 minutes during Quarters 10 and 12.
We used the SAP (Simple Aperture Photometry) time-series data in Data Release 25 retrieved from the {\it Kepler} Data
Archive\footnote{http://archive.stsci.edu/kepler/}. The raw data were detrended by using second-order polynomials that were
separately applied to each quarter. As eclipses influence the detrending process, we fitted the polynomial to
only the outside-eclipse part of the light curve (e.g., Hambleton et al. 2013). The flux measurements were converted to
a magnitude scale by requiring a {\it Kepler} magnitude of +14.355 at maximum light. The detrended {\it Kepler} data are
displayed in Figure 1. The shape of the light curve indicates a significant temperature difference between the binary components
and an ellipsoidal variation due to tidal distortions.
To derive the binary parameters of this system, all {\it Kepler} data were analyzed using the Wilson-Devinney binary program
(Wilson \& Devinney 1971, van Hamme \& Wilson 2007; hereafter W-D). This synthesis was performed in a way similar to that
for the pulsating EBs V404 Lyr (Lee et al. 2014) and KIC 6220497 (Lee et al. 2016a). The effective temperature
of the hotter and more massive star was set to be 7590 K from the {\it Kepler} Input Catalogue (KIC; Kepler Mission Team 2009).
The logarithmic bolometric ($X_{1,2}$) and monochromatic ($x_{1,2}$) limb-darkening coefficients were interpolated from
the values of van Hamme (1993). In Table 1, the parentheses signify the adjusted parameters. In this article, the subscripts
1 and 2 refer to the primary and secondary components being eclipsed at orbital phases 0.0 (Min I) and 0.5 (Min II), respectively.
There has been neither a light-curve solution nor spectroscopic mass ratio ($q$) for KIC 11401845 so far. Thus, we conducted
a photometric $q$-search procedure that calculates a series of models with varying $q$ from 0 to 1. For each assumed
mass ratio, the W-D code was applied for various modes but converged satisfactorily only when detached mode 2 were used.
The weighted sum of the squared residuals, $\sum W(O-C)^2$, reached a minimum around $q$ = 0.07, which was adopted as
the initial value and thereafter adjusted to derive the photometric solutions. The result is given as Model 1 in the second
and third columns of Table 1. The synthetic light curve appears as the blue solid curve in Figure 1, and the corresponding
light residuals are plotted as the gray `x' symbols in the figure. In all the procedures, we considered an orbital eccentricity
($e$) and a third light ($l_3$) as additional free parameters. Both searches led to values for the two parameters which were
zero within their errors, which implies that KIC 11401845 has negligible eccentricity. We obtained the errors for the adjustable
parameters by splitting the {\it Kepler} data into 79 segments at intervals of an orbital period and analyzing them separately
(Koo et al. 2014). In Table 1, the error of each parameter is its standard deviation computed from this process.
Absolute parameters for KIC 11401845 can be computed from our light-curve solutions in Table 1 and from the correlations
between spectral type (temperature) and stellar mass. The surface temperature ($T_1$ = 7590 K) of the primary star corresponds
to a normal dwarf one with a spectral type of $\sim$A8V. Assuming that each component has a temperature error of 200 K,
the primary's mass was estimated to be $M_1$ = 1.70$\pm$0.08 $M_\odot$ from Harmanec's (1988) empirical relation. We calculated
the absolute dimensions for each component given in the last part of Table 1. Here, the luminosity ($L$) and bolometric
magnitudes ($M_{\rm bol}$) were derived by using $T_{\rm eff}$$_\odot$ = 5780 K and $M_{\rm bol}$$_\odot$ = +4.73.
The bolometric corrections (BCs) were obtained from the expression between $\log T_{\rm eff}$ and BC given by Torres (2010).
Considering the temperature error of the primary star, we carried out the light-curve synthesis for 7390 K and 7790 K.
The binary parameters from the two limits are in satisfactory agreement with those from $T_1$ = 7590 K, except for
the secondary's temperature ($T_2$). In the three models, the temperature ratios ($T_2$/$T_1$) of both components
are consistent with each other within their errors. We can see that the adopted $T_1$ does not affect the results presented
in this paper.
\section{LIGHT RESIDUALS AND PULSATIONAL CHARACTERISTICS}
From the temperature ($T_1$) and surface gravity ($\log$ $g_1$) given in Table 1, the primary star of KIC 11401845 resides
within the $\delta$ Sct instability strip and, hence, it would be a candidate for $\delta$ Sct pulsations.
For more reliable frequency analysis, we followed the procedure described by Lee (2016). First, we divided the observed
{\it Kepler} data of KIC 11401845 into 79 subsets as before and modeled each light curve with the W-D code by adjusting
only the ephemeris epoch ($T_0$) in Model 1 of Table 1. Second, the corresponding residuals from the whole datasets were
applied to multiple frequency analyses in the range from 0 to the Nyquist limit of $f_{\rm Ny}$ = 24.4 day$^{-1}$ using
the PERIOD04 program (Lenz \& Breger 2005). Because the binary components block each other's lights during eclipses, we used only
the outside-eclipse residuals (orbital phases 0.07$-$0.43 and 0.57$-$0.93). According to the successive prewhitening procedures,
we detected the frequencies with signal to noise amplitude (S/N) ratios larger than 4.0 (Breger et al. 1993). Third, we solved
the pulsation-subtracted data after removing the pulsations from the observed data. As a result, new binary parameters were
obtained, and they were used to reanalyze the 79 light curves in the first stage.
This process was repeated three times until the detected frequencies were unchanged. Final binary parameters are given as
Model 2 in the fourth and fifth columns of Table 1, and the pulsation-subtracted data and model light curve are plotted in
Figure 2. We can see that the physical parameters for Model 2 are consistent with those for Model 1. Figure 3 shows
the light residuals after removal of the binary effects from the observed {\it Kepler} data. We detected a total of
23 frequencies larger than the empirical threshold of S/N = 4.0. The amplitude spectra before and after prewhitening
the first 10 frequencies and then all 23 frequencies are shown in the first to third panels of Figure 4, respectively.
The detailed result from the frequency analysis is listed in Table 2, where the frequencies are given in order of detection
and the noises are calculated in the range of 5 day$^{-1}$ around each frequency. The uncertainties in the table were obtained
according to Kallinger et al. (2008). The synthetic curve computed from the 23-frequency fit is displayed in the lower panel
of Figure 3.
As listed in Table 2, four in the low-frequency region (0.4$-$3.8 day$^{-1}$) and 19 in the high-frequency region
(13.7$-$23.8 day$^{-1}$) were derived from the multiple frequency analyses of the outside-eclipse light residuals.
Within the frequency resolution of 0.00545 day$^{-1}$ (Loumos \& Deeming 1978), we searched the frequencies for possible
harmonic and combination terms. The result is given in the last column of Table 2. We think that the $f_{11}$ to $f_{23}$
frequencies mainly arise from combination frequencies, some of which can be caused by imperfect removal of the binary effects
in the observed data. On the other hand, the high-frequency signals close to the Nyquist limit can be reflections of
real frequencies (2$f_{\rm Ny}-f_i$) higher than $f_{\rm Ny}$ (Murphy et al. 2013; Lee et al. 2016b). High-cadence photometry
is needed to separate the Nyquist aliases from the detected frequencies of KIC 11401845.
\section{ECLIPSE TIMING VARIATION AND ITS IMPLICATION}
For an orbital period study of KIC 11401845, we determined 150 minimum times and their uncertainties from the observations with
the method of Kwee \& van Woerden (1956). These minima are listed in columns (1)--(5) of Table 3, where we present the cycle
numbers and $O$--$C_1$ residuals calculated with the light elements ($T_0$ and $P$) for Model 2 in Table 1. The resultant eclipse
timing diagram is displayed at the top panel of Figure 5. As shown in the figure, the timing residuals from the primary
(filled circle) and secondary (open circle) eclipses do not agree with each other, which could be caused by the light variations
due to the multiperiodic pulsations of the primary star. Thus, we recalculated the minimum times from the eclipse light curve
after subtracting the 23 frequencies detected in Section 3 from the observed {\it Kepler} data. The results are given in columns
(6)--(8) of Table 3 and are illustrated in the middle panel of Figure 5.
As displayed in Figure 5, the large discrepancy between Min I and Min II is shown more clearly in the pulsation-subtracted data.
This discrepancy can result from the time difference between the primary and secondary eclipses due to LTT in a binary with
unequal masses (Barlow et al. 2012; Parsons et al. 2014). The R{\o}mer delay is given by Kaplan (2010), as follows:
\begin{equation}
\Delta t_{\rm LTT} = {{P K_2} \over {\pi c}} (1-q),
\end{equation}
where $P$ is the orbital period, $K_2$ is the radial velocity (RV) semi-amplitude of the secondary star, and c is the speed of
light. From the Model 2 parameters in Table 1, we derived the velocities ($K_1$ = 13 km s$^{-1}$ and $K_2$ = 187 km s$^{-1}$)
of the binary components (Hilditch 2001). The time delay of $\Delta t_{\rm LTT}$ = 34$\pm$1 s was obtained by applying the values
of $K_2$, $P$, and $q$ to equation (1).
In order to examine this possibility, we computed the secondary eclipses related to one half period after the primary eclipses
in the pulsation-subtracted data and then plotted the difference ($\Delta t_{\rm SE}$) between the measured and computed
secondary times in the bottom panel of Figure 5. As shown in the figure, the mean value is offset from zero and gives
a time delay of $\Delta t_{\rm SE}$ = 56$\pm$17 s in the secondary eclipse. This value is calculated to be $\Delta t_{\rm SE}$
= 37$\pm$27 s in the observed {\it Kepler} data including pulsations. Within their errors, the time delays of $\Delta t_{\rm SE}$
are in satisfactory accord with the predicted delay of $\Delta t_{\rm LTT}$. On the other hand, if KIC 11401845 is in an eccentric orbit,
$\Delta t_{\rm SE}$ might be affected by the time shift of $\Delta t_{\rm e}$ in the secondary eclipse due to a non-zero eccentricity:
\begin{eqnarray}
\Delta t_{\rm e} \simeq {{2Pe} \over {\pi}} \cos {\omega}, \\
\Delta t_{\rm SE} \simeq \Delta t_{\rm e} + \Delta t_{\rm LTT},
\end{eqnarray}
where $e$ and $\omega$ are the eccentricity and the argument of periastron, respectively. Using the equations (2) and (3),
$e \cos {\omega} \simeq$ 0.00003 for the observed data and $e \cos {\omega} \simeq$ 0.00019 for the pulsation-subtracted data.
\section{DISCUSSION AND CONCLUSIONS}
We have studied the physical properties of KIC 11401845, based on the {\it Kepler} data made during Quarters 10 and 12.
The light curve of this system displays multiperiodic pulsations, superimposed on binary effects. To examine whether the binary
parameters are affected by the pulsations, we analyzed individually the observed and pulsation-subtracted {\it Kepler} data
with the W-D code. As listed in Table 1, the photometric solutions for the two datasets are in good agreement with each other,
which means that the pulsations cause little impact on the light-curve parameters. Our light-curve synthesis shows that
KIC 11401845 is a short-period detached EB with a very small mass ratio of about 0.07. The primary and secondary components
fill $F_1$ = 45 \% and $F_2$ = 99 \% of their limiting lobe, respectively, where the filling factor
$F_{1,2} = \Omega_{\rm in} / \Omega_{1,2}$. With its small $q$ and short $P$, our program target closely resembles
the two {\it Kepler} pulsating EBs KIC 10661783 (Lehmann et al. 2013) and KIC 8262223 (Guo et al. 2016), which are detached
binaries with characteristics of the R CMa-type stars (Budding \& Butland 2011; Lee et al. 2016b) among Algols. A comparison of
the KIC 11401845 parameters with the mass-radius, mass-luminosity, and Hertzsprung-Russell (HR) diagrams (Ibano\v{g}lu et al. 2006)
shows that the primary component resides within the main-sequence band. On the contrary, the low mass secondary is highly evolved
and its radius and luminosity are remarkably oversized and overluminous compared to main-sequence stars of the same mass. These
suggest that the initial more massive star becomes the present secondary by losing most of its own mass via mass transfer to
the companion (present primary) and stellar wind (Hong et al. 2015; Guo et al. 2016).
In order to detect the pulsation frequencies of KIC 11401845, multiple frequency analyses were applied to the out-of-eclipse
residuals after removing the binarity effects from the observed {\it Kepler} data. We found 23 frequencies with S/N ratios
larger than 4.0 in two regions: 0.4$-$3.8 day$^{-1}$ and 13.7$-$23.8 day$^{-1}$. Among these, four ($f_4$, $f_6$, $f_9$, $f_{15}$)
in the low frequency region are frequencies at exact multiples of the orbital frequency, $f_{\rm orb}$ = 0.46267 day$^{-1}$.
The orbital harmonics can be attributed to tidally excited modes, which occur when the orbital frequency is close to
a stellar eigenfrequency in a binary star (Welsh et al. 2011; Hambleton et al. 2013). On the contrary, the other frequencies
are strongly reminiscent of the $p$-mode pulsations known in EBs (e.g., Southworth et al. 2011; Lee et al. 2016b). The ratios of
the pulsational to orbital periods in the high-frequency region were calculated to be $P_{\rm pul}/P_{\rm orb}$ = 0.020$-$0.034,
which is within the upper limit of 0.09 for $\delta$ Sct stars in binaries (Zhang et al. 2013). We calculated the pulsation
constants by applying the Model 2 parameters in Table 1 to the equation of
$\log Q_i = -\log f_i + 0.5 \log g + 0.1M_{\rm bol} + \log T_{\rm eff} - 6.456$ (Petersen \& J\o rgensen 1972). The result is
listed in the sixth column of Table 2. The $Q$ values of 0.018$-$0.031 d correspond to $p$ modes of $\delta$ Sct type. The period
ratios, the pulsation constants, and the position on the HR diagram reveal that the primary component is a $\delta$ Sct variable.
The $\delta$ Sct pulsations of KIC 11401845 match well the correlations between the pulsation periods and other parameters
(binary periods, surface gravities of pulsators, and gravitational forces from companions) updated by Liakos \& Niarchos (2016b).
We measured the minimum epochs of the primary and secondary eclipses from the observed and pulsation-subtracted data.
Inspecting them in detail, we found delays of about 37 s and 56 s in the arrival times of the secondary eclipses relative to
the primary eclipses in the same order. The values are consistent with the expected time delay of $\Delta t_{\rm LTT}$ = 34 s
across the binary orbit. This indicates that the LTT delay is the main cause of the time discrepancy between both eclipses,
which is the first detection of this effect in EBs consisting of non-compact components. One might imagine that the time delay
of the secondary eclipse could be apportioned between the LTT delay and a non-zero eccentricity. The result presented in
this paper limits the eccentricity of KIC 11401845 to $e < 0.0002$. When the high-resolution spectra are made, they will help to
determine the RV semi-amplitudes ($K_{1,2}$) and mass ratio ($q$) of the binary star and hence to derive its small eccentricity
($e$). Because the system is a faint pulsating EB with a short orbital period, 8$-$10 m class telescopes are required to
measure its accurate double-lined RVs.
\acknowledgments{ }
This paper includes data collected by the {\it Kepler} mission. {\it Kepler} was selected as the 10th mission of the Discovery Program.
Funding for the {\it Kepler} mission is provided by the NASA Science Mission directorate. We have used the Simbad database maintained
at CDS, Strasbourg, France. This work was supported by the KASI grant 2016-1-832-01. Work by K. Hong was supported by Basic Science
Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number: NRF-2016R1A6A3A01007139).
\newpage
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\section{Introduction}
\label{sec:introduction}
Over the past decade, considerable progress in the field of ultracold
gases has given access to experimentally simulating prototypical
many-body models, e.g., known from condensed-matter physics, with a
high degree of dynamic control
\cite{bloch2008,lewenstein2007,dutta2015}. Accompanied by theoretical
work, this opened up the possibility to study their emergent phases
and nonequilibrium dynamics induced by fast changes of the model
parameters \cite{polkovnikov2011,eisert2015}. As \emph{the}
paradigmatic model for strongly correlated fermions on a lattice, we
consider the Fermi-Hubbard model
\cite{gutzwiller1963,hubbard1963,kanamori1963}. Not only the bosonic
variant \cite{fisher1989,jaksch1998,greiner2002,krutitsky2015} but
also the Fermi-Hubbard model has been realized by recent
optical-lattice experiments
\cite{koehl2005,giorgini2008,esslinger2010}. This promises to provide
insight into fundamental questions related to the Mott transition, to
collective order or even to non-Fermi-liquid physics.
Using standard notations, the Fermi-Hubbard Hamiltonian is given by
\begin{equation}
\label{eq:Hamiltonian}
\op H(t) = \sum_{\langle ij \rangle,\sigma}T_{ij}(t)\cc{i\sigma}\ac{j\sigma} +
U(t) \sum_{i} \n{i\ua}\n{i\da} \,,
\end{equation}
where a fermion at site $i$ and with spin projection $\sigma=\uparrow,
\downarrow$ is annihilated (created) by $c^{(\dagger)}_{i\sigma}$, and
where $\n{i\sigma} = \cc{i\sigma}\ac{i\sigma}$ is the density
operator. Tunneling between neighboring sites $\langle ij \rangle$ is
described by the hopping amplitude $T_{ij}(t)$. On the same site,
fermions are subjected to the repulsive Hubbard interaction $U(t)$. A
nonequilibrium state and nontrivial real-time dynamics of
observables can be initiated by fast changes of the time-dependent
model parameters $T_{ij}(t)$ and $U(t)$.
Despite its conceptual simplicity, the Hubbard model has challenged
theoreticians for more than five decades, and has stimulated the
development of a large number of different methods. Many approaches
which are based on the language of Green's functions \cite{KMS}
have successfully been extended to describe real-time phenomena
\cite{thygesen2007,schmidt2002,freericks2006,aoki2013,munoz2013,jung2012,knap2011,balzer2013,lipavsky1986},
using the Keldysh formalism \cite{keldysh1965}. Of particular
interest in this context are quantum-cluster theories
\cite{maier2005}, which are formulated in the thermodynamic limit, but
nevertheless account for nonlocal correlations on a length scale
defined by the linear extension of a reference cluster, which can have
a profound influence on the nonequilibrium dynamics
\cite{tsuji2013,eckstein2014}. A simple but conceptually appealing
quantum-cluster approach is the cluster-perturbation theory (CPT)
\cite{gros1993,senechal2000,potthoff2011c,gramsch2015}. Within the
CPT, the infinite lattice is tiled into small clusters, which can be
solved exactly by numerical means. This solution is then extended to
the full lattice by infinite-order perturbation theory with respect to
the inter-cluster hopping but neglecting vertex corrections
\cite{potthoff2011c}. This amounts to approximate the self-energy of
the full model by the self-energy of the reference system of
disconnected clusters. A major drawback of the CPT consists in the
fact that, even for a given geometrical tiling of the lattice, the
partitioning of the hopping part and hence the choice of the reference
system, i.e., the starting point of the perturbative expansion is not
unique. However, the nonuniqueness of the CPT construction can be
turned into an advantage, if one can find a variational prescription
for finding the \emph{optimal} starting point for the
cluster-perturbation theory. In fact, this is achieved with the
variational-cluster approach (VCA) \cite{dahnken2004,potthoff2014}
which is best understood in the general framework of self-energy
functional theory \cite{potthoff2003,hofmann2013}.
The main purpose of the present paper is to discuss different
practical issues related to the application of the {\em
nonequilibrium} VCA. While the main theoretical concept is
essentially the same as for the equilibrium VCA, we demonstrate that
the numerical implementation of the nonequilibrium variant of the
approach is by far more complex and requires new techniques. As a
proof of principle, we present a first numerical implementation of the
nonequilibrium VCA, based on simple two-site reference clusters, and
study two types of parameter quenches (or fast ramps) in the
one-dimensional Fermi-Hubbard model with alternating (dimerized)
hopping amplitudes which have attracted experimental attention
recently \cite{greif2015,atala2013}.
The paper is organized as follows: In
Section~\ref{sec:noneq-greens-funct} we briefly review the concept of
nonequilibrium Green's functions used for the nonequilibrium
extension of the self-energy functional theory, as described in
Sec.~\ref{sec:self-energy-funct}. We derive the central Euler
equation in Sec.~\ref{sec:euler} and discuss its numerical
implementation in Sec.~\ref{sec:numer-impl}. Results for the
dimerized Hubbard model are presented in Sec.~\ref{sec:nevca}. A
summary is given in Sec.~\ref{sec:summary-outlook}.
\section{Nonequilibrium Green's functions}
\label{sec:noneq-greens-funct}
The self-energy functional theory relies on functionals that are
formally defined by means of all-order perturbation theory and thus on
the concept of (nonequilibrium) Green's functions
\cite{KMS,keldysh1965}.
Throughout this paper, we make use of the general theory provided by
Refs.~\cite{DLRK}
but in first place closely follow the formal setup by Wagner
\cite{wagner1991}.
For a general fermionic lattice model with Hamiltonian
\begin{equation}
\label{eq:generalHamiltonian}
\op H_{\mat T,\mat U} (t) = \sum_{\alpha\beta}T_{\alpha\beta}(t)\cc{\alpha}\ac{\beta} +
\frac{1}{2}\sum_{\alpha\beta\gamma\delta}
U_{\alpha\beta\delta\gamma}(t)\cc{\alpha}\cc{\beta}\ac{\gamma}\ac{\delta} \, ,
\end{equation}
which is specified by the parameters $\mat T$ and $\mat U$, and which
initially (at time $t=t_{0}$) is prepared in a thermal state with
inverse temperature $\beta$ and chemical potential $\mu$, the elements
of the contour-ordered Green's function $\mat G_{\mat T,\mat U}$ are
defined as
\begin{equation}
\label{eq:DefNEGF}
i G_{\mat T,\mat U}(1,2) = \ev<\ord_{\mathcal C} \ac{\gop H}(1)\cc{\gop H}(2) >_{\mat T, \mat U} \, .
\end{equation}
Here, greek indices combine all one-particle orbitals, and
$c^{(\dagger)}_{\gop H}(i)$ is the annihilator (creator) in the
Heisenberg picture with respect to $\gop H(z) = \op H(z) - \mu N$,
where $N$ is the total particle-number operator. We use the
short-hand notation $i \equiv (\alpha_i,z_i)$, where $z_i$ marks an
arbitrary point on the Keldysh-Matsubara contour $\mathcal C$ (see
Fig.~\ref{fig:contour}). $\ord_{\mathcal C}$ is the contour time ordering.
Further, $\ev<\cdots> = \tr(\rho\,\cdots)$ denotes the expectation
value in the initial state, where $\rho$ is density operator, $\rho =
\exp(-\beta \gop H_{\rm ini})/Z$, and $Z = \tr \exp(-\beta \gop H_{\rm
ini})$ with $\gop H_{\rm ini} = \op H(t_0) - \mu \op N$ is the
partition function.
\begin{figure}[t]
\centering
\includegraphics[width=.58\textwidth]{fig/KMcontour}\hspace{.04\textwidth}%
\begin{minipage}[b]{.38\textwidth}
\caption{The three-branched Keldysh-Matsubara contour $\mathcal C$ in
the complex time plane, extending up to time $t_{\rm max}$; $\mathcal
C_{K_\pm}$ denotes the upper/lower branch and $\mathcal C_M$ the
Matsubara branch.}
\label{fig:contour}
\end{minipage}
\end{figure}
From the Heisenberg equation of motion for the annihilator, we find
the equation of motion
\begin{equation}
\left( i \partial_z - (\mat T(z) - \mu) \right) \mat
G_{\mat T,\mat U}(z,z') = \mat 1 \delta_{\mathcal C}(z,z') + \left(
\mat \Sigma_{\mat T,\mat U} \circ \mat G_{\mat T,\mat U}
\right)(z,z') \, ,
\label{eq:EOM-NEGF}
\end{equation}
where $\delta_{\mathcal C}$ is the contour delta-function and
$\mat\Sigma_{\mat T,\mat U}$ is the self-energy. The circle $\circ$ on
the right-hand side stands for an integration along $\mathcal C$ and an
implicit summation over all orbital indices. By setting $\mat U=0$ in
Eq.~(\ref{eq:EOM-NEGF}), we obtain the inverse of the ``free'' Green's
function $\mat G_{\mat T,0}$ as
\begin{equation}
\label{eq:inverseFreeNEGF}
G^{-1}_{\mat T,0;\alpha\alpha'}(z,z') = \delta_{\mathcal C}(z,z') \left(
\delta_{\alpha\alpha'} i \partial_{z'} -
\left(T_{\alpha\alpha'}(z') - \mu
\delta_{\alpha\alpha'}\right) \right) \, .
\end{equation}
Using this, we can rewrite Eq.~(\ref{eq:EOM-NEGF}) as Dyson's
equation:
\begin{equation}
\label{eq:DysonEqnShort}
\mat G_{\mat T,\mat U} = \mat G_{\mat T,0} + \mat G_{\mat
T,0} \circ \mat \Sigma_{\mat T,\mat U} \circ \mat G_{\mat T,\mat
U} \, .
\end{equation}
\section{Self-energy functional theory}
\label{sec:self-energy-funct}
The (nonequilibrium) self-energy functional theory is based on the
following functional of the nonequilibrium self-energy (see
Refs.~\cite{potthoff2006b,hofmann2013} for details):
\begin{equation}
\label{eq:NESEfcl}
\fcl\Omega_{\mat T,\mat U}[\mat\Sigma]
=
\frac{1}{\beta} \Tr\ln
\left( \mat G_{\mat T,0}^{-1} - \mat\Sigma \right)^{-1}
+
\fcl
F_{\mat U} [\mat \Sigma] \, .
\end{equation}
Here, the first term on the right-hand side is a simple and explicit
functional of $\mat \Sigma$ and depends on the system's one-particle
parameters $\mat T$. The second term is the Legendre transform of the
Luttinger-Ward functional $\fcl\Phi_{\mat U}[\mat G]$
\cite{luttinger1960}. The latter can be defined by means of all-order
perturbation theory or, nonperturbatively, within a path-integral
formalism analogous to the equilibrium case \cite{potthoff2006b}.
Note, that functionals are indicated by a hat, and that the trace is
defined as $\Tr \mat A = \sum_\alpha \int_{\mathcal C} d z\,
A_{\alpha\alpha}(z,z^+)$, where $z^+$ is infinitesimally later than
$z$ on $\mathcal C$.
There are two important properties of the self-energy functional
(\ref{eq:NESEfcl}): First, if $\fcl\Omega_{\mat T,\mat U}[\mat\Sigma]$
is evaluated at the physical self-energy $\mat \Sigma = \mat
\Sigma_{\mat T, \mat U}$, one obtains the physical grand potential of
the system in its initial thermal state. All contributions from the
upper and the lower Keldysh branch cancel in this case. Second, the
self-energy functional is stationary at the physical self-energy,
i.e.,
\begin{equation}
\left.\frac{\delta \fcl{\Omega}_{\mat T, \mat U}[\mat \Sigma]}
{\delta \mat \Sigma}\right|_{\mat \Sigma = \mat \Sigma_{\mat T, \mat
U}}
=
0 \, .
\label{eq:StatPointNESEfcl}
\end{equation}
In fact, using the well-known properties of the Luttinger-Ward
functional and its Legendre transform, it is easy to see that Eq.\
(\ref{eq:StatPointNESEfcl}) is equivalent with Dyson's
equation~(\ref{eq:DysonEqnShort}). Eq.\ (\ref{eq:StatPointNESEfcl})
constitutes a general variational principle by which one could
determine the self-energy (as well as the grand potential) of a system
of correlated fermions. However, the explicit form of the functional
$\fcl F_{\mat U} [\mat \Sigma]$ is generally unknown.
An important observation is that the Luttinger-Ward functional, and
thus also $\fcl F_{\mat U} [\mat \Sigma]$ is universal, i.e., it
depends (besides the argument) on the interaction parameters $\mat U$
only, as made explicit by the subscript. Hence, the functional form
of $\fcl F_{\mat U} [\mat \Sigma]$ remains unchanged for any reference
system with the same interaction $\mat U$ as the original system but
with a different set of one-particle parameters $\mat\lambda'$, i.e.,
for any reference system with a Hamiltonian of the form $\op H' \equiv
\op H_{\mat \lambda',\mat U}$.
Let us choose a reference system in a particular subspace of
one-particle parameters $\mat \lambda'$, which has a sufficiently
simple structure so that its one-particle Green's function, its
self-energy, and its grand potential of the initial thermal state at
$\beta$ and $\mu$ can be computed exactly for any $\mat \lambda'$ of
the subspace. (Here and in the following, primed quantities will
refer to the reference system.) We can write down the self-energy
functional of the reference system,
\begin{equation}
\label{eq:NESEfclRef}
\fcl\Omega_{\mat \lambda',\mat U}[\mat\Sigma]
=
\frac{1}{\beta}
\Tr\ln \left( \mat G_{\mat \lambda',0}^{-1} - \mat\Sigma
\right)^{-1}
+
\fcl F_{\mat U} [\mat \Sigma] \, ,
\end{equation}
and, due to the mentioned universality, eliminate $\fcl F_{\mat
U}[\mat\Sigma]$ by combining Eqs.~(\ref{eq:NESEfcl}) and
(\ref{eq:NESEfclRef}):
\begin{equation}
\label{eq:NESEfclDiff}
\fcl\Omega_{\mat T,\mat U}[\mat\Sigma]
=
\fcl\Omega_{\mat
\lambda',\mat U}[\mat\Sigma] + \frac{1}{\beta} \Tr\ln
\left( \mat G_{\mat T,0}^{-1} - \mat\Sigma \right)^{-1}
-
\frac{1}{\beta} \Tr\ln \left( \mat G_{\mat \lambda',0}^{-1} -
\mat\Sigma \right)^{-1} \,.
\end{equation}
This expression for the self-energy functional, which is still exact,
can be evaluated in practice for a certain subclass of trial
self-energies, namely for the physical self-energies of the reference
system. Inserting $\mat \Sigma = \mat \Sigma_{\mat \lambda', \mat
U}$, we have
\begin{equation}
\label{eq:SFTfcl}
\fcl\Omega_{\mat T,\mat U}[\mat\Sigma_{\mat \lambda',\mat U}]
=
\Omega_{\mat \lambda',\mat U} + \frac{1}{\beta} \Tr\ln \left( \mat
G_{\mat T,0}^{-1} - \mat\Sigma_{\mat \lambda',\mat U} \right)^{-1}
-
\frac{1}{\beta} \Tr\ln \left( \mat G_{\mat \lambda',\mat U}
\right) \,,
\end{equation}
where we used $\fcl\Omega_{\mat \lambda', \mat U}[\mat \Sigma_{\mat
\lambda',\mat U}]= \Omega_{\mat \lambda',\mat U}$ and Dyson's
equation for the reference system, see Eq.\ (\ref{eq:DysonEqnShort}).
Using the stationarity principle Eq.\ (\ref{eq:StatPointNESEfcl}), we
can optimize the self-energy over the restricted subspace of trial
self-energies spanned by the subspace of variational parameters $\mat
\lambda'$ via
\begin{equation}
\label{eq:EulerEqn}
\left. \frac{\delta \fcl\Omega_{\mat T,\mat U}[\mat\Sigma_{\mat \lambda',\mat U}]}{\delta \mat \lambda'(z)} \right|_{\mat\lambda'(z) =
\mat\lambda'_{\rm opt}(z)} = 0 \, .
\end{equation}
This Euler equation yields the optimal parameters $\mat \lambda'_{\rm
opt}(z)$, the optimal initial-state grand potential
$\fcl\Omega_{\mat T, \mat U}[\mat \Sigma_{\mat \lambda'_{\rm opt},\mat
U}]$, the optimal self-energy $\mat \Sigma_{\mat \lambda'_{\rm opt},
\mat U}$, and the related one-particle Green's function
\begin{equation}
\label{eq:gsft}
\mat G^{\rm SFT}
\equiv
(\mat G_{\mat T,0}^{-1} - \mat
\Sigma_{\mat \lambda'_{\rm opt},\mat U})^{-1} \,.
\end{equation}
The type of approximation is specified by the choice of the reference
system. Typically, this is defined by cutting the lattice, which
underlies the original model, into disconnected clusters consisting of
a small number of sites $L_c$ each. For Hubbard-type systems with
local interactions, this generates simple reference systems with a
small Hilbert space, which can be solved, e.g., by exact
diagonalization techniques. The resulting approximation is called the
variational-cluster approximation (VCA)
\cite{dahnken2004,potthoff2014,hofmann2013}. A concrete example will
be given in Sec.~\ref{sec:nevca}. To enlarge the number of variational
degrees of freedom locally, a number $L_{b}$ of uncorrelated ``bath
sites'' can be coupled via a hybridization term to each of the
correlated ones. One may formally re-derive the (nonequilibrium)
dynamical mean-field theory (DMFT)
\cite{georges1996,freericks2006,schmidt2002} by choosing a reference
system of decoupled correlated sites ($L_c=1$) hybridizing with a
continuum of bath sites ($L_b = \infty$). The reference system is
given as a set of decoupled single-impurity Anderson models in this
case. The VCA, the DMFT, and other approximations defined in this way
are nonperturbative by construction and can be improved
systematically by enlarging the space of variational parameters.
Self-energy functional theory in principle provides approximations
which respect the macroscopic conservation laws resulting from the
invariance of the original system under continuous U(1) and SU(2)
symmetry groups. This is similar to the perturbatively defined
``conserving'' approaches in the sense of Baym and Kadanoff
\cite{baym1961,baym1962} which are obtained from the standard recipe
by deriving the self-energy from a truncated Luttinger-Ward
functional. The conserving nature of the nonperturbative
approximations generated within the SFT framework is ensured by a
formal proof \cite{hofmann2013}. This makes use of the fact that
variations of the one-particle parameters of the reference system
comprise gauge transformations of the form $\mat \lambda' \mapsto
\mat{\bar \lambda}'$ with
\begin{equation}
\mat \varepsilon'(z) \mapsto \mat{\bar \varepsilon}'(z) = \mat
\varepsilon'(z) - \partial_z\mat
\chi(z)\,,
\qquad
\mat T'(z) \mapsto \mat{\bar T}'(z) = e^{i\mat\chi(z)}\mat
T'(z)e^{-i\mat\chi(z)} \, ,
\label{eq:gaugetrans1partpar}
\end{equation}
where $\mat\varepsilon'$ denotes the (spatially) diagonal part of
$\mat\lambda'$ and $\mat T'$ its off-diagonal part, and where
$\chi_{ij,\sigma\sigma'}(z)$ is a spatially diagonal contour function.
One shows that stationarity with respect to those gauge
transformations implies that conservation laws, as expressed by
continuity equations, are proliferated from the reference system,
where they hold exactly, to the original system where they are
expressed in terms of the approximate quantities. In fact, this is
unproblematic in practice for the case of the total particle number
and of the $z$-component of the total spin. As discussed in Ref.\
\cite{hofmann2013}, however, respecting total-energy conservation
requires a continuum of variational degrees of freedom. In simple
approximations, such as the VCA, one therefore has to tolerate a
violation of total-energy conservation or use this as a measure for
the quality of the approximation.
\section{Euler equation}
\label{sec:euler}
To proceed to a practical computation, one first has to note that the
trial self-energy depends on variational parameters $\mat \lambda'(z)$
which can be different on the two Keldysh branches and we therefore
distinguish $\mat\lambda'_{+}(t)$ and $\mat\lambda'_{-}(t)$ on $\mathcal
C_{K_+}$ and $\mathcal C_{K_-}$ (cf. Fig.~\ref{fig:contour}). In the end,
we are only interested in physical solutions of the Euler equation
(\ref{eq:EulerEqn}), namely solutions which satisfy
$\mat\lambda'_{+}(t)=\mat\lambda'_{-}(t)$ and hence specify a
\emph{physical} Hamiltonian. To make this more explicit, we perform a
simple rotation in parameter space and define $\mat\lambda'_{\rm
phys}(t) = \frac{1}{2} (\mat\lambda'_{+}(t) + \mat\lambda'_{-}(t))$
and $\mat\lambda'_{\rm trans}(t) = \frac{1}{2} (\mat\lambda'_{+}(t) -
\mat\lambda'_{-}(t))$. The physical parameter manifold is given by
$\mat\lambda'_{\rm trans}(t)=0$. Now, it is easy to see that the
self-energy functional is trivially always stationary with respect to
physical variations when evaluated at a physical parameter set:
\begin{equation}
\left. \frac{\delta \fcl\Omega_{\mat T,\mat U}[\mat\Sigma_{\mat \lambda', \mat U}]}{\delta \mat \lambda_{\rm phys}'(t)} \right|_{\mat\lambda'_{\rm trans}(t) = 0}
= 0 \: .
\label{eq:delphys}
\end{equation}
Hence, stationarity with respect to physical variations simply places
the solution {\em on} the physical manifold. To fix the solution {\em
within} the physical manifold a second condition is needed, namely
stationarity with respect to the transverse variations:
\begin{equation}
\label{eq:nontrivVar}
\left. \frac{\delta \fcl\Omega_{\mat T,\mat U}[\mat\Sigma_{\mat \lambda', \mat U}]}{\delta \mat \lambda_{\rm trans}'(t)} \right|_{\mat\lambda'_{\rm trans}(t) = 0}
= 0 \: .
\end{equation}
This is in fact the central equation of the nonequilibrium SFT.
Hence, variations necessarily go off the physical manifold and thus,
opposed to the standard strategy in the equilibrium case (see e.g.\
Ref.\ \cite{balzer2010}), it is convenient to carry out the functional
derivative analytically before solving the resulting equation
numerically on the physical parameter manifold.
To compute the functional derivative in Eq.~(\ref{eq:EulerEqn}) or
(\ref{eq:nontrivVar}), we use the chain rule:
\begin{equation}
\label{eq:EulerEqnChainRule}
\frac{\delta \fcl\Omega_{\mat T,\mat U}[\mat\Sigma_{\mat\lambda',\mat U}]}{\delta \lambda'_{\alpha_{1}\alpha_{2}}(z)} =
\Tr \left( \frac{\delta \fcl\Omega_{\mat T,\mat U}[\mat\Sigma_{\mat\lambda',\mat U}]}{\delta\mat \Sigma_{\mat
\lambda',\mat U}} \circ \frac{\delta \mat \Sigma_{\mat
\lambda',\mat U}}{\delta \lambda'_{\alpha_{1}\alpha_{2}}(z)} \right) \,.
\end{equation}
The first factor is given by $ \beta \delta \fcl\Omega_{\mat T,\mat
U}[\mat \Sigma_{\mat\lambda',\mat U}]/\delta \mat
\Sigma_{\mat\lambda',\mat U} = \left( \mat G_{\mat T,0}^{-1} -
\mat\Sigma_{\mat\lambda',\mat U} \right)^{-1} - \mat
G_{\mat\lambda',\mat U}$ and can be rewritten in a more convenient
way. With the inter-cluster hopping $\mat V(z) \equiv \mat T(z) -
\mat \lambda'(z)$ we immediately have $G^{-1}_{\mat T,0}(1,2) =
G^{-1}_{\mat \lambda',0}(1,2) - \delta_{\mathcal C}(z_1,z_2)
V_{\alpha_1\alpha_2}(z_{2})$, see Eq.\ (\ref{eq:inverseFreeNEGF}).
Plugging this into the definition of the SFT Green's function, Eq.\
(\ref{eq:gsft}), and using Dyson's equation for the reference system,
we get
\begin{equation}
\label{eq:CPTequation}
\mat G^{\rm SFT} = \mat G_{\mat\lambda',\mat U} + \mat G_{\mat\lambda',\mat U} \mat V \circ \mat G^{\rm SFT} \, .
\end{equation}
This is actually the central equation of nonequilibrium
cluster-perturbation theory \cite{potthoff2011c}. With the associated
Lippmann-Schwinger equation $\mat G^{\rm SFT} = \mat
G_{\mat\lambda',\mat U} + \mat G_{\mat\lambda',\mat U} \circ \mat
Y_{\mat\lambda',\mat T,\mat U} \circ \mat G_{\mat\lambda',\mat U}$,
where $\mat Y_{\mat\lambda',\mat T,\mat U}(z_{1},z_{2}) = \mat
V(z_{1}) \delta_{\mathcal C}(z_{1},z_{2}) + \mat V(z_{1}) \mat G^{\rm
SFT}(z_{1},z_{2}) \mat V(z_{2})$ is the corresponding $T$-matrix, we
eventually obtain
\begin{equation}
\label{eq:first}
\frac{\delta \fcl \Omega_{\mat T,\mat U}[\mat\Sigma_{\mat \lambda',\mat U}]}{\delta\mat \Sigma_{\mat \lambda',\mat U}} = \frac{1}{\beta} \mat G_{\mat\lambda',\mat U} \circ \mat
Y_{\mat\lambda',\mat T,\mat U} \circ \mat G_{\mat\lambda',\mat U} \:
\end{equation}
for the first factor in Eq.\ (\ref{eq:EulerEqnChainRule}). To
evaluate the second factor, we write $\mat\Sigma_{\mat\lambda',\mat U}
= \mat G_{\mat\lambda',0}^{-1} - \mat G_{\mat\lambda',\mat U}^{-1}$.
Apart from the $\mat \lambda'$-dependence of the inverse free Green's
function, see Eq.~(\ref{eq:inverseFreeNEGF}), we have to consider the
derivative of the inverse of the full Green's function: From the
identity $\delta(\mat G_{\mat\lambda',\mat U}\circ\mat
G_{\mat\lambda',\mat U}^{-1})/\delta\mat\lambda' = 0$ we deduce
$\delta \mat G_{\mat\lambda',\mat U}^{-1}/\delta\mat\lambda' = - \mat
G_{\mat\lambda',\mat U}^{-1}\circ\delta \mat G_{\mat\lambda',\mat
U}/\delta\mat\lambda'\circ\mat G_{\mat\lambda',\mat U}^{-1}$.
Finally, making use of the properties of the time ordering operator
and the exponential function, a straightforward calculation yields:
\begin{equation}
\label{eq:dGdlambda}
\frac{\delta G_{\mat\lambda',\mat U}(3,4)}{\delta
\lambda'_{\alpha_1\alpha_2}(z_1)} = G_{\mat\lambda',\mat U}(3,4)
\left. G_{\mat\lambda',\mat U}(2,1^+) \right|_{z_2=z_1} -
\left. G^{(2)}_{\mat\lambda',\mat U}(3,2,1^+,4) \right|_{z_2=z_1} \,,
\end{equation}
where $\mat G_{\mat\lambda',\mat U}^{(2)}$ is the two-particle Green's
function of the reference system. Collecting the results we find:
\begin{equation}
\label{eq:K0}
-\beta \frac{\delta \fcl \Omega_{\mat T,\mat U}[\mat\Sigma_{\mat
\lambda',\mat U}]}{\delta \lambda'_{\alpha_1\alpha_2}(z_1)} =
\dint d3d4 \, Y_{\mat \lambda',\mat T,\mat
U}(4,3) \left. L_{\mat\lambda',\mat U} (3,2,1^+,4)
\right|_{z_2=z_1} =: K^{(0)}[\mat\lambda']_{\alpha_2\alpha_1}(z_1) \,,
\end{equation}
where $L_{\mat\lambda',\mat U}(1,2,3,4) = G_{\mat\lambda',\mat
U}(2,4)G_{\mat\lambda',\mat U}(1,3) - G_{\mat\lambda',\mat
U}(1,4)G_{\mat\lambda',\mat U}(2,3) + G_{\mat\lambda',\mat
U}^{(2)}(1,2,3,4)$ is the two-particle (four-point) vertex function
with external legs. Therewith, we have the Euler equation of the
nonequilibrium SFT:
\begin{equation}
\label{eq:sfteuler}
\left.\mat K^{(0)}[\mat\lambda'](t)\right|_{\mat\lambda'=\mat\lambda'_{\rm opt}} = 0\,,
\end{equation}
which will be the starting point for the numerical determination of
the optimal parameters. Note that those parameters depend on the real
time variable $t$ only rather than on the contour variable $z$,
because after the transverse variations have been carried out
analytically we can restrict the search for optimal parameters to the
physical manifold, see Eq.\ (\ref{eq:nontrivVar}).
\section{Numerical implementation}
\label{sec:numer-impl}
\begin{figure}[t]
\includegraphics[width=.48\textwidth]{fig/2sVCA_Jacdt}\hspace{.04\textwidth}%
\includegraphics[width=.48\textwidth]{fig/2sVCA_param}
\caption{ Test of the numerical implementation for the
time-propapagation of a system in equilibrium. The system is a
one-dimensional chain of $100$ sites at $\beta = 6$ and $U=4$. As
a reference system a two-site cluster is used. \emph{Left}:
$\Delta t$ dependence of the Jacobian of $\mat
K^{(0)}[\mat\lambda'{]}(t)$ (red, left ordinate axis) and of
$\partial_t\mat K^{(0)}[\mat\lambda'{]}(t)$ (blue, right ordinate
axis). Note, that the ordinate axes' scales differ by two orders
of magnitude. Results are independent of the time
$t$. \emph{Right}: Time dependence of the optimal parameters. VCA
results obtained as the roots of $\partial_t\mat
K^{(0)}[\mat\lambda'{]}(t) = 0$ [Eq.~(\ref{eq:dK0dt})] for
different $\Delta t$. The ordinate axis refers to the result for
$\Delta t = 0.01$; results for larger $\Delta t$ are constantly
shifted by multiples of $0.05$.}
\label{fig:K_dtscaling}
\end{figure}
With the \emph{time-dependent} Euler equation~(\ref{eq:sfteuler}) we
are facing a profound root-finding problem of formally infinite
dimensions. However, the SFT variational principle is inherently
causal which allows us to determine the optimal parameters at a given
time on a discrete time grid, without affecting the results at
previous grid times, and then proceed to the next time. This
causality is most easily seen when discussing Eq.~(\ref{eq:K0}): The
integrals over $z_3$ and $z_4$ extend over the entire contour $\mathcal C$
but can be confined to times which are (physically) earlier than
$t_{\rm max}=t_1$ (cf.\ Fig.~\ref{fig:contour}). If one or both times
are located beyond $t_1$, the later time can be shifted from the upper
to the lower branch (or vice versa) without altering the contour
ordering, and the respective contributions to the integral cancel.
Thus, at time $t_{1}$, all quantities in the Euler
equation~(\ref{eq:sfteuler}) and therewith the parameter $\mat
\lambda'_{\rm opt}(t_{1})$ itself depend on parameters at earlier
times only.
Unfortunately, a straightforward numerical solution of
Eq.~(\ref{eq:sfteuler}) turns out to be impossible. The optimal
parameters, determined by standard root-finding techniques, quickly
accumulate a large error after a few time steps only such that a
reasonable solution cannot be found in this way. Generally, for a
functional depending parametrically on time, which is evaluated on a
finite time grid and the parameters of which are varied only at the
very last instant of time, we expect the explicit dependence of the
variation of the functional on the time step $\Delta t$ to scale as
$(\Delta t)^n$ with $n\geq 1$, since for an infinitesimally fine grid
variations at a single instant of time would reduce to variations on a
null set, and thus the variation of the functional must vanish.
In the present case, if one aims to solve Eq.~\eqref{eq:sfteuler} at a
given time $t$, keeping the solution $\mat\lambda'_{\rm opt}$ fixed at
earlier times, the variation $\delta_{\mat\lambda'(t)}\mat
K^{(0)}[\mat\lambda'_{\rm opt}](t)$ of the parameters at the last
time-step will scale as $(\Delta t)^2$, if the whole implementation is
based on an equidistant time-grid $\Delta t$. This is verified
numerically in Fig.~\ref{fig:K_dtscaling} (left) for a special case
(described in the figure caption). This scaling is independent of the
quadrature rules used in solving the time integrals (if the accuracy
of those scales as $(\Delta t)^2$ or better). Hence the observed
$(\Delta t)^2$ scaling is in fact due to the Jacobian
$\delta_{\mat\lambda'(t)}\mat K^{(0)}[\mat\lambda'_{\rm opt}](t)$
itself and turned out to be detrimental for the stability of the
algorithm for practically reasonable choices of $\Delta t$.
Fortunately, this problem can be overcome by requiring the \emph{time
derivative} of $\mat K^{(0)}[\mat\lambda'](t)$ [see
Eq.~(\ref{eq:K0})] to vanish for all times $t>t_0$ and by fixing the
initial conditions at $t_0$ by Eq.~(\ref{eq:sfteuler}). Hence,
instead of solving the Euler equation (\ref{eq:sfteuler}) for all
times, we rather consider
\begin{subequations}
\label{eq:DiffEulerEqn}
\begin{align}
\left.\mat
K^{(0)}[\mat\lambda'](t)\right|_{\mat\lambda'=\mat\lambda'_{\rm
opt}} &= 0 \,, \qquad \text{for } t=
t_0\,, \label{eq:DiffEulerEqn_eq} \\
\left.\partial_t \mat
K^{(0)}[\mat\lambda'](t)\right|_{\mat\lambda'=\mat\lambda'_{\rm
opt}} &= 0 \,, \qquad \text{for } t>
t_0\,. \label{eq:DiffEulerEqn_ne}
\end{align}
\end{subequations}
As can be seen in Fig.~\ref{fig:K_dtscaling} (left), the dependence on
$\Delta t$ is only linear in this case. This greatly improves the
accuracy of the parameter optimization and allows us to trace the
optimal variational parameters as a function of time. As a simple
numerical check, one may consider the {\em equilibrium} problem. The
expected time {\em in}dependence of the optimal variational parameters
is indeed found for sufficiently small $\Delta t$, see
Fig.~\ref{fig:K_dtscaling} (right).
According to Eq.~(\ref{eq:K0}), the time derivative $\partial_t \mat
K^{(0)}[\mat\lambda'](t)$ can be obtained from the corresponding
equations of motion for the four-point vertex function $\mat L_{\mat
\lambda',\mat U}$. There are two qualitatively different
dependencies on $t$: The first one results from the boundary terms due
to the time-ordering operator comprised by all Green's functions.
Contributions from this one cancel out when taking the time
derivative. The second one is the explicit dependence of the
annihilator and the creator on the external time $t_1$. This is
governed by the Heisenberg equation of motion. Commuting the
operators with the one-particle part of the Hamiltonian simply results
in matrix products with $\mat \lambda'$ while commuting with the
interacting part gives rise to higher-order products of annihilators
and creators which we denote by $\psi$ or $\psi^{\dagger}$,
respectively: $\com[\ac{}(1),\op H'_1(1)] \equiv \psi(1)$ and
$\com[\op H'_1(1),\cc{}(1)] \equiv \psi^\dagger(1)$. Thus, the time
derivative of $\mat K^{(0)}[\mat\lambda'](t)$ acquires the form:
\begin{equation}
\partial_t \mat K^{(0)}[\mat\lambda'](t) = \com[\mat K^{(0)}[\mat\lambda'](t),\mat \lambda'(t)] + \mat
K^{(1)}[\mat\lambda'](t) \,, \label{eq:dK0dt}
\end{equation}
where we have used Eq.~(\ref{eq:K0}) and where
\begin{equation}
\label{eq:K1}
K^{(1)}[\mat\lambda']_{\alpha_2\alpha_1}(z_1) =
\dint d3d4\left. Y_{\mat \lambda',\mat T,\mat U}(4,3)
M_{\mat\lambda',\mat U} (3,2,1^+,4) \right|_{z_2=z_1} \,,
\end{equation}
with
\begin{equation}
M_{\mat\lambda',\mat U} (3,2,1^+,4) = L_{\mat\lambda',\mat U} (3,2,1^+_{\psi},4) - L_{\mat\lambda',\mat
U} (3,2_\psi,1^+,4) \,.
\end{equation}
The subscript $\psi$ indicates that $\ac{}$ and $\cc{}$ are replaced
by $\psi$ and $\psi^\dagger$ in the respective correlation function.
For example, $i G_{\mat \lambda',\mat U}(1_\psi,2) = \ev< \ord_{\mathcal C}
\psi(1)\cc{}(2) >_{\mat \lambda', \mat U}$.
\begin{figure}[t]
\centering
\includegraphics[width=.8\textwidth]{fig/prop_scheme}
\caption{Sketch of the propagation algorithm for the optimal
parameters. See text for discussion.}
\label{fig:propscheme}
\end{figure}
The algorithm for the numerical implementation of the VCA in the
general nonequilibrium case [Eq.~(\ref{eq:DiffEulerEqn})] is sketched
in Fig.~\ref{fig:propscheme}. The following steps are performed at
each time step:
\begin{itemize}
\item[(i)] At a given time $t$ and for certain guessed parameters
$\mat\lambda'_{\rm g}(t)$, we propagate $\cc{\alpha}(t)$ and
$\ac{\alpha}(t)$ for all relevant orbitals $\alpha$ by a time step
$\Delta t$ and store their respective representations in the
occupation-number basis. For small cluster sizes, this is
straightforward. Therewith, arbitrary correlation functions can be
calculated for arbitrary times on the contour up to the time $t$.
While the single particle Green's function $\mat G'$ can be updated
from time step to time step and kept in the storage, the higher
correlation functions $\mat L'$ and $\mat M'$ have to be
recalculated on-the-fly for \emph{any} time step and after any
update of $\mat\lambda'_{\rm g}(t)$, because of the external time
$t$. Symmetries can be exploited to reduce the actual number of
elements that have to be calculated.
\item[(ii)] The SFT Green's function $\mat G^{\rm SFT}$ is obtained
from the CPT equation (\ref{eq:CPTequation}), which can be cast in
the form of a Volterra equation of second kind: $(\mat 1 - \mat
G_{\mat\lambda',\mat U} \mat V )\circ\mat G^{\rm SFT} = \mat
G_{\mat\lambda',\mat U}$. The latter is solved up to time $t$ by
standard techniques
(cf. Refs.~\cite{eckstein2009,eckstein2010b,brunner1986}). Here, we
exploit the translational symmetries of $\mat V$ and of $\mat G^{\rm
SFT}$ by Fourier transformation with respect to the super-lattice.
\item[(iii)] The correlation functions $\mat L'$, $\mat M'$ and $\mat
G^{\rm SFT}$, are then used to calculate both $\mat
K^{(0)}[\mat\lambda'](t)$ and $\mat K^{(1)}[\mat\lambda'](t)$ via
Eqs.~(\ref{eq:K0}) and (\ref{eq:K1}). For the initial time $t_0$,
only those integrations contribute where both times are on the
Matsubara branch. For any later time, the mixed and Keldysh
integrations have to be carried out, too. Results for the integrals
from earlier time steps cannot be recycled, due to the external time
$t$ appearing in the correlation functions $\mat L'$ and $\mat M'$.
For both, the integrations involved in the Euler
equation~(\ref{eq:DiffEulerEqn}) as well as in the Volterra
equation, we use high-order integration schemes, like the
Newton-Cotes rules or the Gregory rules
\cite{brunner1986,press2007}, to allow for large $\Delta \tau$ and
large $\Delta t$ steps \cite{CFET}.
\item[(iv)] Generally, the initial guess $\mat \lambda'_{\rm g}(t)$
will not be a root of Eqs.~(\ref{eq:DiffEulerEqn}), and must
therefore be updated. The standard way of doing this, is to apply
Newton's method. This, however, requires the knowledge of the
(inverse of the) functional's Jacobian
$\delta_{\mat\lambda'(t_0)}\mat K^{(0)}[\mat\lambda'](t_0)$ or
$\delta_{\mat\lambda'(t)}\partial_t\mat K^{(0)}[\mat\lambda'](t)$
respectively. Both, the implementation of an analytical expression
for the Jacobian or a direct numerical evaluation via finite
difference methods are rather costly options and thus not feasible.
We thus employ Broyden's method \cite{kelley1987,broyden1965}, which
provides increasingly improved updates for the (inverse of the)
Jacobian during the course of the Newton iteration, starting from an
initial guess for both, the root as well as the Jacobian itself.
However, at the initial time we evaluate the Jacobian numerically by
finite differences at some guessed parameter. This improves the
success and the speed of the method significantly. At later times,
we extrapolate both the optimal parameter as well as the inverse
Jacobian of $\partial_t \mat K^{(0)}[\mat\lambda'](t)$ to again
start the Broyden iteration from an accurate initial guess - with
this, we reliably find the roots of the respective functional after
only one to three iterations per time step.
\end{itemize}
Finally, we would like to add an important remark regarding the
solution of Eq.\ (\ref{eq:DiffEulerEqn_ne}). As Eq.\ (\ref{eq:dK0dt})
contains $\mat\lambda'(t)$ explicitly, a first naive approach would be
to guess an ``optimal'' value $\mat\lambda'_{\rm g}(t)$, calculate
$\mat K^{(0)}[\mat\lambda'_{\rm g}](t)$ and $\mat
K^{(1)}[\mat\lambda'_{\rm g}](t)$, plug these into
Eq.~(\ref{eq:dK0dt}), solve the resulting equation $ 0 = \com[\mat
K^{(0)}[\mat\lambda'_{\rm g}](t),\mat \lambda'(t)] + \mat
K^{(1)}[\mat\lambda'_{\rm g}](t)$ for a new $\mat\lambda'(t)$, update
both functionals and iterate this procedure until convergence.
However, there are two complications. First, close to the optimal
point, $\mat K^{(0)}[\mat\lambda'](t)$, and hence also $\mat
K^{(1)}[\mat\lambda'](t)$ must vanish and therefore solving
Eq.~(\ref{eq:dK0dt}) for $\mat\lambda'(t)$ will get more and more
unstable. Second, for a guessed parameter it is unlikely that the
corresponding functionals will be compatible with
Eq.~(\ref{eq:dK0dt}), i.e., that a solution is existing at all. For
this to be the case, e.g., $\mat K^{(1)}[\mat\lambda'](t)$ has to be
trace-less, since it otherwise could not equal a commutator of two
other matrices. Indeed, in our numerics this requirement is not met in
general. More generally, Eqs.~(\ref{eq:dK0dt}) and
(\ref{eq:DiffEulerEqn}) can be understood as a special case of
Sylvester's equation, namely $\mat A \mat X - \mat X \mat B = \mat C$,
which has a unique solution for any matrix $\mat C$, if and only if
the matrices $\mat A$ and $\mat B$ have distinct spectra (see
Refs.~\cite{sylvester1884,rosenblum1956,bhatia1997}). In our case,
this is clearly not the case, which is why we cannot expect a unique
solution (or any solution at all), for some $\mat
K^{(1)}[\mat\lambda'](t)$ as obtained for an arbitrarily guessed
optimal parameter. A way out would be to not solve
Eq.~(\ref{eq:DiffEulerEqn}) directly, but minimize the norm of the
left-hand side, which then, in any case, again suffers from the first
complication.
\section{Application of the nonequilibrium variational-cluster
approach}
\label{sec:nevca}
In the following, two concrete examples for the application of the
nonequilibrium variational-cluster approach are presented to discuss
some of its characteristics in detail. We consider the
one-dimensional Hubbard model at half-filling and at different
interaction strengths and study quenches (or fast ramps) of the
hopping parameters. Both examples have been in the focus of recent
experiments where the redistribution of antiferromagnetic correlations
between different bonds and for different ramp times \cite{greif2015}
as well as the topological properties of Bloch bands in optical
lattices \cite{atala2013} for the uncorrelated variant of the model
\cite{su1979,rice1982} were studied.
\begin{figure}[b]
\subfloat[dimerized lattice $\to$ homogeneous lattice]
{\label{fig:VCAramp_inifin}\includegraphics[width=.45\textwidth]{fig/2sVCA_Tinteri0f1_inifin}}\hspace{.1\textwidth}%
\subfloat[dimerization change]
{\label{fig:VCAflip_inifin}\includegraphics[width=.45\textwidth]{fig/2sVCA_Tflip_inifin}}
\caption{ Illustration of initial and final states for both (a) the
quench from a dimerized configuration to a homogeneous lattice as
well as (b) change of the dimerization. Black solid lines
indicate a nearest-neighbor intra- or inter-cluster hopping,
$T_{\rm intra}=1$ or $T_{\rm inter}=1$. Black dashed lines:
$T_{\rm inter}=0.2$. Correlated sites are represented by red
filled dots. The reference system used for the VCA calculations
is indicated by a representative two-site reference cluster
highlighted as the blue dashed ellipse.}
\label{fig:VCAinifin}
\end{figure}
Figure~\ref{fig:VCAinifin} provides an illustration of the initial and
of the final states as well as of the reference systems: In both
cases, the system is initially in the thermal state of a dimerized
Hubbard model specified by some inverse temperature $\beta$. In case
\subref{fig:VCAramp_inifin}, this state is generated by an initial
Hamiltonian which consists of decoupled two-site clusters. Hence, the
initial state is a simple valence-bond state with nearest-neighbor
correlations and reduced translational symmetries. The intra-cluster
hopping $T_{\rm intra} = 1$ fixes energy and time units. In case
\subref{fig:VCAflip_inifin}, the clusters with $T_{\rm intra} = 1$ are
weakly coupled by an inter-cluster hopping $T_{\rm inter} =0.2$.
In case \subref{fig:VCAramp_inifin}, the final-state dynamics after
the quench is governed by the full Hubbard Hamiltonian where the
initially disconnected clusters are linked by a final inter-cluster
hopping $T_{\rm inter} = 1$. This is a highly nontrivial example
where the system should build up longer-ranged nonlocal correlations
and entanglement in the course of time. In case
\subref{fig:VCAflip_inifin} the final-state Hamiltonian is basically
the same as the Hamiltonian specifying the initial state but with the
important difference that the nearest-neighbor hopping $T_{\rm
inter}=1$ now connects clusters that are shifted by one lattice
constant. This example is also highly nontrivial as it corresponds
to a sudden switch between Hamiltonians describing states with
well-developed but incompatible valence bonds where the entanglement
and the spin correlations must reorganize between two different local
situations.
The VCA, as a cluster mean-field approximation, can only partly cover
the expected final-state dynamics. Clearly, the quality of the
approximation decisively depends on the size of the cluster used in
the reference system. Note that in both cases one in principle needs
clusters of infinite size to recover the exact solution. Here,
however, our main intention is to discuss some numerical issues and to
demonstrate that the VCA can be implemented successfully and yields
reasonable results (which can in principle be improved systematically
by going to larger cluster sizes). To this end, we have choosen a
simple reference system, namely a system of disconnected clusters
consisting of two sites each, indicated by the blue dashed lines in
Fig.~\ref{fig:VCAinifin}. No additional bath degrees of freedom have
been added. Thus, the only possible variational parameters are the
intra-cluster hopping $T'$ and the on-site energies $\varepsilon'_i$
in the reference system. The latter are fixed by the symmetries of the
model at half-filling and, therefore, only the intra-cluster hopping
$T'$ is left for optimization.
We have tested the computer code in several trivial limits.
Furthermore, for the equilibrium case our data for different $U$ and
$\beta$ and for a \emph{homogeneous} lattice are fully consistent with
those obtained previously for $\beta\ra\infty$ in Ref.\
\cite{balzer2008} using a completely different algorithm.
Calculations for the systems sketched in Fig.~\ref{fig:VCAinifin} have
been performed at inverse temperature $\beta = 10$ which, on the level
of the approximation employed, is already representative of the
zero-temperature limit as has been checked by varying $\beta$. Using
fifth-order integration schemes on the Matsubara branch, converged
results are obtained with $\Delta \tau = 0.1$. On the Keldysh branch
we are limited to the trapezoidal rule only since the implementation
of higher-order schemes for the evaluation of the time-propagation
operator gives rise to numerical instabilities (see
Sec.~\ref{sec:numer-impl} and comment~\cite{CFET}). Converged results
are obtained for a time step of $\Delta t = 0.02$, and a maximum
propagation time of $t_{\max} = 10$ is easily achieved with a desktop
computer. To get converged results with respect to the spatial
extension of the one-dimensional lattice, it is sufficient to consider
systems with $L=100$ sites (using periodic boundary conditions).
Rather than a sudden parameter quench we assume a finite (but short)
ramp within a time interval $\Delta t_{\rm ramp} = 0.5$, using a ramp
profile $r(t) = (1-\cos(\pi t/\Delta t_{\rm ramp}))/2$ with continuous
first-order derivatives at the joints at $t=0$ and $t=\Delta t_{\rm
ramp}$. This has been recognized to stabilize the algorithm
significantly. It is also numerically advantageous to hold the
original system in its initial equilibrium state for a few time steps,
before starting with the ramp at $t=t_0=0$ to build up the desired
integration order.
\begin{figure}[t]
\subfloat[dimerized lattice $\to$ homogeneous
lattice]{\label{fig:VCAramp}\includegraphics[width=.48\textwidth]{fig/2sVCA_Tinteri0f1}}\hspace{.04\textwidth}%
\subfloat[dimerization
change]{\label{fig:VCAflip}\includegraphics[width=.48\textwidth]{fig/2sVCA_Tflip_1_02}}
\caption{ Time dependencies of the optimal hopping $T'_{\rm opt}$ of
the two-site reference cluster, the double occupancy and the total
energy $E_{\rm tot}(t) = \langle H(t) \rangle$ for different $U$ and for the two different ramps of the
hopping parameters of the original system (see
Fig.~\ref{fig:VCAinifin}, a and b).}
\label{fig:VCA}
\end{figure}
Results are shown in Fig.~\ref{fig:VCA}. We first discuss case
\subref{fig:VCAramp_inifin}. For $t=0$, the optimal value of the
variational parameter is found as $T_{\rm opt}'=T_{\rm intra}=1$
(dotted line). In fact, this had to be expected since, for the given
problem, the self-energy of the reference system equals the full
self-energy [cf.\ Eq.~(\ref{eq:gsft})] and thus the VCA is exact in
the initial state. This represents another nontrivial check of our
algorithm.
For $t>0$ we find that $T_{\rm opt}'$ becomes time-dependent, i.e.,
the reference system adjusts itself to the parameter ramp in a
time-dependent way to optimally describe the dynamics of the original
system. In the limit of infinitely large clusters where VCA formally
becomes exact, one would expect $T_{\rm opt}'(t)$ to become constant
after a certain relaxation time. As seen in the figure, this
relaxation of the optimal variational parameter, and also of the
double occupancy, takes place on a short time scale given by one or
two inverse hoppings. With a reference cluster of two sites only,
however, some finite-size effects must be tolerated. These show up
indeed as oscillations of $T_{\rm opt}'(t)$ around an average value
after the relaxation process. For weak $U$, where the physics is
governed by the hopping part of the Hamiltonian, this average is by
about 30\% higher than the initial value $T_{\rm intra}=1$. With
increasing $U$, the final average value of $T_{\rm opt}'(t)$ is seen
to decrease and approaches unity for $U\to \infty$. This is plausible
since for weak $U$ the fermion system is more itinerant and thus an
increased hopping parameter in the reference system is necessary to
(at least partially) compensate for the missing inter-cluster hopping
in the reference system while for strong $U$ this is less important as
the physics is more local.
Similar arguments can be used to explain the time dependence of the
double occupancy. For $t=0$, there is a strong $U$ dependence of
$\langle n_{\uparrow} n_{\downarrow} \rangle$, which is exactly
reproduced by the VCA. For $t>0$, the double occupancy quickly
relaxes to a higher value (apart from significant finite-size
oscillations) as the system becomes more itinerant due to the
additional physical inter-cluster hopping. The effect is strongest
for weak $U$.
For case \subref{fig:VCAflip_inifin}, finite-size effects are
generally somewhat stronger, see Fig.~\ref{fig:VCA} (right). Still
the main trends are clear and plausible: As the inter-cluster hopping
is weak, the initial values of $T_{\rm opt}'$ and of $\langle
n_{\uparrow} n_{\downarrow} \rangle$ are close to those of case
\subref{fig:VCAramp_inifin}. For $t>0$, the relaxation process takes
place on essentially the same time scale of one or two inverse
hoppings. However, one now expects that the optimal intra-cluster
hopping adjusts to a value close to $T_{\rm inter}=0.2$, i.e., close
to the physical hopping parameter (see final state in
Fig.~\ref{fig:VCAinifin}\subref{fig:VCAflip_inifin}). This is in fact
seen (dotted line in Fig.~\ref{fig:VCA}, right). The decrease of the
final average value of the double occupancy with increasing $U$ is
understood in the same way as in case \subref{fig:VCAramp_inifin}.
Note that the geometrical structure of the reference system is the
same for the initial and the final state, see
Fig.~\ref{fig:VCAinifin}. Given this, we expect that the description
of the initial state is better than that of the final state in the
case of the dimerization change. We have also performed calculations
using a reference system that is shifted by one lattice constant for
both, the initial and the final state. In this case, one expects that
the VCA description of the final-state dynamics is more accurate than
for the initial state. The calculations (not shown) yield unphysical
results in this case with a diverging optimal intra-cluster hopping
after one to two inverse hoppings, depending somewhat on $U$. This
demonstrates the crucial importance of an accurate description of the
initial state for the subsequent real-time dynamics. A sudden switch
of the geometrical structure of the reference system at $t=0$, which
follows the dimerization change of the original system, would be a
legitimate choice and would result in a superior approximation. This,
however, requires a substantially higher numerical effort as more than
a single two-site cluster must be considered as a building block in
the calculation which is beyond the scope of the present work.
In both cases, \subref{fig:VCAramp_inifin} and
\subref{fig:VCAflip_inifin}, there is a fairly good conservation of
the total energy right after the ramp (recall that $\Delta t_{\rm
ramp} = 0.5$) with some remaining finite-size oscillations but no
long-time drift. Note that, {\em a priori}, this could not have been
expected as a matter of course, since strict energy conservation
within SFT can only be ensured by variations \emph{nonlocal} in time,
which would correspond to the optimization of infinitely many
bath-degrees of freedom (see Ref.~\cite{hofmann2013} for a detailed
discussion).
In case \subref{fig:VCAramp_inifin} and for all interaction strengths,
we find that the total energy decreases after the ramp (see
Fig.~\ref{fig:VCAramp}). This implies that the increase of the total
energy due to the heating of the system during the ramp is
overcompensated by the energy decrease that is induced by the coupling
of the different isolated clusters via $\op H_{T_{\rm inter},0}$ and
the corresponding lowering of the kinetic energy. In case
\subref{fig:VCAflip_inifin} the total energy increases for all $U$
after the ramp (see Fig.~\ref{fig:VCAflip}). Since the initial and
the final Hamiltonians are identical apart from a translation by one
lattice constant, they have the same ground-state energies. The
observed increase of the total energy must therefore be exclusively
due to the heating of the system during the ramp.
\section{Summary}
\label{sec:summary-outlook}
The self-energy functional theory (SFT) provides a unifying framework
for different types of cluster and impurity approximations and has
recently been extended to nonequilibrium cases \cite{hofmann2013}.
In the present work, we have discussed its numerical implementation in
detail. An important observation is that a straightforward solution of
the central SFT Euler equation is impossible in practice as it suffers
from numerical instabilities which could traced back to an inherent
quadratic dependence of the Jacobian on the time step $\Delta t$.
Fortunately, the Euler equation can equivalently be replaced by its
time derivative (and the appropriate initial condition). The
corresponding Jacobian shows a numerically much more favorable
linear-in-$\Delta t$ scaling. By using high-order integration schemes
and Broyden's method we have put forward and have implemented a stable
propagation algorithm for the optimal parameters of the reference
system.
As a concrete example, the variational cluster approach (VCA) has been
applied to study the dynamics initiated by two different ramps of the
hopping parameters of an initially dimerized one-dimensional Hubbard
model. The time-evolution of the optimal parameters, the double
occupancy and the total energy has been studied at different
interaction strengths. These calculations have been performed with a
two-site reference cluster and demonstrate that plausible and
consistent results can be obtained in fact.
Note that, as compared to plain CPT, the strengths of the VCA are expected to show up when considering situations involving (dynamical) phase transitions or in the possibility to construct approximations involving bath degrees of freedom.
This, however, is beyond the scope of the present paper.
Despite the simple two-site reference system used here, the resulting real-time dynamics is in fact completely different from a mere superposition of oscillations with frequencies that are characteristic for the finite reference system.
Namely, the variational embedding of the cluster rather allows to describe the
relaxation of the system to a new stationary final state. Depending
on the system, on the type of process studied and on the model
parameters, however, the final state does show some unphysical
oscillations which are caused by the small size of the reference
system and which must be tolerated at the given approximation level.
The SFT framework allows to systematically improve the approximation
by consideration of larger clusters, clusters with more variational
parameters or by attaching uncorrelated bath sites. We expect that
the remaining finite-size oscillations can be systematically
controlled in this way.
In practice, calculations for larger clusters are computationally expensive.
Since the exact intra-cluster 4-point correlation function $L(3,2,1^{+},4)$ is required, the practically accessible cluster size is {\em eventually} limited by the exponential growth of the cluster Hilbert-space dimension.
However, for small clusters the computational bottleneck of the algorithm actually consists in the necessity to repeatedly calculate the correlation functions $\mat K^{(0)}[\mat\lambda'](t)$ (and $\mat K^{(1)}[\mat\lambda'](t)$) at each instant of time and for each Newton step via a double contour integration and a double sum over the intra-cluster orbitals (see Eqs.~(\ref{eq:K0}) and (\ref{eq:K1})).
With the present implementation, clusters with 6 or more sites are clearly beyond our capabilities.
Approximations generated within the SFT framework do respect
macroscopic conservation laws in general \cite{hofmann2013}.
Regarding total-energy conservation, however, this would require the
optimization of parameters which are nonlocal in time, i.e., the
allocation of infinitely many bath sites. Since this has not been
considered here, we also have to tolerate superimposed finite-size
oscillations on the total energy. Apart from those, however,
conservation of the total energy is described fairly well, in
particular, there is no long-time drift. However, to answer the
question of whether or not the stationary final state is a thermal
state that can be described by the temperature corresponding to the
total energy after the parameter quench or ramp, in fact requires
improved approximations and larger reference systems.
\ack We would like to thank K.\ Balzer for providing computer code for
setting up the many-body basis and for CPT reference calculations as
well as for helpful discussions. We would like to thank D.\ Gole\v{z},
C.\ Gramsch, H.\ Strand for stimulating discussions. Support of this
work by the Deutsche Forschungsgemeinschaft within the
Sonderforschungsbereich 925 (projects B5 and B4) is gratefully
acknowledged.
\section*{References}
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{"url":"https:\/\/www.electro-tech-online.com\/threads\/the-busy-bee.153513\/","text":"# The busy Bee\n\nStatus\nNot open for further replies.\n\n#### large_ghostman\n\n##### Well-Known Member\nMost Helpful Member\nI know this is 8051, but please can we leave this in general, you will see why when i finish the review. It could help alot of people with projects, its a truly great board at a fantastic price. Yes i got it free to do a review on, but i am under no obligation to be nice about it unless i think its nice. I can say it as i see it so the review is a true reflection of my opinion on it.\n\nFirst off I have had this a while now, i forgot all about it to be honest, since i got given a sample they have brought out others, iam not 100% sure which version mine is, so I wont link datasheets yet. I need to connect it up and sort out which version i have then post the datasheet etc. I do know its one the cheaper ones you can get for around $20-$25.\n\nthe prices do go up and down alot, they often do these kind of boards for $15 on a good day, considering whats on the dev board its worth every cent and alot more. here is a pic of my one. Note it is playing the old space invaders game!! The board has a joystick, buttons and leds etc etc, i will detail all that later, sorry for the pic quality! the next pic look at carefully, it shows the board being powered completely by a single watch battery cell, its actually running a oscilloscope! it comes as a demo program but i havnt tried it much yet. Lets be fair, a$20 oscilloscope on a 8051 is neat.\nI need to reinstall the IDE software, all boards come with a special chip that measures current consumption live on the IDE. This is what will really blow you away, keep in mind all the stk kits for sil labs have a header that can connect directly to it using a stand connector. This means all the different add on boards they do can plug directly onto the kit, including bluetooth, or sub gig radio boards etc etc.\n\nAnything shield wise you can get for an arduino, you can get for the different stk kits and then some!! So for arduino fans take a good look, these are serious kit they use a c compiler, most are a ARM core depending on which board youget, so compiler wise you have from free to various paid for compilers you can use, including Kiel.\n\nThe IDE itself is really nice. So thats just a warm up.\n\nIwill get the IDE reinstalled and we can take this for a serious spin and i will do a review. For those like the Mikes of this world, you are going to love these kits!! I have a fair few if people end up liking the review for this one.\n\nSo tomorrow or Sunday i will be back with a review and datasheet, some pics and a full run down of this babe can do.\n\nEDIT\n\nHad this longer than i thought!! its the first one they did! the model of this one is EFMBB B1\n\nThey do a fair few different busy bees now, the user guide is here for this one\nhttps:\/\/www.silabs.com\/documents\/public\/user-guides\/ug236-efm8bb1-slstk2020a.pdf\n\nI will track down a datasheet as well.\n\nLast edited:\n\n#### large_ghostman\n\n##### Well-Known Member\nMost Helpful Member\nThe other boards i will do at some point are Giant Gecko and the leopard Gecko. The giant i have spoke about loads, they do a stk kit which is these small boards, they are real cheap, then they do the DK kits, those cost alot but they go beyond serious.\n\nI have all kind of these kits from the days i reviewed stuff for them, The leopard kit has a special function, it can run one of those 868\/915 radio boards, this gives them some serious power. There are many IDE's you can use including MD5, but i like the sil labs one and the open source compiler, i did have atolic studio but its expired. Also they have a configuration application, like mplabx but works will all chips and works well.\n\nAll boards are ultra low power and the dev kits all have a special energy monitor chip on, you can watch live power consumption from the IDE, running space invaders including the rgb led and the heart beat lead, runs under 1mA. I will get some screen shots etc.\n\nThe leopard board and a radio plus a wonder gecko DK kit (giant gecko like but Cortex M4 with co processor instead of M3) will power the new hexa copter i am going to do. The DK kit has a color ft screen on the board and all kinds of stuff, it makes a great base station, also has the horse power to do all the image analysis and complex maths without having to use pc. Its ARM Cortex stuff so tool chains are plenty, C\/C++ all kinds of free IDE's and compilers.\n\nFor $25 they blow arduino out the water on every level. The DK kits however or around$300-\\$400 but honestly most wont ever need those, i got them free or i wouldnt have bothered with the DK kits. The radio chip i will link too, 868 for the UK and i have had 6Km easy out of them at very little power. There is some stuff i dont like about them, but i will go through it in the review.\n\nThe Gecko stuff is 32 bit, but they have 8051 chips that blow you away. Forgot to mention ALL test points and connectors etc are gold plated, the quality is second to non\n\nStatus\nNot open for further replies.\n\nLoading","date":"2021-05-15 11:44:16","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\": 0.2512739598751068, \"perplexity\": 2507.9739547898434}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2021-21\/segments\/1620243991801.49\/warc\/CC-MAIN-20210515100825-20210515130825-00131.warc.gz\"}"}
| null | null |
2018 Fender Custom Shop Namm Ltd '51 Nocaster Journeyman Relic Left-Handed, in as new condition. Fantastic quartersawn maple neck with a massice Nocaster U profile. TheHandwound Loaded '51 Nocaster Pickups sound great. Includes OHSC, cert, Shop Floor Traveller spec sheet and all pictured accessories.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 3,794
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Parcul Național Oderul de Jos, parte a ariei protejate transfrontaliere Oderul de Jos, este un parc național din Germania. Se află pe cursul inferior al fluviului Oder, în nord-estul landului Brandenburg, în districtele administrative Barnim și Uckermark. Acoperă o suprafață de 10.323 hectare și a fost inaugurat în septembrie 1995. Pe teritoriul Germaniei, parcul național este mărginit de rezervația peisagistică Regiunea Parcului Național Oderul de Jos (17.774 ha). Din partea poloneză, se învecinează cu rezervația peisagistică Oderul de Jos (Park Krajobrazowy Dolina Dolnej Odry, aproximativ 6.000 ha) și rezervația peisagistică Cedynia (Cedynski Krajobrazowy Park, aproximativ 30.850 ha).
Conform rezoluției Consiliului de mediu germano-polonez din 1992, teritoriul delimitat între canalul Hohensaaten-Friedrichsthaler și Oder în Germania și cel delimitat între Widuchowa și canalul Skosnica în Polonia este recunoscut drept arie protejată transfrontalieră și poartă numele Parcul Internațional Oderul de Jos. Zona de protecție transfrontalieră acoperă o suprafață totală de 1.172 km² și se întinde pe 60 km de-a lungul Oderului, atât pe malul german, cât și pe cel polonez.
Caracteristici
Parcul național se întinde pe o lățime de doi până la opt kilometri. În unele locuri, malul de est al Oderului se ridică abrupt până la o altitudine de 100 m deasupra nivelului mării. Malul de vest și canalul Hohensaaten-Friedrichsthaler, săpat paralel cu fluviul, au un relief mai puțin accidentat, plat în preajma comunei Schwedt/Oder.
În parcul național se află unicul polder intact din Germania. Bazinul fluvial a fost segmentat prin diguri și canale artificiale, în baza modelului olandez. Digurile "de iarnă", amplasate de-a lungul marginii de vest a văii, protejează orașele de inundații nivale, iar digurile "de vară", întinse de-a lungul Oderului, sunt deschise în fiecare an în noiembrie, astfel încât apa din Oder să poată acoperi întreaga vale și să fie absorbită liber. Luncile polderelor sunt inundate iarna și primăvara. Albia râului, fiind spațioasă, micșorează dramatic riscul de inundație a orașului-port Szczecin.
Dacă în aprilie crește nivelul de apă, digurile de vară sunt ridicate, iar apa rămasă este pompată mecanic. Acest lucru permite folosirea pajiștilor pentru pășunat și cosit până toamna.
Lunca întinsă este habitatul preferat al multor specii de plante și animale rare sau protejate. Aceasta găzduiește stoluri de păsări migratoare în popasurile lor. Valea Oderului este mărginită de pante abrupte; pe unii versanți deosebit de abrupți s-au păstrat rămășițe de păduri. Alte zone sunt acoperite de iarbă uscată din cauza pășunatului intensiv care durează de secole.
Faună
Vidrele, castorii, ereții suri și codalbii sunt specii comune în fauna parcului național.
Din 2002, întreaga suprafață a parcului e parte a ariei de importanță aviafaunistică Defileul Oderului de Jos. Printre păsările migratoare care poposesc aici se numără lebedele de iarnă. Pe teritoriul parcului se reproduc specii rare de păsări de pajiște, precum cristelul de câmp, bătăușul și sitarul de mal, dar și de pădure, ca de exemplu grangurul. Aici se află și cea mai mare colonie de chirighițe negre.
Pot fi observați pescărușul albastru și lăcarul de pipirig, ultimul fiind o specie de pasăre cântătoare pe cale de dispariție în Europa. Berzele albe care cuibăresc pe acoperișurile caselor din satele înconjurătoare sunt deosebit de atractive pentru vizitatorii parcului național. De asemenea, deși sunt foarte rare (numai 3-5 perechi la reproducere), pot fi observate berze negre.
În 2006, în aria protejată și-a făcut apariția cea mai mare colonie de chirighițe cu aripi albe din Germania, fiind înregistrate 50 de perechi și 45 de masculi. Ele au fost atrase de inundațiile relativ îndelungate din regiune. Vara, când s-a încălzit, administrația parcului le-a asigurat condiții bune de reproducere. În același an, în parcul național au cuibărit și mai multe exemplare de chighiriță cu obraz alb, fiind înregistrați 15 pui.
În această regiune a albiei Oderului se adună și mulți cocostârci înainte de migrația lor de toamnă; ei pot fi observați din turnul de la Mescherin.
Arar, parcul este traversat de exemplare de elan.
Floră
Pe lângă zonele de polder, în parcul național există habitate importante pentru speciile rare de animale și plante care preferă versanții. Un exemplu este stejarul pufos, o specie de stejar cu frunze catifelate, foarte rar în Europa Centrală, aparținând vegetației mediteraneene.
Valoare turistică
Câmpiile joase ale parcului sunt un obiectiv turistic al regiunii, în 2004 numărând în jur de 150.000 de vizitatori. Potențialul turistic era recunoscut încă în primăvara anului 1997, când ministrul de atunci al mediului din Brandenburg Matthias Platzeck estima că vizitatorii parcului și a regiunii adiacente pot aduce anual venituri de 2,6 milioane de mărci germane (1,3 milioane de euro astăzi).
Punctele de plecare ale multor tururi ghidate pe jos și cu bicicleta sunt comuna Schwedt/Oder și centrul parcului național din Criewen. Un segment de 465 km al traseului de ciclism Oder-Neisse se desfășoară pe creasta unuia dintre diguri. Parcul are o rețea de 200 de kilometri de poteci și 52 de trasee marcate pentru ciclism și drumeție.
Ceva mai la sud sunt amplasate ruinele Castelului Stolpe (cunoscut ca "Grützpott"). De la poalele ruinelor castelului se deschide o priveliște pitorească asupra văii Oderului de Jos. Escaladarea castelului este permisă sporadic.
Statut de protecție
Conform vechii Legi privind parcul național Oderul de Jos (Nationalparkgesetz – NatPUOG) din 1995 și a amendamentelor ulterioare din 2006, exploatarea economică este limitată la cel mult o jumătate din suprafața parcului, iar ecosistemul natural trebuie să rămână neatins cu excepția sistemelor de protecție împotriva inundațiilor.
Note
Bibliografie
Günter Blutke, Ansgar Vössing (Hrsg.): Nationalparksymphonie Unteres Odertal. Eine Bilderreise durch die Jahreszeiten. Nationalparkstiftung Unteres Odertal, Schwedt/Oder 2005, ISBN 3-9810032-1-7.
Wolfgang Dohle: Literatur zu Ökologie des unteren Odertals. In: Ansgar Vössing (Hrsg.): Nationalpark-Jahrbuch Unteres Odertal 2004. Nationalparkstiftung Unteres Odertal, Schwedt/Oder 2004, ISBN 3-9810032-0-9, S. 101–154(Bibliographie v. a. deutscher Arbeiten zur Ökologie des Unteren Odertals)
Wolfgang Dohle, Reinhard Bornkamm, Gerd Weigmann (Hrsg.): Das Untere Odertal. Schweizerbart, Stuttgart 1999, ISBN 3-510-53007-1 (Limnologie aktuell, Band 9).
Mieczyslaw Jasnowski, Michael Succow: Projektstudie für einen deutsch-polnischen Nationalpark "Unteres Odertal". unveröffentlichte Projektstudie im Auftrag des Ministeriums für Umwelt, Naturschutz und Reaktorsicherheit der Bundesrepublik Deutschland, Außenstelle Berlin; gefördert durch die Stiftung Kulturförderung München und die Umweltstiftung WWF-Deutschland, Eberswalde und Szczecin 1991.
Wolfgang Mönninghoff: Nationalpark Unteres Odertal. VEBU, Berlin 1997, (Deutsche Nationalparke, Band 8, Edition Commerzbank).
Ansgar Vössing: Der Internationalpark Unteres Odertal. Ein Werk- und Wanderbuch. Stapp, Berlin 1998, ISBN 3-87776-934-9.
Legături externe
Site oficial
Site-ul web al Fundației Parcului Național
WWF-Oder-Auen-Atlas
Legea parcului național Oderul de Jos (NatPUOG) din 9 noiembrie 2006 la bravors.brandenburg.de
Oderul de Jos
|
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| 2,149
|
Hindu Temple and Cultural Center of Kansas City or HTCC Kansas City, is located on 6330 Lackman Road, Shawnee, KS, 66217 and serves the Hindu & Jain population of the Kansas City Metropolitan Area.
History
In 1982, the need for Hindu Temple was discussed among the local Hindu & Jain population and plans were started to build a Hindu Temple in Johnson County. 4 Families sent requests for establishing an organization to create the Hindu Temple and asked for donations to purchase the lot that the HTCC would be built on. In May 1983, was recognized by the IRS as a religious organization and given tax exempt status. HTCC bought 5 acres of Land in an undeveloped part of Shawnee, Kansas, relatively close to where the majority of Hindus in the Kansas City Area lived. On October 27, 1985, the groundbreaking ceremony began and on May 22, 1988, the Hindu Temple was opened and held an opening ceremony with thousands in attendance. Many murtis, idols of gods, were imported from India and the temple was designed with sculptures of various Hindu & Jain gods and goddesses on the inside and outside. In April 1991, the Temple was complete in its design. Today, HTCCKC receives over 600 visitors a week with certain Hindu festivals such as Diwali drawing crowds over 1,000 people.
Design
The facility is a square building. The entrance has an area to remove shoes before entering the temple. The rest of the facility is a carpeted area dedicated for worship with a middle aisle to bow down to all deities in the temple. The temple has services in 16 languages and has Hindu rituals performed outside the temple for a fee. The Temple also has classrooms, a cafeteria and a kitchen.
Awards
The temple's youth group does several charity events and fund raising for victims of natural disasters around the world. The Youth Group raised over $6,000 for victims of the 2010 Haiti Earthquake.
See also
St. Louis Jain temple
References
Buildings and structures in Johnson County, Kansas
Hinduism in the United States
Jainism in the United States
Religious buildings and structures completed in 1991
1989 establishments in Kansas
Religious organizations established in 1983
Asian-American culture in Kansas
Indian-American culture in Kansas
|
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| 9,491
|
@implementation BILibUtils
#pragma mark - Public Interface
+ (NSString*)saveNameForMethodName:(NSString*)methodName
{
return [NSString stringWithFormat:@"__mi_save_%@", methodName];
}
+ (NSString*)preprocessNameForMethodName:(NSString*)methodName index:(int)index
{
return [NSString stringWithFormat:@"__mi_pre_%d_%@", index, methodName];
}
+ (NSString*)postprocessNameForMethodName:(NSString*)methodName index:(int)index
{
return [NSString stringWithFormat:@"__mi_post_%d_%@", index, methodName];
}
+ (NSString*)superNameForMethodName:(NSString*)methodName
{
return [NSString stringWithFormat:@"__mi_super_%@", methodName];
}
+ (Method)getMethodInClass:(Class)class selector:(SEL)selector
{
return [BILibUtils getMethodInClass:class selector:selector isClassMethod:NULL];
}
+ (Method)getMethodInClass:(Class)class selector:(SEL)selector isClassMethod:(BOOL*)isClassMethod
{
if (isClassMethod) *isClassMethod = NO;
Method method = class_getInstanceMethod(class, selector);
if (!method) {
method = class_getClassMethod(class, selector);
if (method) {
if (isClassMethod) *isClassMethod = YES;
}
}
return method;
}
+ (void)addMethodToClass:(Class)class selector:(SEL)selector imp:(IMP)imp typeEncoding:(const char*)typeEncoding isClassMethod:(BOOL)isClassMethod
{
Method method = [BILibUtils getMethodInClass:class selector:selector];
if (method) {
method_setImplementation(method, imp);
} else {
if (isClassMethod) {
class = object_getClass(class);
}
class_addMethod(class, selector, imp, typeEncoding);
}
}
+ (NSArray*)classesWithRegex:(NSRegularExpression*)regex
{
@autoreleasepool {
NSMutableArray* retClasses = [NSMutableArray array];
int numClasses;
numClasses = objc_getClassList(NULL, 0);
if (0 < numClasses) {
Class* classes = (Class*)malloc(sizeof(Class) * numClasses);
objc_getClassList(classes, numClasses);
for (int i = 0; i < numClasses; ++i) {
Class class = classes[i];
NSString* className = NSStringFromClass(class);
NSTextCheckingResult* match = [regex firstMatchInString:className options:0 range:NSMakeRange(0, className.length)];
if (0 < match.numberOfRanges) {
[retClasses addObject:[NSValue valueWithPointer:(void*)class]];
}
}
free(classes);
}
return retClasses;
}
}
+ (NSArray*)selectorsWithRegex:(NSRegularExpression*)regex forClass:(Class)class
{
@autoreleasepool {
NSArray* instanceMethods = [BILibUtils _selectorsWithRegex:regex forClass:class];
NSArray* classMethods = [BILibUtils _selectorsWithRegex:regex forClass:object_getClass(class)];
NSMutableArray* retSelectors = [NSMutableArray arrayWithArray:instanceMethods];
[retSelectors addObjectsFromArray:classMethods];
return retSelectors;
}
}
#pragma mark - Private Methods
+ (NSArray*)_selectorsWithRegex:(NSRegularExpression*)regex forClass:(Class)class
{
NSMutableArray* retSelectors = [NSMutableArray array];
unsigned int count;
Method* methods = class_copyMethodList(class, &count);
for (int i = 0; i < count; ++i) {
SEL sel = method_getName(methods[i]);
NSString* methodName = NSStringFromSelector(sel);
NSTextCheckingResult* match = [regex firstMatchInString:methodName options:0 range:NSMakeRange(0, methodName.length)];
if (0 < match.numberOfRanges) {
[retSelectors addObject:[NSValue valueWithPointer:(void*)sel]];
}
}
return retSelectors;
}
@end
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 9,332
|
Q: ASP.net Checkbox Pairs I have a page that renders two panels that display the same data differently using a repeater, differently. I then have a javascript function that toggles between the two views. I want each data item in each view to have a checkbox.
<asp:panel id="1" runat="server">
<asp:repeater id="view1" runat="server"/>
</asp:panel>
<asp:panel id="2" runat="server">
<asp:repeater id="view2" runat="server"/>
</asp:panel>
<a onclick="toggle();"/>Toggle</a>
When I transfer between views, I want the checkbox.value to also transfer.
I also want the value of the checkbox to be accessible on a postback. What is the best way to do this?
A: If you checkboxes in each panel have the same name (or other attribute that was the same you could locate the tag with), then it is fairly simple with Jquery:
<script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.4.2/jquery.min.js"></script>
<script type="text/javascript">
$(document).ready(function () {
$(".cb").change(function () {
var name = $(this).attr("name");
var checked = $(this).attr("checked");
$(".cb[name=" + name + "]").attr("checked", checked);
});
});
</script>
<!-- Assuming this is would be the output of your two <asp:Panel /> controls: -->
<div id="1">
<input class="cb" type="checkbox" name="cb1" />
<input class="cb" type="checkbox" name="cb2" />
</div>
<div id="2">
<input class="cb" type="checkbox" name="cb1" />
<input class="cb" type="checkbox" name="cb2" />
</div>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,350
|
Q: Graphical imperfections when plotting a fourier series When running the following I get the plot below. I'm a beginner, can anyone suggest why we see those graphical imperfections in the plot?
ClearAll[P];
F[x_]:=Piecewise[{{1,Mod[x,2Pi]<Pi},{0,Mod[x,2Pi]>Pi}}];
FourierSeries[F[x],x,100];
Plot[%,{x,0,4Pi}]
A: Answer
As pointed out by @Roman, you should increase the default value for PlotPoints. You should also increase the value of MaxRecursion. Play and find a compromise between PlotPoints and MaxRecursion. By increasing these values you will make computation much slower and the plot figure much heavier (if in vector form).
I recommend at least 4 points per period at your highest frequency and a recursion of at least 3.
You can see that the points that come from the recursion are cleverly placed where needed to create a smoother curve, but the ones from plot points are regularly spaced, so irregular curves benefit more from a larger MaxRecursion.
Points in the Mesh grow very fast with the value of MaxRecursion.
Explanation
From the documentation for Plot, which should be the first thing to read if you are experiencing a problem,
Plot initially evaluates $f$ at a number of equally spaced sample points
specified by PlotPoints. Then it uses an adaptive algorithm to choose
additional sample points, subdividing a given interval at most
MaxRecursion times.
Example
ClearAll[P];
F[x_] := Piecewise[{{1, Mod[x, 2 Pi] < Pi}, {0, Mod[x, 2 Pi] > Pi}}];
Plot[
Evaluate[FourierSeries[F[x], x, 100]]
, {x, 0, 4 Pi}
, PlotPoints -> 800
, MaxRecursion -> 4
, PlotTheme -> "Scientific"
]
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 349
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{"url":"https:\/\/chat.stackexchange.com\/transcript\/19167\/2019\/7\/15\/6-7","text":"6:20 AM\n2 days ago, by user193319\nHow exactly is he using the mean value theorem to get, for example, $u(x+s,y+t) - u(x,y+t) = u_x(x+s_1,y+t)s$? It appears that he is treating $s,y,$ and $y$ as constants; but I can't figure out how he is applying it.\n@user193319 If we simply fix $y+t$ and view this as a function of the first variable, we get the expression you posted.\nI mean, consider the function $f(x)=u(x,y+t)$ where $y,t\\in\\mathbb R$ are some fixed constant.\nThen this expression simply says that $f(x+s)-f(x)=f'(x+s_1)s$.\nOr, if you prefer this form, you can denote $x_1=x$ and $x_2=x+s$ and hit is just $f(x_2)-f(x_1)=f'(t)(x_2-x_1)$ for some $t\\in(x_1,x_2)$.","date":"2019-08-22 08:49:10","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\": 0.9336912035942078, \"perplexity\": 160.0759322462552}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027316785.68\/warc\/CC-MAIN-20190822064205-20190822090205-00319.warc.gz\"}"}
| null | null |
\section{Introduction}
\subsection{Critical bond percolation and loop percolation}
Critical bond percolation on $\mathbb{Z}^2$ is the most natural model of a random subgraph of the square lattice: each edge of the lattice is included with probability $1/2$, independently of all other edges. Despite the simplicity of its definition, it is difficult to analyze rigorously. Though the spectacular recent results of Smirnov \cite{smirnov} and Lawler-Schramm-Werner \cite{lawler-etal} related to critical exponents, conformal invariance and the Schramm-Loewner Evolution (SLE) have resolved many of the outstanding open problems for the related model of critical site percolation on the triangular lattice, the same problems are still unsolved in the case of bond percolation on $\mathbb{Z}^2$.
In this paper we will study another percolation model, which is related to critical bond percolation on $\mathbb{Z}^2$ and can be thought of as a variant of it. We will refer to the model as \textbf{loop percolation on $\mathbb{Z}^2$}; an equivalent model has been studied in the statistical physics literature under different names such as the \textbf{dense $O(1)$ loop model} \cite{mitra-etal} or
\textbf{completely packed loops} \cite{zinn-justin}. Another closely related model is the special case of the \textbf{XXZ spin chain} in which the so-called anisotropy parameter $\Delta$ takes the value $-1/2$; see \cite{batchelor-etal, difran-zj-zuber}.
Our motivation in studying this model was twofold: first, to point out, and attempt to systematically exploit, the inherent interest and approachability of the model from the point of view of probability theory---qualities which have not been emphasized in previous studies. Second, to use and enhance some of the deep algebraic tools (referred to under the broad heading of the \textbf{quantum Knizhnik-Zamolodchikov equation}) that were developed in recent years to compute explicit formulas for several quantities of interest in the model.
In this paper we focus on algebraic techniques. The follow-up paper \cite{romik-pipes} will contain additional results of a more probabilistic flavor.
Let us start with the definition of the model. We define loop percolation in the entire plane, but later we will also consider the model on a half-plane and on a semi-infinite cylinder.
\begin{defi}[Loop percolation]
Consider the square lattice $\mathbb{Z}^2$ equipped with a checkerboard coloring of the squares of the dual lattice, such that the square whose bottom-left corner is $(0,0)$ is white. \textbf{Loop percolation} is the random subgraph $\textnormal{LP}=(\mathbb{Z}^2,V(\lpgraph))$ of $\mathbb{Z}^2$ obtained by tossing a fair coin for each white square \raisebox{-2.8pt}{\scalebox{2.0}{$\square$}} in the dual lattice, independently of all other coins, and including in $V(\lpgraph)$ either the bottom and top edges of \raisebox{-2.8pt}{\scalebox{2.0}{$\square$}} or the left and right edges of \raisebox{-2.8pt}{\scalebox{2.0}{$\square$}} according to the result of the coin toss (Fig.~\ref{fig:lp-two-edge-confs}).
\end{defi}
\begin{figure}[h]
\begin{center}
\setlength{\unitlength}{0.18in}
\begin{picture}(10,3)(0,0)
\color{gray}
\put(0,0){\framebox(3,3){}}
\put(7,0){\framebox(3,3){}}
\color{black}
\put(0,0){\circle*{0.35}}
\put(3,0){\circle*{0.35}}
\put(0,3){\circle*{0.35}}
\put(3,3){\circle*{0.35}}
\put(7,0){\circle*{0.35}}
\put(10,0){\circle*{0.35}}
\put(7,3){\circle*{0.35}}
\put(10,3){\circle*{0.35}}
\linethickness{2pt}
\put(0,0){\line(1,0){3}}
\put(0,3){\line(1,0){3}}
\put(7,0){\line(0,1){3}}
\put(10,0){\line(0,1){3}}
\end{picture}
\caption{The two possible edge configurations around a white lattice square.}
\label{fig:lp-two-edge-confs}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\setlength{\unitlength}{1in}
\begin{picture}(3.7,3.4)(0,0.1)
\put(-0.35,1.8){(a)}
\put(0,1.8){$\cdots$}
\put(3.55,1.8){$\cdots$}
\put(1.8,0){$\vdots$}
\put(1.8,3.48){$\vdots$}
\put(0.2,0.15){\scalebox{0.92}{\includegraphics{bondperc20}}}
\end{picture}
\\[38pt]
\setlength{\unitlength}{1in}
\begin{picture}(3.7,3.4)(0,0)
\put(-0.35,1.8){(b)}
\put(0,1.8){$\cdots$}
\put(3.55,1.8){$\cdots$}
\put(1.8,0){$\vdots$}
\put(1.8,3.48){$\vdots$}
\put(0.2,0.15){\scalebox{0.92}{\includegraphics{loopperc20checkerboard}}}
\end{picture}
\end{tabular}
\caption{(a) Critical bond percolation. (b) Loop percolation.}
\label{fig:bondloop}
\end{center}
\end{figure}
Fig.~\ref{fig:bondloop} shows portions of a critical bond percolation configuration and an (unrelated) loop percolation configuration. In fact, the two models are equivalent, in the following sense. The black squares form the vertices of a graph where two black squares are adjacent if their centers differ by one of the vectors $(0,\pm 2), (\pm 2, 0)$. Identifying each black square with its center, this graph decomposes into disjoint components of ``blue'' and ``red'' sites according to parity, each component being isomorphic to $\mathbb{Z}^2$. Define a subgraph $G$ of the blue component whose edges are precisely those that do not cross an edge of the loop percolation graph. Then it is easy to see that $G$ is a critical bond percolation graph; see Fig.~\ref{fig:bond-loop-connection}. Conversely, it is clear that starting from the critical bond percolation graph one can reconstruct the associated loop percolation graph.
\begin{figure}[h]
\begin{center}
\begin{tabular}{ccc}
& $\vdots$ & \\
\raisebox{95pt}{$\cdots$} \hspace{-15pt} &
\scalebox{0.8}{\includegraphics{loopperc-dual.pdf}} &
\hspace{-15pt} \raisebox{95pt}{$\cdots$} \\[-5pt] & $\vdots$
\end{tabular}
\caption{The critical bond percolation graph associated with a loop percolation graph.}
\label{fig:bond-loop-connection}
\end{center}
\end{figure}
As an immediate consequence of the above discussion, we see that the loop percolation graph almost surely decomposes into a disjoint union of cycles, or ``loops.'' Indeed, since each site of $\mathbb{Z}^2$ is incident to precisely two white squares, it has degree $2$, so the connected components of the graph are either loops or infinite paths; however, the presence of an infinite path would imply the existence of an infinite connected component in the ``blue'' critical bond percolation graph containing those blue sites adjacent to an edge in the infinite path, in contradiction to the well-known fact \cite[Lemma~11.12]{grimmett}
that critical bond percolation on $\mathbb{Z}^2$ almost surely has no such infinite component.
Given the equivalence between loop percolation and critical bond percolation described above, one may ask why it is necessary to study loop percolation separately from critical bond percolation. We believe that there is much to be gained in doing so. In particular, the existing research on loop percolation and connections with other natural statistical physics models such as the XXZ spin chain and Fully Packed Loops indicate that studying this type of percolation as a model in its own right suggests a variety of mathematically interesting questions that one might not otherwise be led to consider by thinking directly of critical bond percolation.
Furthermore, the techniques developed for studying loop percolation have shown that it belongs to the (loosely defined) class of so-called ``exactly solvable'' models, for which certain remarkable algebraic properties can be used to get precise formulas for various quantities associated with the model. The new results presented in this paper will provide another strong illustration of this phenomenon and give further credence to the notion that loop percolation is quite worthy of independent study. Ultimately, we hope that this line of investigation may lead to new insights that could be used to attack some of the important open problems concerning critical bond percolation.
\subsection{Noncrossing matchings}
Our discussion will focus on certain combinatorial objects known as \textbf{noncrossing matchings} that are associated with loop percolation configurations. Let us recall the relevant definitions. For two integers $a<b$, let $[a,b]$ denote the discrete interval $\{a,a+1,\ldots,b\}$. Recall that a \textbf{noncrossing matching of order $n$} is a perfect matching of the numbers $1,\ldots,2n$ (which we encode formally as a function $\pi:[1,2n]\to[1,2n]$ such that $\pi\circ\pi = \textrm{id}$ and $\pi(k)\neq k$ for all $k\in[1,2n]$; $\pi(k)$ represents the number matched to $k$), which has the additional property that there do not exist numbers $a<b<c<d$ in $[1,2n]$ such that $\pi(a)=c, \pi(b)=d$.
If $\pi(j)=k$ we say that \textbf{$j$ and $k$ are matched under $\pi$} and denote $j \matched{\pi} k$. Denote by $\noncn{n}$ the set of noncrossing matchings of order $n$. It is well-known that the number $|\noncn{n}|$ of elements in $\noncn{n}$ is $\operatorname{Cat}(n)=\frac{1}{n+1}\binom{2n}{n}$, the $n$th Catalan number.
Similarly, an \textbf{infinite noncrossing matching} is a one-to-one and onto function $\pi:\mathbb{Z}\to\mathbb{Z}$ that satisfies $\pi\circ\pi = \textrm{id}$ and $\pi(k)\neq k$ for all $k\in\mathbb{Z}$, such that there do not exist integers $a<b<c<d$ for which $\pi$ matches $a$ to $c$ and $b$ to $d$. Denote by $\textnormal{NC}_\Z$ the set of noncrossing matchings on $\mathbb{Z}$.
Both finite and infinite noncrossing matchings can be represented graphically in a diagram in which noncrossing edges, or \textbf{arcs}, are drawn between the elements of any matched pair $j\matched{\pi} k$; see Fig.~\ref{fig:example-noncrossing-matching}. As illustrated in the figure, in the case of a finite noncrossing matchings there are two equivalent representations, as a matching of $2n$ points arranged on a line or on a circle.
\begin{figure}[h]
\begin{center}
\setlength{\unitlength}{1in}
\begin{tabular}{cc}
\raisebox{20pt}{\scalebox{0.8}{\includegraphics{noncrossing10-linear}}} &
\scalebox{0.5}{\includegraphics{noncrossing10-circular}} \\ (a) & (b)
\end{tabular}
\medskip
\begin{tabular}{c}
\begin{picture}(4,1)(0,0)
\put(0.2,0){\includegraphics{linkpattern-example}}
\put(0,0.2){\ldots}
\put(3.8,0.2){\ldots}
\end{picture} \\ (c)
\end{tabular}
\caption{(a--b) a finite noncrossing matching shown as a matching of points on a line or on a circle; (c) an infinite noncrossing matching.}
\label{fig:example-noncrossing-matching}
\end{center}
\end{figure}
\subsection{The connectivity pattern of loop percolation: the infinite case}
\label{sec:connectivity-infinite}
We now associate a random noncrossing matching called the \textbf{connectivity pattern} with loop percolation configurations. There will be two variants of the problem according to the type of region being considered, resulting in infinite or finite noncrossing matchings. We start the discussion with the infinite case. To the best of our knowledge, this variant of the problem has not been previously considered in the literature, but nonetheless the rationality phenomenon we will discuss appears most striking in this setting.
\begin{defi}[Half-planar loop percolation]
Denote by $\Z^2_{\textrm{NE}}$ the half-lattice $\Z^2_{\textrm{NE}} = \{ (m,n)\,:\, m+n\ge 0 \}$, and consider a loop percolation graph $\textnormal{LP}_{\textrm{NE}}$ defined as before but only on $\Z^2_{\textrm{NE}}$ instead of on the entire plane. The connectivity pattern associated with it is an infinite noncrossing matching, which we denote by $\Pi_*$. It is defined as follows: for any $j\in\mathbb{Z}$, to compute $\Pi_*(j)$, start at the vertex $(j,-j)$. According to our convention regarding the coloring of the squares of the dual lattice, $(j,-j)$ is at the bottom-left corner of a white square, so $(j,-j)$ is incident to precisely one edge in the configuration. Follow that edge, and, continuing along the path of loop percolation edges leading out from $(j,-j)$, one eventually ends up at a vertex $(k,-k)$ on the boundary of the half-lattice (since if we consider the configuration as part of a configuration on the entire plane, the path must be a loop as noted above). In this case, we say that $j$ and $k$ are matched, and denote $j\matched{\Pi_*}k$; see Fig.~\ref{fig:loopperc-matching} for an illustration.
\end{defi}
\begin{figure}
\begin{center}
\vspace{-20.0pt}
\begin{tabular}{c}
\hspace{-30.0pt}
\scalebox{1}{\includegraphics{loopperc-matching}}
\\ (a) \\[15pt]
\hspace{-25pt} \raisebox{5pt}{$\cdots$}\hspace{-5pt} \scalebox{0.76}{\includegraphics{loopperc-matching2}}
\hspace{-5pt}\raisebox{5pt}{$\cdots$}
\\ (b)
\end{tabular}
\caption{(a) A loop percolation configuration (the paths leading out of the vertices $(n,-n)$ are highlighted and colored to emphasize the connectivities); (b) the associated connectivity pattern.}
\label{fig:loopperc-matching}
\end{center}
\end{figure}
It is clear from elementary topological considerations that $\Pi_*$ is a (random) noncrossing matching. It turns out to have some remarkable distributional properties. In particular, a main motivation for the current work was the observation that the probabilities of many events associated with $\Pi_*$ turn out rather unexpectedly to be rational numbers which can be computed explicitly. We refer to this as the \textbf{rationality phenomenon}. We will prove it rigorously in a few cases, and conjecture it for a large family of events. Although we have not been able to prove the conjecture in this generality, we also derive
several results that establish strong empirical and theoretical evidence for the correctness of the conjecture and reduce it to a much more explicit algebraic conjecture concerning the Taylor coefficients of a certain family of multivariate polynomials.
Given a finite noncrossing matching $\pi_0\in\noncn{n}$ and an infinite noncrossing matching $\pi\in\textnormal{NC}_\Z$, we say that $\pi_0$ \textbf{is a submatching of $\pi$} if $\pi_{\raisebox{2pt}{\big|} [1,2n]} \equiv \pi_0$, and in this case denote $\pi_0 \mathrel{\lhd} \pi$.
We refer to an event of the form $\{ \pi_0 \mathrel{\lhd} \Pi_* \}$ as a \textbf{submatching event}.
We will often represent a submatching event schematically by drawing the diagram associated with the submatching $\pi_0$; for example, the diagram ``\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-34}}}'' corresponds to the submatching event $\{ (\Pi_*)_{[1,4]} = (2,1,4,3) \}$.
\vbox{
\begin{conj}[The rationality phenomenon for submatching events]
\label{conj:rationality}
For any $\pi_0\in\noncn{n}$, the probability $\mathbb{P}( \pi_0 \mathrel{\lhd} \Pi_* )$ of the associated submatching event is a dyadic rational number that is computable by an explicit algorithm (Algorithm~C described in Appendix~A).
\end{conj}
}
\begin{table}[h]
\begin{tabular}{ccc}
\raisebox{51.5pt}{
\begin{tabular}{c|c}
Event & \textrm{Probability} \\
\hline \\[-1ex]
\scalebox{0.14}{\includegraphics{event12}} &
\raisebox{8pt}{$\displaystyle \frac{3}{8}$}
\\[10pt]
\scalebox{0.3}{\includegraphics{event12-34}} &
\raisebox{11pt}{$\displaystyle \frac{97}{512}$}
\\[10pt]
\scalebox{0.3}{\includegraphics{event14-23}} &
\raisebox{12pt}{$\displaystyle \frac{59}{1024}$}
\end{tabular}
}
& &
\begin{tabular}{c|c}
Event & \textrm{Probability} \\
\hline \\
\scalebox{0.3}{\includegraphics{event12-34-56}} &
\raisebox{8pt}{$\displaystyle \frac{214093}{2^{21}}$}
\\[10pt]
\scalebox{0.3}{\includegraphics{event12-36-45}} &
\raisebox{8pt}{$\displaystyle \frac{69693}{2^{21}}$}
\\[10pt]
\scalebox{0.3}{\includegraphics{event14-23-56}} &
\raisebox{8pt}{$\displaystyle \frac{69693}{2^{21}}$}
\\[10pt]
\scalebox{0.3}{\includegraphics{event16-23-45}} &
\raisebox{8pt}{$\displaystyle \frac{37893}{2^{21}}$}
\\[10pt]
\scalebox{0.3}{\includegraphics{event16-25-34}} &
\raisebox{14pt}{$\displaystyle \frac{7737}{2^{21}}$}
\end{tabular}
\end{tabular}
\vspace{10pt}
\caption{The probabilities of some submatching events in the noncrossing matching $\Pi_*$. The first two are established rigorously (Theorem~\ref{thm:two-explicit-cases}), others are conjectured.}
\label{table:prob-submatching}
\end{table}
Table~\ref{table:prob-submatching} lists some of the simplest submatching events and their conjectured probabilities. We can prove the first two cases.
\begin{thm}
\label{thm:two-explicit-cases}
We have
\begin{align}
\mathbb{P}\left(
\raisebox{-8pt}{\scalebox{0.1}{\includegraphics{event12}}}
\mathrel{\lhd} \Pi_*
\right) &= \frac{3}{8}, \label{eq:pistar-three-eighths} \\[3pt]
\mathbb{P}\left(
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-34}}}
\mathrel{\lhd} \Pi_*
\right) &= \frac{97}{512}.
\label{eq:pistar-ninety-seven}
\end{align}
\end{thm}
As we explain in Subsection~\ref{sec:consequences-halfplane}, the relation \eqref{eq:pistar-three-eighths} will follow as an immediate corollary of a result due to Fonseca and Zinn-Justin (Theorem~\ref{thm:fonseca-zinn-justin} in Subsection~\ref{sec:loop-perc-cylinder}). The second relation \eqref{eq:pistar-ninety-seven} can also be derived from the same theorem but in a slightly less trivial manner; see Theorem~\ref{thm:ninety-seven-finiten} in Subsection~\ref{sec:cylinder-new-results}.
We note that the rationality phenomenon is not restricted to submatching events; a similar statement to Conjecture~\ref{conj:rationality} also appears to hold for a larger family of \textbf{finite connectivity events}, which are events of the form $\bigcap_{ (j,k)\in A } \left\{ j\matched{\Pi_*} k \right\}$ for some finite set $A\subset \mathbb{Z}\times \mathbb{Z}$. For example, in Theorem~\ref{thm:event135} in Subsection~\ref{sec:consequences-halfplane} we derive rigorously the value $135/1024$ as the probability of a certain finite connectivity event that is not a submatching event. However, we mostly focus on submatching events since our main results (Theorems~\ref{thm:explicit-formulas-finite-n} and~\ref{thm:submatching-event-expansion}) pertain to them and as a result provide strong theoretical evidence for a rationality phenomenon in this setting.
\subsection{Loop percolation on a cylinder: background}
\label{sec:cylinder-background}
Our study of half-planar loop percolation and its connectivity pattern is based on the fact that the model can be approached in a fairly straightforward manner as a limit of a model with a different geometry of a ``semi-infinite cylinder.'' This corresponds to loop percolation in a semi-infinite diagonal strip whose two infinite boundary edges are identified. The precise definition is as follows.
\begin{defi}[Cylindrical loop percolation] For any $n\ge 1$, the cylindrical loop percolation graph $\textnormal{LP}_n$ has vertex set
\begin{equation} \label{eq:v-lpgraph-n}
V(\textnormal{LP}_n) = \{ (x,y)\in\mathbb{Z}^2\,:\, x+y\ge 0, -n+1 \le x-y\le n \},
\end{equation}
where for any $k\ge 0$ we identify the two vertices $(-n+1+k,n-1+k)$ and $(n+k,-n+k)$. The edges are sampled randomly in the same manner as the usual loop percolation.
\end{defi}
There is a convenient way of representing the cylindrical model as a tiling of a strip of the form $[0,2n]\times[0,\infty)$ with two kinds of square tiles, known as \textbf{plaquettes}, which are shown in Fig.~\ref{fig:plaquettes}. To see the correspondence, first rotate the diagonal strip in the original loop percolation configuration counter-clockwise by 45 degrees, then replace each (rotated) white lattice square with a plaquette whose connectivities correspond to the percolation edges in the white square. Once the plaquette tiling representation is drawn, one can wrap it around a cylinder to obtain a three-dimensional picture; see Fig.~\ref{fig:cylindrical-loop-perc}. The representation using plaquettes is the one traditionally used in most of the existing literature.
\begin{figure}[h]
\begin{center}
\begin{tabular}{ccc}
\scalebox{0.2}{\includegraphics{plaquette0}}
& \hspace{30.0pt} &
\scalebox{0.2}{\includegraphics{plaquette1}}
\end{tabular}
\caption{The two types of plaquettes.}
\label{fig:plaquettes}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\setlength{\unitlength}{0.5in}
\begin{picture}(10,12.5)(1,0)
\put(0,5){\scalebox{1}{\includegraphics{diagonal-strip}}}
\put(8,5){\scalebox{0.76}{\includegraphics{diagonal-strip-plaquettes}}}
\put(9.65,11.7){\circle*{0.05}}
\put(9.65,11.85){\circle*{0.05}}
\put(9.65,12){\circle*{0.05}}
\put(1.5,0.05){\scalebox{0.16}{\includegraphics{cylinder.png}}}
\put(10.3,2.6){\circle*{0.05}}
\put(10.45,2.61){\circle*{0.05}}
\put(10.6,2.62){\circle*{0.05}}
\put(4,4.5){(a)}
\put(9.5,4.5){(b)}
\put(6,0){(c)}
\end{picture}
\caption{(a) Cylindrical loop percolation: the north-west and south-east boundary edges of the infinite diagonal strip are identified along the dashed lines; (b) representation of the same configuration as a tiling of two kinds of plaquettes; (c) wrapping a tiling of plaquettes around a cylinder.
}
\label{fig:cylindrical-loop-perc}
\end{center}
\end{figure}
Let $\Pi_*^{(n)}$ denote the connectivity pattern associated with the cylindrical loop percolation graph $\textnormal{LP}_n$, defined analogously to $\Pi_*$ by following each of the paths originating at the vertices $(-j,j), -n+1\le j\le n$ until it re-emerges in a vertex $(-k,k)$. It is easy to see that $\Pi_*^{(n)}$ is a random \emph{finite} noncrossing matching of order $n$; the only justification needed is the following trivial lemma that shows that it is well-defined.
\begin{lem}
In the graph $\textnormal{LP}_n$, almost surely all paths are finite.
\end{lem}
\begin{proof}
Consider the tiling representation of the model as in Fig.~\ref{fig:cylindrical-loop-perc}(b).
It is easy to see that a row in the tiling in which the plaquettes alternate between the two types of plaquettes forces all paths below it to be finite by ``bouncing back'' any path attempting to cross the row. Any row has probability $1/2^{2n-1}$ to have such structure, independently of other rows, so almost surely there will be infinitely many such rows.
\end{proof}
The cylindrical connectivity pattern $\Pi_*^{(n)}$ has been the subject of extensive research in recent years, and appears in a surprising number of ways that do not seem immediately related to each other. Before presenting our new results on the behavior of $\Pi_*^{(n)}$, let us survey some of the known theory.
For a matching $\pi\in\noncn{n}$ denote $\mu_\pi = \mathbb{P}(\Pi_*^{(n)} = \pi)$. A natural question is how to compute the probability vector $\boldsymbol{\mu}_n = (\mu_\pi)_{\pi\in\noncn{n}}$. It is easy to see that $\boldsymbol{\mu}_n$ is the stationary distribution of a Markov chain on $\noncn{n}$ where the transitions $\pi\to \pi'$ correspond to the operation of extending the cylinder by one additional row of random plaquettes. This operation leaves the distribution $\boldsymbol{\mu}_n$ invariant since it results in a plaquette tiling equal in distribution to the original one. Formally, there is a Markov transition matrix $T_n^{(1/2)} = (t^{(1/2)}_{\pi,\pi'})_{\pi,\pi'\in\noncn{n}}$ such that
$$ \boldsymbol{\mu}_n T_n^{(1/2)} = \boldsymbol{\mu}_n, $$
where $\boldsymbol{\mu}_n$ is considered as a row vector.
(In the statistical physics literature $T_n^{(1/2)}$ is usually referred to as a transfer matrix.)
