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\section{Introduction}
The origin and the nature of ultra-high energy cosmic rays
(UHE CRs), with energies $E\ga10\,$EeV
($1\,{\rm EeV}=10^{18}\,$eV), are still unknown despite several
generations of experiments, most notably the Haverah
Park~\cite{Haverah}, the Akeno Giant Air Shower
Array (AGASA)~\cite{Yoshida1,Hayashida1}, and the Fly's
Eye~\cite{Bird1} experiments. Data from this latter
seem to indicate that the UHE CR component is mainly
composed of protons\cite{Bird1}. At these energies, protons
cannot be confined within the Galactic magnetic field. Thus, the
isotropy of the arrival directions of most of the observed UHE
CRs~\cite{Hayashida1}, or at least the absence of a significant
correlation with the Galactic plane~\cite{Stanev},
suggests that UHE CRs are extra-galactic in origin.
However, such protons would leave a distinct signature in the
energy spectrum, the so-called Greisen-Zatsepin-Kuzmin
high energy cut-off~\cite{GZK} (hereafter GZK cut-off) around
$E\simeq70\,$EeV, due to pion
production on the cosmic microwave background by nucleons with
$E\ga70\,$EeV. There is no strong experimental evidence for this
cut-off and the detection of particles with energies as
high as $E\sim300\,$EeV cannot be easily explained in this
frame. No compeling astrophysical candidate for the
source of the highest energy events could be found
within $\simeq100\,$Mpc \cite{ES,HVSV},
although photopion production limits the range of
nucleons with $E\simeq100\,$EeV to about 30 Mpc. Heavy nuclei
would be disintegrated over similar or slightly larger
distances~\cite{Stecker}, and similar
problems arise for the less likely option of a $\gamma$--ray
\cite{HVSV}, for which the effective attenuation length in
electromagnetic cascades lies between 1 and 20 Mpc, depending on
the poorly known strength of the universal radio
background. Finally, neutrino primaries in general imply too
large a flux because of their small interaction probability in
the atmosphere \cite{SL}.
As to the theoretical models of the origin of UHE CRs, the most
conventional scenario involves
first-order Fermi acceleration of protons in powerful
astrophysical shocks, for instance in the hot spots of
radio-galaxies \cite{RB}. More recently, it was suggested
that protons could be accelerated up to $E\sim10^{21}\,$eV
in fireball models of cosmological $\gamma-$ray bursts
\cite{Waxman1,Waxman2,Vietri1,MU}. In order to reconcile the
observed rates of UHE CRs and cosmological $\gamma-$ray bursts
within $D\sim30\,$Mpc, however, the arrival time of UHE CRs
would have to be spread over $\Delta\tau\ga50\,$yrs, for
instance through deflection in large-scale magnetic
fields~\cite{Waxman1,Waxman2,Waxman3}. As another class of
models, topological defects, possible relics of
early Universe phase transitions, could release supermassive
``X'' particles with mass around the Grand Unification Scale,
through physical processes such as collapse or annihilation
\cite{BHS}. These X particles would subsequently decay to jets
of UHE CRs, with a likely dominance of $\gamma-$rays above
$\simeq50\,$EeV, and an energy spectrum significantly harder
than in the case of shock acceleration~\cite{Hill,Sigl}.
Charged UHE CRs, such as protons or electromagnetic cascades
initiated by a $\gamma-$ray primary, are subject to
energy-dependent deflection, and hence energy-dependent time
delay, in large-scale magnetic fields. The r.m.s. strength
$B_{\rm rms}$ and the coherence length $l_{\rm c}$ of
extra-galactic magnetic fields are thoroughly unknown,
although they are bound by Faraday rotation data to
$B_{\rm rms}l_{\rm c}^{1/2}\la10^{-9}\,{\rm G}\,{\rm
Mpc}^{1/2}$~\cite{Kronberg}. Different authors have proposed to
use UHE CRs
to probe extra-galactic magnetic fields; some rely on
the magnitude of the time delay and the deflection
\cite{Plaga,WC}, or on some features of the angle-time-energy
images of UHE CRs \cite{WM,LSOS}, or even on
synchrotron loss signatures in the energy spectrum of
electromagnetic cascades \cite{LOS}. In a previous
paper, we discussed how information on both the extra-galactic
magnetic field and the origin of UHE CRs could be left in
angle-time-energy images of clusters of proton UHE CRs
\cite{LSOS}. We applied this study to a maximum likelihood
analysis of the three pairs of UHE CRs \cite{SLO},
that were reported by the AGASA experiment \cite{Hayashida1}. In
order to do so, we devised a Monte-Carlo code that follows the
propagation of UHE protons and calculates a likelihood as a
function of the parameters characterizing the origin of these
UHE CRs and the intervening magnetic fields.
Future large scale experiments~\cite{proc}, such as the High
Resolution Fly's Eye \cite{Bird4}, the Telescope Array \cite{Teshima},
and most notably the Pierre Auger project \cite{Cronin}, should
allow to detect clusters of $\ga20$, and possibly more, UHE
CRs per source, if the clustering suggested by the AGASA
results~\cite{Hayashida1}
is real. The recently proposed satellite observatory
concept for an Orbital Wide-angle Collector (OWL)~\cite{OWL}
might even allow to detect clusters of hundreds of events by
watching the Earth's atmosphere from space. In the present
paper, we wish to examine how
magnetic fields could affect the observations of clusters
of UHE CRs by future large-scale experiments. We thus assume
that UHE CRs are indeed dominantly protons, and that the magnetic fields
are strong enough to influence their propagation (see below).
We then simulate the
injection, the propagation, and the detection of UHE CRs
originating from a given source. Finally, we
perform a maximum likelihood analysis on these clusters
of typically $20-50$ particles, and attempt to reconstruct the
physical parameters describing the source and the magnetic
fields.
We describe the simulations in Section~2, and discuss the
reconstruction of the different parameters in Section~3; we briefly
summarize our results in Section~4. We use natural units,
$\hbar=c=1$, throughout the paper.
\section{Simulation of clusters of Ultra-High Energy Cosmic
Rays}
Protons of ultra-high energy are subject to the following
physical processes: energy loss through pair production and
photo-pion production (the latter for $E\ga70\,$EeV) on the cosmic
microwave background, and deflection in the extra-galactic
magnetic field. In photo-pion production, a proton may be
converted to a neutron, that either turns back into proton
through photo-pion production, or decays to a proton on a
distance $\simeq1(E/10^{20}\,{\rm eV})\,$Mpc for a neutron
energy $E$. Pair production is treated as a continuous energy loss
\cite{CZS}. We treat photo-pion production as a stochastic
energy loss; it is important to do so, as the stochastic nature
of this process imprints significant scatter in arrival time and
energy for UHE CRs above the GZK cut-off, as discussed in
Ref.\cite{SLO}. We
model the extra-galactic magnetic field as a gaussian random
field, with zero mean, and a power spectrum given by
$\left\langle B^2(k)\right\rangle\propto k^{n_B}$ for
$k<2\pi/l_c$, and $\left\langle B^2(k)\right\rangle=0$ otherwise.
The cut-off, $l_c$, characterizes the coherence length of the
field. The field is actually calculated on a grid of inter-cell
separation $a$ and is tri-linearly interpolated between the
lattice points such that $l_c\simeq a/\pi$ effectively. The
amplitude of the field is normalized to the r.m.s. strength
$B_{\rm rms}^2\equiv V/(2\pi)^3\int d^3{\bf k}B^2({\bf k})$, and
our model for the extra-galactic magnetic field is thus
described by the three parameters $n_B$, $l_c$, and
$B_{\rm rms}$. Fiducial values for these parameters are
$n_B\simeq0$, $l_c\ga100\,$kpc, and $B_{\rm rms}\la10^{-9}\,$G.
This statistical description of the field allows to treat
deflection of UHE CRs in the most general case, as discussed in
Ref.\cite{SLO}. We also note that any relative motion between
observer and source with relative velocity $v$ would introduce
effects only on timescales larger than $l_c/v$ which is much
larger than delay times and experimental lifetimes. It is,
therefore, justified to assume a stationary situation.
The numerical code that we use to follow the propagation of UHE
protons in an extra-galactic magnetic field is described in
detail in Ref.\cite{SLO}; here, we summarize its main features.
Protons are injected with a flat energy spectrum, and propagated
in a given direction in the extra-galactic magnetic field over
a distance $D$, from the source to the detector. Due to the
stochastic deflection, care has to be taken in how one states
whether different UHE CRs, that have followed different paths,
actually reached the same detector, or not \cite{SLO}.
During their propagation, UHE CRs acquire energy-dependent
deflection $\theta_E$ and time delay $\tau_E$. With a given
sample of nucleons, one can construct different
histograms, in time, angle, and energy, for different values of
the differential injection index $\gamma$, and of the
fluence $N_0$. Histograms are then smeared out in energy with
$\Delta E/E=0.14$, {\it i.e.} a high resolution typical of
future large-scale UHE CR experiments; histograms are also
convolved in time with a top-hat of width $T_{\rm S}$, in order
to simulate emission of particles at the source over a timescale
$T_{\rm S}$. Once the histogram is obtained for different values
of the above parameters, clusters of UHE CRs can be obtained by
picking at random a time window of length
$T_{\rm obs}\simeq5\,$yr, which corresponds to
the lifetime of the experiment, and dialing Poisson statistics over the
histogram. We do so in order to simulate UHE CR clusters of
events.
Conversely, one can use the above code to perform Monte-Carlo
simulations of UHE CR injection, propagation, and detection, and
calculating a likelihood of a given histogram for a given
cluster of events, where the histogram, hence the likelihood, is a
function of the physical parameters described above.
The likelihood is calculated in the standard
way for each observed event cluster, using Poisson statistics,
\begin{equation}
{\cal L}\left(\tau_{100},T_{\rm S},D,\gamma,
N_0,l_c,n_B\right)\equiv\left\langle\prod_{j=1,N}
e^{-\rho_j}\frac{\rho_j^{n(j)}}{n(j){\rm !}}\right\rangle\,,
\label{likelihood}
\end{equation}
where $\rho_j$ is the predicted number of events in cell $j$,
and $n(j)$ is the number of observed events
in cell $j$ for the cluster under consideration.
Each cell is defined by a time coordinate and an
energy. The time-energy histogram is binned to logarithmic
energy bins of size 0.05 in the logarithm to base 10 (as opposed
to 0.1 in Ref.~\cite{SLO} to account for improved energy resolution
of future experiments), and to $0.1\,$yr in linear time bins.
The product in Eq.~(\ref{likelihood}) extends over all energy bins
(from $10^{1.5}\,$EeV to $10^4\,$EeV) and over all time bins
within an observational time window of
length $T_{\rm obs}$; we took $T_{\rm obs}\simeq5\,$yr as a
projected lifetime of a next generation experiment such as the
Pierre Auger Project. The brackets in Eq.~(\ref{likelihood}) indicate
that the likelihood has already been averaged with equal weights
over the position of the observational time window on the
time delay histogram of the UHE CRs, as well as over
different realizations of the extra-galactic magnetic field
between the source and the observer.
The next step is to attempt to reconstruct the parameters in
Eq.~(\ref{likelihood}) from the maximum of the likelihood.
Future experiments are expected to produce as many as
$\ga100$ particles with $E\ga50\,$EeV, if the AGASA pairs
are real. In the present work, we prefer to remain conservative,
and we simulate clusters of $20-50$ particles with
$E\ga30\,$EeV.
In Ref.\cite{LSOS}, we discussed the possible different cases
of UHE proton images in time, angle and energy, and how, in each
case, qualitative information could be gained on the magnetic
field and the origin of UHE CRs. Here, we will discuss how each
physical parameter can be reconstructed, and in which case. The
physical parameters that govern the UHE CRs images are: the time
delay $\tau_{100}$, normalized at 100EeV, the coherence length
$l_c$, the power spectrum index $n_B$, the distance $D$, the
emission timescale $T_{\rm S}$, the differential injection
index $\gamma$, and the fluence $N_0$. The time delay is given
by \cite{WM}:
\begin{equation}
\tau_E\,\simeq\,
1.4\,\left(\frac{3+n_B}{2+n_B}\right)
\left(\frac{D}{30\,{\rm Mpc}}\right)^2
\left(\frac{E}{100\,{\rm EeV}}\right)^{-2}
\left(\frac{B_{\rm rms}}{10^{-11}\,{\rm G}}\right)^2
\left(\frac{l_{\rm c}}{1\,{\rm Mpc}}\right)\;{\rm yr}.
\label{t_delay}
\end{equation}
Hence, information on $B_{\rm rms}$ is contained in
$\tau_{100}$. Both the coherence length and the distance play a
double role. The coherence length not only contributes to the
time delay, it also influences the scatter around the mean of
the $\tau_{100}-E$ correlation \cite{WM}: if
$D\theta_E/l_c\ll1$, all UHE CRs have experienced
the same magnetic field
structure during their propagation, hence the scatter is
expected to be very small in the absence of pion production;
inversely, if $D\theta_E/l_c\gg1$,
the scatter is expected to be significant,
$\Delta\tau_E/\tau_E\sim60$\%, even for negligible energy
loss. The distance also enters the time
delay, and it also governs the amplitude of pion production,
hence the high energy part of the spectrum.
A cluster is seen on the detector as a tri-dimensional image
in angle, time and energy. As the Monte-Carlo likelihood
calculation is very time and memory intensive, we only focus on the
time-energy images in the following. Obviously, information is
also contained in the angular image itself of the cluster.
For instance, in the limit where $D\theta_E/l_c\ll 1$, one
expects to detect a single image,
albeit shifted by a sytematic offset
$\theta_E$ from the true location of the source, where
$\theta_E$ is tied to the time delay through
$\tau_E\simeq D\theta_E^2/4$. Below the GZK cut-off, its
angular size $\Delta\theta/\theta\ll1$.
Note that, provided the cluster
is seen at different energies, and $\theta_E$ is greater than
the angular resolution, the zero-point for $\theta_E$ can be
reconstructed, as $\theta_E\propto E^{-1}$. In the opposite
limit, $D\theta_E/l_c\gg 1$, one expects the image to be
centered on the source, with an r.m.s. angular size $\theta_E$.
In the intermediate limit, one expects to detect several images.
Moreover, if $\theta_E$ can be measured, it provides an estimate
of the combination $DB_{\rm rms}^2l_c$.
Main features of the time-energy images of clusters of UHE
protons are described in detail in Ref.\cite{LSOS}. We summarize
these results briefly, as they are important to the following.
If both $T_{\rm S}<\tau_{100}$, and $\tau_{100}$ is small
compared to
$T_{\rm obs}$, arrival time and energy are correlated according
to $\tau_E\propto E^{-2}$; see Fig.~\ref{F1a}. A source, such that
$\tau_{100}\gg T_{\rm S}$ and $\tau_{100}\gg T_{\rm obs}$, can
be seen only in a limited range of energies, at a given time, as
discussed in Ref.\cite{WM}, as shown in Fig.~\ref{F1b},\ref{F1c}.
Below the GZK cut-off, the width of
this stripe, in the time-energy plane and within the
observational window of length $T_{\rm obs}$, is then tied to
the ratio $D\theta_E/l_c$, as discussed above.
At the other extreme, a source emitting continuously at all
energies of interest here, {\it i.e.} with $T_{\rm S}\gg\tau_{30}$ and
$T_{\rm S}\gg T_{\rm obs}$, yields a time-energy image
in which the distribution of arrival time
{\it vs.} energy is uniform, {\it i.e.} events of any energy can
be recorded at any time, as shown in Fig.~\ref{F1d}.
Finally, for a source, such that $\tau_{100}< T_{\rm S}$ and
$\tau_{30}> T_{\rm S}$, together with $T_{\rm S}\gg T_{\rm obs}$,
there exists an energy $E_{\rm C}$, such
that $\tau_{E_{\rm C}}=T_{\rm S}$. In this case, protons with an
energy lower than $E_{\rm C}$ are not detected, as they could
not have reached us within $T_{\rm obs}$, even if they were
among the first emitted. However, protons with an energy higher
than $E_{\rm C}$ are detected as for a continuously emitting
source, {\it i.e.} with a uniform distribution of arrival times
{\it vs.} energy, see Fig.~\ref{F1e}.
Typical simulated clusters corresponding
to these five main situations are shown in
Figs.~\ref{F1a}-\ref{F1e}. The fluence $N_0$ was normalized in each
case so that $\simeq40$ events are expected within 5 years.
\section{Maximum likelihood reconstruction}
In this section, we discuss, in turn, how each parameter can be
obtained from a likelihood study of UHE CR clusters. Certain
marginalizations of Eq.~(\ref{likelihood}) are used whenever the
focus is only on one or a part of the parameters. The other
parameters are then averaged or integrated over, applying weight
functions ({\it i.e.}, Bayesian priors) that represent the prior
knowledge on their values. As we have currently no
information on the fluence, the emission timescale $T_{\rm S}$ and
the time delay $\tau_E$, the prior chosen would be uniform in the
logarithm of these parameters. However, we note that the time delay
$\tau_E$ is bounded from above by the Faraday rotation data bound on
$B_{\rm rms}l_c^{1/2}$ as combined with Eq.~(\ref{t_delay}). Moreover,
information contained in the angular image should also be included in
the prior on $\tau_E$, as $\tau_E\simeq D\theta_E^2/4$. The
marginalization over the injection spectral index $\gamma$ is achieved
through averaging with equal weights.
Although we focus on only one source, future large-scale
experiments are expected to detect a large number of individual
sources. Obviously, this would considerably increase the sensitivity
to the physical parameters.
\subsection{Time delay $\tau_E$}
Here we assume that the source is a burst, {\it i.e.}
$T_{\rm S}\ll 1\,$yr; we will discuss
the case where $T_{\rm S}\gg1\,$yr in the section concerning
$T_{\rm S}$.
If the time delay is small compared to the length of the
observational window, the time-energy correlation is scanned
through, and, as Fig.~\ref{F1a} reveals, as simple fit of
$\tau_E\propto E^{-2}$ would allow to determine the zero-point
of emission, hence the time delay. This constitutes a
measurement of the combination
$DB_{\rm rms}l_c^{1/2}$. Our likelihood simulations confirm
that, for the cluster shown in Fig.~\ref{F1a} for instance,
$\tau_E$ is obtained within a factor 2. The source
is found to be a burst with a high level of confidence.
When the time delay gets significantly larger than
$T_{\rm obs}$, its actual value cannot be reconstructed
from the maximum of the likelihood. This case
corresponds to the clusters shown in Figs.~\ref{F1b} and~\ref{F1c}.
Indeed, the likelihood is degenerate in the parameters $N_0$ and
$\tau_{100}$, as it depends mainly on the rate of detection
$N_0/\Delta\tau_{100}$, where $\Delta\tau_{100}$ is the scatter
in time around the mean of the $\tau_E-E$ correlation.
As long as $N_0$ is unknown, only a lower
limit to $\tau_{100}$, typically $\tau_{100}\mathrel{\mathpalette\fun >} T_{\rm obs}$, can
be placed. The likelihood, as calculated for the cluster shown in
Fig.~\ref{F1b}, and marginalized over all parameters except
$\tau_{100}$ and $\gamma$, is shown in Fig.~\ref{F2} in order to
illustrate this point. The distance and the coherence length
cannot be readily obtained in this case, as we will discuss further
below. Although only a lower
limit could be placed on the time delay, we note that,
when combined with the Faraday rotation bound, this would still allow
to bracket the strength of the extra-galactic magnetic field, within
less than a few orders of magnitude.
At this point, the information
contained in the angular image of the source becomes important. If the
angular image is not resolved, this translates into an upper limit on
$\tau_E/D$, which may supersede the Faraday rotation bound, see
Eq.~(\ref{t_delay}), and Eq.~(\ref{ang}) below. At the other extreme,
for a sufficiently large time delay, $\theta_E$ should in principle
be measurable, as
\begin{equation}
\theta_E\simeq0.02^{\circ}\left(\frac{D}{10\,{\rm Mpc}}\right)^{-1/2}
\left(\frac{\tau_E}{1\,{\rm yr}}\right)^{1/2}\,.\label{ang}
\end{equation}
Obviously, resolving the angular image would change the prior for $D$
and $\tau_{100}$; it would sharpen the maximum likelihood
reconstruction, notably with respect to the various scenarios
discussed in Section~2. We have not included this angular effect
in a systematic way; a quantitative treatment of the angular images
will be the subject of a future study. We note that the angular
resolutions of future UHE CR experiments are fractions of a degree,
hence the information contained in the angular image becomes
significant for $\tau_{100}\mathrel{\mathpalette\fun >} 10\,$yr.
\subsection{Distance $D$}
As mentioned above, the distance enters the likelihood mainly
through the amplitude of pion production. As long as the
high energy tail of the spectrum, {\it i.e.} $E\ga50\,$EeV,
can be observed, the distance is thus obtained with a reasonably good
accuracy from the likelihood, as marginalized over $T_{\rm S}$,
$\tau_{100}$, $N_0$, and $\gamma$. In particular, the likelihood is
sensitive to the distance if the source has a large emission
timescale, $T_{\rm S}\gg\tau_{100}$. The standard error
is then roughly a
factor $\simeq 2$. For example, the cluster of Fig.~\ref{F1d}
shows a factor $\simeq5$ difference in the marginalized
likelihood for $50\,$Mpc (the true value) and $30\,$Mpc; such a
factor is a typical value. The difference between $50\,$Mpc and
$5\,$Mpc is typically a factor $\simeq15$.
If $T_{\rm S}\ll\tau_{100}\mathrel{\mathpalette\fun <}
T_{\rm obs}$, and the range of energies seen by
the detector is above the GZK cut-off, the distance can still be
evaluated, albeit with a somewhat larger error. Typical
differences in the likelihood between 50 and $30\,$Mpc and 50
and $5\,$Mpc are factors $\simeq2$ and $\simeq6$,
respectively.
In the intermediate case where $\tau_{100}$ and
$T_{\rm S}$ are comparable, so that $\tau_{E_{\rm C}}=T_{\rm S}$
for an $E_{\rm C}$ in the observable energy range, the sensitivity to
$D$ is the better the lower $E_{\rm C}$, albeit not very strong. The
difference in the likelihood between $50\,$Mpc and $30\,$Mpc is
typically a factor 3 or less ({\it e.g.}, for the cluster shown in
Fig.~\ref{F1e}). It quickly rises, however, to a factor $\simeq20$
for clusters of the order of 100 events.
We note that, due to the comparatively limited energy range seen
in this case, there is a partial degeneracy
between $D$ and the injection spectrum parametrized by
$\gamma$. For example, the marginalized likelihood does not
change significantly if $D$ is decreased and $\gamma$ is increased
({\it i.e.} a softer injection spectrum is assumed) at the same time.
Other cases, {\it e.g.}, as shown in Figs.~\ref{F1b}
and~\ref{F1c}, do not allow to reconstruct $D$.
\subsection{Emission timescale $T_{\rm S}$}
If the emission timescale is larger than the width of the
observational window, the likelihood becomes
degenerate in the ratio $N_0/T_{\rm S}$, and only a lower limit
to $T_{\rm S}$ can be obtained, typically
$T_{\rm S}\mathrel{\mathpalette\fun >} T_{\rm obs}$.
However, if the time delay at some intermediate energy, between
say $30\,$EeV and $100\,$EeV, is sufficiently large, and comparable
to the emission timescale, then both the time delay, and the
emission timescale, can be measured as long as a lower energy
cut-off is visible above which the emission appears
continuous. This case corresponds to
the cluster shown in Fig.~\ref{F1e}. Indeed, if the time delay
is sufficiently large, then $\theta_E$ can be observationally
measured according to Eq.~(\ref{ang}).
As discussed above, the likelihood has some sensitivity to the
distance as long as events are observed over a reasonable range of
energies. This sensitivity also depends strongly on the statistics of
the UHE CR cluster. Since
$\tau_E\simeq D\theta_E^2/4$, $\tau_E$ is obtained, and, as
discussed in Ref.\cite{LSOS}, the emission timescale corresponds
to the time delay at the cut-off energy $E_{\rm C}$, below which
no UHE CR are recorded within $T_{\rm obs}$, as follows from the
definition of this lower cut-off energy,
$\tau_{E_{\rm C}}=T_{\rm S}$.
In reality, however, the image observed in such a situation will
appear as a burst with a large time delay most of the time:
for $E<E_{\rm C}$, the image is similar to that of a burst with
a large time delay, as $\tau_E> T_{\rm S}\gg T_{\rm obs}$, {\it
i.e.} only a limited range in energies is detected. Because
$\tau_{30}>\tau_{E_{\rm C}}$, most sources are seen at
$E<E_{\rm C}$ rather than at $E>E_{\rm C}$, where the image is
similar to that of a continuous source. Notably, the likelihood
for a bursting source with $T_{\rm S}\simeq0$ does not exclude
the above intermediate situation, for a cut-off $E_{\rm C}$ above
the observed stripe in the time-energy image of a bursting source.
In the case of Fig.~\ref{F3}, corresponding to the cluster shown in
Fig.~\ref{F1c}, the stripe is observed between $\simeq30\,$EeV and
$\simeq80\,$EeV, and the likelihood does not exclude the above
intermediate case with $E_{\rm C}\ga80\,$EeV.
Needless to say, the best reconstruction of $\tau_{100}$ and
$T_{\rm S}$ takes place when the source is observed above
$E_{\rm C}$, see Fig.~\ref{F4}.
Finally, note that if $T_{\rm S}\gg\tau_{30}$, the
continuous source is hardly mistaken for a burst with a large
time delay, which would be the closest approximation to the
time-energy image of a continuous source. This can be seen in
Fig.~\ref{F5a}, which represents contours of the likelihood in
the $\tau_{100}-T_{\rm S}$ plane. If the likelihood is further
marginalized with respect to $T_{\rm S}$ or $\tau_{100}$ (see
Fig.~\ref{F5b}), a burst with a large time delay is ruled out to
about 95\% confidence level.
Qualitatively speaking, the difference is that for a
burst with a large time delay, the maximum fluence occurs at some
intermediate energy, and the fluence decreases with decreasing
energy below. For a continuous source, in contrast, the fluence
increases with decreasing energy, according to the injection
negative power law spectrum.
\subsection{Injection spectrum index $\gamma$}
The injection spectrum index $\gamma$ can be measured provided
UHE CRs are recorded over a bandpass in energy that is
sufficiently broad. More precisely, in the case of a continuous
source, {\it i.e.} $T_{\rm S}\gg\tau_{30}$, $\gamma$ can be
measured with an absolute accuracy of $\simeq0.3$. This is based
on the likelihood as marginalized over $T_{\rm S}$, $\tau_{100}$,
and $N_0$, albeit for a known distance $D$. For example,
for a continuously emitting source at $D=50\,$Mpc with
$\gamma=2.0$ (see, {\it e.g.}, Fig.~\ref{F1d}), we
obtained a difference in the likelihood for $\gamma=1.5$ and 2.5
of a factor of about 30 and 2, respectively, on average. An
example for this situation is given in Fig.~\ref{F5b}.
In the case of a continuous source with a time delay comparable
to the emission timescale, {\it i.e.} such as shown in Fig.~\ref{F1e},
the respective factors are about 2 and 1, and
therefore hardly significant. For a burst with a small time
delay such as in Fig.~\ref{F1a} these factors are about 10 and
1. A burst with $\tau_{100}\gg T_{\rm obs}$
in which case the signal would be spread over a large
range in energy, is even less sensitive to $\gamma$. In general,
therefore, it is comparably easy to rule out a hard injection
spectrum if the actual $\gamma\ga2.0$, but it is much harder
to distinguish between $\gamma=2.0$ and 2.5.
Our analysis of the sensitivity to $\gamma$ was restricted to a
fixed distance $D$, mainly because of CPU time limitations of
the present serial version of our code. We expect that in the
absence of information on $D$, an additional marginalization
over $D$ would decrease the sensitivity to $\gamma$. In particular,
we already mentioned at the end of Section~3.2, a degeneracy of
the likelihood between $\gamma$
and $D$ for the intermediate case, where $T_{\rm S}$ and
$\tau_{E_{\rm C}}$ are comparable.
\subsection{Fluence $N_0$}
Because of the degeneracy of the likelihood in $N_0/T_{\rm S}$
and/or $N_0/\Delta\tau_{100}$ for large timescales, it is in general
not possible to reconstruct $N_0$. A possible exception is the
case where all the particles are detected, {\it i.e.}
$\tau_{100}\mathrel{\mathpalette\fun <} T_{\rm obs}$, and $T_{\rm S}\mathrel{\mathpalette\fun <} T_{\rm
obs}$.
We take advantage of this section to detail slightly the
marginalization procedure over $N_0$.
In most cases we marginalized over the fluence analytically,
noting that the dependence of the likelihood on $N_0$ can be
written as
\begin{equation}
\ln{\cal L}=a\exp(x)+bx+c\,,\label{margN0}
\end{equation}
with $x\equiv\ln N_0$ and where $a$, $b$, and $c$ depend on all
other parameters except $N_0$. This just follows from the fact that
$\rho_j$ in Eq.~(\ref{likelihood}) is proportional to $N_0$. By
using the approximation
\begin{equation}
\ln{\cal L}(x)\simeq\ln{\cal L}_{\rm max}-
\frac{b}{2}(x-x_{\rm max})^2\label{N)approx}
\end{equation}
in terms of value ${\cal L}_{\rm max}$ and location $x_{\rm
max}$ of the $N_0-$maximized likelihood, marginalizing over
$N_0$ with a uniform prior for $\ln N_0$ then amounts to
computing
\begin{equation}
\int_{-\infty}^{+\infty}dx{\cal L}(x)=
{\cal L}_{\rm max}\left(\frac{2\pi}{b}\right)^{1/2}
\,.\label{N0int}
\end{equation}
\subsection{Coherence length $l_c$}
Our simulations confirm the suggestion of Ref.\cite{WM}, that the
main effect of $l_c$ on the angle-time-energy image comes through the
relative size of the scatter around the $\theta_E-\tau_E-E$
correlations. For a time-energy image, if the source is continuous,
with $T_{\rm S}\gg\tau_{30}$, the correlation between $\tau_E$ and
$E$ is drowned in the uniform emission of particles within the
timescale $T_{\rm S}$, and $l_c$ plays no role. The coherence
length can therefore be estimated only when
$\tau_{100}\gg T_{\rm S}$, and $\tau_{100}\gg T_{\rm obs}$. In
this case, the signal corresponds to that shown in
Figs.~\ref{F1b} and~\ref{F1c}, and the width of the signal is
related to the value of $D\theta_E/l_c$. The
likelihood marginalized over $N_0$ and $\gamma$ for the cluster
Fig.~\ref{F1c}, assuming two different coherence lengths,
$l_c\simeq0.25\,$Mpc and $l_c\simeq1\,$Mpc, is shown in Figs.~\ref{F3}
and~\ref{F6}. The qualitative behavior of these contour plots is the
following. The actual value of $l_c$ used in simulating the cluster is
$l_c\simeq0.25\,$Mpc, so that the likelihood shown in Fig.~\ref{F3} uses the
correct value of $l_c$. The width of a stripe in the time-energy
plance, is tied to the ratio $D\theta_E/l_c$, or, equivalently, to
$(D\tau_E)^{1/2}/l_c$. The likelihood shown in Fig.~\ref{F6},
assuming $l_c\simeq1\,$Mpc, is thus similar to that corresponding to
$l_c\simeq0.25\,$Mpc, although it requires a comparatively larger
({\it i.e.} $\simeq(1/0.25)^2$ times larger) time delay to reproduce
the large scatter in the stripe.
This demonstrates that for a broad observed energy
dispersion, a large coherence length can be ruled out at least
when some information on the distance $D$ and the deflection angle
$\theta_E$, and thus on $\tau_E$, is available. In contrast,
ruling out a small coherence length for a small observed energy
dispersion is much harder, due to the nature of Poisson statistics.
As mentioned previously, provided
$\tau_E\ga10^4\,$yr, $\theta_E$ can be directly measured. Note
that the upper limit on the magnetic field strength, obtained
through Faraday rotation measurements, implies
$\tau_E\mathrel{\mathpalette\fun <} 2\times10^5\,{\rm yr}(D/10\,{\rm Mpc})^2(E/10\,{\rm
EeV})^{-2}$. Whenever $\theta_E$ is measurable, the degeneracy
of the likelihood,
with respect to the width of the signal in energy, thus
concerns only the ratio
$D/l_c$. As we argued above, the likelihood itself is not
sensitive to the distance if the mean energy of the signal lies
below the GZK cut-off. If the mean energy lies above the GZK
cut-off, then not only $\theta_E$ should not be measurable,
but the width itself of the signal now arises from two very
different effects: one due to the different trajectories
followed by different UHE CRs through the magnetic field, another
due to the pion production stochastic broadening of the
signal; these effects cannot be easily disentangled. Hence, the only
way this degeneracy between $D$ and $l_c$ could be broken is
through the observation of an astrophysical counterpart to the
source, or the source host, and the direct measurement of its
distance.
Finally, the likelihood was found extremely insensitive to the index
$n_B$ of the power spectrum of magnetic inhomogeneities.
\section{Conclusions}
We have presented a Monte-Carlo likelihood analysis of the potential
of future large-scale UHE CR experiments to reconstruct the
physical parameters of the source and of intervening magnetic
fields, when the strength of the latter is sufficient ($B_{\rm
rms}\ga10^{-12}\,$G) to affect the propagation of UHE CRs.
We discussed five generic situations of the time-energy images of
UHE CRs, which we classify according to the values of the time delay
$\tau_E$ induced by the magnetic field, the emission timescale of the
source $T_{\rm S}$, as compared to the lifetime of the experiment.
For each case, we simulated clusters of UHE CRs using the
instrumental characteristics typical of future experiments such
as the Telescope Array, the High Resolution Fly's Eye, and most
notably, the Pierre Auger
Project. To this end we have simulated the emission, the propagation in
an extra-galactic magnetic field, and the detection of clusters of UHE
CRs. We then performed a Monte-Carlo likelihood analysis on these
UHE CRs, and tried to reconstruct the physical parameters from the
maximum of the likelihood. We simulated clusters of $\sim40$ events,
as the next generation experiments are expected to detect
$\sim20-100$ events per cluster if the clustering recently
suggested by the AGASA experiment~\cite{Hayashida1} is real.
In summary, the likelihood presents different degeneracies between
different parameters, which complicates the analysis. As an example,
the likelihood is degenerate in the ratios $N_0/T_{\rm S}$, or
$N_0/\Delta\tau_{100}$, where $N_0$ is the total fluence, and
$\Delta\tau_{100}$ is the spread in arrival time: these ratios
represent rates of detection. Another example is given by the
degeneracy between the distance $D$ and the injection energy
spectrum index $\gamma$. Yet another is the ratio
$(D\tau_E)^{1/2}/l_c$,
that controls the size of the scatter around the mean of the
$\tau_E-E$ correlation. Therefore, in most general cases, values for
the different parameters cannot be pinned down, and generally, only
domains of validity are found.
We find that the distance to the source is obtained from the pion
production signature, above the Greisen-Zatsepin-Kuzmin cut-off, when
the emission timescale of the source dominates over the time delay.
Since the time delay decreases with increasing energy, we find
that the lower the
energy $E_{\rm C}$, defined by $\tau_{E_{\rm C}}\simeq T_{\rm S}$, the
higher the accuracy on the distance $D$. The error on $D$ is, in the
best case, typically a factor 2, for one cluster of $\simeq40$ events.
In this case, where the emission timescale dominates over the time
delay at all observable energies,
information on the magnetic field is only contained in the
angular image, which we did not systematically include into our
likelihood analysis due to computational limits. A more detailed
investigation of angular images will be presented separately in
a forthcoming study. Qualitatively, the size of the angular image is
proportional to $B_{\rm rms}(Dl_c)^{1/2}$, whereas the structure
of the image, {\it i.e.} the number of separate images, is
controled by the ratio $D^{3/2}B_{\rm rms}/l_c^{1/2}$. Finally,
the case where the time delay
dominates over the emission timescale, with a time delay shorter than
the lifetime of the experiment, also allows to estimate the distance
with a reasonable accuracy.
The strength of the magnetic field can only be obtained from the
time-energy image in this latter case because the angular image
will not be resolvable. When the time delay
dominates over the emission timescale, and is, at the same time,
larger than the lifetime of the experiment, only a lower limit
corresponding to this latter timescale, can be placed on the time
delay, hence on the strength of the magnetic field. When combined with
the Faraday rotation upper limit,
this would nonetheless allow to bracket the r.m.s. magnetic
field strength within a
few orders of magnitude. Here as well, significant information is
contained in the angular image.
The coherence length enters the ratio $(D\tau_E)^{1/2}/l_c$ that
controls the scatter around the mean of the $\tau_E-E$ correlation in
the time-energy image. It can therefore be estimated from the width of
this image, provided the emission timescale is dominated by $\tau_E$
(otherwise the correlation would not be seen), and some prior
information on $D$ and $\tau_E$ is available.
As a concluding remark, we point out that the magnetic field,
although it 'scrambles' the images of UHE CRs, also brings
extra-information. It not only leaves a signature of its own, it may
also, in the case where the time delay becomes comparable to the
emission timescale at some intermediate energy, allow an evaluation of
the emission timescale of the source itself. There is therefore very
important information hiding in the angle-time-energy images of UHE
CRs, which could be exploited by future large-scale experiments.
\section*{Acknowledgments}
Peter Biermann, Pasquale Blasi, Chris Hill, and
J\"org Rachen are acknowledged for valuable discussions. We
especially thank Jim Cronin for
encouraging us to perform this study and Angela Olinto and David
Schramm for collaboration in earlier work on which it is
partly based. We are grateful to the
Max-Planck Institut f\"ur Physik, M\"unchen, Germany for
providing CPU time. G.S. acknowledges financial support by the Deutsche
Forschungs Gemeinschaft under grant SFB 375 and by the
Max-Planck Institut f\"ur Physik. This work was supported, in
part, by the DoE, NSF, and NASA at the University of Chicago.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,006 |
{"url":"https:\/\/gmatclub.com\/forum\/squares-and-square-roots-clarification-75173.html?kudos=1","text":"It is currently 20 Nov 2017, 01:25\n\n### GMAT Club Daily Prep\n\n#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.\n\nCustomized\nfor You\n\nwe will pick new questions that match your level based on your Timer History\n\nTrack\nYour Progress\n\nevery week, we\u2019ll send you an estimated GMAT score based on your performance\n\nPractice\nPays\n\nwe will pick new questions that match your level based on your Timer History\n\n# Events & Promotions\n\n###### Events & Promotions in June\nOpen Detailed Calendar\n\n# squares and square roots clarification\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\nAuthor Message\nManager\nJoined: 17 Dec 2008\nPosts: 171\n\nKudos [?]: 173 [0], given: 0\n\nsquares and square roots clarification\u00a0[#permalink]\n\n### Show Tags\n\n28 Jan 2009, 06:47\nIn GMAT, is $$sqrt(x)$$ always +ve ?\nThis can get really tricky in some DS questions depending on\nthe equations, example if we simplify equation and use x instead of\n$$sqrt(x)$$ and consider positive\/negative numbers - could get\na wrong answer.\n\nAny suggestions for a clean way to handle this topic is appreciated.\n\nKudos [?]: 173 [0], given: 0\n\nIntern\nJoined: 08 Feb 2014\nPosts: 10\n\nKudos [?]: 25 [2], given: 2\n\nRe: squares and square roots clarification\u00a0[#permalink]\n\n### Show Tags\n\n26 Jun 2014, 23:26\n2\nThis post received\nKUDOS\nQuoting another answer, in essence,\n\n\"In GMAT math, the roots of x are expressed as +\\sqrt{x} and -\\sqrt{x}\n\n\\sqrt{x} itself is ALWAYS positive. I completely understand your logic, but if you don't accept this as a convention, you are bound to either get DS sums wrong or frown on several PS sums.\n\nFor example, the solutions for x in the equation x^2 = 25 are x=5 and x=-5.\n\nHOWEVER, if x = 25, then \\sqrt{x} = 5. PERIOD. Remember, convention not logic!\n\nGood luck!\"\n\nKudos [?]: 25 [2], given: 2\n\nSVP\nJoined: 29 Aug 2007\nPosts: 2471\n\nKudos [?]: 856 [0], given: 19\n\nRe: squares and square roots clarification\u00a0[#permalink]\n\n### Show Tags\n\n30 Jan 2009, 14:45\nConkergMat wrote:\nIn GMAT, is $$sqrt(x)$$ always +ve ?\nThis can get really tricky in some DS questions depending on\nthe equations, example if we simplify equation and use x instead of\n$$sqrt(x)$$ and consider positive\/negative numbers - could get\na wrong answer.\n\nAny suggestions for a clean way to handle this topic is appreciated.\n\nThere are many posts that discussed this issue in details.\nTry finding some of them and go through them. They would, I believe, clear your doubt.\n\nRefer:\n7-t75259\nviewtopic.php?t=39533\n_________________\n\nVerbal: http:\/\/gmatclub.com\/forum\/new-to-the-verbal-forum-please-read-this-first-77546.html\nMath: http:\/\/gmatclub.com\/forum\/new-to-the-math-forum-please-read-this-first-77764.html\nGmat: http:\/\/gmatclub.com\/forum\/everything-you-need-to-prepare-for-the-gmat-revised-77983.html\n\nGT\n\nKudos [?]: 856 [0], given: 19\n\nRe: squares and square roots clarification \u00a0 [#permalink] 30 Jan 2009, 14:45\nDisplay posts from previous: Sort by\n\n# squares and square roots clarification\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\n\nModerator: Bunuel\n\n Powered by phpBB \u00a9 phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT\u00ae test is a registered trademark of the Graduate Management Admission Council\u00ae, and this site has neither been reviewed nor endorsed by GMAC\u00ae.","date":"2017-11-20 08:25:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.3125191330909729, \"perplexity\": 8084.123287737354}, \"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-2017-47\/segments\/1510934805923.26\/warc\/CC-MAIN-20171120071401-20171120091401-00666.warc.gz\"}"} | null | null |
\section{Introduction}
Let $M$ be a smooth manifold endowed with a smooth action of a compact
Lie group $G$. We denote
by $c(G,M)$ the cohomogeneity of the action, i.e. the codimension of
the principal orbits in $M,$ and by $H$ a principal isotropy subgroup.
In \cite[p. 194]{Br} Bredon proved the following inequality
for the dimension of the fixed point set of a maximal torus $T$ in
$G$:
\[
\dim M^T\leqslant c(G,M)-\rk G+\rk H
\]
whenever $M^T$ is nonempty.
Drawing on this fact, P\"uttmann introduced in \cite{Pu} the {\em
homogeneity rank} of $(G,M)$ as the integer
\[
{\rm hrk}(G,M): = \rk G-\rk H-c(G,M).
\]
In this paper we are interested in studying actions on quaternionic
projective spaces and there are at least two reasons to consider
actions with {\em vanishing} homogeneity rank.\\
A first motivation comes from the following proposition which can be deduced
from \cite{Pu}.
\begin{proposition}
\label{chi>0}
Let $M$ be a compact manifold with positive Euler characteristic acted
on by a compact Lie group $G$. Then $\hrk(G,M)\leqslant 0$.
\end{proposition}
Indeed quaternionic projective spaces (and more generally positive
quaternionic-K\"ahler manifolds, see \cite{LBS}) have positive Euler characteristic,
thus the actions we aim to classify are those with {\em maximal}
homogeneity rank and this fact turns out to have remarkable consequences
on the geometry of the action.\\
Furthermore, in the symplectic framework, Hamiltonian actions with vanishing
homogeneity rank have a precise geometric meaning. Let the
compact Lie group $G$ act on the symplectic manifold $(M,\omega)$ in a
Hamiltonian fashion, then $\hrk(G,M)=0$ if and only if every principal
orbit $\mathcal{O}$ of the $G$-action is
coisotropic, i.e. $(T_p {\mathcal{O}})^\omega\subseteq T_p
{\mathcal{O}}$ (such an action is said to be {\em coisotropic}).
If further $M$ is compact and admits a $G$-invariant
$\omega$-compatible complex structure $J$, then $(M,\omega,J)$ turns
out to be a projective algebraic {\em spherical variety}, that is the
Borel subgroup of $G^\C$ has an open orbit in $M$ (see \cite{HW}).
Coisotropic actions on symplectic and K\"ahler manifolds have been
extensively studied starting from \cite{GS} and have been classified
on Hermitian symmetric spaces in \cite{PT2}, \cite{BiG} and
\cite{Bi}.
Linear actions with vanishing homogeneity rank have been considered by
several authors: the classification in
the complex case can be deduced from \cite{Ka} and \cite{BR}, (while
absolutely irreducible real representations with vanishing homogeneity
rank of compact Lie groups have
been classified in \cite{GP}).\\
It is therefore rather natural to look for relations analogous to those found in
the complex/symplectic framework in the quaternionic setting.\\
Let $M$ be a quaternionic K\"ahler manifold with positive scalar curvature and $Z\subset\End TM$
its twistor space. We say that
a submanifold $N$ of $M$ is {\em quaternion}{\em -co\-iso\-trop\-ic} if for every
$p\!\in\! N$ and $J\!\in\!Z_p$ we have $J(T_pN)^\perp\subseteq T_pN$.
The reason to consider the previous definition is the fact that the
principal orbits of {\em polar} actions on compact symmetric quaternionic-K\"ahler manifolds
are indeed quaternion-coisotropic \cite{Te} (in the same way as polar
actions on K\"ahler manifolds have coisotropic principal orbits
\cite{PT}).\\
A first result of this paper is an example of an action on the
quaternionic projective space with vanishing
homogeneity rank which is not quaternion-cosotropic (Example
\ref{controesempio}). We further determine in our main theorem all the compact Lie
subgroups of $\Sp(n)$
acting with vanishing homogeneity rank on the quaternionic projective
space in the following
\begin{theorem}\label{Teoremone}
Let $\rho: G \to {\rm Sp}(V)$ be a $n$-dimensional quaternionic
representation of a compact connected Lie group. Then $\rho$ induces a
minimal vanishing homogeneity rank action of $G$ on $\P_\H(V) \simeq \H\P^{n-1}$ if
and only if one of the following is satisfied:
\begin{enumerate}
\item $G=Sp(1)^{n-1}$ and $\rho=\rho_s\oplus\ldots\oplus\rho_s\oplus 1$,
where $\rho_s: \Sp(1) \to \Sp(\H)$ is the standard representation
and 1 is the trivial representation on $\H$;
\item $G=H \times Sp(1)^r$ and
$\rho=\sigma\oplus\rho_s\oplus\ldots\oplus\rho_s$, where $\sigma: H
\to \Sp(W)$ is one of the following $4(n-r)$-dimensional quaternionic
representation:
\begin{enumerate}
\item $H=S(\U(k)\times \U(n-r-k))\subset \SU(n-r)\subset \Sp(n-r)$ and
$k$ is odd;
\item $H=S(\U(1)\Sp(k)\times \U(n-r-k))\subset S(\U(2k)\times \U(n-r-2k))\subset \Sp(n-r)$;
\item $H\times Sp(1) \curvearrowright W\otimes_\H\H$ is orbit
equivalent to the isotropy representation of a
quaternionic-K\"ahler symmetric space;
\item $H=\Spin(7)\otimes\Sp(1)\subset\SO(8)\otimes\Sp(1)\subset\Sp(8)$.
\end{enumerate}
\end{enumerate}
\end{theorem}
Note that many of these actions turn out to be non polar.\\
The paper is organized as follows.
In Section \ref{1} we prove several lemmas about the homogeneity rank necessary for the
proof of the main theorem.
Results about polar actions on Wolf
spaces and an example of a vanishing homogeneity rank action which is
not quaternion-coisotropic are provided in Section \ref{2}, while Section \ref{class} is devoted to the
classification actions on $\H\P^{n-1}$ with vanishing homogeneity rank.\\
Finally in the appendix one can find some tables we refer to in the
course of the classification.
Most of them are taken from \cite{Ko}.
\section{Homogeneity rank of compact Lie group actions}\label{1}
In this section we are going to prove several results about the homogeneity rank
which will be useful in the classification of actions with vanishing homogeneity rank on
quaternionic projective spaces. On the other hand these statements
have an autonomous interest since they hold in general for actions of
compact Lie groups.
The following lemma allows us to by-pass (sometimes) the computation of the principal isotropy subgroup.
\begin{lemma}\label{Slice}
Let $G$ be a compact connected Lie group acting on a
compact manifold $M$. Take $p\in M$ and denote by $\delta$ the
difference $\rk G-\rk G_p$ and by $\Sigma$ the slice representation at
$p$. Then $\hrk(G,M)=\hrk(G_p,\Sigma)+\delta$. In particular if the
$G$-orbit through $p$ has positive Euler characteristic, then the
action of $G$ on $M$ has vanishing homogeneity rank if and only if the slice representation at $p$ does.
\end{lemma}
\begin{proof}
Since the action of $G$ on $M$ is proper, it is known that at every
point the slice representation has the same
cohomogeneity as that of the action of $G$ on $M$.
Let $\Sigma$ be the slice for the action at $p$. Let $q \in \Sigma$ be
principal both for the $G$-action on $M$ and the $G_p$-action on
$\Sigma$ (which is equivalent to the slice representation).
Obviously $(G_p)_q = G_q =G_{\rm princ}$.
Thus
\begin{eqnarray*}
\hrk (G,\Sigma) & = & \rk G_p -\rk (G_p)_q -c(G_p,\Sigma)\\
& = & \rk G - \delta -\rk G_q -c(G,M)=\hrk(G,M)-\delta\,,
\end{eqnarray*}
and the conclusion follows. The last statement is a consequence of the
well known fact that the homogeneous space
$G/G_p$ has positive Euler characteristic if and only if $ \rk G = \rk G_p$.
\end{proof}
The following is an obvious but important property of the homogeneity
rank which is a consequence of \cite[Proposition 2]{GP}.
\begin{lemma}
\label{sommaranghi}
Let $\rho_i\colon G \to \GL(V_i)$ $(i=1,2)$ be two finite-dimensional
representations of the compact Lie group $G$. Then
$\hrk(G,V_1\oplus V_2)\leqslant\hrk(G,V_1) + \hrk(G,V_2)$.
\end{lemma}
\begin{proof}
Let $v_i$ be a principal point of $(G,V_i)$ for $i=1,2$. Denote by
$\Ocal_i = G/H_i$ the corresponding orbits. \\
Now consider the action of $G$ on $V_1\oplus V_2$. The slice
representation at $(v_1,0)$ is $V_2\oplus U$ where $(G,V_2)$ is the
original action and $U$ is a trivial $G$-module of dimension
$c(G,V_1)$.
Now $(v_ 2,0)$ is obviously principal for the slice representation,
so that a principal isotropy subgroup $H$ of $(G,V_1\oplus V_2)$ is
$(H_1,V_2)_\princ$.
Thus we have $c(G,V)=c(G,V_1)+c(H_1,V_2)$ and therefore
\begin{eqnarray*}
\hrk(G,V) & = & \rk G-\rk H -c(G,V)\\
& = & \rk G-\rk (H_1,V_2)_\princ -c(H_1,V_2)-c(G,V_1)\\
& = & \rk G-\rk H_1 +\hrk(H_1,V_2) -c(G,V_1)\\
& = & \hrk(G,V_1) + \hrk(H_1,V_2) \leqslant \hrk(G,V_1)
+ \hrk(G,V_2)
\end{eqnarray*}
\end{proof}
Another important tool in the classification carried out in Section \ref{class} will be
the following proposition which generalizes, in the case of positive
Euler characteristic, the {\em Restriction
Lemma} given in \cite{HW} for complex $G$-stable orbits of
Hamiltonian isometric actions on compact K\"ahler manifolds.
\begin{proposition}
\label{restriction}
Let $G$ be a compact connected Lie group acting by isometries on a
compact Riemannian manifold $M$
. Let $Y$ be a compact $G$-stable
submanifold of $M$ such that $\chi(Y) >0$. If $\hrk(G,M)=0$, then $\hrk(G,Y)=0$.
\end{proposition}
\begin{proof}
Let $\nu_M Y$ be the normal bundle to $Y$ in $M$.
Since $Y$ is compact, we can use the invariant version of the tubular
neighborhood theorem (see e.g. \cite[p. 306]{Br}) to get a
$G$-equivariant diffeomorphism of an open G-invariant neighborhood $U$ of the zero
section of $\nu_M Y$ onto an open G-invariant neighborhood $W$ of $Y$
in $M$. Now, since $W$ is open in $M$ and $G$ acts with vanishing homogeneity rank
on $M$, the $G$-action on $W$ has vanishing homogeneity rank too, hence
also $\hrk(G,U)=0$.
Consider now the restriction of the natural projection $\pi_{|_U} \colon U
\to Y$. Let $y \in Y$ be such that
\begin{enumerate}
\item $y$ is principal for the action of $G$ on $Y$;
\item $F := \pi_{|_U}^{-1}(y) \subset U$ has non-empty intersection
with $M_{\rm princ}$.
\end{enumerate}
Now consider the action of $G_y$ on $F$ and take $x \in F$ such that
\begin{enumerate}
\item $x\in M_{\rm princ}$;
\item $x$ is principal for the action of $G_y$ on $F$.
\end{enumerate}
Since the action of $G_y$ on $F \cong \nu_M(Y)_y$ is linear, the
homogeneity rank of this action is non-positive (\cite[corollary
1.2]{Pu}), i.e.
\[
\dim F \geqslant \dim G_y - \dim G_x + \rk G_y -\rk G_x.
\]
Thus we can compute
\begin{eqnarray*}
c(G,Y) & = & \dim Y - \dim G +\dim G_y = \dim X-\dim F-\dim G+\dim G_y\\
& \leqslant & \dim X-(\dim G_y -\dim G_x +\rk G_y-\rk G_x)-\dim G+\dim G_y\\
& = & c(G,X)-\rk G_y+\rk G_x = \rk G -\rk G_y,
\end{eqnarray*}
so that $\hrk (G,Y) \geqslant 0$. On the other hand the positive Euler
characteristic of $Y$ obstructs actions with positive homogeneity rank
(Proposition \ref{chi>0}) and the claim follows.
\end{proof}
In the case of the quaternionic projective space we deduce also the following
useful consequence
\begin{corollary}
\label{product}
Let $G_1$ and $G_2$ be closed subgroups respectively of $\Sp(n_1)$ and
$\Sp(n_2)$. Assume that the action of $G=G_1 \times G_2$ on
$\P_\H(\H^{n_1}\oplus\H^{n_2}) \simeq \H\P^{n_1+n_2-1}$ is 3-coisotropic. Then $G_i$
acts 3-coisotropically on $\P_\H(\H^{n_i})\simeq\H\P^{n_i-1}$.
\end{corollary}
\begin{proof}
Simply take two non-zero vectors $v_1$ and $v_2$ respectively in $\H^{n_1}$
and $\H^{n_2}$ and consider the orbits $\mathcal{O}_i=G\cdot[v_i]
\simeq \H\P^{n_i-1}$. Now apply Proposition \ref{restriction} to the
orbits $\mathcal{O}_1$ and $\mathcal{O}_2$.
\end{proof}
\section{Quaternion-coisotropic actions and the vanishing of
homogeneity rank}\label{2}
In order to introduce the right notion of ``coisotropic'' actions in
the quaternionic setting, it is necessary to fix some notation.
Let $(M,g)$ be a Riemannian manifold and $\nabla$ its Levi-Civita connection.
A quaternionic-K\"ahler structure on $M$ is a $\nabla$-parallel rank 3 subbundle $Q$ of ${\rm End}\,TM$,
which is {\em locally} generated by a triple of locally defined anticommuting $g$-orthogonal almost complex
structures $(J_1,J_2,J_3=J_1J_2)$.
Recall that a quaternionic-K\"ahler manifold is automatically
Einstein, hence if its scalar curvature is positive it is
automatically compact. Here we consider only {\em
positive} quaternionic-K\"ahler manifolds.\\
A submanifold $N$ of $M$ will be called {\it quaternion-coisotropic} if for every
$p\in N$ and $J \in Q_p$ we have $J(T_pN)^\perp \subseteq T_pN$.
Trying to seek the analogy with the symplectic context, it is rather natural to consider the following situation.
\begin{definition}
Let $(M,g,Q)$ be a quaternionic-K\"ahler manifold. We say that the action of a compact Lie group of isometries of $M$
is {\em quaternion-coisotropic} if the principal orbits are quaternion-coisotropic submanifolds of $M$.
\end{definition}
Recall that an isometric action of a compact Lie group $G$ on a Riemannian manifold $M$ is said to be polar if
there is an embedded submanifold $\Sigma$ (a {\em section}) which meets all principal orbits orthogonally.
In \cite{Te} it is proved, using the classification results of
\cite{PT} and \cite{Ko2}, that quaternion-coisotropic actions
generalize polar actions on Wolf spaces \cite[Theorem 4.10]{Te} in
the same manner as coisotropic actions generalize polar actions on
compact K\"ahler manifolds (\cite{PT2}).
The classification of polar actions on quaternion
projective space has been obtained by Podest\`a and Thorbergsson.
Here we restate the classification theorem because in the statement of \cite{PT} a (trivial) case is
missing.
\begin{theorem}\label{polari}\cite{PT}
The isometric action of a compact Lie group $G$ on $\H\P^{n-1}$
is polar if and only if it is orbit equivalent to the action induced by
a n-dimensional quaternionic representation $\rho_1 \oplus \ldots
\oplus \rho_k$ where $\rho_i$ is the isotropy representation of a
quaternionic-K\"ahler symmetric space of rank one for $i=1,\ldots,k-1$
and $\rho_k$ is one of the following:
\begin{enumerate}
\item the isotropy representation of a
quaternionic-K\"ahler symmetric space of arbitrary rank;
\item the trivial representation on a 1-dimensional quaternionic
module $\H$.
\end{enumerate}
\end{theorem}
Note that the missing case (this including a trivial module) is easily
seen to be quaternion-coisotropic.\\
In spite of these analogies, the parallel with the symplectic setting
does not go further, indeed we have the following
\begin{example}\label{controesempio}
{\rm Consider the action of $G=\U(k)\times\U(n-k)\subset \U(n) \subset
\Sp(n)$ on $M=\H\P^{n-1}$. It is not hard to see that, for
$k\geqslant 3$, the Lie
algebra of principal isotropy subgroup is isomorphic to
$\u(k-2)\oplus\u(n-k-2)$ whence the cohomogeneity of the action is 4
and the homogeneity rank vanishes. Suppose now that the principal orbits
are quaternion-coisotropic and consider the lifted action of $G$ on
the twistor space $Z=\C\P^{2n-1}$.
In general, when we lift an isometric action with $\hrk=0$ of a compact Lie group
on a positive quaternionic-K\"ahler manifold, three cases may occur
according to the cohomogeneity
of the action of a principal isotropy subgroup $G_p$ on the twistor line
$Z_p\simeq\C\P^1$:
\begin{enumerate}
\item The action of $G_p$ on $Z_p$ is transitive. In this case
$c(G,Z)=c(G,M)$. Furthermore a $G$-principal orbit of $Z$ is
$G/G_z$. If we take into account the homogeneous fibration $G/G_z \to G/G_p$ where $G_p/G_z = S^2$
and the fact that $S^2$ has positive Euler characteristic, we have
$\rk G_z =\rk G_p$, and we can compute
\begin{eqnarray*}
{\rm hrk}(G,Z) & = & \rk G -\rk G_z -c(G,Z) \\
& = & \rk G -\rk G_p -c(G,M) =0
\end{eqnarray*}
\item The action of $G_p$ on $Z_p$ has cohomogeneity one.
Again, if $z$ is principal for the $G_p$-action on $Z_p$, then it is
principal for the $G$-action on $Z$. Furthermore the homogeneous fibration $G/G_z \to G/G_p$ has fibre $S^1=G_p/G_z$,
hence $\rk G_z=\rk G_p - 1$. Now, taking into account that
$c(G,Z)=c(G,M)+1$, by a dimensional computation we obtain that also in this case the $G$-action on $Z$ is coisotropic.
\item The connected component of the identity of $G_p$ acts trivially on $Z_p$. In this case the $G$-action on $Z$
is no more coisotropic.
\end{enumerate}
One easily verifies that in our case the
homogenity rank of the lifted action is $-2$, that is the $G$-action
on $Z$ is not coisotropic. This implies that the connected component
$H$ of $G_p$ acts trivially on the twistor line $Z_p\simeq\C\P^1$.
Now denote by $\nu$ the normal space to the $G$-orbit at $p$ which
is acted on trivially by $H$. On the other hand, if $J_1, J_2, J_3$
are three generators of the algebra $Q_p$, these are fixed by $H$
thanks to the argument above. Thus $H$ pointwisely fixes the three
4-dimensional mutually orthogonal subspaces $J_1\nu, J_2\nu$ and
$J_3\nu$ of $T_p\,G\cdot p$. But this is impossible since we claim that a subspace
of $T_p\,G\cdot p$ fixed by $H$ has dimension 8. Indeed in
correspondence to a principal point we have the following
reductive decomposition:
$$
\u(k)\oplus\u(n-k)=\u(k-2)\oplus\u(n-k-2)\oplus \u(2)\oplus\u(2)\oplus(\C^2\otimes\C^{k-2})\oplus(\C^2\otimes\C^{n-k-2}).
$$
Hence we can identify the tangent space to the principal orbit with
$$\u(2)\oplus\u(2)\oplus(\C^2\otimes\C^{k-2})\oplus(\C^2\otimes\C^{n-k-2})$$
on which $\u(k-2)\oplus\u(n-k-2)$ acts. Then $H$ fixes the two copies of $\u(2)$, thus has dimension
8, as claimed.}
\end{example}
\section{Actions with vanishing homogeneity rank on $\H\P^{n-1}$: proof of Theorem \ref{Teoremone}}
\label{class}
The entire section is devoted to prove Theorem \ref{Teoremone}.
In order to achieve the classification, the following remark will
be useful. Let $G$ be a compact Lie group acting by isometries on
a compact quaternionic-K\"ahler manifold $M$. If $G'$ is a closed
subgroup of $G$ acting on $M$ with $\hrk(G',M)=0$, the same is true
for $G$. Indeed every compact quaternionic-K\"ahler manifold has positive Euler
characteristic (see \cite[Theorem 0.3]{LBS}). As already observed this forces the homogeneity rank
to be non-positive. But $\hrk(G',M)\leqslant
\hrk(G,M)$, by \cite[Proposition 2]{GP}. Thus it is natural to say that a
$G$-action with vanishing homogeneity rank on a manifold $M$ is {\em minimal} if no
closed subgroup $G'$ of $G$ acts on $M$ with vanishing homogeneity rank.\\
From now on we fix $M=\H\P^{n-1}=\Sp(n)/\Sp(1)\Sp(n-1)$.
Since the identity component of ${\rm Iso}(\H\P^{n-1},g)$ is $\Sp(n)$,
we go through all the closed subgroups of it, starting from the
maximal ones and then analysing only the subgroups of those giving
rise to vanishing homogeneity rank actions.\\
We proceed, in some sense, by strata: the first level is made by the maximal
connected subgroups of $\Sp(n)$ listed in Table \ref{maxSp} in the appendix, then we
pass to the maximal connected subgroups of the groups of the previous
level and so on.\\
Before starting the classification we make two remarks:
\begin{enumerate}
\item Cohomogeneity one
$G$-actions on $M$ (with positive Euler characteristic) have
automatically $\hrk(G,M)=0$. Indeed in this case $\rk G-\rk
G_\princ\leqslant 1$ and cannot be zero since otherwise the
homogeneity rank would be odd which is impossible since $M$ has even
dimension.
\item A necessary dimensional condition for an action of $G$ on $M$ to have
vanishing homogeneity rank is that
\begin{equation}\label{dimcond}\dim G+\rk G\geqslant \dim M.
\end{equation}
\end{enumerate}
Finally in order to clarify our procedure we make one more
observation.
When considering the irreducible
representations of simple Lie groups one must often check the
dimensional condition \eqref{dimcond} or a variation of it. This is
made easier by the fact that if $(c_1,\ldots,c_n)$ are the
coefficients of the maximal weights of the representation of a rank
$n$ simple Lie group, then the function
\[
(c_1,\ldots,c_n)\mapsto \deg(\rho_{(c_1,\ldots,c_n)}),
\]
is strictly monotonic, i.e. if $\rho$ and $\rho'$ are two irreducible
representations of a simple compact Lie group with highest weights
$\lambda$ and $\lambda'$, given by $(c_1,\ldots,c_n)$ and
$(c_1',\ldots,c_n')$ respectively, and if $c_i\leqslant c_i'$ for all
$i$ and $c_i<c_i'$ for at least one $i$, then $\deg \rho<\deg\rho'$
(see \cite{On}).
Then, in many cases it is sufficient to test the dimensional
condition for the fundamental representations, and go further only if
the condition is satisfied.
\subsection{Maximal subgroups of $\Sp(n)$}
\subsubsection{$G=\U(n)$} The action of $\U(n)$ on $\H\P^{n-1}$, has
cohomogeneity 1, thus has vanishing homogeneity rank.
\subsubsection{$G=\Sp(k)\times\Sp(n-k)$ ($1\leqslant k \leqslant n$)} These
subgroups act by cohomogeneity 1 on $\H\P^{n-1}$, thus $\hrk=0$.
\subsubsection{$G=\SO(p)\otimes\Sp(q)$ ($n=pq$, $p\geqslant 3, q\geqslant 1$)} For $q\geqslant 2$ we can compute the slice
representation at the quaternionic line $\ell$ spanned by a pure element of
$\R^p\otimes \R^{4q}$. The algebra of the stabilizer is
$\o(p-1)\oplus\sp(1)\oplus\sp(q-1)$ acting on
\begin{equation}
\label{slicetensor}
\R^{p-1}\otimes(U\oplus\R^{4(q-1)})
\end{equation}
where $U$ can be seen as the 3-dimensional vector space of the
imaginary quaternions on which $\Sp(1)$ acts
by conjugation (see \cite[p. 590]{Ko}). Note that $\Sp(1)$ acts also
on $\R^{4(q-1)} \simeq \H^{q-1}$ by right multiplication.
If $p$ is odd the $G$-orbit of $\ell$ has positive Euler characteristic so
we can easily rule out this case by observing that the irreducible
factor $\R^{p-1}\otimes\H^{q-1}$, regarded as a {\em complex} representation, does not
appear in Kac's list \cite{Ka}.\\
So we are left to consider the cases in which $p$ is even.
To get rid of the action on the slice of the unitary quaternions, let
us consider the stabilizer of a principal element of $\R^{p-1}\otimes
U$: such an element is of the form $v_1\otimes i+v_2\otimes
j+v_3\otimes k$, where $v_1,v_2,v_3$ are linear independent elements
of $\R^{p-1}$. Here the algebra of the stabilizer $H$ is
$\o(p-4)\oplus\sp(q-1)$ and the slice contains as a direct summand
the tensor product of the standard representations
$V=\R^{p-4}\otimes\R^{4(q-1)}$. Since at this level $\delta=\rk G-\rk
H=3$, applying Lemma \ref{Slice}, in order to exclude also this case it is enough to show
that $\hrk(H,V)\leqslant -4$. Indeed this is easy to verify once we
subdivide into three more subcases and we compute explicitly the
principal isotropy of $H$ on $V$. If $q\geqslant p-4$ then
$\h_\princ=\sp(q-p+3)$; if $p-6\leqslant q\leqslant p-3$ then
$\h_\princ$ is trivial; if $q\leqslant p-8$ then
$\h_\princ=\sp(p-q-6)$ (see e.g. \cite[p. 202]{HH}) and in all these cases $\hrk(H,V)\leqslant -4$
(note that the equality holds only if $q=1$). \\
For
$q=1$ this is the action on $\H\P^{p-1}$ induced by the isotropy representation of the quaternionic-Kaehler
symmetric space $\SO(p+4)/\SO(p)\times\SO(4)$, thus it is polar by
Theorem \ref{polari}. To determine whether it has vanishing
homogeneity rank or not we have to distinguish according to the parity
of $p$. If $p$ is odd the $G$-orbit of $\ell$ has positive Euler characteristic,
the slice representation at $\ell$ is real and appears in the list of
\cite{GP} since it is orbit equivalent to the isotropy representation
of the real Grassmannian of $3$-planes in $\R^{p+2}$. Thus it has
vanishing homogeneity rank.
When $p$ is even, at the first step the slice is
given by $\R^{p-1}\otimes U$ and with easy computations we find that
the principal isotropy is $\h_\princ=\sp(p-4)$, $c=3$ hence $\hrk=0$.
\subsubsection{$G=\rho(H)$ with $\rho$ complex irreducible representation of quaternionic
type of the simple Lie group $H$} \label{irreducible} In this case the
dimensional condition that should be satisfied becomes $\dim H+\rk
H\geqslant 2\deg \rho -4.$
Going through all the representations of this type, and using the
argument referred to at the beginning of Section \ref{class}, the
following cases remain:
\begin{enumerate}
\item the representation of $\SU(6)$ on $\Lambda^3\C^6$;
\item the representation of $\Sp(3)$ with maximal weight $(0,0,1)$;
\item the spin representation of $\Spin(11)$;
\item the two half-spin representations of $\Spin(12)$;
\item the standard representation of $\E_7$ on $\C^{56}$.
\item the standard representation of $\SU(2)$;
\end{enumerate}
Except for $\SU(2)$, that gives rise to a homogeneity rank zero
action, since it has cohomogeneity one on $\H\P^1\simeq S^4$, the
other cases can be treated using the fact that all of them admit a
totally complex orbit (see \cite{BG} and also \cite{AM}).
These totally complex submanifolds are Hermitian symmetric spaces, and
therefore have positive Euler characteristic.
Thus we compute the slice representation on these orbits, obtaining:
\begin{enumerate}
\item $\SU(3)\times\SU(3)\cdot \U(1)$ on $\C^{3}\otimes\C^3\otimes \C$;
\item $\SU(3)\cdot \U(1)$ on $S^2(\C^3)\otimes \C$;
\item $\SU(5)\cdot \U(1)$ on $\Lambda^2(\C^5)\otimes \C$;
\item $\SU(6)\cdot \U(1)$ on $\Lambda^2(\C^6)\otimes \C$;
\item $E_6\cdot \U(1)$ on $\C^{27}\otimes \C$.
\end{enumerate}
These all give rise to vanishing homogeneity rank actions, since they
are all multiplicity free \cite{Ka}.\\
Let us remark that all of these actions on the quaternionic projective
space are polar.
\subsection{The subgroups of $\U(n)\subset \Sp(n)$.}\label{su(n)} First note that the
maximal compact connected subgroups of $\U(n)$ are $\SU(n)$ and those
of the form $Z\cdot H$ where $Z$ is the center of $\U(n)$ and $H$ is a
maximal compact connected subgroup of $\SU(n)$ (see Table \ref{maxSU}). \\
Certainly $\SU(n)$ has vanishing homogeneity rank on $\H\P^{n-1}$ since it has
the same orbits of $\U(n)$, so let us go through the remaining cases.
\subsubsection{$G=Z\cdot \S(\U(k)\times \U(n-k))=\U(k)\times U(n-k)$.}
We start by computing the slice representation at the class of
the identity in $\Sp(n)/\Sp(1)\Sp(n-1)$. The stabilizer is given by the
intersection of $G$ with $\Sp(1)\Sp(n-1)$. In this way we get
$\U(1)\times\U(k-1)\times\U(n-k)$ acting on the slice
\[
\Sigma=(\C^*\otimes (\C^{k-1})^*)\oplus(\C^*\otimes (\C^{n-k})^*)\oplus(\C^*\otimes \C^{n-k})\,.
\]
Now it is immediate to see that the principal isotropy group is
isomorphic to $\U(k-2)\times\U(n-k-2)$ so that the cohomogeneity is 4
and the action has vanishing homogeneity rank.
\begin{remark}
Observe that the slice representation we just considered is complex,
indecomposable and has vanishing homogeneity rank, though it does not
appear in the classification of Benson and Ratcliff \cite{BR}. In fact
they consider only representations $(G,V)$ which are indecomposable
for the {\em semisimple part} of $G$.
\end{remark}
\subsubsection{$G=Z\cdot \Sp(k)$ with $n=2k$.}\label{sp(n)} Proceeding as before we determine
the orbit through the class of the identity in
$\Sp(n)/\Sp(1)\Sp(n-1)$. Again we get an orbit with positive Euler
characteristic, more precisely the Lie algebra of the isotropy is
$\z \oplus \u(1) \oplus \sp(k-1)$ and the slice representation is given by
$\H^{k-1} \oplus \C\,,$ where the 1-dimensional factor $\z$ acts (non-trivially) only on $\C$
and $\u(1)$ acts by scalar multiplication on $\H^{k-1}$.
Thus the algebra of the principal isotropy is isomorphic to $\u(1)\oplus\sp(k-2)$
and $\hrk(G,\H\P^{n-1})=0$.\\
Note that the action of the center here is essential: Once the action
of $Z$ is removed, there is a trivial module in the slice representation.
Therefore $\Sp(k)\subset\SU(2k)\subset\Sp(2k)$ does not have $\hrk=0$ on $\H\P^{n-1}$.