The reason for the notation $T_n^{(1/2)}$ is that we can generalize the model and define a matrix $T_n^{(p)}$ for any $0< p<1 $, corresponding to a tiling of independently sampled random plaquettes in which each choice between the two types of plaquette is made by tossing a coin with bias $p$. Remarkably, the value of $p$ does not affect the distribution of the connectivity pattern.
\begin{thm}
\label{thm:commuting-matrices}
The transition matrices $(T_n^{(p)})_{0< p <1}$ are a commuting family of matrices. Consequently, since they are all stochastic, they all share the same row eigenvector $\boldsymbol{\mu}_n$ associated with the eigenvalue $1$.
\end{thm}
Note that using the transition matrix $T_n^{(1/2)}$, or $T_n^{(p)}$ for any fixed $p$, is not necessarily the easiest way to compute $\boldsymbol{\mu}_n$, since to compute the entries of $T_n^{(1/2)}$ one has to count the number of possible rows of $n$ plaquettes (out of the $2^{2n}$ possibilities) that would cause a given state transition $\pi\to\pi'$. It turns out that there is a more convenient (and more theoretically tractable) system of linear equations for computing $\boldsymbol{\mu}_n$, which involves a simpler matrix $H_n$ that arises as a limiting case of the $T_n^{(p)}$ as $p\to 0$, or symmetrically as $p\to 1$.
To define $H_n$, first define for each $k\in[1,2n]$ a mapping $e_k:\noncn{n}\to\noncn{n}$ given by
\begin{equation}
\label{eq:temperley-lieb-gens}
e_k(\pi)(m) = \begin{cases}
\pi(m) & \textrm{if }m\notin \{k,k+1,\pi(k),\pi(k+1)\}, \\
k+1 & \textrm{if }m=k, \\
k & \textrm{if }m=k+1, \\
\pi(k+1) & \textrm{if }m=\pi(k), \\
\pi(k) & \textrm{if }m=\pi(k+1),
\end{cases}
\end{equation}
where $k+1$ is interpreted as $1$ if $k=2n$.
In words, $e_k(\pi)$ is the matching $\pi'$ obtained from $\pi$ by unmatching the pairs $k\matched{\pi} \pi(k)$ and $k+1\matched{\pi}\pi(k+1)$ and replacing them with the matched pairs $k\matched{\pi'}k+1$ and $\pi(k)\matched{\pi'}\pi(k+1)$. In the case when $k$ and $k+1$ are already matched under $\pi$, nothing happens and $e_k(\pi)=\pi$. It is easy to see that in general $e_k(\pi)$ is a noncrossing matching. We refer to the $e_k$ as the \textbf{Temperley-Lieb operators}. (They generate an algebra known as the \textbf{Temperley-Lieb algebra} \cite{degier, temperley-lieb}, but this has no bearing on the present discussion.)
\begin{thm}
\label{thm:hamiltonian-eigenvector}
Define a square matrix $H_n$ with rows and columns indexed by elements of $\noncn{n}$ by
\begin{equation} \label{eq:def-h-inf-gen}
(H_n)_{\pi,\pi'} = 2n-\#\left\{ 1\le k\le 2n\,:\, e_k(\pi)=\pi' \right\}.
\end{equation}
Then $\boldsymbol{\mu}_n$ satisfies
\begin{equation*} \label{eq:mun-hn-zero}
\boldsymbol{\mu}_n H_n = 0.
\end{equation*}
\end{thm}
Theorems~\ref{thm:commuting-matrices} and \ref{thm:hamiltonian-eigenvector} seem to be well-known to experts in the field, but their precise attribution is unclear to us. As explained in \cite{zinn-justin} (Sections 2.2.2 and 3.3.2), Theorem~\ref{thm:commuting-matrices} follows from the Yang-Baxter equation. A fact equivalent to Theorem~\ref{thm:hamiltonian-eigenvector} is mentioned without proof in \cite[Section 3]{mitra-etal}. We provide a simple proof of this result in Appendix A.
It is easy to see that the matrix $M_n=I-\frac{1}{2n}H_n$ (where $I$ is the identity matrix) is nonnegative and stochastic, i.e., it is a Markov transition matrix, associated with yet another Markov chain that has $\boldsymbol{\mu}_n$ as its stationary vector. That is, $\boldsymbol{\mu}_n$ can be computed by solving the vector equation $\boldsymbol{\mu}_n M_n = \boldsymbol{\mu}_n$, which can be written more explicitly as the linear system
\begin{equation} \label{eq:razumov-stroganov-linear-system}
\mu_\pi = \frac{1}{2n} \sum_{k=1}^{2n} \sum_{\begin{array}{c} \scriptstyle \pi'\in\noncn{n}, \\[-5pt] \scriptstyle e_k(\pi')=\pi \end{array}} \mu_{\pi'} \qquad (\pi \in\noncn{n}).
\end{equation}
The Markov chain $(\pi_m)_{m\ge 0}$ associated with the transition matrix $M_n$ has the following simple description as a random walk on $\noncn{n}$, known as the \textbf{Temperley-Lieb random walk} or \textbf{Temperley-Lieb stochastic process} \cite{pearce-etal}: start with some initial matching $\pi_0$; at each step, to obtain $\pi_{m+1}$ from $\pi_m$, choose a uniformly random integer $k \in \{1,2,\ldots,2n\}$ (independently of all other random choices), and set $\pi_{m+1}=e_k(\pi_m)$.
The application of the maps $e_k$ can be represented graphically by associating with each $e_k$ a ``connection diagram'' of the form
\begin{center}
\scalebox{0.8}{\includegraphics{pipe-diagram-ek}}
\end{center}
which will be ``composed'' with the diagram of the matching $\pi$ to which $e_k$ is applied by drawing one diagram below the other. (Note that to get the correct picture in the case $k=2n$ one should interpret the diagram as being drawn around a cylinder.) By composing a sequence of such diagrams with a noncrossing matching diagram one can compute the result of the application of the corresponding maps to the matching; see Fig.~\ref{fig:pipe-diagram}. Using this graphical interpretation, it can be seen easily that the stationary distribution $\boldsymbol{\mu}_n$ is realized as the distribution of the connectivity pattern of endpoints in an infinite composition of $e_k$ connection diagrams (where the values of $k$ are i.i.d.\ discrete uniform random variables in $\{1,\ldots,2n\}$), drawn on a semi-infinite cylinder. This is illustrated in Fig.~\ref{fig:tl-randomwalk-graphical}.
\begin{figure}
\begin{center}
\setlength{\unitlength}{0.3in}
\begin{picture}(23,7.5)(2,-1)
\put(3.2,0){\scalebox{0.7}{\includegraphics{pipe-diagram}}}
\put(2.4,4.8){$\left\{ \vphantom{ \begin{array}{c}\ \\[24pt] \ \end{array} }\right.$}
\put(0.2,5.0){\footnotesize Original}
\put(0,4.5){\footnotesize matching $\pi$}
\put(2.4,2.98){$\left\{ \vphantom{ \begin{array}{c}\ \\[-10pt] \ \end{array} }\right.$}
\put(2.4,1.8){$\left\{ \vphantom{ \begin{array}{c}\ \\[-10pt] \ \end{array} }\right.$}
\put(2.4,0.63){$\left\{ \vphantom{ \begin{array}{c}\ \\[-10pt] \ \end{array} }\right.$}
\put(1.7,2.98){\footnotesize $e_4$}
\put(1.7,1.8){\footnotesize $e_1$}
\put(1.7,0.63){\footnotesize $e_5$}
\put(13.3,1){\scalebox{0.7}{\includegraphics{pipe-diagram-transformed}}}
\put(11.8,2){\vector(1,0){1.3}}
\put(6.5,-1){(a)}
\put(17,-1){(b)}
\put(16,0.3){$\pi'=e_5 e_1 e_4 \pi$}
\end{picture}
\caption{Graphical representation of the application of a sequence of operators $e_k$, $1\le k\le 2n$ as a ``composition of diagrams'': (a) the diagrams associated with operators $e_4,e_1,e_5$ are attached to the diagram of the original matching; (b) the lines are ``pulled'' (and any loops are discarded) to arrive at the diagram for the transformed matching. }
\label{fig:pipe-diagram}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\setlength{\unitlength}{0.3in}
\begin{picture}(20,9.4)
\put(-1,0){\scalebox{0.48}{\includegraphics{razumov-stroganov-rw2D}}}
\put(7,1){\scalebox{0.14}{\includegraphics{razumov-stroganov-rw}}}
\put(3.1,9.6){\circle*{0.1}}
\put(3.1,9.85){\circle*{0.1}}
\put(3.1,10.1){\circle*{0.1}}
\put(19.9,5){\circle*{0.1}}
\put(20.1,5.02){\circle*{0.1}}
\put(20.3,5.04){\circle*{0.1}}
\end{picture}
\caption{The connectivity pattern of endpoints in a semi-infinite arrangement of uniformly random i.i.d.\ Temperley-Lieb operator diagrams is invariant under the addition of another operator, hence has $\boldsymbol{\mu}_n$ as its distribution.}
\label{fig:tl-randomwalk-graphical}
\end{center}
\end{figure}
Yet another interpretation of $\boldsymbol{\mu}_n$ is as the ground state eigenvector associated with a certain quantum many-body system, the XXZ spin chain. The Hamiltonian of this spin chain is an operator acting on the space $V=(\mathbb{C}^2)^{\otimes 2n}$, and it has been shown that in the case of ``twisted'' periodic boundary conditions and when a parameter $\Delta$ of the chain, known as the anisotropy parameter, is set to the value $\Delta=-1/2$, the space $V$ will possess a subspace $U$ invariant under the action of the Hamiltonian, such that the restriction of the Hamiltonian to $U$ coincides (under an appropriate choice of basis) with the operator $H_n$ defined in \eqref{eq:def-h-inf-gen}; see \cite[Section 8]{mitra-etal} and \cite[Section 3.2.4]{zinn-justin}.
One additional way in which the probability vector $\boldsymbol{\mu}_n$ makes an appearance is in the study of connectivity patterns associated with a different type of loop model known as the \textbf{fully packed loops} (FPLs). An FPL configuration of order $n$ is a subset of the edges of an $(n-1)\times (n-1)$ square lattice $[0,n-1]\times[0,n-1]$, to which are added $2n$ of the $4n$ ``boundary'' edges connecting the square to the rest of the lattice $\mathbb{Z}^2$, by starting with the edge from $(0,0)$ to $(0,-1)$ and then taking alternating boundary edges as one goes around the boundary in a counter-clockwise direction (e.g., one would take edges incident to $(2,0), (4,0)$, etc.); these extra edges are referred to as \textbf{stubs}. The configuration is subject to the condition that any lattice vertex in the square is incident to exactly two of the configuration edges; see Fig.~\ref{fig:fpl}.
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\scalebox{0.6}{\includegraphics{fpl6-example}}
& &
\scalebox{0.65}{\includegraphics{fpl6-example-matching}}
\end{tabular}
\caption{A fully packed loop configuration of order $6$ and its associated noncrossing matching.}
\label{fig:fpl}
\end{center}
\end{figure}
Let $\textrm{FPL}_n$ denote the set of fully packed loop configurations of order $n$. A well-known bijection \cite{propp} shows that $\textrm{FPL}_n$ is in correspondence with the set of \textbf{alternating sign matrices} (ASMs) of order $n$. These important combinatorial objects also have an interpretation as configurations of the \textbf{six-vertex model} (a.k.a.\ \textbf{square ice}) on an $n\times n$ lattice with prescribed boundary behavior known as the \textbf{domain wall boundary condition}. It was conjectured by Mills, Robbins and Rumsey \cite{mills-robbins-rumsey} and proved by Zeilberger \cite{zeilberger1} (see also \cite{bressoud}, \cite{kuperberg}) that the number $|\textrm{FPL}_n|$ of ASMs of order $n$ is given by the famous sequence of numbers $1,2,7,42,429,\ldots$, defined by
$$ \textnormal{ASM}(n) = \frac{1!4!7!\ldots (3n-2)!}{n!(n+1)!\ldots(2n-1)!}. $$
As the reader will see below, the function $\textnormal{ASM}(n)$ will play a central role in our current investigation of connectivity patterns in loop percolation on a cylinder.
Label the $2n$ stubs around the square $[0,n-1]\times[0,n-1]$ by the numbers $1$ through $2n$, starting with the edge pointing down from $(0,0)$. From the definition of fully packed loop configurations we see that any such configuration induces a connectivity pattern on the stubs, which is a noncrossing matching in $\noncn{n}$, in an analogous manner to the way a loop percolation configuration on the semi-infinite cylinder does. For $\pi\in\noncn{n}$, denote by $A_n(\pi)$ the number of configurations in $\textrm{FPL}_n$ whose connectivity pattern is equal to~$\pi$.
It was first observed by Batchelor, de Gier and Nienhuis \cite{batchelor-etal} that the coordinates of the vector $\boldsymbol{\mu}_n$ are related to the enumeration of alternating sign matrices. They conjectured the following result, which was later proved by Zinn-Justin and Di Francesco \cite{zinn-justin-di-francesco1, zinn-justin-di-francesco2}.
\begin{thm}[Di Francesco-Zinn-Justin]
\label{thm:sum-rules}
\begin{enumerate}
\item The numbers $\mu_\pi$, $(\pi\in\noncn{n})$ are all fractions of the form $\alpha_n(\pi)/\textnormal{ASM}(n)$ where $\alpha_n(\pi)$ is an integer.
\item The minimal value of $\alpha_n(\pi)$ is $1$ and is attained for $\pi=\pi^n_{\textrm{min}}$, the ``minimal''\footnote{according to a certain partial ordering of noncrossing matchings that is discussed in Section~\ref{sec:wheel-polynomials}.} noncrossing matching consisting of $n$ nested arcs (Fig.~\ref{fig:minmax-matchings}(a)).
\item The maximal value of $\alpha_n(\pi)$ is $\textnormal{ASM}(n-1)$ and is attained for $\pi=\pi^n_{\textrm{max}}$, the ``maximal'' noncrossing matching consisting of $n$ nearest-neighbor arcs (Fig.~\ref{fig:minmax-matchings}(b)).
\end{enumerate}
\end{thm}
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\scalebox{0.6}{\includegraphics{minimal-matching}}
& &
\scalebox{0.6}{\includegraphics{maximal-matching}}
\\ (a) & & (b)
\end{tabular}
\caption{(a) The minimal matching $\pi^n_\textrm{min}$; (b) the maximal matching $\pi^n_\textrm{max}$.}
\label{fig:minmax-matchings}
\end{center}
\end{figure}
Shortly after this discovery, Razumov and Stroganov discovered a much more precise conjecture \cite{razumov-stroganov} about the connection between alternating sign matrices and the vector $\boldsymbol{\mu}_n$; it turns out that the correct thing to do is to look at the ASMs as fully packed loop configurations. Their conjecture, known for several years as the Razumov-Stroganov conjecture, was proved in 2010 by Cantini and Sportiello \cite{cantini-sportiello}.
\begin{thm}[Cantini-Sportiello-Razumov-Stroganov theorem]
For any $\pi\in\noncn{n}$ we have $\alpha_n(\pi)=A_n(\pi)$. That is,
$$
\mu_\pi = \frac{A_n(\pi)}{\textnormal{ASM}(n)} \qquad (\pi\in\noncn{n}).
$$
\end{thm}
Probabilistically, this means that the vector $\boldsymbol{\mu}_n$ is realized as the distribution of the connectivity pattern of a \emph{uniformly random} FPL configuration of order $n$. Note that this is a finite probability space, whereas the Markov chain realization would require a potentially unbounded amount of randomness to generate a sample from $\boldsymbol{\mu}_n$.
Cantini and Sportiello's proof of the conjecture of Razumov and Stroganov is highly nontrivial and involves subtle combinatorial and linear algebraic arguments. Note that while the two (\emph{a posteriori} equivalent) definitions of $\boldsymbol{\mu}_n$ as the distribution of connectivity patterns in cylindrical loop percolation and uniformly random fully packed loop configurations are superficially similar, the two models are quite different. In particular, the cylindrical model has an obvious symmetry under rotations of the cylinder by an angle $2\pi/2n$, which induces a symmetry under rotation on $\boldsymbol{\mu}_n$. The fact that the distribution of the connectivity pattern of random fully packed loops---which are defined on a square, not circular, geometry---also has the same symmetry, is far from obvious, and its earlier proof by Wieland \cite{wieland} played an important role in Cantini and Sportiello's analysis.
In this subsection we surveyed several settings in which the probability vector $\boldsymbol{\mu}_n$ appears: as the distribution of the connectivity pattern of cylindrical loop percolation; as the stationary distribution of the Temperley-Lieb random walk (or, equivalently, the distribution of the connectivity pattern of a semi-infinite arrangement of connection diagrams associated with independent, uniformly random Temperley-Lieb operators); as the ground state of the XXZ spin chain; and as the distribution of the connectivity pattern of uniformly random fully packed loop configurations of order $n$. To conclude this discussion, we note that one of the minor results of this paper is an additional characterization of the connectivity probabilities $(\mu_\pi)_{\pi\in\noncn{n}}$ as coefficients in a certain natural linear-algebraic expansion; see Theorem~\ref{thm:lin-alg-expansion} in Subsection~\ref{sec:wheel-polynomials-background}.
\subsection{Loop percolation on a cylinder: connectivity events}
\label{sec:loop-perc-cylinder}
Having described the origins of the investigations into the random noncrossing matching $\Pi_*^{(n)}$ and its distribution $\boldsymbol{\mu}_n$, we are ready to discuss the problem of computing explicitly the probabilities of various events. Our main interest will be with submatching events of the type $\{ \pi_0 \mathrel{\lhd} \Pi_*^{(n)} \}$, where for $\pi_0\in\noncn{k}$ and $\pi\in\noncn{n}$ ($n\ge k$) the notation $\pi_0 \mathrel{\lhd} \pi$ means (as defined earlier when $\pi$ is an infinite matching) that $\pi_0$ is a submatching of $\pi$, i.e., that $\pi_{\raisebox{2pt}{\big|} [1,2k]} \equiv \pi_0$.
It was observed starting with numerical work of Mitra et al.\ \cite{mitra-etal}, Zuber \cite{zuber} and Wilson (unpublished work, cited in \cite{zuber}) that the probabilities of certain events had nice formulas as rational functions in $n$. For example, one has empirically the relations
\begin{align}
\mathbb{P}\left(
\raisebox{-8pt}{\scalebox{0.1}{\includegraphics{event12}}}
\mathrel{\lhd} \Pi_*^{(n)}
\right) &= \frac{3}{2} \cdot \frac{n^2+1}{4n^2-1}, \hspace{115.0pt} (n\ge 1),
\label{eq:three-eights-finite-n} \\[5pt]
\mathbb{P}\left(
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-34}}}
\mathrel{\lhd} \Pi_*^{(n)}
\right) &= \frac18 \cdot \frac{97 n^6 + 82 n^4 - 107 n^2 - 792}{(4n^2-1)^2(4n^2-9)}
\hspace{20.0pt} (n\ge 2),
\label{eq:identity97}
\\
\mathbb{P}\left(
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event14-23}}}
\mathrel{\lhd} \Pi_*^{(n)}
\right) &= \frac1{16} \cdot \frac{59n^6 + 299n^4 +866 n^2 + 576}{(4n^2-1)^2(4n^2-9)}
\hspace{10.0pt} (n\ge 2),
\end{align}
\vspace{-12.0pt}
\begin{align}
& \mathbb{P} \left(
\raisebox{-6pt}{\scalebox{0.25}{\includegraphics{event12-34-56}}}
\mathrel{\lhd} \Pi_*^{(n)}
\right) \nonumber \\ &=
\frac{\scriptstyle 1}{\scriptstyle 512}\cdot
\frac{\scriptscriptstyle 214093n^{12}-980692n^{10}-584436n^8-1887916n^6+1361443n^4 -17432892n^2-316353600}{\scriptstyle (4n^2-1)^3(4n^2-9)^2(4n^2-25)}
\ (n\ge 3),
\label{eq:event12-34-56-prob}
\end{align}
and several other such formulas, which can be discovered by a bit of experimentation after programming the linear equations \eqref{eq:razumov-stroganov-linear-system} into a computer algebra system such as \texttt{Maple} or \texttt{Mathematica}.\footnote{Unfortunately the size of the system is the Catalan number $\operatorname{Cat}(n)$ which grows exponentially with $n$, making it impractical to compute $\boldsymbol{\mu}_n$ for values of $n$ much greater than $n=9$. Zuber \cite{zuber} managed the computation up to $n=11$ by using rotational symmetry to reduce the order of the system.} The discovery of additional conjectural relations of this type is only difficult insofar as it taxes one's patience, programming skill and computational resources, but, as Zuber remarks at the end of \cite{zuber}, ``\textit{More conjectural expressions have been collected for other types of configurations \ldots\ but this seems a gratuitous game in the absence of a guiding principle.}'' It therefore appeared sensible to wait for more theoretical developments before proceeding with the ``gratuitous game.'' Indeed, the first progress (and so far, to our knowledge, the only progress) on this front was made by Fonseca and Zinn-Justin \cite{fonseca-zinn-justin}, who, building on a theoretical framework developed earlier by Zinn-Justin and Di Francesco \cite{zinn-justin-di-francesco1, zinn-justin-di-francesco2} (see also \cite{difran-zj-zuber}) managed to prove explicit formulas for two classes of events.
\begin{thm}[Fonseca-Zinn-Justin \cite{fonseca-zinn-justin}]
\label{thm:fonseca-zinn-justin}
For each $k\ge 1$, Denote by $\textnormal{AC}_k^{(n)}$ the ``anti-cluster'' event that no two of the numbers $1,\ldots,k$ are matched under $\Pi_*^{(n)}$. Denote by $B_k^{(n)}$ the event $\left\{ \pi^k_\textrm{min} \mathrel{\lhd} \Pi_*^{(n)}\right\}$ (the submatching event associated with the minimal matching $\pi^k_\textrm{min}$; see Fig.~\ref{fig:minmax-matchings}). Define a function
\begin{equation}
\label{eq:complicated-product-def}
R_k(n)
= \begin{cases}
\displaystyle \frac{\prod_{j=1}^{(k+1)/2} \prod_{m=j}^{2j-2} (n^2-m^2)}{\prod_{j=0}^{(k-3)/2} (4n^2-(2j+1)^2)^{(k-1)/2-j}}
& \textrm{$k$ odd}, \\[15pt]
\displaystyle \frac{\prod_{j=1}^{k/2} \prod_{m=j}^{2j-1} (n^2-m^2)}{\prod_{j=0}^{k/2-1} (4n^2-(2j+1)^2)^{k/2-j}}
& \textrm{$k$ even}.
\end{cases}
\end{equation}
Then we have
\begin{align}
\mathbb{P}( \textnormal{AC}_k^{(n)} ) &= \frac{1}{\textnormal{ASM}(k)} \frac{R_k(n)}{R_k(k)},
\label{eq:anti-cluster} \\
\mathbb{P}( B_k^{(n)} ) &= \frac{1}{\textnormal{ASM}(n)} \sum_{1\le a_1 < a_2 < \ldots < a_{n-k}} \det \left( \binom{j+k}{a_i-j+ k } \right)_{i,j=1}^{n-k}.
\label{eq:k-nested-arcs}
\end{align}
In particular, we have $\mathbb{P}(\textnormal{AC}_2^{(n)})=\frac52 \frac{n^2-1}{4n^2-1} \vphantom{\begin{array}{c}a\\a\end{array}}$, which implies equation \eqref{eq:three-eights-finite-n} since the event in that identity is the complement of $\textnormal{AC}_2^{(n)}$.
\end{thm}
Note that the approach of Fonseca and Zinn-Justin to the relation \eqref{eq:anti-cluster} started out, similarly to \eqref{eq:k-nested-arcs}, by expressing the probabilities of these events as sums of determinants over a certain family of matrices with binomial coefficients. However, in the case of \eqref{eq:anti-cluster} a family of Pfaffian evaluations due to Krattenthaler \cite{krattenthaler} made it possible to evaluate the sum in closed form. As the authors of \cite{fonseca-zinn-justin} point out, both the formulas \eqref{eq:anti-cluster} and \eqref{eq:k-nested-arcs} have an interesting interpretation in terms of the enumeration of certain families of totally symmetric self-complementary plane partitions.
\subsection{Loop percolation on a cylinder: new results and the rationality phenomenon}
\label{sec:cylinder-new-results}
The identities \eqref{eq:three-eights-finite-n}--\eqref{eq:event12-34-56-prob} generalize in a straightforward way to a conjecture on the form of the dependence on $n$ of probabilities of submatching events.
\begin{conj}[The rationality phenomenon for submatching events; finite $n$ case]
\label{conj:rationality-finite-n}
If $\pi_0\in\noncn{k}$, the probability of the submatching event $\left\{ \pi_0 \mathrel{\lhd} \Pi_*^{(n)} \right\}$ has the form of a rational function in $n$, specifically
\begin{equation} \label{eq:rationality-finite-n}
\mathbb{P}\left( \pi_0 \mathrel{\lhd} \Pi_*^{(n)} \right) = \frac{Q_{\pi_0}(n)}{\prod_{j=1}^k (4n^2-j^2)^{k+1-j}} \qquad (n\ge k),
\end{equation}
where $Q_{\pi_0}(n)$ is an even polynomial of degree $k(k+1)$ with dyadic rational coefficients. $Q_{\pi_0}$ can be computed by polynomial interpolation---see Algorithm~B in Appendix~A.
\end{conj}
In pursuit of an approach that would lead to a proof of Conjecture~\ref{conj:rationality-finite-n}, one of our goals has been to find a ``guiding principle'' of the type alluded to by Zuber, that would reveal the underlying structure behind the empirical phenomena described above. In Section~\ref{sec:wheel-polynomials} we will prove the following result, which can be thought of as one version of such a principle, and is our main result.
\begin{thm}[Explicit formula for submatching event probabilities]
\label{thm:explicit-formulas-finite-n}
For any $\pi_0 \in \noncn{k}$, there exists a multivariate polynomial $F_{\pi_0}(w_1,\ldots,w_k)$, computable by an explicit algorithm (Algorithm~E in Appendix~A), with the following properties:
\begin{enumerate}
\item $F_{\pi_0}$ has integer coefficients.
\item All the monomials in $F_{\pi_0}(w_1,\ldots,w_k)$ are of the form $\prod_{j=1}^k w_j^{2j-a_j}$
where $1\le a_1<\ldots<a_k$ are integers satisfying $a_j\le 2j-1$ for $1\le j\le k$.\footnote{Note that the number of different sequences $(a_1,\ldots,a_k)$ satisfying these conditions is $\operatorname{Cat}(k)$, the $k$th Catalan number, and indeed the sequence $(a_1,\ldots, a_k)$ can be thought of as encoding a noncrossing matching in $\noncn{k}$, a fact that will have a role to play later on---see Section~\ref{sec:wheel-polynomials}.}
\item The probability $ \mathbb{P}\left(\pi_0 \mathrel{\lhd} \Pi_*^{(n)} \right)$ is given for any $n\ge k+1$ by
\begin{align}
\nonumber
& \mathbb{P}\left( \pi_0 \mathrel{\lhd} \Pi_*^{(n)} \right) = \frac{1}{\textnormal{ASM}(n)} [z_1^0 z_2^2 z_3^4 \ldots z_n^{2n-2}] \Bigg( F_{\pi_0}(z_2,\ldots,z_{k+1})
\\ & \hspace{110pt} \times
\prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+ z_i z_j) \prod_{j=k+2}^n (1+z_j) \Bigg),
\label{eq:submatching-polycoeff-formula}
\end{align}
where $[z_1^{m_1} \ldots z_n^{m_n}]g(z_1,\ldots,z_n)$ denotes the coefficient of the monomial $z_1^{m_1} \ldots z_n^{m_n}$ in a polynomial $g(z_1,\ldots,z_n)$.
\end{enumerate}
\end{thm}
The formula \eqref{eq:submatching-polycoeff-formula} can be recast in two equivalent forms which some readers may find more helpful: first, as a constant term identity
\begin{align*}
\nonumber
& \mathbb{P}\left( \pi_0 \mathrel{\lhd} \Pi_*^{(n)} \right) = \frac{1}{\textnormal{ASM}(n)} \textnormal{CT}_{z_1,\ldots,z_n} \Bigg( F_{\pi_0}(z_2,\ldots,z_{k+1})
\!\!\!\!
\\ & \hspace{130.0pt} \left. \times \prod_{1\le i<j\le n} \!\! \!(z_j-z_i)(1+z_j+ z_i z_j)\frac{\prod_{j=k+2}^n (1+z_j)}{
\prod_{j=1}^n z_j^{2j-2}} \right),
\end{align*}
where $\textnormal{CT}_{z_1,\ldots,z_n} g(z_1,\ldots,z_n)$ denotes the constant term of a Laurent polynomial $g(z_1,\ldots,z_n)$; and second, as a multi-dimensional complex contour integral
\begin{align*}
\nonumber
&\mathbb{P}\left( \pi_0 \mathrel{\lhd} \Pi_*^{(n)} \right) = \frac{1}{\textnormal{ASM}(n)} \oint\hspace{-2.0pt}\ldots\hspace{-2.0pt}\oint \Bigg( F_{\pi_0}(z_2,\ldots,z_{k+1})
\\ &
\hspace{60.0pt}
\times
\!\!\!
\! \prod_{1\le i<j\le n} \!\!(z_j-z_i)(1+z_j+ z_i z_j)
\frac{\prod_{j=k+2}^n (1+z_j)}{ \prod_{j=1}^n z_j^{2j-1}}
\Bigg) \prod_{j=1}^n \frac{dz_j}{2\pi i},
\end{align*}
where the contour for each of the $z_j$'s is a circle of arbitrary radius around~$0$.
Table~\ref{table:submatching-polynomials} lists
a few of the simplest submatching events and the multivariate polynomials associated to them. Note that the first case of the empty matching listed in the table (for which the probability of the submatching event is $1$) corresponds to the known identity
\begin{equation}
\label{eq:asm-const-term}
\textnormal{ASM}(n) = [z_1^0 z_2^2 \ldots z_n^{2n-2}] \left( \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j) \prod_{j=2}^n (1+z_j) \right).
\end{equation}
This identity is already a difficult result. It was proved in \cite{zinn-justin-di-francesco2} by Zinn-Justin and Di Francesco, relying conditionally on a conjectural anti-symmetrization identity that they discovered which itself was proved shortly afterwards by Zeilberger \cite{zeilberger2}. Their proof also relies on a highly nontrivial Pfaffian evaluation due to Andrews \cite{andrews} (see also \cite{andrews-burge, krattenthaler}). The following conjecture can be thought of as a natural generalization or ``deformation'' of \eqref{eq:asm-const-term}.
\begin{table}[h]
\begin{center}
\begin{tabular}{ccc}
\raisebox{40pt}{
\begin{tabular}{c|c|c}
k & $\pi_0$ & $\frac{F_{\pi_0}(w_1,\ldots,w_k)}{w_1\ldots w_{2k}}$ \\[3pt]
\hline & & \\[-1ex]
0 & empty
& 1 \\[8pt]
\raisebox{8pt}{1} & \scalebox{0.14}{\includegraphics{event12}} &
\raisebox{8pt}{1}
\\[10pt]
\raisebox{11pt}{2} & \scalebox{0.3}{\includegraphics{event12-34}} &
\raisebox{11pt}{1}
\\[10pt]
\raisebox{11pt}{2} & \scalebox{0.3}{\includegraphics{event14-23}} &
\raisebox{12pt}{$w_1$}
\end{tabular}
}
& &
\begin{tabular}{c|c|c}
k & $\pi_0$ & $\frac{F_{\pi_0}(w_1,\ldots,w_k)}{w_1\ldots w_{2k}}$ \\[3pt]
\hline & & \\
\raisebox{8pt}{3} &
\scalebox{0.3}{\includegraphics{event12-34-56}} &
\raisebox{8pt}{1}
\\[10pt]
\raisebox{8pt}{3} &
\scalebox{0.3}{\includegraphics{event12-36-45}} &
\raisebox{8pt}{$w_2-w_1 w_2$}
\\[10pt]
\raisebox{8pt}{3} &
\scalebox{0.3}{\includegraphics{event14-23-56}} &
\raisebox{8pt}{$w_1$}
\\[10pt]
\raisebox{8pt}{3} &
\scalebox{0.3}{\includegraphics{event16-23-45}} &
\raisebox{8pt}{$w_1 w_2$}
\\[10pt]
\raisebox{8pt}{3} &
\scalebox{0.3}{\includegraphics{event16-25-34}} &
\raisebox{14pt}{$w_1 w_2^2$}
\end{tabular}
\end{tabular}
\vspace{10pt}
\caption{The polynomials $F_{\pi_0}$ associated with some submatching events, after factoring out the product $w_1\ldots w_{2k}$ which always divides $F_{\pi_0}$. (The first case of the ``empty'' matching refers to the trivial matching of order $k=0$, in which case the submatching event has probability $1$.)
}
\label{table:submatching-polynomials}
\end{center}
\end{table}
\vbox{
\begin{conj}[The algebraic rationality phenomenon] \label{conj:perturbed-asm-iden}
Let $k\ge1$, and let $1\le a_1<\ldots<a_k$ be integers satisfying $a_j\le 2j-1$ for all $j$. There exists a rational function of the form $R(n)=P(n)/\prod_{j=1}^k (4n^2-j^2)^{k+1-j}$, where $P(n)$ is an even polynomial of degree at most $k(k+1)/2$ with dyadic rational coefficients, such that for all $n\ge k+1$ we have that
\begin{align}
\nonumber
[ z_1^0 z_2^2 \ldots z_n^{2n-2} ]& \left(\prod_{j=1}^k z_{j+1}^{2j-a_j} \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j)
\prod_{j=k+2}^n (1+z_j) \right)
\\ & \ \ \ = \textnormal{ASM}(n) R(n). \label{eq:perturbed-asm-iden}
\end{align}
\end{conj}
}
It is unclear whether the specific assumption about the sequence $a_1,\ldots,a_k$ that enters the form of the monomial $\prod_{j=1}^k z_{j+1}^{2j-a_j}$ is necessary to imply the rationality of $R(n)$ in \eqref{eq:perturbed-asm-iden}. The main assumption, which may be sufficient to imply the result, is that
we are looking at a Taylor coefficient that is ``a fixed distance away'' from the coefficient in \eqref{eq:asm-const-term}.
An immediate consequence of Theorem~\ref{thm:explicit-formulas-finite-n} is that Conjecture~\ref{conj:perturbed-asm-iden} essentially implies Conjecture~\ref{conj:rationality-finite-n}, although a small gap remains regarding the validity of \eqref{eq:rationality-finite-n} in the case $n=k$.
\begin{thm}
\label{thm:conj-implies-weak-conj}
Conjecture~\ref{conj:perturbed-asm-iden} implies a weaker version of Conjecture~\ref{conj:rationality-finite-n}
in which \eqref{eq:rationality-finite-n} is only claimed to hold for $n\ge k+1$.
\end{thm}
From the above discussion we see that, while we have been unable to get a complete understanding of the probabilities of submatching events, we have reduced the problem to the essentially algebraic question of understanding the form of the dependence of $n$ of the polynomial coefficients appearing on the left-hand side of \eqref{eq:perturbed-asm-iden}. Moreover, the algorithm for finding the polynomial $F_{\pi_0}$ associated with a submatching event, which will be explained in Section~\ref{sec:wheel-polynomials}, stems from a fairly detailed theoretical understanding of the model, and therefore already helps eliminate much of the mystery surrounding identities such as \eqref{eq:three-eights-finite-n}--\eqref{eq:event12-34-56-prob}.
It should be noted as well that the relations \eqref{eq:asm-const-term} and \eqref{eq:perturbed-asm-iden} belong to a large family of identities known as \textbf{constant term identities}. The study of such identities became popular following the discovery by Dyson of
the identity
\begin{equation}
\label{eq:dyson}
\textnormal{CT}_{z_1,\ldots,z_n} \left( \prod_{1\le i\neq j \le n} \left(1-\frac{z_j}{z_i}\right)^{a_j} \right)
= \frac{(a_1+\ldots+a_n)!}{a_1! \ldots a_n!} \qquad (a_1,\ldots,a_n\ge 0),
\end{equation}
which became known as the Dyson conjecture \cite{dyson}. (Dyson's conjecture was proved by Gunson \cite{gunson} and Wilson \cite{wilson}, and a particularly simple proof was later found by Good \cite{good}.) Research on such identities has been an active area that involves a mixture of techniques from algebraic combinatorics and the theory of special functions. In particular, Sills and Zeilberger \cite{sills-zeilberger} proved a deformation of Dyson's identity in which a coefficient near the constant term is shown to equal the multinomial coefficient on the right-hand side of \eqref{eq:dyson} times a rational function in the exponents $a_1,\ldots,a_n$; this is quite similar in spirit to the claim of Conjecture~\ref{conj:perturbed-asm-iden}.
To conclude this section, we show how the relation \eqref{eq:identity97} can be derived in a simple manner from the results of Fonseca and Zinn-Justin, and in addition derive another identity concerning a finite connectivity event which is not a submatching event.
\begin{thm}
\label{thm:ninety-seven-finiten}
The identity \eqref{eq:identity97} holds for $n\ge 2$,\footnote{This proves part of Conjecture 9 in Zuber's paper \cite{zuber}.} and we have the additional identity
\begin{equation} \label{eq:prov-event12-45}
\mathbb{P}\left(
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-45}}}
\mathrel{\lhd} \Pi_*^{(n)}
\right) = \frac{15}{16} \cdot \frac{(n^2-4)(9n^4+38n^2-63)}{(4n^2-1)^2(4n^2-9)}
\hspace{20.0pt} (n\ge 3)
\end{equation}
(we use the notation for submatching events for convenience, but note that this refers to the event that $\Pi_*^{(n)}$ matches the two pairs $1\matched{\Pi_*^{(n)}}2$ and $4\matched{\Pi_*^{(n)}}5$).
\end{thm}
\begin{proof}
By \eqref{eq:anti-cluster}, we know that
\begin{equation} \label{eq:anti-cluster4}
\mathbb{P}(\textnormal{AC}^{(n)}_4) = \frac{33}{8} \cdot \frac{(n^2-1)(n^2-4)(n^2-9)}{(4n^2-1)^2 (4n^2-9)}.
\end{equation}
On the other hand, by the inclusion-exclusion principle, we have that
\begin{align*}
\mathbb{P}(\textnormal{AC}^{(n)}_4) &=
1-
\mathbb{P}\left(\Pi_*^{(n)} \in
\raisebox{-8pt}{\scalebox{0.1}{\includegraphics{event12}}}
\right)
-
\mathbb{P}\left(\Pi_*^{(n)} \in
\raisebox{-8pt}{\scalebox{0.1}{\includegraphics{event23}}}
\right)
-
\mathbb{P}\left(\Pi_*^{(n)} \in
\raisebox{-8pt}{\scalebox{0.1}{\includegraphics{event34}}}
\right)
\\ & \ \ \ +
\mathbb{P}\left(
\Pi_*^{(n)} \in
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-34}}}
\right)
\\ &
= 1 - 3\cdot \frac{3}{2}\cdot\frac{n^2+1}{4n^2-1}
+
\mathbb{P}\left(
\Pi_*^{(n)} \in
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-34}}}
\right)
\end{align*}
(using \eqref{eq:three-eights-finite-n} and the rotation-invariance of $\Pi_*^{(n)}$).
Substituting the probability from \eqref{eq:anti-cluster4} and solving for $\mathbb{P}\left(
\Pi_*^{(n)} \in
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-34}}}
\right)$ gives \eqref{eq:identity97}. For the second identity \eqref{eq:prov-event12-45}, perform a similar inclusion-exclusion computation for the event $\textnormal{AC}^{(n)}_5$, whose probability is given according to \eqref{eq:anti-cluster} by
$$
\mathbb{P}(\textnormal{AC}^{(n)}_5) = \frac{11}{16} \cdot \frac{(n^2-1)(n^2-4)(n^2-9)}{(4n^2-1)^2 (4n^2-9)}.
$$
The details of the computation are easy and left to the reader.
\end{proof}
\subsection{Consequences for loop percolation on a half-plane}
\label{sec:consequences-halfplane}
Let us return to the original setting of loop percolation on a half-plane discussed earlier. The following result allows us to deduce exact results on probabilities of local connectivity events in the half-plane from corresponding results in the cylindrical model.
\begin{lem}
Let $A \subset \mathbb{Z}\times \mathbb{Z}$ be a finite set, and let $E_n$ denote the finite connectivity events
$$ E_n = \bigcap_{(j,k)\in A} \left\{ j\matched{\Pi_*^{(n)}} k \right\} $$
associated with $A$ (which are defined for large enough $n$). Then we have $\mathbb{P}(E_n)\to \mathbb{P}(E)$ as $n\to\infty$, where $E = \bigcap_{(j,k)\in A} \left\{ j\matched{\Pi_*} k \right\}$.
\end{lem}
\begin{proof} Couple the cylindrical and half-planar loop percolation by using the same random bits to select the edge configuration incident to the vertices in the region $V(\textnormal{LP}_n)$ (defined in \eqref{eq:v-lpgraph-n}).
With this coupling, it is easy to see that the symmetric difference $E_n \triangle E$ of the events $E_n$ and $E$ is contained in the event $B_n$ that one of the half-planar loop percolation paths starting at the point $(j,-j)$ for some $j$ belonging to one of the pairs $(j,k)\in A$ reaches the complement $\mathbb{Z}^2\setminus V(\textnormal{LP}_n)$. Since the loop percolation paths are almost surely finite, we have $\lim_{n\to\infty} \mathbb{P}(B_n) = \mathbb{P}\left(\bigcap_{n=1}^\infty B_n\right) = 0$, and therefore we get that
$|\mathbb{P}(E_n)-\mathbb{P}(E)|\le \mathbb{P}(E_n \triangle E) \le \mathbb{P}(B_n) \to 0$ as $n\to\infty$.
\end{proof}
As immediate corollaries from the lemma we get the following facts: first, Conjecture~\ref{conj:rationality-finite-n} (or the weaker version of it mentioned in Theorem~\ref{thm:conj-implies-weak-conj}) implies Conjecture~\ref{conj:rationality}. Second, Theorem~\ref{thm:two-explicit-cases} follows as a limiting case of \eqref{eq:three-eights-finite-n} and \eqref{eq:identity97}. Third, we have the following explicit formulas for events in the half-plane model corresponding to \eqref{eq:anti-cluster} and \eqref{eq:prov-event12-45}.
\begin{thm}
\label{thm:event135}
For $k\ge 1$, let $\textnormal{AC}_k$ denote the anti-cluster event that no two of the numbers $1,\ldots,k$ are matched under $\Pi_*$. We have the formulas
$$
\mathbb{P}\left(\textnormal{AC}_k \right) = \frac{1}{2^{\lfloor k/2\rfloor\cdot \lfloor k/2+1\rfloor}\textnormal{ASM}(k) R_k(k)},
$$
where $R_k(\cdot)$ is defined in \eqref{eq:complicated-product-def} (see Table~\ref{table:anti-cluster}), and
$$
\mathbb{P}\left(
\raisebox{-6pt}{\scalebox{0.2}{\includegraphics{event12-45}}}
\mathrel{\lhd} \Pi_* \right) = \frac{135}{1024}.
$$
\end{thm}
\begin{table}[h]
$$
\begin{array}{c|c|c|c|c|c|c|c}
k & 1&2&3&4&5&6&7 \\ \hline &&&&&&&\\[-1.8ex]
\mathbb{P}\left(\Pi_*\in\textnormal{AC}_k \right) &
\displaystyle \frac58 &
\displaystyle \frac14 &
\displaystyle \frac{33}{512} &
\displaystyle \frac{11}{1024} &
\displaystyle \frac{2431}{2^{21}} &
\displaystyle \frac{85}{2^{20}} &
\displaystyle \frac{126293}{2^{35}}
\displaystyle \end{array}
$$
\caption{Probabilities of the anti-cluster event $\textnormal{AC}_k$ for $k=1,\ldots,7$.}
\label{table:anti-cluster}
\end{table}
\section{The theory of wheel polynomials and the qKZ equation}
\label{sec:wheel-polynomials}
Our goal in this section is to prove Theorem~\ref{thm:explicit-formulas-finite-n}. The proof will be based on an extension of an algebraic theory that was developed in a recent series of papers \cite{fonseca-zinn-justin, zinn-justin, zinn-justin-di-francesco1, zinn-justin-di-francesco2}, where it is shown that a tool from the statistical physics literature known as the \textbf{quantum Knizhnik-Zamolodchikov equation} (or \textbf{qKZ equation}) can be applied to the study of the connectivity pattern of cylindrical loop percolation.
Mathematically, the qKZ equation as applied to the present setting reduces to the analysis of a vector space of multivariate polynomials satisfying a condition known as the \textbf{wheel condition}. (Similar and more general wheel conditions are also discussed in the papers \cite{feigin-etal, kasatani, pasquier}.)
We call such polynomials \textbf{wheel polynomials}.
In Subsections~\ref{sec:wheel-polynomials-background}--\ref{sec:p-nested-matchings} we will survey the known results regarding the theory of wheel polynomials that are needed for our purposes. In Subsection~\ref{sec:gen-submatching} we present a new result (Theorem~\ref{thm:submatching-event-expansion}) on expansions of families of wheel polynomials associated with submatching events, that is another main result of the paper. In Subsection~\ref{sec:proof-main-thm} we show how to derive Theorem~\ref{thm:explicit-formulas-finite-n} from
Theorem~\ref{thm:submatching-event-expansion}.
\subsection{Background}
\label{sec:wheel-polynomials-background}
Let $\noncn{n}$ denote as before the set of noncrossing matchings of $1,\ldots,2n$. It is well-known that elements of $\noncn{n}$ are in canonical bijection with the set of \textbf{Dyck paths} of length $2n$, and with the set of Young diagrams contained in the staircase shape $(n-1,n-2,\ldots,1)$. These bijections will play an important role, and are illustrated in Fig.~\ref{fig:noncn-bijections}; see \cite[Section 2]{fonseca-zinn-justin} for a detailed explanation.
\begin{figure}
\begin{center}
\begin{tabular}{cc}
\scalebox{0.7}{\includegraphics{noncrossing10-dyck}} &
\scalebox{0.7}{\includegraphics{noncrossing10-youngdiag}}
\end{tabular}
\caption{The noncrossing matching from Fig.~\ref{fig:example-noncrossing-matching}(a) represented as a Dyck path and as a Young diagram.}
\label{fig:noncn-bijections}
\end{center}
\end{figure}
We adopt the following notation related to these bijections. If $\pi\in\noncn{n}$, let $\pi_k$ be $1$ if $\pi(k)>k$ (``$k$ is matched to the right'') or $-1$ if $\pi(k)<k$ (``$k$ is matched to the left''). The vector $(\pi_1,\ldots,\pi_{2n})$ is the encoding of $\pi$ as the sequence of steps in the associated Dyck path. Let $\pi^+$ denote the sequence $(a_1,\ldots,a_n)$ of positions where $\pi_k=1$ (the positions of increase of the Dyck path). Encoding $\pi$ in this way maps $\noncn{n}$ bijectively onto the set
\begin{align}
\noncn{n}^+ &= \big\{ \mathbf{a}=(a_1,\ldots,a_n)\,:\, 1\le a_1<\ldots< a_n \le 2n-1
\nonumber\\ & \hspace{160pt} \textrm{and }a_j\le 2j-1 \textrm{ for all }j \big\}.
\label{eq:noncn-plus-def}
\end{align}
The Young diagram associated with a noncrossing matching $\pi\in\noncn{n}$ is denoted~$\lambda_\pi$. For $\pi,\sigma\in\noncn{n}$, denote $\sigma\nearrow \pi$ if $\lambda_\sigma$ is obtained from $\lambda_\pi$ by the addition of a single box, and $\pi \preceq \sigma$ if $\lambda_\pi$ is contained in $\lambda_{\sigma}$. (This partial order on $\noncn{n}$ is precisely the order with respect to which the matchings $\pi^n_{\textrm{min}}$ and $\pi^n_{\textrm{max}}$ defined in Subsection~\ref{sec:cylinder-background} are minimal and maximal, respectively.)
Denote $\pi\nearrow_j \sigma$ if $\lambda_\sigma$ is obtained from $\lambda_\pi$ by adding a box in a position that, in the coordinate system of Fig.~\ref{fig:noncn-bijections}, lies vertically above the positions $j$ and $j+1$ on the horizontal axis.
\begin{defi}[Wheel polynomials]
Let $q=e^{2\pi i/3}$.
A \textbf{wheel polynomial of order $n$} is a polynomial $p(\mathbf{z}) = p(z_1,\ldots,z_{2n})$ having the following properties:
\begin{enumerate}
\item $p$ is a homogeneous polynomial of total degree $n(n-1)$;
\item $p$ satisfies the \textbf{wheel condition}
\begin{equation}
\label{eq:wheel-condition}
p(z_1,\ldots,z_{2n})_{\big| z_k = q^2 z_j = q^4 z_i} = 0 \qquad (1\le i<j<k\le 2n).
\end{equation}
\end{enumerate}
Denote by $\wheelpoly{n}$ the vector space of wheel polynomials of order $n$.
\end{defi}
It is easy to see that the condition \eqref{eq:wheel-condition} depends only on the cyclical ordering of the variables $z_1,\ldots,z_{2n}$ (in other words, the space $\wheelpoly{n}$ is invariant under the rotation action $p(z_1,\ldots,z_{2n})\mapsto p(z_2,\ldots,z_{2n},z_1)$); hence the name ``wheel polynomials.''
The algebraic properties of wheel polynomials are quite elegant and of direct relevance to our study of connectivity patterns in loop percolation. Below we survey some of the known theory.
\begin{example}
\label{example:minimal-wheelpoly}
Let $p(\mathbf{z})=\prod_{1\le i<j\le n} \left(qz_i - q^{-1} z_j \right) \prod_{n+1 \le i<j \le 2n} \left(qz_i - q^{-1} z_j\right)$. It is easy to check that $p$ is a wheel polynomial. Indeed, it is homogeneous of the correct degree, and if $1\le i<j<k\le 2n$ and we make the substitution $z_k=q^2 z_j = q^4 z_i$, then, if $j\le n$ there will be a zero factor in the first product $\prod_{1\le i<j\le n} \left(qz_i - q^{-1} z_j \right)$, and similarly if $j>n$ then there will be a zero factor in the second product. In either case the wheel condition \eqref{eq:wheel-condition} is satisfied.
\end{example}
Two types of pointwise evaluations of a wheel polynomial will be of particular interest: at the point $\mathbf{z}=(1,1,\ldots,1)$ and at a point $\mathbf{z}=(q^{-\pi_1},\ldots,q^{-\pi_{2n}})$ associated with the steps of a Dyck path, where $\pi\in\noncn{n}$. As a shorthand, we denote $p(\mathbf{1})=p(1,\ldots,1)$ and $p(\pi)=p(q^{-\pi_1},\ldots,q^{-\pi_{2n}})$ and refer to these values as the \textbf{$\mathbf{1}$-evaluation} and \textbf{$\pi$-evaluation} of $p$, respectively.
\begin{thm}
\label{thm:sufficient-evaluations}
A polynomial $p\in \wheelpoly{n}$ is determined uniquely by its values $p(\pi)$ as $\pi$ ranges over the noncrossing matchings in $\noncn{n}$. Equivalently, the $\pi$-evaluation linear functionals $(\operatorname{ev}_\pi)_{\pi\in \noncn{n}}$ defined by $\operatorname{ev}_\pi(p)= p(\pi)$ span the dual vector space $(\wheelpoly{n})^*$.
\end{thm}
\begin{proof} See \cite[Appendix C]{fonseca-zinn-justin-doubly}.
\end{proof}
In particular, it follows that $\dim \wheelpoly{n}\le |\noncn{n}|=\operatorname{Cat}(n)$. We shall soon see that this is an equality, that is, the evaluation functionals $(\operatorname{ev}_\pi)_{\pi\in \noncn{n}}$ are in fact a basis of $(\wheelpoly{n})^*$. The proof of this fact involves a remarkable explicit construction of a family of wheel polynomials that will turn out to be a basis of $\wheelpoly{n}$ dual to $(\operatorname{ev}_\pi)_{\pi\in \noncn{n}}$, and will play a central role in our analysis.
For a multivariate polynomial $p\in \mathbb{C}[z_1,\ldots,z_m]$ and $1\le j< m$, define the divided difference operator $\partial_j:\mathbb{C}[z_1,\ldots,z_m]\to \mathbb{C}[z_1,\ldots,z_m]$ by
$$
(\partial_j p)(z_1,\ldots,z_m) =
\frac{p(z_1,\ldots,z_{j+1},z_j,\ldots,z_{2n})-
p(z_1,\ldots,z_j,z_{j+1},\ldots,z_{2n})}{z_{j+1}-z_j}.
$$
The next two lemmas are easy to check; see \cite[Sec.~4.2.1]{zinn-justin} for a related discussion.
\begin{lem}
\label{lem:divided-difference-wheelpoly}
If $p\in \wheelpoly{n}$ and $1\le j\le 2n-1$, then $(q z_j - q^{-1} z_{j+1}) \partial_j p$ is also in $\wheelpoly{n}$.
\end{lem}
\begin{lem}
If for some $1\le j\le 2n-1$, $\pi\in\noncn{n}$ satisfies $j\matched{\pi}j+1$, then the set of preimages $e_j^{-1}(\pi)$ of $\pi$ under $e_j$ (the Temperley-Lieb operator defined in \eqref{eq:temperley-lieb-gens}) consists of:
\begin{enumerate}
\item a noncrossing matching $\hat{\pi}\in \noncn{n}$ satisfying $\pi\nearrow_j \hat{\pi}$, if such a matching exists (i.e., if the associated Young diagram is still contained in the staircase shape $(n-1,\ldots,1)$); and
\item a set of noncrossing matchings $\sigma\in \noncn{n}$ satisfying $\sigma\preceq \pi$.
\end{enumerate}
\end{lem}
The last two lemmas make it possible to construct a family of wheel polynomials indexed by noncrossing matchings in $\noncn{n}$ (or, equivalently, by Young diagrams contained in the staircase shape $(n-1,\ldots,1)$) recursively. We start with an explicit polynomial known to be in $\wheelpoly{n}$---a scalar multiple of the polynomial from Example~\ref{example:minimal-wheelpoly} above---which we associate with the minimal matching $\pi^n_{\textrm{min}}$ (which corresponds to the empty Young diagram). We then define for each noncrossing matching $\pi$ a polynomial obtained from the polynomials of matchings preceding $\pi$ in the order $\preceq$ using linear combinations and the operation of Lemma~\ref{lem:divided-difference-wheelpoly}. The precise definition is as follows.
\begin{defi}[qKZ basis]
The \textbf{qKZ polynomials} are a family of polynomials $(\Psi_\pi)_{\pi\in\noncn{n}}$ defined using the following recursion on Young diagrams:
\begin{align}
\Psi_{\pi^n_{\textrm{min}}}(\mathbf{z}) &= (-3)^{-\binom{n}{2}} \prod_{1\le i<j\le n} \left(qz_i - q^{-1} z_j \right) \prod_{n+1 \le i<j \le 2n} \left(qz_i - q^{-1} z_j\right), \label{eq:qkz1} \\
\Psi_\pi(\mathbf{z}) &= \left(qz_j - q^{-1} z_{j+1}\right) \partial_j \Psi_{\sigma} -
\sum_{\nu \in e_j^{-1}(\sigma) \setminus \{\pi,\sigma\}} \Psi_\nu
\ \ \ \textnormal{ if }\sigma \nearrow_j \pi.
\label{eq:qkz2}
\end{align}
\end{defi}
Since for a given noncrossing matching $\pi$, the choices of $\sigma$ and $j$ such that $\sigma\nearrow_j \pi$ are in general not unique, it is not clear that the above definition makes sense. The fact that it does is a nontrivial statement, which we include as part of the next result.
\begin{thm}
\label{thm:qkz-poly-properties}
The qKZ polynomials satisfy:
\begin{enumerate}
\item $\Psi_\pi$ is well-defined, i.e., the result of the recursive computation does not depend on the order in which boxes are added to the Young diagram.
\item $\Psi_\pi \in \wheelpoly{n}$.
\item Let $\rho(\pi)$ denote the rotation operator acting on $\pi$, defined by $(\rho (\pi))(k)=\pi(k+1)$ for $1\le k<2n$ and $(\rho (\pi))(2n)=\pi(1)$. Then we have
\begin{equation}
\Psi_{\rho(\pi)}(z_1,\ldots,z_{2n}) = \Psi_\pi(z_2,\ldots,z_{2n},z_1).
\label{eq:qkz3}
\end{equation}
\end{enumerate}
\end{thm}
\begin{proof}
See \cite[Section 4.2]{zinn-justin}.
\end{proof}
As explained in \cite[Sections 4.1--4.2]{zinn-justin}, equations \eqref{eq:qkz1}--\eqref{eq:qkz3} together are equivalent to a different system of equations which forms (a special case of) the qKZ equation. A different way of solving the same system, which we will not use here, is described in \cite{de-gier-lascoux-sorrell}.
If $\pi\in\noncn{n}$ and $1\le j\le 2n-1$ is a number such that $j\matched{\pi}j+1$, denote by $\hat{\pi}_j$ the noncrossing matching in $\noncn{n-1}$ obtained by deleting the arc connecting $j$ and $j+1$ from the diagram of the matching and relabelling the remaining elements. This operation makes it possible to perform many computations recursively. The next result provides an important example.
\begin{thm}
\label{thm:qkz-eval-recursion}
If $j\matched{\pi}j+1\in \noncn{n}$ then for any $\sigma\in \noncn{n}$ we have
\begin{equation}
\label{eq:qkz-eval-recursion}
\Psi_\pi(\sigma) = \begin{cases} 3^{n-1} \Psi_{\hat{\pi}_j}(\hat{\sigma}_j)
& \textnormal{if }j\matched{\sigma}j+1, \\ 0 & \textnormal{otherwise}.
\end{cases}
\end{equation}
\end{thm}
\begin{proof}
See \cite[Section 4.2]{zinn-justin}.
\end{proof}
\begin{cor}
\begin{enumerate}
\item For all $\pi,\sigma\in\noncn{n}$, we have
\begin{equation} \label{eq:psi-sigma-evaluations}
\Psi_\pi(\sigma) = \delta_{\pi,\sigma} = \begin{cases}1 & \textrm{if }\pi=\sigma, \\ 0 & \textrm{otherwise}. \end{cases}
\end{equation}
\item The qKZ polynomials $(\Psi_\pi)_{\pi\in\noncn{n}}$ are a basis for $\wheelpoly{n}$.
\item The $\pi$-evaluation linear functionals $(\operatorname{ev}_\pi)_{\pi\in\noncn{n}}$ are the basis of $(\wheelpoly{n})^*$ dual to $(\Psi_\pi)_{\pi\in\noncn{n}}$.
\item $\dim \wheelpoly{n} = \operatorname{Cat}(n)$.
\end{enumerate}
\end{cor}
\begin{proof}
Part 1 follows by induction from \eqref{eq:qkz-eval-recursion}, and parts 2--4 are immediate from part 1.
\end{proof}
\begin{example}[{\cite[Section 4.3.1]{zinn-justin}}]
Let $p(\mathbf{z})=s_\lambda(z_1,\ldots,z_{2n})$, the Schur polynomial associated with the Young diagram $\lambda=(n-1,n-1,\ldots,2,2,1,1)$. That is, we have explicitly
\begin{equation} \label{eq:schur-double-staircase}
p(\mathbf{z}) = \left( \prod_{1\le i<j\le 2n} (z_j-z_i) \right)^{-1} \det \left( z_i^j \right)_{
\!\!\!\begin{array}{l} \scriptstyle 1\le i\le 2n, \\[-1ex]
\scriptstyle 0\le j\le 3n-2, \ j\equiv 0,1\textrm{ (mod $3$)}
\end{array}
}
\end{equation}
Then $p$ is homogeneous of degree $n(n-1)$, and the matrix whose determinant appears in \eqref{eq:schur-double-staircase} has the property that if we make the substitution $z_k=q^2 z_j = q^4 z_i$ for some $1\le i<j<k\le 2n$, the columns with index $i,j,k$ of the matrix become linearly dependent (take the linear combination with coefficients $1,q^2,q^4$), causing $p$ to be $0$. Thus, $p$ is a wheel polynomial.
Since $p$ is also a symmetric polynomial, it satisfies $p(\pi)=p(\pi')$ for any $\pi,\pi'\in \noncn{n}$. By Theorem~\ref{thm:sufficient-evaluations} it follows that $p$ is the unique symmetric wheel polynomial up to scalar multiplication. Note that this Schur function is also known to be the partition function of the six-vertex model with domain wall boundary condition at the ``combinatorial point,'' and played an important role in the study of the enumeration of alternating sign matrices; see \cite{okada, stroganov} and \cite[Section 2.5.6]{zinn-justin}.
\end{example}
\bigskip
Denote $\psi_\pi = \Psi_\pi(\mathbf{1})$ and $\boldsymbol{\psi}_n = (\psi_\pi)_{\pi\in\noncn{n}}$. The vector $\boldsymbol{\psi}_n$ is related to the connectivity pattern $\Pi_*^{(n)}$ in cylindrical loop percolation, as explained in the following result.
\begin{thm}
\label{thm:psi-mu-related}
The vector $\boldsymbol{\psi}_n$ is a solution of the linear system \eqref{eq:razumov-stroganov-linear-system}.
Consequently it is a scalar multiple of the probability distribution $\boldsymbol{\mu}_n$ of $\Pi_*^{(n)}$. That is, for any $\pi\in\noncn{n}$ we have $ \psi_\pi = a_n \mu_\pi $, where $a_n = \sum_{\pi\in\noncn{n}} \psi_\pi$.
\end{thm}
\begin{proof}
See \cite[Sec.~4.3]{zinn-justin}.
\end{proof}
We will see later (see Theorem~\ref{thm:sum-rule} below) that the normalization constant $a_n$ is equal to $\textnormal{ASM}(n)$. Assuming this fact temporarily, Theorem~\ref{thm:psi-mu-related} implies the following interesting and purely algebraic characterization of the numbers $\mu_\pi = \mathbb{P}(\Pi_*^{(n)} = \pi)$, which does not seem to have been noted before.
\begin{thm}
\label{thm:lin-alg-expansion}
For any wheel polynomial $p\in\wheelpoly{n}$, we have that $p(\mathbf{1}) = \textnormal{ASM}(n) \sum_{\pi\in\noncn{n}} \mu_\pi p(\pi)$. In other words, the $\mathbf{1}$-evaluation functional $\operatorname{ev}_\mathbf{1}$ has the expansion
$$ \operatorname{ev}_\mathbf{1} = \textnormal{ASM}(n) \sum_{\pi\in\noncn{n}} \mu_\pi \operatorname{ev}_\pi $$
as a linear combination of the $\pi$-evaluation functionals.
\end{thm}
\begin{proof}
If $p=\sum_{\pi\in\noncn{n}} b_\pi \Psi_\pi$ gives the expansion of $p$ in terms of the qKZ basis polynomials, then
$$ p(\mathbf{1}) = \sum_{\pi\in\noncn{n}} b_\pi \Psi_\pi(\mathbf{1}) =
\sum_{\pi\in\noncn{n}} b_\pi \psi_\pi =
\textnormal{ASM}(n) \sum_{\pi\in\noncn{n}} \mu_\pi b_\pi.
$$
On the other hand, taking the $\pi$-evaluation of the expansion $p=\sum_{\sigma\in\noncn{n}} b_\sigma \Psi_\sigma$ easily implies using \eqref{eq:psi-sigma-evaluations} that $b_\pi = p(\pi)$.
\end{proof}
A referee has pointed out to us that Theorem~\ref{thm:lin-alg-expansion} can also be deduced from some of the results (specifically, Theorem~2 and equations (7) and (8)) of \cite{de-gier-lascoux-sorrell}.
Next, we construct a second important family of wheel polynomials defined using contour integration.
\begin{defi}[Integral wheel polynomials]
Denote
\begin{align*}
\mathcal{A}_n &= \big\{ \mathbf{a}=(a_1,\ldots,a_n)\,:\, 1\le a_1\le \ldots \le a_n \le 2n-1
\\ & \hspace{160pt} \textrm{and }a_j\le 2j-1 \textrm{ for all }j \big\}.
\end{align*}
(These sequences generalize the steps of increase of Dyck paths---compare with \eqref{eq:noncn-plus-def}.)
For a sequence $\mathbf{a}=(a_1,\ldots,a_n)\in\mathcal{A}_n$, denote
\begin{align}
&\Phi_{\mathbf{a}}(z_1,\ldots,z_{2n}) =
(-1)^{\binom{n}{2}} \prod_{1\le j<k\le 2n} (qz_j-q^{-1} z_k)
\nonumber \\ & \hspace{30.0pt} \times \overbrace{\mathlarger{\oint}\ldots\mathlarger{\oint}}^{n}
\frac{\displaystyle \prod_{1\le j<k\le n} (w_k-w_j)(qw_j-q^{-1} w_k)}{\displaystyle \prod_{j=1}^n \left(
\prod_{k=1}^{a_j} (w_j-z_k) \prod_{k=a_j+1}^{2n} (qw_j-q^{-1} z_k)
\right)} \prod_{j=1}^n \frac{dw_j}{2\pi i},
\label{eq:phi-a-def}
\end{align}
where the contour of integration for each variable $w_j$ surrounds in a counterclockwise direction the singularities at $z_k\ (1\le k\le a_j)$, but does not surround the singularities at $q^{-2} z_k\ (a_j+1\le k\le 2n)$.
\end{defi}
We will be mostly interested in $\Phi_\mathbf{a}$ for $\mathbf{a}\in \noncn{n}^+$---the set of such polynomials will turn out to be a basis for $\wheelpoly{n}$---but the case when $\mathbf{a}$ is in the larger set $\mathcal{A}_n$ will play a useful intermediate role.
The above definition of $\Phi_\mathbf{a}$ is somewhat difficult to work with. In particular, it is not even obvious that the $\Phi_\mathbf{a}$'s are polynomials. However, the contour integrals can be evaluated using the residue formula, which gives a more concrete representation of the $\Phi_\mathbf{a}$'s that makes it possible to show that they are in fact wheel polynomials and prove additional properties.
\begin{lem}
\begin{enumerate}
\item
The functions $\Phi_\mathbf{a}$ can be written explicitly as follows.
\begin{align*}
& \hspace{-20.0pt} \Phi_{\mathbf{a}}(\mathbf{z}) =
(-1)^{\binom{n}{2}} \prod_{1\le j<k\le 2n} (qz_j-q^{-1} z_k) \\ &
\times \sum_{\tiny \begin{array}{c} (m_1,\ldots,m_n) \\ 1\le m_j\le a_j, m_j\neq m_k \end{array}}
\frac{\displaystyle \prod_{1\le j<k\le n} (z_{m_k}-z_{m_j})(qz_{m_j}-q^{-1} z_{m_k})}{\displaystyle \prod_{j=1}^n \left(
\prod_{\scriptsize \begin{array}{c} k=1 \\ k\neq a_j \end{array}}^{a_j} (z_{m_j}-z_k) \prod_{k=a_j+1}^{2n} (qz_{m_j}-q^{-1} z_k)
\right) }
\\[3pt] & \hspace{-20.0pt}
\ \ \ \ \ =
(-1)^{\binom{n}{2}}
\\[3pt] &\hspace{-10.0pt} \times \hspace{-15.0pt} \sum_{\tiny \begin{array}{c} M=(m_1,\ldots,m_n) \\ 1\le m_j\le a_j, m_j\neq m_k \end{array}}
\hspace{-15.0pt} (-1)^{\operatorname{inv}(M)}
\frac{\displaystyle \prod_{1\le j<k\le n} (q z_{m_j} - q^{-1} z_{m_k}) \hspace{-20pt} \prod_{\tiny \begin{array}{c}1\le j <k\le 2n\\ j\notin M\textnormal{ or }j=m_p, k\le a_p\end{array}} \hspace{-15.0pt} (qz_j-q^{-1} z_k) }
{\displaystyle \prod_{j=1}^n \prod_{\tiny \begin{array}{c} 1\le k\le a_j \\ k\notin M \textnormal{ or }k>m_j\end{array}} (z_{m_j}-z_k)},
\end{align*}
where for a sequence $M=(m_1,\ldots,m_n)$, $\operatorname{inv}(M)$ denotes the number of inversions of $M$, given by
$\operatorname{inv}(M) =\#\{ 1\le i<j\le n\,:\, m_i>m_j \}$.
\item
$\Phi_\mathbf{a}$ is a wheel polynomial of order $n$.
\end{enumerate}
\end{lem}
\begin{proof} See \cite[Section 3]{zinn-justin-di-francesco2}.
\end{proof}
For each sequence $\mathbf{a}=(a_1,\ldots,a_n)\in\mathcal{A}_n$, let $(C_{\mathbf{a},\sigma})_{\sigma\in\noncn{n}}$ be coefficients such that
\begin{equation} \label{eq:def-c-a-sigma}
\Phi_\mathbf{a} = \sum_{\sigma\in\noncn{n}} C_{\mathbf{a},\sigma} \Psi_{\sigma}.
\end{equation}
For $\pi\in\noncn{n}$ denote $C_{\pi,\sigma} = C_{\mathbf{a},\sigma}$ with $\mathbf{a}=\pi^+$. The coefficients $(C_{\pi,\sigma})_{\sigma\in\noncn{n}}$ have the property that
\begin{equation} \label{eq:def-c-pi-sigma}
\Phi_\pi = \sum_{\sigma\in\noncn{n}} C_{\pi,\sigma} \Psi_{\sigma}.
\end{equation}
\begin{lem} We have the relations
\begin{align*}
\Phi_\mathbf{a}(\sigma) &= C_{\mathbf{a},\sigma} \qquad (\mathbf{a}\in\mathcal{A}_n), \\
\Phi_\pi(\sigma) &= C_{\pi,\sigma} \qquad (\pi\in\noncn{n}).
\end{align*}
\end{lem}
\begin{proof}
This follows immediately from \eqref{eq:psi-sigma-evaluations} by considering the $\pi$-evaluation of both sides of \eqref{eq:def-c-a-sigma}--\eqref{eq:def-c-pi-sigma}.
\end{proof}
The next result is a recurrence relation that makes it possible to recursively compute the quantities $\Phi_\mathbf{a}(\sigma) = C_{\mathbf{a},\sigma}$, analogously to Theorem~\ref{thm:qkz-eval-recursion}.
\begin{thm}
If $\sigma\in\noncn{n}$ has a ``little arc'' $j\matched{\sigma}j+1$ and $\mathbf{a} \in\mathcal{A}_n$, let $p$ be the number of times $j$ appears in $\mathbf{a}$, so that we can write
$$ \mathbf{a} = (a_1,\ldots,a_k,\overbrace{j,\ldots,j}^{p\textrm{ times}},a_{k+p+1},\ldots,a_n)
$$
where $a_k<j<a_{k+p+1}$. In the case when $p>0$, denote
\begin{equation}
\label{eq:hat-a-def}
\hat{\mathbf{a}}_j = (a_1,\ldots,a_k,\overbrace{j,\ldots,j}^{p-1\textrm{ times}},a_{k+p+1}-2,\ldots,a_n-2).
\end{equation}
Then we have
\begin{equation}
\label{eq:phi-a-recurrence}
\Phi_{\mathbf{a}}(\sigma) = \begin{cases}
3^{n-1} \chi(p) \Phi_{\hat{\mathbf{a}}_j}(\hat{\sigma}_j)
& \textrm{if }p>0 \ \ (\textrm{i.e., if }j\in\{a_1,\ldots,a_n\}), \\
0 & \textrm{if }p=0,
\end{cases}
\end{equation}
where
\begin{equation} \label{eq:character-mod-three}
\chi(p) = \frac{\displaystyle q^p-q^{-p}}{\displaystyle q-q^{-1}} = \begin{cases}
0 & \textnormal{if }p \equiv 0 \textnormal{ (mod }3), \\
1 & \textnormal{if }p \equiv 1 \textnormal{ (mod }3), \\
-1 & \textnormal{if }p\equiv 2 \textnormal{ (mod }3).
\end{cases}
\end{equation}
\end{thm}
\begin{proof}
See \cite[Appendix A]{zinn-justin-di-francesco2}.
\end{proof}
Note that if $\mathbf{a}=\pi^+ \in \noncn{n}^+$ then $\hat{\mathbf{a}}_j$ is in $\mathcal{A}_{n-1}$ but not necessarily in $\noncn{n-1}^+$, which is the reason why it was necessary to consider the family of polynomials indexed by the larger set $\mathcal{A}_n$ even though we are mainly interested in the polynomials $\Phi_\pi$.
Like \eqref{eq:qkz-eval-recursion}, the recurrence relation \eqref{eq:phi-a-recurrence} can be solved to give an explicit formula for $C_{\mathbf{a},\sigma}$.
\begin{thm}
For $\mathbf{a}=(a_1,\ldots,a_n)\in\noncn{n}^+$ and $\sigma\in\noncn{n}$, the coefficient $C_{\mathbf{a},\sigma}$ is given by:
\begin{equation}
C_{\mathbf{a},\sigma} = \prod_{j<k,\ j\matched{\sigma}k} \chi(p_{j,k}(\mathbf{a})),
\label{eq:cmatrix-explicit}
\end{equation}
where we denote $p_{j,k}(\mathbf{a}) = \#\{ m\,:\, j\le a_m <k \}-\tfrac12 (k-j-1)$.
\end{thm}
\begin{proof}
See \cite[Appendix A]{zinn-justin-di-francesco2}.
\end{proof}
\begin{thm}
The matrix $(C_{\pi,\sigma})_{\pi,\sigma\in\noncn{n}}$ is triangular with respect to the partial order $\preceq$ on link patterns, and has $1$s on the diagonal (i.e., $C_{\pi,\pi}=1$ for all $\pi\in\noncn{n}$), hence it is invertible.
\end{thm}
\begin{proof} See \cite[Sec.~3.3.2]{fonseca-zinn-justin}.
\end{proof}
\begin{cor}
$(\Phi_\pi)_{\pi\in \noncn{n}}$ form a basis for $\wheelpoly{n}$.
\end{cor}
Denote the coefficients of the inverse matrix of $(C_{\pi,\sigma})_{\pi,\sigma\in\noncn{n}}$ by $\tilde{C}_{\pi,\sigma}$, or $\tilde{C}_{\pi,\mathbf{a}}$ if $\mathbf{a}=\sigma^+$.
In other words, the coefficients $\tilde{C}_{\pi,\sigma}$ are defined by the relation
$$
\Psi_\pi = \sum_{\sigma\in\noncn{n}} \tilde{C}_{\pi,\sigma} \Phi_\sigma.
$$
We have defined two square matrices $\mathbf{C}_n = (C_{\pi,\sigma})_{\pi,\sigma\in\noncn{n}}$ and $\tilde{\mathbf{C}}_n=\mathbf{C}_n^{-1}=(\tilde{C}_{\pi,\sigma})_{\pi,\sigma\in\noncn{n}}$, which are the change of basis matrices transitioning between the two wheel polynomial bases we defined.
Note that $\tilde{C}_{\pi,\sigma}$, like $C_{\pi,\sigma}$, are integers, but there does not seem to be a simple formula for them. The matrices $\mathbf{C}_n$ and $\tilde{\mathbf{C}}_n$ for $n=2,3,4$ are shown in Table~\ref{table:transition-matrices}.
\begin{table}[h]
$$
\begin{array}{c|cc}
n & \mathbf{C}_n & \tilde{\mathbf{C}}_n \\
\hline \\[-1ex]
2 & \begin{pmatrix} 1&0\\0&1 \end{pmatrix} & \begin{pmatrix} 1&0\\0&1 \end{pmatrix}
\\[12pt]
3 &
\begin{pmatrix} 1&0&0&0&0\\0&1&0&1&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1
\end{pmatrix}
&
\begin{pmatrix} 1&0&0&0&0\\0&1&0&-1&0\\0&0&1&0&0\\0&0&0&1&0\\0&0&0&0&1
\end{pmatrix}
\\[35pt]
4 &
\left(
\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{smallmatrix}
\right)
&
\left(
\begin{smallmatrix}
1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 \\
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -1 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\
\end{smallmatrix}
\right)
\end{array}
$$
\caption{The matrices $\mathbf{C}_n, \tilde{\mathbf{C}}_n$ for $n=2,3,4$ (the ordering of the rows and columns corresponds to decreasing lexicographic order of the Young diagrams associated with noncrossing matchings). Note that $\tilde{\mathbf{C}}_n$ can have entries larger in magnitude than $1$; the smallest value of $n$ for which this happens is $n=7$.}
\label{table:transition-matrices}
\end{table}
As we shall see, these matrices will enable us to translate certain sums of the qKZ polynomials (which are associated with certain connectivity events in loop percolation) to linear combinations of the $\Phi_{\pi}$'s. To get back to numerical results about the numbers $\psi_\pi = \Psi_\pi(\mathbf{1})$ (which are related to the probabilities $\mu_\pi$ according to Theorem~\ref{thm:psi-mu-related}), we will want to consider also the $\mathbf{1}$-evaluations of the $\Phi_\pi$'s. Denote $\phi_{\mathbf{a}} = \Phi_\mathbf{a}(\mathbf{1})$,
$\phi_{\pi} = \Phi_\pi(\mathbf{1})$. The coefficients $\phi_\pi$ are related by
\begin{align*}
\phi_\pi &= \sum_{\sigma\in\noncn{n}} C_{\pi,\sigma} \psi_\sigma, \\
\psi_\pi &= \sum_{\sigma\in\noncn{n}} \tilde{C}_{\pi,\sigma} \phi_\sigma.
\end{align*}
\begin{lem}[\cite{zinn-justin-di-francesco2}, Section 3.4]
$\phi_\mathbf{a}$ has the following contour integral formula:
\begin{equation} \label{eq:phi-a-eval1}
\phi_\mathbf{a} = \oint\!\ldots\!\oint \prod_{1\le j<k\le n} (u_k-u_j)(1+u_k+u_j u_k) \prod_{j=1}^n \frac{1}{u_j^{a_j}} \cdot \prod_{j=1}^n \frac{du_j}{2\pi i},
\end{equation}
where the contours are circles of arbitrary radius around $0$. Equivalently, $\phi_{\mathbf{a}}$ can be written as a constant term
\begin{equation}
\label{eq:phi-a-eval1-constantterm}
\phi_\mathbf{a} = \textnormal{CT}_{u_1,\ldots,u_n} \left(
\prod_{1\le j<k\le n} (u_k-u_j)(1+u_k+u_j u_k) \prod_{j=1}^n \frac{1}{u_j^{a_j-1}} \right)
\end{equation}
or as a Taylor coefficient
$$
\phi_\mathbf{a} = \left[u_1^{a_1-1} \ldots u_n^{a_n-1} \right]
\left(
\prod_{1\le j<k\le n} (u_k-u_j)(1+u_k+u_j u_k)\right).
$$
\end{lem}
\begin{proof} In the integral formula \eqref{eq:phi-a-def} set $z_1=\ldots=z_{2n}=1$ and then make the substitution $w_j = \frac{1-q^{-1} u_j}{1-q u_j}$. This gives \eqref{eq:phi-a-eval1} after a short computation.
\end{proof}
\subsection{Product-type expansions and the sum rule}
We are ready to start applying the theory developed above to the problem of computing probabilities of connectivity events for the random noncrossing matching $\Pi_*^{(n)}$. Our methods generalize those of Zinn-Justin and Di-Francesco \cite{zinn-justin-di-francesco2}, so let us recall briefly one of their beautiful discoveries. Define a set $\mathcal{B}_n \subseteq \noncn{n}^+$ consisting of $2^{n-1}$ sequences, given by
$$ \mathcal{B}_n = \{ \mathbf{a}=(a_1,\ldots,a_n)\in\mathcal{A}_n\,:\, a_1=1,\ a_j\in\{2j-2,2j-1\},\ (2\le j\le n) \}. $$
For each $\mathbf{a}=(a_1,\ldots,a_n)\in \mathcal{B}_n$ define
$$ \mathcal{L}(\mathbf{a}) = \{ \pi\in\noncn{n} \,:\, \textrm{for all } 1\le j\le n,\ \ \pi_{2j-1}=1 \textrm{ iff }a_j=2j-1 \}. $$
Note that $\noncn{n} = \displaystyle \bigsqcup_{\mathbf{a}\in\mathcal{B}_n} \mathcal{L}(\mathbf{a})$.
\begin{thm}
\label{thm:phi-a-psi-expansion}
For any $\mathbf{a}\in\mathcal{B}_n$, we have the expansion
\begin{equation} \label{eq:phi-a-psi-expansion}
\Phi_\mathbf{a} = \sum_{\pi\in\mathcal{L}(\mathbf{a})} \Psi_\pi.
\end{equation}
\end{thm}
Theorem~\ref{thm:phi-a-psi-expansion} and its elegant proof served as the inspiration for one of our main results (Theorem~\ref{thm:submatching-event-expansion} in Subsection~\ref{sec:gen-submatching}). As an aid in motivating and understanding our use of Zinn-Justin and Di Francesco's proof technique in a more complicated setting, we include their original proof, taken from \cite[Section 3]{zinn-justin-di-francesco2}, below.
\begin{proof}[Proof of Theorem~\ref{thm:phi-a-psi-expansion}]
By Theorem~\ref{thm:sufficient-evaluations}, it is enough to prove that for any $\sigma\in\noncn{n}$ we have
\begin{equation}
\label{eq:phi-psi-expansion}
\Phi_\mathbf{a}(\sigma) = \sum_{\pi\in\mathcal{L}(\mathbf{a})} \Psi_\pi(\sigma).
\end{equation}
The proof is by induction on $n$. Denote the right-hand side of \eqref{eq:phi-psi-expansion} by $\theta_{\mathbf{a}}(\sigma)$.
Let $k\matched{\sigma}k+1$ be a little arc of $\sigma$. Let $j$ be such that $k\in \{2j-2,2j-1\}$. By the definition of $\mathcal{B}_n$, either $k=a_j$ is the only occurrence of $k$ in $\mathbf{a}$, or $k$ does not occur in $\mathbf{a}$ (in which case $a_j$ is the number in $\{2j-2,2j-1\}$ of the opposite parity from $k$).
For the left-hand side of \eqref{eq:phi-psi-expansion} we have by \eqref{eq:phi-a-recurrence} that
$$ \Phi_\mathbf{a}(\sigma) = \begin{cases}
3^{n-1} \Phi_{\hat{\mathbf{a}}_k}(\hat{\sigma}_k) & \textrm{if }a_j=k, \\
0 & \textrm{if }a_j \neq k.
\end{cases}
$$
From the definition of $\mathcal{B}_n$ it is easy to see that when $k=a_j$, $\hat{\mathbf{a}}_k \in \mathcal{B}_{n-1}$. So, to complete the inductive proof it is enough to show that $\theta_\mathbf{a}(\sigma)$ satisfies the same recurrence. First, assume that $a_j\neq k$. If $k=2j-1$ (so $a_j=2j-2$) then any $\pi\in\mathcal{L}(\mathbf{a})$ will have the property that $\pi_k=-1$, that is, $\pi$ matches $k$ with a number to the left of $k$. In particular $\pi$ does not contain the little arc $(k,k+1)$, and therefore $\Psi_\pi(\sigma)=0$. This shows that
$\theta_{\mathbf{a}}(\sigma) = 0$. Alternatively, if $k=2j-2, a_j=2j-1$, then any $\pi\in\mathcal{L}(\mathbf{a})$ will have the property that $\pi$ matches $a_j=k+1$ with a number to the right of $k+1$, and again the little arc $(k,k+1)$ cannot be in $\pi$ and $\Psi_\pi(\sigma)=0$, so the relation $\theta_{\mathbf{a}}(\sigma) = 0$ also holds.
Finally, in the case $a_j=k$, the noncrossing matchings $\pi\in\mathcal{L}(\mathbf{a})$ can be divided into those for which $(k,k+1)$ is not a little arc of $\pi$, which satisfy $\Psi_\pi(\sigma)=0$, and those for which $(k,k+1)$ is a little arc of $\pi$. The ones in the latter class satisfy $\Psi_\pi(\sigma)=3^{n-1} \Psi_{\hat{\pi}_k}(\hat{\sigma}_k)$, and furthermore the correspondence $\pi\mapsto \hat{\pi}_k$ maps them bijectively onto $\mathcal{L}(\hat{\mathbf{a}}_k)$. Thus we get the relation
$$ \theta_{\mathbf{a}}(\sigma) = 3^{n-1} \theta_{\hat{\mathbf{a}}_k}(\hat{\sigma}_k), $$
which is the same recurrence as the one for $\Phi_\mathbf{a}(\sigma)$ and therefore completes the proof.
\end{proof}
Theorem~\ref{thm:phi-a-psi-expansion} has several immediate corollaries that are directly relevant to the problem of deriving formulas for probabilities of connectivity events. First, summing \eqref{eq:phi-a-psi-expansion} over the sequences $\mathbf{a}\in\mathcal{B}_n$ gives that
$$
\sum_{\pi\in\noncn{n}} \Psi_\pi = \sum_{\mathbf{a}\in \mathcal{B}_n} \Phi_\mathbf{a}.
$$
Taking the $\mathbf{1}$-evaluation of both sides then gives the relation
$$
\sum_{\pi\in\noncn{n}} \psi_\pi = \sum_{\mathbf{a}\in \mathcal{B}_n} \phi_\mathbf{a},
$$
and the right-hand side can then be expanded as a constant term using \eqref{eq:phi-a-eval1-constantterm}, which gives that
\begin{equation} \label{eq:sum-psi-constterm}
\sum_{\pi\in\noncn{n}} \psi_\pi =
\textnormal{CT}_{u_1,\ldots,u_n} \left(
\prod_{1\le j<k\le n} (u_k-u_j)(1+u_k+u_j u_k) \prod_{j=2}^n \frac{1+u_j}{u_j^{2j-2}} \right).
\end{equation}
Note that the key point that makes such an expansion possible is the fact that $\mathcal{B}_n$ decomposes as a Cartesian product, which manifests itself as the product $\prod_{j=2}^n (1+u_j)/u_j^{2j-2}$ in the Laurent polynomial. It is precisely this product structure that our more general result seeks to emulate. Finally, applying the identity \eqref{eq:asm-const-term} gives the following result.
\begin{thm}
\label{thm:sum-rule}
$ \sum_{\pi\in\noncn{n}} \psi_\pi = \textnormal{ASM}(n)$.
\end{thm}
With this result, we are finally able to relate results about the vector $\boldsymbol{\psi}_n$ to statements about its normalized version, the probability vector $\boldsymbol{\mu}_n$. As an example (also taken from \cite{zinn-justin-di-francesco2}), letting $\mathbf{a}=(1,3,5,\ldots,2n-1)$, note that $\mathcal{L}(\mathbf{a})$ is the singleton set $\{\pi^n_{\textrm{max}}\}$, so,
applying \eqref{eq:phi-a-psi-expansion} in this case, then taking the $\mathbf{1}$-evaluation and using \eqref{eq:phi-a-eval1-constantterm} as before, gives the identity
$$
\psi_{\pi^n_{\textrm{max}}} =
\textnormal{CT}_{u_1,\ldots,u_n} \left(
\prod_{1\le j<k\le n} (u_k-u_j)(1+u_k+u_j u_k) \prod_{j=1}^n \frac{1}{u_j^{2j-2}} \right).
$$
This constant term can be evaluated iteratively as
\begin{align*}
\textnormal{CT}_{u_1,\ldots,u_{n-1}} & \textnormal{CT}_{u_n} \left(
\prod_{1\le j<k\le n} (u_k-u_j)(1+u_k+u_j u_k) \prod_{j=1}^n \frac{1}{u_j^{2j-2}} \right)
\\ &=
\textnormal{CT}_{u_1,\ldots,u_{n-1}} \left(
\prod_{1\le j<k\le n-1} (u_k-u_j)(1+u_k+u_j u_k) \prod_{j=1}^n \frac{1+u_j}{u_j^{2j-2}} \right)
\\ &= \textnormal{ASM}(n-1)
\end{align*}
(use \eqref{eq:asm-const-term}, noting that on the right-hand side of that identity, starting the product $\prod_j (1+z_j)$ at $j=1$ instead of $j=2$ does not affect the constant term),
confirming part 3 of Theorem~\ref{thm:sum-rules}.
\subsection{$p$-nested matchings}
\label{sec:p-nested-matchings}
If $\pi\in\noncn{n}$, denote by $\lpbracket{\pi}$ the noncrossing matching of order $n+1$ obtained from $\pi$ by relabeling the elements of $[1,2n]$ as $2,\ldots,2n+1$ and adding a ``large arc'' connecting $1$ and $2n+2$. We call this the \textbf{nesting of $\pi$} or ``$\pi$-nested.'' For an integer $p\ge0$ let $\lpbracket{\pi}^p$ denote the $p$-nesting of $\pi$, defined as the result of $p$ successive nesting operations applied to $\pi$. Similarly, for the associated sequence $\mathbf{a}=\pi^+ \in \noncn{n}^+$, denote $\lpbracket{\mathbf{a}}=(\lpbracket{\pi})^+$ and $\lpbracket{\mathbf{a}}^p=(\lpbracket{\pi}^p)^+$. If $\mathbf{a}=(a_1,\ldots,a_n)$, it is easy to see that this can be written more explicitly as
$$ \lpbracket{\mathbf{a}}^p = (1,\ldots,p,p+a_1,\ldots,p+a_n). $$
The following result is due to Fonseca and Zinn-Justin \cite{fonseca-zinn-justin}.
\begin{thm}
Let $p\ge 0$. Under the $p$-nesting operation, the expansions
\begin{align*}
\Phi_\sigma &= \sum_{\pi\in\noncn{n}} C_{\sigma,\pi} \Phi_{\pi} \qquad (\pi\in\noncn{n}),\\
\Psi_\pi &= \sum_{\sigma\in\noncn{n}} \tilde{C}_{\pi,\sigma} \Phi_{\sigma} \qquad (\sigma\in\noncn{n}),
\end{align*}
turn into
\begin{align*}
\Phi_{\lpbracket{\sigma}^p} &= \sum_{\pi\in\noncn{n}} C_{\sigma,\pi} \Psi_{\lpbracket{\pi}^p} \qquad (\pi\in\noncn{n}), \\
\Psi_{\lpbracket{\pi}^p} &= \sum_{\sigma\in\noncn{n}} \tilde{C}_{\pi,\sigma} \Phi_{\lpbracket{\sigma}^p} \qquad (\sigma\in\noncn{n}).
\end{align*}
Equivalently, we have the recursive relations
\begin{align}
\label{eq:p-nesting-first}
C_{\lpbracket{\sigma}^p, \mu} &= \begin{cases} C_{\sigma,\pi} & \textrm{if }\mu=\lpbracket{\pi}^p, \\ 0 & \textrm{otherwise},
\end{cases} \\[5pt]
\label{eq:p-nesting-second}
\tilde{C}_{\lpbracket{\pi}^p, \mu} &= \begin{cases} \tilde{C}_{\pi,\sigma} & \textrm{if }\mu=\lpbracket{\sigma}^p, \\ 0 & \textrm{otherwise}.
\end{cases}
\end{align}
\end{thm}
\begin{proof}
The relation \eqref{eq:p-nesting-second} is an immediate consequence of \eqref{eq:p-nesting-first}. To prove \eqref{eq:p-nesting-first}, note that the fact that $C_{\lpbracket{\sigma}^p, \mu}=0$ if $\mu$ is not of the form $\lpbracket{\sigma}^p$ for some $\sigma\in\noncn{n}$ follows from the triangularity of the basis change matrix with respect to the order $\preceq$. The relation $C_{\lpbracket{\sigma}^p, \lpbracket{\pi}^p}=C_{\sigma,\pi}$ is easy to prove by induction from~\eqref{eq:phi-a-recurrence}.
\end{proof}
\subsection{General submatching events}
\label{sec:gen-submatching}
Fix $k\ge0$ and $\pi_0\in\noncn{k}$. For $n\ge k+1$ define
$$ \mathcal{E}_n(\pi_0) = \left\{ \pi\in\noncn{n}\,:\, \pi(j)=1+\pi_0(j-1) \textrm{ for }2\le j\le k+1 \right\}. $$
In the terminology of Subsections~\ref{sec:connectivity-infinite} and \ref{sec:loop-perc-cylinder}, this set of matchings is related to the submatching event associated with the submatching $\pi_0$, except that we require the submatching to occur on the sites in the interval $[2,k+1]$ instead of $[1,k]$. Surprisingly, this turns out to be the correct thing to do---see below.
Next, for any $n\ge k+1$ and $\mathbf{b}=(b_{k+2},\ldots,b_n)$ satisfying $b_j\in\{2j-2,2j-1\}$ for all $j$, define
\begin{align*}
\mathcal{E}_n(\pi_0, \mathbf{b}) = \Big\{ &\pi\in\mathcal{E}_n(\pi_0) \,:\,
\pi_{2j-1}=1 \textrm{ iff }b_j=2j-1 \ \ (k+2\le j\le n)
\Big\}.
\end{align*}
Note that
$$
\mathcal{E}_n(\pi_0) = \bigsqcup_{\begin{array}{c} \scriptstyle \mathbf{b}=(b_{k+2},\ldots,b_n) \\[-3pt] \scriptstyle \forall j\,\, b_j\in\{2j-2,2j-1\} \end{array}} \mathcal{E}_n(\pi_0, \mathbf{b}).
$$
The following result will easily imply Theorem~\ref{thm:explicit-formulas-finite-n}, and is one of the main results of this paper.
\begin{thm}
\label{thm:submatching-event-expansion}
For any $\pi_0\in\noncn{k}$, $n\ge k+1$ and vector $\mathbf{b}=(b_{k+2},\ldots,b_n)$ satisfying $b_j\in\{2j-2,2j-1\}$ for all $j$, we have
\begin{align}
\label{eq:submatching-b-expansion}
\sum_{\pi\in\mathcal{E}_n(\pi_0,\mathbf{b})}\Psi_\pi &=
\sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}} \Phi_{(1,1+a_1,\ldots,1+a_k,b_{k+2},\ldots,b_n)}.
\end{align}
It follows that for any $\pi_0\in\noncn{k}$ and $n\ge k+1$,
\begin{equation}
\label{eq:submatching-exansion-ecal}
\sum_{\pi\in\mathcal{E}_n(\pi_0)}\Psi_\pi =
\hspace{-6pt} \sum_{\begin{array}{c} \scriptstyle b_{k+2},\ldots,b_n \\[-3pt] \scriptstyle \forall j\,\,b_j\in\{2j-2,2j-1\} \end{array}}
\hspace{-6pt} \sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}} \Phi_{(1,1+a_1,\ldots,1+a_k,b_{k+2},\ldots,b_n)}.
\end{equation}
\end{thm}
\begin{proof}
By \eqref{eq:p-nesting-second}, the right-hand side of \eqref{eq:submatching-b-expansion} is equal to
$$
\sum_{\mathbf{a}=(a_1,\ldots,a_{k+1})\in\noncn{k+1}^+}
\tilde{C}_{\lpbracket{\pi_0}, \mathbf{a}} \Phi_{(a_1,\ldots,a_{k+1},b_{k+2},\ldots,b_n)}.
$$
So, instead of proving \eqref{eq:submatching-b-expansion} we will show that
\begin{equation}
\label{eq:submatching-b-expansion-secondversion}
\sum_{\pi\in\mathcal{E}_n(\pi_0,\mathbf{b})}\Psi_\pi =
\sum_{\mathbf{a}=(a_1,\ldots,a_{k+1})\in\noncn{k+1}^+}
\tilde{C}_{\lpbracket{\pi_0}, \mathbf{a}} \Phi_{(a_1,\ldots,a_{k+1},b_{k+2},\ldots,b_n)}.
\end{equation}
We will prove by induction on $n$ the following slightly stronger claim: if we have an expansion of the form
\begin{equation} \label{eq:pinested-phi-expansion}
\Psi_{\lpbracket{\pi_0}} = \sum_{\mathbf{a}\in\mathcal{A}_{k+1}} c_{\mathbf{a}} \Phi_{\mathbf{a}}
\end{equation}
for some constants $c_{\mathbf{a}}\ (\mathbf{a}\in\mathcal{A}_{k+1})$, then for any vector $\mathbf{b}=(b_{k+2},\ldots,b_n)$ satisfying $b_j\in\{2j-2,2j-1\}$ as in the theorem, the equality
\begin{equation}
\label{eq:psi-phi-two-sums}
\sum_{\pi\in\mathcal{E}_n(\pi_0,\mathbf{b})} \Psi_{\pi} = \sum_{\mathbf{a}\in\mathcal{A}_{k+1}} c_{\mathbf{a}} \Phi_{(\mathbf{a},\mathbf{b})}
\end{equation}
holds. (Note that this claim implies \eqref{eq:submatching-b-expansion-secondversion} by taking $c_\mathbf{a}=\tilde{C}_{\lpbracket{\pi_0}, \mathbf{a}}$ for $\mathbf{a}\in\noncn{n}^+$ and $c_\mathbf{a}=0$ otherwise, but allowing the sum on the right-hand side of \eqref{eq:pinested-phi-expansion} to range over the bigger set $\mathcal{A}_{k+1}$ helps make the induction work.)
Denote the left-hand side of \eqref{eq:psi-phi-two-sums} by $X_{\pi_0,\mathbf{b}}$ and the right-hand side by $Y_{c,\mathbf{b}}$. We now prove that $X_{\pi_0,\mathbf{b}}=Y_{c,\mathbf{b}}$.
\textbf{The induction base:} here $k=0$, $n=1$, $\pi_0=\emptyset$ (the empty matching), $\mathcal{A}_{k+1}=\mathcal{A}_1=\{ 1\}$ and the expansion necessarily takes the form
$$ \Psi_{ \lpbracket{\emptyset} } = \Phi_1. $$
Also $\mathcal{E}_n(\pi_0,\mathbf{b})=\mathcal{L}((1,b_2,\ldots,b_n))$, so the claim reduces to the relation \eqref{eq:phi-a-psi-expansion} from Theorem~\ref{thm:phi-a-psi-expansion}.
\textbf{The inductive step:} First, if $n=k+1$ then the claim \eqref{eq:psi-phi-two-sums} is the same as the assumption \eqref{eq:pinested-phi-expansion}, since $\mathbf{b}$ is a trivial vector of length $0$ and $\mathcal{E}_n(\pi_0,\mathbf{b})$ is the singleton set $\{ \lpbracket{\pi_0} \}$, so there is nothing to prove. From now on assume $n>k+1$. We need to show that for all $\sigma\in\noncn{n}$, $X_{\pi_0,\mathbf{b}}(\sigma)=Y_{c,\mathbf{b}}(\sigma)$. Let $1\le m\le 2n-1$ be a number such that $m\matched{\sigma}m+1$.
\textbf{Case 1.} Assume $m\ge 2k+2$.
\textbf{Subcase 1a.} Assume $m\notin \mathbf{b}$ (that is, $m\neq b_j$ for all $j$). Then by \eqref{eq:phi-a-recurrence} we have that
$
\Phi_{a_1,\ldots,a_{k+1},b_{k+2},\ldots,b_n}(\sigma)=0
$
for any $\mathbf{a}=(a_1,\ldots,a_{k+1})\in\mathcal{A}_{k+1}$, so $Y_{c,\mathbf{b}}(\sigma)=0$.
\textbf{Subsubcase 1a(i).} Assume $m$ is odd, i.e., $m=2j-1$ for some $k+2\le j\le n$. If $\pi\in\mathcal{E}_n(\pi_0,\mathbf{b})$ then, since we assumed $m\notin \mathbf{b}$, $b_j=2j-2$ and therefore by definition $\pi_m=-1$, that is, $\pi(m)=\pi(2j-1)<m$, so in particular $(m,m+1)$ is not a little arc of $\pi$ and therefore $\Psi_\pi(\sigma)=0$. It follows that $X_{\pi_0,\mathbf{b}}(\sigma)=0=Y_{c,\mathbf{b}}(\sigma)$.
\textbf{Subsubcase 1a(ii).} Assume $m$ is even, i.e., $m=2j-2$ for some $k+2\le j\le n$. If $\pi\in\mathcal{E}_n(\pi_0,\mathbf{b})$ then, since we assumed $m\notin \mathbf{b}$, $b_j=2j-1$ and therefore by definition $\pi(m+1)=\pi(2j-1)>m+1$ and in particular $(m,m+1)$ is not a little arc of $\pi$. Again it follows that $\Psi_\pi(\sigma)=0$ and therefore $X_{\pi_0,\mathbf{b}}(\sigma)=0=Y_{c,\mathbf{b}}(\sigma)$.
\textbf{Subcase 1b.} Assume that $m\in \mathbf{b}$, i.e., $m=b_j$ for some $k+2\le j\le n$. Then by \eqref{eq:phi-a-recurrence}, for any $\mathbf{a}=(a_1,\ldots,a_{k+1})\in\mathcal{A}_{k+1}$ we have
$$ \Phi_{(a_1,\ldots,a_{k+1},b_{k+2},\ldots,b_n)}(\sigma) = 3^{n-1}
\Phi_{(\mathbf{a},\hat{\mathbf{b}}_m)}(\hat{\sigma}_m),
$$
where $\hat{\mathbf{b}}_m$ is obtained from $\mathbf{b}$ by deleting the $m$ and shifting all the numbers to the right of $m$ down by $2$. It follows that
\begin{equation} \label{eq:ycb-sigma-bm-hat}
Y_{c,\mathbf{b}}(\sigma) = 3^{n-1} Y_{c,\hat{\mathbf{b}}_m}(\hat{\sigma}_m).
\end{equation}
On the other hand, we have
\begin{align*}
X_{\pi_0,\mathbf{b}}(\sigma) &= \sum_{\pi\in\mathcal{E}_n(\pi_0,\mathbf{b}),\ m\notmatchedvariant{\pi}m+1}\Psi_\pi(\sigma)
+ \sum_{\pi\in\mathcal{E}_n(\pi_0,\mathbf{b}),\ m\matched{\pi}m+1}\Psi_\pi(\sigma)
\\
&= 0+
\sum_{\pi\in\mathcal{E}_n(\pi_0,\mathbf{b}),\ m\matched{\pi}m+1} 3^{n-1} \Psi_{\hat{\pi}_m}(\hat{\sigma}_m).
\end{align*}
Relabeling $\hat{\pi}_m$ in the last sum as $\pi$ transforms it into
$$
3^{n-1}
\sum_{\pi\in\mathcal{E}_{n-1}(\pi_0,\hat{\mathbf{b}}_m)} \Psi_{\pi}(\hat{\sigma}_m) = 3^{n-1} X_{\pi_0, \hat{\mathbf{b}}_m}(\hat{\sigma}_m),
$$
so by induction we get using \eqref{eq:ycb-sigma-bm-hat} that $X_{\pi_0,\mathbf{b}}(\sigma) = Y_{\pi_0,\mathbf{b}}(\sigma)$.
\textbf{Case 2.} Assume $m\le 2k+1$.
\textbf{Subcase 2(a).} Assume $m\matched{\pi_0}m+1$ (when thinking of $\pi_0$ as ``living'' on the interval $[2,2k+1]$), and note that in this case actually $m$ must be $\le 2k$. Then for any $\pi\in\mathcal{E}_n(\pi_0,\mathbf{b})$ we have
$$ \Psi_\pi(\sigma) = 3^{n-1} \Psi_{\hat{\pi}_m}(\hat{\sigma}_m), $$
and consequently, denoting $\mathbf{b}'=(b_{k+2}-2,\ldots,b_n-2)$, we see that
$$ X_{\pi_0,\mathbf{b}}(\sigma) = 3^{n-1} X_{\widehat{(\pi_0)}_m,\mathbf{b}'}(\hat{\sigma}_m). $$
To evaluate $Y_{c,\mathbf{b}}(\sigma)$, we use \eqref{eq:phi-a-recurrence} to get that
\begin{align}
Y_{c,\mathbf{b}}(\sigma) &= \sum_{\mathbf{a}\in\mathcal{A}_{k+1}} c_{\mathbf{a}} \Phi_{(\mathbf{a},\mathbf{b})}(\sigma)
= \sum_{\mathbf{a}\in\mathcal{A}_{k+1},\,m\in\mathbf{a}} c_{\mathbf{a}} \Phi_{(\mathbf{a},\mathbf{b})}(\sigma)
\nonumber \\ &= \sum_{\mathbf{a}\in\mathcal{A}_{k+1},\,m\in\mathbf{a}} 3^{n-1} \chi(p(\mathbf{a},m))
c_{\mathbf{a}} \Phi_{(\hat{\mathbf{a}}_m,\mathbf{b}')}(\hat{\sigma}_m)
\nonumber \\ &=
\sum_{\mathbf{a}'\in\mathcal{A}_{k}} d_{\mathbf{a}'} \Phi_{(\mathbf{a}',\mathbf{b}')}(\hat{\sigma}_m),
\label{eq:ycb-daprime-epansion}
\end{align}
where $p(\mathbf{a},m)$ denotes the number of times $m$ appears in $\mathbf{a}$, $\hat{\mathbf{a}}_m$ is as in \eqref{eq:hat-a-def}, and in the last line the sum was reorganized as a linear combination with some new coefficients $d_{\mathbf{a}'}$.
(The fact that this sum ranges over $\mathcal{A}_k$ explains why we needed to strengthen the inductive hypothesis at the beginning of the proof.)
Note also that in a similar way, replacing the vector $\mathbf{b}$ with the empty vector $\emptyset$ of length $0$, we have for any $\mu\in\noncn{k+1}$ such that $m\matched{\mu}m+1$,
\begin{align*}
Y_{c,\emptyset}(\mu) &= \sum_{\mathbf{a}\in\mathcal{A}_{k+1}} c_{\mathbf{a}} \Phi_{\mathbf{a}}(\mu)
= \sum_{\mathbf{a}\in\mathcal{A}_{k+1},\,m\in\mathbf{a}} c_{\mathbf{a}} \Phi_{\mathbf{a}}(\mu)
\\ &= \sum_{\mathbf{a}\in\mathcal{A}_{k+1},\,m\in\mathbf{a}} 3^{n-1}
\chi(p(\mathbf{a},m))
c_{\mathbf{a}} \Phi_{\hat{\mathbf{a}}_m}(\hat{\mu}_m)
\\ &=
\sum_{\mathbf{a}'\in\mathcal{A}_{k}} d_{\mathbf{a}'} \Phi_{\mathbf{a}'}(\hat{\mu}_m),
\end{align*}
with exactly the same coefficients $d_{\mathbf{a}'}$ (the main thing to notice is that appending the $\mathbf{b}$ at the end in the subscript of $\Phi$ has no effect on the way the recurrence \eqref{eq:phi-a-recurrence} proceeds, i.e., the $\mathbf{b}$ ``doesn't interact'' with the $\mathbf{a}$). On the other hand, the case $n=k+1$ discussed at the beginning of the proof gives that $Y_{c,\emptyset}(\mu)=X_{\pi_0,\emptyset}(\mu)=3^{n-1} X_{\widehat{(\pi_0)}_m,\emptyset}(\hat{\mu}_m)$. Combining this with \eqref{eq:ycb-daprime-epansion}, we get that
$$ X_{\widehat{(\pi_0)}_m,\emptyset}(\nu) =
3^{-(n-1)} \sum_{\mathbf{a}'\in\mathcal{A}_{k}} d_{\mathbf{a}'} \Phi_{\mathbf{a}'}(\nu)
$$
for any $\nu\in\noncn{k}$, or in other words simply that
$$ X_{\widehat{(\pi_0)}_m,\emptyset} =
3^{-(n-1)} \sum_{\mathbf{a}'\in\mathcal{A}_{k}} d_{\mathbf{a}'} \Phi_{\mathbf{a}'}.
$$
Invoking the inductive hypothesis again, we conclude that
$$
X_{\widehat{(\pi_0)}_m,\mathbf{b}'} =
3^{-(n-1)} \sum_{\mathbf{a}'\in\mathcal{A}_{k}} d_{\mathbf{a}'} \Phi_{(\mathbf{a}',\mathbf{b}')}.
$$
In particular,
$$
3^{(n-1)} X_{\widehat{(\pi_0)}_m,\mathbf{b}'}(\hat{\sigma}_m) =
\sum_{\mathbf{a}'\in\mathcal{A}_{k}} d_{\mathbf{a}'} \Phi_{(\mathbf{a}',\mathbf{b}')}(\hat{\sigma}_m).
$$
But as we showed above, the LHS is equal to $X_{\pi_0,\mathbf{b}}(\sigma)$ and the RHS is equal to $Y_{c,\mathbf{b}}(\sigma)$, so this proves the claim.
\textbf{Subcase 2(b).} Assume $m \notmatched{\pi_0} m+1$. Then for a similar reason as in subcase 2(a) above, $X_{\pi_0,\mathbf{b}}(\sigma)=0$. We need to show that also $Y_{c,\mathbf{b}}(\sigma)=0$. The computation from subcase 2(a) is still valid, so we can write as before $Y_{c,\mathbf{b}}(\sigma) = \sum_{\mathbf{a}'\in\mathcal{A}_k} d_{\mathbf{a}'} \Phi_{(\mathbf{a}',\mathbf{b}')}(\hat{\sigma}_m)$, and also
$$ Y_{c,\emptyset}(\mu)=
\sum_{\mathbf{a}'\in\mathcal{A}_k} d_{\mathbf{a}'} \Phi_{\mathbf{a}'}(\hat{\mu}_m)
$$
for any $\mu\in\noncn{k+1}$ for which $m\matched{\mu}m+1$. But we know (again from the case $n=k+1$) that $Y_{c,\emptyset}(\mu)=X_{\pi_0,\emptyset}(\mu)=0$, so we deduce that
$$ \sum_{\mathbf{a}'\in\mathcal{A}_k} d_{\mathbf{a}'} \Phi_{\mathbf{a}'} = 0. $$
We claim that this implies also that
\begin{equation} \label{eq:no-interaction-claim}
\sum_{\mathbf{a}'\in\mathcal{A}_k} d_{\mathbf{a}'} \Phi_{(\mathbf{a}',\mathbf{b}')} = 0,
\end{equation}
which, if true, would also give in particular that
$$ Y_{c,\mathbf{b}}(\sigma) = \sum_{\mathbf{a}'\in\mathcal{A}_k} d_{\mathbf{a}'} \Phi_{(\mathbf{a}',\mathbf{b}')}(\hat{\sigma}_m) = 0, $$
and finish the proof.
It remains to prove \eqref{eq:no-interaction-claim}. First, for convenience relabel $\mathbf{a}'$ as $\mathbf{a}$, $\mathbf{b}'$ as $\mathbf{b}$ and $n-1$ as $n$. The claim then becomes that if $\mathbf{b}=(b_{k+1},\ldots,b_n)$ satisfies $b_j\in\{2j-2,2j-1\}$ for all $j$ and $(d_\mathbf{a})_{\mathbf{a}\in \mathcal{A}_k}$ are coefficients such that $\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{\mathbf{a}}=0$ then
$\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{(\mathbf{a},\mathbf{b})}=0$.
The proof is by induction on $n$; the idea as before is that the $\mathbf{a}'$ and $\mathbf{b}'$ components ``don't interact'' when we perform recursive computations using \eqref{eq:phi-a-recurrence}, and the proof involves similar ideas to those used above. If $n=k$ there is nothing to prove, so assume that $n>k$. Let $\nu\in \noncn{n}$, and pick some $g$ such that $g\matched{\pi}g+1$. Divide into two cases, when $g\ge 2k$ or when $g\le 2k-1$. We will show that in each case we have $\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{(\mathbf{a},\mathbf{b})}(\nu)=0$, which will imply the claim for the usual reason (Theorem~\ref{thm:sufficient-evaluations}).
In the first case, if $g\notin \{ b_{k+1},\ldots,b_n \}$ then $\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{(\mathbf{a},\mathbf{b})}(\nu) = 0$ by \eqref{eq:phi-a-recurrence}. Alternatively, if $g\in \{ b_{k+1},\ldots,b_n \}$
then, again by \eqref{eq:phi-a-recurrence}, we have that
\begin{equation}
\label{eq:sum-b-overline}
\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{(\mathbf{a},\mathbf{b})}(\nu) =
3^{n-1}
\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{(\mathbf{a},\overline{\mathbf{b}})}(\hat{\nu}_g),
\end{equation}
where we denote $\overline{\mathbf{b}}=(b_{k+1}-2,\ldots,b_n-2)$ (note that the $\chi(p)$ factors in \eqref{eq:phi-a-recurrence} are all equal to $1$ because of the structure of $\mathbf{b}$). The right-hand side of \eqref{eq:sum-b-overline} is equal to $0$ by the inductive hypothesis (applied with the vector $\overline{\mathbf{b}}$ replacing~$\mathbf{b}$).
Finally, in the case where $g\le 2k-1$, we have
\begin{equation}
\label{eq:sum-da-phi-ab}
\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{(\mathbf{a},\mathbf{b})}(\nu)
=
\sum_{\mathbf{a}\in\mathcal{A}_k,\ g\in \{a_1,\ldots,a_k\}}
d_\mathbf{a} 3^{n-1} \chi(p(\mathbf{a},g)) \Phi_{(\hat{\mathbf{a}}_g,\overline{\mathbf{b}})}(\hat{\nu}_g)
\end{equation}
(where the notation $p(\mathbf{a},g)$ was defined immediately after \eqref{eq:ycb-daprime-epansion}),
and this can be rewritten as a sum of the form
\begin{equation} \label{eq:sum-fa-prime}
\sum_{\mathbf{a}'\in\mathcal{A}_{k-1}}
f_{\mathbf{a}'} \Phi_{(\mathbf{a}',\overline{\mathbf{b}})}(\hat{\nu}_g)
\end{equation}
where $(f_{\mathbf{a}'})_{\mathbf{a}'\in\mathcal{A}_{k-1}}$ are some coefficients. On the other hand, by repeating the same computation in the case $n=k$ where $\mathbf{b}$ is replaced by the empty vector of length $0$, we see that for any $\mu\in\noncn{k}$ such that $g\matched{\mu}g+1$, we have
\begin{equation} \label{eq:lin-combin-fa-prime}
\sum_{\mathbf{a}\in\mathcal{A}_k} d_\mathbf{a} \Phi_{\mathbf{a}}(\mu) = \sum_{\mathbf{a}'\in\mathcal{A}_{k-1}} f_{\mathbf{a}'} \Phi_{\mathbf{a}'}(\hat{\mu}_g),
\end{equation}
where the key observation is that the coefficients are the same numbers $f_{\mathbf{a}'}$ as in \eqref{eq:sum-fa-prime}. But we know (by the assumption of the claim we are trying to prove) that the left-hand side of \eqref{eq:lin-combin-fa-prime} is equal to $0$. Since this is true for any $\mu\in\noncn{k}$ for which $g\matched{\mu}g+1$, we conclude that
$\sum_{\mathbf{a}'\in\mathcal{A}_{k-1}} f_{\mathbf{a}'} \Phi_{\mathbf{a}'}=0$. The inductive hypothesis now implies that
$\sum_{\mathbf{a}'\in\mathcal{A}_{k-1}} f_{\mathbf{a}'} \Phi_{(\mathbf{a}',\overline{\mathbf{b}})}=0$. In particular, the expression in \eqref{eq:sum-fa-prime}, which is equal to the left-hand side of \eqref{eq:sum-da-phi-ab}, is $0$.
\end{proof}
\subsection{Proof of Theorem~\ref{thm:explicit-formulas-finite-n}}
\label{sec:proof-main-thm}
We are now in a position to prove Theorem~\ref{thm:explicit-formulas-finite-n}, with an explicit description of the multivariate polynomial $F_{\pi_0}(w_1,\ldots,w_k)$ whose existence is claimed in the theorem. Let $\pi_0 \in\noncn{k}$. The polynomial $F_{\pi_0}$ simply encodes the $\pi_0$th row of the matrix $\tilde{\boldsymbol{C}}_n$; more precisely, we define it as
\begin{equation}
\label{eq:submatching-polynomial-def}
F_{\pi_0}(w_1,\ldots,w_k) = \sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+} \tilde{C}_{\pi_0,\mathbf{a}} \prod_{j=1}^{k} w_j^{2j-a_j}.
\end{equation}
Denote $\Omega_n(\mathbf{z}) = \prod_{1\le i<j\le n} (z_j-z_i)(1+z_j+z_i z_j)$.
Taking the $\mathbf{1}$-evaluation of both sides of \eqref{eq:submatching-exansion-ecal} and combining the resulting equation with Theorems~\ref{thm:psi-mu-related} and \ref{thm:sum-rule}, we get that
\begin{align*}
\textnormal{ASM}(n) \, & \mathbb{P}\left(\Pi_*^{(n)} \in \mathcal{E}_n(\pi_0)\right) = \sum_{\pi\in \mathcal{E}_n(\pi_0)} \psi_\pi
\\ &=
\sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}} \hspace{-14.0pt}
\sum_{\begin{array}{c} \scriptstyle b_{k+2},\ldots,b_n \\[-3pt] \scriptstyle b_j\in\{2j-2,2j-1\} \end{array}}
\hspace{-14.0pt}
\phi_{(1,1+a_1,\ldots,1+a_k,b_{k+2},\ldots,b_n)}.
\end{align*}
The left-hand side is also equal to $\textnormal{ASM}(n) \mathbb{P}\left(\pi_0 \mathrel{\lhd} \Pi_*^{(n)}\right)$ by the rotational symmetry of $\Pi_*^{(n)}$. To evaluate the right-hand side, represent each summand $\phi_{(1,1+a_1,\ldots,1+a_k,b_{k+2},\ldots,b_n)}$ as a constant term using \eqref{eq:phi-a-eval1-constantterm}, to get that the last expression can be written as
\begin{align*}
&
\textnormal{CT}_\mathbf{z} \left[ \vphantom{\sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}}
\Omega_n(\mathbf{z}) \prod_{j=2}^{k+1} \frac{1}{z_j^{a_{j-1}}} \prod_{j=k+2}^n \left(\frac{1}{z_j^{2j-3}}+\frac{1}{z_j^{2j-2}}\right)
} \right.
\sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}} \hspace{-14.0pt}
\sum_{\begin{array}{c} \scriptstyle b_{k+2},\ldots,b_n \\[-3pt] \scriptstyle \forall j\,\,b_j\in\{2j-2,2j-1\} \end{array}}
\Omega_n(\mathbf{z}) \prod_{j=2}^{k+1} \frac{1}{z_j^{a_{j-1}}} \prod_{j=k+2}^n \frac{1}{z_j^{b_j-1}}
\left. \vphantom{\sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}}
\Omega_n(\mathbf{z}) \prod_{j=2}^{k+1} \frac{1}{z_j^{a_{j-1}}} \prod_{j=k+2}^n \left(\frac{1}{z_j^{2j-3}}+\frac{1}{z_j^{2j-2}}\right)
}
\right]
\\ &=
\textnormal{CT}_\mathbf{z} \left[
\sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}}
\Omega_n(\mathbf{z}) \prod_{j=2}^{k+1} \frac{1}{z_j^{a_{j-1}}} \prod_{j=k+2}^n \left(\frac{1}{z_j^{2j-3}}+\frac{1}{z_j^{2j-2}}\right)
\right]
\\ &=
\textnormal{CT}_\mathbf{z} \left[
\sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}}
\Omega_n(\mathbf{z}) \prod_{j=2}^{k+1} \frac{1}{z_j^{a_{j-1}}} \prod_{j=k+2}^n \frac{1+z_j}{z_j^{2j-2}}
\right]
\\ &=
\textnormal{CT}_\mathbf{z} \left[
\Omega_n(\mathbf{z}) \prod_{j=k+2}^n (1+z_j) \prod_{j=1}^{n} \frac{1}{z_j^{2j-2}}
\left( \sum_{\mathbf{a}=(a_1,\ldots,a_k)\in\noncn{k}^+}
\tilde{C}_{\pi_0, \mathbf{a}} \prod_{j=2}^{k+1} z_j^{2j-2-a_{j-1}} \right)
\right]
\\
\\ &=
\textnormal{CT}_\mathbf{z} \left[
\Omega_n(\mathbf{z}) \prod_{j=k+2}^n (1+z_j) \prod_{j=1}^{n} \frac{1}{z_j^{2j-2}}
\cdot F_{\pi_0}(z_2,\ldots,z_{k+1}) \right],
\end{align*}
which is clearly equal to $\textnormal{ASM}(n)$ times the right-hand side of \eqref{eq:submatching-polycoeff-formula}.
\qed
\section*{Appendix A. Summary of algorithms}
The research described in this paper has been significantly aided by computer-aided experimentation and numerical computations performed by the author. We believe use of such methods is likely to continue to play a role in the discovery of new results extending and building on this work. In the hope of stimulating such further research, we summarize the theory presented above in the form of explicit algorithms for computing quantities of interest related to connectivity patterns of loop percolation and the theory of wheel polynomials.
\bigskip \noindent
\textbf{Algorithm A\ \ } Computation of the probability vector $\boldsymbol{\mu}_n=(\mu_\pi)_{\pi\in\noncn{n}}$
\begin{enumerate}[labelwidth=-30pt, label=\textnormal{\textbf{Step \arabic*.}}]
\item
Compute the matrix $S_n = (s_{\pi,\sigma})_{\pi,\sigma\in\noncn{n}}$ where
$$ s_{\pi,\sigma} = \#\{ 1\le j\le 2n\,:\, e_j(\pi)=\sigma \} $$
(with $e_j$ defined by the equation \eqref{eq:temperley-lieb-gens}, interpreted modulo $2n$).
\item
Compute $\boldsymbol{\mu}_n$ as the solution to the vector equation
$$\boldsymbol{\mu}_n (S_n - 2n \mathbf{I})= \boldsymbol{0}$$
(where $\mathbf{I}$ is the identity matrix), normalized to be a probability vector.
\end{enumerate}
\bigskip \noindent
\textbf{Algorithm B \ \ } Computation of the polynomials $Q_{\pi_0}$ (assuming Conjecture~\ref{conj:rationality-finite-n})
\begin{enumerate}[labelwidth=-30pt, label=\textnormal{\textbf{Step \arabic*.}}]
\item Use Algorithm A above to compute the submatching event probabilities $p_n = \mathbb{P}\left( \pi_0 \mathrel{\lhd} \Pi_*^{(n)} \right)$ for $n=k,k+1,\ldots,k(k+3)/2$.
\item Compute the numbers $q_n = p_n \prod_{j=1}^k (4n^2-j^2)^{k+1-j}$.
\item Use the Lagrange interpolation formula to compute the unique polynomial $G_{\pi_0}(m)$ of degree $\le k(k+1)/2$ satisfying
$$ G_{\pi_0}(n^2) = q_n \qquad (k\le n\le k(k+3)/2). $$
\item $Q_{\pi_0}$ is given by $Q_{\pi_0}(n)=G_{\pi_0}(n^2)$.
\end{enumerate}
\bigskip \noindent
\textbf{Algorithm C\ \ } Computation of submatching event probabilities in the half-planar model (assuming Conjecture~\ref{conj:rationality-finite-n})
\smallskip
Use Algorithm B above to compute the polynomial $Q_{\pi_0}$. Let $q_{\pi_0}^*$ be the leading coefficient of $Q_{\pi_0}$.
The probability $\mathbb{P}(\pi_0 \mathrel{\lhd} \Pi_*)$ is $q_{\pi_0}^*/2^{k(k+1)}$.
\vbox{
\bigskip \noindent
\textbf{Algorithm D\ \ } Computation of the matrices $\mathbf{C}_n$, $\tilde{\mathbf{C}}_n$
\smallskip
Translating \eqref{eq:cmatrix-explicit} to a notation appropriate for the computation of $C_{\pi,\sigma}$, we
denote $p_{j,k}(\pi) = \#\{ j\le \ell<k \,:\, \pi_\ell = 1 \}-\tfrac12 (k-j-1)$.
The matrix $\mathbf{C}_n=(C_{\pi,\sigma})_{\pi,\sigma\in\noncn{n}}$ is now computed using the explicit formula
$$
C_{\pi,\sigma} = \prod_{j<k,\ j\matched{\sigma}k} \chi(p_{j,k}(\pi))
$$
(with $\chi(\cdot)$ defined in \eqref{eq:character-mod-three}),
and $\tilde{\mathbf{C}}_n = \mathbf{C}_n^{-1}$ is its inverse matrix.
}
\bigskip \noindent
\textbf{Algorithm E\ \ } Computation of the polynomials $F_{\pi_0}$ (see Theorem~\ref{thm:explicit-formulas-finite-n})
\begin{enumerate}[labelwidth=-30pt, label=\textnormal{\textbf{Step \arabic*.}}]
\item Compute $\tilde{\mathbf{C}}_k$ using Algorithm D above.
\item Use \eqref{eq:submatching-polynomial-def} to compute $F_{\pi_0}$.
\end{enumerate}
\section*{Appendix B. Proof of Theorem~\ref{thm:hamiltonian-eigenvector}}
We start by writing a more explicit formula for the entries of the transition matrices $T_n^{(p)}$. To do so, encode each row of plaquettes as a vector $\mathbf{a}\in\{0,1\}^{2n}$, where a $0$ coordinate corresponds to a ``type $0$'' plaquette, defined as the plaquette shown on the left in Fig.~\ref{fig:plaquettes}, and a $1$ coordinate corresponds to a ``type $1$'' plaquette, which is the one on the right-hand side of the same figure. The parameter $p$ corresponds to the probability of a type $1$ plaquette. For a given plaquette-row vector $\mathbf{a}\in\{0,1\}^{2n}$, denote by $f_\mathbf{a}(\cdot)$ an operator that takes a noncrossing matching $\pi\in\noncn{n}$ and returns a new noncrossing matching $\pi'=f_\mathbf{a}(\pi)$ which is the result of composing the diagram of $\pi$ with the row of plaquettes, in an analogous manner to the composition of matching diagrams with Temperley-Lieb operators shown in Fig.~\ref{fig:pipe-diagram}.
With this notation, it is clear from the definition of the transition matrix $T_n^{(p)}$ that its entries $(T_n^{(p)})_{\pi,\pi'}$ are given by
$$
(T_n^{(p)})_{\pi,\pi'} = \sum_{\begin{array}{c} \scriptstyle \mathbf{a}\in\{0,1\}^{2n} \\[-0.6ex] \scriptstyle f_\mathbf{a}(\pi)=\pi' \end{array}} p^{\sum_j a_j} (1-p)^{\sum_j (1-a_j)}
\qquad (\pi,\pi'\in\noncn{n}).
$$
Differentiate this equation with respect to $p$ at $p=0$. On the right-hand side the nonzero contributions will come from the vector $\mathbf{a}=\mathbf{0}=(0,\ldots,0)$ and from the vectors of the form $\mathbf{a}=\mathbf{a}_k = (0,\ldots,0,1,0,\ldots,0)$ with only one coordinate in some position $k$ equal to $1$. This gives
$$
\frac{d}{dp}_{|p=0} (T_n^{(p)})_{\pi,\pi'} = -2n \delta_{f_{\mathbf{0}}(\pi),\pi'} + \#\{ 1\le k\le 2n\,:\, f_{\mathbf{a}_k}(\pi)=\pi' \}.
$$
It is now not difficult to check (see Fig.~\ref{fig:temperley-lieb-plaquettes}) that $f_\mathbf{0}(\pi)=\rho(\pi)$, where $\rho$ is the rotation operator on matchings (defined in Theorem~\ref{thm:qkz-poly-properties}), and that the operator $f_{\mathbf{a}_k}$ bears a simple relation to the Temperley-Lieb operator $e_k$, namely, we have $f_{\mathbf{a}_k} = \rho\circ e_k = e_{k-1} \circ \rho$ (where $e_{k-1}$ is interpreted with the usual mod $2n$ convention).
Summarizing, since the vector $\boldsymbol{\mu}_n$ is the stationary probability vector for the matrices $T_n^{(p)}$, that is, it satisfies $\boldsymbol{\mu}_n T_n^{(p)} = \boldsymbol{\mu}_n$, the matrix $Q_n = \frac{d}{dp}_{|p=0} T_n^{(p)}$ satisfies $\boldsymbol{\mu}_n Q_n = 0$, and by the above observations it is easy to see that it is related to the operator $H_n$ from \eqref{eq:def-h-inf-gen} by $H_n = - R^{-1} Q_n$, where $R$ is a matrix version of the rotation operator $\rho$, whose entries are given by $ R_{\pi,\pi'} = \delta_{\rho(\pi),\pi'}$. It remains to observe that $\boldsymbol{\mu}_n$ is invariant also under the rotation action, i.e., we have $\boldsymbol{\mu}_n R = \boldsymbol{\mu}_n$, to conclude that $\boldsymbol{\mu}_n H_n = 0$, as claimed.
\qed
\begin{figure}
\begin{center}
\begin{tabular}{ccc}
\scalebox{0.7}{\includegraphics{plaquettes-rotation.pdf}}
& &
\scalebox{0.7}{\includegraphics{plaquettes-rotation-flip.pdf}}
\\ (a)&&(b)
\end{tabular}
\caption{(a) Composing the diagram of a matching $\pi$ with a row of type $0$ plaquettes causes a simple rotation; (b) flipping the $k$th plaquette from type $0$ to type $1$ and leaving all other plaquettes as type $0$ is equivalent to the application of a Temperley-Lieb operator $e_k$ prior to the rotation.}
\label{fig:temperley-lieb-plaquettes}
\end{center}
\end{figure}
\section*{Acknowledgements}
The author thanks Omer Angel, Nathana\"el Berestycki, Jan de Gier, Christina Goldschmidt, Alexander Holroyd, Rick Kenyon, James Martin, Bernard Nienhuis, Ron Peled, David Wilson, Doron Zeilberger, and Paul Zinn-Justin for comments and helpful discussions during the preparation of the paper, and an anonymous referee for helpful suggestions.
The author was supported by the National Science Foundation under grant DMS-0955584, and by grant \#228524 from the Simons Foundation.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 5,924
|
Produced by MFR and the Online Distributed Proofreading
Team at http://www.pgdp.net (This file was produced from
images generously made available by The Internet Archive)
The Miracles of Antichrist
BOOKS BY THE SAME AUTHOR
THE EMPEROR OF PORTUGALLIA
(_Trans. from Swedish by Velma Swanston Howard_)
JERUSALEM, A Novel
(_Trans. from Swedish by Velma Swanston Howard_)
THE STORY OF GÖSTA BERLING
(_Trans. from Swedish by Pauline Bancroft Flach_)
THE WONDERFUL ADVENTURES OF NILS
(_Trans. from Swedish by Velma Swanston Howard_)
THE FURTHER ADVENTURES OF NILS
(_Trans. from Swedish by Velma Swanston Howard_)
THE GIRL FROM THE MARSH CROFT
(_Trans. from Swedish by Velma Swanston Howard_)
THE LEGEND OF THE SACRED IMAGE
(_Trans. from Swedish by Velma Swanston Howard_)
THE MIRACLES OF ANTICHRIST
(_Trans. from Swedish by Pauline Bancroft Flach_)
CHRIST LEGENDS
(_Trans. from Swedish by Velma Swanston Howard_)
FROM A SWEDISH HOMESTEAD
(_Trans. from Swedish by Jessie Brochner_)
INVISIBLE LINKS
(_Trans. from Swedish by Pauline Bancroft Flach_)
LILLIECRONA'S HOME
(_Trans. from Swedish by Anna Barwell_)
THE MIRACLES
_of_ ANTICHRIST
_A NOVEL_
FROM THE SWEDISH OF
SELMA LAGERLÖF
TRANSLATED BY
PAULINE BANCROFT FLACH
[Illustration]
GARDEN CITY NEW YORK
DOUBLEDAY, PAGE & COMPANY
1919
_Copyright, 1899, by_
DOUBLEDAY, PAGE & COMPANY
_All rights reserved, including that of
translation into foreign languages_
CONTENTS
INTRODUCTION:
CHAPTER PAGE
I THE EMPEROR'S VISION 1
II ROME'S HOLY CHILD 9
III ON THE BARRICADE 19
FIRST BOOK
I MONGIBELLO 25
II FRA GAETANO 39
III THE GOD-SISTER 48
IV DIAMANTE 62
V DON FERRANTE 64
VI DON MATTEO'S MISSION 71
VII THE BELLS OF SAN PASQUALE 77
VIII TWO SONGS 113
IX FLIGHT 125
X THE SIROCCO 128
XI THE FEAST OF SAN SEBASTIANO 156
SECOND BOOK
I A GREAT MAN'S WIFE 185
II PANEM ET CIRCENSES 193
III THE OUTCAST 204
IV THE OLD MARTYRDOM 213
V THE LADY WITH THE IRON RING 226
VI FRA FELICE'S LEGACY 229
VII AFTER THE MIRACLE 252
VIII A JETTATORE 255
IX PALAZZO GERACI AND PALAZZO CORVAJA 270
X FALCO FALCONE 286
XI VICTORY 315
THIRD BOOK
I THE OASIS AND THE DESERT 323
II IN PALERMO 329
III THE HOME-COMING 338
IV ONLY OF THIS WORLD 354
V A FRESCO OF SIGNORELLI 373
The Miracles of Antichrist
INTRODUCTION
"_When Antichrist comes, he shall seem as Christ_"
I
THE EMPEROR'S VISION
It was at the time when Augustus was emperor in Rome and Herod was king
in Jerusalem.
It happened once upon a time that a very great and holy night sank down
over the earth. It was the darkest night ever seen by man; it seemed
as if the whole earth had passed under a vault. It was impossible to
distinguish water from land, or to find the way on the most familiar
paths. And it could not be otherwise, for not a ray of light came from
the sky. All the stars stayed in their houses, and the fair moon kept her
face turned away.
And just as intense as the darkness was the silence and the calm. The
rivers stood still in their course; the wind did not stir, and even the
leaves of the aspen ceased to tremble. Any one walking by the sea would
have found that the waves no longer broke on the shore, and the sand of
the desert did not crunch under the wanderer's foot. Everything was as
if turned to stone and without motion, in order not to disturb the holy
night. The grass did not dare to grow, the dew could not fall, and the
flowers feared to exhale their perfume.
During that night the beasts of prey did not hunt, the serpents did not
sting, the dogs did not bay. And what was even more wonderful, none of
the inanimate things would have disturbed the holiness of the night by
lending themselves to an evil deed. No false key could open a lock, and
no knife could shed blood.
In Rome, on that very night, a little group of people came down from the
emperor's palace on the Palatine and made their way over the Forum to
the Capitol. During the day just completed his councillors had asked the
emperor if they might not raise a temple to him on Rome's holy mountain.
But Augustus had not immediately given his consent. He did not know
if it would be pleasing to the gods for him to possess a temple next
to theirs, and he had answered that he wished first to discover by a
nocturnal sacrifice to his genius what their wishes were. Followed by a
few faithful retainers, he was now on his way to perform that sacrifice.
Augustus was carried in his litter, for he was old, and the long stairs
to the Capitol fatigued him. He held the cage of doves which was his
offering. Neither priests, nor soldiers, nor councillors accompanied him;
only his nearest friends. Torch-bearers walked in front of him, as if to
force a way through the darkness of the night, and behind him followed
slaves, carrying the tripod, the charcoal, the knives, the holy fire, and
everything needed for the sacrifice.
On the way the emperor chatted gayly with his retainers, and none of
them noticed the infinite silence and calm of the night. It was only on
reaching the open place on the top of the Capitol, which had been thought
of for the new temple, that it was revealed to them that something
unusual was occurring.
It could not be a night like any other, for on the edge of the cliff they
saw the strangest being. They thought at first that it was an old twisted
olive trunk; then they thought that an ancient statue from the temple of
Jupiter had wandered out on the cliff. At last they saw that it could
only be the old sibyl.
They had never seen anything so old, so weather-beaten, and so gigantic.
If the emperor had not been there, they would have all fled home to their
beds. "It is she," they whispered to each other, "who counts as many
years as there are grains of sand on her native shores. Why has she come
out of her cave to-night? What does she foretell to the emperor and to
the country, she who writes her prophecies on the leaves of trees, and
knows that the wind carries the words of the oracle to him who needs
them?"
They were so terrified that all would have fallen on their knees with
their foreheads to the ground had the sibyl made the slightest movement.
But she sat as still as if she had been without life. Crouched on the
very edge of the cliff, and shading her eyes with her hand, she stared
out into the night. She sat there as if she had gone up on the hill the
better to see something happening far away. She alone could see something
in the black night!
At the same moment the emperor and all his suite perceived how intense
the darkness was. Not one of them could see a hand's-breadth in front
of him. And what a calm, what silence! They could not even hear the
rippling murmur of the Tiber. The air seemed to choke them; a cold sweat
came out on their foreheads, and their hands were stiff and powerless.
They thought that something dreadful must be impending.
But no one liked to show that he was afraid, and everybody told the
emperor that it was a good omen; nature herself held her breath to greet
a new god.
They urged Augustus to hurry, and said that the old sibyl had probably
come up from her cave to greet his genius.
But the truth was that the old sibyl, engrossed in a vision, did not even
know that Augustus had come to the Capitol. She was transported in spirit
to a far distant land, where she thought she was wandering over a great
plain. In the darkness she kept striking her foot against something,
which she thought to be tufts of grass. She bent down and felt with her
hand. No, they were not tufts of grass, but sheep. She was walking among
great sleeping flocks of sheep.
Then she perceived the fire of the shepherds. It was burning in the
middle of the plain, and she approached it. The shepherds were lying
asleep by the fire, and at their sides they had long, pointed staves,
with which they defended their flocks from wild beasts. But the little
animals with shining eyes and bushy tails, which crept forward to the
fire, were they not jackals? And yet the shepherds did not throw their
staves at them; the dogs continued to sleep; the sheep did not flee; and
the wild beasts lay down to rest beside the men.
All this the sibyl saw, but of what was going on behind her on the
mountain she knew nothing. She did not know that people were raising
an altar, lighting charcoal, strewing incense, and that the emperor was
taking one of the doves out of the cage to make a sacrifice to her.
But his hands were so benumbed that he could not hold the bird. With a
single flap of her wings the dove freed herself, and disappeared into the
darkness of the night.
When that happened, the courtiers looked suspiciously at the old sibyl.
They thought that it was she who was the cause of the misfortune.
Could they know that the sibyl still thought she was standing by the
shepherds' fire, and that she was now listening to a faint sound which
came vibrating through the dead silence of the night? She had heard it
for a long time before she noticed that it came from the sky, and not
from the earth. At last she raised her head, and saw bright, glistening
forms gliding about up in the darkness. They were small bands of angels,
who, singing, and apparently searching, flew up and down the wide plain.
While the sibyl listened to the angels' song, the emperor was preparing
for a new sacrifice. He washed his hands, purified the altar, and grasped
the other dove. But although he now made a special effort to hold it
fast, the bird slipped through his fingers, and swung itself up into the
impenetrable night.
The emperor was appalled. He fell on his knees before the empty altar
and prayed to his genius. He called on him for strength to avert the
misfortunes which this night seemed to portend.
Nothing of all this had the sibyl heard. She was listening with her
whole soul to the angels' song, which was growing stronger and stronger.
At last it became so loud that it wakened the shepherds. They raised
themselves on their elbows, and saw shining hosts of silvery angels
moving in the darkness in long, fluttering lines, like birds of passage.
Some had lutes and violins in their hands; others had zithers and harps,
and their song sounded as gay as children's laughter, and as free from
care as the trilling of a lark. When the shepherds heard it they rose up
to go to the village which was their home, to tell of the miracle.
They went by a narrow, winding path, and the sibyl followed them.
Suddenly it became light on the mountain. A great, bright star kindled
over it, and the village on its top shone like silver in the starlight.
All the wandering bands of angels hastened thither with cries of
jubilation, and the shepherds hurried on so fast that they almost ran.
When they had reached the town they found that the angels had gathered
over a low stable near the gate. It was a wretched building, with roof of
straw, and the bare rock for one wall. Above it hung the star, and more
and more angels kept coming. Some of them placed themselves on the straw
roof, or settled down on the steep cliff behind the house; others hovered
over it with fluttering wings. High, high up, the air was lighted by
their shining wings.
At the moment when the star flamed out over the mountain-village all
nature awoke, and the men who stood on the top of the Capitol were
conscious of it. They felt fresh, but caressing breezes; sweet perfumes
streamed up about them; the trees rustled; the Tiber murmured, the stars
shone, and the moon stood high in the heaven and lighted the world.
And out of the sky the two doves flew circling down, and lighted on the
emperor's shoulders.
When this miracle took place Augustus rose up with proud joy, but his
friends and his slaves fell on their knees. "Hail, Cæsar!" they cried.
"Your genius has answered you! You are the god who shall be worshipped on
the heights of the Capitol."
And the tribute which the men in their transport offered the emperor was
so loud that the old sibyl heard it. It waked her from her visions. She
rose from her place on the edge of the cliff, and came forward toward the
people. It seemed as if a dark cloud had risen up from the abyss and sunk
down over the mountain. She was terrifying in her old age. Coarse hair
hung in thin tufts about her head, her joints were thickened, and her
dark skin, hard as bark, covered her body with wrinkle upon wrinkle.
Mighty and awe-inspiring, she advanced towards the emperor. With one hand
she seized his wrist, with the other she pointed towards the distant east.
"Look," she commanded, and the emperor raised his eyes and saw. The
heavens opened before his eyes and he looked away to the far east. And
he saw a miserable stable by a steep cliff, and in the open door some
kneeling shepherds. Within the stable he saw a young mother on her knees
before a little child, who lay on a bundle of straw on the floor.
And the sibyl's big, bony fingers pointed towards that poor child.
"Hail, Cæsar!" said the sibyl, with a scornful laugh. "There is the god
who shall be worshipped on the heights of the Capitol."
Augustus shrank back from her as if from a maniac.
But upon the sibyl fell the mighty spirit of the prophetess. Her dim eyes
began to burn, her hands were stretched towards heaven, her voice did not
seem to be her own, but rang with such strength that it could have been
heard over the whole world. And she spoke words which she seemed to have
read in the stars:--
"On the heights of the Capitol the redeemer of the world shall be
worshipped,
Christ or Antichrist, but no frail mortal."
When she had spoken she moved away between the terrified men, went slowly
down the mountain, and disappeared.
Augustus, the next day, strictly forbade his people to raise him any
temple on the Capitol. In its place he built a sanctuary to the new-born
godchild and called it "Heaven's Altar," Aracoeli.
II
ROME'S HOLY CHILD
On the summit of the Capitol stood a monastery occupied by Franciscan
monks. It was, however, less a monastery than a fortress. It was like a
watch-tower by the seashore, where watch was kept for an approaching foe.
Near the monastery stood the magnificent basilica "Santa Maria in
Aracoeli." The basilica was built because the sibyl had caused Augustus
to see Christ. But the monastery was built because they feared the
fulfilment of the sibyl's prophecy; that Antichrist should come to be
worshipped on the Capitol.
And the monks felt like warriors. When they went to church to sing and
pray, they thought that they were walking on ramparts, and sending
showers of arrows down on the assaulting Antichrist.
They lived always in terror of Antichrist, and all their service was a
struggle to keep him away from the Capitolium.
They drew their hats down over their eyes and sat and gazed out into
the world. Their eyes grew feverish with watching, and they continually
thought they discovered Antichrist. "He is here, he is there!" they
cried. And they fluttered up in their brown robes and braced themselves
for the struggle, as crows gather on a crag when they catch a glimpse of
an eagle.
But some said: "What is the use of prayers and penitence? The sibyl has
said it. Antichrist must come."
Then others said, "God can work a miracle. If it was of no avail to
struggle, He would not have let the sibyl warn us."
Year after year the Franciscans defended the Capitol by penitences, and
works of charity, and the promulgation of God's word.
They protected it century after century, but as time went on, men
became more and more feeble and lacking in force. The monks said among
themselves: "Soon the kingdoms of the earth can stand no longer. A
redeemer of the world is needed as in the time of Augustus."
They tore their hair and scourged themselves, for they knew that he who
was to be born again must be the Antichrist, and that it would be a
regeneration of force and violence.
As a sick man is tormented by his pain, so were they hunted by the
thought of Antichrist. And they saw him before them. He was as rich as
Christ had been poor, as wicked as Christ had been good, as honored as
Christ had been humiliated.
He bore powerful weapons and marched at the head of bloody evil-doers.
He overturned the churches, murdered the priests, and armed people for
strife, so that brother fought against brother, and each feared his
neighbor, and there was no peace.
And for every person of power and might who made his way over the sea of
time, they cried out from the watch-tower on the Capitol: "Antichrist,
Antichrist!"
And for every one who disappeared, and went under, the monks cried:
"Hosanna!" and sang the "Te Deum." And they said: "It is because of our
prayers that the wicked fall before they succeed in scaling the Capitol."
It was a hard punishment that in that beautiful monastery its monks could
never feel at rest. Their nights were heavier than their days. Then they
saw wild beasts come into their cells and stretch themselves out beside
them on their beds. And each wild beast was Antichrist. But some of the
monks saw him as a dragon, and others as a griffin, and others as a
sphinx. When they got up from their dreams they were as weak as after a
severe illness.
The only comfort of these poor monks was the miracle-working image of
Christ, which was kept in the basilica of Aracoeli. When a monk was
frightened to desperation, he went into the church to seek consolation
from it. He would go through the whole basilica and into a well-guarded
chapel at the side of the great altar. There he lighted the consecrated
wax candles, and spoke a prayer, before opening the altar shrine, which
had double locks and doors of iron. And as long as he gazed at the image,
he remained upon his knees.
The image represented a little babe, but he had a gold crown upon his
head, gold shoes upon his feet, and his whole dress shone with jewels,
which were given to him by those in distress, who had called on him for
help. And the walls of the chapel were covered with pictures, which
showed how he had saved from dangers of fire and shipwreck, how he had
cured the sick and helped all those who were in trouble. When the monk
saw it he rejoiced, and said to himself: "Praise be to God! As yet it is
Christ who is worshipped on the Capitol."
The monk saw the face of the image smile at him with mysterious,
conscious power, and his spirit soared up into the holy realms of
confidence. "What can overthrow you in your might?" he said. "What can
overthrow you? To you the Eternal City bends its knees. You are Rome's
Holy Child. Yours is the crown which the people worship. You come in
your might with help and strength and consolation. You alone shall be
worshipped on the Capitol."
The monk saw the crown of the image turn into a halo, which sent out rays
over the whole world. And in whatever direction he followed the rays he
saw the world full of churches, where Christ was worshipped. It seemed
as if a powerful conqueror had shown him all the castles and fortresses
which defended his kingdom. "It is certain that you cannot fall," said
the monk. "Your kingdom will be everlasting."
And every monk who saw the image had a few hours of consolation and
peace, until fear seized him again. But had the monks not possessed the
image, their souls would not have found a moment's rest.
Thus had the monks of Aracoeli, by prayers and struggles, worked their
way through the centuries, and there had never lacked for watchers; as
soon as one had been exhausted by terror and anxiety, others had hurried
forward to take his place.
And although most of those who entered the monastery were struck down by
madness or premature death, the succession of monks never diminished,
for it was held a great honor before God to wage the war on Aracoeli.
So it happened that sixty years ago this struggle still went on, and in
the degenerate times the monks fought with greater eagerness than ever
before, and awaited the certain coming of Antichrist.
At that time a rich Englishwoman came to Rome. She went up to the
Aracoeli and saw the image, and he charmed her so that she thought she
could not live if she did not possess him. She went again and again up to
Aracoeli to see the image, and at last she asked the monks if she might
buy him.
But even if she had covered the whole mosaic floor in the great basilica
with gold coins, the monks would not have been willing to sell her that
image, which was their only consolation.
Still the Englishwoman was attracted beyond measure by the image, and
found no joy nor peace without it. Unable to accomplish her object by any
other means, she determined to steal the image. She did not think of the
sin she was committing; she felt only a strong compulsion and a burning
thirst, and preferred to risk her soul rather than to deny her heart the
joy of possessing the object of her longing. And to accomplish her end,
she first had an image made exactly like the one on Aracoeli.
The image on Aracoeli was carved from olive wood from the gardens of
Gethsemane; but the Englishwoman dared to have an image carved from elm
wood, which was exactly like him. The image on Aracoeli was not painted
by mortal hand. When the monk who had carved him had taken up his
brushes and colors, he fell asleep over his work. And when he awoke,
the image was ,--self-painted as a sign that God loved him. But
the Englishwoman was bold enough to let an earthly painter paint her elm
image so that he was like the holy image.
For the false image she procured a crown and shoes, but they were not of
gold; they were only tin and gilding. She ordered ornaments; she bought
rings, and necklaces, and chains, and bracelets, and diamond suns--but
they were all brass and glass; and she dressed him as those seeking help
had dressed the true image. When the image was ready she took a needle
and scratched in the crown: "My kingdom is only of this world." It was
as if she was afraid that she herself would not be able to distinguish
one image from the other. And it was as if she had wished to appease her
own conscience. "I have not wished to make a false Christ image. I have
written in his crown: 'My kingdom is only of this world.'"
Thereupon she wrapped herself in a big cloak, hid the image under it, and
went up to Aracoeli. And she asked that she might be allowed to say her
prayers before the Christchild.
When she stood in the sanctuary, and the candles were lighted, and the
iron door opened, and the image showed itself to her, she began to
tremble and shake and looked as if she were going to faint. The monk who
was with her hurried into the sacristy after water and she was left alone
in the chapel. And when he came back she had committed the sacrilege.
She had exchanged the holy, miracle-working image, and put the false and
impotent one in his place.
The monk saw nothing. He shut in the false image behind iron doors
and double locks, and the Englishwoman went home with the treasure of
Aracoeli. She placed him in her palace on a pedestal of marble and was
more happy than she had ever been before.
Up on Aracoeli, where no one knew what injury they had suffered, they
worshipped the false Christ image as they had worshipped the true one,
and when Christmas came they built for him in the church, as was the
custom, a most beautiful niche. There he lay, shining like a jewel, on
Maria's knees, and about him shepherds and angels and wise men were
arranged. And as long as he lay there children came from Rome, and the
Campagna, and were lifted up on a little pulpit in the basilica of
Aracoeli, and they preached on the sweetness and tenderness and nobleness
and power of the little Christchild.
But the Englishwoman lived in great terror that some one would discover
that she had stolen the Christ image of Aracoeli. Therefore she confessed
to no one that the image she had was the real one. "It is a copy," she
said; "it is as like the real one as it can be, but it is only copied."
Now it happened that she had a little Italian servant girl. One day when
the latter went through the room she stopped before the image and spoke
to him. "You poor Christchild, who are no Christchild," she said, "if you
only knew how the real child lies in his glory in the niche in Aracoeli
and how Maria and San Giuseppe and the shepherds are kneeling before him!
And if you knew how the children place themselves on a little pulpit just
in front of him, and how they courtesy, and kiss their fingers to him,
and preach for him as beautifully as they can!"
A few days after the little maid came again and spoke to the image. "You
poor Christchild, who are no Christchild," she said, "do you know that
to-day I have been up in Aracoeli and have seen how the true child was
carried in the procession? They held a canopy over him, all the people
fell on their knees, and they sang and played before him. Never will you
see anything so wonderful!"
And mark that a few days later the little maid came again and spoke to
the image: "Do you know, Christchild, who are not a real Christchild,
that it is better for you to stand where you are standing? For the real
child is called to the sick and is driven to them in his gold-laced
carriage, but _he_ cannot help them and they die in despair. And people
begin to say that Aracoeli's holy child has lost his power to do good,
and that prayers and tears do not move him. It is better for you to stand
where you are standing than to be called upon and not to be able to help."
But the next night a miracle came to pass. About midnight a loud ringing
was heard at the cloister gate at Aracoeli. And when the gate-keeper did
not come quickly enough to open, some one began to knock. It sounded
clear, like ringing metal, and it was heard through the whole monastery.
All the monks leaped from their beds. All who had been tortured by
terrible dreams rose at one time, and believed that Antichrist was come.
But when they opened the door--when they opened it! It was the little
Christ image that stood on the threshold. It was his little hand that
had pulled the bell-rope; it was his little, gold-shod foot that had
been stretched out to kick the door.
The gate-keeper instantly took the holy child up in his arms. Then he saw
that it had tears in its eyes. Alas, the poor, holy child had wandered
through the town by night! What had it not seen? So much poverty and so
much want; so much wickedness and so many crimes! It was terrible to
think what it must have experienced.
The gate-keeper went immediately to the prior and showed him the image.
And they wondered how it had come out into the night.
Then the prior had the church bells rung to call the monks to the
service. And all the monks of Aracoeli marched into the great, dim
basilica in order to place the image, with all solemnity, back in its
shrine.
Worn and suffering, they walked and trembled in their heavy homespun
robes. Several of them were weeping, as if they had escaped from some
terrible danger. "What would have happened to us," they said, "if our
only consolation had been taken from us? Is it not Antichrist who has
tempted out Rome's holy child from the sheltering sanctuary?"
But when they came to set the Christ image in the shrine of the chapel,
they found there the false child; him who wore the inscription on his
crown: "My kingdom is only of this world."
And when they examined the image more closely they found the inscription.
Then the prior turned to the monks and spoke to them:--
"Brothers, we will sing the 'Te Deum,' and cover the pillars of the
church with silk, and light all the wax candles, and all the hanging
lamps, and we will celebrate a great festival.
"As long as the monastery has stood it has been a home of terror and a
cursed dwelling; but for the suffering of all those who have lived here,
God has been gracious. And now all danger is over.
"God has crowned the fight with victory, and this that you have seen is
the sign that Antichrist shall not be worshipped on the Capitol.
"For in order that the sibyl's words should be carried out, God has sent
this false image of Christ that bears the words of Antichrist in its
crown, and he has allowed us to worship and adore him as if he had been
the great miracle-worker.
"But now we can rest in joy and peace, for the sibyl's mystic speech is
fulfilled, and Antichrist has been worshipped here.
"Great is God, the Almighty, who has let our cruel fear be dispelled, and
who has carried out His will without the world needing to gaze upon the
false image made by man.
"Happy is the monastery of Aracoeli that rests under the protection of
God, and does His will, and is blessed by His abounding grace."
When the prior had said those words he took the false image in his hands,
went through the church, and opened the great door. Thence he walked
out on the terrace. Below him lay the high and broad stairway with its
hundred and nineteen marble steps that leads down from the Capitol as if
into an abyss. And he raised the image over his head and cried aloud:
"Anathema Antikristo!" and hurled him from the summit of the Capitol down
into the world.
III
ON THE BARRICADE
When the rich Englishwoman awoke in the morning she missed the image and
wondered where she should look for him. She believed that no one but
the monks of Aracoeli could have taken him, and she hurried towards the
Capitol to spy and search.
She came to the great marble staircase that leads up to the basilica of
Aracoeli. And her heart beat wildly with joy, for on the lowest step lay
he whom she sought. She seized the image, threw her cloak about him, and
hurried home. And she put him back on his place of honor.
But as she now sank into contemplation of his beauty, she found that the
crown had been dented. She lifted it off the image to see how great the
damage was, and at the same moment her eyes fell on the inscription that
she herself had scratched: "My kingdom is only of this world."
Then she knew that this was the false Christ image, and that the right
one had returned to Aracoeli.
She despaired of ever again getting it into her possession, and she
decided to leave Rome the next day, for she would not remain there when
she no longer had the image.
But when she left she took the forged image with her, because he reminded
her of the one she loved, and he followed her afterwards on all her
journeys.
She was never at rest and travelled continually, and in that way the
image was carried about over the whole world.
And wherever the image came, the power of Christ seemed to be diminished
without any one rightly understanding why. For nothing could look more
impotent than that poor image of elm wood, dressed out in brass rings and
glass beads.
When the rich Englishwoman who had first owned the image was dead, he
came as an inheritance to another rich Englishwoman, who also travelled
continually, and from her to a third.
Once, and it was still in the time of the first Englishwoman, the image
came to Paris.
As he passed through the great city there was an insurrection. Crowds
rushed wildly screaming through the streets and cried for bread. They
plundered the shops and threw stones at the houses of the rich. Troops
were called out against them, and then they tore up the stones of the
street, dragged together carriages and furniture, and built barricades.
As the rich Englishwoman came driving in her great travelling-carriage,
the mass of people rushed towards it, forced her to leave it, and dragged
the carriage up to one of the barricades.
When they tried to roll the carriage up among all the thousand things of
which the barricade consisted, one of the big trunks fell to the ground.
The cover sprang open, and among other things out rolled the rejected
Christ image.
The people threw themselves upon him to plunder, but they soon saw that
all his grandeur was imitation and quite worthless, and they began to
laugh at him and mock him.
He went from hand to hand among the agitators, until one of them bent
forward to look at his crown. His eyes were attracted by the words which
stood scratched there: "My kingdom is only of this world."
The man called this out quite loudly, and they all screamed that the
little image should be their badge. They carried him up to the summit of
the barricade and placed him there like a banner.
Among those who defended the barricade was one man who was not a poor
working-man, but a man of education, who had passed his whole life in
study. He knew all the want that tortured mankind, and his heart was full
of sympathy, so that he continually sought means to better their lot. For
thirty years he had written and thought without finding any remedy. Now
on hearing the alarm bell he had obeyed it and rushed into the streets.
He had seized a weapon and gone with the insurgents with the thought that
the riddle which he had been unable to solve should now be made clear by
violence and force, and that the poor should be able to fight their way
to a better lot.
There he stood the whole day and fought; and people fell about him, blood
splashed up into his face, and the misery of life seemed to him greater
and more deplorable than ever before.
But whenever the smoke cleared away, the little image shone before his
eyes; through all the tumult of the fight it stood unmoved high up on the
barricade.
Every time he saw the image the words "My kingdom is only of this world"
flashed through his brain. At last he thought that the words wrote
themselves in the air and began to wave before his eyes, now in fire, now
in blood, now in smoke.
He stood still. He stood there with gun in hand, but he had stopped
fighting. Suddenly he knew that this was the word that he had sought
after all his life. He knew what he would say to the people, and it was
the poor image that had given him the solution.
He would go out into the whole world and proclaim: "Your kingdom is only
of this world.
"Therefore you must care for this life and live like brothers. And you
shall divide your property so that no one is rich and no one poor. You
shall all work, and the earth shall be owned by all, and you shall all be
equal.
"No one shall hunger, no one shall be tempted to luxury, and no one shall
suffer want in his old age.
"And you must think of increasing every one's happiness, for there is no
compensation awaiting you. Your kingdom is only of this world."
All this passed through his brain while he stood on the barricade, and
when the thought became clear to him, he laid down his weapon, and did
not lift it again for strife and the shedding of blood.
A moment later the barricade was stormed and taken. The victorious troops
dashed through and quelled the insurrection, and before night order and
peace reigned in the great city.
The Englishwoman sent out her servants to look for her lost possessions,
and they found many, if not all. What they found first of all on the
captured barricade was the image ejected from Aracoeli.
But the man who had been taught during the fight by the image began to
proclaim to the world a new doctrine, which is called Socialism, but
which is an Antichristianity.
And it loves, and renounces, and teaches, and suffers like Christianity,
so that it has every resemblance to the latter, just as the false image
from Aracoeli has every resemblance to the real Christ image.
And like the false image it says: "My kingdom is only of this world."
And although the image that has spread abroad the teachings is unnoticed
and unknown, the teachings are not; they go through the world to save and
remodel it.
They are spreading from day to day. They go out through all countries,
and bear many names, and they mislead because they promise earthly
happiness and enjoyment to all, and win followers more than any doctrine
that has gone through the world since the time of Christ.
FIRST BOOK
"_There shall be great want_"
I
MONGIBELLO
Towards the end of the seventies there was in Palermo a poor boy whose
name was Gaetano Alagona. That was lucky for him! If he had not been one
of the old Alagonas people would have let him starve to death. He was
only a child, and had neither money nor parents. The Jesuits of Santa
Maria i Jesu had taken him out of charity into the cloister school.
One day, when studying his lesson, a father came and called him from the
school-room, because a cousin wished to see him. What, a cousin! He had
always heard that all his relatives were dead. But Father Josef insisted
that it was a real Signora, who was his relative and wished to take him
out of the monastery. It became worse and worse. Did she want to take him
out of the monastery? That she could never do! He was going to be a monk.
He did not at all wish to see the Signora. Could not Father Josef tell
her that Gaetano would never leave the monastery, and that it was of no
avail to ask him? No, Father Josef said that he could not let her depart
without seeing him, and he half dragged Gaetano into the reception-room.
There she stood by one of the windows. She had gray hair; her skin was
brown; her eyes were black and as round as beads. She had a lace veil on
her head, and her black dress was smooth with wear, and a little green,
like Father Josef's very oldest cassock.
She made the sign of the cross when she saw Gaetano. "God be praised, he
is a true Alagona!" she said, and kissed his hand.
She said that she was sorry that Gaetano had reached his twelfth year
without any of his family asking after him; but she had not known that
there were any of the other branch alive. How had she found it out now?
Well, Luca had read the name in a newspaper. It had stood among those who
had got a prize. It was a half-year ago now, but it was a long journey to
Palermo. She had had to save and save to get the money for the journey.
She had not been able to come before. But she had to come and see him.
_Santissima madre_, she had been so glad! It was she, Donna Elisa, who
was an Alagona. Her husband, who was dead, had been an Antonelli. There
was one other Alagona, that was her brother. He, too, lived at Diamante.
But Gaetano probably did not know where Diamante was. The boy drew his
head back. No, she thought as much, and she laughed.
"Diamante is on Monte Chiaro. Do you know where Monte Chiaro is?"
"No."
She drew up her eyebrows and looked very roguish.
"Monte Chiaro is on Etna, if you know where Etna is."
It sounded so anxious, as if it were too much to ask that Gaetano should
know anything about Etna. And they laughed, all three, she and Father
Josef and Gaetano.
She seemed a different person after she had made them laugh. "Will you
come and see Diamante and Etna and Monte Chiaro?" she asked briskly.
"Etna you must see. It is the greatest mountain in the world. Etna is a
king, and the mountains round about kneel before him, and do not dare to
lift their eyes to his face."
Then she told many tales about Etna. She thought perhaps that it would
tempt him.
And it was really true that Gaetano had not thought before what kind of
a mountain Etna was. He had not remembered that it had snow on its head,
oak forests in its beard, vineyards about its waist, and that it stood in
orange groves up to its knees. And down it ran broad, black rivers. Those
streams were wonderful; they flowed without a ripple; they heaved without
a wind; the poorest swimmer could cross them without a bridge. He guessed
that she meant lava. And she was glad that he had guessed it. He was a
clever boy. A real Alagona!
And Etna was so big! Fancy that it took three days to drive round it and
three days to ride up to the top and down again! And that there were
fifty towns beside Diamante on it, and fourteen great forests, and two
hundred small peaks, which were not so small either, although Etna was so
big that they seemed as insignificant as a swarm of flies on a church
roof. And that there were caves which could hold a whole army, and hollow
old trees, where a flock of sheep could find shelter from the storm!
Everything wonderful was to be found on Etna. There were rivers of which
one must beware. The water in them was so cold that any one who drank of
it would die. There were rivers which flowed only by day, and others that
flowed only in winter, and some which ran deep under the earth. There
were hot springs, and sulphur springs, and mud-volcanoes.
It would be a pity for Gaetano not to see the mountain, for it was so
beautiful. It stood against the sky like a great tent. It was as gayly
as a merry-go-round. He ought to see it in the morning and
evening, when it was red; he ought to see it at night, when it was white.
He ought also to know that it truly could take every color; that it could
be blue, black, brown or violet; sometimes it wore a veil of beauty, like
a signora; sometimes it was a table covered with velvet; sometimes it had
a tunic of gold brocade and a mantle of peacock's-feathers.
He would also like to know how it could be that old King Arthur was
sitting there in a cave. Donna Elisa said that it was quite certain that
he still lived on Etna, for once, when the bishop of Catania was riding
over the mountain, three of his mules ran away, and the men who followed
them found them in the cave with King Arthur. Then the king asked the
guides to tell the bishop that when his wounds were healed he would come
with his knights of the Round Table and right everything that was in
disorder in Sicily. And he who had eyes to see knew well enough that
King Arthur had not yet come out of his cave.
Gaetano did not wish to let her tempt him, but he thought that he might
be a little friendly. She was still standing, but now he fetched her a
chair. That would not make her think that he wanted to go with her.
He really liked to hear her tell about her mountain. It was so funny that
it should have so many tricks. It was not at all like Monte Pellegrino,
near Palermo, that only stood where it stood. Etna could smoke like a
chimney and blow out fire like a gas jet. It could rumble, shake, vomit
forth lava, throw stones, scatter ashes, foretell the weather, and
collect rain. If Mongibello merely stirred, town after town fell, as if
the houses had been cards set on end.
Mongibello, that was also a name for Etna. It was called Mongibello
because that meant the mountain of mountains. It deserved to be called so.
Gaetano saw that she really believed that he would not be able to resist.
She had so many wrinkles in her face, and when she laughed, they ran
together like a net. He stood and looked at it; it seemed so strange. But
he was not caught yet in the net.
She wondered if Gaetano really would have the courage to come to Etna.
For inside the mountain were many bound giants and a black castle, which
was guarded by a dog with many heads. There was also a big forge and a
lame smith with only one eye in the middle of his forehead. And worst of
all, in the very heart of the mountain, there was a sulphur sea which
cooked like an oil kettle, and in it lay Lucifer and all the damned. No,
he never would have the courage to come there, she said.
Otherwise there was no danger in living there, for the mountain feared
the saints. Donna Elisa said that it feared many saints, but most Santa
Agata of Catania. If the Catanians always were as they should be to her,
then neither earthquake nor lava could do them any harm.
Gaetano stood quite close to her and he laughed at everything she said.
How had he come there and why could he not stop laughing? It was a
wonderful signora.
Suddenly he said, in order not to deceive her, "Donna Elisa, I am going
to be a monk."--"Oh, are you?" she said. Then without anything more she
began again to tell about the mountain.
She said that now he must really listen; now she was coming to the most
important of all. He was to fellow her to the south side of the mountain
so far down that they were near the castle of Catania, and there he would
see a valley, a quite big and wide oval valley. But it was quite black;
the lava streams came from all directions flowing down into it. There
were only stones there, not a blade of grass.
But what had Gaetano believed about the lava? Donna Elisa was sure that
he believed that it lay as even and smooth on Etna as it lies in the
streets. But on Etna there are so many surprises. Could he understand
that all the serpents and dragons and witches that lay and boiled in
the lava ran out with it when there was an eruption? There they lay and
crawled and crept and twisted about each other, and tried to creep up
to the cold earth, and held each other fast in misery until the lava
hardened about them. And then they could never come free. No indeed!
The lava was not unproductive, as he thought. Although no grass grew,
there was always something to see. But he could never guess what it was.
It groped and fell; it tumbled and crept; it moved on its knees, on its
head, and on its elbows. It came up the sides of the valley and down
the sides of the valley; it was all thorns and knots; it had a cloak of
spider's-web and a wig of dust, and as many joints as a worm. Could it be
anything but the cactus? Did he know that the cactus goes out on the lava
and breaks the ground like a peasant? Did he know that nothing but the
cactus can do anything with the lava?
Now she looked at Father Josef and made a funny face. The cactus was the
best goblin to be found on Etna; but goblins were goblins. The cactus was
a Turk, for it kept female slaves. No sooner had the cactus taken root
anywhere than it must have almond trees near it. Almond trees are fine
and shining signoras. They hardly dare to go out on the black surface,
but that does not help them. Out they must, and out they are. Oh, Gaetano
should see if he came there. When the almond trees stand white with their
blossoms in the spring on the black field among the gray cacti, they are
so innocent and beautiful that one could weep over them as over captive
princesses.
Now he must know where Monte Chiaro lay. It shot up from the bottom of
that black valley. She tried to make her umbrella stand on the floor.
It stood so. It stood right up. It had never thought of either sitting
or lying. And Monte Chiaro was as green as the valley was black. It
was palm next palm, vine upon vine. It was a gentleman in a flowery
dressing-gown. It was a king with a crown on his head. It bore the whole
of Diamante about its temples.
Some time before Gaetano had a desire to take her hand. If he only could
do it. Yes, he could. He drew her hand to him like a captured treasure.
But what should he do with it? Perhaps pat it. If he tried quite gently
with one finger, perhaps she would not notice it. Perhaps she would not
notice if he took two fingers. Perhaps she would not even notice if he
should kiss her hand. She talked and talked. She noticed nothing at all.
There was still so much she wished to say. And nothing so droll as her
story about Diamante!
She said that the town had once lain down on the bottom of the valley.
Then the lava came, and fiery red looked over the edge of the valley.
What, what! was the last day come? The town in great haste took its
houses on its back, on its head, and under its arms, and ran up Monte
Chiaro, that lay close at hand.
Zigzagging up the mountain the town ran. When it was far enough up it
threw down a town gate and a piece of town wall. Then it ran round the
mountain in a spiral and dropped down houses. The poor people's houses
tumbled as they could and would. There was no time for anything else.
No one could ask anything better than crowding and disorder and crooked
streets. No, that you could not. The chief street went in a spiral round
the mountain, just as the town had run, and along it had set down here
a church and there a palace. But there had been that much order that
the best came highest up. When the town came to the top of the mountain
it had laid out a square, and there it had placed the city hall and the
Cathedral and the old palazzo Geraci.
If he, Gaetano Alagona, would follow her to Diamante, she would take him
with her up to the square on the top of the mountain, and show him what
stretches of land the old Alagonas had owned on Etna, and on the plain of
Catania, and where they had raised their strongholds on the inland peaks.
For up there all that could be seen, and even more. One could see the
whole sea.
Gaetano had not thought that she had talked long, but Father Josef seemed
to be impatient. "Now we have come to your own home, Donna Elisa," he
said quite gently.
But she assured Father Josef that at her house there was nothing to see.
What she first of all wished to show Gaetano was the big house on the
corso, that was called the summer palace. It was not so beautiful as the
palazzo Geraci, but it was big; and when the old Alagonas were prosperous
they came there in summer to be nearer the snows of Etna. Yes, as she
said, towards the street it was nothing to see, but it had a beautiful
court-yard with open porticos in both the stories. And on the roof there
was a terrace. It was paved with blue and white tiles, and on every tile
the coat of arms of the Alagonas was burnt in. He would like to come and
see that?
It occurred to Gaetano that Donna Elisa must be used to having children
come and sit on her knees when she was at home. Perhaps she would not
notice if he should also come. And he tried. And so it was. She was used
to it. She never noticed it at all.
She only went on talking about the palace. There was a great state suite,
where the old Alagonas had danced and played. There was a great hall with
a gallery for the music; there was old furniture and clocks like small
white alabaster temples that stood on black ebony pedestals. In the state
apartment no one lived, but she would go there with him. Perhaps he had
thought that she lived in the summer palace. Oh, no; her brother, Don
Ferrante, lived there. He was a merchant, and had his shop on the lower
floor; and as he had not yet brought home a signora, everything stood up
there as it had stood.
Gaetano wondered if he could sit on her knees any longer. It was
wonderful that she did not notice anything. And it was fortunate, for
otherwise she might have believed that he had changed his mind about
being a monk.
But she was just now more than ever occupied with her own affairs. A
little flush flamed up in her cheeks under all the brown, and she made a
few of the funniest faces with her eyebrows. Then she began to tell how
she herself lived.
It seemed as if Donna Elisa must have the very smallest house in the
town. It lay opposite the summer palace, but that was its only good
point. She had a little shop, where she sold medallions and wax candles
and everything that had to do with divine service. But, with all respect
to Father Josef, there was not much profit in such a trade now-a-days,
however it may have been formerly. Behind the shop there was a little
workshop. There her husband had stood and carved images of the saints,
and rosary beads; for he had been an artist, Signor Antonelli. And next
to the workshop were a couple of small rat-holes; it was impossible to
turn in them; one had to squat down, as in the cells of the old kings.
And up one flight were a couple of small hen-coops. In one of them
she had laid a little straw and put up a few hooks. That would be for
Gaetano, if he would come to her.
Gaetano thought that he would like to pat her cheek. She would be sorry
when he could not go with her. Perhaps he could permit himself to pat
her. He looked under his hair at Father Josef. Father Josef sat and
looked on the floor and sighed, as he was in the habit of doing. He did
not think of Gaetano, and she, she noticed nothing at all.
She said that she had a maid, whose name was Pacifica, and a man, whose
name was Luca. She did not get much help, however, for Pacifica was old;
and, since she had grown deaf, she had become so irritable that she could
not let her help in the shop. And Luca, who really was to have been a
wood-carver, and carve saints that she could sell, never gave himself
time to stand still in the workshop; he was always out in the garden,
looking after the flowers. Yes, they had a little garden among the stones
on Monte Chiaro. But he need not think it was worth anything. She had
nothing like the one in the cloister, that Gaetano would understand. But
she wanted so much to have him, because he was one of the old Alagonas.
And there at home she and Luca and Pacifica had said to one another:
"Do we ask whether we will have a little more care, if we can only get
him here?" No, the Madonna knew that they had not done so. But now the
question was, whether he was willing to endure anything to be with them.
And now she had finished, and Father Josef asked what Gaetano thought
of answering. It was the prior's wish, Father Josef said, that Gaetano
should decide for himself. And they had nothing against his going out
into the world, because he was the last of his race.
Gaetano slid gently down from Donna Elisa's lap. But to answer! That
was not such an easy thing to answer. It was very hard to say no to the
signora.
Father Josef came to his assistance. "Ask the signora that you may be
allowed to answer in a couple of hours, Gaetano. The boy has never
thought of anything but being a monk," he explained to Donna Elisa.
She stood up, took her umbrella, and tried to look glad, but there were
tears in her eyes.
Of course, of course he must consider it, she said. But if he had known
Diamante he would not have needed to. Now only peasants lived there,
but once there had been a bishop, and many priests, and a multitude of
monks. They were gone now, but they were not forgotten. Ever since that
time Diamante was a holy town. More festival days were celebrated there
than anywhere else, and there were quantities of saints; and even to-day
crowds of pilgrims came there. Whoever lived at Diamante could never
forget God. He was almost half a priest. So for that reason he ought to
come. But he should consider it, if he so wished. She would come again
to-morrow.
Gaetano behaved himself very badly. He turned away from her and rushed to
the door. He did not say a word of thanks to her for coming. He knew that
Father Josef had expected it, but he could not. When he thought of the
great Mongibello that he never would see, and of Donna Elisa, who would
never come again, and of the school, and of the shut-in cloister garden,
and of a whole restricted life! Father Josef never could expect so much
of him; Gaetano had to run away.
It was high time too. When Gaetano was ten steps from the door, he began
to cry. It was too bad about Donna Elisa. Oh, that she should be obliged
to travel home alone! That Gaetano could not go with her!
He heard Father Josef coming, and he hid his face against the wall. If he
could only stop sobbing!
Father Josef came sighing and murmuring to himself, as he always did.
When he came up to Gaetano he stopped, and sighed more than ever.
"It is Mongibello, Mongibello," said Father Josef; "no one can resist
Mongibello."
Gaetano answered him by weeping more violently.
"It is the mountain calling," murmured Father Josef. "Mongibello is like
the whole earth; it has all the earth's beauty and charm and vegetation
and expanses and wonders. The whole earth comes at once and calls him."
Gaetano felt that Father Josef spoke the truth. He felt as if the earth
stretched out strong arms to catch him. He felt that he needed to bind
himself fast to the wall in order not to be torn away.
"It is better for him to see the earth," said Father Josef. "He would
only be longing for it if he stayed in the monastery. If he is allowed to
see the earth perhaps he will begin again to long for heaven."
Gaetano did not understand what Father Josef meant when he felt himself
lifted into his arms, carried back into the reception-room, and put down
on Donna Elisa's knees.
"You shall take him, Donna Elisa, since you have won him," said Father
Josef. "You shall show him Mongibello, and you shall see if you can keep
him."
But when Gaetano once more sat on Donna Elisa's lap he felt such
happiness that it was impossible for him to run away from her again. He
was as much captured as if he had gone into Mongibello and the mountain
walls had closed in on him.
II
FRA GAETANO
Gaetano had lived with Donna Elisa a month, and had been as happy as a
child can be. Merely to travel with Donna Elisa had been like driving
behind gazelles and birds of paradise; but to live with her was to be
carried on a golden litter, screened from the sun.
Then the famous Franciscan, Father Gondo, came to Diamante, and Donna
Elisa and Gaetano went up to the square to listen to him. For Father
Gondo never preached in a church; he always gathered the people about him
by fountains or at the town gates.
The square was swarming with people; but Gaetano, who sat on the railing
of the court-house steps, plainly saw Father Gondo where he stood on
the curb-stone. He wondered if it could be true that the monk wore a
horse-hair shirt under his robes, and that the rope that he had about his
waist was full of knots and iron points to serve him as a scourge.
Gaetano could not understand what Father Gondo said, but one shiver after
another ran through him at the thought that he was looking at a saint.
When the Father had spoken for about an hour, he made a sign with his
hand that he would like to rest a moment. He stepped down from the steps
of the fountain, sat down, and rested his face in his hands. While the
monk was sitting so, Gaetano heard a gentle roaring. He had never before
heard any like it. He looked about him to discover what it was. And it
was all the people talking. "Blessed, blessed, blessed!" they all said at
once. Most of them only whispered and murmured; none called aloud, their
devotion was too great. And every one had found the same word. "Blessed,
blessed!" sounded over the whole market-place. "Blessings on thy lips;
blessings on thy tongue; blessings on thy heart!"
The voices sounded soft, choked by weeping and emotion, but it was as if
a storm had passed by through the air. It was like the murmuring of a
thousand shells.
That took much greater hold of Gaetano than the monk's sermon. He did
not know what he wished to do, for that gentle murmuring filled him with
emotion; it seemed almost to suffocate him. He climbed up on the iron
railing, raised himself above all the others, and began to cry the same
as they, but much louder, so that his voice cut through all the others.
Donna Elisa heard it and seemed to be displeased. She drew Gaetano down
and would not stay any longer, but went home with him.
In the middle of the night Gaetano started up from his bed. He put on his
clothes, tied together what he possessed in a bundle, set his hat on his
head and took his shoes under his arm. He was going to run away. He could
not bear to live with Donna Elisa.
Since he had heard Father Gondo, Diamante and Mongibello were nothing to
him. Nothing was anything compared to being like Father Gondo, and being
blessed by the people. Gaetano could not live if he could not sit by the
fountain in the square and tell legends.
But if Gaetano went on living in Donna Elisa's garden, and eating peaches
and mandarins, he would never hear the great human sea roar about him.
He must go out and be a hermit on Etna; he must dwell in one of the big
caves, and live on roots and fruits. He would never see a human being; he
would never cut his hair; and he would wear nothing but a few dirty rags.
But in ten or twenty years he would come back to the world. Then he would
look like a beast and speak like an angel.
That would be another matter than wearing velvet clothes and a glazed
hat, as he did now. That would be different from sitting in the shop with
Donna Elisa and taking saint after saint down from the shelf and hearing
her tell about what they had done. Several times he had taken a knife and
a piece of wood and had tried to carve images of the saints. It was very
hard, but it would be worse to make himself into a saint; much worse.
However, he was not afraid of difficulties and privations.
He crept out of his room, across the attic and down the stair. It only
remained to go through the shop out to the street, but on the last step
he stopped. A faint light filtered through a crack in the door to the
left of the stairs.
It was the door to Donna Elisa's room, and Gaetano did not dare to go any
further, since his foster mother had her candle lighted. If she was not
asleep she would hear him when he drew the heavy bolts on the shop door.
He sat softly down on the stairs to wait.
Suddenly he happened to think that Donna Elisa must sit up so long at
night and work in order to get him food and clothes. He was much touched
that she loved him so much as to want to do it. And he understood what a
grief it would be to her if he should go.
When he thought of that he began to weep.
But at the same time he began to upbraid Donna Elisa in his thoughts. How
could she be so stupid as to grieve because he went. It would be such a
joy for her when he should become a holy man. That would be her reward
for having gone to Palermo and fetched him.
He cried more and more violently while he was consoling Donna Elisa. It
was hard that she did not understand what a reward she would receive.
There was no need for her to be sad. For ten years only would Gaetano
live on the mountain, and then he would come back as the famous hermit
Fra Gaetano. Then he would come walking through the streets of Diamante,
followed by a great crowd of people, like Father Gondo. And there would
be flags, and the houses would be decorated with cloths and wreaths. He
would stop in front of Donna Elisa's shop, and Donna Elisa would not
recognize him and would be ready to fall on her knees before him. But so
should it not be; he would kneel to Donna Elisa, and ask her forgiveness,
because he had run away from her ten years ago. "Gaetano," Donna Elisa
would then answer, "you give me an ocean of joy against a little brook of
sorrow. Should I not forgive you?"
Gaetano saw all this before him, and it was so beautiful that he began to
weep more violently. He was only afraid that Donna Elisa would hear how
he was sobbing and come out and find him. And then she would not let him
go.
He must talk sensibly with her. Would he ever give her greater pleasure
than if he went now?
It was not only Donna Elisa, there was also Luca and Pacifica, who would
be so glad when he came back as a holy man.
They would all follow him up to the market-place. There, there would be
even more flags than in the streets, and Gaetano would speak from the
steps of the town hall. And from all the streets and courts people would
come streaming.
Then Gaetano would speak, so that they should all fall on their knees and
cry: "Bless us, Fra Gaetano, bless us!"
After that he would never leave Diamante again. He would live under the
great steps outside Donna Elisa's shop.
And they would come to him with their sick, and those in trouble would
make a pilgrimage to him.
When the syndic of Diamante went by he would kiss Gaetano's hand.
Donna Elisa would sell Fra Gaetano's image in her shop.
And Donna Elisa's god-daughter, Giannita, would bow before Fra Gaetano
and never again call him a stupid monk-boy.
And Donna Elisa would be so happy.
* * * * *
Ah … Gaetano started up, and awoke. It was bright daylight, and Donna
Elisa and Pacifica stood and looked at him. And Gaetano sat on the
stairs with his shoes under his arm, his hat on his head and his bundle
at his feet. But Donna Elisa and Pacifica wept. "He has wished to run
away from us," they said.
"Why are you sitting here, Gaetano?"
"Donna Elisa, I wanted to run away."
Gaetano was in a good mood, and answered as boldly as if it had been the
most natural thing in the world.
"Do you want to run away?" repeated Donna Elisa.
"I wished to go off on Etna and be a hermit."
"And why are you sitting here now?"
"I do not know, Donna Elisa; I must have fallen asleep."
Donna Elisa now showed how distressed she was. She pressed her hands over
her heart, as if she had terrible pains, and she wept passionately.
"But now I shall stay, Donna Elisa," said Gaetano.
"You, stay!" cried Donna Elisa. "You might as well go. Look at him,
Pacifica, look at the ingrate! He is no Alagona. He is an adventurer."
The blood rose in Gaetano's face and he sprang to his feet and struck out
with his hands in a way which astonished Donna Elisa. So had all the men
of her race done. It was her father and her grandfather; she recognized
all the powerful lords of the family of Alagona.
"You speak so because you know nothing about it, Donna Elisa," said the
boy. "No, no, you do not know anything; you do not know why I had to
serve God. But you shall know it now. Do you see, it was long ago. My
father and mother were so poor, and we had nothing to eat; and so father
went to look for work, and he never came back, and mother and we children
were almost dead of starvation. So mother said: 'We will go and look for
your father.' And we went. Night came and a heavy rain, and in one place
a river flowed over the road. Mother asked in one house if we might pass
the night there. No, they showed us out. Mother and children stood in the
road and cried. Then mother tucked up her dress and went down into the
stream that roared over the road. She had my little sister on her arm and
my big sister by the hand and a big bundle on her head. I went after as
near as I could. I saw mother lose her footing. The bundle she carried
on her head fell into the stream, and mother caught at it and dropped
little sister. She snatched at little sister and big sister was whirled
away. Mother threw herself after them, and the river took her too. I was
frightened and ran to the shore. Father Josef has told me that I escaped
because I was to serve God for the dead, and pray for them. And that was
why it was first decided that I was to be a monk, and why I now wish to
go away on Etna and become a hermit. There is nothing else for me but to
serve God, Donna Elisa."
Donna Elisa was quite subdued. "Yes, yes, Gaetano," she said, "but it
hurts me so. I do not want you to go away from me."
"No, I shall not go either," said Gaetano. He was in such a good mood
that he felt a desire to laugh. "I shall not go."
"Shall I speak to the priest, so that you may be sent to a seminary?"
asked Donna Elisa, humbly.
"No; but you do not understand, Donna Elisa; you do not understand. I
tell you that I will not go away from you. I have thought of something
else."
"What have you thought of?" she asked sadly.
"What do you suppose I was doing while I sat there on the stairs? I was
dreaming, Donna Elisa. I dreamed that I was going to run away. Yes,
Donna Elisa, I stood in the shop, and I was going to open the shop door,
but I could not because there were so many locks. I stood in the dark
and unlocked lock after lock, and always there were new ones. I made a
terrible noise, and I thought: 'Now surely Donna Elisa will come.' At
last the door opened, and I was going to rush out; but just then I felt
your hand on my neck, and you drew me in, and I kicked, and I struck you
because I was not allowed to go. But, Donna Elisa, you had a candle with
you, and then I saw that it was not you, but my mother. Then I did not
dare to struggle any more, and I was very frightened, for mother is dead.
But mother took the bundle I was carrying and began to take out what was
in it. Mother laughed and looked so glad, and I grew glad that she was
not angry with me. It was so strange. What she drew out of the bundle
was all the little saints' images that I had carved while I sat with you
in the shop, and they were so pretty. 'Can you carve such pretty images,
Gaetano?' said mother. 'Yes,' I answered. 'Then you can serve God by
it,' said mother. 'Do I not need to leave Donna Elisa, then?' 'No,' said
mother. And just as mother said that, you waked me."
Gaetano looked at Donna Elisa in triumph.
"What did mother mean by that?"
Donna Elisa only wondered.
Gaetano threw his head back and laughed.
"Mother meant that you should apprentice me, so that I could serve God by
carving beautiful images of angels and saints, Donna Elisa."
III
THE GOD-SISTER
In the noble island of Sicily, where there are more old customs left than
in any other place in the south, it is always the habit of every one
while yet a child to choose a god-brother or god-sister, who shall carry
his or her children to be christened, if there ever are any.
But this is not by any means the only use god-brothers and sisters have
of one another. God-brothers and sisters must love one another, serve one
another, and revenge one another. In a god-brother's ear a man can bury
his secrets. He can trust him with both money and sweetheart, and not be
deceived. God-brothers and sisters are as faithful to each other as if
they were born of the same mother, because their covenant is made before
San Giovanni Battista, who is the most feared of all the saints.
It is also the custom for the poor to take their half-grown children to
rich people and ask that they may be god-brothers and sisters to their
young sons and daughters. What a glad sight it is on the holy Baptist's
day to see all those little children in festival array wandering through
the great towns looking for a god-brother or sister! If the parents
succeed in giving their son a rich god-brother, they are as glad as if
they were able to leave him a farm as an inheritance.
When Gaetano first came to Diamante, there was a little girl who was
always coming in and out of Donna Elisa's shop. She had a red cloak and
pointed cap and eight heavy, black curls that stood out under the cap.
Her name was Giannita, and she was daughter of Donna Olivia, who sold
vegetables. But Donna Elisa was her god-mother, and therefore thought
what she could do for her.
Well, when midsummer day came, Donna Elisa ordered a carriage and drove
down to Catania, which lies full twenty miles from Diamante. She had
Giannita with her, and they were both dressed in their best. Donna Elisa
was dressed in black silk with jet, and Giannita had a white tulle dress
with garlands of flowers. In her hand Giannita held a basket of flowers,
and among the flowers lay a pomegranate.
The journey went well for Donna Elisa and Giannita. When at last they
reached the white Catania, that lies and shines on the black lava
background, they drove up to the finest palace in the town.
It was lofty and wide, so that the poor little Giannita felt quite
terrified at the thought of going into it. But Donna Elisa walked bravely
in, and she was taken to Cavaliere Palmeri and his wife who owned the
house.
Donna Elisa reminded Signora Palmeri that they were friends from infancy,
and asked that Giannita might be her young daughter's god-sister.
That was agreed upon, and the young signorina was called in. She was a
little marvel of rose- silk, Venetian lace, big, black eyes, and
thick, bushy hair. Her little body was so small and thin that one hardly
noticed it.
Giannita offered her the basket of flowers, and she graciously accepted
it. She looked long and thoughtfully at Giannita, walked round her, and
was fascinated by her smooth, even curls. When she had seen them, she ran
after a knife, cut the pomegranate and gave Giannita half.
While they ate the fruit, they held each other's hand and both said:--
"Sister, sister, sister mine!
Thou art mine, and I am thine,
Thine my house, my bread and wine,
Thine my joys, my sacrifice,
Thine my place in Paradise."
Then they kissed each other and called each other god-sister.
"You must never fail me, god-sister," said the little signorina, and both
the children were very serious and moved.
They had become such good friends in the short time that they cried when
they parted.
But then twelve years went by and the two god-sisters lived each in her
own world and never met. During the whole time Giannita was quietly in
her home and never came to Catania.
But then something really strange happened. Giannita sat one afternoon
in the room back of the shop embroidering. She was very skilful and was
often overwhelmed with work. But it is trying to the eyes to embroider,
and it was dark in Giannita's room. She had therefore half-opened the
door into the shop to get a little more light.
Just after the clock had struck four, the old miller's widow, Rosa
Alfari, came walking by. Donna Olivia's shop was very attractive from
the street. The eyes fell through the half-open door on great baskets
with fresh vegetables and bright- fruits, and far back in the
background the outline of Giannita's pretty head. Rosa Alfari stopped and
began to talk to Donna Olivia, simply because her shop looked so friendly.
Laments and complaints always followed old Rosa Alfari. Now she was sad
because she had to go to Catania alone that night. "It is a misfortune
that the post-wagon does not reach Diamante before ten," she said. "I
shall fall asleep on the way, and perhaps they will then steal my money.
And what shall I do when I come to Catania at two o'clock at night?"
Then Giannita suddenly called out into the shop. "Will you take me with
you to Catania, Donna Alfari?" she asked, half in joke, without expecting
an answer.
But Rosa Alfari said eagerly, "Lord, child, will you go with me? Will you
really?"
Giannita came out into the shop, red with pleasure. "If I will!" she
said. "I have not been in Catania for twelve years."
Rosa Alfari looked delightedly at her; Giannita was tall and strong, her
eyes gay, and she had a careless smile on her lips. She was a splendid
travelling companion.
"Get ready," said the old woman. "You will go with me at ten o'clock; it
is settled."
The next day Giannita wandered about the streets of Catania. She was
thinking the whole time of her god-sister. She was strangely moved to be
so near her again. She loved her god-sister, Giannita, and she did it not
only because San Giovanni has commanded people to love their god-brothers
and sisters. She had adored the little child in the silk dress; she was
the most beautiful thing she had ever seen. She had almost become her
idol.
She knew this much about her sister, that she was still unmarried and
lived in Catania. Her mother was dead, and she had not been willing to
leave her father, and had stayed as hostess in his house. "I must manage
to see her," thought Giannita.
Whenever Giannita met a well-appointed carriage she thought: "Perhaps it
is my god-sister driving there." And she stared at everybody to see if
any of them was like the little girl with the thick hair and the big eyes.
Her heart began to beat wildly. She had always longed for her god-sister.
She herself was still unmarried, because she liked a young wood-carver,
Gaetano Alagona, and he had never shown the slightest desire to marry
her. Giannita had often been angry with him for that, and not least had
it irritated her never to be able to invite her god-sister to her wedding.
She had been so proud of her, too. She had thought herself finer than the
others, because she had such a god-sister. What if she should now go to
see her, since she was in the town? It would give a lustre to the whole
journey.
As she thought and thought of it, a newspaper-boy came running.
"_Giornale da Sicilia_," he called. "The Palmeri affair! Great
embezzlements!"
Giannita seized the boy by the neck as he rushed by. "What are you
saying?" she screamed. "You lie, you lie!" and she was ready to strike
him.
"Buy my paper, signora, before you strike me," said the boy. Giannita
bought the paper and began to read. She found in it without difficulty
the Palmeri affair.
"Since this case is to be tried to-day in the courts," wrote the paper,
"we will give an account of it."
Giannita read and read. She read it over and over before she understood.
There was not a muscle in her body which did not begin to tremble with
horror when she at last comprehended it.
Her god-sister's father, who had owned great vineyards, had been ruined,
because the blight had laid them waste. And that was not the worst. He
had also dissipated a charitable fund which had been intrusted to him. He
was arrested, and to-day he was to be tried.
Giannita crushed the newspaper together, threw it into the street and
trampled on it. It deserved no better for bringing such news.
Then she stood quite crushed that this should meet her when she came to
Catania for the first time in twelve years. "Lord God," she said, "is
there any meaning in it?"
At home, in Diamante, no one would ever have taken the trouble to tell
her what was going on. Was it not destiny that she should be here on the
very day of the trial?
"Listen, Donna Alfari," she said; "you may do as you like, but I must go
to the court."
There was a decision about Giannita. Nothing could disturb her. "Do you
not understand that it is for this, and not for your sake, that God has
induced you to take me with you to Catania?" she said to Rosa Alfari.
Giannita did not doubt for a moment that there was something supernatural
in it all.
Rosa Alfari must needs let her go, and she found her way to the Palace of
Justice. She stood among the street boys and riff-raff, and saw Cavaliere
Palmeri on the bench of the accused. He was a fine gentleman, with a
white, pointed beard and moustache. Giannita recognized him.
She heard that he was condemned to six months' imprisonment, and Giannita
thought she saw even more plainly that she had come there as an emissary
from God. "Now my god-sister must need me," she thought.
She went out into the street again and asked her way to the Palazzo
Palmeri.
On the way a carriage drove by her. She looked up, and her eyes met those
of the lady who sat in the carriage. At the same moment something told
her that this was her god-sister. She who was driving was pale and bent
and had beseeching eyes. Giannita loved her from the first sight. "It is
you who have given me pleasure many times," she said, "because I expected
pleasure from you. Now perhaps I can pay you back."
Giannita felt filled with devotion when she went up the high, white
marble steps to the Palazzo Palmeri, but suddenly a doubt struck
her. "What can God wish me to do for one who has grown up in such
magnificence?" she thought. "Does our Lord forget that I am only poor
Giannita from Diamante?"
She told a servant to greet Signorina Palmeri and say to her that her
god-sister wished to speak to her. She was surprised when the servant
came back and said that she could not be received that day. Should she
be content with that? Oh, no; oh, no!
"Tell the signorina that I am going to wait here the whole day, for I
must speak to her."
"The signorina is going to move out of the palace in half an hour," said
the servant.
Giannita was beside herself. "But I am her god-sister, her god-sister,
do you not understand?" she said to the man. "I must speak to her." The
servant smiled, but did not move.
But Giannita would not be turned away. Was she not sent by God? He must
understand, understand, she said, and raised her voice. She was from
Diamante and had not been in Catania for twelve years. Until yesterday
afternoon at four o'clock she had not thought of coming here. He must
understand, not until yesterday afternoon at four o'clock.
The servant stood motionless. Giannita was ready to tell him the whole
story to move him, when the door was thrown open. Her god-sister stood on
the threshold.
"Who is speaking of yesterday at four o'clock?" she said.
"It is a stranger, Signorina Micaela."
Then Giannita rushed forward. It was not at all a stranger. It was her
god-sister from Diamante, who came here twelve years ago with Donna
Elisa. Did she not remember her? Did she not remember that they had
divided a pomegranate?
The signorina did not listen to that. "What was it that happened
yesterday at four o'clock?" she asked, with great anxiety.
"I then got God's command to go to you, god-sister," said Giannita.
The other looked at her in terror. "Come with me," she said, as if afraid
that the servant should hear what Giannita wished to say to her.
She went far into the apartment before she stopped. Then she turned so
quickly towards Giannita that she was frightened. "Tell me instantly!"
she said. "Do not torture me; let me hear it instantly!"
She was as tall as Giannita, but very unlike her. She was more delicately
made, and she, the woman of the world, had a much more wild and untamed
appearance than the country girl. Everything she felt showed in her face.
She did not try to conceal it.
Giannita was so astonished at her violence that she could not answer at
first.
Then her god-sister lifted her arms in despair over her head and the
words streamed from her lips. She said that she knew that Giannita had
been commanded by God to bring her word of new misfortunes. God hated
her, she knew it.
Giannita clasped her hands. God hate her! on the contrary, on the
contrary!
"Yes, yes," said Signorina Palmeri. "It is so." And as she was inwardly
afraid of the message Giannita had for her, she began to talk. She did
not let her speak; she interrupted her constantly. She seemed to be so
terrified by everything that had happened to her during the last days
that she could not at all control herself.
Giannita must understand that God hated her, she said. She had done
something so terrible. She had forsaken her father, failed her father.
Giannita must have read the last account. Then she burst out again in
passionate questionings. Why did she not tell her what she wished to
tell her? She did not expect anything but bad news. She was prepared.
But poor Giannita never got a chance to speak; as soon as she began, the
signorina became frightened and interrupted her. She told her story as if
to induce Giannita not to be too hard to her.
Giannita must not think that her unhappiness only came from the fact of
her no longer having her carriage, or a box at the theatre, or beautiful
dresses, or servants, or even a roof over her head. Neither was it enough
that she had now lost all her friends, so that she did not at all know
where she should ask for shelter. Neither was it misfortune enough that
she felt such shame that she could not raise her eyes to any one's face.
But there was something else much worse.
She sat down, and was silent a moment, while she rocked to and fro in
agony. But when Giannita began to speak, she interrupted her.
Giannita could not think how her father had loved her. He had always had
her live in splendor and magnificence, like a princess.
She had not done much for him; only let him think out delightful things
to amuse her. It had been no sacrifice to remain unmarried, for she had
never loved any one like her father, and her own home had been finer than
any one else's.
But one day her father had come and said to her, "They wish to arrest me.
They are spreading the report that I have stolen, but it is not true."
Then she had believed him, and helped him to hide from the _Carabinieri_.
And they had looked for him in vain in Catania, on Etna, over the whole
of Sicily.
But when the police could not find Cavaliere Palmeri, the people began
to say: "He is a fine gentleman, and they are fine gentlemen who help
him; otherwise they would have found him long ago." And the prefect in
Catania had come to her. She received him smiling, and the prefect came
as if to talk of roses, and the beautiful weather. Then he said: "Will
the signorina look at this little paper? Will the signorina read this
little letter? Will the signorina observe this little signature?" She
read and read. And what did she see? Her father was not innocent. Her
father had taken the money of others.
When the prefect had left her, she had gone to her father. "You are
guilty," she said to him. "You may do what you will, but I cannot help
you any more." Oh, she had not known what she said! She had always been
very proud. She had not been able to bear to have their name stamped with
dishonor. She had wished for a moment that her father had been dead,
rather than that this had happened to her. Perhaps she had also said it
to him. She did not rightly know what she had said.
But after that God had forsaken her. The most terrible things had
happened. Her father had taken her at her word. He had gone and given
himself up. And ever since he had been in prison he had not been willing
to see her. He did not answer her letters, and the food that she sent
him he sent back untouched. That was the most dreadful thing of all. He
seemed to think that she wished to kill him.
She looked at Giannita as anxiously as if she awaited her sentence of
death.
"Why do you not say to me what you have to say?" she exclaimed. "You are
killing me!"
But it was impossible for her to force herself to be silent.
"You must know," she continued, "that this palace is sold, and the
purchaser has let it to an English lady, who is to move in to-day. Some
of her things were brought in already yesterday, and among them was a
little image of Christ.
"I caught sight of it as I passed through the vestibule, Giannita. They
had taken it out of a trunk, and it lay there on the floor. It had been
so neglected that no one took any trouble about it. Its crown was dented,
and its dress dirty, and all the small ornaments which adorned it were
rusty and broken. But when I saw it lying on the floor, I took it up and
carried it into the room and placed it on a table. And while I did so,
it occurred to me that I would ask its help. I knelt down before it and
prayed a long time. 'Help me in my great need!' I said to the Christchild.
"While I prayed, it seemed to me that the image wished to answer me. I
lifted my head, and the child stood there as dull as before, but a clock
began to strike just then. It struck four, and it was as if it had said
four words. It was as if the Christchild had answered a fourfold _yes_ to
my prayer.
"That gave me courage, Giannita, so that to-day I drove to the Palace of
Justice to see my father. But he never turned his eyes toward me during
the whole time he stood before his judges.
"I waited until they were about to lead him away, and threw myself on
my knees before him in one of the narrow passages. Giannita, he let the
soldiers lead me away without giving me a word.
"So, you see, God hates me. When I heard you speak of yesterday
afternoon at four o'clock, I was so frightened. The Christchild sends me
a new misfortune, I thought. It hates me for having failed my father."
When she had said that, she was at last silent and listened breathlessly
for what Giannita should say.
And Giannita told her story to her.
"See, see, is it not wonderful?" she said at the end. "I have not been in
Catania for twelve years, and then I come here quite unexpectedly. And I
know nothing at all; but as soon as I set my foot on the street here, I
hear your misfortune. God has sent a message to me, I said to myself. He
has called me here to help my god-sister."
Signorina Palmeri's eyes were turned anxiously questioning towards her.
Now the new blow was coming. She gathered all her courage to meet it.
"What do you wish me to do for you, god-sister?" said Giannita. "Do you
know what I thought as I was walking through the streets? I will ask her
if she will go with me to Diamante, I thought. I know an old house there,
where we could live cheaply. And I would embroider and sew, so that we
could support ourselves. When I was out in the street I thought that it
might be, but now I understand that it is impossible, impossible. You
require something more of life; but tell me if I can do anything for you.
You shall not thrust me away, for God has sent me."
The signorina bent towards Giannita. "Well?" she said anxiously.
"You shall let me do what I can for you, for I love you," said Giannita,
and fell on her knees and put her arms about her.
"Have you nothing else to say?" asked the signorina.
"I wish I had," said Giannita, "but I am only a poor girl."
It was wonderful to see how the features of the young signorina's face
softened; how her color came back and how her eyes began to shine. Now it
was plain that she had great beauty.
"Giannita," she said, low and scarcely audibly, "do you think that it is
a miracle? Do you think that God can let a miracle come to pass for my
sake?"
"Yes, yes," whispered Giannita back.
"I prayed the Christchild that he should help me, and he sends you to me.
Do you think that it was the Christchild who sent you, Giannita?"
"Yes, it was; it was!"
"Then God has not forsaken me, Giannita?"
"No, God has not forsaken you."
The god-sisters sat and wept for a while. It was quite quiet in the room.
"When you came, Giannita, I thought that nothing was left me but to kill
myself," she said at last. "I did not know where to turn, and God hated
me."
"But tell me now what I can do for you, god-sister," said Giannita.
As an answer the other drew her to her and kissed her.
"But it is enough that you are sent by the little Christchild," she said.
"It is enough that I know that God has not forsaken me."
IV
DIAMANTE
Micaela Palmeri was on her way to Diamante with Giannita.
They had taken their places in the post-carriage at three o'clock in the
morning, and had driven up the beautiful road over the lower <DW72>s of
Etna, circling round the mountain. But it had been quite dark. They had
not seen anything of the surrounding country.
The young signorina by no means lamented over that. She sat with closed
eyes and buried herself in her sorrow. Even when it began to grow light,
she would not lift her eyes to look out. It was not until they were quite
near Diamante that Giannita could persuade her to look at the landscape.
"Look! Here is Diamante; this is to be your home," she said.
Then Micaela Palmeri, to the right of the road, saw mighty Etna, that cut
off a great piece of the sky. Behind the mountain the sun was rising,
and when the upper edge of the sun's disc appeared above the line of the
mountain, it looked as if the white summit began to burn and send out
sparks and rays.
Giannita entreated her to look at the other side.
And on the other side she saw the whole jagged mountain chain, which
surrounds Etna like a towered wall, glowing red in the sunrise.
But Giannita pointed in another direction. It was not that she was to
look at, not that.
Then she lowered her eyes and looked down into the black valley. There
the ground shone like velvet, and the white Simeto foamed along in the
depths of the valley.
But still she did not turn her eyes in the right direction.
At last she saw the steep Monte Chiaro rising out of the black,
velvet-lined valley, red in the morning light and encircled by a crown
of shady palms. On its summit she saw a town flanked with towers, and
encompassed by a wall, and with all its windows and weather-vanes
glittering in the light.
At that sight she seized Giannita's arm and asked her if it was a real
town, and if people lived there.
She believed that it was one of heaven's cities, and that it would
disappear like a vision. She was certain that no mortal had ever passed
up the path that from the edge of the valley went in great curves over to
Monte Chiaro and then zigzagged up the mountain, disappearing through the
dark gates of the town.
But when she came nearer to Diamante, and saw that it was of the earth,
and real, tears rose to her eyes. It moved her that the earth still held
all this beauty for her. She had believed that, since it had been the
scene of all her misfortunes, she would always find it gray and withered
and covered with thistles and poisonous growths.
She entered poor Diamante with clasped hands, as if it were a sanctuary.
And it seemed to her as if this town could offer her as much happiness as
beauty.
V
DON FERRANTE
A few days later Gaetano was standing in his workshop, cutting
grape-leaves on rosary beads. It was Sunday, but Gaetano did not feel it
on his conscience that he was working, for it was a work in God's honor.
A great restlessness and anxiety had come over him. It had come into his
mind that the time he had been living at peace with Donna Elisa was now
drawing to a close, and he thought that he must soon start out into the
world.
For great poverty had come to Sicily, and he saw want wandering from
town to town and from house to house like the plague, and it had come to
Diamante also.
No one ever came now to Donna Elisa's shop to buy anything. The little
images of the saints that Gaetano made stood in close rows on the
shelves, and the rosaries hung in great bunches under the counter. And
Donna Elisa was in great want and sorrow, because she could not earn
anything.
That was a sign to Gaetano that he must leave Diamante, go out into the
world, emigrate if there was no other way. For it could not be working to
the honor of God to carve images that never were worshipped, and to turn
rosary beads that never glided through a petitioner's fingers.
It seemed to him that, somewhere in the world, there must be a beautiful,
newly built cathedral, with finished walls, but whose interior yet stood
shivering in nakedness. It awaited Gaetano's coming to carve the choir
chairs, the altar-rail, the pulpit, the lectern, and the shrine. His
heart ached with longing for that work which was waiting.
But there was no such cathedral in Sicily, for there no one ever thought
of building a new church; it must be far away in such lands as Florida or
Argentina, where the earth is not yet overcrowded with holy buildings.
He felt at the same time trembling and happy, and had begun to work with
redoubled zeal in order that Donna Elisa should have something to sell
while he was away earning great fortunes for her.
Now he was waiting for but one more sign from God before he decided on
the journey. And this was that he should have the strength to speak to
Donna Elisa of his longing to go. For he knew that it would cause her
such sorrow that he did not know how he could bring himself to speak of
it.
While he stood and thought Donna Elisa came into the workshop. Then he
said to himself that this day he could not think of saying it to her, for
to-day Donna Elisa was happy. Her tongue wagged and her face beamed.
Gaetano asked himself when he had seen her so. Ever since the famine had
come, it had been as if they had lived without light in one of the caves
of Etna.
Why had Gaetano not been with her in the square and heard the music?
asked Donna Elisa. Why did he never come to hear and see her brother,
Don Ferrante? Gaetano, who only saw him when he stood in the shop with
his tufts of hair and his short jacket, did not know what kind of a man
he was. He considered him an ugly old tradesman, who had a wrinkled face
and a rough beard. No one knew Don Ferrante who had not seen him on
Sunday, when he conducted the music.
That day he had donned a new uniform. He wore a three-cornered hat with
green, red, and white feathers, silver on his collar, silver-fringed
epaulets, silver braid on his breast, and a sword at his side. And when
he stepped up to the conductor's platform the wrinkles had been smoothed
out of his face and his figure had grown erect. He could almost have been
called handsome.
When he had led _Cavalleria_, people had hardly been able to breathe.
What had Gaetano to say to that, that the big houses round the
market-place had sung too? From the black Palazzo Geraci, Donna Elisa had
distinctly heard a love song, and from the convent, empty as it was, a
beautiful hymn had streamed out over the market-place.
And when there was a pause in the music the handsome advocate Favara, who
had been dressed in a black velvet coat and a big broad-brimmed hat and a
bright red necktie, had gone up to Don Ferrante, and had pointed out over
the open side of the square, where Etna and the sea lay. "Don Ferrante,"
he had said, "you lift us toward the skies, just as Etna does, and you
carry us away into the eternal, like the infinite sea."
If Gaetano had seen Don Ferrante to-day he would have loved him. At least
he would have been obliged to acknowledge his stateliness. When he
laid down his baton for a while and took the advocate's arm, and walked
forward and back with him on the flat stones by the Roman gate and the
Palazzo Geraci, every one could see that he could well measure himself
against the handsome Favara.
Donna Elisa sat on the stone bench by the cathedral, in company with the
wife of the syndic. And Signora Voltaro had said quite suddenly, after
sitting for a while, watching Don Ferrante: "Donna Elisa, your brother is
still a young man. He may still be married, in spite of his fifty years."
And she, Donna Elisa, had answered that she prayed heaven for it every
day.
But she had hardly said it, when a lady dressed in mourning came into the
square. Never had anything so black been seen before. It was not enough
that dress and hat and gloves were black; her veil was so thick that it
was impossible to believe that there was a face behind it. Santissimo
Dio! it looked as if she had hung a pall over herself. And she had walked
slowly, and with a stoop. People had almost feared, believing that it was
a ghost.
Alas, alas! the whole market-place had been so full of gayety! The
peasants, who were at home over Sunday, had stood there in great crowds
in holiday dress, with red shawls wound round their necks. The peasant
women on their way to the cathedral had glided by, dressed in green
skirts and yellow neckerchiefs. A couple of travellers had stood by the
balustrade and looked at Etna; they had been dressed in white. And all
the musicians in uniform, who had been almost as fine as Don Ferrante,
and the shining instruments, and the carved cathedral _façade_! And the
sunlight, and Mongibello's snow top--so near to-day that one could
almost touch it--had all been so gay.
Now, when the poor black lady came into the midst of it all, they had
stared at her, and some had made the sign of the cross. And the children
had rushed down from the steps of the town-hall, where they were riding
on the railing, and had followed her at a few feet's distance. And even
the lazy Piero, who had been asleep in the corner of the balustrade, had
raised himself on his elbow. It had been a resurrection, as if the black
Madonna from the cathedral had come strolling by.
But had no one thought that it was unkind that all stared at the black
lady? Had no one been moved when she came so slowly and painfully?
Yes, yes; one had been touched, and that had been Don Ferrante. He had
the music in his heart; he was a good man and he thought: "Curses on all
those funds that are gathered together for the poor, and that only bring
people misfortune! Is not that poor Signorina Palmeri, whose father has
stolen from a charitable fund, and who is now so ashamed that she dares
not show her face?" And, as he thought of it, Don Ferrante went towards
the black lady and met her just by the church door.
There he made her a bow, and mentioned his name. "If I am not mistaken,"
Don Ferrante had said, "you are Signorina Palmeri. I have a favor to ask
of you."
Then she had started and taken a step backwards, as if to flee, but she
had waited.
"It concerns my sister, Donna Elisa," he had said. "She knew your mother,
signorina, and she is consumed with a desire to make your acquaintance.
She is sitting here by the Cathedral. Let me take you to her!"
And then Don Ferrante put her hand on his arm and led her over to Donna
Elisa. And she made no resistance. Donna Elisa would like to see who
could have resisted Don Ferrante to-day.
Donna Elisa rose and went to meet the black lady, and throwing back her
veil, kissed her on both cheeks.
But what a face, what a face! Perhaps it was not pretty, but it had
eyes that spoke, eyes that mourned and lamented, even when the whole
face smiled. Yes, Gaetano perhaps would not wish to carve or paint a
Madonna from that face, for it was too thin and too pale; but it is to be
supposed that our Lord knew what he was doing when he did not put those
eyes in a face that was rosy and round.
When Donna Elisa kissed her, she laid her head down on her shoulder, and
a few short sobs shook her. Then she looked up with a smile, and the
smile seemed to say: "Ah, does the world look so? Is it so beautiful? Let
me see it and smile at it! Can a poor unfortunate really dare to look at
it? And to be seen? Can I bear to be seen?"
All that she had said without a word, only with a smile. What a face,
what a face!
But here Gaetano interrupted Donna Elisa. "Where is she now?" he said. "I
too must see her."
Then Donna Elisa looked Gaetano in the eyes. They were glowing and clear,
as if they were filled with fire, and a dark flush rose to his temples.
"You will see her all in good time," she said, harshly. And she repented
of every word she had said.
Gaetano saw that she was afraid, and he understood what she feared. It
came into his mind to tell her now that he meant to go away, to go all
the way to America.
Then he understood that the strange signorina must be very dangerous.
Donna Elisa was so sure that Gaetano would fall in love with her that she
was almost glad to hear that he meant to go away.
For anything seemed better to her than a penniless daughter-in-law, whose
father was a thief.
VI
DON MATTEO'S MISSION
One afternoon the old priest, Don Matteo, inserted his feet into newly
polished shoes, put on a newly brushed soutane, and laid his cloak in the
most effective folds. His face shone as he went up the street, and when
he distributed blessings to the old women spinning by the doorposts, it
was with gestures as graceful as if he had scattered roses.
The street along which Don Matteo was walking was spanned by at least
seven arches, as if every house wished to bind itself to a neighbor.
It ran small and narrow down the mountain; it was half street and half
staircase; the gutters were always overflowing, and there were always
plenty of orange-skins and cabbage-leaves to slip on. Clothes hung on the
line, from the ground up to the sky. Wet shirt-sleeves and apron-strings
were carried by the wind right into Don Matteo's face. And it felt horrid
and wet, as if Don Matteo had been touched by a corpse.
At the end of the street lay a little dark square, and there Don Matteo
saw an old house, before which he stopped. It was big, and square, and
almost without windows. It had two enormous flights of steps, and two big
doors with heavy locks. And it had walls of black lava, and a "loggia,"
where green slime grew over the tiled floor, and where the spider-webs
were so thick that the nimble lizards were almost held fast in them.
Don Matteo lifted the knocker, and knocked till it thundered. All the
women in the street began to talk, and to question. All the washerwomen
by the fountain in the square dropped soap and wooden clapper, and began
to whisper, and ask, "What is Don Matteo's errand? Why does Don Matteo
knock on the door of an old, haunted house, where nobody dares to live
except the strange signorina, whose father is in prison?"
But now Giannita opened the door for Don Matteo, and conducted him
through long passages, smelling of mould and damp. In several places in
the floor the stones were loose, and Don Matteo could see way down into
the cellar, where great armies of rats raced over the black earth floor.
As Don Matteo walked through the old house, he lost his good-humor. He
did not pass by a stairway without suspiciously spying up it, and he
could not hear a rustle without starting. He was depressed as before some
misfortune. Don Matteo thought of the little turbaned Moor who was said
to show himself in that house, and even if he did not see him, he might
be said to have felt him.
At last Giannita opened a door and showed the priest into a room. The
walls there were bare, as in a stable; the bed was as narrow as a nun's,
and over it hung a Madonna that was not worth three soldi. The priest
stood and stared at the little Madonna till the tears rose to his eyes.
While he stood so Signorina Palmeri came into the room. She kept her head
bent and moved slowly, as if wounded. When the priest saw her he wished
to say to her: "You and I, Signorina Palmeri, have met in a strange old
house. Are you here to study the old Moorish inscriptions or to look for
mosaics in the cellar?" For the old priest was confounded when he saw
Signorina Palmeri. He could not understand that the noble lady was poor.
He could not comprehend that she was living in the house of the little
Moor.
He said to himself that he must save her from this haunted house, and
from poverty. He prayed to the tender Madonna for power to save her.
Thereupon he said to the signorina that he had come with a commission
from Don Ferrante Alagona. Don Ferrante had confided to him that she
had refused his proposal of marriage. Why was that? Did she not know
that, although Don Ferrante seemed to be poor as he stood in his shop,
he was really the richest man in Diamante? And Don Ferrante was of an
old Spanish family of great consideration, both in their native country
and in Sicily. And he still owned the big house on the Corso that had
belonged to his ancestors. She should not have said no to him.
While Don Matteo was speaking, he saw how the signorina's face grew stiff
and white. He was almost afraid to go on. He feared that she was going to
faint.
It was only with the greatest effort that she was able to answer him. The
words would not pass her lips. It seemed as if they were too loathsome
to utter. She quite understood, she said, that Don Ferrante would like
to know why she had refused his proposal. She was infinitely touched and
grateful on account of it, but she could not be his wife. She could
not marry, for she brought dishonor and disgrace with her as a marriage
portion.
"If you marry an Alagona, dear signorina," said Don Matteo, "you need not
fear that any one will ask of what family you are. It is an honorable
old name. Don Ferrante and his sister, Donna Elisa, are considered the
first people in Diamante, although they have lost all the family riches,
and have to keep a shop. Don Ferrante knows well enough that the glory
of the old name would not be tarnished by a marriage with you. Have no
scruples for that, signorina, if otherwise you may be willing to marry
Don Ferrante."
But Signorina Palmeri repeated what she had said. Don Ferrante should
not marry the daughter of a convict. She sat pale and despairing, as if
wishing to practise saying those terrible words. She said that she did
not wish to enter a family which would despise her. She succeeded in
saying it in a hard, cold voice, without emotion.
But the more she said, the greater became Don Matteo's desire to help
her. He felt as if he had met a queen who had been torn from her throne.
A burning desire came over him to set the crown again upon her head, and
fasten the mantle about her shoulders.
Therefore Don Matteo asked her if her father were not soon coming out of
prison, and he wondered what he would live on.
The signorina answered that he would live on her work.
Don Matteo asked her very seriously whether she had thought how her
father, who had always been rich, could bear poverty.
Then she was silent. She tried to move her lips to answer, but could not
utter a sound.
Don Matteo talked and talked. She looked more and more frightened, but
she did not yield.
At last he knew not what to do. How could he save her from that haunted
house, from poverty, and from the burden of dishonor that weighed her
down? But then his eyes chanced to fall on the little image of the
Madonna over the bed. So the young signorina was a believer.
The spirit of inspiration came to Don Matteo. He felt that God had sent
him to save this poor woman. When he spoke again, there was a new ring in
his voice. He understood that it was not he alone who spoke.
"My daughter," he said, and rose, "you will marry Don Ferrante for your
father's sake! It is the Madonna's will, my daughter."
There was something impressive in Don Matteo's manner. No one had ever
seen him so before. The signorina trembled, as if a spirit voice had
spoken to her, and she clasped her hands.
"Be a good and faithful wife to Don Ferrante," said Don Matteo, "and the
Madonna promises you through me that your father will have an old age
free of care."
Then the signorina saw that it was an inspiration which guided Don
Matteo. It was God speaking through him. And she sank down on her knees,
and bent her head. "I shall do what you command," she said.
* * * * *
But when the priest, Don Matteo, came out of the house of the little Moor
and went up the street, he suddenly took out his breviary and began
to read. And although the wet clothes struck him on the cheek, and the
little children and the orange-peels lay in wait for him, he only looked
in his book. He needed to hear the great words of God.
For within that black house everything had seemed certain and sure, but
when he came out into the sunshine he began to worry about the promise he
had given in the name of the Madonna.
Don Matteo prayed and read, and read and prayed. Might the great God in
heaven protect the woman, who had believed him and obeyed him as if he
had been a prophet!
Don Matteo turned the corner into the Corso. He struck against donkeys
on their way home, with travelling signorinas on their backs; he walked
right into peasants coming home from their work, and he pushed against
the old women spinning, and entangled their thread. At last he came to a
little, dark shop.
It was a shop without a window which was at the corner of an old palace.
The threshold was a foot high; the floor was of trampled earth; the door
almost always stood open to let in the light. The counter was besieged by
peasants and mule-drivers.
And behind the counter stood Don Ferrante. His beard grew in tufts; his
face was in one wrinkle; his voice was hoarse with rage. The peasants
demanded an immoderately high payment for the loads that they had driven
up from Catania.
VII
THE BELLS OF SAN PASQUALE
The people of Diamante soon perceived that Don Ferrante's wife, Donna
Micaela, was nothing but a great child. She could never succeed in
looking like a woman of the world, and she really was nothing but a
child. And nothing else was to be expected, after the life she had led.
Of the world she had seen nothing but its theatres, museums, ball-rooms,
promenades, and race courses; and all such are only play places. She had
never been allowed to go alone on the street. She had never worked. No
one had ever spoken seriously to her. She had not even been in love with
any one.
After she had moved into the summer palace she forgot her cares as gayly
and easily as a child would have done. And it appeared that she had the
playful disposition of a child, and that she could transform and change
everything about her.
The old dirty Saracen town Diamante seemed like a paradise to Donna
Micaela. She said that she had not been at all surprised when Don
Ferrante had spoken to her in the square, nor when he had proposed to
her. It seemed quite natural to her that such things should happen in
Diamante. She had seen instantly that Diamante was a town where rich men
went and sought out poor, unfortunate signorinas to make them mistresses
of their black lava palaces.
She also liked the summer-palace. The faded chintz, a hundred years old,
that covered the furniture told her stories. And she found a deep meaning
in all the love scenes between the shepherds and shepherdesses on the
walls.
She had soon found out the secret of Don Ferrante. He was no ordinary
shop-keeper in a side street. He was a man of ambition, who was
collecting money in order to buy back the family estate on Etna and the
palace in Catania and the castle on the mainland. And if he went in short
jacket and pointed cap, like a peasant, it was in order the sooner to be
able to appear as a grandee of Spain and prince of Sicily.
After they were married Don Ferrante always used every evening to put on
a velvet coat, take his guitar under his arm, and place himself on the
stairway to the gallery in the music-room in the summer-palace and sing
canzoni. While he sang, Donna Micaela dreamed that she had been married
to the noblest man in beautiful Sicily.
When Donna Micaela had been married a few months her father was released
from prison and came to live at the summer palace with his daughter. He
liked the life in Diamante and became friends with every one. He liked
to talk to the bee-raisers and vineyard workers whom he met at the Café
Europa, and he amused himself every day by riding about on the <DW72>s of
Etna to look for antiquities.
But he had by no means forgiven his daughter. He lived under her roof,
but he treated her like a stranger, and never showed her affection.
Donna Micaela let him go on and pretended not to notice it. She could
not take his anger seriously any longer. That old man, whom she loved,
believed that he would be able to go on hating her year after year! He
would live near her, hear her speak, see her eyes, be encompassed by
her love, and he could continue to hate her! Ah, he knew neither her
nor himself. She used to sit and imagine how it would be when he must
acknowledge that he was conquered; when he must come and show her that he
loved her.
One day Donna Micaela was standing on her balcony waving her hand to her
father, who rode away on a small, dark-brown pony, when Don Ferrante came
up from the shop to speak to her. And what Don Ferrante wished to say was
that he had succeeded in getting her father admitted to "The Brotherhood
of the Holy Heart" in Catania.
But although Don Ferrante spoke very distinctly, Donna Micaela seemed not
to understand him at all.
He had to repeat to her that he had been in Catania the day before, and
that he had succeeded in getting Cavaliere Palmeri into a brotherhood. He
was to enter it in a month.
She only asked: "What does that mean? What does that mean?"
"Oh," said Don Ferrante, "can I not have wearied of buying your father
expensive wines from the mainland, and may I not sometimes wish to ride
Domenico?"
When he had said that, he wished to go. There was nothing more to say.
"But tell me first what kind of a brotherhood it is," she said.--"What
it is! A lot of old men live there."--"Poor old men?"--"Oh, well, not
so rich."--"They do not have a room to themselves, I suppose?"--"No,
but very big dormitories."--"And they eat from tin basins on a
table without a cloth?"--"No, they must be china."--"But without a
table-cloth?"--"Lord, if the table is clean!"
He added, to silence her: "Very good people live there. If you like to
know it, it was not without hesitation they would receive Cavaliere
Palmeri."
Thereupon Don Ferrante went. His wife was in despair, but also very
angry. She thought that he had divested himself of rank and class and
become only a plain shop-keeper.
She said aloud, although no one heard her, that the summer palace was
only a big, ugly old house, and Diamante a poor and miserable town.
Naturally, she would not allow her father to leave her. Don Ferrante
would see.
When they had eaten their dinner Don Ferrante wished to go to the Café
Europa and play dominoes, and he looked about for his hat. Donna Micaela
took his hat and followed him out to the gallery that ran round the
court-yard. When they were far enough from the dining-room for her father
not to be able to hear them, she said passionately:--
"Have you anything against my father?"--"He is too expensive."--"But you
are rich."--"Who has given you such an idea? Do you not see how I am
struggling?"--"Save in some other way."--"I shall save in other ways.
Giannita has had presents enough."--"No, economize on something for
me."--"You! you are my wife; you shall have it as you have it."
She stood silent a moment. She was thinking what she could say to
frighten him.
"If I am now your wife, do you know why it is?"--"Oh yes."--"Do you also
know what the priest promised me?"--"That is his affair, but I do what
I can."--"You have heard, perhaps, that I broke with all my friends in
Catania when I heard that my father had sought help from them and had not
got it."--"I know it."--"And that I came here to Diamante that he might
escape from seeing them and being ashamed?"--"They will not be coming
to the brotherhood."--"When you know all this, are you not afraid to do
anything against my father?"--"Afraid? I am not afraid of my wife."
"Have I not made you happy?" she asked.--"Yes, of course," he answered
indifferently.--"Have you not enjoyed singing to me? Have you not liked
me to have considered you the most generous man in Sicily? Have you not
been glad that I was happy in the old palace? Why should it all come to
an end?"
He laid his hand on her shoulder and warned her. "Remember that you are
not married to a fine gentleman from the Via Etnea!"--"Oh, no!"--"Up here
on the mountain the ways are different. Here wives obey their husbands.
And we do not care for fair words. But if we want them we know how to get
them."
She was frightened when he spoke so. In a moment she was on her knees
before him. It was dark, but enough light came from the other rooms for
him to see her eyes. In burning prayer, glorious as stars, they were
fixed on him.
"Be merciful! You do not know how much I love him!" Don Ferrante
laughed. "You ought to have begun with that. Now you have made me angry."
She still knelt and looked up at him. "It is well," he said, "for you
hereafter to know how you shall behave." Still she knelt. Then he asked:
"Shall I tell him, or will you?"
Donna Micaela was ashamed that she had humbled herself. She rose and
answered imperiously: "I shall tell him, but not till the last day. And
you _shall_ not let him notice anything."
"No, I _shall_ not," he said, and mimicked her. "The less talk about it,
the better for me."
But when he was gone Donna Micaela laughed at Don Ferrante for believing
that he could do what he liked with her father. She knew some one who
would help her.
* * * * *
In the Cathedral at Diamante there is a miracle-working image of the
Madonna, and this is its story.
Long, long ago a holy hermit lived in a cave on Monte Chiaro. And this
hermit dreamed one night that in the harbor of Catania lay a ship loaded
with images of the saints, and among these there was one so holy that
Englishmen, who are richer than anybody else, would have paid its weight
in gold for it. As soon as the hermit awoke from this dream he started
for Catania. In the harbor lay a ship loaded with images of the saints,
and among the images was one of the holy Madonna that was more holy than
all the others. The hermit begged the captain not to carry that image
away from Sicily, but to give it to him. But the captain refused. "I
shall take it to England," he said, "and the Englishmen will pay its
weight in gold." The hermit renewed his petitions. At last the captain
had his men drive him on shore, and hoisted his sail to depart.
It looked as if the holy image was to be lost to Sicily; but the hermit
knelt down on one of the lava blocks on the shore and prayed to God that
it might not be. And what happened? The ship could not go. The anchor
was up, the sail hoisted, and the wind fresh; but for three long days
the ship lay as motionless as if it had been a rock. On the third day
the captain took the Madonna image and threw it to the hermit, who still
lay on the shore. And immediately the ship glided out of the harbor. The
hermit carried the image to Monte Chiaro, and it is still in Diamante,
where it has a chapel and an altar in the Cathedral.
Donna Micaela was now going to this Madonna to pray for her father.
She sought out the Madonna's chapel, which was built in a dark corner of
the Cathedral. The walls were covered with votive offerings, with silver
hearts and pictures that had been given by all those who had been helped
by the Madonna of Diamante.
The image was hewn in black marble, and when Donna Micaela saw it
standing in its niche, high and dark, and almost hidden by a golden
railing, it seemed to her that its face was beautiful, and that it shone
with mildness. And her heart was filled with hope.
Here was the powerful queen of heaven; here was the good Mother Mary;
here was the afflicted mother who understood every sorrow; here was one
who would not allow her father to be taken from her.
Here she would find help. She would need only to fall on her knees and
tell her trouble, to have the black Madonna come to her assistance.
While she prayed she felt certain that Don Ferrante was even at that
moment changing his mind. When she came home he would come to meet her
and say to her that she might keep her father.
* * * * *
It was a morning three weeks later.
Donna Micaela came out of the summer palace to go to early mass; but
before she set out to the church, she went into Donna Elisa's shop to
buy a wax candle. It was so early that she had been afraid that the shop
would not be open; but it was, and she was glad to be able to take a gift
with her to the black Madonna.
The shop was empty when Donna Micaela came in, and she pushed the door
forward and back to make the bell ring and call Donna Elisa in. At last
some one came, but it was not Donna Elisa; it was a young man.
That young man was Gaetano, whom Donna Micaela scarcely knew. For Gaetano
had heard so much about her that he was afraid to meet her, and every
time she had come over to Donna Elisa he had shut himself into his
workshop. Donna Micaela knew no more about him than that he was to leave
Diamante, and that he was always carving holy images for Donna Elisa
to have something to sell while he was earning great fortunes away in
Argentina.
When she now saw Gaetano, she found him so handsome that it made her
glad to look at him. She was full of anxiety as a hunted animal, but no
sorrow in the world could prevent her from feeling joy at the sight of
anything so beautiful.
She asked herself where she had seen him before, and she remembered that
she had seen his face in her father's wonderful collection of pictures in
the palace at Catania. There he had not been in working blouse; he had
had a black felt hat with long, flowing, white feathers, and a broad lace
collar over a velvet coat. And he had been painted by the great master
Van Dyck.
Donna Micaela asked Gaetano for a wax candle, and he began to look for
one. And now, strangely enough, Gaetano, who saw the little shop every
day, seemed to be quite strange there. He looked for the wax candle in
the drawers of rosaries and in the little medallion boxes. He could not
find anything, and he grew so impatient that he turned out the drawers
and broke the boxes open. The destruction and disorder were terrible. And
it would be a real grief to Donna Elisa when she came home.
But Donna Micaela liked to see how he shook the thick hair back from his
face, and how his gold- eyes glowed like yellow wine when the sun
shines through it. It was a consolation to see any one so beautiful.
Then Donna Micaela asked pardon of the noble gentlemen whom the great Van
Dyck had painted. For she had often said to them: "Ah, signor, you have
been beautiful, but you never could have been so dark and so pale and
so melancholy. And you did not possess such eyes of fire. All that the
master who painted you has put into your face." But when Donna Micaela
saw Gaetano she found that it all could be in a face, and that the
master had not needed to add anything. Therefore she asked the noble old
gentlemen's pardon.
At last Gaetano had found the long candle-boxes that stood under the
counter, where they had always stood. And he gave her the candle, but
he did not know what it cost, and said that she could come in and pay
it later. When she asked him for something to wrap it in he was in such
trouble that she had to help him to look.
It grieved her that such a man should think of travelling to Argentina.
He let Donna Micaela wrap up the candle and watched her while she did it.
She wished she could have asked him not to look at her now, when her face
reflected only hopelessness and misery.
Gaetano had not scrutinized her features more than a moment before he
sprang up on a little step-ladder, took down an image from the topmost
shelf, and came back with it to her. It was a little gilded and painted
wooden angel, a little San Michele fighting with the arch-fiend, which he
had created from paper and wadding.
He handed it to Donna Micaela and begged her to accept it. He wished to
give it to her, he said, because it was the best he had ever carved. He
was so certain that it had greater power than his other images that he
had put it away on the top shelf, so that no one might see and buy it.
He had forbidden Donna Elisa to sell it except to one who had a great
sorrow. And now Donna Micaela was to take it.
She hesitated. She found him almost too daring.
But Gaetano begged her to look how well the image was carved. She saw
that the archangel's wings were ruffled with anger, and that Lucifer was
pressing his claws into the steel plate on his leg? Did she see how San
Michele was driving in his spear, and how he was frowning and pressing
his lips together?
He wished to lay the little image in her hand, but she gently pushed it
away. She saw that it was beautiful and spirited, she said, but she knew
that it could not help her. She thanked him for his gift, but she would
not accept it.
Then Gaetano seized the image and rolled it in paper and put it back in
its place.
And not until it was wrapped up and put away did he speak to her.
But then he asked her why she came to buy wax candles if she was not a
believer. Did she mean to say that she did not believe in San Michele?
Did she not know that he was the most powerful of the angels, and that
it was he who had vanquished Lucifer and thrown him into Etna? Did she
not believe that it was true? Did she not know that San Michele lost a
wing-feather in the fight, and that it was found in Caltanisetta? Did she
know it or not? Or what did she mean by San Michele not being able to
help her? Did she think that none of the saints could help? And he, who
was standing in his workshop all day long, carving saints!--would he do
such a thing if there was no good in it? Did she believe that he was an
impostor?
But as Donna Micaela was just as strong a believer as Gaetano, she
thought that his speech was unjust, and it irritated her to contradiction.
"It sometimes happens that the saints do not help," she said to him. And
when Gaetano looked unbelieving, she was seized by an uncontrollable
desire to convince him, and she said to him that some one had promised
her in the name of the Madonna that, if she was a faithful wife to Don
Ferrante, her father should enjoy an old age free of care. But now her
husband wished to put her father in a brotherhood, which was as wretched
as a poor-house and strict as a prison. And the Madonna had not averted
it; in eight days it would happen.
Gaetano listened to her with the greatest earnestness. That was what
induced her to confide the whole story to him.
"Donna Micaela," he said, "you must turn to the black Madonna in the
Cathedral."
"So you think that I have not prayed to her?"
Gaetano flushed and said almost with anger: "You will not say that you
have turned in vain to the black Madonna?"
"I have prayed to her in vain these last three weeks--prayed to her,
prayed to her."
When Donna Micaela spoke of it she could scarcely breathe. She wanted to
weep over herself because she had awaited help each day, and each day
been disappointed; and yet had known nothing better to do than begin
again with her prayers. And it was visible on her face that her soul
lived over and over again what she had suffered, when each day she had
awaited an answer to her prayer, while the days slipped by.
But Gaetano was unmoved; he stood smiling, and drummed on one of the
glass cases that stood on the counter.
"Have you only _prayed_ to the Madonna?" he said.
Only prayed, only prayed! But she had also promised her to lay aside all
sins. She had gone to the street where she had lived first, and nursed
the sick woman with the ulcerated leg. She never passed a beggar without
giving alms.
Only prayed! And she told him that if the Madonna had had the power
to help her, she ought to have been satisfied with her prayers. She
had spent her days in the Cathedral. And the anguish, the anguish that
tortured her, should not that be counted?
He only shrugged his shoulders. Had she not tried anything else?
Anything else! But there was nothing in the world that she had not tried.
She had given silver hearts and wax candles. Her rosary was never out of
her hand.
Gaetano irritated her. He would not count anything that she had done; he
only asked: "Nothing else? Nothing else?"
"But you ought to understand," she said. "Don Ferrante does not give me
so much money. I cannot do more. At last I have succeeded in getting some
silk and cloth for an altar cloth. You ought to understand!"
But Gaetano, who had daily intercourse with the saints, and who knew
the power and wildness of enthusiasm that had filled them when they
had compelled God to obey their prayers, smiled scornfully at Donna
Micaela, who thought she could subjugate the Madonna with wax candles and
altar-cloths.
He understood very well, he answered. The whole was clear to him. It
was always so with those miserable saints. Everybody called to them
for help, but few understood what they ought to do to get their prayers
granted. And then people said that the saints had no power. All were
helped who knew how they ought to pray.
Donna Micaela looked up in eager expectation. There was such strength and
conviction in Gaetano's words that she began to believe that he would
teach her the right words of salvation.
Gaetano took the candle lying in front of her on the counter and threw it
down into the box again, and told her what she had to do. He forbade her
to give the Madonna any gifts, or to pray to her, or to do anything for
the poor. He told her that he would tear her altar-cloth to pieces if she
sewed another stitch on it.
"Show her, Donna Micaela, that it means something to you," he said, and
fixed his eyes on her with compelling force. "Good Lord, you must be
able to find something to do, to show her that it is serious, and not
play. You must be able to show her that you will not live if you are not
helped. Do you mean to continue to be faithful to Don Ferrante, if he
sends your father away? I know you do. If the Madonna has no need to fear
what you are going to do, why should she help you?"
Donna Micaela drew back. He came swiftly out from behind the counter and
seized her coat sleeve.
"Do you understand? You shall show her that you can throw yourself away
if you do not get help. You shall throw yourself into sin and death if
you do not get what you want. That is the way to force the saints."
She tore herself from him and went without a word. She hurried up the
spiral street, came to the Cathedral, and threw herself down in terror
before the altar of the black Madonna.
That happened one Saturday morning, and on Sunday evening Donna Micaela
saw Gaetano again. For it was beautiful moonlight, and in Diamante it is
the custom on moonlight nights for all to leave their homes and go out
into the streets. As soon as the inhabitants of the summer palace had
come outside their door they had met acquaintances. Donna Elisa had taken
Cavaliere Palmeri's arm, and the syndic Voltaro had joined Don Ferrante
to discuss the elections; but Gaetano came up to Donna Micaela because he
wished to hear if she had followed his advice.
"Have you stopped sewing on that altar-cloth?" he said.
But Donna Micaela answered that all day yesterday she had sewn on it.
"Then it is you who understand what you are doing, Donna Micaela."
"Yes, now there is no help for it, Don Gaetano."
She managed to keep them away from the others, for there was something
she wished to speak to him about. And when they came to Porta Etnea, she
turned out through the gate, and they went along the paths that wind
under Monte Chiaro's palm groves.
They could not have walked on the streets filled with people. Donna
Micaela spoke so the people in Diamante would have stoned her if they had
heard her.
She asked Gaetano if he had ever seen the black Madonna in the Cathedral.
She had not seen her till yesterday. The Madonna perhaps had placed
herself in such a dark corner of the Cathedral so that no one should be
able to see her. She was so black, and had a railing in front of her. No
one could see her.
But to-day Donna Micaela had seen her. To-day the Madonna had had a
festival, and she had been moved from her niche. The floor and walls of
her chapel had been covered with white almond-blossoms, and she herself
had stood down on the altar, dark and high, surrounded by the white glory.
But when Donna Micaela had seen the image she had been filled with
despair; for the image was no Madonna. No, she had prayed to no Madonna.
Oh, a shame, a shame! It was plainly an old heathen goddess. She had a
helmet, not a crown; she had no child on her arm; she had a shield. It
was a Pallas Athene. It was no Madonna. Oh, no; oh, no!
It was like the people of Diamante to worship such an image. It was like
them to set up such a blasphemy and worship it! Did he know what was the
worst misfortune? Their Madonna was so ugly. She was disfigured, and she
had never been a work of art. She was so ugly that one could not bear to
look at her.
And to have been deceived by all the thousand votive offerings that hung
in the chapel; to have been fooled by all the legends about her! To have
wasted three weeks in praying to her! Why had she not been helped? She
was no Madonna, she was no Madonna.
They walked along the path on the town wall running around Monte Chiaro.
The whole world was white about them. A white mist wreathed the base of
the mountain, and the almond-trees on Etna were quite white. Sometimes
they passed under an almond-tree, which arched them over with its
glistening branches, as thickly covered with flowers as if they had been
dipped in a bath of silver. The moonlight shone so bright on the earth
that everything was divested of its color, and became white. It seemed
almost strange that it could not be felt, that it did not warm, that it
did not dazzle the eyes.
Donna Micaela wondered if it was the moonlight that subdued Gaetano,
so that he did not seize her, and throw her down into Simeto, when she
cursed the black Madonna.
He walked silent and quiet at her side, but she was afraid of what he
might do. In spite of her fear, she could not be silent.
What she had still to say was the most dreadful of all. She said that she
had tried all day long to think of the real Madonna, and that she had
recalled to her mind all the images of her she had ever seen. But it had
all been in vain, because as soon as she thought of the shining queen of
heaven, the old black goddess came and placed herself between them. She
saw her come like a dried-up and officious old maid, and stand in front
of the great queen of heaven, so that now no Madonna existed for her any
longer. She believed that the latter was angry with her because she had
done so much for the other, and that she hid her face and her grace from
her. And, on account of the false Madonna, her father was now to suffer
misfortune. Now she would never be allowed to keep him in her home. Now
she would never win his forgiveness. Oh, God! oh, God!
And all this she said to Gaetano, who honored the black Madonna of
Diamante more than anything else in the world.
He now came close up to Donna Micaela, and she feared that it was her
last hour. She said in a faint voice, as if to excuse herself: "I am mad.
Grief is driving me mad. I never sleep."
But Gaetano's only thought had been what a child she was, and that she
did not at all understand how to meet life.
He hardly knew himself what he was doing when he gently drew her to him
and kissed her, because she had gone so astray and was such a helpless
child.
She was so overcome with astonishment that she did not even think of
avoiding it. And she neither screamed nor ran away. She understood
instantly that he had kissed her as he would a child. She only walked
quickly on and began to cry. That kiss had made her feel how helpless and
forsaken she was, and how much she longed for some one strong and good to
take care of her.
It was terrible that, although she had both father and husband, she
should be so forsaken that this stranger should need to feel sympathy for
her.
When Gaetano saw her trembling with silent sobs, he felt that he too
began to shake. A strong and violent emotion took possession of him.
He came close to her once more and laid his hand on her arm. And his
voice, when he spoke, was not clear and loud; it was thick and choked
with emotion.
"Will you go with me to Argentina if the Madonna does not help you?"
Then Donna Micaela shook him off. She felt suddenly that he no longer
talked to her as to a child. She turned and went back into the town.
Gaetano did not follow her; he remained standing in the path where he had
kissed her, and it seemed as if never again could he leave that place.