\subsubsection{$G=Z\cdot \SO(n)$} \label{qualcosa2} First consider the totally complex orbit
of $\U(n) \supset \SO(n)$ which is $\C\P^{n-1}$
canonically embedded in $\H\P^{n-1}$. This orbit in its turn
contains a Lagrangian $G$-orbit ($\R\P^{n-1}$ canonically embedded).
Here the 1-dimensional factor of the isotropy $\z\oplus\o(n-1)$ acts
on the slice $\R^{n-1} \oplus \C^{2(n-1)}\otimes \C^*$ only on the
second module. From this one easily sees that $\g_\princ \simeq
\o(n-4)$ and the cohomogeneity is therefore 5.
Thus $\hrk(G,\H\P^{n-1})=-2$ and the action has non-zero homogeneity rank.
\subsubsection{$G=Z\cdot \SU(p)\otimes\SU(q)$ ($n=pq$ and
$p,q\geqslant2$)} Here $G$ acts on
$\P_\H(\C^p\otimes\C^q\oplus(\C^p\otimes\C^q)^*)$.
The orbit through the quaternionic line spanned by a pure element of
$\C^p\otimes\C^q$ is the product of two complex projective spaces
$\C\P^{p-1}\times\C\P^{q-1}$ and therefore has positive Euler
characteristic. So we are in a position to apply the criterion
deriving from Lemma \ref{Slice}. The slice
representation contains the module $\C^{p-1}\otimes\C^{q-1}\oplus(\C^{p-1}\otimes\C^{q-1})^*$
on which $\z\oplus \u(1)\oplus \u(p-1)\oplus \u(1) \oplus \u(q-1)$
acts. If $p \geqslant 3$ this module does not appear in the
classification of \cite{BR}, thus the corresponding action has
non-zero homogeneity rank. The case $p=2$ is left to consider: If $q
\leqslant 5$ the dimensional condition \eqref{dimcond} is not even satisfied, if $q \geqslant 6$
it is easy to find directly that the principal isotropy is $\su(q-4)$,
so that the homogeneity rank is $-2$.
\subsubsection{$G=Z\cdot \rho(H)$ with $\rho$ irreducible representation of complex
type of the simple Lie group $H$}\label{irrcompl}
If $G$ acts with vanishing homogeneity rank on $\H\P^{n-1}$ then, by Proposition
\ref{restriction}, it acts {\em coisotropically} on the $G$-invariant
totally complex submanifold
$L=\C\P^{n-1}=\U(n)/\U(1)\times\U(n-1)$ and, since $Z$ acts trivially
on $L$ this is in turn equivalent to the fact that the representation
of $\rho(H)^\C \times \C^*$ on $\C^{n}$ is multiplicity free. Using
Kac's list \cite{Ka}, and taking only the representations of complex
type we get the standard representation of $\SU(n)$, the
representations of $\SU(n)$ on $\Lambda^2(\C^n)$ with $n\geqslant 5$
and on $S^2_0(\C^n)$,
the half-spin representation of $\Spin(10)$, the
standard representation of $\E_6$ on $\C^{26}$.
We have to consider those representations of complex type satisfying
the dimensional condition that in this case becomes $\dim H+\rk
H\geqslant 4 \deg \rho -6$. The only remaining case is the first
one and has already been
treated in subsection \ref{su(n)}.
\subsection{The subgroups of $\U(k)\times\U(n-k)\subset \U(n)$}
Except the diagonal subgroup (when $2k=n$), the maximal compact
connected subgroups of $\U(k)\times\U(n-k)$ are $\S(U(k)\times\U(n-k))$ and those
of the form $H\times \U(n-k)$ where $H$ is a maximal compact connected
subgroup of $\U(k)$.
For the subgroups of this form we can apply Corollary \ref{product}
arguing that $H$ must necessarily act with vanishing homogeneity rank on $\H\P^{k-1}$.
Thus for $H$ we have only two possibilities: either
$H=\U(k_1)\times\U(k_2)$ (with $k_1+k_2=k$) or $H=Z\cdot\Sp(k/2)$
(when $k$ is even).
\subsubsection{$H=\U(k_1)\times\U(k_2)$} We can exploit the previous
computations and consider the orbit
$\C\P^{k_1-1}\subset\C\P^{k-1}\subset\C\P^{n-1}\subset\H\P^{n-1}$;
so the slice representation is given by
\[
\C^*\otimes ((\C^{k_1-1})^*\oplus \C^{k_2} \oplus (\C^{k_2})^* \oplus \C^{n-k} \oplus (\C^{n-k})^*)\,.
\]
on which $\U(1)\times\U(k_1-1)\times\U(k_2)\times\U(n-k)$ acts.
Analogously to a previous case it is easy to see that the principal isotropy group is
isomorphic to $\U(k_1-2)\times\U(k_2-2)\times\U(n-k-2)$ so that the cohomogeneity is 8
and the action has homogeneity rank equal to $-2$.
\subsubsection{$H=Z\cdot\Sp(k/2)$} We can compute the slice representation at the class of
the identity in $\Sp(n)/\Sp(1)\Sp(n-1)$. The intersection of $\g$ with
$\sp(1)\oplus\sp(n-1)$ is $\u(1)\oplus\u(1)\oplus\sp(k/2-1)\oplus\u(n-k)$
acting on the slice
\[
\C^{n-k}\oplus (\C^{n-k})^*\oplus\H^{k/2-1}\oplus\C\,,
\]
where one of the two 1-dimensional copies of $\u(1)$ acts on every
module and the other only on the first two modules.
Now it is immediate to see that the principal isotropy subalgebra is
isomorphic to $\sp(k/2-2)\oplus\u(n-k-2)$ so that the cohomogeneity is 5
and the action has vanishing homogeneity rank.
\subsubsection{$G=\U(k)_\Delta\subset \U(k) \times \U(k)$ with $n=2k$} In order to conclude that
$\U(k)_\Delta$ has non-zero homogeneity rank on $\H\P^{n-1}$ it is
sufficient to observe that $\U(k)_\Delta\subset \Sp(k)_\Delta \subset
\Sp(k)\times\Sp(k)$ and that the action of $\Sp(k)_\Delta$ on
$\H\P^{n-1}$ is equivalent to that of $\Sp(k)\subset U(2k)$ since the
standard representation of $\Sp(k)$ on $\C^{2k}$ is self-dual.
\subsection{The subgroups of $G=Z(\U(n))\cdot\Sp(k)\subset \U(n)$ (with
$n=2k$).} Now we are going to show that the action of
$Z(\U(n))\cdot\Sp(k)\subset \U(n)$ is minimal as vanishing homogeneity
rank action.
The maximal compact connected subgroups of $G$ other than $\Sp(k)$ (that we have considered in a
previous step) are of the form $Z\cdot H$ where $H$ is a
maximal compact connected subgroup of $\Sp(k)$.
\subsubsection{$H=U(k)$} As for this subgroup the conclusion follows immediately from the observation that
$Z\cdot \U(k)$ is contained in $Z\cdot \SO(2k)$ which does not act
with vanishing homogeneity rank on $\H\P^{n-1}$.
\subsubsection{$H=\SO(p)\otimes\Sp(q)$ with $2pq=n$} If $Z\cdot H$ acts
with vanishing homogeneity rank on $\H\P^{n-1}$, then it should act coisotropically
on the totally complex $\U(2pq)$-orbit $\C\P^{2pq-1}$, but this is not
the case as one can deduce from the list of \cite{Ka} and \cite{BR}.
\subsubsection{$H=\rho(H')\subset \Sp(k)$ where $\rho$ is an
irreducible representation of quaternionic type of the simple Lie
group $H'$}
We can argue as in subsection \ref{irrcompl}, that is we apply our
version of restriction lemma combined with the classification of
Kac's. In this way we find no proper subgroup $H$ of $\Sp(k)$.
\subsubsection{$H=\Sp(r)\times\Sp(k-r)$ with $1\leqslant r\leqslant
k-1$} Here it is sufficient to note that $Z(U(2k)) \cdot H$ is a
subgroup of $Z(\U(r))\cdot\Sp(r)\times Z(\U(k-r))\cdot\Sp(k-r)$ whose
action on $\H\P^{n-1}$ has non-zero homogeneity rank.
\subsection{The subgroups of $Z(U(k))\cdot\Sp(r)\times\U(n-k)\subset
\U(n)$ with $k=2r$} Now we prove that the vanishing homogeneity rank action of
$Z(U(2r))\cdot\Sp(r)\times\U(n-2r)$ is minimal. Since the action of
$Z(U(2r))\cdot\Sp(r)$ is minimal, by Proposition \ref{restriction},
the only subgroups we need to consider are of the form
$Z(U(2r))\cdot\Sp(r)\times H$, where $H$ is a maximal compact connected
subgroup of $\U(n-2r)$ acting with vanishing homogeneity rank on $\H\P^{n-2r-1}$.
There are three possibilities for $H$: $H_1=\U(k_1)\times\U(k_2)$ with
$k_1+k_2=n-2r$, $H_2=Z(U(n-2r))\cdot Sp(\frac{n-2r}{2})$ (when $n$ is
even), $H_3=\SU(n-2r)$.
The subgroup $Z(U(2r))\cdot\Sp(r)\times H_1$ is contained in
$\U(2r)\times\U(k_1)\times\U(k_2)$, hence its action has non-zero
homogeneity rank.\\
The subgroup $Z(U(2r))\cdot\Sp(r)\times H_2$ need to be treated explicitly,
finding the intersection of it with $\Sp(1)\Sp(n-1)$. In this way we get the isotropy
subalgebra $\mathfrak{l}=\u(1)\oplus\u(1)\oplus\u(1)\oplus\sp(r-1)\oplus\sp(n/2-r)$ acting on
the slice
\[
\H^{r-1}\oplus \H^{n/2-r} \oplus \H^{n/2-r} \oplus \C \oplus \C\,.
\]
Since the abelian subalgebra of $\mathfrak{l}$ acts on the
1-dimensional modules, this action has vanishing homogeneity rank on
each {\em irreducible} submodule, nevertheless it is easy to see that
the principal isotropy is $\sp(r-2)\oplus\sp(n/2-r-2)$. Therefore the
cohomogeneity is 8 and $\hrk(Z(U(2r))\cdot\Sp(r)\times H_2,\H\P^{n-1})=-2$.
As for $Z(U(2r))\cdot\Sp(r)\times H_3$ it is sufficient to observe
that it induces on the quaternionic projective space the same action
of $\Sp(r)\times\U(n-2r)$, which has non-zero homogeneity rank.\\
This concludes the analysis of the subgroups of $\U(n)\subset\Sp(n)$.
\subsection{The subgroups of $G=\rho(H)$ with $\rho$ irreducible representation of quaternionic
type of the simple Lie group $H$} We have to examine only those
subgroups that in case \ref{irreducible} give rise to vanishing
homogeneity rank actions. We exclude all of them simply noting that none of the
subgroups of maximal dimension satisfy the dimensional condition
\eqref{dimcond}. The list of subgroups of
maximal dimension is given in \cite{Mann} and can be found also in \cite{Ko}.
\subsection{The subgroups of $G=\SO(n)\otimes \Sp(1)$.}\label{sosp(1)} Now we prove that the
action of $\SO(n)\otimes \Sp(1)$ is minimal except for $n=8$.\\
A maximal compact connected subgroups of $G$ is conjugate to one of the form $H_1
\otimes H_2$ where $H_1$ is either a compact connected maximal subgroup of
$\SO(n)$ or $\SO(n)$ itself, and $H_2$ is either $\Sp(1)$ or $\U(1)$.
The subgroup $\SO(n) \otimes U(1)$ is the same as $Z(\U(n))\cdot
\SO(n)\subset \U(n)$ that we have already excluded (see
case \ref{qualcosa2}), so let us turn to the case $H_1 \otimes \Sp(1)$
and look at Table \ref{maxSO} for maximal subgroups of $\SO(n)$.
\subsubsection{$H_1=\U(k)$ where $n=2k$} It is easy to find the slice representation at
the quaternionic line $\ell$ spanned by a pure element of
$\R^k\otimes \R^{4}$ starting from \eqref{slicetensor}. The
stabilizer subalgebra is
$\u(k-1)\oplus\sp(1)$ acting on
\[
\C^{k-1}\otimes_\R\R^3 \oplus \R^3
\]
where $\R^3$ stands for the adjoint representation of $\o(3) \simeq \sp(1)$.
It follows immediately that the principal isotropy subalgebra is
isomorphic to $\u(k-4)$ if $n \geqslant 5$, otherwise it is trivial.
In any case the homogeneity rank is -4.
\subsubsection{$H_1=\S(\O(k)\times\O(n-k))$} The isotropy subalgebra at
$\ell\in\H\P^{n-1}$ is $\o(k-1)\oplus\o(n-k)\oplus\sp(1)$ acting on
\[
\R^{k-1}\otimes\R^3 \oplus \R^{n-k}\otimes\R^3 \oplus \R^{n-k}\,.
\]
Here, in the general case, we are not allowed to skip the computation of the
principal isotropy subalgebra. Nevertheless it is not hard to find
that it is isomorphic to $\o(k-4)\oplus\o(n-k-4)$ for $k, n-k
\geqslant 6$ so that $c=13$ and $\hrk=-8$. If either $k$ or $n-k$ are
smaller than $6$, a similar argument leads to the same conclusion.
The remaining low-dimensional cases can be excluded
using \eqref{dimcond}.
\subsubsection{$H_1=\SO(p)\otimes\SO(q)$ with $n=pq$} The isotropy subalgebra at
$\ell\in\H\P^{n-1}$ is $\o(p-1)\oplus\o(q-1)\oplus\sp(1)$ acting on
\[
\Sigma=
(\R^{p-1}\otimes\R^{q-1}) \oplus (\R^{p-1}\otimes\R^{q-1}\otimes \R^3) \oplus (\R^{p-1}\otimes\R^3) \oplus (\R^{q-1}\otimes\R^3) \,.
\]
Let us distinguish three subcases according to the parity of $p$ and
$q$.
If $p$ and $q$ are odd then the orbit through $\ell$ has positive
Euler characteristic but the real irreducible module
$\R^{p-1}\otimes\R^{q-1}\otimes \R^3$
has negative homogeneity rank (it does not appear in the
classification of \cite{GP}).\\
If only one among $p$ and $q$ is even (say $p$), then the orbit has no more
positive Euler characteristic but, with the notations of Lemma \ref{Slice}, we have $\delta=1$. Thus it is sufficient to show
that $\hrk(G_\ell,\Sigma)\leqslant -2$. Thanks to Lemma
\ref{sommaranghi}
\begin{eqnarray*}
\hrk(G_\ell,\Sigma) & \leqslant &\hrk(\O(p-1)\times\O(q-1)\times\O(3),\R^{p-1}\otimes\R^{q-1}\otimes \R^3)+\\
& & \hrk(\O(p-1)\times\O(3),\R^{p-1}\otimes \R^3)\leqslant -2\,.
\end{eqnarray*}
If both $p$ and $q$ are even, we have $\delta=2$, but
\begin{eqnarray*}
\hrk(G_\ell,\Sigma) & \leqslant &\hrk(\O(p-1)\times\O(q-1)\times\O(3),\R^{p-1}\otimes\R^{q-1}\otimes\R^3)+ \\
& & \hrk(\O(p-1)\times\O(3),\R^{p-1}\otimes \R^3)+\\
& & \hrk(\O(q-1)\times\O(3),\R^{q-1}\otimes \R^3)\leqslant -3\,.
\end{eqnarray*}
\subsubsection{$H_1=\Sp(p)\otimes\Sp(q)$ with $n=4pq \geqslant 8$} This action has no
orbit of positive Euler characteristic. If $p,q\geqslant 2$ the isotropy subalgebra at
$\ell\in\H\P^{n-1}$ is $\sp(p-1)\oplus\sp(q-1)\oplus\sp(1)$ acting on
\[
(U\otimes\R^3) \oplus (\H^{p-1}\otimes \R^3) \oplus (\H^{q-1}\otimes
\R^3) \oplus M\otimes \R^3) \oplus M \oplus U \,,
\]
where $M=\mathcal{M}(p-1,q-1,\H)$ and $U$ is the adjoint
representation of $\sp(1)$. Here $\delta=2$ but
$\hrk(\Sp(p-1)\times\Sp(1),\H^{p-1}\otimes \R^3)=-8$
. Thus the action has non-zero homogeneity rank.\\
Obviously this module appears in the slice even when $q=1$, so we get
no new vanishing homogeneity rank actions.
\subsubsection{$H_1=\rho(K)$ with $\rho$ irreducible representation of
real type of the simple Lie group $K$} We here use again the dimensional
condition \eqref{dimcond} that in this situation becomes
\[
\dim K+\rk K\geqslant 4\deg\rho -8.
\]
Kollross in lemma 2.6 in \cite{Ko} lists all the representations
$\sigma$ of real type of
Lie groups $L$ such that $2\dim L\geqslant \deg \sigma-2$. This
condition is always looser than ours.
Counting the dimensions for the groups and the representations from
this list, we have that
only the spin representation of $K=\Spin(7)$ and the standard
representations of $\SO(n)$ satisfy the condition. The latter
correspond to the case treated in subsection \ref{sosp(1)}.
Let us compute $\hrk(\Spin(7)\times\Sp(1),\H\P^7)$.
As usually we consider the orbit through the quaternionic line $\ell$
spanned by a pure tensor of $\R^8\otimes\R^4$. It turns out to be the
seven-dimensional sphere $\Spin(7)/{\rm G}_2$ and the slice
representation is the tensor product of the standard representation of
${\rm G}_2$ with the adjoint representation of $\Sp(1)$. It is well
known (see e.g. \cite[p. 11]{GP}) that this irreducible representation has trivial principal
isotropy and from this follows that $\hrk(\Spin(7)\times\Sp(1),\H\P^7)=0$.
\subsection{The subgroups of $\Sp(k)\times \Sp(n-k)$.} We analyse this case with
the aid of the following lemma:
\begin{lemma} \label{prodSp} Let $G\subseteq \Sp(N)$ be a compact Lie group acting
with vanishing homogeneity rank on $\H\P^{N-1}$. Then $\widetilde{G}=G\times \Sp(n)$
acts with vanishing homogeneity rank on $\H\P^{N+n-1}=\P_\H(\H^N\oplus \H^n)$.
\end{lemma}
\begin{proof}
If $v$ is taken in $\H^n$, the $\widetilde{G}$-orbit through $[v]$ in
$\H\P^{N+n-1}=\P_\H(\H^N\oplus \H^n)$ is $\H\P^{n-1}$. Therefore the
action of $\widetilde{G}$ has homogeneity rank zero if and only if the slice representation
at this quaternionic orbit has vanishing homogeneity rank. Note that
the last
factor of the isotropy subgroup $G\times\Sp(1)\cdot \Sp(n-1)$ acts
trivially on the slice $\Sigma_{[v]}\simeq \H^N$.
Consider now the natural projection of $\H^N\setminus\{0\}$ on
$\H\P^{N-1}$. This is an equivariant fibration with fiber $\H$.
Thus arguing as in Proposition \ref{restriction} we deduce that
\[
\hrk(G\times \Sp(1),\H^N)=\hrk(G,\H\P^{N-1})+\hrk(\Sp(1),\H)
\]
and the claim follows since both the homogeneity ranks in the right hand side
of the equality vanish.
\end{proof}
As a consequence, combining the previous lemma with Proposition
\ref{restriction} we obtain the following
\begin{corollary}The group $G\subseteq \Sp(n)$ acts on $\H\P^{n-1}$ with
vanishing homogeneity rank if and only if $G\times \Sp(N)\subseteq
\Sp(n)\times \Sp(N)$ on $\H\P^{n+N-1}$ does.
\end{corollary}
The previous corollary avoid the analysis of those subgroups of
$\Sp(k)\times\Sp(n-k)$ of the form $H_1\times H_2$ where either $H_1$
or $H_2$ equals $\Sp(k)$ or $\Sp(n-k)$. Except for the diagonal action
of $\Sp(k)_\Delta$ when $k=n-k$ (which has already been excluded), it is therefore sufficient to
analyse all the subgroups $H_1\times H_2$, where $H_1\subsetneq \Sp(k)$ acts
on $\H\P^{k-1}$ and $H_2 \subsetneq \Sp(n-k)$
acts on $\H\P^{n-k-1}$ both with vanishing homogeneity rank.\\
The cases that we shall consider are given by all possible
combinations of the following:
\begin{eqnarray*}
H_1 & = & \U(k),\Sp(k_1)\times \Sp(k_2)\,\textmd{with}\,k_1+k_2=k, \\
& & \SO(k)\otimes\Sp(1),\Spin(7)\otimes \Sp(1), \rho(H_1)\\
\vspace{0.2cm}
H_2 & = & \U(n-k),\Sp(l_1)\times \Sp(l_2)\,\textmd{with}\,l_1+l_2=n-k, \\
& & \SO(n-k)\otimes\Sp(1),\Spin(7)\otimes \Sp(1), \rho(H_2)
\end{eqnarray*}
Where $\rho(H_1)\otimes \sigma$ and $\rho(H_2)\otimes \sigma$ are orbit equivalent to isotropy
representations of a quaternionic-K\"ahler symmetric space, where
$\sigma$ is the standard representation of $\Sp(1)$.\\
The case $\U(k)\times\U(n-k)$ has already been treated, the cases in
which one of the factor is either $\Sp(k_1)\times \Sp(k_2)$ or
$\Sp(l_1)\times \Sp(l_2)$
give rise to vanishing homogeneity rank actions thanks to Lemma \ref{prodSp}.\\
The remaining cases can be all excluded with a common argument: we
treat explicitly one of them and then we explain how to generalize.\\
Consider for example
$G=\E_7\times \Spin(11)$ acting on $\P_\H(\H^{28}\oplus \H^{16})$. Let $\E_7/\E_6\cdot \U(1)\subseteq
\H\P^{27}\subseteq \H\P^{43}$ be the maximal totally
complex orbit of $G$. The factor
$\U(1)\times \Spin(11)$ of the isotropy acts on the second
module of the slice $\C^{27}\oplus\H^{16}$ with non
vanishing homogeneity rank, since it is neither the isotropy
representation of a symmetric space of inner type nor it appears in the
list of \cite{GP}.\\ Observe now
that all of the factors of the products $H_1\times H_2$ we are considering admit a totally complex
orbit (see \cite{BG}). All the cases can therefore be excluded in the same
manner taking at a first step a maximal totally complex orbit for
the group $H_1$, and then observing that the slice representation
contains a module on which the isotropy acts with non vanishing homogeneity rank.\\
The classification is now complete. In fact once one goes further the
only possibility that can occur is the product of three factors $G_1\times G_2\times
G_3$ where all of $G_i\neq \Sp(n_i)$ (otherwise this case can be
treated with the aid of Lemma \ref{prodSp}), where each $G_i$ gives rise to
vanishing homogeneity rank action on $\H\P^{n_i-1}$. This case can be
easily excluded applying Proposition \ref{restriction} to the product of two
of the factors.
\section{Appendix: Tables}
\begin{table}[h]
\caption{Maximal subgroups of $\SO(n)$}
\centering
\begin{tabular}{|r|c|l|}
\hline i) & $\SO(k)\times \SO(n-k)$ & $1 \leq k \leq n-1$ \\
\hline ii) & $\U(m)$ & $2m=n$ \\
\hline iii) & $\SO(p)\otimes \SO(q)$ & $pq=n,\ 3 \leq p\leq q$ \\
\hline iv) & $\Sp(p)\otimes \Sp(q)$ & $4pq=n$ \\
\hline v) & $\rho(H)$ & $H$ simple, $\rho \in \Irr_{\R}$, $\deg\rho=n$ \\
\hline
\end{tabular}
\label{maxSO}
\end{table}
\begin{table}[h]
\caption{Maximal subgroups of $\SU(n)$}
\centering
\begin{tabular}{|r|c|l|}
\hline i) & $\SO(n)$ & \\
\hline ii) & $\Sp(m)$ & $2m=n$ \\
\hline iii) & $\S(\U(k)\times \U(n-k))$ & $1 \leq k \leq n-1$ \\
\hline iv) & $\SU(p) \otimes \SU(q)$ & $pq=n,\ p \geq 3,\ q \geq 3$ \\
\hline v) & $\rho(H)$ & $H$ simple, $\rho \in \Irr_{\C}$, $\deg\rho=n$ \\
\hline
\end{tabular}
\label{maxSU}
\end{table}
\begin{table}[h]
\caption{Maximal subgroups of $\Sp(n)$}
\centering
\begin{tabular}{|r|c|l|}
\hline i) & $\U(n)$ & \\
\hline ii) & $\Sp(k) \times Sp(n-k)$ & $1 \geq k \geq n-1$ \\
\hline iii) & $\SO(p) \otimes \Sp(q)$ & $pq=n,\ p \geq 3,\ q\geq 1$ \\
\hline iv) & $\rho(H)$ & $H$ simple, $\rho \in\Irr_{\H}$, $\deg\rho=2n$ \\
\hline
\end{tabular}
\label{maxSp}
\end{table}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,521 |
Q: How to set supported localization programmatically When using nib files I can set the supported localizations here:
When this is done and the pop up "Paste" button will be translated into the system language like this:
But I'm not using a nib file now, and the translate will not happen, so how can I programmatically set the supported localizations just like what I did to the nib file?
A: If you are not using a nib file, I am guessing you are creating the UIView's in the code. I would just use the NSLocalizedString(string,comment) function and would assign it's translation on the moment of the UIView's creation.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,730 |
Broadway will remain shuttered for the rest of 2020 and yes, that sound you can hear is a million gay hearts breaking
Nick Duffy June 30, 2020
Broadway will remain closed until 2021 due to the ongoing coronavirus pandemic. (Cindy Ord/Getty Images)
Broadway theatres will not be reopening in 2020 due to the Coronavirus pandemic, it has been announced.
New York industry trade association The Broadway League confirmed that all 41 venues will remain closed until the end of 2020, as theatre bosses struggle to conceive of a feasible way to make theatres comply with social distancing rules.
Broadway performances will remain suspended until January 3, 2021 at the earliest, as work continues "with city and state officials as well as leaders in science, technology, and medicine to formulate the best plan to restart the industry".
'We'll be back': Broadway theatres will remain shuttered until 2021.
Broadway League board chair Thomas Schumacher said: "The Broadway experience can be deeply personal but it is also, crucially, communal.
"The alchemy of 1000 strangers bonding into a single audience fuelling each performer on stage and behind the scenes will be possible again when Broadway theatres can safely host full houses.
"Every single member of our community is eager to get back to work sharing stories that inspire our audience through the transformative power of a shared live experience.
"The safety of our cast, crew, orchestra and audience is our highest priority and we look forward to returning to our stages only when it's safe to do so. One thing is for sure, when we return we will be stronger and more needed than ever."
Broadway League president Charlotte St. Martin added: "We are focused on identifying and implementing necessary measures that will enable us to resume performances safely for Broadway audiences and employees.
"We are determined to bring back the people who rely on this industry for their livelihood, and to welcome back all those who love this vital part of New York City, as soon as it is safe to do so.
"As so many of us in the Broadway community have been saying during this time, we'll be back, and we have so many more stories to tell."
West End shows to test temperature checks and drive-in performances amid battle to re-open.
The approach taken on Broadway is different to that on London's West End, where Andrew Lloyd Webber has announced hopes to test live performances this year using thermal imaging cameras to run temperature checks.
The composer told the BBC earlier this month: "What I hope to do is to demonstrate what has happened in South Korea at the London Palladium… we've just had the final piece of equipment delivered and it's just clearing customs. Then we're going to do a series of tests to see if it's going to work.
More from PinkNews
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"The reason that we've chosen the Palladium is that it's a very big theatre, just under 2,300 seats. It's the biggest theatre we have and in one way the most problematic. We want to be able to demonstrate there that this can work.
"All we can do is continue to be positive and demonstrate we can open. If we do that and we fail, then at least we've tried."
Meanwhile, West End concert musical Six is planning to hold distanced drive-in performances across the UK.
All West End theatres will remain closed until at least August 2, with a "roadmap" published by UK culture secretary Oliver Dowden not including firm dates for when theatres may be allowed to resume.
Dowden said: "I desperately want to raise the curtain on live performances in theatres and music venues as soon we can — they are the soul of our nation and a lynchpin of our world-beating creative industries.
"We know the challenges — theatres must be full to make money, and performers need to be safe on stage as they sing, dance and play instruments — but I am determined to ensure the performing arts do not stay closed longer than is absolutely necessary to protect public health."
More: Broadway, Theatre
Bridgerton star Jonathan Bailey was warned not to come out as gay by powerful queer TV figures
Josh Milton - December 31, 2020
Travis Alabanza's gripping new play teaches a powerful lesson about bathrooms, transphobia and female friendship
Vic Parsons - December 16, 2020
The Prom's Ariana DeBose used 'all the pressures' of growing up queer to tell a lesbian love story the right way
Matilda Davies - December 11, 2020
West End chemsex drama star Jimmy Essex is determined to tell queer stories beyond 'the coming out thing' | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,020 |
{"url":"https:\/\/mathoverflow.net\/questions\/368966\/cj-topology-considered-by-greene-and-krantz\/369014","text":"# $C^j$-topology considered by Greene and Krantz\n\nMy question is about the $$C^j$$ topology used by Greene and Krantz in their paper \"Deformations of Complex Structures, Estimates for the $$\\bar{\\partial}$$-equation, and stability of the Bergman kernel\". As it is not clear to me that this topology is the same as the usual Whitney strong (or weak) $$C^j$$ topology and it is not explicitly defined for functions on manifolds.\n\n1. First they describe the $$C^j$$ topology for maps $$f:U \\to \\mathbb{C}$$ for any open set $$U \\subset \\mathbb{C}^n$$. This is done in a quite standard way, for instance:\n\n$$||f||_{C^j(U)}:= \\sum_{|\\alpha|+|\\beta| \\leq j} \\left|\\left|\\left(\\frac{\\partial}{\\partial z}\\right)^{\\alpha}\\left(\\frac{\\partial}{\\partial \\bar{z}}\\right)^{\\beta}f\\right|\\right|_{\\infty}$$\n\nWhere $$\\alpha$$ and $$\\beta$$ are taken as multidices and $$||\\cdot||_{\\infty}$$ denotes the supremum norm.\n\n1. Inmediately after, they define another $$C^\\infty$$-norm. And say that it extends to a smooth manifold \"via a fixed coordinate atlas\". How is this extension performed? I guess you have to take a locally finite coordinate atlas and sum over all charts the previously defined norm (?). Moreover, they make a remark saying that two functions defined on $$U$$ are $$C^\\infty$$ close if they are $$C^k$$ close for $$k$$ big enough and they say that this remark extends trivially to the manifold case. So implicitly they are considering a $$C^j$$ norm on the space of $$C^\\infty$$ functions defined on a manifold. What is this norm?\n\n2. A very similar problem arises later in page 35 when they define a topology in the space of almost-complex structures of a smooth manifold. And they claim that there are neighborhoods of the form $$S_j(\\prod_{1,0},\\epsilon):=\\{\\prod_{1,0}': \\text{where }\\prod_{1,0} - \\prod_{1,0}' \\text{ is less than } \\epsilon \\text{ with respect to a } C^j \\text{norm}\\}.$$ So again, it seems that they are considering a norm on the space of $$(1,1)$$ tensors on manifolds (rather than open sets) and they are taking the topology induced by this norm.\n\nMy question is, what is the precise definition of this norm that induces the $$C^j$$ topology for complex valued smooth functions on manifolds and how does it relate to usual Whitney topologies? (references appreciated). It looks like this topology can't be the same as the Whitney topology (otherwise the Whitney topology would be usually defined using this norm rather than the more intricate usual definition). But of course this is just a moral argument.\n\nIt is pretty much as you guessed in 1.) Instead of summing, you take the maximum. Although maybe in some settings both describe equivalent norms (I am no expert). The key point here is that the target space is $$\\mathbb{C}$$ and not a general manifold. In general the Whitney topology does not come from a norm, although it does come from a metric when $$M$$ is compact (The distance Greene and Krantz talk about does generalize to manifolds).\nIn general one can define a norm on the space of sections of a vector bundle. In this particular case you are dealing with sections of the trivial vector bundle $$M \\times \\mathbb{C}$$ where $$M$$ is a compact manifold. For the general case of sections of any vector bundle, look at Section 3 of \"The Banach manifold $$C^k(M,N)$$ by Johannes Wittmann.\nFix a finite coordinate atlas $$\\{(U_i, \\phi_i)\\}_{1, \\ldots, \\ell}$$ such that $$\\bar{U}_i$$ is compact and is still contained in a coordinate atlas. Then define:\n$$||f||_{C^j(M)}:=\\max_{1\\leq i \\leq \\ell} ||f||_{C^j(U_i)}$$\nThe $$||\\cdot||_{C^j(U)}$$ norm defined in Wittmann's work is slightly different but I think they are equivalent. (Witmann does not sum the norms for all partial derivatives but rather takes the maximum of them).\n\u2022 Thank you. I guess the only piece missing is that the norm resulting from summing the sup of all partial derivatives up to order $k$ is equivalent to the norm taking the maximum of those supremums. Do you know a reference for that? \u2013\u00a0Pita Aug 13 at 12:31","date":"2020-12-05 10:02:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 29, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9232861399650574, \"perplexity\": 130.40465992049056}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141747323.98\/warc\/CC-MAIN-20201205074417-20201205104417-00717.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.crypto.ethz.ch\/publications\/abstract.html?label=BaCaMa05","text":"Publications: Abstract\n\nEfficient Proofs of Knowledge of Discrete Logarithms and Representations in Groups with Hidden Order\n\nEndre Bangerter and Jan Camenisch and Ueli Maurer\n\n For many one-way homomorphisms used in cryptography, there exist efficient zero-knowledge proofs of knowledge of a preimage. Examples of such homomorphisms are the ones underlying the Schnorr or the Guillou-Quisquater identification protocols. In this paper we present, for the first time, efficient zero-knowledge proofs of knowledge for exponentiation $\\psi(x_1) = h_1^{x_1}$ and multi-exponentiation homomorphisms $\\psi(x_1, \\ldots, x_l) \\doteq h_1^{x_1} \\cdots h_l^{x_l}$ with $h_1, \\ldots,h_l \\in H$ (i.e., proofs of knowledge of discrete logarithms and representations) where $H$ is a group of hidden order, e.g., an RSA group.","date":"2017-09-21 17:47:50","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\": 0.7018608450889587, \"perplexity\": 1475.4519518485902}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818687834.17\/warc\/CC-MAIN-20170921172227-20170921192227-00093.warc.gz\"}"} | null | null |
<div class="user-details">
<h1> My Projects </h1>
</div>
<div class="user-projects">
<div class="images-right">
<img alt="onyx" src="{{ "/assets/img/onyx.png" | prepend: site.baseurl }}" />
</div>
<div class="contents">
<h3> Onyx </h3>
<p>Onyx is an open source voice assistant. It can be used everywhere and help you. It is open so it can be remixed, improved...</p>
<p>I lead this project since 2014, it is open source and I always keep it up to date.</p>
<a class="project-link" href="https://onyxlabs.fr" target="_blank">More</a>
</div>
</div>
<div class="user-projects">
<div class="images-left">
<img alt="dashyx" src="{{ "/assets/img/dashyx.png" | prepend: site.baseurl }}" />
</div>
<div class="contents-right">
<h3> Dashyx </h3>
<p>Dashyx is an application that allows you to group all your applications into one for better productivity.</p>
<p>This project is not open source and I develop it since June 2018.</p>
<a class="project-link" href="https://dashyx.com" target="_blank">More</a>
</div>
</div> | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,969 |
A Primeira pré-eliminatória da Liga Europa da UEFA de 2017–18 foi disputada entre os dias 29 de junho e 6 de julho de 2017. Um total de 100 equipes competiram nesta fase para decidir 50 das 66 vagas na Segunda pré-eliminatória.
Todas as partidas seguem o Horário de Verão da Europa Central (UTC+2).
Primeira pré-eliminatória
O sorteio para esta fase foi realizado em 19 de junho de 2017. As partidas de ida foram disputadas em 29 de junho e as partidas de volta em 9 de julho de 2017.
|}
Jogo 1
Jogo 2
Jogo 3
Jogo 4
Jogo 5
Jogo 6
Jogo 7
Jogo 8
Jogo 9
Jogo 10
Jogo 11
Jogo 12
Jogo 13
Jogo 14
Jogo 15
Jogo 16
Jogo 17
Jogo 18
Jogo 19
Jogo 20
Jogo 21
Jogo 22
Jogo 23
Jogo 24
Jogo 25
Jogo 26
Jogo 27
Jogo 28
Jogo 29
Jogo 30
Jogo 31
Jogo 32
Jogo 33
Jogo 34
Jogo 35
Jogo 36
Jogo 37
Jogo 38
Jogo 39
Jogo 40
Jogo 41
Jogo 42
Jogo 43
Jogo 44
Jogo 45
Jogo 46
Jogo 47
Jogo 48
Jogo 49
Jogo 50
Notas
A. O Maccabi Tel Aviv disputou sua partida como mandante no Netanya Stadium em Netanya, ao invés do seu estádio regular, o Bloomfield Stadium em Tel Aviv.
B. Shkëndija disputou sua partida como mandante no Stadion Mladost em Strumica, ao invés do seu estádio regular, o Ecolog Arena em Tetovo.
C. Chikhura Sachkhere disputou sua partida como mandante no Boris Paichadze Dinamo Arena em Tbilisi, ao invés do seu estádio regular, o Central Stadium em Sachkhere.
D. Zira disputou sua partida como mandante no Dalga Arena em Baku, ao invés do seu estádio regular, o Zira Olympic Sport Complex Stadium.
E. Beitar Jerusalem disputou sua partida como mandante no HaMoshava Stadium em Petah Tikva, ao invés do seu estádio regular, o Teddy Stadium em Jerusalém, devido a disputa das Macabíadas.
F. Mladost Podgorica disputou sua partida como mandante no Podgorica City Stadium em Podgorica, ao invés do seu estádio regular, o Stadion FK Mladost em Podgorica.
G. Pyunik disputou sua partida como mandante no Vazgen Sargsyan Republican Stadium em Yerevan, ao invés do seu estádio regular, o Yerevan Football Academy Stadium em Yerevan.
H. Dinamo Batumi disputou sua partida como mandante no Ramaz Shengelia Stadium em Kutaisi, ao invés do seu estádio regular, o Chele Arena em Kobuleti.
I. Videoton disputou sua partida como mandante no Pancho Aréna em Felcsút, ao invés do seu estádio regular, o Sóstói Stadion em Székesfehérvár, devido a reformas.
J. Zaria Bălți disputou sua partida como mandante no Zimbru Stadium em Chișinău, ao invés do seu estádio regular, o Stadionul Orășenesc em Bălți.
K. St Joseph's disputou sua partida como mandante no Estádio Algarve em Loulé, ao invés do seu estádio regular, o Victoria Stadium em Gibraltar.
L. Bala Town disputou sua partida como mandante no Belle Vue em Rhyl, ao invés do seu estádio regular, o Maes Tegid em Bala.
M. Domžale disputou sua partida como mandante na Arena Petrol em Celje, ao invés do seu estádio regular, o Domžale Sports Park em Domžale.
N. Connah's Quay Nomads disputou sua partida como mandante no Nantporth em Bangor, ao invés do seu estádio regular, o Deeside Stadium em Connah's Quay.
O. Nõmme Kalju disputou sua partida como mandante na A. Le Coq Arena em Tallinn, ao invés do seu estádio regular, o Hiiu Stadium em Tallinn.
P. Levadia Tallinn disputou sua partida como mandante no Pärnu Rannastaadion em Pärnu, ao invés do seu estádio regular, o Kadriorg Stadium em Tallinn.
Q. KÍ disputou sua partida como mandante no Gundadalur em Tórshavn, ao invés do seu estádio regular, o Við Djúpumýrar em Klaksvík.
R. Inter Baku disputou sua partida como mandante na Dalga Arena em Baku, ao invés do seu estádio regular, o Inter Arena em Baku.
S. Pelister disputou sua partida como mandante no Stadion Mladost em Strumica, ao invés do seu estádio regular, o Stadion Tumbe Kafe em Bitola.
T. Vasas disputou sua partida como mandante no Szusza Ferenc stadion em Budapest, ao invés do seu estádio regular, o Illovszky Rudolf Stadion em Budapest, devido a reformas.
U. Milsami Orhei disputou sua partida como mandante no Zimbru Stadium em Chișinău, ao invés do seu estádio regular, o CSR Orhei em Orhei.
V. Gandzasar FC disputou sua partida como mandante no Vazgen Sargsyan Republican Stadium em Yerevan, ao invés do seu estádio regular, o Gandzasar Stadium em Kapan.
X. Ordabasy disputou sua partida como mandante no Central Stadium em Almaty, ao invés do seu estádio regular, o Kazhymukan Munaitpasov Stadium em Shymkent.
Y. Botev Plovdiv disputou sua partida como mandante no Lazur Stadium em Burgas, ao invés do seu estádio regular, o Botev 1912 Football Complex em Plovdiv.
W. Floriana disputou sua partida como mandante no Hibernians Stadium em Paola, ao invés do seu estádio regular, o Independence Ground em Floriana.
Z. FK Rabotnički disputou sua partida como mandante no Training Centre Petar Miloševski em Skopje, ao invés do seu estádio regular, o Philip II Arena em Skopje.
AA. Zeta disputou sua partida como mandante no City Stadium em Podgorica, ao invés do seu estádio regular, o Stadion Trešnjica em Golubovci.
AB. Lincoln Red Imps disputou sua partida como mandante no Estádio Algarve em Loulé, ao invés do seu estádio regular, o Victoria Stadium em Gibraltar.
AC. Derry City disputou sua partida como mandante no The Showgrounds em Sligo, ao invés do seu estádio regular, o Brandywell Stadium em Derry, devido a reformas.
AD. NSÍ Runavík disputou sua partida como mandante no Svangaskarð em Toftir, ao invés do seu estádio regular, o Runavík Stadium em Runavík.
AE. Ballymena United disputou sua partida como mandante no Seaview em Belfast, ao invés do seu estádio regular, o The Showgrounds em Ballymena.
Ver também
Liga dos Campeões da UEFA de 2017–18
Liga Europa da UEFA de 2017–18
Liga Europa da UEFA de 2017–18 – Segunda pré-eliminatória
Liga Europa da UEFA de 2017–18 – Terceira pré-eliminatória
Liga Europa da UEFA de 2017–18 – Rodada de play-off
Ligações externas
Liga Europa da UEFA de 2017–18
2017 no futebol | {
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} | 7,447 |
Asier del Horno Cosgaya (* 19. Januar 1981 in Barakaldo) ist ein ehemaliger spanischer Fußballspieler.
Karriere
Verein
Asier del Horno stammt aus Barakaldo im Baskenland, durchlief sämtliche Jugendmannschaften des baskischen Vorzeigeklubs Athletic Bilbao und debütierte am 9. September 2000 in der ersten Mannschaft.
Im Juli 2005 wechselte der spiel- und kampfstarke Linksverteidiger für zwölf Millionen Euro zum englischen Klub FC Chelsea nach London. Gleich in seiner ersten Saison gewann er die englische Meisterschaft und den englischen Ligapokal.
Nur ein Jahr später verließ del Horno Chelsea wieder und wechselte zurück nach Spanien zum FC Valencia. Er unterschrieb einen Sechsjahresvertrag und kostete wiederum zwölf Millionen Euro. In seiner ersten Saison hatte er mit Verletzungen zu kämpfen, so dass er nur sechs Spiele bestreiten konnte. Zur Saison 2007/08 wechselte del Horno zunächst auf Leihbasis zurück zu seinem Stammverein Athletic Bilbao. Bis zum Ende der Saison besaß der Klub zudem eine Kaufoption in Höhe von drei Millionen Euro. In Bilbao wurden große Hoffnungen in del Horno gesetzt, sollte er doch die in den letzten beiden Saisons akute Abwehrschwäche beheben.
Da Athletic Bilbao mit seinen Leistungen nicht zufrieden war, wurde die Kaufoption nicht gezogen. Deswegen musste der gebürtige Baske wieder zum FC Valencia zurück, wo er nur sehr selten zum Einsatz kam. Am 30. Januar 2010 wechselte er für das Ende der Saison 2009/10 auf Leihbasis zu Real Valladolid.
Für die Saison 2010/11 war er an UD Levante verliehen.
Nationalmannschaft
Seit dem 3. September 2004, als er gegen Schottland debütierte, spielte del Horno auf der linken Außenbahn im Dress der spanischen Nationalmannschaft. Er bestritt zehn Länderspiele und stand ursprünglich auch im Kader für die WM 2006 in Deutschland. Eine Verletzung zwang ihn jedoch zur Absage der Teilnahme. Für ihn wurde Mariano Andrés Pernía nachnominiert. Seit seiner Verletzung bestritt er kein Spiel mehr für die spanische Nationalelf.
Erfolge
Englischer Meister: 2005/06
Englischer Ligapokalsieger: 2005
Weblinks
Spielerprofil bei BDFutbol
Einzelnachweise
Fußballnationalspieler (Spanien)
Fußballspieler (Athletic Bilbao)
Fußballspieler (FC Chelsea)
Fußballspieler (FC Valencia)
Fußballspieler (Real Valladolid)
Fußballspieler (UD Levante)
Person (Baskenland)
Englischer Meister (Fußball)
Spanier
Geboren 1981
Mann | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,790 |
package org.elasticsearch.search.aggregations.metrics;
import org.apache.lucene.index.LeafReaderContext;
import org.apache.lucene.search.ScoreMode;
import org.elasticsearch.common.lease.Releasables;
import org.elasticsearch.common.util.BigArrays;
import org.elasticsearch.common.util.DoubleArray;
import org.elasticsearch.index.fielddata.SortedNumericDoubleValues;
import org.elasticsearch.search.DocValueFormat;
import org.elasticsearch.search.aggregations.Aggregator;
import org.elasticsearch.search.aggregations.InternalAggregation;
import org.elasticsearch.search.aggregations.LeafBucketCollector;
import org.elasticsearch.search.aggregations.LeafBucketCollectorBase;
import org.elasticsearch.search.aggregations.pipeline.PipelineAggregator;
import org.elasticsearch.search.aggregations.support.ValuesSource;
import org.elasticsearch.search.internal.SearchContext;
import java.io.IOException;
import java.util.List;
import java.util.Map;
class SumAggregator extends NumericMetricsAggregator.SingleValue {
private final ValuesSource.Numeric valuesSource;
private final DocValueFormat format;
private DoubleArray sums;
private DoubleArray compensations;
SumAggregator(String name, ValuesSource.Numeric valuesSource, DocValueFormat formatter, SearchContext context,
Aggregator parent, List<PipelineAggregator> pipelineAggregators, Map<String, Object> metaData) throws IOException {
super(name, context, parent, pipelineAggregators, metaData);
this.valuesSource = valuesSource;
this.format = formatter;
if (valuesSource != null) {
sums = context.bigArrays().newDoubleArray(1, true);
compensations = context.bigArrays().newDoubleArray(1, true);
}
}
@Override
public ScoreMode scoreMode() {
return valuesSource != null && valuesSource.needsScores() ? ScoreMode.COMPLETE : ScoreMode.COMPLETE_NO_SCORES;
}
@Override
public LeafBucketCollector getLeafCollector(LeafReaderContext ctx,
final LeafBucketCollector sub) throws IOException {
if (valuesSource == null) {
return LeafBucketCollector.NO_OP_COLLECTOR;
}
final BigArrays bigArrays = context.bigArrays();
final SortedNumericDoubleValues values = valuesSource.doubleValues(ctx);
return new LeafBucketCollectorBase(sub, values) {
@Override
public void collect(int doc, long bucket) throws IOException {
sums = bigArrays.grow(sums, bucket + 1);
compensations = bigArrays.grow(compensations, bucket + 1);
if (values.advanceExact(doc)) {
final int valuesCount = values.docValueCount();
// Compute the sum of double values with Kahan summation algorithm which is more
// accurate than naive summation.
double sum = sums.get(bucket);
double compensation = compensations.get(bucket);
for (int i = 0; i < valuesCount; i++) {
double value = values.nextValue();
if (Double.isFinite(value) == false) {
sum += value;
} else if (Double.isFinite(sum)) {
double corrected = value - compensation;
double newSum = sum + corrected;
compensation = (newSum - sum) - corrected;
sum = newSum;
}
}
compensations.set(bucket, compensation);
sums.set(bucket, sum);
}
}
};
}
@Override
public double metric(long owningBucketOrd) {
if (valuesSource == null || owningBucketOrd >= sums.size()) {
return 0.0;
}
return sums.get(owningBucketOrd);
}
@Override
public InternalAggregation buildAggregation(long bucket) {
if (valuesSource == null || bucket >= sums.size()) {
return buildEmptyAggregation();
}
return new InternalSum(name, sums.get(bucket), format, pipelineAggregators(), metaData());
}
@Override
public InternalAggregation buildEmptyAggregation() {
return new InternalSum(name, 0.0, format, pipelineAggregators(), metaData());
}
@Override
public void doClose() {
Releasables.close(sums, compensations);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,101 |
Derry Irish dance teacher sentenced for sexual offenses
John Gerard Roddy, who was a "well-known" Irish dancing teacher for many years, was sentenced for sexual offenses, including sexual activity with a child.
Kerry O'Shea
@kerry_oshea
John Gerard Roody will serve six years in custody for his crimes. Getty Images
John Gerard Roddy was sentenced at Antrim Crown Court in Northern Ireland on January 18 for indecent assault on a male, sexual activity with a child, and sexual assault.
Roddy, a 63-year-old from the Derry area, will serve six years in custody and three years on license, the Police Service of Northern Ireland (PSNI) said in a statement on January 18.
Roddy will be placed on the Sex Offenders Register indefinitely and will be subjected to a Sexual Offences Prevention Order (SOPO) for ten years.
Today, a 63 year old man from the Derry/Londonderry area was sentenced at Antrim Crown Court for indecent assault on a male, sexual activity with a child and sexual assault. Read more here: https://t.co/aKZkuO5ZgJ pic.twitter.com/6Q0906cpvW
— Police Derry City and Strabane (@PSNIDCSDistrict) January 18, 2023
Derry Daily reported that, according to court papers, the offenses took place between July 1, 2007, and December 31, 2012, and that three of the indecent assault charges took place outside of the UK.
A report in the January 22 print edition of the Sunday World newspaper said that Roddy, an "Irish dancing teacher," was "well-known for many years within Irish dancing circles in Northern Ireland."
The Sunday World noted that Roddy's crimes were all committed against the same victim.
A source told IrishCentral on January 23 that "it is understood that Mr. Roddy has not been registered with CLRG [An Coimisiún Le Rincí Gaelacha, the oldest and largest Irish dance organization in the world] in the last 15 years."
(Editor's Note: On January 25, CLRG told IrishCentral that these dates were incorrect and that it is reviewing the dates of Roddy's registration.)
Irish dance org CLRG announces tender to recruit change management consultancy
PSNI Constable Jason McMorris said after Roddy's sentencing: "Roddy breached his position of trust in the victim's life, coercing and manipulating him.
"He took what should have been carefree, happy, teenage years away from him.
"No person, let alone a child, should ever be exploited in this heinous way.
"The victim showed immense courage and bravery to report these crimes to us, which have had a lasting impact on him for many years.
"I hope that his strength and confidence in the Police to conduct a thorough investigation encourages other victims of child abuse to come forward and report.
"We take a robust stance against targeting sex offenders, particularly those who target and abuse children, and will work tirelessly to get take these people off our streets.
"Time is no barrier to reporting offences and I would urge anyone who has been victim of a sexual offence at any time to come forward. Do not suffer in silence. Call 101 or 999 in case of emergency."
Related: Crime, Irish Dance, Northern Ireland
Irish dance org CLRG "reviewing" sex offender teacher's past registration
GAA club to lodge formal appeal after controversial Croke Park final
Sacked teacher continues to show up to Co Westmeath school | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,167 |
package com.emel.alert;
import com.emel.alert.*;
import android.app.Activity;
import android.os.Bundle;
public class ServerResponseActivity extends Activity {
/** Called when the activity is first created. */
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
setContentView(R.layout.main);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,365 |
# DOVER BOOKS ON FASHION
1920s FASHIONS FROM B. ALTMAN & COMPANY, B. Altman & Co. (0-486-40293-2)
HATS: A HISTORY OF FASHION IN HEADWEAR, Hilda Amphlett. (0-486-42746-3)
BRITISH COSTUME FROM EARLIEST TIMES TO 1820, Mrs. Charles H. Ashdown. (0-486-41813-8)
EVERYDAY FASHIONS OF THE THIRTIES AS PICTURED IN SEARS CATALOGS, Edited by Stella Blum. (0-486-25108-X)
FASHIONS AND COSTUMES FROM GODEY'S LADY'S BOOK: INCLUDING 8 PLATES IN FULL COLOR, Stella Blum. (0-486-24841-0)
EVERYDAY FASHIONS OF THE TWENTIES AS PICTURED IN SEARS AND OTHER CATALOGS, Edited by Stella Blum. (0-486-24134-3)
VICTORIAN FASHIONS AND COSTUMES FROM HARPER'S BAZAR, 1867-1898, Stella Blum. (0-486-22990-4)
HISTORIC COSTUME IN PICTURES, Braun & Schneider. (0-486-23150-X)
A PICTORIAL HISTORY OF COSTUME FROM ANCIENT TIMES TO THE NINETEENTH CENTURY: WITH OVER 1900 ILLUSTRATED COSTUMES, INCLUDING 1000 IN FULL COLOR, Wolfgang Bruhn and Max Tilke. (0-486-43542-3)
AMERICAN DRESS PATTERN CATALOGS, 1873-1909: FOUR COMPLETE REPRINTS, Edited by Nancy Villa Bryk. (0-486-25654-5)
ENGLISH WOMEN'S CLOTHING IN THE NINETEENTH CENTURY: A COMPREHENSIVE GUIDE WITH 1,117 ILLUSTRATIONS, C. Willett Cunnington. (0-486-26323-1)
THE HISTORY OF UNDERCLOTHES, C. Willett Cunnington and Phillis Cunnington. (0-486-27124-2)
AMERICAN VICTORIAN COSTUME IN EARLY PHOTOGRAPHS, Priscilla Harris Dalrymple. (0-486-26533-1)
WOMEN'S HATS, HEADDRESSES AND HAIRSTYLES: WITH 453 ILLUSTRATIONS, MEDIEVAL TO MODERN, Georgine de Courtais. (0-486-44850-9)
WOMEN'S COSTUME OF THE ANCIENT WORLD: 700 FULL-COLOR ILLUSTRATIONS, Paul Louis de Giafferri. (0-486-44527-5)
HISTORIC COSTUMES AND HOW TO MAKE THEM, Mary Fernald and E. Shenton. (0-486-44906-8)
VICTORIAN AND EDWARDIAN FASHION: A PHOTOGRAPHIC SURVEY, Alison Gernsheim. (0-486-24205-6)
TURN-OF-THE-CENTURY FASHION PATTERNS AND TAILORING TECHNIQUES, S. S. Gordon. (O-486-41241-5)
WHAT PEOPLE WORE: 1,8OO ILLUSTRATIONS FROM ANCIENT TIMES TO THE EARLY TWENTIETH CENTURY, Douglas Gorsline. (0-486-28162-0)
AUTHENTIC VICTORIAN DRESSMAKING TECHNIQUES, Edited by Kristina Harris. (0-486-40485-4)
59 AUTHENTIC TURN-OF-THE-CENTURY FASHION PATTERNS, Kristina Harris. (0-486-28357-7)
VICTORIAN FASHION IN AMERICA: 264 VINTAGE PHOTOGRAPHS, Edited by Kristina Harris. (0-486-41814-6)
AUTHENTIC VICTORIAN FASHION PATTERNS: A COMPLETE LADY'S WARDROBE, Edited by Kristina Harris. (0-486-40721-7)
MEDIEVAL COSTUME AND HOW TO RECREATE IT, Dorothy Hartley. (0-486-42985-7)
THE "KEYSTONE" JACKET AND DRESS CUTTER: AN 1895 GUIDE TO WOMEN'S TAILORING, Chas. Hecklinger. Preface by Kristina Seleshanko. (0-486-45105-4)
COSTUMES OF THE GREEKS AND ROMANS, Thomas Hope. (0-486-20021-3)
At Dover Publications we're committed to producing books in an
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Everyday Fashions of the Twenties as Pictured in Sears and Other Catalogs is a new work, first published by Dover Publications, Inc., in 1981.
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# Table of Contents
DOVER BOOKS ON FASHION
Title Page
Copyright Page
INTRODUCTION
Part One: 1919—1924
Part Two: 1925-1931
# INTRODUCTION
The first mail-order catalog was issued by Aaron Montgomery Ward in 1872; Sears, Roebuck and Co. produced their first in 1896. During the early part of the twentieth century innumerable firms joined the mail-marketing business and the volume of sales was prodigious. By the end of World War I, buying through the mails had grown into a firmly established American institution and the mail-order catalog had become the "Farmer's Bible" and the "Nation's Wishbook."
People living in isolated hamlets, on far-flung farms or in the less-affluent sections of the cities, awaited each new catalog with excited anticipation. Its arrival meant hours of entertainment, a fund of information, some dreams to be realized and others to be kept warm by the hope of being fulfilled in the future. Even those too poor to succumb to the temptations of the fashion pages could, through the purchase of some thread, a length of yard goods or a meager article of farm equipment, be assured of receiving the next catalog and news of the latest fashions. If there was no money to buy them ready-made, they could somehow be copied and sewn at home.
In 1925 Sears announced, "We have become the world's largest store" and stated that nine million families bought from them. Based on this claim, and since all mail-order houses included wearing apparel, the fashion sections of the mail-order catalogs must have been America's most popular and best-read fashion magazines.
In the 1920s the entire family could be dressed via the United States Postal Service system. The mail-order catalogs not only showed women's clothes but also consistently included fashions for children, teenagers and men. Although small children's fashions resembled in some measure those of the adults, there were now clothes specially designed for little boys and girls. Changes in men's fashions during this period were relatively slow and subtle. Nevertheless they were there and become quite obvious if one compares, for example, the fashions of 1919 with those of 1927.
Although today there is a great deal of buying through the mails on even the highest levels, it has become a popular notion that mail-order clothes of the past were purely utilitarian, having little flair of design or quality. While it is true that many pages were devoted to cheap, practical wearing apparel, and none to ballgowns or white ties and tails, the most impressive segment of these catalogs was the one devoted to fashions — clothes to be worn as Sunday best, for going out, sports, leisure times and for everyday wear. Even housedresses and workshirts had a modicum of style and were in tune with the times. Placed at the beginning of the catalog, carefully delineated drawings and photographs, many in color, gave the book excitement, life and eye appeal.
Mail-order merchandisers did not attempt to project fashion trends. What they promised was rapid delivery. As Sears, Roebuck boasted, "We are proud of our merchandise and proud of the service we give our customers. REAL 24-hour service. 99 out of every 100 orders we receive are shipped in less than 24 hours." These firms had to have a ready supply of whatever they offered for a specified time at the listed prices. Their investment was enormous and they could not afford to gamble with the untried or untested, especially in areas as unpredictable as next season's fashions. Yet, if the styles they featured did not have the elan or the ultra-chic avant-garde appearance of the latest fashions shown in New York or Paris, they did inform their readers of what was currently espoused and accepted. Fads and unsuccessful projections are minimal in these catalogs; anyone interested in knowing how the majority of Americans dressed during the period can feel secure in the knowledge that what was illustrated was pretty much what was generally worn.
Few periods demonstrate with such clarity the way fashions reflect their own times as do the 1960s and the 1920s; both periods have much in common and present many parallels. Each focused on social realignments and youth; each involved feminine liberation. In both cycles, wars and technological developments produced rapid changes that led to a quest for excitement, to restlessness and even to violence and destruction.
How was life in the 1920s reflected in the era's fashions? Improved production methods enabled a rapidly growing middle class, even those at its lower level, to participate in the world of fashion that previously had been the sole realm of a privileged few. Another important factor in the democratization of fashions came as a result of the interest among the members of the leisure group in more casual daytime wear. The war years had brought on harsh realities and evoked a desire to do one's bit that touched all levels of society. Even those who formerly had lived in glossy cocoons, where a minimum of physical exertion was a symbol of high station and wealth, sought to become involved in the war effort. After the war many of these people found their prior sedentary life boring and had little desire to return to it. The taste of activity as a release of energy appealed to them. To fit into the pattern of this new version of the good life, fashions became more informal and less complicated. Clothing manufacturers could now easily produce cheap versions that were within the price range of their fashion-hungry customers who had to work for a living.
All fashions reflected the new spirit as youthful, more carefree, ideals gradually replaced the earlier, more staid, models. But it was in fashions for women that the changes were most obvious. Feminine liberation found freedom in discarding the corset. For the first time in centuries women's legs were exposed and freed for mobility and action. To gain equality with men and to resemble them, women flattened their breasts and hips and cut their hair. The 1920s bob and boyish ideal were the period's own version of unisex. Aesthetically, the fashions mirrored the abstract elements of art movements of the period. Geometric in form, they relied on the motion of living women to breathe a shape and a sensuality into an otherwise sterile silhouette. All this was totally sympathetic with an era of pulsating dynamism bent on breaking down remaining restrictions based on the social, economic, political and moral concepts of the past century.
These were the forces that helped to create the fashions of the twenties. Yet, for all the violence, excesses, the licentiousness we have come to associate with this flamboyant period, as one looks through the pages of mail-order catalogs one finds that, although the changes are there, the progression is smooth and orderly. Long hair gave way to bobbed hair. Skirts gradually rose to the knees. Underwear diminished to accommodate the new mood and look. More and more space was devoted to cosmetics, and here and there pants for women were featured. Nowhere are there examples of the revealing, extravagantly low-cut gowns tantalizingly covered with fringes or sparkling beading. The few party dresses shown are very modest and demure, pictures of naive innocence (page 90). The "flapper" dresses, as they appear on page 92, are merely a timid, decorous reflection of the sophisticated sexuality of the so-called "Roaring Twenties," only a muted echo of the ragtime rhythms of the Jazz Age.
Men's coats and suits on the whole remained fairly conservative, only occasionally showing the collegiate "rah-rah" or the "razzmatazz" flashiness we have come to associate with the decade. The new concept found expression mainly in casual and sports clothes, accessories and in a wider range of designs available in work clothes.
Apparently revolution was the choice and privilege of a minority. The majority chose the safer path and course of evolution.
In the 1920s motion pictures exerted an ever-increasing impact on the American scene. Movie stars brought the viewers adventure, a shimmering aura of wealth, beauty and romance. Films gave a semblance of reality to fantasies and aroused the public to new hopes, tastes and appetites. Sensing this, Sears, Roebuck and Co. began to include fashions endorsed by such stars as Clara Bow (page 105), Gloria Swanson (page 56) and Joan Crawford (page 127). For men there were Western-style hats and boots (page 14); little boys could playact in cowboy and Indian costumes (page 118). Although the movies during this decade kept their audiences informed of the latest fashions, the female stars' most significant influence was on the face and figure, coiffure, posture and grooming. As a result beauty parlors and reducing regimens abounded, and the field of cosmetics became a major industry.
Women may have looked to Hollywood for goddesses to emulate, but the direction of fashion was set in Paris. As glamorous as the clothes appeared in the movies, they were in the main versions of what the French couture had proposed. This seems to have been no mystery to the staffs responsible for the fashions to be featured in the catalogs. References to movie stars were primarily to such details as hats and shoes. The bulk of the fashion merchandise, coats, suits and dresses claim to have their origins in New York or Paris. This is probably quite true because a well-trained eye can readily spot elements of the inventive creations of various French designers. The greatest single influence throughout the 1920s, however, was that of Coco Chanel. Her use of supple jerseys, simple ensembles consisting of jacket, blouse and skirt, costume accessories such as scarves and inexpensive jewelry, had a natural appeal for the active, practical American woman. The workmanship or quality certainly could not compare with that of the French couture. The degree of downgrading, however, is difficult to ascertain. The clientele of mail-order purchasing was hardly the sort to stockpile a wardrobe and leave it for posterity; they were more apt to wear out their clothes.
Sunday-best clothes would be updated through alterations or be downgraded for everyday wear. And when they were replaced, they would be passed on to younger or less-fortunate members of the family or community, if they were still useable. The Puritan ethic was still in force; to do otherwise would have been considered sinful. In any case, not enough examples have survived to make a valid comparison.
The common practice of placing time and life into tidy compartments by decades is a handy device used to pull together loosely the peak essences of an era. Actually, the fashions of the twenties can be split into two phases. As usual, changes occur somewhat earlier in the world of the haute couture (in this case 1919—1923 and 1924—1929). Those shown in mail-order catalogs break into the following two time spans: 1919—1924 and 1925—1931. Although neither period remained static, there is something homogenous about each segment and it is interesting to view them in that context.
# Part One: 1919—1924
In 1919, people in Europe and America, exhausted and depleted by World War I, longed to return to what they considered normalcy, to the way of life they had known before the war. Fashions reverted to those of 1913—1914, as though they had only been dropped for the duration. A new view of how women should dress had begun around 1909 and the course toward freedom, youth and equality was established even before 1914. By 1920, after a few steps backwards, the movement was accelerated by the experience and changes brought on by the war. During the next several years, the fashion ideal became younger and younger and proceeded to divest itself of many of the physical and mental trappings of the nineteenth century.
Growing urbanization, increased affluence, shorter working hours and paid vacations allowed for more leisure time and extra energy. As a result, interest in sports escalated, necessitating a whole range of special clothes designed for active and spectactor sports. Gradually this freer concept of dressing crept into daywear. Clothes became simpler and lighter in weight. Feminine curves, long a symbol of a woman's frailty, were negated by the fashion for the new streamlined vertical lines. These six years were essentially a transitional period in women's fashions. The new style was to emerge fully in 1925.
By 1919 pregnancy was no longer veiled in gowns for déshabillé or at-home robes. Maternity dresses designed in the styles of the period, along with maternity corsets, were illustrated graphically with explicit text explaining their function and virtues (page 7).
When one compares the fashions shown by Sears, Roebuck and Co. during this period with those in a French magazine such as L'Art et la Mode or with the American Vogue or Harper's Bazaar, it is interesting to note that there is only about a one-year lag in the overall aspect of the mail-order fashions. Yet, although the styles were not exactly the dernier cri or as handsomely presented as those in the high-fashion magazines, many fashions shown by Sears in 1919—24 reveal a surprising amount of chic and elegance. Not all of the clothes were inexpensive. Some coats and suits sold for almost $50, while some "better" dresses were priced over $30. Considering the purchasing power of a dollar in those days, it is apparent that those who could afford these prices were not confined to large cities and that mail-order catalogs catered not only to the rustic needs of farmers or the meager purses of the poorer classes. During these six years, the range of cost and taste was rather wide; the fashions presented must have been aimed at a broad spectrum of Americans.
# 1919
(pages 5-14)
The hobble skirt of the prewar period took on the "peg-top" look (pages 5 and 6) and the 1913 "barrel form" was shown along with pyramid shapes popular in 1915—16. The waistlines were either high or undefined. The bust retained the earlier low monobosom look. By our standards, the figure was quite full. The use of decorations, such as a proliferation of buttons, tassels and braid, was also a holdover from past fashions.
Although the current silhouette actually required little constriction, women, except for the most liberated, continued to wear corsets. There were even corsets for "children up to 12 years" (page 8). White cotton, trimmed with eyelet and lace, was popular for lingerie. Very pointed high shoes, laced or buttoned, with solid or spat tops and Louis or military heels were preferred. Stockings, which showed only when pumps were occasionally worn, were generally black or dark gray, although white was sometimes worn with white shoes. Hats, which had large crowns to accommodate long hair, were worn low, just above the eyebrows.
Male fashions reminiscent of the Edwardian styles are shown on models with large, square-jawed heads, and hair neatly plastered down. Their clothes had narrow shoulders and were slightly high-waisted, like the women's fashions. For sports there were Norfolk jackets and knickerbockers for golf or hiking and suits for riding.
The cosmetics available were limited to rouge, face powder and discreet lip rouge. One could buy a pencil to darken eyebrows, lashes and, for the men, beards and mustaches. For the nails there were cuticle removers, nail whites and polishing pastes.
# 1920
(pages 15-34)
Skirts became a little shorter, figures somewhat slimmer, bosoms smaller and the waistline was more naturally placed. Suits appeared sleeker and more tailored. Middy and over-blouses, now an important item, figured prominently in modified forms into the 1930s. Lingerie—petticoats, chemises, bloomers — were shown in a profusion of colored silks in purple, flesh, blue, green, plum and black. Bandeaux or brassieres began to displace the camisole.
Automobile dusters were included in the menswear section. Although jackets for youths and boys were similar to those for men, suits for boys 9 to 17 were shown with knickerbockers.
# 1921
(pages 35-48)
There was a further simplicity this year. Dresses on page 38 were designed to fall in an unbroken line from shoulder to hem. Worn loose, slightly belted at the normal waist, this was to be the silhouette of most of the decade. Although dresses remained below calf length, coats became shorter. Heavy trimming began to disappear. Some hair was obviously cut but was kept soft-looking with side curls (page 38). High shoes and spats were still worn but there was in increase in the popularity of pumps and oxfords. Stockings remained dark. The Japanese-style kimono as well as sleeping suits (pajamas for women) made their appearance. Men's outerwear included chesterfields, town ulsters and reversible rubber interlined raincoats. Shirts with detachable collars were popular. The separate collars could be either stiff or soft, and some, called "rubber collars," were made of celluloid.
# 1922
(pages 49-66)
Skirts reached mid-calf length. Coats continued to hold to the earlier style with full or dolman sleeves and were trimmed with some braid, tassels, embroidery and buttons. Suits, however, generally had a more male look. They were worn either beltless or with belts placed a little below the waist. Dresses showed the effects of the styles by the French designer, Paul Poiret — especially his use of peasant-type embroidery (page 50). Touches of Jeanne Lanvin can also be detected in the dresses worn by the two center figures on page 50. The echoes of Chanel's designs are too numerous to mention since much of the knitwear and classically simple clothes of the 1920s must be attributed to her influence.
Moving toward the new slimness, foundations began to accent hip and bust flattening. As hemlines rose, footwear became decorative: T-strap slippers and fashions for gaiters, galoshes and "arctics." Stockings, though still on the dark side, developed clocks and fancy heels. Rayon stockings made the silken look for legs available at a low price (78¢ as opposed to $2.69 for a pair of silk stockings with clocks.)
Sweaters for men were featured in a wide range of colors, patterns and details, such as shawl collars and turtlenecks. Sports clothes received added attention. For bathing, men were offered one piece knit suits with attached skirts while women could choose from several dressmaker-type costumes that were worn over an undergarment. There were also suits for football, hockey, skiing, golf and shooting. Underwear for men took on an athletic tone in the form of boxer shorts.
# 1923
(pages 67-76)
The waistline now has slipped down to the top of the hips. But, as though there was still some doubt or uneasiness about the future, this year's fashions harked back not so much to those of 1913 but all the way to those of 1909. A matronly silhouette — with wide sleeves, tassel and braid trimming, lower hemlines nearly ankle length — seems to have come back. In dresses, Lanvin's robe de style, with its low-waisted bodice and long full skirt, was shown in many adopted versions (page 69, right figure). Accessories now included mesh purses and silver-plated compacts.
# 1924
(pages 77-84)
Fashions this year were a blend of the old and the new (page 78). The waistline descended to the hips. There was a hint of the surface decoration and geometric insertions that would serve to break up the stark simplicity of the coming rectangular silhouette (page 77). Beltless jackets were shorter and worn with slim untrimmed skirts. Page 83 shows sports pants outfits; page 82 advertises "Bob" hats for women with bobbed hair.
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# Part Two: 1925-1931
Beginning in 1925, the standards and range of women's fashions offered in mail-order catalogs started to decline. The available selection diminished. The most expensive coats and dresses offered were nearly half the price of those offered in 1919. The same was true of men's dress clothes. One of the reasons for this, although by no means the only one, was the lure of the automobile.
In the mid-1920s, through technological advances and because of an unprecedented growth of prosperity, the automobile came within the reach of the average middle-class American. Quite naturally it was a class symbol to own a car. What was more important was the freedom of movement it provided. Within a short time America was in the grip of a full-blown love affair with the car. No sacrifice seemed too great for this new infatuation. Installment buying had become an accepted practice and now millions of Americans were buying automobiles on time. The impact of this development was enormous and touched every facet of life in America, including fashions and the way they were marketed.
Since many of their rural customers could now drive into town to shop, mail-order houses found themselves in competition with city stores. The larger organizations tried to meet this challenge by opening up their own retail stores. The catalogs of the latter part of the 1920s reveal, however, that in the area of wearing apparel, this move met with limited success. Articles such as denim coveralls, long woolen underwear, corsets for older women who from habit found them indispensable, remained fairly constant throughout the decade. But for the fashion-minded, there was less variety, generally duller-looking offerings with a strong accent on economy. Profitable sales in mail orders now lay primarily in their appeal to the isolated, the thrifty or the poor. Those with money, the more discriminating customers, preferred buying in department stores or in specialty shops which had mushroomed all over the country. Not only did they find a richer selection there, but they could also try on and examine the clothes and, having paid for them or charged them, walk out of the store with their purchases. For a great many Americans this was an attractive new experience.