For two days Gaetano dreamed of Donna Micaela, but on the third he came
to the summer palace to speak to her.
He found her on the roof-garden, and instantly told her that she must
flee with him.
He had thought it out since they parted. He had stood in his workshop and
considered everything that had happened, and now it was all clear to him.
She was a rose which the strong sirocco had torn from its stem and
roughly whirled through the air, that she might find so much the better
rest and protection in a heart which loved her. She must understand
that God and all the saints wished and desired that they should love
one another, otherwise these great misfortunes would not have brought
her near to him. If the Madonna refused to help her, it was because she
wished to set her free from her promise of faithfulness to Don Ferrante.
For all the saints knew that she was his, Gaetano's. She was created for
him; for him she had grown up; for him she was alive. When he kissed
her in the path in the moonlight he had been like a lost child who had
wandered long in the desert and now at last had come to the gate of his
home. He possessed nothing; but she was his home and his hearth; she was
the inheritance God had apportioned to him, the only thing in the world
that was his.
Therefore he could not leave her behind. She must go with him; she must,
she must!
He did not kneel before her. He stood and talked to her with clenched
hands and blazing eyes. He did not ask her, he commanded her to go with
him, because she was his.
It was no sin to take her away; it was his duty. What would become of her
if he deserted her?
Donna Micaela listened to him without moving. She sat silent a long time,
even after he had ceased speaking.
"When are you going?" she asked at length.
"I leave Diamante on Saturday."
"And when does the steamer go?"
"It goes on Sunday evening from Messina."
Donna Micaela rose and walked away towards the terrace stairs.
"My father is to go to Catania on Saturday," she said. "I shall ask Don
Ferrante to be allowed to go with him." She went down a few steps, as if
she did not mean to say anything more. Then she stopped. "If you meet me
in Catania, I will go with you whither you will."
She hurried down the steps. Gaetano did not try to detain her. A time
would come when she would not run away from him. He knew that she could
not help loving him.
Donna Micaela passed the whole of Friday afternoon in the Cathedral. She
had come to the Madonna and thrown herself down before her in despair.
"Oh, Madonna mia, Madonna mia! Shall I be to-morrow a fugitive wife? Will
the world have the right to say all possible evil of me?" Everything
seemed equally terrible to her. She was appalled at the thought of
fleeing with Gaetano, and she did not know how she could stay with Don
Ferrante. She hated the one as much as the other. Neither of them seemed
able to offer her anything but unhappiness.
She saw that the Madonna would not help. And now she asked herself if it
really would not be a greater misery to go with Gaetano than to remain
with Don Ferrante. Was it worth while to ruin herself to be revenged on
her husband?
She suffered great anguish. She had been driven on by a devouring
restlessness the whole week. Worst of all, she could not sleep. She no
longer thought clearly or soundly.
Time and time again she returned to her prayers. But then she thought:
"The Madonna cannot help me." And so she stopped.
Then she came to think of the days of her former sorrows, and remembered
the little image that once had helped her, when she had been in despair
as great as this.
She turned with passionate eagerness to the poor little child. "Help me,
help me! Help my old father, and help me myself that I may not be tempted
to anger and revenge!"
When she went to bed that night, she was still tormented and distressed.
"If I could sleep only one hour," she said, "I should know what I wanted."
Gaetano was to start on his travels early the next morning. She came at
last to the decision to speak to him before he left, and tell him that
she could not go with him. She could not bear to be considered a fallen
woman.
She had hardly decided that before she fell asleep. She did not wake
till the clock struck nine the next morning. And then Gaetano was already
gone. She could not tell him that she had changed her mind.
But she did not think of it either. During her sleep something new and
strange had come over her. It seemed to her that in the night she had
lived in heaven and was filled with bliss.
* * * * *
What saint is there who does more for man than San Pasquale? Does it not
sometimes happen to you to stand and talk in some lonely place in the
woods or plains, and either to speak ill of some one or to make plans
for something foolish? Now please notice that just as you are talking
and talking you hear a rustling near by, and look round in wonder to see
if some one has thrown a stone. It is useless to look about long for the
thrower of the stone. It comes from San Pasquale. As surely as there is
justice in heaven, it was San Pasquale who heard you talking evil, and
threw one of his stones in warning.
And any one who does not like to be disturbed in his evil schemes may not
console himself with the thought that San Pasquale's stones will soon
come to an end. They will not come to an end at all. There are so many of
them that they will hold out till the last day of the world. For when San
Pasquale lived here on the earth, do you know by chance what he did, do
you know what he thought about more than anything else? San Pasquale gave
heed to all the little flint-stones that lay in his path, and gathered
them up into his bag. You, signor, you will scarcely stoop to pick up a
soldo, but San Pasquale picked up every little flint-stone, and when he
died, he took them all with him up to heaven, and there he sits now, and
throws them at everybody who thinks of doing anything foolish.
But that is not by any means the only use that San Pasquale is to man.
It is he, also, who gives warning if any one is to be married, or if any
one is to die; and he even gives the sign with something besides stones.
Old Mother Saraedda at Randazzo sat by her daughter's sick bed one night
and fell asleep. The daughter lay unconscious and was about to die, and
no one could summon the priest. How was the mother waked in time? How was
she waked, so that she could send her husband to the priest's house? By
nothing else than a chair, which began to rock forward and back, and to
crack and creak, until she awoke. And it was San Pasquale who did it. Who
else but San Pasquale is there to think of such a thing?
There is one thing more to tell about San Pasquale. It was of big
Cristoforo from Tre Castagni. He was not a bad man, but he had a bad
habit. He could not open his mouth without swearing. He could not say two
words without one of them being an oath. And do you think that it did
any good for his wife and neighbors to admonish him? But over his bed he
had a little picture representing San Pasquale, and the little picture
succeeded in helping him. Every night it swung forward and back in its
frame, swung fast or slow, as he had sworn that day. And he discovered
that he could not sleep a single night until he stopped swearing.
In Diamante San Pasquale has a church, which lies outside the Porta
Etnea, a little way down the mountain. It is quite small and poor, but
the white walls and the red roof stand beautifully embedded in a grove of
almond-trees.
Therefore, as soon as the almond-trees bloom in the spring, San
Pasquale's church becomes the most beautiful in Diamante. For the
blossoming branches arch over it, thickly covered with white, glistening
flowers, like the most gorgeous garment.
San Pasquale's church is very miserable and deserted, because no service
can be held there. For when the Garibaldists, who freed Sicily, came to
Diamante, they camped in San Pasquale's church and in the Franciscan
monastery beside it. And in the church itself they stabled brute beasts,
and led such a wild life with women and with gambling that ever since it
has been considered unhallowed and unclean, and has never been opened for
divine service from that time.
Therefore it is only when the almond-trees are in bloom that strangers
and fine people pay attention to San Pasquale. For although the whole of
the <DW72>s of Etna are white then with almond-blossoms, still the biggest
and the most luxuriant trees stand about the old, condemned church.
But the poor people of Diamante come to San Pasquale the whole year
round. For although the church is always closed, people go there to get
advice from the saint. There is an image of him under a big stone canopy
just by the entrance, and people come to ask him about the future. No one
can foretell the future better than San Pasquale.
Now it happened that the very morning when Gaetano left Diamante the
clouds had come rolling down from Etna, as thick as if they had been
dust from innumerable hosts, and they filled the air like dark-winged
dragons, and vomited forth rain, and breathed mists and darkness. It grew
so thick over Diamante that one could scarcely see across the street.
The dampness dripped from everything; the floor was as wet as the roof,
the doorposts and balustrades were covered with drops, the fog stood and
quivered in the passage-ways and rooms, until one would have thought them
full of smoke.
That very morning, at an early hour, before the rain had begun, a rich
English lady started in her big travelling-carriage to make the trip
round Etna. But when she had driven a few hours a terrible rain began,
and everything was wrapped in mist. As she did not wish to miss seeing
any of the beautiful district through which she was travelling, she
determined to drive to the nearest town and to stay there until the storm
was over. That town was Diamante.
The Englishwoman was a Miss Tottenham, and it was she who had moved into
the Palazzo Palmeri at Catania. Among all the other things she brought
with her in her trunks was the Christ image, upon which Donna Micaela had
called the evening before. For that image, which was now both old and
mishandled, she always carried with her, in memory of an old friend who
had left her her wealth.
It seemed as if San Pasquale had known what a great miracle-worker
the image was, for it was as if he wished to greet him. Just as Miss
Tottenham's travelling-carriage drove in through Porta Etnea, the bells
began to ring on San Pasquale's church.
They rang afterwards all day quite by themselves.
San Pasquale's bells are not much bigger than those that are used on
farms to call the work people home; and like them, they are hung under
the roof in a little frame, and set in motion by pulling a rope that
hangs down by the church wall.
It is not heavy work to make the bells ring, but nevertheless they are
not so light that they can swing quite by themselves. Whoever has seen
old Fra Felice from the Franciscan monastery put his foot in the loop of
the rope and tread up and down to start them going, knows well enough
that the bells cannot begin to ring without assistance.
But that was just what they were doing that morning. The rope was
fastened to a cleat in the wall, and there was no one touching it. Nor
did any one sit crouching on the roof to set them going. People plainly
saw how the bells swung backwards and forwards, and how the tongues hit
against the brazen throats. It could not be explained.
When Donna Micaela awoke, the bells were already ringing, and she lay
quiet for a long time, and listened, and listened. She had never heard
anything more beautiful. She did not know that it was a miracle, but she
lay and thought how beautiful it was. She lay and wondered if real bronze
bells could sound like that.
No one will ever know what the metal was that rang in San Pasquale's
bells that day.
She thought that the bells said to her that now she was to be glad; now
she was to live and love; now she was to go to meet something great and
beautiful; now she was never again to have regrets and never be sad.
Then her heart began to dance in a kind of stately measure, and she
marched solemnly to the sound of bells into a great castle. And to whom
could the castle belong, who could be lord of such a beautiful place, if
not love?
It can be hidden no longer: when Donna Micaela awoke she felt that she
loved Gaetano, and that she desired nothing better than to go with him.
When Donna Micaela drew back the curtain from the window and saw the gray
morning, she kissed her hand to it and whispered: "You, who are morning
to the day when I am going away, you are the most beautiful morning I
have ever seen; and gray as you are, I will caress and kiss you."
But she still liked the bells best.
By that you may know that her love was strong, for to all the others it
was torture to hear those bells, that would not stop ringing. No one
asked about them during the first half-hour. During the first half-hour
people hardly heard any ringing, but during the second and the third!!!
No one need believe that San Pasquale's little bells could not make
themselves heard. They are always loud and their clang seemed now to grow
and grow. It soon sounded as if the fog were filled with bells; as if the
sky hung full of them, although no one could see them for the clouds.
When Donna Elisa first heard the ringing she thought that it was San
Giuseppe's little bell, and then that it was the bell of the Cathedral
itself. Then she thought she heard the bell of the Dominican monastery
chime in, and at last she was certain that all the bells in the town rang
and rang all they could, all the bells in the five monasteries and the
seven churches. She thought that she recognized them all, until finally
she asked, and heard that it was only San Pasquale's little bells that
were ringing.
During the first hours, and before people generally knew that the bells
were ringing all by themselves, they noticed that the raindrops fell
in time to the sound of the bells, and that every one spoke with a
metallic voice. People also noticed that it was impossible to play on
mandolin and guitar, because the bells blended with the music and made
it ear-splitting; neither could any one read, because the letters swung
to and fro like bell-clappers, and the words acquired a voice, and read
themselves out quite audibly.
Soon the people could not bear to see flowers on long stalks, because
they thought that they swung to and fro. And they complained that sound
came from them, instead of fragrance.
Others insisted that the mist floating through the air moved in time with
the sound of the bells, and they said that all the pendulums conformed to
it, and that every one who went by in the rain tried to do likewise.
And that was when the bells had only rung a couple of hours, and when the
people still laughed at them.
But at the third hour the ringing seemed to increase even more, and then
some stuffed cotton into their ears, while others buried themselves under
pillows. But they felt just as distinctly how the air quivered with the
strokes, and they thought that they perceived how everything moved in
time. Those who fled up to the dark attic found the sound of the bells
clear and ringing there, as if they came from the sky; and those who fled
down into the cellar heard them as loud and deafening there as if San
Pasquale's church stood under ground.
Every one in Diamante began to be terrified except Donna Micaela, whom
love protected from fear.
And now people began to think that it must mean something, because it was
San Pasquale's bells that rang. Every one began to ask himself what the
saint foretold. Each had his own dread, and believed that San Pasquale
gave warning to him of what he least wished. Each had a deed on his
conscience to remember, and now thought that San Pasquale was ringing
down a punishment for him.
Toward noon, when the bells still rang, everybody was sure that San
Pasquale was ringing such a misfortune upon Diamante that they might all
expect to die within the year.
Pretty Giannita came terrified and weeping to Donna Micaela, and lamented
that it was San Pasquale who was ringing. "God, God, if it had been any
other than San Pasquale!"
"He sees that something terrible is coming to us," said Giannita. "The
mist does not prevent him from seeing as far as he will. He sees that an
enemy's fleet is approaching in the bay! He sees that a cloud of ashes is
rising out of Etna which will fall over us and bury us!"
Donna Micaela smiled, and thought that she knew of what San Pasquale
was thinking. "He is tolling a passing-bell for the beautiful
almond-blossoms, that are destroyed by the rain," she said to Giannita.
She let no one frighten her, for she believed that the bells were ringing
for her alone. They rocked her to dream. She sat quite still in the
music-room and let joy reign in her. But in the whole world about her was
fear and anxiety and restlessness.
No one could sit at his work. No one could think of anything but the
great horror that San Pasquale foretold.
People began to give the beggars more gifts than they had ever had; but
the beggars did not rejoice, because they did not believe they would
survive the morrow. And the priests could not rejoice, although they had
so many penitents that they had to sit in the confessional all day long,
and although gift upon gift was piled up on the altar of the saint.
Not even Vicenzo da Lozzo, the letter-writer, was glad of the day,
although people besieged his desk under the court-house loggia, and were
more than willing to pay him a soldo a word, if they only might write a
line of farewell on this their last day to their dear ones far away.
It was not possible to keep school that day, for the children cried the
whole time. At noon the mothers came, their faces stiff with terror, and
took their little ones home with them, so that they might at least be
together in misfortune.
The apprentices at the tailors and shoe-makers had a holiday. But the
poor boys did not dare to enjoy it; they preferred to sit in their places
in the workshops, and wait.
In the afternoon the ringing still continued.
Then the old gate-keeper of the palazzo Geraci, where now no one lives
but beggars, and who is himself a beggar, and goes dressed in the most
miserable rags, went and put on the light-green velvet livery that he
wears only on saints' days and on the king's birthday. And no one could
see him sitting in the gateway dressed in that array without being
chilled with fear, for people understood that the old man expected that
no other than destruction would march in through the gate he was guarding.
It was dreadful how people frightened one another.
Poor Torino, who had once been a man of means, went from house to house
and cried that now the time had come when every one who had cheated and
beggared him would get his punishment. He went into all the little shops
along the Corso and struck the counter with his hand, saying that now
every one in the town would get his sentence, because all had connived to
cheat him.
It was also terrifying to hear of the game of cards at the Café Europa.
There the same four had played year after year at the same table, and
no one had ever thought that they could do anything else. But now they
suddenly let their cards fall, and promised each other that if they
survived the horror of this day they would never touch them again.
Donna Elisa's shop was packed with people; to propitiate the saints and
to avert the menace, they bought all the sacred things that she had to
sell. But Donna Elisa thought only of Gaetano, who was away, and believed
that San Pasquale was warning her that he would be lost during the
voyage. And she took no pleasure in all the money that she was earning.
When San Pasquale's bells went on ringing the whole afternoon people
could hardly hold out.
For now they knew that it was an earthquake which they foretold, and that
all Diamante would be wrecked.
In the alleys, where the very houses seemed afraid of earthquakes, and
huddled together to support one another, people moved their miserable old
furniture out on the street into the rain, and spread tents of bed-quilts
over them. And they even carried out their little children in their
cradles, and piled up boxes over them.
In spite of the rain, there was such a crowd on the Corso that it was
almost impossible to pass through. For every one was trying to go out
through Porta Etnea to see the bells swinging and swinging, and to
convince themselves that no one was touching the rope,--that it was
firmly tied. And all who came out there fell on their knees in the road,
where the water ran in streams, and the mud was bottomless.
The doors to San Pasquale's church were shut, as always, but outside the
old gray-brother, Fra Felice, went about with a brass plate, among those
who prayed, and received their gifts.
In their turn the frightened people went forward to the image of San
Pasquale beneath the stone canopy, and kissed his hand. An old woman came
carefully carrying something under a green umbrella. It was a glass with
water and oil, in which floated a little wick burning with a faint flame.
She placed it in front of the image and knelt before it.
Though many thought that they ought to try to tie up the bells, no one
dared to propose it. For no one dared to silence God's voice.
Nor did any one dare to say that it might be a device of old Fra Felice
to collect money. Fra Felice was beloved. It would fare badly with
whoever said such things as that.
Donna Micaela also came out to San Pasquale and took her father with her.
She walked with her head high and quite without fear. She came to thank
him for having rung a great passion into her soul. "My life begins this
day," she said to herself.
Don Ferrante did not seem to be afraid either, but he was grim and angry.
For every one had to go in to him in his shop, and tell him what they
thought, and hear his opinion, because he was one of the Alagonas, who
had governed the town for so many years.
All day terrified, trembling people came into his shop. And they all came
up to him and said: "This is a terrible ringing, Don Ferrante. What is to
become of us, Don Ferrante?"
Even Ugo Favara, the splenetic advocate, came into the shop, and took
a chair, and sat down behind the counter. And Don Ferrante had him
sitting there all day, quite livid, quite motionless, suffering the most
inconceivable anguish without uttering a word.
Every five minutes Torino-il-Martello came in and struck the counter,
saying that the hour had come in which Don Ferrante was to get his
punishment.
Don Ferrante was a hard man, but he could no more escape the bells than
any other. And the longer he heard them, the more he began to wonder why
everybody streamed into his shop. It seemed as if they meant something
special. It seemed as if they wished to make him responsible for the
ringing, and the evil it portended.
He had not spoken of it to any one, but his wife must have spread it
about. He began to believe that everybody was thinking the same, although
they did not dare to say it. He thought that the advocate was sitting and
waiting for him to yield. He believed that the whole town came in to see
if he would really dare to send his father-in-law away.
Donna Elisa, who had so much to do in her own shop that she could not
come herself, sent old Pacifica continually to him to ask what he thought
of the bell-ringing. And the priest too came to the shop for a moment and
said, like all the others: "Did you ever hear such a terrible ringing,
Don Ferrante?"
Don Ferrante would have liked to know if the advocate and Don Matteo
and all the others came only to reproach him because he wished to send
Cavaliere Palmeri away.
The blood began to throb in his temples. The room swam now and then
before his eyes. People came in continually and asked: "Have you ever
heard such a terrible ringing?" But one never came and asked, and that
was Donna Micaela. She could not come when she felt no fear. She was
merely delighted and proud that the passion which was to fill her whole
life had come. "My life is to be great and glorious," she said. And she
was appalled that till now she had been only a child.
She would travel with the post-carriage that went by Diamante at ten
o'clock at night. Towards four, she thought, she must tell her father
everything, and begin his packing.
But that did not seem hard to her. Her father would soon come to her in
Argentina. She would beg him to be patient for a few months, until they
could have a home to offer him. And she was sure that he would be glad to
have her leave Don Ferrante.
She moved in a delicious trance. Everything that had seemed dreadful
appeared so no longer. There was no shame, no danger; no, none at all.
She only longed to hear the rattling of the post-carriage.
Then she heard many voices on the stairs leading from the court-yard to
the second floor. She heard a multitude of heavy feet tramping. She saw
people passing through the open portico that ran round the court-yard,
and through which one had to go to come into the rooms. She saw that they
were carrying something heavy between them, but she could not see what it
was, because there was such a crowd.
The pale-faced advocate walked before the others. He came and said to her
that Don Ferrante had wished to drive Torino out of his shop; Torino had
cut him with his knife. It was nothing dangerous. He was already bandaged
and would be well in a fortnight.
Don Ferrante was carried in, and his eyes wandered about the room, not in
search of Donna Micaela, but of Cavaliere Palmeri. When he saw him, he
let his wife know without a word, only by a few gestures, that her father
never would need to leave his house; never, never.
Then she pressed her hands against her eyes. What, what! her father need
not go? She was saved. A miracle had come to pass to help her!
Ah, now she must be glad, be content! But she was not. She felt the most
terrible pain.
She could not go. Her father was allowed to remain, and so she must be
faithful to Don Ferrante. She struggled to understand. It was so. She
could not go.
She tried to change it in some way. Perhaps it was a false conclusion.
She had been so confused. No, no, it was so, she could not.
Then she became tired unto death. She had travelled and travelled the
whole day. She had been so long on the way. And she would never get
there. She sank down. A torpor and faintness came over her. There was
nothing to do but to rest after the endless journey she had made. But
that she could never do. She began to weep because she would never reach
her journey's end. Her whole life long she would travel, travel, travel,
and never reach the end of her journey.
VIII
TWO SONGS
It was the morning after the day when San Pasquale's bells had rung;
and Donna Elisa sat in her shop and counted her money. The day before,
when everyone had been afraid, there had been an incredible sale in the
shop, and the next morning, when she had come down, she had at first
been almost frightened. For the whole shop was desolate and empty; the
medallions were gone, the wax candles were gone, and so were all the
great bunches of rosaries. All Gaetano's beautiful images had been taken
down from the shelves and sold, and it was a real grief to Donna Elisa
not to see the host of holy men and women about her.
She opened the money-drawer, and it was so full that she could hardly
pull it out. And while she counted her money she wept over it as if it
had all been false. For what good did it do her to possess all those
dirty lire and those big copper coins when she had lost Gaetano!
Alas! she thought that if he had stopped at home one day more he would
not have needed to go, for now she was laden down with money.
While she was counting she heard the post-carriage stop outside her door.
But she did not even look up; she did not care what happened, since
Gaetano was gone. Then the door opened, and the bell rang violently. She
only wept and counted. Then some one said: "Donna Elisa, Donna Elisa!"
And it was Gaetano!
"But heavens! how can you be at home?" she cried.--"You have sold all
your images. I had to come home to carve new ones for you."--"But how
did you find out about it?"--"I met the post-carriage at two o'clock in
the night. Rosa Alfari was in it, and she told me everything."--"What
luck that you went down to the post-carriage! What luck that you happened
to think of going down to the post-carriage!"--"Yes; was it not good
fortune?" said Gaetano.
In less than an hour Gaetano was again standing in his workshop; and
Donna Elisa, who had nothing at all to do in her empty shop, came
incessantly to the door to look at him. No, was he really standing there
and carving? She could not let five minutes pass without coming to look
at him.
But when Donna Micaela heard that he was back she felt no joy, rather
anger and despair. For she was afraid that Gaetano would come to tempt
her.
She had heard that a rich Englishwoman had come to Diamante the day the
bells rang. She was deeply affected when she heard that it was the lady
with the Christ image. He had therefore come as soon as she had called on
him. The rain and the bell-ringing were his work!
She tried to rejoice her soul with the thought that there had been a
miracle for her sake. It would be more to her than all earthly happiness
and love to feel that she was surrounded by God's grace. She did not
wish anything earthly to come and drag her down from that blessed rapture.
But when she met Gaetano on the street he hardly looked at her; and when
she met him at Donna Elisa's he did not take her hand and did not speak
to her at all.
For the truth was that, although Gaetano had come home because it had
been too hard to go without Donna Micaela, he did not wish to tempt or to
persuade her. He saw that she was under the protection of the saints, and
she had become so sacred to him that he scarcely dared to dream of her.
He wished to be near her, not in order to love her, but because he
believed that her life would blossom with holy deeds. Gaetano longed for
miracles, as a gardener longs for the first rose in the spring.
But when weeks went by and Gaetano never tried to approach Donna Micaela,
she began to doubt, and to think that he had never loved her. She said
to herself that he had won the promise from her to flee with him only in
order to show her that the Madonna could work a miracle.
If that were true, she did not know why he had not continued his journey
without turning back.
That caused her anxiety. She thought that she could conquer her love
better if she knew whether Gaetano loved her. She weighed the pros and
cons, and she was more and more sure that he had never loved her.
While Donna Micaela was thinking of this, she had to sit and keep Don
Ferrante company. He had lain sick a long time. He had had two strokes of
paralysis, and had risen from his sick-bed a broken man. All at once he
had become old and dull and afraid, so that he never dared to be alone.
He never worked in the shop; he was in every way a changed man.
He had been seized with a great desire to be aristocratic and
fashionable. It looked as if poor Don Ferrante's head was turned with
pride.
Donna Micaela was very good to him, and sat hour after hour and chatted
with him.
"Who could it be," she used to ask, "who once stood in the market-place
with plumes on his hat, and braid on his coat, and sword at his side,
and who played so that people said that his music was as uplifting as
Etna, and as strong as the sea? And who caught sight of a poor signorina
dressed in black, who did not dare to show her face to the world, and
went forward to her and offered his arm? Who could it be? Could it be Don
Ferrante, who stands the whole week in his shop and wears a pointed cap
and a short jacket? No; that cannot be possible. No old merchant could
have done such a thing."
Don Ferrante laughed. That was just the way he liked to have her talk to
him. She would also tell him how it would be when he came to court. The
king would say this, and the queen would say that. "The old Alagonas have
come up again," they would say at court. And who has brought up the race?
People will wonder and wonder. The Don Ferrante, who is a Sicilian prince
and Spanish grandee, is that the same man who stood in a shop in Diamante
and shouted at the teamsters? No, people will say, it cannot be the same.
It is impossible for it to be the same.
Don Ferrante liked that, and wished to hear her talk so day in and day
out. He was never tired of listening, and Donna Micaela was very patient
with him.
But one day while she was chatting, Donna Elisa came in. "Sister-in-law,
if you happen to own the 'Legend of the Holy Virgin of Pompeii,' will you
lend it to me?" she asked.--"What, are you going to begin to read?" asked
Donna Micaela.--"The saints preserve us! you know very well that I cannot
read. Gaetano is asking for it."
Donna Micaela did not own the "Legend of the Holy Virgin at Pompeii." But
she did not say so to Donna Elisa; she went to her book-shelf and took a
little book, a collection of Sicilian love-songs, and gave it to Donna
Elisa, who carried the little book over to Gaetano.
But Donna Micaela had no sooner done so before a lively regret seized
her. And she asked herself what she had meant by behaving so,--she who
had been helped by the little Christchild?
She blushed with shame as she thought that she had marked one of the
little songs, one that ran thus:--
"For one single question's answer longing,
Night I asked, and asked the daytime's burning;
Watched the flight of birds, and swift clouds thronging,
In water strove to read the hot lead's turning;
Leaves I counted plucked from many flowers,
Lured dark prophets forth, and sought their powers,
Till at last I called on Heaven above me:
'Doth he love me still, as once he loved me?'"
She had hoped to get an answer to it. But it would serve her right if
no answer came. It would serve her right if Gaetano despised her and
thought her forward.
Yet she had meant no harm. The only thing she had desired had been to
find out if Gaetano loved her.
Several weeks again passed and Donna Micaela still sat with Don Ferrante.
But one day Donna Elisa had tempted her out. "Come with me into my
garden, sister-in-law, and see my big magnolia-tree. You have never seen
anything so beautiful."
She had gone with Donna Elisa across the street and had come into her
court-yard. And Donna Elisa's magnolia was like the shining sun, so that
people were aware of it even before they saw it. At a great distance the
fragrance lay and rocked in the air, and there was a murmuring of bees,
and a twittering of birds.
When Donna Micaela saw the tree she could hardly breathe. It was very
high and broad, with a beautifully even growth, and its large, firm
leaves were of a fresh, dark green. But now it was entirely covered with
great, bright flowers, that lighted and adorned it so that it looked
as if dressed for a feast, and one felt an intoxicating joy streaming
forth from the tree. Donna Micaela almost lost consciousness, and a
new and irresistible power took possession of her. She drew down one
of the stiff branches, and without breaking it spread out the flower
that it bore, took a needle and began to prick letters on the flower
leaf. "What are you doing, sister-in-law?" asked Donna Elisa.--"Nothing,
nothing."--"In my time young girls used to prick love-letters on the
magnolia-blossoms."--"Perhaps they do it still."--"Take care; I shall
look at what you have written when you are gone."--"But you cannot
read."--"I have Gaetano."--"And Luca; you had better ask Luca."
When Donna Micaela came home, she repented of what she had done. Would
Donna Elisa really show the flower to Gaetano? No, no; Donna Elisa was
too sensible. But if he had seen her from the window of his workshop?
Well, he would not answer. She had made herself ridiculous.
No, never, never again would she do such a thing. It was best for her not
to know. It was best for her that Gaetano did not ask after her.
Nevertheless she wondered what answer she would get. But none came.
So another week passed. Then it came into Don Ferrante's mind that he
would like to go out for a drive in the afternoon.
In the carriage-house of the summer palace there was an ancient state
carriage, which was certainly more than a hundred years old. It was very
high; it had a small, narrow body, which swung on leather straps between
the back wheels, which were as big as the water-wheels of a mill. It was
painted white, with gilding; it was lined with red velvet, and had a coat
of arms on its doors.
Once it had been a great honor to ride in that carriage; and when the
old Alagonas had passed in it along the Corso, people had stood on their
thresholds, and crowded to their doors, and hung over balconies to see
them. But then it had been drawn by spirited barbs; then the coachman
had worn a wig, and the footman gold braid, and it had been driven with
embroidered silk reins.
Now Don Ferrante wished to harness his old horses before the gala
carriage and have his old shopman take the place of coachman.
When Donna Micaela told him that it could not be, Don Ferrante began to
weep. What would people think of him if he did not show himself on the
Corso in the afternoon? That was the last thing a man of position denied
himself. How could anyone know that he was a nobleman, if he did not
drive up and down the street in the carriage of the old Alagonas?
The happiest hour Don Ferrante had enjoyed since his illness was when
he drove out for the first time. He sat erect and nodded and waved very
graciously to every one he met. And the people of Diamante bowed, and
took off their hats, so that they swept the street. Why should they not
give Don Ferrante this pleasure?
Donna Micaela was with him, for Don Ferrante did not dare to drive alone.
She had not wished to go, but Don Ferrante had wept, and reminded her
that he had married her when she was despised and penniless. She ought
not to be ungrateful; she ought not to forget what he had done for her,
and ought to come with him. Why did she not wish to drive with him in his
carriage? It was the finest old carriage in Sicily.
"Why will you not come with me?" said Don Ferrante. "Remember that I am
the only one who loves you. Do you not see that not even your father
loves you? You must not be ungrateful."
In this way he had forced Donna Micaela to take her place in the gala
carriage.
But it was not at all as she had expected. No one laughed. The women
courtesied, and the men bowed as solemnly as if the carriage had been a
hundred years younger. And Donna Micaela could not detect a smile on any
face.
No one in all Diamante would have wished to laugh; for every one knew
how Don Ferrante treated Donna Micaela. They knew how he loved her, and
how he wept if she left him for a single minute. They knew, too, that he
tormented her with jealousy, and that he trampled her hats to pieces,
if they became her, and never gave her money for new dresses, because
no other was to find her beautiful, and love her. But all the time he
told her that she was so ugly that no one but he could bear to look at
her face. And because every one in Diamante knew it all, no one laughed.
Laugh at her, sitting and chatting with a sick man! They are pious
Christians in Diamante, and not barbarians.
So the gala-carriage in its faded glory drove up and down the Corso in
Diamante during the hour between five and six. And in Diamante it drove
quite alone, for there were no other fine carriages there; but people
knew that at that same time all the carriages in Rome drove to Monte
Pincio, all those in Naples to the Via Nazionale, and all in Florence to
the Cascine, and all in Palermo to La Favorita.
But when the carriage approached the Porta Etnea for the third time, a
merry sound of horns was heard from the road outside.
And through the gate swung a big, high coach in the English style.
It was meant to look old-fashioned also. The postilion riding on the off
leader had leather trousers, and a wig tied in a pig-tail. The coach was
like an old diligence, with the body behind the coach box and seats on
the roof.
But everything was new; the horses were magnificent, powerful animals,
carriage and harness shone, and the passengers were some young gentlemen
and ladies from Catania, who were making an excursion up Etna. And they
could not help laughing as they drove by the old gala-carriage. They
leaned over from where they sat on the high roof to look at it, and their
laughter sounded very loud and echoed between the high, silent houses of
Diamante.
Donna Micaela was very unhappy. They were some of her old circle of
friends. What would they not say when they came home? "We have seen
Micaela Palmeri in Diamante." And they would laugh and talk, laugh and
talk.
Her life seemed so squalid. She was nothing but the slave of a fool. Her
whole life long she would never do anything but chat with Don Ferrante.
When she came home she was quite exhausted. She was so tired and weak
that she could scarcely drag herself up the steps.
And all the time Don Ferrante was rejoicing in his good fortune at having
met all those fine people, and having been seen in his state. He told her
that now no one would ask whether she was ugly, or whether her father had
stolen. Now people knew that she was the wife of a man of rank.
After dinner Donna Micaela sat quite silent, and let her father talk to
Don Ferrante. Then a mandolin began to sound quite softly in the street
under the window of the summer palace. It was a single mandolin with
no accompaniment of guitar or violin. Nothing could be more light and
airy; nothing more captivating and affecting. No one could think that
human hands were touching the strings. It was as if bees and crickets and
grasshoppers were giving a concert.
"There is some one again who has fallen in love with Giannita," said Don
Ferrante. "That is a woman, Giannita. Any one can see that she is pretty.
If I were young I should fall in love with Giannita. She knows how to
love."
Donna Micaela started. He was right, she thought. The mandolin-player
meant Giannita. That evening Giannita was at home with her mother, but
otherwise she always lived at the summer palace. Donna Micaela had
arranged it so since Don Ferrante had been ill.
But Donna Micaela liked the mandolin playing, for whomever it might be
meant. It came sweet, and soft, and comforting. She went gently into her
room to listen better in the dark and loneliness.
A sweet, strong fragrance met her there. What was it? Her hands began to
tremble before she found a candle and a match. On her work-table lay a
big, widely opened magnolia-blossom.
On one of the flower petals was pricked: "Who loves me?" And now stood
under it: "Gaetano."
Beside the flower lay a little white book full of love-songs. And there
was a mark against one of the little verses:--
"None have known the love that I have brought thee,
Silent, secret, born in midnight's measure.
All my dreams have stolen forth and sought thee;
Miser-like, the while, I watched my treasure:
Tho' the priest shall seek to shrive me, dying,
Silent I, nor needing him to speed me,
Bar the door, fling forth the key, and lying
Thus unshriven, go where death shall lead me."
The mandolin continued to play. There is something of open air and
sunlight in a mandolin; something soothing and calming; something of the
cheering carelessness of beautiful nature.
IX
FLIGHT
At that time the little image from Aracoeli was still in Diamante.
The Englishwoman who owned it had been fascinated by Diamante. She had
not been able to bring herself to leave it.
She had hired the whole first floor of the hotel, and had established
herself there as in a home. She bought for large sums everything she
could find in the way of old pots and old coins. She bought mosaics, and
altar-pictures, and holy images. She thought that she would like to make
a collection of all the saints of the church.
She heard of Gaetano, and sent him a message to come to her at the hotel.
Gaetano collected what he had carved during the last few days and took
them with him to Miss Tottenham. She was much pleased with his little
images, and wished to buy them all.
But the rich Englishwoman's rooms were like the lumber-rooms of a museum.
They were filled with every conceivable thing, and there was confusion
and disorder everywhere. Here stood half-empty trunks; there hung cloaks
and hats; here lay paintings and engravings; there were guide-books,
railway time-tables, tea-sets, and alcohol lamps; elsewhere halberds,
prayer-books, mandolins, and escutcheons.
And that opened Gaetano's eyes. He flushed suddenly, bit his lips, and
began to repack his images.
He had caught sight of an image of the Christchild. It was the outcast,
who was standing there in the midst of all the disorder, with his
wretched crown on his head and brass shoes on his feet. The color was
worn off his face; the rings and ornaments hanging on him were tarnished,
and his dress was yellowed with age.
When Gaetano saw that, he would not sell his images to Miss Tottenham; he
meant simply to go his way.
When she asked him what was the matter with him he stormed at her, and
scolded her.
Did she know that many of the things she had about her were sacred?
Did she know, or did she not know, that that was the holy Christchild
himself? And she had let him lose three fingers on one hand, and let the
jewels fall out of his crown, and let him lie dirty, and tarnished, and
dishonored! And if she had so treated the image of God's own son, how
would she let everything else fare? He would not sell anything to her.
When Gaetano burst out at her in that way Miss Tottenham was enraptured,
enchanted.
Here was the true faith and the righteous, holy wrath. This young man
must become an artist. To England, he should go to England! She wished to
send him to the great master, her friend, who was trying to reform art;
to him who wished to teach people to make beautiful house-furnishings,
beautiful church-fittings, who wished to create a whole beautiful world.
She decided and arranged, and Gaetano let her go on, because he would
rather now go away from Diamante.
He saw that he could no longer endure to live there. He believed that it
was God leading him out of temptation.
He went away quite unobserved. Donna Micaela scarcely knew anything of it
until he was gone. He had not dared to come and bid her good-bye.
X
THE SIROCCO
After that two years passed quietly. The only thing that happened at
Diamante and in all Sicily was that the people grew ever poorer and
poorer.
Then there came an autumn, and it was about the time when the wine was to
be harvested.
At that time songs generally rise full-fledged to the lips; at that time
new and beautiful melodies stream from the mandolins.
Then crowds of young people go out to the vineyards, and there is work
and laughter all day, dance and laughter all night, and no one knows what
sleep is.
Then the bright ocean of air over the mountain is more beautiful than
at any other time. Then the air is full of wit; sparkling glances flash
through it; it gets warmth not only from the sun, but also from the
glowing faces of the young women of Etna.
But that autumn all the vineyards were devastated by the phylloxera. No
grape-pickers pushed their way between the vines; no long lines of women
carrying heaped-up baskets on their heads wound up to the presses, and at
night there was no dancing on the flat roofs.
That autumn no clear, light October air lay over the Etna region. As
if it had been in league with the famine, the heavy, weakening wind
from the Sahara came over from Africa, and brought with it dust and
exhalations that darkened the sky.
Never, as long as that autumn lasted, was there a fresh mountain breeze.
The baleful Sirocco blew incessantly.
Sometimes it came dry and heavy with sand, and so hot that they had to
shut doors and windows, and keep in their rooms, not to faint away.
But oftener it came warm and damp and enervating. And the people felt no
rest; trouble left them neither by day nor by night, and cares piled upon
them like snow-drifts on the high mountains.
And the restlessness reached Donna Micaela as she sat and watched with
her old husband, Don Ferrante.
During that autumn she never heard any one laugh, nor heard a song.
People crept by one another, so full of anger and despair that they were
almost choked. And she said to herself that they were certainly dreaming
of an insurrection. She saw that they had to revolt. It would help no
one, but they had no other resource.
In the beginning of the autumn, sitting on her balcony, she heard the
people talk in the street. They always talked of the famine: We have
blight in wheat and wine; there is a crisis in sulphur and oranges; all
Sicily's yellow gold has failed. How shall we live?
And Donna Micaela understood that it was terrible. Wheat, wine, oranges,
and sulphur, all their yellow gold!
She began to understand, too, that the misery was greater than men could
bear long, and she grieved that life should be made so hard. She asked
why the people should be forced to bear such enormous taxes. Why should
the salt tax exist, so that a poor woman could not go down to the shore
and get a pail of salt water, but must buy costly salt in the government
shops? Why should there be a tax on palm-trees? The peasants, with anger
in their hearts, were felling the old trees that had waved so long over
the noble isle. And why should a tax be put on windows? What did they
want? Was it that the poor should take away their windows, move out of
their rooms, and live in cellars?
In the sulphur-mines there were strikes and turbulence, and the
government was sending troops to force the people back to work. Donna
Micaela wondered if the government did not know that there was no
machinery in those mines. Perhaps it had never heard that children
dragged the ore up from the deep shafts. It did not know that these
children were slaves; it could not imagine that parents had sold them to
overseers. Or if the government did know it, why did it wish to help the
mine-owners?
At one time she heard of a terrible number of crimes. And she began again
with her questions. Why did they let the people become so criminal? And
why did they let them be so poor and so ragged? Why must they all be so
ragged? She knew that any one living in Palermo or Catania did not need
to ask. But he who lived in Diamante could not help fearing and asking.
Why did they let the people be so poor that they died of hunger?
As yet the summer was hardly over; it was no later in the autumn than the
end of October, and already Donna Micaela began to see the day when the
insurrection would break out. She saw the starved people come rushing
along the street. They would plunder the shops and they would plunder
the few rich men there were in the town. Outside the summer palace the
wild horde would stop, and they would climb up to the balcony and the
glass doors. "Bring out the jewels of the old Alagonas; bring out Don
Ferrante's millions!" That was their dream,--the summer palace! They
believed that it was as full of gold as a fairy palace.
But when they found nothing, they would put a dagger to her throat, to
make her give up the treasures that she had never possessed, and she
would be killed by the bloodthirsty crowds.
Why could not the great land-owners stop at home? Why must they irritate
the poor by living in grand style in Rome and Paris? The people would not
be so bitter against them if they stayed at home; they would not swear
such a solemn and sacred oath to kill all the rich when the time should
come.
Donna Micaela wished that she could have escaped to one of the big towns.
But both her father and Don Ferrante fell ill that autumn, and for their
sakes she was forced to remain where she was. And she knew that she would
be killed as an atonement for the sins of the rich against the poor.
For many years misfortunes had been gathering over Sicily, and now they
could no longer be held back. Etna itself began to menace an eruption. At
night sulphurous smoke floated red as fire, and rumblings were heard as
far away as Diamante. The end of everything was coming. Everything was to
be destroyed at once.
Did not the government know of the discontent? Ah, the government had
at last heard of it, and it had appointed a committee. It was a great
comfort to see the members of the committee come driving one fine day
along the Corso in Diamante. If only the people had understood that they
wished them well! If the women had not stood in their doorways and spat
at the fine gentlemen from the mainland; if the children had not run
beside the carriages and cried: "Thief, thief!"
Everything they did only stirred up the revolt, and there was no one who
could control the people and quiet them. They trusted no officials. They
despised those least who only took bribes. But people said that many
belonged to the society of Mafia; they said that their one thought was to
extort money and acquire power.
As time went on, several signs showed that something terrible was
impending. In the papers they wrote that crowds of working-men were
gathering in the larger towns and wandering about the streets. People
read also in the papers how the socialist leaders were going through the
country, and making seditious speeches. All at once it became clear to
Donna Micaela whence all the trouble came. The socialists were inciting
the revolt. It was their firebrand speeches that set the blood of the
people boiling. How could they let them do it? Who was king in Sicily?
Was his name Don Felice, or Umberto?
Donna Micaela felt a horror which she could not shake off. It was as if
they had conspired especially against her. And the more she heard of the
socialists, the more she feared them.
Giannita tried to calm her. "We have not a single socialist in
Diamante," she said. "In Diamante no one is thinking of revolt." Donna
Micaela asked her if she did not know what it meant when the old distaff
spinners sat in their dark corners, and told of the great brigands and
of the famous Palermo fisherman, Giuseppe Alesi, whom they called the
Masaniello of Sicily.
If the socialists could once get the revolt started, Diamante would also
join in. All Diamante knew already that something dreadful was impending.
They had seen the ghost of the big, black monk on the balcony of the
Palazzo Geraci; they heard the owls scream through the night, and some
declared that the cocks crowed at sunset, and were silent at daybreak.
One day in November Diamante was suddenly filled with terrible people.
They were men with the faces of wild beasts, with bushy beards, and
with big hands set on enormously long arms. Several of them wore wide,
fluttering linen garments, and the people thought that they recognized in
them famous bandits and newly freed galley-slaves.
Giannita related that all these wild people lived in the mountain wastes
inland and had crossed Simeto and come to Diamante, because a rumor had
gone about that revolt had already broken out. But when they had found
everything quiet, and the barracks full of soldiers, they had gone away.
Donna Micaela thought incessantly of those people, and expected them to
be her murderers. She saw before her their fluttering linen garments and
their brute faces. She knew that they were lurking in their mountain
holes, and waiting for the day when they should hear shots and the noise
of an outbreak in Diamante. Then they would fall upon the town with
fire and murder, and march at the head of all the starving people as the
generals and leaders in the plundering.
All that autumn Donna Micaela had to nurse both her father and Don
Ferrante; for they lay sick month after month. People had told her,
however, that their lives were in no danger.
She was very glad to be able to keep Don Ferrante alive, for it was her
only hope that at the last the people would spare him, who was of such an
old and venerated race.
As she sat by their sick-beds, her thoughts went often in longing to
Gaetano, and many were the times when she wished that he were at home.
She would not feel such terror and fear of death if he stood once more in
his workshop. Then she would have felt nothing but security and peace.
Even now, when he was so far away, it was to him her thoughts turned when
fear was driving her mad. Not a single letter had come from him since he
had gone away, so that sometimes she believed that he had forgotten her
entirely. At other times she was quite sure that he loved her, for she
felt herself compelled to think of him, and knew that he was near her in
thought, and was calling to her.
That autumn she at last received a letter from Gaetano. Alas, such a
letter! Donna Micaela's first thought was to burn it.
She had gone up to the roof-garden in order to be alone when she read the
letter. She had once heard Gaetano's declaration of love there. That had
not moved her. It had neither warmed her nor frightened her.
But this letter was different. He prayed that she would come to him, be
his, give him her life. When she read it she was frightened at herself.
She felt how she longed to cry out into the air, "I am coming, I am
coming," and set out. It drew her, carried her away.
"Let us be happy!" he wrote. "We are losing time; the years are passing.
Let us be happy!"
He described to her how they would live. He told her of other women who
had obeyed love and been happy. He wrote as temptingly as convincingly.
But it was not the contents; it was the love that glowed and burned
in the letter which overcame her. It rose from the paper like an
intoxicating incense, and she felt it penetrate her. It was burning,
longing, speaking, in every word.
Now she was no longer a saint to him, as she had been before. It came so
unexpectedly, after two years' silence, that she was stunned. And she was
troubled because it delighted her.
She had never thought that love was like this. Should she really like it?
She found with dismay that she did like it.
And so she punished both herself and him by writing a severe reply. It
was moral, moral; it was nothing but moral! She was proud when she had
written it. She did not deny that she loved him, but perhaps Gaetano
would not be able to find the words of love, they were so buried in
admonitions. He could not have found them, for he wrote no more letters.
But now Donna Micaela could no longer think of Gaetano as a shelter and a
support. Now he was more dangerous than the men from the mountains.
Every day graver news came to Diamante. Everybody began to get out their
weapons. And although it was forbidden, they were carried secretly by
every one.
All travellers left the island, and in their place one regiment after
another was sent over from Italy.
The socialists talked and talked. They were possessed by evil spirits;
they could not rest until they had brought on the disaster!
At last the ringleaders had decided on the day on which the storm was to
break loose. All Sicily, all Italy, was to rise. It was no longer menace;
it was reality.
More and more troops came from the mainland. Most of them were
Neapolitans, who live in constant feud with the Sicilians. And now the
news came that the island had been declared in a state of siege. There
were to be no more courts of justice; only court-martials. And the people
said that the soldiers would be free to plunder and murder as they
pleased.
No one knew what was to happen. Terror seemed to make every one mad. The
peasants raised ramparts in the hills. In Diamante men stood in great
groups on the market-place, stood there day after day, without going to
their work. There was something terrible in those groups of men dressed
in dark cloaks and slouch hats. They were all probably dreaming of the
hour when they should plunder the summer palace.
The nearer the day approached when the insurrection was to break out, the
sicker Don Ferrante became; and Donna Micaela began to fear that he would
die.
It seemed to her a sign that she was predestined to destruction, that she
was also losing Don Ferrante. Who would have any regard for her when he
was no longer alive?
She watched over him. She and all the women of the quarter sat in silent
prayer about his bed.
One morning, towards six o'clock, Don Ferrante died. And Donna Micaela
mourned him, because he had been her only protector, and the only one who
could have saved her from destruction; and she wished to honor the dead,
as is still the custom in Diamante.
She had them drape the room where the body was lying with black, and
close all the shutters, so that the glad sunlight should not enter. She
had all the fires put out on the hearths, and sent for a blind singer to
come to the palace every day and sing dirges.
She let Giannita care for Cavaliere Palmeri, so that she herself might
sit quiet in the death-room, among the other women.
It was evening on the day of death before all preparations were
completed, and they were waiting only for the White Brotherhood to come
and take away the corpse. In the death-chamber there was the silence of
the grave. All the women of the quarter sat there motionless with dismal
faces.
Donna Micaela sat pale with her great fear, and stared involuntarily at
the pall that was spread over the body. It was a pall which belonged to
the family; their coat of arms was heavily and gorgeously embroidered on
the centre, and it had silver fringes and thick tassels. The pall had
never been spread over any one but an Alagona. It seemed to lie there so
that Donna Micaela should not for a moment forget that her last support
had fallen, and that she was now alone, and without protection from the
infuriated people.
Some one came in and announced that old Assunta had come. Old Assunta;
what did old Assunta want? Yes, it was she who came to sing the praises
of the dead.
Donna Micaela let Assunta come into the room. She appeared just as she
looked every day, when she sat and begged on the Cathedral steps; the
same patched dress, the same faded headcloth, and the same crutch.
Little and bent, she limped forward to the coffin. She had a shrivelled
face, a sunken mouth, and dull eyes. Donna Micaela said to herself that
it was incarnate helplessness and feebleness who had come into the room.
The old woman raised her voice and began to speak in the wife's name.
"My lord is dead, and I am alone! He who raised me to his side is
dead! Is it not terrible that my home has lost its master?--Why are
the shutters of your windows closed? say the passers-by.--I answer, I
cannot bear to see the light, because my sorrow is so great; my grief is
three-fold.--What, are so many of your race carried away by the White
Brethren?--No, none of my race is dead, but I have lost my husband, my
husband, my husband!"
Old Assunta needed to say no more. Donna Micaela burst into lamentations.
The whole room was filled with the sound of weeping from the sympathetic
women; for there is no grief like losing a husband. Those who were
widows thought of what they had lost, and those who were not as yet
widows thought of the time when they would not be able to go on the
street, because no husband would be with them; when they would be left to
loneliness, poverty, oblivion; when they would be nothing, mean nothing;
when they would be the world's outcast children because they no longer
had a husband; because nothing any longer gave them the right to live.
* * * * *
It was late in December, the days between Christmas and the New Year.
There was still the same danger of insurrection, and people still heard
terrifying rumors. It was said that Falco Falcone had gathered together
a band of brigands in the quarries, and that he was only waiting for the
appointed day to break into Diamante and plunder it.
It was also whispered that the people in several of the small mountain
towns had risen, torn down the custom's offices at the town-gates, and
driven away the officials.
People said too that troops were passing from town to town, arresting all
suspicious people, and shooting them down by hundreds.
Every one said that they must fight. They could not let themselves be
murdered by those Italians without trying to make some resistance.
During all this, Donna Micaela sat tied to her father's sick-bed, just as
she had sat before by Don Ferrante's. She could not escape from Diamante,
and terror so grew within her that she was nothing but one trembling fear.
The last and worst of all the messages of terror that reached her had
been about Gaetano.
For when Don Ferrante had been dead a week Gaetano had come home. And
that had not caused her dismay; it had only made her glad. She had
rejoiced in at last having some one near her who could protect her.
At the same time she decided that she could not receive Gaetano if he
came to see her. She felt that she still belonged to the dead. She would
rather not see Gaetano until after a year.
But when Gaetano had been at home a week without coming to the summer
palace, she asked Giannita about him. "Where is Gaetano? Has he perhaps
gone away again, since no one speaks of him?"
"Alas, Micaela," answered Giannita, "the less people speak of Gaetano,
the better for him."
She told Donna Micaela, as if she was telling of a great shame, that
Gaetano had become a socialist.
"He has been quite transformed over there, in England," she said. "He no
longer worships either God or the saints. He does not kiss the priest's
hand when he meets him. He says to every one that they shall pay no more
duties at the town-gates. He encourages the peasants not to pay their
rent. He carries weapons. He has come home to start a rebellion, to help
the bandits."
She needed to say no more to chill Donna Micaela with a greater terror
than she had ever felt before.
It was this that the sultry days of the autumn had portended. It would be
he who would shake the bolt from the clouds. Why had she not understood
it long ago?
It was a punishment and a revenge. It would be he who would bring the
misfortune!
During those last days she had been calmer. She had heard that all the
socialists on the island had been put in prison, and all the little
insurrection fires lighted in the mountain towns had been quickly choked.
It looked almost as if the rebellion would come to nothing!
But now the last Alagona was come, and him the people would follow. Life
would enter into those black groups on the market-place. The men in the
linen garments would climb up out of the quarries.
* * * * *
The next evening Gaetano spoke in the market-place. He had sat by the
fountain, and had seen how the people came to get water. For two years
he had foregone the pleasure of seeing the slender girls lift the heavy
water-jars to their heads and walk away with firm, slow step.
But it was not only the young girls who came to the fountain; there were
people of all ages. And when he saw how poor and unhappy most of them
were, he began to talk to them of the future.
He promised them better times soon. He said to old Assunta that she
hereafter should get her daily bread without needing to ask alms of any
one. And when she said that she did not understand how that could be,
he asked her almost with anger if she did not know that now the time
had come when no old people and no children should be without care and
shelter.
He pointed to the old chair-maker, who was as poor as Assunta, and
moreover very sick, and he asked if she believed that the people would
endure much longer having no support for the poor, and no hospitals.
Could she not understand that it was impossible for such things to
continue? Could they not all understand that hereafter the old and the
sick should be cared for?
He also saw some children who, as he knew, lived on cresses and sorrel,
which they gathered on the river-banks and by the roadside, and he
promised that henceforward no one should need to starve. He laid his hand
on the children's heads, and swore as solemnly as if he were prince of
Diamante, that they should never again want for bread.
They knew nothing in Diamante, he said; they were ignorant; they did not
understand that a new and blessed time had come; they believed that this
old misery would continue forever.
While he was thus consoling the poor, more and more had gathered about
him, and he suddenly sprang up, placed himself on the steps of the
fountain, and began to speak.
How could they, he said, be so foolish as to believe that nothing
better would come? Should the people, who possessed the whole earth, be
content to let their parents starve, and their children grow up to be
good-for-nothings and criminals?
Did they not know that there were treasures in the mountains, and in the
sea, and in the ground? Had they never heard that the earth was rich? Did
they think that it could not feed its children?
They should not murmur among themselves, and say that it was impossible
to arrange matters differently. They should not think that there must be
rich and poor. Alas, they understood nothing! They did not know their
Mother Earth. Did they think that she hated any of them? They had lain
down on the ground and heard the earth speak? Perhaps they had seen her
make laws? They had heard her pass sentence? She had commanded some to
starve, and some to die of luxury?
Why did they not open their ears and listen to the new teachings pouring
through the world? Would they not like to have a better life? Did they
like their rags? Were they satisfied with sorrel and cresses? Did they
not wish to possess a roof over their heads?
And he told them that it made no difference, no difference, if they
refused to believe in the new times that were coming. They would come in
spite of it. They did not need to lift the sun up from the sea in the
morning. The new times would come to them as the sun came, but why would
they not be ready to meet them? Why did they shut themselves in, and fear
the new light?
He spoke long in the same strain, and more and more of the poor people of
Diamante gathered about him.
The longer he continued, the more beautiful became his speech and the
clearer grew his voice.
His eyes were full of fire, and to the people looking up at him, he
seemed as beautiful as a young prince.
He was one of the race of once powerful lords, who had possessed means
to shower happiness and gold on everybody within their wide lands. They
believed him when he said that he had happiness to give them. They felt
comforted, and rejoiced that their young lord loved them.
When he had finished speaking they began to shout, and call to him that
they wished to follow him and do what he commanded.
He had gained ascendency over them in a moment. He was so beautiful and
so glorious that they could not resist him. And his faith seized and
subdued.
That night there was not one poor person in Diamante who did not believe
that Gaetano would give him happy days, free from care. That night
they called down blessings on him, all those who lived in sheds and
out-houses. That night the hungry lay down with the sure belief that the
next day tables groaning under many dishes would stand spread for them
when they awoke.
For when Gaetano spoke, his power was so great that he could convince
an old man that he was young, and a freezing man that he was warm. And
people felt that what he promised must come.
He was the prince of the coming times. His hands were generous, and
miracles and blessings would stream down over Diamante, now that he had
come again.
* * * * *
The next day, towards sunset, Giannita came into the sick-room and
whispered to Donna Micaela: "There is an insurrection in Paternó. They
have been shooting for several hours, and you can hear them as far away
as here. Orders for troops have already gone to Catania. And Gaetano says
that it will break out here, too. He says that it will break out in all
the towns of Etna at one time."
Donna Micaela made a sign to Giannita to stay with her father, and she
herself went across the street and into Donna Elisa's shop.
Donna Elisa sat behind the counter with her frame, but she was not
working. The tears fell so heavy and fast that she had ceased to
embroider.
"Where is Gaetano?" said Donna Micaela, without any preamble. "I must
speak to him."
"God give you strength to talk to him," answered Donna Elisa. "He is in
the garden."
She went out across the court-yard and into the walled garden.
In the garden there were many narrow paths winding from terrace to
terrace. There was also a number of arbors and grottos and benches. And
it was so thick with stiff agaves, and close-growing dwarf palms, and
thick-leaved rubber-plants, and rhododendrons, that it was impossible to
see two feet in front of one. Donna Micaela walked for a long time on
those innumerable paths before she could find Gaetano. The longer she
walked, the more impatient she became.
At last she found him at the farther end of the garden. She caught sight
of him on the lowest terrace, built out on one of the bastions of the
wall of the town. There sat Gaetano at ease, and worked with chisel and
hammer on a statuette. When he saw Donna Micaela, he came towards her
with outstretched hands.
She hardly gave herself time to greet him. "Is it true," she said, "that
you have come home to be our ruin?" He began to laugh. "The syndic has
been here," he said. "The priest has been here. Are you coming too?"
It wounded her that he laughed, and that he spoke of the priest and the
syndic. It was something different, and more, that she came.
"Tell me," she said, stiffly, "if it is true that we are to have an
uprising this evening."--"Oh, no," he answered; "we shall have no
uprising." And he said it in such a voice that it almost made her sorry
for him.
"You cause Donna Elisa great grief," she burst out.--"And you too, do I
not?" he said, with a slight sneer. "I cause you all sorrow. I am the
lost son; I am Judas. I am the angel of justice who is driving you from
that paradise where people eat grass."
She answered: "Perhaps we think that what we have is better than
being shot by the soldiers."--"Yes, of course; it is better to starve
to death. We are used to that."--"Nor is it pleasant to be murdered
by bandits."--"But why for Heaven's sake have any bandits, if you
do not want to be murdered by them?"--"Yes, I know," she said, more
passionately, "that you want all the rich to perish."
He did not answer immediately; he stood and bit his lips, so as not to
lose his temper. "Let me talk with you, Donna Micaela!" he said at last.
"Let me explain it to you!"
At the same time he put on a patient expression. He talked socialism with
her, so clear and simple that a child could have understood.
But she was far from being able to follow it. Perhaps she could have, but
she did not wish to. She did not wish just then to hear of socialism.
It had been so wonderful to her to see him. The ground had rocked under
her; and something glorious and blessed had passed through and quite
overcome her. "God, it is he whom I love!" she said to herself. "It is
really he."
Before she had seen him she had known very well what she would say to
him. She would have led him back to the faith of his childhood. She
would have shown him that those new teachings were detestable and
dangerous. But then love came. It made her confused and stupid. She could
not answer him. She only sat and wondered that he could talk.
She wondered if he was much handsomer now than formerly. Formerly she had
not been confused at all when she saw him. She had never been attracted
to that extent. Or was it that he had become a free, strong man? She was
frightened when she felt how he subdued her.
She dared not contradict him. She dared not even speak, for fear of
bursting into tears. Had she dared to speak, she would not have talked
of public affairs. She would have told him what she had felt the day the
bells rang. Or she would have prayed to be allowed to kiss his hand. She
would have told him how she had dreamed of him. She would have said that
if she had not had him to dream of she could not have borne her life. She
would have begged to be allowed to kiss his hand in gratitude, because he
had given her life all these years.
If there was to be no uprising, why did he talk socialism? What had
socialism to do with them, sitting alone in Donna Elisa's garden? She sat
and looked along one of the paths. Luca had put up wooden arches on both
sides of it, and up these climbed garlands of light rose-shoots, full of
little buds and flowers. One always wondered whither one was coming when
one went along that path. And one came to a little weather-beaten cupid.
Old Luca understood things better than Gaetano.
While they sat there the sun set, and Etna grew rosy-red. It was as if
Etna flushed with anger at what was going on in Donna Elisa's garden.
It was at sunset, when Etna glowed red, that she had always thought of
Gaetano. It seemed as if they both had been waiting for it. And they had
both arranged how it would be when Gaetano came. She had only feared that
he would be too fiery, and too passionately wild. And he talked only of
those dreadful Socialists, whom she detested and feared.
He talked a long time. She saw Etna grow pale and become bronze-brown,
and then the darkness came. She knew that there would be moonlight. There
she sat quite still, and hoped for help from the moonlight. She herself
could do nothing. She was entirely in his power. But when the moonlight
came, it did not help either. He continued to talk of capitalists and
working-men.
Then it seemed to her as if there could be but one explanation for all
this. He must have ceased to love her.
Suddenly she remembered something. It was a week ago. It was the same
day that Gaetano had come home. She had come into Giannita's room, but
she had walked so softly that Giannita had not heard her. She had seen
Giannita stand as if in ecstasy, with up-stretched arms and up-turned
face. And in her hands she held a picture. First she carried it to her
lips and kissed it, then she lifted it up over her head and looked up to
it in rapture. And the picture had been of Gaetano.
When Donna Micaela had seen that, she had gone away as silently as she
had come. She had only thought then that Giannita was to be pitied if
she loved Gaetano. But now, when Gaetano only talked socialism, now she
remembered it.
Now she began to think that Gaetano also loved Giannita. She remembered
that they were friends from childhood. He had perhaps loved her a long
time. Perhaps he had come home to marry her. Donna Micaela could say
nothing; she had nothing to complain of. It was scarcely a month since
she wrote to Gaetano that it was not right of him to love her.
He now leaned towards her, enchained her glance, and actually compelled
her to listen to what he was saying.
"You shall understand; you shall see and understand, Donna Micaela! What
we need here in the South is a regeneration, a pulling up by the roots,
such as Christianity was in its time. Up with the slaves; down with the
masters! A plow which turns up new social furrows! We must sow in new
earth; the old earth is impoverished. The old surface furrows bear only
weak, miserable growth. Let the deep earth come up to the light, and we
shall see something different!
"See, Donna Micaela, why does socialism live; why has it not gone under?
Because it comes with a new word. 'Think of the earth,' it says, just as
Christianity came with the word, 'Think of heaven.' Look about you! Look
at the earth; is it not all that we possess? Let us therefore establish
ourselves here so that we shall be happy. Why, why, has no one thought of
it before? Because we have been so busy with that Hereafter. Let us leave
the Hereafter! The earth, the earth, Donna Micaela! Ah, we socialists, we
love her! We worship the sacred earth,--the poor, despised mother, who
wears mourning because her children yearn for heaven.
"Believe me, Donna Micaela," he said, "it will be accomplished in less
than seven years. In the year nineteen hundred it will be ready. Then
martyrs will have bled; then apostles will have spoken; then shall crowds
upon crowds have been won over! We, the rightful sons of the earth, shall
have the victory! And she shall lie before us in all her loveliness; she
shall bring us beauty, bring us pleasure, bring us knowledge, bring us
health!"
Gaetano's voice began to tremble, and tears quivered in his eyes. He went
forward to the edge of the terrace, and he stretched out his arms as if
to embrace the moonlit earth. "You are so dazzlingly beautiful," he said,
"so dazzlingly beautiful!"
And Donna Micaela for a moment thought she felt his grief over all
the sorrow that lay under the surface of beauty. She saw life full
of vice and suffering, like a dirty river filled with the stench of
uncleanliness, wind through the glistening world of beauty.
"And no one can enjoy you," said Gaetano; "no one can dare to enjoy you.
You are untamed, and full of whims and anger. You are uncertainty and
peril; you are sorrow and pain; you are want and shame; you are the force
that grinds; you are everything terrible that can be named, because the
people have not wished to make you better.
"But your day will come," he said, triumphantly. "Some day they will turn
to you with all their love; they will not turn to a dream, which gives
nothing and is good for nothing."
She interrupted him roughly. She began to fear him more and more.
"So it is true that you have had no success in England?"
"What do you mean?"
"People say that the great master, to whom Miss Tottenham sent you, has
said that you--"
"What has he said?"
"That you and your images suited Diamante, but nowhere else."
"Who says such things?"
"People think so, because you are so changed."
"Since I am a socialist."
"Why should you be one if you had been successful?"
"Ah, why--? You do not know," he continued, with a laugh, "that my master
in England himself was a socialist. You do not know that it was he who
taught me these opinions--"
He paused, and did not go on with the controversy. He went over to
the bench where he had been sitting when she came, and brought back a
statuette. He handed it to Donna Micaela. He seemed to wish to say: "See
for yourself if you are right."
She took it, and held it up in the moonlight. It was a Mater Dolorosa in
black marble. She could see it quite plainly.
She could also recognize it. The image had her own features. It
intoxicated her for a moment. In the next she was filled with horror. He,
a socialist; he, an unbeliever; he dared to create a Madonna! And he had
given the image her features! He entangled her in his sin!
"I have done it for you, Donna Micaela," he said.
Ah, since it was hers! She threw it out over the balustrade. It struck
against the steep mountain side; fell deeper and deeper; broke loose
stones, and certainly shattered itself to pieces. At last a splash was
heard down in Simeto.
"What right have you to carve Madonnas?" she asked Gaetano.
He stood silent. He had never seen Donna Micaela thus.
In the moment when she rose up before him she had become tall and
stately. The beauty that always came and went in her, like an uneasy
guest, was enthroned in her face. She looked cold and inflexible; a woman
to win and conquer.
"Then you still believe in God, since you carve Madonnas?" she said.
He breathed hurriedly. Now it was he who was paralyzed. He had been a
believer himself. He knew how he had wounded her. He saw that he had
forfeited her love. He had made a terrible, infinite chasm between them.
He must speak, must win her over to his side.
He began again, but feebly and falteringly.
She listened quietly for a while. Then she interrupted him almost
compassionately.
"How did you become so?"
"I thought of Sicily," he said submissively.
"You thought of Sicily," she repeated thoughtfully. "And why did you come
home?"
"I came home to cause an insurrection."
It was as if they had spoken of an illness, a chill, that he had
contracted, and that could quite easily be cured.
"You came home to be our ruin," she said, sternly.
"As you will; as you will," he said, complying. "You can call it so. As
everything is going now, you are certainly right to call it so. Ah, if
they had not given me false information; if I had not come a week too
late! Is it not like us Sicilians to let the government anticipate us?
When I came the leaders were already arrested, the island garrisoned with
forty thousand men. Everything lost!"
It sounded strangely blank when he said that "everything lost." And for
that which never could be anything, he had lost happiness. His opinions
and principles seemed to him now to be dry cobwebs, which had captured
him. He wished to tear himself away to come to her. She was the only
reality, the only thing that was his. So he had felt before. It came back
now. She was the only thing in the world.
"They are, however, fighting to-day in Paternó."
"There has been a disagreement by the town-gate," he said. "It is
nothing. If I had been able to inflame all Etna, the whole circle of
towns round about Etna! Then they would have understood us! they would
have listened to us! Now they are shooting down a few hungry peasants to
make a few hungry mouths the less. They do not yield an inch to us."
He strove to break through his cobwebs. Could he venture to go up to her,
to tell her that all that was of no importance? He did not need to think
of politics. He was an artist; he was free! And he wanted to possess her!
Suddenly it seemed as if the air trembled. A shot echoed through the
night, then another and another.
She came forward to him and grasped his wrist. "Is that the uprising?"
she asked.
Shot upon shot came thundering. Then were heard the cries and din of a
crowd rushing down the street.
"It is the uprising; it must be the uprising! Ah, long live socialism!"
He was filled with joy. Entire faith in his belief came back to him. He
would win her too. Women have never refused to belong to the victor.
They both hurried without another word through the garden to the door.
There Gaetano began to swear and call. He could not get out. There was no
key in the lock. He was shut into the garden.
He looked about. There were high walls on three sides, and on the fourth
an abyss. There was no way out for him. But from the town came a terrible
noise. The people were rushing up and down; there were shots and cries.
And they heard them yell: "Long live freedom! Long live socialism!" He
threw himself against the door, and almost shrieked. He was imprisoned;
he could not take part.
Donna Micaela came up to him as quickly as she could. Now, since she had
heard him, she no longer thought of keeping him back.
"Wait, wait!" she said. "I took the key."
"You, you!" he said.
"I took it when I came. It occurred to me that I could keep you shut in
here if you should want to cause an uprising. I wished to save you."
"What folly!" he said, and snatched the key from her.
While he stood and fumbled to find the key-hole, he still had time to say
something.
"Why do you not want to save me now?"
She did not answer.
"Perhaps so that your God may have a chance to destroy me."
She was still silent.
"Do you not dare to save me from His wrath?"
"No, I do not dare," she said quietly.
"You believers are terrible!" he said.
He felt that she threw him aside. It froze him, and took away his
courage, that she did not make a single attempt to persuade him to stay.
He turned the key forward and back without being able to open the door,
paralyzed by her standing there pale and cold behind him.
Then he suddenly felt her arms about his neck and her lips seeking his.
At the same moment the door flew open and he rushed away. He would not
have her kisses, which only consecrated him to death. She was as terrible
as a spectre to him with her ancient faith. He rushed away like a
fugitive.
XI
THE FEAST OF SAN SEBASTIANO
When Gaetano rushed away, Donna Micaela stood for a long time in Donna
Elisa's garden. She stood there as if turned to stone, and could neither
feel nor think.
Then suddenly the thought came that Gaetano and she were not alone in the
world. She remembered her father lying sick, whom she had forgotten for
so many hours.
She went through the gate of the court-yard out to the Corso, which lay
deserted and empty. Tumult and shots were still audible far away, and she
said to herself that they must be fighting down by Porta Etnea.
The moon shed its clear light on the façade of the summer-palace, and it
amazed her that at such an hour, and on such a night, the balcony doors
stood open, and the window shutters were not closed. She was still more
surprised that the gate was standing ajar, and that the shop-door was
wide open.
As she went in through the gate, she did not see the old gate-keeper,
Piero, there. The lanterns in the court-yard were not lighted, and there
was not a soul to be seen anywhere.
She went up the steps to the gallery, and her foot struck against
something hard. It was a little bronze vase, which belonged in
the music-room. A few steps higher up she found a knife. It was a
sheath-knife, with a long, dagger-like blade. When she lifted it up a
couple of dark drops rolled down from its edge. She knew that it must be
blood.
And she understood too that what she had feared all the autumn had now
happened. Bandits had been in the summer-palace for plunder. And everyone
who could run away had run away; but her father, who could not leave his
bed, must be murdered.
She could not tell whether the brigands were not still in the house. But
now, in the midst of danger, her fears vanished; and she hurried on,
unheeding that she was alone and defenceless.
She went along the gallery into the music-room. Broad rays of moonlight
fell upon the floor, and in one of those rays lay a human form stretched
motionless.
Donna Micaela bent down over that motionless body. It was Giannita. She
was murdered; she had a deep, gaping wound in her neck.
Donna Micaela laid the body straight, crossed the hands over the breast,
and closed the eyes. In so doing, her hands were wet with the blood; and
when she felt that warm, sticky blood, she began to weep. "Alas, my dear,
beloved sister," she said aloud, "it is your young life that has ebbed
away with this blood. All your life you have loved me, and now you have
shed your blood defending my house. Is it to punish my hardness that God
has taken you from me? Is it because I did not allow you to love him whom
I loved that you have gone from me? Alas, sister, sister, could you not
have punished me less severely?"
She bent down and kissed the dead girl's forehead. "You do not believe
it," she said. "You know that I have always been faithful to you. You
know that I have loved you."
She remembered that the dead was severed from everything earthly, that it
was not grief and assurances of friendship she needed. She said a prayer
over the body, since the only thing she could do for her sister was to
support with pious thoughts the flight of the soul soaring up to God.
Then she went on, no longer afraid of anything that could happen to
herself, but in inexpressible terror of what might have happened to her
father.
When she had at last passed through the long halls in the state apartment
and stood by the door to the sick-room, her hands groped a long time for
the latch; and when she had found it, she had not the strength to turn
the key.
Then her father called from his room and asked who was there. When she
heard his voice and knew that he was alive, everything in her trembled,
and burst, and lost its power to serve her. Brain and heart failed her
at once, and her muscles could no longer hold her upright. She had still
time to think that she had been living in terrible suspense. And with a
feeling of relief, she sank down in a long swoon.
Donna Micaela regained consciousness towards morning. In the meantime
much had happened. The servants had come out of their hiding-places, and
had gone for Donna Elisa. She had taken charge of the deserted palace,
had summoned the police, and sent a message to the White Brotherhood.
And the latter had carried Giannita's body to her mother's house.
When Donna Micaela awoke, she found herself lying on the sofa in a room
next her father's. No one was with her, but in her father's room she
heard Donna Elisa talking.
"My son and my daughter," said Donna Elisa, sobbing; "I have lost both my
son and my daughter."
Donna Micaela tried to raise herself, but she could not. Her body still
lay in a stupor, although her soul was awake.
"Cavaliere, Cavaliere," said Donna Elisa, "can you understand? The
bandits come here from Etna, creeping down to Diamante. The bandits
attack the custom-house and shout: 'Long live Socialism!' They do it only
to frighten people away from the streets and to draw the Carabiniere down
to Porta Etnea. There is not a single man from Diamante who has anything
to do with it. It is the bandits who arrange it all, to be able to
plunder Miss Tottenham and Donna Micaela, two women, Cavaliere! What did
those officers think at the court-martial? Did they believe that Gaetano
was in league with the bandits? Did they not see that he was a nobleman,
a true Alagona, an artist? How could they have sentenced him?"
Donna Micaela listened with horror, but she tried to imagine that she was
still dreaming. She thought she heard Gaetano ask if she was sacrificing
him to God. She thought she answered that she did. Now she was dreaming
of how it would be in case he really had been captured. It could be
nothing else.
"What a night of misfortune!" said Donna Elisa. "What is flying about
in the air, and making people mad and confused? You have seen Gaetano,
Cavaliere. He has always been passionate and fiery, but it has not been
without intelligence; he has not been without sense and judgment. But
to-night he throws himself right into the arms of the troops. You know
that he wanted to cause an uprising; you know that he came home for
that. And when he hears the shooting, and some one shouting, 'Long live
Socialism!' he becomes wild, and beside himself. He says to himself,
'That is the insurrection!' and he rushes down the street to join it.
And he shouts the whole time, 'Long live Socialism!' as loud as he can.
And so he meets a great crowd of soldiers, a whole host. For they were
on their way to Paternó, and heard the shooting as they passed by, and
marched in to see what was going on. And Gaetano can no longer recognize
a soldier's cap. He thinks that they are the rebels; he thinks that they
are angels from heaven, and he rushes in among them and lets them capture
him. And they, who have already caught all the bandits sneaking away with
their booty, now lay hands on Gaetano too. They go through the town and
find everything quiet; but before they leave, they pass sentence on their
prisoners. And they condemn Gaetano like the others, condemn him like
those who have broken in and murdered women. Have they not lost their
senses, Cavaliere?"
Donna Micaela could not hear what her father answered. She wished to ask
a thousand questions, but she was still paralyzed and could not move. She
wondered if Gaetano had been shot.
"What do they mean by sentencing him to twenty-nine years'
imprisonment?" said Donna Elisa. "Do you think that he can live so long,
or that any one who loves him can live so long? He is dead, Cavaliere; as
dead for me as Giannita."
Donna Micaela felt as if strong fetters bound her beyond escape. It was
worse, she thought, than to be tied to a pillory and whipped.
"All the joy of my old age is taken from me," said Donna Elisa. "Both
Giannita and Gaetano! I have always expected them to marry each other.
It would have been so suitable, because they were both my children, and
loved me. For what shall I live now, when I have no young people about
me? I was often poor when Gaetano first came to me, and people said to me
that I should have been better off alone. But I answered: 'It makes no
difference, none, if only I have young people about me.' And I thought
that when he grew up he would find a young wife, and then they would have
little children, and I would never need to sit a lonely and useless old
woman."
Donna Micaela lay thinking that she could have saved Gaetano, but had
not wished to do so. But why had she not wished? It seemed to her quite
incomprehensible. She began to count up to herself all her reasons for
permitting him to rush to destruction. He was an atheist; a socialist; he
wished to cause a revolt. That had outweighed everything else when she
opened the garden gate for him. It had crushed her love also. She could
not now understand it. It was as if a scale full of feathers had weighed
down a scale full of gold.
"My beautiful boy!" said Donna Elisa, "my beautiful boy! He was already
a great man over there in England, and he came home to help us poor
Sicilians. And now they have sentenced him like a bandit. People say that
they were ready to shoot him, as they shot the others. Perhaps it would
have been better if they had done so, Cavaliere. It had been better to
have laid him in the church-yard than to know that he was in prison. How
will he be able to endure all his suffering? He will not be able to bear
it; he will fall ill; he will soon be dead."
At these words, Donna Micaela roused herself from her stupor, and got up
from the sofa. She staggered across the room and came in to her father
and Donna Elisa, as pale as poor murdered Giannita. She was so weak that
she did not dare to cross the floor; she stood at the door and leaned
against the door-post.
"It is I," she said; "Donna Elisa, it is I--"
The words would not come to her lips. She wrung her hands in despair that
she could not speak.
Donna Elisa was instantly at her side. She put her arm about her to
support her, without paying any attention to Donna Micaela's attempt to
push her away.
"You must forgive me, Donna Elisa," she said, with an almost inaudible
voice. "I did it."
Donna Elisa did not heed much what she was saying. She saw that she had
fever, and thought that she was delirious.
Donna Micaela's lips worked; she plainly wished to say something, but
only a few words were audible. It was impossible to understand what she
meant. "Against him, as against my father," she said, over and over. And
then she said something about bringing misfortune on all who loved her.
Donna Elisa had got her down on a chair, and Donna Micaela sat there and
kissed her old, wrinkled hands, and asked her to forgive her what she had
done.
Yes, of course, of course, Donna Elisa forgave her.
Donna Micaela looked her sharply in the face with great, feverish eyes,
and asked if it were true.
It was really true.
Then she laid her head on Donna Elisa's shoulder and sobbed, thanked her,
and said that she could not live if she did not obtain her forgiveness.
She had sinned against no one so much as against her. Could she forgive
her?
"Yes, yes," said Donna Elisa again and again, and thought that the other
was out of her head from fever and fright.
"There is something I ought to tell you," said Donna Micaela. "I know it,
but you do not know it. You will not forgive me if you hear it."
"Yes, of course I forgive you," said Donna Elisa.
They talked in that way for a long time without understanding each other;
but it was good for old Donna Elisa to have some one that night to put to
bed, comforted and dosed with strengthening herbs and drops. It was good
for her to still have some one to come and lay her head on her shoulder
and cry away her grief.
* * * * *
Donna Micaela, who had loved Gaetano for nearly three years without a
thought that they could ever belong to each other, had accustomed herself
to a strange kind of love. It was enough for her to know that Gaetano
loved her. When she thought of it, a tender feeling of security and
happiness stole through her. "What does it matter; what does it matter?"
she said, when she suffered adversity. "Gaetano loves me." He was always
with her, cheering and comforting her. He took part in all her thoughts
and undertakings. He was the soul of her life.
As soon as Donna Micaela could get his address, she wrote to him. She
acknowledged to him that she had firmly believed that he had gone to
misfortune. But she had been so much afraid of what he proposed to
accomplish in the world that she had not dared to save him.
She also wrote how she detested his teachings. She did not dissemble at
all to him. She said that even if he were free she could not be his.
She feared him. He had such power over her that, if they were united, he
would make her a socialist and an atheist. Therefore she must always live
apart from him, for the salvation of her soul.
But she begged and prayed that in spite of everything he would not cease
to love her. He must not; he must not! He might punish her in any way he
pleased, if only he did not cease to love her.
He must not do as her father had. He had perhaps reason to close his
heart to her now, but he must not. He must be merciful.
If he knew how she loved him! If he knew how she dreamed of him!
She told him that he was nothing less than life itself to her.
"Must I die, Gaetano?" she asked.
"Is it not enough that those opinions and teachings part us? Is it not
enough that they have carried you to prison? Will you also cease to love
me, because we do not think alike?
"Ah, Gaetano, love me! It leads to nothing; there is no hope in your
love, but love me; I die if you do not love me."
Donna Micaela had hardly sent off the letter before she began to wait for
the answer. She expected a stormy and angry reply, but she hoped that
there would be one single word to show her that he still loved her.
But she waited several weeks without receiving any letter from Gaetano.
It did not help her to stand and wait every morning for the
letter-carrier out on the gallery, and almost break his heart because he
was always obliged to say that he did not have anything for her.
One day she went herself to the post-office, and asked them, with the
most beseeching eyes, to give her the letter she was expecting. It
must be there, she said. But perhaps they had not been able to read
the address; perhaps it had been put into the wrong box? And her soft,
imploring eyes so touched the postmaster that she was allowed to look
through piles of old, unclaimed letters, and to turn all the drawers in
the post-office upside down. But it was all in vain.
She wrote new letters to Gaetano; but no answer came.
Then she tried to believe what seemed impossible. She tried to make her
soul realize that Gaetano had ceased to love her.
As her conviction increased, she began to shut herself into her room. She
was afraid of people, and preferred to sit alone.
Day by day she became more feeble. She walked deeply bent, and even her
beautiful eyes seemed to lose their life and light.
After a few weeks she was so weak that she could no longer keep up, but
lay all day on her sofa. She was prey to a suffering that gradually
deprived her of all vital power. She knew that she was failing, and she
was afraid to die. But she could do nothing. There was only one remedy
for her, but that never came. While Donna Micaela seemed to be thus
quietly gliding out of life, the people of Diamante were preparing to
celebrate the feast of San Sebastiano, that comes at the end of January.
It was the greatest festival of Diamante, but in the last few years it
had not been kept with customary splendor, because want and gloom had
weighed too heavily on their souls.
But this year, just after the revolt had failed, and while Sicily was
still filled with troops, and while the beloved heroes of the people
languished in prison, they determined to celebrate the festival with all
the old-time pomp; for now, they said, was not the time to neglect the
saint.
And the pious people of Diamante determined that the festival should be
held for a week, and that San Sebastiano should be honored with flags and
decorations, and with races and biblical processions, illuminations, and
singing contests.
The people bestirred themselves with great haste and eagerness. There
was polishing and scrubbing in every house. They brought out the old
costumes, and they prepared to receive strangers from all Etna.
The summer-palace was the only house in Diamante where no preparations
were made. Donna Elisa was deeply grieved at it, but she could not induce
Donna Micaela to have her house decorated. "How can you ask me to trim a
house of mourning with flowers and leaves?" she said. "The roses would
shed their petals if I tried to use them to mask the misery that reigns
here."
But Donna Elisa was very eager for the festival, and expected much good
to result from honoring the saint as in the old days. She could talk of
nothing but of how the priests had decorated the façade of the Cathedral
in the old Sicilian way, with silver flowers and mirrors. And she
described the procession: how many riders there were to be, and what high
plumes they were to have in their hats, and what long, garlanded staves,
with wax candles at the end, they were to carry in their hands.
When the first festival day came, Donna Elisa's house was the most
gorgeously decorated. The green, red, and white standard of Italy waved
from the roof, and red cloths, fringed with gold, bearing the saint's
initials, were spread over the window-sills and balcony railings. Up
and down the wall ran garlands of holly, shaped into stars and arches,
and round the windows crept wreaths made of the little pink roses from
Donna Elisa's garden. Just over the entrance stood the saint's image,
framed in lilies, and on the threshold lay cypress-branches. And if one
had entered the house, one would have found it as much adorned on the
inside as on the outside. From the cellar to the attic it was scoured and
covered with flowers, and on the shelves in the shop no saint was too
small or insignificant to have an everlasting or a harebell in his hand.
Like Donna Elisa, every one in penniless Diamante had decorated along
the whole street. In the street above the house of the little Moor there
was such an array of flags that it looked like clothes hung out to dry
from the earth to the sky. Every house and every arch carried flags, and
across the streets were hung ropes, from which fluttered pennant after
pennant.
At every tenth step the people of Diamante had raised triumphal arches
over the street. And over every door stood the image of the saint, framed
in wreaths of yellow everlastings. The balconies were covered with red
quilts and bright- table-cloths, and stiff garlands wound up the
walls.
There were so many flowers and leaves that no one could understand how
they had been able to get them all in January. Everything was crowned
and wreathed with flowers. The brooms had crowns of crocuses, and each
door-knocker a bunch of hyacinths. In windows stood pictures with
monograms, and inscriptions of blood-red anemones.
And between those decorated houses the stream of people rolled as mighty
as a rising river. It was not the inhabitants of Diamante alone who were
honoring San Sebastiano. From all Etna came yellow carts, beautifully
ornamented and painted, drawn by horses in shining harness, and loaded
down with people. The sick, the beggars, the blind singers came in great
crowds. There were whole trains of pilgrims, unhappy people, who now,
after their misfortunes, had some one to pray to.
Such numbers came that the people wondered how they all would ever find
room within the town walls. There were people in the streets, people
in the windows, people on the balconies. On the high stone steps sat
people, and the shops were full of them. The big street-doors were thrown
wide, and in the openings chairs were arranged in a half-circle, as in a
theatre. There the house-owners sat with their guests and looked at the
passers-by.
The whole street was filled with an intoxicating noise. It was
not only the talking and laughter of the people. There were also
organ-grinders standing and turning hand-organs big as pianos. There
were street-singers, and there were men and women who declaimed Tasso in
cracked, worn-out voices. There were all kinds of criers, the sound of
organs streamed from all the churches, and in the square on the summit
of the mountain the town band played so that it could be heard over all
Diamante.
The joyous noise, and the fragrance of the flowers, and the flapping of
the flags outside Donna Micaela's window had power to wake her from her
stupor. She rose up, as if life had sent for her. "I will not die," she
said to herself. "I will try to live."
She took her father's arm and went out into the street. She hoped
that the life there would mount to her head so that she might forget
her sorrow. "If I do not succeed," she thought, "if I can find no
distraction, I must die."
Now in Diamante there was a poor old stone-cutter, who had thought of
earning a few soldi during the festival. He had made a couple of small
busts out of lava, of San Sebastiano and of Pope Leo XIII. And as he knew
that many in Diamante loved Gaetano, and grieved over his fate, he also
made a few portraits of him.
Just as Donna Micaela came out into the street she met the man, and he
offered her his wretched little images.
"Buy Don Gaetano Alagona, Donna Micaela," said the man; "buy Don Gaetano,
whom the government has put in prison because he wished to help Sicily."
Donna Micaela pressed her father's arm hard and went hurriedly on.
In the Café Europa the son of the innkeeper stood and sang canzoni. He
had composed a few new ones for the festival, and among others some about
Gaetano. For he could not know that people did not care to hear of him.
When Donna Micaela passed by the café and heard the singing, she stopped
and listened.
"Alas, Gaetano Alagona!" sang the young man. "Songs are mighty. I shall
sing you free with my songs. First I will send you the slender canzone.
He shall glide in between your prison-gratings, and break them. Then I
will send you the sonnet, that is fair as a woman, and which will corrupt
your guards. I will compose a glorious ode to you, which will shake the
walls of your prison with its lofty rhythms. But if none of these help
you, I will burst out in the glorious epos, that has hosts of words. Oh,
Gaetano, mighty as an army it marches on! All the legions of ancient Rome
would not have had the strength to stop it!"
Donna Micaela hung convulsively on her father's arm, but she did not
speak, and went on.
Then Cavaliere Palmeri began to speak of Gaetano. "I did not know that he
was so beloved," he said.
"Nor I," murmured Donna Micaela.
"To-day I saw some strangers coming into Donna Elisa's shop, and begging
her to be allowed to buy something that he had carved. She had left only
a couple of old rosaries, and I saw her break them to pieces and give
them out bead by bead."
Donna Micaela looked at her father like a beseeching child. But he did
not know whether she wished him to be silent or to go on speaking.
"Donna Elisa's old friends go about in the garden with Luca," he said,
"and Luca shows them Gaetano's favorite places and the garden beds
that he used to plant. And Pacifica sits in the workshop beside the
joiner's-bench, and relates all sorts of things about him, ever since he
was--so big."
He could tell no more; the crush and the noise became so great about him
that he had to stop.
They meant to go to the Cathedral. On the Cathedral steps sat old
Assunta, as usual. She held a rosary in her hands and mumbled the same
prayer round the whole rosary. She asked the saint that Gaetano, who had
promised to help all the poor, might come back to Diamante.
As Donna Micaela walked by her, she distinctly heard: "San Sebastiano,
give us Gaetano! Ah, in your mercy; ah, in our misery, San Sebastiano,
give us Gaetano!"
Donna Micaela had meant to go into the church, but she turned on the
steps.
"There is such a crowd there," she said, "I do not dare to go in."
She went home again. But while she had been away, Donna Elisa had watched
her opportunity. She had hoisted a flag on the roof of the summer-palace;
she had spread draperies on the balconies, and as Donna Micaela came
home, she was fastening up a garland in the gateway. For Donna Elisa
could not bear to have the summer-palace underrated. She wished no honor
to San Sebastiano omitted at this time. And she feared that the saint
would not help Diamante and Gaetano if the palace of the old Alagonas did
not honor him.
Donna Micaela was pale as if she had received her death warrant, and bent
like an old woman of eighty years.
She murmured to herself: "I make no busts of him; I sing no songs about
him; I dare not pray to God for him; I buy none of his beads. How can he
believe that I love him? He must love all these others, who worship him,
but not me. I do not belong to his world, he can love me no longer."
And when she saw that they wished to adorn her house with flowers, it
seemed to her so piteously cruel that she snatched the wreath from Donna
Elisa and threw it at her feet, asking if she wished to kill her.
Then she went past her up the stairs to her room. She threw herself on
the sofa and buried her face in the cushions.
She now first understood how far apart she and Gaetano were. The idol of
the people could not love her.
She felt as if she had prevented him from helping all those poor people.
How he must detest her; how he must hate her!
Then her illness came creeping back over her. That illness which
consisted of not being loved! It would kill her. She thought, as she lay
there, that it was all over.
While she lay there, suddenly the little Christchild stood before her
inward eye. He seemed to have entered the room in all his wretched
splendor. She saw him plainly.
Donna Micaela began to call on the Christchild for help. And she was
amazed at herself for not having turned before to that good helper. It
was probably because the image did not stand in a church, but was carried
about as a museum-piece by Miss Tottenham, that she remembered him only
in her deepest need.
* * * * *
It was late in the evening of the same day. After dinner Donna Micaela
had given all her servants permission to go to the festival, so that she
and her father were alone in the big house. But towards ten o'clock her
father rose and said he wished to hear the singing-contest in the square.
And as Donna Micaela did not dare to sit alone, she was obliged to go
with him.
When they came to the square they saw that it was turned into a theatre,
with lines upon lines of chairs. Every corner was filled with people, and
it was with difficulty that they found places.
"Diamante is glorious this evening, Micaela," said Cavaliere Palmeri. The
charm of the night seemed to have softened him. He spoke more simply and
tenderly to his daughter than he had done for a long time.
Donna Micaela felt instantly that he spoke the truth. She felt as she had
done when she first came to Diamante. It was a town of miracles, a town
of beauty, a little sanctuary of God.
Directly in front of her stood a high and stately building made of
shining diamonds. She had to think for a moment before she could
understand what it was.
Yet it was nothing but the front of the Cathedral, covered with flowers
of stiff silver and gold paper and with thousands of little mirrors stuck
in between the flowers. And in every flower was hung a little lamp with a
flame as big as a fire-fly. It was the most enchanting illumination that
Donna Micaela had ever seen.
There was no other light in the market-place, nor was any needed. That
great wall of diamonds shone quite sufficiently. The black Palazzo Geraci
was flaming red, as if it had been lighted by a conflagration.
Nothing of the world outside of the square was visible. Everything below
it was in the deepest darkness, and that made her think again that she
saw the old enchanted Diamante that was not of the earth, but was a
holy city on one of the mounts of heaven. The town-hall with its heavy
balconies and high steps, the long convent and the Roman gate were again
glorious and wonderful. And she could hardly believe it was in that town
that she had suffered such terrible pain.
In the midst of the great crowd of people, no chill was felt. The winter
night was mild as a spring morning; and Donna Micaela began to feel
something of spring in her. It began to stir and tremble in her in a way
which was both sweet and terrible. It must feel so in the snow-masses on
Etna when the sun melts them into sparkling brooks.
She looked at the people who filled the market-place, and was amazed at
herself that she had been so tortured by them in the forenoon. She was
glad that they loved Gaetano. Alas, if he had only continued to love her,
she would have been unspeakably proud and happy in their love. Then she
could have kissed those old callous hands that made images of him and
were clasped in prayers for him.
As she was thinking this, the church-door was thrown open and a big, flat
wagon rolled out of the church. Highest on the red-covered wagon stood
San Sebastiano by his stake, and below the image sat the four singers,
who were to contest.
There was an old blind man from Nicolosi; a cooper from Catania, who
was considered to be the best improvisatore in all Sicily; a smith
from Termini, and little Gandolfo, who was son to the watchman in the
town-hall of Diamante.
Everybody was surprised that Gandolfo dared to appear in such a difficult
contest. Did he do it perhaps to please his betrothed, little Rosalia? No
one had ever heard that he could improvise. He had never done anything in
his whole life but eat mandarins and stare at Etna.
The first thing was to draw lots among the competitors, and the lots fell
so that the cooper should come first and Gandolfo last. When it fell so
Gandolfo turned pale. It was terrible to come last, when they all were to
speak on the same subject.
The cooper elected to speak of San Sebastiano, when he was a soldier of
the legion in ancient Rome, and for his faith's sake was bound to a stake
and used as a target for his comrades. After him came the blind man, who
told how a pious Roman matron found the martyr bleeding and pierced with
arrows, and succeeded in bringing him back to life. Then came the smith,
who related all the miracles San Sebastiano had worked in Sicily during
the pest in the fifteenth century. They were all much applauded. They
spoke many strong words of blood and death, and the people rejoiced in
them. But every one from Diamante was anxious for little Gandolfo.
"The smith takes all the words from him. He must fail," they said.
"Ah," said others, "little Rosalia will not take the engagement ribbon
out of her hair for that."
Gandolfo shrunk together in his corner of the wagon. He grew smaller
and smaller. Those sitting near could hear how his teeth chattered with
fright.
When his turn came at last, and he rose and began to improvise, he was
very bad. He was worse than any one had expected. He faltered out a
couple of verses, but they were only a repetition of what the others had
said.
Then he suddenly stopped and gasped for breath. In that moment the
strength of despair came to him. He straightened himself up, and a slight
flush rose to his cheeks.
"Oh, signori," said little Gandolfo, "let me speak of that of which I am
always thinking! Let me speak of what I always see before me!"
And he began unopposed and with wonderful power to tell what he himself
had seen.
He told how he who was son to the watchman of the town-hall had crept
through dark attics and had lain hidden in one of the galleries of the
court-room the night the court-martial had been held to pass sentence on
the insurgents in Diamante.
Then he had seen Don Gaetano Alagona on the bench of the accused with a
lot of wild fellows who were worse than brutes.
He told how beautiful Gaetano had been. He had seemed like a god to
little Gandolfo beside those terrible people about him. And he described
those bandits with their wild-beast faces, their coarse hair, their
clumsy limbs. He said that no one could look into their eyes without a
quiver of the heart.
Yet, in all his beauty, Don Gaetano was more terrible than those people.
Gandolfo did not know how they dared to sit beside him on the bench.
Under his frowning brows his eyes flashed at his fellow-prisoners with
a look which would have killed their souls, if they like others had
possessed such a thing.
"'Who are you,' he seemed to ask, 'who dare to turn to plundering and
murder while you call on sacred liberty? Do you know what you have done?
Do you know that on account of your devices I am now a prisoner? And it
was I who would have saved Sicily!'" And every glance he cast at them was
a death warrant.
His eyes fell on all the things that the bandits had stolen and that
were now piled up on a table. He recognized them. Could he help knowing
the clocks and the silver dishes from the summer-palace? could he help
knowing the relics and coins that had been stolen from his English
patroness? And when he had recognized the things, he turned to his
fellow-prisoners with a terrible smile. "'You heroes! you heroes!' said
the smile; 'you have stolen from two women!'"
His noble face was constantly changing. Once Gandolfo had seen it
contracted by a sudden terror. It was when the man sitting nearest to him
stretched out a hand covered with blood. Had he perhaps had a sudden idea
of the truth? Did he think that those men had broken into the house where
his beloved lived?
Gandolfo told how the officers who were to be the judges had come in,
silent and grave, and sat down in their places. But he said when he had
seen those noble gentlemen his anxiety had diminished. He had said to
himself that they knew that Gaetano was of good birth, and that they
would not sentence him. They would not mix him up with the bandits. No
one could possibly believe that he had wished to rob two women.
And see, when the judge called up Gaetano Alagona his voice was without
hardness. He spoke to him as to an equal.
"But," said Gandolfo, "when Don Gaetano rose, he stood so that he could
see out over the square. And through the square, through this same
square, where now so many people are sitting in happiness and pleasure, a
funeral procession was passing.
"It was the White Brotherhood carrying the body of the murdered Giannita
to her mother's house. They walked with torches, and the bier, carried
on the bearers' shoulders, was plainly visible. As the procession passed
slowly across the market-place, one could recognize the pall spread over
the corpse. It was the pall of the Alagonas adorned with a gorgeous coat
of arms and rich silver fringes. When Gaetano saw it, he understood that
the corpse was of the house of Alagona. His face became ashy gray, and
he reeled as if he were going to fall.
"At that moment the judge asked him: 'Do you know the murdered woman?'
And he answered: 'Yes.' Then the judge, who was a merciful man,
continued: 'Was she near to you?' And then Don Gaetano answered: 'I love
her.'"
When Gandolfo had come so far in his story, people saw Donna Micaela
suddenly rise, as if she had wished to contradict him, but Cavaliere
Palmeri drew her quickly down beside him.
"Be quiet, be quiet," he said to her.
And she sat quiet with her face hidden in her hands. Now and then her
body rocked and she wailed softly.
Gandolfo told how the judge, when Gaetano had acknowledged that, had
shown him his fellow-prisoners and asked him: "'If you loved that woman,
how can you have anything in common with the men who have murdered her?'"
Then Don Gaetano had turned towards the bandits. He had raised his
clenched hand and shaken it at them. And he had looked as if he had
longed for a dagger, to be able to strike them down one after another.
"'With those!'" he had shouted. "'Should I have anything in common with
those?'"
And he had certainly meant to say that he had nothing to do with robbers
and murderers. The judge had smiled kindly at him, as if he had only
waited for that answer to set him free.
But then a divine miracle had happened.
And Gandolfo told, how among all the stolen things that lay on the table,
there had also been a little Christ image. It was a yard high, richly
covered with jewels and adorned with a gold crown and gold shoes. Just at
that moment one of the officers bent down to draw the image to him; and
as he did so, the crown fell to the floor and rolled all the way to Don
Gaetano.
Don Gaetano picked up the Christ-crown, held it a moment in his hands and
looked at it carefully. It seemed as if he had read something in it.
He did not hold it more than one minute. In the next the guard took it
from him.
Donna Micaela looked up almost frightened. The Christ image! He was there
already! Should she so soon get an answer to her prayer?
Gandolfo continued: "But when Don Gaetano looked up, every one trembled
as at a miracle, for the man was transformed.
"Ah, signori, he was so white that his face seemed to shine, and his eyes
were calm and tender. And there was no more anger in him.
"And he began to pray for his fellow-prisoners; he began to pray for
their lives.
"He prayed that they should not kill those poor fellow-creatures. He
prayed that the noble judges should do something for them that they might
some day live like others. 'We have only this life to live,' he said.
'Our kingdom is only of this world.'
"He began to tell how those men had lived. He spoke as if he could read
their souls. He pictured their life, gloomy and unhappy as it had been.
He spoke so that several of the judges wept.
"The words came strong and commanding, so that it sounded as if Don
Gaetano had been judge and the judges the criminals. 'See,' he said,
'whose fault is it that these poor men have gone to destruction? Is it
not you who have the power who ought to have taken care of them?'
"And they were all dismayed at the responsibility he forced upon them.
"But suddenly the judge had interrupted him.
"'Speak in your own defence, Gaetano Alagona,' he said; 'do not speak in
that of others!'
"Then Don Gaetano had smiled. 'Signor,' he said, 'I have not much more
than you with which to defend myself. But still I have something. I have
left my career in England to make a revolt in Sicily. I have brought over
weapons. I have made seditious speeches. I have something, although not
much.'
"The judge had almost begged him. 'Do not speak so, Don Gaetano,' he had
said. 'Think of what you are saying!'
"But he had made confessions that compelled them to sentence him.
"When they told him that he was to sit for twenty-nine years in prison,
he had cried out: 'Now may her will be done, who was just carried by. May
I be as she wished!'
"And I saw no more of him," said little Gandolfo, "for the guards placed
him between them and led him away.
"But I, who heard him pray for those who had murdered his beloved, made a
vow that I would do something for him.
"I vowed to recite a beautiful improvisation to San Sebastiano to induce
him to help him. But I have not succeeded. I am no improvisatore; I could
not."
Here he broke off and threw himself down, weeping aloud before the image.
"Forgive me that I could not," he cried, "and help him in spite of it.
You know that when they sentenced him I promised to do it for his sake
that you might save him. But now I have not been able to speak of you,
and you will not help him."
Donna Micaela hardly knew how it happened, but she and little Rosalia,
who loved Gandolfo, were beside him at almost the same moment. They
drew him to them, and both kissed him, and said that no one had spoken
like him; no one, no one. Did he not see that they were weeping? San
Sebastiano was pleased with him. Donna Micaela put a ring on the boy's
finger and round about him the people were waving many- silk
handkerchiefs, that glistened like waves of the sea in the strong light
from the Cathedral.
"Viva Gaetano! viva Gandolfo!" cried the people.
And flowers and fruits and silk handkerchiefs and jewels came raining
down about little Gandolfo. Donna Micaela was crowded away from him
almost with violence. But it never occurred to her to be frightened. She
stood among the surging people and wept. The tears streamed down her
face, and she wept for joy that she could weep. That was the greatest
blessing.
She wished to force her way to Gandolfo; she could not thank him enough.
He had told her that Gaetano loved her. When he had quoted the words,
"Now may her will be done who was just carried by," she had suddenly
understood that Gaetano had believed that it was she lying under the pall
of the Alagonas.
And of that dead woman he had said: "I love her."
The blood flowed once more in her veins; her heart beat again; her tears
fell. "It is life, life," she said to herself, while she let herself be
carried to and fro by the crowd. "Life has come again to me. I shall not
die."
They all had to come up to little Gandolfo to thank him, because he had
given them some one to love, to trust in, to long for in those days of
dejection, when everything seemed lost.
SECOND BOOK
"_Antichrist shall go from land to land and give bread to the poor_"
I
A GREAT MAN'S WIFE
It was in February, and the almond-trees were beginning to blossom on the
black lava about Diamante.
Cavaliere Palmeri had taken a walk up Etna and had brought home a big
almond branch, full of buds and flowers and put it in a vase in the
music-room.
Donna Micaela started when she saw it. So they had already come, the
almond-blossoms. And for a whole month, for six long weeks, they would be
everywhere.
They would stand on the altar in the church; they would lie on the
graves, and they would be worn on the breast, on the hat, in the hair.
They would blossom over the roads, in the heaps of ruins, on the black
lava. And every almond-flower would remind her of the day when the bells
rang, when Gaetano was free and happy, and when she dreamed of passing
her whole life with him.
It seemed to her as if she never before fully understood what it meant
that he was shut in and gone, that she should never see him again.
She had to sit down in order not to fall; her heart seemed to stop, and
she shut her eyes.
While she was sitting thus she had a strange experience.
She is all at once at home in the palace in Catania. She is sitting in
the lofty hall reading, and she is a happy young girl, Signorina Palmeri.
A servant brings in a wandering salesman to her. He is a handsome young
fellow with a sprig of almond-blossoms in his button-hole; on his head he
carries a board full of little images of the saints, carved in wood.
She buys some of the images, while the young man's eyes drink in all the
works of art in the hall. She asks him if he would like to see their
collections. Yes, that he would. And she herself goes with him and shows
him.
He is so delighted with what he sees that she thinks that he must
be a real artist, and she says to herself that she will not forget
him. She asks where his home is. He answers: "In Diamante."--"Is
that far away?"--"Four hours in the post-carriage."--"And with the
railway?"--"There is no railway to Diamante, signorina."--"You must build
one."--"We! we are too poor. Ask the rich men in Catania to build us a
railway!"
When he has said that he starts to go, but he turns at the door and
comes and gives her his almond-blossoms. It is in gratitude for all the
beautiful things she has let him see.
When Donna Micaela opened her eyes she did not know whether she had been
dreaming or whether perhaps once some such thing had really happened.
Gaetano could really have been some time in the Palazzo Palmeri to sell
his images, although she had forgotten it; but now the almond-blossoms
had recalled it.
But it was no matter, no matter. The important thing was that the young
wood-carver was Gaetano. She felt as if she had been talking to him. She
thought she heard the door close behind him.
And it was after that that it occurred to her to build a railway between
Catania and Diamante.
Gaetano had surely come to her to ask her to do it. It was a command from
him, and she felt that she must obey.
She made no attempt to struggle against it. She was certain that Diamante
needed a railway more than anything else. She had once heard Gaetano say
that if Diamante only possessed a railway, so that it could easily send
away its oranges and its wine and its honey and its almonds, and so that
travellers could come there conveniently, it would soon be a rich town.
She was also quite certain that she could succeed with the railway. She
must try at all events. It never occurred to her not to. When Gaetano
wished it, she must obey.
She began to think how much money she herself could give. It would not go
very far. She must get more money. That was the first thing she had to do.
Within the hour she was at Donna Elisa's, and begged her to help her
arrange a bazaar. Donna Elisa lifted her eyes from her embroidery.
"Why do you want to arrange a bazaar?"--"I mean to collect money for
a railway."--"That is like you, Donna Micaela; no one else would have
thought of such a thing."--"What, Donna Elisa? What do you mean?"--"Oh,
nothing."
And Donna Elisa went on embroidering.
"You will not help me, then, with my bazaar?"--"No, I will not."--"And
you will not give a little contribution towards it?"--"One who has so
lately lost her husband," answered Donna Elisa, "ought not to trifle."
Donna Micaela saw that Donna Elisa was angry with her for some reason
or other, and that she therefore would not help her. But there must be
others who would understand; and it was a beautiful plan, which would
save Diamante.
But Donna Micaela wandered in vain from door to door. However much she
talked and begged, she gained no partisans.
She tried to explain, she used all her eloquence to persuade. No one was
interested in her plans.
Wherever she came, people answered her that they were too poor, too poor.
The syndic's wife answered no. Her daughters were not allowed to sell
at the bazaar. Don Antonio Greco, who had the marionette theatre, would
not come with his dolls. The town-band would not play. None of the
shop-keepers would give any of their wares. When Donna Micaela was gone
they laughed at her.
A railroad, a railroad! She did not know what she was thinking of. There
would have to be a company, shares, statutes, concessions. How should a
woman manage such things?
While some were content to laugh at Donna Micaela, some were angry with
her.
She went to the cellar-like shop near the old Benedictine monastery,
where Master Pamphilio related romances of chivalry. She came to ask him
if he would come to her bazaar and entertain the public with Charlemagne
and his paladins; but as he was in the midst of a story, she had to sit
down on a bench and wait.
Then she noticed Donna Concetta, Master Pamphilio's wife, who was sitting
on the platform at his feet knitting a stocking. As long as Master
Pamphilio was speaking, Donna Concetta's lips moved. She had heard his
romances so many times that she knew them by heart, and said the words
before they had passed Master Pamphilio's lips. But it was always the
same pleasure to her to hear him, and she wept, and she laughed, as she
had done when she heard him for the first time.
Master Pamphilio was an old man, who had spoken much in his day, so that
his voice sometimes failed him in the big battle-scenes, when he had to
speak loud and fast. But Donna Concetta, who knew it all by heart, never
took the word from Master Pamphilio. She only made a sign to the audience
to wait until his voice came back. But if his memory failed him, Donna
Concetta pretended that she had dropped a stitch, raised the stocking to
her eyes, and threw him the word behind it, so that no one noticed it.
And every one knew that although Donna Concetta perhaps could have told
the romances better than Master Pamphilio, she would never have been
willing to do such a thing, not only because it was not fitting for a
woman, but also because it would not give her half so much pleasure as
to listen to dear Master Pamphilio.
When Donna Micaela saw Donna Concetta, she fell to dreaming. Oh, to sit
so on the platform, where her beloved was speaking; to sit so day in and
day out and worship. She knew whom that would have suited.
When Master Pamphilio had finished speaking Donna Micaela went forward
and asked him to help her. It was hard for him to say no, on account of
the thousand prayers that were written in her eyes. But Donna Concetta
came to his rescue. "Master Pamphilio," she said, "tell Donna Micaela of
Guglielmo the Wicked." And Master Pamphilio began.
"Donna Micaela," he said, "do you know that once there was a king in
Sicily whose name was Guglielmo the Wicked? He was so covetous that he
took all his subjects' money. He commanded that every one possessing gold
coins should give them to him. And he was so severe and so cruel that
they all had to obey him.
"Well, Donna Micaela, Guglielmo the Wicked wished to know if any one had
gold hidden in his house. Therefore he sent one of his servants along the
Corso in Palermo with a beautiful horse. And the man offered the horse
for sale, and cried loudly: 'Will be sold for a piece of gold; will be
sold for a piece of gold!' But there was no one who could buy the horse.
"Yet it was a very beautiful horse, and a young nobleman, the Duke of
Montefiascone, was much taken by him. 'There is no joy for me if I cannot
buy the horse,' said he to his steward. 'Signor Duca,' answered his
steward, 'I can tell you where you can find a piece of gold. When your
noble father died and was carried away by the Capucins, according to the
ancient custom I put a piece of gold in his mouth. You can take that,
signor.'
"For you must know, Donna Micaela, that in Palermo they do not bury the
dead in the ground. They carry them to the monastery of the Capucins, and
the monks hang them up in their vaults. Ah, there are so many hanging in
those vaults!--so many ladies, dressed in silk and cloth of silver; so
many noble gentlemen, with orders on their breasts; and so many priests,
with cloak and cap over skeleton and skull.
"The young duke followed his advice. He went to the Capucin monastery,
took the piece of gold from his father's mouth and bought the horse with
it.
"But you understand that the king had only sent his servant with the
horse in order to find out if any one still had any money. And now the
duke was taken before the king. 'How does it happen that you still have
gold pieces?' said Guglielmo the Wicked.--'Sire, it was not mine; it was
my father's.' And he told how he had got the piece of gold. 'It is true,'
said the king. 'I had forgotten that the dead still had money.' And he
sent his servants to the Capucins and had them take all the gold pieces
out of the mouths of the dead."
Here old Master Pamphilio finished his story. And now Donna Concetta
turned to Donna Micaela with wrathful eyes. "It is you who are out with
the horse," she said.
"Am I? am I?"
"You, you, Donna Micaela! The government will say: 'They are building
a railway in Diamante. They must be rich.' And they will increase our
taxes. And God knows that we cannot pay the tax with which we are already
loaded down, even if we should go and plunder our ancestors."
Donna Micaela tried to calm her.
"They have sent you out to find out if we still have any money. You
are spying for the rich; you are in league with the government. Those
bloodsuckers in Rome have paid you."
Donna Micaela turned away from her.
"I came to talk to you, Master Pamphilio," she said to the old man.
"But I shall answer you," replied Donna Concetta; "for this is a
disagreeable matter, and such things are my affair. I know what is the
duty of the wife of a great man, Donna Micaela."
Donna Concetta became silent, for the fine lady gave her a look which was
so full of jealous longing that it made her sorry for her. Heavens, yes,
there had been a difference in their husbands; Don Ferrante and Master
Pamphilio!
II
PANEM ET CIRCENSES
In Diamante travellers are often shown two palaces that are falling into
ruins without ever having been completed. They have big window-openings
without frames, high walls without a roof, and wide doors closed with
boards and straw. The two palaces stand opposite each other on the
street, both equally unfinished and equally in ruins. There are no
scaffoldings about them, and no one can enter them. They seem to be only
built for the doves.
Listen to what is told of them.
What is a woman, O signore? Her foot is so little that she goes through
the world without leaving a trace behind her. For man she is like his
shadow. She has followed him through his whole life without his having
noticed her.
Not much can be expected of a woman. She has to sit all day shut in like
a prisoner. She cannot even learn to spell a love-letter correctly. She
cannot do anything of permanence. When she is dead there is nothing to
write on her tombstone. All women are of the same height.
But once a woman came to Diamante who was as much above all other women
as the century-old palm is above the grass. She possessed lire by
thousands, and could give them away or keep them, as she pleased. She
turned aside for no one. She was not afraid of being hated. She was the
greatest marvel that had ever been seen.
Of course she was not a Sicilian. She was an Englishwoman. And the first
thing she did when she came was to take the whole first floor of the
hotel for herself alone. What was that for her? All Diamante would not
have been enough for her.
No, all Diamante was not enough for her. But as soon as she had come she
began to govern the town like a queen. The syndic had to obey her. Was it
not she who made him put stone benches in the square? Was it not at her
command that the streets were swept every day?
When she woke in the morning all the young men of Diamante stood waiting
outside her door, to be allowed to accompany her on some excursion. They
had left shoemaker's awl and stone-cutter's chisel to act as guides to
her. Each had sold his mother's silk dress to buy a side-saddle for his
donkey, so that _she_ might ride on it to the castle or to Tre Castagni.
They had divested themselves of house and home in order to buy a horse
and carriage to drive her to Randazzo and Nicolosi.
We were all her slaves. The children began to beg in English, and the
old blind women at the hotel door, Donna Pepa and Donna Tura, draped
themselves in dazzlingly white veils to please her.
Everything moved round her; industries and trades grew up about her.
Those who could do nothing else dug in the earth for coins and pottery
to offer her. Photographers moved to the town and began to work for her.
Coral merchants and hawkers of tortoise-shell grew out of the earth about
her. The priests of Santa Agnese dug up the old Dionysius theatre, that
lay hidden behind their church, for her sake; and every one who owned a
ruined villa unearthed in the darkness of the cellar remains of mosaic
floors and invited her by big posters to come and see.
There had been foreigners before in Diamante, but they had come and
gone, and no one had enjoyed such power. There was soon not a man in the
town who did not put all his trust in the English signorina. She even
succeeded in putting a little life into Ugo Favara. You know Ugo Favara,
the advocate, who was to have been a great man, but had reverses and came
home quite broken. She employed him to take care of her affairs. She
needed him, and she took him.
There has never been a woman in Diamante who has done so much business
as she. She spread out like green-weed in the spring. One day no one
knows that there is any, and the next it is a great clump. Soon it was
impossible to go anywhere in Diamante without coming on her traces. She
bought country houses and town houses; she bought almond-groves and
lava-streams. The best places on Etna to see the view were hers as well
as the thirsting earth on the plain. And in town she began to build two
big palaces. She was to live in them and rule her kingdom.
We shall never see a woman like her again. She was not content with
all that. She wished also to fight the fight with poverty, O signore,
with Sicilian poverty! How much she gave out each day, and how much she
gave away on feast-days! Wagons, drawn by two pairs of oxen, went down
to Catania and came back piled up with all sorts of clothing. She was
determined that they should have whole clothes in the town where she
reigned.
But listen to what happened to her; how the struggle with poverty ended
and what became of the kingdom and the palace.
She gave a banquet for the poor people of Diamante, and after the banquet
an entertainment in the Grecian theatre. It was what an old emperor might
have done. But who has ever before heard of a woman doing such a thing?
She invited all the poor people. There were the two blind women from the
hotel-door, and old Assunta from the Cathedral steps. There was the man
from the post-house, who had his chin bound up in a red cloth on account
of cancer of the face; and there was the idiot who opens the iron doors
of the Grecian theatre. All the donkey-boys were there, and the handless
brothers, who exploded a bomb in their childhood and lost their fingers;
and the man with the wooden leg, and the old chair-maker who had grown
too old to work, both were there.
It was strange to see them creep out of their holes, all the poor in
Diamante. The old women who sit and spin with distaffs in the dark
alleys were there, and the organ-grinder, who has an instrument as big
as a church-organ, a wandering young mandolinist from Naples with a body
full of all possible deviltries. All those with diseased eyes and all
the decrepit; those without a roof over their heads; those who used to
collect sorrel by the roadside for dinner; the stone-cutter, who earned
one lira a day and had six children to provide for,--they had all been
invited and were present at the feast.
It was poverty marshalling its troops for the English signorina. Who has
such an army as poverty? But for once the English signorina could conquer
it.
She had something to fight with too and to conquer with. She filled the
whole square with loaded tables. She had wine-skins arranged along the
stone bench that lines the wall of the Cathedral. She had turned the
deserted convent into a larder and kitchen. She had all the foreign
colony in Diamante dressed in white aprons, to serve the courses. She had
all of Diamante who are used to eating their fill, wandering to and fro
as spectators.
Ah, spectators, what did she not have for spectators? She had great Etna
and the dazzling sun. She had the red peaks of the inland mountains and
the old temple of Vulcan, that was now consecrated to San Pasquale. And
none of them had ever seen a satisfied Diamante. None of them had ever
before happened to think how much more beautiful they themselves would be
if the people could look at them without hunger hissing in their ears and
trampling on their heels.
But mark one thing! Although that signorina was so wonderful and so
great, she was not beautiful. And in spite of all her power, she was
neither charming nor attractive. She did not rule with jests, and she did
not reward with smiles. She had a heavy, clumsy body, and a heavy, clumsy
disposition.
The day she gave food to the poor she became a different person. A
chivalrous people live in our noble island. Among all those poor people
there was not one who let her feel that she was exercising charity. They
worshipped her, but they worshipped her as a woman. They sat down at the
table as with an equal. They behaved to her as guests to their hostess.
"To-day I do you the honor to come to you; to-morrow you do me the
honor to come to me. So and not otherwise." She stood on the high steps
of the town-hall and looked down at all the tables. And when the old
chair-maker, who sat at the head of the table, had got his glass filled,
he rose, bowed to her and said: "I drink to your prosperity, signorina."
So did they all. They laid their hands on their hearts and bowed to her.
It would have perhaps been good for her if she had met with such chivalry
earlier in life. Why had the men in her native land let her forget that
women exist to be worshipped?
Here they all looked as if they were burning with a quiet adoration. Thus
are women treated in our noble island. What did they not give in return
for the food and the wine that she had offered them? They gave youth and
light-heartedness and all the dignity of being worth coveting. They made
speeches for her. "Noble-hearted signorina, you who have come to us from
over the sea, you who love Sicily," and so on, and so on. She showed that
she could blush. She no longer hid her power to smile. When they had
finished speaking, the lips of the English signorina began to tremble.
She became twenty years younger. It was what she needed.
The donkey-boy was there, who carries the English ladies up to Tre
Castagni, and who always falls in love with them before he parts from
them. Now his eyes were suddenly opened to the great benefactress. It is
not only a slender, delicate body and a soft cheek that are worthy to be
adored, but also strength and force. The donkey-boy suddenly dropped
knife and fork, leaned his elbows on the table, and sat and looked
at her. And all the other donkey-boys did the same. It spread like a
contagion. It grew hot with burning glances about the English signorina.
It was not only the poor people who adored her. The advocate, Ugo Favara,
came and whispered to her that she had come as a providence to his poor
land and to him. "If only I had met such a woman as you before," he said.
Fancy an old bird which has sat in a cage for many years and become rough
and lost all the gloss of his feathers. And then some one comes and
straightens them out and smooths them back. Think of it, signore!
There was that boy from Naples. He took his mandolin and began to sing
his very best. You know how he sings; he pouts with his big mouth and
says ugly words. He usually is like a grinning mask. But have you seen
the angel in his eyes? An angel which seems to weep over his fall and is
filled with a holy frenzy. That evening he was only an angel. He raised
his head like one inspired by God, and his drooping body became elastic
and full of proud vitality. Color came into his livid cheeks. And he
sang; he sang so that the notes seemed to fly like fireflies from his
lips and fill the air with joy and dance.
When it grew dark they all went over to the Grecian theatre. That was
the finishing touch to the entertainment. What did she not have to offer
there!