As the price level dropped, mail-order fashions began to fall behind those of Paris and by 1930 the lag increased to about two years. Late and somewhat diluted, the style of the period nevertheless touched even the cheapest wearing apparel. The art movements in Paris and the Exposition Internationale des Arts Décoratifs of 1925 managed eventually to make their influence felt on the farms of Iowa, Nebraska and Kansas, and in the ghettos of the large cities.
# 1925-26
# (pages 87-98)
In the fashions of the second half of the 1920s, in the silhouette, the hair styles, hats, shoes, gloves and the jewelry, as in the paintings of Picasso, Braque, Léger and Matisse, the accent was on the hard edge of geometric forms and the clean beauty of pure line. In clothing design, to relieve the monotony of the spacial forms of rectangles, squares and circles and simple linear outlines, inner planes were broken up with abstract rhythmic patterns of appliques, piecings, tucks and formalized embroidered patterns.
The focus now was on "the slender mode of youth." The boyish look, totally flat, rectangular, mid-calf in length had arrived. Advertisements for foundations claiming "new freedom in corsetry" actually implied freedom primarily for the waistline. Women endowed with what were formerly considered feminine charms — a full bosom and wide hips — could now correct these "faults" with bust and hip constrictors.
As clothes became more casual, there was less restriction on what was to be worn at different times of the day or for special occasions. Many pieces of apparel once considered purely i men's wear and some that were looked on as work clothes were absorbed into women's fashions. Sears, Roebuck copy said, "Being exceedingly smart and practical from every standpoint, the 'Collegiate' slicker or 'Fisherman's Oilskin' has become one of the most popular models in raincoats. It comes in the attractive natural yellow shade and is absolutely waterproof."
Primarily through the influence of the movies, cosmetics were now offering a wide range of powders, rouges, lipsticks, black and brown mascaras and eyelash curlers. Although liquid nail polish "for highly tinted brilliance" was for sale, the average American woman did not sport scarlet nails until the next decade.
# 1927-28
# (pages 99-120)
There were no dramatic changes this year. Coats remained the same as before. The belts on dresses tightened at the hips to produce a blousing above. Skirt portions, diminished in size, were designed for the swinging motion of easy movement through pleats, gathers, shirrings and insets.
The quest to look like an underdeveloped youth continued — so much so that, except for the fact that they were scaled for smaller figures, fashions for schoolgirls were much the same as those designed for their mothers.
Rayon had become an accepted substitute for silk in hosiery, women's dresses and underwear. Now artificial effects such as alligator patterned rubber and synthetic materials such as leatherette were also used. "New automatic fasteners" (zippers) appeared on overshoes. Footwear in general became more imaginative. Even some sneakers were decorated. Women who wished to elongate their legs and to look taller and more slender could buy shoes with spike heels. Gloves and other accessories grew in variety and embellishment, and were an important part of a total ensemble.
In men's wear, suits, although reflecting the new slimness and straighter lines, retained traditional styles. Informal and work clothes, however, showed a new burst of creativity and design.
# 1928-29
# (pages 121-134)
In Paris, hemlines had begun to dip downwards and waistlines started to climb up to the natural level. This step toward a new cycle in fashion was not yet in evidence in mail-order catalogs where the cresting of the hem at the top of the knee that was seen in the haute couture of 1927 finally occurred. Except for further detailed treatment of flat planes and the addition of asymmetry, the styles differed little from the year before.
This year, however, more adventuresome young men could buy "Broadway's favorite" or "Collegiate" style suits. Made of boldly striped wool, some had contrasting waistcoats. Others had double-breasted vests that were either collared or collarless. "Black bottom" cuffs on trousers could also be ordered.
# 1930-31
# (pages 135-152)
"The trend is toward femininity." The adult female figure returned to fashion. Hemlines dropped below the knee and the waistline became defined at its normal position. In France, Madeleine Vionnet, by using material cut on the bias, was creating beautiful figure-molding gowns. However, because bias construction, or using fabric on the cross, was expensive and called for great skill in handling, its interpretation was limited and relegated mainly to skirts and minor details.
For this new silhouette, foundation garments and underclothes were shaped to conform to the body. Reflecting the stress on natural form, men's suits also began to curve in at the waist.
The inclusion of slacks or "gob outfits," as they were called, anticipated the oncoming popularity of long pants for women. Shorts were listed, but only for little girls. Pumps became fashionable again and were available with different heels, including four-inch spike heels. The men's section added tuxedos to the selection of suits. Trimmings and surface decorations in most clothes began to fall away as the lure of a totally different look came on the horizon.
With the end of the 1920s came the end of the reign of the preadolescent ideal. The Depression and changing times were forging new fashions.
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HISTORIC ENGLISH COSTUMES AND How TO MAKE THEM, Talbot Hughes. Introduction by Kristina Seleshanko. (0-486-46985-9)
MEDIEVAL AND RENAISSANCE FASHION: 90 FULL-COLOR PLATES, Raphaël Jacquemin. (0-486-45776-1)
PICTORIAL ENCYCLOPEDIA OF HISTORIC COSTUME: 1200 FULL-COLOR FIGURES, Albert Kretschmer and Karl Rohrbach. (0-486-46142-4)
COSTUME DESIGN IN THE MOVIES: AN ILLUSTRATED GUIDE TO THE WORK OF 157 GREAT DESIGNERS, Elizabeth Leese. (0-486-26548-X)
ACCESSORIES OF DRESS: AN ILLUSTRATED ENCYCLOPEDIA, Katherine Lester and Bess Viola Oerke. (O-486-43378-1)
THE CORSET AND THE CRINOLINE: AN ILLUSTRATED HISTORY, W. B. Lord. (O-486-46186-6)
JAPANESE KIMONO DESIGNS, Shôjirô Nomura and Tsutomu Ema. (0-486-44426-0)
TUDOR COSTUME AND FASHION, Herbert Norris. (0-486-29845-0)
MEDIEVAL COSTUME AND FASHION, Herbert Norris. (0-486-40486-2)
EVERYDAY FASHIONS, 1909-1920, As PICTURED IN SEARS CATALOGS, Edited by JoAnne Olian. (0-486-28628-2)
CHILDREN'S FASHIONS 1900-1950 As PICTURED IN SEARS CATALOGS, Edited by JoAnne Olian. (0-486-42325-5)
VICTORIAN AND EDWARDIAN FASHIONS FROM "LA MODE ILLUSTRÉE", JoAnne Olian. (0-486-29711-X)
EVERYDAY FASHIONS OF THE SIXTIES As PICTURED IN SEARS CATALOGS, Edited by JoAnne Olian. (0-486-40120-O)
EVERYDAY FASHIONS OF THE FORTIES As PICTURED IN SEARS CATALOGS, Edited by JoAnne Olian. (0-486-26918-3)
80 GODEY'S FULL-COLOR FASHION PLATES: 1838-1880, JoAnne Olian. (0-486-40222-3)
EVERYDAY FASHIONS OF THE FIFTIES As PICTURED IN SEARS CATALOGS, Edited by Joanne Olian. (O-486-42219-4)
FULL-COLOR SOURCEBOOK OF FRENCH FASHION : 15TH TO 19TH CENTURIES, Pauquet Frères. (O-486-42838-9)
A DICTIONARY OF COSTUME AND FASHION: HISTORIC AND MODERN, Mary Brooks Picken. (0-486-40294-0)
AN ILLUSTRATED DICTIONARY OF HISTORIC COSTUME, James Robinson Planché. (0-486-42323-9)
60 CIVIL WAR-ERA FASHION PATTERNS, Kristina Seleshanko. (0-486-46176-9)
THE MODE IN HATS AND HEADDRESS: A HISTORICAL SURVEY WITH 198 PLATES, R. Turner Wilcox. (O-486-46762-7)
THE MODE IN FOOTWEAR: A HISTORICAL SURVEY WITH 53 PLATES, R. Turner Wilcox. (O-486-46761-9)
THE MODE IN COSTUME: A HISTORICAL SURVEY WITH 202 PLATES, R. Turner Wilcox. (O-486-46820-8)
EVERYDAY DRESS OF RURAL AMERICA, 1783-1800: WITH INSTRUCTIONS AND PATTERNS, Merideth Wright. (O-486-27320-2)
See every Dover book in print at www.doverpublications.com
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\section{Introduction}\label{1}
It is a matter of great importance to study wireless communication channels since they include numerous challenges caused by noise and fading. The wireless environment also allows any uninvited signal to enter the channel. These signals can act as interference or in the worse case as a malicious jammer whose aim is to disrupt the communication between the legitimate users. In this work, we explore how these various signals interact with each other to restrict the overall capacity of the channel.
Goldsmith and Varaiya in \cite{Varaiya} studied the point-to-point Gaussian channel with fast fading in which the fading coefficients are drawn i.i.d. from a distribution known to all parties. They determined the capacity of the channel where the fading coefficients are available at the receiver and possibly the transmitter. When the fading coefficients are not available at the transmitter, the capacity is equal to the expected value of the capacity of the corresponding Gaussian channel with the received signal-to-noise ratio affected by the fading gains. If both the transmitter and receiver know the exact channel gains, the transmitter can maximize the capacity by constructing its signal power as a function of the channel gains.
Another line of work studies channels in the presence of a malicious adversary. The adversary can be an active attacker who sends its signal to the channel in order to cease or restrict the communication between the legitimate users. If the adversary's signal is arbitrarily chosen or is drawn from an unknown distribution, then the channel is called as an arbitrary-varying channel (AVC). We focus on the variant of the problem wherein the adversary's knowledge is limited to the code of the legitimate user, but it does not have access to the user's transmitted messages, neither exact nor noisy. Csisz\'ar and Narayan established the capacity of discrete AVC with input and jammer power constraints in \cite{AVCNarayan,AVCNarayan2}. They also derived the capacity for a continuous version of the AVC with input and state (jammer) power constraints in \cite{Csiszar}.
Csisz\'ar and Narayan in \cite{Csiszar} characterized the capacity of the Gaussian arbitrarily-varying channel under the average probability of error and the average power constraints for input and state. It is assumed that the adversary does not have any information about the legitimate signal except the code. It is shown that if the adversary has enough power to forge a message and send it to the channel, then the receiver gets confused and cannot distinguish between the true message and the malicious one. This occurrence is called \emph{symmetrizability} , causing the capacity to drop to zero. In \cite{Csiszar}, it is shown that the adversary can symmetrize the channel if and only if it has greater power that the legitimate transmitter. However, if the jammer does not have enough power, then the capacity is equal to the capacity of a standard Gaussian channel with the noise variance increased by the power of the jammer.
The problem of AVC capacity has been also studied for discrete and continuous multi-user channels. The authors in \cite{144713} considered the discrete arbitrarily-varying multiple-access memoryless channel. We characterized lower and upper bounds for the capacity of arbitrarily-varying Gaussian interference channel in \cite{FatemehAllerton2016}. List-decoding is also investigated for the discrete and Gaussian AVCs in \cite{Hughes,6157083} and in \cite{FatemehISIT2018}, respectively. The list capacity is derived by using list-decoding which decodes a list of messages instead of unique message at the receiver.
The three elements of Gaussian noise, fading, and adversary have been previously combined to study problems with a (passive) eavesdropping adversary, rather than an (active) transmitting adversary. A secrecy capacity problem with slow fading, in which the fading gains are constant over each block length, is considered in \cite{EavsFading}. The authors determined the secrecy capacity of the channel where both of transmitter and receiver know the channel state information (CSI) of the main path, but do not have any information about the eavesdropper channel. Furthermore, the capacity is generalized to multiple eavesdroppers in \cite{MultiEavsFading}. The problem is also studied in \cite{RayleighFading} for a specific case of a fast Rayleigh fading eavesdropper channel and a standard Gaussian main channel where the CSI are only known to the eavesdropper.
In this paper, we consider a Gaussian AVC with fast fading on the main path, as illustrated in Fig. \ref{fig:GAVFC}; we refer to this channel as the Gaussian arbitrarily-varying fading channel (GAVFC). We characterize the capacity of the GAVFC under the average probability of error criterion. Similar to the Gaussian fading channel, we also assume that everyone knows the fading gain distribution including the adversary, but they may or may not know the realization of the gain sequence. Note that the ``arbitrarily-varying'' aspect of the channel is the adversary's signal, not the channel gains, which we assume to be random from a known distribution. The receiver always needs the exact fading gains to decode the message, while the adversary and the transmitter may or may not know the exact values of fading gains. Therefore, we derive the capacity of the GAVFC for four cases wherein the channel gains are available at the transmitter and/or adversary as follows:
\begin{itemize}
\item Neither the transmitter nor the adversary knows the channel gains.
\item Only the transmitter knows the channel gains.
\item Only the adversary knows the channel gains.
\item Both the transmitter and the adversary knows the channel gains.
\end{itemize}
If the jammer does not know the channel gains, we show that the capacity is equal to the capacity of the corresponding fading channel with increased noise variance by the power of the jammer. If the jammer knows the fading gains, then it can choose its signal as a function of the gains, and under some power constraints it can symmetrize the channel and make the capacity zero.
Note that if the channel gains are not available at the adversary, it does not have the required channel information to symmetrize the channel. Moreover, all the results still hold if the adversary and the encoder have the channel gains causally or non-causally, except one situation. If the adversary knows the channel gains causally while the encoder knows them non-causally, then the adversary cannot symmetrize the channel since the encoder possesses some extra information that the adversary does not.
The rest of the paper is organized as follows. We describe the GAVFC model and define the various capacities in Sec. \ref{2}. We state our main theorem including the capacity of the GAVFC in all cases in Sec. \ref{3}. Before giving the proof of our main theorem, we need some auxiliary lemmas and tools which are presented in Sec. \ref{4}. Later, in sections \ref{secV}, \ref{6}, \ref{7}, and \ref{8}, we provide the converse and achievability proofs of each of the main results in Theorem \ref{Thrm}. Finally in the Appendix, we provide a brief proof for the auxiliary results.
\emph{Notation:} We use bold letters to indicate $n$-length vectors. We employ $\langle \cdot,\cdot \rangle $ and $\circ$ to denote inner product and Hadamard product (element-wise multiplication), respectively. We indicate the positive-part function, 2-norm and the expectation by $|\cdot|^+$, $\|\cdot\|$ and $\mathbb{E}[\cdot]$, respectively. Also, for an integer $N$, $[N]$ stands for the set $\{1,2,3,\ldots,N\}$. Notation $\mathbf{I}_n$ represents the identity matrix of size $n$. Each of $\log(\cdot)$ and $\exp(\cdot)$ functions has base 2. Moreover, $C(x)=\frac{1}{2}\log(1+x)$, and $X\sim\mathcal{N} (\mu, \sigma^2)$ denotes Gaussian random variable $X$ with mean $\mu$ and variance $\sigma^2$
\vspace{1ex}
\section{Problem Statement} \label{2}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\linewidth]{GAVFC_Gains.pdf}
\caption{Gaussian Arbitrarily-Varying Fading Channel.}
\label{fig:GAVFC}
\end{figure}
The Gaussian arbitrarily-varying fading channel (GAVFC) in Fig. \ref{fig:GAVFC} is a point-to-point fading channel with additive Gaussian noise and an intelligent adversary who does not have any information about the transmitted signal except the code. The received signal is given by
\begin{align}\label{eq:1}
\begin{split}
\mathbf{Y} = \mathbf{G} \circ \mathbf{x} + \mathbf{s} + \mathbf{V}
\end{split}
\end{align}
where $\mathbf{G}$ is a random sequence of identical and independently distributed (i.i.d.) fast fading channel gains from the legitimate transmitter to the receiver drawn from continuous distribution $f_G(g)$ assumed to have positive and finite variance, $\mathbf{x}$ is the $n$-length deterministic vector representing the user's signal, $\mathbf{s}$ is the adversary signal chosen arbitrarily, and $\mathbf{V}$ is a random $n$-length noise vector distributed as a sequence of i.i.d. zero mean Gaussian random variables with variance $\sigma^2$, independent of $\mathbf{x}$, $\mathbf{G}$ and $\mathbf{s}$. Note that the receiver always knows the exact fading coefficients $\mathbf{g}$ while the transmitter and the adversary may not know the gains, know them causally, or know them non-causally.
Define an $\left(N,n\right)$ code for the GAVFC by a message set, an encoding function and a decoding function as follows:
\begin{itemize}
\item Message set $\mathcal{M}=[N]$,
\item Encoding function (one of the following)
\begin{itemize}
\item (No knowledge) $\mathbf{x}(m):\mathcal{M}\to \mathbb{R}^n$ where $\mathbf{x} = (x_1,\ldots,x_n)$,
\item (Causal) $x_i(m,\mathbf{g}^i)\!\!:\!\mathcal{M}\times \mathbb{R}^i \to \mathbb{R}$ where $\mathbf{g}^i\!=\!(g_1,\ldots,g_i)$ and $\mathbf{x} \!=\! (x_1,\ldots,x_n)$ for $i\in[n]$,
\item (Non-causal) $x_i(m,\mathbf{g})\!\!:\!\mathcal{M}\times \mathbb{R}^n\!\to\! \mathbb{R}$ where $\mathbf{g}\!=\!(g_1,\ldots,g_n)$ and $\mathbf{x} \!=\! (x_1,\ldots,x_n)$~for~$i\!~\in~[n]$,
\end{itemize}
\item Decoding function $\Theta(\mathbf{y},\mathbf{g}):\mathbb{R}^n\times \mathbb{R}^n\to\mathcal{M}$,
\end{itemize}
where the rate of the code is $R = \frac{1}{n}\log (N)$. The message $m$ is drawn uniformly from the set $\mathcal{M}$. If the encoder does not know the channel gains, it maps the message to $\mathbf{x}(m)\in \mathbb{R}^n$. If the encoder knows the channel gains causally, then it maps the message to $x_i(m,\mathbf{g}^i)\in \mathbb{R}$, and if the encoder knows the channel gains non-causally, then it maps the message to $x_i(m,\mathbf{g})\in \mathbb{R}$ where $\mathbf{x} = (x_1,\ldots, x_n)$. Given channel gains $\mathbf{g}$ at the receiver, the signal $\mathbf{y}$ is decoded by function $\Theta(\mathbf{y},\mathbf{g})$ to the message $\hat{m}$. Moreover, we assume that if the channel gains are available at the transmitter then the transmitter's signal satisfies the expected power constraints $\mathbb{E} \left[\|\mathbf{X}(m,\mathbf{G})\|^2\right]\leq nP$ for any message $m\in \mathcal{M}$. Otherwise, the power constraint is $\|\mathbf{x}(m)\|^2\leq nP$. The same definition applies to the adversary's signal power constraint, i.e. if the adversary knows the channel gains, the constraint is $\mathbb{E} \left[\|\mathbf{S}(\mathbf{G})\|^2\right]\leq n\Lambda$; otherwise, it is $\|\mathbf{s}\|^2\leq n\Lambda$. The three parameters $P$, $\Lambda$, and $\sigma^2$ as well as the distribution of fading gains $f_G(g)$ are known to all parties.
The probability of error $e(\mathbf{s},m)$ for the message $m \in \mathcal{M}$ in the presence of adversary signal $\mathbf{s}\in \mathbb{R}^n$ is now given by the probability that $\hat{m}\neq m$. Thus, the average probability of error for a specific $\mathbf{s}\in \mathbb{R}^n$ is
\begin{align}
\bar{e}(\mathbf{s}) = \frac{1}{N}\sum_{m=1}^N e(\mathbf{s},m).
\end{align}
If the adversary knows the channel gains non-causally then his signal is denoted by $s_i(\mathbf{g})$ for $i\in [n]$. Alternatively, if the adversary knows the gains causally, then the adversary's action is given by functions $s_i(\mathbf{g}^i)$ for $i\in [n]$ where $\mathbf{s} = (s_1,\cdots,s_n)$ and $\mathbf{g}^i = (g_1,\cdots,g_i)$. Therefore, the average probability of error for this specific $\mathbf{s}(G)\in \mathbb{R}^n$ is
\begin{align}
\bar{e} (\mathbf{s}(\cdot)) = \frac{1}{N}\sum_{m=1}^N \mathbb{E} e(\mathbf{s} (\mathbf{G}),m).
\end{align}
Finally, the overall probability of error $P_e^{(n)}$ is maximized over all possible choices of jammers' sequences $\mathbf{s}$ which satisfy either $\mathbb{E} \left[\|\mathbf{S}\|^2\right]\leq n\Lambda$ or $\|\mathbf{s}\|^2\leq n\Lambda$. Rate $R$ is \emph{achievable} if there exists a sequence of $\left(2^{nR},n\right)$ codes where $\underset{n \rightarrow \infty}{\lim}P_e^{(n)} = 0$. The capacity is the supremum of all achievable rates. We denote the capacity of the GAVFC as $C_{\alpha,\beta}$ where $\alpha$ denotes the transmitter's knowledge, and $\beta$ denotes the adversary's knowledge; $\alpha$ and $\beta$ can be U, C, or N depending on whether the transmitter or adversary does not know the gains (U $=$ unknown), knows the gains causally (C), or knows the gains non-causally (N). For example, $C_{\text{U,N}}$ is the capacity where the transmitter does not know the gains and the adversary knows the gains non-causally.
\section{Main Results}\label{3}
We present our results for the capacity of GAVFC whether the fading channel gains $\mathbf{G}$ are available causally or non-causally at the encoder and/or the adversary (the decoder always knows the gains) in the following theorems.
\begin{theorem}\label{Thrm} The capacities of the GAVFC are given by
\begin{align}
&C_{\text{U,U}} = \mathbb{E}_G \left[C\left(\frac{G^2P}{\Lambda+\sigma^2}\right)\right],\label{C1Capacity}\\
&C_{\text{N,U}} = C_{\text{C,U}} =
\underset{\substack{\varphi(g):\mathbb{E} \varphi(G)\leq P}}{\max}\mathbb{E}_G \left[C\left(\frac{G^2\varphi(G)}{\Lambda+\sigma^2}\right)\right],\label{C2Capacity}\\
&C_{\text{U,N}} = C_{\text{U,C}} =\begin{cases}
\underset{\psi(g):\mathbb{E} \psi(G)\leq \Lambda}{\min}\mathbb{E}_G \left[C\left(\frac{G^2P}{\psi(G)+\sigma^2}\right)\right],\! & \mathbb{E} G^2P\!>\!\Lambda\\
0,\! & \mathbb{E} G^2P\!\le\!\Lambda
\end{cases}\label{C3Capacity}\\
&C_{\text{N,N}} =\! C_{\text{C,C}}\! =\! C_{\text{C,N}} =\nonumber\\
&\begin{cases}
\underset{\substack{\varphi(g):\mathbb{E} \varphi(G)\leq P,\\ \mathbb{E} G^2\varphi(G)\geq\Lambda}}{\max} \ \ \underset{\psi(g):\mathbb{E} \psi(G)\leq \Lambda}{\min}\mathbb{E}_G \left[C\!\left(\frac{G^2\varphi(G)}{\psi(G)+\sigma^2}\right)\!\right], & \text{if}\underset{\varphi(g):\mathbb{E} \varphi(G)\leq P}{\max}\mathbb{E} G^2\varphi(G)>\Lambda \\
0, & \text{if} \underset{\varphi(g):\mathbb{E} \varphi(G)\leq P}{\max}\mathbb{E} G^2\varphi(G)\le \Lambda
\end{cases}\label{C4Capacity}\\
&C_{\text{N,C}} =
\underset{\substack{\varphi(g):\mathbb{E} \varphi(G)\leq P}}{\max} \ \ \underset{\psi(g):\mathbb{E} \psi(G)\leq \Lambda}{\min}\mathbb{E}_G \left[C\!\left(\frac{G^2\varphi(G)}{\psi(G)+\sigma^2}\right)\!\right].\label{C42Capacity}
\end{align}
\end{theorem}
Note that when the encoder knows the gains \eqref{C2Capacity}, \eqref{C4Capacity}, and \eqref{C42Capacity}, the capacity expression includes a maximization of the input power as a function $\varphi(\cdot)$ of the gain, similar to the result in \cite{Varaiya}. Similarly, when the jammer knows the gains \eqref{C3Capacity}--\eqref{C42Capacity}, the capacity expression includes a minimization that represents the jammer's choice of noise power as a function $\psi(\cdot)$ of the gain. Moreover, when the jammer knows the gains, with enough power it can symmetrize the channel by mimicking the legitimate signal, thus reducing the capacity to zero. However, in \eqref{C42Capacity} we have assumed that the adversary knows the gains causally and the encoder and the decoder know the gains non-causally. Thus, the encoder and decoder effectively share a secret (the channel gains at the end of the block) unknown to the adversary, so the adversary cannot symmetrize the channel. It is also worth mentioning that for the other cases (except \eqref{C42Capacity}) our proof works exactly the same whether the transmitter and/or the adversary know the gain sequence causally, non-causally, or even memorylessly (i.e., at time $i$, you only know the gain value at time $i$).
While we have stated the theorem by writing the capacities in terms of optimization over the $\varphi$ and/or $\psi$ functions, these expressions can be computed by solving for the optimizing functions. In particular, the optimum value of $\varphi^*(g)$ in \eqref{C2Capacity} is $\left|\lambda-\frac{\Lambda+\sigma^2}{g^2}\right|^+$ where $\lambda$ is obtained by $\mathbb{E} [\varphi^*(G)]=P$, and
the optimum value of $\psi^*(\cdot)$ in \eqref{C3Capacity} is a function of gain $g$ as follows
\begin{align}
\psi^*(g) = \left|\frac{-2\sigma^2-g^2P+\sqrt{g^4P^2+\frac{2g^2P}{\lambda}}}{2}\right|^+,\label{fi_g3}
\end{align}
and $\lambda$ is obtained by solving $\mathbb{E} \psi^*(G) = \Lambda$. Moreover, the optimum values of $\varphi^*(g)$ and $\psi^*(g)$ in \eqref{C4Capacity} are
\begin{align}
\varphi^*(g) &= \left| \frac{1}{2(\lambda_1-g^2\lambda_3) \left(1+\frac{\lambda_1-g^2\lambda_3}{g^2\lambda_2}\right)}\right|^+\\
\psi^*(g) &= \left|\frac{g^2}{2g^2\lambda_2+2(\lambda_1-g^2\lambda_3)}-\sigma^2\right|^+,
\end{align}
where $\lambda_1$, $\lambda_2$ and $\lambda_3$ are found by solving $\mathbb{E} \varphi(G)= P$, $\mathbb{E} \psi(G)= \Lambda$ and $\mathbb{E} G^2\varphi(G)=\Lambda$, respectively.
Finally, the optimum values of $\varphi^*(g)$ and $\psi^*(g)$ in \eqref{C42Capacity} are
\begin{align}
\varphi^*(g) &= \left| \frac{1}{2\lambda_1 \left(1+\frac{\lambda_1}{g^2\lambda_2}\right)}\right|^+\\
\psi^*(g) &= \left|\frac{g^2}{2g^2\lambda_2+2\lambda_1}-\sigma^2\right|^+,
\end{align}
where $\lambda_1$ and $\lambda_2$ can be obtained by solving $\mathbb{E} \varphi(G)= P$ and $\mathbb{E} \psi(G)= \Lambda$, respectively.
In the Fig. \ref{fig:GAVFC_Simu}, the capacity of GAVFC with Rayleigh fading is shown for $P=1, \sigma^2=0.25, 0<\Lambda<5$ and the Rayleigh distribution scale parameter $\sigma_R = 1$ whether the channel gains are available at the encoder and/or the adversary. However, $C$ is the capacity of Gaussian arbitrary-varying channel with no fading. Note that when the transmitter's knowledge increases, the capacity increases, whereas when the adversary's knowledge increases, the capacity decreases. On the other hand, the knowledge of adversary about the channel gains may decrease the capacity, and in this case if the adversary's power exceeds $2$, the capacity will be zero by the symmetrizability.
The proofs of the different capacity variants all follow a similar pattern, so we have attempted to reduce the redundancy in our presentation. We provide essentially one converse proof for each capacity expression in the theorem:
\begin{enumerate}
\item Sec. \ref{subsecV-A}: Converse for $C_{U,U}$.
\item Sec. \ref{subsecVI-A}: Converse for $C_{N,U}$. This also bounds $C_{C,U}$ which is trivially upper bounded by $C_{N,U}$.
\item Sec. \ref{subsecVII-A}: Converse for $C_{U,C}$. This also bounds $C_{U,N}$ which is trivially upper bounded by $C_{U,C}$.
\item Sec. \ref{subsecVIII-A}: Converse for $C_{C,C}$. This also bounds $C_{C,N}$ which is trivially upper bounded by $C_{C,C}$. Essentially the same proof also works for $C_{N,N}$. The case $C_{N,C}$ requires a different proof also covered in this section.
\end{enumerate}
We provide essentially three achievability proofs:
\begin{enumerate}
\item Sec. \ref{subsecVI-B}: Achievability for $C_{C,U}$. This also bounds the $C_{N,U}$, which is trivially lower bounded by $C_{C,N}$. It also provides a bound for $C_{U,U}$, because the same proof works assuming $\varphi(g)=P$ (i.e., the encoder's power is independent of the channel gain).
\item Sec. \ref{subsecVIII-B}: Achievability for $C_{C,N}$. This also bounds $C_{N,N}$ and $C_{C,C}$, which are trivially lower bounded by $C_{C,N}$. It also provides a bound for $C_{U,N}$ and $C_{U,C}$, because the same proof works again assuming $\varphi(g)=P$.
\item Sec. \ref{subsecVIII-C}: Achievability for $C_{N,C}$. This case is different from all others in that there is effectively a shared secret between encoder and decoder.
\end{enumerate}
\begin{figure}[t]
\centering
\includegraphics[width=0.7\linewidth]{FadingCapacity3.pdf}
\vspace{-1em}\caption{GAVFC capacities for $P=1, \sigma^2=0.25, 0<\Lambda<5$ with Rayleigh fading. $C$ is the capacity of standard Gaussian channel without fading.}
\label{fig:GAVFC_Simu}
\end{figure}
\section{Auxiliary Results and Tools}\label{4}
Before proceeding to the proofs, we first define the typical set for continuous random variables $X_1,\ldots,X_k$ with probability density function $f_{X_1,\ldots,X_k}(x_1,\ldots,x_k)$ as follows:
\begin{multline}
\mathcal{T}_{\epsilon}^{(n)}(X_1,\ldots,X_k)=
\bigg\{(x_1,\ldots,x_k)\!:
\left|-\frac{1}{n}\log f_{X_A}(x_{A})
-h(X_A)\right| \leq \epsilon
\text{ for all }A\!\subset\![k]\!\bigg\}\label{TypicalSet}
\end{multline}
where $h(X_A)$ is the differential entropy of $(X_i:i\in A)$. Next, we define the typical set for continuous random variables $X_1,\ldots,X_k$ with probability density function $f_{X_1,\ldots,X_k}(x_1,\ldots,x_k)$ and a discrete random variable $\tilde{G}$ with probability mass function $P_{\tilde{G}(\tilde{g})}$ as follows:
\begin{multline}
\mathcal{T}_{\epsilon}^{(n)}(X_1,\ldots,X_k,\tilde{G})=
\bigg\{(x_1,\ldots,x_k,\hat{g})\!:
\left|-\frac{1}{n}\log P_{\tilde{G}}(\tilde{g})-H(\tilde{G})\right| \leq \epsilon ,\\
\left|-\frac{1}{n}\log f_{X_A}(x_{A})
-h(X_A)\right| \leq \epsilon,
\left|-\frac{1}{n}\log f_{X_A
|\tilde{G}}(x_{A}|\tilde{g})
-h(X_A|\tilde{G})\right| \leq \epsilon ,
\text{ for all }A\!\subset\![k]\!\bigg\},\label{TypicalSet}
\end{multline}
where $H(\tilde{G})$ and $h(X_A|\tilde{G})$ denote the entropy of $G$ and the conditional differential entropy of $X_A$ given $\tilde{G}$.
Throughout the achievability proofs, we will utilize several lemmas including the \emph{joint typicality lemma} and \emph{conditional typicality lemma} for Gaussian random variables given in \cite{FatemehISIT2018}. In addition, we will need the following two lemmas; they show that with high probability a Gaussian codebook satisfies several desirable properties. The proof is given in the Appendix.
\begin{lemma}\label{lem5_0} Fix $\epsilon'>0$. There exists $\gamma>0$ such that the following holds. Let $\mathbf{X}(m)$ for $m\in[N]$, $N=2^{nR}$ be a zero mean Gaussian codebook with variance $1-\gamma$. Let $G$ be drawn from probability density function $f_G(g)$. With probability approaching 1 as $n\to\infty$, for any $\mathbf{s},\mathbf{g}$ where $\|\mathbf{s}\|^2\le n\Lambda$,
there exists a function $\delta(\epsilon')>0$ such that
\begin{align}
\frac{1}{N} \left|\left\{m: (\mathbf{x}(m),\mathbf{s},\mathbf{g})\notin \hspace{-1.6em} \bigcup_{\substack{X\text{ independent of } (S,G): \\EX^2=1, ES^2\le\Lambda}}\hspace{-2.1em}\mathcal{T}^{(n)}_{\epsilon'}(X,S,G)\right\}\right|\le \exp(-n\delta(\epsilon')),\label{eq:two_codebooks_c0}
\end{align}
where the union is over zero mean conditionally Gaussian random vectors $(X,S)$ given $G$.
\end{lemma}
\begin{lemma}\label{lem5} Fix $\epsilon>0$. There exists $\gamma>0$ such that the following holds. Let $\mathbf{X}(m)$ for $m\in[N]$, $N=2^{nR}$ be a zero mean Gaussian codebook with variance $1-\gamma$. Let $G$ be drawn from probability density function $f_G(g)$. With probability approaching 1 as $n\to\infty$, for any
\begin{itemize}
\item zero-mean conditionally Gaussian random vector $(X,X',S)$ given $G$ where $\mathbb{E} X^2=\mathbb{E} X'^2=1$ and $\mathbb{E} S^2\le \Lambda$,
\item $\mathbf{x},\mathbf{s},\mathbf{g}$ where $\|\mathbf{s}\|^2\le n\Lambda$,
\end{itemize}
there exists a function $\delta(\epsilon)>0$ such that
\begin{align}
&\mathbb{P} \left\{\big|\big\{(\mathbf{x}(m'),\mathbf{s},\mathbf{G})\!\in\!\mathcal{T}_{\epsilon}^{(n)}\!(X',S,G)\text{ for some }m'\big\}\big|\right\} \le 2 \exp\{\!-n\delta(\epsilon)/2\},\nonumber \\& \hspace{20em}\text{if }I(G;X'S)\!\ge\! |R\!-\! I(X';S)|^+\!\!+\! \delta(\epsilon),\label{eq:Hughes_c4}\\
&\big|\big\{m':(\mathbf{x}(m'),\mathbf{s})\in\mathcal{T}_{\epsilon}^{(n)}(X',S)\big\}\big| \le \exp\big\{n\big[|R-I(X';S)|^++\delta(\epsilon)\big]\big\},\label{eq:Hughes_c5}
\\
&\big|\big\{m':(\mathbf{x},\mathbf{x}(m'),\mathbf{s},\mathbf{g})\in\mathcal{T}_{\epsilon}^{(n)}(X,X',S,G)\big\}\big|\le \exp\big\{n\big[|R-I(X';XSG)|^++\delta(\epsilon)\big]\big\},\label{eq:22Huz_c5}\\
&\frac{1}{N}\big|\big\{\! m\!:\!(\mathbf{x}(m),\mathbf{x}(m'),\mathbf{s},\mathbf{g})\!\in\!\mathcal{T}_{\epsilon}^{(n)}\text{ for some }m'\!\ne\! m\!\big\}\big|\le \!2\exp\{\!-n\delta(\epsilon)/2\},\nonumber\\
&\hspace{19em}\text{if }I(X;X'SG)\!\ge\! |R\!-\! I(X';SG)|^+\!\!+\! \delta(\epsilon).\label{eq:21Huz_c4}
\end{align}
\end{lemma}
\section{Capacity Proof with Gains Available at Decoder}\label{secV}
\subsection{Converse Proof}\label{subsecV-A}
We initially assume that for any arbitrary adversary strategy there is a sequence of $(2^{nR},n)$ codes with vanishing probability of error. The adversary can generate a Gaussian sequence with variance $\Lambda-\gamma$ for any $\gamma>0$; if this sequence has power less than $\Lambda$, it is transmitted, otherwise, the adversary sends the all-zero sequence. Note that the power of this Gaussian sequence exceeds $\Lambda$ only with small probability by the law of large numbers. With this choice of adversary, the channel corresponds to a standard Gaussian fading channel with the noise variance $\Lambda+\sigma^2-\gamma$ where the channel gains are available only at the decoder. Therefore, using capacity of a non-adversarial Gaussian fading channel \cite{ElGamal} for arbitrarily small $\gamma$, we may upper bound the capacity by
\begin{equation}
C\le \mathbb{E}_G \left[C\left(\frac{G^2P}{\Lambda+\sigma^2}\right)\right].
\end{equation}
\subsection{Achievability Proof}\label{subsecV-B}
The achievability proof of this case can be counted as a special case of $C_{N,U}$ in Sec. \ref{subsecVI-B} where both encoder and decoder know the channel gains. However, in this case since the encoder does not know the channel gains, we do not have any $\varphi(g)$ function at the encoder. In other words, the achievability proof for this case is identical to that in Sec. \ref{subsecVI-B} with $\varphi(g)=P$.
\section{Capacity Proof with Gains Available at Encoder and Decoder}\label{6}
\subsection{Converse Proof}\label{subsecVI-A}
As in the previous case, the adversary can simply send Gaussian send noise with variance $\Lambda-\gamma$. By the law of large numbers, the resulting channel is equivalent to a standard Gaussian fading channel with the knowledge of gains at both encoder and decoder and noise variance $\Lambda+\sigma^2-\gamma$ with high probability. Thus, since $\gamma$ can be chosen arbitrarily small, from the capacity of a non-adversarial Gaussian fading channel \cite{Varaiya}, we have
\begin{equation}
C\le \underset{\varphi(g):\mathbb{E} \varphi(G)\leq P}{\max} \mathbb{E}_G \left[C\left(\frac{G^2\varphi(G)}{\Lambda+\sigma^2}\right)\right].
\end{equation}
\subsection{Achievability Proof}\label{subsecVI-B}
For simplicity we assume $P=1$. Suppose any arbitrary function $\varphi(G)$ that satisfies $\mathbb{E} \varphi(G)\leq 1$ and $\var(G\sqrt{\varphi(G)})>0$. We further assume that $G^2 \varphi(G)$ has a positive variance. Note that this is only a concern if the optimum $\varphi^*(G)=\frac{c}{G^2}$; in this case, we can instead take $\varphi(G)=\frac{c}{(G-d)^2}$ where $c,d$ are two positive constants and $d$ can be chosen arbitrarily small. Let
\begin{align}
R<\mathbb{E}_G \left[C\left(\frac{G^2\varphi(G)}{\Lambda+\sigma^2}\right)\right]. \label{eq:R_assumption2}
\end{align}
We now propose a $(2^{nR},n)$ code sequence, and prove that using this code the probability of error tends to zero as $n\to \infty$.
\emph{Codebook generation:} Fix $\epsilon>\epsilon'>\gamma>0$. We generate $2^{nR}$ i.i.d zero mean Gaussian sequences $\mathbf{X}(m)$ with variance $(1-\gamma)$ for each $m \in [2^{nR}]$. By Lemma~\ref{lem5_0} and Lemma \ref{lem5}, we assume that the deterministic codebook satisfies \eqref{eq:two_codebooks_c0}--\eqref{eq:21Huz_c4}.
\emph{Encoding:} Since the transmitter knows the channel gains, it sends $\sqrt{\varphi(\mathbf{g})}\circ \mathbf{x}(m)$ (at time $i$ signal $\sqrt{\varphi(g_i)}x_i(m)$ is sent) if its power is less than $1$, otherwise it sends zero.
\emph{Decoding:} Given $\mathbf{y}$, let $\mathscr{S}$ be the set of messages $\hat{m}$ such that $(\mathbf{x}(\hat{m}),\mathbf{g}, \mathbf{y})\in \mathcal{T}_\epsilon^{(n)}(X',G,Y)$ for some random variables $X'\sim \mathcal{N}(0,1)$, $G\sim f_G(g)$ and zero mean Gaussian $Y-G\sqrt{\varphi(G)}X'$ where $(X',G,Y-G\sqrt{\varphi(G)}X')$ are mutually independent.
Now, we define the decoding function as
\begin{align}
\Theta(\mathbf{y},\mathbf{g}) &= \argmin_{\hat{m} \in \mathscr{S}}
\left\| \mathbf{y} - \mathbf{g} \circ \sqrt{\varphi(\mathbf{g})}\circ \mathbf{x}(\hat{m}) \right\|^2.
\end{align}
\emph{Analysis of the probability of error:}
Suppose the true message sent by the legitimate user is message $M$ with the power constraint $\|\mathbf{x}(M)\|^2\le n(1-\gamma)$. Then, the overall probability of error is upper bounded by $P_e^{(n)}\leq P_0+P_1$ where
\begin{align}
P_0&=\mathbb{P}\left\{M \notin\mathscr{S} \right\},\label{firstPe}\\
P_1&=\mathbb{P}\bigg\{\left\| \mathbf{Y} - \mathbf{G}\circ\sqrt{\varphi(\mathbf{G})}\circ\mathbf{x}(\hat{m}) \right\|^2
\leq \left\| \mathbf{s}+\mathbf{V} \right\|^2 \text{ for some } \hat{m}\in \mathscr{S}\setminus \{M\} \bigg\}.\label{secondPe}
\end{align}
Consider any state sequence $\mathbf{s}$. By \eqref{eq:two_codebooks_c0}, with high probability $(\mathbf{x}(M),\mathbf{s},\mathbf{G})\in \mathcal{T}_{\epsilon'}^{(n)}(X,S,G)$ where $(X,S,G)$ are independent, and $\mathbb{E} X^2=1,\allowbreak \mathbb{E} S^2 \le \Lambda$. By the conditional typicality lemma, for every $\epsilon>\epsilon'$ with high probability $(\mathbf{x}(M),\mathbf{s},\mathbf{G},\mathbf{V})\in \mathcal{T}_{\epsilon}^{(n)}(X,S,G,V)$ where $(X,S,G,V)$ are mutually independent, and $\mathbb{E} V^2=\sigma^2$.
Thus, according to the definition of $\mathscr{S}$, with high probability $M\in \mathscr{S}$ and $P_0$ tends to zero as $n\to \infty$.
Define the shorthand $\vec{X}=(XX'SGV)$. Let $\mathcal{V}$ be a finite $\epsilon$-dense subset in the set of all distributions of random vectors $\vec{X}$ that are determined by $f_G(g)$ and jointly zero mean Gaussian vector $(XX'SV)$ independent of $G$ with bounded covariances at most $(1,1,\Lambda,\sigma^2)$. Note that because the distribution of $f_G(g)$ is fixed, the overall distribution of $\vec{X}$ can be determined by the covariance matrix of $(XX'SV)$, so $\mathcal{V}$ only needs to cover a compact set. Now, we may upper bound $P_1$ by
\begin{equation}
\sum_{\vec{X}\in\mathcal{V}} \frac{1}{N} \sum_{m=1}^N \mathbb{E}_{G}[e_{\vec{X}}(m,\mathbf{s},\mathbf{G})]
\end{equation}
where
\begin{multline}\label{eq:eX1_def2}
e_{\vec{X}}(m,\mathbf{s},\mathbf{g})
=\mathbb{P}\bigg\{ (\mathbf{x}(m),\mathbf{x}(\hat{m}),\mathbf{s},\mathbf{g},\mathbf{V})\in\mathcal{T}_\epsilon^{(n)}(\vec{X}),\\
\|\mathbf{g} \circ \sqrt{\varphi(\mathbf{g})} \circ \mathbf{x}(m)+\mathbf{s}+\mathbf{V}-\mathbf{g} \circ \sqrt{\varphi(\mathbf{g})}\circ \mathbf{x}(\hat{m})\|^2 \le \|\mathbf{s}+\mathbf{V}\|^2 \text{ for some }\hat{m}\in\mathscr{S}\setminus \{m\}\bigg\}.
\end{multline}
We will show that $\frac{1}{N} \sum_{m=1}^N e_{\vec{X}}(m,\mathbf{s},\mathbf{g})\to 0$ for all vectors $\mathbf{g}$ and all vectors $(XX'SV)$ which are Gaussian given $G$ (whether or not they are in $\mathcal{V}$). Let $Z =G\sqrt{\varphi(G)}X+S+V-G\sqrt{\varphi(G)}X'$. We may restrict ourselves to $\vec{X}$ where
\begin{gather}
(X,S,G,V)\text{ are mutually independent},\label{XSV_indep}\\
(X,X',S,V)\text{ are zero mean Gaussian},\label{zero_mean}\\
\mathbb{E} X^2=\mathbb{E} X'^2=1,\quad
\mathbb{E} V^2=\sigma^2,\quad \mathbb{E} S^2\le\Lambda,\label{signalpower}\\
\left(X',G,Z\right)\!\text{ are independent},\label{XZ1_indep}\\
\mathbb{E} \left[Z^2\right] \le \Lambda+\sigma^2,\label{Zles}
\end{gather}
where \eqref{XSV_indep} holds since the input $X$, adversary $S$, fading gains $G$ and noise $V$ are all generated independently, \eqref{zero_mean}--\eqref{signalpower} follows from $m,\hat{m}\in\mathscr{S}$, and $\vec{X}\in\mathcal{V}$, \eqref{XZ1_indep} holds since we have $(X',G,Y-GX')$ are mutually independent using $\mathbf{x}(\hat{m})\in\mathscr{S}$, and \eqref{Zles} corresponds to $\mathbb{E} \left[\left(Y-G\sqrt{\varphi(G)}X'\right)^2\right]$ which is less than $\Lambda+\sigma^2$ from \eqref{eq:eX1_def2}.
Observe that if $I(X,V,G;X',S)=0$, then we would have
\begin{align}
0&=\mathbb{E} [X'Z]\label{G_indep2}\\
&=\mathbb{E} [X'(G\sqrt{\varphi(G)}X+S+V-G\sqrt{\varphi(G)}X')]\\
&= \mathbb{E} [X'(S-G\sqrt{\varphi(G)}X')]\label{28_2}\\
&= \mathbb{E} [X'S] - \mathbb{E} G\sqrt{\varphi(G)},\label{contr3}
\end{align}
where \eqref{G_indep2} follows from \eqref{XZ1_indep}, \eqref{28_2} holds because $(X',G,X,V)$ are all mutually independent by the assumption $I(X,V,G;X',S)=0$ and \eqref{XSV_indep}, and the last equality holds since $X'$ is independent of $G$ and because $\mathbb{E} [X'^2]=1$. Therefore, $\mathbb{E} [X'S] = \mathbb{E} G\sqrt{\varphi(G)}$.
Moreover, from \eqref{eq:eX1_def2} we have
\begin{align}
\mathbb{E} (S+V)^2&\geq \mathbb{E}(G\sqrt{\varphi(G)}X\!+\!S\!+\!V\!-\!G\sqrt{\varphi(G)}X')^2
\\&=\mathbb{E} G^2\varphi(G)(X-X')^2+2\mathbb{E} G\sqrt{\varphi(G)}(X-X')(S+V)+\mathbb{E}(S+V)^2
\\&=\mathbb{E} G^2\varphi(G)\mathbb{E} X^2+\mathbb{E} G^2\varphi(G)\mathbb{E} X'^2-2\mathbb{E} G\sqrt{\varphi(G)}X'S+\mathbb{E}(S+V)^2\label{lotwork}
\\&=2\mathbb{E} G^2\varphi(G)-2\mathbb{E} G\sqrt{\varphi(G)}\mathbb{E} X'S+\mathbb{E}(S+V)^2,\label{cancel1}
\end{align}
where \eqref{lotwork} holds because $\mathbb{E} X = \mathbb{E} X' = \mathbb{E} V = 0$, $(X,X',G)$ are mutually independent, $(X,S,V)$ are mutually independent, and $(X',V)$ are independent by \eqref{XSV_indep}, \eqref{zero_mean} and the assumption $I(X,V,G;X',S)=0$. Canceling $\mathbb{E}(S+V)^2$ from both sides of \eqref{cancel1} gives us
\begin{align}
\mathbb{E} G^2\varphi(G)-\mathbb{E} G\sqrt{\varphi(G)}\mathbb{E} X'S\leq 0.\label{contra2}
\end{align}
Now, if we apply the result from \eqref{contr3} to \eqref{contra2}, we get
\begin{align}
\mathbb{E} G^2\varphi(G)-\mathbb{E} G\sqrt{\varphi(G)}\mathbb{E} X'S&=\mathbb{E} G^2\varphi(G)-\mathbb{E} G\sqrt{\varphi(G)}\mathbb{E} G\sqrt{\varphi(G)}\\
&=\mathbb{E} G^2\varphi(G)-\mathbb{E}^2 G\sqrt{\varphi(G)}\\
&=\var{G\sqrt{\varphi(G)}}\\
&\leq 0.
\end{align}
which is a contradiction since we assume $\var{(G\sqrt{\varphi(G)})}$ is always positive. Thus, there exists an $\eta>0$ such that
\begin{equation}\label{eq:eta_bound12}
\eta\le I(XVG;X'S).
\end{equation}
Also, by \eqref{eq:21Huz_c4}, we may restrict ourselves to distributions where
\begin{equation}\label{eq:R1R212}
I(X;X'SG)< |R-I(X';SG)|^++\delta(\epsilon)
\end{equation}
and
\begin{equation}\label{eq:RI(X'S)2}
I(G;X'S)< |R-I(X';S)|^++\delta(\epsilon).
\end{equation}
Note that $I(X;X'SG)=I(X;X'|SG)$. We also have the upper bound
\begin{align}
e_{\vec X}(m,\mathbf{s},\mathbf{g})&\leq\sum_{\hat{m}:(\mathbf{x}(m),\mathbf{x}(\hat{m}),\mathbf{s},\mathbf{g})\in\mathcal{T}_{\epsilon}^{(n)}(X,X',S,G)}
\hspace{-2em}\mathbb{P}\left\{\!(\mathbf{x}(m),\mathbf{x}(\hat{m}),\mathbf{s},\mathbf{g},\mathbf{V})\!\in\! \mathcal{T}_{\epsilon}^{(n)} (X,X',S,G,V)\!\right\}
\\& \leq \exp\big\{n\big[|R\!-\! I(X';XSG)|^+\!\!-I(V;X'|XSG)+\delta(\epsilon)\big]\label{e(s,i)_I2}
\end{align}
where \eqref{e(s,i)_I2} follows from $I(V;XSG)=0$, \eqref{eq:22Huz_c5} and the joint typicality lemma.
Now, let us consider three cases as follows:
Case (a): $R<I(X';S)$ that implies $R<I(X';XSG)$.
From \eqref{e(s,i)_I2}, for any $m, \mathbf{s},\mathbf{g}$
\begin{align}
e_{\vec X}(m,\mathbf{s},\mathbf{g})&\le \exp\left\{-n\left( I(V;X'|XSG)-\delta(\epsilon)\right)\right\}\label{eq:two_codebook_ebound11}\\
&= \exp\{- n(I(XV;X'|SG)-I(X;X'|SG)-I(XV;S|G)-\delta(\epsilon))\}\\
&= \exp\{- n(I(XV;X'S|G)\!-\! I(X;X'|SG)- \delta(\epsilon))\}\\
&= \exp\{- n(I(XVG;X'S)-I(G;X'S)-I(X;X'|SG)-\delta(\epsilon))\}\\
&\le \exp\{-n(\eta-\delta(\epsilon)-\delta'(\epsilon))\}\label{casearesult2}
\end{align}
where \eqref{casearesult2} follows from \eqref{eq:eta_bound12}, \eqref{eq:R1R212} and \eqref{eq:RI(X'S)2}. Therefore, $e_{\vec X}(m,\mathbf{s},\mathbf{g})$ vanishes exponentially fast if $\delta(\epsilon)$ is sufficiently small.
Case (b): $I(X';S)\leq R$. Since $R\geq I(X';S)$ and $I(G;S)=0$, from \eqref{eq:RI(X'S)2} we have
\begin{align}
R &> I(G;X'S)+I(X';S)-\delta(\epsilon)\\
& =I(G;S)+I(G;X'|S)+I(X';S)-\delta(\epsilon)\\
& =I(X';SG)-\delta(\epsilon).
\end{align}
Using this result in \eqref{eq:R1R212}, we have
\begin{align}
I(X;X'SG)<R-I(X';SG)+\delta(\epsilon)+\delta(\epsilon).
\end{align}
Therefore,
\begin{align}
R &>I(X;X'SG)+I(X';SG)-2\delta(\epsilon)\\
&\geq I(X';XSG)-2\delta(\epsilon).\label{neweq}
\end{align}
Now, from \eqref{e(s,i)_I2}, we have for any $m, \mathbf{s},\mathbf{g}$
\begin{align}
e_{\vec X}(m,\mathbf{s},\mathbf{g})&\leq \exp\big\{n\big[|R\!-\! I(X';XSG)|^+\!\!-I(V;X'|XSG)+\delta(\epsilon)\big]\\
&\leq \exp\big\{n\big[R\!- I(X';XSG)+2\delta(\epsilon)-\! I(V;X'|XSG)\!+\!\delta(\epsilon)\big]\label{abslt2}\\
& = \!\exp (n[R-I(X';XSGV)+3\delta(\epsilon)])\label{result222}
\end{align}
where \eqref{abslt2} follows from \eqref{neweq}.
We now lower bound $I(X';XSVG)$ as follows:
\begin{align}
I(X';XSVG)&=I(X';XSV|G)+I(X';G)\\
& \geq I(X';G\sqrt{\varphi(G)}X\!+\!S\!+\!V|G)\\
& = I(X';Z+G\sqrt{\varphi(G)}X'|G)\\
& = h(Z+G\sqrt{\varphi(G)}X'|G)-h(Z+G\sqrt{\varphi(G)}X'|G,X')\\
& = \mathbb{E} \bigg[\frac{1}{2} \log 2\pi e\left(G^2\varphi(G)+\mathbb{E} [Z^2|G]\right)-\frac{1}{2} \log 2\pi e \mathbb{E} [Z^2|G] \bigg] \\
& = \mathbb{E} \left[C\left(\frac{G^2\varphi(G)}{\mathbb{E} [Z^2|G]} \right) \right] \\
& \geq \mathbb{E} \left[C\left(\frac{G^2\varphi(G)}{\Lambda+\sigma^2}\right)\right]\label{ZleLambda}
\end{align}
where \eqref{ZleLambda} follows from \eqref{XZ1_indep} and \eqref{Zles}. Replacing this result in \eqref{result222}, we obtain
\begin{equation}
e_{\vec X}(m,\mathbf{s},\mathbf{g})\le\exp\left\{n\left[R-\mathbb{E} \left[C\left(\frac{G^2\varphi(G)}{\Lambda+\sigma^2}\right)\right]+3\delta(\epsilon)\right]\right\}\label{e_1result}
\end{equation}
meaning that $e_{\vec X}(m,\mathbf{s},\mathbf{g})$ is exponentially vanishing if $\delta(\epsilon)$ is sufficiently small, and \eqref{eq:R_assumption2} holds.
\section{Capacity Proof with Gains Available at Decoder and Jammer}\label{7}
\subsection{Converse Proof}\label{subsecVII-A}
Consider a sequence of $(2^{nR},n)$ codes with vanishing probability of error that must function for arbitrary jamming signals. Because we are proving the converse, we may assume the best case scenario from the legitimate user's perspective; in particular, that the adversary only knows the channel gains causally.
We begin with the case that $\Lambda\leq\mathbb{E} G^2P$. Given any function $\psi(g)$ satisfying $\mathbb{E} \psi(G) \le \Lambda$, we may obtain an upper bound by assuming that the jammer transmits a random sequence $\mathbf{S}=(S_1,\cdots,S_n)$ where $S_i$ is Gaussian with mean zero and variance $\psi(G_i)$ for $i=1,\cdots,n$. Note that
\begin{align}
\mathbb{E} [\|\mathbf{S}\|^2]& = \mathbb{E} \sum_{i=1}^n S_i^2\\
& = \sum_{i=1}^n \mathbb{E} S_i^2\\
& = \sum_{i=1}^n \mathbb{E} \psi(G_i)\\
& \le n\Lambda.
\end{align}
The resulting channel is equivalent to a standard Gaussian fading channel with the knowledge of gains only at the decoder and noise variance $\psi(g)+\sigma^2$. From the capacity of a non-adversarial Gaussian fading channel
\begin{equation}
C\le \mathbb{E}_G \left[C\left(\frac{G^2P}{\psi(G)+\sigma^2}\right)\right].
\end{equation}
Therefore, the capacity is also less than the minimum over all $\psi(G)$ that satisfies $\mathbb{E} \psi(G)\leq \Lambda$.
\begin{equation}
C\le \min_{\psi(G):\mathbb{E} \psi(G)\leq \Lambda} \mathbb{E}_G \left[C\left(\frac{G^2P}{\psi(G)+\sigma^2}\right)\right].
\end{equation}
For the case $\Lambda>\mathbb{E} G^2P$, we first show that the adversary has enough power to choose a codeword and send it to the channel, thereby symmetrizing the channel. Let $\tilde{M}$ be a uniformly chosen message by the adversary and $M$ be the true message sent by the legitimate transmitter. Suppose the adversary chooses $\mathbf{S}=\mathbf{G} \circ \mathbf{x}(\tilde{M})$ then the adversary's power constraint is satisfied as follows:
\begin{align}
\mathbb{E} \left[\|\mathbf{S}\|^2\right] & =\mathbb{E} \left[\|\mathbf{G} \circ \mathbf{x}(\tilde{M})\|^2\right] \\
& = \mathbb{E} \left[\sum_{i=1}^n G_i^2 x_i^2(\tilde{M})\right]\\
& <\sum_{i=1}^n x_i^2(\tilde{M}) \frac{\Lambda}{P}\label{last1}\\
& \le n\Lambda\label{last2}
\end{align}
where \eqref{last1} follows from the assumption $\Lambda > \mathbb{E} G^2 P$, and \eqref{last2} follows from the codebook power constraint $\|\mathbf{x}^2\|\le nP$.
Given this choice of $\mathbf{S}$, $\mathbf{Y} = \mathbf{G}\circ\mathbf{x}(M)+ \mathbf{G}\circ \mathbf{x}(\tilde{M})+\mathbf{V}$. Thus, with high probability the decoder cannot decode the message since it does not know whether the true message is $M$ or $\tilde{M}$.
\subsection{Achievability Proof}\label{subsecVII-B}
The achievability proof of this case is very similar to the achievability proof of Sec.~\ref{subsecVIII-B} where the encoder, the decoder and the adversary all know the channel gains.
Here, the transmitter does not know the channel gain so it cannot leverage this knowledge to choose its transmit power. However, the achievability proof for this case is identical to that in Sec.~\ref{subsecVIII-B} except that the transmitter's power function is constant; i.e., $\varphi(g)=1$.
\section{Capacity Proof with Gains Available at Encoder, Decoder, and Jammer}\label{8}
In this section, we first provide the converse proof for the case that the channel gains are available at the encoder, the decoder and the adversary in Sec. \ref{subsecVIII-A}. The converse proof includes all the four cases in which each of the adversary and the encoder knows the fading gains causally or non-causally. In Sec. \ref{subsecVIII-B}, we show the achievability proof of the case that the channel gains are available non-causally at the adversary and causally at the encoder. This proof also works for the two cases of channel gains being available causally at both the adversary and the encoder or non-causally at both ends. Finally, we provide the achievability proof for the last case when the channel gains are causally available at the adversary and non-causally available at the encoder in Sec. \ref{subsecVIII-C}.
\subsection{Converse Proof}\label{subsecVIII-A}
Consider a sequence of $(2^{nR},n)$ codes with vanishing probability of error. Since in this case both the encoder and the adversary know the channel gains, we consider four cases to prove the converse whether each of them knows the fading gains causally or non-causally.
First assume that both the encoder and adversary know the channel gains causally. Let $\varphi_i(g)=\frac{1}{N}\sum_{m=1}^N \mathbb{E} [ X_i^2(m,G^i)|G_i=g]$ and $\varphi(g)=\frac{1}{n}\sum_{i=1}^n\varphi_i(g)$ where $G^i=(G_1,\ldots,G_i)$, for $i\in[n]$. Thus, $\varphi(g)$ satisfies $\mathbb{E} \varphi(G) \leq P$ as follows:
\begin{align}
\mathbb{E} \varphi(G) & = \mathbb{E} \left[\frac{1}{n}\sum_{i=1}^n \varphi_i(G)\right] \\
& =\frac{1}{N}\sum_{m=1}^N\frac{1}{n}\mathbb{E} \left[\sum_{i=1}^n X_i^2(m,G^i)\right] \\
& \le P\label{powassum}
\end{align}
where \eqref{powassum} follows by the power constraint for the input signal.
Now, similar to the previous case, where the adversary and decoder know the channel gains, we also have symmetrizability and non-symmetrizability cases, but with different conditions. We first show the symmetrizability case, that is if $\Lambda\ge \mathbb{E} G^2\varphi(G)$, then the jammer can symmetrize the channel. Suppose the adversary chooses a message $\tilde{M}$ uniformly at random and sends $S_i = G_i X_i(\tilde{M},G^i)$ where $G^i = (G_1,\cdots,G_i)$ for $i\in[n]$. Note that this selection of jamming signal is a causal function of the channel gains. Then we have
\begin{align}
\mathbb{E} \left[\|\mathbf{S}\|^2\right]&= \mathbb{E} \left[\sum_{i=1}^n S_i^2 \right]\\
&= \frac{1}{N}\sum_{\tilde{m}=1}^N\mathbb{E} \left[\sum_{i=1}^n G_i^2X_i^2 (\tilde{m},G^i)\right]\\
&= \sum_{i=1}^n \mathbb{E}_{G} \left[G^2 \frac{1}{N}\sum_{\tilde{m}=1}^N \mathbb{E} \left[X_i^2 (\tilde{m},G^i)|G_i=G\right]\right]\\
&= \sum_{i=1}^n \mathbb{E}_{G} \left[G^2 \varphi_i(G)\right]\\
&= \mathbb{E}_{G} \left[G^2 \sum_{i=1}^n \varphi_i(G)\right]\\
&= n\mathbb{E}_{G} \left[G^2 \varphi(G)\right]\\
&\le n \Lambda.\label{Spowerconstrn}
\end{align}
Therefore, this choice of jammer satisfies the adversary power constraint.
Given $\mathbf{Y} = \mathbf{g}\circ\mathbf{x}(M,\mathbf{g})+ \mathbf{g}\circ \mathbf{x}(\tilde{M},\mathbf{g})+\mathbf{V}$, the decoder cannot determine the correct message between true message $M$ or the adversary message $\tilde{M}$ with high probability. Thus, the probability of error is bounded away from zero. By the above argument, if $ \mathbb{E} G^2 \varphi(G) \leq \Lambda$ for all $\varphi(g)$ where $\mathbb{E} \varphi(G)\le P$, then the capacity cannot be positive; the adversary can always symmetrize the channel, so the capacity is 0.
On the other hand, consider the case where there exists some function $\varphi(g)$ where $\mathbb{E}G^2 \varphi(G)>\Lambda$ and $\mathbb{E} \varphi(G)\le P$.
Let $\psi_i(g)$ be given by
\begin{align}
\psi_i(g)= \argmin_{\psi(g): \mathbb{E} \psi(G)\le \Lambda} \mathbb{E} \left[C\left(\frac{G^2\varphi_i(G)}{\sigma^2+\psi(G)}\right)\right].
\end{align}
Since the transmitted codes should work for arbitrary jamming signals, an outer bound may be obtained by assuming the adversary sends $S_i\sim \mathcal{N}(0,\psi_i(G))$. By the assumption that $\mathbb{E} \psi_i(G)\le~\Lambda$, the jammer's expected power constraint is satisfied. Therefore, the rate is upper bounded by
\begin{align}
nR &\le \sum_{i=1}^n I(X_i;Y_i|G_i)\\
& = \sum_{i=1}^n I(X_i;G_i X_i+S_i+V_i|G_i)\\
& \le \sum_{i=1}^n \mathbb{E}_{G_i}\left[ C\left(\frac{G_i^2\varphi_i(G_i)}{\psi_i(G_i)+\sigma^2}\right)\right]\label{GFC1}\\
& = \sum_{i=1}^n \min_{\psi(g):\mathbb{E}\psi(g)\le \Lambda}\mathbb{E}_{G}\left[ C\left(\frac{G^2\varphi_i(G)}{\psi(G)+\sigma^2}\right)\right]\label{GFC2}\\
& \le n \min_{\psi(g):\mathbb{E}\psi(g)\le \Lambda}\mathbb{E}_{G} \left[C\left(\frac{G^2\frac{1}{n}\sum_{i=1}^n\varphi_i(G)}{\psi(G)+\sigma^2}\right)\right]\label{GFC3}\\
& \le n \underset{\substack{\varphi(g):\mathbb{E} \varphi(G)\leq P\\ \mathbb{E} G^2\varphi(G)\geq\Lambda}}{\max} \ \min_{\psi(g):\mathbb{E}\psi(g)\le \Lambda}\mathbb{E}_{G} \left[C\left(\frac{G^2\varphi(G)}{\psi(G)+\sigma^2}\right)\right]\label{GFC5}
\end{align}
where \eqref{GFC1} follows since the mutual information is less than the capacity of equivalent standard fading channel with noise variance $\psi_i(g_i)+\sigma^2$, and the gains being available at both encoder and decoder, \eqref{GFC2} follows by the definition of $\psi_i(g)$, \eqref{GFC3} follows by the concavity of $C(\cdot)$ with respect to $\varphi_i(g)$ and Jensen's inequality, and \eqref{GFC5} follows since we have established that $\varphi(g)=\frac{1}{n}\sum_{i=1}^n\varphi_i(g)$ satisfies $\mathbb{E} \varphi(G)\le P$ and $\mathbb{E} G^2\varphi(G)\ge \Lambda$.
Moreover, if the encoder knows the channel gains causally, and the adversary knows them non-causally, then the adversary is stronger than in the previous case, so exactly the same bound holds. If both encoder and adversary know the channel gains non-causally, then we instead assume
\begin{align}
\varphi_i(g)=\frac{1}{N}\sum_{m=1}^N \mathbb{E} \left[ X_i^2(m,\mathbf{G})|G_i=g\right]
\end{align}
where $\mathbf{G} = (G_1,\ldots,G_n)$ and $S_i = G_i X_i(\tilde{m},\mathbf{G})$, so we get the same upper bound.
However, the case wherein the encoder knows the channel gains non-causally, and the adversary knows them causally is somewhat different. In this case, the encoder may send $ X_i^2(m,\mathbf{G})$ while the adversary does not have any access to $(G_{i+1}, \ldots, G_n)$ to construct $S_i = G_i X_i(\tilde{m},\mathbf{G})$. Thus, it cannot do better than sending Gaussian noise. In this case, we derive only a converse bound based on the adversary sending Gaussian noise. Hence, we obtain the following bound:
\begin{align}
R \le \underset{\substack{\varphi(g):\mathbb{E} \varphi(G)\leq P}}{\max} \ \min_{\psi(g):\mathbb{E}\psi(g)\le \Lambda}\mathbb{E}_{G} \left[C\left(\frac{G^2\varphi(G)}{\psi(G)+\sigma^2}\right)\right].
\end{align}
Note that here we are not making the assumption that $ \mathbb{E} G^2\varphi(G)\geq\Lambda$.
\subsection{Achievability Proof (Gains Available Non-causally at Adversary and Causally at Encoder)}\label{subsecVIII-B}
We first quantize $G$ in the following way. Fix $\nu>0$. Given the assumption that $G$ has finite variance, there exists a real-valued random variable $\tilde{G}$ with a finite support such that $\tilde{G}$ is a deterministic function of $G$ and $\mathbb{E} [(G-\tilde{G})^2|\tilde{G}=\tilde{g}]\le \nu$ for each $\tilde{g}$. We further assume that $\tilde{G}$ is the expected value of $G$ within each quantization set; that is, $\mathbb{E}[G|\tilde{G}]=\tilde{G}$.
Without loss of generality, assume $P=1$. Let $R$ be a rate and $\varphi(\tilde{g})$ be any function satisfying
\begin{align}
& \mathbb{E} \varphi(\tilde{G}) \le 1, \label{firstPowCond}\\
&\Lambda<\mathbb{E} \tilde{G}^2\varphi(\tilde{G}), \label{eq:lambda_assumption4} \\
&R<\underset{\psi(\tilde{g}):\mathbb{E} \psi(\tilde{G})\leq \Lambda}{\min}\mathbb{E}_{\tilde{G}} \left[C\left(\frac{\tilde{G}^2\varphi(\tilde{G})}{\psi(\tilde{G})+\sigma^2}\right)\right]. \label{eq:R_assumption4}
\end{align}
We construct a $(2^{nR},n)$ code as follows:
\emph{Codebook generation:}
Fix $\epsilon >\ \epsilon''> \epsilon ' >\lambda >0$. Generate $2^{nR}$ i.i.d. zero mean Gaussian
sequences $\mathbf{X}(m)$ with variance $(1-\gamma )$ for each $m \in [2^{nR}]$. By Lemmas \ref{lem5} and Lemma \ref{lem5_0}, we may assume that the deterministic codebook satisfies \eqref{eq:two_codebooks_c0}--\eqref{eq:21Huz_c4}.
\emph{Encoding:} Given message $m$ and gain sequence $\mathbf{g}$, the transmitter computes $\tilde{g}$ from the quantization function, and then sends $\sqrt{\varphi(\tilde{\mathbf{g}})}\circ\mathbf{x}(m)$ (at time $i$ signal $\sqrt{\varphi(\tilde{g_i})}x_i(m)$ is sent) if $\|\sqrt{\varphi(\tilde{\mathbf{g}})}\circ\mathbf{x}(m)\|^2\le n$; otherwise, it sends zero. Note that here we assume that the encoder knows the channel gains causally.
\emph{Decoding:} Given $\mathbf{y}$ and $\mathbf{g}$, let $\nu< \epsilon$ and $\mathscr{S}$ be the set of messages $\hat{m}$ such that $(\mathbf{x}(\hat{m}),\tilde{\mathbf{g}}, \mathbf{y})\in \mathcal{T}_\epsilon^{(n)}(X',\tilde{G},Y)$ where $\tilde{G}$ is the quantized random variable from $G$ and $(X',Y)$ are some random variables that are conditionally Gaussian given $\tilde{G}=\tilde{g}$ with zero mean and covariance
\begin{align}\label{sdefinition2}
\cov\left(X',Y\Big|\tilde{G}=\tilde{g}\right)=\left[\begin{array}{cccc}1&\tilde{g}\sqrt{\varphi(\tilde{g})} \\\tilde{g}\sqrt{\varphi(\tilde{g})}&a_{\tilde{g}}\end{array}\right]
\end{align}
where $a_{\tilde{g}}\ge \tilde{g}^2 \varphi(\tilde{g})+\sigma^2$. Note that the following can be shown from \eqref{sdefinition2}.
\begin{gather}
X' \text{ is independent of } \tilde{G}, \label{s15}\\
\mathbb{E} X'^2 = 1,\label{s25}\\
Y-\tilde{G}\sqrt{\varphi(\tilde{G})}X' \text{ is independent of } X' \text{ given } \tilde{G},\label{s35}\\
\var\left(Y-\tilde{G}\sqrt{\varphi(\tilde{G})}X'\bigg|\tilde{G}\right) \ge \sigma^2. \label{s45}
\end{gather}
Now, we define the decoding function as
\begin{align}
\Theta(\mathbf{y},\tilde{\mathbf{g}}) &= \argmin_{\hat{m} \in \mathscr{S}}
\left\| \mathbf{y} - \tilde{\mathbf{g}} \circ \sqrt{\varphi(\tilde{\mathbf{g}})} \circ \mathbf{x}(\hat{m}) \right\|^2.