She had the Russian singer and the German variété artists. She had the
English wrestlers and the American magician. But what was that compared
to all the rest: the silvery moonlight and the place and its memories?
Those poor people seemed to feel like the Greeks and leaders of fashion
when they once more took their places on the stone-benches of their own
old theatre and from between the tottering pillars looked out at the most
beautiful panorama.
Those poor people did not stint; they shared all the pleasure they
received. They did not spare jubilation; there was no stopping their
hand-clapping. The performers left the platform with a wealth of praise.
Some one begged the English signorina to appear. All the adoration was
meant for her. She ought to stand face to face with it and feel it. And
they told her how intoxicating it was, how elevating, how inflaming.
She liked the proposal. She immediately agreed. She had sung in her
youth, and the English never seem to be afraid to sing. She would not
have done it if she had not been in a good mood, and she wished to sing
for those who loved her.
She came as the last number. Fancy what it was to stand on such an old
stage! It was where Antigone had been buried alive and Iphigenia had been
sacrificed. The English signorina stepped forward there to receive every
conceivable honor.
It stormed to meet her as soon as she showed herself. They seemed to wish
to stamp the earth to pieces to honor her.
It was a proud moment. She stood there with Etna as a background and
the Mediterranean as wings. Before her on the grass-grown benches was
sitting conquered poverty, and she felt that she had all Diamante at her
feet.
She chose "Bellini," our own "Bellini." She too wished to be amiable and
so she sang "Bellini," who was born here under Etna; "Bellini" whom we
know by heart, note for note.
Of course, O signore, of course she could not sing. She had mounted the
tribune only to receive homage. She had come in order to let the love of
the people find an outlet. And now she sang false and feebly. And the
people knew every note.
It was that mandolinista from Naples. He was the first to grimace and to
take a note as false as that of the English signorina. Then it was the
man with the cancer, who laughed till he laughed his neckcloth off. Then
it was the donkey-boy, who began to clap his hands.
Then they all began. It was madness, but that they did not understand.
It is not in the land of the old Greeks that people can bear barbarians
who sing false. Donna Pepa and Donna Tura laughed as they had never
done before in their lives. "Not one true note! By the Madonna and San
Pasquale, not one true note!"
They had eaten their fill for once in their lives. It was natural that
intoxication and madness should take hold of them. And why should they
not laugh? She had not given them food in order to torture their ears
with files and saws. Why should they not defend themselves by laughing?
Why should they not mimic and hiss and scream? Why should they not lean
backward and split their sides with laughter? They were not the English
signorina's slaves, I suppose.
It was a terrible blow to her. It was too great a blow for her to
understand. Were they hissing her? It must be something happening among
them; something that she could not see. She sang the aria to its end.
She was convinced that the laughter was for something with which she had
nothing to do.
When she had finished a sort of storm of applause roared over her. At
last she understood. Torches and the moonlight made the night so bright
that she could see the rows of people twisting with laughter. She heard
the scoffs and the jests now, when she was not singing. They were for
her. Then she fled from the stage. It seemed to her that Etna itself
heaved with laughter, and that the sea sparkled with merriment.
But it grew worse and worse. They had had such a good time, those poor
people; they had never had such a good time before, and they wished
to hear her once again. They called for her; they cried: "Bravo! Bis!
Da capo!" They could not lose such a pleasure. She, she was almost
unconscious. There was a storm about her. They screamed; they roared to
get her in. She saw them lift their arms and threaten her to get her in.
All at once it was all turned into an old circus. She had to go in to be
devoured by monsters.
It went on; it went on; it became wilder and wilder. The other performers
were frightened and begged her to yield. And she herself was frightened.
It looked as if they would have killed her if she did not do what they
wished.
She dragged herself on the stage and stood face to face with the crowd.
There was no pity. She sang because they all wished to be amused. That
was the worst. She sang because she was afraid of them and did not dare
not to. She was a foreigner and alone, and she had no one to protect her,
and she was afraid. And they laughed and laughed.
Screams and cries, crowing and whistling accompanied the whole aria. No
one had mercy on her. For the first time in her life she felt the need of
mercy.
Well, the next day she resolved to depart. She could not endure Diamante
any longer. But when she told the advocate, Favara, he implored her to
stay for his sake and made her an offer of marriage.
He had chosen his time well. She said yes, and was married to him. But
after that time she built no more on her palaces; she made no struggle
against poverty; she cared nothing to be queen in Diamante. Would you
believe it? She never showed herself on the street; she lived indoors
like a Sicilian.
Her little house stood hidden away behind a big building, and of herself
no one knew anything. They only knew that she was quite changed. No
one knew whether she was happy or unhappy; whether she shut herself in
because she hated the people, or because she wished to be as a Sicilian
wife ought to be.
Does it not always end so with a woman? When they build their palaces
they are never finished. Women can do nothing that has permanence.
III
THE OUTCAST
When Donna Micaela heard how the poor people had hooted Miss Tottenham
out, she hurried to the hotel to express her condolence. She wished to
beg her not to judge those poor creatures by what they had done when they
had been put out of their heads with pleasure and wine. She would beg her
not to take her hand from Diamante. She herself did not care very much
for Miss Tottenham, but for the sake of the poor--She would say anything
to pacify her.
When she came to the hotel Etna, she saw the whole street filled with
baggage-wagons. So there was no hope. The great benefactress was going
away.
Outside the hotel there was much sorrow and despair. The two old blind
women, Donna Pepa and Donna Tura, who had always sat in the hotel
court-yard, were now shut out, and they were kneeling before the door.
The young donkey-driver, who loved all young English ladies, stood with
his face pressed against the wall and wept.
Inside the hotel the landlord walked up and down the long corridor,
raging at Providence for sending him this misfortune. "Signor Dio," he
mumbled, "I am beggared. If you let this happen, I will take my wife by
the hand and my children in my arms and throw myself with them down into
Etna."
The landlady was very pale and humble. She scarcely dared to lift her
eyes from the ground. She would have liked to creep about on her knees to
prevail upon the rich signorina to remain.
"Do you dare to speak to her, Donna Micaela?" she said. "May God help
you to speak to her! Alas! tell her that the Neapolitan boy, who was the
cause of the whole misfortune, has been turned out of the town. Tell her
that they all wish to make amends. Speak to her, signora!"
The landlady took Donna Micaela to the Englishwoman's drawing-room and
went in with her card. She came back immediately and asked her to wait a
few minutes. Signorina Tottenham was having a business talk with Signor
Favara.
It was the very moment when the advocate Favara asked Miss Tottenham's
hand in marriage; and while Donna Micaela waited she heard him say quite
loud: "You must not go away, signorina! What will become of me if you go
away? I love you; I cannot let you go. I should not have dared to speak
if you had not threatened to go away. But now--"
He lowered his voice again, but Donna Micaela would hear no more and
went away. She saw that she was superfluous. If Signor Favara could not
succeed in keeping the great benefactress, no one could.
When she went out again through the gateway the landlord was standing
there quarrelling with the old Franciscan, Fra Felice. He was so
irritated that he not only quarrelled with Fra Felice, he also drove him
from his house.
"Fra Felice," he cried, "you come to make more trouble with our great
benefactress. You will only make her more angry. Go away, I tell you! You
wolf, you man-eater, go away!"
Fra Felice was quite as enraged as the landlord, and tried to force
his way past him. But then the latter took him by the arm, and without
further notice marched him down the steps.
Fra Felice was a man who had received a great gift from his Creator. In
Sicily, where everybody plays in the lottery, there are people who have
the power to foretell what numbers will win at the next drawing. He who
has such second sight is called "polacco," and is most often found in
some old begging monk. Fra Felice was such a monk. He was the greatest
polacco in the neighborhood of Etna.
As every one wished him to tell them a winning tern or quartern, he was
always treated with great consideration. He was not used to be taken by
the arm and be thrown into the street, Fra Felice.
He was nearly eighty years old and quite dried-up and infirm. As he
staggered away between the wagons, he stumbled, trod on his cloak, and
almost fell. But none of the porters and drivers that stood by the door
talking and lamenting had time that day to think of Fra Felice.
The old man tottered along in his heavy homespun cloak. He was so thin
and dry that there seemed to be more stiffness in the cloak than in the
monk. It seemed to be the old cloak that held him up.
Donna Micaela caught up with him and gently drew the old man's arm
through her own. She could not bear to see how he struck against the
lamp-posts and fell over steps. But Fra Felice never noticed that she
was looking after him. He walked and mumbled and cursed, and did not know
but that he was as much alone as if he sat in his cell.
Donna Micaela wondered why Fra Felice was so angry with Miss Tottenham.
Had she been out to his monastery and taken down frescos from the walls,
or what had she done?
Fra Felice had lived for sixty years in the big Franciscan monastery
outside the Porta Etnea, wall to wall with the old church San Pasquale.
Fra Felice had been monk there for thirty years, when the monastery was
given up and sold to a layman. The other monks moved away, but Fra Felice
remained because he could not understand what selling the house of San
Francisco could mean.
If laymen were to come there, it seemed to Fra Felice almost more
essential that at least one monk should remain. Who else would attend to
the bell-ringing, or prepare medicines for the peasant women, or give
bread to the poor of the monastery? And Fra Felice chose a cell in a
retired corner of the monastery, and continued to go in and out as he had
always done.
The merchant who owned the monastery never visited it. He did not care
about the old building; he only wanted the vineyards belonging to it. So
Fra Felice still reigned in the old monastery, and fastened up the fallen
cornices and whitewashed the walls. As many poor people as had received
food at the monastery in former days, still received it. For his gift of
prophecy Fra Felice got such large alms as he wandered through the towns
of Etna that he could have been a rich man; but every bit of it went to
the monastery.
Fra Felice had suffered an even greater grief than for the monastery on
account of the monastery church. It had been desecrated during war, with
bloody fights and other atrocities, so that mass could never be held
there. But that he could not understand either. The church, where he had
made his vows, was always holy to Fra Felice.
It was his greatest sorrow that his church had fallen entirely into ruin.
He had looked on when Englishmen had come and bought pulpit and lectern
and choir chairs. He had not been able to prevent collectors from Palermo
coming and taking the chandeliers and pictures and brass hooks. However
much he had wished it, he had not been able to do anything to save his
church. But he hated those church-pillagers; and when Donna Micaela saw
him so angry, she thought that Miss Tottenham had wished to take some of
his treasures from him.
But the fact was that now, when Fra Felice's church was emptied, and
no one came any more to plunder there, he had begun to think of doing
something to embellish it once more, and he had had his eye on the
collection of images of the saints in the possession of the rich English
lady. At her entertainment, when she had been kind and gentle towards
every one, he had dared to ask her for her beautiful Madonna, who had a
dress of velvet and eyes like the sky. And his request had been granted.
That morning Fra Felice had swept and dusted the church, and put flowers
on the altar, before he went to fetch the image. But when he came to the
hotel, the Englishwoman had changed her mind; she had not been at all
willing to give him the valuable Madonna. In its stead she had given him
a little ragged, dirty image of the Christchild, which she thought she
could spare without regret.
Ah, what joy and expectation old Fra Felice had felt, and then had been
so disappointed! He could not be satisfied; he came back time after time
to beg for the other image. It was such a valuable image that he could
not have bought it with all that he begged in a whole year. At last the
great benefactress had dismissed him; and it was then that Donna Micaela
had found him.
As they went along the street, she began to talk to the old man and won
his story from him. He had the image with him, and right in the street
he stopped, showed it to her, and asked her if she had ever seen a more
miserable object.
Donna Micaela looked at the image for a moment with stupefaction. Then
she smiled and said: "Lend me the image for a few days, Fra Felice!"
"You can take it and keep it," said the old man. "May it never come
before my eyes again!"
Donna Micaela took the image home and worked on it for two days. When she
then sent it to Fra Felice it shone with newly polished shoes; it had a
fresh, clean dress; it was painted, and in its crown shone bright stones
of many colors.
He was so beautiful, the outcast, that Fra Felice placed him on the empty
altar in his church.
* * * * *
It was very early one morning. The sun had not risen, and the broad sea
was scarcely visible. It was really very early. The cats were still
roaming about the roofs; no smoke rose from the chimneys; and the mists
lay and rolled about in the low valley round the steep Monte Chiaro.
Old Fra Felice came running towards the town. He ran so fast that he
thought he felt the mountain tremble beneath him. He ran so fast that the
blades of grass by the roadside had no time to sprinkle his cloak with
dew; so fast that the scorpions had no time to lift their tails and sting
him.
As the old man ran, his cloak flapped unfastened about him, and his rope
swung unknotted behind. His wide sleeves waved like wings, and his heavy
hood pounded up and down on his back, as if it wished to urge him on.
The man in the custom-office, who was still asleep, woke and rubbed his
eyes as Fra Felice rushed by, but he had no time to recognize him. The
pavements were slippery with dampness; beggars lay and slept by the high
stone steps with their legs heedlessly stretched out into the street;
exhausted domino-players were going home from the Café reeling with
sleep. But Fra Felice hastened onward regardless of all obstructions.
Houses and gateways, squares and arched-over alleys disappeared behind
old Fra Felice. He ran half-way up the Corso before he stopped.
He stopped in front of a big house with many heavy balconies. He seized
the door-knocker and pounded until a servant awoke. He would not be quiet
till the servant called up a maid, and the maid waked the signora.
"Donna Micaela, Fra Felice is downstairs. He insists on speaking to you."
When Donna Micaela at last came down to Fra Felice, he was still panting
and breathless, but there was a fire in his eyes, and little pale roses
in his cheeks.
It was the image, the image. When Fra Felice had rung the four-o'clock
matins that morning he had gone into the church to look at him.
Then he had discovered that big stones had loosened from the dome just
over the image. They had fallen on the altar and broken it to pieces, but
the image had stood untouched. And none of the plaster and dust that had
tumbled down had fallen on the image; it was quite uninjured.
Fra Felice took Donna Micaela's hand and told her that she must go with
him to the church and see the miracle. She should see it before any one,
because she had taken care of the image.
And Donna Micaela went with him through the gray, chilly morning to his
monastery, while her heart throbbed with eagerness and expectation.
When she arrived and saw that Fra Felice had told the truth, she said to
him that she had recognized the image as soon as she had caught sight of
it, and that she knew that it could work miracles. "He is the greatest
and gentlest of miracle-workers," she said.
Fra Felice went up to the image and looked into its eyes. For there is a
great difference in images, and the wisdom of an old monk is needed to
understand which has power and which has not. Now Fra Felice saw that
this image's eyes were deep and glowing, as if they had life; and that on
its lips hovered a mysterious smile.
Then old Fra Felice fell on his knees and stretched his clasped hands
towards the image, and his old shrivelled face was lighted by a great
joy.
It seemed to Fra Felice all at once as if the walls of his church were
covered with pictures and purple hangings; candles shone on the altar;
song sounded from the gallery; and the whole floor was covered with
kneeling, praying people.
All imaginary glory would fall to the lot of his poor old church, now
that it possessed one of the great miracle-working images.
IV
THE OLD MARTYRDOM
From the summer-palace in Diamante many letters were sent during
that time to Gaetano Alagona, who was in prison in Como. But the
letter-carrier never had a letter in his bag from Gaetano addressed to
the summer-palace.
For Gaetano had gone into his life-long imprisonment as if it had been a
grave. The only thing he asked or desired was that it should give him the
grave's forgetfulness and peace.
He felt as if he were dead; and he said to himself that he did not wish
to hear the laments and wails of the survivors. Nor did he wish to be
deceived with hopes, or be tempted by tender words to long for family and
friends. Nor did he wish to hear anything of what was happening in the
world, when he had no power to take part and to lead.
He found work in the prison, and carved beautiful works of art, as he
had always done. But he never would receive a letter, nor a visitor. He
thought that in that way he could cease to feel the bitterness of his
misfortunes. He believed that he would be able to teach himself to live a
whole life within four narrow walls.
And for that reason Donna Micaela never had a word of answer from him.
Finally she wrote to the director of the prison and asked if Gaetano was
still alive. He answered that the prisoner she asked about never read a
letter. He had asked to be spared all communications from the outside
world.
So she wrote no more. Instead she continued to work for her railway. She
hardly dared to speak of it in Diamante, but nevertheless she thought
of nothing else. She herself sewed and embroidered, and she had all her
servants make little cheap things that she could sell at her bazaar. In
the shop she looked up old wares for the tombola. She had Piero, the
gate-keeper, prepare lanterns; she persuaded her father to paint
signs and placards; and she had her maid, Lucia, who was from Capri,
arrange coral necklaces and shell boxes.
She was not at all sure that even one person would come to her
entertainment. Every one was against her; no one would help her. They did
not even like her to show herself on the streets or to talk business. It
was not fitting for a well-born lady.
Old Fra Felice tried to assist her, for he loved her because she had
helped him with the image.
One day, when Donna Micaela was lamenting that she could not persuade any
one that the people ought to build the railway, he lifted his cap from
his head and pointed to his bald temples.
"Look at me, Donna Micaela," he said. "So bald will that railway make
your head if you go on as you have begun."
"What do you mean, Fra Felice?"
"Donna Micaela," said the old man, "would it not be folly to start on a
dangerous undertaking without having a friend and helper?"
"I have tried enough to find friends, Fra Felice."
"Yes, men!" said the old man. "But how do men help? If any one is going
fishing, Donna Micaela, he knows that he must call on San Pietro; if any
one wishes to buy a horse, he can ask help of San Antonio Abbate. But if
I want to pray for your railway, I do not know to whom I shall turn."
Fra Felice meant that the trouble was that she had chosen no patron saint
for her railway. He wished her to choose the crowned child that stood out
in his old church as its first friend and promoter. He told her that if
she only did that she would certainly be helped.
She was so touched that any one was willing to stand by her that she
instantly promised to pray for her railway to the child at San Pasquale.
Fra Felice got a big collection-box and painted on it in bright, distinct
letters: "Gifts for the Etna Railway," and he hung it in his church
beside the altar.
It was not more than a day after that that Don Antonio Greco's wife,
Donna Emilia, came out to the old, deserted church to consult San
Pasquale, who is the wisest of all the saints.
During the autumn Don Antonio's theatre had begun to fare ill, as was to
be expected when no one had any money.
Don Antonio thought to run the theatre with less expense than before. He
had cut off a couple of lamps and did not have such big and gorgeously
painted play-bills.
But that had been great folly. It is not at the moment when people are
losing their desire to go to the theatre that it will answer to shorten
the princesses' silk trains and economize on the gilding of the king's
crowns.
Perhaps it is not so dangerous at another theatre, but at a marionette
theatre it is a risk to make any changes, because it is chiefly
half-grown boys who go to the marionette theatre. Big people can
understand that sometimes it is necessary to economize, but children
always wish to have things in the same way.
Fewer and fewer spectators came to Don Antonio, and he went on
economizing and saving. Then it occurred to him that he could dispense
with the two blind violin-players, Father Elia and Brother Tommaso, who
also used to play during the interludes and in the battle-scenes.
Those blind men, who earned so much by singing in houses of mourning,
and who took in vast sums on feast-days, were expensive. Don Antonio
dismissed them and got a hand-organ.
That caused his ruin. All the apprentices and shop-boys in Diamante
ceased to go to the theatre. They would not sit and listen to a
hand-organ. They promised one another not to go to the theatre till Don
Antonio had taken back the fiddlers, and they kept their promise. Don
Antonio's dolls had to perform to empty walls.
The young boys who otherwise would rather go without their supper than
the theatre, stayed away night after night. They were convinced that they
could force Don Antonio to arrange everything as before.
But Don Antonio comes of a family of artists. His father and his brother
have marionette theatres; his brothers-in-law, all his relations are
of the profession. And Don Antonio understands his art. He can change
his voice indefinitely; he can manœuvre at the same time a whole army
of dolls; and he knows by heart the whole cycle of plays founded on the
chronicles of Charlemagne.
And now Don Antonio's artistic feelings were hurt. He would not be forced
to take back the blind men. He wished to have the people come to his
theatre for his sake, and not for that of the musicians.
He changed his tactics and began to play big dramas with elaborate
mountings. But it was futile.
There is a play called "The Death of the Paladin," which treats of
Roland's fight at Ronceval. It requires so much machinery that a puppet
theatre has to be kept shut for two days for it to be set up. It is so
dear to the public that it is generally played for double price and to
full houses for a whole month. Don Antonio now had that play mounted, but
he did not need to play it; he had no spectators.
After that his spirit was broken. He tried to get Father Elia and Brother
Tommaso back, but they now knew what their value was to him.
They demanded such a price that it would have been ruin to pay them. It
was impossible to come to any agreement.
In the small rooms back of the marionette theatre they lived as in a
besieged fortress. They had nothing else to do but to starve.
Donna Emilia and Don Antonio were both gay young people, but now they
never laughed. They were in great want, but Don Antonio was a proud man,
and he could not bear to think that his art no longer had the power to
draw.
So, as I said, Donna Emilia went down to the church of San Pasquale to
ask the saint for good advice. It had been her intention to repeat nine
prayers to the great stone-image standing outside of the church, and
then to go; but before she had begun to pray she had noticed that the
church-door stood open. "Why is San Pasquale's church-door open?" said
Donna Emilia. "That has never happened in my time,"--and she went into
the church.
The only thing to be seen there was Fra Felice's beloved image and the
big collection-box. The image looked so beautiful in his crown and his
rings that Donna Emilia was tempted forward to him, but when she came
near enough to look into his eyes, he seemed to her so tender and so
cheering that she knelt down before him and prayed. She promised that
if he would help her and Don Antonio in their need, she would put the
receipts of a whole evening in the big box that hung beside him.
After her prayers were over, Donna Emilia concealed herself behind the
church-door, and tried to catch what the passers-by were saying. For if
the image was willing to help her, he would let her hear a word which
would tell her what to do.
She had not stood there two minutes before old Assunta of the Cathedral
steps passed by with Donna Pepa and Donna Tura. And she heard Assunta say
in her solemn voice: "That was the year when I heard 'The Old Martyrdom'
for the first time." Donna Emilia heard quite distinctly. Assunta really
said "The Old Martyrdom."
Donna Emilia thought that she would never reach her home. It was as if
her legs could not carry her fast enough, and the distance increased as
she ran. When she finally saw the corner of the theatre with the red
lanterns under the roof and the big illustrated play-bills, she felt as
if she had gone many miles.
When she came in to Don Antonio, he sat with his big head leaning on his
hand and stared at the table. It was terrible to see Don Antonio. In
those last weeks he had begun to lose his hair; on the very top of his
head it was so thin that the skin shone through. Was it strange, when he
was in such trouble? While she had been away he had taken all his puppets
out and inspected them. He did that now every day. He used to sit and
look at the puppet that played Armida. Was she no longer beautiful and
beguiling? he would ask. And he tried to polish up Roland's sword and
Charlemagne's crown. Donna Emilia saw that he had gilded the emperor's
crown again; it was for at least the fifth time. But then he had stopped
in the midst of his work and had sat down to brood. He had noticed it
himself. It was not gilding that was lacking; it was an idea.
As Donna Emilia came into the room, she stretched out her hands to her
husband.
"Look at me, Don Antonio Greco," she said. "I bear in my hands golden
bowls full of ripe figs!"
And she told how she had prayed, and what she had vowed, and what she had
been advised.
When she said that to Don Antonio, he sprang up. His arms fell stiffly
beside his body, and his hair raised itself from his head. He was seized
with an unspeakable terror. "'The Old Martyrdom'!" he screamed, "'The Old
Martyrdom'!"
For "The Old Martyrdom" is a miracle-play, which in its time was given
in all Sicily. It drove out all other oratorios and mysteries, and was
played every year in every town for two centuries. It was the greatest
day of the year, when "The Old Martyrdom" was performed. But now it is
never played; now it only lives in the people's memory as a legend.
In the old days it was also played in the marionette theatres. But now it
has come to be considered old-fashioned and out-of-date. It has probably
not been played for thirty years.
Don Antonio began to roar and scream at Donna Emilia, because she
tortured him with such folly. He struggled with her as with a demon, who
had come to seize him. It was amazing; it was heartrending, he said. How
could she get hold of such a word? But Donna Emilia stood quiet and let
him rave. She only said that what she had heard was God's will.
Soon Don Antonio began to be uncertain. The great idea gradually took
possession of him. Nothing had ever been so loved and played in Sicily,
and did not the same people still live on the noble isle? Did they
not love the same earth, the same mountains, the same skies as their
forefathers had loved? Why should they not also love "The Old Martyrdom"?
He resisted as long as he could. He said to Donna Emilia that it would
cost too much. Where could he get apostles with long hair and beards? He
had no table for the Last Supper; he had none of the machinery required
for the entry, and carrying of the cross.
But Donna Emilia saw that he was going to give in, and before night
he actually went to Fra Felice and renewed her vow to put the receipts
of one evening in the box of the little image, if it proved to be good
advice.
Fra Felice told Donna Micaela about the vow, and she was glad, and at the
same time anxious how it would turn out.
Through all the town it was known that Don Antonio was mounting "The Old
Martyrdom," and every one laughed at him. Don Antonio had lost his mind.
The people would have liked well enough to see "The Old Martyrdom," if
they could have seen it as it was played in former days. They would have
liked to see it given as in Aci, where the noblemen of the town played
the kings and the servants, and the artisans took the parts of the Jews
and the apostles; and where so many scenes from the Old Testament were
added that the spectacle lasted the whole day.
They would have also liked to see those wonderful days in Castelbuoco,
when the whole town was transformed into Jerusalem. There the mystery was
given so that Jesus came riding to the town, and was met with palms at
the town-gate. There the church represented the temple at Jerusalem and
the town-hall Pilate's palace. There Peter warmed himself at a fire in
the priest's court-yard; the crucifixion took place on a mountain above
the town; and Mary looked for the body of her son in the grottoes of the
syndic's garden.
When the people had such things in their memory how could they be content
to see the great mystery in Don Antonio's theatre?
But in spite of everything, Don Antonio worked with the greatest
eagerness to prepare the actors and to arrange the elaborate machinery.
And behold, in a few days came Master Battista, who painted placards, and
presented him with a play-bill. He had been glad to hear that Don Antonio
was going to play "The Old Martyrdom;" he had seen it in his youth, and
had great pleasure in it.
So there now stood in large letters on the corner of the theatre: "'The
Old Martyrdom' or 'The Resurrected Adam,' tragedy in three acts by
Cavaliere Filippo Orioles."
Don Antonio wondered and wondered what the people's mood would be. The
donkey-boys and apprentices who passed by his theatre read the notice
with scoffs and derision. It looked very black for Don Antonio, but in
spite of it he went on faithfully with his work.
When the appointed evening came, and the "Martyrdom" was to be played, no
one was more anxious than Donna Micaela. "Is the little image going to
help me?" she asked herself incessantly.
She sent out her maid, Lucia, to look about. Were there any groups of
boys in front of the theatre? Did it look as if there were going to be a
crowd? Lucia might go to Donna Emilia, sitting in the ticket-office, and
ask her if it looked hopeful.
But when Lucia came back she had not the slightest hope to offer. There
was no crowd outside the theatre. The boys had resolved to crush Don
Antonio.
Towards eight o'clock Donna Micaela could no longer endure sitting
at home and waiting. She persuaded her father to go with her to the
theatre. She knew well that a signora had never set her foot in Don
Antonio's theatre, but she needed to see how it was going to be. It would
be such a dizzily great success for her railway if Don Antonio succeeded.
When Donna Micaela came to the theatre it was a few minutes before eight,
and Donna Emilia had not sold a ticket.
But she was not depressed; "Go in, Donna Micaela!" she said; "we shall
play at any rate, it is so beautiful. Don Antonio will play it for you
and your father and me. It is the most beautiful thing he has ever
performed."
Donna Micaela came into the little hall. It was hung with black, as the
big theatres always were in the old days when "The Old Martyrdom" was
given. There were dark, silver-fringed curtains on the stage, and the
little benches were covered with black.
Immediately after Donna Micaela came in, Don Antonio's bushy eyebrows
appeared in a little hole in the curtain. "Donna Micaela," he cried, as
Donna Emilia had done, "we shall play at any rate. It is so beautiful, it
needs no spectators."
Just then came Donna Emilia herself, and opened the door, and
courtesying, held it back. It was the priest, Don Matteo, who entered.
"What do you say to me, Donna Micaela?" he said, laughing. "But you
understand; it is 'The Old Martyrdom.' I saw it in my youth at the big
opera in Palermo; and I believe that it was that old play that made me
become a priest."
The next time the door opened it was Father Elia and Brother Tommaso,
who came with their violins under their arms and felt their way to their
usual places, as quietly as if they had never had any disagreement with
Don Antonio.
The door opened again. It was an old woman from the alley above the house
of the little Moor. She was dressed in black, and made the sign of the
cross as she came in.
After her came four, five other old women; and Donna Micaela looked at
them almost resentfully, as they gradually filled the theatre. She knew
that Don Antonio would not be satisfied till he had his own public back
again,--till he had his self-willed, beloved boys to play for.
Suddenly she heard a hurricane or thunder. The doors flew open,--all at
the same time! It was the boys. They threw themselves down in their usual
places, as if they had come back to their home.
They looked at one another, a little ashamed. But it had been impossible
for them to see one old woman after another go into their theatre to see
what was being played for them. It had been quite impossible to see the
whole street full of old distaff-spinners in slow procession toward the
theatre, and so they had rushed in.
But hardly had the gay young people reached their places before
they noticed that they had come under a severe master. Ah, "The Old
Martyrdom," "The Old Martyrdom!"
It was not given as in Aci and in Castelbuoco; it was not played as at
the opera in Palermo; it was only played with miserable marionettes with
immovable faces and stiff bodies; but the old play had not lost its
power.
Donna Micaela noticed it already in the second act during the Last
Supper. The boys began to hate Judas. They shouted threats and insults at
him.
As the story of the Passion went on, they laid aside their hats and
clasped their hands. They sat quite still, with their beautiful brown
eyes turned towards the stage. Now and then a few tears dropped. Now and
then a fist was clenched in indignation.
Don Antonio spoke with tears in his voice; Donna Emilia was on her knees
at the entrance. Don Matteo looked with a gentle smile at the little
puppets and remembered the wonderful spectacle in Palermo that had made
him a priest.
But when Jesus was cast into prison and tortured, the young people were
ashamed of themselves. They too had hated and persecuted. They were like
those pharisees, like those Romans. It was a shame to think of it. Could
Don Antonio forgive them?
V
THE LADY WITH THE IRON RING
Donna Micaela often thought of a poor little dressmaker whom she had
seen in her youth in Catania. She dwelt in the house next to the Palazzo
Palmeri, sitting always in the gateway with her work, so that Donna
Micaela had seen her a thousand times. She always sat and sang, and she
had certainly only known a single canzone. Always, always she sang the
same song.
"I have cut a curl from my black hair," she had sung. "I have unfastened
my black, shining braids, and cut a curl from my hair. I have done it
to gladden my friend, who is in trouble. Alas, my beloved is sitting in
prison; my beloved will never again twine my hair about his fingers. I
have sent him a lock of my hair to remind him of the silken chains that
never more will bind him."
Donna Micaela remembered the song well. It seemed as if it had sounded
through all her childhood to warn her of the suffering that awaited her.
* * * * *
Donna Micaela often sat at that time on the stone steps of the church of
San Pasquale. She saw wonderful events take place far off on that Etna so
rich in legends.
Over the black lava glided a railway train on newly laid shining rails.
It was a festival train; flags waved along the road; there were wreaths
on the carriages; the seats were covered with purple cushions. At the
stations the people stood and shouted: "Long live the king! long live the
queen! long live the new railway!"
She heard it so well; she herself was on the train. Ah, how honored, how
honored she was! She was summoned before the king and queen; and they
thanked her for the new railway. "Ask a favor of us, princess!" said the
king, giving her the title that the ladies of the race of Alagona had
formerly borne.
"Sire," she answered, as people answer in stories, "give freedom to the
last Alagona!"
And it was granted to her. The king could not say no to a prayer from her
who had built that fine railway, which was to give riches to all Etna.
* * * * *
When Donna Micaela lifted her arm so that her dress-sleeve slid up, one
saw that she wore as a bracelet a ring of rusty iron. She had found it
in the street, forced it over her hand, and now she always wore it.
Whenever she happened to see or touch it, she grew pale, and her eyes
no longer saw anything of the world about her. She saw a prison like
that of Foscari in the doge's palace in Venice. It was a dark, narrow,
cellar-like hole; light filtered in through a grated aperture; and from
the wall hung a great bunch of chains, which wound like serpents round
the prisoner's legs and arms and neck.
May the saint work a miracle! May the people work! May she herself soon
have such praise that she can beg freedom for her prisoner! He will die
if she does not hurry. May the iron ring eat incessantly into her arm, so
that she shall not forget him for a second.
VI
FRA FELICE'S LEGACY
When Donna Emilia opened the ticket-office to sell tickets for the second
performance of "The Old Martyrdom," the people stood in line to get
places; the second evening the theatre was so overcrowded that people
fainted in the crush, and the third evening people came from both Adernó
and Paternó to see the beloved tragedy. Don Antonio foresaw that he
would be able to play it a whole month for double price, and with two
performances every evening.
How happy they were, he and Donna Emilia, and with what joy and gratitude
they laid twenty-five lire in the collection-box of the little image!
In Diamante the incident caused great surprise, and many came to Donna
Elisa to find out if she believed that the saint wished them to support
Donna Micaela.
"Have you heard, Donna Elisa," they said, "that Don Antonio Greco has
been helped by the Christchild in San Pasquale, because he promised to
give the receipts of one evening to Donna Micaela's railway?"
But when they asked Donna Elisa about it, she shut her mouth and looked
as if she could not think of anything but her embroidery.
Fra Felice himself came in and told her of the two miracles the image had
already worked.
"Signorina Tottenham was very stupid to let the image go, if it is such a
miracle-worker," said Donna Elisa.
So they all thought. Signorina Tottenham had owned the image many years,
and she had not noticed anything. It probably could not work miracles; it
was only a coincidence.
It was unfortunate that Donna Elisa would not believe. She was the only
one of the old Alagonas left in Diamante, and the people followed her,
more than they themselves knew. If Donna Elisa had believed, the whole
town would have helped Donna Micaela.
But Donna Elisa could not believe that God and the saints wished to aid
her sister-in-law.
She had watched her since the festival of San Sebastiano. Whenever any
one spoke of Gaetano, she turned pale, and looked very troubled. Her
features became like those of a sinful man, when he is racked with the
pangs of conscience.
Donna Elisa sat and thought of it one morning, and it was so engrossing
that she let her needle rest. "Donna Micaela is no Etna woman," she said
to herself. "She is on the side of the government; she is glad that
Gaetano is in prison."
Out in the street at that same moment people came carrying a great
stretcher. On it lay heaped up a mass of church ornaments; chandeliers
and shrines and reliquaries. Donna Elisa looked up for a moment, then
returned to her thoughts.
"She would not let me adorn the house of the Alagonas on the festival
of San Sebastiano," she thought. "She did not wish the saint to help
Gaetano."
Two men came by dragging a rattling dray on which lay a mountain of red
hangings, richly embroidered stoles, and altar pictures in broad, gilded
frames.
Donna Elisa struck out with her hand as if to push away all doubts. It
could not be an actual miracle which had happened. The saint must know
that Diamante could not afford to build a railway.
People now came past driving a yellow cart, packed full of music-stands,
prayer-books, praying-desks and confessionals.
Donna Elisa woke up. She looked out between the rosaries that hung
in garlands over the window panes. That was the third load of church
furnishings that had passed. Was Diamante being plundered? Had the
Saracens come to the town?
She went to the door to see better. Again came a stretcher, and on it lay
mourning-wreaths of tin, tablets with long inscriptions, and coats of
arms, such as are hung up in churches in memory of the dead.
Donna Elisa asked the bearers, and learned what was happening. They were
clearing out the church of Santa Lucia in Gesù. The syndic and the town
council had ordered it turned into a theatre.
After the uprising there had been a new syndic in Diamante. He was a
young man from Rome, who did not know the town, but nevertheless wished
to do something for it. He had proposed to the town-council that Diamante
should have a theatre like Taormina and other towns. They could quite
easily fit up one of the churches as a play-house. They certainly had
more than enough, with five town churches and seven monastery churches;
they could easily spare one of them.
There was for instance the Jesuits' church, Santa Lucia in Gesù. The
monastery surrounding it was already changed to a barracks, and the
church was practically deserted. It would make an excellent theatre.
That was what the new syndic had proposed, and the town-council had
agreed to it.
When Donna Elisa heard what was going on she threw on her mantilla and
veil, and hurried to the Lucia church, with the same haste with which one
hurries to the house where one knows that some one is dying.
"What will become of the blind?" thought Donna Elisa. "How can they live
without Santa Lucia in Gesù?"
When Donna Elisa reached the silent little square, round which the
Jesuits' long, ugly monastery is built, she saw on the broad stone
steps that extend the whole length of the church front, a row of ragged
children and rough-haired dogs. All of them were leaders of the blind,
and they cried and whined as loud as they could.
"What is the matter with you all?" asked Donna Elisa. "They want to take
our church away from us," wailed the children. And thereupon all the dogs
howled more piteously than ever, for the dogs of the blind are almost
human.
At the church-door Donna Elisa met Master Pamphilio's wife, Donna
Concetta. "Ah, Donna Elisa," she said, "never in all your life have you
seen anything so terrible. You had better not go in."
But Donna Elisa went on.
In the church at first she saw nothing but a white cloud of dust. But
hammer-strokes thundered through the cloud, for some workmen were busy
breaking away a big stone knight, lying in a window niche.
"Lord God!" said Donna Elisa, and clasped her hands together; "they are
tearing down Sor Arrigo!" And she thought how tranquilly he had lain in
his niche. Every time she had seen him she had wished that she might be
as remote from disturbance and change as old Sor Arrigo.
In the church of Lucia there was still another big monument. It
represented an old Jesuit, lying on a black marble sarcophagus with a
scourge in his hand and his cap drawn far down over his forehead. He was
called Father Succi, and the people used to frighten their children with
him in Diamante.
"Would they also dare to touch Father Succi?" thought Donna Elisa. She
felt her way through the plaster dust to the choir, where the sarcophagus
stood, in order to see if they had dared to move the old Jesuit.
Father Succi still lay on his stone bed. He lay there dark and hard,
as he had been in life; and one could almost believe that he was still
alive. Had there been doctors and tables with medicine-bottles and
burning candles beside the bed, one would have believed that Father Succi
lay sick in the choir of his church, waiting for his last hour.
The blind sat round about him, like members of the family who gather
round a dying man, and rocked their bodies in silent grief. There were
both the women from the hotel court-yard, Donna Pepa and Donna Tura;
there was old Mother Saraedda, who ate the bread of charity at the house
of the Syndic Voltaro; there were blind beggars, blind singers, blind of
all ages and conditions. All the blind of Diamante were there, and in
Diamante there is an incredible number who no longer see the light of the
sun.
They all sat silent most of the time, but every now and then one of them
burst into a wail. Sometimes one of them felt his way forward to the
monk, Father Succi, and threw himself weeping aloud across him.
It made it all the more like a death-bed that the priest and Father Rossi
from the Franciscan monastery were there and were trying to comfort the
despairing people.
Donna Elisa was much moved. Ah, so often she had seen those people happy
in her garden, and now to meet them in such misery! They had won pleasant
tears from her when they had sung mourning-songs over her husband, Signor
Antonelli, and over her brother, Don Ferrante. She could not bear to see
them in such need.
Old Mother Saraedda began to speak to Donna Elisa.
"I knew nothing when I came, Donna Elisa," said the old woman. "I left
my dog outside on the steps and went in through the church door. Then I
stretched out my arm to push aside the curtain over the door, but the
curtain was gone. I put my foot down as if there were a step to mount
before the threshold, but there was no step. I stretched out my hand
to take the holy water; I courtesied as I went by the high altar; and I
listened for the little bell that always rings when Father Rossi comes to
the mass. Donna Elisa, there was no holy water, no altar, no bell; there
was nothing!"
"Poor thing, poor thing," said Donna Elisa.
"Then I hear how they are hammering and pounding up in a window. 'What
are you doing with Sor Arrigo?' I cry, for I hear instantly that it is in
Sor Arrigo's window.
"'We are going to carry him away,' they answer me.
"Just then the priest, Don Matteo, comes to me, takes me by the hand, and
explains everything. And I am almost angry with the priest when he says
that it is for a theatre. They want our church for a theatre!
"'Where is Father Succi?' I say instantly. 'Is Father Succi still here?'
And he leads me to Father Succi. He has to lead me, for I cannot find
my way. Since they have taken away all the chairs and praying-desks and
carpets and platforms and folding steps, I cannot find my way. Before, I
found my way about here as well as you."
"The priest will find you another church," said Donna Elisa. "Donna
Elisa," said the old woman, "what are you saying? You might as well say
that the priest can give us sight. Can Don Matteo give us a church where
we see, as we saw in this? None of us needed a guide here. There, Donna
Elisa, stood an altar; the flowers on it were red as Etna at sunset, and
we saw it. We counted sixteen wax-lights over the high altar on Sundays,
and thirty on festival days. We could see when Father Rossi held the
mass here. What shall we do in another church, Donna Elisa? There we
shall not be able to see anything. They have extinguished the light of
our eyes anew."
Donna Elisa's heart grew as warm as if molten lava had run over it. It
was certainly a great wrong they were doing to those blind unfortunates.
So Donna Elisa went over to Don Matteo.
"Your Reverence," she said, "have you spoken to the syndic?"
"Alas, alas, Donna Elisa," said Don Matteo, "it is better for you to try
to talk to him than for me."
"Your Reverence, the syndic is a stranger; perhaps he has not heard of
the blind."
"Signor Voltaro has been to him; Father Rossi has been to him; and I too,
I too. He answers nothing but that he cannot change what is decided in
the town Junta. We all know, Donna Elisa, that the town Junta cannot take
back anything. If it has decided that your cat shall hold mass in the
Cathedral, it cannot change it."
Suddenly there was a movement in the church. A large blind man came in.
"Father Elia!" the people whispered, "Father Elia!"
Father Elia was the head man of the company of blind singers, who always
collected there. He had long white hair and beard, and was beautiful as
one of the holy patriarchs.
He, like all the others, went forward to Father Succi. He sat down beside
him, and leaned his head against the coffin.
Donna Elisa went up to Father Elia and spoke to him. "Father Elia," she
said, "_you_ ought to go to the syndic."
The old man recognized Donna Elisa's voice, and he answered her, in his
thick, old-man's tones:--
"Do you suppose that I have waited to have you say that to me? Don't you
know that my first thought was to go to the syndic?"
He spoke with such a hard and distinct voice that the workmen stopped
hammering and listened, thinking some one had begun to preach.
"I told him that we blind singers are a company, and that the Jesuits
opened their church for us more than three hundred years ago, and gave us
the right to gather here to select new members and try new songs.
"And I said to him that there are thirty of us in the company; and that
the holy Lucia is our patroness; and that we never sing in the streets,
only in courts and in rooms; and that we sing legends of the saints and
mourning-songs, but never a wanton song; and that the Jesuit, Father
Succi, opened the church for us, because the blind are Our Lord's singers.
"I told him that some of us are _recitatori_, who can sing the old songs,
but others are _trovatori_, who compose new ones. I said to him that we
give pleasure to many on the noble isle. I asked him why he wished to
deprive us of life. For the homeless cannot live.
"I said to him that we wander from town to town through all Etna, but the
church of Lucia is our home, and mass is held here for us every morning.
Why should he refuse us the comfort of God's word?
"I told him that the Jesuits once changed their attitude towards us and
wished to drive us away from their church, but they did not succeed. We
received a letter from the Viceroy that we might hold our meetings in
perpetuity in Santa Lucia in Gesù. And I showed him the letter."
"What did he answer?"
"He laughed at me."
"Can none of the other gentlemen help you?"
"I have been to them, Donna Elisa. All the morning I have been sent from
Herod to Pilatus."
"Father Elia," said Donna Elisa with lowered voice, "have you forgotten
to call on the saints?"
"I have called on both the black Madonna and San Sebastiano and Santa
Lucia. I have prayed to as many as I could name."
"Do you think, Father Elia," said Donna Elisa, and lowered her voice
still more, "that Don Antonio Greco was helped, because he promised money
to Donna Micaela's railway?"
"I have no money to give," said the old man, disconsolately.
"Still, you ought to think of it, Father Elia," said Donna Elisa,
"since you are in such straits. You ought to try if, by promising the
Christ-image that you yourself and all who belong to your company will
speak and sing of the railway, and persuade people to give contributions
to it, you may keep your church. We do not know if it can help, but one
ought to try every possible thing, Father Elia. It costs nothing to
promise."
"I will promise anything for your sake," said the old man.
He laid his old blind head again against the black coffin, and Donna
Elisa understood that he had given the promise in his desire to be left
in peace with his sorrow.
"Shall I present your vow to the Christ-image?" she said.
"Do as you will, Donna Elisa," said the old man.
* * * * *
That same day old Fra Felice had risen at five o'clock in the morning
and begun to sweep out his church. He felt quite active and well; but
while he was working it seemed as if San Pasquale, sitting with his
bag of stones outside the church-door, had something to say to him. He
went out, but there was nothing the matter with San Pasquale; quite the
contrary. Just then the sun glided up from behind Etna, and down the dark
mountain-sides the rays came hurrying, many- as harp-strings. When
the rays reached Fra Felice's old church they turned it rosy red; rosy
red were also the old barbaric pillars that held up the canopy over the
image, and San Pasquale with his bag of stones, and Fra Felice himself.
"We look like young boys," thought the old man; "we have still long years
to live."
But as he was going back into the church, he felt a sharp pressure at
his heart, and it came into his mind that San Pasquale had called him
out to say farewell. At the same time his legs became so heavy that he
could hardly move them. He felt no pain, but a weariness which could mean
nothing but death. He was scarcely able to put his broom away behind the
door of the sacristy; then he dragged himself up the choir, lay down on
the platform in front of the high altar, and wrapped his cloak about him.
The Christ-image seemed to nod to him and say: "Now I need you, Fra
Felice." He lay and nodded back: "I am ready; I shall not fail you."
It was only to lie and wait; and it was beautiful, Fra Felice thought.
He had never before in all his life had time to feel how tired he was.
Now at last he might rest. The image would keep up the church and the
monastery without him.
He lay and smiled at the thought that old San Pasquale had called him out
to say good-morning to him.
Fra Felice lay thus till late in the day, and dozed most of the time. No
one was with him, and a feeling came over him that it would not do to
creep in this way out of life. It was as if he had cheated somebody of
something. That woke him time after time. He ought of course to get the
priests, but he had no one to send for them.
While he lay there he thought that he shrank together more and more.
Every time he awoke he thought that he had grown smaller. He felt as if
he were quite disappearing. Now he could certainly wind his cloak four
times about him.
He would have died quite by himself if Donna Elisa had not come to ask
help for the blind of the little image. She was in a strange mood when
she came, for she wished of course to get help for the blind, but yet she
did not wish Donna Micaela's plans to be promoted.
When she came into the church she saw Fra Felice lying on the platform
under the altar, and she went forward and knelt beside him.
Fra Felice turned his eyes towards her and smiled quietly. "I am going
to die," he said, hoarsely; but he corrected himself and said: "I am
permitted to die."
Donna Elisa asked what the matter was, and said that she would fetch help.
"Sit down here," he said, and made a feeble attempt to wipe away the dust
on the platform with his sleeve.
Donna Elisa said that she wished to fetch the priests and sisters of
charity.
He seized her skirt and held her back.
"I want to speak to you first, Donna Elisa."
It was hard for him to talk, and he breathed heavily after each word.
Donna Elisa sat down beside him and waited.
He lay for a while and panted; then a flush rose to his cheeks; his eyes
began to shine, and he spoke with ease and eagerness.
"Donna Elisa," said Fra Felice, "I have a legacy to give away. It has
troubled me all day. I do not know to whom I shall give it."
"Fra Felice," said Donna Elisa, "do not concern yourself with such a
thing. There is no one who does not need a good gift."
But now when Fra Felice's strength had returned, he wished, before he
made up his mind about the legacy, to tell Donna Elisa how good God had
been to him.
"Has not God been great in his grace to make me a _polacco_?" he said.
"Yes, it is a great gift," said Donna Elisa.
"Only to be a little, little _polacco_ is a great gift," said Fra Felice;
"it is especially useful since the monastery has been given up, and when
my comrades are gone or dead. It means having a bag full of bread before
one even stretches out one's hand to beg. It means always seeing bright
faces, and being greeted with deep reverences. I know no greater gift
for a poor monk, Donna Elisa."
Donna Elisa thought how revered and loved Fra Felice had been, because he
had been able to predict what numbers would come out in the lottery. And
she could not help agreeing with him.
"If I came wandering along the road in the heat," said Fra Felice, "the
shepherd came to me and went with me a long way, and held his umbrella
over me as shelter against the sun. And when I came to the laborers in
the cool stone-quarries, they shared their bread and their bean-soup
with me. I have never been afraid of brigands nor of _carabinieri_. The
official at the custom-house has shut his eyes when I went by with my
bag. It has been a good gift, Donna Elisa."
"True, true," said Donna Elisa.
"It has not been an arduous profession," said Fra Felice. "They spoke to
me, and I answered them; that was all. They knew that every word has its
number, and they noticed what I said and played accordingly. I never knew
how it happened, Donna Elisa; it was a gift from God."
"You will be a great loss to the poor people, Fra Felice," said Donna
Elisa.
Fra Felice smiled. "They care nothing for me on Sunday and Monday, when
there has just been a drawing," he said. "But they come on Thursday
and Friday and on Saturday morning, because there is a drawing every
Saturday."
Donna Elisa began to be anxious, because the dying man thought of nothing
but that. Suddenly there flashed across her memory thoughts of one and
another who had lost in the lottery, and she remembered several who had
played away all their prosperity. She wished to turn his thoughts from
that sinful lottery business.
"You said that you wished to speak of your will, Fra Felice."
"But it is because I have so many friends that it is hard for me to know
to whom I shall give the legacy. Shall I give it to those who have baked
sweet cakes for me, or to those who have offered me artichokes, browned
in sweet oil? Or shall I bequeath it to the sisters of charity who nursed
me when I was ill?"
"Have you much to give away, Fra Felice?"
"It will do, Donna Elisa. It will do."
Fra Felice seemed to be worse again; he lay silent with panting breast.
"I had also wished to give it to all poor, homeless monks, who had lost
their monasteries," he whispered.
And then after thinking for a while: "I should also have liked to give it
to the good old man in Rome. He, you know, who watches over us all."
"Are you so rich, Fra Felice?" said Donna Elisa.
"I have enough, Donna Elisa; I have enough."
He closed his eyes, and rested for a while; then he said:--
"I want to give it to everybody, Donna Elisa."
He acquired new strength at the thought; a slight flush was again visible
in his cheeks, and he raised himself on his elbow.
"See here, Donna Elisa," he said, while he thrust his hand into his cloak
and drew out a sealed envelope, which he handed to her, "you shall go and
give this to the syndic, to the syndic of Diamante.
"Here, Donna Elisa," said Fra Felice, "here are the five numbers that win
next Saturday. They have been revealed to me, and I have written them
down. And the syndic shall take these numbers and have them fastened up
on the Roman Gate, where everything of importance is published. And he
shall let the people know that it is my testament. I bequeath it to the
people. Five winning numbers, a whole quintern, Donna Elisa!"
Donna Elisa took the envelope and promised to give it to the syndic. She
could do nothing else, for poor Fra Felice had not many minutes left to
live.
"When Saturday comes," said Fra Felice, "there will be many who will
think of Fra Felice. 'Can old Fra Felice have deceived us?' they will ask
themselves. 'Can it be possible for us to win the whole quintern?'
"On Saturday evening there is a drawing on the balcony of the town-hall
in Catania, Donna Elisa. Then they carry out the lottery-wheel and
table, and the managers of the lottery are there, and the pretty little
poor-house child. And one number after another is put into the lucky
wheel until they are all there, the whole hundred.
"All the people stand below and tremble in expectation, as the sea
trembles before the storm-wind.
"Everybody from Diamante will be there, and they will stand quite pale
and hardly daring to look one another in the face. Before, they have
believed, but not now. Now they think that old Fra Felice has deceived
them. No one dares to cherish the smallest hope.
"Then the first number is drawn, and I was right. Ah, Donna Elisa, they
will be so astonished they will scarcely be able to rejoice. For they
have all expected disappointment. When the second number comes out, there
is the silence of death. Then comes the third. The lottery managers will
be astonished that everything is so quiet. 'To-day they are not winning
anything,' they will say. 'To-day the state has all the prizes.' Then
comes the fourth number. The poor-house child takes the roll from the
wheel; and the marker opens the roll, and shows the number. Down among
the people it is almost terrible; no one is able to say a word for joy.
Then the last number comes. Donna Elisa, the people scream, they cry,
they fall into one another's arms and sob. They are rich. All Diamante is
rich--"
Donna Elisa had kept her arm under Fra Felice's head and supported him
while he had panted out all this. Suddenly his head fell heavily back.
Old Fra Felice was dead.
* * * * *
While Donna Elisa was with old Fra Felice, many people in Diamante had
begun to trouble themselves about the blind. Not the men; most of the men
were in the fields at work; but the women. They had come in crowds to
Santa Lucia to console the blind, and finally, when about four hundred
women had gathered together, it occurred to them to go and speak to the
syndic.
They had gone up to the square and called for the syndic. He had come out
on the balcony of the town-hall, and they had prayed for the blind. The
syndic was a kind and handsome man. He had answered them pleasantly,
but had not been willing to yield. He could not repeal what had been
decided in the town Junta. But the women were determined that it should
be repealed, and they remained in the square. The syndic went into the
town-hall again, but they stayed in the square and called and prayed.
They did not intend to go away till he yielded.
While this was going on, Donna Elisa came to give the syndic Fra Felice's
testament. She was grieved unto death at all the misery, but at the same
time she felt a bitter satisfaction, because she had received no help
from the Christchild. She had always believed that the saints did not
wish to help Donna Micaela.
It was a fine gift she had received in San Pasquale's church. Not only
could it not help the blind, but it was in a fair way to ruin the whole
town. Now what little the people still possessed would go to the lottery
collector. There would be a borrowing and a pawning.
The syndic admitted Donna Elisa immediately, and was as calm and polite
as always, although the women were calling in the square, the blind were
bemoaning themselves in the waiting-room, and people had run in and out
of his room all day.
"How can I be at your service, Signora Antonelli?" he said. Donna Elisa
first looked about and wondered to whom he was speaking. Then she told
about the testament.
The syndic was neither frightened nor surprised. "That is very
interesting," he said, and stretched out his hand for the paper.
But Donna Elisa held the envelope fast and asked: "Signor Sindaco, what
do you intend to do with it? Do you intend to fasten it to the Roman
Gate?"
"Yes; what else can I do, signora? It is a dead man's last wish."
Donna Elisa would have liked to tell him what a terrible testament it
was, but she checked herself to speak of the blind.
"Padre Succi, who directed that the blind should always be allowed in his
church, is also a dead man," she interposed.
"Signora Antonelli, are you beginning with that too?" said the syndic,
quite kindly. "It was a mistake; but why did no one tell me that the
blind frequent the church of Lucia? Now, since it is decided, I cannot
annul the decision; I cannot."
"But their rights and patents, Signor Sindaco?"
"Their rights are worth nothing. They have to do with the Jesuits'
monastery, but there is no longer such a monastery. And tell me, Signora
Antonelli, what will become of me if I yield?"
"The people will love you as a good man."
"Signora, people will believe that I am a weak man, and every day I shall
have four hundred laborers' wives outside the town-hall, begging now for
one thing, now for another. It is only to hold out for one day. To-morrow
it will be forgotten."
"To-morrow!" said Donna Elisa; "we shall never forget it."
The syndic smiled, and Donna Elisa saw that he thought that he knew the
people of Diamante much better than she.
"You think that their hearts are in it?" he said.
"I think so, Signor Sindaco."
Then the syndic laughed softly. "Give me that envelope, Signora."
He took it and went out on the balcony.
He began to speak to the women. "I wish to tell you," he said, "that I
have just now heard that old Fra Felice is dead, and that he has left a
legacy to you all. He has written down five numbers that are supposed
to win in the lottery next Saturday, and he bequeaths them to you. No
one has seen them yet. They are lying here in this envelope, and it is
unopened."
He was silent a moment to let the women have time to think over what he
had said.
Instantly they began to cry: "The numbers, the numbers!"
The syndic signed to them to be silent.
"You must remember," he said, "that it was impossible for Fra Felice
to know what numbers will be drawn next Saturday. If you play on these
numbers, you may all lose. And we cannot afford to be poorer than we
are already here in Diamante. I ask you therefore to let me destroy the
testament without any one seeing it."
"The numbers," cried the women, "give us the numbers!"
"If I am permitted to destroy the testament," said the syndic, "I promise
you that the blind shall have their church again."
There was silence in the square. Donna Elisa rose from her seat in the
hall of the court-house and seized the back of her chair with both hands.
"I leave it to you to choose between the church and the numbers," said
the syndic.
"God in heaven!" sighed Donna Elisa, "is he a devil to tempt poor people
in such a way?"
"We have been poor before," cried one of the women, "we can still be
poor."
"We will not choose Barabbas instead of Christ," cried another.
The syndic took a match-box from his pocket, lighted a match, and brought
it slowly up to the testament.
The women stood quiet and let Fra Felice's five numbers be destroyed. The
blind people's church was saved.
"It is a miracle," whispered old Donna Elisa; "they all believe in Fra
Felice, and they let his numbers burn. It is a miracle."
* * * * *
Later in the afternoon Donna Elisa again sat in her shop with her
embroidery frame. She looked old as she sat there, and there was
something shaken and broken about her. It was not the usual Donna Elisa;
it was a poor, elderly, forsaken woman.
She drew the needle slowly through the cloth, and when she wished to take
another stitch she was uncertain and at a loss. It was hard for her to
keep the tears from falling on her embroidery and spoiling it.
Donna Elisa was in such great grief for to-day she had lost Gaetano
forever. There was no more hope of getting him back.
The saints had gone over to the side of the opponent, and worked miracles
in order to help Donna Micaela. No one could doubt that a miracle had
happened. The poor women of Diamante would never have been able to stand
still while Fra Felice's numbers burned if they had not been bound by a
miracle.
It made a poor soul so old and cross to have the good saints help Donna
Micaela, who did not like Gaetano.
The door-bell jingled violently, and Donna Elisa rose from old habit.
It was Donna Micaela. She was joyful, and came toward Donna Elisa with
outstretched hands. But Donna Elisa turned away, and could not press her
hand.
Donna Micaela was in raptures. "Ah, Donna Elisa, you have helped my
railway. What can I say? How shall I thank you?"
"Never mind about thanking me, sister-in-law!"
"Donna Elisa!"
"If the saints wish to give us a railway, it must be because Diamante
needs it, and not because they love _you_."
Donna Micaela shrank back. At last she thought she understood why Donna
Elisa was angry with her. "If Gaetano were at home," she said. She stood
and pressed her hand to her heart and moaned. "If Gaetano were at home he
would not allow you to be so cruel to me."
"Gaetano?--would not Gaetano?"
"No, he would not. Even if you are angry with me because I loved him
while my husband was alive, you would not dare to upbraid me for it if he
were at home."
Donna Elisa lifted her eyebrows a little. "You think that he could
prevail upon me to be silent about such a thing," she said, and her voice
was very strange.
"But, Donna Elisa," Donna Micaela whispered in her ear, "it is
impossible, quite impossible not to love him. He is beautiful; don't you
know it? And he subjugates me, and I am afraid of him. You must let me
love him."
"Must I?" Donna Elisa kept her eyes down and spoke quite shortly and
harshly.
Donna Micaela was beside herself. "It is I whom he loves," she said. "It
is not Giannita, but me, and you ought to consider me as a daughter;
you ought to help me; you ought to be kind to me. And instead you stand
against me; you are cruel to me. You do not let me come to you and talk
of him. However much I long, and however much I work, I may not tell you
of it."
Donna Elisa could hold out no longer. Donna Micaela was nothing but a
child, young and foolish and quivering like a bird's heart,--just one to
be taken care of. She had to throw her arms about her.
"I never knew it, you poor, foolish child," she said.
VII
AFTER THE MIRACLE
The blind singers had a meeting in the church of Lucia. Highest up in the
choir behind the altar sat thirty old, blind, men on the carved chairs
of the Jesuit fathers. They were poor, most of them; most of them had a
beggar's wallet and a crutch beside them.
They were all very earnest and solemn; they knew what it meant to be
members of that holy band of singers, of that glorious old Academy.
Now and then below in the church a subdued noise was audible. The blind
men's guides were sitting there, children, dogs, and old women, waiting.
Sometimes the children began to romp with one another and with the dogs,
but it was instantly suppressed and silenced.
Those of the blind who were _trovatori_ stood up one after another and
spoke new verses.
"You people who live on holy Etna," one of them recited, "men who live
on the mountain of wonders, rise up, give your mistress a new glory! She
longs for two ribbons to heighten her beauty, two long, narrow bands of
steel to fasten her mantle. Give them to your mistress, and she will
reward you with riches; she will give gold for steel. Countless are the
treasures that she in her might will give them who assist her."
"A gentle worker of miracles has come among us," said another. "He stands
poor and unnoticed in the bare old church, and his crown is of tin, and
his diamonds of glass. 'Make no sacrifices to me, O ye poor,' he says;
'build me no temple, all ye who suffer. I will work for your happiness.
If prosperity shines from your houses, I shall shine with precious
stones; if want flees from the land, my feet will be clothed in golden
shoes embroidered with pearls.'"
As each new verse was recited, it was accepted or rejected. The blind men
judged with great severity.
The next day they wandered out over Etna, and sang the railway into the
people's hearts.
* * * * *
After the miracle of Fra Felice's legacy, people began to give
contributions to the railway. Donna Micaela soon had collected about a
hundred lire. Then she and Donna Elisa made the journey to Messina to
look at the steam-tram that runs between Messina and Pharo. They had no
greater ambition; they would be satisfied with a steam-tram.
"Why does a railway need to be so expensive?" said Donna Elisa. "It
is just an ordinary road, although people do lay down two steel rails
on it. It is the engineer and the fine gentlemen who make a railway
expensive. Don't trouble yourself about engineers, Micaela! Let our good
road-builders, Giovanni and Carmelo, build your railway."
They carefully inspected the steam-tramway to Pharo and brought back all
the knowledge they could. They measured how wide it ought to be between
the rails, and Donna Micaela drew on a piece of paper the way the rails
ran by one another at the stations. It was not so difficult; they were
sure they would come out well.
That day there seemed to be no difficulties. It was as easy to build a
station as an ordinary house, they said. Besides, more than two stations
were not needed; a little sentry-box was sufficient at most of the
stopping-places.
If they could only avoid forming a company, taking fine gentlemen into
their service, and doing things that cost money, their plan of the
railway would be realized. It would not cost so much. The ground they
could certainly get free. The noble gentlemen who owned the land on Etna
would of course understand how much use of the railway they would have,
and would let it pass free of charge over their ground.
They did not trouble themselves to stake out the line beforehand. They
were going to begin at Diamante and gradually build their way to Catania.
They only needed to begin and lay a little piece every day. It was not so
difficult.
After that journey they began the attempt to build the road at their own
risk. Don Ferrante had not left a large inheritance to Donna Micaela,
but one good thing that he had bequeathed her was a long stretch of
lava-covered waste land off on Etna. Here Giovanni and Carmelo began to
break ground for the new railway.
When the work began, the builders of the railway possessed only one
hundred lire. It was the miracle of the legacy that had filled them with
holy frenzy.
What a railway it would be, what a railway!
The blind singers were the share-collectors, the Christ-image gave the
concession, and the old shop woman, Donna Elisa, was the engineer.
VIII
A JETTATORE
In Catania there was once a man with "the evil eye," a _jettatore_. He
was almost the most terrible _jettatore_ who had ever lived in Sicily.
As soon as he showed himself on the street people hastened to bend their
fingers to the protecting sign. Often it did not help at all; whoever met
him could prepare himself for a miserable day; he would find his dinner
burned, and the beautiful old jelly-bowl broken. He would hear that his
banker had suspended payments, and that the little note that he had
written to his friend's wife had come into the wrong hands.
Most often a _jettatore_ is a tall, thin man, with pale, shy eyes and
a long nose, which overhangs and _hacks_ his upper lip. God has set
the mark of a parrot's beak upon the _jettatore_. Yet all things are
variable; nothing is absolutely constant. This _jettatore_ was a little
fellow with a nose like a San Michele.
Thereby he did much more harm than an ordinary _jettatore_. How much
oftener is one pricked by a rose than burned by a nettle!
A _jettatore_ ought never to grow up. He is well off only when he is a
child. Then he still has his little mamma, and she never sees the evil
eye; she never understands why she sticks the needle into her finger
every time he comes to her work-table. She will never be afraid to kiss
him. Although she has sickness constantly in the house, and the servants
leave, and her friends draw away, she never notices anything.
But after the _jettatore_ has come out into the world, he often has a
hard time enough. Every one must first of all think of himself; no one
can ruin his life by being kind to a _jettatore_.
There are several priests who are _jettatori_. There is nothing strange
in that; the wolf is happy if he can tear to pieces many sheep. They
could not very well do more harm than by being priests. One need only ask
what happens to the children whom he baptizes, and the couples whom he
marries.
The _jettatore_ in question was an engineer and wished to build railways.
He had also a position in one of the state railway buildings. The
state could not know that he was a _jettatore_. Ah, but what misery,
what misery! As soon as he obtained a place on the railway a number of
accidents occurred. When they tunnelled through a hill, one cave-in after
another; when they tried to lay a bridge, breach upon breach; when they
exploded a blast, the workmen were killed by the flying fragments.
The only one who was never injured was the engineer, the _jettatore_.
The poor fellows working under him! They counted their fingers and limbs
every evening. "To-morrow perhaps we will have lost you," they said.
They informed the chief engineer; they informed the minister. Neither
of them would listen to the complaint. They were too sensible and too
learned to believe in the evil eye. The workmen ought to mind better what
they were about. It was their own fault that they met with accidents.
And the gravel-cars tipped over; the locomotive exploded.
One morning there was a rumor that the engineer was gone. He had
disappeared; no one knew what had become of him. Had some one perhaps
stabbed him? Oh, no; oh, no! would any one have dared to kill a
_jettatore_?
But he was really gone; no one ever saw him again.
It was a few years later that Donna Micaela began to think of building
her railway. And in order to get money for it, she wished to hold a
bazaar in the great Franciscan monastery outside Diamante.
There was a cloister garden there, surrounded by splendid old pillars.
Donna Micaela arranged little booths, little lotteries, and little places
of diversion under the arcades. She hung festoons of Venetian lanterns
from pillar to pillar. She piled up great kegs of Etna wine around the
cloister fountain.
While Donna Micaela worked there she often conversed with little
Gandolfo, who had been made watchman at the monastery since Fra Felice's
death.
One day she made Gandolfo show her the whole monastery. She went through
it all from attic to cellar, and when she saw those countless little
cells with their grated windows and whitewashed walls and hard wooden
seats, she had an idea.
She asked Gandolfo to shut her in in one of the cells and to leave her
there for the space of five minutes.
"Now I am a prisoner," she said, when she was left alone. She tried the
door; she tried the window. She was securely shut in.
So that was what it was to be a prisoner! Four empty walls about one, the
silence of the grave, and the chill.
"Now I can feel as a prisoner feels," she thought.
Then she forgot everything else in the thought that possibly Gandolfo
might not come to let her out. He could be called away; he could be
taken suddenly ill; he could fall and kill himself in some of the dark
passage-ways. Many things could happen to prevent him from coming.
No one knew where she was; no one would think of looking for her in that
out-of-the-way cell. If she were left there for even an hour she would go
mad with terror.
She saw before her starvation, slow starvation. She struggled through
interminable hours of anguish. Ah, how she would listen for a step; how
she would call!
She would shake the door; she would scrape the masonry of the walls with
her nails; she would bite the grating with her teeth.
When they finally found her she would be lying dead on the floor, and
they would find everywhere traces of how she had tried to break her way
out.
Why did not Gandolfo come? Now she must have been there a quarter of an
hour, a half-hour. Why did he not come?
She was sure that she had been shut in a whole hour when Gandolfo came.
Where had he been such a long time?
He had not been long at all. He had only been away five minutes.
"God! God! so that is being a prisoner; that is Gaetano's life!" She
burst into tears when she saw the open sky once more above her.
A while later, as they stood out on an open _loggia_, Gandolfo showed her
a couple of windows with shutters and green shades.
"Does any one live there?" she asked.
"Yes, Donna Micaela, some one does."
Gandolfo told her that a man lived there who never went out except at
night,--a man who never spoke to any one.
"Is he crazy?" asked Donna Micaela.
"No, no; he is as much in his right mind as you or I. But people say that
he has to conceal himself. He is afraid of the government."
Donna Micaela was much interested in the man. "What is his name?" she
said.
"I call him Signor Alfredo."
"How does he get any food?" she asked.
"I prepare it for him," said Gandolfo.
"And clothes?"
"I get them for him. I bring him books and newspapers, too."
Donna Micaela was silent for a while. "Gandolfo," she said, and gave him
a rose which she held in her hand, "lay this on the tray the next time
you take food to your poor prisoner."
After that Donna Micaela sent some little thing almost every day to the
man in the monastery. It might be a flower, a book or some fruit. It was
her greatest pleasure. She amused herself with her fancies. She almost
succeeded in imagining that she was sending all these things to Gaetano.
When the day for the bazaar came, Donna Micaela was in the cloister early
in the morning. "Gandolfo," she said, "you must go up to your prisoner
and ask him if he will come to the entertainment this evening."
Gandolfo soon came back with the answer. "He thanks you very much, Donna
Micaela," said the boy. "He will come."
She was surprised, for she had not believed that he would venture out.
She had only wished to show him a kindness.
Something made Donna Micaela look up. She was standing in the cloister
garden, and a window was thrown open in one of the buildings above her.
Donna Micaela saw a middle-aged man of an attractive appearance standing
up there and looking down at her.
"There he is, Donna Micaela," said Gandolfo.
She was happy. She felt as if she had redeemed and saved the man. And it
was more than that. People who have no imagination will not understand
it. But Donna Micaela trembled and longed all day; she considered how she
would be dressed. It was as if she had expected Gaetano.
Donna Micaela soon had something else to do than to dream; the livelong
day a succession of calamities streamed over her.
The first was a communication from the old Etna brigand, Falco Falcone:--
DEAR FRIEND, DONNA MICAELA,--As I have heard that you intend
to build a railway along Etna, I wish to tell you that with my
consent it will never be. I tell you this now so that you need
not waste any more money and trouble on the matter.
Enlightened and most nobly born signora, I remain
Your humble servant,
FALCO FALCONE.
Passafiero, my sister's son, has written this letter.
Donna Micaela flung the dirty letter away. It seemed to her as if it were
the death sentence of the railway, but to-day she would not think of it.
Now she had her bazaar.
The moment after, her road-builders, Giovanni and Carmelo, appeared. They
wished to counsel her to get an engineer. She probably did not know what
kind of ground there was on Etna. There was, first, lava; then there was
ashes; and then lava again. Should the road be laid on the top layer of
lava, or on the bed of ashes, or should they dig down still deeper? About
how firm a foundation did a railway need? They could not go ahead without
a man who understood that.
Donna Micaela dismissed them. To-morrow, to-morrow; she had no time to
think of it to-day.
Immediately after, Donna Elisa came with a still worse piece of news.
There was a quarter in Diamante where a poverty-stricken and wild people
lived. Those poor souls had been frightened when they heard of the
railway. "There will be an eruption of Etna and an earthquake," they had
said. Great Etna will endure no fetters. It will shake off the whole
railway. And people said now that they ought to go out and tear up the
track as soon as a rail was laid on it.
A day of misfortune, a day of misfortune! Donna Micaela felt farther from
her object than ever.
"What is the good of our collecting money at our bazaar?" she said
despondingly.
The day promised ill for her bazaar. In the afternoon it began to rain.
It had not rained so in Diamante since the day when the clocks rang.
The clouds sank to the very house-roofs, and the water poured down from
them. People were wet to the skin before they had been two minutes in the
street. Towards six o'clock, when Donna Micaela's bazaar was to open, it
was raining its very hardest. When she came out to the monastery, there
was no one there but those who were to help in serving and selling.
She felt ready to cry. Such an unlucky day! What had dragged down all
these adversities upon her?
Donna Micaela's glance fell on a strange man who was leaning against a
pillar, watching her. Now all at once she recognized him. He was the
_jettatore_--the _jettatore_ from Catania, whom people had taught her to
fear as a child.
Donna Micaela went quickly over to him. "Come with me, signor," she said,
and went before him. She wished to go so far away that no one should hear
them, and then she wished to beg of him never to come before her eyes
again. She could do no less. He must not ruin her whole life.
She did not think in what direction she went. Suddenly she was at the
door of the monastery church and turned in there.
Within, it was almost dark. Only by the Christ-image a little oil lamp
was burning.
When Donna Micaela saw the Christ-image she was startled. Just then she
had not wished to see him.
He reminded her of the time when his crown had rolled to Gaetano's feet,
when he had been so angry with the brigands. Perhaps the Christ-image did
not wish her to drive away the _jettatore_.
She had good reason to fear the _jettatore_. It was wrong of him to come
to her entertainment; she must somehow be rid of him.
Donna Micaela had gone on through the whole church, and now stood and
looked at the Christ-image. She could not say a word to the man who
followed her.
She remembered what sympathy she had lately felt for him, because a
prisoner, like Gaetano. She had been so happy that she had tempted him
out to life. What did she now wish to do? Did she wish to send him back
to captivity?
She remembered both her father and Gaetano. Should this man be the third
that she--
She stood silent and struggled with herself. At last the _jettatore_
spoke:--
"Well, signora, is it not true that now you have had enough of me?"
Donna Micaela made a negative gesture.
"Do you not desire me to return to my cell?"
"I do not understand you, signor."
"Yes, yes, you understand. Something terrible has happened to you to-day.
You do not look as you did this morning."
"I am very tired," said Donna Micaela, evasively.
The man came close up to her as if to force out the truth. Questions and
answers flew short and panting between them.
"Do you not see that all your festival is likely to be a failure?"--"I
must arrange it again to-morrow."--"Have you not recognized me?"--"Yes,
I have seen you before in Catania."--"And you are not afraid of the
_jettatore_?"--"Yes, formerly, as a child."--"But now, now are you not
afraid?" She avoided answering him. "Are you yourself afraid?" she said.
"Speak the truth!" he said, impatiently. "What did you wish to say to me
when you brought me here?"
She looked anxiously about her. She had to say something; she must have
something to answer him. Then a thought occurred to her which seemed to
her quite terrible. She looked at the Christ-image. "Do you require it?"
she seemed to ask him. "Shall I do it for this strange man? But it is
throwing away my only hope."
"I hardly know whether I dare to speak of what I wish of you," she said.
"No, you see; you do not dare."--"I intend to build a railway; you know
that?"--"Yes, I know."--"I want you to help me."--"I?"
Now that she had made a beginning, it was easier for her to continue. She
was surprised that her words sounded so natural.
"I know that you are a railroad builder. Yes, you understand of course
that with my railroad no pay is given. But it would be better for you
to help me work than to sit shut in here. You are making no use of your
time."
He looked at her almost sternly. "Do you know what you are saying?"--"It
is of course a presumptuous request."--"Just so, yes, a presumptuous
request."
Thereupon the poor man began to try to terrify her.
"It will go with your railway as with your festival." Donna Micaela
thought so too, but now she thought that she had closed all ways of
escape for herself; now she must go on being good. "My festival will soon
be in full swing," she said calmly.
"Listen to me, Donna Micaela," said the man. "The last thing a man ceases
to believe good of is himself. No one can cease to have hope for himself."
"No; why should he?"
He made a movement as if he were impatient with her confidence.
"When I first began to think about the thing," he said, "I was easily
consoled. 'There have been a few unfortunate occurrences,' I said to
myself, 'so you have the reputation, and it has become a belief. It is
the belief that has made the trouble. People have met you, and people
have believed that they would come to grief, and come to grief they did.
It is a misfortune worse than death to be considered a _jettatore_, but
you need not yourself believe it.'"
"It is so absurd," said Donna Micaela.
"Yes, of course, whence should my eyes have got the power to bring
misfortune? And when I thought of it I determined to make a trial. I
travelled to a place where no one knew me. The next day I read in the
paper that the train on which I had travelled had run over a flagman.
When I had been one day in the hotel, I saw the landlord in despair, and
all the guests leaving. What had happened? I asked. 'One of our servants
has been taken with small-pox.' Ah, what a wretched business!
"Well, Donna Micaela, I shut myself in and drew back from all intercourse
with people. When a year had passed I had found peace. I asked myself why
I was shut in so. 'You are a harmless man,' I said; 'you wish to hurt no
one. Why do you live as miserably as a criminal?' I had just meant to go
back to life again, when I met Fra Felice in one of the passages. 'Fra
Felice, where is the cat?'--'The cat, signor?'--'Yes, the monastery cat,
that used to come and get milk from me; where is he now?'--'He was caught
in a rat-trap.'--'What do you say, Fra Felice?'--'He got his paw in a
steel trap and he could not get loose. He dragged himself to one of the
garrets and died of starvation.' What do you say to that, Donna Micaela?"
"Was it supposed to be your fault that the cat died?"
"I am a _jettatore_."
She shrugged her shoulders. "Ah, what folly!"
"When some time had passed, again the desire to live awoke within me.
Then Gandolfo knocked on my door, and invited me to your festival. Why
should I not go? It is impossible to believe that one brings misfortune
only by showing one's self. It was a festival in itself, Donna Micaela,
only to get ready and to take out one's black clothes, brush them, and
put them on. But when I came down to the scene of the festival, it was
deserted; the rain streamed in torrents; your Venetian lanterns were
filled with water. And you yourself looked as if you had suffered all
life's misfortunes in a single day. When you looked at me you became
ashy gray with terror. I asked some one: 'What was Signora Alagona's
maiden name?'--'Palmeri.'--'Ah, Palmeri; so she is from Catania. She has
recognized the _jettatore_.'"
"Yes, it is true; I recognized you."
"You have been very friendly, very kind, and I am distressed to have
spoiled your festival. But now I promise you that I shall keep away both
from your entertainment and your railway."
"Why should you keep away?"
"I am a _jettatore_."
"I do not believe it. I cannot believe it."
"I do not believe it either. Yes, yes, I believe. Do you see, people say
that no one can have power over a _jettatore_ who is not as great in evil
as he. Once, they say, a _jettatore_ looked at himself in the glass,
and then fell down and died. Well, I never look at myself in the glass.
Therefore I believe it."
"I do not believe it. I think I almost believed it when I saw you out
there. Now I do not believe it."
"Perhaps you will let me work on your railway?"
"Yes, yes, if you only will."
He came again close up to her, and they exchanged a few short sentences.
"Come forward to the light; I wish to see your face!"--"You think that
I am dissembling."--"I think that you are polite."--"Why should I be
polite to you?"--"That railway means something to you?"--"It means life
and happiness to me."--"How is that?"--"It will win one who is dear to
me."--"Very dear?"
She did not reply, but he read the answer in her face.
He bent his knee to her, and sank his head so low that he could kiss the
hem of her dress. "You are good; you are very good. I shall never forget
it. If I were not who I am, how I would serve you!"
"You _shall_ serve me," she said. And she was so moved by his misfortunes
that she felt no more fear of his injuring her.
He sprang up. "I will tell you something. You cannot go across the floor
without stumbling if I look at you."
"Oh!" she said.
"Try!"
And she tried. She was very much frightened, and had never felt so
unsteady as when she took her first step. Then she thought: "If it were
for Gaetano's sake, I could do it." And then it was easy.
She walked to and fro on the church floor. "Shall I do it again?" He
nodded.
As she was walking, the thought flashed through her brain: "The
Christchild has taken the curse from him, because he is to help me." She
turned suddenly and came back to him.
"Do you know, do you know? you are no _jettatore_!"
"Am I not?"
"No, no!" She took him by the shoulders and shook him. "Do you not see?
do you not understand? It is taken from you."
Little Gandolfo's voice was heard in the path outside the church. "Donna
Micaela, Donna Micaela, where are you? There are so many people, Donna
Micaela. Do you hear; do you hear?"
"Is it no longer raining?" said the _jettatore_, in an uncertain voice.
"It is not raining; how could it be raining? The Christ-image has taken
the curse from you because you are going to work for his railway."
The man reeled and grasped at the air with his hands. "It is gone. Yes, I
think it is gone. Just now it was there. But now--"
He wished again to fall on his knees before Donna Micaela.
"Not to me," she said; "to him, to him." She pointed to the Christ-image.
But nevertheless he fell down before her. He kissed her hands, and with a
voice broken by sobs he told her how every one had hated and persecuted
him, and how much misery life had brought him hitherto.
The next day the _jettatore_ went out on Etna and staked out the road.
And he was no more dangerous than any one else.
IX
PALAZZO GERACI AND PALAZZO CORVAJA
At the time when the Normans ruled in Sicily, long before the family of
Alagona had come to the island, the two magnificent buildings, Palazzo
Geraci and Palazzo Corvaja, were built in Diamante.
The noble Barons Geraci placed their house in the square, high up on the
summit of Monte Chiaro. The Barons Corvaja, on the other hand, built
their home far down the mountain and surrounded it with gardens.
The black-marble walls of Palazzo Geraci were built round a square
court-yard, full of charm and beauty. A long flight of steps, passing
under an arch adorned with an escutcheon, led to the second story. Not
entirely round the court-yard, but here and there in the most unexpected
places, the walls opened into little pillared loggias. The walls were
covered with bas-reliefs, with speckled slabs of Sicilian marble and with
the coats of arms of the Geraci barons. There were windows also, very
small, but with exquisitely carved frames; some round, with panes so
small that they could be covered with a grape leaf; some oblong, and so
narrow that they let in no more light than a slit in a curtain.
The Barons Corvaja did not try to adorn the court-yard of their palace,
but on the lower floor of the house they fitted up a magnificent hall.
In the floor was built a basin for gold-fish; in niches in the walls
fountains covered with mosaic, in which clear water spouted into gigantic
shells. Over it all, a Moorish vaulted roof, supported on slender
pillars, with twining vines in mosaic. It was a hall whose equal is only
to be seen in the Moorish palace in Palermo.
There was much rivalry and emulation during all the time of building.
When Palazzo Geraci put forth a balcony, Palazzo Corvaja acquired its
high Gothic bay-windows; when the roof of Palazzo Geraci was adorned with
richly carved battlements, a frieze of black marble, inlaid with white a
yard wide, appeared on Palazzo Corvaja. The Geraci house was crowned by
a high tower; the Corvaja had a roof garden, with antique pots along the
railing.
When the palaces were finished the rivalry began between the families who
had built them. The houses seemed to breed hostility and strife for all
who lived in them. A Baron Geraci could never agree with a Baron Corvaja.
When Geraci fought for Anjou, Corvaja fought for Manfred. If Geraci
changed sides, and supported Aragoni, Corvaja went to Naples, and fought
for Robert and Joanna.
But that was not all. It was an understood thing that when Geraci found a
son-in-law, Corvaja had to increase his power by a rich marriage. Neither
of the families could rest. They had to vie with each other while eating,
while amusing themselves, while working. The Geraci came to the court of
the Bourbons in Naples, not out of desire of distinction, but because the
Corvaja were there. The Corvaja on the other hand had to grow grapes and
mine sulphur, because the Geraci were interested in agriculture and the
working of mines. When a Geraci received an inheritance some old relative
of the Corvaja had to lie down and die, so that the honor of the family
should not be hazarded.
Palazzo Geraci was always kept busy counting its servants, in order not
to let Palazzo Corvaja lead. But not only the servants, but the braid
on the caps, the harnesses and the horses. The pheasant feather on the
heads of the Corvaja leaders must not be an inch higher than that on
the Geraci. Their goats must increase in the same proportion, and the
Geraci's oxen must have just as long horns as the Corvaja's.
In our time one might have expected an end to the enmity between the two
palaces. In our time there are just as few Corvaja in the one palace as
there are Geraci in the other.
The Geraci court-yard is now a dirty hole, which contains donkey-stalls
and pig-styes and chicken houses. On the high steps rags are dried and
the bas-reliefs are broken and mouldy. In one of the passage-ways a
trade in vegetables is carried on, and in the other shoes are made. The
gate-keeper looks like the most ragged of beggars, and from cellar to
attic live none but poor and penniless people.
It is no better in Palazzo Corvaja. There is not a vestige of the mosaic
left in the big hall; only bare, empty arches. No beggars live there,
because the palace is principally in ruins. It no longer raises its
beautiful façade with the carved windows to the bright Sicilian sky.
But the enmity between Geraci and Corvaja is not over. In the old days
it was not only the noble families themselves who competed with one
another; it was also their neighbors and dependents. All Diamante is
to this day divided into Geraci and Corvaja. There is still a high,
loop-holed wall running across the town, dividing the part of Diamante
which stands by the Geraci from that which has declared itself for the
Corvaja.
Even in our day no one from Geraci will marry a girl from Corvaja. And a
shepherd from Corvaja cannot let his sheep drink from a Geraci fountain.
They have not even the same saints. San Pasquale is worshipped in Geraci,
and the black Madonna is Corvaja's patron saint.
A man from Geraci can never believe but that all Corvaja is full of
magicians, witches, and werewolves. A man from Corvaja will risk his
salvation that in Geraci there are none but rogues and pick-pockets.
Donna Micaela lived in the Geraci district, and soon all that part of the
town were partisans of her railway. But then Corvaja could do no less
than to oppose her.
The inhabitants of Corvaja specially disliked two things. They were
jealous of the reputation of the black Madonna, and therefore did not
like to have another miracle-working image come to Diamante. That was
one thing. The other was that they feared that Mongibello would bury all
Diamante in ashes and fire if any one tried to encircle it with a railway.
A few days after the bazaar Palazzo Corvaja began to show itself hostile.
Donna Micaela one day found on the roof-garden a lemon, which was so
thickly set with pins that it looked like a steel ball. It was Palazzo
Corvaja, that was trying to bewitch as many pains into her head as there
were pins in the lemon.
Then Corvaja waited a few days to see what effect the lemon would have.
But when Donna Micaela's people continued to work on Etna and stake out
the line, they came one night and pulled everything up. And when the
stakes were set up again the next day, they broke the windows in the
church of San Pasquale and threw stones at the Christ-image.
* * * * *
There was a long and narrow little square on the south side of Monte
Chiaro. On both the long sides stood dark, high buildings. On one of
the short sides was an abyss; on the other rose the steep mountain. The
mountain wall was arranged in terraces, but the steps were crumbled and
the marble railings broken. On the broadest of the terraces rose the
stately ruins of Palazzo Corvaja.
The chief ornament of the square was a beautiful, oblong water-basin
which stood quite under the terraces, close to the mountain wall. It
stood there white as snow, covered with carvings, and full of clear, cold
water. It was the best preserved of all the former glories of the Corvaja.
One beautiful and peaceful evening two ladies dressed in black came
walking into the little square. For the moment it was almost empty. The
two ladies looked about them, and when they saw no one they sat down on
the bench by the fountain, and waited.
Soon several inquisitive children came forward and looked at them, and
the older of the two began to talk to the children. She began to tell
them stories: "It is said," and "It is told," and "Once upon a time,"
she said.
Then the children were told of the Christchild who turned himself into
roses and lilies when the Madonna met one of Herod's soldiers, who had
been commanded to kill all children. And they were told the legend of how
the Christchild once had sat and shaped birds out of clay, and how he
clapped his hands and gave the clay pigeons wings with which to fly away
when a naughty boy wished to break them to pieces.
While the old lady was talking, many children gathered about her, and
also big people. It was a Saturday evening, so that the laborers were
coming home from their work in the fields. Most of them came up to the
Corvaja fountain for water. When they heard that some one was telling
legends they stopped to listen. Both the ladies were soon surrounded by a
close, dark wall of heavy, black cloaks and slouch hats.
Suddenly the old lady said to the children: "Do you like the
Christchild?" "Yes, yes," they said, and their big, dark eyes
sparkled.--"Perhaps you would like to see him?"--"Yes, we should indeed."
The lady threw back her mantilla and showed the children a little
Christ-image in a jewelled dress, and with a gold crown on his head and
gold shoes on his feet. "Here he is," she said. "I have brought him with
me to show you."
The children were in raptures. First they clasped their hands at the
sight of the image's grave face, then they began to throw kisses to it.
"He is beautiful, is he not?" said the lady.
"Let us have him! Let us have him!" cried the children.
But now a big, rough workman, a dark man with a bushy, black beard,
pushed forward. He wished to snatch away the image. The old lady had
barely time to thrust it behind her back.
"Give it here, Donna Elisa, give it here!" said the man.
Poor Donna Elisa cast one glance at Donna Micaela, who had sat silent and
displeased the whole time by her side. Donna Micaela had been persuaded
with difficulty to go to Corvaja and show the image to the people there.
"The image helps us when it wills," she said. "We shall not force
miracles."
But Donna Elisa had been determined to go, and she had said that the
image was only waiting to be taken to the faithless wretches in Corvaja.
After everything that he had done, they might have enough faith in him to
believe that he could win them over also.
Now she, Donna Elisa, stood there with the man over her, and she did not
know how she could prevent him from snatching the image away.
"Give it to me amicably, Donna Elisa," said the man, "otherwise, by God,
I will take it in spite of you. I will hack it to small pieces, to small,
small pieces. You shall see how much there will be left of your wooden
doll. You shall see if it can withstand the black Madonna."
Donna Elisa pressed against the mountain wall; she saw no escape. She
could not run, and she could not struggle. "Micaela!" she wailed,
"Micaela!"
Donna Micaela was very pale. She held her hands against her heart, as she
always did when anything agitated her. It was terrible to her to stand
opposed to those dark men. These were they of the slouch hats and short
cloaks of whom she had always been afraid.
But now, when Donna Elisa appealed to her, she turned quickly, seized the
image and held it out to the man.
"See here, take it!" she said defiantly. And she took a step towards him.
"Take it, and do with it what you can!"
She held the image on her outstretched arms, and came nearer and nearer
to the dark workman.
He turned towards his comrades. "She does not believe that I can do
anything to the doll," he said, and laughed at her. And the whole group
of workmen slapped themselves on the knee and laughed.
But he did not take the image; he grasped instead the big pick-axe, which
he held in his hand. He drew back a few steps, lifted the pick over his
head, and stiffened his whole body for a blow which was to crush at once
the entire hated wooden doll.
Donna Micaela shook her head warningly. "You cannot do it," she said, and
she did not draw the image back.
He saw that nevertheless she was afraid, and he enjoyed frightening her.
He stood longer than was necessary with uplifted pick.
"Piero!" came a cry shrill and wailing.
"Piero! Piero!"
The man dropped his pick without striking. He looked terrified.
"God! it is Marcia calling!" he said.
At the same moment a crowd of people came tumbling out of a little
cottage which was built among the ruins of the old Palazzo Corvaja.
There were about a dozen women and a carabiniere, who were fighting.
The carabiniere held a child in his arms, and the women were trying to
drag the child away from him. But the policeman, who was a tall, strong
fellow, freed himself from them, lifted the child to his shoulder, and
ran down the terrace steps.
The dark Piero had looked on without making a movement. When the
carabiniere freed himself, he bent down to Donna Micaela and said
eagerly: "If _the little one_ can prevent that, all Corvaja shall be his
friend."
Now the carabiniere was down in the square. Piero made a sign with his
hand. Instantly all his comrades closed in a ring round the fugitive. He
turned squarely round. Everywhere a close ring of men threatened him with
picks and shovels.
All at once there was terrible confusion. The women who had been
struggling with the carabiniere came rushing down with loud cries. The
little girl, whom he held in his arms, screamed as loud as she could and
tried to tear herself away. People came running from all sides. There
were questionings and wonderings.
"Let us go now," said Donna Elisa to Donna Micaela. "Now no one is
thinking of us."
But Donna Micaela had caught sight of one of the women. She screamed
least, but it was instantly apparent that it was she whom the matter
concerned. She looked as if she was about to lose her life's happiness.
She was a woman who had been very beautiful, although all freshness now
was gone from her, for she was no longer young. But hers was still an
impressive and large-souled face. "Here dwells a soul which can love and
suffer," said the face. Donna Micaela felt drawn to that poor woman as to
a sister.
"No, it is not the time to go yet," she said to Donna Elisa.
The carabiniere asked and asked if they would not let him come out.
No, no, no! Not until he let the child go!
It was the child of Piero and his wife, Marcia. But they were not the
child's real parents. The trouble arose from that.
The carabiniere tried to win the people over to his side. He tried to
convince, not Piero nor Marcia, but the others. "Ninetta is the child's
mother," he said; "you all know that. She has not been able to have the
child with her while she was unmarried; but now she is married, and
wishes to have her child back. And now Marcia refuses to give her the
boy. It is hard on Ninetta, who has not been able to have her child with
her for eight years. Marcia will not give him up. She drives Ninetta away
when she comes and begs for her child. Finally Ninetta had to complain
to the syndic. And the syndic has told us to get her the child. It is
Ninetta's own child," he said appealingly.
But it had no great effect on the men of Corvaja.
"Ninetta is a Geraci," burst out Piero, and the circle stood fast round
the carabiniere.
"When we came here to fetch the child," said the latter, "we did not
find him. Marcia was dressed in black, and her rooms were draped with
black, and a lot of women sat and mourned with her. And she showed us the
certificate of the child's death. Then we went and told Ninetta that her
child was in the church-yard.
"Well, well, a while afterwards I went on guard here in the square. I
watched the children playing there. Who was strongest, and who shouted
the loudest, if not one of the girls? 'What is your name?' I asked her.
'Francesco,' she answered instantly.
"It occurred to me that that girl, Francesco, might be Ninetta's boy,
and I stood quiet and waited. Just now I saw Francesco go into Marcia's
house. I followed, and there sat the girl Francesco and ate supper with
Marcia. She and all the mourners began to scream when I appeared. Then
I seized Signorina Francesco and ran. For the child is not Marcia's.
Remember that, signori! He is Ninetta's. Marcia has no right to him."
Then at last Marcia began to speak. She spoke in a deep voice which
compelled every one to listen, and she made only a few, but noble
gestures. Had she no right to the child? But who had given him food and
clothing? He had been dead a thousand times over if she had not been
there. Ninetta had left him with La Felucca. They knew La Felucca. To
leave one's child to her was the same as saying to it: "You shall die."
And, moreover, right? right? What did that mean? The one whom the boy
loved had a right to him. The one who loved the boy had a right to him.
Piero and she loved the boy like their own son. They could not be parted
from him.
The wife was desperate, the husband perhaps even more so. He threatened
the carabiniere whenever he made a movement. Yet the carabiniere seemed
to see that the victory would be his. The people had laughed when he
spoke of "Signorina Francesco." "Cut me down, if you will," he said to
Piero. "Does it help you? Will you retain the child for that? He is not
yours. He is Ninetta's."
Piero turned to Donna Micaela. "Pray to him to help me." He pointed to
the image.
Donna Micaela instantly went forward to Marcia. She was shy and trembled
for what she was venturing, but it was not the time for her to hold back.
"Marcia," she whispered, "confess! Confess,--if you dare!" The startled
woman looked at her. "I see it so well," whispered Donna Micaela; "you
are as alike as two berries. But I will say nothing if you do not wish
it." "He will kill me," said Marcia. "I know one who will not let him
kill you," said Donna Micaela. "Otherwise they will take your child from
you," she added.
All were silent, with eyes fixed on the two women. They saw how Marcia
struggled with herself. The features of her strong face were distorted.
Her lips moved. "The child is mine," she said, but in so low a voice that
no one heard it. She said it again, and now it came in a piercing scream:
"The child is mine!"
"What will you do to me when I confess it?" she said to the man. "The
child is mine, but not yours. He was born in the year when you were at
work in Messina. I put him with La Felucca, and Ninetta's boy was there
too. One day when I came to La Felucca she said, 'Ninetta's boy is
dead.' At first I only thought: 'God! if it had been mine! Then I said
to La Felucca: 'Let my boy be dead, and let Ninetta's live.' I gave La
Felucca my silver comb, and she agreed. When you came home from Messina
I said to you: 'Let us take a foster child. We have never been on good
terms. Let us try what adopting a child will do.' You liked the proposal,
and I adopted my own child. You have been happy with him, and we have
lived as if in paradise."
Before she finished speaking the carabiniere put the child down on the
ground. The dark men silently opened their ranks for him, and he went his
way. A shiver went through Donna Micaela when she saw the carabiniere
go. He should have stayed to protect the poor woman. His going seemed to
mean: "That woman is beyond the pale of the law; I cannot protect her."
Every man and woman standing there felt the same: "She is outside of the
law."
One after another went their way.
Piero, the husband, stood motionless without looking up. Something fierce
and dreadful was gathering in him. Rage and suffering were gathering
within him. Something terrible would happen as soon as he and Marcia were
alone.
The woman made no effort to escape. She stood still, paralyzed by the
certainty that her fate was sealed, and that nothing could change it. She
neither prayed nor fled. She shrank together like a dog before an angry
master. The Sicilian women know what awaits them when they have wounded
their husbands' honor.
The only one who tried to defend her was Donna Micaela. Never would she
have begged Marcia to confess, she said to Piero, if she had known what
he was. She had thought that he was a generous man. Such a one would
have said: "You have done wrong; but the fact that you confess your sin
publicly, and expose yourself to my anger to save the child, atones for
everything. It is punishment enough." A generous man would have taken the
child on one arm, put the other round his wife's waist, and have gone
happy to his home. A signor would have acted so. But he was no signor; he
was a bloodhound.
She talked in vain; the man did not hear her; the woman did not hear her.
Her words seemed to be thrown back from an impenetrable wall.
Just then the child came to the father, and tried to take his hand.
Furious, he looked at the boy. As the latter was dressed in girl's
clothes, his hair smoothly combed and drawn back by the ears, he saw
instantly the likeness to Marcia, which he had not noticed before. He
kicked Marcia's son away.
There was a terrible tension in the square. The neighbors continued to go
quietly and slowly away. Many went unwillingly and with hesitation, but
still they went. The husband seemed only to be waiting for the last to go.
Donna Micaela ceased speaking; she took the image instead and laid it in
Marcia's arms. "Take him, my sister Marcia, and may he protect you!" she
said.
The man saw it, and his rage increased. It seemed as if he could no
longer contain himself till he was alone. He crouched like a wild beast
ready to spring.
But the image did not rest in vain in the woman's arms. The outcast moved
her to an act of the greatest love.
"What will Christ in Paradise say to me, who have first deceived my
husband, and then made him a murderer?" she thought. And she remembered
how she had loved big Piero in the days of her happy youth. She had not
then thought of bringing such misery upon him.
"No, Piero, no, do not kill me!" she said eagerly. "They will send you to
the galleys. You shall be relieved of seeing me again without that."
She ran towards the other side of the square, where the ground fell away
into an abyss. Every one understood her intention. Her face bore witness
for her.
Several hurried after her, but she had a good start. Then the image,
which she still carried, slipped from her arms and lay at her feet. She
stumbled over it, fell, and was overtaken.
She struggled to get away, but a couple of men held her fast. "Ah, let me
do it!" she cried; "it is better for him!"
Her husband came up to her also. He had caught up her child and placed
him on his arm. He was much moved.
"See, Marcia, let it be as it is," he said. He was embarrassed, but his
dark, deep-set eyes shone with happiness and said more than his words.
"Perhaps, according to old custom, it ought to be so, but I do not care
for that. Look, come now! It would be a pity for such a woman as you,
Marcia."
He put his arm about Marcia's waist, and went towards his house in the
ruins of Palazzo Corvaja. It was like a triumphal entry of one of the
former barons. The people of Corvaja stood on both sides of the way and
bowed to him and Marcia.
As they went past Donna Micaela, they both stopped, bowed deep to her,
and kissed the image which some one had given back to her. But Donna
Micaela kissed Marcia. "Pray for me in your happiness, sister Marcia!"
she said.
X
FALCO FALCONE
The blind singers have week after week sung of Diamante's railway, and
the big collection-box in the church of San Pasquale has been filled
every evening with gifts. Signor Alfredo measures and sets stakes on
the <DW72>s of Etna, and the distaff-spinners in the dark alleys tell
stories of the wonderful miracles that have been performed by the little
Christ-image in the despised church. From the rich and powerful men who
own the land on Etna comes letter after letter promising to give ground
to the blessed undertaking.
During these last weeks every one comes with gifts. Some give building
stone for the stations, some give powder to blast the lava blocks,
some give food to the workmen. The poor people of Diamante, who have
nothing, come in the night after their work. They come with shovels and
wheelbarrows and creep out on Etna, dig the ground, and ballast the road.
When Signor Alfredo and his people come in the morning they believe that
the Etna goblins have broken out from their lava streams and helped on
the work.
All the while people have been questioning and asking: "Where is the king
of Etna, Falco Falcone? Where is the mighty Falco who has held sway on
the <DW72>s of Etna for five and twenty years? He wrote to Don Ferrante's
widow that she would not be allowed to construct the railway. What did
he mean by his threat? Why does he sit still when people are braving his
interdiction? Why does he not shoot down the people of Corvaja when they
come creeping through the night with wheelbarrows and pickaxes? Why does
he not drag the blind singers down into the quarry and whip them? Why
does he not have Donna Micaela carried off from the summer-palace, in
order to be able to demand a cessation in the building of the railway as
a ransom for her life?"
Donna Micaela says to herself: "Has Falco Falcone forgotten his promise,
or is he waiting to strike till he can strike harder?"
Everybody asks in the same way: "When is Etna's cloud of ashes to fall on
the railway? When will Mongibello cataracts tear it away? When will the
mighty Falco Falcone be ready to destroy it?"
While every one is waiting for Falco to destroy the railway, they talk a
great deal about him, especially the workmen under Signor Alfredo.
Opposite the entrance to the church of San Pasquale, people say, stands
a little house on a bare crag. The house is narrow, and so high that
it looks like a chimney left standing on a burnt building site. It is
so small that there is no room for the stairs inside the house; they
wind up outside the walls. Here and there hang balconies and other
projections that are arranged with no more symmetry than a bird's nest on
a tree-trunk.
In that house Falco Falcone was born, and his parents were only poor
working-people. In that miserable hut Falco learned arrogance.
Falco's mother was an unfortunate woman, who during the first years of
her marriage brought only daughters into the world. Her husband and all
her neighbors despised her.
The woman longed continually for a son. When she was expecting her fifth
child she strewed salt every day on the threshold and sat and watched who
should first cross it. Would it be a man or a woman? Should she bear a
son or a daughter?
Every day she sat and counted. She counted the letters in the month
when her child was to be born. She counted the letters in her husband's
name and in her own. She added and subtracted. It was an even number;
therefore she would bear a son. The next day she made the calculation
over again. "Perhaps I counted wrong yesterday," she said.
When Falco was born his mother was much honored, and she loved him on
account of it more than all her other children. When the father came
in to see the child he snatched off his cap and made a low bow. Over
the house-door they set a hat as a token of honor, and they poured
the child's bath water over the threshold, and let it run out into
the street. When Falco was carried to the church he was laid on his
god-mother's right arm; when the neighbors' wives came to look after his
mother they courtesied to the child sleeping in his cradle.
He was also bigger and stronger than children generally are. Falco had
thick hair when he was born, and when he was a week old he already had
a tooth. When his mother laid him to her breast he was so wild that she
laughed and said: "I think that I have brought a hero into the world."
She was always expecting great achievements from Falco, and she put
pride into him. But who else hoped anything of him? Falco could not
even learn to read. His mother tried to take a book and teach him the
letters. She pointed to A, that is the big hat; she pointed to B, that
is the spectacles; she pointed to C, that is the snake. That he could
learn. Then his mother said: "If you put the spectacles and the big hat
together, it makes Ba." That he could not learn. He became angry and
struck her, and she let him alone. "You will be a great man yet," she
said.
Falco was dull and bad-tempered in his childhood and youth. As a child,
he would not play; as a youth, he would not dance. He had no sweetheart,
but he liked to go where fighting was to be expected.
Falco had two brothers who were like other people, and who were much
more esteemed than he. Falco was wounded to see himself eclipsed by his
brothers, but he was too proud to show it. His mother was always on his
side. After his father's death she had him sit at the head of the table,
and she never allowed any one to jest with him. "My oldest son is the
best of you all," she said.
When the people remember it all they say: "Falco is proud. He will make
it a point of honor to destroy the railway."
And they have hardly terrified themselves with one story before they
remember another about him.
For thirty long years, people say, Falco lived like any other poor
person on Etna. On Monday he went away to his work in the fields with
his brothers. He had bread in his sack for the whole week, and he made
soup of beans and rice like every one else. And he was glad on Saturday
evening to be able to return to his home. He was glad to find the table
spread, with wine and macaroni, and the bed made up with soft pillows.
It was just such a Saturday evening. Falco and Falco's brothers were on
their way home; Falco, as usual, a little behind the others, for he had a
heavy and slow way of walking. But look, when the brothers reached home,
no supper was waiting, the beds were not made, and the dust lay thick
on the threshold. What, were all in the house dead? Then they saw their
mother sitting on the floor in a dark corner of the cottage. Her hair was
drawn down over her face, and she sat and traced patterns with her finger
on the earth floor. "What is the matter?" said the brothers. She did not
look up; she spoke as if she had spoken to the earth. "We are beggared,
beggared." "Do they want to take our house from us?" cried the brothers.
"They wish to take away our honor and our daily bread."
Then she told: "Your eldest sister has had employment with Baker Gasparo,
and it has been good employment. Signor Gasparo gave Pepa all the bread
left over in the shop, and she brought it to me. There has been so much
that there was enough for us all. I have been happy ever since Pepa found
that employment. It will give me an old age free from care, I thought.
But last Monday Pepa came home to me and wept; Signora Gasparo had turned
her away."
"What had Pepa done?" asked Nino, who was next younger to Falco.
"Signora Gasparo accused Pepa of stealing bread. I went to Signora
Gasparo and asked her to take Pepa back. 'No,' she said, 'the girl is
not honest.' 'Pepa had the bread from Signor Gasparo,' I said; 'ask him.'
'I cannot ask him,' said the signora; 'he is away, and comes home next
month.' 'Signora,' I said, 'we are so poor. Let Pepa come back to her
place.' 'No,' she said; 'I myself will leave Signor Gasparo if he takes
that girl back.' 'Take care,' I said then; 'if you take bread from me, I
will take life from you.' Then she was frightened and called others in,
so that I had to go."
"What is to be done about it?" said Nino. "Pepa must find some other
work."
"Nino," said Mother Zia, "you do not know what that woman has said to the
neighbors about Pepa and Signor Gasparo."
"Who can prevent women from talking?" said Nino.
"If Pepa has nothing else to do, now she might at least have cooked
dinner for us," said Turiddo.
"Signora Gasparo has said that her husband let Pepa steal bread that she
should--"
"Mother," interrupted Nino, red as fire, "I do not intend to have myself
put in the galleys for Pepa's sake."
"The galleys do not eat Christians," said Mother Zia.
"Nino," said Pietro, "we had better go to the town to get some food."
As they said it they heard some one laugh behind them. It was Falco who
laughed.
A while later Falco entered Signora Gasparo's shop and asked for bread.
The poor woman was frightened when Pepa's brother came into the shop.
But she thought: "He has just come from his work. He has not been home
yet. He knows nothing."
"Beppo," she said to him, for Falco's name was not then Falco, "is the
harvest a good one?" And she was prepared not to have him answer.
Falco was more talkative than usual, and immediately told her how
many grapes had already been put through the press. "Do you know," he
continued, "that a farmer was murdered yesterday."--"Alas, yes, poor
Signor Riego; I heard so." And she asked how it had happened.
"It was Salvatore who did it. But it is too dreadful for a signora to
hear!"--"Oh, no, what is done can be and is told."
"Salvatore went up to him in this way, signora." And Falco drew his knife
and laid his hand on the woman's head. "Then he cut him across the throat
from ear to ear."
As Falco spoke, he suited the action to the word. The woman did not even
have time to scream. It was the work of a master.
After that, Falco was sent to the galleys, where he remained five years.
When the people tell of that, their terror increases. "Falco is brave,"
they say. "Nothing in the world can frighten him away from his purpose."
That immediately made them think of another story.
Falco was taken to the galleys in August, where he became acquainted with
Biagio, who afterwards followed him through his whole life. One day he
and Biagio and a third prisoner were ordered to go to work in the fields.
One of the overseers wished to construct a garden around his house. They
dug there quietly, but their eyes began to wander and wander. They were
outside the walls; they saw the plain and the mountains; they even saw up
to Etna. "It is the time," whispered Falco to Biagio. "I will rather die
than go back to prison," said Biagio. Then they whispered to the other
prisoner that he must stand by them. He did not wish to do so, because
his time of punishment was soon up. "Else we will kill you," they said,
and then he agreed.
The guard stood over them with his loaded rifle in his hand. On account
of their fetters, Falco and Biagio hopped with feet together over to the
guard. They swung their shovels over him, and before he had time to think
of shooting he was thrown down, bound, and had a clump of earth in his
mouth. Thereupon the prisoners pried open their chains with the shovels,
so that they could take a step, and crept away over the plain to the
hills.
When night came Falco and Biagio abandoned the prisoner whom they had
taken with them. He was old and feeble, so that he would have hindered
their flight. The next day he was seized by the carabinieri, and shot.
They shudder when they think of it. "Falco is merciless," they say. They
know that he will not spare the railway.
Story after story comes to frighten the poor people working on the
railway on the <DW72>s of Etna.
They tell of all the sixteen murders that Falco has committed. They tell
of his attacks and plunderings.
There is one story more terrifying than all the others together.
When Falco escaped from the galleys he lived in the woods and caves, and
in the big quarry near Diamante. He soon gathered a band about him, and
became a wonderful and famous brigand hero.
All his family were held in much greater consideration than before. They
were respected, as the mighty are respected. They scarcely needed to
work, for Falco loved his relations and was generous to them. But he was
not lenient towards them; he was very stern.
Mother Zia was dead, and Nino was married and lived in his father's
cottage. It happened one day that Nino needed money, and he knew no
better way than to go to the priest,--not Don Matteo, but to old Don
Giovanni. "Your Reverence," said Nino to him, "my brother asks you for
five hundred lire." "Where shall I find five hundred lire?" said Don
Giovanni. "My brother needs them; he must have them," said Nino.
Then old Don Giovanni promised to give the money, if he only were given
time to collect it. Nino was hardly willing to agree to that. "You can
scarcely expect me to take five hundred lire from my snuff-box," said
Don Giovanni. And Nino granted him three days' respite. "But beware of
meeting my brother during that time," he said.
The next day Don Giovanni rode to Nicolosi to try to claim a payment. Who
should he meet on the way but Falco and two of his band. Don Giovanni
threw himself from his donkey and fell on his knees before Falco. "What
does this mean, Don Giovanni?"--"As yet I have no money for you, Falco,
but I will try to get it. Have mercy upon me!"
Falco asked, and Don Giovanni told the whole story. "Your Reverence,"
said Falco, "he has been deceiving you." He begged Don Giovanni to go
with him to Diamante. When they came to the old house Don Giovanni rode
in behind the wall of San Pasquale, and Falco called Nino out. Nino came
out on one of the balconies. "Eh, Nino!" said Falco, and laughed. "You
have cheated the priest out of money?" "Do you know it already?" said
Nino. "I was just going to tell it to you."
Now Falco became sterner. "Nino," he said, "the priest is my friend, and
he believes that I have wished to rob him. You have done very wrong." He
suddenly put his gun to his shoulder and shot Nino down, and when he had
done so he turned to Don Giovanni, who had almost fallen from his donkey
with terror. "You see now, your Reverence, that I had no part in Nino's
designs on you!"
And that happened twenty years ago, when Falco had not been a brigand for
more than five years.
"Will Falco spare the railway," people say, as they tell it, "when he did
not spare his own brother?"
There was yet more.
After Nino's murder there was a vendetta over Falco. Nino's wife was
so terrified when she found her husband dead that half her body became
paralyzed, and she could no longer walk. But she took her place at the
window in the old cottage. There she has sat for twenty years with a gun
beside her, and waited for Falco. And of her the great brigand has been
afraid. For twenty years he has not gone past the home of his ancestors.
The woman has not deserted her post. No one ever goes to the church of
San Pasquale without seeing her revengeful eyes shining behind the panes.
Who has ever seen her sleep? Who has seen her work? She could do nothing
but await her husband's murderer.
When people hear that, they are even more afraid. Falco has luck on his
side, they think. The woman who wishes to kill him cannot move from her
place. He has luck on his side. He will also succeed in destroying the
railway. Fortune has never failed Falco. The carabinieri have hunted, but
have never been able to catch him. The carabinieri have feared Falco more
than Falco has feared the carabinieri.
People tell a story of a young carabiniere lieutenant who once pursued
Falco. He had arranged a line of beaters and hunted Falco from one
thicket to another. At last the officer was certain that he had Falco
shut in in a grove. A guard was stationed round the wood, and the
officer searched the covert, gun in hand. But however much he searched,
he saw no Falco. He came out, and met a peasant. "Have you seen Falco
Falcone?"--"Yes, signor; he just went by me, and he asked me to greet
you."--"_Diavolo!_"--"He saw you in the thicket, and he was just going
to shoot you, but he did not do so, because he thought that perhaps it
was your duty to prosecute him."--"_Diavolo! Diavolo!_"--"But if you try
another time--"--"_Diavolo! Diavolo! Diavolo!_"
Do you think that lieutenant came back? Do you not think that he
instantly sought out a district where he did not need to hunt brigands?
And the workmen on Etna asked themselves: "Who will protect us against
Falco? He is terrible. Even the soldiers tremble before him."
They remember that Falco Falcone is now an old man. He no longer plunders
post-wagons; he does not carry off land-owners. He sits quiet generally
in the quarry near Diamante, and instead of robbing money and estates, he
takes money and estates under his protection.
He takes tribute from the great landed proprietors and guards their
estates from other thieves, and it has become calm and peaceful on Etna,
for he allows no one to injure those who have paid a tax to him.
But that is not reassuring. Since Falco has become friends with the
great, he can all the more easily destroy the railway.
And they remember the story of Niccola Galli, who is overseer on the
estate of the Marquis di San Stefano on the southern side of Etna. Once
his workmen struck in the middle of the harvest time. Niccola Galli was
in despair. The wheat stood ripe, and he could not get it reaped. His
workmen would not work; they lay down to sleep at the edge of a ditch.
Niccola placed himself on a donkey and rode down to Catania to ask his
lord for advice. On the way he met two men with guns on their shoulders.
"Whither are you riding, Niccola?"
Before Niccola had time to say many words they took his donkey by the bit
and turned him round. "You must not ride to the Marquis, Niccola?"--"Must
I not?"--"No; you must ride home."
As they went along, Niccola sat and shook on his donkey. When they were
again at home the men said: "Now show us the way to the fields!" And
they went out to the laborers. "Work, you scoundrels! The marquis has
paid his tribute to Falco Falcone. You can strike in other places, but
not here." That field was reaped as never before. Falco stood on one side
of it and Biagio on the other. The grain is soon harvested with such
overseers.
When the people remember that, their terror does not decrease. "Falco
keeps his word," they say. "He will do what he has threatened to do."
No one has been a robber chief as long as Falco. All the other famous
heroes are dead or captives. He alone keeps himself alive and in his
profession by incredible good fortune and skill.
Gradually he has collected about him all his family. His brothers-in-law
and nephews are all with him. Most of them have been sent to the galleys,
but not one of them thinks whether he suffers in prison; he only asks if
Falco is satisfied with him.
In the newspapers there are often accounts of Falco's deeds. Englishmen
thrust a note of ten lire into their guide's hand if he will show them
the way to Falco's quarry. The carabinieri no longer shoot at him,
because he is the last great brigand.
He so little fears to be captured that he often comes down to Messina
or Palermo. He has even crossed the sound and been in Italy. He went to
Naples when Guglielmo and Umberto were there to christen a battle-ship.
He travelled to Rome when Umberto and Margherita celebrated their silver
wedding.
The people think of it all, and tremble. "Falco is loved and admired,"
the workmen say. "The people worship Falco. He can do what he will."
They know too that when Falco saw Queen Margherita's silver wedding, it
pleased him so much that he said: "When I have lived on Etna for five and
twenty years, I shall celebrate my silver wedding with Mongibello."
People laughed at that and said that it was a good idea of Falco's. For
he had never had a sweetheart, but Mongibello with its caves and forests
and craters and ice-fields had served and protected him like a wife. To
no one in the world did Falco owe such gratitude as to Mongibello.
People ask when Falco and Mongibello are going to celebrate their silver
wedding. And people answer that it will be this spring. Then the workmen
think: "_He is coming to destroy our railway on the day of Mongibello_."
They are filled with doubt and terror. They soon will not dare to work
any more. The nearer the time approaches when Falco is to celebrate his
union with Mongibello, the more there are who leave Signor Alfredo. Soon
he is practically alone at the work.
* * * * *
There are not many people in Diamante who have seen the big quarry on
Etna. They have learned to avoid it because Falco Falcone lives there.
They have been careful to keep out of range of his gun.
They have not seen the great hole in Mongibello's side from which their
ancestors, the Greeks, took stone in remote times. They have not seen the
beautifully walls, and the mighty rocks that look like ruined
pillars. Perhaps they do not know that on the bottom of the quarry grow
more magnificent flowers than in a conservatory. There it is no longer
Sicily; it is India.
In the quarry are mandarin trees, so yellow with fruit that they look
like gigantic sun-flowers; the camellias are as big as tambourines; and
on the ground between the trees lie masses of magnificent figs and downy
peaches embedded in fallen rose-leaves.
One evening Falco is sitting alone in the quarry. Falco is busy making a
wreath, and he has beside him a mass of flowers. The string he is using
is as thick as a rope; he holds his foot on the ball so that it shall not
roll away from him. He wears spectacles, which continually slip too far
down his hooked nose.
Falco is swearing horribly, for his hands are stiff and callous from
incessantly handling a gun, and cannot readily hold flowers. The fingers
squeeze them together like steel tongs. Falco swears because the lilies
and anemones fall into little pieces if he merely looks at them.
Falco sits in his leather breeches and in the long, buttoned-up coat,
buried in flowers like a saint on a feast-day. Biagio and his nephew,
Passafiore, have gathered them for him. They have piled up in front of
him an Etna of the most beautiful flowers of the quarry. Falco can choose
among lilies and cactus-flowers and roses and pelargoniums. He roars at
the flowers that he will trample them to dust under his leather sandals
if they do not submit themselves to his will.
Never before has Falco Falcone had to do with flowers. In the whole
course of his life he has never tied a nosegay for a girl, or plucked a
rose for his button-hole. He has never even laid a wreath on his mother's
grave.
Therefore the delicate flowers rebel against him. The flower sprays are
entangled in his hair and in his hat, and the petals have caught in his
bushy beard. He shakes his head violently, and the scar in his cheek
glows red as fire as it used to do in the old days, when he fought with
the carabinieri.
Still the wreath grows, and thick as a tree-trunk it winds round Falco's
feet and legs. Falco swears at it as if it were the steel fetters that
once dragged between his ankles. He complains more, when he tears himself
on a thorn or burns himself on a nettle, than he did when the whip of the
galley guard lashed his back.
Biagio and Passafiore, his nephew, do not dare to show themselves; they
lie concealed in a cave till everything is ready. They laugh at Falco
with all their might, for such wailings as Falco's have not sounded in
the quarry since unhappy prisoners of war were kept at work there.
Biagio looks up to great Etna, which is blushing in the light of the
setting sun. "Look at Mongibello," he says to Passafiore; "see how it
blushes. It must guess what Falco is busy with down in the quarry." And
Passafiore answers: "Mongibello has probably never thought that it would
ever have anything on its head but ashes and snow."
But suddenly Biagio stopped laughing. "It is not well, Passafiore," he
said. "Falco has become too proud. I am afraid that the great Mongibello
is going to make a fool of him."
The two bandits look one another in the eyes questioningly. "It is well
if it is only pride," says Passafiore.
But now they look away at the same moment, and dare say no more. The same
thought, the same dread has seized them both. Falco is going mad. He is
already mad at times. It is always so with great brigand chiefs; they
cannot bear their glory and their greatness; they all go mad.
Passafiore and Biagio have seen it for a long time, but they have borne
it in silence, and each has hoped that the other has seen nothing. Now
they understand that they both know it. They press each other's hands
without a word. There is still something so great in Falco. Both of them,
Passafiore and Biagio, will take care that no one shall perceive that he
is no longer the man he was.
Finally Falco has his wreath ready; he hangs it on the barrel of his gun
and comes out to the others. All three climb out of the quarry, and at
the nearest farm-house they take horses in order to come quickly to the
top of Mongibello.
They ride at full gallop so that they have no chance to talk, but as they
pass the different farms they can see the people dancing on the flat
roofs. And from the sheds, where the laborers sleep at night, they hear
talk and laughter. There happy, peaceful people are sitting, guessing
conundrums and matching verses. Falco storms by, such things are not for
him. Falco is a great man.
They gallop towards the summit. At first they ride between almond-trees
and cactus, then under plane-trees and stone-pines, then under oaks and
chestnut-trees.
The night is dark; they see nothing of the beauty of Mongibello. They do
not see the vine-encircled Monte Rosso; they do not see the two hundred
craters that stand in a circle round Etna's lofty peak like towers round
a town; they do not see the endless stretches of thick forest.
In Casa del Bosco, where the road ends, they dismount. Biagio and
Passafiore take the wreath and carry it between them. As they walk along,
Falco begins to talk. He likes to talk since he has grown old.
Falco says that the mountain is like the twenty-five years of his life
that he has passed there. The years that founded his greatness had
blossomed with deeds. To be with him then had been like going through
an endless arbor, where lemons and grapes hung down overhead. Then his
deeds had been as numerous as the orange-trees round Etna's base. When
he had come higher the deeds had been less frequent, but those he had
executed had been mighty as the oaks and chestnut-trees on the rising
mountain. Now that he was at the summit of greatness, he scorned to act.
His life was as bald as the mountain top; he was content to see the
world at his feet. But people ought to understand that, if he should now
undertake anything, nothing could resist him. He was terrible, like the
fire-spouting summit.
Falco walks before and talks; Passafiore and Biagio follow him in silent
terror. Dimly they see the mighty <DW72>s of Mongibello with their towns
and fields and forests spread out beneath them. And Falco thinks that he
is as mighty as all that!
As they struggle upwards they are beset with a growing feeling of
dread. The gaping fissures in the ground; the sulphur smoke from the
crater, which rolls down the mountain, too heavy to rise into the air;
the explosions inside the mountain; the incessant, gently rumbling
earthquake; the slippery, rough ice-fields crossed by gushing brooks;
the extreme cold, the biting wind,--make the walk hideous. And Falco
says that it is like him! How can he have such things in his soul? Is it
filled with a cold and a horror to be compared to Etna's?
They stumble over blocks of ice, and they struggle forward through snow
lying sometimes a yard deep. The mountain blast almost throws them down.
They have to wade through slush and water, for through the day the sun
has melted a mass of snow. And while they grow stiff with cold, the
ground shakes under them with the everlasting fire.
They remember that Lucifer and all the damned are lying under them. They
shudder because Falco has brought them to the gates of Hell.
But nevertheless beyond the ice-field they reach the steep cone of ashes
on the very summit of the mountain. Here they drag themselves up, walking
on sliding ashes and pumice-stone. When they are half way up the cone
Falco takes the wreath, and motions to the others to wait. He alone will
scale the summit.
The day is just breaking, and as Falco reaches the top the sun is
visible. The glorious morning light streams over Mongibello and over the
old Etna brigand on its summit. The shadow of Etna is thrown over the
whole of Sicily, and it looks as if Falco, standing up there, reached
from sea to sea, across the island.
Falco stands and gazes about him. He looks across to Italy; he fancies
he sees Naples and Rome. He lets his glance pass over the sea to the land
of the Turk to the east and the land of the Saracen to the south. He
feels as if it all lay at his feet and acknowledged _his_ greatness.
Then Falco lays the wreath on the summit of Mongibello.
When he comes down to his comrades he solemnly presses their hands. As
he leaves the cone they see that he picks up a piece of pumice-stone,
and puts it in his pocket. Falco takes with him a souvenir of the most
beautiful hour of his life. He has never before felt himself so great as
on the top of Mongibello.
On that day of happiness Falco will do no work. The next day, he says, he
will begin the undertaking of freeing Mongibello from the railway.
* * * * *
There is a lonely farm-house on the road between Paternó and Adernó. It
is quite large, and it is owned by a widow, Donna Silvia, who has many
strong sons. They are bold people who dare to live alone the whole year
in the country.
It is the day following the one when Falco crowned Mongibello. Donna
Silvia is sitting on the grass-plot with her distaff; she is alone; there
is no one else at home on the farm. A beggar comes softly creeping in
through the gate.
He is an old man with a long, hooked nose which hangs down over his upper
lip, a bushy beard, pale eyes with red eyelids. They are the ugliest eyes
imaginable; the whites are yellowish, and they squint. The beggar is
tall and very thin; he moves his body when he walks, so that it looks as
if he wriggled forward. He walks so softly that Donna Silvia does not
hear him. The first thing she notices is his shadow, which, slender as a
snake, bends down towards her.
She looks up when she sees the shadow. Then the beggar bows to her and
asks for a dish of macaroni.
"I have macaroni on the fire," says Donna Silvia. "Sit down and wait; you
shall have your fill."
The beggar sits down beside Donna Silvia, and after a while they begin to
chat. They soon talk of Falco.
"Is it true that you let your sons work on Donna Micaela's railway?" says
the beggar.
Donna Silvia bites her lips together, and nods an assent.
"You are a brave woman, Donna Silvia. Falco might be revenged on you."
"Then he can take revenge," says Donna Silvia. "But I will not obey one
who has killed my father. He forced him to escape from prison in Augusta,
and my father was captured and shot."
And so saying she rises and goes in to get the food.
As she stands in the kitchen she sees the beggar through the window,
sitting and rocking on the stone-bench. He is not quiet for a moment. And
in front of him writhes his shadow, slender and lithe as a snake.
Donna Silvia remembers what she had once heard Caterina, who had been
married to Falco's brother, Nino, say. "How will you recognize Falco
after twenty years?" people had asked her. "Should I not recognize the
man with the snake-shadow?" she answered. "He will never lose it, long
as he may live."
Donna Silvia presses her hand on her heart. There in her yard Falco
Falcone is sitting. He has come to be revenged because her sons work on
the railway. Will he set fire to the house, or will he murder her?
Donna Silvia is shaking in every limb as she serves up her macaroni.
Falco begins to find the time long as he sits on the stone-bench. A
little dog comes up to him and rubs against him. Falco feels in his
pocket for a piece of bread, but he finds only a stone, which he throws
to the dog.
The dog runs after the stone and brings it back to Falco. Falco throws it
again. The dog takes the stone again, but now he runs away with it.
Falco remembers that it is the stone he picked up on Mongibello, and goes
after the dog to get it back. He whistles to the dog, and it comes to him
instantly. "Drop the stone!" The dog puts its head on one side and will
not drop it. "Ah, give me the stone, rascal!" The dog shuts its mouth. It
has no stone. "Let me see; let me see!" says Falco. He bends the dog's
head back and forces it to open its mouth. The stone lies far in under
the gums, and Falco tries to force it out. Then the dog bites him, till
the blood flows.
Falco is terrified. He goes in to Donna Silvia. "I hope your dog is
healthy," he says.
"My dog? I have no dog. It is dead."--"But the one running outside?"--"I
do not know which one you mean," she says.
Falco says nothing more, nor does he do Donna Silvia any harm. He simply
goes his way, frightened; he thinks that the dog is mad, and he fears
hydrophobia.
* * * * *
One evening Donna Micaela sits alone in the music-room. She has put out
the lamp and opened the balcony doors. She likes to listen to the street
in the evening and at night. No more smiths and stone-cutters and criers
are heard. There is song, laughter, whispering, and mandolins.
Suddenly she sees a dark hand laid on the balcony railing. The hand drags
up after it an arm and a head; within a moment a whole human being swings
himself into the balcony. She sees him plainly, for the street-lamps are
still burning. He is a small, broad-shouldered, bearded fellow, dressed
like a shepherd, with leather sandals, a slouch hat, and an umbrella tied
to his back. As soon as he is on his feet he snatches his gun from his
shoulder and comes into the room with it in his hands.
She sits still without giving a sign of life. There is no time either to
summon help or to escape. She hopes that the man will take what he wishes
to take, and go away without noticing her, sitting back in the dark room.
The man puts his gun down between his legs, and she hears him scratching
with a match. She shuts her eyes. He will believe that she is asleep.
When the robber gets the match lighted, he sees her instantly. He coughs
to wake her. As she remains motionless, he creeps over to her and
carefully stretches out a finger towards her arm. "Do not touch me! do
not touch me!" she screams, and can no longer sit still. The man draws
back instantly. "Dear Donna Micaela, I only wanted to wake you."
There she sits and shakes with terror, and he hears how she is sobbing.
"Dear signora, dear signora!" he says. "Light a candle that I can see
where you are," she cries. He scratches a new match, lifts the shade and
chimney off the lamp, and lights it as neatly as a servant. He places
himself again by the door, as far from her as possible. Suddenly he goes
out on the balcony with his gun. "Now the signora cannot be afraid any
longer."
But when she does not cease weeping he says: "Signora, I am Passafiore;
I come with a message to you from Falco. He no longer wishes to destroy
your railway."
"Have you come to jest with me?" she says.
Then the man answers, almost weeping: "Would God that it were a jest!
God! that Falco were the man he has been!"
He tells her how Falco went up Mongibello and crowned its top. But the
mountain had not liked it; it had now overthrown Falco. A single little
piece of pumice-stone from Mongibello had been enough to overthrow him.
"It is all over with Falco," says Passafiore. "He goes about in the
quarry, and waits to fall ill. For a week he has neither slept nor eaten.
He is not sick yet, but the wound in his hand does not heal either. He
thinks that he has the poison in his body. 'Soon I shall be a mad dog,'
he says. No wine nor food tempt him. He takes no pleasure in my praising
his deeds. 'What is that to talk about?' he says. 'I shall end my life
like a mad dog.'"
Donna Micaela looked sharply at Passafiore. "What do you wish me to do
about it? You cannot mean that I am to go down into the quarry to Falco
Falcone?"
Passafiore looks down and dares not answer anything.
She explains to him what that same Falco has made her suffer. He has
frightened away her workmen. He has set himself against her dearest wish.
All of a sudden Passafiore falls on his knees. He dares not go a step
nearer to her than he is, but he falls on his knees.
He implores her to understand the importance of it. She does not know,
she does not understand who Falco is. Falco is a great man. Ever since
Passafiore was a little child he has heard of him. All his life long he
has longed to come out to the quarry and live with him. All his cousins
went to Falco; his whole race were with him. But the priest had set his
heart that Passafiore should not go. He apprenticed him to a tailor; only
think, to a tailor! He talked to him, and said that he should not go. It
was such a terrible sin to live like Falco. Passafiore had also struggled
against it for many years for Don Matteo's sake. But at last he had not
been able to resist; he had gone to the quarry. And now he has not been
with Falco more than a year before the latter is quite destroyed. It is
as if the sun had gone out in the sky. His whole life is ruined.
Passafiore looks at Donna Micaela. He sees that she is listening to him,
and understands him.
He reminds Donna Micaela that she had helped a _jettatore_ and an
adulteress. Why should she be hard to a brigand? The Christ-image in San
Pasquale gave her everything she asked for. He was sure that she prayed
to the Christchild to protect the railway from Falco. And he had obeyed
her; he had made Mongibello's pumice-stone break Falco's might. But now,
would she not be gracious, and help them, that Falco might get his health
again, and be an honor to the land, as he had been before?
Passafiore succeeds in moving Donna Micaela. All at once she understands
how it is with the old brigand in the dark caves of the quarry. She sees
him there, waiting for madness. She thinks how proud he has been, and how
broken and crushed he now is. No, no; no one ought to suffer so. It is
too much, too much.
"Passafiore," she exclaims, "tell me what you wish. I will do whatever I
can. I am no longer afraid. No, I am not at all afraid."
"Donna Micaela, we have begged Falco to go to the Christchild and ask for
grace. But Falco will not believe in the image. He will not do anything
but sit still and wait for the disaster. But to-day, when I implored him
to go and pray, he said: 'You know who sits and waits for me in the old
house opposite the church. Go to her, and ask her if she will give me the
privilege to go by her into the church. If she gives her permission, then
I shall believe in the image, and say my prayers to him.'"
"Well?" questions Donna Micaela.
"I have been to old Caterina, and she has given her permission. 'He shall
be allowed to go into San Pasquale without my killing him,' she said."
Passafiore is still on his knees.
"Has Falco already been to the church?" asks Donna Micaela.
Passafiore moves somewhat nearer. He wrings his hands in despair. "Donna
Micaela, Falco is very ill. It is not alone that about the dog; he was
ill before." And Passafiore struggles with himself before he can say it
out. At last he acknowledges that although Falco is a very great man,
he sometimes has attacks of madness. He had not spoken of old Caterina
alone; he had said: "If Caterina will let me go into the church, and if
Donna Micaela Alagona comes down into the quarry and gives me her hand,
and leads me to the church, I will go to the image." And from that no one
had been able to move him. Donna Micaela, who was greatest and holiest of
women, must come to him, or he would not go.
When Passafiore has finished, he remains kneeling with bowed head. He
dares not look up.
But Donna Micaela does not hesitate a second, since there has been
question of the Christ-image. She seems not to think of Falco's being
already mad. She does not say a word of her terror. Her faith in the
image is such that she answers softly, like a subdued and obedient
child:--
"Passafiore, I will go with you."
She follows him as if walking in her sleep. She does not hesitate to go
with him up Etna. She does not hesitate to climb down the steep cliffs
into the quarry. She comes, pale as death, but with shining eyes, to the
old brigand in his hole in the cliff and gives him her hand. He rises up,
ghastly pale as she, and follows her. They do not seem like human beings,
but like spectres. They move on towards their goal in absolute silence.
Their own identity is dead, but a mightier spirit guides and leads them.
Even the day after it seems like a fairy tale to Donna Micaela that she
has done such a thing. She is sure that her own compassion, or pity, or
love could never have made her go down into the brigands' cave at night
if a strange power had not led her.
While Donna Micaela is in the robber's cave, old Caterina sits at her
window, and waits for Falco. She has consented, almost without their
needing to ask her.
"He shall go in peace to the church," she says. "I have waited for him
twenty years, but he shall go to the church."
Soon Falco comes by, walking with Donna Micaela's hand in his. Passafiore
and Biagio follow him. Falco is bent; it is plain that he is old and
feeble. He alone goes into the church; the others remain outside.
Old Caterina has seen him very plainly, but she has not moved. She sits
silent all the time Falco is inside the church. Her niece, who lives with
her, believes that she is praying and thanking God because she has been
able to conquer her thirst for revenge.
At last Caterina asks her to open a window. "I wish to see if he still
has his snake shadow," she says.
But she is gentle and friendly. "Take the gun, if you wish," she says.
And her niece moves the gun over to the other side of the table.
At last Falco comes from the church. The moonlight falls on his face,
and Caterina sees that he is unlike the Falco she remembered. The
terrible moroseness and arrogance are no longer visible in his face. He
comes bent and broken; he almost inspires her with pity.
"_He_ helps me," he says aloud to Passafiore and Biagio. "He has promised
to help me."
The brigands wish to go, but Falco is so happy that he must first tell
them of his joy.
"I feel no buzzing in my head; there is no burning, no uneasiness. He is
helping me."
His comrades take him by the hand to lead him away.
Falco goes a few steps, then stops again. He straightens himself up, and
at the same time moves his body so that the snake shadow writhes and
twists on the wall.
"I shall be quite well, quite well," he says.
The men drag him away, but it is too late.
Caterina's eyes have fallen on the snake shadow. She can control herself
no longer; she throws herself across the table, takes the gun, shoots
and kills Falco. She had not intended to do it, but when she saw him it
was impossible for her to let him go. She had cherished the thought of
revenge for twenty years. It took the upper hand over her.
"Caterina, Caterina," screams her niece.
"He only asked me to be allowed to go in peace _into_ the church,"
answers the old woman.
Old Biagio lays Falco's body straight, and says with a grim look:--
"He would be quite well; quite well."
XI
VICTORY
Far back in ancient days the great philosopher Empedokles lived in
Sicily. He was the most beautiful and the most perfect of men; so
wonderful and so wise that the people regarded him as an incarnate god.
Empedokles owned a country-place on Etna, and one evening he prepared a
feast there for his friends. During the repast he spoke such words that
they cried out to him: "Thou art a god, Empedokles; thou art a god!"
During the night Empedokles thought: "You have risen as high as you can
rise on earth. Now die, before adversity and feebleness take hold of
you." And he wandered up to the summit of Etna and threw himself into the
burning crater. "When no one can find my body," he thought, "the people
will say that I have been taken up alive to the gods."
The next morning his friends searched for him through the villa and on
the mountain. They too came up to the crater, and there they found by the
crater's mouth Empedokles' sandal. They understood that Empedokles had
sought death in the crater in order to be counted among the immortals.
He would have succeeded had not the mountain cast up his shoe.
But on account of that story Empedokles' name has never been forgotten,
and many have wondered where his villa could have been situated.
Antiquaries and treasure-seekers have looked for it; for the villa of the
wonderful Empedokles was naturally filled with marble statues, bronzes,
and mosaics.
Donna Micaela's father, Cavaliere Palmeri, had set his heart on solving
the problem of the villa. Every morning he mounted his pony, Domenico,
and rode away to search for it. He was armed as an investigator, with a
scraper in his belt, a spade at his side, and a big knapsack on his back.
Every evening, when Cavaliere Palmeri came home, he told Donna Micaela
about Domenico. During the years that they had ridden about on Etna,
Domenico had become an antiquary. Domenico turned from the road as soon
as he caught sight of a ruin. He stamped on the ground in places where
excavations should be made. He snorted scornfully and turned away his
head if any one showed him a counterfeit piece of old money.
Donna Micaela listened with great patience and interest. She was sure
that in case that villa finally did let itself be found Domenico would
get all the glory of the discovery.
Cavaliere Palmeri never asked his daughter about _her_ undertaking. He
never showed any interest in the railway. It seemed almost as if he were
ignorant that she was working for it.
It was not singular however; he never showed interest in anything that
concerned his daughter.
One day, as they both sat at the dining-table, Donna Micaela all at once
began to talk of the railway.
She had won a victory, she said; she had finally won a victory.
He must hear what news she had received that day. It was not merely to be
a railway between Catania and Diamante, as she first had thought; it was
to be a railway round the whole of Etna.
By Falco's death she had not only been rid of Falco himself, but now the
people believed also that the great Mongibello and all the saints were on
her side. And so there had arisen an agitation of the people to make the
railway an actuality. Contributions were signed in all the towns of Etna.
A company was formed. To-day the concession had come; to-morrow the work
was to begin in earnest.
Donna Micaela was excited; she could not eat. Her heart swelled with joy
and thankfulness. She could not help talking of the tremendous enthusiasm
that had seized the people. She spoke with tears in her eyes of the
Christchild in the church of San Pasquale.
It was touching to see how her face shone with hope. It was as if she
had, besides the happiness of which she was speaking, a whole world of
bliss in expectation.
That evening she felt that Providence had guided her well and happily.
She perceived that Gaetano's imprisonment had been the work of God to
lead him back to faith. He would be set free by the miracles of the
little image, and that would convert him so that he would become a
believer as before. And she might be his. How good God was!
And while this great bliss stirred within her, her father sat opposite
her quite cold and indifferent.
"It was very extraordinary," was all he said.
"You will come to-morrow to the ceremony of the laying of the
foundations?"
"I do not know; I have my investigations."
Donna Micaela began to crumble her bread rather hastily. Her patience was
exhausted. She had not asked him to share her sorrows, but her joys; he
must share her joys!
All at once the shackles of submission and fear, which had bound her ever
since the time of his imprisonment, broke.
"You who ride so much about Etna," she said with a very quiet voice,
"must have also come to Gela?"
The cavaliere looked up and seemed to search his memory. "Gela, Gela?"
"Gela is a village of a hundred houses, which is situated on the southern
side of Monte Chiaro, quite at its foot," continued Donna Micaela, with
the most innocent expression. "It is squeezed in between Simeto and the
mountain, and a branch of the river generally flows through the principal
street of Gela so that it is very unusual to be able to pass dry-shod
through the village. The roof of the church fell in during the last
earthquake, and it has never been mended, for Gela is quite destitute.
Have you really never heard of Gela?"
Cavaliere Palmeri answered with inexpressible solemnity: "My
investigations have taken me up the mountain. I have not thought of
looking for the great philosopher's villa in Gela."
"But Gela is an interesting town," said Donna Micaela, obstinately. "They
have no separate out-houses there. The pigs live on the lower floor, the
people one flight up. There is an endless number of pigs in Gela. They
thrive better than the people, for the people are almost always sick.
Fever is always raging there; malaria never leaves it. It is so damp that
the cellars are always under water, and it is wrapped in swamp mists
every night. In Gela there are no shops and no police, nor post-office,
nor doctor, nor apothecary. Six hundred people are living there forgotten
and brutalized. You have never heard of Gela?" She looked honestly
surprised.
Cavaliere Palmeri shook his head. "Of course I have heard the name--"
Donna Micaela cast a questioning glance on her father. She then bent
quickly forward towards him, and drew out of his breastpocket a small,
bent knife, such a knife as is used to prune grape-vines.
"Poor Empedokles," she said, and all at once her whole face sparkled
with fun. "You may believe you have mounted to the gods, but Etna always
throws up your shoe."
Cavaliere Palmeri sank back as if shot.
"Micaela!" he said, feebly fencing like some one who does not know how he
shall defend himself.
But she was instantly as serious and innocent as before. "I have been
told," she said, "that Gela a few years ago was on the way to ruin. All
the people there grow grapes, and when the phylloxera came and destroyed
their vineyards, they almost starved to death. The Agricultural Society
sent them some of those American plants that are not affected by the
phylloxera. The people of Gela set them out, but all the plants died. How
could the people of Gela know how to tend American vines? Well, some one
came and taught them."
"Micaela!"--it came almost like a wail. Donna Micaela thought that her
father already looked like a conquered man, but she continued as if she
had noticed nothing.
"_Some one came_," she said with strong emphasis, "and he had had new
vines sent out. He began to plant them in their vineyards. They laughed
at him; they said that he was mad. But look, his vines grew and lived;
they did not die. And he has saved Gela."
"I do not think that your story is entertaining, Micaela," said Cavaliere
Palmeri with an attempt to interrupt her.
"It is quite as entertaining as your investigations," she said, calmly.
"But I will tell you something. One day I went into your room to get a
book on antiquities. Then I found that all your bookshelves were full of
pamphlets about the phylloxera, about the cultivation of grapes, about
wine-making."
The cavaliere twisted on his chair like a worm. "Be silent; be silent!"
he said feebly. He was more embarrassed than when he was accused of theft.
Now all the suppressed fun shone once more in her eyes.
"I sometimes looked at the letters you sent off," she continued. "I
wished to see with what learned men you corresponded. It surprised me
that the letters were always addressed to presidents and secretaries of
Agricultural Societies."
Cavaliere Palmeri was unable to utter a word. Donna Micaela enjoyed his
helplessness more than can be described.
She looked him steadily in the eyes. "I do not believe that Domenico
has yet learned to recognize a ruin," she said with emphasis. "The
dirty children of Gela play with him every day, and feed him with
water-cresses. Domenico seems to be a god in Gela, to say nothing of
his--"
Cavaliere Palmeri seemed to have an idea.
"Your railway," he said; "what did you say about your railway? Perhaps I
really can come to-morrow."
Donna Micaela did not listen to him. She took up her pocket-book.
"I have here a counterfeit old coin," she said,--"a 'Demarata' of nickel.
I bought it to show Domenico. He is going to snort."
"Listen, child!"
She did not answer his attempts to make amends. Now the power was hers.
It would take more than that to pacify her.
"Once I opened your knapsack to look at your antiquities. The only thing
there was an old grape-vine."
She was full of sparkling gayety.
"Child, child!"
"What is it to be called? It does not seem to be investigating. Is it
perhaps charity; is it perhaps atonement--"
Cavaliere Palmeri struck with his clenched fist on the table so that
the glasses and plates rang. It was unbearable. A dignified and solemn
old gentleman could not endure such mockery. "As surely as you are my
daughter, you must be silent now."
"Your daughter!" she said, and her gayety was gone in an instant; "am
I really your daughter? The children in Gela are allowed to caress at
least Domenico, but I--"
"What do you wish, Micaela, what do you want?"
They looked at one another, and their eyes simultaneously filled with
tears.
"I have no one but you," she murmured.
Cavaliere Palmeri opened his arms unconditionally to her. She rose
hesitatingly; she did not know if she saw right.
"I know how it is going to be," he said, grumblingly; "not one minute
will I have to myself."
"To find the villa?"
"Come here and kiss me, Micaela! To-night is the first time since we left
Catania that you have been irresistible."
When she threw her arms about him it was with a hoarse, wild cry which
almost frightened him.
THIRD BOOK
"_And he shall win many followers_"
I
THE OASIS AND THE DESERT
In the spring of 1894 the Etna railway was begun; in the autumn of 1895
it was finished. It went up from the shore, made a circuit round the
mountain in a wide half-circle, and came down again to the shore.
Trains come and go every day, and Mongibello lies subdued and makes no
sign. Foreigners pass with amazement through the black, distorted lava
streams, through the groves of white almond-trees, through the dark old
Saracen towns. "Look, look! is there such a land on earth!" they say.
In the railway carriages there is always some one telling of the time
when the Christ-image was in Diamante.
What a time! What a time! Each day new miracles were performed. They
cannot tell of them all, but he brought as much happiness to Diamante as
if the hours of the day had been dancing maidens. People thought that
Time had filled his hour-glass with shining sands of gold.
If any one had asked who reigned in Diamante at that time, the answer
would have been that it was the Christ-image. Everything was done
according to his will. No one took a wife, or played in a lottery, or
built himself a house without consulting him.
Many knife-thrusts were spared for the image's sake, many old feuds
settled, and many bitter words were never uttered.
The people had to be good, for they observed that the image helped those
who were peaceable and helpful. To them he granted the pleasant gifts of
happiness and riches.
If the world had been as it ought to be, Diamante would soon have become
a rich and powerful town. But instead, that part of the world which did
not believe in the image destroyed all his work. All the happiness he
scattered about him was of no avail.
The taxes were constantly increased, and took all their money. There was
the war in Africa. How could the people be happy when their sons, their
money, and their mules had to go to Africa? The war did not go well; one
defeat followed another. How could they be happy when their country's
honor was at stake?
Especially after the railway had been finished was it manifest that
Diamante was like an oasis in a great desert. An oasis is exposed to the
drifting sands of the desert and to robbers and wild beasts. So was also
Diamante. The oasis would have to spread over the whole desert to feel
secure. Diamante began to believe that it could never be happy until the
whole world worshipped its Christ-image.
It now happened that everything that Diamante hoped and strove for was
denied it.
Donna Micaela and all Diamante longed to get Gaetano back. When the
railway was ready Donna Micaela went to Rome and asked for his release,
but it was refused her. The king and the queen would have liked to help
her, but they could not. You know who was minister then. He ruled Italy
with a hand of iron; do you think that he allowed the king to pardon a
rebellious Sicilian?
The people also longed that the Christchild of Diamante should have the
adoration that was his due, and Donna Micaela sought an audience for
his sake with the old man in the Vatican. "Holy Father," she said, "let
me tell you what has been taking place in Diamante on the <DW72>s of
Etna!" And when she had told of all the miracles performed by the image,
she asked the pope to have the old church of San Pasquale purified and
consecrated, and to appoint a priest for the worship of the Christchild.
"Dear Princess Micaela," said the pope, "those incidents of which you
speak, the church dares not consider miracles. But you need not at all
despair. If the Christchild wishes to be worshipped in your town, he will
give one more sign. He will show Us his will so plainly that We shall not
need to hesitate. And forgive an old man, my daughter, because he has to
be cautious!"
A third thing the people of Diamante had hoped. They had expected at last
to hear something from Gaetano. Donna Micaela journeyed also to Como,
where he was held prisoner. She had letters of recommendation from the
highest quarters in Rome, and she was sure that she would be allowed
to speak to him. But the director of the prison sent her to the prison
doctor.
The latter forbade her to speak to Gaetano.
"You wish to see the prisoner?" he said. "You shall not do it. Do you say
that he loves you and believes you to be dead? Let him think it! Let him
believe it! He has bowed his head to Death. He suffers no longing. Do you
wish him to know that you are alive, so that he may begin to long? You
wish, perhaps, to kill him? I will tell you something; if he begins to
long for life, he will be dead within three months."
He spoke so positively that Donna Micaela understood that she must give
up seeing Gaetano. But what a disappointment, what a disappointment!
When she came home, she felt like one who has dreamt so vividly that he
cannot, even after he is awake, rouse himself from his visions. She could
not realize that all her hopes had been a mockery. She surprised herself
time after time thinking: "When I have saved Gaetano." But now she no
longer had any hope of saving him.
She thought now of one, now of another enterprise, on which she wished to
embark. Should she drain the plain, or should she begin to quarry marble
on Etna. She hesitated and wondered. She could not keep her mind on
anything.
The same indolence that had taken possession of Donna Micaela crept
through the whole town. It was soon plain that everything that depended
on people who did not believe in the Christchild of Diamante was badly
managed and unsuccessful. Even the Etna railway was conducted in the
wrong way. Accidents were happening constantly on the steep inclines;
and the price of the tickets was too high. The people began to use the
omnibuses and post wagons again.
Donna Micaela and others with her began to think of carrying the
Christ-image out into the world. They would go out and show how he
gave health and subsistence and happiness to all who were quiet and
industrious and helped their neighbor. If people could once see, they
would certainly be converted.
"The image ought to stand on the Capitol and govern the world," said the
people of Diamante.
"All those who govern us are incapable," said the people. "We prefer to
be guided by the holy Christchild."
"The Christchild is powerful and charitable; if he ruled us, the poor
would be rich, and the rich would have enough. He knows who wish to do
right. If he should come to power, they who now are ruled would sit in
the parliament. He would pass through the world like a plough with a
sharp edge, and that which now lies unprofitable in the depths would then
bear harvests."
Before their longed-for plans came to pass, however, in the first days
of March, 1896, the news of the battle at Adna arrived. The Italians
had been defeated, and several thousands of them were killed or taken
prisoners.
A few days later there was a change of ministry in Rome. And the man who
came to power was afraid of the rage and despair of the Sicilians. To
pacify them he pardoned out several of the imprisoned socialists. The
five for whom he thought the people longed most were set free. They were
Da Felice, Bosco, Verro, Barbato and Alagona.
Ah, Micaela tried to be glad when she heard it. She tried not to weep.
She had believed that Gaetano was in prison because the Christ-image was
to break down the walls of his cell. He was sent there by the grace of
God, because he had to be forced to bow his head before the Christchild
and say: "My Lord and my God."
But now it was not the image which had freed him; he would come out the
same heathen as before; the same yawning chasm would still exist between
them.
She tried to be glad. It was enough that he was free. What did she or her
happiness matter in comparison to that!
But it happened so with everything for which Diamante had hoped and
striven.
The great desert was very cruel to the poor oasis.
II
IN PALERMO
At last, at last, it is one o'clock at night. Those who are afraid to
oversleep rise from their beds, dress themselves and go out into the
street.
And those who have sat and hung over a café table till now start up when
they hear steps echo on the stone pavements. They shake the drowsiness
from their bodies and hurry out. They mingle in the swiftly increasing
stream of people, and the heavy feet of Time begin to move a little
faster.
Mere acquaintances press each other's hands with heartfelt warmth. It
is plain that the same enthusiasm fills all souls. And the most absurd
people are out; old university professors, distinguished noblemen and
fine ladies, who otherwise never set their foot in the street. They are
all equally joyous.
"God! God! that he is coming, that Palermo is to have him back again!"
they say.
The Palermo students, who have not moved from their usual headquarters in
Quattro Canti all night, have provided torches and lanterns. They
were not to be lighted till four o'clock, when the man they expected was
to come; but about two o'clock one or two of them begin to try whether
their torches burn well. Then they light everything and greet the flames
with cheers. It is impossible to stand in darkness when so much joy is
burning within them.
In the hotels the travellers are waked and urged to get up. "There is a
festival in Palermo to-night, O signori!"
The travellers ask for whom. "For one of the socialists whom the
government has pardoned out of prison. He is coming now in the steamer
from Naples."--"What kind of a man is he?"--"His name is Bosco, and the
people love him."
There are preparations everywhere in the night for his sake. One of the
goatherds on Monte Pellegrino is busy tying little bunches of blue-bells
for his goats to wear in their collars. And as he has a hundred goats,
and they all wear collars--But it must be done. His goats could not
wander into Palermo the next morning without being adorned in honor of
the day.
The dressmakers have had to sit at their work till midnight to finish all
the new dresses that are to be worn that morning. And when such a little
dressmaker has finished her work for others, she has to think of herself.
She puts a couple of plumes in her hat and piles up bunches of ribbon a
yard high. To-day she must be beautiful.
The long rows of houses begin to be illuminated. Here and there a rocket
whizzes up. Fire-crackers hiss and snap at every street corner.
The flower shops along Via Vittorio Emanuele are emptied again and again.
Always more, more of the white orange-blossoms! All Palermo is filled
with the sweet fragrance of the orange-blossoms.
The gate-keeper in Bosco's house has no peace for a moment. Magnificent
cakes and towerlike bouquets are incessantly passing up the stairway, and
poems of welcome and telegrams of congratulation are constantly coming.
There is no end to them.
The poor bronze emperor on the Piazza Bologna, poor, ugly Charles the
Fifth, who is forlorn and thin and wretched as San Giovanni in the
desert, has in some inscrutable manner got a bunch of flowers in his
hand. When the students standing on Quattro Canti, quite near by, hear of
it, they march up to the emperor in a procession, light him with their
torches, and raise a cheer for the old despot. And one of them takes his
bunch of flowers to give it to the great socialist.
Then the students march down to the harbor.
Long before they get there their torches are burnt out, but they do not
care. They come with arms about each other's necks, singing loudly, and
sometimes breaking off in their song to shout: "Down with Crispi! Long
live Bosco!" The song begins again, but it is again broken off, because
those who cannot sing throw their arms round the singers and kiss them.
Guilds and corporations swarm out of the quarters of the town where the
same trade has been carried on for more than a thousand years. The masons
come with their band of music and their banner; there come the workers in
mosaic; here come the fishermen.
When the societies meet, they salute one another with their banners.
Sometimes they take time to stop and make speeches. Then they tell of
the five released prisoners, the five martyrs whom the government at
last has given back to Sicily. And all the people shout: "Long live
Bosco! Long live Da Felice! Long live Verro! Long live Barbato! Long live
Alagona!"
If any one who has had enough of the life in the streets comes down to
the harbor of Palermo, he stops and asks: "What place is this? Madonna
Santissima, where am I?"
For he has expected to find the harbor still deserted and dark.
All the boats and skiffs in the harbor of Palermo have been taken by
different societies and unions. They are floating about in the harbor,
richly hung with Venetian lights, and every minute great bunches
of rockets are sent up from them.
Over the heavy thwarts priceless rugs and hangings have been spread, and
on them sit ladies, the beautiful Palermo ladies, dressed in light silks
and shaded velvets.
The small craft glide about on the water, now in big groups, now
separately. From the big ships rise masts and oars covered with pennants
and lights, and the little harbor steam-launches dart about with funnels
wreathed in flowers.
Beneath it all the water lies and shines and mirrors and reflects, so
that the light from one lantern becomes a stream of brightness, and the
drops that fall from the oars are like a rain of gold.
Round about the harbor stand a hundred thousand, a hundred and fifty
thousand people, quite delirious with joy. They kiss one another; they
raise shouts of rapture, and they are happy, happy. They are beside
themselves with joy. Many of them cannot keep from weeping.
Fire, that is joy. It is good that fires can be lighted. Suddenly a great
blaze flames up on Monte Pellegrino, just over the harbor. Mighty flames
burst from all the pointed mountain walls surrounding the town. There are
fires on Monte Falcone, on San Martino, on the mountain of The Thousands,
where Garibaldi passed.
Far out on the sea comes the big Naples steamer. And on the steamer is
Bosco, the socialist.
He cannot sleep that night. He has gone up from his cabin, and paces to
and fro on the deck. And then his old mother, who has journeyed to Naples
to meet him, comes from her cabin to keep him company. But he cannot
talk with her. He is thinking that he will soon be at home. Ah, Palermo,
Palermo!
He has been in prison over two years. They have been two years of
suffering and longing, and has it been of any good? That is what he
wishes to know. Has it been of benefit that he has been faithful to
the cause, and gone to prison? Has Palermo thought of him? Have his
sufferings won the cause a single follower?
His old mother sits crouched on the gangway, and shivers in the chill
of the night. He has asked her, but she knows nothing of such things.
She speaks of little Francesco and little Lina, how they have grown. She
knows nothing of what he is struggling for.
Now he comes to his mother, takes her by the wrist, leads her to the
railing, and asks her if she sees anything far away to the south. She
looks out over the water with her dim eyes, and sees only the night, only
the black night on the water. She does not see at all that a cloud of
fire is floating on the horizon.
Then he begins to walk again, and she creeps down under cover. He does
not need to talk to her; it is joy enough to have him home again after
only two years' absence. He was condemned to be away for twenty-four.
She had not expected ever to see him again. But now the king has showed
grace. For the king is a good man. If only he were allowed to be as good
as he wished!
Bosco walks across the deck, and asks the sailors if they do not see the
golden cloud on the horizon.
"That is Palermo," say the seamen. "There is always a bright light
floating over it at night."
It cannot be anything that concerns him. He tries to persuade himself
that nothing is being done for him. He can hardly expect every one all at
once to have become socialists.
But after a while he thinks: "Still there must be something unusual going
on. All the sailors are gathering forward at the bow."
"Palermo is burning," say the seamen.
Yes, that is what it must be.--It is because he has suffered so terribly
that he expects something should be done for him.
Then the sailors see the fires on the mountains.
It cannot be a conflagration. It must be some saint's day. They ask one
another what day it is.
He, too, tries to believe that it is some such thing. He asks his mother
if it is a feast-day. They have so many of them.
They come nearer and nearer. The thundering sound of the festival in the
great city meets them.
"All Palermo is singing and playing to-night," says one.
"A telegram must have come of a victory in Africa," says another.
No one has a thought that it can be for his sake. He goes and places
himself at the stern in order not to see anything. He will not deceive
himself with false hopes. Would all Palermo be illuminated for a poor
socialist?
Then his mother comes and fetches him. "Do not stand there! Come and see
Palermo! It must be a king who is coming there to-day. Come and look at
Palermo!"
He considers a moment. No, he does not think that any king is visiting
Sicily just now. But he cannot dare to think, when no one else, not even
his mother--
All at once every one on the steamer gives a loud cry. It sounds almost
like a cry of distress. A big cutter has steered right down on them and
now glides along by the steamer's side.
The cutter is all flowers and lights; over the railing hang red and white
silken draperies, everybody on board is dressed in red and white. Bosco
stands on the steamer and looks to see what that beautiful messenger
brings. Then the sail turns, and on its white surface shines to meet him:
"Long live Bosco!"
It is his name. Not a saint's, not a king's, not the victorious
general's! The homage is for no other on the steamer. His name, his name!
The cutter sends up some rockets; a whole cloud of stars rain down, and
then it is gone.
He enters the harbor, and there is jubilation and enthusiasm and cheering
and adoration. People say: "We do not know how he will be able to live
through it."
But as soon as he realizes the homage, he feels that he does not at all
deserve it. He would like to fall on his knees before those hundred and
fifty thousand people who pay him homage and pray to them for forgiveness
that he is so powerless, that he has done nothing for them.
* * * * *
As though by a special fate, Donna Micaela is in Palermo that night. She
is there to start one of those new undertakings which she thinks she
ought to organize in order to retain life and reason. She is probably
there either on account of the draining or of the marble quarry.
She is down at the harbor; like all the others. People notice her as she
pushes her way forward to the edge of the water: a tall, dark woman, with
an air of being some one, a pale face with marked features and imploring,
longing, passionate eyes.
During the reception in the harbor, Donna Micaela is fighting out a
strange struggle. "If it were Gaetano," she thinks, "could I, could I--
"If it were for him all these people were rejoicing, could I--"
There is so much joy--a joy the like of which she has never seen. The
people love one another and are like brothers. And that not only because
a socialist is coming home, but because they all believe that the earth
will soon be happy. "If he were to come now, while all this joy is
roaring about me," she thinks. "Could I, could I--"
She sees Bosco's carriage trying to force a way through the crowd. It
moves forward step by step. For long moments it stands quite still. It
will take several hours to come up from the harbor.
"If it were he, and I saw every one crowding round him, could I forbear
from throwing myself into his arms? Could I?"
* * * * *
As soon as she can work her way out of the crowd she takes a carriage,
drives out of Palermo, and passes through the plain of Conca d'Oro to the
big Cathedral of the old Norman kings in Monreale.
She goes in, and stands face to face with the most beautiful image
of Christ that human art has created. High up in the choir sits the
blessing-giving Christ in glowing mosaic. He is mighty and mysterious and
majestic. Without number are they who make a pilgrimage to Monreale in
order to feel the consolation of gazing upon his face. Without number are
they who in far distant lands long for him.
The ground rocks under any one who sees him for the first time. His eyes
compel the knees of the foreigner to bend. Without being conscious of it
the lips falter: "Thou, God, art God."
About the walls of the temple glow the great events of the world in
wonderful mosaic pictures. They only lead to him. They are only there to
say: "All the past is his; all the present belongs to him, and all the
future."
The mysteries of life and death dwell within that head.
There lives the spirit which directs the fate of the world. There glows
the love which shall lead the world to salvation.
And Donna Micaela calls to him: "Thou son of God, do not part me from
thee! Let no man have power to part me from thee!"
III
THE HOME-COMING
It is a strange thing to come home. While yet on the journey, you cannot
at all realize how strange it will be.
When you come down to Reggio on the Strait of Messina, and see Sicily
emerge from the sea like a bank of fog, you are at first almost
impatient. "Is it nothing else?" you say. "It is only a land like all
others."
And when you disembark at Messina you are still impatient. Something
ought to have happened while you have been away. It is dreadful to be met
by the same poverty, the same rags, the same misery as when you went away.
You see that the spring has come. The fig-trees are again in leaf; the
grape-vines send out tendrils which grow yards long in a few hours, and a
mass of peas and beans are spread out on the fruit-stands by the harbor.
If you glance towards the heights above the town, you see that the gray
cactus plants that climb along the edges of the cliffs are covered with
blood-red flowers. They have blossomed everywhere like little, glowing
flames. It looks as if the flower cups had been filled with fire, which
now is breaking out.
But, however much the cactus blossoms, it is still gray and dusty and
cobwebby. You say to yourself that the cactus is like Sicily. However
many springs it may blossom, it is still the gray land of poverty.
It is hard to realize that everything has remained quiet and the same.
Scylla and Charybdis ought to have begun to roar as in former days. The
stone giant in the Girgenti temple should have risen with reconstructed
limbs. The temple of Selinunto ought to have raised itself from its
ruins. All Sicily should have awakened.
If you continue your journey from Messina down the coast, you are still
impatient. You see that the peasants are still ploughing with wooden
ploughs and that their horses are just as thin and broken and jaded.
Yes, everything is the same. The sun sheds its light over the earth like
a rain of color; the pelargoniums bloom at the roadside; the sea is a
soft pale blue, and caresses the shore.
Wild mountains with bold peaks line the coast. Etna's lofty top shines in
the distance.
You notice all at once that something strange is taking place. All your
impatience is gone. Instead you rejoice in the blossoming earth and in
the mountains and in the sea. You are reclaimed by the beautiful earth
as a bit of her lost property. There is no time to think of anything but
tufts and stones.
At last you approach your real home, the home of your childhood. What
wicked thoughts have filled your mind while you have been away! You never
wished to see that wretched home again, because you had suffered too much
there. And then you see the old walled town from afar, and it smiles at
you innocently, unconscious of its guilt. "Come and love me once more,"
it says. And you can only be happy and grateful because it is willing to
accept your love.
Ah, when you go up the zigzag path that leads to the gate of the town!
The light shade of the olive-tree falls over you. Was it meant as a
caress? A little lizard scampers along a wall. You have to stop and
look. May not the lizard be a friend of your childhood who wishes to say
good-day?
Suddenly a fear strikes you. Your heart begins to throb and beat. You
remember that you do not know what you may be going to hear when you come
home. No one has written letters; you have received none. Everything that
recalled home you have put away. It seemed the most sensible way, since
you were never to come home again. Up to that moment your feelings for
your home have been dead and indifferent.
But in that moment you do not know how you can bear it if everything is
not exactly the same on the mountain of your birth. It will be a mortal
blow if there is a single palm missing on Monte Chiaro or if a single
stone has loosened from the town wall.
Where is the big agave at the turn of the cliff? The agave is not
there; it has blossomed and been cut down. And the stone bench at the
street-corner is broken. You will miss that bench; it has been such a
pleasant resting-place. And look, they have built a barn on the green
meadow under the almond-trees. You will never again be able to stretch
out there in the flowering clover.
You are afraid of every step. What will you meet next?
You are so moved that you feel that you could weep if a single old
beggar-woman has died in your absence.
No, you did not know that to come home was so strange.
You came out of prison a few weeks ago, and the torpor of the prison
still has possession of you. You hardly know if you will take the trouble
to go home. Your beloved is dead; it is too terrible to tear your longing
from its grave. So you drift aimlessly about, and let one day pass like
the next. At last you pluck up courage. You must go home to your poor
mother.
And when you are there, you feel that you have been longing for every
stone, every blade of grass.
* * * * *
Ever since he came into the shop Donna Elisa has thought: "Now I will
tell him of Micaela. Perhaps he does not even know that she is alive."
But she puts it off from minute to minute, not only because she wishes to
have him for a while to herself alone, but also because as soon as she
mentions Micaela's name he will fall into the anguish and misery of love.
For Micaela will not marry him; she has said so to Donna Elisa a thousand
times. She would like to free him from prison, but she will not be the
wife of an atheist.
Only for one half-hour will Donna Elisa keep Gaetano for herself; only
for one half-hour.
But even so long she may not sit with his hand in hers, asking him a
thousand questions, for the people have learned that he has come. All at
once the whole street is full of those who wish to see him. Donna Elisa
has bolted the door, for she knew that she would not have him in peace a
moment after they had discovered him, but it was of little avail. They
knock on the windows, and pound on the door.
"Don Gaetano," they cry; "Don Gaetano!"
Gaetano comes laughing out to the steps. They wave their caps and cheer.
He hurries down into the crowd, and embraces one after another.
But that is not what they wish. He must go up on the steps and make a
speech. He must tell them how cruel the government has been to him, and
how he has suffered in prison.
Gaetano laughs still, and stations himself on the steps. "Prison," he
says; "what is it to talk about? I have had my soup every day, and that
is more than many of you can say."
Little Gandolfo swings his cap and calls to him: "There are many more
socialists in Diamante now than when you went away, Don Gaetano."
"How else could it be?" he laughs. "Everybody must become a socialist. Is
socialism anything dreadful or terrible? Socialism is an idyl. It is an
idyl of one's own home and happy work, of which every one dreams from his
childhood. A whole world filled with--"
He stops, for he has cast a glance towards the summer-palace. There
stands Donna Micaela on one of the balconies, and looks down at him.
He does not think for a moment that it is an illusion or a hallucination.
He sees instantly that she is flesh and blood. But just for that
reason--and also because the prison life has taken all his strength from
him, so that he cannot be considered a well person--
He feels a terrible difficulty in holding himself upright. He clutches
in the air with his hands, tries to get support from the door-post, but
nothing helps. His legs give way under him; he slides down the steps and
strikes his head on the stones.
He lies there like one dead.
Every one rushes to him, carries him in, runs after surgeon and doctor,
prescribes, talks, and proposes a thousand ways to help him.
Donna Elisa and Pacifica get him finally into one of the bedrooms. Luca
drives the people out and places himself on guard before the closed door.
Donna Micaela, who came in with the others, was taken first of them all
by the hand and led out. She was not allowed to stay in at all. Luca had
himself seen Gaetano fall as if from a blow on the temple when he caught
sight of her.
Then the doctor comes, and he makes one attempt after another to rouse
Gaetano. He is not successful; Gaetano lies as if turned to stone. The
doctor thinks that he received a dangerous blow on the head when he fell.
He does not know whether he will succeed in bringing him to life.
The swoon in itself was nothing, but that blow on the hard edge of the
stone steps--
In the house there is an eager bustle. The poor people outside can only
listen and wait.
There they stand the livelong day outside Donna Elisa's door. There stand
Donna Concetta and Donna Emilia. No love has been lost between them in
former times, but to-day they stand beside one another and mourn.
Many anxious eyes peer in through the windows of Donna Elisa's house.
Little Gandolfo and old Assunta from the Cathedral steps, and the poor
old chair-maker, stand there the whole afternoon without tiring. It is so
terrible that Gaetano is going to die just when they have got him back
again.
The blind stand and wait as if they expected him to give them their
sight, and the poor people, both from Geraci and Corvaja, are waiting to
hear how it will turn out for their young lord, the last Alagona.
He wished them well, and he had great strength and power. If he could
only have lived--
"God has taken his hand from Sicily," they say. "He lets all those perish
who wish to help the people."
All the afternoon and evening, and even till midnight, the crowd of
people are still outside Donna Elisa's house. At precisely twelve o'clock
Donna Elisa throws open the shop-door and comes out on the steps. "Is he
better?" they all cry at the sight of her.--"No, he is not better."
Then there is silence; but at last a single trembling voice asks: "Is he
worse?"--"No, no; he is not worse. He is the same. The doctor is with
him."
Donna Elisa has thrown a black shawl over her head and carries a lantern
in her hand. She goes down the steps to the street, where the people are
sitting and lying, closely packed one beside one another. She makes her
way quietly through them.
"Is Gandolfo here?" she asks. "Yes, Donna Elisa." And Gandolfo comes
forward to her.
"You must come with me and open your church for me."
Every one who hears Donna Elisa say that, understands that she wishes
to go to the Christchild in the church of San Pasquale and pray for
Gaetano. They rise and wish to go with her.
Donna Elisa is much touched by their sympathy. She opens her heart to
them.
"I will tell you something," she says, and her voice trembles
exceedingly. "I have had a dream. I do not know how I could sleep
to-night. But while I was sitting at the bedside, and was most anxious,
I did fall asleep. I had scarcely closed my eyes before I saw the
Christchild before me in his crown and gold shoes, as he stands out in
San Pasquale. And he spoke in this way to me: 'Make the unhappy woman who
is on her knees praying in my church your son's wife, then Gaetano will
be well.' He hardly had time to say it before I awoke, and when I opened
my eyes, I seemed to see the Christchild disappearing through the wall.
And now I must go out and see if any one is there.
"But now you all hear that I vow that if there is any woman out in the
church of San Pasquale, I shall do what the image commanded me. Even if
it is the poorest girl from the street, I shall take charge of her and
make her my son's wife."
When Donna Elisa has spoken, she and all those who have waited in the
street go out to San Pasquale. The poor people are filled with shuddering
expectation. They can scarcely contain themselves from rushing by Donna
Elisa, in order to see if there is any one in the church.
Fancy if it is a gypsy girl who has sought shelter there for the night!
Who can be in the church at night except some poor, homeless wanderer?
Donna Elisa has made a terrible vow.
At last they come to Porta Etnea, and from there they go quickly,
quickly down the hill. The saints preserve us, the church door is open!
Some one really is there.
The lantern shakes in Donna Elisa's hand. Gandolfo wishes to take it from
her, but she will keep it. "In God's name, in God's name," she murmurs as
she goes into the church.
The people crowd in after her. They almost crush one another to death in
the door, but their excitement keeps them silent, no one says a word. All
gaze at the high altar. Is any one there? Is any one there? The little
hanging-lamp over the image shines pitifully faint. Is any one there?
Yes, some one is there. There is a woman there. She is on her knees,
praying, and her head is so deeply bent that they cannot see who she is.
But when she hears steps behind her she lifts her long, bowed neck and
looks up. It is Donna Micaela.
At first she is frightened and starts up as if she wished to escape.
Donna Elisa is also frightened, and they look at one another as if they
had never met before. Then Donna Micaela says in a very low voice: "You
have come to pray for him, sister-in-law." And the people see her move a
little way along so that Donna Elisa may have room directly in front of
the image.
Donna Elisa's hand trembles so that she has to set the lantern down on
the floor, and her voice is quite hoarse as she says: "Has none other but
you been here to-night, Micaela?"--"No, none other."
Donna Elisa has to support herself against the wall to keep from falling,
and Donna Micaela sees it. She is instantly beside her and puts her
arm about her waist. "Sit down, sit down!" She leads her to the altar
platform and kneels down in front of her. "Is he so ill? We will pray for
him."
"Micaela," says Donna Elisa, "I thought that I should find help
here."--"Yes, you shall see, you will."--"I dreamed that the image
came to me, that he came to me and said that I was to come here."--"He
has also helped us many times before."--"But he said this to me: 'Make
the unhappy woman who is on her knees praying before my altar your
son's wife, then your son will be well.'"--"What do you say that he
said?"--"I was to make her who was kneeling and praying out here my son's
wife."--"And you were willing to do it? You did not know whom you would
meet!"
"On the way I made a vow--and those who followed me heard it--that
whoever it might be, I would take her in my arms and lead her to my home.
I thought that it was some poor woman whom God wished to help."--"It is
one indeed."--"I was in despair when I saw that there was no one here but
you."
Donna Micaela does not answer; she gazes up at the image. "Is it your
will? Is it your will?" she whispers anxiously.
Donna Elisa continues to bemoan herself. "I saw him so plainly, and
he has never deceived before. I thought that some poor girl who had
no marriage portion had prayed to him for a husband. Such things have
happened before. What shall I do now?"
She laments and bewails; she cannot get away from the thought that it
ought to be a poor woman. Donna Micaela grows impatient. She takes her
by the arm and shakes her. "But Donna Elisa, Donna Elisa!"
Donna Elisa does not listen to her; she continues her laments. "What
shall I do? what shall I do?"
"Why, make the poor woman who was kneeling and praying here your son's
wife, Donna Elisa!"
Donna Elisa looks up. Such a face as she sees before her! So bewitching,
so captivating, so smiling!
But she may not look at it for more than a second. Donna Micaela hides it
instantly in Donna Elisa's old black dress.
* * * * *
Donna Micaela and Donna Elisa go together into the town. The street
winds so that they cannot see Donna Elisa's house until they are quite
near. When it at last comes into view they see that the shop windows are
lighted up. Four gigantic wax-candles are burning behind the bunches of
rosaries.
Both the women press each other's hands. "He lives!" one whispers to the
other. "He lives!"
"You must not tell him anything about what the image commanded you to
do," says Donna Micaela to Donna Elisa.
Outside the shop they embrace one another and each goes her own way.
In a little while Gaetano comes out on the steps of the shop. He stands
still for a moment and breathes in the fresh night air. Then he sees how
lights are burning in the dark palace across the street.
Gaetano breathes short and panting; he seems almost afraid to go further.
Suddenly he dashes across like some one going to meet an unavoidable
misfortune. He finds the door to the summer-palace unlocked, takes the
stairs in two bounds, and bursts open the door to the music-room without
knocking.
Donna Micaela is sitting there, wondering if he will come now in the
night or the next morning. Then she hears his step outside in the
gallery. She is seized with terror; how will he be? She has longed so
unspeakably for him. Will he really be so that all that longing will be
satisfied?
And will no more walls rise between them? Will they for once be able to
tell each other everything? Will they speak of love, and not of socialism?
When he opens the door she tries to go to meet him, but she cannot; she
is trembling in every limb. She sits down and hides her face in her hands.
She expects him to throw his arms about her and kiss her, but that he
does not do. It is not Gaetano's way to do what people expect of him.
As soon as he could stand upright he has thrown on his clothes to come
to see her. He is apparently wildly gay when he comes now. He would
have liked her to take it lightly also. He will not be agitated. He had
fainted in the forenoon. He could stand nothing.
He stands quietly beside her until she regains her composure. "You have
weak nerves," he says. That is actually all he says.
She and Donna Elisa and every one is convinced that he has come to clasp
her in his arms and say that he loves her. But just for that reason it is
impossible for Gaetano. Some people are malicious; it is their nature
never to do just what they ought to do.
Gaetano begins to tell her of his journey; he does not speak even of
socialism, but talks of express-trains and conductors and curious
travelling companions.
Donna Micaela sits and looks at him; her eyes beg and implore more and
more eagerly. Gaetano seems to be glad and happy to see her, but why can
he not say what he has to say?
"Have you been on the Etna railway?" she asks.
"Yes," he answers, and begins quite unconstrainedly to speak of the
beauty and usefulness of the road. He knows nothing of how it came to be.
Gaetano is saying to himself that he is a brute. Why does he not speak
the words for which she is longing? But why is she sitting there so
humbly? Why does she show that he needs only to stretch out his hand and
take her? He is desperately, stormily happy to be near her, but he feels
so sure of her, so certain. It is so amusing to torture her.
The people of Diamante are still standing outside in the street, and they
all feel as great a happiness as if they had given away a daughter in
marriage.
They have been patient till now in order to give Gaetano time to declare
himself. But now it surely must be accomplished. And they begin to
shout:--
"Long live Gaetano! long live Micaela!"
Donna Micaela looks up with inexpressible dismay. He surely must
understand that she has nothing to do with it.
She goes out to the gallery and sends Luca down with the request that
they will be silent.
When she comes back, Gaetano has risen. He offers her his hand; he wishes
to go.
Donna Micaela puts out her hand almost without knowing what she is doing.
But then she draws it back; "No, no," she says.
He wishes to go, and who knows whether he will come again on the morrow.
She has not been able to talk to him; she has not been able to say a word
to him of all that she wished to say.
Surely there was no need for them to be like ordinary lovers. That man
had given her life all its life for many years. Whether he spoke to her
of love or not was of no importance; yet she wishes to tell him what he
has been to her.
And now, just now. One has to make the most of one's opportunities when
Gaetano is in question. She dares not let him go.
"You must not go yet," she says. "I have something to say to you."
She draws forward a chair for him; she herself places herself a little
behind him. His eyes are too gay to-night, they trouble her.
Then she begins to speak. She lays before him the great, hidden treasures
of her life. They were all the words he had said to her and all the
dreams he had set her to dreaming. She had not lost one. She had
collected and saved them up. They had been the only richness in her poor
life.
In the beginning she speaks fast, as if repeating a lesson. She is afraid
of him; she does not know whether he likes her to speak. At last she
dares to look at him. He is serious now, no longer malicious. He sits
still and listens as if he would not lose a syllable. Just now his face
was sickly and ashen, but now it suddenly changes. His face begins to
shine as though transfigured.
She talks and talks. She looks at him, and now she is beautiful. How
could she help being beautiful? At last she can speak out to him, she
can tell him how love came to her and how it has never left her since.
Finally she can tell him how he has been all the world to her.
Words cannot say enough; she takes his hand and kisses it.
He lets her do it without moving. The color in his cheeks grows no
deeper, but it becomes clearer, more transparent. She remembers Gandolfo,
who had said that Gaetano's face was so white that it shone.
He does not interrupt her. She tells him about the railway, speaks of one
miracle after another. He looks at her now and then. His eyes glow at the
sight of her. He is not by any means making fun of her.
She wonders exceedingly what is passing in him. He looks as if what
she said was nothing new to him. He seems to recognize everything she
says. Could it be that his love for her was the same as that she felt
for him? Was it connected with every noble feeling in him? Had it been
the elevating power in his life? Had it given wings to his artistic
powers? Had it taught him to love the poor and the oppressed? Is it once
more taking possession of him, making him feel that he is an artist, an
apostle, that nothing is too high for him?
But as he is still silent she thinks that perhaps he will not be tied to
her. He loves her, but possibly he wishes to be a free man. Perhaps he
thinks that she is not a suitable wife for a socialist.
Her blood begins to boil. She thinks that he perhaps believes that she is
sitting there and begging for his love.
She has told him almost everything that has happened while he has been
away. Now she suddenly breaks off in her story.
"I have loved you," she says. "I shall always love you, and I think that
I should like you to tell me once that you love me. It would make the
parting easier to bear."
"Would it?" he says.
"Can I be your wife?" she says, and her voice trembles with indignation.
"I no longer fear your teachings as I did; I am not afraid of your poor;
I wish to turn the world upside down, I, as well as you. But I am a
believer. How can I live with you if you do not agree with me in that? Or
perhaps you would win me to unbelief? Then the world would be dead for
me. Everything would lose its meaning, its significance. I should be a
miserable, destitute creature. We must part."
"Really!" he turns towards her. His eyes begin to glow with impatience.
"You may go now," she says quietly; "I have said to you everything I
wished to say. I should have wished that you had something to say to me.
But perhaps it is better as it is. We will not make it harder to part
than it need be."
One of Gaetano's hands holds her hands firmly and closely, the other
holds her head still. Then he kisses her.
Was she mad, that she could think that he would let anything, anything in
the world, part them now?
IV
ONLY OF THIS WORLD
As she grew up everybody said of her: "She is going to be a saint, a
saint."
Her name was Margherita Cornado. She lived in Girgenti on the south side
of Sicily, in the great mining district. When she was a child her father
was a miner; later he inherited a little money, so that he no longer
needed to work.
There was a little, narrow, miserable roof-garden on Margherita Cornado's
house in Girgenti. A small and steep stairway led up to it, and one had
to creep out through a low door. But it was well worth the trouble. When
you reached the top you saw not only a mass of roofs, but the whole
air over the town was gaily crowded with the towers and façades of all
Girgenti's churches. And every façade and every tower was a quivering
lace-work of images, of loggias, of glowing canopies.
And outside the town there was a wide plain which sloped gently down
towards the sea, and a semicircle of hills that guarded the plain. The
plain was glittering red; the ocean was blue as enamel; the hillsides
were yellow; it was a whole orient of warmth and color.
But there was even more to be seen. Ancient temples were dotted about
the valley. Ruins and strange old towers were everywhere, as in a fairy
world.
As Margherita Cornado grew up, she used to spend most of her days there;
but she never looked out over the dazzling landscape. She was occupied
with other things.
Her father used to tell her of the life in the sulphur mines at Grotte,
where he had worked. While Margherita Cornado sat on the airy terrace,
she thought that she was incessantly walking about the dark mine veins,
and finding her way through dim shafts.
She could not help thinking of all the misery that existed in the mines;
especially she thought of the children, who carried the ore up to the
surface. "The little wagons," they called them. That expression never
left her mind. Poor, poor little wagons, the little mine-wagons!
They came in the morning, and each followed a miner down into the mine.
As soon as he had dug out enough ore, he loaded the mine-wagon with a
basket of it, and then the latter began to climb. Several of them met on
the way, so that there was a long procession. And they began to sing:--
"One journey made in struggling and pain,
Nineteen times to be travelled again."
When they finally reached the light of day, they emptied their baskets
of ore and threw themselves on the ground to rest a moment. Most of them
dragged themselves over to the sulphurous pools near the shaft of the
mine and drank the pestiferous water.
But they soon had to go down again, and they gathered at the mouth of
the mine. As they clambered down, they cried: "Lord and God, have mercy,
have mercy, have mercy!"
Every journey the little wagons made, their song grew more feeble. They
groaned and cried as they crawled up the paths of the mine.
The little wagons were bathed in perspiration; the baskets of ore ground
holes in their shoulders. As they went up and down they sang:--
"Seven more trips without pause for breath,
The pain of living is worse than death."
Margherita Cornado had suffered for those poor children all her own
childhood. And because she was always thinking of their hardships, people
believed that she would be a saint.
Neither did she forget them as she grew older. As soon as she was grown,
she went to Grotte, where most of the mines are, and when the little
wagons came out into the daylight, she was waiting for them by the shaft
with fresh, clean water. She wiped the perspiration from their faces, and
she dressed the wounds on their shoulders. It was not much that she could
do for them, but soon the little wagons felt that they could not go on
with their work any day that Margherita Cornado did not come and comfort
them.
But unfortunately for the little wagons, Margherita was very beautiful.
One day one of the mining-engineers happened to see her as she was
relieving the children, and instantly fell very much in love with her.
A few weeks after, Margherita Cornado stopped coming to the Grotte mines.
She sat at home instead and sewed on her wedding outfit. She was going
to marry the mining-engineer. It was a good match, and connected her with
the chief people of the town, so she could not care for the little wagons
any longer.
A few days before the wedding the old beggar, Santuzza, who was
Margherita's god-mother, came and asked to speak to her. They betook
themselves to the roof-garden in order to be alone.
"Margherita," said the old woman, "you are in the midst of such happiness
and magnificence that perhaps there is no use speaking to you of those
who are in need and sorrow. You have forgotten all such things."
Margherita reproved her for speaking so.
"I come with a greeting to you from my son, Orestes. He is in trouble,
and he needs your advice."
"You know that you can speak freely to me, Santuzza," said the girl.
"Orestes is no longer at the Grotte mines; you know that, I suppose. He
is at Racalmuto. And he is very badly off there. Not that the pay is so
bad, but the engineer is a man who grinds down the poor to the last drop
of blood."
The old woman told how the engineer tortured the miners. He made them
work over time; he fined them if they missed a day. He did not look after
the mines properly; there was one cave-in after another. No one was
secure of his life as long as he was under earth.
"Well, Margherita, Orestes had a son. A splendid boy; just ten years old.
The engineer came and wished to buy the boy from Orestes, and set him to
work with the little wagons. But Orestes said no. His boy should not be
ruined by such work.
"Then the engineer threatened him, and said that Orestes would be
dismissed from the mine."
Santuzza paused.
"And then?" asked Margherita.
"Yes, then Orestes gave his son to the engineer. The next day the boy
got a whipping from him. He beat him every day. The boy grew more and
more feeble. Orestes saw it, and asked the engineer to spare the boy,
but he had no mercy. He said that the boy was lazy, and he continued to
persecute him. And now he is dead. My grandson is dead, Margherita."
The girl had quite forgotten all her own happiness. She was once more
only the miner's daughter, the protector of the little wagons, the poor
child who used to sit on the bright terrace and weep over the hardships
of the black mines.
"Why do you let the man live?" she cried.
The old woman looked at her furtively. Then she crept close to her with a
knife. "Orestes sends you this with a thousand questions," she said.
Margherita Cornado took the knife, kissed the blade, and gave it back
without a word.
It was the evening before the wedding. The parents of the bridegroom were
awaiting their son. He was to come home from the mines towards night; but
he never came. Later in the night a servant was sent to the Grotte mines
to look for him, and found him a mile from Girgenti. He lay murdered at
the roadside.
A search for the murderer was immediately instituted. Strict examinations
of the miners were held, but the culprit could not be discovered. There
were no witnesses; no one could be prevailed upon to betray a comrade.
Then Margherita Cornado appeared and denounced Orestes, who was the son
of her god-mother, Santuzza, and who had not moved to Racalmuto at all.
She did it although she had heard afterwards that her betrothed had
been guilty of everything of which Santuzza had accused him. She did it
although she herself had sealed his doom by kissing the knife.
She had hardly accused Orestes before she repented of it; she was filled
with the anguish of remorse.
In another land what she had done would not have been considered a crime,
but it is so regarded in Sicily. A Sicilian would rather die than be an
informer.
Margherita Cornado enjoyed no rest either by night or by day. She had a
continual aching feeling of anguish in her heart, a great unhappiness
dwelt in her.
She was not severely judged, because every one knew that she had loved
the murdered man and thought that Santuzza had been too cruel towards
her. No one spoke of her disdainfully, and no one refused to salute her.
But it made no difference to her that others were kind to her. Remorse
filled her soul and tortured her like an aching wound. Orestes had
been sentenced to the galleys for life. Santuzza had died a few weeks
after her son's sentence had been passed, and Margherita could not ask
forgiveness of either of them.
She called on the saints, but they would not help her. It seemed as if
nothing in the world could have the power to free her from the horror of
remorse.
At that time the famous Franciscan monk, Father Gondo, was sojourning in
the neighborhood of Girgenti. He was preaching a pilgrimage to Diamante.
It did not disturb Father Gondo not to have the pope acknowledge the
Christ-image in the church of San Pasquale as a miracle-worker. He had
met the blind singers on his wanderings and had heard them tell of the
image. Through long, happy nights he had sat at the feet of Father Elia
and Brother Tommaso, and from sunset to sunrise they had told him of the
image.
And now the famous preacher had begun to send all who were in trouble
to the great miracle-worker. He warned the people not to let that holy
time pass unheeded. "The Christchild," he said, "had not hitherto been
much worshipped in Sicily. The time had come when he wished to possess
a church and followers. And to effect it he let his holy image perform
miracle after miracle."
Father Gondo, who had passed his novitiate in the monastery of Aracoeli
on the Capitol, told the people of the image of the Christchild that
was there, and of the thousand miracles he had performed. "And now that
good little child wishes to be worshipped in Sicily," said Father Gondo.
"Let us hesitate no longer, and hasten to him. For the moment heaven is
generous. Let us be the first to acknowledge the image! Let us be like
the shepherds and wise men of the East; let us go to the holy child
while he is still lying on his bed of straw in the miserable hut!"
Margherita Cornado was filled with a new hope when she heard him. She was
the first to obey Father Gondo's summons. After her others joined him
also. Forty pilgrims marched with him through the plateaus of the inland
to Diamante.
They were all very poor and unhappy. But Father Gondo made them march
with song and prayer. Soon their eyes began to shine as if the star of
Bethlehem had gone before them.
"Do you know," said Father Gondo, "why God's son is greater than all the
saints? Because he gives the soul holiness; because he forgives sins;
because he grants to the spirit a blessed trust in God; because his
kingdom is not of this world."
When his little army looked tired, he gave them new life by telling them
of the miracles the image had performed. The legends of the blind singers
were like cooling drinks and cheering wine. The poor wanderers in the
barren lands of Sicily walked with a lighter step, as if they were on
their way to Nazareth to see the carpenter's son.
"He will take all our burdens from us," said Father Gondo. "When we come
back our hearts will be freed from every care."
And during the wandering through the scorched, glowing desert, where no
trees gave cooling shade, and where the water was bitter with salt and
sulphur, Margherita Cornado felt that her heart's torments were relieved.
"The little king of heaven will take away my pain," she said.
At last, one day in May, the pilgrims reached the foot of the hill of
Diamante. There the desert stopped. They saw about them groves of
olive-trees and fresh green leaves. The mountain shone; the town shone.
They felt that they had come to a place in the shadow of God's grace.
They toiled joyfully up the zigzag path, and with loud and exultant
voices sang an old pilgrims' song.
When they had gone some way up the mountain, people came running from
Diamante to meet them. When the people heard the monotonous sound of the
old song, they threw aside their work and hurried out. And the people of
Diamante embraced and kissed the pilgrims.
They had expected them long ago; they could not understand why they
had not come before. The Christ-image of Diamante was a wonderful
miracle-worker; he was so compassionate, so loving that every one ought
to come to him.
When Margherita Cornado heard them she felt as if her heart was already
healed of its pain. All the people of Diamante comforted her and
encouraged her. "He will certainly help you; he helps every one," they
said. "No one has prayed to him in vain."
At the town-gate the pilgrims parted. The townspeople took them to their
homes, so that they might rest after their journey. In an hour they were
all to meet at the Porta Etnea in order to go out to the image together.
But Margherita had not the patience to wait a whole hour. She asked her
way out to the church of San Pasquale and went there alone before all the
others.
When Father Gondo and the pilgrims came out to San Pasquale an hour
later, they saw Margherita Cornado sitting on the platform by the high
altar. She was sitting still and did not seem to notice their coming. But
when Father Gondo came close up to her, she started up as if she had lain
in wait for him and threw herself upon him. She seized him by the throat
and tried to strangle him.
She was big, splendidly developed and strong. It was only after a severe
struggle that Father Gondo and two of the pilgrims succeeded in subduing
her. She was quite mad, and so violent that she had to be bound.
The pilgrims had come in a solemn procession; they sang, and held burning
candles in their hands. There was a long line of them, for many people
from Diamante had joined them. Those who came first immediately stopped
their singing; those coming after had noticed nothing and continued their
song. But then the news of what had happened passed from file to file,
and wherever it came the song stopped. It was horrible to hear how it
died away and changed into a low wail.
All the weary pilgrims realized that they had failed in their coming. All
their laborious wanderings had been in vain. They were disappointed in
their beautiful hopes. The holy image would have no consolation to offer
them.
Father Gondo himself was in despair. It was a more severe blow to him
than to any one else, for each one of the others had only his own sorrow
to think of, but he bore the sorrows of all those people in his heart.
What answer could he give to all the hopes he had awakened in them?
Suddenly one of his beautiful, child-like smiles passed over his face.
The image must wish to test his faith and that of the others. If only
they did not fail, they would certainly be helped.
He began again to sing the pilgrim song in his clear voice and went up to
the altar.
But as he came nearer to the image, he broke off in his song again. He
stopped and looked at the image with staring eyes. Then he stretched out
his hand, took the crown and brought it close to his eyes. "It is written
there; it is written there," he murmured. And he let the crown fall from
his hand and roll down on the stone floor.
From that moment Father Gondo knew that the outcast from Aracoeli was
before him.
But he did not immediately cry it out to the people, but said instead,
with his usual gentleness,--
"My friends, I wish to tell you something strange."
He told them of the Englishwoman who had wished to steal the Christ-image
of Aracoeli. And he told how the image had been called Antichrist and had
been cast out into the world.
"I still remember old Fra Simone," said Father Gondo. "He never showed me
the image without saying: 'It was this little hand that rang. It was this
little foot that kicked on the door.'
"But when I asked Fra Simone what had become of the other image, he
always said: 'What should have become of him? The dogs of Rome have
probably dragged him away and torn him to pieces.'"
When Father Gondo had finished speaking, he went, still quite slowly and
quietly, and picked up the crown that he had just let fall to the floor.
"Now read that!" he said. And he let the crown go from man to man. The
people stood with their wax-candles in their hands and lighted up the
crown with them. Those who could read, read; the others saw that at least
there was an inscription.
And each one who had held the crown in his hand instantly extinguished
his candle.
When the last candle was put out, Father Gondo turned to his pilgrims
who had gathered about him. "I have brought you here," he said to them,
"that you might find one who gives the soul peace and an entry to God's
kingdom; but I have brought you wrong, for this one has no such thing to
give. His kingdom is only of this world.
"Our unfortunate sister has gone mad," continued Father Gondo, "because
she came here and hoped for heavenly benefits. Her reason gave way when
her prayers were not heard. He could not hear her, for his kingdom is
only of this world."
He was silent a moment, and they all looked up at him to find out what
they ought to think of it all.
He asked as quietly as before: "Shall an image which bears such words in
its crown any longer be allowed to desecrate an altar?"
"No, no!" cried the pilgrims. The people of Diamante stood silent.
Father Gondo took the image in his hands and carried it on his
outstretched arms through the church and towards the door.
But although the Father had spoken gently and humbly, his eyes had rested
the whole time sternly and with compelling force on the crowd of people.
There was not one there whom he had not subdued and mastered by the
strength of his will. Every one had felt paralyzed and without the power
of thinking independently.
As Father Gondo approached the door, he stopped and looked around. One
last commanding glance fell on the people.
"The crown also," said Father Gondo. And the crown was handed to him.
He set the image down and went out under the stone canopy that protected
the image of San Pasquale. He whispered a word to a couple of pilgrims,
and they hurried away. They soon came back with their arms full of
branches and logs. They laid them down before Father Gondo and set them
on fire.
All who had been in the church had crowded out. They stood in the yard
outside the church, still subdued, with no will of their own. They saw
that the monk meant to burn their beloved image that helped them so, and
yet they made no resistance. They could not understand themselves why
they did not try to save the image.
When Father Gondo saw the fire kindle and therefore felt that the image
was entirely in his power, he straightened himself and his eyes flashed.
"My poor children," he said gently, and turned to the people of Diamante.
"You have been harboring a terrible guest. How is it possible for you not
to have discovered who he is?
"What ought I to believe of you?" he continued more sternly. "You
yourselves say that the image has given you everything for which you
have prayed. Has no one in Diamante in all these years prayed for the
forgiveness of sins and the peace of the soul?
"Can it be possible? The people of Diamante have not had anything to pray
for except lottery numbers and good years and daily bread and health and
money. They have asked for nothing but the good of this world. Not one
has needed to pray for heavenly grace.
"Can it really be? No, it is impossible," said Father Gondo joyfully,
as if filled with a sudden hope. "It is I who have made a mistake. The
people of Diamante have understood that I would not lay the image on the
fire without asking and investigating about it. You are only waiting for
me to be silent to step forward and give your testimony.
"Many will now come and say: 'That image has made me a believer;' and
many will say: 'He has granted me the forgiveness of sins;' and many will
say: 'He has opened my eyes, so that I have been able to gaze on the
glory of heaven.' They will come forward and speak, and I shall be mocked
and derided and compelled to bear the image to the altar and acknowledge
that I have been mistaken."
Father Gondo stopped speaking and smiled invitingly at the people. A
quick movement passed through the crowd of listeners. Several seemed to
have the intention of coming forward and testifying. They came a few
steps, but then they stopped.
"I am waiting," said the Father, and his eyes implored and called on the
people to come.
No one came. The whole mass of people was in wailing despair that they
would not testify to the advantage of their beloved image. But no one did
so.
"My poor children," said Father Gondo, sadly. "You have had Antichrist
among you, and he has got possession of you. You have forgotten heaven.
You have forgotten that you possess a soul. You think only of this world.
"Formerly it was said that the people of Diamante were the most religious
in Sicily. Now it must be otherwise. The inhabitants of Diamante are
slaves of the world. Perhaps they are even infidel socialists, who love
only the earth. They can be nothing else. They have had Antichrist among
them."
When the people were accused in such a way, they seemed at last to be
about to rise in resistance. An angry muttering passed through the ranks.
"The image is holy," one cried. "When he came San Pasquale's bells rang
all day."
"Could they ring for less time to warn you of such a misfortune?"
rejoined the monk.
He went on with his accusations with growing violence. "You are
idolaters, not Christians. You serve him because he helps you. There is
nothing of the spirit of holiness in you."
"He has been kind and merciful, like Christ," answered the people.
"Is not just that the misfortune?" said the Father, and now all of a
sudden he was terrible in his wrath. "He has taken the likeness of Christ
to lead you astray. In that way he has been able to weave his web about
you. By scattering gifts and blessings over you, he has lured you into
his net and made you slaves of the world. Or is it not so? Perhaps some
one can come forward and say the contrary? Perhaps he has heard that some
one who is not present to-day has prayed to the image for a heavenly
grace."
"He has taken away the power of a _jettatore_," said one.
"Is it not he who is as great in evil as the _jettatore_ who has power
over him?" answered the father, bitterly.
They made no other attempts to defend the image. Everything that they
said seemed only to make the matter worse.
Several looked round for Donna Micaela, who was also present. She stood
among the crowd, heard and saw everything, but made no attempt to save
the image.
When Father Gondo had said that the image was Antichrist she had been
terrified, and when he showed that the people of Diamante had only asked
for the good of this world, her terror had grown. She had not dared to do
anything.
But when he said that she and all the others were in the power of
Antichrist, something in her rose against him. "No, no," she said, "it
cannot be so." If she should believe that an evil power had governed her
during so many years, her reason would give way. And her reason began to
defend itself.
Her faith in the supernatural broke in her like a string too tightly
stretched. She could not follow it any longer.
With infinite swiftness everything of the supernatural that she herself
had experienced flashed through her mind, and she passed sentence on it.
Was there a single proven miracle? She said to herself that there were
coincidences, coincidences.
It was like unravelling a skein. From what she herself had experienced
she passed to the miracles of other times. They were coincidences. They
were hypnotism. They were possibly legends, most of them.
The raging monk continued to curse the people with terrible words. She
tried to listen to him to get away from her own thoughts. But all she
thought was that what he said was madness and lies.
What was going on in her? Was she becoming an atheist?
She looked about for Gaetano. He was there also; he stood on the church
steps quite near the monk. His eyes rested on her. And as surely as if
she had told him it, he knew what was passing in her. But he did not
look as if he were glad or triumphant. He looked as if he wished to stop
Father Gondo, to save a little vestige of faith for her.
Donna Micaela's thoughts had no mercy. They went on and robbed her soul.
All the glowing world of the supernatural was destroyed, crushed. She
said to herself that no one knew anything of celestial matters, nor could
know anything. Many messages had gone from earth to heaven. None had gone
from heaven to earth.
"But I will still believe in God," she said, and clasped her hands as if
still to hold fast the last and best.
"Your eyes, people of Diamante, are wild and evil," said Father Gondo.
"God is not in you. Antichrist has driven God away from you."
Donna Micaela's eyes again sought Gaetano's. "Can you give a poor,
doubting creature something on which to live?" they seemed to ask. His
eyes met hers with proud confidence. He read in her beautiful, imploring
eyes how her trembling soul clung to him for support. He did not doubt
for a moment that he would be able to make her life beautiful and rich.
She thought of the joy that always met him wherever he showed himself.
She thought of the joy that had roared about her that night in Palermo.
She knew that it rose from the new faith in a happy earth. Could that
faith and that joy take possession of her also?
She wrung her hands in anguish. Could that new faith be anything to her?
Would she not always feel as unhappy as now?
Father Gondo bent forward over the fire.
"I say to you once more," he cried, "if only one person comes and says
that this image has saved his soul, I will not burn it."
Donna Micaela had a sudden feeling that she did not wish the poor image
to be destroyed. The memory of the most beautiful hours of her life was
bound to it.
"Gandolfo, Gandolfo," she whispered. She had just seen him beside her.
"Yes, Donna Micaela."
"Do not let him burn the image, Gandolfo!"
The monk had repeated his question once, twice, thrice. No one came
forward to defend the image. But little Gandolfo crept nearer and nearer.
Father Gondo brought the image ever closer to the fire.
Involuntarily Gaetano had bent forward. Involuntarily a proud smile
passed over his face. Donna Micaela saw that he felt that Diamante
belonged to him. The monk's wild proceedings made Gaetano master of
their souls.
She looked about in terror. Her eyes wandered from face to face. Was
the same thing going on in all those people's souls as in her own? She
thought she saw that it was so.
"Thou, Antichrist," said Father Gondo, threateningly, "dost thou see that
no one has thought of his soul as long as thou hast been here? Thou must
perish."
Father Gondo laid the outcast on the pyre.
But the image had not lain there more than a second before Gandolfo
seized him.
He caught him up, lifted him high above his head, and ran. Father Gondo's
pilgrims hurried after him, and there began a wild chase down Monte
Chiaro's precipices.
But little Gandolfo saved the image.
Down the road a big, heavy travelling-carriage came driving. Gandolfo,
whose pursuers were close at his heels, knew nothing better to do than to
throw the image into the carriage.
Then he let himself be caught. When his pursuers wished to hurry after
the carriage, he stopped them. "Take care; the lady in the carriage is
English."
It was Signora Favara, who had at last wearied of Diamante and was
travelling out into the world once more. And she was allowed to go away
unmolested. No Sicilian dares to lay hands on an Englishwoman.
V
A FRESCO OF SIGNORELLI
A week later Father Gondo was in Rome. He was granted an interview with
the old man in the Vatican and told him how he had found Antichrist
in the likeness of Christ, how the former had entangled the people of
Diamante in worldliness, and how he, Father Gondo, had wished to burn
him. He also told how he had not been able to lead the people back to
God. Instead, all Diamante had fallen into unbelief and socialism. No one
there cared for his soul; no one thought of heaven. Father Gondo asked
what he should do with those unfortunate people.
The old pope, who is wiser than any one now living, did not laugh at
Father Gondo's story; he was deeply distressed by it.
"You have done wrong; you have done very wrong," he said.
He sat silent for a while and pondered; then he said: "You have not seen
the Cathedral in Orvieto?"--"No, Holy Father."--"Then go there now and
see it," said the pope; "and when you come back again, you shall tell me
what you have seen there."
Father Gondo obeyed. He went to Orvieto and saw the most holy Cathedral.
And in two days he was back in the Vatican.
"What did you see in Orvieto?" the pope asked him.
Father Gondo said that in one of the chapels of the Cathedral he
had found some frescoes of Luca Signorelli, representing "The Last
Judgment." But he had not looked at either the "Last Judgment" or at the
"Resurrection of The Dead." He had fixed all his attention on the big
painting which the guide called "The Miracles of Antichrist."
"What did you see in it?" asked the pope.
"I saw that Signorelli had painted Antichrist as a poor and lowly man,
just as the Son of God was when he lived here on earth. I saw that he had
dressed him like Christ and given him Christ's features."
"What more did you see?" said the pope.
"The first thing that I saw in the fresco was Antichrist preaching so
that the rich and the mighty came and laid their treasures at his feet.
"The second thing I saw was a sick man brought to Antichrist and healed
by him.
"The third thing I saw was a martyr proclaiming Antichrist and suffering
death for him.
"The fourth thing I saw in the great wall-picture was the people
hastening to a great temple of peace, the spirit of evil hurled from
heaven, and all men of violence killed by heaven's thunderbolts."
"What did you think when you saw that?" asked the pope.
"When I saw it, I thought: 'That Signorelli was mad. Does he mean that in
the time of Antichrist evil shall be conquered, and the earth become holy
as a paradise?'"
"Did you see anything else?"
"The fifth thing I saw depicted in the painting was the monks and priests
piled up on a big bonfire and burned.
"And the sixth and last thing I saw was the Devil whispering in
Antichrist's ear, and suggesting to him how he was to act and speak."
"What did you think when you saw that?"
"I said to myself: 'That Signorelli is not mad; he is a prophet.
Antichrist will certainly come in the likeness of Christ and make a
paradise of the world. He will make it so beautiful that the people will
forget heaven. And it will be the world's most terrible temptation.'"
"Do you understand now," said the pope, "that there was nothing new in
all that you told me? The Church has always known that Antichrist would
come, armed with the virtues of Christ."
"Did you also know that he had actually come, Holy Father?" asked Father
Gondo.
"Could I sit here on Peter's chair year after year without knowing that
he has come?" said the pope. "I see starting a movement of the people,
which burns with love for its neighbor and hates God. I see people
becoming martyrs for the new hope of a happy earth. I see how they
receive new joy and new courage from the words 'Think of the earth,' as
they once found them in the words 'Think of heaven.' I knew that he whom
Signorelli had foretold had come."
Father Gondo bowed silently.
"Do you understand now wherein you did wrong?"
"Holy Father, enlighten me as to my sin."
The old pope looked up. His clear eyes looked through the veil of chance
which shrouds future events and saw what was hidden behind it.
"Father Gondo," he said, "that little child with whom you fought in
Diamante, the child who was merciful and wonder-working like Christ, that
poor, despised child who conquered you and whom you call Antichrist, do
you not know who he is?"
"No, Holy Father."
"And he who in Signorelli's picture healed the sick, and softened the
rich, and felled evil-doers to the earth, who transformed the earth to a
paradise and tempted the people to forget heaven. Do you not know who he
is?"
"No, Holy Father."
"Who else can he be but the Antichristianity, socialism?"
The monk looked up in terror.
"Father Gondo," said the pope, sternly, "when you held the image in your
arms you wished to burn him. Why? Why were you not loving to him? Why did
you not carry him back to the little Christchild on the Capitolium from
whom he proceeded?
"That is what you wandering monks could do. You could take the great
popular movement in your arms, while it is still lying like a child
in its swaddling clothes, and you could bear it to Jesus' feet; and
Antichrist would see that he is nothing but an imitation of Christ, and
would acknowledge him his Lord and Master. But you do not do so. You
cast Antichristianity on the pyre, and soon he in his turn will cast you
there."
Father Gondo bent his knee. "I understand, Holy Father. I will go and
look for the image."
The pope rose majestically. "You shall not look for the image; you shall
let him go his way through the ages. We do not fear him. When he comes
to storm the Capitol in order to mount the throne of the world, we shall
meet him, and we shall lead him to Christ. We shall make peace between
earth and heaven. But you do wrong," he continued more mildly, "to hate
him. You must have forgotten that the sibyl considered him one of the
redeemers of the world. 'On the heights of the Capitol the redeemer of
the world shall be worshipped, Christ or Antichrist.'"
"Holy Father, if the miseries of this world are to be remedied by him,
and heaven suffers no injury, I shall not hate him."
The old pope smiled his most subtle smile.
"Father Gondo, you will permit me also to tell you a Sicilian story. The
story goes, Father Gondo, that when Our Lord was busy creating the world,
He wished one day to know if He had much more work to do. And He sent San
Pietro out to see if the world was finished.
"When San Pietro came back, he said: 'Every one is weeping and sobbing
and lamenting.'
"'Then the world is not finished,' said Our Lord, and He went on working.
"Three days later Our Lord sent San Pietro again to the earth.
"'Everyone is laughing and rejoicing and playing,' said San Pietro, when
he came back.
"'Then the world is not finished,' said Our Lord, and He went on working.
"San Pietro was dispatched for the third time.
"'Some are weeping and some are laughing,' he said, when he came back.
"'Then the world is finished,' said Our Lord.
"And so shall it be and continue," said the old pope. "No one can save
mankind from their sorrows, but much is forgiven to him who brings new
courage to bear them."
THE END
End of Project Gutenberg's The Miracles of Antichrist, by Selma Lagerlöf
***
|
{
"redpajama_set_name": "RedPajamaBook"
}
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\section{Introduction}
Long-baseline interferometry consists in recombining the light from several telescopes to measure interference fringes on an astronomical object \citep{LAW00}. The key observable is the visibility, a complex number containing the contrast and phase of fringes obtained on a telescope pair. It was first introduced in radioastronomy \citep{BRA58} to generalise Michelson's term \citep[a synonym for contrast in][]{MIC21}. In the ideal case, the visibility is the Fourier transform of the object's image taken at a spatial frequency related to the telescope separation \citep{BRA58,HON75,LAB75}. In the infrared and optical, though, the atmospheric turbulence shifts the fringes in milliseconds, so that only the square visibility amplitude and partial phase information can be retrieved \citep{ROD84b}, for instance via the closure phase, the sum of phases over a telescope triplet \citep{ROD86,COR87}. A robust determination of the uncertainties on these observables is paramount to ensure confidence on the physical parameters derived from model fitting.
\textit{``Statistical'' errors}---deviations of expected mean zero---on interferometric observables are produced by fast-varying intrinsic, atmospheric, and instrumental effects. In addition to the detector and photon noises, a few sources of errors that have been observed at the Very Large Telescope Interferometer (VLTI) are the differential atmospheric piston \citep{COL99,ESP00}, imperfect fibre injection \citep{KOT03}, mechanical vibrations \citep[][Sect.~5.6]{PIONIER}, background fluctuations \citep{ABS04}, detector efficiency variations due to cooling cycle \citep{ABS04}, and 50\,Hz electronic noise from the power grid \citep{ABS04}. The relatively large number of sources of uncertainties made it difficult to obtain a reliable assessment of the precision of interferometric measurements. In particular, the theoretical estimate using photon and detector noises is unrealistically low, so most processing software tools need heuristics to provide a better one, for instance using the dispersion of a given data set. One of the common pitfalls is the assumption that the statistical errors on visibility measurements are uncorrelated \citep{MEI05} and follow a Gaussian distribution. In particular, most public data processing software tools do not determine these correlations, in particular those for the VLTI instruments MIDI\footnote{MID-infrared Interferometric instrument} \citep{MIDI}, AMBER\footnote{Astronomical Multi-BEam combineR} \citep{AMBER}, PIONIER\footnote{Precision Integrated-Optics Near-infrared Imaging ExpeRiment} \citep{PIONIER}, and GRAVITY \citep{GRAVITYpipe}. The same happens with popular model-fitting tools \citep[e.g. Litpro, see][]{TAL08} or image reconstruction programmes \citep[e.g. MIRA, see][]{THI08}. Also, the Optical Interferometric FITS \citep[OIFITS v. 1, see][]{OIFITS1} format did not provide a codified way to document correlations in its first and most used version. Only very recently has an update to the standard given specifications for a covariance matrix \citep[OIFITS v. 2, see][]{OIFITS2}.
The assumption of uncorrelated Gaussian measurement errors could not be further from the truth. For instance, closure phases are not independent \citep{MON07}. Also, several random effects impact the different spectral channels of a same observation in the same way, such as the blurring of fringes due to the turbulent atmosphere \citep[][Sect.~7.5 ``Atmospheric Biases'']{LAW00}. Finally, all observations of an observing sequence are impacted in the same way by the errors on the calibrators, virtually leading to correlations between all data points \citep{PER03}, even collected in different runs at different facilities. The assumption that the fringe contrasts or phases follow a Gaussian distribution is not confirmed by experience either \citep[][in the case of AMBER]{SCH14}. Also, when deriving the instrumental transfer function, a weighted average of square visibilities and closure/differential phases is obtained, using a few calibrators observed close to the science targets. In most cases this average does not follow Gaussian distribution, as \citet{PER03} note.
For these reasons, \citet{PER03} proposed an analytic formalism to propagate the non-Gaussian correlated uncertainties of the square visibility amplitudes in an approximate, yet relatively accurate way. Several authors have applied these results to the FLUOR\footnote{Fiber Linked Unit for Optical Recombination} instrument \citep[at IOTA\footnote{Infrared and Optical Telescope Array}, then CHARA\footnote{Center for High Angular Resolution Array}, see for instance][]{PER04,ABS06,BER06} to our knowledge the only one for which correlations have been regularly determined.
In addition to the statistical errors that one can infer from the data and/or noise modelling, there are \emph{``systematic'' errors} that typically plague interferometric data and impact all the data of a given set of observations in a similar and poorly understood way. Part of the biases are removed either theoretically \citep[e.g. group-delay dispersion, see][]{ZYV03} or calibrated out \citep[e.g. polarisation,][]{HAG00} by the measurement of the ``instrumental visibility'' on stars of known geometry, ideally unresolved ones, with the underlying assumption that it varies slowly enough to be interpolated to science observations with sufficient precision \citep{HAN74,PER03}.
However, complete removal does not happen. \citet{COL03} measured $\approx 5\%$ systematic errors on the calibrated squared visibility amplitudes at the Keck Interferometer by observing binaries of known orbital parameters. More recently, high-precision diameter measurements using sufficiently well resolved stars with CHARA \citep{WHI18,KAR18} were shown to significantly differ (several $\sigma$ and up to 15\%) from values previously obtained from under-resolved interferometric observations, leading the authors to conclude that they were plagued with undiagnosed systematics. At VLTI, \citet{LEB09} and \citet{KER04} also tried to identify the origins of the large systematics, discarding the uncertainty on central wavelength \citep[calibration errors are reported to be 0.35 to 0.50\%{} at PIONIER at VLTI and PAVO\footnote{Precision Astronomical Visible Observations} at CHARA, respectively by][]{HUB12,GAL18}, atmospheric jitter and injection efficiency due to seeing, and instrumental variations during the observation ($\kappa$ matrix). Possible sources are a differential polarisation effect \citep[several percents,][]{LEB12} and the way the bias is removed in Fourier space during the PIONIER data processing \citep[see][in the case of FINITO\footnote{Fringe-tracking Instrument of NIce and TOrino}]{LEB09}. We also stress that calibrators may be an additional source of systematics: for instance, an unsuspected binary with a flux ratio of 1:100 would likely go undetected in closure phase with PIONIER ($\lesssim 2.3$\,deg), yet could account for a bias in the squared visibility amplitudes of up to 2\%.
While some authors deal with the uncertainty on the calibrators' diameters as a systematic error \citep[e.g.][but we include them in the correlated ``statistical'' errors for practical reasons]{CRE15,PER16}, few take into account other systematics, despite all the evidence. \citet{HUB12} introduce systematic errors of the order of one percent in order to account for errors in the spectral calibration, but they are still much smaller than the observed and unexplained errors.
In this paper, we present a model that make use of a technique that is relatively easy to implement into existing pipelines, is more accurate than the classical error propagation, and requires no additional analytic developments. Known as the bootstrap method \citep{BOOTSTRAP}, it consists in randomly selecting interferograms to feed the data reduction software. Repeating it enough times, we generate a sampling of the multivariate probability density function of the square visibility amplitudes and closure phases. Bootstrapping was originally introduced in interferometry by \citet{KER04drs} in order to determine the statistical errors for VINCI\footnote{\change{VLT Interferometer commissioning instrument}} at the VLTI. In addition, we treat the systematic errors as additional correlated term whose magnitude is left as a free parameter.
In Sect.~\ref{sec:obs}, we present the data set, acquired for the companion paper by \citet{RAB18}, that has led us to undertake this work. Sect.~\ref{sec:dataproc} details our modelling of the correlated statistical errors and systematic errors. We then show (Sect.~\ref{sec:ud}) how the estimate uniform disc diameters with PIONIER at the VLTI is significantly impacted by the level of detail in the error modelling. We summarise and conclude in Sect.~\ref{sec:conclusion}.
\section{Data set}
\label{sec:obs}
\input{obslog.textable}
\begin{table}
\caption{Tuning of the parameters in the interpolation of the transfer function for all setups (Sect.~\ref{sec:tf}, Eqs.~\ref{eq:tf},~\ref{eq:tfw}). MJD: Modified Julian Day; \#: Number of spectral setups; DIT: total integration time during a scan; \Nfowl: number of Fowler (i.e. non-destructive) reads of the infrared detector per scan position; \Nopd: number of scan positions; \timescale: the time-scale to determine the weight of calibrators taken close in time; \angscale: the alt+az angular distance scale to determine the weight of calibrators close by in the sky; \minerr: the minimum relative error in the calibrator visibility considered when determining the weight of a calibrator. The first setup is a good example of an unstable night when only calibrators very close in time to the science observation ($\lesssim 20$\,min and 15\,deg) have a significant weight. Many nights are stable enough and free of alt+az polarisation effects (e.g. MJD 56384--56850).}
\label{tab:setups}
\centering
\begin{tabular}{lrccccc}
\hline
MJD & \# & DIT & \Nfowl$\times$\Nopd & \timescale & \angscale & \minerr\\
$56000 +$ & & [s] & & [h] & [deg] & [\%]\\
\hline
210 & 1 & 0.39 & 4$\times$512 & 0.3 & 15 & 2\\
& 1 & 0.60 & 1$\times$1024 & 0.8 & 20 & 1\\
252 & 4 & all & all & 0.8 & $+\infty$ & 1\\
253 & 1 & 0.23 & 1$\times$512 & 0.8 & 20 & 1\\
325--326 & 5 & all & all & 0.8 & $+\infty$ & 1\\
384--850 & 21 & all & all & 0.8 & 20 & 1\\
889 & 3 & all & all & 0.8 & $+\infty$ & 1\\
890 & 3 & all & all & 0.8 & 20 & 1\\
\hline
\end{tabular}
\end{table}
In a companion paper by \citet{RAB18}, we have needed to obtain accurate stellar diameters of marginally resolved M~dwarfs in order to calibrate the mass-radius relation down to the fully convective regime. While the present paper focuses on the data processing method that leads to reliable uncertainties, we find useful to show our main findings on actual data. Our sample consists of 20 under-resolved late-type stars of the solar neighbourhood, 13 M dwarfs of the original programme by Rabus et al. plus 7 backup targets acquired during the observing campaign.
Table~\ref{tab:obslog} summarises the science and calibrator observations that we have carried out. The observation strategy was to observe each science target with different calibrators and on different nights. Each science observation was bracketed by calibrator observations close in time ($\approx 10$\,min) and altitude + azimuth (alt + az) positions (a few degrees when possible). Calibrators were chosen so that as to minimise uncertainties on the calibration of the transfer function, i.e. either unresolved or resolved with a small diameter uncertainty. We performed the selection with the \texttt{searchCal} tool \citep{searchCal} provided by the Jean-Marie Mariotti Center (JMMC). All calibrators have indirect (i.e. non-interferometric) diameter determinations, so they are immune to the biases investigated in this paper. We used the Calibrator stars for 200\,m baseline interferometry \change{by} \citet[][hereafter MER05, \change{using the} method: absolute spectro-photometric calibration]{MER05} for 15 large ($\sim 1$\,mas) K0III-K5III calibrators with a typical precision of the order of 1--2\% on diameter and $\lesssim 1$\% in visibility calibration, the Catalogue of Calibrator Stars for LBSI by \citet[][\change{using the} same method]{BOR02} for one K1III star with a similar level of precision, and the JMMC Stellar Diameters Catalogue \citep[JSDC,][\change{using the} method: photometric calibration]{JSDC17} for 65 smaller ($\lesssim 0.5$\,mas) calibrators with a typical precision of 10--20\% in diameter and $\lesssim 2$\% in visibility. \citet{SWI17} have showed that the indirect spectro-photometric calibration method for large calibrators is not biased, as their catalogue is consistent both with interferometric measurements and catalogue by MER05. For most of our science stars (15 of 20) we observed small calibrators from JSDC or ones smaller than the target from MER05. For 4 of our science targets (GJ~1, GJ~54.1, GJ~86, GJ~370) we used one or more calibrators from MER05 that are more resolved than the target, but we also included several smaller ones from JSDC to mitigate the possible impact of a large, unexpected error in a calibrator's diameter: GJ~1 and GJ~54.1, main targets of \citet{RAB18}, have been observed together with 8 and 9 different JSDC calibrators, respectively, in addition to the 3 and 1 from MER05. While the bracketing calibrators have the largest impact on the calibration of a given science observation, other calibrators taken for other targets on the same night and with the same instrumental setup contribute to some extent to the transfer function, typically if they were taken within an hour of the science target and relatively close in the sky. For any given target and observing night, Table~\ref{tab:obslog} lists all relevant calibrators with their relative weight (see Eq.~\ref{eq:tfw} in Sect.~\ref{sec:tf} for the weighing as a function of distance in time and position) as well as sky conditions.
Table~\ref{tab:setups} gives an overview of the instrumental setups used and the fine-tuning of calibration parameters.
\section{Data processing}
\label{sec:dataproc}
We call ``data set'', with index $d$ ($d$ in $1 \ldots \Nobs$), a set of $\Nint \sim 10^2$ interferograms \interf{}, indexed by \inti{} ($1 \ldots \Nint$), taken in quick succession with a single telescope pair in a single spectral channel. An interferogram is the temporal scan as function of optical path difference (OPD). Each data set of a science target will result in exactly one calibrated squared visibility amplitude \calvis, which we will refer to as ``visibility''. The interferograms of a data set are assumed to share the projected baseline length $\base = \telsep\cdot\pos / \lambda_{d}$, where $\lambda_{d}$ is the effective wavelength of the spectral channel, $\telsep$ is the mean separation between the telescopes, and $\pos$ the radial unit vector representing the target's location in the horizontal coordinate system (elevation and azimuth).
An ``observation'' consists of several data sets taken at the same time, with different values of \base, for several baselines are used at the same time, sometimes also different wavelengths. In the present case, we have used PIONIER with six telescope pairs and one to three spectral channels, so each observations consist of 6 or 18 data sets. In a single telescope pointing, five observations are usually performed in a row (for a total of $5$--10\,min). Over the observing runs, we have acquired data for a few dozens of pointing positions per scientific target, collecting of the order of one thousand data sets per star.
A ``setup'' is a unique combination of configuration of the telescope array (stations used), instrumental setup (spectral dispersion, readout mode, scanning speed) and observing night. Because of the span in stellar brightness among the sources and the varying observing conditions, a few setups are used each night. In a setup, several calibrator stars and one or more scientific targets are observed. All data take\change{n} with the same setup can show some level of correlated systematics due to the wavelength calibration error.
We shall call ``\emph{baseline}'' a unique combination of a telescope pair and a setup. In particular, we will consider that data sets taken with the same stations with different spectral configurations or on different nights originate from different baselines as they are likely to have different systematic error terms.
The PIONIER data reduction software \citep[hereafter \pndrs, see Sect. 5 of][]{PIONIER}, that we have modified, determines uncalibrated visibilities, computes the instrumental transfer function for calibrators of known diameters, interpolates it for scientific targets taken with the same setup, and derives the calibrated visibilities for these sources.
Our treatment of uncertainties proceeds in four steps: we determine the ``statistical'' errors that can be inferred from the noise in the data and the uncertainty on the diameters of the calibrators (Sect.~\ref{sec:staterror}), we model an additional error term to account for the dispersion of the reduced visibilities at each baseline (Sect.~\ref{sec:bl-staterr}), introduce a highly correlated, systematic error term to account for the discrepancy between the reduced visibilities of different baselines (Sect.~\ref{sec:syserr}), and model the wavelength calibration error as an additional systematic error term (Sect.~\ref{sec:waveerr}). Sect.~\ref{sec:covmat} gives the resulting covariance matrix.
\subsection{``Statistical'' errors}
\label{sec:staterror}
\subsubsection{Raw visibilities}
\label{sec:rawvis}
The ``raw'' visibility (uncalibrated visibility) is determined by correcting the fringe contrast of the interferograms from different atmospheric and instrumental effects such as the finite bandwidth and the flux imbalance between the beams. For the sake of clarity, we will assume that it is obtained separately on each interferogram and then averaged. However, the details of how \pndrs{} computes visibility amplitudes may vary according to setup and processing mode \citep[see Sect.~5 of][for further details of the data processing]{PIONIER}. The bootstrap method will work as long as the resulting visibility is a function of a significant number of scans or frames. For instance, it fails for the science detector of GRAVITY with long exposures \citep{GRAVITYpipe}, but it should work for all other VLTI instruments.
Uncertainties are determined by the bootstrap method. The bootstraps $\interfboot$ ($\booti$ in $1\ldots\Nboot$) are $\Nboot$ sets of $\Nint$ interferograms picked at random with repeats from the original set $\interf$. With the exception that the first bootstrap is the original set, i.e. $\interfboot[1] = \interf$, we pick $\Nint(\Nboot - 1)$ independent random numbers $r_{\inti\booti}$ uniformly in $1\ldots\Nint$, so that $\interfboot = \interfname_{d r_{\inti\booti}}$ for $\inti \ge 2$. The random numbers $r_{\inti\booti}$ are the same for all data sets of the same observation, so that cross-channel and cross-baseline correlations are correctly measured.
We then obtain $\Nboot$ raw visibilities $\rawvisboot$ ($1 \le \booti \le \Nboot$) by averaging the visibility $\visibproc(\interfboot)$ obtained for each interferogram. Together with its uncertainty, it is given by
\begin{subequations}
\begin{align}
\rawvisboot &= \left < \visibproc(\interfboot) \right>_{\inti} \\
\drawvis &= \frac{1}{\Nboot} \sqrt{
\sum_{\booti}
\left(\rawvisboot - \left< \rawvisboot[\bootj] \right>
_{\bootj} \right)^2
}
\end{align}
\label{eq:rawvisboot}
\end{subequations}
In this work, we picked $\Nboot = 5\times10^3$ so that we can derive a covariance matrix for a few thousands of data sets and use the (very slightly biased) sample variance to avoid numerical issues. The original version of \pndrs{} published by \citet{PIONIER} also bootstraps the interferograms (with $\Nboot \sim 10^2$) to determine the uncertainty, but discards them afterwards. It keeps the value $\rawvisboot[1] \pm \drawvis$ of Eq.~(\ref{eq:rawvisboot}) and propagates the errors assuming an uncorrelated multivariate Gaussian distribution. We have modified the software to keep all the bootstraps down to the final product and get an empirical sampling of the calibrated visibility distribution.
\subsubsection{Instrumental transfer function}
\label{sec:tf}
There are still instrumental effects, difficult to compute, in the raw visibilities, so that the fringe contrast is lower than expected from a theoretical point of view. To remove them, the transfer function (also known as instrumental visibility) is calculated on unresolved sources or targets of precisely known geometry, observed in the vicinity of the scientific targets. Then it is interpolated for science targets. Some care has to be taken to include the uncertainties on the calibrators' diameters and variations of the transfer function due to changing atmospheric conditions.
\defd{c}
If the star is a calibrator with known uniform disc diameter $\ud \pm \dud$, then $\Nboot$ diameters $\udboot$ are picked at random assuming a Gaussian distribution $\Norm(\ud, \dud^2)$, with the exception that $\udboot[1] = \ud$. This is done once per calibrator star for all the observing runs, so that correlations from calibrator errors are correctly propagated to the final data.
For data set $d$ corresponding to a calibrator and bootstrap number $\booti$, the ratio of the raw visibility $\rawvisboot$ to the theoretical uniform disc visibility $\udvisboot$ yields the transfer function $\tfboot$:
\begin{subequations}
\begin{align}
\udvisboot &= \udvisfunc\left(\pi\base\udboot\right)\\
\dudvis &= \deriv{\udvisfunc}{x} (\pi\base\ud) \pi\base\dud\\
\tfboot &= \frac{\rawvisboot}{\udvisboot }\\
\dtf &= \tfboot[1] \sqrt{
\left(\frac{\drawvis}{\rawvis}\right)^2
+ \left(\frac{\dudvis}{\udvisboot[1]}\right)^2
}
\end{align}
\end{subequations}
where the theoretical visibility amplitude of a uniform disc is given by the function $\udvisfunc(x) = |2 J_1(x) / x|^2$ with $J_1$ the Bessel function of the first kind.
\defd{d}
If the star is a scientific target, the reduction software interpolates the transfer function using calibrator observations $c_1, \ldots, c_n$ obtained on the baseline (same telescope pair and setup). Calibrator observations close in time and/or position in the sky and with smaller error bars are given more weight. If, for bootstrap number $\booti$, the transfer functions $\tfname_{c_1\booti}, \ldots, \tfname_{c_n\booti}$ are determined at time $t_{c_1}, \ldots, t_{c_n}$ and horizontal coordinates $\posname_{c_1}, \ldots, \posname_{c_n}$, then the estimated transfer function for a science observation $d$ at time $t_d$ and position $\pos$ is given by
\begin{subequations}
\begin{align}
\tfboot &= \frac{\sum_k w_k \tfname_{c_k\booti}}{\sum_k w_k}
\label{eq:tf}\\
w_k &=
\max\left(\minerr, \frac{\Delta\tfname_{c_k\booti}}
{ \tfname_{c_k\booti}} \right) ^ {-2}
\mathrm{e} ^ {-\frac{(t_{c_k} - t_{d})^2 }{\timescale^2}}
\mathrm{e} ^ {-\frac{\angdist (\posname_{c_k}, \pos)^2}{\angscale^2}} \label{eq:tfw}
\end{align}
\end{subequations}
where $\alpha(\posname_1, \posname_2)$ is the alt + az difference between telescope pointing positions $\posname_1$ and $\posname_2$. $\timescale$, $\angscale$, and $\minerr$ are constants within a given setup. $\timescale$ is the time-scale of variations of the transfer function, typically of the order of 1~hour in the original \pndrs{} software, but we have shortened it for a few very ``agitated'' nights (see Table~\ref{tab:setups}). $\angscale$ is the angular size on the sky over which the transfer function varies. In the original software, $\angscale = +\infty$ so that calibrators have a weight independent of their position relative to the science target. However, in some nights, polarisation effects led us to use a finite value (see Table~\ref{tab:setups}). $\minerr = 0.01$ (also used in the original \pndrs) is the minimum relative uncertainty we consider in the determination of the weight of calibrator observations, in order to avoid that a calibrator with an unexpectedly low uncertainty biases the transfer function.
The calculation of $\dtf$ includes three terms\change{:}
\begin{enumerate}
\item the error propagation using Eq.~ (\ref{eq:tf}), which includes the uncertainty on the diameter of the calibrators;
\item an interpolation uncertainty taking into account the varying atmospheric conditions between target and calibrator, which we measure by comparing the interpolated transfer function for calibrator observations with the measured one;
\item an extrapolation uncertainty for data not bracketed by calibrators (which we have avoided).
\end{enumerate}
An example for the first two uncertainty terms of the transfer function is given in Sect.~3 of \citet{NUN17} in the case of weights linear in time. The full expression for $\dtf$ is not given here for it follows exactly the same steps as in the original software.
\subsubsection{Calibrated visibilities}
\label{sec:calvis}
The calibrated visibilities \calvisboot, their uncertainties \dcalvis, and their covariances \covar{} are given by:
\begin{subequations}
\begin{align}
\calvisboot{} &= \frac{\rawvisboot}{\tfboot} \label{eq:calvisboot},\\
\dcalvis &= \calvisboot[1]{} \sqrt{
\left(\frac{\drawvis}{\rawvis}\right)^2
+ \left(\frac{\dtf}{\tfboot[1]}\right)^2
} \label{eq:dV}\\
\covar &= \frac{
\sum_{\booti}
\left(\calvisiboot - \left<\calvisiboot[\bootj]\right>_\bootj \right)
\left(\calvisjboot - \left<\calvisjboot[\bootj]\right>_\bootj \right)
}{\Nboot}\label{eq:covar}.\\
\intertext{In some cases, the uncertainties on the calibrated visibility obtained from the standard deviation of the calibrated bootstraps}
\dcalvisboot &= \sqrt{\covaridiag} \label{eq:var}
\end{align}
\end{subequations}
differ to some extent from the ones derived by error propagation ($\dcalvis$ in Eq.~\ref{eq:dV}) in the original software. The reasons are correlations, departure from the Gaussian distribution, and non-linearity in the calculations in the reduction software (in particular the divisions).
\subsection{Baseline-dependent ``statistical'' errors}
\label{sec:bl-staterr}
We have noted that within a given baseline, it frequently happens that the dispersion of the data points is higher than what our carefully deduced uncertainties suggest. Because most of the stars in our programme are \textit{bona fide} centro-symmetric targets, under-resolved in the $H$ band at VLTI, the uniform disc model \citep[or any symmetric model, see][]{LAC03} should fit the data correctly. Thus, we expect the reduced chi squared of a least squares fit to the data of a single baseline to be close to unity. While it is the case on some baselines, it might be significantly higher ($\approx 2$) on others.
It is quite possible that the determination of the instrumental transfer function in quickly changing conditions is not perfect. In particular, the interpolation of the transfer function (Eq.~\ref{eq:tf}) would fail if conditions changed abruptly. In general, our knowledge on the variation of the transfer function is very limited so we rely on an imperfect, generic smoothing law (Eq.~\ref{eq:tfw}).
We also considered that time-correlations may impact the bootstrap method. In Sect.~\ref{sec:rawvis}, the determination assumes that the $\visibproc(\interf)$, for $\inti$ in $1 \cdots \Nint$, are not time-correlated. Unfortunately, the data are too noisy to infer a meaningful auto-correlation function and model its impact on the result. However, we know that the observing cadence ($\sim 1$\,s) is significantly slower than the typical interferometric coherence time \citep[$\sim 100$\,ms in the IR, see][]{PER97,DI03,GLI11} and atmospheric turbulence time $\tau_0$ ($\sim 10$\,ms in the IR). So, we expect little correlation from the differential atmospheric piston and fibre injection variations. In the case unsuspected sources of temporal correlation did show up, we would expect to underestimate uncertainties. However, we have checked that, even with highly correlated visibilities, the impact is very small: in the case of a correlation coefficient of 0.9 between consecutive interferograms, we simulated batches of 100 measurements, which yielded a bootstrap estimate of $\drawvis$ within 10\% of the correct value.
In the absence of a clear understanding of these errors, we decide to model them as an uncorrelated additive term $\sysuncorr$. It has expectancy 0, standard deviation $\sigmauncorr\calvis$, and correlations 0. For each baseline where the reduced chi squared $\chisqr$ differs significantly from 1\footnote{$\chisqr$ is expected to be $1.00 \pm 0.22$ with a 3-$\sigma$ confidence interval for an observation with two telescope pointings and three spectral channels, but we have found values of 2 to 4 on some baselines.}, the value of $\sigmauncorr$ is adjusted so that a least-squares fit to the data of this baseline (and only these data) yields a reduced $\chisqr = 1$.
\begin{table*}
\caption{The different uncertainty models (Sect.~\ref{sec:ud}). The first four models used uncorrelated statistical errors, the three following ones correlated statistical errors, and the last ones include correlated systematic errors.
\emph{Data uncertainties} can be obtained from \pndrs{} error propagation, the bootstrap variances, the bootstrap variances plus a noisy correlation matrix representing uncorrelated data (``corr. noise''), or the bootstrap covariances.
\emph{Baseline-dependent statistical errors} of uncertain origin may be ``ignored'' or ``fit'' on each baseline so that baseline $\chisqr = 1$.
\emph{Systematic errors} can either be modelled with ``correlation'' matrix or one can just perform a ``rescaling'' of all statistical errors to a global $\chi^2 = 1$. For the sake of comparison we also include the incorrect correlation matrix (``PPP corr.'') that may lead to Peelle's Pertinent Puzzle.
\emph{Fitted visibilities} can either be the mean of the
bootstraps (``mean'') or all bootstraps (``bootstraps''). In the bootstrap case, the diameter value and uncertainty are obtained from the median and 1-$\sigma$ confidence interval of the obtained distribution.
\emph{Mathematical expression for uncertainties}: either an error vector or a covariance matrix.
}
\label{tab:err-model}
\begin{tabular}{l lll lll}
\hline\hline
Model & Data uncertainties & Baseline-dependent & Systematic & Fitted & \multicolumn{2}{l}{Mathematical expression}\\
& & statistical errors & errors & visibilities & \multicolumn{2}{l}{used for the uncertainties}\\
\hline
\modelpndrs & propagation & ignored & rescaling & mean & Eq.~(\ref{eq:dV}) & $\dcalvis$ \\
\modelvar & variances & ignored & rescaling & mean & Eq.~(\ref{eq:var}) & $\dcalvisboot$ \\
\modelvarboot & variances & ignored & rescaling & bootstraps & Eq.~(\ref{eq:var}) & $\dcalvisboot$ \\
\modelvarnoisy & variances & ignored & rescaling & mean & Eq.~(\ref{eq:noisycovar}) & $\noisycovar$ \\
& + corr. noise \\
\hline
\modelcov & covariances & ignored & rescaling & mean & Eq.~(\ref{eq:covar}) & $\covar$ \\
\modelcovbl & covariances & fit & rescaling & mean & Eq.~(\ref{eq:covar}) & $\ubcovar$ with $\sigmasys = \sigmawave = 0$ \\
\modelcovboot & covariances & fit & rescaling & bootstraps & Eq.~(\ref{eq:covar}) & $\ubcovar$ with $\sigmasys = \sigmawave = 0$ \\
\hline
\modelppp & covariances & fit & PPP corr. & mean & Eq.~(\ref{eq:covar}) & $\ubcovar$ with $\meancalvis=\calvis$\\
\modelsys & covariances & fit & correlation & mean & Eq.~(\ref{eq:covar}) & \ubcovar \\
\modelsysboot & covariances & fit & correlation & bootstraps & Eq.~(\ref{eq:covar}) & \ubcovar \\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{Uniform disc diameters of the sample using different model fitting routines of Sect.~\ref{sec:ud}: $\diampndrs$ (uncorrelated least squares using the error propagation of the reduction pipeline), $\diamvar$ (uncorrelated least squares using observed variances), $\diamcovar$ (least squares using observed covariances), $\diamcovbl$ (least squares using observed covariance and fitting baseline-dependent statistical errors), and $\diamsys$ (correlated least squares with baseline systematics). For each model, the following relative errors on the visibilities are given: $\sigmastat$ is obtained from the data, $\sigmauncorr$ is derived by modelling an additional baseline-dependent statistical (i.e. uncorrelated) error term, $\sigmasys$ is obtained by fitting the strength of the systematics (fully correlated on the same baseline). The reduced $\chisqr$ of the fits is also given. For GJ~581, the reduced chi squared of the \modelcovbl{} model is below one, so systematics adjust to zero and $\sigmasys = 0$ for \modelsys.}
\label{tab:ud-comp}
\tabcolsep=1.5pt
\small
\input diameters.textable
\end{table*}
\subsection{``Systematic'' errors}
\label{sec:syserr}
We have tried to avoid the commonest systematic error sources with PIONIER. Any calibrator showing hints of binarity, thus being able to skew the transfer function for baselines parallel to the binary separation, was excluded. Calibrators were chosen close (a few degrees when possible) to the science targets so that the differential polarisation effect is minimised. On nights were this effect was impacting the data processing with the default \pndrs{} parameters, the instrumental transfer function was interpolated in a way that calibrators distant in alt+az position are filtered out (see Sect.~\ref{sec:tf} and Table~\ref{tab:setups}).
In spite of these efforts, there is strong evidence of systematics in our data. Visually (see Fig.~\ref{fig:visib}, in particular GJ~86, GJ~1061, GJ~1), the data of some baselines do not align with those at other baselines. This first look is confirmed by a reduced $\chi^2 > 1$ on most of our fits in spite of a detailed modelling of statistical errors (Sects.~\ref{sec:staterror}~\& \ref{sec:bl-staterr}).
We model these systematic errors by a multiplicative term $\sys$ with high correlations along the same baselines. It has expectancy 1, standard deviation $\sigmasys$, and correlation coefficients $\corr \approx 1$ for all observations $d$ of the same baseline, and 0 for observations on different baselines. The value of $\sigmasys$ is adjusted so that a least squares fit to the full data set yields a reduced $\chi^2 = 1$.
\subsection{Wavelength calibration errors}
\label{sec:waveerr}
\citet{GAL18} showed that the PIONIER wavelength calibration has a relative uncertainty of $\approx 0.35$\% by alternating observations of a well know binary with PIONIER and wavelength-stabilised second-generation VLTI instrument GRAVITY \citep{GRAVITY}. The uncertainty on the central wavelength of spectral channels of a given spectral configuration, $\sigmawave = \Delta\lambda/\lambda$ produces an uncertainty on the projected baseline $\base$ of the same amount (0.35\%). To model it exactly, we should introduce correlated uncertainties along the $x$-axis ($\base \pm \sigmawave\base$) in addition to those on the $y$-axis ($V^2 \pm \Delta V^2$). Since the model is a continuously differentiable function of baseline, we decide instead to translate this small error in baseline into a small error in visibility. For under-resolved objects, $1 - \calvis \propto \base^2$ so that we obtain:
\begin{equation}
\Delta V^2 \approx 2(1 - V^2) \sigmawave.