\end{align}
\emph{Analysis of the probability of error:} Assume the legitimate transmitter sends message $M$. Then, we can upper bound the probability of error by the summation of the following error probabilities:
\begin{align}
P_0&=\mathbb{P}\left\{M \notin\mathscr{S} \right\},\label{firstPe}\\
P_1&=\mathbb{P}\bigg\{\left\| \mathbf{Y} -\tilde{ \mathbf{G}}\circ \sqrt{\varphi(\tilde{\mathbf{G}})}\circ \mathbf{x}(\hat{m}) \right\|^2
\leq \left\| \mathbf{s}+\mathbf{V} \right\|^2 \text{ for some } \hat{m}\in \mathscr{S}\setminus \{M\} \bigg\}.\label{secondPe}
\end{align}
We can prove with high probability that
\begin{align}
\frac{1}{n}\left\|\mathbf{x}\circ \sqrt{\varphi(\tilde{\mathbf{G}})}\circ(\mathbf{G}-\tilde{\mathbf{G}})\right\|^2 &= \frac{1}{n} \sum_{i=1}^n \left(x_i \sqrt{\varphi(\tilde{G}_i)} \left(G_i - \tilde{G}_i\right)\right)^2\\
& \le \frac{1}{n} \sum_{i=1}^n x_i^2 \mathbb{E}_{\tilde{G}_i} \left[\mathbb{E}_{G_i} \left[\varphi(\tilde{G}_i)\left(G_i - \tilde{G}_i\right)^2\Big| \tilde{G}_i \right]\right] + \nu\label{bynu111}\\
& \le \frac{1}{n} \sum_{i=1}^n x_i^2 \mathbb{E}_{\tilde{G}_i} \left[\varphi(\tilde{G}_i)\right] \nu + \nu\label{bynu211}\\
& \le 2\nu\label{bynu123}
\end{align}
where \eqref{bynu111} follows from the law of large numbers for non-identical independent random variables $x_i^2\varphi(\tilde{G}_i)\left(G_i-\tilde{G}_i\right)^2$, \eqref{bynu211} follows from the assumption $\mathbb{E} \left[\left(G_i - \tilde{G}_i\right)^2\Big| \tilde{G}_i=\tilde{g}_i\right] \leq \nu$, and \eqref{bynu123} follows from the assumption $\frac{1}{n} \|\mathbf{x}\|^2\leq 1$ and $\mathbb{E} \varphi(\tilde{G}) \leq 1$.
Consider any jammer sequence $\mathbf{s}$. We may assume sequence $\mathbf{G}$ is typical since it is drawn i.i.d. from the distribution $f_G(g)$. Similarly, $\tilde{\mathbf{G}}$ is also typical because it is from the corresponding discrete distribution $P_{\tilde{G}}(\tilde{g})$. Thus, $(\mathbf{s},\tilde{\mathbf{G}})$ is also typical with respect to some distribution $P_{\tilde{G}}(\tilde{g})f_{S|\tilde{G}}(s|\tilde{g})$ where $f_{S|\tilde{G}}(s|\tilde{g})$ is conditionally Gaussian. Note that we can make no assumptions about the conditional variances defining $f_{S|\tilde{G}}$, because the adversary is assumed to know $G$ in its choice of $s$. By \eqref{eq:two_codebooks_c0}, with high probability $(\mathbf{x}(M),\mathbf{s},\tilde{\mathbf{G}})\in \mathcal{T}_{\epsilon'}^{(n)}(X,S,\tilde{G})$ where $X$ is independent of $(S,\tilde{G})$, and $\mathbb{E} X^2=1 ,\allowbreak \mathbb{E} S^2 \le \Lambda$.
Thus, by the conditional typicality lemma, with high probability $(\mathbf{x},\mathbf{s},\tilde{\mathbf{G}},\mathbf{V})\in \mathcal{T}_{\epsilon''}^{(n)}(X,S,\tilde{G},V)$ where $X,S,\tilde{G}$ are independent of $V$, and $\mathbb{E} V^2=\sigma^2$. Hence, using \ref{bynu123}, we have $\left(\mathbf{x},\mathbf{s},\tilde{\mathbf{G}},\mathbf{V} +\mathbf{x} \circ (\mathbf{G} - \tilde{\mathbf{G}}) \circ \sqrt{\varphi(\tilde{\mathbf{G}})}\right)\in \mathcal{T}_{\epsilon}^{(n)}(X,S,\tilde{G},V)$ where $\nu$ is sufficiently small compared to $\epsilon$. Also, since $\mathbf{Y}-\mathbf{x} \circ \tilde{\mathbf{G}} \circ \sqrt{\varphi(\tilde{\mathbf{G}})}-\mathbf{s} -\mathbf{V} = \mathbf{x} \circ (\mathbf{G} - \tilde{\mathbf{G}} )\circ \sqrt{\varphi(\tilde{\mathbf{G}})}$, by \eqref{bynu123} we may roughly assume $\mathbf{Y}=\mathbf{x} \circ \tilde{\mathbf{G}} \circ \sqrt{\varphi(\tilde{\mathbf{G}})}+\mathbf{s} +\mathbf{V}$ and obtain $\left(\mathbf{x},\mathbf{s},\tilde{\mathbf{G}},\mathbf{Y}\right)\in \mathcal{T}_{\epsilon}^{(n)}(X,S,\tilde{G},Y)$. Moreover, to completely show that with high probability $M\in\mathscr{S}$, we also need to compute the covariance matrix of $(X,Y)$ given $\tilde{G}=\tilde{g}$, where $Y=\tilde{G}\sqrt{\varphi(\tilde{G})} X+S+V$, and show that it is in the form of \eqref{sdefinition2}.
First, $\mathbb{E} \left(X^2|\tilde{G}=\tilde{g}\right) = \mathbb{E} X^2 = 1$ since $X$ is independent of $\tilde{G}$,
\begin{align}
\mathbb{E} \left(\!X\!\left(\!\tilde{G}\sqrt{\varphi(\tilde{G})}X\!+\!S\!+\!V\right)\Big|\tilde{G}\!=\!\tilde{g}\right) \!
& =\tilde{g}\sqrt{\varphi(\tilde{g})}\mathbb{E} X^2\! +\! \mathbb{E} \left(XS|\tilde{G}\! =\! \tilde{g}\right)\! +\! \mathbb{E} \left(XV|\tilde{G}\! =\! \tilde{g}\right)\! \\
& =\tilde{g}\sqrt{\varphi(\tilde{g})}\label{cov_22}
\end{align}
where $ \mathbb{E} \left(XS\Big|\tilde{G}=\tilde{g}\right)=0$ follows from the weak union rule since $X$ is independent of $(S,G)$.
Furthermore,
\begin{align}
\mathbb{E} &\left(\left(\tilde{G}\sqrt{\varphi(\tilde{G})}X+S+V\right)^2\bigg|\tilde{G}=\tilde{g}\right)
= \mathbb{E} \left(\tilde{G}^2{\varphi(\tilde{G})}X^2\bigg|\tilde{G}=\tilde{g}\right)+\mathbb{E} \left(S^2\Big|\tilde{G}=\tilde{g}\right)\nonumber\\
&+\mathbb{E} \left(V^2\Big|\tilde{G}=\tilde{g}\right) +\!2\mathbb{E} \left(\tilde{G}\sqrt{\varphi(\tilde{G})}XS\bigg|\tilde{G}\!=\!\tilde{g}\right)\!+\!2\mathbb{E} \left(\tilde{G}X\sqrt{\varphi(\tilde{G})}V\bigg|\tilde{G}\!=\!\tilde{g}\right)\!+\! 2\mathbb{E} \left(SV\Big|\tilde{G}\!=\!\tilde{g}\right)\\
&= \tilde{g}^2{\varphi(\tilde{g})}\!+\!\mathbb{E} \left(S^2\Big|\tilde{G}\!=\!\tilde{g}\right)\!+\!\sigma^2\!+\!2\tilde{g}\sqrt{\varphi(\tilde{g})}\mathbb{E} \left(XS\Big|\tilde{G}\!=\!\tilde{g}\right)\!+\!2\tilde{g}\sqrt{\varphi(\tilde{g})}\mathbb{E} \left(XV\Big|\tilde{G}\!=\!\tilde{g}\right)\!\nonumber\\
&\quad{}+\! 2\mathbb{E} \left(SV\Big|\tilde{G}\!=\!\tilde{g}\right)\\
&= \tilde{g}^2{\varphi(\tilde{g})}+\mathbb{E} \left(S^2\Big|\tilde{G}=\tilde{g}\right)+\sigma^2\label{weekunion3} \\
&\geq \tilde{g}^2{\varphi(\tilde{g})}+\sigma^2\label{cov_23}
\end{align}
where \eqref{weekunion3} follows from the weak union rule for $X$ independent of $(S,\tilde{G})$ and $V$ independent of $(S,\tilde{G})$. Therefore, the conditional covariance matrix of $(X,Y)$ can be obtain from $\mathbb{E} X^2 = 1$, \eqref{cov_22} and \eqref{cov_23}, and is the same as \eqref{sdefinition2}. Now, since $(\mathbf{x}(\hat{M}),\tilde{\mathbf{g}}, \mathbf{y})\in \mathcal{T}_\epsilon^{(n)}(X,\tilde{G},Y)$ and the conditional covariance matrix of $(X(M),Y)$ satisfies \eqref{sdefinition2}, with high probability $M\in \mathscr{S}$, and $P_0$ vanishes as $n\to\infty$.
Using \eqref{bynu123} and triangle inequality, we may upper bound $P_1$ by the following:
\begin{multline}
P_1\leq \\
\mathbb{P}\left\{ \!\left\|\mathbf{x}(m)\!\circ\! \tilde{\mathbf{G}}\sqrt{\varphi(\tilde{\mathbf{G}})}\!+\!\mathbf{s}\!+\!\mathbf{V}\!-\!\mathbf{x}(\hat{m})\!\circ\! \tilde{\mathbf{G}}\sqrt{\varphi(\tilde{\mathbf{G}})}\right\|^2\!\!\!\le\! \|\mathbf{s}\!+\!\mathbf{V}\|^2 \!+\!2n \nu
\text{ for some }\! \hat{m}\!\in\!\mathscr{S}\setminus\! \{m\}\!\right\}.
\end{multline}
Define the shorthand $\vec{X}=(XX'S\tilde{G}V)$. Let $\mathcal{V}$ denote a finite $\epsilon$-dense subset in the set of all distributions of random vectors $\vec{X}$ that are determined by $P_{\tilde{G}}(\tilde{g})$ and a random vector $(XX'SV)$ distributed conditionally zero mean Gaussian given $\tilde{G}$ with bounded covariances at most $(1,1,\Lambda,\sigma^2)$. Note that because the distribution of $P_{\tilde{G}}(\tilde{g})$ is completely known, the overall distribution of $\vec{X}$ can be determined by the conditional covariance matrix of $(XX'SV)$ given $\tilde{G}=\tilde{g}$ for each of the finitely many $\tilde{g}$ realizations, so $\mathcal{V}$ only needs to cover a compact set. Now, we may upper bound $P_1$ by
\begin{equation}
\sum_{\vec{X}\in\mathcal{V}} \frac{1}{N} \sum_{m=1}^N \mathbb{E}_{\tilde{G}}\left[e_{\vec{X}}(m,\mathbf{s},\tilde{\mathbf{G}})\right]
\end{equation}
where
\begin{multline}
e_{\vec{X}}\left(m,\mathbf{s},\tilde{\mathbf{g}}\right)
= \mathbb{P}\bigg\{ \left(\mathbf{x}(m),\mathbf{x}(\hat{m}),\mathbf{s},\tilde{\mathbf{g}},\mathbf{V}\right)\in\mathcal{T}_\epsilon^{(n)}\left(\vec{X}\right),\\
\left\|\tilde{\mathbf{g}}\!\circ\! \sqrt{\varphi(\tilde{\mathbf{g}})}\!\circ\! \mathbf{x}(m)\!+\!\mathbf{s}\!+\!\mathbf{V}\!-\!\tilde{\mathbf{g}}\!\circ\! \sqrt{\varphi(\tilde{\mathbf{g}})}\!\circ\! \mathbf{x}(\hat{m})\right\|^2\!\le \!\|\mathbf{s}\!+\!\mathbf{V}\|^2 \!+\!2n \nu
\text{ for some }\ \hat{m}\in\mathscr{S}\setminus \{m\}\bigg\}.\label{nu_eq5}
\end{multline}
We will show that $\frac{1}{N} \sum_{m=1}^N e_{\vec{X}}(m,\mathbf{s},\tilde{\mathbf{g}})\to 0$ for all vectors $\tilde{\mathbf{g}}$ and all vectors $(XX'SV)$ which are Gaussian given $\tilde{G}$ (whether or not they are in $\mathcal{V}$).
Let $Z = \tilde{G}\sqrt{\varphi(\tilde{G})}X+S+V-\tilde{G}\sqrt{\varphi(\tilde{G})}X'$.
We may restrict ourselves to $\vec{X}$ where
\begin{gather}
I(X;X'S\tilde{G})< |R-I(X';S\tilde{G})|^++\delta(\epsilon),\label{eq:R1R25}\\
\tilde{G}\sim P_{\tilde{G}}(\tilde{g}),\label{signalpower5}\\
(X,X',S,V)\text{ are zero mean Gaussian given }\tilde{G},\label{defofS5}\\
X,(S,\tilde{G}),V\text{ are mutually independent},\label{XSV_indep5}\\
X', \tilde{G}\text{ are independent}, \label{X'indG5}\\
\mathbb{E} X^2=\mathbb{E} X'^2=1, \mathbb{E} S^2 \le \Lambda, \mathbb{E} V^2=\sigma^2,\label{codgenr5}\\
X',Z \text{ are independent given } \tilde{G},\label{XZ1_indep5}\\
\mathbb{E} \left[Z^2\Big|\tilde{G}\right]\ge \sigma^2,\label{Zles5}\\
\var( Z)\le \sigma^2+\Lambda+2\nu,\label{Zless5}.
\end{gather}
Note that using \eqref{eq:21Huz_c4}, we only need to consider the distributions that satisfies \eqref{eq:R1R25}. in addition, \eqref{signalpower5}--\eqref{defofS5} are obtained by the definition of $\mathscr{S}$, \eqref{XSV_indep5} holds since the codebook $X$, Gaussian noise $V$ and fading gains $\tilde{G}$ are generated independently, and the adversary signal $S$ may depend on $\tilde{G}$ but not the others, \eqref{X'indG5} follows from \eqref{s15}, \eqref{codgenr5} follows from the power constraints of the codebook, the adversary and the distribution of noise, \eqref{XZ1_indep5}-\eqref{Zles5} follows from \eqref{s35}-\eqref{s45}, and \eqref{Zless5} follows from \eqref{nu_eq5}. Let $\psi(\tilde{g})=\mathbb{E}\left[Z^2\Big|\tilde{G}=\tilde{g}\right]-\sigma^2$. Therefore, using \eqref{Zles5} we have $\psi(\tilde{g})\ge 0$, and by \eqref{Zless5} we get $\mathbb{E} \psi(\tilde{G})=\var(Z)-\sigma^2\le \Lambda+2\nu$.
Observe that if $I(XV;X'S|\tilde{G})=0$, then we would have
\begin{align}
0&= \mathbb{E} \left[X'Z|\tilde{G}\right]\label{XZindependent}\\
&= \mathbb{E} \left[X'\left(\tilde{G}\sqrt{\varphi(\tilde{G})}X+S+V-\tilde{G}\sqrt{\varphi(\tilde{G})}X'\right)\bigg|\tilde{G}\right]\\
&= \mathbb{E} \left[X'S\Big|\tilde{G}\right] -\mathbb{E} \left[\tilde{G}\sqrt{\varphi(\tilde{G})}X'^2\bigg|\tilde{G}\right]\label{mutual1}\\
&= \mathbb{E} \left[X'S\Big|\tilde{G}\right] -\tilde{G}\sqrt{\varphi(\tilde{G})}\label{XindepntG}
\end{align}
where \eqref{XZindependent} follows from \eqref{XZ1_indep5}, \eqref{mutual1} follows from the assumption $I(XV;X'S|\tilde{G})=0$ in which $X'$ is independent of $(X,V)$, and \eqref{XindepntG} holds since $X'$ is independent of $\tilde{G}$. Therefore, $\mathbb{E} \left[X'S\Big|\tilde{G}\right] = \tilde{G}\sqrt{\varphi(\tilde{G})}$ and the covariance matrix of $S,X'$ given $\tilde{G}$ is equal to
\begin{align}
\cov\left(S,X'\Big|\tilde{G}\right)=\begin{bmatrix}
\mathbb{E} \left[S^2\Big|\tilde{G}\right]& \tilde{G}\sqrt{\varphi(\tilde{G})} \\
\tilde{G}\sqrt{\varphi(\tilde{G})}& 1
\end{bmatrix}.
\end{align}
The determinant of $\cov\left(S,X'\Big|\tilde{G}\right)$ is $\mathbb{E}\left[S^2\Big|\tilde{G}\right] -\tilde{G}^2\varphi(\tilde{G})$ that should be non-negative since the covariance matrix must be positive semi-definite. Thus, its expectation is also non-negative:
\begin{align}\label{psd}
0 & \le \mathbb{E} S^2 - \mathbb{E} \tilde{G}^2\varphi(\tilde{G}).
\end{align}
However, since $\mathbb{E}S^2 \leq \Lambda$, \eqref{psd} contradicts the initial assumption on $\varphi$ in \eqref{eq:lambda_assumption4}. Thus, there exists $\eta>0$ such that
\begin{equation}\label{eq:eta_bound5}
\eta\le I(XV;X'S|\tilde{G})=I(XV;X'|S\tilde{G})
\end{equation}
where we have used the fact that $I(XV;S)=0$.
Probability $e_{\vec X}$ may be upper bounded by
\begin{align}
e_{\vec X}(m,\mathbf{s},\tilde{\mathbf{g}})&\leq\sum_{\hat{m}:(\mathbf{x}(m),\mathbf{x}(\hat{m}),\mathbf{s},\tilde{\mathbf{g}})\in\mathcal{T}_{\epsilon}^{(n)}(X,X',S,\tilde{G})}\hspace{-2em}\mathbb{P}\left\{\!(\mathbf{x}(m),\mathbf{x}(\hat{m}),\mathbf{s},\tilde{\mathbf{g}},\mathbf{V})\!\in\! \mathcal{T}_{\epsilon}^{(n)} (X,X',S,\tilde{G},V)\!\right\}
\\&\leq \exp\big\{n\big[|R\!-\! I(X';XS\tilde{G})|^+\!\!-I(V;X'|XS\tilde{G})\!+\!\delta(\epsilon)\big]\label{e(s,i)5}
\end{align}
where \eqref{e(s,i)5} follows from \eqref{eq:22Huz_c5} and the joint typicality lemma.
We consider the following two cases.
Case (a): $R<I(X';S\tilde{G})$. Applying this condition to \eqref{eq:R1R25}, we get
\begin{align}
\delta(\epsilon)& >I(X;X'S\tilde{G}) \\
& = I(X;X'|S\tilde{G}).\label{IXX'2}
\end{align}
Since $I(X';S\tilde{G})\le I(X';XS\tilde{G})$ then $R-I(X';XS\tilde{G})< 0$. Considering \eqref{e(s,i)5}, for any $m, \mathbf{s},\tilde{\mathbf{g}}$ we have
\begin{align}
e_{\vec X}(m,\mathbf{s},\tilde{\mathbf{g}})&\le \exp\left\{-n\left( I(V;X'|XS\tilde{G})-\delta(\epsilon)\right)\right\}\label{eq:two_codebook_ebound5}\\
&= \exp\{- n(I(XV;X'|S\tilde{G})-I(X;X'|S\tilde{G})-\delta(\epsilon))\}\\
&\le \exp\{-n(\eta-2\delta(\epsilon))\}\label{casearesult5}
\end{align}
where \eqref{casearesult5} follows from \eqref{eq:eta_bound5} and \eqref{IXX'2}. Therefore, $e_{\vec X}(m,\mathbf{s},\tilde{\mathbf{g}})$ vanishes exponentially fast if $\delta(\epsilon)$ is sufficiently small.
Case (b): $R\geq I(X';S\tilde{G})$. Then we may apply this condition to \eqref{eq:R1R25} as
\begin{align}
R &> I(X;X'S\tilde{G})+I(X';S\tilde{G})-\delta(\epsilon)\\
& \ge I(X;X'|S\tilde{G})+I(X';S\tilde{G})-\delta(\epsilon)\\
& =I(X';XS\tilde{G})-\delta(\epsilon).
\end{align}
Since $R-I(X';XS\tilde{G})+\delta(\epsilon)>0$, we may upper bound \eqref{e(s,i)5} by
\begin{align}
e_{\vec X}(m,\mathbf{s},\tilde{\mathbf{g}})
& \leq \exp \left(n\left[R\!-\! I(X';XS\tilde{G})\!-\! I(V;X'|XS\tilde{G})\!+\!2\delta(\epsilon)\right]\right)\label{formula1}\\
& = \!\exp (n[R-I(X';XS\tilde{G}V)+2\delta(\epsilon)])\\
& \le \!\exp (n[R-I(X';XSV|\tilde{G})+2\delta(\epsilon)])\label{result335}
\end{align}
where by \eqref{X'indG5}, we have $I(X';\tilde{G})=0$. In the following, we find a lower bound for the mutual information in \eqref{result335}.
\begin{align}
I\left(X';XSV\Big|\tilde{G}\right)
& \geq I\left(X';\tilde{G}\sqrt{\varphi(\tilde{G})}X\!+\!S\!\!+\!\!V\Big|\tilde{G}\right)\label{dataproces2}\\
& = I\left(X';Z+\tilde{G}\sqrt{\varphi(\tilde{G})}X'\Big|\tilde{G}\right)\\
& = h\left(Z+\tilde{G}\sqrt{\varphi(\tilde{G})}X'\Big|\tilde{G}\right)-h\left(Z+\tilde{G}\sqrt{\varphi(\tilde{G})}X'\Big|\tilde{G},X'\right)\\
& = \mathbb{E}_{\tilde{G}} \left[\frac{1}{2} \log 2\pi e\left(\tilde{G}^2{\varphi(\tilde{G})}+\mathbb{E} \left[Z^2\Big|\tilde{G}\right]\right)-\frac{1}{2} \log 2\pi e \mathbb{E} \left[Z^2\Big|\tilde{G}\right] \right] \\
& = \mathbb{E}_{\tilde{G}} \left[C\left(\frac{\tilde{G}^2\varphi(\tilde{G})}{\mathbb{E} \left[Z^2\Big|\tilde{G}\right]} \right) \right] \label{GaussCap5}\\
& = \mathbb{E}_{\tilde{G}} \left[C\left(\frac{\tilde{G}^2\varphi(\tilde{G})}{\psi(\tilde{G})+\sigma^2+2\nu}\right)\right]\label{psiDef}
\end{align}
where \eqref{dataproces2} follows from data processing inequality, \eqref{GaussCap5} follows from standard argument for the capacity of Gaussian channel, and \eqref{psiDef} follows from the definition of $\psi$. Therefore, by the assumptions about $R$ and $\Lambda$ in \eqref{eq:lambda_assumption4}--\eqref{eq:R_assumption4}, $R<I(X';XSV|\tilde{G})$, so by \eqref{result335} $e_{\vec X}(m,\mathbf{s},\tilde{\mathbf{g}})$ is exponentially vanishing if $\delta(\epsilon)$ and $\nu$ are sufficiently small.
It is worth mentioning that this achievability proof also works for the case where both the adversary and encoder know the channel gains causally, or both know the gains non-causally. Since in all three cases the knowledge of the encoder is not more than the knowledge of the adversary, the jammer is able to impersonate the legitimate transmitter, and thereby symmetrize the channel, depending on the power allocation.
\subsection{Achievability Proof (Gains Available Causally at Adversary and Non-causally at Encoder)}\label{subsecVIII-C}
In this case, both the encoder and the decoder know the channel gains non-causally meaning that they know the whole $\mathbf{g}$ string including $(g_1,g_2,\cdots,g_n)$. However, the adversary only knows the gains causally, so at time $i$ it only has access to $(g_1,g_2,\cdots,g_i)$. Therefore, both the encoder and the decoder have some extra common information $(g_{i+1},g_{i+2},\cdots,g_n)$ that the adversary does not know. In particular, the encoder and the decoder immediately know $g_n$ which the adversary knows only at time $n$. Hence, we can leverage this common knowledge between the encoder and the decoder as common randomness that is unknown to the jammer. Moreover, by the assumption that $G$ is a continuous random variable with positive variance, in fact just $G_n$ has infinite entropy, and thus can be considered a source of an infinite number of bits of common randomness. Therefore, we proceed to provide an achievability proof where the encoder and decoder are assumed to share an infinite source of common randomness. However, note that implementing this approach would require measuring $G_n$ to an arbitrarily level of precision, which is not practical. Even so, the random code reduction technique of, for example, \cite[Lemma 12.8]{CsiszarKorner}, can be used to show that only $O(\log n)$ bits of common randomness need to be extracted from $G_n$ (or perhaps $G_{n-k},\ldots, G_n$ for some $k$) in order to achieve the same rate.
A large amount of the achievability proof is identical to the Sec. \ref{subsecVIII-B}. The main difference is that the codebook is based on common randomness between encoder and decoder, so we denote the codebook as random variables $\mathbf{X}(m)$ which are independent from the jammer signal. As a consequence, the symmetrizability condition $\Lambda<\mathbb{E} \tilde{G}^2\varphi(\tilde{G})$ is not needed, and the proof is somewhat simpler. In particular, we fix a $\varphi(\cdot)$ satisfying \eqref{firstPowCond}, and a rate satisfying \eqref{eq:R_assumption4}, and we prove achievability as follows.
\emph{Codebook generation:}
Let $\mathbf{X}(m)$ be a Gaussian codebook with variance $1\!-\!\gamma$ satisfying \eqref{eq:two_codebooks_c0}. This random codebook is generated from the infinite source of common randomness, so it is unknown to the adversary.
\emph{Encoding:} Given message $m$ and gain sequence $\mathbf{g}$, the transmitter first computes $\tilde{g}$ from the quantization function, and then sends $\sqrt{\varphi(\tilde{\mathbf{g}})}\circ\mathbf{X}(m)$ (at time $i$ signal $\sqrt{\varphi(\tilde{g_i})}X_i(m)$ is sent) if $\|\sqrt{\varphi(\tilde{\mathbf{g}})}\circ\mathbf{X}(m)\|^2 \le n$; otherwise, it sends zero.
\emph{Decoding:} Given $\mathbf{y}$ and $\mathbf{g}$, let $\nu< \epsilon$ and let $\mathscr{S}$ be the set of messages $\hat{m}$ such that $(\mathbf{X}(\hat{m}),\tilde{\mathbf{g}}, \mathbf{y})\in \mathcal{T}_\epsilon^{(n)}(X',\tilde{G},Y)$ where $\tilde{G}$ is the quantized random variable from $G$ and $(X',Y)$ are conditionally Gaussian given $\tilde{G}=\tilde{g}$ with zero mean and covariance matrix $\Sigma_{\tilde{g}}$ as follows:
\begin{align}\label{sdefinition}
\Sigma_{\tilde{g}}=\cov\left(X',Y\Big|\tilde{G}=\tilde{g}\right)=\left[\begin{array}{cccc}1&\tilde{g}\sqrt{\varphi(\tilde{g})} \\\tilde{g}\sqrt{\varphi(\tilde{g})}&a_{\tilde{g}}\end{array}\right]
\end{align}
where $a_{\tilde{g}}\ge {\tilde{g}}^2 \varphi({\tilde{g}})+\sigma^2$. Note that the following can be shown from \eqref{sdefinition}:
\begin{gather}
X' \text{ is independent of } {\tilde{G}}, \label{s1}\\
\mathbb{E} X'^2 = 1,\label{s2}\\
Y-{\tilde{G}}\sqrt{\varphi({\tilde{G}})}X' \text{ is independent of } X' \text{ given } {\tilde{G}},\label{s3}\\
\var\left(Y-{\tilde{G}}\sqrt{\varphi({\tilde{G}})}X'\bigg|{\tilde{G}}\right) \ge \sigma^2.\label{s4}
\end{gather}
Now, we define the decoding function as
\begin{align}
\Theta(\mathbf{y},\tilde{\mathbf{g}}) &= \argmin_{\hat{m} \in \mathscr{S}}
\left\| \mathbf{y} - \tilde{\mathbf{g}} \circ \sqrt{\varphi(\tilde{\mathbf{g}})} \circ \mathbf{X}(\hat{m}) \right\|^2.
\end{align}
\emph{Analysis of the probability of error:} Assume the legitimate transmitter sends message $M$. Then, we can upper bound the probability of error by the summation of the following error probabilities:
\begin{align}
P_0&=\mathbb{P}\left\{M \notin\mathscr{S} \right\},\label{firstPe}\\
P_1&=\mathbb{P}\bigg\{\left\| \mathbf{Y} - \tilde{\mathbf{G}}\circ \sqrt{\varphi(\tilde{\mathbf{G}})}\circ \mathbf{X}(\hat{m}) \right\|^2
\leq \left\| \mathbf{s}+\mathbf{V} \right\|^2 \text{ for some } \hat{m}\in \mathscr{S}\setminus \{M\} \bigg\}.
\label{secondPe}
\end{align}
Using the same argument in Sec. \ref{subsecVIII-B}, we can prove with high probability that $P_0$ tends to zero as $n\to \infty$. Using \eqref{bynu123} and triangle inequality, we may upper bound $P_1$ by the following:
\begin{multline}
P_1\leq \\
\mathbb{P}\left\{ \!\left\|\mathbf{X}(m)\!\circ\! \tilde{\mathbf{G}}\sqrt{\varphi(\tilde{\mathbf{G}})}\!+\!\mathbf{s}\!+\!\mathbf{V}\!-\!\mathbf{X}(\hat{m})\!\circ\! \tilde{\mathbf{G}}\sqrt{\varphi(\tilde{\mathbf{G}})}\right\|^2\!\!\!\le\! \|\mathbf{s}\!+\!\mathbf{V}\|^2 \!+\!2n \nu
\text{ for some }\! \hat{m}\!\in\!\mathscr{S}\setminus\! \{m\}\!\!\right\}.
\end{multline}
Defining $\vec{X}=(XX'S\tilde{G}V)$ and $\mathcal{V}$ the same as Sec. \ref{subsecVIII-B}, we may now upper bound $P_1$ by
\begin{equation}
\sum_{\vec{X}\in\mathcal{V}} \frac{1}{N} \sum_{m=1}^N \mathbb{E}_{{\tilde{G}}}\left[e_{\vec{X}}(m,\mathbf{s},\tilde{\mathbf{G}})\right]
\end{equation}
where
\begin{multline}\label{eq:eX1_def}
e_{\vec{X}}(m,\mathbf{s},\tilde{\mathbf{g}})
=\!\bigg\{\! (\mathbf{X}(m),\mathbf{X}(\hat{m}),\mathbf{s},\tilde{\mathbf{g}},\mathbf{V})\!\in\!\mathcal{T}_\epsilon^{(n)}(\vec{X}),\\
\quad{} \left\|\tilde{\mathbf{g}}\!\circ\! \sqrt{\varphi(\tilde{\mathbf{g}})}\!\circ\! \mathbf{X}(m)\!+\!\mathbf{s}\!+\!\mathbf{V}\!-\!\tilde{\mathbf{g}}\!\circ\! \sqrt{\varphi(\tilde{\mathbf{g}})}\!\circ\! \mathbf{X}(\hat{m})\right\|^2\!\le\! \|\mathbf{s}+\!\mathbf{V}\|^2 + 2n \nu\text{ for some } \hat{m}\!\in\! \mathscr{S}\setminus \{M\}\!\bigg\},
\end{multline}
and $\vec{X}$ satisfies the same properties as in \eqref{eq:R1R25}--\eqref{Zless5}.
Now, it suffices to show that $\frac{1}{N} \sum_{m=1}^N e_{\vec{X}}(m,\mathbf{s},\tilde{\mathbf{g}})$ vanishes for all typical vectors $\mathbf{g}$ and all vectors $(XX'SV)$ which are Gaussian given ${\tilde{G}}$ (whether or not they are in $\mathcal{V}$).
Using the joint typicality lemma in \cite[Remark 2.2]{ElGamal} we may upper bound $ e_{\vec X}$ in \eqref{nu_eq5} (with the codewords $\mathbf{X}(m)$ and $\mathbf{X}(\hat{m})$) as follows:
\begin{align}
e_{\vec X}(m,\mathbf{s},\tilde{\mathbf{g}})
&\leq\sum_{\hat{m}\in \mathscr{S}\setminus \{m\}} \mathbb{P}\left\{\!(\mathbf{X}(m),\mathbf{X}(\hat{m}),\mathbf{s},\tilde{\mathbf{g}},\mathbf{V})\!\in\! \mathcal{T}_{\epsilon}^{(n)} (X,X',S,{\tilde{G}},V)\!\right\}
\\& \le \exp \{n(R-I(X';XSV{\tilde{G}})+\epsilon)\}\label{eq:174}\\
& = \!\exp \{n(R-I(X';XSV|{\tilde{G}})+\epsilon)\}\label{result333}
\end{align}
where in \eqref{eq:174} we have used the fact that $\mathbf{X}(\hat{m})$ is independent of $(\mathbf{X}(m),\mathbf{s},\tilde{\mathbf{g}},\mathbf{V})$, and \eqref{result333} follows from \eqref{s1}.
We now lower bound the mutual information in \eqref{result333} by the following.
\begin{align}
I(X';XSV|{\tilde{G}})
& \geq I\left(X';{\tilde{G}}\sqrt{\varphi({\tilde{G}})}X+S+V\Big|{\tilde{G}}\right)\label{dataProcSS}\\
& = I\left(X';Z+{\tilde{G}}\sqrt{\varphi({\tilde{G}})}X'\Big|{\tilde{G}}\right)\\
& = h\left(Z+{\tilde{G}}\sqrt{\varphi({\tilde{G}})}X'\Big|{\tilde{G}}\right)-h\left(Z+{\tilde{G}}\sqrt{\varphi({\tilde{G}})}X'\Big|{\tilde{G}},X'\right)\\
& = \mathbb{E}_{\tilde{G}} \bigg[\frac{1}{2} \log 2\pi e\left({\tilde{G}}^2\varphi({\tilde{G}})\mathbb{E} \left[X'^2\Big|{\tilde{G}}\right]+\mathbb{E} \left[Z^2\Big|{\tilde{G}}\right]\right)-\frac{1}{2} \log 2\pi e \mathbb{E} \left[Z^2\Big|{\tilde{G}}\right] \bigg] \\
& = \mathbb{E}_{\tilde{G}} \left[C\left(\frac{{\tilde{G}}^2\varphi({\tilde{G}})}{\mathbb{E} \left[Z^2\Big|{\tilde{G}}\right]} \right) \right] \label{GaussCap}\\
& = \mathbb{E}_{\tilde{G}} \left[C\left(\frac{{\tilde{G}}^2\varphi({\tilde{G}})}{\psi({\tilde{G}})+\sigma^2+2\nu}\right)\right]\label{psiDef6}
\end{align}
where \eqref{dataProcSS} follows from data processing inequality, \eqref{GaussCap} follows from standard argument for the capacity of Gaussian channel, and \eqref{psiDef6} follows from the definition of $\psi$. Therefore, by the assumptions about $R$ and $\Lambda$ in \eqref{firstPowCond} and \eqref{eq:R_assumption4}, $R<I(X';XSV|{\tilde{G}})$, so by \eqref{result333} $e_{\vec X}(m,\mathbf{s},\tilde{\mathbf{g}})$ is exponentially vanishing if $\delta(\epsilon)$ and $\nu$ are sufficiently small.
\section{Conclusion}\label{9}
This paper studied two phenomena together which are usually studied separately: namely, active adversaries and fading. We derived the capacity of Gaussian arbitrarily-varying fading channels where the adversary knows the transmitter's code but not the exact transmitted signal. The adversary affects the capacity by increasing the noise variance by an amount related the adversary's power. The capacity also depends on whether the transmitter and/or the adversary know the fading gains or not. The transmitter uses its knowledge to maximize the channel capacity while the adversary tries to minimize the capacity by its knowledge of the channel gains. Furthermore, if the adversary's knowledge is at least that of the transmitter's knowledge, then the adversary is able to make the capacity zero with enough power. In this paper, we have focused on the scenario where fading applies to the transmitter-to-receiver path, but not the adversary-to-receiver path. Future work could including considering fading along both paths. Such an alternatively model would present somewhat different challenges. Alternative directions include considering fading and adversaries in network settings, or an adversary with some direct control of the fading gains.
\section*{Acknowledgment}
This material is based upon work supported by the National Science Foundation under Grant No. CCF-1453718.
| {
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William Addison Ireland (1880 – May 29, 1935), a native of Chillicothe, Ohio, was a self-taught cartoonist well known throughout Ohio. The Billy Ireland Cartoon Library & Museum was named in his honor in 2009.
Career
Shortly after his 1898 high school graduation, Ireland was hired by The Columbus Dispatch in Columbus, Ohio. Ireland worked his entire life for the Dispatch, drawing four to seven editorial cartoons each week in addition to his weekly feature, The Passing Show.
Ireland was best known for The Passing Show, which debuted on February 9, 1908, with its title inspired by George Lederer's The Passing Show (1894), the first successful American revue-format entertainment. In Ireland's full-page color Sunday strip, he commented on everything from local politics and visiting celebrities to the trials and tribulations of the Ohio State University football team. For the September 30, 1923 Passing Show page, Ireland created a character inspired by Ohio State's 1902 school song, "Carmen Ohio". The Passing Show came to an end on June 2, 1935, the Sunday following his death on May 29.
Legacy
Comic strip historian Allan Holtz commented:
The feature was a much-beloved fixture of the Sunday Dispatch, both for its graphic inventiveness (the mastheads alone are worth the price of admission) and all the local color. Ireland seemingly knew everyone and everything in Columbus, and he lovingly lampooned it all each Sunday. The pages were always jam-packed... filled to the brim with local happenings, oddball news and personal anecdotes... The creator took a vacation every summer, during which substitutes would be called upon to keep the Show rolling, as it were... Billy Ireland was noted for his kindnesses to aspiring cartoonists. He was Dudley Fisher's mentor in the 1910s, and later gave Milton Caniff his first pro cartooning job at the Dispatch. Noel Sickles also acknowledges a great debt to Ireland for giving him early encouragement. After Ireland's death The Passing Show was continued by Harry Keys, who renamed it We Folks in 1938. Others who took a crack at the Ireland legacy page were Bob Vittur and Myron Dixon. As We Folks, the feature ran until at least 1944.
In 2003, his work was exhibited at Ohio State University's Cartoon Library & Museum, since renamed the Billy Ireland Cartoon Library & Museum. Another exhibition of Ireland's work was mounted in 2010, as announced by the Museum's founding curator, Lucy Shelton Caswell:
Billy Ireland was a Columbus celebrity during his lifetime. He enjoyed a national reputation, and his work is still delightful to read. This is a fitting honor for a great cartoonist. We look forward to sharing his work with a new generation of readers... Few Ohioans have celebrated their affection for their home state as consistently and creatively as cartoonist Billy Ireland. William Addison Ireland was a child of rural Ohio. He remembered this geography in his art, reflected this point of view in his editorial cartoons, and refused to abandon his Ohio roots for the increased money and fame that might have been his had he worked in New York or Chicago.
The Elizabeth Ireland Graves Foundation is managed by the cartoonist's granddaughter, Sayre Graves. In September 2009, it was announced that the Ohio State University Board of Trustees approved a new name, Billy Ireland Cartoon Library & Museum, in recognition of a $7 million gift from the Elizabeth Ireland Graves Foundation to support the renovation of the University's Sullivant Hall. The $20.6 million project was completed in 2013, and the Sullivant Hall houses both the Billy Ireland Cartoon Library and Museum and Ohio State's Department of Dance.
While at the Dispatch, Ireland mentored notable cartoonists Dudley Fisher, Milton Caniff and Ray Evans.
Books
Some of his work is collected in Lucy Caswell's book, Billy Ireland, first published by the Museum in 1980 and expanded in a 2008 edition. Promoting the book, Caswell gave a January 24, 2008 lecture which was reviewed by J. Caleb Mozzocco:
Each Passing Show had a unique title strip across the top, with the words The Passing Show, "by" and a shamrock representing Ireland, all arranged into little scenes, like the letters all playing baseball or football, or forming a bridge, or baby birds in a nest, or captured German soldiers or whatever. Below that would be a dozen or so little mini-features or cartoons. There were more or less regular features within the page, like one-panel strip The Jedge and Jerry and caricatures highlighting local people and their interests and accomplishments, but the bulk of the page were standalone text and cartoon pieces dealing with nature, corn on the cob, OSU football, city politics, fashion and whatever the hell Ireland felt like drawing that week. The dozen or two little pieces didn't really interact with one another, and Caswell said the page was designed to be read over the course of the day, with some people returning to it throughout their reading experience, while others sat there and read the whole thing. Appearing fairly regularly was Ireland himself. Not just as the shamrock-headed caricature in the title panel, but also as a little, fat white-haired guy in a janitor's outfit; the page was his page, and he saw himself as in charge of its maintenance.
See also
Noel Sickles, another comic strip artist from Chillicothe, Ohio
References
Listen to
"Carmen Ohio"
External links
The Ohio State University Billy Ireland Cartoon Library & Museum: "Ireland of the Dispatch" digital exhibit
The Ohio State University Billy Ireland Cartoon Library & Museum: Ohio Cartoonists: Billy Ireland
1880 births
1935 deaths
American comic strip cartoonists
American editorial cartoonists
People from Chillicothe, Ohio | {
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{"url":"https:\/\/zbmath.org\/?q=an%3A1146.39012","text":"## Dynamical properties for a class of fourth-order nonlinear difference equations.(English)Zbl\u00a01146.39012\n\nSummary: We consider the dynamical properties for a kind of fourth-order rational difference equations. The key is for us to find that the successive lengths of positive and negative semicycles for nontrivial solutions of this equation periodically occur with same prime period 5. Although the period is same, the order for the successive lengths of positive and negative semicycles is completely different. The rule is $$\\dots,3^+,2^-,3^+,2^-,3^+,2^-,3^+,2^-,\\dots,$$ or $$\\dots,2^+,1^-,1^+,1^-,2^+,1^-,1^+,1^-,\\dots,$$ or $$\\dots,1^+,4^-,1^+,4^-,1^+,4^-,1^+,4^-,\\dots$$. By the use of the rule, the positive equilibrium point of this equation is proved to be globally asymptotically stable.\n\n### MSC:\n\n 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations\nFull Text:\n\n### References:\n\n [1] Kulenovi\u0107 MRS, Ladas G: Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures. Chapman & Hall\/CRC, Boca Raton, Fla, USA; 2002:xii+218. \u00b7 Zbl\u00a00981.39011 [2] Agarwal RP: Difference Equations and Inequalities. Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 155. Marcel Dekker, New York, NY, USA; 1992:xiv+777. [3] Koci\u0107 VL, Ladas G: Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and Its Applications. Volume 256. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xii+228. [4] Amleh, AM; Grove, EA; Ladas, G; Georgiou, DA, On the recursive sequence [inlineequation not available: see fulltext.], Journal of Mathematical Analysis and Applications, 233, 790-798, (1999) \u00b7 Zbl\u00a00962.39004 [5] Xianyi, L; Deming, Z, Global asymptotic stability in a rational equation, Journal of Difference Equations and Applications, 9, 833-839, (2003) \u00b7 Zbl\u00a01055.39014 [6] Li, X; Zhu, D, Global asymptotic stability of a nonlinear recursive sequence, Applied Mathematics Letters, 17, 833-838, (2004) \u00b7 Zbl\u00a01068.39014 [7] Li, X, Qualitative properties for a fourth-order rational difference equation, Journal of Mathematical Analysis and Applications, 311, 103-111, (2005) \u00b7 Zbl\u00a01082.39004 [8] Xi, H; Sun, T, Global behavior of a higher-order rational difference equation, No. 2006, 7, (2006) \u00b7 Zbl\u00a01140.39312 [9] Rhouma, MB; El-Sayed, MA; Khalifa, AK, On a [inlineequation not available: see fulltext.]-rational recursive sequence, Advances in Difference Equations, 2005, 319-332, (2005) \u00b7 Zbl\u00a01095.39003\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2022-11-30 20:14:02","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\": 0.5662646889686584, \"perplexity\": 2238.694148300061}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710771.39\/warc\/CC-MAIN-20221130192708-20221130222708-00227.warc.gz\"}"} | null | null |
CALGARY, Alberta and SAN DIEGO, Aug. 03, 2018 (GLOBE NEWSWIRE) -- Oncolytics Biotech® Inc. (Nasdaq: ONCY) (TSX: ONC), currently developing REOLYSIN® (pelareorep), an intravenously delivered immuno-oncolytic virus turning cold tumors hot, today announced financial results and operational highlights for the quarter ended June 30, 2018. All dollar amounts are Canadian unless otherwise noted.
Reached agreement with the U.S. Food and Drug Administration (FDA) under a Special Protocol Assessment (SPA) for the protocol design, clinical endpoints and statistical analysis approach for the company's phase 3 study evaluating pelareorep for the treatment of metastatic breast cancer.
Investigating pelareorep in combination with Keytruda to treat second line pancreatic cancer patients. The study, run by Dr. Devalingham Mahalingam, will plan to enroll approximately 40 patients with advanced pancreatic cancer and will be conducted at the Robert H. Lurie Comprehensive Cancer Center of Northwestern University.
Investigating pelareorep in combination with Keytruda, Velcade® and dexamethasone to treat multiple myeloma patients. The study, facilitated by Dr. Kevin Kelly, Associate Professor of Clinical Medicine, will be conducted at the USC Norris Comprehensive Cancer Center.
Presented poster highlights from pelareorep studies at the American Society of Clinical Oncology (ASCO) 2018 Annual Meeting. The presentation demonstrated that pelareorep promotes the expression of gene signatures predictive of a response to immunotherapy in breast cancer and hepatocellular carcinoma and that the tumor inflammation promoting effects in breast cancer models provide a compelling explanation for the significant overall survival benefit in hormone receptor positive metastatic breast cancer patients in the phase 2, IND 213, study.
Presented posters highlighting data from pelareorep studies at the American Association for Cancer Research (AACR) Annual Meeting 2018. The presentations showed preclinical models demonstrating pelareorep increased PD-L1 expression in microsatellite stable (MSS) colorectal cancer cells (CRC) and demonstrated efficacy for pelareorep and anti-PD1 agent combination.
Presented positive pelareorep data in combination with Keytruda and anti-CD73 at the International Oncolytic Virus Conference 2018. The poster highlighted the effectiveness of pelareorep in combination with Keytruda and/or an anti-CD73 immunotherapy in prostate cancer cell lines.
Announced a share consolidation on the basis of 1 new common share for every 9.5 outstanding common shares.
Announced the listing of the company's shares of common stock on the Nasdaq Capital Market and commenced trading on June 1, 2018, under the symbol "ONCY".
Closed an underwritten public share offering of 1,532,278 common shares at a purchase price of USD $5.83 for gross proceeds of approximately USD $8.9 million.
Expanded the clinical development team in San Diego, including Senior Medical Personnel.
Initiate a phase 2 window of opportunity study of pelareorep in combination with a checkpoint inhibitor and/or the standard of care in the neoadjuvant breast cancer setting in 2H 2018.
Initiate a phase 2 study in combination with Merck's Keytruda in multiple myeloma in 2H 2018.
Initiate a phase 2 study in combination with Merck's Keytruda in advanced pancreatic cancer in 2H 2018.
Data from window of opportunity study in mBC in 1H 2019.
Data from Keytruda combination study in multiple myeloma in 2H 2019.
Preliminary data from Keytruda combination study in advanced pancreatic cancer in 1H 2020.
At June 30, 2018, the company reported $18.7 million in cash and cash equivalents.
As at August 2, 2018, the company had an unlimited number of authorized common shares with 16,531,956 common shares issued and outstanding, 16,443,500 warrants exercisable into 1,730,894 common shares with a $9.025 strike price and 1,153,080 options and share units.
To view the Company's Fiscal 2018 Second Quarter Consolidated Financial Statements, related Notes to the Consolidated Financial Statements, and Management's Discussion and Analysis, please see the Company's filings, which will be available at www.sedar.com, www.sec.gov and on Oncolytics' website at http://www.oncolyticsbiotech.com/investor-centre/financials/.
Oncolytics is a biotechnology company developing REOLYSIN®, also known as pelareorep, an intravenously delivered immuno-oncolytic virus. The compound induces selective tumor lysis and promotes an inflamed tumor phenotype -- turning "cold" tumors "hot" -- through innate and adaptive immune responses to treat a variety of cancers. Oncolytics' clinical development program emphasizes three pillars: chemotherapy combinations to trigger selective tumor lysis and immuno-therapy and immune modulator (IMiD) combinations to produce innate and adaptive immune responses. Oncolytics is currently conducting and planning additional studies in combination with checkpoint inhibitors and targeted and IMiD therapies in solid and hematological malignancies, as it prepares for a phase 3 registration study in metastatic breast cancer. For further information, please visit: www.oncolyticsbiotech.com. | {
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So where are the Democrat leaders on this virulent anti Semitism emanating from their party? Why were there boos from their side of the aisle during the State Of The Union speech when Pres. Trump mentioned moving the embassy to Jerusalem? Where are Pelosi and Schumer to call out this vile behavior? And if they're silent on this, why the outrage at a governor who dressed up 35 years ago in black face? Double standard??
Hey, who left all these post-it notes of support outside Rep. Ilhan Omar's office?
Posted at 5:51 pm on February 11, 2019 by Doug P.
Democrat Rep. Ilhan Omar has apologized-but-not-really after Nancy Pelosi and other Dems issued a strongly-worded but toothless statement about her most recent anti-Semitic remarks.
Democrat Jews are starting to see the light. As for the ones who aren't, keep an eye on who they are for the future.
It's all about the Benjamins baby. After this and the women's March, can the liberal amothers admit there is an antisemitic problem in the democratic party?
The leadership put Ilhan Omar on the education and foreign affairs committee. It shows you how much they value Jews.
Don't forget Kate Perry's blackface shoes.
I wouldn't worry too much about white genocide.
I would worry when disgusting and offensive clothing like that becomes mainstream. Fortunately, people object to that ... and to antisemitic clothing.
Unless, of course, that doesn't bother you either.
Big flipping deal. This statement came after the Jews in Congress condemned her. This isn't a statement originating from them. They waited instead of reacting right away.
Democratic leaders are in bed with antisemites. Despite her prior antisemetic statements, they put her on the Foreign Affairs Committee. Do you think Israel can get a fair shake from her?
DEMOCRATS HAVEN'T CONDEMNED ILHAN OMAR!
Sure they have. Here you go.
UMMM. NOW I'M GOING TO COME UP WITH ANOTHER EXCUSE TO SAY BAD THINGS ABOUT DEMOCRATS.
Rep. Omar seems to be apologizing now for the "all about the Benjamins" tweet after being condemned by the House leadership.
While I tend to believe there should be a statute of limitations on stupidity, Governor Northam's ever-changing response troubles me. First, this was not high school; he was 25 years old at the time. Second, while 1984 may seem like a long time ago to many of you, I can assure you that people were quite well aware in 1984 that blackface and KKK robes were too hot to handle. In fact, I recall while I was in college around the same time, the Panhellenic Association, the governing body of the fraternities and sororities, gave annual sensitivity training to officers of the member organizations.
However, these are mere quibbles. I'd be happy for Rep. Omar to peddle her anti-Semitic worldview and for Gov. Northam to dress up however he wants were it not for the complicity of the legacy media.
Rep. Omar's beliefs about Jews are no secret. Laura Loomer did plenty of reporting on this before the 2018 elections, and she was roundly ignored on both sides of the aisle. Everyone -- Republicans and Democrats alike -- were so eager not to be accused of Islamaphobia that they literally ignored the overwhelming evidence.
Gov. Northam's misdeeds only got attention when his remarks about infanticide disturbed not only pro-life supporters but many pro-choice advocates as well. Suddenly, the media switched the focus almost exclusively to the yearbook photo.
Just keep in mind that the "independent press" would never have uncovered or reported any of these stories on their own. Were it not for social media and independent journalists and activists, we would never know.
The Jews in Europe also didn't worry too much about Jewish genocide and look where that led..
Anti-semitic clothing is different than regular clothing deemed to be "blackface". Ski masks are common and not considered blackface so the sweater with half a ski mask shouldn't offend anyone as it is clearly not blackface. And the shoes are adorable, I don't understand the blackface part of it at all. That is especially since there have been complaints that some things arent made in the black version leaving black people left out of things white people enjoy. For example, bandages aren't black and dolls werent black etc. So now that black versions are included and being sold they complain about that as well. If they wouldn't have been included then there would be complaints about that as well. White people are walking on eggshells. Damned if they do and damned if they don't. How far is it going to go before there is white genocide?
They appointed her to several prestigious committees. "It's all about the Benjamins, baby" is not a change in sentiment for her. It's her being her.
There is not one Democrat on this site who admitted that they were wrong to participate in the antisemitic lead Womens March.
I don't need to say bad things about democrats. They are doing it themselves. Their hypocrisy and prejudice is blatantly obvious even with the media on their side.
If you don't see a problem with that Gucci sweater, I'd suggest that you do some reading up on blackface.
I also suggest that you take your head out of the sand and learn about white privilege in this country. Simply put, there is no risk of "white genocide." In the meantime, however, black parents still have to talk to their children about the risks of walking or driving while black.
Ok, here we go again. All the women's marches were on shabbat. No one on this site participated. Ergo they can't admit they're wrong. "When did you stop beating your wife, Mr Smith?"
I still haven't seen you admit that you're wrong about the hats. That they are indeed cats, and not genitalia.
He never made remarks about infanticide. That's just a Republican fabrication.
And while he should be called out for blackface, its the height of hypocrisy for Republicans -- who have repeatedly re-elected Steven King (and put up with his racism in Congress for 15 years without so much as a peep), and who elected a Republican who traded on the blatantly racist "birther" lied about President Obama, not to mention his comments about Judge Curiel, about Latin Americans, about Muslims, ad nauseum -- to be talking about it.
And I'm looking for equal condemnation of Thomas K. Norment Jr. And Tate Reeves. And others.
But of course you're changing the subject, since the Democrats have condemned Omar.
As you're not white you don't have to worry.
Orange Amother, you're out of line.
First of all, you are in violation of Imamother TOS when you make inflammatory accusations under the guise of Amother.
Second of all, my post contained absolutely nothing regarding Republican versus Democratic divisions. You are the one who brought that up. I regularly condemn Republicans, and you just are regularly ignore it because it doesn't fit with the image you've created of a fire-breathing, troglodyte conservative. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,234 |
My sister is going to be moving in and living with me.
The last time I lived in the same house as my sis was when I was 16 and she was 18. I recall desperately hoping that she would pass her A levels so that she could get into university and leave home. As teenage siblings are wont to do (or not, I don't know about other families), we were arguing every single day and I just wanted her gone.
She duly passed her exams, left home (never to return) and I found myself missing her terribly! Since then, we've always got on and been close as adults, but part of the reason might be that even when we were living in the same country, she was down in London so we weren't in each other's faces and only saw each other for relatively short periods of time.
Of course, we are no longer hormonal teenagers (although verging on hormonal middle-aged, which could be another issue!), we're reasonable adults but I've been living a carefree, quiet, (mostly) singleton life, with no responsibilities (except for myself) and all that will be changing.
Speaking of hormonal teenagers, my sis will not be moving in alone as she will have my tweenager nephew in tow.
In one sense, I'm very much looking forward to spending a lot of time with them both (in my mind, I've already planned days out, board game nights etc), as I love them both dearly and it'll be good to finally have close members of the family living in the same country again.
I just don't know how this might affect my journey to FIRE (positively or negatively).
Or my blogging – my family don't know about this blog!
How will I be able to keep it secret from my eagle-eyed sis or inquisitive nephew!?
Long-standing readers of my blog may recall that I live in the family home. When I parted ways with my ex, we sold our house – I used my 50% to purchase a small BTL flat and moved into the family home which had been empty for the best part of ten years, my parents having returned to their country of birth when they retired early.
I pay my folks rent, which is below the market rate as I am also responsible for the upkeep and maintenance of their house.
The good news is that with my sis moving in, household bills will be shared. The bad news is that household bills will go up with more people living in the house. I can already picture myself running round the house turning off taps, timing their showers and switching off lights….
How long will they be living with me?
I have been really selfish thus far with this post because I'm only really seeing how this will affect me and my plans for FIRE, without really considering the huuuuge life change for my sister and my nephew.
I must consider that although my life will veer from its comfortable little norm, it will be nothing compared to the massive upheaval they will be experiencing.
It's highly likely that she's not exactly relishing having to live with me – my sis is an extremely organised person and although I'm organised at work, I'll admit to being probably at the other end of the spectrum at home!
Also, she has suffered from mental health problems in the past, so I need to think of her and be supportive during this stressful period of her life.
My nephew is chatty and well behaved. I have a great relationship with him, so I need to do my utmost to ensure that he settles in and remains a happy boy.
So there we have it. I've got a lot of work to do around the house, decluttering (sis has already been dropping big hints about Marie Kondo) and making space for when they move in.
I have until the summer to do this, which sounds like plenty of time but isn't, knowing me and my procrastination!
How would you cope with living with a sibling again?
This entry was posted in Family, Well-being by weenie. Bookmark the permalink.
I see this as an excellent opportunity to pass on the principles of fi to the next generation!
In that sense, yes, it is an opportunity to shape a young mind regarding finances!
My own space will be my bedroom I guess, haha!
Oh wow! Oh wow, oh wow.
Yup that definitely counts as a huge life change. Don't feel about about being selfish on this blog. We're all on your side!
I have a sister that's almost two years older than me as well. Like you guys we get on better as adults than kids but we aren't that close. We stay with each other 3-4 times a year for a few days at a time and that's probably as much as we can cope with.
Equally huge to be living with a child as that's another dynamic again.
Whatever happens I think you're going to grow a lot over the next couple of years. One huge positive is that I think that you're likely to build an even stronger relationship with your nephew. As he gets older that's going to be invaluable.
I agree, I will grow a lot over the next couple of years, hopefully for the better and yes, I can see a stronger relationship building with my nephew.
Woah – didn't see that one coming weenie!
I have a sister 2 years older than me, and funnily enough we live very close to each other (her back garden backs on to mine!) which was just totally random really, but we probably don't see each other all that much more than when we lived right across town from each other. We get on well and always have done, even as kids and teens, but I don't think I could hack living with her again haha. Although if push came to shove and it was needed for any reason of course I would do it… that's what family is for right!?
It will be a big change but as usual you are already looking at the positive side of the situation which is why it will all work out for the 3 of you I'm sure.
I didn't see it coming either! My sis mentioned something in passing a year ago and I dismissed/forgot about it. However, she confirmed her plan during my recent trip to HK and I was like, bloody hell, it's really going to happen!
You're right – family's there for when you need them and I have to say I'm in a better position than many others, who are estranged from their familes.
The change is both scary though exciting – it will take getting used to.
I'm not sure he'd be doing anything as 'old-fashioned' as a blog – it's all about vlogs, until the next thing!
Oohh big changes indeed. I guess you will have a few months to get used to the idea but it may take a little time to make adjustments to your routines etc when they move in. Look upon it as an opportunity for personal development!
Seriously, good luck when the time comes.
Yes, as @Caveman said, I will grow and there will undoubtedly be some personal development, which will do me some good, I'm sure.
Providing I have my own quiet space, I quite like a full house. Before getting married and even for a few years after, I lived in a shared house and enjoyed meeting and doing things with people from a variety of different backgrounds. The one thing I cannot stand, however, is loud music. But perhaps I'm being nostalgic. My missus is a late riser and it means I almost never have to share any facilities with anyone in the morning.
Overall, providing your sister is contributing to the bills it should be cheaper – especially if you can share cooking, don't mind doing stuff like watch TV etc together. Council tax should be a nice saving too.
I grew up in a full house and loved living in a shared house as a student. But I've gone a long time living in the house on my own (apart from odd members of the family staying for short periods). I like my music loud so might have to use anti-social headphones!
All bills will be shared so yes, savings there. She's a better cook so has already mentioned 'taking over the kitchen' (and probably rearrange it again).
Wow quite a change there Weenie – good luck! As FBA says, a good opportunity to pass on sensible habits to the next generation!
I dread to think if we were complete opposites, it definitely wouldn't work. We're different but alike enough in some ways I guess.
Thanks – I'll see how I get on with keeping the blog quiet and will surely say on here if I've been rumbled!
WOW such an exciting time! I guess it will take some time to get used to the extra people and noise in the house.
It is exciting but that's not the only emotion I'm feeling right now! It will take a long time I think – when I have family staying with me, although it's lovely to have them around, I often can't wait for them to leave me in peace. Hmm…that doesn't bode well, haha!
Lovely news! I have a sister three years younger than me and like you we've become much closer as adults. She's got a new flat with a spare bedroom so I've been staying with her and her partner a lot this past month. Not sure I could live with them – the flat typically looks like a bomb site – and they are terrible with food and energy waste- but then I could see it as a challenge to spread minimalism and sustainability.
I stay with my sis all the time when I go over to HK which is fine, but like you say, it's very different actually living with each other long term.
The good news is I think my sis can be quite frugal when she wants to be, certainly she'll be watching her pennies when she first moves over, unless she can secure a job transfer.
Wow! That must be very exciting after so many years of living separately. It Is a big change and it probably needs time for you and them to readapt, so I would take it easy.
In my case, I don't have any siblings but always wanted to. Unfortunately my parents couldn't afford it… I ve always wished I had at least one to support each other during difficult times. Luckily I've always had good friend, but I don't think it carries the same deep feelings as a brother or sister would.
Good luck accommodating the house!
Tony recently posted…Paying yourself first – how long it takes to become a millionaire?
No, friends are not family so it's a very different kind of feeling, closeness etc.
I think there will be good and bad points about living together – I'm a pretty calm person so let's home I continue to be so, haha!
Agree, I may not know how long they will be with me so will try to make the most of it.
I think that people who are working towards FI are those who have decided to take some control of their lives – certainly that's me – and when you know that you are going to lose some of that control it can be a bit scary. As you say, will they continue to follow your frugal habits – maybe not, but sharing the bills has got to be a good thing. Even if they go up a bit, dividing them between the two of you has got to mean less money for you to fork out overall. My mum always taught us good spending habits and I am sure, as others have said, this will be a great opportunity to pass your wisdom on to your nephew, even if it is just by being a good role model. He may not necessarily adopt these things now, but I am sure he will be able to put it to good use in the future and look back and realise how wise his auntie was.
Think you might have hit the nail on the head. I've been in control of my life, certainly since I've been on the path to FIRE and this is something that isn't in my control. However, what I perhaps should do is just try to do a bit of planning (better than nothing) so that there is an element of control.
As I said above, I think my sister can be frugal when she wants to be and I will certainly try to pass on some good wisdom to my nephew. I so want to tell him about investing for starters!
A big change indeed, but i'm sure that there'll be far more positives than negatives for all three of you over the next couple of years. Perhaps the bigger question will be how you'll cope with things once they move on, rather than how you'll handle them moving in. No worries though, you'll still have your big extended family here. Take care kid sis.
Yes, I think there will be more positives than negatives, it's just me stressing about change!
And you're right, I could get used to having them around and then find that I can't live on my own…there's a thought!
Thanks for your kind words and for your continued support.
I have a sister who's a few years younger than me. I went travelling with her for a few months once and I'm not sure there's anyone else whose company I could have tolerated to the same degree 24/7 for that period. What helped was that if we got annoyed with each other we'd just snap at each other but a few minutes later it would be forgotten, whereas if I was travelling with a friend I think it would have become a big, friendship-ruining deal. Not sure if the same principles would apply to living with her for a couple of years though. I'm a bit weird about mixing my family life with my London life. My parents always say that they probably wouldn't know I'm in a relationship until I turn up one day married – and they're only half joking.
Yes, if there end up some sharp words between us, it'll tend to blow over. If this happened with friendships, it would be a lot harder to mend. Still, if it got so bad and I needed to get out of the house, I can just jump into my car, go to the gym (or sympathetic friends) to get some space!
Haha, funny comment about your relationships – it's not too dissimilar to me!
What a big change Weenie – such exciting times. I'm sure you'll all settle into a routine of living together in no time and you'll start to enjoy each other's company.
At the moment, I just don't feel excited, more like a bit anxious but I think I should be fine when we start getting used to each other.
Hehe… I think it's pretty natural to at times quarrel with loved ones, especially when your young and rebellious. I'm sure this will be a positive experience for you and your sister and you'll enjoy the company.
Let's just hope we don't end up quarrelling since we are no longer young or rebellious!
Hard for me to imagine while I sit in my quiet house but I think I will enjoy the company.
Couldn't do it myself. But, what a great opportunity to shape a young mind of tomorrow! Teach that boy, Mrs Miyagi.
If me and my bruv moved in together we'd probably become greater degenerates than we already are! Praise the Lord for The Wife keeping me in check.
Thanks for the best wishes and I shall do my best to shape a young mind of tomorrow!
Nice to hear your big news. That sounds like a lovely opportunity to get closer to your sister and nephew, no doubt you will look back on it as a time you cherished. I think if you set some ground rules from the start about being clean/tidy/energy efficient that everyone agrees to, it will help with the status quo. Also it's super important to have a private space (like your bedroom as you have mentioned) to escape to when you need it, and for no-one to be offended if the other disappears off for their "quiet time".
I have just bought 2 Marie Kondo books and have been watching the Netflix series, I think there's something in this decluttering/striving for minimalism approach!
Hoping your news doesn't impact on the blog we've all come to love.
I do hope that I'm not so stuck in my ways that I forget to cherish this time with my sister and nephew!
I am aware of aspects of Marie Kondo's ways and hope to implement some of them soon, though not to such extreme lengths as I really couldn't live like a minimalist!
Thanks for your continued support, it's much appreciated!
Wow ! the news has broken.
Quite a change for you, it will be exciting as the time gets nearer.
I have a younger sister and we have lived together in the past. She lived with me while waiting for her house purchase to complete.
I then lived with her when I left my ex and was waiting for my house sale/purchase to go through.
She enjoys living on her own and has done for years and is very set in her ways, she has become more insular over time. I go on holiday with her periodically and after a week, I am glad to be back home so I can do my own thing.
I live on my own and have got use to being on my own after living with someone else, I miss the company some time; coming back after a hard day at work to an empty house can be sad.
In summary : Good for sharing and talking. Bad for 'me time' and space.
You can make it work I'm sure, I know that if my sister and I had to live together again we would. Just setting ground rules and having some accepted split of activities will help, time together and time apart to balance the shared living.
Time to start sorting the house out ready for their arrival.
Now that I've broken the news, it doesn't seem as daunting as it was before – I think I just needed to get it off my chest!
Like you though, I've lived a long time on my own (which in itself took a while to get used after the break up of 15-year relationship) and I like my own company. I'm not sure that I feel alone, my social life is enough that I don't feel like I'm stuck in the house all the time on my own.
However, I think it's the 'doing my own thing' thing which will take some getting used to.
I'm quite looking forward to sharing out the household chores however, haha! They can do the stuff I hate doing!
Wow! I think it'll take some getting used to – sharing your space as an established singleton. But it'll certainly be great to connect with family on such a close level. There'll be little frictions, I'm sure, but you'll get to know them in a way that you can't just by visiting and meeting and on such a different level than when you were teens.
And it'll definitely good to share household chores, I reckon. I'm a reluctant cleaner (although I love a clean and tidy house) so I'm the kind of person who'll clean like a mad thing while cursing all my messy co-habitants as I go. In my really-fed-up-with-all-the-messy-kids moments I dream about a retirement where there will just be me and Mr Fu in a tiny bungalow which will take an hour to clean, tops, which is where we started our married life.
It's a pipe dream. With our brood I reckon we'll always have someone with us.
I think I will get used to it and adapt and you're right, just visiting and meeting for short spaces of time is not the same as spending a lot more time and living with them, so in that respect, I'm looking forward to them moving in.
Thanks, I've started clearing a few things out but probably not doing as much as I should!
Escaping to the supermarket sounds like a good idea if things got too much, haha!
I honestly would probably kill my brother if we lived together again assuming he didn't kill me first! I love him but I think we both have our own ideas of what a comfortable home is and we would clash. I would be turning lights off and he would be turning them on. I would be cooking home made meals and he would be ordering take aways.
Haha, thanks for sharing, Mrs Chai!
Everyone's relationship with their siblings is different – speaking to my friends, they all say that they couldn't live with theirs again!
The good thing is that my sis is more into home made meals rather than takeaways so in that respect, nothing will change! | {
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To try this example, you need GNU `rebar3` , `git` and `Erlang` in your PATH.
```
rebar3 release
```
To start the example server in a interactive way do this:
```
_build/default/rel/example/bin/example console
```
Now, if we point our favorite web browser at [http://localhost:8080/api-docs](http://localhost:8080/api-docs),
we should see the swagger API doc.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,419 |
Альдо Антоніо Бобаділья Авалос — парагвайський футболіст
Енріке Авалос — парагвайський футболіст
Марсіаль Авалос — парагвайський футболіст
Родольфо Норберто Паес Авалос — аргентинський рок-музикант | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,567 |
Kiril Strélnikov –en ruso, Кирилл Стрельников– (29 de mayo de 1992) es un deportista ruso que compite en natación. Ganó una medalla de oro en el Campeonato Mundial de Natación en Piscina Corta de 2021, en la prueba de 4 × 50 m estilos.