\end{equation}
We model this systematic error source as an additive term \syswave{} of zero expectancy, standard deviation $2 \sigmawave (1 - \calvis)$ and correlation $\approx 1$ for all observations taken with the same spectral setup on the same night. Because \citet[][see their Fig.~1]{GAL18} show that the wavelength error varies from one night to the other, we have assumed that correlations are zero \change{for} data taken on different nights and/or different spectral configurations.
For observations taken on a single night in a single spectral setup, we can expect a diameter uncertainty of $0.35\%$. However, according to our observation strategy, most of the stars in our sample were observed on different nights, so we expect these systematics to have a lower impact on the final uncertainty determination (0.15 to 0.25\%).
\subsection{Final uncertainty determination}
\label{sec:covmat}
The final quantity we measure and fit is therefore
\begin{equation}
\ubvis = \sys(\calvis + \sysuncorr) + \syswave.
\label{eq:ubvis}
\end{equation}
The determination of the uncertainties and the covariance matrix must be done with some care. It is well known that a na\"\i{}ve error propagation in presence of high correlations can lead, and indeed leads as we discovered in our data, to a least-squares fit that falls far off the data and yields a parameter estimation inconsistent with a more careful analysis. The paradox, known as Peelle's pertinent puzzle \citep{PEE87}, has an easy remedy in the case of a fit by a constant value \citep{NEU12} or a set of constant values \citep{NEU14}. In their analytic derivation of the covariances, the weighted average of the measurements replaces the individual measurements. We need to generalise their results to suit our needs, since the different visibilities of the same baseline do not have the same expected value ($\udvisfunc$ is not constant). As \citet{ZHA92} already noted, there is no longer an obvious (analytic) candidate for the weighted average in the case of a non-linear model. We decided to use the most ``natural'' estimate: we replace the visibilities $\calvis$ by their estimator $\meancalvis$ obtained from a least-squares model fit to the data of the given baseline and setup. In the case of a constant model, it would give the same result as \citet{NEU14}.
\def\mathcal{V}{\mathcal{V}}
\def\ensuremath{\varname^\mathrm{sys}}{\ensuremath{\mathcal{V}^\mathrm{sys}}}
\def\ensuremath{\varname^\mathrm{bl}_b}{\ensuremath{\mathcal{V}^\mathrm{bl}_b}}
\def\ensuremath{\varname^\mathrm{wl}}{\ensuremath{\mathcal{V}^\mathrm{wl}}}
We introduce the following variances related to the baseline-dependent statistical errors \sysuncorr, the systematic errors \sys{}, and the wavelength calibration errors \syswave:
\begin{subequations}
\begin{align}
\ensuremath{\varname^\mathrm{sys}} = \Var(\sysuncorr)\phantom{\calvis} \,\, &= \big( \sigmasys \meancalvis \big)^2\\
\ensuremath{\varname^\mathrm{bl}_b} = \Var(\sys\calvis) &= \big( \sigmauncorr \calvis \big)^2\\
\ensuremath{\varname^\mathrm{wl}} = \Var(\syswave)\phantom{\calvis} \, &= \big( 2\sigmawave (1 - \meancalvis) \big)^2
\end{align}
\end{subequations}
The final error and correlation matrix are given by
\begin{subequations}
\begin{align}
(\dubvis)^2 &=
(\dcalvis)^2
+ \ensuremath{\varname^\mathrm{sys}}
+ \ensuremath{\varname^\mathrm{bl}_b}
+ \ensuremath{\varname^\mathrm{wl}}\label{eq:dubvis}\\
\ubcovar &=
\begin{cases}
\covar + \phantom{\corr}\ensuremath{\varname^\mathrm{wl}} + \phantom{\corr}\ensuremath{\varname^\mathrm{sys}} + \ensuremath{\varname^\mathrm{bl}_b}
\text{ if $d = \obsj$}
\\
\covar + \corr\ensuremath{\varname^\mathrm{wl}} + \rho\ensuremath{\varname^\mathrm{sys}}\phantom{\null + \ensuremath{\varname^\mathrm{bl}_b} }
\text{ if $\baselinei = \baselinej$}
\\
\covar + \corr\ensuremath{\varname^\mathrm{wl}}\phantom{\null + \rho\ensuremath{\varname^\mathrm{sys}} + \ensuremath{\varname^\mathrm{bl}_b} }
\text{ if $\confi = \confj$}
\\
\covar\phantom{\null + \corr\ensuremath{\varname^\mathrm{wl}} + \rho\ensuremath{\varname^\mathrm{sys}} + \ensuremath{\varname^\mathrm{bl}_b} }
\text{ otherwise}
\end{cases}
\label{eq:syscovar}
\end{align}
\end{subequations}
where $\baselinei$ and $\confi$ are the baseline and spectral setup of data set $d$. In order to prevent a (numerically) singular matrix, we use $\corr = 0.95$ instead of 1. Since errors are relatively small (a few percents) the second-order terms ($< 0.1$\%) arising when propagating the errors from Eq.~\ref{eq:ubvis} have been ignored in Eqs.~\ref{eq:dubvis}~\&~\ref{eq:syscovar}.
\section{Impact of the uncertainty model}
\label{sec:ud}
\begin{figure*}
\centering
\includegraphics{GL1-GL370-diameter-by-error-type.pdf}
\caption{Comparison of uniform disc diameters in milliarcseconds (mas) (left axis) and their deviation from the robust \modelsys{} model estimate in \% (right axis) obtained with different error models (Sect.~\ref{sec:ud}). For each star, two graphs are displayed on the same line. In each graph, the least squares fits include models with uncorrelated statistical and systematic errors (green, points on the left), correlated statistical errors and uncorrelated systematics (black, points in the middle), and correlated statistical and systematic errors (blue, points on the right.) \emph{Left graph:} influence of correlations and systematic errors. \emph{Right graph:} robustness with respect to baseline systematics, by comparing fits where a baseline has been removed.}
\label{fig:ud-comp}
\end{figure*}
\begin{figure*}
\centering
\includegraphics{GL406-GJ667C-diameter-by-error-type.pdf}
\contcaption{Comparison of uniform disc diameters using different fitting methods.}
\end{figure*}
\begin{figure*}
\includegraphics{GJ674-GL1061-diameter-by-error-type.pdf}
\contcaption{Comparison of uniform disc diameters using different fitting methods.}
\end{figure*}
We assess the respective importance of the correlation and the level of detail used in the determination of uncertainties by comparing the results obtained with different error models. These error models, explained below, are briefly summarised in Table~\ref{tab:err-model}. The values of the uniform disc diameters obtained from these models are given in Table~\ref{tab:ud-comp} and Fig.~\ref{fig:ud-comp}. For the most discrepant error models, the deviation between estimates is shown in Fig.~\ref{fig:estim-comp}.
In all models, a least-squares fit of the value of the diameter is performed on the calibrated visibilities $\calvis$, sometimes for each bootstraps $\booti$ using $\calvisboot$. When the fits are performed on each bootstrap, the diameter estimate is the median of obtained values and the uncertainty is obtained with the 1-$\sigma$ confidence interval.
We have gathered the different models into three groups. In the first one uncorrelated errors (uncertainties $\dcalvis$ or $\dcalvisboot$) are assumed and no systematics are taken into account. They are called \modelvar{} (``variance''). The second group, uses correlated errors (covariance matrix $\covar$ or $\ubcovar$) but no systematics, they are called \modelcov{} (``covariance''). The last group of models use both correlated statistical errors and systematics, they are called \modelsys{} (``systematics''). Within these groups small differences in the error handling have been considered, they are specified after a slash.
\begin{figure*}
\includegraphics{estimate-comparison.pdf}
\caption{Comparison between diameter estimates for the four error models that most differ \modelpndrs, \modelvar, \modelcov, \modelsys{} (see Sect.~\ref{sec:ud}).
\textit{Left:} histogram of the number of standard deviations between the estimates. As a reference, we plot the expected distribution if the two estimates had the same expected value and were independent Gaussian random variables.
\textit{Right:} direct comparison of the diameters estimates. The dashed line indicates the place where the estimates are equal.}
\label{fig:estim-comp}
\end{figure*}
\subsection{Error propagation}
\label{sec:influence-prop}
The two first error models we introduce and compare take neither correlations nor systematics into account. They differ in the way the error propagation is performed from the raw to calibrated visibilities. In the first model, \modelpndrs{}, the standard quadratic addition of error terms along the way is performed (fit to $\calvis \pm \dcalvis$). The second one, \modelvar, determines the error from the bootstraps (fit to $\calvis \pm \dcalvisboot$). \modelpndrs{} and \modelvar{} are displayed as the first two points in the left panels of Fig.~\ref{fig:ud-comp}.
If errors are not strictly Gaussian, as expected from the quotient in Eq.~(\ref{eq:calvisboot}), the bootstrap method samples the real distribution of the calibrated visibilities and estimates correctly its average and confidence intervals, while the standard propagation may not.
In most cases (see Fig.~\ref{fig:estim-comp}, top-left panel), the estimates for the diameters using \modelpndrs{} and \modelvar{} are consistent with each other and differ by at most 2-$\sigma$. For two stars (10\% of the sample), GJ~785\change{,} and GJ~86\change{,} the results are at least 3-$\sigma$ apart. We conclude that departure from a Gaussian distribution has not a significant impact in most cases.
\subsection{Correlation of statistical errors}
\label{sec:influence-corr}
The \modelvar{} error model group assumes that errors on the visibilities are not correlated ($\calvis \pm \dcalvis$ is fit), while the \modelcov{} model group does ($\calvis$ is fit with correlation matrix $\covar$). We expect that positive correlations increase the errors bars on the diameter, as the different data points become partially redundant. They can also shift its estimated value, as groups of correlated data lose their weight relative to uncorrelated data.
This is indeed what we observe for most of our stars (in the left panel of Fig.~\ref{fig:ud-comp}, 2nd and 5th points from the left labelled \modelvar{} and \modelcov{}). The difference is quite significant as 40\% of the stars show a discrepancy of at least 2-$\sigma$ (see Fig.~\ref{fig:estim-comp}, middle-left panel).
\subsection{Baseline-dependent statistical errors}
As explained in Sect.~\ref{sec:bl-staterr}, the data at some baselines show more dispersion than the statistical errors determined during the data reduction. An additional baseline-dependent error term has been introduced to account for this discrepancy in model \modelcovbl.
As it can be seen in Fig.~\ref{fig:estim-comp} (for star GJ~785), including these errors can significantly modify the diameter estimate. When a few baselines are impacted by very high noise of unknown origin, most of the time bad and fast varying atmospheric conditions, the inclusion of an additional error term decreases the weight of these baselines in the fit (for GJ~785, see two baselines around 50 and 70--80\,M$\lambda${} in Fig.~\ref{fig:visib}, right column, 3rd panel from the bottom, which show higher dispersion than error bars predict as well as some bias towards a less resolved star). GJ~785 is a K star with no hint of specially high activity, so it is unlikely that this $V^2$ dispersion is produced by a short-term variability. A relatively fast atmospheric turbulence ($\tau_0 = 5$--7\,ms in $H$ with a decent seeing of 0.7--0.9) may explain part of it. Two other stars of the sample show a high, unexplained dispersion of measurements at several baselines which could be imputed to an intrinsic short-term variability: for M-type flare stars GJ~54.1 and GJ~447, $\sigmauncorr=4.3$\% and 6.1\%, respectively but, in their case, the dispersion only impacts the uncertainty, not the diameter estimate. For the remaining 85\% of the sample, the unexplained dispersion is moderate ($\sigmauncorr = 2.3\pm0.9$\%) and the \modelcov{} \& \modelcovbl{} diameter estimates are consistent with each other.
\subsection{Systematic errors}
\label{sec:influence-syserror}
\begin{figure}
\includegraphics[width=\linewidth]{relative-diameter-error.pdf}
\caption{Relative uncertainty on the diameter versus the resolution factor $r = \vartheta/(\lambda/B)$, with the standard processing technique (\modelvar{} model, green) and our determination with covariances and systematics (\modelsys{} model, Sect.~\ref{sec:influence-syserror}). Stars observed close to, or below, the nominal sensitivity limit of PIONIER are represented as hollow markers.}
\label{fig:delta-ud}
\end{figure}
Usually, error models assume that the uncertainties on data points are underestimated and rescale them so that a reduced chi squared of 1 is obtained. This is indeed what we do in our models that exclude systematic errors (\modelvar{} and \modelcov{} model groups).
In the \modelsys{} model group, however, no rescaling of error bars occurs. Instead a relative error term $\sigmasys$ is introduced, as explained in Sect.~\ref{sec:syserr} and the fit is performed with a correlated covariance matrix ($\ubcovar$ in Eq.~\ref{eq:syscovar}). The value of $\sigmasys$ is fit to a reduced chi squared of one.
The difference in estimated diameter is often significant as 25\% of the stars show a discrepancy of at least 1-$\sigma$ (see e.g.\ Fig.~\ref{fig:estim-comp}, bottom-left panel) between the \modelcov{} and \modelsys{} models.
In Fig.~\ref{fig:ud-comp}, one can also see (left panel, 6th and 9th points from the left, labelled \modelcovbl{} and \modelsys) that systematics tend to increase the diameter uncertainties (as positive correlation usually do). A comparison of uncertainties in the \modelvar{} and \modelsys{} models is given in Fig.~\ref{fig:delta-ud} as a function of the resolution factor, which we define as the ratio of the UD diameter of the star to the nominal resolution of the interferometer $\lambda/B_\mathrm{max}$. The typical error in \modelsys{} is 4.8\% for a resolution factor of 0.2 ($\approx 0.45$\,mas diameter at 140\,m baselines in the $H$ band) with respect to 1.2\% in the standard \modelvar{} model. Since the \modelvar{} model is widely used in the literature, we conclude that diameter uncertainties may be significantly underestimated. The relative uncertainty scales approximately as the inverse of the 2.5th power of the resolution, making determinations for under-resolved objects ($\lesssim 0.4$\,mas at VLTI in $H$) difficult. Since the fringe contrast loss $1-V^2$ scales as the square of the diameter, we expected the diameter uncertainty to scale as the inverse of the second power of the resolution factor. The additional drop in precision may be attributed to a loss of precision as the stars get fainter. In Fig.~\ref{fig:ud-comp}, we can see that 3 of the 4 most under-resolved targets have been observed very close to the sensitivity limit of PIONIER.
For reference, Fig.~\ref{fig:ud-comp} shows another error model, \modelppp, that uses the na\"\i{}ve but erroneous correlation matrix leading to Peelle's pertinent puzzle. Indeed, the diameter estimates are significantly off for about half of the sample.
\subsection{Noise in the covariance matrix}
\label{sec:influence-noise}
With model \modelvarnoisy{} we aim to assess the influence of the noise in the correlation to disentangle noise systematics from actual correlation influences. We measure the noise level in the covariances $\covar$, generate noisy matrices for uncorrelated data, and perform a data fit using these matrices.
Most data are only lightly correlated, for different nights and instrument configurations are loosely correlated by the small uncertainty on the calibrators diameter. So, the histogram of the correlation matrix values features a central component, a Gaussian-like distribution around a small positive constant, corresponding to the mostly uncorrelated data. In addition it has a bump with large positive correlations corresponding to the small fraction of highly correlated data (spectral channels of the same observation, for instance). We fit the width of the central component to get the noise level.
Then, we generate $\Nboot$ covariance matrices for noisy uncorrelated data
\begin{subequations}
\begin{align}
\correl &= \delta_{d\obsj}
+ \varepsilon \sum_k M_{kd}^{(\booti)} M_{k\obsj}^{(\booti)}\\
\noisycovar^{(\booti)} &= \correl \dcalvisi \dcalvisj
\label{eq:noisycovar}
\end{align}
\end{subequations}
where $M_{kd}^{(\booti)}$ are picked using independent normal Gaussian distributions and $\varepsilon$ is adjusted to the noise level. The largest dimension along $k$ that allows \smash{$\correl$} to be positive definite is used.
For each of these noisy covariance matrices, we perform a least-squares fit to all calibrated visibilities $\calvis{}$ taken at baselines $\base$. Let \smash{$\diamnoisy$} ($\booti$ in $1 \cdots \Nboot$) be the values of the UD diameters. The UD diameter estimate $\diamnoisy[]$ is obtained from their average. Its uncertainty $\ddiamnoisy$ is the quadratic sum of the model uncertainty $\ddiamvar$ and the dispersion of \smash{$\diamnoisy$}.
As we can see from the data in Fig.~\ref{fig:ud-comp} (left panels, second and fourth point), there is no significant difference between the \modelvar{} and \modelvarnoisy{} models for any of our stars. We conclude that the impact of correlations is a real effect, not a bias introduced by noisy data.
\subsection{Non-Gaussian uncertainties}
\label{sec:influence-gauss}
In order to assess the influence of non Gaussian uncertainties, we performed least squares fit to each bootstrap and look at the distribution of the estimates for the uniform disc diameter. We expect that a distribution of visibilities with significant skewness, kurtosis, or long tails would be reflected in the distribution of diameter estimates, and, in turn, in its mean value and/or uncertainty.
For model \modelvarboot, $\calvisboot \pm \dcalvis$ ($1 \le b \le \Nboot$) is fitted, yielding a set of uniform disc diameter estimates $\diamvarboot$. The median value and 1-$\sigma$ confidence interval of $\diamvarboot$ yields the \modelvarboot{} diameter estimate $\diamvar \pm \ddiamvar$. The same is performed for error models with correlated statistical errors (\modelcovboot) and correlated statistical and systematic errors (\modelsysboot).
Figure~\ref{fig:ud-comp} (left panel) clearly shows that there is no significant difference in the diameter estimate for \modelvar{} and \modelvarboot, \modelcovbl{} and \modelcovboot, and \modelsys{} and \modelsysboot.
\subsection{``Bad baseline'' scenario}
\label{sec:influence-badbase}
We additionally check for the scenario that a ``bad baseline'', e.g. tainted with a strong systematic, may significantly alter the value of the uniform disc estimate. Given that baseline, we fit the data that are taken at any other baseline and compare the diameter estimates with the one obtained with all baselines. The process is repeated for each baseline. The diameter estimates with one baseline removed are reported on the right side of Fig.~\ref{fig:ud-comp}.
Two examples o. ``bad baselines'' can be seen in Fig.~\ref{fig:visib}. For GJ~876, a baseline around 75 to 80 megacycles displays very high dispersion ($V^2$ between 0.5 and 1.2) despite small error bars ($\sim 0.05$). This issue does not originate from a bad calibrator, because this kind of dispersion would only be seen with a well resolved, low contrast binary. That would clearly been seen (1) in closure phases which was not the case and (2) as a strong bias towards a more resolved object with most points below the fit. The inclement weather during this particular observation is a likely explanation with a seeing below average (1.05 arcsec in $H$) and a short coherence time (6 ms in $H$). Interestingly, this ``bad baseline'' has very little impact on the diameter estimates as GJ~876 has all its estimates (\modelvar, \modelcov, \modelcovbl, \modelsys{} models) within less than 1\% \change{dispersion}, probably because it shows little bias (the data approximately average the $V^2 \approx 0.85$ estimate). Another example of ``bad baseline'' has already been mentioned for GJ~785 (less resolved around 50 and 70-80\,M$\lambda${} in Fig.~\ref{fig:visib}, left column, 3rd panel from the bottom), this time with a clear bias. Its origin is unknown, but it is unlikely to be a calibrator issue either: if there was a large error on the diameter of the unique, large calibrator HD~196387, there would be an unseen error on GJ~785's diameter but very little additional dispersion between baselines. The calibrator has no measured closure phase ($\lesssim 0.02$\,rad) and no detectable near-infrared excess \citep[no evidence from JHK modelling in][]{MCD12}, so binarity or circumstellar material are very unlikely to account for a 5\% error in the visibility calibration. The bias induced by the ``bad baseline'' reflects in Fig.~\ref{fig:ud-comp}, right panel: with this baseline removed (3rd point of the 6), the estimates of the \modelvar, \modelcovbl, and \modelsys{} models are consistent, but with it (left panel) they differ by several $\sigma$.
These two ``bad baselines'' are extreme cases that could be solved easily by removing the data from the fit. However, there are targets (GJ~86, GJ~229, GJ~447, GJ~551) where the visibility plot (Fig.~\ref{fig:visib}) does not show anything obviously wrong, but the analysis of removing one baseline in the fits produces a significant difference in the \modelvar{} and/or \modelcovbl{} diameter estimates (right panel of Fig.~\ref{fig:ud-comp}).
When systematic errors are included, the impact of ``bad baselines'' disappears completely (right panel of Fig.~\ref{fig:ud-comp}): all diameter fits with one baseline removed are consistent with each other. The price to pay for this stability is larger error bars as we have shown in Sect.~\ref{sec:influence-syserror} and Fig.~\ref{fig:delta-ud}.
It appears, therefore, that observations with a few configurations over a few nights are enough to prevent a single baseline to significantly bias the diameter estimate as long as systematic errors are modelled.
\subsection{Bad calibrator scenario}
For several of our targets (GJ 1, GJ 54.1, GJ 86, GJ 370) one or more calibrators from MER05 more resolved than the target were used together to smaller, unresolved ones. In the bad luck scenario that a resolved calibrator's diameter is off by several standard deviations, it would bias the transfer function at the two or three large baselines it is observed with. Fortunately, for these four stars, there is no evidence that a few baselines skew the diameter estimate. The difference in the COV estimates when one baseline removed from the fit is $< 1$\% (right panel of Fig.~\ref{fig:ud-comp}) and the COV and SYS estimate are within 1\% of each other. We are confident that the diameter estimates of these stars are not significantly impacted by a bad calibrator.
\begin{figure*}
\centering
\includegraphics{GL1-GL1061-visibilities.pdf}
\caption{Squared visibility amplitudes versus baseline length. \textit{Vertical error bars:} reduced data, using one shade of grey and red per baseline. \textit{Lines:} best model fits \modelvar{} (uncorrelated, green dashed line), \modelcovbl{} (correlated, black solid line), \modelsys{} (with systematics blue dotted line). In many models they can hardly be distinguished from one another.}
\label{fig:visib}
\end{figure*}
\section{Conclusion}
\label{sec:conclusion}
Many astronomical objects (young stellar objects, late-type dwarfs) are too faint in the visible to be observed by optical interferometers and require kilometric baselines to be fully resolved in the infrared. For this reason, we are bound to infer the object's properties from under-resolved observations. In the case of stellar diameters, we are fortunate enough that there are no degeneracies in the (unique) parameter estimation, but the under-resolved character has a strong impact in terms of precision. While the geometric size of a fully resolved object can be estimated within a fraction of a percent, uncertainties and systematics of 5--10\% on a single observation are common when the target is under-resolved.
Our study had the main objective to partially overcome these limitations by a careful observation layout and to better quantify the remaining uncertainties. To that end, our \modelsys{} model fully takes into account correlated statistical uncertainties and systematics.
The main results are
\begin{itemize}
\item The error model has a significant impact on the estimate of the uniform disc diameter and its uncertainty. Correlations between visibilities have a strong impact (Sects.~\ref{sec:influence-corr}) as well as baseline systematics (Sect.~\ref{sec:influence-syserror}). In both cases, the estimates may differ by more than three standard deviations. The additional errors atop the statistical errors determined by the data processing software are $3.3 \pm 1.2$\,\% on the square visibilities (mean and dispersion in our 20 surveyed stars), which is in line with the generally accepted value of 5\% \citep{COL03}. However, the purely systematic term (highly correlated errors) is only $1.8 \pm 0.9$\,\%.
\item A few observations with different configurations and/or nights are usually enough to avoid a significant bias by a single baseline and instrumental configuration (Sect.~\ref{sec:influence-badbase}) provided that systematic errors are taken into account. It confirms the usual observation strategy by, for instance, \citet{BOY12}, \citet{GAL12}, and \citet{RAB18} in the case of stellar diameters of under-resolved stars.
\item Departure from a Gaussian distribution has no significant impact (Sect.~\ref{sec:influence-gauss}) except indirectly in the error propagation (Sect.~\ref{sec:influence-prop}). It needs not to be modelled provided that the uncertainties on the calibrated visibilities are obtained from the bootstraps (this work) or from an analytic determination of the probability density function \citep{PER03}.
\item The uncertainty on the diameter is four times larger than the ones modelled with a standard least-squares fit to uncorrelated data. For a diameter of 0.45\,mas in $H$ band at 140\,m baselines (typical of VLTI/PIONIER), our typical uncertainty is 4.8\% (\modelsys{} model) with respect to 1.2\% with the usual determination (\modelpndrs{} and \modelvar{} models).
\end{itemize}
We have offered here a relatively easy way, albeit numerically intensive, to obtain correlations between observables, by means of bootstrapping. Even if most reduction pipelines do not propagate correlations, it is possible to run them a large number of times on randomised data (interferograms and calibrator diameters are picked at random) to obtain the multivariate probability density function of the interferometric observables. Systematics errors such as a bad calibrator not bad enough to be detected, rapidly varying atmospheric conditions between science target and calibrators, or instrumental systematics (e.g. differential polarisation) have typically cast a doubt on the robustness of parameter estimation. We have provided a method to deal with these systematics in a relatively inexpensive way: the covariance matrix of the least-squares fit is modified to include a relative systematic error term. The price to pay for a more robust estimation (accurate, i.e. non-biased) is a significantly larger uncertainty.
Therefore, we strongly recommend that future interferometric studies take into account correlated errors and take time to model systematics.
\section*{Acknowledgements}
Based on observations collected at the European Organisation for Astronomical Research in the Southern Hemisphere under ESO IDs 090.D-0917, 091.D-0584, 092.D-0647, and 093.D-0471. R.L., M.R., A.J., and R.B. acknowledge support from CONICYT project Basal AFB-170002. A.J. acknowledges support from FONDECYT project 1171208, BASAL CATA PFB-06, and project IC120009 ``Millennium Institute of Astrophysics (MAS)'' of the Millenium Science Initiative, Chilean Ministry of Economy. R.B. acknowledges additional support from project IC120009 ``Millenium Institute of Astrophysics (MAS)'' of the Millennium Science Initiative, Chilean Ministry of Economy. This work made use of the Smithsonian/NASA Astrophysics Data System (ADS) and of the Centre de Données astronomiques de Strasbourg (CDS). This research made use of Astropy, a community-developed core Python package for Astronomy (Price-Whelan et al. 2018). We would like to thank the referee for carefully reading our manuscript and for giving constructive comments which substantially helped improving the paper.
\bibliographystyle{mnras}
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{
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Дря́новець () — село в Добрицькій області Болгарії. Входить до складу общини Добричка.
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Тихоокеанський кубок з хокею із шайбою серед жінок () — міжнародний жіночий хокейний турнір, що пройшов двічі в перерві між чемпіонатами світу 1995 та 1996 року. Чотири збірні брали участь у турнірі: Канади, Китаю, США та Японії.
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| 3,651
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{"url":"https:\/\/www.hackmath.net\/en\/math-problem\/93","text":"# Computer revolution\n\nWhen we started playing with computers, the first processor, which I remember was the Intel 8080 from 1974, with the performance of 0.5\u00a0MIPS.\n\nCalculate how much percent a year rose CPU performance when Intel 486DX from 1992 has 54\u00a0MIPS.\n\nWhat would be the consumption of normal car now (2018), if consumption declined at the same rate from 10 liters per 100 km (round to 3 decimal places)?\n\nResult\n\nyear performance growth:: \u00a029.7 %\ncar comsumption 2018: \u00a00 l\/100 km\n\n#### Solution:\n\n$p = 100 \\% \\cdot e^{\\dfrac{ (54\/0.5)}{ (1992-1974)}-1} = 29.7 \\%$\n$s = 10 \\cdot \\left(2 - 1.29708365423\\right)^{ 2018-1974 } \\doteq 0$\n\nOur examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!\n\nLeave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):\n\nBe the first to comment!\n\nTips to related online calculators\n\n## Next similar math problems:\n\n1. Car value\nThe car loses value 15% every year. Determine a time (in years) when the price will be halved.\n2. Theorem prove\nWe want to prove the sentence: If the natural number n is divisible by six, then n is divisible by three. From what assumption we started?\n3. Annual pension\nCalculate the amount of money generating an annual pension of EUR 1000, payable at the end of the year and for a period of 10 years, shall be inserted into the bank to account with an annual interest rate of 2%\n4. Intercept with axis\nF(x)=log(x+4)-2, what is the x intercept\n5. Exponential equation\nIn the set R solve the equation: ?\n6. The city 3\nThe city has 22,000 residents. How long it is expected to have 25,000 residents if the average annual population growth is 1.4%?\n7. Researchers\nResearchers ask 200 families whether or not they were the homeowner and how many cars they had. Their response was homeowner: 14 no car or one car, two or more cars 86, not homeowner: 38 no car or one car, two or more cars 62. What percent of the families\n8. Logarithm\nDetermine the number whose decimal logarithm is -3.8.\n9. Log\nif ?, what is b?\n10. Coordinate\nDetermine missing coordinate of the point M [x, 120] of the graph of the function f bv rule: y = 5x\n11. Demographics\nThe population grew in the city in 10 years from 42000 to 54500. What is the average annual percentage increase of population?\n12. Sequence\nCalculate what member of the sequence specified by ? has value 86.","date":"2020-05-26 09:44:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 2, \"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\": 0.4668511748313904, \"perplexity\": 2210.8543457775195}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-24\/segments\/1590347390755.1\/warc\/CC-MAIN-20200526081547-20200526111547-00561.warc.gz\"}"}
| null | null |
Dorn-Dürkheim is an Ortsgemeinde – a municipality belonging to a Verbandsgemeinde, a kind of collective municipality – in the Mainz-Bingen district in Rhineland-Palatinate, Germany.
Geography
Location
Dorn-Dürkheim lies between Mainz and Worms, in the "Heart of Rhenish Hesse". The municipality belongs to the Verbandsgemeinde Rhein-Selz.
History
In 767, Dorn-Dürkheim had its first documentary mention in a document from the Lorsch Abbey. The municipality belonged from the 10th to 12th century to the Bishopric of Worms and passed thereafter as a fief to the Lords of Bolanden. Assigned to the Oberamt of Alzey beginning in 1457, Dorn-Dürkheim was temporarily occupied by the French, before the community, along with the whole province of Rhenish Hesse passed to the Grand Duchy of Hesse. In 1897, Dorn-Dürkheim acquired a railway link on the Osthofen–Gau-Odernheim line.
Since the Second World War, Dorn-Dürkheim has belonged to the newly founded federal state of Rhineland-Palatinate, at first in the Alzey-Worms district. The municipality was incorporated into the Verbandsgemeinde of Guntersblum in 1972 and was also assigned to the Mainz-Bingen district.
Politics
Municipal council
The council is made up of 13 council members, counting the part-time mayor, with seats apportioned thus:
(as at municipal election held on 13 June 2004)
Coat of arms
The municipality's arms might be described thus: Per fess sable a demi-lion rampant Or armed, langued and crowned gules, and azure a crozier from base issuant, the crook ending in a rose argent.
Economy and infrastructure
Transport
The nearest Autobahn interchange is Biebelnheim on the A 63, some 10 km away.
In the neighbouring centre of Hillesheim a railway connection on the Osthofen–Gau Odernheim line was once available. Service ended on 29 September 1974.
Palaeontology
In 1972, one of Europe's richest mammalian fossil fields was discovered through pedological investigation at Dorn-Dürkheim, with many species from the Miocene. In an oxbow of the ancient Rhine bone and tooth fragments were recovered from more than 70 mammalian species, among others sabre-toothed cats, hyenas, tapirs, muntjacs, dwarf deer, forest antelopes, forerunners of today's horses, and proboscideans from the time about 8.5 million years ago
References
External links
Municipalities in Rhineland-Palatinate
Rhenish Hesse
Mainz-Bingen
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И́ван Пе́ха (; 23 января 1986, Братислава, Чехословакия) — словацкий футболист, защитник.
Карьера
В сезоне 2004/05 числился в составе клуба «Слован» из города Братислава, провёл два матча в чемпионате Словакии. В 2005 году перешёл в словацкий клуб «Сенец», за два с половиной сезона провёл не менее 37 матчей, забил один гол.
В феврале 2008 года перешёл в румынский клуб «Чахлэул». В августе 2008 года подписал контракт с белорусским клубом БАТЭ, сыграл за клуб 4 матча в чемпионате Белоруссии.
В дальнейшем выступал снова в чемпионатах Белоруссии и Румынии, а также Азербайджана, Латвии и Эстонии. В последние годы выступал в низших дивизионах Польши и Испании.
Примечания
Ссылки
Профиль на официальном сайте ФК «Неман» (Гродно)
Футболисты Словакии
Игроки ФК «Слован» Братислава
Игроки ФК «Сенец»
Игроки ФК «Чахлэул»
Игроки ФК БАТЭ
Игроки ФК «Хазар-Ленкорань»
Игроки ФК «Неман» Гродно
Игроки ФК «Ряван»
Игроки ФК «Лиепая»
Игроки ФК «Оцелул»
Игроки ФК «ФКИ Левадия»
Игроки ФК «Мотор» Люблин
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<?xml version="1.0" encoding="UTF-8" standalone="yes"?>
<feed><tipo>Rua</tipo><logradouro>Dez</logradouro><bairro>Vale Verde</bairro><cidade>Cubatão</cidade><uf>SP</uf><cep>11542130</cep></feed>
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{
"redpajama_set_name": "RedPajamaGithub"
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<?php
namespace Ds\Component\Filter;
use Symfony\Component\HttpKernel\Bundle\Bundle;
/**
* Class DsFilterBundle
*
* @package Ds\Component\Filter
*/
final class DsFilterBundle extends Bundle
{
}
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"redpajama_set_name": "RedPajamaGithub"
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Home /TV
How Did Prince William and Kate Middleton Meet Rose Hanbury?
by Michelle Kapusta twitter linkedin | More Articles: TV
By now you probably know the name Rose Hanbury as it's dominated the headlines of multiple news outlets recently. While most people know better than to believe everything they read online, especially when it comes to the royal family, a salacious claim involving Hanbury and Prince William has threatened to shatter the image many had of the Duke of Cambridge as a family man.
Rumors have been spreading on social media and several gossip sites about the reason why William's wife, Kate Middleton, no longer speaks to Hanbury and some of what's being alleged has turned this into an all-out royal scandal. Almost immediately, the Daily Mail debunked the speculation and said that both women "have considered legal action but, because none of the reports have been able to offer any evidence about what the so-called dispute is about, they have chosen to ignore it."
(L) Prince William | Richard Stonehouse – WPA Pool/Getty Images (R) David M. Benett/Dave Benett/Getty Images for Chris Levine
Now though there are plenty of questions about Hanbury's relationship with the royal couple and how they know each other. Here's how William and Middleton met the Marchioness of Cholmondeley.
How they met
The Duke and Duchess of Cambridge have known Hanbury for quite some time as she and her husband, David Rocksavage, were at William and Middleton's wedding in 2011. However, they reportedly became good friends while staying at their Anmer Hall country home full-time in 2014 as Hanbury and Rocksavage live on a property close to the royals. Like William and Middleton, the couple also had three young children.
Hanbury is a former fashion model and is 23 years younger than Rocksavage, who is a filmmaker and 7th Marquess of Cholmondeley. He inherited more than $200 million and the Houghton Hall estate in Norfolk in 1990.
A post shared by Celebitchy (@celebitchyofficial)
While it's not known if there really is any bad blood between Middleton and Hanbury now don't look to the palace for answers.
The palace will not comment on rumors
It shouldn't come as too much of a surprise that Kensington Palace is not commenting on the rumors since the rarely make statements confirming or denying reports about the royal family's personal lives.
So while the palace is staying mum, the Australian tabloid New Idea noted that a friend of the duchess and Rosebury claimed that the rumors of William having an affair are false. According to that source, the fact that they were out there at all though was enough for Middleton to cut Hanbury out of her life.
"The rumors started around Christmas that William had become just a bit too close to Rose and was infatuated with her," the insider claimed. "She is a naturally flirtatious and beautiful woman and William is only human. The rumors are untrue but they got back to Kate and she hit the roof … It's very sad because they were close friends and Rose's children played with Kate's George and Charlotte."
Read more: Is Prince William 'Controlling' and 'Standoffish' In Person?
Tags:Kate Middleton Prince William
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 3,963
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Q: Django template filter - one line I'm looking for a Django template filter that turns a multi-line construction into one big line. Has anyone implemented it?
The reason is - I have a form, {{form.as_p}} creates a multi-line html fragment, I want to create a javascript variable which is an html fragment, but when I do like this:
var new_div_text = '{{form.as_p}}';
it doesn't work. The reason is obvious, in javascript constructions like
var hello = 'Hello
world';
are invalid!
A: Reading your use case, it doesn't appear that you just want to remove lines. What if one of your form labels contains a ' character? Oops, your javascript is now invalid.
Django comes with a filter called escapejs which us used for precisely this problem.
With escapejs, you would type:
var newDivText = '{{ form.as_p|escapejs }}'
and you won't have to worry about any characters destroying your javascript.
|
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| 4,115
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https://warriorswire.usatoday.com/2021/01/29/santa-cruz-warriors-to-open-season-against-g-league-ignite-on-feb-10/
Santa Cruz Warriors to open season against G League Ignite on Feb. 10
Russell Isabella-USA TODAY Sports
January 29, 2021 11:35 pm PT
For the next couple of weeks, the Golden State Warriors bench is going to look a little different. Instead of suiting up for Golden State, Nico Mannion, Jordan Poole and Alen Smailagic are slated to join the Santa Cruz Warriors for the G League's Orlando Bubble.
Along with the trio of young Warriors, the Santa Cruz lineup will feature Jeremy Lin. The nine-year NBA veteran spent last season in the Chinese Basketball Association playing for the Beijing Ducks. Lin last played in the NBA for the Toronto Raptors in the 2018-19 season.
Golden State training camp invites Axel Toupane, Kaleb Wesson, Ryan Taylor and Dwayne Sutton will help round out Santa Cruz's rotation at Disney World.
With the roster set, the G League edition of the Orlando Bubble will begin a 15 game run on Feb. 10. First on Santa Cruz's schedule is a highly anticipated matchup with the newly formed G League Ignite.
The Ignite is a developmental team built with draft-eligible prospects, including projected top 2021 picks Jalen Green and Jonathan Kuminga. Along with draft prospects, veterans Jarrett Jack, Amir Johnson and Bobby Brown will suit up for the Ignite.
The G League battle between the Ignite and Warriors is scheduled to get underway on Feb. 10 at 8 a.m. PST in Orlando.
Jordan Poole, Nico Mannion and Alen Smailagic joining Santa Cruz Warriors at G League Orlando bubble
Warriors land Gonzaga's Jalen Suggs with No. 4 overall pick in latest Bleacher Report 2021 mock draft
Veteran guard Jeremy Lin officially signs with Warriors G League affiliate
Breaking down the Golden State Warriors' future draft picks
Rookie Report: Looking at Jordan Poole's top performances from his first season with the Warriors
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"redpajama_set_name": "RedPajamaCommonCrawl"
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package test.quote.backend.config;
import com.google.common.base.CaseFormat;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import org.springframework.amqp.core.Binding;
import org.springframework.amqp.core.Queue;
import org.springframework.amqp.rabbit.listener.SimpleMessageListenerContainer;
import org.springframework.amqp.remoting.service.AmqpInvokerServiceExporter;
import org.springframework.beans.BeansException;
import org.springframework.beans.factory.config.BeanDefinition;
import org.springframework.beans.factory.config.ConfigurableListableBeanFactory;
import org.springframework.beans.factory.support.BeanDefinitionBuilder;
import org.springframework.beans.factory.support.BeanDefinitionRegistry;
import org.springframework.beans.factory.support.BeanDefinitionRegistryPostProcessor;
import org.springframework.context.annotation.ClassPathScanningCandidateComponentProvider;
import test.quote.common.rpc.RemoteImplementation;
import java.util.Set;
import java.util.stream.Collectors;
/**
* Created by Daniil Molchanov on 17.02.17.
*/
public class RemoteExporterConfigurer implements BeanDefinitionRegistryPostProcessor {
private String packagesToScan;
public RemoteExporterConfigurer(String packagesToScan) {
this.packagesToScan = packagesToScan;
}
private static final Logger log = LoggerFactory.getLogger(RemoteExporterConfigurer.class);
@Override
public void postProcessBeanDefinitionRegistry(BeanDefinitionRegistry registry) throws BeansException {
ClassPathScanningCandidateComponentProvider scanner = new ClassPathScanningCandidateComponentProvider(false);
scanner.addIncludeFilter((metadataReader, metadataReaderFactory) ->
metadataReader.getAnnotationMetadata().hasAnnotation(RemoteImplementation.class.getCanonicalName()));
Set<BeanDefinition> beans = scanner.findCandidateComponents(packagesToScan);
log.info("Found {} remote bean candidates: {}",
beans.size(),
beans.stream().map(BeanDefinition::getBeanClassName).collect(Collectors.toSet()));
for (BeanDefinition bean : beans) {
try {
String beanClassName = bean.getBeanClassName();
Class beanClass = Class.forName(beanClassName);
Class interfaceClass = beanClass.getInterfaces()[0];
String interfaceName = interfaceClass.getCanonicalName();
String interfaceSimpleName = interfaceClass.getSimpleName();
String beanName = CaseFormat.UPPER_CAMEL.to(CaseFormat.LOWER_CAMEL, beanClass.getSimpleName());
String queueBeanName = beanName + "Queue";
String bindingHelperBeanName = beanName + "BindingHelper";
String bindingBeanName = beanName + "Binding";
String remoteBeanName = beanName + "Remote";
String listenerBeanName = beanName + "Listener";
registerQueue(registry, queueBeanName, interfaceName);
registerBindingHelper(registry, bindingHelperBeanName, queueBeanName, interfaceSimpleName);
registerBinding(registry, bindingBeanName, bindingHelperBeanName);
registerServiceExporter(registry, remoteBeanName, interfaceClass, beanName);
registerMessageListener(registry, listenerBeanName, queueBeanName, remoteBeanName);
log.info("Successfully registered RPC export service for {}", interfaceClass);
} catch (ClassNotFoundException e) {
log.error("Class not found", e);
}
}
}
private void registerMessageListener(BeanDefinitionRegistry registry, String beanName, String queueBean, String serviceExporterBean) {
BeanDefinition listener = BeanDefinitionBuilder.rootBeanDefinition(SimpleMessageListenerContainer.class)
.addConstructorArgReference("connectionFactory")
.addPropertyReference("queues", queueBean)
.addPropertyReference("messageListener", serviceExporterBean)
.getRawBeanDefinition();
registry.registerBeanDefinition(beanName, listener);
log.debug("Registered message listener bean: {}", beanName);
}
private void registerServiceExporter(BeanDefinitionRegistry registry, String beanName, Class interfaceClass, String serviceBean) {
BeanDefinition amqpServiceExporter = BeanDefinitionBuilder.rootBeanDefinition(AmqpInvokerServiceExporter.class)
.addPropertyReference("service", serviceBean)
.addPropertyValue("serviceInterface", interfaceClass)
.addPropertyReference("amqpTemplate", "template")
.getRawBeanDefinition();
registry.registerBeanDefinition(beanName, amqpServiceExporter);
log.debug("Registered service exporter bean: {}", beanName);
}
private void registerBinding(BeanDefinitionRegistry registry, String beanName, String bindingHelper) {
BeanDefinition bindingDefinition = BeanDefinitionBuilder.rootBeanDefinition(Binding.class)
.setFactoryMethodOnBean("getBinding", bindingHelper)
.getRawBeanDefinition();
registry.registerBeanDefinition(beanName, bindingDefinition);
log.debug("Registered binding bean: {}", beanName);
}
private void registerBindingHelper(BeanDefinitionRegistry registry, String beanName, String queueBean, String routingKey) {
BeanDefinition bindingBean = BeanDefinitionBuilder.rootBeanDefinition(BindingHelper.class)
.addConstructorArgReference("exchange")
.addConstructorArgReference(queueBean)
.addConstructorArgValue(routingKey)
.getRawBeanDefinition();
registry.registerBeanDefinition(beanName, bindingBean);
log.debug("Registered binding helper bean: {}", beanName);
}
private void registerQueue(BeanDefinitionRegistry registry, String beanName, String queueName) {
BeanDefinition queueBeanDefinition = BeanDefinitionBuilder.rootBeanDefinition(Queue.class)
.addConstructorArgValue(queueName)
.getRawBeanDefinition();
registry.registerBeanDefinition(beanName, queueBeanDefinition);
log.debug("Registered queue bean: {}", beanName);
}
@Override
public void postProcessBeanFactory(ConfigurableListableBeanFactory beanFactory) throws BeansException {}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
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| 9,142
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Le thulium (Tm) possède 35 isotopes connus, de nombre de masse variant entre 145 et 179, ainsi que 26 isomères nucléaires. Parmi eux, un seul est stable, 169Tm, et représente l'intégralité du thulium naturel, faisant du thulium un élément monoisotopique et un élément mononucléidique. La masse atomique standard du thulium est donc la masse isotopique de 169Tm, soit .
Parmi les 34 radioisotopes qui ont été décrits, les plus stables sont 171Tm, avec une demi-vie de , 170Tm (), 168Tm () et 167Tm (). Des isomères nucléaires, le plus stable est 164mTm (t1/2 = ).
Les radioisotopes plus légers que 169Tm se désintègrent principalement par émission de positron (β+) en isotopes de l'erbium, à plusieurs exceptions :
les deux radioisotopes les plus légers se désintègrent principalement par émission de proton, également en isotopes de l'erbium ;
pour les isotopes allant de 153Tm à 156Tm, la désintégration β+ est concurrencée par la désintégration α, produisant des isotopes de l'holmium ; ce dernier mode peut d'ailleurs être ultra-majoritaire (91 % pour 153Tm).
167Tm se désintègre par capture électronique en 167Er.
Les isotopes plus lourds que 169Tm se désintègrent eux principalement par désintégration β− en isotopes de l'ytterbium.
Tableau des isotopes
Notes
Les valeurs notées # ne viennent pas uniquement de données expérimentales, mais sont au moins partiellement extrapolées à partir de tendances observées. Les spins dont la détermination est fragile sont entre parenthèses.
Les incertitudes sont données en forme courte entre parenthèses après les derniers chiffres significatifs correspondant. Les valeurs d'incertitude sont données pour un écart-type, sauf pour la composition isotopique et la masse atomique standard venant de l'IUPAC, qui utilise les incertitudes étendues.
Références
Masse isotopiques issues de :
Compositions isotopiques et masses atomiques standards issues de :
Demi-vies, spin, et données isomériques issues des sources suivantes :
Thulium
Thulium
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{"url":"http:\/\/mathhelpforum.com\/calculus\/43109-recursive-sequence.html","text":"# Thread: recursive sequence\n\n1. ## recursive sequence\n\nSuppose a sequence is defined by $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = \\frac{1}{2}\\left(a_{n}+a_{n-1}\\right)$ for $n \\geq 2$. Prove $a_n$ converges and determine its limit.\n\nSo $a_n = \\frac{1}{2}\\left(a_{n-1}+a_{n-2} \\right)$. This means that for all $\\varepsilon > 0$, there exists $N \\in \\mathbb{N}$, such that whenever $n \\geq N$, then $|a_{n}-L| < \\varepsilon$ and $|a_{n+1}-L| < \\varepsilon$.\n\nIs this the right way to go about proving it?\n\n2. Originally Posted by particlejohn\nSo $a_n = \\frac{1}{2}\\left(a_{n-1}+a_{n-2} \\right)$. This means that for all $\\varepsilon > 0$, there exists $N \\in \\mathbb{N}$, such that whenever $n \\geq N$, then $|a_{n}-L| < \\varepsilon$ and $|a_{n+1}-L| < \\varepsilon$.\n\nIs this the right way to go about proving it?\nYes I believe thats ok... Its induction isnt it?\n\nOriginally Posted by particlejohn\nSuppose a sequence is defined by $a_0 = 0$, $a_1 = 1$, and $a_{n+1} = \\frac{1}{2}\\left(a_{n}+a_{n-1}\\right)$ for $n \\geq 2$. Prove $a_n$ converges and determine its limit.\n\nWell there could be different ways to the limit, i tried the hardest way out there. I actually found out the sequence explicitly\n\nThere are different approaches, one of them is using generating functions and the other is by redefining the sequence and simple algebraic tricks.\n\nI will do it using generating functions,\n\nLet $f(x) = \\sum_{k=0}^{k=\\infty} a_k x^k$\n\nThis means $f(x) - \\frac{xf(x)+x^2f(x)}2 = a_0 + \\left(a_1 - \\frac{a_0}{2}\\right)x+ \\sum_{k=2}^{k=\\infty} \\left(a_k - \\frac{a_{k-1} + a_{k-2}}2\\right) x^k$\n\nBut for our series, $\\forall k \\ge 2, a_k - \\frac{a_{k-1} + a_{k-2}}2 = 0$. And $a_1 = 1, a_0 = 0$.\n\nThus,\n$f(x) - \\frac{xf(x)+x^2f(x)}2 = x$\n\n$f(x) = \\dfrac{2x}{2 - x - x^2} = \\dfrac{2x}{(1-x)(x+2)} = \\frac23\\left(-\\dfrac{1}{1 + \\frac{x}2} + \\dfrac{1}{1-x}\\right)$\n\nClearly both can be represented as geometric progressions. And also the above function exists for some values of x(its called the region of convergence)\n\n$-\\dfrac{1}{1 + \\frac{x}2} = -\\sum_{k=0}^{k=\\infty} \\left(\\frac{-x}{2}\\right)^k$\n\n$\\dfrac{1}{1-x} = \\sum_{k=0}^{k=\\infty} x^k$\n\nThus:\n\n$f(x)=-\\sum_{k=0}^{k=\\infty}\\frac23 \\left(\\frac{-x}{2}\\right)^k + \\sum_{k=0}^{k=\\infty} \\frac23 x^k$\n\n$f(x)=\\frac23\\sum_{k=0}^{k=\\infty} \\left(1-\\left(\\frac{-1}{2}\\right)^k\\right)x^k = \\sum_{k=0}^{k=\\infty} a_k x^k$\n\nBy polynomial equality,\n\n$a_k = \\frac23\\left(1-\\left(\\frac{-1}{2}\\right)^k\\right)$\n\nNow very clearly, $\\lim_{k \\to \\infty} a_k = \\frac23$.","date":"2016-08-30 08:41:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 34, \"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\": 0.9850866198539734, \"perplexity\": 258.7298584251042}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2016-36\/segments\/1471982974951.92\/warc\/CC-MAIN-20160823200934-00237-ip-10-153-172-175.ec2.internal.warc.gz\"}"}
| null | null |
Keri Lynn Russell (Fountain Valley, 23 maart 1976) is een Amerikaanse actrice en danseres. Nadat ze in de jaren 90 een aantal televisiefilms en series speelde werd ze uiteindelijk bekend van de titelrol van Felicity Porter in de televisieserie Felicity, een serie die liep van 1999 tot 2002. Russell heeft sindsdien in meerdere bekende films gespeeld, waaronder We Were Soldiers, The Upside of Anger en Mission: Impossible III. Voor haar televisiewerk kreeg ze in 2017 een ster op de Hollywood Walk of Fame.
Levensloop
Russell werd geboren in Fountain Valley, gelegen in de Amerikaanse staat Californië. Haar ouders waren David Russell, een medewerker van Nissan Motors, en Stephanie Stevens. Ze heeft een oudere broer, Todd, en een jongere zus, Julie. Russell verhuisde vanwege de baan van haar vader vaak. Ze woonde onder andere in Dallas, Mesa en Denver.
Haar eerste televisierol had ze in de televisieserie New Mickey Mouse Club. Van 1991 tot en met 1993 speelde ze in de serie en was ze te zien samen met Christina Aguilera, Britney Spears, JC Chasez en Justin Timberlake, die later allen zouden uitgroeien tot popsterren. In 1992 had ze een rol in de televisiefilm Honey, I Blew Up the Kid naast acteur Rick Moranis. Ook had ze rollen in de televisiefilm The Babysitter's Seduction uit 1996 en in de soapserie Malibu Shores. In 1994 had ze een rol in de muziekvideo "Always" van Bon Jovi. Het jaar erna had ze ook een rolletje in Married... with Children.
Van 1998 tot 2002 speelde Russell de hoofdrol in de dramaserie Felicity van J.J. Abrams. Ze won in 1999 voor haar rol een Golden Globe. Ondanks haar grote rol in de televisieserie speelde ze ook nog in de films Eight Days a Week, The Curve en Mad About Mambo, die allen enkel in Noord-Amerika werden uitgebracht. Haar grootste filmrol tot dan toe had ze in de oorlogsfilm We Were Soldiers, geregisseerd door Mel Gibson. De film kwam twee maanden voor het einde van Felicity uit en de bioscopen.
Toen Felicity aan z'n einde kwam nam Russell een korte acteerpauze. Ze verhuisde naar New York en hield een pauze van twee jaar, waarin ze tijd met vrienden doorbracht. Ze overwoog er zelfs te stoppen met acteren. In 2004 speelde Russell in de New Yorkse theaterproductie Fat Pig van Neil LaBute, met onder andere Jeremy Piven, Andrew McCarthy en Ashlie Atkinson. In 2005 keerde ze terug naar film en televisie; ze had een rol in de televisiefilm The Magic of Ordinary Days en in de bioscoopfilm The Upside of Anger met Kevin Costner, Joan Allen en Evan Rachel Wood. In datzelfde jaar speelde ze ook in de miniserie 'Into the West van Steven Spielberg, over de Verenigde Staten aan het begin van de negentiende eeuw. Hoewel een aantal van de acteurs van Felicity ook te zien waren in Alias, een televisieserie geproduceerd door J.J. Abrams, heeft Russell nooit een rolletje gehad in die serie; elke uitnodiging weigerde ze. In 2005 vroeg Abrams of ze mee wilde spelen in Mission: Impossible III, een film die hij zou regisseren, en ze accepteerde het aanbod. Op 5 mei 2006 werd de film uitgebracht en was een groot succes. Russell werd ook gezien als de geschikte persoon om "Lois Lane" te spelen in Superman Returns, maar ze werd niet goed genoeg bevonden en de rol ging naar Kate Bosworth.
In 2007 kwam de film Waitress uit, een onafhankelijke film waarin ze het personage Jenna speelt, een zwangere serveerster in het zuiden van Amerika. De rol van Russell werd goed ontvangen door critici.
Eveneens in 2007 had ze een gastrol in twee televisieseries van het televisieprogramma Scrubs van NBC. In juni 2007 was Russell te zien in de sitcom The Keri Kronicles, een serie die enkel te bekijken was via de website MySpace. De serie werd opgenomen in haar eigen huis en ging over haar eigen leven. Ook in 2007 speelde ze een rol in de film August Rush.
Privé
Russell woont in Manhattan. Op 14 februari 2007 trouwde ze met timmerman Shane Deary in New York. Op 9 juni 2007 kregen ze een zoon en op 27 december 2011 een dochter. In de zomer van 2013 scheidden ze. Ze hertrouwde in 2013 met Matthew Rhys, met wie ze samen de hoofdrol speelde in de serie The Americans. Eind mei 2016 kregen ze samen een kind, genaamd Sam.
Filmografie
|-
|-
|align="center"| 1996 || Malibu Shores || Chloe Walker
|-
|align="center"| 1998-2002 || Felicity || Felicity Porter
|-
|align="center"| 2005 || Into the West || Naomi Wheeler
|-
|align="center"| 2010-2011 || Running Wilde || Emmy Kadubic
|-
|align="center"| 2013-2018 || The Americans || Elizabeth Jennings
|-
|-
|align="center"| 1992 || Honey, I Blew Up the Kid || Mandy Park
|-
|align="center"| 1996 || The Babysitter's Seduction || Michelle Winston || Televisiefilm
|-
||| The Lottery || Felice Dunbar || Televisiefilm
|-
|align="center"| 1997 || Eight Days a Week || Erica
|-
||| When Innocence Is Lost || Erica French || Televisiefilm
|-
|align="center"| 1998 || Dead Man's Curve || Emma
|-
|align="center"| 1999 || Cinderelmo || Prinses || Televisiefilm
|-
|align="center"| 2000 || Mad About Mambo || Lucy McLoughlin
|-
|align="center"| 2002 || We Were Soldiers || Barbara Geoghegan
|-
|align="center"| 2005 || The Upside of Anger || Emily Wolfmeyer
|-
||| The Magic of Ordinary Days || Livy || Televisiefilm
|-
|align="center"| 2006 || Mission: Impossible III || Lindsey Farris
|-
||| Rohtenburg || Katie Armstrong || aka Grimm Love
|-
|align="center"| 2007 || Waitress || Jenna Hunterson
|-
||| The Girl in the Park || Celeste
|-
||| August Rush || Lyla Novacek
|-
|align="center"| 2008 || Bedtime Stories || Jill
|-
|align="center"| 2009 || Wonder Woman || Wonder Woman || Direct-naar-videofilm
|-
||| Leaves of Grass || -
|-
||| Crowley || Aileen Crowley
|-
|align="center"| 2013 || Austenland || Jane Hayes/Erstwhile
|-
| || Dark Skies || Lacy Barrett
|-
|align="center"| 2014|| Dawn of the Planet of the Apes || Ellie
|-
|align="center"|2019 || Star Wars: The Rise of Skywalker || Zorii Bliss
|-
|align="center"|2023 || Cocaine Bear || Sari McKendry
|-
|}
Externe link
Amerikaans acteur
|
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The 1985 Campeonato Nacional was Chilean football league top tier's 53rd season. Cobreloa was the tournament's champion, winning its third title.
League table
Results
Topscorer
Liguilla Pre-Copa Libertadores
Pre-Copa Libertadores play-off
Played between 1985 League champions and 1984 League runners-up.
Cobresal qualified for the 1986 Copa Libertadores
See also
1985 Copa Polla Gol
References
External links
ANFP
RSSSF Chile 1985
Primera División de Chile seasons
Chile
Primera
|
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| 6,101
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Founding a successful technology company has been the highlight of my professional career. And doing so away from my home country means I have a unique perspective on the journey of a founder in the United States.
When I was a manager in a big corporate finance department, I saw how much organizations wasted processing bills, so I started a company to make that process more efficient. But getting to this point was no walk in the park. Just immigrating to the U.S. took a lot of effort.
Professionals in Silicon Valley and the broader United States respect innovation like nowhere else in the world — at least, not in the places I've done business before.
So don't be shy about sharing your passion. If you do, people will listen and help. Innovation is highly valued here. There are no dumb questions or bad ideas. People pay attention to ideas. They believe that absolutely any business venture can turn out to be the next big thing. Be proud of how your new business will make things better for everyone.
The U.S. business community is uniquely focused on new ideas. If you offer innovation, it helps build a brand that endures beyond just the product you are immediately offering. Innovation equals value. People will be willing to work with you as a result.
I identified a problem that was causing pain for people in the accounting industry. At the time, it was perceived as a small problem, but our technology spoke to the challenges in a way that demonstrated how things could be better.
We showed the market how we could positively impact their lives, and they listened. Today, we continue to innovate by adding technology capabilities that speak to that audience. If you present a solution to a big problem, people will listen.
When I arrived in the U.S., I wish I'd known just how eager people were to listen to and talk about new ideas. Feedback from clients, users, and customers made our product and company better, and it can help you with your own product or service. Learning from your users is how you innovate.
When we first started, we spent a lot of time on product development before we went to market. In hindsight, I wish we'd started distributing early, gotten feedback sooner, and made our platform better faster.
Focus on distribution of your product or service first. Just get it out there. If you offer something to the marketplace, even if it's imperfect, potential users will give you a shot if your solution has the potential to improve their pain points. Their feedback will tell you what to do next.
When you're starting a company, there are a lot of things outside of your control. However, distribution of your product or service is not one of them. It's how you can build momentum in the marketplace.
When you have users, customers or an audience, you need to give your user community something to talk about. As well, distribution creates interaction with potential investors, business partners, and employees. You gain traction as a result. Luckily, there are fewer barriers to distribution in the U.S. than in many other parts of the world.
Traction validates your efforts and accelerates your development. More people will talk with you about funding and partnerships — making it easier to secure both faster.
In my experience, the American business community is more tolerant of well-intentioned missteps than other communities. Failure is a way to find out what works, so embrace it. People talk about "failing fast," which helps you avoid a grand failure that would close your company. So make sure you can recover, admit you were wrong, and build on the knowledge you've gained. Build a company that can survive even if your feature set didn't quite serve customer needs, your marketing campaign didn't yield the projected results, or the pricing matrix didn't incentivize conversions as you'd expected.
Fail fast and early, then have a look at what's left. Figure out what works, draw success from that, and then replicate it over and over again. It's cheaper and quicker if you don't have to shutter the doors every time you fail.
Finally, your peers need your support too, so encourage your fellow entrepreneurs. Starting a company can become a lonely process sometimes, so take every opportunity to provide guidance to those who need it.
I'm grateful for what my mentors taught me, and I hope what I've shared can benefit others on their journey. For anyone starting a business from scratch, the wisdom of those who went first is invaluable. It certainly was to me. I saved a ton of time and avoided numerous pitfalls thanks to my mentors.
Overall, I'd say the key is to be ready when the right partners come along. If I were to start now, I would focus on being prepared when opportunities for funds, partners and hiring are available.
These six points are also what make the U.S. such an appealing place to develop new ideas. The love of innovation and collaboration inherent in the business community here can provide the turbo charge for the engine of your new business.
|
{
"redpajama_set_name": "RedPajamaC4"
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| 1,660
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\section{Introduction}
Recall that a permutation group $G$ acting on a set $X$ is sharply $2$-transitive if for any two pairs $(x,y)$ and $(x',y')$ of distinct elements of $X$ there is a unique $g\in G$ with $gx=x'$ and $gy=y'$. Then $G$ has involutions, and either involutions have fixed points and $G$ is of permutation characteristic $2$, or the action on $X$ is equivalent to the conjugation action on the set $I$ of involutions. In that case, all translations, i.e.\ products of two distinct involutions, are also conjugate and have the same order $p$, which is either an odd prime or $\infty$; the number $p$ (or $0$ if $p=\infty$) is the permutation characteristic of $G$. We say that $G$ splits if it has a regular normal subgroup $N$; in that case $G=N\rtimes C_G(i)$ for any involution $i\in I$. Note that Tent, Rips and Segev have constructed non-split sharply $2$-transitive permutation groups of characteristic $0$ and $2$.
V. D. Mazurov asked in the Kourovka Notebook (question 12.48):\\
Let G be a sharply $2$-transitive permutation group.\begin{enumerate}
\item Does G possess a regular normal subgroup if a point stabilizer is locally finite?
\item Does G possess a regular normal subgroup if a point stabilizer has an abelian
subgroup of finite index?\end{enumerate}
We shall answer question (b) affirmatively in permutation charateristic $0$. In fact, we shall show a more general result for near-domains.
\section{Near-domains and near-fields.}
Instead of working with sharply $2$-transitive groups, we shall work in the equivalent setting of near-domains.
\begin{definition}
$(K,0,1,+,\cdot)$ is a {\em near-domain} if for all $a,b,c\in K$
\begin{enumerate}
\item $(K,0,+)$ is a {\em loop}, i.e.\ $a+x=b$ and $y+a=b$ have unique solutions, with $a+0=0+a=a$;
\item $(K\setminus\{0\},1,\cdot)$ is a group, and $0\cdot a=a\cdot 0=0$;
\item left distributivity holds: $a\cdot(b+c)=a\cdot b+a\cdot c$;
\item for all $a,b\in K$ there is $d_{a,b}\in K$ such that $a+(b+x)=(a+b)+d_{a,b}\cdot x$ for all $x$.\end{enumerate}
A near-domain is a {\em near-field} if addition is associative.\end{definition}
Hence a near-field is a skew field iff right distributivity holds.
\begin{fact}[Tits, Karzel]
A sharply $2$-transitive permutation group $G$ is isomorphic to the group of affine transformations of some near-domain $K$, i.e.\ of the set of permutations $\{x\mapsto a+bx:a,b\in K,\,b\not=0\}$; the centraliser of any involution is isomorphic to the multiplicative group $K^\times$. It is split iff $K$ is a near-field.\end{fact}
Let $E$ be the set $\{d\in K:1+d=d+1\}$. Since the additive loop of $K$ is power- associative, it is easy to see that $1$ generates a subfield of $K$ contained in $E$, which is either $\mathbb Q$ or $\mathbb F_p$. Thus $K$ has a characteristic, which is easily seen to be equal to the permutation characteristic of $G$. Note that in characteristic $>2$ there is a unique maximal sub-near-field, which is equal to $E$.
\begin{fact}[\cite{Ker74}] For all $a,b,c\in K$ we have:\begin{enumerate}
\item $d_{a,a}=1$.
\item $d_{a,b}(b+a)=a+b$.
\item $cd_{a,b}c^{-1}=d_{ca,cb}$.
\item $d_{a,b}=d_{a,c}d_{c+a,-c+b}d_{-c,b}$.
\item If $a,b\in E$ then $(a+b)\,2\in E$.
\item $|K^\times:C_{K^\times}(d_{a,b})|=\infty$ if $d_{a,b}\not=1$.
\end{enumerate}
\end{fact}
Let now $A$ be any subgroup of finite index in $K^\times$ which avoids all non-trivial coefficients $d_{a,b}$ for $a,b\in K$.
Kerby \cite[Theorem 8.26]{Ker74} has shown that $K$ must be a near-field in the following cases:\begin{enumerate}
\item $\mbox{char} K=0$ and $|K^\times:A|=2$,
\item $\mbox{char} K=2$, $|K^\times:A|=2$ and $|E|>2$,
\item $\mbox{char} K=p>2$ and $|K^\times:A|<|E|$.
\end{enumerate}
We shall adapt the proof of (3) to characteristic $0$.
\begin{lmm}\label{lemma} Suppose $d_{a,1/k}=1$. Then $d_{a,n/k}=1$ for all $n\in\mathbb N$.\end{lmm}
\begin{proof} By induction on $n$. This is clear for $n=0$ and $n=1$. So suppose it holds for $n$, and consider
$$\begin{aligned}\frac{n+1}k +a&=\Big(\frac nk+\frac1k\Big)+a=\frac nk+\Big(\frac1k+a\Big)=\frac nk+\Big(a+\frac 1k\Big)\\
&=\Big(\frac nk+a\Big)+\frac 1k=\Big(a+\frac nk\Big)+\frac1k=a+\Big(\frac nk+\frac 1k\Big)=a+\frac{n+1}k.\qedhere\end{aligned}$$
\end{proof}
\begin{proposition}\label{propn} If $A\le K^\times$ is a subgroup of finite index avoiding all nontrivial $d_{a,b}$ and $\mbox{char}(K)=0$, then $K$ is a near-field.\end{proposition}
\begin{proof} Recall that $\mathbb Q\subseteq E$. If $K=E$, then $d_{a,b}=1$ for all $a,b\in K$ and $K$ is a near-field. So assume $E\subsetneq K^\times$, and take $a\in K\setminus E\,2^{-1}$. Let $n=|K^\times:A|$. Then there are distinct $i>j$ in $\{0,1,2,\ldots,n\}/n!$ with $d_{a,i}A=d_{a,j}A$; since $d_{-j,i}=1$ we obtain
$$d_{a,i}=d_{a,j}d_{j+a,-j+i}d_{-j,i}=d_{a,j}d_{j+a,-j+i}.$$
Hence $d_{j+a,-j+i}\in A$, and $d_{j+a,-j+i}=1$ by assumption.
Now $d_{(i-j)^{-1}(j+a),1}=d_{j+a,-j+i}=1$, so $(i-j)^{-1}(j+a)\in E$. Since $-(i-j)^{-1}j\in\mathbb Q\subseteq E$, we have
$$[-(i-j)^{-1}j+(i-j)^{-1}(j+a)]\,2=(i-j)^{-1}a\,2\in E,$$
and $d_{a2,i-j}=1$. But $0<(i-j)\,n!\le n$ is integer, and there is an integer $k>0$ with $i-j=\frac1k$. By Lemma \ref{lemma} we obtain $d_{a2,1}=1$ and $a\,2\in E$, a contradiction.
\end{proof}
\begin{cor} Let G be a sharply doubly transitive permutation group of characteristic $0$ whose point stabilizer is virtually abelian. Then $G$ is split.\end{cor}
\begin{proof} If $K$ is the associated near-domain, $K^\times$ has an abelian subgroup $A$ of finite index. Now any non-trivial $d_{a,b}$ has a centralizer of infinite index in $K^\times$, so $d_{a,b}\notin A$. We finish by Proposition \ref{propn}.\end{proof}
|
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| 4,828
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\subsection*{Abstract}
The dynamics of two planar elastic pendula mounted on the horizontally
excited platform have been studied. We give evidence that the pendula
can exhibit synchronous oscillatory and rotation motion and show that
stable in-phase and anti-phase synchronous states always co-exist. The complete bifurcational scenario leading from synchronous to asynchronous motion is shown.
We argue that our results are robust as they exist in the wide range
of the system parameters.
\noindent \textit{Keywords:} coupled oscillators, elastic pendulum,
synchronization
\vspace{0.2cm}
\noindent \rule[0.5ex]{1\columnwidth}{0.5pt}
\vspace{0.5cm}
\section{Introduction}
The elastic pendulum is a simple mechanical system which comprises
heavy mass suspended from a fixed point by a light spring which can
stretch but not bend when moving in the gravitational field. The state
of the system is given by three (spherical elastic pendulum) or two
(planar elastic pendulum) coordinates of the mass, i.e. the system
has three (spherical case) or two (planar case) degrees of freedom.
The equations of motion are easy to write but, in general, impossible
to solve analytically, even in the Hamiltonian case. The elastic pendulum
exhibits a wide and surprising range of highly complex dynamic phenomena
\cite{Anicin1993,Breitenberger1981,Carretero-Gonzalez1994,
Cayton1977,Cuerno1992,Davidovic1996,Holm2002,Kuznetsov1999,Lai1984,Lynch2002,
Lynch2002a,Lynch2004,Nunez-Yepez1990,Olsson1976,Pokorny2008,Rusbridge1980}.
For small amplitudes perturbation techniques can be applied, the system
is integrable and approximate analytical solutions can be found. The
first known study of the elastic pendulum was made by Vitt and Gorelik
\cite{Vitt1933}. They considered small oscillations of the planar
pendulum and identified the linear normal modes of two distinct types,
vertical or springing oscillations in which the elasticity is the
restoring force and quasi-horizontal swinging oscillations in which
the system acts like a pendulum. When the frequency of the springing
and swinging modes are in the ratio $2:1$, an interesting non-linear
phenomenon occurs, in which the energy is transferred periodically
back and forth between the springing and swinging motions \cite{Anicin1993,Breitenberger1981,Carretero-Gonzalez1994,Cayton1977,Cuerno1992,Davidovic1996}.
The most detailed treatment of small amplitude oscillations of both
plane and spherical elastic pendula is presented in the works of Lynch
and his collaborators \cite{Holm2002,Lynch2002,Lynch2002a,Lynch2004}.
For large finite amplitudes the system exhibits different dynamical
bifurcations and can show chaotic behavior \cite{Kuznetsov1999,Lai1984,Nunez-Yepez1990,Olsson1976,Pokorny2008,Rusbridge1980}.
The dynamics of elastic pendulum attached to linear forced oscillator has been studied by Sado \cite{sado2004}. She has shown a one parameter bifurcation diagrams showing different behaviour of the systems (periodic, quasiperiodic and chaotic). According to our knowledge this is the only study of considered systems, but one can find a
lot of papers concerning dynamics of classical pendulum attached to
linear or non-linear oscillator. Hatwal et al. \cite{Hatwal1982153,hatwal:657,hatwal:663}
gives approximate solutions in the primary parametric instability
zone, which allows calculation of the separate regions of periodic solutions.
Further analysis enables us to understand the dynamics around primary
and secondary resonances \cite{Bajaj1994,Cartmell1994173,Balthazar20011075,kecik2005,Song2003747}.
Then the analysis was extended to systems with non-linear base where
non-linearity is usually introduced by changing the linear spring
into nonlinear one \cite{Warminski2009612,WARMINSKI2001363,4723450,BVG2008}
or magnetorheological damper \cite{ISI:000289102700001}. Recently
the complete bifurcation diagram of oscillating and rotating
solutions has been presented \cite{Brzeski2012}. Dynamics of two coupled single-well Duffing oscillators forced by the common signal has been investigated in our previous papers \cite{Perlikowski2008,Perlikowski2008c}. We have shown the
detailed analysis of synchronization phenomena and compare different
methods of synchronization detection.
In this paper we study the dynamics of two planar elastic pendula
mounted on the horizontally excited platform. Our aim is to identify
the possible synchronous states of two pendula. We give evidence that
the pendula can synchronize both in the oscillatory and rotational motion
moreover in-phase and anti-phase synchronizations co-exist. Our calculations
have been performed using software Auto--07p \cite{Doedel2011} developed
for numerical continuation of the periodic solutions and verified
by the direct integration of the equations of motion. We argue that
our results are robust as they exist in the wide range of the system
parameters.
The paper is organized as follows. Sec. 2 describes the considered
model. We derive the equations of motion and identify the possible
synchronization states. In Sec. 3 we study the stability of different
types of synchronous motion. Finally Sec. 4 summarizes our results.
\section{Model of the system}
The analyzed system is shown in Fig. \ref{fig:Model-of-the}. It consists
of two identical elastic pendula of length $l_{0}$, spring stiffness
$k_{2}$ and masses $m$, which are suspended on the oscillator. The
oscillator consists of a bar, suspended on linear spring with stiffness
$k_{1}$ and linear viscous damper with damping coefficient $c_{1}$.
The system has five degrees of freedom. Mass $M$ is constrained to
move only in vertical direction and thus is described by the coordinate
$y$. The motion of the first pendulum is described by angular displacement
$\varphi$ and its mass by coordinate $x_{2}$, that represent the
elongation of the elastic pendulum. Similarly the second pendulum
is described by angular displacement $\phi$ and its mass by coordinate
$x_{3}$. Both pendula are damped by torques with identical damping
coefficient $c_{2}$, that depend on their angular velocities (not
shown in Fig. \ref{fig:Model-of-the}). The small damping, with damping
coefficient $c_{3}$ is also taken for pendula masses. The system
is forced parametrically by vertically applied force $F(t)=F_{0}\cos\nu t$,
acting on the bar of mass $M$, that connects the pendula. Force $F_{0}$
denotes the amplitude of excitation and $\nu$ the excitation frequency.
\begin{figure}
\begin{centering}
\includegraphics{fig1.eps}
\par\end{centering}
\caption{\label{fig:Model-of-the}Model of the system}
\end{figure}
The equations of motion can be derived using Lagrange equations of
the second type. The kinetic energy $T$, potential energy $V$ and
Rayleigh dissipation $D$ are given respectively by:
\begin{spacing}{0.5}
\begin{flalign}
\hspace{1cm}\begin{array}{c}
T=\frac{1}{2}(M+2m)\dot{y}{}^{2}+\frac{1}{2}m\dot{x}_{3}^{2}+\frac{1}{2}m(l_{0}+y_{wst2}+x_{3})^{2}\dot{\phi}^{2}+m\dot{y}\dot{x}_{3}\cos\phi-m\dot{y}\dot{\phi}(l_{0}+y_{wst2}+x_{3})\sin\phi+\end{array} & & {}
\end{flalign}
\begin{flalign*}
\hspace{1cm}+\frac{1}{2}m\dot{x}_{2}^{2}+\frac{1}{2}m(l_{0}+y_{wst2}+x_{2})^{2}\dot{\varphi}^{2}+m\dot{y}\dot{x}_{2}\cos\varphi-m\dot{y}\dot{\varphi}(l_{0}+y_{wst2}+x_{2})\sin\varphi & & {}
\end{flalign*}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{1cm}\begin{array}{c}
V=-mg(l_{0}+y_{wst2}+x_{2})\cos\varphi-mg(l_{0}+y_{wst2}+x_{3})\cos\phi+mg(l_{0}+y_{wst2})+mg(l_{0}+y_{wst2})+\end{array} & & {}
\end{flalign}
\begin{flalign*}
\hspace{1cm}+\frac{1}{2}k_{1}(y+y_{wst1})^{2}+\frac{1}{2}k_{2}(y_{wst2}+x_{2})^{2}+\frac{1}{2}k_{2}(y_{wst2}+x_{3})^{2}-(M+2m)gy & & {}
\end{flalign*}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{1cm}D & =\frac{1}{2}C_{2}\dot{\varphi}^{2}+\frac{1}{2}C_{2}\dot{\phi}^{2}+\frac{1}{2}C_{3}\dot{x}_{2}^{2}+\frac{1}{2}C_{3}\dot{x}_{3}^{2} & {}
\end{flalign}
\end{spacing}
\vspace{0.5cm}
\noindent where $c_{3}$ is the damping coefficient of the pendulum
mass and $y_{wst1}=\frac{(M+2m)g}{k_{1}}$, $y_{wst2}=\frac{mg}{k_{2}}$
represent static deflation of mass $M$ and pendulums' mass $m$ respectively.
The system is described by five second order differential equations
given in the following form:
\begin{spacing}{0.5}
\begin{flalign}
\hspace{1cm}\begin{array}{c}
m(l_{0}+y_{wst2}+x_{2})^{2}\ddot{\varphi}+2m(l_{0}+y_{wst2}+x_{2})\dot{\varphi}\dot{x}_{2}-m\ddot{y}(l_{0}+y_{wst2}+x_{2})\sin\varphi+\end{array} & & {}
\end{flalign}
\begin{flalign*}
\hspace{1cm}+mg(l_{0}+y_{wst2}+x_{2})\sin\varphi+C_{2}\dot{\varphi} & =0 & {}
\end{flalign*}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\begin{array}{c}
\hspace{1cm}m(l_{0}+y_{wst2}+x_{3})^{2}\ddot{\phi}+2m(l_{0}+y_{wst2}+x_{3})\dot{\phi}\dot{x}_{3}-m\ddot{y}(l_{0}+y_{wst2}+x_{3})\sin\phi+\end{array} & & {}
\end{flalign}
\begin{flalign*}
\hspace{1cm}+mg(l_{0}+y_{wst2}+x_{3})\sin\phi+C_{2}\dot{\phi} & =0 & {}
\end{flalign*}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{1cm}m\ddot{x}_{3}+m\ddot{y}\cos\phi-m\dot{\phi}^{2}(l_{0}+y_{wst2}+x_{3})-mg\cos\phi+k_{2}(y_{wst2}+x_{3})+C_{3}\dot{x}_{3} & =0 & {}
\end{flalign}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{1cm}m\ddot{x}_{2}+m\ddot{y}\cos\varphi-m\dot{\varphi}^{2}(l_{0}+y_{wst2}+x_{2})-mg\cos\varphi+k_{2}(y_{wst2}+x_{2})+C_{3}\dot{x}_{2} & =0 & {}
\end{flalign}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\begin{array}{c}
\hspace{1cm}(M+2m)\ddot{y}+m\ddot{x}_{3}\cos\phi-2m\dot{x}_{3}\dot{\phi}\sin\phi-m(l_{0}+y_{wst2}+x_{3})\ddot{\phi}\sin\phi-m(l_{0}+y_{wst2}+x_{3})\dot{\phi}^{2}\cos\phi+\end{array} & & {}
\end{flalign}
\begin{flalign*}
\hspace{1cm}+m\ddot{x}_{2}\cos\varphi-2m\dot{x}_{2}\dot{\varphi}\sin\varphi-m(l_{0}+y_{wst2}+x_{2})\ddot{\varphi}\sin\varphi-m(l_{0}+y_{wst2}+x_{2})\dot{\varphi}^{2}\cos\varphi+ & & {}
\end{flalign*}
\begin{flalign*}
\hspace{1cm}-(M+2m)g+k_{1}(y+y_{wst1})+C_{1}\dot{y}-F_{0}\cos\nu t & =0 & {}
\end{flalign*}
\end{spacing}
\vspace{0.5cm}
\noindent In the numerical calculations we use the following values
of parameters: $M=10\hspace{0.05in}[\textrm{kg]}$, $m=0.2\hspace{0.05in}[\textrm{kg]}$,
$l_{0}=0.24849\hspace{0.05in}[\textrm{m]}$, $k_{1}=1642.0\hspace{0.05in}[\textrm{\ensuremath{\frac{N}{m}}]}$,
$k_{2}=19.7\hspace{0.05in}[\textrm{\ensuremath{\frac{N}{m}}]}$, $c_{1}=13.1\hspace{0.05in}[\textrm{\ensuremath{\frac{Ns}{m}}]}$
, $c_{2}=0.00776\hspace{0.05in}[\textrm{Nms]}$, $c_{3}=0.49\hspace{0.05in}[\textrm{\ensuremath{\frac{Ns}{m}}]}$,
$y_{wst1}=0.062\hspace{0.05in}[\mathrm{m}]$, $y_{wst2}=0.1\hspace{0.05in}[\mathrm{m}]$.
Introducing dimensionless time $\tau=\omega_{1}t$, where $\omega_{1}^{2}=\frac{k_{1}}{M+2m}$
is the natural frequency of mass $M$ with the attached pendula,
we obtain dimensionless equations of motion written as:
\begin{spacing}{0.5}
\begin{flalign}
\hspace{0.7cm}\ddot{\Psi}+\frac{2\beta_{2}}{(1+y_{2st}+\chi_{2})}\dot{\Psi}\dot{\chi}_{2}-\frac{\beta_{1}^{2}}{(1+y_{2st}+\chi_{2})}\ddot{\gamma}\sin\Psi+\frac{\sin\Psi}{(1+y_{2st}+\chi_{2})}+\frac{\alpha_{2}}{(1+y_{2st}+\chi_{2})^{2}}\dot{\Psi} & =0 & {}\label{eq:eq1}
\end{flalign}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{0.7cm}\ddot{\Phi}+\frac{2\beta_{2}}{(1+y_{2st}+\chi_{3})}\dot{\Phi}\dot{\chi}_{3}-\frac{\beta_{1}^{2}}{(1+y_{2st}+\chi_{3})}\ddot{\gamma}\sin\Phi+\frac{\sin\Phi}{(1+y_{2st}+\chi_{3})}+\frac{\alpha_{2}}{(1+y_{2st}+\chi_{3})^{2}}\dot{\Phi} & =0 & {}\label{eq:eq2}
\end{flalign}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{0.7cm}\ddot{\chi}_{3}+\frac{\beta_{1}^{2}}{\beta_{2}^{2}}\ddot{\gamma}\cos\Phi-\frac{1+y_{2st}+\chi_{3}}{\beta_{2}^{2}}\dot{\Phi}^{2}-\frac{1}{\beta_{2}^{2}}\cos\Phi+y_{st2}+\chi_{3}+\alpha_{3}\dot{\chi}_{3} & =0 & {}\label{eq:eq3}
\end{flalign}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{0.7cm}\ddot{\chi}_{2}+\frac{\beta_{1}^{2}}{\beta_{2}^{2}}\ddot{\gamma}\cos\Psi-\frac{1+y_{2st}+\chi_{2}}{\beta_{2}^{2}}\dot{\Psi}^{2}-\frac{1}{\beta_{2}^{2}}\cos\Psi+y_{st2}+\chi_{2}+\alpha_{3}\dot{\chi}_{2} & =0 & {}\label{eq:eq4}
\end{flalign}
\end{spacing}
\vspace{0.5cm}
\begin{spacing}{0.5}
\begin{flalign}
\hspace{0.7cm}\begin{array}{c}
\ddot{\gamma}+\frac{\beta_{2}^{2}a}{\beta_{1}^{2}}\ddot{\chi}_{3}\cos\Phi-\frac{2\beta_{2}a}{\beta_{1}^{2}}\dot{\chi}_{3}\dot{\Phi}\sin\Phi-\frac{(1+y_{2st}+\chi_{3})a}{\beta_{1}^{2}}\ddot{\Phi}\sin\Phi-\frac{(1+y_{2st}+\chi_{3})a}{\beta_{1}^{2}}\dot{\Phi}^{2}\cos\Phi+\frac{\beta_{2}^{2}a}{\beta_{1}^{2}}\ddot{\chi}_{2}\cos\Psi+\end{array} & & {}\label{eq:eq5}
\end{flalign}
\begin{flalign*}
\hspace{0.7cm}-\frac{2\beta_{2}a}{\beta_{1}^{2}}\dot{\chi}_{2}\dot{\Psi}\sin\Psi-\frac{(1+y_{2st}+\chi_{2})a}{\beta_{1}^{2}}\ddot{\Psi}\sin\Psi-\frac{(1+y_{2st}+\chi_{2})a}{\beta_{1}^{2}}\dot{\Psi}^{2}\cos\Psi-\frac{1}{\beta_{1}^{2}}+\gamma+y_{1st}+\alpha_{1}\dot{\gamma}-q\cos\mu\tau & =0 & {}
\end{flalign*}
\end{spacing}
\vspace{0.5cm}
\noindent where $\omega_{2}^{2}=\frac{k_{2}}{m}$, $\omega_{4}^{2}=\frac{g}{l_{0}}$,
$\mu=\frac{\nu}{\omega_{1}}$, $\beta_{1}=\frac{\omega_{1}}{\omega_{4}}$,
$\beta_{2}=\frac{\omega_{2}}{\omega_{4}}$, $a=\frac{m}{M+2m}$, $q=\frac{F_{0}}{\omega_{1}^{2}l_{0}(M+2m)}$,
$\alpha_{1}=\frac{C_{1}}{\omega_{1}(M+2m)}$, $\alpha_{2}=\frac{C_{2}}{m\omega_{4}l_{0}^{2}}$,
$\alpha_{3}=\frac{C_{3}}{ml_{0}\omega_{2}^{2}}$, $y_{1st}=\frac{y_{wst1}}{l_{0}}$,
$y_{2st}=\frac{y_{wst2}}{l_{0}}$, $\gamma=\frac{y}{l_{0}}$, $\dot{\gamma}=\frac{\dot{y}}{l_{0}\omega_{4}}$,
$\ddot{\gamma}=\frac{\ddot{y}}{l_{0}\omega_{4}^{2}}$, $\chi_{3}=\frac{x_{3}}{l_{0}}$,
$\dot{\chi}_{3}=\frac{\dot{x}_{3}}{l_{0}\omega_{2}}$, $\ddot{\chi}_{3}=\frac{\ddot{x}_{3}}{l_{0}\omega_{2}^{2}}$,$\chi_{2}=\frac{x_{2}}{l_{0}}$,
$\dot{\chi}_{2}=\frac{\dot{x}_{2}}{l_{0}\omega_{2}}$, $\ddot{\chi}_{2}=\frac{\ddot{x}_{2}}{l_{0}\omega_{2}^{2}}$,
$\Psi=\varphi$, $\dot{\Psi}=\frac{\dot{\varphi}}{\omega_{4}}$, $\ddot{\Psi}=\frac{\ddot{\varphi}}{\omega_{4}^{2}}$,
$\Phi=\phi$, $\dot{\Phi}=\frac{\dot{\phi}}{\omega_{4}}$, $\ddot{\Phi}=\frac{\ddot{\phi}}{\omega_{4}^{2}}$
The dimensionless parameters of the system have the following values:
$\beta_{1}=2$, $\beta_{2}=1.58$, $\alpha_{1}=0.1$, $\alpha_{2}=0.01$,
$\alpha_{3}=0.1$, $a=0.0192$, $y_{1st}=0.25$, $y_{2st}=0.4$.
We study system (9-13) in order to detect possible synchronization
ranges. There are two basic types of synchronous motion, which are
depicted in Fig. 2(a,b). The pendula can synchronize either in-phase
or in anti-phase with each other, i.e., $\Psi=\Theta$ or $\Psi=-\Theta$.
In both mentioned cases the forces acting in vertical direction on
mass $M$ are identical (there are no forces in horizontal direction),
hence the energy transmitted between mass $M$ and pendula in in-phase
and anti-phase motion is also identical. If there is an in-phase synchronization,
the anti-phase also coexists in the same range of parameters. The
accessibility of in-phase and anti-phase motion is governed only by
initial conditions. The pendula's masses are always synchronized in
the in-phase with each other, i.e., $\chi_{2}=\chi_{3}$. The anti-phase
configuration of the masses is not observed $(\chi_{2}=-\chi_{3})$ with
the oscillating pendula. The anti-phase synchronization of masses
is possible when the pendula are in equilibrium positions, then the
sum of forces transmitted to mass $M$ is equal to zero.
\begin{figure}[H]
\begin{centering}
\includegraphics{fig2}
\par\end{centering}
\caption{Possible synchronization (a) in-phase, (b) in anti-phase}
\end{figure}
\section{Stability of synchronous motion}
\subsection{Synchronous solutions in two dimensional parameters space}
In this section we study the stability of the observed synchronous
oscillations and rotations of the pendula. We present the bifurcation
diagrams calculated in two-parameter space: amplitude $q$ versus
frequency $\mu$ of excitation. We focus our attention on determining
the regions of synchronous stable motion and bifurcations that lead
to its destabilization. We consider the state of the system in the
following range $q\in[0.0,\:1.2]$ of forcing amplitudes and frequency
of excitation belonging to the range $\mu\in[0.3,\:1.2]$, which cover
the possible resonances in the system. Resonance should be observed
when the frequency of excitation comes close to the natural frequencies:
of mass $M$ equal to $\mu_{M}=1$, of pendula $\mu_{p}=0.50$ and
pendulum mass $\mu_{pm}=0.79$. We describe synchronous solutions
with respect to the forcing period according to ratio $r:1-s:1$, where
$r$ and $s$ denotes number of forcing periods, for which pendula
and pendula masses perform period one motion. Fig. \ref{fig:runge} presents
two parameter bifurcation diagram, obtained by direct integration of (9-13).
It shows the existence of synchronous, asynchronous motion and equilibrium
solutions. As soon as we have a lot of coexisting solutions to hold
clearance of Fig. \ref{fig:runge} we do not distinguish which type
of synchronous or asynchronous we find. By synchronous solution we
mean, that both pendula are in complete synchronization state, i.e.,
their amplitudes and frequencies are identical. For low amplitudes
of excitation, the only solution is equilibrium, which turns into
synchronous or asynchronous solution as the frequency of excitation
increases. The detailed analysis of synchronous solutions (shaded in grey color) was performed using continuation software Auto--07p \cite{Doedel2011}. We calculate the stability borders of each identified case, i.e., the ranges inside
which the given motion is stable. The first periodic solution is observed
for frequency of excitation equal to $\mu=0.406$ and for amplitudes
of excitation above $q=0.709$. This periodic solution is shown in
Fig. \ref{fig:auto}(a) is identified as synchronous oscillations
of pendula and pendula masses locked $1:1-1:1$ with forcing. This solution
is destabilized by saddle-node (green line), period doubling (blue
line) and Neimark-Sacker (red line) bifurcations curves. The continuation
reveals that for small range of parameters, around the frequency of
excitation close to the natural frequency of pendula, this solution
coexists with synchronous $2:1-1:1$ oscillations. Synchronous oscillations
$2:1-1:1$ are destabilized by saddle-node bifurcation curve then
by Neimark-Sacker and pitchfork symmetry breaking (SB2) bifurcations.
In the investigated system we distinguish two different symmetry breaking
pitchfork bifurcations one of them (SB2) brokes symmetry between the
pendula, the second one (SB1) brokes the symmetry of each pendula
but their motion remains identical \cite{Miles:1988:RSB:51797.51803}.
As the frequency of excitation increases we observe either asynchronous
motion or equilibrium. With further increase of excitation frequency
we observe asynchronous behavior, which change into two small regions
of synchronous rotations $3:1-1:1$. We show it in Fig. \ref{fig:auto}(f)
and this area is bounded by Neimark-Sacker, period doubling and saddle-node
bifurcations. This solution coexists with synchronous $2:1-1:1$ rotations,
presented also in Fig. \ref{fig:auto}(f). The stability region for
this solution is bounded by pitchfork SB1 bifurcation from the left
and right, Neimark-Sacker from above, and saddle-node and Neimark-Sacker
bifurcations from the right. Both these solutions coexist in small
range of considered parameters with another synchronous rotations
$4:1-1:1$, presented in Fig. \ref{fig:auto}(f). The synchronous
motion destabilizes from the right by saddle-node and pitchfork SB2
curves, from above by Neimark-Sacker, and from the left by Neimark-Sacker,
saddle-node and pitchfork SB2 curves.
Around $\mu\approx 0.8$, where the resonance of pendulum masses occur,
the system possesses rich dynamics, which results in the coexistence
of different synchronous together with asynchronous solutions. This
includes synchronous rotations $2:1-1:1$ depicted in Fig. \ref{fig:auto}(c,d)
and synchronous $3:1-3:1$ rotations of pendula and pendula masses
presented in Fig. \ref{fig:auto}(e). The third which was found solution is synchronous half-rotations $1:1-1:1$ (Fig. \ref{fig:auto}(b)), for which both
pendula stop before approaching stable and unstable equilibrium transferring
the whole energy into displacement of pendula masses. This multistability
causes that it is hard to compare the bifurcation diagrams from
the direct integration and Auto-07p. In the case
of rotations $2:1-1:1$ the synchronous motion is destabilzed from
the right by saddle-node and pitchfork SB2 curves, from above by Neimark-Sacker,
and from the left by saddle-node, Neimark-Sacker and pitchfork SB2
bifurcations. Synchronous rotations $1:1-1:1$ loose stability by
pitchfork SB2 from the right, and by Neimark-Sacker and period doubling
from the left. The synchronous rotations $3:1-3:1$ are mainly destabilized
by pitchfork SB2 from the right and by pitchfork SB2 and period doubling
from the bottom, and by period doubling, pitchfork SB2, saddle-node
and Neimark-Sacker from the left. From this solution, through period-doubling
bifurcation we find synchronized rotations $6:1-6:1$, shown in Fig.
\ref{fig:auto}(e). This solution is destabilized from above by period-doubling
bifurcation, from the left through pitchfork SB2 bifurcation, and
from below through saddle-node bifurcation (not visible, since coincides
with period-doubling boundary for rotations $3:1-3:1$).
As we pass through the resonance frequency of mass $M$ equal to $\mu=1$,
for higher amplitudes of excitation the only synchronous solution
includes $2:1-1:1$ synchronous rotations depicted in Fig. \ref{fig:auto}(c,d).
After the resonance, for amplitudes of excitation above $q=0.141$,
only asynchronous solutions are observed. Below this value, many small
synchronous regions were found. This includes synchronous $1:1-1:1$
oscillations and two regions of synchronous $2:1-1:1$ oscillations,
together with two regions of synchronous $2:1-1:1$ rotations. The
region of $1:1-1:1$ oscillations is enclosed by saddle-node, Neimark-Sacker
and pitchfork SB2 bifurcation curves. Oscillatory $2:1-1:1$ motion destabilizes
through pitchfork SB1 from above and Neimark-Sacker curves from below.
This solution coexists for small range of parameters with $2:1-1:1$ rotations,
which motion is destabilized by period doubling and Neimark-Sacker
from the left, and from the right by pitchfork SB2, Neimark-Sacker
and period doubling curves. We observe the excellent correlation in
these regions between the results from numerical continuation and
direct integration.
\noindent
\begin{figure}[H]
\begin{centering}
\includegraphics{fig3}
\par\end{centering}
\caption{\label{fig:runge}(color online) The synchronous (black dots), asynchronous
(red dots) and equilibrium (small grey crosses) solutions of system
(9-13) in two parameters space: $\mu$ frequency and $q$ amplitude
of excitation. We calculate this plot by direct integration using
4th order Runge-Kutta algorithm. In rectangles (a-f) we highlighted
regions of synchronous motion calculated in Auto-07p (see Fig. \ref{fig:auto}).}
\end{figure}
\noindent
\begin{figure}[p]
\begin{centering}
\includegraphics{fig4}
\end{centering}
\caption{\label{fig:auto}(color) Stable ranges of synchronous motion calculated
in Auto-07p (see rectangles in Fig. \ref{fig:runge}). Color of lines
stand for different types of bifurcation: Neimark-Sacker (red), saddle-node
(green), pitchfork SB1 (violet), pitchfork SB2 (yellow) and period
doubling (blue). In region inside lines synchronous solutions are
periodic and stable. }
\end{figure}
\subsection{One parameter continuation}
In this subsection we show one parameter continuation of four periodic
solutions (two oscillating and two rotational) for fixed amplitude
of excitation, as a bifurcation parameter we choose the frequency
of excitation $\mu$. We start each path-following on the periodic solution and continue in two directions (forward and backward). In Fig. \ref{fig:1-parameter-continuation2}(a-d)
we present the synchronized oscillating periodic solutions, their
regions of stability are shown in Fig. \ref{fig:auto} (a). System (\ref{eq:eq1}-\ref{eq:eq5}) is given by five second order ODEs,
hence the phase space is ten dimensional and at least five figures
(amplitude of each degree of freedom) are necessary to show its complete
dynamics. To decrease it we focus on the dynamics of first pendula (second pendula has the same amplitude in the synchronized state) and mass $M$.
The first presented branch of periodic solutions in Fig. \ref{fig:1-parameter-continuation2}(a,b)
is synchronous $2:1-1:1$ oscillations, in previous subsection we
show that this family is destabilized by Neimark-Sacker bifurcation
from the right and from the left by pitchfork symmetry braking SB2.
Switching the branch at pitchfork bifurcation enables us to find another
stable branch of asynchronized oscillations $2:1-1:1$, that
looses its stability through the saddle-node bifurcation. After pitchfork
symmetry breaking SB2 bifurcation the solutions of one pendulum is
located at upper branch (see Fig. \ref{fig:1-parameter-continuation2}(b))
and the second pendulum on lower branch or vice-versa. The branch synchronous
oscillations $2:1-1:1$, shown in Fig. \ref{fig:1-parameter-continuation2}
(c,d), present much richer scenario than other. These oscillations
destabilize from both sides through pitchfork SB2 bifurcation. When
we switch branch in left SB2 point, we find family of stable
asynchronized oscillations $2:1-1:1$. Finally, when the amplitudes of pendula
reach zero their motion stops. When we continue in the opposite direction the stability
is lost in pitchfork SB2 bifurcation. Another change of branch allows
us to observe another asynchronous periodic solutions, for which first pendulum
oscillates $2:1-1:1$, second pendulum is at rest (not shown here) and pendula
masses oscillate $1:1-1:1$ in asynchronized manner. One end of this stable
branch destabilizes through saddle-node bifurcation and the second
one by pitchfork SB2 bifurcation. As the frequency of excitation increases
the stability of this solutions is regained through pitchfork SB2 bifurcation
and lost again through saddle-node bifurcation. Note that for the
mass $M$, the bifurcation points that are responsible for the destabilization
of periodic solutions for synchronized oscillations $2:1-1:1$ and
asynchronized oscillations $2:1-1:1$, are placed very close to
each other. When we switch the branch in the right SB2 bifurcation
point of the synchronized oscillations $2:1-1:1$, we find asynchronized
solution of oscillations $2:1-1:1$ that persists for small interval
of excitation frequency. It destabilizes from above and below through
period-doubling bifurcations. Switching the branch in lower
period doubling bifurcation point enables us to observe asynchronized
oscillations $4:1-2:1$, that destabilize through Neimark-Sacker bifurcation.
The bifurcation diagram, shown in Fig. \ref{fig:1-parameter-continuation}(a-b),
shows $3:1-3:1$ rotational periodic solutions for $q=0.654$. In
contrary to the previous cases, to hold a physical meaning, on the
horizontal axis we plot the amplitude of velocity. The stability region
of this branch is shown in Fig. \ref{fig:auto}(e). This family of solutions
looses its stability through period-doubling and pitchfork SB2 bifurcations.
Switching the branch in the period-doubling bifurcation point, allows
to observe another period doubling, which leads us to two branches
of synchronized $6:1-6:1$ rotational solution. This branch
looses its stability once again through period-doubling bifurcation. After another switch of branch in period doubling bifurcation point, we
reach two synchronized periodic $12:1-12:1$ rotational
solutions. They are stable in very narrow range of excitation frequency
and loosing stability via Neimark-Sacker bifurcations. After switching
the branch in right pitchfork SB2 bifurcation point, we observe asynchronized
rotations $3:1-3:1$, that are stable in very small interval, finally
loosing its stability through saddle-node bifurcation. In Fig. \ref{fig:1-parameter-continuation}(c-d) we present synchronized
rotations $4:1-1:1$, that loose stability through pitchfork symmetry
braking from the right and left. Switching the branch in both SB2
bifurcation points let us to find asynchronous rotations $4:1-4:1$,
that are stable in very narrow interval of excitation frequency, loosing
finally stability through saddle-node bifurcation.
\begin{figure}
\begin{centering}
\includegraphics{fig5}
\par\end{centering}
\caption{\label{fig:1-parameter-continuation2}1 parameter continuation of
fully synchronized: 2:1-1:1 oscillations ((a) mass $M$, (b) pendulum
1, \foreignlanguage{polish}{$q=0.899$, $\mu=0.455$}), 2:1-1:1 oscillations
((c) mass $M$, (d) pendulum 1, \foreignlanguage{polish}{$q=0.654$,
$\mu=0.5$}). The continuous and dashed lines correspond to stable
and unstable periodic solutions respectively. Abbreviations depicted
following bifurcations: NS (Neimark-Sacker), PD (period doubling),
SB1 (pitchfork SB1) and SB2 (pitchfork SB2). Other changes of the
stability take place through the saddle-node bifurcations.}
\end{figure}
\begin{figure}
\begin{centering}
\includegraphics{fig6}
\par\end{centering}
\caption{\label{fig:1-parameter-continuation}1 parameter continuation of fully
synchronized: $3:1-3:1$ rotations ((a) mass $M$, (b) pendulum 1,
\foreignlanguage{polish}{$q=0.654$, $\mu=0.8$}), $4:1-1:1$ rotations
((c) mass $M$, (e) pendulum 1,\foreignlanguage{polish}{ $q=0.8$,
$\mu=0.7$}). The continuous and dashed lines correspond to stable
and unstable periodic solutions respectively. Abbreviations depicted
following bifurcations: NS (Neimark-Sacker), PD (period doubling),
SB1 (pitchfork SB1) and SB2 (pitchfork SB2). Other changes of the
stability take place through the saddle-node bifurcations.}
\end{figure}
\section{Conclusions}
In the system of two planar elastic pendula suspended on the excited
linear oscillator one can observe both in-phase and anti-phase synchronization
of the elastic pendula. In-phase and anti-phase synchronous states always co-exist.
Pendula can synchronize during the oscillatory and rotational motion
but only when their behaviour is periodic. We have not observed the
synchronization of the chaotically behaving pendula. This result is
contrary to the great number of chaos synchronization examples \cite{Stefanski2003,Stefanski2000,Maistrenko1997}
but confirms the results obtained in \cite{Czolczynski2007} where
it has been shown that the forced Duffing\textquoteright{}s oscillators
mounted to the elastic beam can synchronize only after motion become
periodic. The synchronization of the chaotic motion of the pendula
is impossible as the excited oscillator transfers the same signal
to both pendula which cannot differently modify the pendula\textquoteright{}s
motion. We also have not observed in-phase or anti-phase synchronization
of the pendula when masses $m_{2}$ and $m_{3}$ are in anti-phase. With one parameter bifurcation diagrams we present a bifurcational scenario of synchronous solutions. We show a route from synchronous via asynchronous periodic solutions to quasiperiodic and chaotic behaviour.
In this case the pendula in-phase or anti-phase synchronization is
impossible as the pendula have different and non-constant lengths. In this case one
can expect some kind of generalized synchronization but this problem
will be addressed elsewhere \cite{Kapitaniak}.
We show two dimensional bifurcation diagrams with the most representative
periodic solutions in the considered system. In the neighbourhood of
the linear resonances of subsystems we have rich dynamics with both
periodic and chaotic attractors\cite{ChudzikPSK11}. Our results are
robust as they exit in the wide range of system parameters, especially
two dimensional bifurcation diagram can be used as a scheme of bifurcations
in the class of systems similar to investigated in this paper.
\section*{Acknowledgement}
This work has been supported by the Foundation for Polish Science,
Team Programme (Project No TEAM/2010/5/5).
\bibliographystyle{elsarticle-num}
\nocite{*}
|
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Q: store images of different dimension in numpy array I have two images , image 1 of dimension (32,43,3) and image2 of dimension (67,86,3) . How can i store this in a numpy array , Whenever i try to append the array
image=cv2.imread(image1,0)
image=cv2.resize(image,(32,43))
x_train=np.array(image.flatten())
x_train=x_train.reshape(-1,3,32,43)
X_train =np.append(X_train,x_train) #X_train is my array
image=cv2.imread(image2,0)
image=cv2.resize(image,(67,86))
x_train=np.array(image.flatten())
x_train=x_train.reshape(-1,3,67,86)
X_train =np.append(X_train,x_train)
Value Error: total size of new array must be unchanged.
i want the X_train in shape (-1,depth,height,width).So that i can feed it into my neural network. Is there any way to store images of different dimension in array and feed into neural network ?
A: Don't use np.append. If you must join arrays, start with np.concatenate. It'll force you to pay more attention to the compatibility of dimensions.
You can't join 2 arrays with shapes (32,43,3) (67,86,3) to make a larger array of some compatible shape. The only dimension they share is the last.
These reshapes don't make sense either: (-1,3,32,43), (-1,3,67,86).
It works, but it also messes up the 'image'. You aren't just adding a 4th dimension. It looks like you want to do some axis swapping or transpose as well. Practice with some small arrays so you can see what's happening, e.g. (2,4,3).
What final shape do you expect for Xtrain?
You can put these two images in a object dtype array, which is basically the same as the list [image1, image2]. But I doubt if your neuralnet can do anything practical with that.
If you reshaped the (32,43,3) array to (16,86,3) you could concatenate that with (67,86,3) on axis=0 to produce a (83,86,3) array. If you needed the 3 to be first, I'd use np.transpose(..., (2,0,1)).
Conversely reshape (67,86,3) to (2*67,43,3).
Passing the (32,43,3) to (32,86,3) is another option.
Joining them on a new 4th dimension, requires that the number of 'rows' match as well as the number of 'columns'.
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{
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/siviltaram.github.io\/publication\/2020-04-11-you","text":"# You Impress Me: Dialogue Generation via Mutual Persona Perception\n\nPublished:\n\n[PAPER] [CODE] [SLIDES] [MEDIA]\n\n# Introduction\n\nDespite the continuing efforts to improve the engagingness and consistency of chit-chat dialogue systems, the majority of current work simply focus on mimicking human-like responses, leaving understudied the aspects of modeling understanding between interlocutors. The research in cognitive science, instead, suggests that understanding is an essential signal for a high-quality chit-chat conversation. Motivated by this, we propose P^2 Bot, a transmitter-receiver based framework with the aim of explicitly modeling understanding. Specifically, P^2 Bot incorporates mutual persona perception to enhance the quality of personalized dialogue generation. Experiments on a large public dataset, Persona-Chat, demonstrate the effectiveness of our approach, with a considerable boost over the state-of-the-art baselines across both automatic metrics and human evaluations.\n\n# Cite\n\n@inproceedings{liu-etal-2020-personachat,\ntitle = \"You Impress Me: Dialogue Generation via Mutual Persona Perception\",\nauthor = \"Liu, Qian and Chen, Yihong and Chen, Bei and Lou, Jian-Guang and Chen, Zixuan and Zhou, Bin and Zhang, Dongmei\",\nbooktitle = \"Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics\",\nyear = \"2020\",\npublisher = \"Association for Computational Linguistics\"\n}","date":"2020-08-04 04:55:13","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\": 0.4127446711063385, \"perplexity\": 10892.731697964962}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"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-2020-34\/segments\/1596439735860.28\/warc\/CC-MAIN-20200804043709-20200804073709-00491.warc.gz\"}"}
| null | null |
{"url":"https:\/\/tex.stackexchange.com\/questions\/433219\/including-math-environment-in-tick-labels-from-newcommand-with-tikz","text":"# Including math environment in tick labels from newcommand with tikz\n\nI am writing a book, and it will include many figures with the same x-axis. In order to keep each figure consistent across chapters I am trying to define a series of commands to typeset the axes. I very new to extending LaTeX, and even more new to pgf\/tikz.\n\nWhen I try to put symbols or math commands in the tick labels argument of my axis I get the following error:\n\nMissing \\endcsname inserted.\n\\begingroup\nl.14 ]\n\n\nHere is a MWE:\n\n\\documentclass{article}\n\\usepackage{pgfplots}\n\n\\newcommand{\\xpiaxis}[0]{\nxtick={0,1},%\nxticklabels={$0$,$\\frac{\\pi}{4}$}%\n}\n\\pgfplotsset{compat=1.14}\n\\begin{document}\n\\begin{figure}\n\\begin{tikzpicture}\n\\begin{axis}[%\n\\xpiaxis\n]\n\\end{axis}\n\\end{tikzpicture}\n\\end{figure}\n\\end{document}\n\n\ntry:\n\n\\documentclass{article}\n\\usepackage{pgfplots}\n\n\\pgfplotsset{xpiaxis\/.style={% instead of command define style ...\nxtick={0,1},%\nxticklabels={,$0$,$\\frac{\\pi}{4}$}%\n}\n}\n\\pgfplotsset{compat=1.14}\n\n\\begin{document}\n\\begin{figure}\n\\begin{tikzpicture}\n\\begin{axis}[\nxpiaxis % here commands are not alowed\n]\n\\end{axis}\n\\end{tikzpicture}\n\\end{figure}\n\\end{document}\n\n\n\u2022 For my own edification where can I use new commands vs your method? \u2013\u00a0dbjergaard May 24 '18 at 15:37\n\u2022 commands cannot be used as styles in tikz and pgfplots. anywhere else they should work as expected. even inside of tikz or pgfplots as element of picture (but not as style). \u2013\u00a0Zarko May 24 '18 at 17:48","date":"2019-05-23 19:48:22","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\": 0.9932152032852173, \"perplexity\": 2061.545588317525}, \"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-2019-22\/segments\/1558232257361.12\/warc\/CC-MAIN-20190523184048-20190523210048-00430.warc.gz\"}"}
| null | null |
\section{Introduction}
Leibniz algebras are a generalized version of Lie algebras, without the antisymmetry property. They were introduced by J.-L. Loday in 1993 and they turned out to be useful both in mathematics and physics. In \cite{FMM} the authors develop the versal deformation theory for Leibniz algebras. The existence of a versal deformation under certain cohomology condition follows from a general theorem of Schlessinger \cite{Sch}. The construction of a versal deformation is essential to solve the basic deformation question, as it is a deformation which induces all nonequivalent deformations of a given Leibniz algebra.
In this paper we give an explicit example on which we demonstrate the general construction and computations. For this, after recalling some definitions and results in Section \ref{lcohomology}, we describe and prove the relationship between Massey brackets and obstructions for Leibniz algebra deformations in Section \ref{Massey Brackets and Obstructions}.
Our example is the following.
Consider a three dimensional vector space $L$ spanned by $\{e_1,~e_2,~e_3\}$ over $\mathbb{C}$. Define a bilinear map $[~,~]: L\times L \longrightarrow L$ by $[e_1,e_3]=e_2$ and $[e_3,e_3]= e_1$, all other products of basis elements being $0$. Then $(L,[~,~])$ is a Leibniz algebra over $\mathbb{C}$ of dimension $3$. The Leibniz algebra $L$ is nilpotent and is denoted by $\lambda_6$ in the classification of three dimensional nilpotent Leibniz algebras, see \cite{A3}. We compute cohomologies necessary for our purpose, Massey brackets and construct a versal deformation of our example in Section \ref{computation}.
\section{Leibniz Algebra, Cohomology and Deformations} \label{lcohomology}
Leibniz algebras were introduced by J.L.-Loday \cite{L1,L3} and their cohomology was defined in \cite{LP,L2}. Let us recall some basic definitions. Let $\mathbb{K}$ be a field.
\begin{defn}
A Leibniz algebra is a $\mathbb{K}$-module $L$, equipped with a bracket operation that satisfies the Leibniz identity:
$$[x,[y,z]]= [[x,y],z]-[[x,z],y],~~\mbox{for}~x,~y,~z \in L.$$
\end{defn}
Any Lie algebra is automatically a Leibniz algebra, as in the presence of antisymmetry, the Jacobi identity is equivalent to the Leibniz identity. More examples of Leibniz algebras were given in \cite {L1,LP}, and recently for instance in \cite{A3,A1, A2}.
Let $L$ be a Leibniz algebra and $M$ a representation of $L$. By definition, $M$ is a $\mathbb{K}$-module equipped with two actions (left and right) of $L$,
$$[-,-]:L\times M\longrightarrow M~~\mbox{and}~[-,-]:M \times L \longrightarrow M ~~\mbox{such that}~$$
$$[x,[y,z]]=[[x,y],z]-[[x,z],y]$$
holds, whenever one of the variables is from $M$ and the two others from $L$.
Define $CL^n({L}; {M}):= \mbox{Hom} _\mathbb{K}({L}^{\otimes n}, {M}), ~n\geq 0.$ Let
$$\delta^n : CL^n({L}; {M})\longrightarrow CL^{n+1}(L; M)$$
be a $\mathbb{K}$-homomorphism defined by
\begin{equation*}
\begin{split}
&\delta^nf(x_1,\cdots,x_{n+1})\\
&:= [x_1,f(x_2,\cdots,x_{n+1})] + \sum_{i=2}^{n+1}(-1)^i[f(x_1,\cdots,\hat{x}_i,\cdots,x_{n+1}),x_i]\\
&+ \sum_{1\leq i<j\leq n+1}(-1)^{j+1}f(x_1,\cdots,x_{i-1},[x_i,x_j],x_{i+1},\cdots,\hat{x}_j,\cdots, x_{n+1}).
\end{split}
\end{equation*}
Then $(CL^*(L; M),\delta)$ is a cochain complex, whose cohomology is called the cohomology of the Leibniz algebra $L$ with coefficients in the representation $M$. The $n$ th cohomology is denoted by $HL^n(L; M)$. In particular, $L$ is a representation of itself with the obvious action given by the bracket in $L$. The $n$ th cohomology of $L$ with coefficients in itself is denoted by $HL^n(L; L).$
Let $S_n$ be the symmetric group of $n$ symbols. Recall that a permutation $\sigma \in S_{p+q}$ is called a $(p,q)$-shuffle, if $\sigma(1)<\sigma(2)<\cdots<\sigma(p)$, and $\sigma(p+1)<\sigma(p+2)<\cdots<\sigma(p+q)$. We denote the set of all $(p,q)$-shuffles in $S_{p+q}$ by $Sh(p,q)$.
For $\alpha \in CL^{p+1}(L;L)$ and $\beta \in CL^{q+1}(L;L)$, define $\alpha \circ \beta \in CL^{p+q+1}(L;L)$ by
\begin{equation*}
\begin{split}
&\alpha \circ \beta (x_1,\ldots,x_{p+q+1} )\\
=&~\sum_{k=1}^{p+1}(-1)^{q(k-1)}\{\sum_{\sigma \in Sh(q,p-k+1)}sgn(\sigma)\alpha(x_1,\ldots,x_{k-1},\beta(x_k,x_{\sigma(k+1)},\ldots,x_{\sigma(k+q)}),\\
&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~x_{\sigma(k+q+1)},\ldots,x_{\sigma(p+q+1)}) \}.
\end{split}
\end{equation*}
The graded cochain module
$CL^{*}(L;L)=\bigoplus_{p} CL^p(L;L)$ equipped with the bracket $\nu$ as defined by
$$[\alpha,\beta]=\alpha \circ \beta + (-1)^{pq+1} \beta \circ \alpha
~~\mbox{for}~ \alpha \in CL^{p+1}(L;L)~~\mbox{and}~\beta \in CL^{q+1}(L;L)$$
and the differential map $d$ by $d \alpha =(-1)^{|\alpha|}\delta \alpha~\mbox{for}~\alpha \in CL^{*}(L;L) $ is a differential graded Lie algebra \cite{B}.
Let now $\mathbb{K}$ a field of zero characteristic and the tensor product over $\mathbb{K}$ will be denoted by $\otimes$. We recall the notion of deformation of a Leibniz algebra $L$ over a local algebra base $A$ with a fixed augmentation $\varepsilon:{A}\rightarrow
\mathbb{K}$ and maximal ideal $\mathfrak{M}$. Assume $dim(\mathfrak{M}^k/\mathfrak{M}^{k+1})<\infty$ for every $k$ (see \cite{FMM}).
\begin{defn}
A deformation $\lambda$ of ${L}$ with base
$({A},\mathfrak{M})$, or simply with base ${A}$ is an ${A}$-Leibniz
algebra structure on the tensor product
${A}\otimes {L}$ with the bracket $[,]_\lambda$ such that
\[
\varepsilon\otimes id:{A}\otimes {L}\rightarrow \mathbb{K}\otimes {L}
\]
is a ${A}$-Leibniz algebra homomorphism (where the $A$-Leibniz algebra structure on $\mathbb{K}\otimes {L}$ is given via $\varepsilon$).
\end{defn}
A deformation of the Leibniz algebra $L$ with base $A$ is called {\it infinitesimal}, {or \it first order}, if in addition to this $\mathfrak{M}^2=0$. We call a deformation of {\it order k}, if $\mathfrak{M}^{k+1}=0$.
Suppose $A$ is a complete local algebra ( $A=\mathop{\varprojlim}\limits_{n\rightarrow
\infty}({A}/{\mathfrak{M}^n})$), where $\mathfrak{M}$ is the maximal
ideal in $A$. Then a deformation of $L$ with base $A$ which is obtained as the projective limit of deformations of $L$ with base $A/\mathfrak{M}^{n}$ is called a {\it formal deformation} of $L$.
Observe that for $l_1,l_2 \in L$ and $a,b \in A$ we have $$[a\otimes l_1,b\otimes l_2]_\lambda = ab[1\otimes l_1,1\otimes l_2]_\lambda$$ by $A$- linearity of $[,]_\lambda$.
Thus to define a deformation $\lambda$ it is enough to specify the brackets $[1\otimes l_1,1\otimes l_2]_\lambda$ for $l_1,l_2 \in L$.
Moreover, since $\varepsilon\otimes id:{A}\otimes {L}\rightarrow \mathbb{K}\otimes {L}$
is a ${A}$-Leibniz algebra homomorphism,
$$(\varepsilon\otimes id)[1\otimes l_1,1\otimes l_2]_\lambda =[l_1,l_2]=(\varepsilon\otimes id)(1\otimes[l_1,l_2])$$
which implies $$ [1\otimes l_1,1\otimes l_2]_\lambda -1\otimes[l_1,l_2] \in ker (\varepsilon \otimes id).$$
Hence we can write
$$[1\otimes l_1,1\otimes l_2]_\lambda =1\otimes[l_1,l_2]+\sum_{j} c_j \otimes y_j ,$$ where $\sum_{j} c_j \otimes y_j$ is a finite sum with $c_j \in ker(\varepsilon)=\mathfrak{M}$ and $y_j \in L$.
\begin{defn}
Suppose $\lambda_1$ and $\lambda_2$ are two deformations of a Leibniz algebra $L$ with finite dimensional local algebra base $A$. We call them equivalent if there exists a Leibniz algebra isomorphism $$ \phi:(A\otimes L,[,]_{\lambda_1})\rightarrow (A\otimes L,[,]_{\lambda_2})$$ such that $(\varepsilon\otimes id)\circ \phi=\varepsilon\otimes id$.
\end{defn}
The definition naturally generalizes to deformations complete local algebra base.
We write $\lambda_1 \cong \lambda_2$ if $\lambda_1$ is equivalent to $\lambda_2$.
\begin{Exm}
If $A= \mathbb{K}[[t]]$ then a formal deformation of a Leibniz algebra $L$ over $A$ is precisely a formal $1$-parameter deformation of $L$(see \cite{B}).