Palmarés internacional
Referencias
Nadadores de Rusia
Campeones mundiales de natación | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,453 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/302818\/complete-blocks-of-projections-and-spectral-theorem\/302845","text":"# Complete blocks of projections and spectral theorem\n\nThis question was asked in a slightly different way here, and then I realized that it could be of higher interest in the physics community.\n\nLet $\\mathcal{P}(\\mathcal{H})$ be an (atomic) lattice of projections on a separable Hilbert space $\\mathcal{H}$. Let $\\mathcal{B}$ be a maximal Boolean subalgebra (block) of $\\mathcal{P}(\\mathcal{H})$. Then, by spectral theorem, there exists a measure space $(X,\\mu)$ and unitary operator $U:\\mathcal{H}\\rightarrow L^2(X,\\mu)$ such that for each $P\\in\\mathcal{B}$, it holds that $UPU^{-1}$ is a multiplication by a measurable function $f_P:X\\rightarrow\\mathbb{R}$.\n\nThe interesting implication is that if $\\mathcal{B}$ is a complete** block, then $\\mathcal{B}\\simeq\\mathcal{B_0}$, where $\\mathcal{B_0}$ is a measure algebra of $(X,\\mu)$. Obviously, whenever $\\mathcal{B}$ is an atomic BA, $\\mathcal{B_0}$ is also atomic. (Details can be found e.g. in Takeuti's \"Two applications of logic to mathematics\" pg. 63.)\n\nI have several questions to that issue:\n\n\u2022 minor one: to what extent $(X,\\mu)$ is unique in the spectral theorem?\n\u2022 major one 1: my intuition is that if $\\mathcal{B}$ is a complete block generated (and maximized by Zorn's lemma) by spectral measure of a self-adjoint operator with a continuous spectrum, then $\\mathcal{B}$ is atomless. On the other hand, if a complete $\\mathcal{B}$ comes similarly from a self-adjoint operator with a pure point spectrum, then $\\mathcal{B}$ is atomic (must $\\mu$ be discrete then?) - is this correct?*\n\u2022 major one 2: what can be said about atomicity of a complete block $\\mathcal{B}$ generated by a self-adjoint operator with mixed (both continuous and point) spectrum?\n\u2022 aside: could this be related to the presence of minimal projections in von Neumann algebras? I suspect the answer is no, since from the very beginning every block in $\\mathcal{P}(\\mathcal{H})$ being a BA is commutative, hence not factor.\n\n*This would come from the fact that projections $P$ under the isomorphism are mapped to characteristic functions $\\chi_P$; if a projection's range is a one-dimensional subspace of $\\mathcal{H}$, then $\\chi_P=\\chi_{\\{p\\}}$ for some $p\\in X$.\n\n**Edit: I realized that the completeness is a subtle point here: a complete BA of projections is not only complete as a BA (i.e. containing all the sup's), but also in the following, stronger sense: if $P=\\mathrm{sup}(P_a)$, then $P(\\mathcal{H})$ (the range of $P$) is the closure of the linear space spanned by $\\bigcup P_a(\\mathcal{H})$.\n\nFirst of all I am not sure to understand well your terminology, but I think that maximality implies completeness (the $\\sup$ of a set of projectors, viewed as the strong limit of a class of projectors, commutes with all the projectors of the block using the strong operator topology and so it belongs to the block as it is maximal). So your blocks are complete.\n\nRegarding your first question actually I do not know, I think it is unique but I should check my conjecture and I do not have much time now. Sorry.\n\nRegarding your second question. I assume that atomic means that for every $P\\neq 0$ in the block there is an atom $Q$ with $Q \\leq P$. If the spectrum has only continuous component the lattice not only is not atomic, but it does not contain atoms because the atoms are associated to the single eigenvalues. If the spectrum of $A$ is a pure point spectrum, then $P_{\\sigma_p(A)}=I$ by definition (notice that $\\sigma_c(A)\\neq \\emptyset$ is still possible, think of a self-adjoint compact operator whose eigenvectors are accumulated by $0$ which, in turn, is not an eigenvalue). By difference, $P_{\\sigma_c(A)}=0$. Therefore, every Borel set $E\\subset \\sigma(A)$ either intersects $\\sigma_p(A)$ or produces the zero projector. In the first case an atom $Q \\leq P_E$ exists since it is associated to some eigenvalue of $A$ included in $E$.\n\nRegarding your third question, as the atoms are all of the form $P_{\\{\\lambda\\}}$ with $\\lambda \\in \\sigma_p(A)$, then the lattice may be atomic or not. As I pointed out above, pure point spectrum does not mean that $\\sigma_c(A)$ is empty, but only that $P_{\\sigma_p(A)}=I$, in this case however the lattice is atomic. In more complicated cases where the continuous part of the spectrum is full open segment for instance and the point spectrum includes also some isolated points, the lattice is not atomic.\n\nI do not understand well your last question. Think of a Hermitian matrix $A$ so that its commutative lattice is generated by the atoms $P_\\lambda$, where $\\lambda$ is every eigenvalue of $A$. The commutative von Neumann algebra generated by the lattice coincides to the algebra of all complex functions $f(A)$ spectrally defined. It is clear that, even if the algebra is not a factor, the projectors $P_\\lambda$ are minimal projectors of that algebra.\n\n\u2022 Well, I had to edit the question to explain completeness, it is not obvious that every block is complete also in the stronger sense. You elaborated well on the spectra. However, I have trouble to think of an example of atomless BA of projections that in the process of maximizing and completing remains atomless (i.e. 1-dim projections are not included). This is the only unclear point to me. Last point: lack of atoms in some BA of projections and lack of minimal projections in type II, III factors have to be accidental, since BA of projections by definition can not by a factor of any type. \u2013\u00a0krzysiekb Jan 4 '17 at 16:13\n\u2022 \" I have trouble to think of an example of atomless BA of projections that in the process of maximizing and completing remains atomless (i.e. 1-dim projections are not included)\" Could you be more explicit? In particular what do you mean by \"process of maximizing and completing\"? \u2013\u00a0Valter Moretti Jan 4 '17 at 16:32\n\u2022 Regarding completeness of a maximal block, I was actually referring to $\\sigma$-completeness since the Hilbert space is supposed to be sparable. In that case, given an at most countable sequence of elements $P_n$ in the block, $\\sup_n P_n$ can always be written as a strong limit sum $P= \\sum_m Q_n$ where $Q_n Q_m =0$ if $n\\neq m$ and each $Q_n$ is in the block (essentially by means of Gramm-Schmidt procedure). Therefore $P$ commutes with every projector commuting with each $Q_m$, that is with every element of the block. By maximality $P$ belongs to the block. \u2013\u00a0Valter Moretti Jan 4 '17 at 16:47\n\u2022 An example in $H= L^2(\\mathbb R, dx)$ is the spectral measure of the position operator. It is known that if an orthogonal projector $Q$ (actually a bounded self-adjoint operator) commutes with every element of that spectral measure then it is a function of it. The only possibility for a projector it is that $Q$ belongs to the spectral measure (in other words we have a maximal and also $\\sigma$ complete block). Since the spectrum of the operator is continuous, the block is atomless. \u2013\u00a0Valter Moretti Jan 4 '17 at 18:02\n\u2022 Another example is the joint spectral measure of $P_1,P_2,P_3$ in $L^2(\\mathbb R^3, d^3p)$. Those three operators define a maximal set of compatible observables: a bounded self-adjoint operator commuting with their joint spectral measure is a bounded measurable function of it (an element of the von Neumann algebra generated by that spectral measure)...the lattice is atomless \u2013\u00a0Valter Moretti Jan 4 '17 at 18:12","date":"2021-05-13 12:13:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.8857355117797852, \"perplexity\": 199.57817788461966}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243989916.34\/warc\/CC-MAIN-20210513111525-20210513141525-00217.warc.gz\"}"} | null | null |
{"url":"https:\/\/hal.archives-ouvertes.fr\/hal-00101417","text":"# Adaptive density estimation for general ARCH models\n\nAbstract : We consider a model $Y_t=\\sigma_t\\eta_t$ in which $(\\sigma_t)$ is not independent of the noise process $(\\eta_t)$, but $\\sigma_t$ is independent of $\\eta_t$ for each $t$. We assume that $(\\sigma_t)$ is stationary and we propose an adaptive estimator of the density of $\\ln(\\sigma^2_t)$ based on the observations $Y_t$. Under various dependence structures, the rates of this nonparametric estimator coincide with the minimax rates obtained in the i.i.d. case when $(\\sigma_t)$ and $(\\eta_t)$ are independent, in all cases where these minimax rates are known. The results apply to various linear and non linear ARCH processes.\nKeywords :\nType de document :\nPr\u00e9-publication, Document de travail\n2006\nDomaine :\n\nLitt\u00e9rature cit\u00e9e [31 r\u00e9f\u00e9rences]\n\nhttps:\/\/hal.archives-ouvertes.fr\/hal-00101417\nContributeur : Marie-Luce Taupin <>\nSoumis le : mercredi 27 septembre 2006 - 10:25:57\nDerni\u00e8re modification le : jeudi 31 mai 2018 - 09:12:01\nDocument(s) archiv\u00e9(s) le : lundi 5 avril 2010 - 23:13:06\n\n### Citation\n\nFabienne Comte, J\u00e9r\u00f4me Dedecker, Marie-Luce Taupin. Adaptive density estimation for general ARCH models. 2006. \u3008hal-00101417\u3009\n\n### M\u00e9triques\n\nConsultations de la notice\n\n## 210\n\nT\u00e9l\u00e9chargements de fichiers","date":"2018-06-23 02:47:34","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\": 0.47944697737693787, \"perplexity\": 2994.9665171872207}, \"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-2018-26\/segments\/1529267864919.43\/warc\/CC-MAIN-20180623015758-20180623035758-00269.warc.gz\"}"} | null | null |
David duChemin
When the Life Creative
Becomes the Life Created
**A Beautiful Anarchy:**
**When the Life Creative Becomes the Life Created**
David duChemin
www.davidduchemin.com
Project editor: Ted Waitt
Project manager: Lisa Brazieal
Marketing manager: Jessica Tiernan
Layout and type: Kim Scott, Bumpy Design
Cover design: Kim Scott, Bumpy Design
Cover title illustration: James Victore
Back cover author photo: Cynthia Haynes
ISBN: 978-1-68198-234-2
1st Rocky Nook Edition (1st printing, December 2016)
© 2017 David duChemin
Rocky Nook Inc.
1010 B Street, Suite 350
San Rafael, CA 94901
USA
www.rockynook.com
Distributed in the U.S. by Ingram Publisher Services
Distributed in the UK and Europe by Publishers Group UK
Library of Congress Control Number: 2016957327
All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without written permission of the publisher.
Many of the designations in this book used by manufacturers and sellers to distinguish their products are claimed as trademarks of their respective companies. Where those designations appear in this book, and Rocky Nook was aware of a trademark claim, the designations have been printed in caps or initial caps. All product names and services identified throughout this book are used in editorial fashion only and for the benefit of such companies with no intention of infringement of the trademark. They are not intended to convey endorsement or other affiliation with this book.
While reasonable care has been exercised in the preparation of this book, the publisher and author assume no responsibility for errors or omissions, or for damages resulting from the use of the information contained herein or from the use of the discs or programs that may accompany it.
This book is printed on acid-free paper.
Printed in the USA
"Life isn't about finding yourself.
Life is about creating yourself."
~ George Bernard Shaw
_For Cynthia,
My Beautiful Anarchist_
## CONTENTS
Foreword
A Beautiful Anarchy
Life Is Short
Ex Nihilo
The Artist's Journey
An Act of Creativity
This Might Not Work
Choosing Your Risk
Living Above the 45
Pretending to Be Brave
Listening to Voices
Inspiration
Incubation
Begin
Embracing the Constraints
Process Vs. Product
More Bad Ideas
The Starving Artist
The Art of Exclusion
Waiting for the Knock
Know Your Rhythm
The Myth of Originality
Ruts & Grooves
Winning at Yoga
Art As Gift
Now
Toward Mastery
I Will
Ripples
About the Author
## FOREWORD
"When was the last time you failed?" This is a frequent question asked of me. I don't know if my answer is jacked up on hubris . . . but I do know it's honest—"Every day."
As an artist and designer, it's my job to tempt failure every day. It's not that I'm looking to stumble or am seeking financial ruination, but not tempting failure would lead to me finding a gimmick, a trick that people like—then rolling that trick out again and again and again. Society tells us this is good—the _goal_ even. But "people pleasing" leads to losing your own way and, ultimately, boredom. It's my job to make myself happy, to make work I love and, if I do a good job, others will love it as well. Or they won't—hence, tempting failure every day, but I'd rather risk it all on love every day because love always pays off.
With love,
James Victore
## A BEAUTIFUL ANARCHY
The word "anarchy" literally means "without a ruler." In popular use, it's a political word with heavy baggage, a bloody history, and occasional car bombs. This book is not about that: it's about freedom.
This is a book about the freedom to create—to live a life of unapologetic, passionate, daring creation—in whatever arena resonates best for you. Parents create when they raise a child, entrepreneurs create when they begin a business, and teachers create when they design a lesson plan. Some people identify with the urge to create more than others, and it's to them I speak in this book, not because others can't benefit, but because anyone who persists in the idea that "I'm just not really creative" is unlikely to read this book, believing instead that the die has been cast and they've been excluded. They, of all people, need most to read it, and I hope they do.
This book is for people who have a sense of their own urge to create, or those who don't but long to look under the hood and find it waiting there. But to its bones, this is a book about art and the process of making it, because what is our life but a chance to make the greatest art of all? Whether you ever set your paintbrush on actual canvas isn't remotely the point, though I hope you will. What is very much the point is that each of us is given a canvas—from one edge to the other the span of our life—and each of us has a chance to do something brilliant with it. Each of us has the chance to fill that canvas with wild, achingly beautiful swirls of colour, and if you're reading this there's a chance that you feel right now that your canvas is empty, or dotted here and there with hesitant, half-hearted stops and starts, the brush pulled up before you could even gain momentum, for fear of doing it wrong.
As I write this introduction to a now half-written book, the sun is rising in Bali. It's August 2013 and a trailer's just come out for a movie version of one of my favourite short stories, _The Secret Life of Walter Mitty_. Mitty's a dreamer, working as a photo editor at _LIFE_ magazine, allowing daydreams to be a substitute for actually doing what he longs to do. Working at _LIFE_ , but never having one. And then he looks at a photograph of a photojournalist in a refugee camp (played by Sean Penn), and the photograph comes to life long enough for Penn's character to gesture an invitation to Mitty. Come. Stop observing. Stop abdicating your life. Live a great story instead of just watching, telling, or dreaming them.
I'm self-conscious about saying it, but I want to be that man; I want to be someone, among many, I hope, who calls to the dreamer and says, "Wake up." I want to invite others to begin living now, not later, and to ask them with a straight face to step out of their comfort zones and face the fear. I want to see every person in my life doing what they long to do, free from the things that hold them back. Life is not in the dreaming, but in the doing. Don't you dare get to the end of your life, your canvas clean and unmarked. There is no prize for the one who leaves his canvas clean, his scribbled signature in the corner the only thing to differentiate his own off-white rectangle of a life from all the million others who—too paralyzed by fear—have done the same.
My massage therapist once told me the stars were aligning auspiciously. She told me it was a good time to dream big dreams. Not one with a particular reverence for the opinions of the stars, I told her I dream big dreams every day; it's up to the universe to keep up. I wasn't being sarcastic, and she knew it. I also wasn't being cocky. I was just being honest. I do believe in dreams; the bigger the better. But I also believe in action. I believe in great ideas, too, but I don't believe coming up with great ideas is the same as being creative. Being creative is about creating. It's about doing. And so too is living, because life is an act of creation. Day by day, whatever else we make, our first act of creation is our own lives. We must first make the artist before we make the art. Out of nothing, nothing comes.
This book is a call to colour outside the lines, in both art and life. It is a book about living free from the rule of everything that holds us back from being the humans we were created to become. It is about living free of the rule, or tyranny, of fear and shame, of debt and obligation, and every "should" or "should not" that we have not willingly signed off on. Your art, the thing that stirs from your heart, mind, and soul, the thing that moves you (and hopefully, others), is a free agent, and the moment you begin to ask, "What _should_ I do?" or, "How _should_ I do this?" you allow your art to teeter, to lean towards conformity and away from authentic expression.
To do what we _should_ in art is bondage. To tell others, with our art, what they _should_ think or feel or do is propaganda. And to tell other artists how they _should_ do their art, whether that's visual art, the written word, or creating a business, is presumptuous, and unkind, and tells the muse we've learned nothing at all under her influence.
All very well for the artist, but what of the rest of the world, those working a regular job, whatever that means? I think it all applies equally, if not differently, and that there is room (there _must_ be room) to live our lives increasingly on our terms, as engaged and intentional as possible, as creatively as possible, with the freedom to follow the muse, or our own curiosity, down the road that's unique to us. I think almost any endeavour undertaken on those terms can be art.
I'm a photographer and author, a publisher, and former comedian. I've made a living from my own creativity since I worked my way through college as a comedian, and while making a living in the arts in no way means my art is _good_ , per se, it does mean I've relied on it a little more than I might have otherwise, and I think that dependence on my muse has made us more familiar with each other than I might have been otherwise. To write that my muse and I are familiar, however, is to understate what's happened between us. My muse and I have worked closely together over the last twenty years, and the uneasy relationship has become less turbulent over time. And while I'm never quite sure how she feels about me, I think it's fair—if not overly anthropomorphic and unnecessarily romantic—to say I've fallen in love with her, and the life my creativity has made possible.
There is nothing I would rather do than work creatively and, in so doing, to make a living. Whether making a business, making a photograph, or writing a book, the urge to create has always been central to who I am. I believe it is central to who we all are, which is one of the reasons I get twitchy when I hear someone tell me they "aren't really creative." We're _all_ creative, but we've allowed the arts to co-opt that word while making every other area of human creativity feel a little too self-conscious about using it. And I think we've misunderstood the creative process, which if it's anything at all is messy, each successful endeavour hardwon, each masterpiece the result of a hundred failed sketches and many tears.
It's the fundamental creative urge within us that makes otherwise rational people take complete leave of our senses and have children. It's that same urge that compels people to build houses, find cures for diseases, build companies and products, solve complicated mathematical problems, or write music. The same creative urge that compelled the first cave man to draw animals on cave walls is the same urge that compelled him to carve obsidian into arrowheads and hunt those same animals for food and clothing. It is that impulse within us to follow the whispers of our curiosity, or the urgency of our needs, around dark corners and into the unknown, that is responsible for every astonishing advancement in our history—the discovery of fire, the law of gravity, the revolution of the earth around the sun, the evolution of the species, and the creation of Michelangelo's _David_ , Picasso's _Guernica_ , and Handel's _Messiah_.
For every advancement that has taken place, every creation of some new beauty in some new field, that advancement has taken place as a movement from the known and accepted into the unknown and, at times, unaccepted—to the point of outright rejection. Galileo was declared a heretic for his idea that the earth revolved around the sun and not the other way around. Darwin's never been popular with the same crowd, either. Picasso's cubism was revolutionary and took several years to overcome initial negative reception, even from fellow artists, people who should know better than to hang on so tightly to convention.
Of course, public reaction, either negative or positive, is not the point. The point is that the long history of creativity—in every imaginable field—takes us inevitably into places where we have to pour new wine into old wineskins, and that invites criticism, which in turn invites fear, and soon we're back to hiding in the shadows, letting others take the risk while we abdicate the responsibility to do the one great thing we can do with our lives—be fully ourselves and make art of our lives.
This book is an invitation to celebrate the life creative, and in so doing to embrace its essential and beautiful anarchy. I use the word "anarchy" metaphorically rather than politically, as a call to live our lives on our own terms, which is the only way we can fully be ourselves. It's a call to live our lives free from the bondage of _should_ and _ought to_ : the only way to be truly alive. We need more of these kinds of anarchists, more people who understand the extraordinary beauty and brevity of life, and who daily find the courage to follow that voice that calls them to something more, even when they don't know what that _something more_ is. Even when that voice calls them to places beyond the points that other voices say they should turn back.
What we do not need is anarchy for anarchy's sake. I value this thin metaphor of anarchy because I think, as a metaphor, it represents a beautiful way of life, and of thinking about the creative process and our lives-as-art. When I chose to use this metaphor I did so because it works for me. This beautiful anarchy is not about freedom from law, nor my own desire to live with, or without, certain rules. But where I do embrace rules, they are rules I have myself signed off on because they resonate with me, they make the world a better place and are, I hope, descriptive of my life, not prescriptive. I've chosen to live by the rule of love, kindness, and respect—toward others and myself. I believe in forgiveness and grace and, so far as it aligns with my conscience, the law of the land. I pay my taxes. I believe in being a responsible citizen. In fact, I believe the life this metaphor describes makes me a better global citizen, and a better human being. If the metaphor fails for you, then find one that works, so long as it leads you to the freedom and joy of creation, and of doing so in the most authentic way possible.
Whatever metaphor you choose, I hope this book gives you the courage to begin filling your canvas again, and if you reach down for the brush only to find it hardened from lack of use, then throw it away and plunge your hands into the paint. However you have to do it, don't leave your canvas blank. Don't deprive your soul, and the people around you, of the chance to see you fill every inch of that canvas—messy and wildly human as it might be—with every flaming colour.
## LIFE IS SHORT
That life is short is so blindingly obvious to most of us that it's become a cliché. I'm not sure where the line between truth and cliché is, but it's thin. And I'm not sure that we can be free of the truth of things, or free from the chance to act on them, merely by calling them cliché. Sometimes I wonder if we call things cliché in order to excuse ourselves from thinking about them.
Our days are numbered, folks. Not only are they limited, we have no idea exactly how many days remain in the storehouse. Our time here is not merely a resource to be managed. It is _all_ we have, and it's insanely beautiful at times, but it's _short_.
One of the gifts of photography is the way it makes us conscious of time. Time is one of our raw materials. Our exposures are measured, in part, in fractions of a second. Sometimes so fast the shutter is closed before you know it's open. The best photographs also rely on the strength, beauty, and universality of a particular _moment_. Blink and it's gone, but when photographed it remains, frozen in space and time, to consider for as long as the print remains. Photography helps those who are willing to see the moments we'd otherwise miss. And moments are important because the way we live our moments is the way we live our days, our lives. Photographs—the best of them, at any rate—honour the moments, and they speak to us because we know how limited these moments are. Time is limited and we've no idea how much of it we have, so the sooner we cherish and redeem it, the better.
Time is not money. If time were money we could borrow it. We could steal it. We could bank it and see our days compounded. We can't. We can live it. We can use it to do the thing we are here for, or we can waste it. But we can do little else with it because it's not ours to control. It's given to us in unspecified measure to wring from it what we can.
I am strongly motivated by the brevity of life, not because I fear its end (though like Woody Allen, I'd rather achieve immortality by not dying than through my work), but because, simply, it will end. What I control is how deeply I live my days, not how long. But I think somewhere along the way the urge to live deeply gets subverted.
We settle. We find a path of lesser resistance and we take the deal, because it's easier to be safe. It's easier to fit in. We'd rather tiptoe through life and make it safely through, because we seem to have willingly forgotten that there's no reward in making it unscathed to our funeral.
Why we take the deal in the first place is another discussion to be had later, because it's got a lot to do with fear and the voices we listen to, but it's important to realize we've settled. There's not a week goes by that someone doesn't tell me they envy my life (and by that they usually mean the good bits, the public bits of my story; few have told me they desperately want to take the path I've taken to get here) and that they wish they could do what I do. And I get it. I really do. But what they seem to mean is that they want what I have without paying the price I've paid to get here. I know people want to change the world and create great art and live the dream and so on. So do I. But some want to do so without giving up what is demanded, by life, in exchange. Were life longer I might have time to do it all, but I don't, and so I make choices: do I do this, or do I do that? Seldom am I given a choice to do both. I don't own a home. My freedom from mortgage payments and maintenance issues frees me to travel. Some can do both. I can't. I've made a choice. I'd rather have a plane ticket to Bali than a big screen television. But to the one who sees no choice but to keep up with their neighbors, the television comes first and the plane ticket remains a dream. For me, if owning the latest car or appliance means I give up the experience of travel, and the freedom to do my work, it comes at too high a cost. Most of us love the idea of having a choice until we're told that choice means giving up one to have another. Some don't realize it is a choice.
Life will go by so fast it'll make our aged heads spin when we get to the end. But it's not only short, it's uncertain. When I graduated from high school we were already talking about careers and what we'd do when we retired. Not once did we say, " _If_ we retire"—we treated it as a given. We would retire and in health enough to enjoy the dreams we'd set aside for that retirement. But life has this way of getting in the way of deferred dreams. Leukemia arrives uninvited. A headache becomes a brain tumour that becomes a fight against a possibility we never imagined until it's clear the dreams we saved for later will never happen. I don't mean to be morbid but we live, many of us, in a culture that lives in perpetual denial of the inevitable, and it's costing us our dreams. You can't bank your time. The time, as it has always been, is only now.
That my days are numbered forces me to choose carefully how I spend them. And because my life—even if I live to 120—will seem so heartbreakingly short, I will choose not only _what_ I do with my life, but _how_ I do it.
I know. It sounds so selfish. We've been taught to keep our heads down. We've been taught not to be selfish. Many of us are also taught to respect the choices of others, and to give them the dignity of living their lives on their own terms. We're taught to extend forgiveness, to be kind. We're taught to love others as we love ourselves. But the moment we try to love ourselves the way we're taught to love others we're chastised: "Don't be so selfish."
And yes, sometimes they're right. But often they aren't, and the admonition against selfishness has become a perverse reversal of things. The most loving people I know find that love _first_ within themselves. It is the self-loathing who abhor others. It's the ones who won't respect themselves that don't respect others. It is the ones who don't allow themselves to risk and dream and live extraordinary, unconventional lives who discourage others from the same.
At the risk of being misunderstood, I think it's time we took back a healthy regard for selfishness. In fact, I'll go one better (in for a penny, in for a pound, right?): I think it's time we made ourselves a priority. To do otherwise is to expect a bountiful yield from a garden we've neither planted nor tended. I'm not suggesting we allow ourselves to become egomaniacs, just that we extend the same love and grace to ourselves that we do to others, and to do so _first_ so we have a place from which to love and respect others. That we respect ourselves and allow ourselves the same chance to live our dreams as we allow others. Only when we take back the responsibility to make our own choices—to live on our terms—do we have a place for extraordinary generosity, profound kindness, and the acts of heroism of which we're capable, and of which others will one day call _selfless_.
I'm not looking to justify a life of what I would have once called selfishness; I'm looking for a healthy place to put myself in this world. A place to stand. A place from which to love and do what I have been called, by Life, to do. A place to do good, to love boldly and without fear. A place to be generous and hospitable, and to create my art without shame in the days I have allotted to me. A place to become everything I can be, without settling for anything less. A place from which I can find the leverage to make the same things happen in the lives of those I love. Life is too short to do anything else, and too beautiful not to fight hard to be a part of it.
## EX NIHILO
There is an old Latin saying that gets thrown around in the theological circles from which I emerged as a young man when I left college: _ex nihilo nihil fit_. "Out of nothing, nothing comes." Its use, as far as doctrine goes, is to enforce the idea of a Prime Mover. Nothing comes from nothing, so before there was something, there must have been Something Else to create the something. Or something like that. The years have taken most of the details from the dogma of which I was once fond, and smoothed my edges a little. But the idea remains sound, at least as a metaphor.
I'm listening to Miles Davis's _Kind of Blue_ album right now, the cover of which was shot by photographer Jay Maisel. That's neither here nor there, but the following story is. As the story goes, a student approached Jay and asked him, "How do I make more interesting photographs?" Without pausing, Jay replied, "Become a more interesting person." Indeed.
We are the source of our own creations, whether that's a story, a child, a photograph, or a business. That work of art, if it's to be art at all (and I think all of those can be), will reflect the artist in some fashion. So then the act of creation that is our first concern is ourselves. Before we create art, we must create the artist. I think it's fair to talk in these terms, forgetting for a moment that on the surface it sounds profoundly narcissistic, because I don't believe we're just passive victims of fate. Yes, life happens to us in ways we never expected, and luck, or serendipity, has a way about it that's hard not to see as wondrous and mysterious (as well as cruel and malign) much of the time. But we live and create in reaction to these events, and it is those reactions over which we have control.
When the potter is given a lump of clay, he creates something of it—either passively, by doing nothing and letting it harden into a useless block, or actively, by putting it on the wheel and shaping it to his desire. We are what we are, flaws and all—and I'll talk about the power of constraints later—but what we are _not_ is powerless. As our history on this planet too well illustrates, the human will is powerful, and the decision to react to what life brings us is either a creative force or a destructive one in our all-too-short lives. It's by virtue of the will to react and make choices—even in the light of some very dark, or paralyzing, circumstances—that we create our own lives.
As a photographer, I am a vocal advocate of a very intentional approach to making photographs. There are a lot of decisions that affect the final outcome of the image, and I think abdicating those decisions is a lost opportunity to create something that more clearly expresses ourselves. I believe the same about life.
In fact, I believe this so strongly that I would like to try one more metaphor. As a race we've found meaning in stories for millennia. We consume stories at an astonishing rate. What stories we choose to read, watch, or listen to become a part of us. Sadly, because they do give meaning, I suspect many of these stories have become a substitute for living an interesting life. Stripped of all risk, it's easier to _watch_ great stories than to _live_ them. But choose to live a great story, and we open ourselves to all the possibilities the human drama has always drawn on. Exciting, heady stuff to find the love of your life, but it comes with the risk of heartache and loss. Easier, perhaps, to curl up with whatever movie in which Tom Hanks and Meg Ryan's characters are falling in love than to do so ourselves. Amazing to jump on a plane to Africa or Southeast Asia for an adventure, but you risk all the uncertainties that have kept thousands from doing the same thing, safe at home on the couch instead with their Lonely Planet book and BBC travel documentaries.
We have the choice to actively write a more interesting story or passively accept the one that comes our way. I'd be contradicting myself to say we _ought_ to choose one over the other. Part of being human is having the dignity to choose. But if our lives are stories then it's the more interesting one that I'd rather both read and write. And it's the person living that more interesting story that is going to create the most interesting, meaningful art with their life. That kind of life happens intentionally. We may not choose the things that happen to us—few of us do—but we control our own reactions and, in that way, shape the clay we've been given.
So much of our raw material lies outside our grasp. None of us controls what is behind us in our past. We don't control the parents to whom we were born, or the place or income bracket in which we grew up. We went to this or that school, and by the time we turn 18 we've had a childhood of victories and defeats, joys and sorrows, and enough traumas, either real or imagined, to fuel a lifetime of angst-ridden dreams or novels, should we decide to pay it forward and inflict those on future generations. We will, because this is life, continue to collect these experiences. But they are raw materials only, and what we do with them is a part of the choices we make in the creation of ourselves. It's a collaborative effort with Life, an unpredictable partner to be sure, but it's our _reactions_ that form the people we become.
To those reactions we add our choices about the stories we listen to, the books we read, the people with whom we surround ourselves, and the jobs in which we choose to remain too long. We choose the ones to whom we give our hearts, our time, our money. We choose to continue learning or not. We choose to buy that new stereo instead of the ticket to Australia for the year in the outback we always wanted. And in so doing, we create the person we become, piece by piece. It's a good argument for making those decisions with greater care and intention.
If I've got a tendency to oversimplify, forgive me; I know life is profoundly complicated at times. But I also know that "it's complicated" is a poor excuse for resigning ourselves to our fate, as though it's our lot in life. It would be easy to allow overwhelming debt, bankruptcy, divorce, a diabetes diagnosis, or a near-fatal fall that shatters both your feet, to sideline you. Or me. I've lived through all of those, and there have been times I'd have thrown all this right back in the face of anyone who told me excitedly that I was "living a really great story." But they'd be right all the same, and at the end of it, what have we got but to make the best of it, and write the best damn story we can? Self-pity makes an interesting scene in the movie, especially when it leads to broken furniture, a bar fight, or preferably both—but it gets old fast, and after a few minutes it's neither a story we want to keep watching, nor one we want to be a part of. The best stories are never the easy ones.
I keep using the word _react_ but it's only half the story; living in reaction, even mindful reaction, is not living intentionally. Take your favourite story: the hero usually resists the initial call to adventure, or love. Then something comes along to force him into the fray—he reacts and embarks. But at a certain point the story becomes his own, and it is his desire that drives him forward, not just circumstances. He eventually risks it all because the calling now comes not from outside voices or forces, but from deep inside. Now might be a good time to start unabashedly asking yourself, " _What do I want?_ "
Some of the best stories don't really begin until the hero grapples with that question. And for some of us the grappling will come hard because we've been taught not to ask the question. It seems selfish. But I think the things we do in life that are motivated by desire and love are the most powerful, and I don't believe that our happiness has to come at the expense of others. I believe we're connected and my happiness has to include that of others. I'll talk more about this, but I want to plant the question. The first question is not, "What should I be doing with my life?" It's, "What do I want to do with my life?" And if that sounds selfish to some, I can only say that it's in identifying the deepest desires of my heart or mind that I find my calling. It's my hardwiring, and I believe it was put there by Something or Someone good.
Knowing, deep down, what you want to do with your life leads to a ruthless prioritization of resources. Knowing your time, among other resources, is limited, and knowing what you want to do with that time, allows you a profound freedom, even if that freedom is not always easy.
If you want to create more interesting, meaningful, beautiful songs, paintings, businesses, or meals, become a more interesting, meaningful, and beautiful person. Gather the best raw materials you can (or the only ones you've got), work within the constraints you're handed, and make something new. The art comes from the will of the artist, but first there has to be an artist, and—out of nothing, nothing comes. Reaction only takes us so far: it's more a posture of daily life, one we assume while we go about intentionally pursuing, or creating, the things we most long for.
What do you want to do with your life? I don't mean this abstractly or obliquely. Find some silence and sit down with a cup of coffee. Mindfully consider your life now, the things you've done with your life that have made you the most content or fulfilled. What do you want to be remembered for, one day long in the future when your moments run out? Write it down. What does your heart long for? What dreams can you not let go of? What obstacles stand in the way now? What steps could you take to clear them? If those desires are worth pursuing, they're worth recognizing and clearing a path for.
## THE ARTIST'S JOURNEY
One of the great revelations of my life came when I discovered Joseph Campbell and his book _The Hero with a Thousand Faces_. I read it while studying films and screenplays and trying to understand what makes a great story. And I read it while in the middle of a divorce and unsure I would survive the trauma of that.
While challenging to read, I found one idea that stuck with me and resonated so completely that it's still a rare month that I haven't gone back to it, to sift through the paradigm and find meaning. That idea is the Hero's Journey—the idea that in the mythology of every culture through time there are common elements in the stories that give our lives meaning. Books like _The Writer's Journey_ by Christopher Vogler and _Story_ by Robert McKee explore this idea in order to help writers create stories that are more powerfully resonant. I think we can use the idea to see the creative process from a new perspective, and to write our own story in more intentional ways, and with greater understanding.
The Hero's Journey is not a formula, and it's not prescriptive. It's descriptive. It looks into the stories we've told each other since the dawn of time and asks why they have such a ring of truth, and what elements they have in common. The hero takes many forms, and is not gender specific. She can be a warrior, to be sure, but as often can be an explorer, an inventor, a mother, a tradesman, or an artist. What he or she is is not important; the Hero's Journey is more concerned about the path the hero takes.
Borrowing from the language of Vogler's _The Writer's Journey_ (because he makes the often difficult language of Campbell much more accessible), the Hero's Journey begins in the Ordinary World. It is the place in which we live our ordinary lives, waking daily to our ordinary tasks, and from which the hero is about to be shaken by the Call to Adventure (which I'm capitalizing to make it a little easier to see the structure). That call comes in a million ways, but is almost always an awakening, a desire, a longing, or a crazy idea. For me it's often expressed as a "What if . . . ?" However it comes, it's a call away from the mundane: a call to rock a boat, to change a status that's been quo for too long, a call to make—to create—a change.
If you're like me or like most others (including the protagonists in millennia of stories), you're likely to refuse the call at first. The Refusal of the Call is common, and though there are times it's taken up quickly and without thought, it's the human tendency to prefer the familiar and the safe over the unknown. And so we turn our attention elsewhere; we make excuses; we procrastinate. Even when the call itself seems so right and fills a longing, it's our nature to count to three a few times to build the bravado needed to jump into water we know will be cold and dark. Once we own that call, once we accept and dive in, pack our bags, sit down at the typewriter, or get out a new canvas, the journey truly begins. Acceptance of the call is, in most movies, where the action begins and the protagonist crosses the First Threshold into the special world of the adventure, the new world in which the story unfolds and after which nothing will be the same. Dorothy steps onto the Yellow Brick Road, Luke joins the Rebel forces, and Peter Parker puts on the Spider-Man costume for the first time. Here the hero meets allies and enemies, encounters tests and trials, and begins his approach to the Inmost Cave.
The Inmost Cave is where the hero endures the ordeal that is central to the story, the conflict around which the story revolves. Without conflict there is no story. There is no catalyst to change, and no reason for us to keep reading. Who wants to read a story in which there is no conflict? No one. Because life is not like that. We seldom get what we want without struggle and loss. We do not become who we want to become—the best versions of ourselves—without passing through the fire. We do not write the book without tearing up a few pages, nor paint the masterpiece without ruining a few canvases. We know that. It's why the truly epic stories have so endured; we know the best stories, the ones that most resonate with our humanity and fuel our hopes, are the ones that remind us that the harder the struggle, the greater the reward. And it's no token struggle, either. The losses we accumulate on the way can be almost catastrophic, and there are times we're not sure the hero will make it. We don't know the book will get finished, let alone feel right when it does. We don't know our marriage will make it, our business won't go under, or that the year we spent chasing the thing we long for will pay off.
And it doesn't always. Sometimes the hero just doesn't make it, and the story never gets told. You know that. I know that. What doesn't kill us gives us something to tell stories about. But when she does make it, and she takes hold of the Reward for which she struggled so hard, and takes the Road Back to the ordinary world, even still pursued by the villain, the enemy of our souls, or her own doubts, she experiences something of a rebirth, a Resurrection. She is changed, has faced her fears, her demons, and she has won, Returning Home, back to the tribe, the kingdom, the family, with the magic sword or elixir.
The dragons have only ever been metaphors for us, the swords now only symbols, but this pattern in one form or another has played out in a million eras on a million faces. And I think it's worth the time to write here and to think about, because I think it makes it easier when we wake up hearing that call to do something new, to venture past the threshold of the mundane (even if that's an everyday effort) to remember what's coming our way. It's easier to deal with the approach to our inmost cave, or deepest fears, when we can brace ourselves for it, and when we know that every artist through every age, whether their art was in raising monuments, raising money, or raising kids, has fought tooth and nail to get there, and that struggle has changed them before it's allowed them to go home and do it all again the next day.
Our own hero's journey, just as easily seen as the Artist's Journey or the Human's Journey, will both span our lives and repeat in smaller cycles, with each new book, each song we try to write, each canvas we fill, or each new initiative we begin when the words "what if . . ." wake us from our slumber and call us to the adventure. We will go through the same challenges, whether our conflict is with ourselves, the project we're working on, or something else. It won't make the struggle easier, but perhaps we'll have less fear, and blame ourselves less, when we know we're fighting on the same battleground. And it should make us nervous and raise red flags when the journey doesn't take us through those battlefields. If we haven't struggled with it, we will—or there's a good chance the work isn't worth the effort. I'm not saying it's always this way, I'm just saying that's the way it seems to be most of the time. Our art is only worth as much as it cost us.
Living a great story is much harder than watching a great story, but it's why we watch at all. The great movies steel our nerves and give us hope. They remind us that no story worth the telling, or the living, comes without conflict and struggle; they remind us that the necessary price we pay is transformation. The promise of any great story is that the hero never returns the same. Story is not about entertainment. It's about change, and it is change that makes us the artists—the humans—that we are and gives us the place from which we make our art and write our own stories; to live our lives with greater intention instead of allowing our stories merely to write themselves.
## AN ACT OF CREATIVITY
This book began as a book about the creative process. And because the creative process is what it is, and part of that is it's reliable unpredictability, the book is becoming something else, something bigger, as I write it: a book about life as creative process. In part that has happened accidentally.
The more I've written—a chapter here or there—the more I've found unexpected connections between things I didn't initially see. The biggest connection has been the way the principles of an intentionally nurtured creative process mirror those of a life lived intentionally and creatively. So while I started out to write a book about the creative life, I've written something much more about the created life, which is fitting because the more I discuss creativity with others the more I see it touching every aspect of our lives, in every discipline, and if it can so permeate our lives I think it deserves a conversation that is larger than just how being creative relates to the life of the artist.
The creative process, in broad terms, is relatively predictable. We know something of how it works and we know _that_ it works, in part because the human brain is just so truly good at what it does when it's all working as it should. But being able to rely on it doesn't remotely mean that we know exactly where it's going to take us. There are too many authors, musicians, or inventors who testify to the fact that the muse has dragged them into unexpected places, to ignore the serendipitous nature of the creative journey. If the creative process is predictable in broad strokes, it's wildly unpredictable in the details. But that doesn't mean we can't be as intentional about the life we're creating as we are about the work that fills our days.
Being intentional begins with a difficult question for most of us, difficult because we seem to have learned not to ask it. That question, asked in several different ways, is simply this: What do I want? What do I want to create? What kind of person do I want to become? What kind of people do I want to be surrounded by? What kind of legacy do I want to leave? And, conversely, what do I not want? For most of us it just seems easier to let life come as it does, to react to what comes, and one day find our lives have formed themselves into an ad hoc collection of all the decisions we chose not to make, all the ways we settled for things we never wanted. We find our house has been built of the flotsam and jetsam that's washed up on the beach and been banged together over the years, more from reaction to what's come our way than from an effort to build something specific.
I'm not sure if it's because we don't trust ourselves to be both ambitious and generous, but we've come to equate ambitious people with people who hunger for power. Ambition is a fuel and the machine it powers can be used for good or evil. But it's not the fault of ambition when people choose to aspire to less than noble things.
It takes great ambition to do the things of which Mother Teresa or Gandhi were capable. It takes focused intent to dedicate your days to perfecting a heart implant or the discovery of a cure for cancer, but those are not the only efforts to which ambition is nobly applied. It takes ambition to do your art, to write your book, to raise your children to be everything you hope for them. And yes, it takes ambition and some difficult honesty to admit that what you want is to make an astonishing amount of money in order to do astonishing things that cost money. Someone has to pay for cancer research. It's easy for artists to say they don't think about these things, but try being broke or on the edge of bankruptcy and you'll see that you think about money much, much more than you ever imagined.
Being hungry and dodging creditors as you pursue your art is not noble, nor is it likely (necessarily) to lead to great art. Of course, money is not the point here. Desire it or don't, but you can't ignore it because too many good, beautiful, world-changing things come at a cost and unless you find someone willing to cure cancer or teach your children by bartering for a chicken or a goat, cash will be preferred, because they too have bills to pay. If you want it, money or otherwise, it'll be easier to come by if you're honest and intentional about making it. Sure, people discover things by accident, but for the most part even those accidental discoveries were made while they were intentionally looking for _something_. Search for nothing and you're bound to find it. Strive for nothing, hope for nothing, and desire nothing, and there's a better chance than not that you'll get exactly what you strive for, hope for, and desire: nothing.
This is not the same sermon as the one that starts with "If you can dream it, you really can do it." A wonderful notion, but not one based in any reality I'm familiar with. I've failed at plenty of things I've been able to dream of. But I'm not sure I'd have known one way or the other for certain unless I tried. Anyone who hopes to spend any time doing anything creative will become more familiar than they'd like to be with the reality that this, whatever this is at the time, just might not work. But it sure as hell won't work if you don't try.
## THIS MIGHT NOT WORK
I took a forced hiatus in 2011 from my work as a photographer for humanitarian organizations, the result of an accident in Italy that left me unable to walk. The last assignment I did, in the far reaches of northern Kenya among nomadic Rendille, Turkana, and Samburu tribes, changed my life.
As I write this chapter it's January 2013 and I'm on my way back to Kenya, my head full of ideas and the expectations I should know well enough not to have. I have all kinds of notions about what my work—my resulting photographs—will be like, which sits uneasily beside the knowledge that what I will accomplish, and what images I return with, will not be that. Whatever I create there will come from that heady, addictive mix of my own hopes for this work, and serendipity. Unforeseen, and uncontrollable, the days ahead will come without needing my permission but open to my collaboration; my receptivity and a hungry willingness to say Yes to what comes my way will take the place of these expectations very quickly. They always do.
On top of all that, and the way my own expectations seem to constantly be getting in the way of seeing things as they truly are, a couple days into this I'll learn again to be receptive to _what is_ instead of try to force _what may never be_. It is these expectations that define what it will be for me to fail. So I'll be worried about that too, until, as I said, I adjust those expectations and redefine what it means to succeed. I fight this battle so often, it's astonishing I'm still surprised when it happens.
That said, I know all too well that part of the life creative, of doing what's not been done before and trying new ways of accomplishing the familiar, is the promise of failure. Maybe I should use the word "risk" instead of "promise," but that's unkindly optimistic. Failure in the creative life is not only a risk, a possibility to be avoided, but an eventuality to be embraced. Worrying we _might_ fail leads to fear and paralysis; it leads to making "safe" decisions instead of the ones demanded by our art, our longings. Knowing failure is part of our process leads to new ideas, stronger work, and more honest questions that liberates us to peer, a little less frightened, into the unknown.
I think much of this fear of failure is a question of expectation and definition. If we define success in our creative efforts as "getting it right the first time," then failure is, as I said, a promise. The alternative is mediocrity, and I'll take failure on these terms any day over mediocrity and first-effort results. If, instead, we define failure in terms of risks untaken, questions unasked, or work untried, then we go into those risks, questions, and work knowing there's a good chance we'll fall down a few times before we find our stride. To use a metaphor from childhood, failure is not in falling off the bike, it's in not getting back on and learning to ride the damn thing. Falling is assumed. It's our best, and most faithful, teacher. That doesn't mean, of course, that it doesn't hurt like hell; our pride especially smarts from this.
But really, who the hell do we think we are that we should accomplish something new without first bumping around in the darkness a little? Arrogance and a teachable spirit are mutually exclusive; as much as we all wish we could experience mastery after reading a couple books, we'd be crazy to expect it.
The words "this might not work" are probably some of the healthiest in the lexicon of anyone who wants to live creatively. They indicate a certain humility and openness to what comes next. Whether that is failure or success very much depends on how you define it. "This won't work" is defeatist. "This probably won't work" is self-fulfilling. "Let's see what happens" is rife with possibility.
I was a guest lecturer at an arts school in Vancouver recently. I taught for a couple hours, by which I mean I talked for a couple hours in hopes that some of my rambling words would find purchase in their beautiful young minds. I wanted an honest conversation about creativity and fear, and knowing this would be difficult, was surprised when the discussion turned meaningfully candid. Several times the fear of failure came up.
"I'm afraid I'll fail and it'll be too hard." It's always hard, and always has been. The difficulty is the measure of its worth, and is proportionately offset by the joy of creation. If it's easy, find something better to dedicate your too-few days to.
"I'm afraid I'll fail and have to get a _real job_." That might happen too. Hell, I went bankrupt, but no real job, especially a temporary one while you get back on your feet, will stop you from doing that one creative thing you long to do. You can still paint, write, sculpt, and start your business. Your schedule might be different than you planned, but it always is.
"I'm afraid I'll fail and my work won't be received." It won't be, and if getting negative critical feedback on your work is, for you, a failure, then you most certainly will fail. And you will fail over and over—unless you change your definition. What about making work that pleases you, in the full knowledge that there are millions out there that won't like what you create? The Beatles did. So did Warhol, Picasso, Steve Jobs, every artist you've ever admired, and, God help us, there are people that don't like what Mother Teresa did either, and I'd call her life a work art if ever there was such a thing.
To be blunt (and this is directed at myself first): Suck it up, princess. The only failure is to not do. The real failure is to rob this world of the contribution only you can make, and to fail to make work that truly gives you that "this is what I was created to do" feeling that has no equal—not drugs, alcohol, shiny stuff, or a lover's kiss. Having children might rival it, but I'd argue that's an act of creation, and one of the most primal. To do otherwise is to have failed, to admit that the praise of others is more important than your own work. It's a betrayal of your own voice in favour of the voices of the critics, and there is nothing I can think of that will send the muse packing, or kill her outright, faster than this.
Other failures sound more legitimate than this. They aren't.
"I'm scared I'll fail and my work won't be what I hope it to be." It won't. Get okay with that now. It will be different. Sometimes it will be more than you ever imagined; it will surprise you to the depths of your being and the muse will whisper, "See? I told you so." Sometimes it will be less than you hoped and you'll learn something while the muse, still whispering, says, "One more time, this time with _feeling_." And you'll do it again. Have you ever noticed how some novelists or songwriters seem to tell the same story, in different ways, over their entire career? My favourite writer, Chaim Potok, did so by refining his story through different plots and characters until the day he died. He didn't repeat himself; he built on his previous work. He evolved, and changed, and so did his writing. Nothing is _perfect_ the first time. Or ever. Nothing.
A friend of mine used to run the Canadian studio of a brilliant and well-known story factory. You've seen their movies. We spent a long Thanksgiving weekend away last year, camping in a Jeep on the west coast of British Columbia's Vancouver Island, talking long and late about creativity. There might have been wine. One of the long conversations we had that weekend was about the Too Perfect Theory, here capitalized for emphasis, not because it's necessarily a real thing with a real name. But it should be. I once studied sleight of hand and the art, and artifice, of illusion. One of the notions that informed this art—and when done right, it is art—was the idea that an illusion could be _too_ perfect. When the illusion was too perfect, the audience rescinded their willing suspension of disbelief. In fact the most sustained moments of wonder I have ever experienced have occurred during Cirque du Soleil shows, their wires visible for all to see, but my wonder at their floating in air so much greater. And my friend echoed this. In the world of animation, anything too perfect was disbelieved. Why? The real world doesn't work like this. We find beauty, for example, not in perfect symmetry, but _near_ symmetry. So not only is _nothing_ perfect, there's little reason to aim at it. Perfection, so free of what it means to be human, resonates with few of us, and the things that do resonate as _perfect_ are imperfectly so.
Failure? It's a question of definition, but it's going to happen, and it's going to take us, kicking and screaming, no doubt, and with broken bones and bruised egos, to better work. It will, to take us back to the idea that creating the artist is the first order of business, make us better, more human, creators. It'll make us happier too. Not to get too Zen about it, but if it's true that _what's in the way is the way_ , then accepting the essential failures is a beautiful part of, and not separate from, the creative life.
The only failure is not bouncing back, not learning from the thing. And I don't mean necessarily the big stuff. I mean the song that you've been pouring your heart into that just didn't land with the audience the way you'd hoped, or the exhibit of photographs that fell flat, or whatever creative project you took a risk on only to find it taking a turn you didn't expect. Expectations are a dangerous thing, and if we can allow our creativity to take us on an adventure (the definition of which includes detours and the shuttling of plans to the wayside), we're likely to find that those unexpected roadblocks, the ones that looked so much like failure when we were approaching them, look in hindsight like course corrections. Painful course corrections, if we're honest, but how many of us really learn the most valuable lessons the easy way? We label things as failures when our view is too shortsighted to see the whole picture.
You won't hear applause at every practice, nor get kudos and positive reviews on your crappy first drafts. Those are places we allow ourselves to fail safely, privately, and that's the only difference. You don't get to do all your failing in some dark corner, far from the world's eye. And when you do fail, assuming you get up and try again, it is only your pride that gets hurt, not your art. When you feel you've failed and never find the courage or the will to get up again, then it is your art that gets hurt at the expense of your pride, and that's a high price to pay for something so naturally part of the creative process as failure.
Failure is the testing of ideas that have yet to find their best expression. Buckminster Fuller said, "There is no such thing as a failed experiment, only experiments with unexpected outcomes." Miles Davis said, "Do not fear mistakes; there are none," which I take to mean that while not everything we do has an intended outcome, if it does not stop us and it gets us to an unexpected place of creativity, it is not to be feared. It still comes back to a fear of the unknown, which I believe we can learn to accept and embrace if we hold on to things a little less tightly.
## CHOOSING YOUR RISK
There are no guarantees as we head into the unknown. There can't be.
If you want to create something new, whether that's a novel or an unconventional life, there is no getting around the risk, and anyone anywhere that sells you something using the word "risk-free" is lying. There is always, always, a risk. Risk of doing. Risk of not doing. The question is not whether or not there is a risk, but what the risks are. A friend of mine has always been risk-averse. Now that I think about it, many of my friends have been. They don't travel. They don't quit the jobs that are quietly killing their souls. They don't step out and follow their dreams of becoming a musician, or novelist, or that guy that sells his house to sail around the world. It's too risky.
Too risky? Life is risky! And lest you think selling your house and sailing around the world is risky, how about the risk of not doing the one thing you've dreamed of since you were seven years old reading _The Kon-Tiki Expedition_ , and dying with regrets instead? How about the risk of teaching your children that following your dreams is less important than remaining safe, going to college, and dying unfulfilled? Sure, there's a risk in taking your kids out of school and teaching them yourself while you travel. But is it greater or worse than the risk of leaving them in a class of 30 other students, with an exhausted teacher, surrounded by homogeny? That's for you to decide.
The chronically employed see a life of self-employment and entrepreneurialism as too risky, not safe enough. Never having had a real job as an adult, I've seen friends lose their jobs, betrayed by the safety they felt they had by betting on the nameless, faceless man running a corporation created to make money, not to care for them, and I see chronic employment as too great a risk for me. I'd rather have the freedom to change with the economy, a freedom most companies don't have, and by the time they change course it's too late: time for layoffs. I'd rather bet on my own ability to learn, to succeed, and yes—to fail and bounce back. I control the amount of my bet, and I know what I'm betting on. I can't avoid risk, but I can intentionally choose the risks that come with the life I desire.
It's risky to leave the job you hate. It's also risky, and at so high a cost, to stay there at the expense of sanity and soul. How many nights can you lie in bed staring at the ceiling, replaying conflicts and demeaning conversations with the boss? How long is your life that you can wait another five years before you cut your hours back so you can invest them instead in your own business?
It's risky, too, to write a book, cut an album, or put your things into the back of your truck and set off across the continent, which I did in 2010, unable to reconcile myself any longer to taking the risk of it never happening. So I sold every piece of furniture I had, relinquished my lease on a nice apartment, and piled my cameras, laptops, and clothing into a 1993 Land Rover Defender with a rooftop tent, and set off down the west coast from Vancouver to take a year and circle the U.S. and Canada.
I wanted to experience things I would never experience at home, find some new stories, and meet new people. I wanted to camp out in places like Monument Valley and the coast of Texas along the Gulf of Mexico. So many people told me they wished they could do what I did, and said so with such longing, never seeing that they could do it as readily as I could, but would have to make similar choices, taking similar risks, as I did. In the end I suffered no major breakdown, met no horrible end in some dark corner of the continent, and didn't get sick. Nothing I was warned about happened.
Instead I was in Italy, the Land Rover parked in Atlanta for a month, when I fell off a 30-foot wall onto concrete below. I landed, like a cat—or a ninja, if you prefer—on my feet. And then I crumpled, which is what you do when you've fallen that far and landed on your feet, shattering both of them and cracking your pelvis. I ended up in the hospital in Pisa after a dramatic rescue and an ambulance ride, where I spent four days before my medical evacuation could take place and I was finally (and very heavily) sedated and put on a medical jet home to Canada.
After 40 days and nights, and surgery on my feet, I was sent home, able only to crawl, to recover at my family home, learn to walk again, and finish my fourth book. A hundred warnings about the risks of taking a journey on which I was happier than I'd been in years, and not a soul told me to be careful in Tuscany.
You just can't know. I've done assignments for clients in places like Haiti, Bosnia, Democratic Republic of Congo, Ethiopia, and El Salvador. All places the U.S. State Department counselled its citizens not to visit. I'm Canadian, so I guess I get a free pass. And in all of them I was fine. I go to Tuscany to teach photography and I'm painfully broken. I will never walk the same again. What arrogance to assume we can know the true cost of anything we do. What loss to put aside the things we long—with all our hearts—to do, fearful of the risks, as if we have the first idea what those risks might even be.
The imagined risks may never happen. And we're sidelined by the ones we never in a million years might have anticipated. I'd rather take the risk of being broken all over again than to sit safely at home only to be diagnosed, far too early in life, with cancer and be surrounded in my final days with family, friends, and bitter regret.
It's not about avoiding risk. It never is. Because the risk is always there and always truly unknown. There are no safe bets. What there is, and always has been, is choice in the face of the unknown. You follow your heart and the best wisdom you can find in the light available to you, and then you choose. Intentionally, wholeheartedly, and knowing there will be fears and doubts, and parts of it will be scary as all hell at times. To do otherwise—to play it safe—is delusional, because safety is an illusion. "I can't risk it" is the way we talk when we've abdicated responsibility for making our lives extraordinary, a thing we can create intentionally. It's what we say when we lack the balls to say we choose to do A over B, knowing there are risks inherent in both. Life, and art, is about choosing.
The best things in life are discovered after walking through gates clearly warning of so-called risks ahead. Love. Art. Investments. Adventure. Upon seeing those signs, it's harder to turn back, when you know that the moment you turn your back on those risks, you see signs warning of the risks of walking away. And you don't always know. And you won't always make the right choice. But you have one life to live and one chance to live it to its fullest, and to teach your children and friends to do the same. Don't you dare wait until it's crystal clear; it will never be. Nor should you choose to ignore your heart for the sake of what others expect or because it's a little easier to do so. It will kill your soul. The soil in which you plant your seeds will be mediocre, and the fruit will be bland to others and bitter to you.
## LIVING ABOVE THE 45
My friend Dylan is a gifted animator. He's extremely talented in that Renaissance Man kind of way that would make you want to hate him if he weren't so damn likeable. A few years ago he was speaking at a creative event in Vancouver and I popped in to see his talk. He talked about living past our comfort zone and it kept me awake and thinking, all the way to Bosnia where I was heading to photograph for a client in the rural areas around Sarajevo.
The thing about Dylan's talk, the thing that shook me up, was that I thought I knew better. I thought I was living the life he was advocating. Hell, I was about to sell everything I owned and spend a year driving around North America in my 20-year-old Land Rover. If anyone was living above the 45, and I'll explain that in a moment, it was me. Turns out you have to put yourself there every day, because the 45 is relative to who we are, and that keeps changing.
Dylan described "Living Above the 45" in the following way. Imagine your life on a graph. On one axis you've got the opportunities and challenges we undertake or submit ourselves to. On the other axis are your abilities and comfort level—essentially your perceived ability to handle the opportunities and challenges. When the two are equal to each other, the line bisecting the graph is at 45 degrees. Everything at that line on the 45 is in balance. More bluntly, it represents stagnation, because growth only happens when the opportunities we create or seize outpace our talent, ability, or comfort. To further abuse an already ill-fitting metaphor, biting off more than we feel we can chew is the only way to grow in our capacity to chew more.
It's above the 45, and only above the 45, where growth happens and where we stop repeating ourselves and create something beautiful, important, or good. When creatives and artists get stalled on the idea of making money with every project and paying the rent, they abandon the muse, because the muse doesn't give a hot damn whether you make money. She cares about making something beautiful and honest, about creating something that will outlast us. And while there are too many people that will put down a few bucks for something mediocre, there are as many people willing to put down more for something amazing, something beautiful, something that took risk and honesty to create. So take the risk and trust that it'll pay off.
It's when we live above the 45 that we begin creating things for the very reasons for which we stayed below the 45. It applies to more than just creation in the artistic sense. It applies to raising children, growing a business, and keeping the flames of a relationship lit and raging. Life is lived most vitally above the 45. To hell with balance. Leave that to the mediocre, the uninspired, and the uninspiring.
At the risk of flirting with presumptuous inspirational nonsense, are you living consistently above the 45? Are you one step ahead of your fear or has it been a while since you even considered its presence? Are you growing or stagnating? Moving forward or back? I ask because my own answer is not always Yes.
On the day I heard Dylan talk I was, as I said, heading out on assignment to Bosnia. I was packing up my home to live the life nomadic when I returned. To all appearances I was above the 45, but in significant ways I wasn't. I wasn't pushing my craft. I wasn't leaning into the fear. Not the way I thought I was, anyway. What looked difficult and fearless to others was comfortable to me. What is above and below that 45-degree line is relative to each of us. And it's ever-changing, which is why it's so easy to wake one morning to find ourselves stagnant. We didn't move an inch, but the line shifts, slowly, ever higher, and one day we realize we've been living below the line instead of above it.
The magic rarely happens within our comfort zone, but outside it, on the ragged, scary edge, where we have to fight like hell to keep from drowning in the unknown. This is where most of us create our best stuff, have our most adventurous thoughts, and feel the most alive. No one lives above the 45 by accident. You wake up every day and decide, not to wait for inspiration, but to work, to do the best work of your life, even your life's work. You don't sit around waiting for your real life to begin, because those that do will find it never comes, or some other unexpected horror comes first to wake us and our waited-for dreams slip away.
Now is the time to feed your hunger for freedom, for beauty, for meaningful work. It's not, Seneca said, that we live for too short a time, but that we waste so much of it.
## PRETENDING TO BE BRAVE
A handful of years ago you couldn't swing a dead cat, forgive the expression, without hitting a kid wearing a t-shirt that read, optimistically, NO FEAR. You don't see a lot of those shirts anymore, probably because the people wearing them died doing things their fear might otherwise have prevented or, more likely, they realized the propaganda didn't live up to the reality. Ironically, I suspect more people wore the shirt out of a desire to fit in, a backhanded way of saying they were _afraid_ they wouldn't. No fear, indeed.
Fear has a place in our lives, much the same as pain. Put your hand in the fire as a child, you get burned. Unless you're a very slow learner, you won't do that many more times before it occurs to you: I should stop doing that. Pain is a good teacher, fear makes the lesson stick; it stops us from getting burned again.
As we grow up, the same thing happens over and over again with different flames. In second grade we sing a song for show-and-tell and are rewarded with ridicule. It'll be a long time before we sing in public again. We venture to give an opinion in class and are called out for being stupid. We take our first risk in love and have our heart broken. We try a game we've never played, fail the first time in front of others, and discover what our fear will tell us for the rest of our lives: it's safer not to try.
Fear wants nothing more for us than our own safety, and if safe is all you long for, then your fears will serve you well. But I know very few people who—once safe—long to remain there. Stay there long enough and paralysis sets in.
Because the creative life is lived, necessarily, on the edges of "I haven't done this before," it is lived in risk. Step outside the safety of the rules, into the beautiful anarchy of creativity, invention, and expression, and we step, with both feet, into the unknown. We step into the unfamiliar and our fears rush in with panicked voices of warning. Stop! Remember the last time you did that? Rejection! Shame! Failure!
The greatest obstacle to the creative life is not fear itself, but what we do with it. I know no one, creative or otherwise, who lives without fear. The challenge is not to find a place free from fear, but a way to put fear in its proper place. For some of us that means first acknowledging the fear and calling it out of the shadows.
When I was 24 I hastily married, one-half of a desperate union of two ill-fitting people trying earnestly to be years more mature than we were. Six years later we were divorcing, and the friction and hurt had made me, and probably both of us, very angry. A friend at the time suggested anger was connected to fear and suggested I make a list of my fears and see if any of them were really as bad as they (my fears) kept promising, usually late at night when my world seemed to be collapsing on me with greater consequence than it did in the daytime. I don't know if he had in mind an actual list, but I grabbed a pad of paper, a pen, and a cup of coffee.
The coffee was cold by the time my list, a couple pages long, was complete. It could have been shorter but I've never been known for brevity. Just seeing my fears called by name was helpful. But more helpful was the moment I pulled them, one by one, into the light to size them up. In each case I discovered their voices more frightening than the thing they represented. Each of them preyed on my past in some way, and found their power in what I did not know. I was losing my marriage. What else would I lose? My friends? The respect of my family? My financial stability? I had no idea, and it was not knowing that scared me so much.
Fear isn't easily turned back, but it loses its power quickly when you shine a light into the dark areas from which it whispers. Leave it whispering in the shadows and it grows, in part because it's always rooted in some truth, some actual possibility. We've been hurt before, and we remember the very real sting of the pain. Our fear only reminds us of that pain and points out the possibility of being burned by the same flame again. It's because these fears find their fulcrum on a small corner of truth that they get such leverage and are so hard to ignore. If they were complete bullshit, we'd ignore them outright. But they aren't. They might be right. What if . . . ?
After I made my list I looked at each fear. I asked questions like: Really? So what? What's the alternative? In each case the fear became manageable, helped by knowing none of this would actually kill me, and that I truly had no choice but to move forward, at least not if I wanted to survive emotionally. And where the fears still loomed, I discovered a courage I didn't know I had. Yes, I might lose friends, but I'd keep those who mattered. Yes, I might have a long financial recovery ahead, but I wouldn't lose the only real asset with which I made my living: my creativity. It was in that single word, "but," that I found my courage.
Whatever our fears, I know no one who has found a way to live without them. Courage is not an absence of fear, but an act of the will to move forward in the presence of fear. Fear whispers, "You might . . ." Courage rebuffs it with, "Sure, but. . . ." To seek a fearless life is not the same as seeking a life of courage.
If we're talking about story, which is about nothing if not life, no one gives a damn about fearlessness. Very few great stories move forward with a fearless hero. Why would they? Not only is there nothing to gain from a hero without fear, there's not a single one of us out here in the real world who resonates with that. Instead, we resonate with courage and the very real struggle to find it. There's a great line in the movie _The Ghost and the Darkness_ , which is on my mind now because at this moment I'm sitting in a tent in Tsavo, Kenya, where the story about the infamous man-eating lions took place. Michael Douglas's character, the hunter Remington, on the eve of the hunt in which they hope to kill the lions, walks off to join the Maasai warriors dancing around the fire, telling his friends, "I'm going to join them now, maybe try to convince each other we're still brave." One of the men says, "I wouldn't have thought bravery would be a problem for you." He replies, "Well, you hope each time it won't be . . . but you never know."
The world resonates with courage, and the will to press on, not with fearlessness. Because we resonate with it so much we've come to glorify it, making it something we feel somewhat self-conscious about attributing to ourselves, like humility. It's the same with the word "Art." Ask an artist if they have courage and they'll say, "No. And I'm not really an artist." But they do. And they are. We've made more of it than we should, so it feels a little unattainable. Call it determination if you feel more comfortable being determined. Call it stubborn if you're more the self-deprecating type. Call it whatever you want. What matters is that you acknowledge your fears, hear them out, and when they point to some painful possibility, you harness your will and move forward, regardless, into the unknown, which is the only place in which we do our work.
Fear is speculative. Nothing more. It uses the word "might" a lot. As in, "You might die a horrible death." Fear is a little melodramatic and is very poor at looking forward with anything but guesses. If it did look forward with any clarity it would also see regret for the things it holds us back from: things untried, work unaccomplished, words unsaid. And we would, I think, fear those much more. We put off writing a book, starting a new business, or singing in front of others because we fear the same sting of rejection or failure we've felt before. Fear holds us back. Fear does not pull us forward with the same strength when there's as much reason to regret not doing those same things. What of the fear of regret? Regret from missed opportunity, missed glory, missed chances to make of our lives, even our smallest moments, something astonishing, important, or helpful?
Fear is better at looking back at past hurts. They're over. Done. They aren't coming back except when we bid them to. If you want a voice that looks forward, I think you'll only hear it in the brave whispering of courage when it says, "Yes, but. . . ."
If you want to make fear a more positive force, then words credited to Lao Tse Tung might be helpful: "What's in the way, is the way." Steven Pressfield, the author of _The War of Art_ , less obliquely says our fear points us towards the very thing we ought to be doing. The greater our fears over some new venture, the more urgently we should be walking in that very direction. "Yes, I know, but I'm scared . . ." Exactly. Do it anyway. Rewards are greatest where the risk is greatest. If you can't be brave, pretend you are. No one's looking. We're all too busy pretending we're brave ourselves.
## LISTENING TO VOICES
We all hear voices. We're surrounded by them, and as social media becomes more a part of our lives, they're becoming more numerous, less personal, and much, much louder. Besides the constant squeaking of media, there are voices from the past, from peers and colleagues, from loved ones and that place inside, out of which pours an endless stream of conflicting voices, alternately cheering us on and pulling us back. Hearing voices, barring some psychotic episode, is not the problem; it's choosing which ones to listen to and which ones to ignore.
It would be so much easier if we could simply box these voices into categories, embracing the good voices, ignoring the bad. But life isn't often so easily dichotomized. In fact, a constant stream of encouragement from the wrong sources can be as hindering to us as giving no heed to well-meaning but critical voices in our lives. Both can keep us well and truly anchored in mediocrity.
My mother told me, from as far back as my earliest memories can take me, that I can do anything I want to. She encouraged me to let that belief inform my actions, and I still believe, with some well-founded exceptions, that I can in fact do anything I want to. But when my hopes, brief as they were, to become a doctor collided with a complete inability to make sense of math, she was equally encouraging: I could do _anything else_ I wanted. By that time I'd lost the desire to go into medicine, in part because I didn't enjoy anything associated with that path: not biology, not chemistry. What I loved was art and language, and anything that allowed me to think outside the box in a way that math didn't seem to allow.
Our parents' desires for us can as easily hold us back. Without seeing that my gifts and aptitudes lay here instead of there, my mother might have insisted I keep at it, told me not to "waste my time" on my photography or drawing, or whatever creative outlet was keeping my attention at the time.
The words "Keep at it!" can push us forward or hold us back. I think the same is true of more critical voices, like the teacher, father, or coach for whom no performance is good enough. As a child it's hard to filter these voices out so rationally, but as adults hearing those voices now, they can either help us aim higher or discourage us to the point of resignation. It is not the voices that mean a damn thing; it's our reaction to them. It's not what people say; it's what we hear.
"What did he mean by that?" is a great question when we hear those voices. But perhaps more important is, "What do I do with that?" In the case of a friend who constantly praises your work, the intention is clear, but if you're honest about wanting to be better at what you do, you need to ask if that voice is really helpful. Perhaps an enthusiastic friend will make you feel better about your work when what you need is someone who's able to give you actual feedback. In other words, listen to Mom when she puts your art on the fridge and tells you it's the most beautiful thing she's ever seen, but don't let that stop you from hearing your art teacher when he tells you it's time to move past the stick figures. One voice tells you to keep doing it, the other tells you to keep doing it better.
So what do we do when the voices aren't so kind? Ignore them! To hell with the naysayers! Not so fast. Even if it were possible to so easily ignore the critics, I think we need to hear what they say before we so easily dismiss them. Even jerks and morons have a point once in a while. Each time one of my books has come out, I've waited nervously to read the Amazon reviews that eventually get written. Most of them are great, written by very obviously kind and intelligent people with great taste. Those are the positive reviews. A very few of them are less positive. One guy called me "fat, bald, and ugly," to which I took great exception, because I'm really not that bald yet. The personal attacks are easy to dismiss. But to receive the praise of strangers as unthinkingly as I dismiss their criticism is, I think, a touch on the hypocritical side. Or at very least it's missing a chance to learn something.
Once the book's written, it's true that I can't exactly make the changes some of these critical voices call for. But if over the course of several books these critical voices (the sane ones, at any rate) all point to a similar weakness in my writing, then they've given me something the cheerleaders have not: a chance to learn. If what you want is to be praised, then read the good and leave the bad. If what you want is to beat yourself up, then do the reverse. But if you want to learn, give the voices a chance, and sift the helpful from the harmful before you take them too personally.
On the extreme ends, there will always be sycophants and assholes. I suggest we ignore them both, but the moderate voices should be given a hearing. Unless they're all sycophants and assholes, in which case you're in trouble. Cheerleaders have one job, and critics have another. It's wise to know which are which. But in the end, there is one voice that you must listen to above all others, and that is your own. The only problem is this: our own voice can be as negatively critical, or as blindly positive, as any voice you'll hear outside your own head. Still. . . .
Sylvia Plath is quoted as saying the worst enemy of creativity is self-doubt. More than likely a result of hearing so many voices for so long, self-doubt paralyzes like little else. It's fear in a different guise, and I don't know of a way to get rid of it. We could spend a lifetime trying to unravel the voices that led us to this place, but who's got the time or the emotional resources to do so? We can do all the self-affirmations we want, and for some that might help, but I've tried, and while one part of me is saying, "You can do this!" the other part of me, the one that's supposed to be listening to the affirmation, is saying, "I doubt it." So when the emotional impasse has me completely exhausted, I ask myself a better question than "What can you do?" I ask, "What do you _want_ to do?" Or on better days, "What _will_ you do?" And then I try.
And maybe I'm right—maybe I can't do it. But then maybe I can and it won't turn out the way I expect anyway. Maybe people will love what I create, but it will fall short of my own hopes. Or maybe my own hopes will be met and the critics will abhor it. The outcome is a guess, at best. What is guaranteed is that if I never do the work, I'll never have a chance to enjoy all the things that process brings to me. The photograph I make might thrill me, or it might disappoint, but the joy of holding the camera, or learning something new, or allowing the disappointing image to lead me to a better one will never be mine if I allow my doubts to stop me from trying.
The difference between someone who serves their muse and their work and someone who serves their ego is that the one is willing to listen to self-doubt and move forward despite it, and the other stops short. One will do everything to protect the work; the other will do anything to protect themselves. And the life creative is never—ever—lived without frightening, intoxicating risk.
We all wrestle with self-doubt. It's a natural companion to the humility that keeps us teachable and honest. It's when self-doubt spirals downward into self-pity that things begin to go dangerously off the rails for the creative spirit. Self-pity isn't humility: it's indulgent and arrogant. I have seen it enough in myself to know its toxic effect.
Self-pity gets its momentum from the assumption that all this—life and art—should be easier for us than it is, that creating anything of any value should come without effort or cost. It comes from the assumption that our own creative process should come without the same struggle faced by every other person who tried to live creatively and make something the world's never seen before. We should get it right the first time. Why? No one else does it that way. What makes us so special? We've paid our dues longer? We've got fancy business cards with the word "professional" on them?
At the risk of being too direct, the muse doesn't give a damn about anything but your willingness to feed her and honour the process with your sweat. Life is not easy, or even fair. The muse doesn't stick around longer or give you inspiration that burns with a brighter flame because of past success or how much you had to give up to get to this point. She cares about one thing alone: the art you make with your life. She doesn't care how bruised you become, how much sleep you have to lose, or how much criticism you have to bear in order to get there.
Maybe I'm being too harsh; it could be that some of us just don't know that creation comes out of chaos, and with tears, frustration, and too many do-overs. It could be that our self-pity comes not from pride but from ignorance. We just didn't know. Everyone else makes it look so fucking easy. I get that. But now you know.
Seeing only the final work of the so-called pros in any field of endeavour can leave us feeling inadequate. Reading the best-seller and not seeing the years of rewriting and the piles of rejection letters beside that now-empty bottle of gin makes it seem easy. We seldom see the final painting with any clue as to the frustration, boredom, and hours spent scraping the canvas and beginning again that it took to get there. We're presented with a final product so beautiful that it retains no trace of the difficult process. It's little wonder we spend too much time in the emotional darkness of self-pity, preferring the voice of sympathy to the voice of the muse.
It's important to understand we're not alone. It's important to know that what we do when we conjure something from nothing is hard for us all. None of us has a sterile process, and I suspect if we do, we're not creating much of value. A sterile process creates sterile art, which moves no one. It's the difficulty of the process itself that makes us who we are, and from which flows our best work. It's that struggle that give us the best chance of getting past the obvious, the cliché, and the done-before, and into more honest work. Among the voices to which we need to listen are those of other artists. Not only the voices of triumph that come when some created thing is born, finally, but also the curses and mutterings and frustrated sighs that come during the labour. Imagine the terror of a new mother who has no idea that childbirth is going to take so long, and hurt so much, and that this is the way it's always been. It hurts no less when we know what's coming, when we know that this is the way of things, but it takes away some of the fear.
There is a difference between pain and harm and I think that being unable to see that difference sits at the bottom of why we fear some things so unnecessarily, and allow that pain to paralyze us. We fear the pain of losing something we love, of humiliation, of failure, and all those million other things that jolt us awake at night, because not only does it hurt, and hurt is unpleasant, but we've come to know hurt and harm as the same thing, which it is not.
Lying on my massage therapist's table with tears pooling on my pillow as she strips apart layers of scar tissue in my legs, I feel pain, but no fear. Fear is absent because I trust my therapist and I know that the pain is temporary and leading me somewhere better. With every visit I leave her office straighter, and walk with less pain; I know it will be the same this time, even if the sensations in my legs at this very moment are sharply painful. I think the key word there is "trust." I trust her to heal, and not to harm.
But when we equate hurt with harm, and where trust is absent, we fear. Our survival instinct kicks in, and it's either fight or flight. A lifetime of equating hurt with harm, and responding accordingly, has created well-worn pathways in our brains, pathways that have to be consciously re-mapped if we're ever to see hurt and harm as different things and take Nietzsche at his word when he tells us, "That which doesn't kill us makes us stronger."
For most of us, it is not the pain that hurts us. It is the fear that arises from that pain, and the ways in which we live from that fear. Our hearts are broken, we feel great pain, and the first wall around our heart goes up almost overnight unless we make a conscious decision to leave the battlements unbuilt. We try, we fail, and we do not try again for fear of another failure. And in not trying again we lose a chance to succeed at the thing that—like most things in life—doesn't come to anyone on the first or second try. If we're not careful, our responses to the things in life that hurt us can cause a wound far more harmful than the initial hurt ever could. If we're to fear anything, it should be that.
Art is hard. So is life. I think one of the reasons it's so hard is that making art, or creating anything of worth in life, reveals who we are. It tells us about ourselves. Much as we long for our work to be a stained-glass window, it proves just as often to be a mirror, and it's unflinching. That's hard. It offers all kinds of reasons for fear, for procrastination. But there is also the knowledge that our work reveals who we are to others. Hard as it is, vulnerability is powerful. It changes our relationships and gives depth to anything we touch, and our art will be best where we allow ourselves the trauma of transparency. The sooner we care more for what we create than for the opinions of others, the sooner our art will become bigger than us.
Back to the voices. As Shakespeare's play _King Lear_ comes to an end, the next to final words are from Edgar. Almost everyone else is dead, a result of all the lies and insincere words spoken to please the ego of a king, and he says: "The weight of this sad time we must obey: speak what we feel, not what we ought to say." Not one to miss a chance at pointing out the moral of the story, Shakespeare is urging us to speak, not from obligation, but from our hearts, and his final curtain falls on a stage full of corpses to underline the point.
As creatives, if not simply as human beings, among the voices we fail to listen to are our own. And when we do, we often tell ourselves the things we feel we _should_ say, and not what we long to say. It might just be time to sit down for an hour, or a day's retreat somewhere, to pour yourself a cup of coffee or maybe a stiff drink, and be honest with yourself. No one's listening, so please, get it out.
Get what out? I have no idea. That's up to you. For me it was an admission of all the things I truly wanted, but for one reason or another was ashamed to admit I wanted. It was an admission that I _wanted_ to write a best-selling book. I _wanted_ to make a difference with my work. I _wanted_ to travel the world. I _wanted_ to make the work I _wanted_ to make and not the work a client asked me to make. And yes, I _wanted_ to make money doing what I loved. In smaller ways, they are admissions of what I want from my creative process—and allowing that to channel my work according to my desires and curiosities instead of allowing my marketing niche to become a creative rut. So many of us have been taught not to speak like this, not even to think honestly like this. Just who does he think he is, anyway? What makes him so special? Like crabs in a bucket, we tend to pull the ambitious ones back in.
This book comes from my honest search for a life that is genuine, passionate, generous, and full of the things, activities, and relationships I want. My own list of desires and hopes and ambitions for my life and work will be different than yours, as will my skills, talents, and life experiences. So the way in which I live my life, and the path I bash out of the jungle to do so, will be different than anyone else's. It begins with being honest about what I want.
Whether you're Mother Teresa or a Machiavellian opportunist is not my concern; the world is full of people who long for good things, and those who seem bent on making it to the top at the expense of others. It is what you long for and what price you're willing to pay, or allow others to pay, to get there for which others might have good cause to judge you, not that you hunger deeply. That you long for something more is the fire that will fuel your efforts to find it; denying that longing will only dilute the fuel. Deny it long enough and you'll find the fuel has no capacity to ignite. Or else it will have completely evaporated. The desire will still be there, but your strength to do anything about it will have long dried up.
To hear your own voice—your longing, your ambition—you must first give it permission to speak. I think this permission comes to us naturally, but I suspect it's been gently (and sometimes not so) beaten out of us. Children can seem surprisingly abrupt to adults; they speak their minds, and give voice to their desires and opinions, until they get taught that it's better to be polite than honest. We learn early to play from what Ralph Waldo Emerson called "the rulebook of nice." That rulebook, unwritten and unspoken, but known well by those of us that aren't simply sociopathic or completely oblivious to others, has one purpose: to stop the boat from rocking. The most obvious side effect is the silence of our voices, especially those that dissent.
And that's the problem. We've learned to keep our mouths shut on anything that takes us away from normal, or the status quo. We avoid rocking the boat at all costs. And if we keep those mouths shut long enough they either lose their power to speak at all, or they can remain shut no longer and when they finally open they do so with such unrestrained power that they harm more than they help. We don't need more voices that parrot the things already said, or that reinforce the mantra, "But we've always done it this way." We need voices willing to ask, "What if?" We need voices, both as a society and as individual human beings, to dissent to what is not working, to refuse to be part of the suffocating homogeny. And, to be more positive, we need voices unashamed to say, "I love you," or point out what is right, and beautiful, about the people around us.
That's the voice of anarchy I'm arguing for: the voice that refuses to sing from the rulebook of nice and instead chooses words that are more honest. In so doing we wind up speaking something better than "nice." We speak words that are kind, because if our words aren't honest, then they really aren't nice at all. I'm not talking about being "brutally honest." I don't believe in brutality, honest or otherwise. One of the kindest things we can do for those around us is to live honestly, free from hypocrisy, and give others the permission to do the same. From that place of honest, respectful existence is where it becomes possible to live the lives most closely aligned with our ambitions and talents. It's there that freedom and contentment and gratitude find their best voice and loudest volume.
I know I'm repeating myself, but: what do you want? What do you truly want? Getting what you want may seem impossible. It might, in fact, be impossible. The way may not seem obvious right now. But how can we know without first admitting our hunger? This might be one of the hardest things we ever do; for some it's opening a box we've fought hard to keep closed. Let this one out and you may never get the box closed again. You might fight for the rest of your life to meet those longings, fulfill those desires, realize those ambitions. God knows there are voices enough telling us we can't, or we shouldn't: the price is too high. The failure too humiliating. And what about the kids?
What about the kids? I'm not a parent, but there has to be some value in children growing up with parents that keep telling them to live their dreams and are, themselves, not too scared to live their own. Too risk averse. Or worse, keep using the kids to hold them back. No one is suggesting you leave home and abandon the kids to pursue your dream of being a professional NASCAR driver. Just admit that the dreams are there. Listen to that voice. Maybe, for now, it's enough that you share that dream with your kids, and take them go-karting on weekends. We rarely have any idea where things will lead, what possibilities will open up to us. But if you keep that voice muted, refuse to hear it, then the possibilities remain closed.
All of us hear voices, and those voices fall anywhere on the spectrum from dark and harmful to being full of light and love. Hearing those voices is not the problem; the problem is listening to them without sifting the truth from the lies, and listening to them at such volume that we drown out the voice most important in creating something beautiful of our lives: our own.
## INSPIRATION
Inspiration comes from work, or so it is said. But what if we've got this whole "inspiration" metaphor wrong? If there's something to the language of this, then the word "inspire" means to breathe in. A deep breath is invigorating, to be sure, but I don't know anyone who takes one deep breath, then goes to work while holding it. "I was inspired!" we say. Great. But you're turning blue. Breathe out.
Creative people look for inspiration like it's the Holy Grail. It is not. It's a practice, and that practice is directly tied to work. If an inflow of ideas and thoughts and what-ifs are our act of breathing in, it is at work we breathe out, clear the lungs, and make the repetition of the process possible.
The most creative people exhibit this ebb and flow of constantly inhaling and exhaling. They put their thoughts into sketchbooks, they make photographs with their cameras, they sit down and write, or sit at the wheel to throw a pot, understanding that one thing leads to another. It is not so simple as waiting for a great idea to come out of the blue, the gift of the muse we call "inspiration." It's no one's job but yours to breathe—not the muses, angels, or anyone else. Only you can do that.
I think there's something good about this idea of our ideas coming from a source beyond us. I think it shows a certain humility. But it comes from a time when we credited the gods with everything. On one hand, it allows us to avoid hubris, but on the other it allows us the luxury of avoiding responsibility for our own creativity. After all, if the gods don't deliver, it's hardly my fault.
Our brains are astonishing organs. They do things we don't yet remotely understand, though we're coming to that knowledge, and the more we know, the more amazing it is. Steve Jobs said that creativity is merely connecting the dots, which is a good picture of what the brain does. Our thoughts and ideas are the connections between dots in our brain. Those dots are there because we've collected them in a lifetime of experiences, and the more dots we collect, the more connections are possible. That collection of experiences is our inspiration, our breathing in. Playing with those connections—with paint, words, clay, programming languages, wood, or whatever medium we prefer—is the way we exhale. In and out. Repeat.
It's a cycle. We may not find genius in each cycle; it might take many cycles before the best of our ideas are revealed; A+B might lead to C, and C+D might lead to E. But you might need a good deal more of the alphabet before you've got the raw materials for the poem within you. And you might not know the poem is even there until you've unearthed all the letters, played with the words, and discovered it there. And when you do, it won't be a bolt out of the blue, a sudden gift of the gods, but a result of that astonishing brain making unforeseen connections out of the dots you've worked so hard to connect. That doesn't make it any less miraculous, and you can thank Whomever you like for the gift of that brain, but had you not fed that brain by breathing—both physically and figuratively—you'd still be waiting. Or you'd pass out.
If creativity is, in part, about connecting dots, then the more divergent (or unlikely) those dots are, the better, and it is the daily task of the creative to be curious and collect dots. The most creative people I know fill their brains, their idea factories, with as much raw material as they can. They have voracious appetites. They connect with other people. They read books. They watch films and take classes and go to new places and try new things. The more we increase our inputs, the more we increase possibilities. The more we cut ourselves off, insulated from new people, new ideas, new flavours and experiences, the more we limit our creative potential.
They say that if you wait until you're thirsty before you drink water, you've already started to dehydrate. The same is true of ideas. It's no good waiting until you're dry before you go looking for new input. But when you're dry, and riding the wave of your creative rhythm off the peaks and into the valleys, increasing your inputs is the only way to make sure you've got the momentum to ride that wave from its lowest point back to the peak.
We're not islands. We don't self-generate ideas. "Out of nothing, nothing comes," also applies here. Seed your garden with everything you can, soak it in, and don't try to force the dots together: just let them sit there. If ideation is about collecting and connecting dots, then it's the collecting which we can do intentionally. The connecting is a passive act of incubation, and that takes time.
## INCUBATION
Understanding inspiration as work, as intentional effort, flies in the face of popular meaning, which has more to do with fairy dust and implies that ideas just come out the blue, which they do not. At the best of times, when everything is going well and the cogs are oiled, it might feel that way, as if one moment, with our heads vacant, we were looking around, glassy-eyed, for an idea, and the next moment our minds are swimming with ideas that will change the world. In these moments, thank the gods you've been ingesting new ideas and increasing your inputs, because those, combined with the incubation of time, are what got you here. Fairy dust has nothing to do with it.
This is why so many artists have weighed in on the power of setting to our work to find so-called inspiration. It was the poet Charles Baudelaire who said inspiration comes from working. The painter Chuck Close said inspiration was for amateurs, the rest of us get to work. Because when we get to work we set the cogs in motion and our brains, like boats, are easier to steer while in motion. It's setting ourselves to work that combines the two halves of a messy process—the generation of ideas and the execution of them.
Surprisingly, while we're taught that ideas lead to execution, it's more complicated. Closer to true is that it's cyclical, and a couple of things need to be happening simultaneously. First, we need to be putting as much fuel on the fire as possible. That's increasing our inputs. Then we wait, and those divergent thoughts and ideas—the dots—incubate and eventually, unpredictably, connect and become new ideas. That's ideation. But if we understand incubation to mean we sit around watching the wallpaper peel, we'll be waiting a long time. Instead, while we wait, we work. We keep adding new dots. We work with the ideas already in play. If it's painting, we sit down in front of the canvas and start moving the brush. If it's writing, we put our bodies in the chair and do the work. We show up. We put in the time. That's the execution. And while we faithfully execute, the unborn ideas incubate. They come when they're ready.
The most creative people I know are also the hardest working. I don't think that's a coincidence. Our brains work best when the machinery in the front is working hard and letting the machinery in the back do what it does without too many eyes on it. Doing the work leads to new ideas and on those rare occasions when the idea comes out in the shower, it's only because it couldn't wait until you got around to the day's work.
Give your ideas time. Don't rush them. Let them come. Be patient. But don't wait passively because the brain that is going to connect the dots is only as strong as you make it with the day-to-day effort of your work. It's an astonishing thing, this brain of ours, but it's not magic. And on the flip side, the most creative people I know also understand that time spent staring out the window, napping, having a glass of wine, and being with the ones we love instead of working is not time wasted. This eight-hour workday that we've adopted, and one I've mercifully avoided for most of my life, doesn't serve most of us very well, especially those of us in creative work. It's true that I sometimes work 12 hours instead, so distracted by what I'm doing, so in the momentum of my flow that time passes without my notice. It's also true that there are days I sit down to write and after two hours of productive time, my brain won't go any further, and I need to go for a walk or take a nap, and those activities restore me where pushing through would only shut me down further. It's why ultimately only you can know how you work best, and the sooner you get a sense of how you work most productively, and the sooner you give yourself permission to work this way, the better.
## BEGIN
While so-called creatives are often talked about as dreamers with their heads in the clouds, I find it hard to reconcile that with my belief that creativity is about actual creation, not merely ideation. Sure, it's important to have time for great ideas to come your way, and there's a lot you can do to make sure that happens. But mistaking ideation for creation isn't unlike mistaking an architect for a contractor. Both are necessary, but neither does the job of the other.
In order to create, you have to actually make something. In _Linchpin_ , Seth Godin calls it "shipping," and he places a great deal of importance on it. Scott Belsky, author of _Making Ideas Happen_ , talks about it in terms of living a life with a strong bias towards action. And if the attributions are correct, the poet Goethe says simply, "Begin."
Few things will block the creative process like procrastination, and the more "creative" we are, the easier it is to rationalize that procrastination, to find reasons why it's not only justified, but necessary.
We do it because to honour the work means to do it with excellence, and (this is the voice of procrastination speaking,) if I spend more time sharpening my pencils, and polishing my ideas, the work will be much better. I'll start tomorrow. To quote Ze Frank, "My pencils are sharp enough. Even the dull ones will make a mark."
We do it because "we're not inspired today," so we put it off until tomorrow, never twigging to the reality that the longer we put off our work, the longer the muse stays away, watching from the sidelines while we talk ourselves out of the one thing guaranteed to bring her to our side: work.
We do it because the kids are calling or the dishes need washing. The measure of how desperate we are to avoid beginning our work lies in what otherwise onerous tasks we perform eagerly instead of just sitting down at the potter's wheel, the laptop, the canvas, and starting.
When pushed, we throw the blame at genetics. "I'm just a procrastinator," we say, which is, of course, total crap because most of us just substitute one activity for another. We're substitutors, and if we were more honest with ourselves we'd admit that we're just looking for one more thing to stand in the way so we don't have to look ourselves in the mirror and admit that we're scared of starting yet another thing that comes with the promise of failure, and effort, and the great trauma of being transparent.
I've mentioned this once, but it bears repeating. In his book _The War of Art_ , Steven Pressfield talks about fear as though it's a compass, faithfully pointing in the direction of the thing we most need to be doing. So too, our procrastination points to the lodestone of our fears, and as Pressfield points out, the very thing on which we ought to be working.
That one thing may be terrifying. You may have more to lose by attempting this than by spending an hour on Facebook, but all the friends and followers in the world aren't going to make your work better, or get it started on your behalf. And where you have the most to lose, you have the most to gain, so get started. Begin despite the fear. Begin despite your unreadiness. Begin despite your lack of inspiration or growing pile of dishes. Start.
The most prolific writers know that this is not an option. They sit down, often at the same time, in the same place, for the same amount of time, every single day, and they write. It is not optional and there is nothing more important they should be doing right at that moment. Giving yourself the _choice_ is where it all unravels for most of us.
I carve out blocks of time and put a line through my calendar, and that time becomes non-negotiable. When I'm writing, I get up at the same time, go to the same coffee shop, listen to the same music, and do my work. There's something comfortable to me about the ritual of it, but it's more than that: it's the self-imposed constraint of non-negotiability. So on days I don't feel well, I get up and write, even if it's crap. On days I feel uninspired, I get up and write, and trust that the muse, if she's out there at all, will show up. On days I've got important things nagging at me, I remind myself that they can wait, and my work cannot. If I put it off today, I will put it off tomorrow, and there will always, always, be something that begs for my attention in the peripheries of my life. Facebook can wait. So can my friends and my family. I'm not writing all day, I'll get to them later, and when I do, they can have my full attention. If I were a brain surgeon the world out there wouldn't be able to interrupt my work, the dry cleaning wouldn't suddenly need to be picked up, and online social media channels would continue to churn in my absence, while I, barely missed, did the one thing no one else can do for me: my work.
Conceiving a child is nothing remotely like carrying it to term and giving birth. They are two different, if not connected, things. Conceiving it, if the growth of our race is any indication, is pretty easy for most. Some have even called it fun. There are people that so enjoy that part of it, that they do it often. But the act of getting the seed planted, when everything works as it should, is nothing like the effort required to do the rest. Make love as often as you've got the stamina for, but don't mistake it for the whole process of creating a child. Ideas are fun to play with, but they're the easy part; they incubate on their own and require little from us but the fertile ground and the, uh, inputs. It's getting that idea birthed into the world from which we run, which is a shame, because by the time it's ready to come out and become truly something, the work becomes the most rewarding. Stop now and what little efforts we give those ideas will be born gasping for air and clutching to life.
Of course most of us know this. We don't come up with ideas that excite us to simply shelve them. Not intentionally. But life happens, and those ideas sit for a while, waiting for the perfect time to begin. The time when we've got all the details figured out, or when we just have more time. The longer those ideas sit, the greater the chance they'll never happen. It's not enough that we simply begin.
We need, most of us, to begin now. When the idea comes and you've got the seed of something that excites you, start right now. While you have some momentum, harness it. While there's fresh energy there, ride that wave over some of the fears that'll settle in when you've had time to sit on it a little. Common sense and the so-called realities of life will drown that spark, and it's the ones quenched by those voices that are the ones our own souls, and the world, most need.
Make a checklist. Sketch it out. Put a line on your calendar. Call a collaborator. Do _something_. The longer you wait, the greater the chance your idea will seem less brilliant, and that's a shame because the light of day, after you've waited a while, is not the time to evaluate your idea. Ideas rarely come out whole. They change as they get brought to life. New constraints appear, new directions suggest themselves, and new influences come to bear. The creative process is a place where evolutions and juxtapositions and unexpected mash-ups occur, but _only once we begin_. It's here the ideas show their true potential to us, often so much more than what we expected. Had we not started when we did there's a chance it never would have happened. It's not that getting to work immediately is our only chance of bringing the idea into the world, it's also that it's often our only chance of making the idea better, proving it and seeing its true genius. The initial idea is a big "what if?" But getting to work on it—and not letting it die out—is where we begin to listen to the answers and find new questions. It's there that the real work of creation begins.
## EMBRACING THE CONSTRAINTS
Creativity has always happened best within constraints. In fact, creativity has often been a response to constraints, not much more, really, than problem solving. For the artist, a lack of constraints would be paralyzing. When I teach my students to not only embrace their constraints, but to pursue them, by choosing only one lens, only one subject or theme, they find it immediately liberating, and their work becomes more creative.
Most of us do not have to create constraints; we all live and create with enough limitations. Not enough money. Not enough time. Not tall enough. Not smart enough. But there's a fine line between a constraint and an excuse, though often the only difference between them is perspective, and how we act on them. A constraint seen as a help to the creative process gets us closer to accomplishing our work. Seen as a barrier, that constraint becomes an excuse. Like so much in life it's a matter of attitude, though it's taken me over 40 years to see it.
The word "attitude" has always stuck in my throat. Flogged with the word all through school because my attitude never met expectations, it's taken me a long time to get comfortable with the idea that our attitudes matter. Psychology has a lot to say about the effect of the way we think about every aspect of our lives, not because there is magic in it, but because our thinking directs us. Positive thinking, as abused as it is by hucksters and charlatans, is a powerful force, and negative thinking is devastating.
Refusing to embrace our constraints and use them as a creative force can very quickly turn us from artists and creatives into victims. Victims don't make art; they make excuses. We don't make art in the absence of fear, doubt, hurt, financial ruin, broken hearts, sickness and violence, but as a reaction to them. It's the way of the human heart and mind to respond to what life has thrown at us and to keep processing even after the tears have run out and there's no light at the end of the tunnel. For millennia our suffering has been the forge in which great art has been made and great lives have been lived. Returning to the metaphor of the Artist's Journey, it is here we find ourselves in the Innermost Cave, facing the darkness and demons, that we are changed and it's here where we either win or lose. In my experience the innermost cave is no place for excuses and self-pity. They feel good at the time but they work against us.
We all have reasons for not creating the art—or the life—we fantasize about. Some of those reasons are grounded in reality. They're not excuses, they're just life. But they are not reasons to stop creating art at all, or for not living an exceptional life. "I can't paint like Picasso" is just an excuse not to paint to the best of your ability. If Helen Keller, blind, deaf, and mute, could lead an extraordinary life, so can you. If artists can paint with their feet, so can you. If paraplegics can climb some of the world's highest mountains, so can you. Your biggest handicap is a constraint, not an excuse.
If all this sounds like so much chest pounding, then perhaps we could all use a little more chest pounding. I don't subscribe to the idea that our limitations and circumstances stop us from living lives that become art and inspire others, because art isn't always in the finishing but in the trying. You can live a great story without ever reaching your goal, because the journey is in the trying. It's the struggle itself, and while you may not make it to the top of Everest, for most of us it's inspiring enough that you tried, and the story—your story—will still play out in unexpected ways. The story that ends exactly as predicted is the story of the one that doesn't try, for whom their constraints are defining limitations that keep them from ever beginning their art, starting their adventure, or writing their story. All of us make our art in the context of a tough and unpredictable world, full of heartbreak and unfairness. There is no such thing as safety; none of us emerges unscathed. We make our art despite, and even aided by, our constraints, never in the absence of them.
## PROCESS VS. PRODUCT
Read much about the art of writing and at some point you're bound to wonder if a certain amount of masochism is required to enter the field. One reader on my blog, commenting on the way I'd been so transparent about my own frustrations (in this case concerning my photography), told me he was no longer interested in reading my blog because my angst exhausted him. And here I thought I was making it look easy. Making art is hard.
One writer said writing was easy: it only required sitting in front of a typewriter and opening a vein. Indeed. I think most things in life, the things that really matter, require us to bleed a little. To put ourselves so thoroughly into something can't be easy. But if it's so hard that we find no joy in it, why bother? For the joy of completion? For the praise of others? Those seem like poor incentives, given how often a work—whatever it is—never gets completed, and how seldom our finished work finds truly honest and lasting praise.
True, there's a thrill in finishing the work, and seeing it published, or hanging on a wall, but too much time spent nurturing an addiction to completion risks rushing through the process of creation itself, sabotaging the very work we aim to see done. It is the process itself where we discover new things, and find the first hints about new directions. Very little work becomes, in the end, the thing we imagined it to be at the beginning. Like the artist that makes it, it evolves, reacts, and becomes something more than we once expected. Unless, in rushing to the end, we miss those chances to not only take the work in new directions, but to enjoy the process, and savour the challenge. It might not be enjoyment in the same way we enjoy a good glass of wine, but it can be a deep-down sense of being alive, of being stretched, of knowing you can do this without having the foggiest idea exactly how.
The same sabotage of process happens when we create merely for the praise of others. It's true: even as adults, most of us long in some way to have our art put on the fridge and praised. There's a thrill to knowing something we've done has struck a chord with others, and means something to someone outside our own heads. Who doesn't long to be relevant, to be noticed? But if that's where you find the joy, and not in the creative process itself, then it's as likely as not that you'll sabotage your own work.
Creativity carries with it, necessarily, that sense of "this might not work." And freed from that, and from the frisson that comes with risk, putting yourself out there, and into this thing you're making, whatever it is, the work loses its spark. There can be no guarantee that anything we create will be praised, or even understood, so to labour through a process you do not love, and in which you find no joy, only to create something that may never bring you the adulation you want, or need, seems a waste of the few, uncertain days we have on this earth. Better to find something you love doing, and do it for the love of it, than to work so hard for an insatiable ego making something that might never feed it. Even when we do make something that strikes a chord, praise fades quickly and has diminishing returns.
Creation is work, at times hard work, and the product of our creative process often yields a low return on the investment. We sure as hell better love the process, and find some joy in the struggle itself because we'll spend much more of our lives actively creating than we ever will looking at the final piece, or hearing how good it is from the lips of others. Pragmatically I'm arguing for more than just a feel-good love of the labour, though that's reason enough to create. A preoccupation with the end product of our efforts takes us from the present moment in which we need to give ourselves over to the process, and robs us of the very thing we long for: finished work that's bigger or better than we dared hope for. It is this way whether that work is a story, a painting, or raising a child. Art is created in the present, where nothing is guaranteed to us but the process of making it. If we stay in that moment and enjoy the full experience of it—if not because of the challenge then despite it—our work will be better for it.
## MORE BAD IDEAS
There might be a reason why some writers prefer to exercise their craft while following Hemingway's advice to "Write drunk; edit sober." Whether the approach is healthy or not, I can't say, but the ability of a stiff drink to lower our inhibitions has been celebrated by many in their search for good ideas. Now, let me state categorically, that very few of my best ideas have come from a bottle. I wrote one of my books while recovering from two broken feet, and I can tell you the morphine only made things more challenging for my editor. However, alcohol does more than just keep the liver distracted. It has a way of quieting the inner censor, which I think was the point Hemingway was making.
Almost all of us have an inner voice whose self-appointed role in our lives is protecting us from ourselves and our bad ideas. The moment it catches wind that we're working on something new it begins dropping the filters into place, trying desperately to stop our half-baked ideas from getting out into the world and embarrassing us.
Without this voice we might be saved from some small social faux pas and a few awkward moments, but also from some truly great ideas. But we'll never fully know because those ideas never had a chance to become what they might have been. That inner censor may be very self-protective, but it's rubbish at understanding how ideas work. Free of this understanding, it unknowingly kills a great many beautiful notions before they ever have a chance.
Consider this truly trite but relevant metaphor: you love butterflies and want one very badly. Without understanding how metamorphosis works, but knowing very well that you do not like caterpillars, you step on each one you find. It could be a long time indeed before you get what you're after. Dead caterpillars lead to very few butterflies. I warned you it was trite. But I'd rather my metaphor work for you than be particularly poetic or profound. Bad ideas are like caterpillars.
Good ideas, on balance, do not simply appear from the ether. Ignoring for a moment the role of work and so-called inspiration, they come from the relentless generation of many ideas, and of those, most are mediocre, if not outright lousy. I'm not suggesting all ideas are good ideas, but they are useful. Because it is the mind that is encouraged to take the bad ideas, suspend judgment for a time, and follow the trail to see where they lead in order to find more good ideas. An idea may in fact be completely unworkable, but combine it with another impractical idea and you might have something astonishing.
It's important to understand how metamorphosis works. Creativity has been described as the simple ability to make unlikely connections. These are the dots that Steve Jobs talked about, and here it is quantity that matters, not quality. The quality comes, not from the dots themselves, but from your ability to see, and follow, the combinations of connections. It comes from a relentless ability to collect these dots and keep them around, without judging them good or bad. One combination may be less workable than another, but if working through the less workable solutions is what got you to a connection between others, changing them from two or three so-called bad ideas into one beautiful idea, then the bad ideas aren't bad at all, just necessary raw materials.
It would be a mistake, however, to assume the inner censor is purely negative. It's often the voice saying, "That's a dumb idea, stop now." But get on a roll and it'll be the one saying, "That's a good idea, maybe you should quit while you're ahead." Either way, it's just a voice and there's no reason in the world not to tell it to shut the hell up. To return to Hemingway, you're drunk (metaphorically), you're writing, you're on a roll. Don't you dare stop now; you can stop later when you sober up and begin to edit. But right now, order another (metaphorical) drink and keep going. Connect those dots like a man on fire. Now is not the time to be tentative, to worry about your reputation, or to fear failure.
In my younger days I did improv comedy, from which, at the expense of paying audiences to whom I now feel somewhat apologetic, I learned more about the creative process than I did about acting. One of the guiding principles of improv is "Say yes." Go with it. Whatever you do, don't say no. No stops things dead in their tracks. No blocks possibility and makes for very short, very unfunny improv. Ideas move _forward_ very well. It is their natural state to move quickly, to change when exposed to new ideas, and become something else. Without the forward momentum of Yes, they die.
I carry with me a Moleskine notebook. It goes everywhere with me, along with my camera, and both the camera and the notebook are full to busting with some very bad ideas. My process with the camera is to create visual notes, or sketch images. They're the ones people will sift through after my death, shaking their heads at the vast amounts of mediocrity I was able to create in only one lifetime. But they aren't really mediocre, just misunderstood. They're raw materials. They're the record of me playing with lines, light, balance, tension, moments, and the way my lenses change all of those. In my notebooks the so-called bad ideas are the record of my playing with thoughts and words. That play almost always begins with old, repeated, or obvious ideas. But it often leads to the favourite question of every curious person: What if I . . . ?
You can't judge the raw materials the way you evaluate a finished product. As a novel, it might stink, but as a first draft it's full of possibility and questions. As a painting it might not work, but as a series of sketches, there's a new idea that merits exploration. As a meal it might be truly offensive, but if there's a new combination of flavours and textures that will work brilliantly in something else, then it's part of your creative process. We ought not be so hard on things unfinished, and that includes ourselves.
It's because the raw materials differ so much from the final product, and are so necessarily undeveloped, half-baked, and yes, even stupid, that we need to silence the censors who would stop their creation entirely. If it takes a glass of whisky, so be it.
## THE STARVING ARTIST
Embracing our constraints (and refusing to be a victim of them) does not mean passive acceptance. Debt—and other financial issues—is one of those constraints. A great deal of art has been made by the prototypical starving artist, but that doesn't mean you have to turn yourself into one. You can live an extraordinary life under the weight of debt, but getting out from that weight is an even better story.
On February 14, 2006, I walked into a trustee's office and with one signature became bankrupt. Years of debt had spiralled out of control, to the point where even with a good living at the top of my career as a comedian I couldn't get any further than the minimum payments, and the interest was piling up. I was given mountains of paperwork to fill out and on the line that said "Reason for Bankruptcy," I put one word: optimism.
Going bankrupt wasn't my first choice. I'd gotten on top of my spending, but even without adding to the debt, the interest was accumulating. I'd gone in for credit counselling and, after laying my embarrassing financial situation out, was told I had two options: a credit proposal that would allow me to renegotiate my debt, or bankruptcy. I chose the credit proposal. I was then sent to a trustee to deal with the paperwork and she also told me that I had two options, but this time the options were different. Because I was self-employed (as a comedian, no less), I had no chance of getting my credit proposal approved, so my new options were these: go bankrupt this month, or go bankrupt _next_ month. Happy Valentine's Day.
Years later I can see two things with much greater clarity. The first is how much I allowed the clamour of a culture of conspicuous consumption to dictate my spending habits, which led me into uncontrolled credit spending, as I knew my prospects were improving imminently. They weren't—at least not faster than my spending. The second is how much freedom I now have, and will always have, because I will never go into debt again. I know I was told as a kid that borrowing money made you beholden to the lender but clearly I didn't absorb the lesson in ways that were anything but theoretical. I'd somehow accumulated an astonishing amount of credit debt with nothing to show for it, and that debt had begun to severely limit what I could do. I had traded against my future for the shiny toys of the present, and when the future showed up I was still paying for the toys, now broken or forsaken in favour of something newer, also paid for by my future self.
Talking about money is not easy. But it has to be said, though I'll keep it short. If it helps, think of this as a bit of a distracted sidebar. In the broadest possible sense, the appetites of our culture have outgrown what we're able to swallow, yet alone afford. Though time is not money, it's like it in the sense that most of us have only so much of it. We can't afford everything. But the things we hunger for—the bucket list items and the bigger dreams—always seem just a little less immediately accessible than the larger television or the new sofa. When people say they wish they could do the things I do with my time (usually travel), they usually mean they wish they could do what I do while still maintaining the lifestyle they've chosen, a lifestyle I couldn't afford if I wanted to do what I do. I don't have unlimited money any more than anyone else forced to choose between one dream and another: we want it all. But as the prophet Jagger said, "You can't always get what you want, but if you try sometimes, you might find you get what you need." A wise man, that guy.
I read somewhere that there are two ways to get what you want. The first is to acquire more and more. The second is to desire less and less. Why this matters in the context of living our lives creatively and intentionally is that, in light of the fact that we can't have it all, every dream we have requires a choice. That choice is removed from us in the present when we're paying for the decisions we made in the past. Without exaggeration, the greatest freedom I've ever experienced is the sudden loss of the debt that was draining me. I only wish I'd had that freedom much sooner.
The problem is not debt. Debt is the symptom. The problem is our appetite and an overly optimistic hope for tomorrow's finances, against which we're borrowing. Of course the money we've got doesn't feed today's appetites, so there's no way—short of a windfall, and that's the voice of optimism, not reality—it will be enough tomorrow, never mind enough to also pay for the debt we're collecting. To say no to debt and "later" to our appetites is an act of anarchy that leads to greater freedom. But it's not easy. We measure ourselves—often _define_ ourselves—by our acquisitions. We so identify with brand messaging that we don't feel we're being true to ourselves unless we drive the right car, wear the right clothes, or use the latest, and right, computer.
The problem: it's not sustainable. It's not sustainable on a global level, and it's not sustainable personally. If you want to live with greater freedom you need to control your appetite. And if you can't do that, you need to control the spending that feeds your appetite. Buy the new iPhone, but do it with the money you have today because the debt you're adding to will last much longer than the iPhone will. And then you'll need another iPhone you still can't afford.
How many generations of iPhones can you afford to pay for in the future? Wouldn't you rather spend that money on a trip to Nepal or Paris? Wouldn't you rather have money in savings to take a year off and write the novel you long to write? Wouldn't most of us be happier with a smaller television or an older car (or neither), and a chance to swim with the dolphins or get to Everest Base Camp instead? I'm going to go with "yes," because if the answer's "no," then you aren't reading this book right now. You've long given up and are watching reruns of _Lost_ on TV right now. But none of those things are going to happen on their own.
Where were the courses on how not to be a moron with my money when I was in school? Why did I never absorb the lessons that were there? If I'm honest, it's not because I wasn't taught; I just didn't learn. What I learned was it would be wise to save my money. It would be wise to remain out of debt. But I wasn't interested in wisdom. I was interested in experiences, and I wonder what would have happened if I'd been told I could be driving a Ferrari when I was 18 if I wanted to. No kid wants a savings account. Every boy wants a Ferrari, even if that particular Ferrari isn't a red Italian sports car.
It comes back to the question, "What do I want?" In my forties I have clearer answers to that question. I don't want a Ferrari. But I want freedom. Freedom to travel. To create. To live my so-called bucket list. To do good in this world, and to help the people closest to me to do the same. I can't do that with debt.
Debt cripples me where I should otherwise be free, and I will never again be bound by it. Where money is concerned, that is my biggest desire and I work hard to remain in that place. It means paying cash for things, and never carrying a balance on my credit card. It means making choices, and though it can be argued that my income is now consistent enough that it's an easy sermon for me to preach, it should be pretty obvious that this is all relative. People of every income bracket seem to find it difficult, regardless of the size of their paycheque, to live within their means. In fact, it seems the larger the paycheque, the greater the gulf between what's getting earned and what's getting spent. Our appetites grow exponentially if we allow them to.
Whatever else it is, this is not a book about finances. But it is a book about the freedom to live your creative life to the fullest, and you'd be insane to believe that living in obedience to debt and under the tyranny of bill collectors is your best possible creative space. I also think this particular discussion dovetails well with the discussion on fear because so many of us spend the way we do to fill empty spaces in our lives, or to placate the fears: fear of not fitting in; fear of missing out; fear of the way others see us. If we're willing to face those fears for what they are and stop feeding the hunger those fears create, it becomes a little easier to get our financial house in order and stop leveraging the future to pay for our past and present lust.
If you're looking for more, all I can suggest is a path similar to the one I took. What you do with your debt is up to you, and I'm not necessarily suggesting you go bankrupt. I'm suggesting you learn as much as you can about money. Take a deep breath, swallow your pride, and call a debt counsellor. Go to the library and check out titles on personal financial management. Read books like _The Richest Man in Babylon_ , which is aging and a little contrived but contains solid wisdom. Find someone you respect who has their financial house in order and ask them how they did it. Look for wisdom. Renegotiate your debt, get rid of the credit cards that charge you criminal rates of interest, and put the other credit card in a block of ice in the freezer. Learn to live within your means: the freedom it brings you will be worth the overhaul it requires to get there.
Whatever you choose to do, don't let your optimism get the best of you. This is one area where being conservative more than you are optimistic will pay solid dividends. Things might be getting better, your prospects might be improving, and you might truly be about to "make it." Just because there's suddenly more food on the table doesn't mean you should eat more. Use the growth to be more aggressive about eliminating your debt, keep the taxman happy, and save, but not as an excuse to go out and buy something. Celebrate when the debt is gone, not now.
## THE ART OF EXCLUSION
If life is short, then it follows that there is more to do in our brief, beautiful days than we could ever accomplish.
It is said that photography is the art of exclusion. What the photographer leaves out of the frame is as important as what he leaves in. Within the frame of the photograph every element pulls the eye, and though not every element demands our attention in the same way, there is only so much impact a frame can contain; the more the included elements compete for that impact, the less impact any one part of the frame can have. That is to say, the photograph is stronger for the photographer's ability to say no.
The photographer who wants to move hearts or change minds knows that saying yes to all the possible elements and letting them all into the frame is not an act of generosity: it's a refusal to allow the most important elements to play with the strength with which they're capable.
No one can do it all, but the pressure to try is paralyzing. And so we say yes to a million efforts that pull us in a million directions, and say no to the most important things in our lives by our refusal to give them the time they need.
But we keep saying yes because we want our lives to have impact. We want to make a difference. We want to love others. But what impact can we have if, by diluting ourselves over a thousand trivialities and the tasks of others, we leave our work undone, or done with less than our full attention and energy? Saying no seems so selfish, but so be it. I don't think there are many lives so full that we can't extend ourselves generously to others, so perhaps there's a middle ground. If you have trouble saying no to others, I want you to look around, make sure no one's listening, and repeat after me: "I would love to help you. I'm in the middle of a project right now that requires all of my attention. I can give you an hour tomorrow." Now say it again (this time say it for real, because I know most of you just cheated). Does it have to be tomorrow? No. But give it some time because if it's really, truly important they'll be glad for the help, even tomorrow, and if it's really urgent, they'll find someone else before then. Do your work first.
Imagine having unlimited funds. You'd give money to almost anyone that asked, wouldn't you? But now imagine you've got money in your pocket but have no idea how much is there. How carefully then would you give it away? What would you choose to spend it on, and to whom or what would you say no? You can say no with a smile. You can apologize. You can be kind about it. But every time soneone places a demand on your time that does not serve you, your work, and the people and causes that mean the most in this world to you, they are asking an audacious thing when they ask, unblinking, to take a piece of your most precious commodity.
Guard your time fiercely. Be generous with it, but be intentional about it. Guard it the same way you guard your money. It's the one resource with which we have increasingly less to do our life's work, and to be with those we love. Say yes to those things first.
## WAITING FOR THE KNOCK
A friend of mine is a comedian. He's been making people around the world laugh for over thirty years. He's very good and very funny. And he goes home after every gig, these days after long weeks of entertaining on large cruise ships, and when his wife asks him how it went he tells her the same thing—"I still didn't get the knock." "The knock?" I once asked. "Yeah, you know, the knock. When someone comes to your dressing room and knocks and tells you they've discovered you have no talent and want you to leave." Ah yes, the knock.
How many of us feel like we're faking it? In those moments when I'm totally transparent and feeling brave, I'll tell you it's one of the two fears with which I wrestle daily: the first that one day I'll wake to find my muse has abandoned me and that I've shot my last good photograph, written my last decent sentence; and the second is that one day everyone will all wake up to the collective revelation that I'm just faking it. The fear, at least the second one, is like most fear: it's bullshit. But there's also a little truth in it.
Because I _am_ faking it. We all are. We're making it up as we go. That's what creative people in any pursuit do. It's not a part of what we do, it's the very _nature_ of what we do. We try new things, go where we've never gone, and do things for which there are neither rules nor established ways we _should_ do what we do. In the process we make a lot of mistakes, fall on our faces, and—in the case of photographers—we make a lot of really bad photographs, or sketch images, in pursuit of the good ones. The public, whoever that is, only sees the good stuff. We see it all: the crap, the dross, the chaff, and it's often the flotsam and jetsam of the creative process that we get hung up on, forgetting that every artist creates the same waste as they chase their own muse. The more creative we are (or endeavour to be), the more of it—the crap, the evidence of our faking it—we produce. It's part of the process. The alternative is following patterns, colouring within the lines, covering territory we've covered a million times before, and taking no risks.
It's easy to look at someone who fits our own understanding of what it means to have succeeded, and to assume they no longer battle these demons, but your view of any artist, or any human in any field of endeavour, is as muddled as his or her own view of themselves. You see a photographer who's made it. Published books. Worked for great clients. Created something for which they've received accolades or, God help them, awards. And you assume he no longer fears the knock, the one where they come to tell him he's finally been fingered for faking it. The thing is, he fears it _more_. Because most often the artist (and I've yet to meet one so well adjusted that I can say there are exceptions to this) just sees his success as a string of flukes, hard work, and probably a little mistaken identity. It isn't the faking it–artist I worry about; it's the one who thinks he isn't.
Why I think this matters is because when we begin to see this as normal, as part of the inner life of the artist, we can stop beating ourselves up for it. We can take the proper place of the artist—a posture of humility before the muse—knowing we are dependent not on our gifts or talents or painfully waiting for inspiration from above, but on hard work and circumstance and the mystery of the creative process.
There is great freedom in knowing we might never _make_ it, even when in the eyes of so many we already have. Or we might already be there and never see it. I suspect it's a little of both, depending on what day it is, because "to make it" is so subjective. For me the goal is not to make it but to _be making it_. To live a life of daily creation, where the "it" changes often. To every day find new ways to "fake it" and see if that leads to something beautiful, knowing that if the knock comes, the imagined accusers on the other side of the door can tell us nothing we've not told the world already and tried, every day, to own. We're faking it. Of course we are. You should expect nothing else but that we do so honestly, intentionally, and with our whole hearts.
I think there's also another sense in which we feel a fraud, and that's the issue of qualification. In most fields of endeavour there is a standard of experience or education that you either meet, or not. And no matter what anyone tells you there is no such standard of qualification to be a human being or an artist. It is enough that you are human, that you experience this life in ways common to us all and unique to yourself, and that you do your work.
Qualification matters in one sense only: the acceptance of peers. And I'd wager in the grand scheme of things, our art—whatever that means to you—is better created outside those circles than within them. Perhaps you feel talent is the qualification you lack. And you know what, maybe you're right. The fact is there are some very talented human beings on this planet—people who, through some astonishing genetic fluke, can do what they do in ways most of us never imagined. Lucky them.
Most of us will get by just fine without half their talent, and many of us will get further, by doing our work and leaving the question of talent to the philosophers. You are more talented than some, and if you're like me, you're less talented (whatever that means) than many. Fine. Now that that's out of the way, do your work. Talent doesn't qualify you to do your art. Doing your art is what qualifies you to do your art.
Of course there's a sense in which we're not faking it at all, and I know those words make some feel uncomfortable. I'm talking about the honest admission that we're making this up as we go, even if that improvisation comes on twenty years of experience and a certain expertise. Whatever you call it, improvisation makes us all feel a little nervous, and feeling like a fraud is the natural result of not only improvising, but also getting away with it. When the world at large and the prevailing culture honours road maps and implied rules, those who succeed by following their intuition and a voice that won't stop asking "What if?" are bound to feel a little like they've cheated. It'll pass. Or it won't. But don't let it sideline you. The real fraud isn't asking the question or being haunted by the doubts. Waiting for the knock keeps us honest.
## KNOW YOUR RHYTHM
Every person I know—whether they identify as creative or not—goes through ups and downs, though I think the self-identifying creative or artist can feel it more acutely, as though our creative life rides on top of the water and rises and falls with the waves. We experience brilliant highs and depressing lows. When the wind kicks up and the ocean is wild, the highs are higher, and we feel glorious, unstoppable, and they crash harder, the glory gone. Stopped.
What helps is not looking too closely at the wave, but at the ocean itself. Pull back, look at the water from a hill ten miles distant and the water looks smooth as glass—as your creative life does, or will, from a distance. The dips and peaks evened out. This helps not because it makes one bit of difference when you're at the bottom of a wave cycle and you feel like you've made your last good, beautiful, photograph or written your last honest word. It helps because it allows us to understand the cycle, to use it, to ride out the waves, even building momentum.
Our creative life, the very nature of how most of us work internally, is rhythmic. Brilliant creativity is unsustainable day to day. A wave that has a high, but is not flanked by lows, is not a wave: it's placid water. No lows, but no highs, either. We have a word for it in the creative world: mediocrity. In his book _The Accidental Creative_ , Todd Henry says, "Mediocrity is a high price to pay for a lifetime of safety." You can't have this creative life, ask for the highs, and never get the lows. That doesn't make the lows easier, but it's nice to feel normal, isn't it?
Creativity happens in the space between taking in and incubating as many influences as the world allows us, and the sudden rush of a newborn idea that comes into the world in a mix of hard work and joy, sweat, and tears. The birth of that idea, and the execution of it, are often on the crest of the wave. They are the high points for which we live. If the high point of that wave is adoration and praise, then you're missing out. Singer/songwriter Josh Ritter sings, "I'm singing for the love of it, have mercy on the man who sings to be adored." Russian actor (and originator of Method acting) Konstantin Stanislavski, said, "Love the art in yourself, not yourself in the art." But that's a digression, not really my point.
My long-winded point is this: it's in the lows of the wave where we feed inspiration. If we are conscious of the shape of the wave and the way our process works, we know that wave will crest again. What we do at the bottom of the wave determines how much momentum we have at the top. We can spend that time being depressed and feeling sorry for ourselves, or we can feed the muse, take our Sabbath rest. We can go to the museum, the gallery, the coffee shop, the library, the theatre, or wherever it is you find your own paint stirred. Forget how you've suddenly lost your brilliance. Go find the brilliance of others and let it feed your soul. Go be with your family, read a book, and then, most importantly, do the work. Don't set your camera down simply because inspiration hasn't yet come.
Riding these waves gets more predictable the longer you do it; you see the rhythm in it, you begin to know your process. I will often mumble this to myself in the lows, when I am doing the work and my muse (wretched, unreliable, prodigal muse, where the hell is she?!) is nowhere in sight. "Trust your process, David. It'll come." And I keep working, mumbling other things, less savory and less family-rated things, but I keep at it, and the movement of the wave carries me forward, pulls me upward, as it always does, and I begin to get excited about what I might find at the top, and I get more grateful for the muse (wonderful, reliable, always-present muse!).
Be conscious of the highs and lows and give yourself the grace to learn to ride those waves. It's easy to write about it, sitting here myself when the wave feels high and strong. But when we are in the lowest parts, thrashing about and choking on the surf, it doesn't feel like an inevitable part of our rhythm. It sure as hell doesn't feel like part of a process that will again pull us back to the crest of the wave. It feels lonely and dark and uninspired, and every single person I know goes through it; those creating work that is the most personal, that feels the most as if everything is on the line, feel it the most. There's no way around it but through it. But if you can hold on to a little perspective, recall the way this cycle has resolved in the past, it can give you hope. And when the lows are so low you feel your soul is about to drown, it helps a little to know that you're in the innermost cave again, and this is where you do the hard battle. Will it help if I tell you now that your art will be better for it, and your story stronger? I doubt it. But it will be. And the surge will pull you out the way it always does. Chances are we're both in a valley, separated by only one wave. We'll make it. Let's try not to swallow too much water.
## THE MYTH OF ORIGINALITY
_"Millions of men have lived to fight, build palaces and boundaries, shape destinies and societies; but the compelling force of all times has been the force of originality and creation profoundly affecting the roots of human spirit."_
~ Ansel Adams
The word "originality" has too many meanings in play for me to really trust it. The way Ansel Adams says it makes me want to stand up and cheer, but I'm barely out my seat when, out of the side of his mouth, M.C. Escher whispers, "Originality is merely an illusion," and I sit back down, glad I didn't embarrass myself.
Then Herman Melville takes umbrage, turns to Escher and says, "It is better to fail in originality than to succeed in imitation," before Ezra Pound joins the fray and shuts it all down, dismissively stating that "Utter originality is, of course, out of the question."
I leave and C.S. Lewis passes me a note on the way out. Scribbled, it says, "Even in literature and art, no man who bothers about originality will ever be original: whereas if you simply try to tell the truth (without caring twopence how often it has been told before) you will, nine times out of ten, become original without ever having noticed it."
The pursuit of originality has never interested me much, in part because until we define our terms, it seems like a moving target at which I haven't yet found a good reason to aim. By _it_ , do we mean that it's unlike anything else, a completely new thing? For some people this means more than it does to others. For the artist, it's often more important they create something apparently original than that they create something honest or beautiful. Few people, I think, are deeply moved by novelty.
But free from the need to create something novel, there's a sense in which we are free to create something faithful to its origins—which is to say, faithful to us. If "original" can be taken to mean "honest" or "authentic," then I think that's a pursuit worth discussing. "Is it Art?" isn't remotely as interesting to me as "Is it _your_ art?" Is it you? Is it what you want? Is it worth so much to you that you're willing to spend your time, effort, and probably a few tears on it? Authenticity trumps novelty every time. But it's so, so much easier said than done.
Authenticity is about one thing: honesty. Is it you? Tough question, I know. Who we are is always changing. Martin Luther once said that this life is not about being but becoming. It's splitting hairs, I know, but it's helpful to remember that we are growing into ourselves as humans, and as artists. I used to snicker when artists said their work was an exploration of this or that. But now I get it. All our work is an exploration. And exploration changes us. It opens our eyes, changes our minds, and makes us think new thoughts. I am not the person I was. My vision changes. And then, necessarily, so does my voice.
Imagine a writer. The stories he writes in his childhood will be about different struggles than in his teens, young adulthood, and the late years of his life. So too will the words change. He may, in his thirties, switch for a while to poetry. A different voice, to be sure, but not necessarily less authentic. In fact, he could discover in poetry his most authentic voice because it allows him to say things in ways he never could in his novels. My favourite writers, like the characters in their stories, change with the arc of their lives, and so too do their voices. But the best of them, the ones that resonate, remain authentic. Genuine. To be authentic is not to be homogenous.
Chasing authenticity is like chasing originality. Spend too much time doing it and you'll lose sight of the thing you were aiming for. It helps not to be too self-conscious. It's the kind of thing that's seen peripherally; it's not seen so easily when you look at it head on. Don't overthink it. Explore. Play. Follow your gut. You'll know when it's you and when it's not. But don't mistake the goal. The goal is to make work that is consistent with who you are and are becoming, not who you once were. Repetition is not the same thing as consistency. It's as easy to be inauthentic by not keeping up with who you are, as it to be so by copying others. Copying others is helpful, even necessary, as we learn, but not so helpful when it comes to making our art. You just have to know the difference. Learn to write like Hemingway. But when you write your novel, adapt what you learn to who you are, not the other way around. Learn to play guitar like Bruce Cockburn, but then write and play your own songs, which will be better for what you've learned.
Don't let the pressure to be original paralyze you or steal your joy. Our highest art is making a life that aligns with who we are, real and whole, if not messy and a little rough around the edges. Originality is a chimera. It's hard enough being you, and being vulnerable enough to be so without hiding behind masks and walls. Put that into your work and your relationships, and the art of your life, and it will be original in the truest sense: it will be you. If that's not good enough for the critics, to hell with them. You can't please everyone.
## RUTS & GROOVES
There is a state of creative process, a place in the rhythm of our lives that psychologist Mihaly Csikszentmihalyi calls _flow_. It's a way of working, a state of being when we're fully immersed in what we're doing, fully present, and the usual barriers to our work just seem to step aside. When flow happens it feels like time stops, only catching up to us again when flow stops and it comes hurtling back at us, and we find it's hours later than we thought it was.
Flow is an enviable state, one in which most of us wish we spent more time, but if flow happens only at the crest of the wave, then as much as it's possible we should stack the deck and make sure that wave comes around again by increasing the inputs, exposing ourselves to divergent influences, having more conversations, allowing for incubation time, and most importantly, doing the work; it's impossible to remain there or force it to come around again according to our own timing. Flow doesn't happen when we chase it down. It seems to prefer to be wooed. Flow is not just the coming of the muse; it's when she arrives with such force that she blows the doors off.
Flow is a groove that channels our energy and allows us to put our foot a little harder on the gas without spending quite so much energy worrying about the steering, which in a state of flow feels intuitive, subconscious. But flow comes and goes and it's good it does because the only thing it takes for a groove to become a rut is repeatedly taking the same path, which is tempting when we long for a return to flow more than we long to do the hard work. Ruts require no steering, and the deeper they become the harder it is to climb out. Ruts happen when we look to what's been done before instead of what's untried. They happen when we rest on past victories instead of going out to meet the enemy on whatever battlefield we find him today as we begin our work.
In ruts there is no risk to our creative pursuits, except the risk of repetition, to which I've never known the muse to call me. Life is about change. Art is about change. And if our work changes us, it takes us to new places. Ruts are a sign that we're in the same safe place. No, it wasn't always safe. When we first got there, there was risk, adventure, and that energy translated to some of the best work we've ever done. We made bold moves. We felt the flow. And we made camp, not noticing that the flow, as it does, moved on. We went to bed in a groove, and unwilling to move on, woke up in a rut.
To tie this loosely to the theme of the book, ruts become a controlling force to which we either submit and allow our creative efforts to suffer, or in whose face we rebel, turning the wheel hard in another direction, refusing its rule and jumping the rut. I wish I could tell you how to capture flow and keep it in a bottle, but that's against its very nature. The only thing is to keep moving, don't look back, and do your work. And don't spin your wheels, it'll only dig you in deeper.
## WINNING AT YOGA
My immediate world, the world of photography, has a strange obsession with competition. Wherever I look there's a new competition promising big shiny prizes to the best young photographer, travel photographer, wedding photographer, portrait photographer, and on and on. It's not a phenomenon unique to photographers, I know. I'm not sure what the word "best" means in the context of art, but there's no doubt that artists of all stripes have been competing in one way or another since the first two cave men compared the size of their pigment-stained handprints on the wall of the rock hole in which they lived: mine's bigger than yours. Indeed. We're a competitive race. No doubt it comes from the necessarily competitive nature of survival; it's probably thanks to that competitive streak that we've come so far. Go, team, go.
But the creative life is something different and I'm not sure it benefits from that more primal need to be the best. I'm pretty sure anchoring art—which calls us to something more as a species—in competition only holds back the art. It binds us in the very areas our art should be liberating us.
The most enduring art has so endured, in part, because it has the indelible fingerprints of the artist on it, or within it. It is produced from somewhere within a unique person, expressed through some level of competence in their chosen craft, and pushed into the world because the artist could do no other. It may lie unrecognized for generations, may never find public acclaim, may never earn a dollar, but it's uncompromisingly their own. Competition encourages us not to look in but look out, to create not first to please ourselves or express some ineffable thing, but to please another. What magic, I wonder, do we attribute to these judges that they can pin a ribbon, a score out of ten, on this work and not that? What criteria can they possibly be using? Imagine Monet and his impressionist colleagues enduring the mockery of the artistic elite, the Paris Salon (which they did). Imagine our loss had they listened to the harsh condemnation of this new way of seeing and expressing beauty and packed it in, gave up.
Whatever benefit competition has in fuelling a hunger in us to move forward and do our best, it is outweighed by its power to be self-sabotaging. The artist longing to create, the mother longing to raise a child, or the entrepreneur beginning a new enterprise all do well to cast a suspicious eye at competition because making great art is hard enough without comparing ourselves to others. Living an intentional life and making art of our lives has nothing whatsoever to do with what others are doing. That's a distorted mirror in which we'll never look anything but misproportioned.
You don't win at art or life any more than you can win at yoga. They are incompatible paradigms. Comparing ourselves with others plants seeds of envy, jealousy, discouragement, doubt, and fear. It gives way in one direction to self-loathing and in the other to arrogance. In neither of those directions is our best work created.
If competition is natural, part of a long-residual DNA, it does not follow that it is what is best for us. I'm not sure that giving in to every one of our most fundamental urges is healthy. Revenge is one of those urges. It's why forgiveness is so hard. One is instinct; the other is a grace. To be human is to choose, and choosing to compare ourselves with others and base both our beliefs about ourselves, and our actions, on those comparisons is a waste of creative energy—energy that could be used to create something.
Physically, we tend to move in the direction we look. Comparisons are made looking to the side or backwards, but not forward in the direction we long to move. It's a good way to trip over something, or worse, to find ourselves in a rut we never chose on our way to a destination we never set our hearts on, while the thing we most long to do drifts off, divergent, into the distance.
Singer/songwriter Bruce Cockburn put it well when he sang, "Can it be so hard / to love yourself without thinking / someone else holds a lower card?" Let it go. Right now there are millions of people creating astonishing lives and accomplishing things neither you nor I can imagine. I will never be Einstein, or Mother Teresa, or Richard Branson. But wishing I were is the fastest way to make sure I will never be the one thing at which I have a real shot—being fully and completely me, someone who no one else on this planet can be.
I don't yet know the man I am capable of being or the things I am capable of achieving. I do know that the answer won't be found out there on the path of another, no matter how much I covet their talent, their health, their money. They have their own battles to wage, their own constraints to keep them moving forward.
Make your art, and allow yourself to be inspired by theirs. Life is too short to worry about how you stack up: it's not a race. The reward is in the work itself, and the discovery in that work of the person you're becoming.
## ART AS GIFT
One of my favourite acts of rebellion in the great narratives of history is the Hebrew flight from Egypt thousands of years ago, under the leadership of Moses. The biblical version goes like this: tired of the tyranny of Pharaoh and the effects of slavery on the people he comes to lead, Moses says let my people go, Pharaoh says no, and then God wipes out every firstborn as the final act of an escalating series of plagues. Gruesome stuff.
Fast forward to the desert. The escaped Hebrews—up to a million strong—are now wandering aimlessly, and hungry. So God provides a food they call manna, a word meaning, "What the heck is this stuff?" Seriously, that's what it means. Though I doubt the word "heck" is a literal translation.
Manna was a flaky food, and while it's called bread, it seems that might only have been the closest thing to compare it to. God provides it daily. Enough for everyone. But there's a catch. With the exception of the Sabbath they're told they can't store it. They have to trust that it will be there the next day. And the next. And the next. Eat it while you have it, because it turns putrid overnight.
I think there are some intangible things in life that come to us from beyond ourselves, that are meant to be exercised and used as we're given them, with no stockpiling allowed. I think faith is like this, whether its object is God or other people. Love, too. Hope, certainly. And creativity.
Creativity is like manna: it doesn't store well. It comes from somewhere outside ourselves, a gift where every ounce of it is meant to be used, without thought for tomorrow. Don't pace yourself, don't stockpile, and don't hoard it. Use it while you have it. Act on it while the idea makes sense. Remember that creative inhalation (inspiration) can only happen with the same frequency as exhalation (execution). Creativity grows with the expenditure and shrinks with the hoarding. Try to hang on to it without doing the work and it'll turn to dust. Good ideas build on each other and lead to new ideas, but keep them in a box and even those ideas fade away. Use it (now) or lose it.
The source of our creativity is like water. From a spring or deep well, the water keeps flowing. We take what we need; it flows on. It'll be there when we need it. But put the water into a cistern, fearful the well will dry up, and that water becomes stagnant. It's got to keep moving. The gift comes to us, we make something of it, and in turn it makes something of us, and what we make touches others and keeps moving, accumulating the fingerprints of those that touch it. At some point my analogies fail me, but if you want to be more creative, don't hold too tight. If you want to live an extraordinary life (whatever that means for you), let the gift keep moving. A closed heart is the same thing as a closed mind.
## NOW
It was Annie Dillard who said the way we live our moments is the way we live our lives, a reminder for which I'm grateful because when we splice our lives into units like years, months, and weeks, it's easy to let the smaller moments that make up the now slip past us.
Every act of creativity is both cause and effect in our lives—it is the thing we make and a thing (among other influences in our lives) that makes us—it is a very present thing. Past failures from those efforts, especially, are helpful; in many ways they teach us better than success ever could. But together with our triumphs, they are over and the person we were is changed; in effect, gone. Replaced by the person we are now. Failures belong to the person we once were, and so in effect, they belong to another. We take responsibility, we own our flaws, we learn from our mistakes, but we move on. I do not mean that we forget the past, simply that we do not hang on to it, which I'm the first to acknowledge is a fine line. My past is the part of the story I've already written, and I'm grateful for the memories. It's part of my life that no one can take from me, even the failures. I wonder, if we placed more value in our missteps and mistakes, would we live more freely from guilt about them? How would our relationships and our creative lives change with the freedom to be more present?
Neither do we live in the future. Though I don't mean by that that we do not plan, dream, or hope as we look into that direction. The idea that we live as if today were our last has some merits, but if it turns out that we make it into tomorrow, we'll wish we'd given it a little thought. What we do today prepares us for tomorrow, and living an intentional life includes intentional planning. But just as looking back with regret or guilt to a past we can do nothing about is a waste of energy we might better apply to our lives and relationships in the present, so too does worrying about a future that has not come take us out of the moment that is now. If we're not careful, worry and regret will keep us from ever having a present moment, and if Annie Dillard is right about our lives being nothing more than an accumulation of our present moments, then there's a good chance many of us could go our whole lives with very few present moments, and find at the end that we've lived lives that were long but not deep; alive but not living.
In practical terms, the libraries and internet are full of ideas about how to live in the present. But as helpful as all these tips are, they commonly miss a discussion of the very things that stop us from living now. I wish I had something easier to tell you, but I know of only two ideas that take us there: forgiveness and faith. These words carry some baggage, and are often used in a religious context. That's not how I mean them. I think both ideas stand on their own regardless of whether you identify yourself as religious, spiritual, or neither.
Forgiveness is about freedom. You can choose to hang onto guilt or your anger, and you may be justified in doing so. But you can be right or you can be happy. You can live in the present or the past. You can't do both. It's easy to reject forgiveness, easy to hang on because the hurt goes too deep, the offence unforgivable. But that's what forgiveness is for: the unforgivable stuff, the things too painful and evil to imagine. Find a way to forgive (not forget) or don't, but know that clinging to that past prevents us from living in the present. It's our choice. The chains that bind us to our past are only as strong as our grip upon them. Is it easy? Of course not. But forgiveness, whatever it does for others, is a key we use first for our freedom from the tyranny of our past.
Faith is harder to talk about. By faith, I do not mean blind faith that it'll all work out, because it just might not. It might end in tears. It might end worse than that. I used to joke that if 99% of the things we worry about never happen, then clearly worrying works. But it doesn't. It fills us with fear and paralyzes us. It often creates a self-fulfilling loop that's hard to escape. It poisons our art, our relationships, and our lives. Perhaps by faith what I mean is an openness to the possibility that we can handle whatever comes our way. That what doesn't kill us makes us stronger or gives us something to blog about. Faith that while not everything happens for a reason, we can find or make meaning in it, and make our lives and art stronger for it. Faith that pain does not equal harm and that what has been true of every hero in every story since the dawn of time is true for us and that if we triumph we will return a new person. I say "if" because if faith is to have any value at all, it must be honest. And we don't always triumph. And we don't always make it out alive. Even if we live 100 years, there will be one final day in which we breathe our last. But it might come sooner: it's inevitable, inescapable. The moment we accept our mortality we are free from the fear of it, and thus free to live fully in the moment. Our story had a beginning; it will have an end. Have faith that until that end comes, we are not powerless in the writing of our own story.
But what if something horrible happens? I won't be the guy that tells you it won't. It might. Something horrible happens in every great story. That's life. It's what we've got. We've also got the will and the creativity to fight that something horrible, to make art of our struggle, to laugh honest laughter and cry honest tears in its presence. Wishing for safety won't bring it. The present is the only time guaranteed to us to live our lives and make our art. All we have is now.
## TOWARD MASTERY
I'm uncomfortable with the word "mastery" in the same way some are uncomfortable with calling themselves an artist. The word itself implies a pinnacle, and I get nervous about heights these days. But as an artist and a human being, I hope all the same that I'm on a _journey_ towards mastery of the tools of my craft, and of my own life. Whether I get there is, I think, a matter of perspective, but it's not what matters most to me. It's the journey that matters. There may not even be a destination.
I do not practice Zen, but I see something beautiful in Zen teaching, particularly where that teaching offers wisdom about living in the present and learning to see. There's a word used in Zen teaching: shoshin. It means "beginners mind" and reflects an internal posture towards life, a way of thinking. It says that the mind of the one who believes himself to have arrived, to have become an expert or master in something, is closed to possibility because by definition he believes there remains nothing to learn. It is the learner—the acolyte—who believes that no matter how much he learns, there is always more. So if we walk this long path to mastery only to come to a place where we learn there is much more to learn, the journey is more a cycle than a line. That gives me great hope.
As a photographer, my art doesn't come from the camera. It comes when I'm open and receptive enough to life that I see moments for what they are instead of what I wish them to be. There's a kind of humility towards life that is necessary, a willingness to suspend my own expectations of what will happen, and watch it as it unfolds. The expectations and sense of what _should_ happen only blind me to what is. The more I let go, the more I see, and with greater clarity when not seen through the filter of those expectations.
When we think we know what's coming, we prepare to react, and often set ourselves into motion prematurely. We commit to a course of action that's hard to back out of, mentally or emotionally, and we lose the freedom to react with clear intent to what actually comes our way. Being present and receptive—keeping a beginner's mind—are the eyes with which we see as we walk toward mastery.
There's a beautiful Zen tale told of a farmer who, together with his horse, had worked the land for years. One day that horse disappeared and his neighbours came to him to console him. "This is terrible," they said.
"Maybe," was the farmer's reply.
The next day the horse returned, and with it brought three wild horses. The farmer's friends were astonished. "This is amazing!" they said, "How wonderful!"
The farmer replied, "Maybe."
The next day the farmer's son fell from one of the wild horses while trying to tame it, and in his fall broke his leg. "This is terrible," the friends lamented.
Again, the farmer's reply was, "Maybe."
The next day the army marched past, looking to draft young men. They passed by the boy, unable to serve because of his broken leg. Again the friends chimed in and told him how lucky he was.
"Maybe," said the farmer.
This story makes the farmer seem emotionally indifferent, but I think we can chalk that up to a cultural difference and the reason for which the story was told. The lesson is that we see what we want (or expect) to see, and if we suspend our expectations, we see more clearly. The first work of the photographer is not to use the camera, but to see; it's the same for the artist, the poet, the father, and the entrepreneur. Our lives, and our art, are more free and intentional when we hold our expectations with an open hand and see with greater receptivity.
## I WILL
I told her I had a few things left on my bucket list. She told me my life _was_ a bucket list. I pulled my pen from my pocket, scribbled that down in my dog-eared little notebook. It seemed clever at the time. Like that one time when a friend told me she thought of me as Indiana Jones with a camera, a thought that made me smile for a week. Who doesn't love Indiana Jones?
What makes me recall these details so fondly is that I, of course, want these fantasies to be true as much as anyone else does. I'm surrounded by friends who are one martini away from being James Bond, or one adventure short of being Ernest Shackleton. My friends are hard to connect with. It's recently taken me a couple weeks to get something I needed from a friend because he's been in Antarctica, and the window was tight because he was on his way to Iceland. And when it's not them, it's me; there's as much a chance of us meeting in an airport somewhere on the other side of the globe than here at home. Home is a transferable concept, most of the time. But it's an intoxicating way to live when that same poison doesn't kill you. I don't know one of us who, living this kind of life, wouldn't claim to be the luckiest man or woman on the planet, and I don't know one of us who hasn't been told so a hundred times.
So it makes it hard, when you feel so lucky, so truly kissed by fortune, to have to waver a little and place credit where credit is due: not entirely with fortune, as grateful as we are for her brilliant, meddling ways, but squarely on our own shoulders. It makes it hard because no one likes feeling cocky or ungrateful.
But putting aside for a moment the fact that all of us live our lives floating on a raft that rides the waves and currents of circumstances over which we do not have full, if any, control, it's the foolish traveler who doesn't fashion a rudder to do whatever he can to get where he longs to go. The great American thinker Henry David Thoreau said the mass of men lead lives of quiet desperation. I think they just feel lost on the raft, desperate because the current is too strong, the waves too big, and it's never occurred to them to make a rudder and push across the current. It's never occurred to them to instead live a life of desperate intention.
When you awaken to the truly heartbreaking brevity of life, your heart quickens. It's easy to tape Carpe Diem to your wall, or jot in your journal Mark Twain's encouragement to throw off the bowlines and sail away from safe harbour. But without intention, it will fade, and the boat will stay safe in the harbour. Without action, we don't so much seize the day as slap it on the shoulder as it passes us by.
I don't think we're fundamentally lazy. I don't think it's that so many of us don't make it from re-Tweeting extraordinary quotes to living extraordinary lives just because we never got around to it. I think many of us don't know what's possible, let alone permitted. Raised in a culture that honours the patterns, we went to school, got a job, got married and had kids, and it's only somewhere after that that our souls begin to ask if there weren't other options we might have taken. And now we feel stuck, constrained. If only people would live their lives as creatively and intentionally as they create their art. But then maybe it feels safer to take a risk or two with art than it does with our families, our finances, our futures.
No, the reason so many never got around to it, I think, is because they never knew they had the option. And now they feel stuck. If only they had some notion of how easily we could all climb out of debt if our appetites shrunk. It's not the want of big screen televisions and new cars that is the problem, it's that we want them more than the life we're always telling others we wished we could live. We want to go to India but it seems so far off. Easier to spend $2000 on a new laptop, the one we think we'll need to write our novel when we one day get around to going to India. Except that the new laptop just put India so much further away. It's not wanting this bigger life that is the problem, it's that we're choosing a smaller one every day by the accumulation of little compromises and thoughtless activities that, by now, feel like a rut, and we can't have both. We never knew the other was so possible.
But it is. Most of my friends right now are somewhere amazing on the planet, doing something astonishing, and they're just like you and I. Like you, they struggle sometimes to pay the bills. They have kids and families. They have the same fears and health concerns. Yet there they are. Mongolia. Antarctica. Venice. In the coming year and a half I'll spend time with grizzly bears in British Columbia, and polar bears on Hudson Bay. I'll photograph the snowbound landscapes of Hokkaido, Japan, and the nomadic pastoralists in northern Kenya before coming home to spend a week dogsledding on Baffin Island, then diving with whale sharks in Mexico. And the dirty little secret is—you can do this too, or whatever your version is. You can. And you're either choosing not to, for a million reasons, or you're choosing to do something else. But it's a choice all the same.
There is so much power in the human will. I don't want to make this seem so simple that I'm called out for being a dreamer, but it's amazing what happens when you look at the calendar a year from now and, with a red marker, put a line through three weeks and make a plan. Set a budget. Save your money. Make sacrifices. Sell something you don't use. Put off replacing that iPhone (tech is the worst way to spend money). Put 10% aside faithfully. Tighten the belt. Book the flights on points or start looking for seat sales. Cut the grocery bill by 20% and get creative with lentils and beans. Much of the world lives on a fraction of what we do.
I know. Before you say it, I know. This isn't realistic. You're right, it's not. And it's not realistic for me, either. It never has been. Not for any of us. And most people that have lived their short lives on their own terms have been told so time and time again. But still they do it. They find a way. They find creative ways around the troubling persistence of reality, their doubts, fears, and whatever constraints they have. They don't wait until conditions are perfect.
Conditions are never perfect. It could be they fail nine out of ten times every time they decide to cross the threshold of that next brilliant adventure. But they do it, don't they? And when they don't, when it all goes south, they have stories, not excuses. The year after my accident in Italy I still managed to set foot on six continents while nursing broken feet in the process of healing. I wasn't brave, and I wasn't stupid. I was just stubborn. I want to see this astonishingly beautiful planet and all her surprises, and I'll crawl to see them if I have to.
Whatever your next big adventure—raising your kids, launching your business, seeing the world—do it intentionally. Do it with boldness. Sacrifice what you have to. Cut off the prevailing voices that tell you how impossible it is. It's why I stopped watching television: too many advertisements telling me what I needed to live the good life. Those voices will tell you that you can have anything you want, as long as it's a car, something shiny that the neighbours don't have yet, or anything else, as long as it keeps you shackled to the job that's slowly digesting your soul. I know what I need to live a good life, and it won't be found on television. The story that means the most to me will not be acted, but lived. I don't want to watch _Indiana Jones_. I want to _be_ Indiana Jones (or my aging version of it). I promise you, it is possible, but it won't happen accidentally. It happens intentionally, with many a failure. As beautiful as the words "I wish" are, they're impotent next to the words "I will. . . ."
## RIPPLES
I could have called this book _A Beautiful Democracy_ , but as beautiful as the idea of democracy is, it's a poor fit for the creative life; who we are and what we create is not a matter of majority rule. No one else gets to cast a deciding vote. Our lives together, in community, rely on interdependence and kindness, respect, and often, compromise. But who we are at our core, and what we believe about our calling—our life's true work—is something only we can know, subject as it is to a voice only we can hear. Democracy _depends_ on a population that lives like this; it is the only way a vote is meaningful. To merely be a sheep and vote as you're told is not democracy; it's tyranny.