\end{Exm}
\begin{defn}
Suppose $\lambda$ is a given deformation of $L$ with base $(A,\mathfrak{M})$ and augmentation $\varepsilon:{A}\rightarrow \mathbb{K}$, where $A$ is a finite dimensional local algebra. Let $A^\prime$ be another commutative local algebra with identity and augmentation $\varepsilon^{\prime}:{A^\prime}\rightarrow \mathbb{K}$. Suppose $\phi:A \rightarrow A^{\prime} $ is an algebra homomorphism with $\phi(1)=1$ and $\varepsilon^{\prime} \circ \phi =\varepsilon$. Let $ker(\varepsilon^{\prime})= \mathfrak{M}^\prime$. Then the push-out $\bf{\phi_{*} \lambda}$ is the deformation of $L$ with base $(A^\prime,\mathfrak{M}^\prime)$ and bracket
$$[{a_1 }^\prime \otimes_A (a_1\otimes {l_1}),a_2 ^\prime
\otimes_A(a_2\otimes l_2) ]_{\phi_* \lambda}=a_1 ^\prime a_2 ^\prime
\otimes_A[a_1\otimes l_1,a_2\otimes l_2]_\lambda $$
where $a_1^\prime,a_2 ^\prime \in {A}^\prime,~ a_1,a_2 \in A$ and $l_1,l_2 \in
L$. Here $A^\prime$ is considered as an $A$-module by the map $a^\prime \cdot a=a^\prime \phi(a)$ so that $$A^\prime \otimes L=(A^\prime {\otimes}_{A} A)\otimes L =A^\prime {\otimes}_{A}(A \otimes L).$$
\end{defn}
The same definition holds for complete algebra base by taking projective limit.
\begin{rem}\label{push-out}
If the bracket $[,]_\lambda$ is given by
\begin{equation*
[1\otimes l_1,1\otimes l_2]_\lambda =1\otimes[l_1,l_2]+\sum_{j} c_j \otimes y_j~\mbox{for}~ c_j \in \mathfrak{M}~\mbox{and}~y_j \in L
\end{equation*}
then the bracket $[,]_{\phi_* \lambda}$ can be written as
\begin{equation*
[1\otimes l_1,1\otimes l_2]_ {\phi_* \lambda}=1\otimes [l_1,l_2]+\sum_{j}\phi(c_j) \otimes y_j.
\end{equation*}
\end{rem}
Let us recall the construction of a specific infinitesimal deformation of a Leibniz algebra $L$, which is universal in the class of all infinitesimal deformations from \cite{FMM}.
Assume that $dim (HL^2(L;L)) < \infty$. Denote the space $HL^2(L;L)$ by $\mathbb{H}$. Consider the algebra $C_1=\mathbb{K}\oplus \mathbb{H}^\prime$ where $\mathbb{H}^\prime$ is the dual of $\mathbb{H}$~, by setting $$(k_1,h_1)\cdot(k_2,h_2)=(k_1 k_2,k_1 h_2+k_2 h_1)~\mbox{for}~(k_1,h_1), (k_2,h_2)\in C_1.$$ Observe that the second summand is an ideal of $C_1$ with zero multiplication. Fix a homomorphism $$\mu: \mathbb{H} \longrightarrow
CL^2(L;L)=Hom(L^{\otimes 2};L) $$
which takes a cohomology class into a cocycle representing it. Notice that there is an isomorphism $\mathbb{H}^\prime \otimes L \cong Hom(\mathbb{H}~;L)$,
so we have
$$C_1 \otimes L
=(\mathbb{K} \oplus \mathbb{H}^\prime)\otimes L
\cong (\mathbb{K}\otimes L) \oplus (\mathbb{H}^\prime \otimes L)
\cong L \oplus Hom(\mathbb{H}~;L).$$
Using the above identification, define a Leibniz bracket on $C_1 \otimes L$ as follows.
For $(l_1,\phi_1),(l_2,\phi_2) \in L \oplus Hom(\mathbb{H}~;L)$ let
$$[(l_1,\phi_1),(l_2,\phi_2)]=([l_1,l_2],\psi)$$ where the map $\psi:\mathbb{H} \longrightarrow L$ is given by
$$\psi(\alpha)=\mu(\alpha)(l_1,l_2)+[\phi_{1}(\alpha),l_2]+[l_1,\phi_2(\alpha)]~\mbox{for}~\alpha \in \mathbb{H}~.$$
It is straightforward to check that $C_1\otimes L$ along with the above bracket is a Leibniz algebra over $C_1$. The Leibniz identity is a consequence of the fact that $\delta \mu(\alpha)=0~ \mbox{for}~ \alpha \in \mathbb{H}$~. Thus $\eta_1$ is an infinitesimal deformation of $L$ with base $C_1=\mathbb{K}\oplus \mathbb{H}^\prime $. It is proved in \cite{FMM}:
\begin{prop}\label{up toisomorphism}
Up to an isomorphism, the deformation $\eta_1$ does not depend on
the choice of $\mu$.
\end{prop}
\begin{rem}\label{exp of inf}
Suppose $\{h_i \}_{1\leq i \leq n}$ is a basis of $\mathbb{H}$ and $\{g_i\}_{1\leq i \leq n}$ is the dual basis. Let $\mu(h_i)=\mu_i \in CL^2(L;L)$. Under the identification $C_1 \otimes L = L \oplus Hom(\mathbb{H}~;L)$, an element $(l,\phi)\in L \oplus Hom(\mathbb{H}~;L)$ corresponds to $1\otimes l +\sum_{i=1}^{n}{g_i\otimes \phi(h_i)}$.
Then for $(l_1,\phi_1),(l_2,\phi_2) \in L \oplus Hom(\mathbb{H};L)$ their bracket $([l_1,l_2],\psi)$
corresponds to
$$1\otimes [l_1,l_2]+ \sum_{i=1}^{n} g_i\otimes (\mu_i(l_1,l_2)+[\phi_1(h_i),l_2]+[l_1,\phi_2(h_i)]).$$
In particular, for $l_1,l_2 \in L$ we have
$$[1\otimes l_1,1\otimes l_2]_{\eta_1}=1\otimes [l_1,l_2]+\sum_{i=1}^{n}g_i \otimes \mu_i(l_1,l_2).$$
\end{rem}
The main property of $\eta_{1}$ is the universality in the class of infinitesimal deformations with a finite dimensional base.
\begin{prop}\label{couniversal}
For any infinitesimal deformation $\lambda$ of a Leibniz algebra
$L$ with a finite dimensional base $A$ there exists a unique
homomorphism $\phi:C_1=({\mathbb{K}}\oplus
\mathbb{H}^\prime)\longrightarrow A$ such that $\lambda$ is
equivalent to the push-out $\phi_{*}\eta_1$.
\end{prop}
Suppose $A$ is a local algebra with the unique maximal ideal $\mathfrak{M}$ and $\pi:A\rightarrow A/{\mathfrak{M}^{2}}$ the corresponding quotient map. The algebra $A/{\mathfrak{M}^{2}}$ is obviously local with maximal ideal ${\mathfrak{M}}/{\mathfrak{M}^{2}}$ and $({\mathfrak{M}}/{\mathfrak{M}^{2}})^{2}=0$. If $\lambda$ is a deformation of $L$ with base $A$ then $\pi_{*} \lambda$ is a deformation with base $A/{\mathfrak{M}^{2}}$ and it is clearly infinitesimal. Therefore, by the previous proposition, we have a map
$$a_{\pi *\lambda}:({\mathfrak{M}}/{\mathfrak{M}^{2}})^\prime \rightarrow \mathbb{H}~. $$
\begin{defn}
The dual space $({\mathfrak{M}}/{\mathfrak{M}^{2}})^\prime$ is called the tangent space of A and is denoted by $TA$. The map $a_{\pi *\lambda}$ is called the differential of $\lambda$ and is denoted by $d{\lambda}$.
\end{defn}
It follows from Proposition \ref{couniversal} that equivalent deformations have the same differential (see \cite{FMM}).
\begin{defn}
Let $C$ be a complete local algebra. A formal deformation $\eta$ of a Leibniz algebra $L$ with base $C$ is called versal, if\\
(i)~for any formal deformation $\lambda$ of $L$ with base $A$ there exists a homomorphism $f:C \rightarrow A$ such that the deformation $\lambda$ is equivalent to $f_{*}\eta$; \\
(ii)~if $A$ satisfies the condition ${\mathfrak{M}}^2=0$, then $f$ is unique.
\end{defn}
In \cite{FMM} a construction for a versal deformation of a Leibniz algebra was given. The construction involves realizing obstructions to extend a deformation with base $A$ to a deformation with base $B$ for a given extension
$$0\longrightarrow
{M}\stackrel{i}{\longrightarrow} B
\stackrel{p}{\longrightarrow}A\longrightarrow 0.$$
Suppose a deformation $\lambda$ of $L$ is given with base $A$. If we try to extend it to a deformation with base $B$, it gives rise to a cohomology class in
$$ HL^3(L;M\otimes L)=M\otimes HL^3(L;L).$$
The above assignment yields the {\it obstruction map} for this extension
$$ \theta_{\lambda}:H_{Harr}^2(A;M) \longrightarrow M \otimes HL^3(L;L),~(\mbox{see}~\cite{FMM}).~$$
(Here $H_{Harr}^2(C_k;\mathbb{K})$ denotes the two dimensional Harrison cohomology space.)
Let us recall the main steps of the construction.
Consider the Leibniz algebra $L$ with $dim(\mathbb{H})<\infty$ and the extension
$$0\longrightarrow
\mathbb{H}^\prime \stackrel{i}{\longrightarrow} C_1
\stackrel{p}{\longrightarrow} C_0 \longrightarrow 0,$$
where $C_0=\mathbb{K}$ and
$C_1=\mathbb{K}\oplus \mathbb{H}^\prime$ as before.
Let $\eta_1$ be the universal infinitesimal deformation with base $C_1$.
We proceed by induction. Suppose for some $k\geq 1$ we have constructed a finite dimensional local algebra $C_k$ and a deformation $\eta_k$ of $L$ with base $C_k$.
Let
$$\mu:H_{Harr}^2(C_k;\mathbb{K})\longrightarrow (Ch_{2} (C_k))^\prime$$
be a homomorphism sending a cohomology class to a cocycle representing the class. Let
$$f_{C_k}:Ch_{2} (C_k) \longrightarrow H_{Harr}^2(C_k;\mathbb{K})^\prime$$
be the dual of $\mu$. Then we have the following extension of $C_k$:
\begin{equation}\label{universal extension}
0\longrightarrow
H_{Harr}^2(C_k;\mathbb{K})^\prime \stackrel{\bar {i}_{k+1}}{\longrightarrow} {\bar C}_{k+1}
\stackrel{\bar{p}_{k+1}}{\longrightarrow} C_k \longrightarrow 0.
\end{equation}
The corresponding {\it obstruction} $\theta_{\eta_{k}}([f_{C_k}]) \in H_{Harr}^2(C_k;\mathbb{K})^\prime \otimes HL^3(L;L)$ gives a linear map
$\omega_k:H_{Harr}^2(C_k;\mathbb{K}) \longrightarrow HL^3(L;L)$
with the dual map
$${\omega_k}^\prime:HL^3(L;L)^\prime \longrightarrow H_{Harr}^2(C_k;\mathbb{K})^\prime .$$
We have an induced extension
$$ 0\longrightarrow coker (\omega'_{k})\longrightarrow \bar{C}_{k+1}/\bar{i}_{k+1}\circ \omega'_{k}(HL^3(L;L)')\longrightarrow C_k \longrightarrow 0.$$
Since $coker (\omega'_k)\cong (ker (\omega_k))^\prime$,
it yields an extension
\begin{equation}\label{yields an extension}
0\longrightarrow (ker(\omega_k))^\prime \stackrel{i_{k+1}}{\longrightarrow} C_{k+1}
\stackrel{p_{k+1}}{\longrightarrow} C_k \longrightarrow 0
\end{equation}
where $C_{k+1}= { \bar{C}_{k+1}}/{\bar{i}_{k+1}\circ~ \omega_{k}^\prime (HL^3(L;L)')}$ and $i_{k+1}$, $p_{k+1}$ are the mappings induced by $\bar{i}_{k+1}$ and $\bar{p}_{k+1}$, respectively.
It turns out that the obstruction associated to the extension (\ref{yields an extension}) is $\omega|_{ker(\omega_k)}$.
As a consequence it is proved in \cite{FMM}
\begin{prop}
The deformation $\eta_k$ with base $C_{k}$ of a Leibniz algebra $L$ admits an extension to a deformation with base $C_{k+1}$, which is unique up to an isomorphism and an automorphism of the extension
$$0\longrightarrow
(ker(\omega_k))^\prime \stackrel{i_{k+1}}{\longrightarrow} C_{k+1}
\stackrel{p_{k+1}}{\longrightarrow} C_k \longrightarrow 0.$$
\end{prop}
By induction, the above process yields a sequence of finite dimensional local algebras $C_{k}$ and deformations $\eta_{k}$ of the Leibniz algebra $L$ with base $C_{k}$
$$ \mathbb{K} \stackrel{p_{1}}{\longleftarrow} C_{1} \stackrel{p_{2}}{\longleftarrow} C_{2}\stackrel{p_{3}}{\longleftarrow} \ldots \ldots \stackrel{p_{k}}{\longleftarrow} C_{k}\stackrel{p_{k+1}}{\longleftarrow} C_{k+1}\ldots$$
such that $ {p_{k+1}}_{*} \eta_{k+1}=\eta_{k}$.
Thus by taking the projective limit we obtain a formal deformation $\eta$ of $L$ with base $C=\mathop{\varprojlim}\limits_{k\rightarrow
\infty} C_{k}$.
\section{Massey Brackets and Obstructions}\label{Massey Brackets and Obstructions}
After constructing the universal infinitesimal deformation, one would like to extend it to higher order deformation. For this we need to compute obstructions. The standard procedure is to relate obstructions to Massey brackets. The connection between these two notions was first noticed in \cite{D}. A general approach to treat Massey brackets is given in \cite{FuL}. This approach is used to establish connection between Massey brackets and obstructions arising from Lie algebra deformations.
The aim of this section is to apply results in \cite{FuL} to relate Massey brackets to obstructions in the deformation of Leibniz algebras. A special case of the general definition is an inductive definition of Retakh (\cite{R, FuL}) which is useful for computational purposes.
Suppose $(\mathcal{L},\nu,d)$ is a differential graded Lie algebra. We denote by $\mathcal{H}= \bigoplus_i \mathcal{H}^i$, the cohomology of $\mathcal{L}$ with respect to the differential $d$.
Let $F$ be a graded cocommutative coassociative coalgebra, that is a graded vector space with a degree $0$ mapping (comultiplication) $\Delta:F\longrightarrow F\otimes F$ satisfying the conditions $S\circ \Delta=\Delta $ and $(1\otimes \Delta)\circ \Delta= (\Delta \otimes 1)\circ \Delta$, where
$$S:F\otimes F\longrightarrow F\otimes F $$ is defined as $$S(\phi \otimes \psi)=(-1)^{|\phi||\psi|}(\psi \otimes \phi).$$
Suppose also that a filtration $F_0 \subset F_1\subset F$ is given in $F$, such that $F_0 \subset ker (\Delta)$ and $ Im (\Delta) \subset F_1\otimes F_1$. We need the following result (see \cite{FuL}.)
\begin{prop}\label{horizontal}
Suppose a linear mapping $\alpha:F_1 \longrightarrow \mathcal{L}$ of degree $1$ satisfies the condition
\begin{equation}\label{for Massey brackets}
d \alpha =\nu \circ (\alpha \otimes \alpha)\circ \Delta.
\end{equation}
Then $\nu \circ (\alpha \otimes \alpha)\circ \Delta (F)\subset ker (d) $.
\end{prop}
\begin{defn}\label{definition of Massey Bracket}
Let $a:F_0\longrightarrow \mathcal{H}$, $b:F/F_1\longrightarrow \mathcal{H}$ be two linear maps of degree $1$. We say that $b$ is contained in the Massey $F$-bracket of $a$, and write $b \in [a]_F$, or $b \in [a]$, if there exists a degree $1$ linear mapping $\alpha:F_1 \longrightarrow \mathcal{L}$ satisfying condition (\ref{for Massey brackets}) and such that the following diagrams are commutative, where the vertical maps labeled by $\pi$ denote the projections of each space onto the quotient space.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=8.5cm]{masseyF1.epsi}
\end{center}
\caption{}
\end{figure}
\end{defn}
Note that the upper horizontal maps of the above diagrams are well defined, since $\alpha(F_0)\subset \alpha(ker \Delta)\subset ker (d)$ by virtue of (\ref{for Massey brackets}), and $\nu \circ (\alpha \otimes \alpha)\circ \Delta(F)\subset ker (d)$ by Proposition \ref{horizontal}.
The definition makes sense even if $F_1=F$. In that case $Hom (F/F_1,~\mathbb{K})=0$, and $[a]_F$ may either be empty or contain 0. In that case we say that $a ~ satisfies~ the~ condition ~of~ triviality~ of~ Massey ~F\mbox{-}brackets$.
Let $A$ be a complete local algebra with $1$ and augmentation $\varepsilon$. Let $\mathfrak{M}=ker(\varepsilon)$.
Let $\rho: (A \otimes L)\times (A \otimes L)\longrightarrow (A \otimes L)$ be a $A$-bilinear operation on $A \otimes L$ ($\rho$ need not satisfy the Leibniz identity) such that $\varepsilon \otimes id :A \otimes L \longrightarrow L$ is a homomorphism with respect to the operation $\rho$ on $A \otimes L$ and the usual bracket operation on $L$ in other words,
$$(\varepsilon \otimes id )\circ \rho(a_1 \otimes l_1, a_2 \otimes l_2)= \varepsilon (a_1 a_2) [l_1,l_2].$$
Note that for $1\otimes l_1,~1\otimes l_2 \in A \otimes L$ we have
$$(\varepsilon \otimes id)\circ \rho(1\otimes l_1,1\otimes l_2)= \varepsilon(1)[l_1,l_2]=\varepsilon \otimes id (1 \otimes [l_1,l_2])$$
Therefore \begin{equation}\label{rho}
\rho(1\otimes l_1,1 \otimes l_2)-1 \otimes [l_1,l_2] \in ker (\varepsilon \otimes id )=ker (\varepsilon) \otimes L=\mathfrak{M}\otimes L.
\end{equation}
We consider the differential graded Lie algebra $(CL^{*}(L;L),\nu, d)$. Let $F=F_1=\mathfrak{M}'$, the dual of $\mathfrak{M}$ and $F_0=(\mathfrak{M}/\mathfrak{M}^2)^\prime$. Let $\Delta: F \longrightarrow F\otimes F$ be the comultiplication in $F$ which is the dual of the multiplication in $\mathfrak{M}$. Then $F$ is a cocommutative coassociative coalgebra.
For a linear functional $\phi:{\mathfrak{M}}\longrightarrow \mathbb{K}$ define a map $\alpha_{\phi}:L\otimes L \longrightarrow L$ by
$$ \alpha_{\phi}(l_1,l_2)= (\phi \otimes id)(\rho(1\otimes l_1,1 \otimes l_2)-1 \otimes [l_1,l_2] ).$$
This gives $\alpha:{\mathfrak{M}}'\longrightarrow CL^2(L;L)$ by $\phi \mapsto \alpha_{\phi}$. From the definition it is clear that $\rho$ and $\alpha$ determine each other.
Then we have
\begin{prop}\label{Leibniz identity iff}
The operation $\rho$ satisfies the Leibniz identity if and only if $\alpha$ satisfies the equation $d\alpha -\frac{1}{2}\nu\circ(\alpha\otimes \alpha)\circ \Delta=0$.
\end{prop}
\begin{proof}
Let $\{m_i\}$ be a basis of $\mathfrak{M}$. Using (\ref{rho}) we can write $$\rho(1\otimes l_1,1 \otimes l_2)=1 \otimes [l_1,l_2]+\sum_{i}m_i\otimes \psi_i(l_1,l_2)$$ where $\psi_i \in CL^2(L;L)$ is given by $\psi_i=\alpha_{m^\prime_i}$.
\begin{equation*}
\begin{split}
\mbox{Thus}~~&\rho(1\otimes l_1 ,\rho(1\otimes l_2,1\otimes l_3))\\
=&~\rho(1 \otimes l_1,1\otimes [l_2,l_3]+\sum_{i}m_i\otimes \psi_i(l_2,l_3))\\
=&~\rho(1\otimes l_1,1\otimes[l_2,l_3])+\sum_{i}m_i\rho(1\otimes l_1,1\otimes \psi_i(l_2,l_3))\\
=&~1\otimes [l_1,[l_2,l_3]]+\sum_{i}m_i \otimes \psi_i(l_1,[l_2,l_3])+\sum_{i}m_i \otimes [l_1,\psi_i(l_2,l_3)]\\
&~~~+\sum_{i,j}m_im_j \otimes \psi_j(l_1,\psi_i(l_2,l_3)).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{Similarly}~~ &\rho(\rho(1\otimes l_1,1\otimes l_2),1\otimes l_3)\\
=&~1\otimes [[l_1,l_2],l_3]
+\sum_{i}m_i \otimes \psi_i([l_1,l_2],l_3)+\sum_{i}m_i \otimes [\psi_i(l_1,l_2),l_3]\\
&~~~+\sum_{i,j}m_im_j \otimes \psi_j(\psi_i(l_1,l_2),l_3)
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{and}~~&\rho(\rho(1\otimes l_1,1\otimes l_3),1\otimes l_2)\\
=&~1\otimes [[l_1,l_3],l_2]
+\sum_{i}m_i \otimes \psi_i([l_1,l_3],l_2)+\sum_{i}m_i \otimes [\psi_i(l_1,l_3),l_2]\\
&~~~+\sum_{i,j}m_im_j \otimes \psi_j(\psi_i(l_1,l_3),l_2).
\end{split}
\end{equation*}
For any linear functional $\phi:\mathfrak{M} \longrightarrow \mathbb{K}$, let $\phi(m_i)=x_i \in \mathbb{K}$. Then by (\ref{rho})
\begin{equation*}
\begin{split}
\alpha_{\phi}(l_1,l_2)=&(\phi \otimes id)(\sum_{i}m_i\otimes \psi_i(l_1,l_2))\\
=&\sum_{i}x_i \otimes \psi_i(l_1,l_2)\\
=&1\otimes (\sum_{i}x_i \psi_i)(l_1,l_2).
\end{split}
\end{equation*}
So, $\alpha_{\phi}$ can be expressed as $\sum_{i}x_i \psi_i$.
Let $\Delta(\phi)=\sum_{p}\xi_p \otimes \eta_p~~~\mbox{for some}~\xi_p,\eta_p \in \mathfrak{M}'$. We set $\xi_p(m_i)=\xi_{p,i}~~\mbox{and}~\eta_p(m_i)=\eta_{p,i}$. Thus
\begin{equation*}
\begin{split}
\phi(m_i~ m_j)=&~\Delta(\phi)(m_i\otimes m_j)
=(\sum_{p}\xi_p\otimes \eta_p)(m_i\otimes m_j)
=\sum_{p}\xi_{p,i}~\eta_{p,j}.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{Now}~~~&(\phi \otimes id)( \sum_{i,j}m_im_j \otimes \psi_j(l_1,\psi_i(l_2,l_3))\\
=&~\sum_{i,j,p}\xi_{p,i}~\eta_{p,j}~\psi_j(l_1,\psi_i(l_2,l_3))\\
=&~\sum_{p}(\sum_{i}\xi_{p,i}(\sum_{j}\eta_{p,j}\psi_j(l_1,\psi_i(l_2,l_3))))\\
=&~\sum_{p}(\sum_{i}\xi_{p,i}\alpha_{\eta_p}(l_1,\psi_i(l_2,l_3)))\\
=&~\sum_{p}\alpha_{\eta_p}(l_1,\sum_i\xi_{p,i}\psi_i(l_2,l_3))\\
=&~\sum_{p}\alpha_{\eta_p}(l_1,\alpha_{\xi_p}(l_2,l_3)).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{Therefore}~~&(\phi \otimes id)( \rho(1\otimes l_1 ,\rho(1\otimes l_2,1\otimes l_3)))\\
=&~ \sum_{i}\phi(m_i)\otimes \psi_i(l_1,[l_2,l_3])+\sum_{i}\phi(m_i)\otimes [l_1,\psi_i(l_2,l_3)] \\
&~~~+\sum_{p}\alpha_{\eta_p}(l_1,\alpha_{\xi_p}(l_2,l_3))\\ =&~\alpha_{\phi}(l_1,[l_2,l_3])+[l_1,\alpha_{\phi}(l_2,l_3)]
+\sum_{p}\alpha_{\eta_p}(l_1,\alpha_{\xi_p}(l_2,l_3)).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{Similarly}~~ &(\phi \otimes id)(\rho(\rho(1\otimes l_1,1\otimes l_2),1\otimes l_3))\\
=&~\alpha_{\phi}([l_1,l_2],l_3)+[\alpha_{\phi}(l_1,l_2),l_3]
+\sum_{p}\alpha_{\eta_p}(\alpha_{\xi_p}(l_1,l_2),l_3).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{and}~~&(\phi \otimes id)(\rho(\rho(1\otimes l_1,1\otimes l_3),1\otimes l_2))\\
=&~\alpha_{\phi}([l_1,l_3],l_2)+[\alpha_{\phi}(l_1,l_3),l_2]
+\sum_{p}\alpha_{\eta_p}(\alpha_{\xi_p}(l_1,l_3),l_2).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{Hence we get,}~&(\phi \otimes id)(\rho(1\otimes l_1 ,\rho(1\otimes l_2,1\otimes l_3))-\rho(\rho(1\otimes l_1, 1\otimes l_2),1\otimes l_3 )\\
&~~~+\rho(\rho(1\otimes l_1, 1\otimes l_3),1\otimes l_2 ))\\
=&~\delta \alpha_{\phi}(l_1,l_2,l_3)+\frac{1}{2}\sum_{p}[\alpha_{\eta_p},\alpha_{\xi_p}](l_1,l_2,l_3)\\
=&~(-d\alpha +\frac{1}{2}\nu\circ(\alpha\otimes \alpha)\circ \Delta )\phi(l_1,l_2,l_3).
\end{split}
\end{equation*}
Thus it follows that $\rho$ satisfies the Leibniz identity if and only if $\alpha$ satisfies the equation $d\alpha -\frac{1}{2}\nu \circ(\alpha\otimes \alpha)\circ \Delta=0$.
\end{proof}
It follows from Proposition \ref{Leibniz identity iff} that for a deformation $\rho$ of $L$, $\alpha(F_0)\subset ker(d)$ as $F_0 \subset ker(\Delta)$.
Let $a$ denote the composition
$$a:F_0 \stackrel{\alpha}\longrightarrow ker(d) \stackrel{\pi}\longrightarrow \mathbb{H}~~\mbox{where}~\mathbb{H}=HL^2(L;L).$$
Then the following is a consequence of Proposition \ref{Leibniz identity iff} and definition of Massey $F$ bracket.
\begin{cor}
A linear map $a:F_0 \longrightarrow \mathbb{H}$ is a differential of some deformation with base $A$ if and only if $\frac{1}{2}a$ satisfies the condition of triviality of Massey $F$-brackets.
\end{cor}
Next we relate the obstruction $\omega_k$ at the $k$th stage in the construction of versal deformation to Massey brackets.
Consider the sequence of finite dimensional local algebras $C_{k}$ with maximal ideals $\mathfrak{M}_k$ and deformations $\eta_{k}$ of the Leibniz algebra $L$ with base $C_{k}$ yielding an inverse system
$$ \mathbb{K} \stackrel{p_{1}}{\longleftarrow} C_{1} \stackrel{p_{2}}{\longleftarrow} C_{2}\stackrel{p_{3}}{\longleftarrow} \ldots \ldots \stackrel{p_{k}}{\longleftarrow} C_{k}\stackrel{p_{k+1}}{\longleftarrow} C_{k+1}\ldots$$
$$\mbox{where}~~ {p_{k+1}}_{*} \eta_{k+1}=\eta_{k}.$$
Taking the dual we get the direct system
$$ \mathbb{K} \stackrel{p' _{1}}{\longrightarrow} C'_{1} \stackrel{p' _{2}}{\longrightarrow} C'_{2}\stackrel{p' _{3}}{\longrightarrow} \ldots \ldots \stackrel{p' _{k}}{\longrightarrow} C'_{k}\stackrel{p'_{k+1}}{\longrightarrow} C'_{k+1}\ldots.$$
Also, by considering the maximal ideals $\mathfrak{M}_k$ we get another system
$$ \mathbb{K} \stackrel{p' _{1}}{\longrightarrow} \mathfrak{M}'_{1} \stackrel{p' _{2}}{\longrightarrow} \mathfrak{M}'_{2}\stackrel{p' _{3}}{\longrightarrow} \ldots \ldots \stackrel{p' _{k}}{\longrightarrow} \mathfrak{M}'_{k}\stackrel{p'_{k+1}}{\longrightarrow} \mathfrak{M}'_{k+1}\ldots$$
where each $p'_k$ is injective.
In the induction process we get an extension of $C_k$ given by
$$0\longrightarrow
H_{Harr}^2(C_k;\mathbb{K})^\prime \stackrel{\bar {i}_{k+1}}{\longrightarrow} {\bar C}_{k+1}
\stackrel{\bar{p}_{k+1}}{\longrightarrow} C_k \longrightarrow 0$$
where the obstruction for extending $\eta_k$ to a deformation of $L$ with base $\bar{C}_{k+1}$ is given by
$\omega_k:H_{Harr}^2(C_k;\mathbb{K}) \longrightarrow HL^3(L;L).$
To make this obstruction zero we consider $$C_{k+1}= { \bar{C}_{k+1}}/{\bar{i}_{k+1}\circ~ \omega_{k}^\prime (HL^3(L;L))} .$$
Let $F=(\bar {\mathfrak{M}}_{k+1})';~F_1=\mathfrak{M}'_k~~\mbox{and}~F_0=\mathfrak{M}'_{1}=\mathbb{H}$.\\
Thus $F/F_1=H^2_{Harr}(C_k;\mathbb{K})$ and $\omega_k$ can be viewed as a map
$$\omega_k:F/F_1\longrightarrow HL^3(L;L).$$
\begin{thm}\label{obs at each stage}
The obstruction $\omega_k$ has the property,
$2\omega_k \in [id]_F$. Moreover, an arbitrary element of $[id]_F$ is equal to $2\omega_k$ for an appropriate extension of the deformation $\eta_1$ of $L$ with base $C_1$ to a deformation $\eta_k$ of $L$ with base $C_k$.
\end{thm}
\begin{proof}
As before we define a map
$$ \alpha: \mathfrak{M}'_k\longrightarrow CL^2(L;L)$$
by $\alpha_{\phi}(l_1,l_2)=(\phi\otimes id)([1\otimes l_1,1\otimes l_2]_{\eta_k}-1\otimes [l_1,l_2])
~~\mbox{for}~\phi \in \mathfrak{M}'_k~~\mbox{and}~l_1,l_2 \in L$, using the deformation $\eta_k$ with base $C_k$.
Since $\eta_k$ is a Leibniz algebra structure on $C_k\otimes L$, Proposition \ref{Leibniz identity iff} implies $d\alpha=\frac{1}{2}\nu \circ(\alpha\otimes \alpha)\circ\Delta$. It is clear that different $\alpha$ with these properties corresponds to different extensions $\eta_k$ of $\eta_1$.
Observe that $\alpha|_{F_0}:F_0\longrightarrow CL^2(L;L)$ is given by $\alpha|_{F_0}(h_i)=\mu(h_i)$, a representative of the cohomology class $h_i$. So
$$\alpha|_{F_0}:F_0\longrightarrow CL^2(L;L)\stackrel{\pi}\longrightarrow \mathbb{H}$$
gives $a:F_0\longrightarrow \mathbb{H}$, the identity map.\\
In the definition of Massey $F$-bracket, the map $b:F/F_1\longrightarrow HL^3(L;L)$ is represented by the map $\nu \circ(\alpha\otimes \alpha)\circ \Delta:F\longrightarrow CL^3(L;L)$. In our case the obstruction is given by $\omega_k:H_{Harr}^2(C_k;\mathbb{K}) \longrightarrow HL^3(L;L)$.
Consider a basis $\{m_i\}_{1\leq i \leq r}$ of $\mathfrak{M}_k$ and extend it to a basis $\{\bar{m}_i \}_{1\leq i \leq r+s}$ of $\bar{\mathfrak{M}}_{k+1}$. Now we can write $$[1\otimes l_1,1\otimes l_2]_{\eta_k}=1\otimes [l_1,l_2]+\sum_{i=1}^r m_i \otimes \psi_i(l_1,l_2) .$$
Then by definition of $\alpha$ we have $\alpha(m'_i)(l_1,l_2)=\psi_i(l_1,l_2)~~\mbox{for}~i\geq r$.\\
For arbitrary cochains $\psi_i \in CL^2(L;L)~~\mbox{for}~r+1 \leq i\leq s$ the $\bar{C}_{k+1}$-bilinear map $\{,\}$ on $\bar{C}_{k+1}\otimes L$ is given by
$$\{1\otimes l_1,1\otimes l_2\}=1\otimes[l_1,l_2]+\sum_{i=1}^{r+s} \bar{m}_i \otimes \psi_i(l_1,l_2).$$
Let the multiplication in $\bar{\mathfrak{M}}_{k+1}$ be defined (on the basis) as $$\bar{m}_i~\bar{m}_j=\sum_{p=1}^{r+s}c_{i~j}^p \bar{m}_p.$$
Then $\Delta:(\bar{\mathfrak{M}}_{k+1})^\prime \longrightarrow \mathfrak{M}_{k}^\prime \otimes \mathfrak{M}_k ^\prime$ is given by $\Delta(\bar{m}'_{p})=\sum_{i,j=1}^{s}c_{ij}^p m'_i \otimes m'_j$.
Now
\begin{equation*}
\begin{split}
&\{\{1\otimes l_1,1\otimes l_2 \},1\otimes l_3\}\\
=&~\{1\otimes [l_1,l_2]+\sum_{i=1}^{r+s}\bar{m}_i \otimes \psi_i(l_1,l_2),1\otimes l_3\}\\
=&~1\otimes [[l_1,l_2],l_3]+\sum_{i=1}^{r+s}\bar{m}_i\otimes \psi_i ([l_1,l_2],l_3)
+\sum_{i=1}^{r+s}\bar{m}_i\otimes [\psi_i(l_1,l_2),l_3]\\
&~~~+\sum_{i,j=1}^{r}\bar{m}_j \bar{m}_i \otimes \psi_j(\psi_i(l_1,l_2),l_3)\\
=&~1\otimes [[l_1,l_2],l_3]+\sum_{i=1}^{r+s}\bar{m}_i\otimes \psi_i ([l_1,l_2],l_3)
+\sum_{i=1}^{r+s}\bar{m}_i\otimes [\psi_i(l_1,l_2),l_3]\\
&~~~+\sum_{i,j=1}^{r}\sum_{p=1}^{r+s}c_{ij}^p \bar{m}_p \otimes \psi_j(\psi_i(l_1,l_2),l_3).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{Similarly}~~&\{\{1\otimes l_1,1\otimes l_3 \},1\otimes l_2\}\\
=&~1\otimes [[l_1,l_3],l_2]+\sum_{i=1}^{r+s}\bar{m}_i\otimes \psi_i ([l_1,l_3],l_2)
+\sum_{i=1}^{r+s}\bar{m}_i\otimes [\psi_i(l_1,l_3),l_2]\\
&~~~+\sum_{i,j=1}^{r}\sum_{p=1}^{r+s}c_{ij}^p \bar{m}_p \otimes \psi_j(\psi_i(l_1,l_3),l_2)
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{and}~~&\{1\otimes l_1,\{1\otimes l_2,1\otimes l_3\}\}\\
=&~1\otimes [l_1,[l_2,l_3]]+\sum_{i=1}^{r+s}\bar{m}_i\otimes \psi_i(l_1,[l_2,l_3])
+\sum_{i=1}^{r+s}\bar{m}_i\otimes [l_1,\psi_i(l_2,l_3)]\\
&~~~+ \sum_{i,j=1}^{r}\sum_{p=1}^{r+s}c_{ij}^p \bar{m}_p \otimes \psi_j(l_1,\psi_i(l_2,l_3)).
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\mbox{Therefore}~~&(\bar{m}'_p \otimes id)(\{1\otimes l_1,\{1\otimes l_2,1\otimes l_3\}\}-\{\{1\otimes l_1,1\otimes l_2 \},1\otimes l_3\}\\
&~~~+\{\{1\otimes l_1,1\otimes l_3 \},1\otimes l_2\} )\\
=&~\delta \psi_p (l_1,l_2,l_3)+\frac{1}{2}\sum_{i,j=1}^{r}c_{ij}^p [\psi_j,\psi_i](l_1,l_2,l_3)\\
=&~\delta \psi_p (l_1,l_2,l_3)+\frac{1}{2}\nu\circ(\alpha \otimes \alpha)\circ \Delta (\bar{m}'_{p})(l_1,l_2,l_3).
\end{split}
\end{equation*}
Taking $b=2\omega_k$ and $a=id|_\mathbb{H}$ in Definition \ref{definition of Massey Bracket} the result follows.
\end{proof}
\section{Computations for the Leibniz algebra $\lambda_6$}\label{computation}
To construct a versal deformation of $\lambda_6$, we need to compute the second and third cohomology space of $\lambda_6=L$. First consider $HL^2(L;L)$.
Our computation consists of the following steps:\\
(i) To determine a basis of the space of cocycles $ZL^2(L;L)$,\\
(ii) to find out a basis of the coboundary space $BL^2(L;L)$,\\
(iii) to determine the quotient space $HL^2(L;L)$.\\
(i) Let $\psi$ $\in$ $ZL^2(L;L)$. Then $\psi :L\otimes L\longrightarrow L$ is a linear map and $\delta \psi =0$, where
\begin{equation*}
\begin{split}
\delta \psi(e_i, e_j, e_k)
&=[e_i,\psi(e_j, e_k)]+[\psi (e_i, e_k), e_j]-[\psi(e_i, e_j), e_k] -\psi([e_i, e_j], e_k) \\
&~ +\psi(e_i,[e_j,e_k])+\psi([e_i, e_k], e_j) ~\mbox{for}~0\leq i,j,k \leq 3.
\end{split}
\end{equation*}
Suppose $\psi(e_i,e_j)=\sum_{k=1} ^{3} a_{i,j}^{k} e_k$ where $a_{i,j}^{k} \in \mathbb C$
; for $1\leq i,j,k\leq3$.
Since $\delta \psi =0$ equating the coefficients of $e_1, e_2
~\mbox{and}~ e_3 $ in $\delta \psi(e_i, e_j, e_k)$ we get the following relations:
\begin{equation*}
\begin{split}
&(i)~ a_{1,1}^1 =a_{1,1}^3=0 ;\\
&(ii)~ a_{1,2}^1=a_{1,2}^3=0 ;\\
&(iii)~ a_{2,1}^1 =a_{2,1}^2=a_{2,1}^3=0 ;\\
&(iv)~a_{2,2}^1=a_{2,2}^2=a_{2,2}^3=0 ;\\
&(v)~a_{3,1}^2=a_{3,1}^3=0 ;\\
&(vi)~ a_{3,2}^2=a_{3,2}^3=0;\\
&(vii)~ a_{2,3}^3=0 ;\\
&(viii)~a_{1,1}^2=a_{3,1}^1=-a_{3,3}^3;\\
&(ix)~a_{1,2}^2=-a_{1,3}^3=a_{3,2}^1 .
\end{split}
\end{equation*}
Observe that there
is no relation among $a_{1,3}^1$,$a_{1,3}^2$, $a_{2,3}^1$, $a_{2,3}^2$, $a_{3,3}^1 $ and
$a_{3,3}^2$. Therefore, in terms of the ordered basis $\{e_1\otimes e_1, e_1\otimes e_2, e_1\otimes e_3, e_2\otimes e_1, e_2\otimes e_2, e_2\otimes e_3, e_3\otimes e_1, e_3\otimes e_2, e_3\otimes e_3\}$ of $L\otimes L$ and $\{e_1, e_2, e_3\}$ of $L$, the matrix corresponding to $\psi $ is of the form
$$M= \left( \begin{array}{llrllllll}
0& 0 & x_3 &0 &0 &x_5 &x_1 &x_2 &x_7 \\
x_1& x_2& x_4 &0 &0 &x_6 &0 &0 &x_8 \\
0& 0 & -x_2 &0 &0 & 0 &0 &0 &-x_1
\end{array} \right)$$
where $$x_1=a_{1,1}^2 ; x_2=a_{1,2}^2 ; x_3=a_{1,3}^1; x_4=a_{1,3}^2;
x_5=a_{2,3}^1; x_6=a_{2,3}^2; x_7=a_{3,3}^1; x_8=a_{3,3}^2$$ are in $\mathbb C$~.
Let $\phi_i \in ZL^2(L;L)$ for $1 \leq i\leq 8$, be the cocycle with
$x_i=1$ and $x_j=0$ for $i\neq j$ in the above matrix of $\psi$. It is easy to check that $\{\phi_1,\cdots, \phi_8\}$ forms a basis of $ZL^2(L;L)$.
(ii)
Let $ \psi_0 \in BL^2(L;L)$. We have $\psi_0=\delta g$ for some $1$-cochain $g \in CL^1(L;L)=Hom(L;L)$. Suppose the matrix associated to $\psi_0$ is same as the above matrix $M$.
Let $g(e_i)=g_i ^1 e_1 +g_i ^2 e_2+g_i ^3 e_3$ for $i=1,2,3$.
The matrix associated to $g$ is given by
\begin{center} $\left(\begin{array}{lll}
g_1 ^1& g_2 ^1 & g_3 ^1\\
g_1 ^2& g_2 ^2 & g_3 ^2\\
g_1 ^3& g_2 ^3 & g_3 ^3
\end{array} \right).$ \end{center}
From the definition of coboundary we get $$\delta g(e_i,e_j)=[e_i,g(e_j)]+[g (e_i),e_j]-g([e_i,e_j])$$ for $0\leq i,j \leq 3$. The matrix $\delta g$ can be written as
\begin{center}$
\left( \begin{array}{rrrrrrrll}
0 & 0 & (g_{1}^3-g_{2}^1) &0 &0 & g_{2}^3 &g_{1}^3&g_{2}^3 &(2g_{3}^3-g_{1}^1) \\
g_{1}^3 & g_{2}^3 & (g_{3}^3 +g_{1}^1-g_{2}^2) &0 &0 & g_{2}^1 &0 &0 &(g_{3}^1-g_{1}^2) \\
0 & 0 & -g_{2}^3 &0 &0 & 0 &0 &0 &-g_{1}^3
\end{array} \right).$ \end{center}
Since $\psi_0=\delta g$ is also a cocycle in $CL^2(L;L)$, comparing matrices $\delta g$ and $M$ we conclude that the matrix of $\psi_0$ is of the form
\begin{center}$
\left( \begin{array}{rrrrrrrll}
0 & 0 & x_3 &0 &0 &x_2 &x_1 &x_2&x_7 \\
x_1 & x_2 & x_4 &0 &0 &(x_1-x_3) &0 &0 &x_8 \\
0 & 0 & -x_2 &0 &0 & 0 &0 &0 &-x_1
\end{array} \right).$ \end{center}
Let ${\phi_i}^\prime \in BL^2(L;L)~\mbox{for}~i=1,2,3,4, 7,8$ be the coboundary with $x_i=1$ and $x_j=0$ for $i\neq j$ in the above matrix of
$\psi_0$. It follows that $\{\phi_1^\prime,\phi_2^\prime,\phi_3^\prime,\phi_4^\prime,\phi_7^\prime,\phi_8^\prime\}$ forms a basis of the coboundary space $BL^2(L;L)$.
(iii)
It is straightforward to check that $[\phi_2]$ and $[\phi_3]$ span $HL^2(L;L)$ where $[\phi_i]$ denotes the cohomology class represented by the cocycle $\phi_i$.
Thus $dim(HL^2(L;L))=2$.
Next let us consider $HL^3(L;L)$.
If $\psi \in ZL^3(L;L)$, then a computation similar to $2$-cocycles shows that the transpose of the matrix of $\psi$ is
\[ \left( \begin{array}{rrr}
0 &x_1 & 0 \\
0 &x_2 & 0 \\
x_3 &x_4 & (x_2+x_5) \\
0 &x_5 & 0 \\
0 &0 &0 \\
x_6 &x_{17} &0 \\
x_7 &x_8 & -x_5 \\
\frac{1}{5}(2x_2-3x_{6}+2x_{11}) &(x_{13} - x_{10}+2x_7+x_3-2x_{1}) & 0 \\
(2x_{16}-x_{14}) &x_9 &x_1 \\
0 &0 &0 \\
0 &0 &0 \\
\frac{1}{5}(3x_{2}+3x_6-2x_{11})-x_5 &x_{10} &0 \\
0 &0 &0 \\
0 &0 &0 \\
0 &x_{11} &0 \\
x_5 & (x_1-x_7) &0 \\
0 &\frac{1}{5}(3x_{2}+3x_6-2x_{11}) &0 \\
(x_1-x_7) &(3x_{16}-x_{14}-x_{8}) &x_5 \\
x_1 &0 &0 \\
x_2 &0 &0 \\
x_{12} &x_{18} &x_{13}\\
x_5 &0 &0 \\
0 &0 &0 \\
(x_{17}-x_{13}-x_{10}+3x_7+2x_3) &x_{19} &\frac{1}{5}(6x_{2}+x_{6}+x_{11}) \\
x_{14} &x_{15} &-x_1 \\
(2x_{13}-2x_{1}-x_3-x_7) & (x_{14}+x_{12}-x_8-x_4) &-x_2 \\
(x_9+x_{15}) &x_{20} &x_{16}\end{array} \right).\]
Let ${\tau_i} \in ZL^3(L;L)$ for $1\leq i \leq 20$ be the cocycle with $x_i=1$ and $x_j=0$ for $i\neq j$ in the
above matrix.
Then one can check that $\{\tau_i\}_{1\leq i\leq 20}$ forms a basis
of $ZL^3(L;L)$. So $dim(ZL^3(L;L))=20$.
On the other hand suppose $\psi \in CL^3(L;L)$ is a coboundary with $\psi=\delta g$. Let $g(e_i,e_j)=g_{i,j} ^1 e_1 +g_{i,j} ^2 e_2+g_{i,j} ^3 e_3$; for $1 \leq i,j \leq 3$. Then the transpose of the matrix of $\psi=\delta g$ is
$$ \left(\begin{array}{ccc}
0 & g_{1,1}^{3} & 0 \\
0 & g_{1,2}^{3} & 0 \\
(g_{2,1}^{1}+g_{1,2}^{1}-g_{1,1}^{3}) &( g_{2,1}^{2}+g_{1,2}^{2}-g_{1,1}^{1}+g_{1,3}^{3}) &( g_{2,1}^{3}+g_{1,2}^{3}) \\
0 & g_{2,1}^{3} & 0 \\
0 & g_{2,2}^{3} & 0 \\
(g_{2,2}^{1}-g_{1,2}^{3}) &(g_{2,2}^{2}+g_{2,3}^{3}-g_{1,2}^{1}) &g_{2,2}^{3} \\
(g_{1,1}^{3}-g_{2,1}^{1})& (g_{1,1}^{1}+g_{3,1}^{3}-g_{2,1}^{2}) &-g_{2,1}^{3} \\
(g_{1,2}^{3}-g_{2,2}^{1}) & (g_{1,2}^{1}+g_{3,2}^{3}-g_{2,2}^{2}) & -g_{2,2}^{3} \\
g_{1,1}^{1} & (g_{3,3}^{3}+g_{1,1}^{2}) & g_{1,1}^{3} \\
0 & 0 & 0 \\
0 & 0 & 0 \\
(g_{2,2}^{1}-g_{2,1}^{3})& (g_{2,2}^{2}-g_{2,1}^{1}) & g_{2,2}^{3} \\
0 & 0 & 0 \\
0 & 0 & 0 \\
-g_{2,2}^{3}&-g_{2,2}^{1} & 0 \\
g_{2,1}^{3} &g_{2,1}^{1} & 0 \\
g_{2,2}^{3} & g_{2,2}^{1} & 0 \\
g_{2,1}^{1} &g_{2,1}^{2} & g_{2,1}^{3} \\
g_{1,1}^{3}& 0 & 0 \\
g_{1,2}^{3} & 0 & 0 \\
(g_{1,1}^{1}+g_{3,2}^{1}-g_{3,1}^{3}+g_{1,3}^{3}) & (g_{1,1}^{2}+g_{3,2}^{2}-g_{3,1}^{1}) & (g_{1,1}^{3}+g_{3,2}^{3}) \\
g_{2,1}^{3}& 0 & 0 \\
g_{2,2}^{3}& 0 & 0 \\
(g_{2,3}^{3}-g_{3,2}^{3}+g_{1,2}^{1})&(g_{1,2}^{2}-g_{3,2}^{1}) & g_{1,2}^{3} \\
(2g_{3,1}^{3}-g_{1,1}^{1}) & (g_{3,1}^{1}-g_{1,1}^{2}) & -g_{1,1}^{3} \\
(2g_{3,2}^{3}-g_{1,2}^{1}) & (g_{3,2}^{1}-g_{1,2}^{2}) & -g_{1,2}^{3} \\
(g_{3,1}^{1}+g_{3,3}^{3}) & g_{3,1}^{2} &g_{3,1}^{3} \\
\end{array}
\right)$$
Since $\delta \psi$ is also zero, the transpose of the matrix of $\psi$ is of the previous form as well. Thus a coboundary $\psi$ has the following transpose matrix.
\[ \left( \begin{array}{rrr}
0 &x_1 & 0 \\
0 &x_2 & 0 \\
x_3 &x_4 & (x_2+x_5) \\
0 &x_5 & 0 \\
0 &0 &0 \\
-(x_2+x_{11}) &x_{17} &0 \\
x_7 &x_8 & -x_5 \\
(x_{2}+x_{11}) &(x_{13} - x_{10}+2x_7+x_3-2x_{1}) & 0 \\
(2x_{16}-x_{14}) &x_9 &x_1 \\
0 &0 &0 \\
0 &0 &0 \\
-(x_{11}+x_{5}) &x_{10} &0 \\
0 &0 &0 \\
0 &0 &0 \\
0 &x_{11} &0 \\
x_5 & (x_1-x_7) &0 \\
0 &-x_{11} &0 \\
(x_1-x_7) &(3x_{16}-x_{14}-x_{8}) &x_5 \\
x_1 &0 &0 \\
x_2 &0 &0 \\
x_{12} &x_{18} &x_{13}\\
x_5 &0 &0 \\
0 &0 &0 \\
(x_{17}-x_{10}+3x_7+2x_3-x_{13}) &(x_4 +x_8-x_{12}-x_{14}) &x_2 \\
x_{14} &x_{15} &-x_1 \\
(2x_{13}-2x_{1}-x_3-x_7) & (x_{14}+x_{12}-x_8-x_4) &-x_2 \\
(x_9+x_{15}) &x_{20} &x_{16}\end{array} \right).\]
This implies that $dim(BL^3(L;L))=18$. Consequently $dim(HL^3(L;L))=2$.
Since $HL^3(L;L)$ is nontrivial, it is necessary to compute possible obstructions in order to extend an infinitesimal deformation to a higher order one.
First we describe the universal infinitesimal deformation for our Leibniz algebra.
To make our computation simpler, we choose the representative cocycles $\mu_1,\mu_2$ where $\mu_1=\phi_2 - \phi^\prime_2$ and $\mu_2=\phi_3$. Let us denote a dual basis in $HL^2(L;L)^\prime$ by $\{t,s\}$. By Remark \ref{exp of inf} the universal infinitesimal deformation of $L$ can be written as
$$[1\otimes e_i,1\otimes e_j]_{\eta_1}=1\otimes [e_i,e_j]+ t \otimes \mu_1(e_i,e_j)+s \otimes \mu_2(e_i,e_j).$$
with base $C_1 =\mathbb{C}~\oplus \mathbb{C}~t ~ \oplus ~ \mathbb{C}~s$.
Let us describe a simpler version of the inductive definition of Massey brackets by Retakh \cite{R}(see \cite{F}), relevant for Leibniz algebra deformations. These $n$ th order operations are partially defined and they are well defined modulo the $(n-1)$ th order ones. The second order operation is the superbracket in the cochain complex. More precisely, if $y_1=[x_1], y_2=[x_2]$ are $2$- cohomology classes, then the second order operation $<y_1,y_2>$ is represented by the superbracket $[x_1,x_2]$.
Suppose that $y_i \in HL^2(L;L)$, $1\leq i\leq 3$ such that $<y_i,y_j>=0$ for every $i$ and $j$. This means that for a cocycle $x_i$ representing $y_i$ we have $[x_i,x_j]=d x_{ij}$ for some $2$- cochain $x_{ij}$. Then the third order Massey operation $<y_1,y_2,y_3>$ is defined and is represented by
$$ [x_{12},x_3]+[x_1,x_{23}]+[x_{13},x_2].$$
The cohomology class is independent of the choice of $x_{ij}$. The higher order Massey operations are defined inductively.
Now we compute the Massey brackets using the above definition.
\begin{description}
\item[(i)]By definition $<[\mu_1],[\mu_1]>$ is represented by $[\mu_1,\mu_1]=2 (\mu_1 \circ \mu_1).$\\
Now $(\mu_1 \circ \mu_1)(e_i,e_j,e_k)\\
=\mu_1(\mu_1(e_i,e_j),e_k)-\mu_1(\mu_1(e_i,e_k),e_j)-\mu_1(e_i,\mu_1(e_j,e_k))$ for $1\leq i,j,k\leq 3$.
Since $\mu_1(e_2,e_3)=-e_1$ and takes value zero on all other basis element of $L\otimes L$, it follows that $\mu_1 \circ \mu_1=0$.
\item[(ii)] Similarly $<[\mu_1],[\mu_2]>$ is represented by $[\mu_1,\mu_2]= \mu_1\circ \mu_2 +\mu_2\circ \mu_1$.
Since $\mu_2(e_1,e_3)=e_1$ and takes value zero on all other basis element of $L\otimes L$ it follows that $<[\mu_1],[\mu_2]>=0$.
\item[(iii)] The bracket $<[\mu_2],[\mu_2]>$ is represented by $[\mu_2,\mu_2]=2(\mu_2 \circ \mu_2)=0$.
\end{description}
Since $\{[\mu_1],[\mu_2]\}$ form a basis for $HL^2(L;L)$, it follows that all the Massey $2$- brackets are trivial. So all the Massey $3$- brackets are defined.
From the definition of Massey $3$- bracket it follows that all the Massey $3$- brackets $<[\mu_i],[\mu_j],[\mu_k]>$ are trivial and represented by the $0$-cocycle. By induction it follows that any $<[\mu_1],[\mu_2],\cdots,[\mu_k]>=0$ for $[\mu_i]\in HL^2(L;L)$ and moreover, they are represented by the $0$-cocycle.
By Theorem \ref{obs at each stage} and considering the inductive definition of Massey brackets in \cite{FuL} it follows that the possible obstruction at each stage in extending $\eta_1$ to a versal deformation with base $\mathbb{C}[[t,s]]$ can be realised as the Massey brackets of $\mu_1$ and $\mu_2$. So the possible obstruction vanishes.
As there are no obstructions to extending the universal infinitesimal deformation $\eta_1$, it means that $\eta_1$ extends to a versal deformation with base $\mathbb{C}[[t,s]]$. Moreover, observe that by our choice of $\mu_1$ and $\mu_2$ every Massey brackets is represented by the $0$- cochain, and so $\eta_1$ is itself a Leibniz bracket with base $\mathbb{C}[[t,s]]$. It follows by the construction in \cite{FMM} that $\eta_1$ is a versal deformation.
Let us write out the versal deformation we have constructed:
\begin{equation*}
\begin{split}
& [e_1,e_3]_{t,s}=e_2+e_1s,~~ [e_3,e_3]_{t,s}=e_1,~~ [e_2,e_3]_{t,s}=-e_1 t\\
&\mbox{with all the other brackets of basis elements being 0}.
\end{split}
\end{equation*}
Thus we obtain the following two nonequivalent $1$-parameter deformations for the Leibniz algebra $\lambda_6$.
\begin{equation*}
\begin{split}
&(i)~[e_1,e_3]_t=e_2,~~ [e_2,e_3]_t=-e_1 t,~~ [e_3,e_3]_t=e_1\\
&\mbox{ all the other brackets of basis elements are zero,}~
\\
&(ii)~ [e_1,e_3]_s=e_2+e_1 s,~~ [e_3,e_3]_s=e_1\\
&\mbox{ all the other brackets of basis elements are zero.}
\end{split}
\end{equation*}
{\bf \large Conclusions:}
In this paper we computed a versal deformation of a $3$- dimensional nilpotent Leibniz algebra. For computing obstructions we introduced the notion of Massey brackets and proved the relationship between Massey brackets and obstructions. It turned out that in our example there are no obstructions in extending an infinitesimal deformation to a formal base, and so the universal infinitesimal deformation itself is versal with base $\mathbb{C}[[t,s]]$. From the computation it follows that our Leibniz algebra has two nonequivalent $1$- parameter family of deformations which are both infinitesimal and formal. We gave this deformation in an explicit form.
\vspace{.5cm}
{\bf \large Acknowledgements:}
The author would like to thank Professor A. Fialowski and Professor G. Mukherjee for their useful comments.
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{
"redpajama_set_name": "RedPajamaArXiv"
}
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News / Alina Baraz - The Color of You (Album)
Alina Baraz - The Color of You (Album)
Instagram: @alinabaraz Write up via HNHH
Alina Baraz most recently dropped off her Color Of You surprise project equipped with appearances from Jada and Khalid, who pops up twice in the track listing. The finished product is a nine-track effort that follows up 2015's Urban Flora, a collaborative debut released alongside Galimatias. The arrival is a well-received outing from the budding siren, with cuts that all provide an ethereal R&B soundtrack to help you breeze through the romances of spring, summer, and beyond. "I have always been infatuated by colors," she explained of the inspiration behind the project. "I better understand myself in color. For example If I can't understand my mood I'll switch through a color light until it matches up with my thoughts. But in this case, something came into my life and I couldn't define what it was and what I felt. It felt like a color that didn't exist. So it's me trying to interpret and understand a color I'm seeing for the first time."
Tagged: alina baraz album, alina baraz cleveland, alina baraz cleveland singer, alina baraz ohio, alina baraz the color of you, Category_Mixtapes, music
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
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Fürst Dmitri Iwanowitsch Dolgorukow (; * in Moskau; † ebenda) war ein russischer Diplomat und Dichter.
Leben
Dolgorukows Eltern waren der Dichter und Schriftsteller Fürst Iwan Dolgorukow und seine Frau Jewgenija geborene Smirnowa. Dolgorukow absolvierte das Adelspensionat der Universität Moskau und wurde 1816 Kanzlist in der Moskauer Gouvernementsregierung. 1819 wurde er Beamter des Kollegiums für auswärtige Angelegenheiten.
Dolgorukow war Mitglied der 1819–1820 existierenden Literatengesellschaft Seljonnaja Lampa (Grüne Lampe), die sich regelmäßig in ihrem Versammlungsraum mit grüner Lampe im Hause Nikita Wsewoloschskis traf und zum Wohltätigkeitsverband der Dekabristen gehörte. Mitglieder der Grünen Lampe waren Jakow Tolstoi (Vorsitz), Alexander Puschkin, Anton Delwig und die Dekabristen Sergei Trubezkoi, Fjodor Glinka und Pjotr Kawerin. An den Versammlungen nahmen auch Nikolai Gneditsch, Alexander Ulybyschew, Dmitri Barkow, Arkadi Rodsjanko und Wassili Engelhardt teil. Dolgorukow trug dort seine Gedichte vor. 1822 wurden einige seiner Gedichte in den Nowosti literartury veröffentlicht.
Dolgorukow wurde Gesandtschaftssekretär in Konstantinopel (1820), Rom (1822) und Madrid (1826), Im Mai und Juni 1827 wanderte er mit dem US-amerikanischen Schriftsteller Washington Irving von Sevilla nach Granada und besuchte die Alhambra. Es folgten Dolgorukows Versetzungen nach London (1830), Den Haag (1831) und Neapel (1838). 1843 wurde er zum Gesandtschaftsrat in Konstantinopel ernannt.
1845 wurde Dolgorukow bevollmächtigter Minister am Persischen Hof in Teheran als Nachfolger Alexander von Medems. Nach Beginn des Krimkriegs erreichte Dolgorukow von Schah Nāser ad-Din Schāh die Neutralität Persiens. In seinen Berichten an die Vorgesetzten in St. Petersburg beschrieb er die grausame Verfolgung der Babi und beklagte die Grausamkeiten der Folter und Hinrichtungen. Auch überreichte er dem Schah eine Protestnote und beantragte, für das Gefängnis des Bab einen Ort weit entfernt von der russischen Grenze zu wählen. Als 1853 der Religionsstifter des Bahaitums Bahāʾullāh von der persischen Regierung ins Ausland verbannt wurde, bot ihm Dolgorukow namens der russischen Behörden Asyl auf russischem Territorium an. Bahāʾullāh wählte Bagdad im Osmanischen Reich als Verbannungsort, wohin ihn während der dreimonatigen Reise ein Vertreter der russischen Gesandtschaft begleitete.
1854 verließ Dolgorukow aufgrund familiärer Umstände Persien und wurde Senator und Geheimer Rat (3. Rangklasse). Sein Nachfolger in Teheran war Nikolai Anitschkow.
Dolgorukow baute sich eine reichhaltige Sammlung persischer Handschriften auf und sammelte mit Begeisterung Autographen und Ikonen. Er schrieb Reiseskizzen, Tagebücher und historische Notizen. Er veröffentlichte seine Gedichtsammlungen in Moskau (1856, 1857, 1859, 1860 und 1863), von denen die ersten beiden anonym erschienen. Seine Briefe aus den 1820er Jahren an Vater, Bruder und Schwiegersohn, die auch Gedichte und Reiseschilderungen enthielten, wurden 1914 im Russki archiw veröffentlicht.
Dolgorukow war zweimal verheiratet und hatte drei Töchter. Er starb in Moskau und wurde auf dem Friedhof des Donskoi-Klosters begraben.
Der Musiker Pawel Dolgorukow und der Dichter Alexander Dolgorukow waren Dolgorukows Brüder.
Ehrungen, Preise
Russischer Orden der Heiligen Anna I. Klasse
Sankt-Stanislaus-Orden I. Klasse
Sonnen- und Löwenorden I. Klasse
Weblinks
Katalog der Russischen Nationalbibliothek: Долгоруков, Дмитрий Иванович
Einzelnachweise
Botschafter (Russisches Kaiserreich)
Botschafter im Iran
Träger des Ordens der Heiligen Anna
Träger des Sankt-Stanislausordens (Russland)
Träger des Sonnen- und Löwenordens
Fürst (Russland)
Russe
Geboren 1797
Gestorben 1867
Mann
Rurikide
|
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James McCanny is featured in the Planet X Video, produced in 2000/2001. Archives of his past pronouncements are online.
James McCanny has an audio file up at CyberSpaceOrbit. He is stating [in August 2002] that Nibiru will not come in from Orion but in the opposite direction. He is stating that the Egyptians referred to Nibiru in association with Orion at its Perihelion, a point also made by Sitchin. He is claiming that the window of opportunity for spotting this will be over by October, 2002 when (since it is opposite from where Zetatalk says it will be) it will go behind the sun until appearing in April of 2003. Strangely enough he gives the ZetaTalk date in the spring of 2003 for when it is going to appear to the naked eye!
The sci.astro Usenets had some things to say about these statements.
Spoke about PX, and the fake date of May 15 (given by the Gov) so that when that day passes, everyone will not believe, .... another Y2K. But he is saying, there is something! "don´t throw the baby out with the bathwater" :-) He said the above, like 5 times, he sounded nervous (to me) or ruffled, as he kept on stammering with the quote. Then right at the end. He said, do not be put off by nothing happening on the 15th. He expects something huge to unfold on the planet in the latter half of May. This came out of nowhere, and I was like wow! HE DID NOT SAY, what it would be. HE DID NOT SAY, Planet X.
Let´s call any new planet to come into the solar system "Planet X," because that is a term that is used for objects that have not been named yet. These objects are the ones that have affected Earth in the past. We are now expecting another object, which possibly has been here before, to come into the inner solar system within possibly the next 10 years -- and there is a good reason to believe that it might be as early as May 2003. Other books that would be pertinent might be Mark Hazlewood´s book, Blindsided, which talks about the possibility that this object could be here as early as May 2003.
He clearly says that a large object is coming into the solar system from the South. He repeats this on many shows thereafter.
I heard James McCanney trashing your website twice last week, on coasttocoastam.com and his own radio show and seriously backpedaling on his previous statements about Planet X entering the solar system, which he had made late in 2004. The archives of James McCanney's previous statements are still on his website, he can hardly deny them, but he is. He now says, he NEVER said that it had been seen and that there are hundreds of Planet Xs out there - it just means that they haven't been named yet. However, it seems that he either has a mental/emotional problem because he doesn't want anyone repeating what he says, OR someone in one of the alphabet agencies has told him to shut up and he is. Something is definitely wrong there.
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{
"redpajama_set_name": "RedPajamaC4"
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Q: Can I use Code Connection tool with non-educational version of Minecraft (iPad)? It is not quite clear for me - can I use Code Connection tool (installed on my Mac) with the standard (non-educational) version of Minecraft (installed on my iPad Pro)? When I try to connect from iPad to Code Connect, I get an error - connection failed. Both devices are in the same local network. 'Cheats' and 'Education Edition' features are enabled when the new World is created on the iPad.
A: Sadly, while I'm not 100% sure of this, I think code connection attempts a check against the version of Minecraft that connects and immediately disconnects if it isn't Education Edition or Win 10 Edition. My Android device connects to my code connection on my Win 10 machine, but then immediately disconnects. Connecting from another machine with Win10 edition works, so sadly it looks like code connection is version locked.
Seems like a poor decision on Microsoft's part.
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Oberea subteraurea is a species of beetle in the family Cerambycidae. It was described by Stephan von Breuning in 1961. It is known from Borneo.
References
Beetles described in 1961
subteraurea
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{"url":"https:\/\/www.physicsforums.com\/threads\/integral-of-f-times-cosine-as-period-of-cosine-goes-to-zero.395720\/","text":"# Integral of f times cosine, as period of cosine goes to zero.\n\n1. Apr 16, 2010\n\n### Hoblitz\n\n1. The problem statement, all variables and given\/known data\nThe problem is as follows:\n\nLet f be a real valued function that is Riemann integrable on [a,b]. Show that\n\n$$\\lim_{\\lambda \\rightarrow \\infty} \\int_{a}^{b} f(x)\\cos(\\lambda x)dx = 0$$.\n\n2. Relevant equations\nI am freely able to use the fact that the product of Riemann integrable functions is Riemann integrable, and that the integral of cosine is sine (+ constant).\n\n3. The attempt at a solution\nThe first thing I tried was just to let f be the indicator function on the interval [a,b]. In that case, it wasn't so bad because then I could find $\\lambda > 0$ so large that (for given epsilon > 0) $\\frac{1}{\\lambda} < \\frac{1}{2}\\epsilon$.\n\nThen for that lambda I can find a unique integer k such that\n$$\\frac{2\\pi k}{\\lambda} \\leq (b-a) < \\frac{2\\pi (k + 1)}{\\lambda}$$.\n\nThen\n$$|\\int_{a}^{b} \\cos(\\lambda x)dx| = |\\int_{a}^{a + \\frac{2 \\pi k}{\\lambda}} \\cos(\\lambda x) \\,dx + \\int_{a + \\frac{2\\pi k}{\\lambda} }^{b} \\cos(\\lambda x)\\,dx| = |\\frac{\\sin(b)}{\\lambda} - \\frac{\\sin(a)}{\\lambda}|$$\n\nFinally,\n$$|\\frac{\\sin(b)}{\\lambda} - \\frac{\\sin(a)}{\\lambda}| \\leq \\frac{2}{\\lambda} < \\epsilon.$$\n\nFrom there on it is clear that for $\\lambda^{'} > \\lambda > 0$, we are still less than epsilon. (Find a new unique integer as above, and run through the integrals again).\n\nI've been trying to think of a way to apply this fact to the case with general $f$. What I've tried to do is trap\n$$\\int_{a}^{b} f(x)\\cos(\\lambda x)dx$$\nbetween plus\/minus some constant times $\\int_{a}^{b} \\cos(\\lambda x)dx$, where the constant might be something like the max of the absolute values of the supremum and infemum of f over the interval [a,b], but I cant get it work out (problems arise with negatives multiplying to become positive, etc...). Any hints or comments would be very helpful, thank you!\n\n2. Apr 16, 2010\n\n### Office_Shredder\n\nStaff Emeritus\nI think a bigger concern should be why f(x) is bounded on that interval (it doesn't have to be)\n\n3. Apr 16, 2010\n\n### Hoblitz\n\nIndeed I can think a few unbounded functions f which are integrable on an interval [a,b], so I can see why my idea would certainly fail. I'll try looking at this from a new perspective and see what I get. What I understand to be happening is that the rapidly quickening oscillations of cosine are going to force f to \"cancel itself out\" in the integral, eventually bringing that sucker close to zero...\n\nThanks, I'll give this more thought.","date":"2018-03-24 10:47:13","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\": 0.8377017974853516, \"perplexity\": 321.95767027178243}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257650188.31\/warc\/CC-MAIN-20180324093251-20180324113251-00718.warc.gz\"}"}
| null | null |
- I waited for the final phone-call (after months of interviewing) on Jan 9th. I answered nervously and
slumped over, awaiting the voice on the other end's message of "unfortunately...we decided...".
I could already feel a large shadow of myself towering over my body. I decided that I'd sit by the window of the bus
and cry on the way home to get over my rejection.
- This was not the day for that. I received good news that I will be receiving an offer for a Spring internship.
They picked me. I'm puzzled, but say yes. First I'm sitting on the couch. By the end of the phone call, I am doing
a signature happy dance, barely holding the phone. Did I remember to put my shoes back on? Will my phone battery on
my flip-phone die before the voice on the other end hangs up? That *would* be super awkward.
- I am finalizing everything and figuring out how and when to tie up some other loose ends local to LA before
heading out. I will have to move soon. It will also give me the opportunity to look for housing for my Summer
internship, which I uh..haven't really started looking for yet. I've sort of thought about a lot of things,
but not fully, which is exciting because things that you don't plan that fail horribly because you didn't plan
for them are mini-adventures and mini-tests of your patience! :D
- Most of all, I'm really excited for the mentorship and the opportunity. My team makes me so happy. So much
of it feels like such a fit in a way I can't explain. I am happy for the opportunity to grow and broaden
my skills, as well as grow as a person. I'm excited to be in a place of people *just like me*. So I won't
feel as lonely (in my thoughts about certain things I'm passionate about) anymore. I won't have to live through
the internet alone anymore.
- Unfortunately, I will also not be able to speak about this opportunity in detail, but I believe that my growth will
become self-evident. It will be a fruitful experience.
## LA is a strange place
- I joke often that there is force like the heliopause; it has kept me here for so long that I could not break
through. I could not break beyond the black hole of LA and the limited opportunities for growth. This year is
different.
## The Bay area is my space
- Something resonates with me about the people and the place. I love it in a way I can't explain. It energizes me.
I have so many friends there, and I've received thoughtful mentorship there. I've grown so much from my experiences
there.
## My birthday
- I'll be celebrating my birthday over there. I'm excited about that. I haven't thought about what I'll be doing,
but it might start with an H and end with an L (clue: seven letters) :D
## Heartbreak
- One of my really good friends in LA is experiencing a bit of heartbreak because of me. I've moved around so much
in the US, that I think it doesn't affect me as much anymore, but I'm also terrible at goodbyes. I'd like to think
that everyone I've touched (and who has touched me) will forever stay positively with me as I move forward to the
next adventure. I can't hope for too much more than that.
## That's about all I have to say for now
- I've been staying up to get work done, and have a bit of a flu at the moment. I may also check out a talk on
Randomization if I'm up for it tomorrow evening.
## And that's about it
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In 1957 Gabe and Jackie Milanese started their family's home remodeling business with a great idea. The idea was to take modern materials and apply them to the exterior of homes and businesses to make them maintenance free and energy efficient.
Gabe and Jackie proved that being honest and working hard to offer the best products, installation and service at the lowest possible price is the best way to succeed in business.
The second generation of the Milanese family has the same dedication to offering only the best to their clients.
Gabe Milanese Jr. served as the President of Milanese Remodeling for over 25 years and he treated employees, clients, colleagues and all he dealt with honestly, fairly and respectfully. He sadly passed away, in July 2013, but his memory lives on through the Gabe Milanese Jr. Foundation. Gabe's wife,Trish, is the Milanese Remodeling "Director of First Impressions" and a big reason Milanese Remodeling is still the home of good old-fashioned customer service. Contact Trish to schedule an appointment, installation, or service for any project and you will find a friend who is happy to help.
Michael and Mark Milanese both grew up in the business. They have the benefit of learning from the example of their older brother, their parents and by "hands-on" experience from an early age.
A third generation of the Milanese family is continuing the remodeling tradition.
Jacqueline Milanese loves home remodeling and watching home improvement shows on TV. She graduates Bishop Shanahan High School in 2014.
Cecelia Milanese enjoys supporting clients with friendly service. She is a senior at West Chester Henderson.
Nicholas Milanese has 3 years experience. He worked on more challenging projects this summer before returning to Pitt University to continue his studies.
Mark Anthony Milanese has been installing for 4 years and has learned the skills to install retractable awnings, storm doors and replacement windows. He is pursuing a business degree at DCCC.
Gabriel Milanese III is an eager learner who takes pride in a job well done. We expect great things from Gabe III.
Michael Angelo Milanese Jr. has been helping his dad for many years. This year he often accompanied his dad on service calls and learned that service after the sale is very important.
From 1957-1985 our business was called Milanese Aluminum.
After nearly 30 years operating under that name, we changed our business name to Milanese Remodeling to better reflect our commitment to using a variety of modern materials and handling a complete assortment of exterior remodeling projects.
On March 28, 2017, we celebrated our 60th birthday. We were thrilled to be recognized by the Pennsylvania House of Representatives for our services to Chester County and the Commonwealth of Pennsylvania.
Jackie Milanese received a citation, presented by PA State Rep. Tim Hennessey and PA State Rep. Harry Lewis. Check out our blog post for more information about the award.
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| 2,492
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Йо-мобіль () — амбітний російський проект послідовного гібридного автомобіля, в конструкції якого передбачалося використання електричної трансмісії з комбінованим живленням від генератора, що обертається газо-бензиновим роторно-лопатевим двигуном внутрішнього згоряння, і від ємнісного накопичувача енергії. За рахунок застосування такої схеми автомобіль мав би базовий повний привод.
Проект був анонсований в грудні 2010 року як «гібридний автомобіль Прохорова». Передбачалось, що потужність двигуна становитиме 60 к.с., а енергетична установка в складі «двигун + генератор» з системою накопичення енергії мала забезпечити енергоозброєність, аналогічну 2-літровому 150-сильному ДВЗ. На момент анонсу дата випуску була визначена як «початок 2012 року».
На кінець 2012 року було анонсовано підписання довгострокового контракту на будівництво Йо-мобілів у В'єтнамі. Потужність заводу на перших етапах мала складати 100 000 авто на рік, а в подальшому більше одного мільйона.
Промислове виробництво машин було заплановано на початок 2015. В 2014 році проект був закритий. Причиною закриття назвали «різке зростання курсу євро». У травні 2021 року в Мінську було знайдене звалище «Йо-мобілів», які не вдалося продати через завищену ціну.
Загальні витрати на проект склали € 150 млн. По закриттю він проданий державному підприємству НАМІ за 1 євро разом з проектом «Marussia Motors».
Див. також
Marussia Motors
Примітки
Гібридні автомобілі
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| 4,845
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layout: page
title: Noah Bass's 34th Birthday
date: 2016-05-24
author: Linda Johnston
tags: weekly links, java
status: published
summary: Cras eget auctor elit. Fusce.
banner: images/banner/meeting-01.jpg
booking:
startDate: 03/04/2019
endDate: 03/06/2019
ctyhocn: FLONOHX
groupCode: NB3B
published: true
---
Nullam mollis sed felis non iaculis. Donec sit amet aliquet elit. Aliquam augue ligula, viverra quis convallis quis, egestas sit amet enim. Mauris eleifend elementum felis imperdiet porttitor. Praesent vel finibus augue, eu pretium lacus. Phasellus id pulvinar lectus. Aenean erat leo, aliquet quis mauris sed, tincidunt malesuada dui. Vestibulum lobortis ultricies commodo. Duis molestie neque sit amet nulla suscipit, ac molestie ligula auctor. Proin vitae lacinia metus. Suspendisse commodo lacus lectus, id tempus augue semper nec. Aliquam et dolor nunc. Suspendisse eu fringilla elit. Vestibulum vel feugiat justo.
* Ut molestie odio id iaculis imperdiet
* Fusce ornare enim eget porta pretium
* Proin elementum eros eu porta finibus
* Fusce pulvinar nisi vel ligula fringilla, at posuere justo laoreet
* Nulla hendrerit risus vitae risus ornare rutrum.
Nullam ultrices lacinia nisl quis feugiat. In eget lectus ut tortor iaculis volutpat vitae aliquet neque. Vestibulum interdum leo enim, non ultrices tellus lacinia vel. Cras et varius ante. Morbi et rutrum tellus. Cras bibendum, ipsum sit amet congue venenatis, tortor nunc posuere risus, eu dapibus eros eros nec augue. Cras at nunc orci. Etiam sed laoreet mauris. Duis malesuada elit vitae risus ultrices, vitae pellentesque metus fermentum. Nullam ligula velit, tempor non nunc ac, lobortis porttitor urna. Nullam scelerisque quis felis sit amet faucibus. Nunc ex dolor, laoreet sed sagittis at, porttitor ac tortor. Phasellus bibendum magna dui, ac rutrum nibh rutrum ut. Quisque mattis eu neque sit amet suscipit. Integer sagittis massa quis leo sagittis, id tristique justo dictum.
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Leslie Williams
Leslie Williams's first book, Success of the Seed Plants, won the 2010 Bellday Poetry Prize. Her poems have appeared in Poetry, Image, Southern Review, Gulf Coast, and many other journals. She has received the Robert H. Winner Memorial Award from the Poetry Society of America and grants from the Illinois Arts Council and the Massachusetts Cultural Council.
Items per page 10 items per page 25 items per page 50 items per page 100 items per page Sort by Title - A to Z Title - Z to A Price - Low to High Price - High to Low Date - Newest to Oldest Date - Oldest to Newest
Even the Dark
Book prices
E-book $15.95 978-0-8093-3750-7
Paperback $15.95 978-0-8093-3749-1
SIU Press News
Sign Up for Email Announcements
Conference on Illinois History
Writing English Across Borders
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{
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| 3,459
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Raymond Perry Stanford plus connu sous le nom de R. J. Stanford (né le à Chino) est un joueur américain de football américain.
Carrière
Université
Stanford fait ses études dans le lycée de sa ville natale, le Chino High School. Il entre à l'université d'Utah et joue avec les Utes. Durant sa carrière, il joue cinquante-et-un matchs depuis ses débuts officiels en 2006. Il commence comme running back mais se tourne vers le poste de cornerback peu de temps après.
Professionnel
Stanford est drafté lors du septième tour du draft de 2010 de la NFL par les Panthers de la Caroline au . Il entre dans l'équipe active mais ne joue aucun match lors de la saison 2010. En 2011, il entre au cours de douze matchs et intercepte la première passe de sa carrière.
En 2012, il signe avec les Dolphins de Miami où il reste à un poste de remplaçant.
Naissance à Chino (Californie)
Naissance en mai 1988
Joueur américain de football américain
Joueur des Panthers de la Caroline
Joueur des Dolphins de Miami
Joueur des Bengals de Cincinnati
Cornerback
Joueur de football américain des Utes de l'Utah
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| 5,026
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De Soto est un village situé dans le comté de Jackson, dans l'est de l'État de l'Illinois, aux États-Unis.
Démographie
Liens externes
Village en Illinois
Comté de Jackson (Illinois)
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{
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| 9,127
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Q: jQuery ajax forms with json response Im having troubles trying to figure out why my JSON data.email response is returning null. Can anyone advise?
//javascript
$.ajax(
{
type: 'POST',
url: 'process.php',
dataType: 'json',
data: { email : "me@home.com" },
success: function(data)
{
alert("result = "+data.email);
}
});
//php (process.php)
if ($_POST['email'])
$return['email'] = $_POST['email'];
else
$return['email'] = "no email specified";
echo json_encode($return);
A: whoops, missed the call to json_encode.
Still, you need to set the Content-Type of the response to 'application/json' in the php
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delanceyplace.com 8/8/11 - cinema stops taking itself quite so seriously
In today's excerpt - after World War II, movies about spies and war were taken seriously—a movie told a story and the audience was expected to believe it. Ian Fleming's wildly popular books about James Bond continued that tradition—Fleming's Bond was a humorless, high-born, and unquestioningly patriotic creature. (In fact, Fleming dismissed the idea of Alfred Hitchcock directing his films because he he felt he would not treat them seriously enough). Sean Connery—who was relatively unknown, was climbing up from a working class background, and was paid a relative pittance for the role—knew intuitively to imbue his Bond with insolence and an amoral humor. Thus 1962's Dr. No, the first of the Bond movies, marks the beginning of decadence in post-war cinema—the first time the audience is in on the joke:
"Sean Connery was nothing like the Bond so plainly envisaged in the Dr. Noscreenplay. That screenplay, finally knocked together by [director] Terence Young and his assistant Joanna Harwood in a week-long session at the Dorchester Hotel, was written for precisely the kind of old-fashioned public-school hero Connery's instinctively insolent creation was about to do away with. ...
" 'But of course!' Connery's Bond is required to say at one point in the action (as he will be throughout his pictures in the series; Roger Moore—on whom such stuffed-shirt inanities might have sat rather better—was never asked to utter the line). Hitherto, such stockbroker Sheridan would have been sayable only if dressed in the exclamatory high camp, British cinema specialized in. Connery, though, played the line for laughs by uttering it in the mocking, unshockable, dignified yet dressed-down drawl that would come to be one of the hallmarks of his Bond. It is impossible to overemphasise how central to the movie those democratically satirical inflections were. Without them, the Bond of Dr. No would have been as insufferably snobbish as the Bond of Fleming's original novels. Without them, there would have been no From Russia with Love, let alone any Quantum of Solace. ... The movie [version of] Bond owed much ... to Connery's languorously insurrectionary take on what he saw as the jumped-up imperialist bore.
"Both Young and Connery have claimed authorship for the movie Bond's flip, amoral humour. 'When I flew out [to Jamaica] with Sean, before anyone else came,' Young told a TV documentary, 'I said, 'For Christ's sake, Sean, we've got to make this picture a little bit amusing—it's the only way we're going to get away with murder.' Because a lot of the sex and violence, I think, (a), is objectionable and, (b), will never get past the censor.
"Connery, for his part, was always insistent that humour was essential to the movie Bond. Indeed so, though it should be pointed out that ... the Bond of the novels never cracks a joke, and Fleming himself was so humourless that fancy the idea of Alfred Hitchcock turning his novels into movies though he did, he also worried that Hitch might treat them with insufficient seriousness. ...
"While patriotic to the point of self-mockery, Connery's Bond seemed not at all tied to notions of history and tradition, but merely to the self-aggrandising conspicuous consumption that was the true hallmark of the [1960s] decade which spawned him. Bond's affectlessness—when he learns the detail of Dr No's plans he neither laughs nor cries but merely sounds weary at what he calls 'the same old dream: world domination'- chimed with a post-Suez British public wary of imperial overreach. Hence the movie's pop art elements, which tipped the wink to the audience that they weren't meant to take this stuff seriously. Everything they were watching, Connery's sly, sideways-on performance kept reminding them, was all part of a big joke.
"Nobody saw this more clearly than [French auteur] Francois Truffaut. 'For me,' the director told an auteurist worshipper, 'the film that marks the beginning of the period of decadence in the cinema is the first James Bond—Dr. No. Until then the role of the cinema had been by and large to tell a story in the hope that the audience would believe it. There had been a few minority films which were parodies of this narrative tradition, but in the main a film told a story and the audience wanted to believe that story.' (Don Allen interviewing Truffaut, Sight and Sound, Autumn 1979.)
Christopher Bray
Sean Connery: A Biography
Pegasus Books, LLC
Copyright 2011 by Christopher Bray
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 3,059
|
About DDCRW Law
Landlord/ Tenant Disputes
The Death of "Tax-Free" Internet Shopping: Buyer AND Seller Beware
by dcdclaw | Aug 22, 2018 | Business Law
The recent decision from the Supreme Court of the United States in South Dakota Department of Revenue v. Wayfair, Inc. et al. is a major win for a State's ability to extend its taxing authority beyond its physical borders and a potential pitfall for an unwary online...
Part VI: What Employers Can Do To Prepare
by dcdclaw | Jun 20, 2018 | Business Law, Employment Law
On July 1, 2018, An Act to Establish Pay Equity (the "Act") goes into effect in Massachusetts. The Act prohibits discrimination in payment of wages on the basis of gender. Below is the final article in our series concerning the Act, which provides a list of "Action...
Part V: Safe Harbor-Voluntary Employer Salary Self-Evaluations
On July 1, 2018, An Act to Establish Pay Equity (the "Act") goes into effect. The Act prohibits discrimination in payment of wages on the basis of gender. Below is the fifth article in our series concerning the Act. Employers can protect themselves against...
Part IV: Restrictions on Inquiries by Employers and Other Important Provisions of the Act
by dcdclaw | Jun 6, 2018 | Business Law, Employment Law
On July 1, 2018 an Act to Establish Pay Equity (the "Act") goes into effect. The Act prohibits discrimination in payment of wages on the basis of gender. Below is the fourth article in our series concerning the Act, which examines restrictions on salary inquiries by...
Part III: Permissible Pay Disparities
by dcdclaw | May 30, 2018 | Business Law, Employment Law
On July 1, 2018, An Act to Establish Pay Equity (the "Act") goes into effect. The Act prohibits discrimination in payment of wages on the basis of gender. In our second article about the Act we examined the complicated definition of "comparable work". Below is...
Part II: What is Comparable Work Under the Pay Equity Act?
Part II: What is Comparable Work Under the Pay Equity Act? On July 1, 2018, An Act to Establish Pay Equity (the "Act") goes into effect. The Act prohibits discrimination in payment of wages on the basis of gender. In our first article about the Act, we examined the...
Does Child Support End in Massachusetts When a Child Turns 18?
How Do Children from Another Relationship Affect the Child Support You Pay or Receive?
Homestead or not to Homestead?
Paid Family and Medical Leave-Notice to Employees
State Leaders Delay Start of Payroll Deductions for Paid Family and Medical Leave
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 4,692
|
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|
{
"redpajama_set_name": "RedPajamaC4"
}
| 8,098
|
package com.facebook.buck.halide;
import com.facebook.buck.io.ProjectFilesystem;
import com.facebook.buck.model.BuildTarget;
import com.facebook.buck.model.BuildTargets;
import com.facebook.buck.rules.AbstractBuildRule;
import com.facebook.buck.rules.AddToRuleKey;
import com.facebook.buck.rules.BuildContext;
import com.facebook.buck.rules.BuildRuleParams;
import com.facebook.buck.rules.BuildableContext;
import com.facebook.buck.rules.SourcePathResolver;
import com.facebook.buck.rules.Tool;
import com.facebook.buck.step.Step;
import com.facebook.buck.step.fs.MakeCleanDirectoryStep;
import com.google.common.collect.ImmutableList;
import java.nio.file.Path;
public class HalideCompile extends AbstractBuildRule {
@AddToRuleKey
private final Tool halideCompiler;
@AddToRuleKey
private final String targetPlatform;
public HalideCompile(
BuildRuleParams params,
SourcePathResolver pathResolver,
Tool halideCompiler,
String targetPlatform) {
super(params, pathResolver);
this.halideCompiler = halideCompiler;
this.targetPlatform = targetPlatform;
}
@Override
public ImmutableList<Step> getBuildSteps(
BuildContext context,
BuildableContext buildableContext) {
Path outputDir = getPathToOutput();
String shortName = getBuildTarget().getShortName();
buildableContext.recordArtifact(objectOutputPath(getBuildTarget()));
buildableContext.recordArtifact(headerOutputPath(getBuildTarget()));
ImmutableList.Builder<Step> commands = ImmutableList.builder();
ProjectFilesystem projectFilesystem = getProjectFilesystem();
commands.add(new MakeCleanDirectoryStep(projectFilesystem, outputDir));
commands.add(
new HalideCompilerStep(
projectFilesystem.getRootPath(),
halideCompiler.getEnvironment(getResolver()),
halideCompiler.getCommandPrefix(getResolver()),
outputDir,
shortName,
targetPlatform));
return commands.build();
}
@Override
public Path getPathToOutput() {
return pathToOutput(getBuildTarget());
}
private static Path pathToOutput(BuildTarget buildTarget) {
return BuildTargets.getGenPath(buildTarget, "%s");
}
public static Path objectOutputPath(BuildTarget buildTarget) {
return pathToOutput(buildTarget).resolve(buildTarget.getShortName() + ".o");
}
public static Path headerOutputPath(BuildTarget buildTarget) {
return pathToOutput(buildTarget).resolve(buildTarget.getShortName() + ".h");
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,783
|
\section{Introduction}
Cataclysmic variables (CVs) are binary systems where a white dwarf
(WD) accretes material from a secondary star which is filling its
Roche lobe (see Warner 1995 for a review).
Recent theoretical works (e.g. Howell et al. 2001, and Kolb 2001)
predict that the majority of the CVs should be very old and evolved
systems. These systems are characterized by short orbital periods
(i.e. $P<$2 hr), low intrinsic luminosity ($M_V<11$ mag), and evolved low
mass secondary stars (see also Howell et al. 1995). However, the
number of currently known/observed short orbital period systems does
not match the expectation. Thus, in recent years, many surveys have
taken place to fill such a gap (e.g. SDSS by Szkody et al. 2002, HQS
by Gansicke et al. 2002, FSVS by Groot et al. 2003, etc). At a
same time, new ideas (e.g., the need to go even fainter) and different
accretion scenarios (e.g. Spruit and Taam 2001, Dubus et al. 2002),
have been presented and analyzed.
H$\alpha$0242-2802 (hereafter H$\alpha$0242) was a CV candidate within
the UK-Schmidt survey (Davenhall et al. 2001). It was confirmed
as CV by Howell et al. (2002), who observed it spectroscopically and
reported a spectrum very similar to that of WZ~Sge. Indeed, the
authors also advanced the hypothesis that H$\alpha$0242 belongs to the
same dwarf nova subclass of Tremendous Outburst Amplitude Dwarf novae
(TOADs, see Howell et al. 1995 for a review on TOADs). An interesting
implication of this would be that H$\alpha$0242 is quite high above
the galactic plane ($z=250$ pc), and resides in the hold disk - halo
population.
Here, we present time-series spectroscopy obtained with VLT+FORS2 with
the aim of determining the orbital period, the system parameters, the
emission lines characteristics, and comparing such values with those
of WZ~Sge, the prototype object of the very old evolved CV population
(see Howell et al., 2004). We present our spectroscopic observations
in Sec.~2, the data analysis in Sec.~3 and our conclusions in Sec.~4
\section{Observations and data reduction}
H$\alpha$0242 was observed at VLT+FORS2 using a series of 5 min
exposures over 4 consecutive hours. An 8-m class telescope is
mandatory in order to perform time resolved spectroscopy of faint
targets (H$\alpha0242$ B mag is $\sim$19, see Howell et al. 2002), and
determine the radial velocity curve of systems having orbital period
around 2 hr. We used FORS2 in Multi Object Spectroscopy (MOS)
mode with the holographic grism 1400V. We preferred the MOS mode to
the long slit spectroscopy (LSS), in order to cover a bluer wavelength
range\footnote{Namely $\lambda\lambda$ 4270-5550, rather than the
standard $\lambda\lambda$4560-5860.}. Indeed, the bluer wavelength
coverage allowed us to observe the Doppler broadened Balmer lines
H$\beta$ and H$\gamma$ (see Fig.~1). The slit width was set to 1''
and together with the grism 1400V provided a dispersion of 0.63
\AA/pix.
\begin{table}
\label{log}
\begin{center}
\scriptsize
\caption{The log. of observation.}
\begin{tabular}{cc}
date of obs. & 2002/09/10 \\
UT start & 05:41\\
UT end & 09:52 \\
average seeing & 0.7 \\
average transparency & CLR-THN\\
number of spectra & 40 \\
exptime per spectrum & 300 sec \\
instrument & FORS2\\
instr. set up & MOS/SR/no filter/GRIS 1400V\\
read out mode & 200kHz, 2x2, 1.25 (speed, binning, gain)\\
\end{tabular}
\end{center}
\end{table}
\begin{figure*}
\centering
\rotatebox{-90}{\includegraphics[width=14cm,]{2349fig1.eps}}
\caption{The average spectrum of H$\alpha$0242. The flux is
given in erg sec${-1}$ cm$^{-2}$ \AA$^{-1}$, but has not been slit or
sky corrected thus does not correspond to an absolute calibration.}
\label{f1}
\end{figure*}
The log. of observation is presented in Table~1 . A standard star
with the same instrument set up was observed one month later allowing
us to place the spectra on a relative flux scale. Fortunately,
absolute flux measurements are not needed for the analysis presented
in this paper.
The data reduction was performed through standard IRAF routines within
the packages {\it ccdproc}, {\it apex}, and {\it onedspec}.
\section{Data analysis}
\subsection{Overview/General properties}
We plot in Fig.1 the average spectrum of H$\alpha$0242. The
continuum shape is not as blue as in the previous observation of
Howell et al. (2002) and this is probably an artifact of the
calibration. The emission lines which are visible in the spectrum are
the two Balmer lines H$\beta$ and H$\gamma$ and the FeII lines from
multiplets 37, 38, 42 (the strongest), and 49. Visible are also the
HeI lines $\lambda$4471 and $\lambda$4713. The observation of optical
FeII emission has already been reported in the literature but never
with sufficient attention. We will develop a brief discussion about
FeII emission and their formation mechanism in CVs in Sec.~3.6. Here,
we note that all the emission lines have a similar shape. They are
double peaked with a deep central absorption core which extends below
the continuum. This is signature of high orbital inclination. The
B/R ratio of the {\it average spectrum} (Fig.~1) appears greater than
1 for the Balmer lines and $<$1 for the FeII. This is, at least in
part, an artifact of the flux calibration which introduces a red
continuum. Indeed, on one hand, the {\it normalized average spectrum}
shows that B/R$>$1 in the Balmer lines and B/R$\sim$1 in the FeII (42)
emission lines. On the other hand, the analysis of the {\it single
spectra} shows that the Balmer, HeI, and the FeII lines follow a
similar modulation throughout the orbit. This is well shown by the
trailed spectrograms that we present further below (see Fig.~5). A
possible explanation for the different B/R ratio between the Balmer
and the FeII lines in the {\it average spectrum}, may be a larger
fractional contribution of the hot spot in the Balmer lines, when it
is blue shifted.
We also observed a weak emission from the HeII $\lambda$4684. The HeII
line is not readily visible in the single spectra, nor in the
average one. It is visible only in the trailed spectrogram where
it produces a pure S-wave (see Fig.~5 and Sec.~3.4).
In the following sections we will analyze the spectra searching for
the orbital period of the binary system, and applying the usual
analysis for the accretion disk emission lines (RV measurements,
trailed spectra etc). We will pay particular attention to the
comparison between WZ~Sge and H$\alpha$0242.
\subsection{Period search}
\begin{figure*}
\centering
\rotatebox{-90}{\includegraphics[width=14cm,]{2349fig2.eps}}
\caption{The continuum and Balmer emission line light
curves. From top to bottom: the continuum flux measured at
5500\AA, H$\beta$ emission lines flux, and H$\gamma$ emission
line flux. }
\label{f2}
\end{figure*}
We searched for the binary system orbital period applying the
Phase Dispersion Minimization method (PDM method, Stellingwerf 1987)
to different emission line features\footnote{We measured in
particular: the position of the red and the blue peak in the emission
line, the position of the central absorption, and the line flux
barycenter.}. We did not discover any statistically significant
period. However, we found clear evidence for the orbital period by
plotting the light curve of the continuum flux measured at 5500\AA \
(see Fig.~2). Indeed, the light curve is characterized by a deep
eclipse of almost 2 magnitudes with periodicity of 107 min. Woudt et
al. (2004), determined the same orbital period through time resolved
photometry.
We also found a periodicity of 106 min from the radial velocity
measurements of the Balmer emission lines through the double Gaussian
fit method which was developed by Shafter (1983, and reference
therein).
Within this paper, we will adopt the period of 107 min and will phase
our spectra assuming as $T_0$ the time of the observed mid-eclipse
minimum\footnote{Due to the time resolution of our spectra, the
``true'' minimum could be deeper.}. Our approximate ephemeris is:
$HJD=2452527.89275(\pm 0.00395)+0.0743055(\pm 0.0017361)E $
where the uncertainty on the time $T_0$ corresponds to the time
resolution of our data points which is 341 sec, i.e. the sum of the
exposure time and the readout time of the mosaic CCD. Thus, the
uncertainty on the period (150 sec), was derived both from measuring
the width of the period peak in the standard PDM ($\theta$, P) plot
and an estimation from the eclipse light curves in Fig. 2.
\begin{figure}[h]
\centering
\rotatebox{-90}{\includegraphics[width=6.8cm]{2349fig3.eps}}
\caption{The diagnostic diagram. Squared symbols are for the
H$\gamma$ data points; circles are for H$\beta$ data points.
Filled symbols mark the best fit data points. }
\label{f1}
\end{figure}
\subsection{Radial velocity curves}
Knowing the orbital period we can phase the spectra and the
radial velocity measurements to derive a few parameters characterizing
the binary system. We can then produce trailed spectrograms and
Doppler maps to qualitatively analyze the evolution of the line
profile and the emission components.
In principle, radial velocity measurements of the accretion disk
emission lines provide valid information on the white dwarf orbital
motion. However, this does not appear to be the case for many CVs and,
in particular, the short orbital period systems (e.g. WZ~Sge, Mason
et al. 2000 and reference therein, and V893 Sco, Mason et al. 2001).
The exact causes of such behavior is not well known, nor is there a
model capable to explain why the radial velocity curves from different
emission lines yield discordant system parameters. Since we are not
yet able to quantify the discrepancy between the derived
quantities and the corresponding real values, it is still worthwhile
measuring and fitting the radial velocity curves of the emission lines
and determining coarse values which reasonably constraint the WD
Keplerian velocity and the systemic velocity. Of course, within our
uncertainties, the readers should be cautionary in their
interpretation.
We measured the radial velocity of the Balmer lines using the double
Gaussian fit method as defined by Shafter (1983, and references
therein). We used narrow Gaussian FWHM (4-2\AA), and Gaussian
separation steps of equal size. The diagnostic diagram in Fig.~3 shows
the fitting parameters corresponding to steps of 4 \AA. The best fit
for the H$\beta$ radial velocity curve is found at a Gaussian
separation of 60 \AA, while the best fit for the H$\gamma$ radial
velocity curve is at a separation of 47 \AA. We list in Table~2 the
best fit parameters and plot in Fig.~4 (two top panels) the radial
velocity curves. As expected and already observed in other short
orbital period systems, the two Balmer lines do not produce consistent
values for the WD Keplerian velocity and the systemic velocity, which
indeed differs by more than $3\sigma$. In addition we may note that
the two radial velocity curves consist of largely scattered points.
We believe that one possible explanation can be found in the fact that
the line profile is largely variable across different orbital cycles
(see Fig.~5 and Sec.~3.4). Also, the larger scatter of the velocity
measurements in the H$\gamma$ lines are probably due to the smaller
Gaussian separation, which, in principle, can be biased by the hot
spot S-wave. However, a larger separation does not produce a better
result due to the noisy wings of the H$\gamma$ emission line. We will
thus adopt a white dwarf Keplerian velocity of 99 km/sec (from just
the H$\beta$ line) throughout the present work. Since the systemic
velocity $\gamma$ resulting by the application of the Gaussian fit
method is the most un-reliable parameter (see Tappert 1999 for
details), we will adopt the value of $\gamma=22$ km/sec derived
below.
The HeII emission is visible only as an S-wave in the trailed
spectrogram (see Fig.5), thus, the only way we had to measure its
radial velocities was by cursor position and visual inspection.
\begin{figure*}
\centering
\rotatebox{-90}{\includegraphics[width=14cm,]{2349fig4.eps}}
\caption{The radial velocity curves. From top to bottom: 1)
H$\beta$ accretion disk emission, 2) H$\gamma$ accretion disk
emission, 3) HeII hot spot emission. }
\label{f4}
\end{figure*}
\begin{table}[h]
\label{log}
\begin{center}
\scriptsize
\caption{Best fit radial velocity curve parameters for the H Balmer
lines and HeII 4686.}
\begin{tabular}{cccc}
em. line & $\gamma$ (km sec$^{-1}$) & K$_1$ (km sec$^{-1}$) & $\phi_{R/B}$ \\
& & & \\
H$\beta$ & 53$\pm$6 & 99$\pm$9 & 0.056$\pm$0.014 \\
H$\gamma$ & 18$\pm$7 & 71$\pm$10 & 0.053$\pm$0.025 \\
HeII & 22$\pm$3 & 680$\pm$4 & 0.44$\pm$1E-3 \\
\end{tabular}
\end{center}
\end{table}
The radial velocity curve is plotted in Fig.~4, while its best fit
parameters are reported in Table~2. The derived $K_1$ value
corresponds to the Keplerian velocity of the outer edge of the
accretion disk, to a first approximation. Indeed, the real value
should be larger as the WD (instantaneous) radial velocity subtracts
off from each spectrum/measurement. However, the derived value
$K_1=680$ km/sec matches fairly well the half peak separation (HPS)
measured on the Balmer lines: $\langle HPS \rangle = 650 \pm 46$
km/sec. The HPS is also a good estimate of the Keplerian velocity at
the outer accretion disk edge. We used the value of K=680 km/sec to
determine the accretion disk size. Assuming a WD mass of 0.64
$M_\odot$ (see Sec.~3.5) we find $r_d\sim 1.8\times 10^{10}$
km/sec\footnote{The slightly smaller velocity derived from the HPS
produces a larger accretion disk radius of $r_d\sim 1.98\times
10^{10}$ km/sec. However, we believe that the average spectrum is
affected by both the orbital motion and the hot spot emission which
reduce the peak separation.}.
The derived value of $\phi_{R/B}=0.44$ is consistent with the standard
hot spot position at an angle of $\sim$50-60 deg from the line
connecting the two star centers of mass. While, the HeII $\gamma$
value corresponds to just the systemic velocity, within our
assumption. This is the value adopted in our next analysis.
\subsection{Trailed spectra and Doppler maps}
\begin{figure*}
\centering
\rotatebox{-90}{\includegraphics[width=13cm,]{2349fig5.ps}}
\rotatebox{-90}{\includegraphics[width=13cm,]{2349fig6.ps}}
\caption{Trailed spectrograms: top panel from left to right:
H$\gamma$, H$\beta$, and HeII $\lambda$4686 emission lines;
bottom panel: FeII 42 emission lines. }
\label{f5}
\end{figure*}
Trailed spectrograms and Doppler maps of the observed emission
lines are a common tool to qualitatively analyze the line forming
region. We thus plot in Fig.~5 the trailed spectrograms of the Balmer
lines and the HeII $\lambda$4686 (top panels of Fig.~5), as well as of
the FeII (42) lines (bottom panels of Fig.~5). All the emission lines
clearly show evidence for an eclipse, which implies a high orbital
inclination of the binary system (with possibly total eclipse of the
white dwarf itself). The Balmer and the FeII lines are double peaked,
i.e. form in the upper atmosphere/corona of the accretion disk. They
also show only weak evidence of the hot spot emission which indeed is
visible only in the phase ranges 0.25$\div$0.30 and 0.75$\div$0.85,
i.e. when it is seen from the ``outside'' and the ``inside''
respectively (see Mason et al. 2000). We also note that the
Balmer emission lines vary in width and intensity both across the
orbit and different orbital cycles (alternate spectra in the trailed
spectrograms belong to two distinct orbital cycles). The same behavior
is not evident in the FeII lines but we cannot say whether this is an
effect of the smaller S/N or is rather and an intrinsic properties of
the system.
On the other side, the HeII emission shows no evidence for the
accretion disk contribution and produces just a clear S-wave which is
signature of pure hot spot emission. This can be explained with the
fact that the high ionization emission lines can form only at high
temperatures, as in the hot spot region where the stream of in-falling
gas from the secondary star hits the outer edge of the accretion disk.
The Balmer, FeII, and HeII lines were also used to produce back
projected Doppler maps. The maps confirm our previous conclusions
about the emission lines from the Balmer series and low ionization
elements. They mostly form in the accretion disk while the hot spot
fractional contribution is small and washed out by the disk
emission\footnote{We may stress that this was not at all the case in
WZ~Sge where the hot spot emission was contributing up to 50\% of the
line flux.}. The high ionization emission line from HeII is confirmed
to form in the impact region within the accretion disk.
We plot on top of each Doppler map the Roche lobe geometry and the
stream trajectory derived for H$\alpha$0242 (see section 3.5). Also,
the circle drawn on top of the H and FeII Doppler maps fits by eye the
bulk of the accretion disk emission and corresponds to 0.55 of the
white dwarf Roche lobe radius ($r_{L1}$). In the case of the HeII
Doppler map we draw the circle at 0.76 $r_{L1}$, as derived from the
computation in Sec.~3.5.
\begin{figure*}
\centering
\rotatebox{-90}{\includegraphics[width=4.3cm,]{2349fig7.ps}}
\rotatebox{-90}{\includegraphics[width=4.3cm,]{2349fig8.ps}}
\rotatebox{-90}{\includegraphics[width=4.3cm,]{2349fig9.ps}}
\rotatebox{-90}{\includegraphics[width=4.3cm,]{2349fig10.ps}}
\rotatebox{-90}{\includegraphics[width=4.3cm,]{2349fig11.ps}}
\rotatebox{-90}{\includegraphics[width=4.3cm,]{2349fig12.ps}}
\caption{Back projected Doppler maps: from left to right, top to
bottom: H$\gamma$, H$\beta$, HeII $\lambda$4686, FeII(42)
$\lambda\lambda$4923, 5018, and 5169. See text for details. }
\label{f5}
\end{figure*}
\subsection{System parameters and geometry}
In the present section we will make use of the previous observations
and results to constraint the system geometry.
The continuum light curve at 5500 \AA \ shows a 2 mag deep eclipse
which is comparable to that observed in Z~Cha and OY~Car (e.g. Ritter
and Kolb 2004). We infer, to first approximation, that the orbital
inclination of H$\alpha$0242 must be similar to these two eclipsing
systems, hence $i\sim$82$^o$.
Howell and Skidmore (2002) present a $M_2$-$P$ relation which can be
used to predict the mass of the secondary star in both the hypothesis
of a pre- and post-orbital period minimum system. In particular, we
found a secondary star of mass $M_2=0.17 M_\odot$ and radius $R_2=0.19
R_\odot$ in the case of pre- orbital period minimum system; while, a
secondary star of mass $M_2=0.03 M_\odot$ and radius $R_2=0.11
R_\odot$, in case of a system which has already evolved past the
orbital period minimum.
Knowing the orbital inclination and the mass of the secondary star, we
can solve for $M_1$ (the white dwarf mass), the secondary star mass
function (i.e. equation 2.79 in Warner 1995). We derive a primary mass
of either 0.64 $M_\odot$ or $0.03 M_\odot$, depending on H$\alpha$0242
being a pre- or post-orbital period minimum system. The case $M_2=M_1=
0.03 M_\odot$ seems unlikely both because of standard stellar
evolution time scales (a star able to form a low mass WD of just 0.03
$M_\odot$ has not evolved off the main sequence, yet, within our
galaxy), and the standard scenario for interacting binaries. Thus, we
discard it and conclude that H$\alpha$0242 has not reached the orbital
period minimum yet and is likely to have a white dwarf of mass 0.64
$M_\odot$.
In the case of $M_1=0.64 M_\odot$, the mass ratio will be $q=0.27$,
the binary separation is $\sim 4.8\times 10^{10}$ cm and the radius of
the primary star Roche lobe is $2.4 \times 10^{10}$ cm. Thus, the
accretion disk radius (see Sec.~3.3) is $\sim 0.76$ times the size of
the Roche lobe, and 0.38 times the star separation $a$.
\subsection{The iron lines and the Balmer decrement}
As already pointed out, FeII emission are probably quite common
in CVs, though, they have never been studied carefully. In particular,
many times they have likely been identified either as HeI (due to
limited spectral rage) or a combination of FeII and HeI. In the case
of H$\alpha$0242, we see a multitude of FeII emissions and just weak
HeI lines. Thus, it is reasonable to conclude that (at least for this
system) the emission lines at 4924\AA \ and 5018\AA \ consist mostly
of FeII transitions.
FeII emission in the optical were observed since the 70s in Seyfert 1
galaxies (see Osterbrock 1975 and Phillips 1977). The two mechanisms
which are believed to produce optical FeII emissions in Seyfert 1
galaxies are: {\it 1)} the resonance fluorescence (e.g. Wampler \& Oke
1967), and {\it 2)} the collisional excitation (e.g. Boksenber et
al. 1975). In the first case the UV photons emitted by a hot source
($T_{eff}\leq25000$ K) are absorbed by the iron peak elements mostly
in the wavelength range 2300$\div$2800\AA. These UV absorption would
be followed by downward transitions in the optical. In the second
case the UV spectrum should be characterized by {\it emission}
resonance lines in the UV region.
Now, the optical spectrum of H$\alpha$0242 shows emission lines from
FeII 42, 49, 37 and 38 similar to what is seen in many Seyfert 1
galaxies. We do not have UV observation of H$\alpha$0242, still
several DN systems have been observed by HST, and for at least some of
them (OY Car, Horne et al. 1994; Z~Cha, WZ~Sge and V2051 Oph, Catalan
et al. 1998) the observations of an iron curtain (the UV absorption of
the iron peak elements) has been reported. It is thus reasonable to
infer that H$\alpha$0242 UV spectrum is characterized by iron peak
absorptions, and that the mechanism responsible for the observed
optical FeII emission is indeed resonance fluorescence possibly from a
disk wind. It would be interesting, now, to observe/recover high S/N
spectra of DNe to search for FeII emission in order to both derive the
fraction of DNe which show optical FeII emission lines and verify the
formation of these FeII lines through the fluorescence mechanism. We
are currently analyzing our database with such a purpose. Here, we can
comment that the optical spectrum of WZ~Sge in 2002 was showing FeII
optical emissions (see Fig.~4 of Howell et al. 2002), while the same
system was not showing optical FeII emission lines in 1996. This is
perfectly consistent with the fact that WZ~Sge underwent an outburst
in 2001 and the idea that WD became hotter as a consequence of the
accretion, exciting the FeII in an increased disk wind. It also
matches the observation by Catalan et al. (1998) who report the
weakest signature of iron curtain in WZ~Sge (before the 2001
outburst).
\begin{table}
\begin{center}
\scriptsize
\caption{The Balmer decrement as derived from the intensities of the
emission lines during times when they are not effected by the hot spot
emission. The value for the WZ~Sge 1996 spectra is from Mason et
al. (2000). The 2002 NTT WZ Sge values (see Howell et al. 2002) are
from a single spectrum covering 0.07 of the orbital period, while the
values for H$\alpha0242$ in 2002 (Howell et al., 2002) span 0.2 of an
orbital period. The intensity ratio of the blue and red peak for the
2003 spectra of H$\alpha$0242 (this work) has been derived from the
spectra within the phase ranges 0.02-0.11 and 0.43-0.56.}
\begin{tabular}{ccccc}
object & date & H$\alpha$/H$\beta$ & H$\gamma$/H$\beta$ &
H$\delta$/H$\beta$ \\
& & & & \\
WZ~Sge & 1996 & 3.82 & - & - \\
WZ~Sge & 08/2002 & 1.33 & 0.75 & 0.61 \\
H$\alpha$0242 & 08/2002 &
1.10 & 0.77 & 0.74 \\
H$\alpha$0242 & 11/2003 & - & 0.59 & - \\
\end{tabular}
\end{center}
\end{table}
In order to complete our comparison between H$\alpha$0242 and WZ~Sge,
we also compare the Balmer decrement in the two systems. Direct
comparison is not simple/possible because of the different wavelength
and phase coverages\footnote{Some of the data are in time resolved
mode, while others consist of just a single spectrum.} of the spectra
in our hand. Still, we report in Table~3 the Balmer decrements
measured in WZ~Sge before the outburst (spectra from Mason et al.
2000) and after the 2001 outburst (spectrum from Howell et al. 2002)
with the Balmer decrement of H$\alpha$0242-28 measured in 2001 (Howell
et al. 2002) and in this work. From Table~3, it is clear that: {\it
i)} the Balmer decrement in WZ~Sge before the 2001 outburst was larger
(and probably steeper) than after its outburst; {\it ii)} the Balmer
decrement in H$\alpha$0242 is flatter than in WZ~Sge (at least in
2002). We also note that in the 2003 spectra of H$\alpha$0242 there is
little or no difference in the Balmer decrement resulting from the
average spectrum and/or the time-resolved spectra (which exclude the
hot spot). This implies that the opacity and probably the gas physics
within the disk and the hot spot are similar. The conclusion is that
the line forming region in the accretion disk of H$\alpha$0242
consists of gas which is optically thicker, thus, probably denser
and/or warmer, than that in WZ~Sge.
\section{Summary and conclusions}
We have presented time series spectra of the CV H$\alpha$0242. The
object was identified as a CV candidate from an H$\alpha$-R band
survey (Davenhall et al. 2001) and further suspected as a candidate
TOAD by Howell et al. (2002). We determined the system orbital period
of 107 min and showed evidence for a very deep eclipse in the light
curve (Sec.~3.1), both reported also by Woudt et al. (2004). We infer
an orbital inclination of 82$^o$ from the observed eclipse depth. We
measured radial velocities of the Balmer lines and derived an
approximate value for the white dwarf Keplerian velocity
(Sec.~3.3). We constrained the secondary star on the basis of current
evolution theory and derived the mass of the white dwarf through
geometrical considerations (Sec.3.5). We found $M_2=0.17 M_\odot$ and
$M_1=0.64 M_\odot$ for the mass of the secondary star and the white dwarf,
respectively. We examined the line forming region within the
accretion disk (Sec.~3.4 and 3.6), observing a multitude of FeII
emission lines and a flat Balmer decrement.We note strong FeII
emission in the optical spectrum and postulate that many other
cataclysmic variables show it as well, but previous observations often
lacked of either the proper wavelength coverage and/or sufficient S/N.
We believe that the FeII lines are produced by the resonance
fluorescence mechanism and, therefore, that the white dwarf has a
relatively high effective temperature. From the Balmer decrement we
also conclude that the gas both in the accretion disk and the hot spot
is optically thick. The hot spot was found to not contribute
significantly to both the Balmer and the FeII lines. On the contrary,
we show evidence for pure hot spot emission in the HeII line
$\lambda$4684. We interpret the HeII emission as a signature of high
temperature in the impact region, thus of high density gas within the
accretion disk.
The results listed above clearly indicate that H$\alpha$0242 is a
short orbital period system which has not yet evolved past the orbital
period minimum. The accretion disk appearance (see trailed
spectrograms, Doppler maps and Table~3) resemble that of a normal
SU~UMa star.
\begin{acknowledgements}
The authors wish to thank the ESO Director for the generous allocation
of time allowing these observations to be made.
\end{acknowledgements}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,864
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Q: SSH from Linux to Windows does not execute commands in windows environment(only as background process) I am somewhat out of my dept here, so please go easy on me. I installed Openssh on both linux vm and Windows vm in vmware. As far as I can tell this implements some form of cygwin on windows. I can ssh to the user folder:
C:\Program Files\OpenSSH\home\ATV>
, like this:
ssh -p22 -t ATV@DESKTOP-CGHF9HU bash
or like this:
ssh -p22 -t ATV@DESKTOP-CGHF9HU cmd OpenExcelFile.vbs
or similar with cmd or sh. I can cd and DIR to my hearts content, and I can run .exe's, scripts you name it. It is just that they all start as background processes.
I want to be able to trigger ACTUAL execution of applications in the foreground.
The basics of it, I can access the command line and my windows os system through my linux terminal. I can dir file system and cd everything there. But I cannot run anything as foreground application, only as background processes, and I would like to trigger the actual opening of Microsoft Excel in the windows vm gui?
EDIT:
The problem is the display.... I can start Microsoft Excel, and see it running in background processes. So I guess the question would be how to be able to :
*
*Either start an application in the foreground over ssh.
*Or trigger a script that would open Excel in a "window".
Right now I am transferring the "file" via scp from python
os.system('scp CalculatedOutput/Opera.csv ATV@DESKTOP-CM5F9HU:')
os.system('ssh -t ATV@DESKTOP-CGHF9HU RunIf.vbs')
, trying to open the script through ssh.
I also managed to find version 1.1 of winexe, but there seem to be a lot of problems with this client. I have scoured the net, but no combinations of the syntax seems to work:
os.system('./winexe --user=\DESKTOP-CGHF9HU/ATV%Mypassword //windows.domain.local --interactive=1 --uninstall --system OpenExcelFile.vbs')
My guess would be that some people know how to do(hack) this. Why you would want to ? Unless you are a unix purist I would think it obvious...
I didn't have the opportunity to give up on this, so I continued digging. As mentioned in the comments below there are indeed access restrictions, and I guess microsoft wouldn't want to allow access to their os. Nevertheless, seeing as your other windows settings allow sharing and your syntax is correct:
./winexe -U ATV%password //DESKTOP-CGHF9HU 'cmd.exe'
, you can edit your registry as per https://tommynation.com/enable-remote-access-administrative-shares-windows-10/
, and the winexe1.1 will indeed allow you access. Now I know this isn't safe, but all my vm's are physically under my control so I DO NOT CARE.
Moving on the problem still persists though, as my code above actually gives you access to the windows command line, but still opens applications in the background. I can not bring them to the foreground( as in -fg), and the purpose is defeated. Question remains, how do I get them to display ?
I can do the same thing with openssh and cygwin so what is then the purpose of the winexe application ?
Edit2:
Well, I'm not going to call it quits here, but The best way for me to do this at the moment, is transferring the file in question, then running the "if(file exists)" script at the same time over ssh(From python):
os.system('scp CalculatedOutput/Opera.csv ATV@DESKTOP-CGHF9HU:')
os.system('ssh -p22 -t ATV@DESKTOP-CGHF9HU "cscript RunIf.vbs"')
The vbscript for automating Excel is automatically run in the background with Excel, and I guess the fact that I manually have to open Excel for visualization is something I will have to live with for the time being. That WAS my original question though, so I will still leave this unanswered.
A: Not sure if it will work for all combinations of windows/linux
I wanted for logged windows user to see what ssh-ed user is doing, and vice versa, I needed for ssh linux session to "see" programs that it started in foreground.
So only way i got it to work was for that logged windows user and linux ssh user are the same, and that windows user started its own ssh with command
C:\OpenSSH\sshd.exe -f C:\wherever\sshd_config
If sshd was started as a service, or as a scheduled task, it didn't work.
Hope it helps someone
A: When you login as an ssh client to cygwin ssh server it seems like ssh session does not have access to all of the windows commands.
But there is a workaround, the idea is
*
*create service to execute windows command and launch it from startup script
*use this service by adding command to execute at file at cygwinhome.
Here is the implementation.
Service reads commands from ~/winexec.bat file one line single time and passes this line to cmd.exe:
tail -f ~/winexec.bat|xargs -l cmd /c
Here is how to setup it in simple way (you can copy and paste following commands to cygwin terminal)
Use cygwin shell to setup service at windows:
# findout dos path of bash
bash=`which bash|xargs cygpath -d`
# findout full path to startup folder - works for windows 10
startup_dir="$APPDATA/Microsoft/Windows/Start Menu/Programs/Startup/"
# create batch which launch cygwin service
# to execute windows commands
cmd=">~/winexec.bat;tail -f ~/winexec.bat|xargs -l cmd /c"
# and put it to startup folder
service_fn="$startup_dir/winexec-service.bat"
echo $bash -l -c '"$cmd"'>"$service_fn"
This service will start at Windows startup or you can start it immediately by double click $startup_dir/winexec-service.bat
Now to execute some troubled command using ssh from linux just add new line with command to ~/winexec.bat like this:
ssh 192.168.2.107 'echo troubled-command>>~/winexec.bat'
Voilà, you have just launched full blown windows command at cygwin host from linux shell prompt!
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,044
|
Q: Abandon cart or basket recovery Any one aware if Salesforce Marketing Cloud provide basket recovery or Abandon cart off the shelf.
A: Andrew is, not surprisingly, spot on. We've seen this done a few ways. On the one hand, data is sent into Marketing Cloud and the emails triggered there. The other is to monitor the basket outside of Marketing Cloud via a 3rd party (with a custom built solution) and have it trigger the email in Marketing Cloud. In our experience, we've seen the latter (3rd party) far more frequently than the former and, in general, think it is easier to set-up and manage over time. It does, however, come with some restrictions.
Regarding 3rd parties to look at, WindsorCircle is definitely a good place to start. Without trying to be too self serving, we'd also recommend taking a look at ourselves (getstride.com). It is a different approach to Windsor (not necessarily better or worse) and may give you some perspective on options. Based on your other questions on SE, either Windsor or Stride may also be able to address some of the other workflows you are currently handling manually. Hope that helps and good luck!
A: Might be too late for this but Salesforce has released a solution kit for Cart Abandonment
https://help.salesforce.com/articleView?id=cross_cloud_marketing_commerce_kit_abandoned_cart.htm&type=5
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 532
|
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