That said, if all these many words have encouraged you to set out, the world be damned, on your own, without others, then I've placed too great an emphasis on the individual aspects of this beautiful anarchy, and I owe you both an apology and a clarification. None of us can do it all on our own. Owning your own soul, and living in freedom and intent, is not the same thing as living only for, and by, yourself. You can create a life with and for others without losing your will and your soul, but I'm not sure the opposite is true. The life created purely for ourselves withers. Life is for sharing. We are better when we work together, when we help—and allow ourselves to be helped by—others. We are better when what we create ripples outward to touch other lives. And we are richer when we allow our shores to be touched by the ripples of others.
Whatever your work, make it art. Do it with all your heart. Fill your canvas with paint. Let it be messy. Make it bold. Make it unapologetically yours. Most of all: make it.
## ABOUT THE AUTHOR
David duChemin is a photographer, writer, and adventurer based in Victoria, Canada, on those rare months he's home. His best-selling books on the art of photography have been translated into a dozen languages, almost none of which he speaks. David is a passionate advocate of the intentional life. When he was a kid his mother told him to leave the world a better place. He's still working on that.
You can find David online at davidduchemin.com.
Join us on social!@rocky_nook
# Table of Contents
1. Foreword
2. A Beautiful Anarchy
3. Life Is Short
4. Ex Nihilo
5. The Artist's Journey
6. An Act of Creativity
7. This Might Not Work
8. Choosing Your Risk
9. Living Above the 45
10. Pretending to Be Brave
11. Listening to Voices
12. Inspiration
13. Incubation
14. Begin
15. Embracing the Constraints
16. Process Vs. Product
17. More Bad Ideas
18. The Starving Artist
19. The Art of Exclusion
20. Waiting for the Knock
21. Know Your Rhythm
22. The Myth of Originality
23. Ruts & Grooves
24. Winning at Yoga
25. Art As Gift
26. Now
27. Toward Mastery
28. I Will
29. Ripples
30. About the Author
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\section{Introduction}
\label{sec:introduction}
In the standard model (SM) of elementary particle physics, the interactions involving the Higgs boson take the following simple form
\begin{equation} \label{eq:LHiggs}
{\cal L} \supset \left |D_\mu H \right |^2 - \sum_f \left ( y_f \hspace{0.25mm} \bar f_L H f_R + {\rm h.c.} \right ) - V \,, \qquad V = -\mu \left |H \right |^2 + \lambda \left |H \right |^4 \,,
\end{equation}
where $D_\mu$ denotes the $S\hspace{-0.5mm}U(2)_L \times U(1)_Y$ covariant derivative, $H$ is the Higgs doublet, the subscripts~$L,R$ indicate the chirality of fermionic fields and $y_f$ are the so-called Yukawa couplings.
An obvious question that one may ask concerning (\ref{eq:LHiggs}) is: what is presently known about the interactions of the Higgs boson? The ATLAS and CMS combination of the LHC~Run-I Higgs measurements~\cite{Khachatryan:2016vau} imply that the gauge-Higgs interactions, which are encoded by~$|D_\mu H|^2$, agree with the SM predictions at the level of $10\%$. The Yukawa interactions $y_f \hspace{0.25mm} \bar f_L H f_R + {\rm h.c.}$, on the other hand, have been tested with this accuracy only in the case of the tau lepton, while the constraints on the top and bottom Yukawa couplings just reach the $20\%$ level. Apart from the muon Yukawa coupling, which is marginally constrained by the existing LHC data, first and second generation Yukawa couplings are not directly probed at present. In the case of the Higgs potential~$V$, the vacuum expectation value (VEV) of $H$ is known since the discovery of the $W$ and $Z$ bosons, while the LHC discovery of a scalar with a mass of around $125 \, {\rm GeV}$ tells us about the second derivative of $V$ around its VEV, because this quantity determines the Higgs mass. The $h^3$ and $h^4$ Higgs self-interactions that result from (\ref{eq:LHiggs}) are in contrast essentially untested at the moment.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.975 \textwidth]{multihxsecs.pdf}
\vspace{0mm}
\caption{\label{fig:multihsxsec} Left: Total production cross section for $pp \to h$ (red), $pp \to hh$ (blue) and $pp \to hhh$~(yellow) as a function of $\sqrt{s}$. Right: Dependence of the cross section ratio $\sigma (pp \to h)/\sigma (pp \to hh)$ (green) and $\sigma (pp \to hh)/\sigma (pp \to hhh)$ (purple) on the collider CM energy. The shown predictions are based on the state-of-the-art SM calculations of single-Higgs~\cite{Anastasiou:2015ema,Anastasiou:2016cez,Contino:2016spe}, double-Higgs~\cite{Borowka:2016ehy, Borowka:2016ypz,Heinrich:2017kxx,Grazzini:2018bsd} and triple-Higgs~\cite{Maltoni:2014eza} production.}
\end{center}
\end{figure}
Given our limited knowledge of the properties of the discovered $125 \, {\rm GeV}$ resonance, constraining or measuring as many of its so far poorly known or unknown couplings will be an important part of any future high-energy programme. In the case of the Higgs self-couplings the most obvious way to get access to the cubic and quartic interactions consists in searching for multi-Higgs production. Unfortunately, all multi-Higgs production rates are quite small in the SM, as can be seen from~Figure~\ref{fig:multihsxsec}, making already LHC measurements of double-Higgs production a formidable task. As a result, at best ${\cal O} (1)$ determinations of the cubic Higgs self-coupling seem to be possible at the LHC~(cf.~for instance~\cite{Goertz:2014qta,Azatov:2015oxa,ATL-PHYS-PUB-2015-046,Kling:2016lay,ATL-PHYS-PUB-2016-024,DiVita:2017eyz,ATL-PHYS-PUB-2017-001}). Significantly improved prospects in extracting the $h^3$ coupling would be offered by a high-energy upgrade of the LHC~(HE-LHC) to $27 \, {\rm TeV}$~\cite{Goncalves:2018qas} or a future circular collider~(FCC-pp) operating at a centre-of-mass~(CM)~energy of $100 \, {\rm TeV}$~\cite{Azatov:2015oxa,Barr:2014sga,He:2015spf,Contino:2016spe,Mangano:2016jyj,Banerjee:2018yxy,Chang:2018uwu}. A~$100 \, {\rm TeV}$ $pp$ machine, in particular, may ultimately allow one to determine the cubic Higgs self-coupling with a statistical precision of the order of a few percent. Even a $100 \, {\rm TeV}$ FCC-pp collider is, however, not powerful enough to determine the SM triple-Higgs production rate to an accuracy better than just order one~\cite{Contino:2016spe,Mangano:2016jyj,Papaefstathiou:2015paa,Chen:2015gva,Fuks:2015hna,Kilian:2017nio,Fuks:2017zkg}. The resulting bounds on the quartic Higgs self-coupling turn out to be weak, in general allowing for ${\cal O} (10)$ modifications of the $h^4$ vertex with respect to the~SM prediction.
Motivated by the above observations, we apply in this work the general idea of testing the~$h^3$ interaction indirectly~\cite{McCullough:2013rea,Gorbahn:2016uoy,Degrassi:2016wml,Bizon:2016wgr, Degrassi:2017ucl,Kribs:2017znd,DiVita:2017eyz,Maltoni:2017ims,DiVita:2017vrr, Maltoni:2018ttu,Liu:2018peg,Borowka:2018pxx,Gorbahn:2019lwq} to the case of the $h^4$ vertex. Specifically, we consider the constraints on the quartic Higgs self-coupling that future precision measurements of double-Higgs production in gluon-fusion may provide. In order to determine the dependence of the $gg \to hh$ distributions on the value of the $h^4$ coupling, we calculate the relevant electroweak~(EW) two-loop amplitudes and combine them with the exact ${\cal O} (\alpha_s^2)$ matrix elements~\cite{Borowka:2016ehy,Borowka:2016ypz,Heinrich:2017kxx}. This allows us to predict the cross section and various distributions for double-Higgs production at the next-to-leading~order~(NLO) in QCD, including arbitrary modifications of the cubic and quartic Higgs self-couplings. We then perform an exploratory study of the synergy and complementarity of double-Higgs and triple-Higgs production in constraining the $h^3$ and $h^4$ interactions, considering both the HE-LHC and a FCC-pp machine as an example. A similar analysis has very recently also been performed~in~\cite{Borowka:2018pxx}. For completeness, we add that the indirect constraints on the quartic Higgs self-coupling that high-energy $e^+ e^-$ machines may be able to set have been studied in~\cite{Maltoni:2018ttu,Liu:2018peg}. In these articles it has been shown that future lepton colliders can in general only provide coarse bounds on possible modifications of the $h^4$ vertex, if one makes no assumption about how ultraviolet~(UV) physics alters the cubic and quartic Higgs self-interactions. We will compare our limits to those obtained in the publications~\cite{Maltoni:2018ttu,Liu:2018peg,Borowka:2018pxx}.
This article is structured as follows. In Section~\ref{sec:preliminaries} we introduce our parameterisation of the Higgs potential and discuss how the coefficients entering it are related
to the Wilson coefficients of two higher-dimensional operators of the SM effective field theory (SMEFT). The calculation of the two-loop corrections to the $g g \to hh$ scattering amplitude resulting from a modified quartic Higgs self-coupling is described in Section~\ref{sec:calculation}. In Section~\ref{sec:numerics} we determine the hypothetical reach of a $27 \, {\rm TeV}$ HE-LHC and a $100 \, {\rm TeV}$ FCC-pp in constraining the cubic and quartic Higgs self-couplings by measurements of double-Higgs and triple-Higgs production in gluon-fusion. Section~\ref{sec:conclusions} contains our conclusions.
\section{Preliminaries}
\label{sec:preliminaries}
After EW symmetry breaking, the cubic and quartic self-interactions of the Higgs field $h$ can be parameterised in a model-independent fashion by
\begin{equation} \label{eq:V}
V \supset \kappa_3 \hspace{0.25mm} \lambda v h^3 + \kappa_4 \hspace{0.25mm} \frac{\lambda}{4} \hspace{0.25mm} h^4 \,.
\end{equation}
Here $\lambda = m_h^2/(2v^2)$ with $m_h \simeq 125 \, {\rm GeV}$ the Higgs-boson mass and $v \simeq 246 \, {\rm GeV}$ the EW VEV. Notice that the normalisation of the terms in~(\ref{eq:V}) has been chosen such that within the SM one has~$\kappa_{3} = \kappa_{4} = 1$.
In the presence of physics beyond the SM~(BSM) the coefficients $\kappa_3$ and $\kappa_4$ will in general deviate from 1. As an illustrative example, let us consider the following two terms
\begin{equation} \label{eq:LSMEFT}
{\cal L}_{\rm SMEFT}\supset {\cal O}_6 + {\cal O}_8 = -\frac{\lambda \hspace{0.5mm} \bar c_6}{v^2} \hspace{0.5mm} \left |H \right |^6 -\frac{\lambda \hspace{0.5mm} \bar c_8}{v^4} \hspace{0.5mm} \left |H \right |^8 \,,
\end{equation}
in the SMEFT. In such a case the parameters $\kappa_3$ and~$\kappa_4$ are related at tree level to the Wilson coefficients $\bar c_6$ and $\bar c_8$ via
\begin{equation} \label{eq:kappa3kappa4}
\kappa_3= 1 + \Delta \kappa_{3} = 1 + \bar c_6 + 2 \hspace{0.25mm} \bar c_8 \,, \qquad
\kappa_4 = 1 + \Delta \kappa_{4} = 1 + 6 \hspace{0.25mm} \bar c_6 + 16 \hspace{0.25mm} \bar c_8 \,.
\end{equation}
The relations~(\ref{eq:kappa3kappa4}) imply that if the dimension-six operator ${\cal O}_6$ represents the only numerically relevant modification in the SMEFT, the shifts in the cubic and quartic Higgs self-couplings are strongly correlated as they obey $\Delta \kappa_4 = 6 \Delta \kappa_3$. This correlation is, however, broken by the dimension-eight contribution ${\cal O}_8$, if this operator receives a non-zero Wilson coefficient. Notice that the initial conditions of the Wilson coefficients $\bar c_6$ and $\bar c_8$ can be found in any UV complete BSM model by a suitable matching calculation. If the new interactions that lead to ${\cal O}_6$ and ${\cal O}_8$ are weakly coupled and the new-physics scale $\Lambda$ is in the TeV range, one expects on general grounds that the dimension-eight and dimension-six contributions have the following hierarchy~$\bar c_8/\bar c_6 = {\cal O} (v^2/\Lambda^2) \ll 1$. The Wilson coefficients $\bar c_8$ and $\bar c_6$ can however be of the same order of magnitude if the underlying UV theory is strongly coupled or the new-physics scale $\Lambda$ is at (or not far~above) the~EW scale. To~achieve the inverted hierarchy $\bar c_8/\bar c_6 \gg 1$ the new particles that give rise to (\ref{eq:LSMEFT}) have to have masses of ${\cal O} (v)$ and have to have interactions with the Higgs doublet $H$ that are strong --- SM extensions with colourless $S\!U(2)$ quadruplets $\Theta$~\cite{deBlas:2014mba} can for instance lead to such an inverted hierarchy if the quadruplet is sufficiently light and the Higgs portal coupling $|\Theta|^2 \, |H|^2$ is sufficiently large.
In our work we choose to be agnostic about how UV dynamics modifies the Higgs self-interactions, and hence make no assumption about the actual size of $\bar c_6$ and $\bar c_8$. In this case, the cubic and quartic Higgs self-couplings can deviate independently from the SM predictions. The important point is now that even if $\Delta \kappa_{3}$ and $\Delta \kappa_{4}$ are treated as free parameters, quantum processes such as $gg \to h$ or loop corrections to $e^+ e^- \to hhZ$ can still be calculated consistently as long as the SMEFT is used to perform the computations~(see~\cite{McCullough:2013rea,Gorbahn:2016uoy,Degrassi:2016wml,Bizon:2016wgr, Degrassi:2017ucl,Kribs:2017znd,Maltoni:2018ttu,Liu:2018peg,Borowka:2018pxx,Gorbahn:2019lwq} for non-trivial one-loop and two-loop examples and further explanations). Since modifications in the cubic and quartic Higgs self-coupling are most commonly parametrised by $\Delta \kappa_{3}$ and $\Delta \kappa_{4}$, we will also use this parameterisation in what follows. We however emphasise that constraints on the latter parameters can always be translated into bounds on the Wilson coefficients $\bar c_6$ and $\bar c_8$ by means of~(\ref{eq:kappa3kappa4}).
\section{Calculation}
\label{sec:calculation}
The scattering amplitude describing the process $g(p_1) + g(p_2) \to h(p_3) + h(p_4)$ can be written as
\begin{equation} \label{eq:Agghh}
{\cal A} \left (gg\to hh \right ) = \delta^{a_1 a_2} \hspace{0.5mm} \epsilon_{1}^{\mu} (p_1) \hspace{0.5mm} \epsilon_{2}^\nu (p_2) \hspace{1mm} {\cal A}_{\mu \nu} \,,
\end{equation}
where $a_{1}$ and $a_{2}$ denote colour indices while $\epsilon_{1}^\mu (p_{1})$ and $\epsilon_{2}^\nu (p_{2})$ are the polarisation vectors of the two initial-state gluons. Using Lorentz symmetry, parity conservation and gauge invariance, one can show that the amplitude tensor ${\cal A}_{\mu \nu}$ that appears in~(\ref{eq:Agghh}) can be expressed in terms of two form factors as follows
\begin{equation} \label{eq:Amunu}
{\cal A}_{\mu \nu} = \sum_{m=1}^2 T_{m\, \mu \nu} \, {\cal F}_m \,,
\end{equation}
where~\cite{Glover:1987nx}
\begin{equation} \label{eq:Tmmunu}
\begin{split}
T_{1\, \mu \nu} & = \eta_{\mu \nu} - \frac{p_{1 \, \nu} \hspace{0.5mm} p_{2 \, \mu}}{p_1 \cdot p_2} \,, \\[2mm]
T_{2\, \mu \nu} & = \eta_{\mu \nu} + \frac{1}{p_T^2 \left ( p_1 \cdot p_2 \right)} \,
\left ( \, m_h^2 \hspace{0.5mm} p_{1 \, \nu} \hspace{0.5mm} p_{2 \, \mu}
- 2 \left ( p_1 \cdot p_3 \right ) p_{2 \, \mu} \hspace{0.5mm} p_{3 \, \nu} \right. \\[1mm]
& \hspace{3.4cm} \left . - 2 \left ( p_2 \cdot p_3 \right ) p_{1 \, \nu} \hspace{0.5mm} p_{3 \, \mu}
+ 2 \left ( p_1 \cdot p_2 \right ) p_{3 \, \mu} \hspace{0.5mm} p_{3 \, \nu} \, \right ) \,,
\end{split}
\end{equation}
with $\eta_{\mu \nu} = {\rm diag} \left ( 1, -1, -1, -1 \right)$ the Minkowski metric and $p_T$ denoting the Higgs transverse momentum. In terms of the partonic Mandelstam variables
\begin{equation} \label{eq:mandelstam}
\hat s = \left (p_1 + p_2 \right )^2 \,, \qquad
\hat t = \left (p_1 - p_3 \right )^2 \,, \qquad
\hat u = \left (p_2 - p_3 \right )^2 \,,
\end{equation}
one can write
\begin{equation} \label{eq:pT2}
p_T^2 = \frac{\hat t \hspace{0.75mm} \hat u - m_h^4}{\hat s} \,.
\end{equation}
Notice furthermore that $p_1^2=p_2^2 = 0$ while $p_3^2=p_4^2 = m_h^2$ and that the Mandelstam variables fulfill the relation $\hat s + \hat t + \hat u = 2 m_h^2$.
The form factors entering~(\ref{eq:Amunu}) are most conveniently extracted by using a projection procedure. The appropriate projectors read~(see~\cite{Borowka:2016ehy} for example)
\begin{equation} \label{eq:projectors}
\begin{split}
P_1^{\mu \nu} & = \frac{1}{4 \left (d-3 \right)} \, \left [ \, \left ( d-2 \right ) T_1^{\mu \nu} - \left ( d-4 \right ) T_2^{\mu \nu} \, \right ] \,, \\[2mm]
P_2^{\mu \nu} & = \frac{1}{4 \left (d-3 \right)} \, \left [ \, -\left ( d-4 \right ) T_1^{\mu \nu} + \left ( d-2 \right ) T_2^{\mu \nu} \, \right ] \,,
\end{split}
\end{equation}
where $d = 4 - 2 \epsilon$ denotes the number of space-time dimensions.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.8 \textwidth]{diagrams.pdf}
\vspace{4mm}
\caption{\label{fig:diagrams} Prototypes of two-loop Feynman diagrams with an insertion of an effective quartic Higgs self-coupling~(black square) that contribute to the process $gg \to hh$.}
\end{center}
\end{figure}
After applying the projectors~(\ref{eq:projectors}) each of the two $gg \to hh$ form factors can be calculated separately. Since the form factors are independent of the external polarisation vectors, all the standard techniques employed in multi-loop computations can be applied. In practice, we proceed in the following way. We generate the relevant two-loop Feynman diagrams with {\tt FeynArts}~\cite{Hahn:2000kx}. Representative examples of two-loop graphs are shown in Figure~\ref{fig:diagrams}. The projection onto form factors as well as the colour and Dirac algebra is performed with the help of~{\tt FORM}~\cite{Vermaseren:2000nd}. The resulting two-loop integrals are then evaluated numerically using the {\tt pySecDec}~\cite{Borowka:2012yc,Borowka:2015mxa,Borowka:2017idc} package. Including all two-loop diagrams leads to UV-finite results for the form factors, and we have checked that the double and single $1/\epsilon$ poles cancel to a relative accuracy of at least a permyriad for all calculated phase-space points. Since in addition the quartic Higgs self-coupling does not result in a finite one-loop correction of the Higgs wave function, it follows that the calculation of the ${\cal O} (\kappa_4)$ contributions to the $gg \to hh$ form factors arising from the Feynman diagrams depicted in Figure~\ref{fig:diagrams} does not require renormalisation.
As a further check of our numerical results, we have performed a systematic expansion of the two-loop form factors in the limit $m_t^2 \gg m_h^2, \hat s, \hat t, \hat u$ by employing the method of asymptotic expansions~(see~\cite{Smirnov:2002pj}~for a review and~\cite{Gorbahn:2019lwq} an application in a similar context). Our analytic calculation made use of~{\tt MATAD}~\cite{Steinhauser:2000ry},~{\tt LiteRed}~\cite{Lee:2013mka}, the tensor reduction procedures described in~\cite{Tarasov:1995jf,Tarasov:1996br,Tarasov:1997kx} and the results of massive two-loop vacuum integrals first given in~\cite{Avdeev:1994db}. The agreement of the final results in the limit $\hat s < m_t^2$ between the two approaches serves as a non-trivial cross-check of our numerical computations.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.875 \textwidth]{refimf.pdf}
\vspace{-4mm}
\includegraphics[width=0.875 \textwidth]{regimg.pdf}
\vspace{-4mm}
\includegraphics[width=0.875 \textwidth]{rehimh.pdf}
\vspace{-4mm}
\caption{\label{fig:refimf} Real part (left panel) and imaginary part (right panel) of the function $f (\hat s)$ (upper row), $g (\hat s)$~(middle row) and $h (\hat s)$~(lower row) as introduced in~(\ref{eq:DeltaF1F2}), (\ref{eq:gfunction}) and (\ref{eq:hfunction}), respectively.}
\end{center}
\end{figure}
Using the results of our numerical two-loop calculation, we find that the ${\cal O} (\kappa_4)$ corrections to the two $gg \to hh$ form factors that arise from the graphs shown in Figure~\ref{fig:diagrams} can be written as
\begin{equation} \label{eq:DeltaF1F2}
\Delta {\cal F}_{1,n}^{\kappa_4} = \frac{\alpha_s}{4 \pi} \frac{\lambda \, \kappa_4}{(4 \pi)^2} \; y_t^2 \, f ( \hat s ) \,, \qquad
\Delta {\cal F}_{2,n}^{\kappa_4} = 0 \,.
\end{equation}
Here $\alpha_s = g_s^2/(4 \pi)$ is the strong coupling constant, $y_t = \sqrt{2} m_t/v$ denotes the SM top-quark Yukawa coupling and the subscript $n$ indicates that the above corrections arise from non-factorisable two-loop diagrams. Two features of the expressions~(\ref{eq:DeltaF1F2}) are worth emphasising. First, the correction to the spin-0 form factor~${\cal F}_1$ depends only on $\hat s$ but not on the other two Mandelstam variables~$\hat t, \hat u$ $\big($or the combination $p_T^2$ introduced in~(\ref{eq:pT2})$\big)$. Second, the correction to the spin-2 form factor ${\cal F}_2$ turns out to be identical to zero. The first feature is readily understood by noticing that the momentum routing in the two diagrams in~Figure~\ref{fig:diagrams} can be chosen such that the external momenta only enter in the combination~$p_1 + p_2$. Due to Lorentz invariance the corresponding Feynman integrals can thus only depend on $\hat s = (p_1+p_2)^2 = 2 p_1 \cdot p_2$. The vanishing of the correction $\Delta {\cal F}_{2,n}$ to the spin-2 form factor follows for instance from the observation that the vertex $h^4$ can effectively be generated via the $s$-channel exchange of a heavy scalar $S$ that interacts with the Higgs boson through a term of the form $S h^2$. As a result the graphs in Figure~\ref{fig:diagrams} are mathematically equivalent to the off-shell production of a heavy CP-even spin-0 state that subsequently decays to $hh$. The corresponding scattering amplitude has evidently no spin-2 component.
The real and imaginary parts of the function $f(\hat s)$ that appears in~(\ref{eq:DeltaF1F2}) are depicted in the upper row of Figure~\ref{fig:refimf}. The shown results correspond to $m_h = 125 \, {\rm GeV}$ and $m_t = 173 \, {\rm GeV}$. From the left panel one sees that the real part of~$f(\hat s)$ changes its slope at the top-quark threshold $\sqrt{\hat s} = 2 m_t \simeq 375 \, {\rm GeV}$ and has a pronounced global maximum at $\sqrt{\hat s} = 2 \left ( m_h + m_ t \right ) \simeq 600 \, {\rm GeV}$. As illustrated on the right-hand side and expected from the optical theorem, the imaginary part of~$f(\hat s)$ instead vanishes at the threshold for double-Higgs production,~i.e.~$\sqrt{\hat s} = 2 m_h \simeq 250 \, {\rm GeV}$. It then decreases rapidly, developing a distinct minimum in the vicinity of $\sqrt{\hat s} \simeq 400 \, {\rm GeV}$. We will see in Section~\ref{sec:numerics2} that the non-trivial $\hat s$ dependence of the real and imaginary parts of~$f(\hat s)$ leads to distortions in the~kinematic $gg \to hh$ distributions such as the invariant mass $m_{hh}$ of the di-Higgs~system.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.8 \textwidth]{diagramsadditional.pdf}
\vspace{4mm}
\caption{\label{fig:diagramsadditional} Two-loop graph with an effective cubic and quartic (left diagram) and quintic (right diagram) Higgs self-coupling that gives rise to $gg \to hh$ production. The effective interactions are indicated by black squares. For further details consult the text.}
\end{center}
\end{figure}
Additional corrections to the $gg \to hh$ form factors~(\ref{eq:Amunu}) that are proportional to the self-coupling modifier $\kappa_4$ arise from the two types of Feynman diagrams displayed in Figure~\ref{fig:diagramsadditional}. Since the operators ${\cal O}_6$ and ${\cal O}_8$ do not generate couplings between two Higgs and two would-be Goldstone fields only graphs with Higgs-boson exchange contribute to the class of diagrams with a cubic and~a~quartic self-interaction. As a result this contribution is of ${\cal O} (\kappa_3 \kappa_4)$. In the case of the graphs with a quintic scalar self-coupling instead both Higgs and would-be Goldstone loops are present, and this contribution turns out to be proportional to ${\cal O} (\kappa_4)$. Besides the two-loop diagrams shown in the latter figure one-loop counterterm contributions associated to the renormalisation of the Higgs tadpole $T_h$, its wave function $Z_h$ and its mass $m_h$ as well as corrections associated to operator renormalisation have to be included to obtain a gauge-invariant result. We perform the renormalisation of $T_h$, $Z_h$ and $m_h$ in the on-shell scheme, while the fine structure constant $\alpha$ is renormalised in the so-called $G_F$ scheme (cf.~\cite{Denner:1991kt} for a review of the renormalisation of the EW sector of the SM). To~guarantee that the Wilson coefficients or effective couplings obey the usual~SMEFT renormalisation group equations~\cite{Jenkins:2013zja}, the~operator renormalisation is performed in the~$\overline{\rm MS}$ scheme with the renormalisation scale set to $m_h$. Our renormalisation procedure therefore resembles the one employed in the article~\cite{Maltoni:2018ttu} to which we relegate the interested reader for further technical details.
After renormalisation, we find that the ${\cal O} (\kappa_3 \kappa_4)$ correction associated to the Feynman diagrams of Figure~\ref{fig:diagramsadditional} can be written as
\begin{equation} \label{eq:DeltaF1F2additional}
\Delta {\cal F}_{1,f}^{\kappa_3 \kappa_4} = \frac{\alpha_s}{4 \pi} \, \lambda \, \kappa_3 \, \frac{\lambda \, \kappa_4}{(4 \pi)^2} \, g ( \hat s ) \,. \qquad
\Delta {\cal F}_{2,f}^{\kappa_3 \kappa_4} = 0 \,,
\end{equation}
where
\begin{equation} \label{eq:gfunction}
g(\hat s) = \frac{18 \hspace{0.25mm} \hat s}{\hat s - m_h^2} \; \tau_t \left [ 1+ (1- \tau_t) \arctan^2 \left ( \frac{1}{\sqrt{\tau_t - 1}} \right ) \right ] \left [ \sqrt{1 - \tau_h} \, \ln \left ( \frac{ \sqrt{1 - \tau_h} + 1}{\sqrt{1 - \tau_h} - 1} \right ) + \frac{2 \pi}{\sqrt{3}} - 6 \right ] \,,
\end{equation}
and $\tau_a = 4m_a^2/\hat s$ with $a = t, h$ and it is understood that $\tau_a \to \tau_a - i0$ for analytic continuation. The subscript $f$ in (\ref{eq:DeltaF1F2additional}) indicates that these corrections arise from two-loop diagrams that factorise into two one-loop graphs. We add that the above result is identical to the one that one obtains if the renormalised $h^3$ vertex is defined via a subtraction at vanishing external momenta as done in~\cite{Liu:2018peg}.
Under the assumption that the operators ${\cal O}_6$ and ${\cal O}_8$ introduced in~(\ref{eq:LSMEFT}) represent the only modifications of the effective one-loop~$h^3$~vertex, we find that the ${\cal O} (\kappa_4)$ correction arising from the graphs displayed in Figure~\ref{fig:diagramsadditional} takes the following form
\begin{equation} \label{eq:DeltaF1F2further}
\Delta {\cal F}_{1,f}^{\kappa_4} = \frac{\alpha_s}{4 \pi} \, \frac{\lambda \, \kappa_4}{(4 \pi)^2} \, \lambda \, h ( \hat s ) \,. \qquad
\Delta {\cal F}_{2,f}^{\kappa_4} = 0 \,,
\end{equation}
where
\begin{equation} \label{eq:hfunction}
h(\hat s) = -\frac{42 \hspace{0.25mm} \hat s}{\hat s - m_h^2} \; \tau_t \left [ 1+ (1- \tau_t) \arctan^2 \left ( \frac{1}{\sqrt{\tau_t - 1}} \right ) \right ] \,.
\end{equation}
We emphasise that the result~(\ref{eq:DeltaF1F2further}) unlike (\ref{eq:DeltaF1F2additional}) is model dependent. This model dependence arises because $\Delta {\cal F}_{1,f}^{\kappa_4}$ receives contributions from Feynman diagrams with quintic scalar self-interactions such as the one shown on the right-hand side of Figure~\ref{fig:diagramsadditional}. The expressions given in~(\ref{eq:DeltaF1F2further}) and (\ref{eq:hfunction}) correspond to a quintic Higgs self-interaction of the form $V \supset \kappa_5/v \hspace{0.5mm} h^5$ with $\kappa_5 = \lambda \left ( 3 \hspace{0.25mm} \bar c_6 + 14 \hspace{0.25mm} \bar c_8 \right )/4$. Like in~\cite{Maltoni:2018ttu,Borowka:2018pxx} other possible contributions to the coupling modifier $\kappa_5$ such as those that arise for instance from the higher-dimensional operator ${\cal O}_{10} = -\lambda \hspace{0.5mm} \bar c_{10}/v^6 \, |H|^{10}$ have instead been neglected in our calculation of $\Delta {\cal F}_{1,f}^{\kappa_4}$. This should be contrasted with the analysis~\cite{Liu:2018peg} where it has been assumed that the contributions from higher-dimensional operators other than ${\cal O}_6$ and ${\cal O}_8$ are such that the correction $\Delta {\cal F}_{1,f}^{\kappa_4}$ effectively vanishes. Since (\ref{eq:DeltaF1F2further}) is compared to~(\ref{eq:DeltaF1F2}) parametrically suppressed by a factor of $\lambda/y_t^2 \simeq 0.13$ it turns out that neglecting the $\bar c_{10}$ contribution in the calculation of $\Delta {\cal F}_{1,f}^{\kappa_4}$ is satisfied in BSM models in which the ratios of Wilson coefficient scale as $\bar c_D/\bar c_{D+2} = {\cal O} (v^2/\Lambda^2) \ll 1$ or $\bar c_D/\bar c_{D+2} = {\cal O} (1)$ with $D = 6, 8$. SM extensions that give rise to $\bar c_{10}/\bar c_{8} \gg 1$ are, on the other hand, not well described by (\ref{eq:DeltaF1F2further}). To be directly sensitive to the Wilson coefficient $\bar c_{10}$ or equivalently $\kappa_5$, one would need to evaluate quadruple-Higgs production at the tree level or study one-loop effects in triple-Higgs production, and combine the obtained constraints with the measurements of double-Higgs and triple-Higgs production considered in~Section~\ref{sec:numerics}. Such an analysis is, however, beyond the scope of this article.
The two panels in the middle and lower row of Figure~\ref{fig:refimf} show the real and imaginary parts of~(\ref{eq:gfunction}) and~(\ref{eq:hfunction}), respectively. One sees that the $\hat s$-dependence of both $g(\hat s)$ and $h(\hat s)$ is non-trivial with extrema at $\sqrt{\hat s} = 2 m_t \simeq 375 \, {\rm GeV}$ and $\sqrt{\hat s} = 2 \left ( m_h + m_ t \right ) \simeq 600 \, {\rm GeV}$. The observed features will again distort the~kinematic distributions in $gg \to hh$ production.
\section{Numerics}
\label{sec:numerics}
In this section we derive limits on the parameters $\Delta \kappa_3$ and $\Delta \kappa_4$ that describe possible modifications of the cubic and quartic Higgs self-couplings with respect to the SM. We consider both double-Higgs and triple-Higgs production at a $27 \, {\rm TeV}$ HE-LHC with an integrated luminosity of $15 \, {\rm ab}^{-1}$ as well as a $100 \, {\rm TeV}$ FCC-pp collider assuming $30 \, {\rm ab}^{-1}$ of data.
\subsection{Inclusive double-Higgs and triple-Higgs production}
\label{sec:numerics1}
We begin our study by presenting results for the relevant inclusive production cross sections. In the case of double-Higgs production, we find the following expressions {
\begin{equation} \label{eq:hhxsecs}
\begin{split}
\sigma \left ( pp \to hh \right )_{\text{HE-LHC}} & = 131 \hspace{1mm} \Big [ 1 - 0.73 \hspace{0.25mm} \Delta \kappa_3 + 1.9 \cdot 10^{-3} \hspace{0.25mm} \Delta \kappa_4 \\[1mm]
& \hspace{1.15cm} + 0.24 \left ( \Delta \kappa_3 \right )^2 + 4.9 \cdot 10^{-4} \hspace{0.25mm} \Delta \kappa_3 \hspace{0.25mm} \Delta \kappa_4 + 2.7 \cdot 10^{-5} \left ( \Delta \kappa_4 \right)^2 \\[1mm]
& \hspace{1.15cm} - 1.3 \cdot 10^{-3} \left ( \Delta \kappa_3 \right )^2 \hspace{0.25mm} \Delta \kappa_4 - 1.8 \cdot 10^{-5} \hspace{0.25mm} \Delta \kappa_3 \left ( \Delta \kappa_4 \right )^2 \\[1mm]
& \hspace{1.15cm} + 8.8 \cdot 10^{-6} \left ( \Delta \kappa_3 \right )^2 \left ( \Delta \kappa_4 \right )^2\Big ] \, {\rm fb} \,, \\[2mm]
\sigma \left ( pp \to hh \right )_{\text{FCC-pp}} & = 1151 \hspace{1mm} \Big [ 1 - 0.76 \hspace{0.25mm} \Delta \kappa_3 + 2.1 \cdot 10^{-3} \hspace{0.25mm} \Delta \kappa_4 \\[1mm]
& \hspace{1.15cm} + 0.23 \left ( \Delta \kappa_3 \right )^2 + 6.1 \cdot 10^{-4} \hspace{0.25mm} \Delta \kappa_3 \hspace{0.25mm} \Delta \kappa_4 + 3.1 \cdot 10^{-5} \left ( \Delta \kappa_4 \right)^2 \\[1mm]
& \hspace{1.15cm} - 1.3 \cdot 10^{-3} \left ( \Delta \kappa_3 \right )^2 \hspace{0.25mm} \Delta \kappa_4 - 2.1 \cdot 10^{-5} \hspace{0.25mm} \Delta \kappa_3 \left ( \Delta \kappa_4 \right )^2 \\[1mm]
& \hspace{1.15cm} + 8.9 \cdot 10^{-6} \left ( \Delta \kappa_3 \right )^2 \left ( \Delta \kappa_4 \right )^2\Big ] \, {\rm fb} \,.
\end{split}
\end{equation} }
These formulas have been obtained with a customised version of the {\tt POWHEG BOX}~\cite{Alioli:2010xd} implementation of the NLO QCD calculation of double-Higgs production~\cite{Borowka:2016ehy,Borowka:2016ypz,Heinrich:2017kxx} using {\tt PDF4LHC15\_nlo} parton distribution functions~(PDFs)~\cite{Butterworth:2015oua}. Our scale choice is $\mu_R = \mu_F = m_{hh}/2$ with $\mu_R$ and $\mu_F$ denoting the renormalisation and factorisation scale, respectively. As a cross-check we have also derived similar expressions using {\tt MCFM}~\cite{Campbell:2010ff} and {\tt MadGraph5\_aMC@NLO}~\cite{Alwall:2014hca}, finding numerical agreement between all results at~leading order (LO) in QCD. We note that the SM cross sections that follow from (\ref{eq:hhxsecs}) agree with the central values of the NLO QCD~results quoted in~\cite{Borowka:2016ehy,Borowka:2016ypz,Grazzini:2018bsd} within a few percent. The observed small differences are in part due to a slightly different treatment of~$\alpha_s$~in {\tt POWHEG BOX} and the latter~calculations.
In the case of triple-Higgs production, the dependence of the total production cross sections on $\Delta \kappa_3$ and $\Delta \kappa_4$ instead takes the form
\begin{align} \label{eq:hhhxsecs}
\sigma \left ( pp \to hhh \right )_{\text{HE-LHC}} & = 0.44 \hspace{1mm} \Big [ 1 - 0.79 \hspace{0.25mm} \Delta \kappa_3 - 0.10 \hspace{0.25mm} \Delta \kappa_4 \nonumber \\[1mm]
& \hspace{1.25cm} + 0.81 \left ( \Delta \kappa_3 \right)^2 - 0.16 \hspace{0.25mm} \Delta \kappa_3 \hspace{0.25mm} \Delta \kappa_4 + 1.6 \cdot 10^{-2} \left ( \Delta \kappa_4 \right)^2 \nonumber \\[1mm]
& \hspace{1.25cm} - \, 0.23 \left ( \Delta \kappa_3 \right )^3 + 4.5 \cdot 10^{-2} \left ( \Delta \kappa_3 \right )^2 \hspace{0.25mm} \Delta \kappa_4 \nonumber \\[1mm]
& \hspace{1.25cm} + 3.5 \cdot 10^{-2} \left ( \Delta \kappa_3 \right )^4 \Big ] \, {\rm fb} \,, \hspace{6mm} \nonumber \\[2mm]
\sigma \left ( pp \to hhh \right )_{\text{FCC-pp}} & = 5.1 \hspace{1mm} \Big [ 1 - 0.67 \hspace{0.25mm} \Delta \kappa_3 - 0.11 \hspace{0.25mm} \Delta \kappa_4 \nonumber \\[1mm]
& \hspace{1.05cm} + 0.72 \left ( \Delta \kappa_3 \right)^2 - 0.14 \hspace{0.25mm} \Delta \kappa_3 \hspace{0.25mm} \Delta \kappa_4 + 1.6 \cdot 10^{-2} \left ( \Delta \kappa_4 \right)^2 \nonumber \\[1mm]
& \hspace{1.05cm} - \, 0.20 \left ( \Delta \kappa_3 \right )^3 + 4.0 \cdot 10^{-2} \left ( \Delta \kappa_3 \right )^2 \hspace{0.25mm} \Delta \kappa_4 \nonumber \\[1mm]
& \hspace{1.05cm} + 3.0 \cdot 10^{-2} \left ( \Delta \kappa_3 \right )^4 \Big ] \, {\rm fb} \,.
\end{align}
These expressions have been obtained at LO in QCD with the help of {\tt MadGraph5\_aMC@NLO}, taking into account the NLO QCD corrections calculated in~\cite{Maltoni:2014eza} in the form of an overall normalisation. The used PDF set is again {\tt PDF4LHC15\_nlo}. We add that the $\Delta \kappa_3$ and $\Delta \kappa_4$ dependence of our FCC-pp result as given in~(\ref{eq:hhhxsecs}) agrees with a similar formula presented in~\cite{Papaefstathiou:2015paa} for the special case~$\Delta \kappa_4 = 6 \Delta \kappa_3$.
In order to estimate the precision of future hadron colliders in measuring the inclusive double-Higgs production cross section, we consider the $b \bar b \gamma \gamma$ final state. This channel has been identified in the literature~\cite{Contino:2016spe,Goncalves:2018qas,Azatov:2015oxa,Barr:2014sga,He:2015spf,Mangano:2016jyj,Chang:2018uwu} to lead to the best SM signal significance and the highest precision in the measurement of the cubic Higgs self-coupling. At the $27 \, {\rm TeV}$ HE-LHC with $15 \, {\rm ab}^{-1}$ of integrated luminosity the statistical precision of $pp \to hh \to b \bar b \gamma \gamma$ is expected to be around~$14\%$~\cite{Goncalves:2018qas}, while at a $100 \, {\rm TeV}$ FCC-pp collider with $30 \, {\rm ab}^{-1}$ statistical uncertainties in the ballpark of $3\%$ are anticipated~\cite{Contino:2016spe,Goncalves:2018qas,Azatov:2015oxa,Barr:2014sga,He:2015spf,Mangano:2016jyj,Chang:2018uwu}. Estimating the theoretical uncertainties on the prediction of the signal and the systematic uncertainty on the overall determination of the background rates is more difficult and necessarily has to rely on assumptions. The study of double-Higgs production at approximate next-to-next-to-leading order (NNLO) in QCD~\cite{Grazzini:2018bsd} finds that the inclusive production cross section at $27 \, {\rm TeV}$~($100 \, {\rm TeV}$) is plagued by scale uncertainties of $2.6\%$~($2.1\%$) and uncertainties of $3.4\%$~($4.6\%$) due to unknown top-quark mass effects. Given these numbers and envisioning that the understanding of top-quark mass effects at NNLO QCD will be drastically improved in the years to come, it seems not implausible that a total theoretical uncertainty on $\sigma\left ( pp \to hh \right )$ of order~$3\%$ ($2\%$) may ultimately be achievable at the HE-LHC~(FCC-pp). A detailed analysis of the systematic uncertainty on the overall determination of the SM background rates at a FCC-pp has been performed in~\cite{Contino:2016spe}. From the results presented in this work one can conclude that the experimental systematic uncertainties may amount to only about~$2\%$, making them subleading compared to other sources of uncertainty. Treating all quoted uncertainties as uncorrelated Gaussian errors then leads to total uncertainties of around $15\%$ and~$5\%$ on the double-Higgs production cross section at the HE-LHC and FCC-pp, respectively. The latter uncertainty estimates will be used in our numerical analysis.
In the case of triple-Higgs production, $pp \to hhh \to b \bar b b \bar b \gamma \gamma$ is the channel that has obtained the most attention~\cite{Contino:2016spe,Mangano:2016jyj,Papaefstathiou:2015paa,Chen:2015gva,Fuks:2015hna,Kilian:2017nio}. Under optimistic assumptions about the detector performance~(see~\cite{Contino:2016spe,Papaefstathiou:2015paa} for details) these analyses concur that a $100 \, {\rm TeV}$ FCC-pp collider with $30 \, {\rm ab}^{-1}$ of data should be able to exclude triple-Higgs production cross sections that are larger by a factor of~2 than the SM prediction. It may be possible to further improve this 95\%~CL upper limit by considering for instance the $b \bar b b \bar b \tau^+ \tau^-$ final state~\cite{Fuks:2017zkg}, but we will not entertain such a possibility here. A~sensitivity study of triple-Higgs production at the HE-LHC does to the best of our knowledge not exist. To estimate the sensitivity that a measurement of triple-Higgs production in the $b \bar b b \bar b \gamma \gamma$ channel can achieve at a $27 \, {\rm TeV}$ machine with $15 \, {\rm fb}^{-1}$ of integrated luminosity, we proceed as follows. We generate the dominant background channels, i.e.~$b \bar b b \bar b \gamma \gamma$ and $hhb \bar b$, as well as the triple-Higgs signal at LO in QCD using {\tt MadGraph5\_aMC@NLO}. Our analysis follows the articles~\cite{Contino:2016spe,Papaefstathiou:2015paa} for what concerns $K$-factors, systematic uncertainties, selection cuts and detector performances such as the $b$-tagging efficiency and the jet-to-photon mis-identification rate. Based on our simulations, we expect $0.2$ and $0.2$ background events from the $b \bar b b \bar b \gamma \gamma$ and $hh b \bar b$ channel, respectively, while for the $pp \to hhh \to b \bar b b \bar b \gamma \gamma$ signal we predict $0.5$ events within the SM. Using these numbers and calculating the significance from a Poisson ratio of likelihoods modified to incorporate systematic uncertainties on the background, we find that a $27 \, {\rm TeV}$ HE-LHC with $15 \, {\rm ab}^{-1}$ of data is expected to exclude triple-Higgs production cross sections that are larger than the SM prediction by a factor of~approximately~11.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.95 \textwidth]{xsecprojections.pdf}
\vspace{-2mm}
\caption{\label{fig:xsecprojections}
Hypothetical constraints in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane. The red and green contours correspond to the allowed regions in parameter space that arise from double-Higgs and triple-Higgs production, respectively, while the yellow regions are obtained by a combination of the two constraints requiring $\Delta \chi^2 = 5.99$. In~both panels the SM is indicated by the black point and the black dashed line corresponds to $\Delta \kappa_4 = 6 \Delta \kappa_3$. In BSM models that lead to the hierarchy $\bar c_6/\bar c_8 \gg 1$ of Wilson coefficients only $\Delta \kappa_3$ and $\Delta \kappa_4$ values close to the black dashed line can be obtained. The results in the left (right) panel have been obtained by assuming that the double-Higgs production cross section has been measured with an accuracy of $15\%$ ($5\%$) at the HE-LHC (FCC-pp). In the case of triple-Higgs production, our projection is instead based on the assumption that cross section values that are larger by a factor of 11~(2) than the SM value are experimentally disfavoured by the HE-LHC~(FCC-pp) at 95\%~CL. See text for further explanations.}
\end{center}
\end{figure}
The two panels in Figure~\ref{fig:xsecprojections} display the expected exclusion sensitivity in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane for the $27 \, {\rm TeV}$~HE-LHC~(left) and a $100 \, {\rm TeV}$ FCC-pp collider~(right) with $15 \, {\rm ab}^{-1}$ and $30 \, {\rm ab}^{-1}$ of integrated luminosity, respectively. The red and green curves illustrate the limits from measurements of the double-Higgs and triple-Higgs production cross sections with the accuracy discussed above, while the yellow regions are the $\Delta \chi^2 = 5.99$ contours (corresponding to a 95\%~CL for a Gaussian distribution) that derive from a combination of the two measurements in the form of a~$\chi^2$~fit. The SM point is indicated by the black dots. One observes that the constraints that arise from the hypothetical measurements of double-Higgs production are bands that form ear-shaped exclusion regions. The widths of the bands are determined by the accuracy of the associated measurement of the inclusive $pp \to hh$ cross section, and as a result the band is narrower by a factor of around~3 for the FCC-pp than the HE-LHC. The shape of the constraints from triple-Higgs production instead depends on whether a future hardron collider can set an~${\cal O} (10)$ or an~${\cal O} (1)$ bound on the cross section. If, like in the case of the HE-LHC, only rough limits can be obtained the triple-Higgs constraint has the form of a banana that extends along the $\Delta \kappa_3$ axis, while the allowed region turns out to be oval-shaped, if a future hadron collider such as the FCC-pp is able to probe triple-Higgs production cross sections at the SM level.
Combining the two constraints, two regions of parameter space remain viable at the HE-LHC that are centred around $\{0,0\}$ and $\{3,4\}$, respectively. In the case of~$\kappa_3 = 1$, we find that the range $\kappa_4 \in [-21, 29]$ is allowed at~95\%~CL. Also notice that the family of solutions $\Delta \kappa_3 = 6 \Delta \kappa_4$~(dashed black line) goes right through the non-SM region of viable parameters. This implies that measurements of the inclusive double-Higgs and triple-Higgs production cross sections at the HE-LHC are unlikely to be able to tell apart scenarios in which large modifications of both the $h^3$ and $h^4$ vertices arise from the single operator~${\cal O}_6$ or the two operators ${\cal O}_6$ and ${\cal O}_8$ $\big($cf. the discussion after~(\ref{eq:kappa3kappa4})$\big)$. The FCC-pp should instead be able to disentangle these two possibilities since it is expected to almost entirely rule out parameters choices in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane that are located close to the point~$\{3,4\}$. Large modifications of the quartic Higgs self-coupling could in such a case only arise from the simultaneous presence of ${\cal O}_6$ and~${\cal O}_8$. One also sees that the allowed region around the SM-point~$\{0,0\}$ will be largely reduced at the FCC-pp compared to the HE-LHC. Numerically, the following 95\%~CL range $ \kappa_4 \in [-5,14]$ is obtained under the assumption that $\kappa_3 = 1$. The quoted range agrees with the FCC-pp bound on the quartic Higgs self-coupling reported in~\cite{Contino:2016spe}.
Before discussing how modified cubic and quartic Higgs self-couplings impact the kinematic distributions in double-Higgs production, we briefly comment on the maximal size that the parameters $\Delta \kappa_3$ and $\Delta \kappa_4$ may take. Under plausible assumptions about the UV structure of the Higgs potential, it has been shown in~\cite{Falkowski:2019tft} that values $\left | \Delta \kappa_3 \right | \lesssim 4$ are compatible with Higgs and EW precision measurements, direct LHC searches and vacuum stability. The corresponding limit on the modifications of the quartic Higgs self-coupling reads $\left | \Delta \kappa_4 \right | \lesssim 40$, if one assumes hat there is a parametric separation between the EW and new-physics scales, leading to a suppression of operators with dimension higher than six. The quoted upper bounds fall into the same ballpark than the limits obtained in~\cite{Maltoni:2018ttu} from perturbativity considerations (see also \cite{DiLuzio:2017tfn,Chang:2019vez} for related discussions). Since the mentioned theoretical bounds on the parameters $\Delta \kappa_3$ and $\Delta \kappa_4$ are neither sharp nor model independent, we do not explicitly indicate them in Figures~\ref{fig:xsecprojections}, \ref{fig:distcprojections} and \ref{fig:allprojections} when reporting the results of our phenomenological $\Delta \chi^2$ fits in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane.
\subsection{Kinematic distributions in double-Higgs production}
\label{sec:numerics2}
In the previous section we have seen that combining the information on the inclusive double-Higgs and triple-Higgs production cross section may not be able to resolve all ambiguities in the $\Delta \kappa_3$ and $\Delta \kappa_4$ determination --- a feature that is nicely illustrated by the left panel in Figure~\ref{fig:xsecprojections}. It is well-known~\cite{Goncalves:2018qas, Azatov:2015oxa,Barr:2014sga,He:2015spf,Contino:2016spe,Mangano:2016jyj,Banerjee:2018yxy,Chang:2018uwu} that precise measurements of differential distributions in double-Higgs production can be used to resolve ambiguities and/or flat directions in the extraction of coupling modifiers or Wilson coefficients, and in the following we will apply this general idea to the case of a simultaneous determination of the cubic and quartic Higgs self-couplings.
\begin{figure}[!t]
\begin{center}
\includegraphics[clip, trim=2cm 2cm 2cm 2cm,width= \textwidth]{mhhspectra.pdf}
\vspace{-2mm}
\includegraphics[clip, trim=2cm 2cm 2cm 2cm,width= \textwidth]{pthspectra.pdf}
\vspace{-4mm}
\caption{\label{fig:distributions} Normalised predictions for the $m_{hh}$ (upper row) and the leading $p_{T,h}$ (lower row) spectrum in $pp \to hh$ production at the $100 \, {\rm TeV}$ FCC-pp. The black histograms represent the SM distributions, while the coloured curves correspond to different choices of the Higgs boson self-coupling modifiers $\kappa_3$ and~$\kappa_4$. Consult the main text for additional details.}
\end{center}
\end{figure}
In Figure~\ref{fig:distributions} we depict the differential distributions of two relevant kinematic variables, namely the invariant mass $m_{hh}$ of the di-Higgs system (upper row) and the leading transverse momentum $p_{T,h}$ of the two Higgs bosons (lower row) in $pp \to hh$. The shown results are NLO accurate and, as before, have been obtained with a modified version of {\tt POWHEG BOX} using {\tt PDF4LHC15\_nlo}~PDFs. They assume $pp$ collisions at a CM energy of $100 \, {\rm TeV}$. The coloured histograms represent the four choices $\{\kappa_3, \kappa_4 \} = \{0.7,0\}, \{1,60\}, \{1.05,0\}, \{1,-30\}$. The former two parameter combinations lead to enhancements of the inclusive $pp \to hh$ cross section by roughly $30\%$ with respect to the SM, while the latter two choices reduce the double-Higgs production rate by about $-5\%$. Based on measurements of $\sigma \left (pp \to hh \right )$ the choices $ \{0.7,0\}$ and $\{1,60\}$ $\big( \{1.05,0\}$ and $\{1,-30\} \big)$ are therefore not distinguishable. As can be seen from the four panels in Figure~\ref{fig:distributions}, the predictions for the normalised $m_{hh}$ and $p_{T,h}$ spectra are however not the same for the two types of $\{\kappa_3, \kappa_4 \}$ sets. Since the distortions in the $p_{T,h}$ distribution turn out to be typically smaller than those in the~$m_{hh}$ spectrum, we will use the latter kinematic observable in our shape analysis.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.95 \textwidth]{distprojections.pdf}
\vspace{-2mm}
\caption{\label{fig:distcprojections}
Hypothetical constraints in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane following from a shape analysis of the $m_{hh}$ spectrum in $pp \to hh$ production at the HE-LHC (left panel) and FCC-pp (right panel). The green (yellow) contours correspond to 68\%~CL (95\%~CL) regions. In~both figures the SM is indicated by the black point and the black dashed line represents the family of solutions that satisfy $\Delta \kappa_4 = 6 \Delta \kappa_3$. In SM extensions that give rise to the hierarchy $\bar c_6/\bar c_8 \gg 1$ of dimension-eight and dimension-six contributions only $\Delta \kappa_3$ and $\Delta \kappa_4$ values close to the black dashed line can be accommodated. For further details consult the text.}
\end{center}
\end{figure}
The signal needed to perform the shape analysis is generated at NLO in QCD with {\tt POWHEG~BOX} matched to {\tt Pythia~8}~\cite{Sjostrand:2007gs,Sjostrand:2014zea} to include parton-shower effects (we use a customised version of the computer code presented in~\cite{Heinrich:2017kxx}). {\tt PDF4LHC15\_nlo}~PDFs are employed and jets ($j$) are reconstructed with {\tt FastJet}~\cite{Cacciari:2011ma} using an anti-$k_t$ algorithm~\cite{Cacciari:2008gp}. Our analysis then follows~\cite{Goncalves:2018qas}. We demand two $b$-tagged jets ($b$) and two isolated photons ($\gamma$) with the following minimal cuts on the transverse momentum, pseudorapidity and radius separation: $p_{T,x} > 30 \, {\rm GeV}$, $|\eta_x| > 2.5$ and $\Delta R_{xy} > 0.4$ for $x,y = j,b,\gamma$. A flat $b$-tagging efficiency of~$70\%$, and mis-tag rates of $15\%$ for charm quarks and $0.3\%$ for light flavours are adopted. Events with more than three jets are vetoed, and the requirements $|m_{b \bar b} - m_h| < 25 \, {\rm GeV}$, $|m_{\gamma \gamma} - m_h| < 1 \, {\rm GeV}$ and $m_{hh} > 400 \, {\rm GeV}$ are imposed as a final selection. The obtained $m_{hh}$ distributions have then been binned into bins of $25 \, {\rm GeV}$. Our shape fit includes the statistical uncertainties in each bin as well as theoretical and experimental systematic uncertainties of $3\%$ ($2\%$) and $2\%$ ($2\%$) at HE-LHC (FCC-pp), respectively. The quoted uncertainties have been treated as uncorrelated Gaussian errors in the $\chi^2$ fit. We emphasise that our fit does not consider the impact of backgrounds, but we have verified that with the described methodology we are able to reproduce the CL-level curves presented in~\cite{Goncalves:2018qas} for both the HE-LHC and FCC-pp quite well. This agreement gives us confidence that our simplified approach is able to mimic the more sophisticated analysis~\cite{Goncalves:2018qas} that includes a simulation of all relevant SM backgrounds.
The results of our $m_{hh}$ shape analysis are shown in Figure~\ref{fig:distcprojections}. The green (yellow) regions are the $\Delta \chi^2 = 2.28$ ($\Delta \chi^2 = 5.99$) contours, corresponding to 68\%~CL (95\%~CL) limits for a Gaussian distribution. In~both panels the SM point is indicated by a black dot and the black dashed line illustrates the equality $\Delta \kappa_4 = 6 \Delta \kappa_3$. From the panel on the left-hand side one sees that at the HE-LHC a shape analysis of the $m_{hh}$ distribution in $pp \to hh$ will not allow one to exclude choices in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane around $\{3, 4\}$,~i.e.~parameters that survive a combination of the measurements of the inclusive double-Higgs and triple-Higgs production cross sections (see the left panel in Figure~\ref{fig:xsecprojections}). For~$\kappa_3 =1$ we find the following 95\%~CL range $\kappa_4 \in [-82, 37]$. As shown in the right panel in Figure~\ref{fig:distcprojections}, at the FCC-pp the constraints in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane that follow from a $m_{hh}$ shape analysis are expected to improve noticeably compared to the corresponding HE-LHC limits. Differential measurements of $pp \to hh$ at the FCC-pp alone will in consequence be able to distinguish scenarios in which large modifications of both the $h^3$ and $h^4$ interactions arise from the operator ${\cal O}_6$ or a combination of ${\cal O}_6$ and ${\cal O}_8$ $\big($cf.~the text after (\ref{eq:kappa3kappa4})$\big)$. Assuming again that $\kappa_3 =1$, the 95\%~CL range for the parameter $\kappa_4$ reads $\kappa_4 \in [-22, 15]$. Profiling over $\kappa_3$ by means of the profile likelihood ratio~\cite{Cowan:2010js}, we obtain the following 95\%~CL bound $\kappa_4 \in [-89, 159]$ and $\kappa_4 \in [-19, 21]$ at the HE-LHC and the FCC-pp, respectively. Our FCC-pp constraints on $\kappa_4$ are comparable to those that have been derived in the analysis~\cite{Borowka:2018pxx}.
\subsection{Global fit at the HE-LHC and a FCC-pp}
\label{sec:numerics3}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.95 \textwidth]{allprojections.pdf}
\vspace{-2mm}
\caption{\label{fig:allprojections}
Hypothetical constraints in the $\Delta \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \Delta \kappa_4$ plane following from a combination of a shape analysis of the $m_{hh}$ spectrum in $pp \to hh$ production and a measurement of the inclusive production cross section of $pp \to hhh$. The green (yellow) contours correspond to 68\%~CL (95\%~CL) regions and the left (right) panel shows the HE-LHC (FCC-pp) projections. The SM solution is indicated by the black point and the black dashed line represents the parameter choices satisfying $\Delta \kappa_4 = 6 \Delta \kappa_3$. Only $\Delta \kappa_3$ and $\Delta \kappa_4$ values with $\Delta \kappa_4 \simeq 6 \Delta \kappa_3$ can be obtained in BSM models that give rise to $\bar c_6/\bar c_8 \gg 1$. See text for additional details.}
\end{center}
\end{figure}
The full potential of the HE-LHC and the FCC-pp in constraining simultaneously the coupling modifications $\kappa_3$ and $\kappa_4$ can be assessed by combining the information on the differential measurements of $pp \to hh$ with the expected accuracies in the determination of the inclusive $pp \to hhh$ production cross section. The outcome of such an exercise is presented in Figure~\ref{fig:allprojections}. Here the green~(yellow) contours correspond to 68\%~CL (95\%~CL) regions, while the black dots represent the SM point and the black dashed lines illustrate parameter choices of the form $\Delta \kappa_4 = 6 \Delta \kappa_3$. {Numerically, we find that for $\kappa_3 = 1$, the 95\%~CL bounds on $\kappa_4$ from a global analysis of differential double-Higgs and inclusive triple-Higgs data at the HE-LHC (FCC-pp) is $ \kappa_4 \in [-21, 27]$ ($ \kappa_4 \in [-5, 12]$). Notice that these limits represent an improvement of the bounds derived in Section~\ref{sec:numerics1} based on inclusive measurements alone. Profiling instead over $\kappa_3$, the following 95\%~CL bounds are obtained $\kappa_4 \in [-17, 26]$ and $\kappa_4 \in [-3, 13]$.
\subsection[Comparison between sensitivities of future $pp$ and $e^+ e^-$ machines]{Comparison between sensitivities of future $\bm{pp}$ and $\bm{e^+ e^-}$ machines}
\label{sec:numerics4}
The constraints on the cubic and quartic Higgs self-couplings that high-energy $e^+ e^-$ machines may be able to set have been studied recently in~\cite{Maltoni:2018ttu,Liu:2018peg}. Both articles have performed global fits assuming various ILC and CLIC setups to determine the allowed modifications of the cubic and quartic Higgs self-couplings. Under the assumption of $\kappa_3 = 1$, the article~\cite{Maltoni:2018ttu} finds for instance for an ILC with a CM energy of $500 \, {\rm GeV}$, polarisations of $P(e^-,e^+) = (-0.8, 0.3)$ and an integrated luminosity of $4 \, {\rm ab}^{-1}$ (ILC-500) the following 95\%~CL bound $\kappa_4 \in [-11, 13]$. At CLIC with a CM energy of $3000 \, {\rm GeV}$, polarisations of $P(e^-,e^+) = (-0.8, 0.0)$ and an integrated luminosity of $2 \, {\rm ab}^{-1}$~(CLIC-3000) the corresponding limit is said to be $\kappa_4 \in [-5, 7]$. The constraints presented in~\cite{Liu:2018peg} are less stringent than those obtained in~\cite{Maltoni:2018ttu}. Taking the limits given in~\cite{Maltoni:2018ttu} at face value and comparing them to the bounds presented in the last subsection, suggests that the HE-LHC has a weaker sensitivity to modifications of the quartic Higgs self-coupling than the ILC-500. On the other hand, the FCC-pp reach seems to be better than that of the ILC-500, and roughly comparable to the CLIC-3000 potential.
\section{Conclusions}
\label{sec:conclusions}
In this work, we have investigated the possibility to constrain the quartic Higgs self-interactions indirectly through precise measurements of double-Higgs production at future hadron colliders. We have first presented the results of a calculation of the two-loop contributions to the $gg \to hh$ amplitudes that involve a modified $h^4$ vertex. Our results have been obtained in numerical form with the help of {\tt pySecDec}~\cite{Borowka:2012yc,Borowka:2015mxa,Borowka:2017idc} and have been implemented into {\tt POWHEG BOX}~\cite{Alioli:2010xd}. Combining the two-loop EW corrections calculated here with the~${\cal O} (\alpha_s^2)$~matrix elements computed in~\cite{Borowka:2016ehy,Borowka:2016ypz,Heinrich:2017kxx}, we are able to predict the cross section and the most important distributions for double-Higgs production at NLO in QCD, including arbitrary modifications of both the $h^3$ and the~$h^4$ coupling.
Based on our results, we have then performed an exploratory study of the sensitivity of the $27 \, {\rm TeV}$~HE-LHC and a $100 \, {\rm TeV}$ FCC-pp in constraining simultaneously the cubic and quartic Higgs self-couplings by measurements of double-Higgs and triple-Higgs production in gluon-fusion. In a first step, we have considered only measurements of total rates. In the case of the HE-LHC with $15 \, {\rm ab}^{-1}$ of integrated luminosity, we have found that a combined fit to $\sigma \left ( pp \to hh \right )$ and $\sigma \left ( pp \to hhh \right )$ will have a two-fold ambiguity in the $ \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \kappa_4$ plane with a family of solutions located either around $\{1,1\}$ or in the vicinity of~$\{4,5\}$.} The resulting bounds on possible modifications of the quartic Higgs self-coupling turn out to be generically weak. For instance, for~$\kappa_3 = 1$ we found that $\kappa_4$ values in the range $\kappa_4 \in [-21, 29]$ are allowed at~95\%~CL. Due to its significantly improved sensitivity to triple-Higgs production, a FCC-pp with $30 \, {\rm ab}^{-1}$ of data should be able to resolve the aforementioned degeneracy by reducing the viable parameter space to a stripe in the $ \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \kappa_4$ plane with SM-like cubic Higgs self-couplings. Numerically, we found that for $\kappa_3 = 1$, the range of $ \kappa_4 \in [-5,14]$ is allowed at 95\%~CL. Our limit agrees with the FCC-pp bound on the $h^4$ coupling quoted in~\cite{Contino:2016spe}.
Given that precise measurements of differential distributions in double-Higgs production are known~\cite{Goncalves:2018qas, Azatov:2015oxa,Barr:2014sga,He:2015spf,Contino:2016spe,Mangano:2016jyj,Banerjee:2018yxy,Chang:2018uwu} to be able to resolve degeneracies in the extraction of coupling modifiers or Wilson coefficients, we have in a second step performed a shape analysis to determine the allowed regions in the $ \kappa_3 \hspace{0.25mm}$--$ \hspace{0.25mm} \kappa_4$ plane. We have considered both the $m_{hh}$ spectrum and the leading $p_{T,h}$ distribution and found the former observable to have more discriminating power in a simultaneous extraction of the cubic and quartic Higgs self-couplings. From our $m_{hh}$ shape analysis it follows that at the HE-LHC with $15 \, {\rm ab}^{-1}$ of data it should be possible to constrain $\kappa_4$ to the 95\%~CL range $\kappa_4 \in [-82, 37]$, if one assumes that $\kappa_3 = 1$. The corresponding constraint at a FCC-pp with $30 \, {\rm ab}^{-1}$ of integrated luminosity turns out to be $\kappa_4 \in [-22, 15]$. The 95\%~CL bounds $\kappa_4 \in [-89, 159]$ and $\kappa_4 \in [-19, 21]$ instead apply if one profiles over $\kappa_3$. The obtained limits show that differential measurements in the $pp \to hh$ channel alone are expected to lead compared to measurements of the inclusive $pp \to hhh$ cross sections to notable weaker determinations of the quartic Higgs self-coupling at both the HE-LHC and a~FCC-pp. The same conclusion has been drawn in~\cite{Borowka:2018pxx} for what concerns the FCC-pp.
To assess the full potential of the HE-LHC and the FCC-pp in constraining simultaneously the coupling modifiers $\kappa_3$ and $\kappa_4$, we have combined the differential measurements of $pp \to hh$ with the inclusive measurements of $pp \to hhh$. Our global analysis demonstrates that under the assumption $\kappa_3 = 1$, one can expect to obtain a 95\%~CL bound on $\kappa_4$ at the HE-LHC (FCC-pp) of $ \kappa_4 \in [-21, 27]$ ($ \kappa_4 \in [-5, 12]$). By profiling over $\kappa_3$, we arrived at $\kappa_4 \in [-17, 26]$ and $\kappa_4 \in [-3, 13]$. The former bounds can be compared to the hypothetical constraints of the Higgs self-couplings that high-energy $e^+ e^-$ machines might be able to set~\cite{Maltoni:2018ttu,Liu:2018peg}. For example, the ILC-500 (CLIC-3000) is expected to be able to set the 95\%~CL bound $ \kappa_4 \in [-11,13]$ ($ \kappa_4 \in [-5,7]$)~\cite{Maltoni:2018ttu}, assuming that $\kappa_3 =1$. These numbers indicate that the HE-LHC should have a weaker sensitivity to modified quartic Higgs self-interactions than ILC-500. A FCC-pp and CLIC-3000 can, however, be expected to have roughly similar potentials in constraining the coupling modifier $\kappa_4$.
\acknowledgments
We thank Fady Bishara for useful discussions about profile likelihood ratios, Sophia Borowka for her patient help with {\tt pySecDec} and are grateful to Paolo~Torrielli and Marco~Zaro for their assistance with {\tt MadGraph5\_aMC@NLO}. We express gratitude to Fady Bishara and Marek Sch\"onherr for trying to be good {\tt Sherpa}s~\cite{Gleisberg:2008ta} (so that we could lean back and enjoy). We~finally would like to express our gratitude to Kunfeng Lyu for drawing our attention to the ${\cal O} (\kappa_3 \kappa_4)$ contributions~(\ref{eq:DeltaF1F2additional}) that have not been considered in the first version of this article. WB~has been supported by the ERC Consolidator Grant HICCUP (614577), UH acknowledges the hospitality and support of the CERN Theoretical Physics Department and LR has been supported by the ERC Starting Grants PDF4BSM~(335260) and REINVENT (714788).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,963 |
La subfamilia Spiraeoideae es una subfamilia de plantas de flores perteneciente al orden Rosales.
La mayoría son arbustos, pero también hay algunas hierbas. La mayoría tiene hojas simples, pero el género Aruncus y Sorbaria tienen hojas pinadas. Los carpelos son distintos y pocos (2-5). Casi todos los géneros de la tribu Spiraeoideae producen flores conteniendo semillas que maduran a frutos que son agregados de folículos.
Enlaces externos
University of Illinois 2002-05-29
Rosaceae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,119 |
#ifndef _linux_POSIX_TIMERS_H
#define _linux_POSIX_TIMERS_H
#include <linux/spinlock.h>
#include <linux/list.h>
#include <linux/sched.h>
#include <linux/timex.h>
#include <linux/alarmtimer.h>
union cpu_time_count {
cputime_t cpu;
unsigned long long sched;
};
struct cpu_timer_list {
struct list_head entry;
union cpu_time_count expires, incr;
struct task_struct *task;
int firing;
};
/*
* Bit fields within a clockid:
*
* The most significant 29 bits hold either a pid or a file descriptor.
*
* Bit 2 indicates whether a cpu clock refers to a thread or a process.
*
* Bits 1 and 0 give the type: PROF=0, VIRT=1, SCHED=2, or FD=3.
*
* A clockid is invalid if bits 2, 1, and 0 are all set.
*/
#define CPUCLOCK_PID(clock) ((pid_t) ~((clock) >> 3))
#define CPUCLOCK_PERTHREAD(clock) \
(((clock) & (clockid_t) CPUCLOCK_PERTHREAD_MASK) != 0)
#define CPUCLOCK_PERTHREAD_MASK 4
#define CPUCLOCK_WHICH(clock) ((clock) & (clockid_t) CPUCLOCK_CLOCK_MASK)
#define CPUCLOCK_CLOCK_MASK 3
#define CPUCLOCK_PROF 0
#define CPUCLOCK_VIRT 1
#define CPUCLOCK_SCHED 2
#define CPUCLOCK_MAX 3
#define CLOCKFD CPUCLOCK_MAX
#define CLOCKFD_MASK (CPUCLOCK_PERTHREAD_MASK|CPUCLOCK_CLOCK_MASK)
#define MAKE_PROCESS_CPUCLOCK(pid, clock) \
((~(clockid_t) (pid) << 3) | (clockid_t) (clock))
#define MAKE_THREAD_CPUCLOCK(tid, clock) \
MAKE_PROCESS_CPUCLOCK((tid), (clock) | CPUCLOCK_PERTHREAD_MASK)
#define FD_TO_CLOCKID(fd) ((~(clockid_t) (fd) << 3) | CLOCKFD)
#define CLOCKID_TO_FD(clk) ((unsigned int) ~((clk) >> 3))
/* POSIX.1b interval timer structure. */
struct k_itimer {
struct list_head list; /* free/ allocate list */
spinlock_t it_lock;
clockid_t it_clock; /* which timer type */
timer_t it_id; /* timer id */
int it_overrun; /* overrun on pending signal */
int it_overrun_last; /* overrun on last delivered signal */
int it_requeue_pending; /* waiting to requeue this timer */
#define REQUEUE_PENDING 1
int it_sigev_notify; /* notify word of sigevent struct */
struct signal_struct *it_signal;
union {
struct pid *it_pid; /* pid of process to send signal to */
struct task_struct *it_process; /* for clock_nanosleep */
};
struct sigqueue *sigq; /* signal queue entry. */
union {
struct {
struct hrtimer timer;
ktime_t interval;
} real;
struct cpu_timer_list cpu;
struct {
unsigned int clock;
unsigned int node;
unsigned long incr;
unsigned long expires;
} mmtimer;
struct {
struct alarm alarmtimer;
ktime_t interval;
} alarm;
struct rcu_head rcu;
} it;
};
struct k_clock {
int (*clock_getres) (const clockid_t which_clock, struct timespec *tp);
int (*clock_set) (const clockid_t which_clock,
const struct timespec *tp);
int (*clock_get) (const clockid_t which_clock, struct timespec * tp);
int (*clock_adj) (const clockid_t which_clock, struct timex *tx);
int (*timer_create) (struct k_itimer *timer);
int (*nsleep) (const clockid_t which_clock, int flags,
struct timespec *, struct timespec __user *);
long (*nsleep_restart) (struct restart_block *restart_block);
int (*timer_set) (struct k_itimer * timr, int flags,
struct itimerspec * new_setting,
struct itimerspec * old_setting);
int (*timer_del) (struct k_itimer * timr);
#define TIMER_RETRY 1
void (*timer_get) (struct k_itimer * timr,
struct itimerspec * cur_setting);
};
extern struct k_clock clock_posix_cpu;
extern struct k_clock clock_posix_dynamic;
void posix_timers_register_clock(const clockid_t clock_id, struct k_clock *new_clock);
/* function to call to trigger timer event */
int posix_timer_event(struct k_itimer *timr, int si_private);
void posix_cpu_timer_schedule(struct k_itimer *timer);
void run_posix_cpu_timers(struct task_struct *task);
void posix_cpu_timers_exit(struct task_struct *task);
void posix_cpu_timers_exit_group(struct task_struct *task);
void set_process_cpu_timer(struct task_struct *task, unsigned int clock_idx,
cputime_t *newval, cputime_t *oldval);
long clock_nanosleep_restart(struct restart_block *restart_block);
void update_rlimit_cpu(struct task_struct *task, unsigned long rlim_new);
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,114 |
Q: .NET Entity framework for java i would like to ask, if there is some framework in Java which can automaticly generate entities (objects) from database?
There is a similar possibility in visual studio to "generate model from database".
Thanks for replies
A: Take a look at Java Persistence API. You can generate JPA entities from an existing database in Eclipse.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,802 |
Produced by Tor Martin Kristiansen, Joseph Cooper and the
Online Distributed Proofreading Team at http://www.pgdp.net
UNIVERSITY OF CALIFORNIA PUBLICATIONS
AMERICAN ARCHAEOLOGY AND ETHNOLOGY
Vol. 4 No. 6
THE RELIGION OF THE INDIANS OF CALIFORNIA
BY
A. L. KROEBER
BERKELEY
THE UNIVERSITY PRESS
SEPTEMBER, 1907
Facsimile Reprint by
Coyote Press
P.O. Box 3377
Salinas, CA 93912
http://www.CoyotePress.com
UNIVERSITY OF CALIFORNIA PUBLICATIONS
IN
AMERICAN ARCHAEOLOGY AND ETHNOLOGY
VOL. 4 NO. 6
THE RELIGION OF THE INDIANS OF CALIFORNIA[1]
BY
A. L. KROEBER.
Fundamentally the religion of the Indians of California was very
similar to that of savage and uncivilized races the world over. Like
all such peoples, the California Indians were in an animistic state of
mind, in which they attributed life, intelligence, and especially
supernatural power, to virtually all living and lifeless things. They
lacked no less the ideas and practices of shamanism, the universal
accompaniment of animism: namely, the belief that certain men, through
communication with the animate supernatural world, had the power to
accomplish what was contrary to, or rather above, the events of daily
ordinary experience, which latter in so far as they were distinguished
from the happenings caused by supernatural agencies, were of natural,
meaningless, and, as it were, accidental origin. As in most parts of
the world, belief in shamanistic power was centered most strongly about
disease and death, which among most tribes were not only believed to be
dispellable but to be entirely caused by shamans. In common with the
other American Indians, those of California made dancing, and with it
always singing, a conspicuous part of nearly all their ceremonies that
were of a public or tribal nature. They differed from almost all other
tribes of North America in showing a much weaker development of the
ritualism, and symbolism shading into pictography, that constitute
perhaps the most distinctive feature of the religion of the Americans
as a whole. Practically all the approaches to a system of writing
devised in North America, whether in Mexico, Yucatan, or among the
tribes of the United States and Canada, are the direct outcome of a
desire of religious expression. The California Indians however were
remarkably free from even traces of this tendency, equally in their
religion and in the more practical aspects of their life. In many parts
of North America, and more often where the culture was considerably
developed than where it was rude, there was a considerable amount of
fetishism, not of the crass and so to speak superstitious type of
Africa, but rather as an accompaniment and result of over-symbolism.
This fetishistic tendency was very slightly developed in California,
and this in spite of--or as an Americanist could more properly say on
account of--the generally rude and primitive condition of culture. By
contrast, as the action and the visible symbol were a less important
means of religious expression, the word, both spoken and sung, was of
greater significance in California. The weakness of the ritualistic
tendency is however again marked in the circumstance that the exact
form of religious speech was frequently less regarded than its
substance. In this aspect the Indians of California differed widely
from such nations as the Egyptians and the peoples of Asia, where the
efficacy of the word and speech used for a religious purpose was
usually directly dependent upon the accuracy of their external and
audible rendering, even to their pronunciation and intonation.
[1] This paper may be cited as Univ. Calif. Publ. Am. Arch.
Ethn., Vol. 4, No. 6.
As an ethnographic province the greater part of California plainly
forms a unit. There are, however, two portions of the present political
state that showed much cultural distinctness in times of native life
and that must usually be kept apart in all matters of ethnological and
religious consideration. One of these divergent culture areas comprised
the extreme northwestern corner of the state, in the drainage of the
lower Klamath and about Humboldt Bay. The other consisted of what is
now usually known as Southern California, extending from the Tehachapi
pass and mountains in the interior, and from Point Conception on the
coast, southward to the Mexican boundary. The religion of the Indians
of the peninsula of Lower California is very little known from
literature, and the people themselves are almost extinct. It is
probable that it was more or less different from the forms of religion
occurring in Southern California, that is to say, the southern part of
the American state of California. Ethnographically Southern California
was considerably diversified. The tribes of the plains and mountains
near the sea must be distinguished on the one hand from those of the
desert interior and of the valley of the Colorado river, and on the
other from those of the Santa Barbara archipelago and the adjacent
coast of the mainland to the north. The latter island group of tribes
has become entirely extinct without leaving more than the merest trace
of records of its religion. The two other groups, the sea-ward and the
interior, apparently presented a much greater uniformity in religion
than in their material and social life, so much so that in the present
connection all the tribes of Southern California of whom anything is
known may be regarded as constituting a single ethnographic province.
The culture of the small Northwestern area was in every way, and that
of the larger Southern province at least in some respects, more highly
organized and complex than that of the still larger and principal
Central region, which comprised at least two-thirds of the state and
which, if such a selection is to be made, must be considered as the
most typically Californian.
The religious practices of the Indians of California fall into three
well marked divisions: (1) such observances as are followed and
executed by individuals, although their perpetuation is traditionary
and tribal; that is to say, customary observances; (2) individual
practices resting upon a direct personal communication of an individual
with the supernatural world; in other words, shamanism; (3) observances
and practices which are not only the common property of the tribe by
tradition, but in which the entire tribe or community directly or
indirectly participates; in other words, ceremonies.
CUSTOMARY OBSERVANCES BY INDIVIDUALS.
Customary observances are as strongly developed as farther north along
the Pacific <DW72>. This entire western coast region thus forms a unit
that differs from the interior and eastern parts of the continent, in
which such observances are usually a less conspicuous feature than
public and tribal ceremonies. By far the most important of the
customary observances in California are those relating to death. Next
come those connected with birth and sexual functions. Beliefs and
practices centering about the individual's name are of importance
particularly in so far as they are connected with the customs relating
to death. There are restrictions and superstitions as to food, but
these are not more numerous than seems generally to have been the case
among the North American Indians, and certainly of much less importance
than in the Pacific island world and Australia.
Death was considered to cause defilement and almost everywhere brought
after it purification ceremonies. In the Northwestern region these were
particularly important, and among such tribes as the Hupa and Yurok the
observance of religious purification from contact with the dead, the
most essential part of which was the recitation of a certain formula,
was the most stringently exacted religious custom. The method of
disposing of the dead varied locally between burial and cremation,
cremation being practiced over at least half of the state. Air burial
and sea burial were nowhere found. Mourning, which consisted primarily
of singing and wailing, began immediately upon death and continued for
about a day, sometimes longer by the immediate relatives of the
deceased. Among some tribes this mourning commenced with full vigor
some time before impending death, often during the full consciousness
of the patient and with his approval. Mutilations on the part of the
mourners were not practiced to any great degree, except that the hair
was almost universally cut more or less, especially by the women. Among
many tribes the widow, but she only, cut or burned off all her hair.
Mourning observances were almost always carried further by the women
than men. Among some tribes of the Sierra Nevada the widow did not
speak from the time of her husband's death until the following annual
tribal mourning ceremony, except to one attendant, or, in cases of
actual necessity, to women only. In the Sierra Nevada was found also
the custom of the widow smearing her face and breast with pitch, which
was not washed or removed until this annual ceremony. Except in the
case of the Northwestern tribes, who possessed more elaborately
constructed houses of wood, the house in which a death had occurred was
not used again, but was burned. Objects that had been in personal
contact or associated with the deceased were similarly shunned and
destroyed. The name of the dead was not spoken. Even the word which
constituted his name was not used in ordinary discourse, a
circumlocution or newly coined word being employed. It is certain that
this stringently observed custom has been a factor in the marked
dialectic differentiation of the languages of California. The mention
of the name of the dead, whether intentionally or accidentally, in some
cases aroused feelings of fear connected with his spirit, but more
generally was objected to as causing grief, which appears to have been
actually and often intensely felt on such occasions. In Northwestern
California the naming of the dead could be compensated for only by the
payment of a considerable sum. Practically the only form of curse or
malediction known, other than an occasional indirect allusion to the
object of the malediction as being in the condition of a corpse, was a
reference to his dead relatives. Some property, but more rarely food,
was buried with the dead. The idea that such articles were for his use
in the world of the dead was not so strong a motive for such acts as,
on the one hand, the feeling that the objects had been defiled by
association with him, and on the other, the desire to give expression
to the sincerity of the mourning by the destruction of valuables. On
the whole, however, the immediate observances of death paled in
importance before the annual communal mourning ceremony, which was
everywhere, except in the Northwestern region, one of the most deeply
rooted and spectacular acts of worship.
Observances connected with sexual functions, including birth, are next
in importance after those relating to death. The menstruating woman was
everywhere regarded as unclean, and excluded especially from acts of
worship. Not infrequent was the conception that she contaminated food,
especially meat; in other words those varieties of food which were at
once more highly prized and at the same time, through being obtained
with less regularity and only through special and skilled exertions,
regarded as most directly under the control and influence of
supernatural powers. Among many tribes, as elsewhere in America and
other continents, she was excluded from the living-house as well as
from the ceremonial chamber, and confined to the menstrual hut. As
elsewhere in North America, the custom in this regard however varied
from tribe to tribe, the menstrual hut not having been used in some
localities even in purely aboriginal times. Not only was seclusion, as
a means of preventing contact and association, frequently required of
the woman for the protection of others, but her refraining from all but
the most necessary activity was sometimes deemed essential for her own
good.
All these observances were greatly intensified at the time of a girl's
first menstruation, a condition for which most of the languages of
California possess a distinctive and often unanalyzed word. The girl at
this period was thought to be possessed of a particular degree of
supernatural power, and this was not always regarded as entirely
defiling or malevolent. Often, however, there was a strong feeling of
the power of evil inherent in her condition. Not only was she secluded
from her family and the community, but an attempt was made to seclude
the world from her. One of the injunctions most strongly laid upon her
was not to look about her. She kept her head bowed and was forbidden to
see the world and the sun. Some tribes covered her with a blanket. Many
of the customs in this connection resembled those of the North Pacific
Coast most strongly, such as the prohibition to the girl to touch or
scratch her head with her hand, a special implement being furnished her
for the purpose. Sometimes she could eat only when fed and in other
cases fasted altogether. Some form of public ceremony, often
accompanied by a dance and sometimes by a form of ordeal for the girl,
was practiced nearly everywhere. Such ceremonies were well developed in
Southern California, where a number of actions symbolical of the girl's
maturity and subsequent life were performed. Certain tribes, however,
including at least one in the Northwestern area and certain of those in
the Sierra region, did not practice public ceremonies of this type.
Religious customs connected with birth consisted in part of observances
before the birth of the child, in part of observances relating to it
after birth, and especially of restrictions imposed on one or both of
its parents after birth. Practices affecting the child itself, or the
mother before its birth, related in great part to food. In the
Northwest the newly born child was fed for a number of days only on a
soup of vegetable substance resembling milk. The newly born child was
washed, often repeatedly, among many tribes. The mother after a birth
was regarded as more or less defiled, though this feeling usually did
not approach in intensity those connected with either death or the
woman's periodical functions. Either the mother or both father and
mother were usually enjoined from activity for some time after a birth,
the motive being not only protection of the child but of themselves.
This idea is especially developed among the Yokuts of the southern San
Joaquin valley. The couvade in its strict form, with restrictions and
observances which are imposed entirely upon the father to the exclusion
of the mother, does not seem to be found.
Observances regulating or restricting the use of food were in the main
connected with the customs relating to death, sexual functions, and
birth. That is to say it was primarily the persons affected by these
occurrences, and next to these such as were engaged in acts of intense
worship or shamanistic practices, who were prohibited from using
certain or all foods. As already stated, animal food rather than
vegetable, and meat rather than fish, and among meat that of the deer
and elk, the largest of the game animals, were particularly subjected
to restriction. In Northwestern California the idea was very deeply
rooted that the deer when killed and eaten are not destroyed, but come
to life again and report to their fellows their treatment in the hands
of the hunter. Any violation of the numerous stringent observances
regarding deer meat are therefore known to all the deer, who, as their
capture is always a voluntary act on their part, are in position to
utterly destroy his luck in the chase if not placated by certain spoken
formulae. In Southern California young people, or in some cases the
hunter himself, must not eat his game. Fasting is less frequently and
less rigorously practiced by the California tribes than by those of
most other parts of North America. This is in keeping with the
generally lower pitch of intensity of their religious feeling. Many
public ceremonies are not accompanied by any requirement of abstention
from food. In the Northwestern region it is only the principal priest,
in whom the most sacred part of the ceremony is vested, who fasts. On
the other hand there is a general feeling in this region that not only
acts of a religious nature but ordinary work cannot be well performed
after eating. Among the men of Northwestern California breakfast was
therefore habitually slight or entirely omitted. Perhaps the greatest
development of the practice of fasting in North America occurs in
connection with the acquisition of shamanistic power. Shamanism is
fully as important among the California Indians as elsewhere, but
differs in that it is more frequently regarded as an obsession,
something that of its own accord comes upon a man rather than something
that it is sought to acquire by actions. Much of the incentive for
fasting among other Indians is therefore lacking, and when the practice
is observed it is usually less rigorous. In Northwestern California,
for instance, a person engaged in almost any supernatural or religious
practice abstains from drinking water; but as to practical effect this
provision is done away with through his being allowed to drink thin
acorn soup at will.
In Northwestern California there is a special development of spoken
formulae, whose content is little else than a myth and which constitute
not only the basis and essential element of public ceremonies but are
connected with almost all customary observances. To such an extent have
these formulae, locally called "medicines," grown into the mind of
these Indians as being what is most sacred and most efficacious in all
aspects of religion, that they partly supplant shamanism, which is a
less important feature of religious life here than elsewhere in the
state, where the characteristic features of this peculiar ritual by
formula are almost absent. Not only purification from death and other
defilement, but luck in hunting and fishing, in gambling, escape from
danger, success in felling trees and making baskets, in the acquisition
of wealth, in short the proper achievement of every human wish, were
thought to be accompanied by the proper knowledge and recitation of
these traditional myth-formulae, usually accompanied by only the
smallest amount of ritualistic action.
SHAMANISM.
Shamanism, the supposed individual control of the supernatural through
a personally acquired power of communication with the spirit world,
rests upon much the same basis in California as elsewhere in North
America. In general among uncivilized tribes the simpler the stage of
culture the more important the shaman. It is as if he constituted an
element that remained nearly constant in quantity of effect, as it is
fundamentally unvarying in form, through all successive periods of
civilization to the highest; but that as increase in degree of
civilization brought with it ever more and more new elements, religious
and otherwise, and these unfolded in ever expanding complexity, he
became, relatively to the total mass of thought and action of a people,
less and less important. Certainly the difference is marked between the
Eskimo, whose religion consists of little else than shamanism, and the
much more highly organized Indians of the North Pacific Coast, where
shamanism is but one of several and by no means the most important
religious factor, even though it may be the most deep seated. The same
contrast is found between the rude simple-minded Indians of California
as compared with those of the Plains and of the Southwest, where the
supremacy of the shaman is rather obscured by that of the priest
conversant with a ceremony. Even within California the difference holds
good. In the Northwest, where the native civilization reached on the
whole its greatest complexity, the shaman is less prominent than
anywhere else in the state. In the south, where the culture is also
more developed than in the Central part of the state, the shaman is
certainly as much dreaded as there; but that his province is more
restricted is shown by the fact that in Southern California the shamans
in their capacity as such do not seem to form associations, perform
public ceremonies, or directly participate in the tribal dances.
The power of the shaman being directly dependent upon his personal
acquisition of a connection with the supernatural world, an
understanding of the method by which this acquisition takes place
generally furnishes also a pretty accurate idea of the nature of his
functions and influence. The most common way of acquiring shamanistic
power in California, as in so many other parts of the world, is by
dreaming. A spirit, be it that of an animal, a place, the sun or
another natural object, a deceased relative, or an entirely unimbodied
spirit, visits the future medicine-man in his dreams, and the
connection thus established between them is the source and basis of the
latter's power. This spirit becomes his guardian spirit or "personal."
From it he receives the song or rite or knowledge of the charm and the
understanding which enable him to cause or remove disease and to do and
endure what other men cannot. In California, with a few special
exceptions, the idea does not seem so prevalent as elsewhere that this
guardian spirit is an animal. Occasionally it is the ghost of a person
who has once lived, usually a relative. Perhaps most frequently it is
merely a spirit as such, not connected with any tangible embodiment or
form, either human, animate, or inanimate. The belief that the shaman
acquires the spirits most frequently in dreaming is prevalent through
the whole Sierra Nevada region and in many other parts of the state.
In certain regions another important method, that of the waking vision
and trance, is recognized. The person is in a wild desolate place,
perhaps hunting. Suddenly there is an appearance before him. He becomes
unconscious and while in this state receives his supernatural power. On
his return to his people he is for a time demented or physically
affected. After he again becomes normal he has control of his
supernatural influences. Such beliefs prevail in part among the Yuki
and Athabascans of the Coast Range and the Maidu of the Sacramento
valley, and no doubt occur more or less sporadically in other regions.
Finally, the shaman sometimes acquires his powers through seeking for
them rather than by having them thrust upon him during a dream or
vision. This of course is a common procedure in the Plains and in part
on the North Pacific Coast. Among the Yurok of the lower Klamath, for
instance, the person whom the spirits have visited in dreams, ascends
high peaks where he spends one or more nights until he has acquired his
powers. Among the Wiyot of Humboldt Bay there are similar beliefs. In
the same Northwestern region a man who wishes to be fierce, strong, and
invulnerable swims at night in lakes inhabited by monsters or thunders.
From these, if his courage is sufficient to await and endure their
presence, he receives the desired powers. This practice of bathing in
lonely lakes closely recalls the custom prevalent along the Pacific
<DW72> for some distance northward, and within California it is probably
not strictly confined to the Northwestern culture area. On the whole,
however, this deliberate method of acquiring shamanistic power is not
common, nor, as has already been stated, would it be in accord with the
generally lower intensity of religious feeling among the California
Indians as compared with those of most other parts of the continent.
The Northwestern area is not only exceptional in being the principal
one within the state where this deliberate seeking of shamanistic power
is prevalent. The conception of a guardian spirit is much less clearly
defined among the Northwestern tribes, with whom the possession of
"pains," the small material objects which cause disease, rather than of
true spirits, seems to be what is generally associated with shamanistic
power. As already stated, shamanism forms a much less important part of
religion as a whole in the Northwestern area than elsewhere, and it is
in accord with this fact that the majority of the shamans, and those
supposed to be most powerful, are women.
In parts of Southern California also the idea of the guardian spirit
does not seem to be well developed. Here the method of acquiring
shamanistic power is almost exclusively by dreams; but among the Mohave
and probably other Colorado river tribes, myths, and not a personal
meeting or communion with an individual spirit, constitute the subject
of the dreams. The Mohave shamans believe that they were present at the
beginning of the world, before mankind had separated into tribes. They
were with the great leader and almost creator, Mastamho. They saw him
singing, blowing, and rubbing over the body of a sick man, if their own
power be that of curing disease, and from Mastamho they thus learned
the actions and speeches which constitute their power. Before him they
showed what they had learned from him, and by him were designated those
who had seen and learned most and those of less power. Each man saw
only the shamanistic actions relating to his particular power, whether
these had reference to the curing of disease, to love, to war, or to
some other activity. The Mohave universally speak of having dreamed
these scenes, just as each narrator affirms his knowledge of
non-shamanistic myths and of ceremonies to have been individually
derived from dreaming them. It is probable that to a certain extent
this is true. That it is not entirely true becomes evident when the
Mohave with equal unanimity state that these dreams were dreamed by
them before birth. In other words, their statement that they have
dreamed such experiences is to be interpreted mainly as a belief that
they as individuals were present in spirit form at the beginning of the
world, at the time when it took shape and everything was ordained, and
when all power, shamanistic and otherwise, was established and
allotted. It is obvious that with this conception as the basis of their
whole religion, there is but little room for any beliefs as to guardian
spirits of the usual form.
Of course there is nothing that limits the shaman to one spirit, and
among many or most tribes, such as the Maidu, a powerful medicine-man
may possess a great number.
Frequently in Central and Northwestern California there is some more or
less public ceremony at which a new shaman is, so to speak, initiated
before he practices his powers. The body of initiated shamans do not
form a definite society or association. The ceremony is rather an
occasion that marks the first public appearance of the novice, in which
he receives for his own good, and presumably for that of the community
also, the assistance of the more experienced persons of his profession.
Commonly it is thought that the novice cannot receive and exercise the
full use of his powers without this assistance. The ceremony is usually
held in the ceremonial chamber and is accompanied by dancing. The
efforts of the older shamans are directed toward giving the initiate a
firm and permanent control of the spirits which have only half attached
themselves to him and which are thought to be still more or less
rebellious. Of course exhibitions of magic and of the physical effects
of the presence of the spirits are a prominent feature of these
ceremonies. This initiation of doctors is found among the Northwestern
tribes and in the Central region among the Maidu and Wintun and
probably other groups.
A special class of shamans found to a greater or less extent among
probably all the Central tribes, though they are wanting both in the
Northwest and the South, are the so-called bear doctors, shamans who
have received power from grizzly bears, often by being taken into the
abode of these animals--which appear there in human form,--and who
after their return to mankind possess many of the qualities of the
grizzly bear, especially his apparent invulnerability to fatal attack.
The bear shamans can not only assume the form of bears, as they do in
order to inflict vengeance on their enemies, but it is believed that
they can be killed an indefinite number of times when in this form and
each time return to life. In some regions, as among the Pomo and Yuki,
the bear shaman was not thought as elsewhere to actually become a bear,
but to remain a man who clothed himself in the skin of a bear to his
complete disguisement, and by his malevolence, rapidity, fierceness,
and resistance to wounds to be capable of inflicting greater injury
than a true bear. Whether any bear shamans actually attempted to
disguise themselves in this way to accomplish their ends is doubtful.
It is certain that all the members of some tribes believed it to be in
their power.
The rattlesnake doctor, who cured or prevented the bite of the
rattlesnake, was usually distinct from other medicine-men. Among the
Yuki his power, as that of the rattlesnake, was associated with the
sun; among the Maidu with the thunder. Among the Yokuts the rattlesnake
shamans annually held a public ceremony designed to prevent rattlesnake
bites among the tribe. On this occasion they displayed their power over
the snakes by handling them in a manner analogous to that of the Hopi,
and by even allowing themselves to be bitten.
As everywhere else, the practice of shamanism in California centers
about disease and death. It is probably more narrowly limited to this
phase than in most other portions of North America. Being an
essentially unwarlike even though a revengeful people, it is natural
that the supernatural power personally acquired by the California
Indian should not often be directed toward success in battle. Success
in love is also less often the result of such personal power than for
instance on the Plains, perhaps because in the latter region the custom
which made virtually every young man seek shamanistic power, resulted
in a condition where those whose proclivities were not toward medicine
or war, desired and received their powers in this direction. Influence
over game and over nature's yield of vegetable products was sometimes
attributed to shamans in California, but on the whole their powers in
this respect were not very much insisted upon except in Southern
California, favorable or adverse conditions of this kind being
attributed rather to the tribal ceremonies, and in the Northwest
connected with the all-important formulae. The causing and prevention
of disease and death were therefore even more largely the predominant
functions of the person who had acquired personal supernatural power in
California than elsewhere in America.
That the medicine-men who could cure diseases were also the ones who
must cause it, unless it were the direct consequence of an infraction
of some religious observance or prohibition, was the almost universal
belief, which was probably adhered to with greater definiteness than in
most portions of North America. The killing of medicine-men was
therefore of frequent occurrence. Among some tribes, as the Yokuts, the
medicine-man who had lost several patients was held responsible for
their death by their relatives. Among the Mohave also murder seems to
have been the normal end of the medicine-man. In the Northwestern
region the shaman who failed to cure was forced to return the fee
received in advance. If he refused to attend a patient when summoned,
he was compelled to pay, in the event of the latter's death, an amount
of property equal that proffered him for his services. So completely
was the shaman regarded as the cause of disease and death, as well as
of their prevention, that one hears very little among the California
Indians of witchcraft, that is to say, of malevolent practices
performed by persons, often very old or very young people, who are not
believed to be endowed with the shaman's power of curing.
Disease, as among most primitive peoples the world over, was usually
held to be caused by small material objects which had in a supernatural
way been caused to enter the body. The determination and extraction of
these was the principal office of the medicine-man and, also as
elsewhere, was most frequently accomplished by sucking. In certain
regions, especially the South, the tubular pipe was brought into
requisition for this purpose, the disease-object being supposed to be
sucked into the doctor's mouth through it. Among such tribes the pipe
was also smoked by the medicine-man as part of his ritual. In other
cases the sucking was performed directly with the mouth, but, just as
the disease-causing object had by supernatural means entered the body
without causing or leaving an opening, so it was extracted by the
medicine-man without an incision or a trace of its passage. This object
might be a bit of hair, a stick, an insect or small reptile, a piece of
bone, deer sinew, or almost any other material. In the greater part of
northern California, including the Northwestern region, it was not an
ordinary physical object working mischief by its mere presence in the
body or by the supernatural properties with which the shaman or his
spirits had endowed it, but an object itself supernatural and called a
"pain." These pains are variously described, frequently as being sharp
at both ends and clear as ice. They possessed the power of moving even
after extracted, and were able to fly through the air to the intended
victim at the command of the person who had sent them. The medicine-man
after extracting the disease-object or pain almost always exhibited it.
It was then either destroyed by him or kept by him for his own use. In
Northwestern California he sometimes swallowed it, the degree of his
power being thought to be dependent upon the number of pains he kept in
his body, both those which he received upon his becoming a shaman, when
they were "cooked" before a great fire in the doctor-initiation dance,
and those which he subsequently secured in doctoring his patients. The
rattlesnake's bite was regarded as being dangerous on account of its
injection into the victim's body of a material animate object, which
the rattlesnake shaman must extract if death was not to ensue. Among
the Yuki this object was a small snake; among the Yokuts a rodent's
tooth or other object supposed to have formed part of the animals upon
which the snake subsisted. In some cases two classes of medicine-men
were distinguished, one diagnosing, the other treating the patient.
Among the Wiyot or Wishosk the former by dancing before the patient saw
in a vision the nature and location of the disease-object and
determined what had caused it to enter the body. Somewhat similar
though varying distinctions between shamans whose power consists of
knowledge, and those who have practical capacity as well, occurred
among other tribes. Sucking is not always resorted to. The Mohave
principally blow or spit over their patients and stroke or rub or knead
their bodies, which actions are supposed by them to drive out the
disease. Medicines and drugs are but little used, or if so, in a manner
that gives no opportunity for their physiological efficacy. Four or
five drops--the number varying according to the ceremonial number of
the tribe--of a weak decoction may be given to the patient or even only
applied to him externally. It is natural that where the magic effect of
the drug as used in a certain ritual is believed in, the quantity so
used is not an essential consideration. It is the supernatural
qualities connected with the plant that bring about the desired result,
and these are as inherent in a drop placed upon the forehead as in a
basketful taken internally. Perhaps the most-used medicinal plant
throughout the state is the angelica root, probably principally on
account of its fragrance. Tobacco is considerably employed by shamans,
but is of equal importance in other aspects of religion.
PUBLIC CEREMONIES.
After the exclusion of such public observances as the shaman
initiation, menstrual dance, and victory celebration, which, while
generally participated in, are performed primarily for the benefit of
individuals, the ceremonies of the California Indians which are of a
really public or communal purpose and character fall into three
classes: (1) mourning ceremonies; (2) initiation ceremonies connected
with a secret society; and (3) a more varied group of dances and other
observances which all, however, have in common the benefit either of
the community or of the world at large, in that they cause a good crop
of acorns and natural products, make the avoidance of rattlesnake bites
possible, or prevent the occurrence of disease, earthquake, flood, and
other calamities.
Of these three classes of ceremonies the mourning ceremonies are at
least as important as the others and by far the most distinctive of the
state as an ethnographic province, although neither they nor the secret
society are found in the specialized Northwestern area. The mourning
ceremonies further do not occur among the Athabascan, Yuki, and Pomo
tribes to the south of the Northwestern tribes as far as the bay of San
Francisco; but outside of this strip in the northern coast region they
are universal in the state. Among the Maidu they are usually known as
"burning," among the Miwok as "cry." Among the Yokuts they have been
called "dance of the dead," and among the Mohave and Yuma "annual."
These ceremonies are usually participated in by a number of visiting
communities or villages. They last for one or more nights, during which
crying and wailing, sometimes accompanied by singing and exhortation,
are indulged in, and find their climax in a great destruction of
property. While those who have recently lost relatives naturally take a
prominent part, the ceremony as a whole is not a personal but a tribal
one. Among the Yokuts and probably other groups it is immediately
followed by a dance of a festive nature, and usually there is a
definitely expressed idea that this general ceremony puts an end to all
individual mourning among the participants. A typical form of the
mourning ceremony is found among the Maidu, who call it oestu. Each
village or political unit possesses its burning ground. Participation
in the ceremony is effected by receipt of a membership-string or
necklace, both the receipt and return of which are marked by payments
or presents. The ceremony is held in autumn in a circular brush
enclosure. Property to be destroyed is tied to poles which are erected
on the ground. After an opening exhortation by the chief or shaman in
charge of the ceremony, the wailing begins, to continue throughout the
night, many exclamations to the dead being uttered. Toward morning the
numerous articles displayed on the poles are taken down and burned.
When everything has been destroyed the assembly breaks up for gambling
and feasting. The purpose of the ceremony is to supply the ghosts of
the dead with clothing, property, and food. Although its general tenor
is communal, each family offers only to its own relatives. In some
cases elaborate images of stuffed skins ornamented with dancing apparel
are made to represent important people who have died. These are burned
with the property offered to the dead.
Initiation ceremonies which result in something analogous to a secret
society are found in the whole state except in the Northwestern region
and among the agricultural tribes at the extreme southeast in the
Colorado valley. They are apparently as well developed among the Yuki
and Pomo, who do not practice tribal mourning ceremonies, as among
their neighbors who do. In a strict sense there is no secret society,
even though the precepts taught boys at initiation are not made public.
There are usually no paraphernalia or insignia of a society, no degrees
or ranks, no membership or other organization, nor is there a definite
purpose for the society. The great majority of the males of the tribes
are made to undergo the initiation, and in many cases there is a
distinct desire to force it upon every man, whether he be willing or
unwilling. In so far as a society may therefore be said to exist at
all, its principal purpose and public function are the initiation of
new members. There is however often a special name for those who have
been initiated, such as yeponi among the Maidu and pumal among the
Luiseno, and to a certain extent the initiates are regarded as a class
or council having a more or less indefinite decision over religious
matters affecting the community. The precepts imparted to the
initiates, other than the ritualistic knowledge relating to the
initiation ceremony itself, seems to be of the most general kind and
pertains principally to daily life and the most obvious maxims of
native morality. In some ways this initiation is a puberty ceremony for
boys corresponding to the first-menstruation-ceremony of girls. The
initiates are however not limited as to age, men being sometimes
included. Among at least the Yokuts in Central California and the
Mission Indians of Southern California the initiation was accompanied
by the drinking of toloache or jimson-weed, datura meteloides, the
stupor and visions produced by which were regarded as supernatural. In
Southern California the idea of an ordeal and instruction was specially
developed. Boys were made to undergo severe tests of pain and endurance
and were given numerous injunctions regarding their adult life. Among
the Maidu of the Sacramento valley instruction both in the myths of the
tribe and in the more important ceremonies was imparted. Among certain
of the Maidu the secret society, in so far as it comprises the more
adult men, is difficult to distinguish from an association of shamans.
The public ceremonies other than mourning and initiation observances,
in other words the tribal dances of California, differ thoroughly in
the three culture regions, which must therefore be considered
separately.
In Central California these dances, like the initiation ceremonies,
have disappeared to a much greater extent than the mourning ceremonies,
and where they survive have often been more or less influenced by
modern ideas. As a rule they were held in the large assembly or
ceremonial chamber, more often at night than during the day, and either
lasted for a number of nights or consisted of a series of successive
dances extending over a considerable period. Some of the dances, though
a minority, were named after animals, and in such there was usually
some imitation of the actions of animals. Sometimes rude paraphernalia
were used to represent the animal itself, but this was not very common
and masks were never employed. At least in the Sacramento valley and
northern Coast Range region there was some impersonation of mythical
characters, as of Taikomol, creator among the Yuki, and of the mythical
being Kuksu among the Pomo and Maidu. Such impersonators usually wore
either the "big head," an enormous head-dress of feathers attached to
radiating sticks, or a large cape of feathers fastened to a network,
which concealed both body and face, or both pieces of apparel. There
seems to have been nothing corresponding to an altar. The dancers were
painted but crudely, and such symbolism as was denoted by the painting
was of the simplest. One or more of the posts that supported the roof
of the assembly chamber were usually of ceremonial importance. The
dancers frequently entered and left the house by a hole above instead
of the door at the ground. A rude drum consisting of a hollow slab
placed on the ground and stamped with the feet was often used. An
important character in most ceremonies was the clown or buffoon, part
of whose duties was to caricature the more serious performance. In some
cases shamanistic exhibitions of magic were included in the ceremony.
At times an exchange or compulsory giving of property formed part of
the ceremony. The participants were rarely if ever called upon to
undergo severe trials of endurance, pain, or courage, as among so many
other Indians. The whole ritual was comparatively simple.
The exact nature and relation of the various dances are very little
known among most of the tribes of the Central region. Probably a
typical example of these dances is furnished by the Maidu of the
Sacramento valley, who declare that their ceremonies were obtained from
their neighbors, the Wintun. This statement is borne out by indirect
evidence. Among the Maidu the ceremonies were performed in winter and
constituted a series of fifteen or more distinct dances, coming for the
most part in a definite order. So far as known they were the following:
Hesi, Luyi, Loli, Salalu-ngkasi, Duck, Bear, Coyote, Creeper, Turtle,
Aloli-ngkasi, Yokola-ngkasi, Moloko-ngkasi, Deer, Aki, Hesi. The
majority of these dances were performed by men, but some by women only.
There is no evidence that participation in these dances was dependent
upon anything like membership in an association. Each had its
characteristic paraphernalia or combinations of paraphernalia. In
several there are participants with special apparel and with a
distinctive name. At least some of these seem to represent mythical
characters. In several instances these performers enact ceremonial
operations, largely in the nature of complex approaches and departures
which take place outside the assembly chamber. The names of several of
these ceremonies occur also among neighboring Indians speaking entirely
different languages, and thus give proof of the transmission of the
ceremonies from one locality to another. The Hesi, the most important
of the Maidu series, is danced also by the Wintun. The Loli is an
important ceremony among the Maidu, Miwok, and Pomo. The performer
called Kuksu, who refers to important myths, is found among the Maidu,
Wintun, Pomo, and either the Miwok or Costanoan Indians formerly at
Mission San Jose. There is every reason to believe that a fuller
acquaintance with the tribes whose ceremonies are as yet least known
will reveal other instances of ceremonies held in common and known
under the same name. Farther to the south, among the Yokuts of the
Tulare basin, these ceremonies do not seem to have penetrated. Here the
majority of the public ceremonies, like the rattlesnake ceremony that
has been mentioned, were of the nature of shamanistic performances.
Throughout the Central region the dances, while they might be held only
in structures of certain kinds, were never rigorously attached to a
specific locality.
In Northwestern California the more important ceremonies can always be
held only at certain spots, and the performance of ceremonies of the
same name always varies somewhat at different places. The performers do
not represent mythological or other characters and do not imitate
animals. The more important dances last at least a number of days, not
infrequently as many as ten. The dances are held either out-doors or in
certain sacred houses, which are however not different from the
ordinary living-house of the region except through their traditionary
and ceremonial associations. The essential religious portion of the
ceremony consists of the actions gone through by a priest, with
sometimes one or two assistants. The more important part of his
procedure is the recital of one of the sacred formulae so
characteristic of the region. This formula relates specifically to the
exact locality at which the dance is held, and therefore often varies
considerably from spot to spot. The formula is regarded as it were as
private property, and its knowledge is sufficient to institute the
priest in his capacity. The public portions of the ceremony, such as
the dancing, are practically dissociated from this purely religious
element. The dancers are mostly young men without any knowledge of the
ceremony other than of the simple dance-step and songs. The
paraphernalia which they wear belong neither to them nor to the
priests, but to wealthy men of the tribe, for whom the occasion is an
all-important opportunity for the display of their wealth, which
consists in large part of the dancing regalia, and the possession of
which is the chief factor toward their social prominence. The dancers
appear in from two to five parties, representing neighboring villages,
each of which is aided by the wealthy men of other villages; and these
parties vie with each other primarily in the display of their regalia.
The most important ceremonies are the Deerskin dance and the Jumping
dance, which are held either annually or biennially, the former always
out-doors, the latter at some places out-doors, sometimes in boats, at
others in-doors. The purpose of both dances, which where both are
practiced are usually given in close succession, is the good of the
world. Earthquake and disease are prevented and a food supply insured.
Very little of the sacred formulae and accompanying ritual, and nothing
in the remainder of the dance, has however any specific reference to
this purpose. A third, minor ceremony, the Brush dance, completes the
series of public ceremonies in this region, the remaining dances being
held only on occasion of war, a girl's puberty, or the initiation of a
shaman. Even the Brush dance is not fully of a tribal character,
inasmuch as it is performed for the benefit of a single individual, a
sick child, although it is participated in by an entire village with
the assistance of visitors from others, and though there seems to be a
desire to perform the ceremony at least once a year in each of the
larger villages.
In Southern California mourning ceremonies are everywhere the most
prominent. In the coast region, among the various groups of Mission
Indians, initiation ceremonies make up most of the public rituals that
are not connected with mourning. In the interior the Mohave possess no
initiation ceremonies. In both regions such ceremonies as partake
neither of the nature of mourning nor initiation are conspicuous by the
prominence of the myth element. They consist essentially of long series
of songs, occupying one or more nights in the recital, which recount,
in part directly but more often by allusion, an important myth. At
times the myth is actually related in the intervals between the songs.
In some cases dancing by men or women accompanies the singing, but this
is never spectacular and in many cases is entirely lacking. Being only
ceremonial recitations of myths, these ceremonies are not attached in
their performance to specific localities, and when dancing regalia are
used they are of the simplest character; nor is there opportunity for
either altar or ritual. The predominance of the mourning element in the
ceremonies of this region is further shown by the fact that among some
tribes, as the Mohave, these same singing ceremonies, besides being
performed independently, are also sung for many hours at every death.
The series of songs selected for each individual on this occasion is
that with which he is acquainted. In accord with what has been said of
the dream as the basis of Mohave religious life, these singing
ceremonies are almost always believed by each person to have been
dreamed by himself.
CEREMONIAL STRUCTURES AND PARAPHERNALIA.
The ceremonial chamber is also of distinctive character in the three
culture areas. In the Central region it is a large, circular,
dome-shaped structure, partly underground and with a covering of earth.
It serves also as place of assembly and at least at times as sudatory,
whence its popular name of sweat-house. In the Northwest the
sweat-house is quite small, almost entirely underground, and its roof
consists of boards without a covering of earth. It is constantly used
for sweating and is the regular sleeping place of all adult males. It
is not used for public ceremonies except in the case of the dance
initiating shamans. In the South the ceremonial structure is not a
house, but either a mere enclosure of brush, as among the Mission
tribes, or a simple shade of brush on upright posts, as among the
Mohave. This type of ceremonial structure is also found in the southern
part of the Central region among the Yokuts.
In the matter of dancing apparel the Northwest differs fundamentally
from all the remainder of the state. Some of the most important of the
regalia, such as long obsidian knives and albino deerskins, are not
worn on the body or used ritually but merely carried for display, being
primarily objects of great value. Large forehead-bands entirely covered
with brilliant red woodpecker feathers more nearly resemble ordinary
dancing apparel, but are also articles of value, the unmounted
woodpecker feathers virtually constituting one form of currency. Other
objects used in dancing are dresses, cloaks, and head-bands of skin and
fur, head-dresses of network, and carefully ornamented plumes and head
feathers. All these, while worn on the body, and decorative, also
possess considerable commercial value. The drum is not used, the
whistle employed at times, and the rattle, which consists of deer
hoofs, but sparingly.
In the Central region objects made of feathers greatly predominate over
all others, and are mostly made to be worn actually on the body.
Head-dresses are particularly conspicuous and of many forms. In the
Sacramento and San Joaquin valleys and the adjacent region cloaks of
large feathers attached to a network are worn. In the Tulare basin
these are replaced by skirts consisting of strings of eagle-down. With
these down-skirts are worn large upright head-dresses of crow and
magpie feathers. This combination of costume was used also by the
Mission Indians in Southern California and by the Washo of Nevada, and
at least the head-dress is found as far north as the Sacramento valley.
Network caps filled with down, and forehead bands of down, are frequent
in various parts. Perhaps the most typical single object of ceremonial
apparel is a flat band, usually worn on the forehead, and consisting of
the trimmed red quills of the yellow-hammer sewed side by side. This
head-band occurs through the whole of Central California and is used
also by the tribes east of the Sierra Nevada, in the state of Nevada,
and south of Tehachapi pass in Southern California. The large foot drum
of the Central region has already been mentioned. Whistles are also
used and there are two forms of rattle, one consisting of silk cocoons
containing gravel, the other of a split stick. The cocoon rattle is
usually associated with the shaman, the clap-stick with dancing.
In the South, especially among the Mission Indians, the dancing
apparel, as is evident from the instances already mentioned, is of much
the same type as in the Central area. On the Colorado river feather
ornaments of the same general character are used, though they are of a
simpler type and head-dresses predominate. The whistle is but little
used in the South, the drum occasionally, baskets and other objects
being chiefly employed for this purpose. The rattle is the
all-important musical instrument in this region. It is made most
frequently from a gourd or a turtle-shell.
MYTHOLOGY AND BELIEFS.
In mythology a deep-going difference between the three culture areas
again appears. The Northwestern mythologies are characterized primarily
by a very deeply impressed conception of a previous, now vanished,
race, who by first living the life and performing the actions of
mankind were the producers of all human institutions and arts as well
as of some of the phenomena of nature. Second in importance in the
Northwest are myths dealing with culture-heroes more or less of the
trickster type familiar from so many other parts of North America.
In Central California there is always a true creation of the world, of
mankind, and of its institutions. The conception of the creator is
often quite lofty, and tricky exploits or defeats are usually not
connected with him. Often there is an antithesis between this
beneficent and truly divine creator and a second character, usually the
Coyote, who in part cooeperates with the creator but in part thwarts
him, being responsible for the death of mankind and other imperfections
in the world-scheme. In the northern half of the Central region the
creator is generally anthropomorphic; if not, he is merged into one
personage with the more or less tricky Coyote. In the southern half of
the region the creators seem always to be animals with the dignified
and wise eagle as the chief. The myths of the Central region not
directly concerned with creation are mostly stories of adventure, of
much the same type as European folk and fairy tales. They do not
explain the origin of phenomena except in a casual, isolated way, and
but rarely are of ceremonial import.
In Southern California there is no creation. The various animate and
inanimate existences in the world are born from heaven and earth as the
first parents. Sometimes heaven and earth are regarded as the first
concrete existences, who were, however, preceded by a series of psychic
beings grouped in pairs. The bulk of the Southern origin myth consists
of a history of mankind, at first as a single tribe and later centered
in the tribe which tells the story. In the successive experiences of
this body of people, which are accompanied by more or less journeying,
the world is gradually brought to its present stage, and all the
institutions of mankind, particularly of the narrating tribe but also
of others, are developed. The people are under the leadership of one or
two great leaders, at least one of whom always dies or departs after
his beneficent directions. The thoroughly Southwestern and Pueblo
character of this long origin myth is obvious. It is usually followed
to a greater or less extent by migration legends recounting the
wandering and conflicts of different tribes or clans. The remaining
myths are in plot essentially not very different from the adventure
stories of the Central region, but both much longer and more elaborate,
and at the same time distinctively ritualistic in that they form the
basis or framework of the singing ceremonies that have been described.
As these ceremonies themselves are nothing but myths, there is neither
need nor room for traditionary accounts explaining the origin of the
ceremonies.
An identification of myth and ceremony that is in many ways similar to
that prevalent in Southern California is characteristic also of the
Northwestern region, where the formulae which constitute the essential
religious elements, as well as being the direct means, of most
supernatural accomplishment, are nothing but myths. The Northwestern
formula is a myth, rarely a direct prayer, and practically every more
serious myth is either in whole or in part a also a formula. In
purpose, however, as well as in rendering, the spoken myth-formulae of
the Northwest and the sung myth-ceremonies of the South are different,
the former having always a definite practical result in view, whereas
the latter have no aim other than their own recital.
Thus the mythology of Southern California resembles that of the
Southwest rather than that of the remainder of the state. That of the
Northwestern region shows affinities to the North Pacific Coast in its
prevalence of the culture-hero and trickster over the creator. The most
marked special characteristic of the Northwestern mythology, other than
its practical use of myths for religious purposes in the shape of
formulas, is its strong and definite, though inconsistently carried
out, idea of the previous race which is parallel to but distinct from
mankind, and which is the originator, not by any act of creation but by
merely living its life, of everything human except mankind itself, the
origin of which is never accounted for. This idea of a previous
supernatural race analogous to mankind crops out to some extent in
almost all North American mythologies, and particularly in other parts
of California: but it seems nowhere to be so deep-seated and so freely
expressed as in this region. The members of this vanished race are
almost always strictly human, in Northwestern California, and not
animals or personifications. They are nothing but men, living the life
of the Indians, transposed into a mythic supernatural age, and by the
fact of their mere existence regarded as the originators of the present
condition of the world. They therefore leave no room for a creator, and
but little for the culture hero, whose exploits, when not of purely
personal significance, consequently consist mainly of the destruction
of evil beings.
If the mythology of Northwestern California in spite of its partial
northern affinities accordingly has a dominant character all its own,
the same is also true of the larger, more representative Central
region. A true creator, and a full and consistent attempt at an account
of the creation, are found nowhere else in North America, or at least
only sporadically and carried out with an apparently much less degree
of thoroughness. The remainder of the Central Californian mythology
however scarcely presents any unique qualities, even some of the
specific myth-episodes, such as the favorite one of the bear and deer
children, being found over considerable territories outside of
California. Even the important characteristic of the presence of
creation-myths is in a measure a negative one, for from a world view
some approach to such a myth may be expected among most peoples,
whether primitive or civilized, and it is primarily only in America
that special bents of mind and of religious thought have supplanted the
idea of creation by the culture hero, the tribal history, and other
conceptions. We are therefore not far from right if we regard the
unique development of creation myths over the greater part of
California as merely a part of a general tendency of the California
Indians towards simplicity and lack of strongly marked peculiar and
American qualities in any one direction, a tendency which has already
been emphasized in other aspects of their religion, and which must be
said to characterize their whole life and culture.
Ideas as to the world and the existence of the dead vary from tribe to
tribe but present nothing specially distinctive. The world is usually
regarded as surrounded by water, sometimes as floating upon it. It is
often secured by four or five pillars, ropes, or other supports. Beyond
where earth and sky meet there is often another land. The dead
sometimes go below, sometimes above, sometimes across the ocean to the
west, and sometimes to more or less distant parts of this earth. The
entrance to the world of the dead is pointed out by some tribes. People
who have temporarily died have been there and returned to describe it.
Dances constitute the principal occupation of the dead. No ideas of
future rewards and punishments based on conduct in this life have yet
been found. If such ideas exist they must be very scantily developed.
As in other parts of the world, there are occasional ideas of
transmigration of souls into animals, but these conceptions are nowhere
systematically worked out or of any religious importance.
SPECIAL CHARACTERISTICS OF DIFFERENT TRIBES.
Such are the principal characteristics of the religion of the Indians
of California as a whole, and of the larger ethnographical areas of the
state. It is obvious that with so great a linguistic and political
diversification as existed among these Indians, there must have been
many local modifications of the scheme which has been outlined. The
most conspicuous or best known of these special modifications it is the
purpose of the remainder of this paper to consider. In this review the
groups to be taken up will, for the sake of greatest convenience of
classification, be the linguistic families. These numerous families are
territorially so restricted, and usually so small in numbers, that they
almost form the equivalent of the tribe in other regions of North
America, that is to say, of a subdivision of the family. Strictly there
are no tribes in the greater part of California. The families or stocks
are the largest linguistic units, usually subdivided into several
dialectic areas, each of which contains a number of small village
communities that are the only units of political or social
organization.
In the Northwestern region, in spite of the excessive limitation of
this territory, a distinction must be made between three tribes which
occupy the heart of the region and show the culture in its most extreme
form, and a fringe of surrounding tribes where the Northwestern culture
is either less developed or subject to greater extraneous influences.
The three more characteristic groups are the Yurok and Karok, small
independent linguistic families, and the Hupa division of the
Athabascan family. These alone practice the Deerskin dance and the "New
Year's" or world-making ceremonies. With them also the peculiar
mythological and shamanistic conceptions typical of the region are
found in the purest form. The surrounding tribes are the Wishosk or
Wiyot, perhaps the Chimariko and some of the Shasta, the Athabascan
Tolowa, and the Athabascans southwest of the Hupa.
The Yurok held the Deerskin and Jumping dances at three places along
the Klamath river, and the Jumping dance alone at three points on the
coast, to the south. At the mouth of the river an annual spring
ceremony to cause or regulate the ascent of the salmon was made. Until
this ceremony had been made salmon were not eaten. The shamans of the
Yurok were almost all women. Alone of all the tribes in the
Northwestern region the Yurok held no dance or public ceremony on the
occasion of a girl's puberty. Their traditions seem to have the
peculiar Northwestern qualities perhaps more deeply impressed upon them
than even those of their neighbors, the Karok and Hupa, especially in
regard to the underlying conception of a previous race and its
function. In accord with the development of this conception, the
mythical heroes of the Yurok show less approximation to being creators
than those of the other tribes, and animals are mentioned in the
mythology surprisingly little.
The Karok, who live immediately upstream from the Yurok on the Klamath,
held the Deerskin and Jumping dances at three places. At each of these
the dances were conducted in connection with a sacred ceremony called
"New Year's" by the whites and "making the world" by the Indians. This
ceremony was performed early in autumn, practically by one man, the
priest who knew the formula and ritual. A similar ceremony was held at
a fourth locality in spring, in connection with the coming of the
salmon. The Karok regard the Deerskin and Jumping dances of the Yurok
and Hupa as the equivalents of these ceremonies of their own, reckoning
altogether ten places in the world at which they are performed. Karok
mythology is of the Northwestern type, but shows more animal characters
than that of the Yurok.
The territory held by the Hupa was much less extended than that of
their neighbors, and this was no doubt the occasion of their making
only one Deerskin and Jumping dance in their valley. They held a New
Year's ceremony in autumn which had distinct reference to the acorn
crop. Ceremonials and restrictions connected with menstruation were
considerably developed, much more than among the neighboring Yurok. It
was thought dangerous to speak to a dog, as he might be provoked to
answer, which would be a fatal portent.
The religion of the other Athabascans in this part of the state is very
little known, but it is certain that before the southern end of
Humboldt county is reached, in other words, in the Eel river drainage,
a totally distinct set of conceptions and practices is encountered,
which are allied to those of the Central region.
The Wiyot or Wishosk, who adjoin the Yurok on the south, did not
practice the Jumping dance, other ceremonies, which are very little
known, taking its place among them. One dance was performed by women
standing up to the hips in water. Shamanism is of more prominence among
them than with their neighbors the Yurok, and men as well as women are
affected with supernatural powers. The sex of the guardian spirit is
usually the opposite of that of the shaman. It is possible that on
account of the almost complete disappearance of their tribal life and
communal religious practices, shamanism, which has been retained with
greater vigor among the Wiyot, now appears relatively more important,
as the only remnant of the religious side of their culture. An
elaborate hanging feather head-dress, a belt, a pipe for smoking, and
another for sucking, are the constant paraphernalia of the
medicine-man. Two shamans often support each other in curing disease,
one diagnosing, the other removing the pain. The mythology of the Wiyot
resembles that of the Yurok chiefly through possessing certain specific
narrative episodes in common with it. But the idea of a previous
parallel race is very little developed, and there is a true creator,
Above-Old-Man. Most of the other mythical characters are animals. The
whole mythology therefore is of the Central rather than of the
Northwestern type.
With the Yuki of Mendocino county a pure form of the Central culture
obtains. The creator is Taikomol, "he who goes alone." His companion,
who supplements his work, especially as regards the culture of man, is
Coyote. There is a Taikomol ceremony in which this character is
impersonated, and which is shamanistic at least to the degree of being
performed to cure an individual of sickness. There is no trace of the
sacred formulae of the Northwest. The shaman, who is usually a man,
receives his power either by dreaming or in a vision in a desolate
place. His power is not sought by him and he possesses definite
guardian spirits. Bear shamans are much feared. All the Yuki possess a
sacred society initiation ceremony, in which performances of magic are
prominent. Among the northern Yuki and neighboring Wailaki this is
called Flint ceremony, and the initiates display magic powers in
handling and swallowing flint points. Among the southern Yuki, as among
the neighboring Pomo and Athabascan Kato, the ceremony relates to
ghosts and is popularly known as Devil dance. The members possess power
of causing sickness and contend against each other much like the
shamans of the Maidu and Yokuts.
One of the most conspicuous features of the religion of the Pomo, who
are south of the Yuki, is their shamanistic fetishes. The medicine-man
possesses a number of objects, stones, parts of animals, and other
articles, which he treasures and with which his power is largely bound
up. Pomo mythology is characterized by the importance of Coyote, who
comes nearer than any other personage to playing the part of creator.
In certain ceremonies there are exhibitions of fire-eating and the
clown occurs.
The Wintun occupy a territory which is of much greater extent from
north to south than from east to west. The northern and southernmost
members of the family therefore differ considerably. In the north there
is a well defined conception of a creator who dwells above, and to whom
Coyote forms an antithesis. In the south, where everything shows the
Wintun and Pomo to have influenced each other considerably, he is
replaced by Coyote. In both regions a world-fire is prominent in the
mythology. In the north the shaman is inaugurated in his career in a
ceremony in which he is assisted by his older colleagues. The southern
Wintun may prove to have been the people who largely developed the
dances and ceremonies characteristic of a large part of the Sacramento
valley. They show much in common with their western neighbors the Pomo,
and with the Maidu who adjoin them on the east and who themselves
declare that they have derived the Hesi and other dances from them.
None of the groups so far discussed, with the possible exception of
part of the Wintun, practiced any distinct mourning ceremony. On the
other hand, all that follow, with the possible doubtful exception of
one or two tribes on the outskirts of the state, held mourning
ceremonies as among the most important of all their religious
practices.
The Maidu everywhere possessed a secret society. Their system of dances
becomes less and less developed as one proceeds farther from Wintun
influence. Among the mountain tribes almost all ceremonies were much
less developed than in the Sacramento valley. Shamanistic beliefs and
practices also varied, although there was everywhere a clear idea of
spirits personally acquired and controlled by the medicine-man. Among
the northeastern Maidu every shaman's son invariably became a shaman,
although only through his own acquisition of spirits, which might be
those of his father. In the Sacramento valley spirits were acquired by
involuntary dreaming without much regard to heredity. Puberty
ceremonies for girls were performed both among the northwestern and
northeastern Maidu, perhaps among those of the south also. The
mythology of the several Maidu divisions is much more uniform than
their religious practices. The creator is always opposed and his
beneficent work rendered incomplete by Coyote. It is clear that the
mythology of the Maidu is distinctive and much less under Wintun
influence than their ceremonies.
Among the Miwok the Coyote largely takes the place of the creator. As
among their northern neighbors the Maidu, the mourning ceremony was
important, and the two stocks held at least certain dances in common.
The individual mourning practices and restrictions of the widow were
elaborate and severe. Nothing is as yet known of a secret society, but
as both the southern and northern neighbors of the Miwok performed
initiation ceremonies, it is likely that they also possessed them.
Among the Yokuts, who occupied the head of the San Joaquin-Tulare
valley south of the Miwok, there are no traces of the ceremonial system
of the Sacramento valley, which is replaced by public shamanistic
ceremonies, in which contests and exhibitions of magic were
conspicuous. The annual rattlesnake ceremony which has been described
is of this type, as is the Ohowish, a ceremony in which medicine-men
from different villages or districts; directed their powers against
each other. There seem to have been also certain animal dances among
the Yokuts. Medicine-men usually acquired their power by dreaming,
sometimes by visions while alone. Bear shamans were known, but were not
so much dreaded as farther north. Rain doctors, who could control the
weather, were important. Their power was bound up with certain stone
amulets evidencing a fetishistic development. Formulae, some with
ritualistic accompaniment, were spoken, but differed from those of the
Northwest in being short direct prayers or supplications instead of
mythical narratives. The creators in Yurok mythology are several
animals, the chief of whom is the eagle and among whom Coyote always
finds a place. A favorite mythological personage is the prairie-falcon,
and a myth which has found a particular development relates the visit
of a husband to the world of the dead in pursuit of his wife.
Very little is known of the ethnology of the coast tribes west of the
Miwok and Yokuts. Among the Southern Costanoan peoples creation myths
resembling those of the Yokuts are found. Coyote is at once a trickster
and a giver of civilization and arts to man. Similar ideas probably
prevailed among the Salinan tribes. As regards the Esselen and Chumash
nothing is known.
Tribes belonging to the great Shoshonean family held almost all the
eastern border of the state as well as a large part of the southern
desert and coast region. The former inhabited the Great Basin, and are
culturally entirely distinct from those of Southern California, of whom
alone is there any considerable knowledge extant as regards religion.
Certain of the northern groups, such as the Mono, lived on the western
or California <DW72> of the Sierra Nevada, in contact with the Yokuts
and Miwok, and partook more largely of the culture and presumably
religion of these people than of the tribes of the Basin.
Among the Shoshoneans of Southern California, such as the Gabrielino
and Luiseno, the so-called Mission Indians, mourning ceremonies were
more important than any others, and were held both on the death of a
person, sometime afterwards, and again in a still more public manner at
large gatherings. At some of these ceremonies images representing the
dead, and recalling those of the Maidu far to the north, were burned.
One form of mourning ceremony was the Eagle dance, performed with an
eagle that was slowly killed as the ceremony went on through the night.
Many of the songs of the mourning ceremonies are of mythological
content, referring to the great leader or culture-hero Wiyot. The
puberty ceremonial for girls was elaborate and contained symbolic
actions. The initiation of males was intended for boys, and therefore
also took on largely the character of a puberty ceremony. This
character was heightened by the presence of numerous ordeals. Part of
the initiation of boys consisted of the drinking of jimson-weed. Sand
paintings of a very simple type, evidently influenced by basket
patterns, but thoroughly symbolic in meaning and therefore essentially
of the same nature as those of the Pueblos and Navaho, were made in
connection with this initiation. On the whole religious symbolism was
more developed than in Central California or even among the Yuman
tribes to the east, who are geographically so much nearer the Indians
of the Southwest. The shaman acquired his power by dreaming, and the
pipe with which he sucked as well as smoked was of the utmost
importance to him. Paraphernalia were much used by the shamans,
especially boards or wooden swords, which were swallowed and worn as
head-dresses. These, however, were not purely fetishistic objects, but
of potency rather through symbolism and association. The mythology of
the Shoshonean Mission Indians was not essentially different from that
of the other Indians of Southern California.
The Yuman family, which is so much represented in Arizona and Lower
California, occupied the southernmost portion of Southern California.
The Diegueno in the coast mountains and on the coast were culturally
similar to the Shoshonean Luiseno, with whom they are generally
included as the present Mission Indians. Along the Colorado river the
physical and ethnic environment was quite different, but as has already
been said, there was much closer resemblance to the Mission Indians in
matters of religion than in almost any other phase of culture. The
principal Yuman tribes in this Colorado region are the Mohave and the
Yuma. The religion of only the former is known, but the two give every
evidence of having been very similar. The religion of the Shoshonean
Paiute or Chemehuevi in the desert adjoining the Mohave has been
largely by the influence of the latter. The most distinctive
feature of Mohave religion is the insistence upon dreaming as the
source of everything religious, although this dreaming must be
interpreted rather as a belief in the presence of the individual in
spirit form at the great events of mythic times. All myths that are at
all of sacred character are believed not to be handed down by
tradition, but to be dreamed by each narrator. The shaman receives his
power by dreaming ritualistic myths, which reveal to him his practices.
The lengthy series of songs which are the essence of all ceremonies,
and the mythical narratives connected with them, are also learned in
dreams. It is probably a result of this importance of the dream-world
and of the identification of myth and ceremony, of religious belief and
religious practice, that ritualism is so slightly developed among the
Mohave. Their geographical nearness and intercourse with the Hopi and
other southwestern tribes, among whom ritualism and symbolism find
perhaps their highest development on the continent north of Mexico,
would certainly justify a contrary expectation. Both ceremonial actions
and ceremonial paraphernalia and dress are developed only to a very
slight extent. There is no initiation or society. The singing
ceremonies, which with the exception of a few minor observances such as
that for a girl's puberty, constitute all the Mohave ceremonies other
than mourning ceremonies, are quite numerous, more than twenty being
known. Some of these ceremonies are acknowledged to have been borrowed
from other Yuman tribes, especially the Yuma, and these Indians no
doubt have also acquired Mohave ceremonies. Some of the ceremonies are
primarily mythical in character, others somewhat shamanistic. All are
also sung in mourning. In addition there is a distinctive mourning
ceremony held annually for important men.
BIBLIOGRAPHY.
Much of the material on which the statements in the preceding essay are
based is information collected by the University of California's
Ethnological and Archaeological Survey of California since 1901 and as
yet unpublished. Of old accounts dealing with the religion of the
Indians of California, the best is by the Franciscan missionary
Boscana, entitled Chinigchinich and published in the 1846 edition of a
volume by A. Robinson called Life in California. It deals with the
Shoshonean Indians of Mission San Juan Capistrano. An occasional
reference of value may be found in other works, such as Venegas'
History of California. The series of translations and republications of
early explorers in California and the Southwest, published in the Land
of Sunshine, later Out West, beginning in 1899, is also convenient,
though naturally it deals but incidentally with religion. Reid's
account of the Indians of Los Angeles county, published in an early Los
Angeles newspaper and republished by Alexander Taylor in the fourteenth
volume of California Farmer in 1861, is particularly good, though less
so on the side of religion than on most others. Stephen Powers' Tribes
of California, issued in 1877 as the third volume of the Contributions
to North American Ethnology, a government series, deals with the
Indians of the greater part of the state and contains many references
to their religious life. Powers is however often very inexact, and the
value of his work is in its comprehensiveness rather than in its
reliability. An important work is Creation Myths of Primitive America,
by Jeremiah Curtin, which consists of a collection of myths from the
Wintun and Yana of Northern California. The differences of form which
these myths show from most Indian myths that have been published in
translation are apparently chiefly due to the method of their
presentation by the author. Curtin's introduction is very suggestive
but exaggerated. Professor R. B. Dixon has brought out a paper on Maidu
Myths, and another, a great part of which is devoted to religion, on
the Northern Maidu, both in the seventeenth volume of the Bulletin of
the American Museum of Natural History. These two contributions are
among the most careful studies as yet made by a trained observer in any
part of the state. The same author has also published briefer articles
on Some Coyote Stories from the Maidu Indians of California, System and
Sequence in Maidu Mythology, and Some Shamans of Northern California,
in recent volumes of the Journal of American Folk-Lore, and on The
Mythology of the Shasta-Achomawi in the American Anthropologist for
1905. Professor P. E. Goddard has published Life and Culture of the
Hupa, the last portion of which refers to religion; and Hupa Texts
(with both interlinear and current translations), almost all of which
are religious in character. These two papers constitute Volume I of the
University of California Publications in American Archaeology and
Ethnology. In the Journal of American Folk-Lore for 1906 is a paper by
the same author on Lassik Tales. Miss Constance Goddard DuBois has
published a number of valuable papers on the Mission Indians, mainly
concerning the mythology of the Diegueno, in the volumes of the Journal
of American Folk-Lore for 1901, 1904, and 1906. In the American
Anthropologist for 1905 Miss DuBois has an article on the Religious
Ceremonies and Myths of the Mission Indians, while another paper on The
Mythology of the Dieguenos appears in the Proceedings of the Thirteenth
International Congress of Americanists. From the present author there
have appeared, in the second and fourth volumes of the series of
American Archaeology and Ethnology, of the University of California
Publications, Types of Indian Culture in California, in part treating
of religion, and Indian Myths from South-Central California; in the
Journal of American Folk-Lore between 1904 and 1906, A Ghost Dance in
California, Wishosk Myths, and Two Myths of the Mission Indians; in the
American Anthropologist for 1902, A Preliminary Sketch of the Mohave
Indians. In the American Anthropologist for 1905 and 1906 the late
Major H. N. Rust has two brief articles on The Obsidian Blades of
California and A Puberty Ceremony of the Mission Indians. The Journal
of American Folk-Lore has contained a rather confused article on The
Cosmogony and Theogony of the Mojave Indians, by Capt. J. G. Bourke, in
1889, and others by G. W. James, on myths of the Mission Indians of
Southern California, in 1902 and 1903. In the same Journal appeared in
1902 An Indian Myth from the San Joaquin Basin by J. W. Hudson, and A
Composite Myth of the Pomo Indians by S. A. Barrett in 1906. Since 1906
the Journal has contained a series of Notes on California Folk-Lore.
End of the Project Gutenberg EBook of The Religion of the Indians of
California, by A. L. Kroeber
*** | {
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<?php
namespace OC\PlatformBundle\Entity;
use Doctrine\Common\Collections\ArrayCollection;
use Doctrine\ORM\Mapping as ORM;
//use Gedmo\Mapping\Annotation as Gedmo;
/**
* Advert
*
* @ORM\Table(name="advert")
* @ORM\Entity(repositoryClass="OC\PlatformBundle\Entity\AdvertRepository")
* @ORM\HasLifecycleCallbacks()
*/
class Advert
{
/**
* @var integer
*
* @ORM\Column(name="id", type="integer")
* @ORM\Id
* @ORM\GeneratedValue(strategy="AUTO")
*/
private $id;
/**
* @ORM\OneToMany(targetEntity="OC\PlatformBundle\Entity\Application", mappedBy="advert")
*/
private $applications; // Notez le « s », une annonce est liée à plusieurs candidatures
/**
* @ORM\Column(name="updated_at", type="datetime", nullable=true)
*/
private $updatedAt;
/**
* @ORM\Column(name="nb_applications", type="integer")
*/
private $nbApplications = 0;
/**
* @ORM\OneToOne(targetEntity="OC\PlatformBundle\Entity\Image", cascade={"persist"})
*/
private $image;
/**
* @ORM\ManyToMany(targetEntity="OC\PlatformBundle\Entity\Category", cascade={"persist"})
*/
private $categories;
/**
* @ORM\Column(name="published", type="boolean")
*/
private $published = true;
/**
* @var \DateTime
*
* @ORM\Column(name="date", type="datetime")
*/
private $date;
/**
* @var string
*
* @ORM\Column(name="title", type="string", length=255)
*/
private $title;
/**
* @var string
*
* @ORM\Column(name="author", type="string", length=255)
*/
private $author;
/**
* @var string
*
* @ORM\Column(name="content", type="text")
*/
private $content;
/**
* Get id
*
* @return integer
*/
public function getId()
{
return $this->id;
}
/**
* Set date
*
* @param \DateTime $date
* @return Advert
*/
public function setDate($date)
{
$this->date = $date;
return $this;
}
/**
* Get date
*
* @return \DateTime
*/
public function getDate()
{
return $this->date;
}
/**
* Set title
*
* @param string $title
* @return Advert
*/
public function setTitle($title)
{
$this->title = $title;
return $this;
}
/**
* Get title
*
* @return string
*/
public function getTitle()
{
return $this->title;
}
/**
* Set author
*
* @param string $author
* @return Advert
*/
public function setAuthor($author)
{
$this->author = $author;
return $this;
}
/**
* Get author
*
* @return string
*/
public function getAuthor()
{
return $this->author;
}
/**
* Set content
*
* @param string $content
* @return Advert
*/
public function setContent($content)
{
$this->content = $content;
return $this;
}
/**
* Get content
*
* @return string
*/
public function getContent()
{
return $this->content;
}
/**
* Set published
*
* @param boolean $published
* @return Advert
*/
public function setPublished($published)
{
$this->published = $published;
return $this;
}
/**
* Get published
*
* @return boolean
*/
public function getPublished()
{
return $this->published;
}
/**
* Set image
*
* @param \OC\PlatformBundle\Entity\Image $image
* @return Advert
*/
public function setImage(\OC\PlatformBundle\Entity\Image $image = null)
{
$this->image = $image;
return $this;
}
/**
* Get image
*
* @return \OC\PlatformBundle\Entity\Image
*/
public function getImage()
{
return $this->image;
}
/**
* Constructor
*/
public function __construct()
{
$this->date = new \Datetime();
$this->categories = new \Doctrine\Common\Collections\ArrayCollection();
$this->applications = new \Doctrine\Common\Collections\ArrayCollection();
}
/**
* Add categories
*
* @param \OC\PlatformBundle\Entity\Category $categories
* @return Advert
*/
public function addCategory(\OC\PlatformBundle\Entity\Category $categories)
{
$this->categories[] = $categories;
return $this;
}
/**
* Remove categories
*
* @param \OC\PlatformBundle\Entity\Category $categories
*/
public function removeCategory(\OC\PlatformBundle\Entity\Category $categories)
{
$this->categories->removeElement($categories);
}
/**
* Get categories
*
* @return \Doctrine\Common\Collections\Collection
*/
public function getCategories()
{
return $this->categories;
}
/**
* Add applications
*
* @param \OC\PlatformBundle\Entity\Application $applications
* @return Advert
*/
public function addApplication(\OC\PlatformBundle\Entity\Application $applications)
{
$this->applications[] = $applications;
// On lie l'annonce à la candidature
$application->setAdvert($this);
return $this;
}
/**
* Remove applications
*
* @param \OC\PlatformBundle\Entity\Application $applications
*/
public function removeApplication(\OC\PlatformBundle\Entity\Application $applications)
{
$this->applications->removeElement($applications);
// Et si notre relation était facultative (nullable=true, ce qui n'est pas notre cas ici attention) :
// $application->setAdvert(null);
}
/**
* Get applications
*
* @return \Doctrine\Common\Collections\Collection
*/
public function getApplications()
{
return $this->applications;
}
public function updateDate()
{
$this->setUpdatedAt(new \Datetime());
}
public function increaseApplication()
{
$this->nbApplications++;
}
public function decreaseApplication()
{
$this->nbApplications--;
}
/**
* Set updatedAt
*
* @param \DateTime $updatedAt
* @return Advert
*/
public function setUpdatedAt($updatedAt)
{
$this->updatedAt = $updatedAt;
return $this;
}
/**
* Get updatedAt
*
* @return \DateTime
*/
public function getUpdatedAt()
{
return $this->updatedAt;
}
/**
* Set nbApplications
*
* @param integer $nbApplications
* @return Advert
*/
public function setNbApplications($nbApplications)
{
$this->nbApplications = $nbApplications;
return $this;
}
/**
* Get nbApplications
*
* @return integer
*/
public function getNbApplications()
{
return $this->nbApplications;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,793 |
{"url":"https:\/\/math.stackexchange.com\/questions\/1446170\/a-conjectured-continued-fraction-for-tan-left-fracz-pi4z2n-right","text":"# a conjectured continued fraction for $\\tan\\left(\\frac{z\\pi}{4z+2n}\\right)$\n\nGiven a complex number \\begin{aligned}\\frac{z}{n}=x+iy\\end{aligned} and a gamma function $$\\Gamma(z)$$ with $$x\\gt0$$, it is conjectured that the following continued fraction for $$\\displaystyle\\tan\\left(\\frac{z\\pi}{4z+2n}\\right)$$ is true\n\n$$\\begin{split}\\displaystyle\\tan\\left(\\frac{z\\pi}{4z+2n}\\right)&=\\frac{\\displaystyle\\Gamma\\left(\\frac{z+n}{4z+2n}\\right)\\Gamma\\left(\\frac{3z+n}{4z+2n}\\right)}{\\displaystyle\\Gamma\\left(\\frac{z}{4z+2n}\\right)\\Gamma\\left(\\frac{3z+2n}{4z+2n}\\right)}\\\\&=\\cfrac{2z}{2z+n+\\cfrac{(n)(4z+n)} {3(2z+n)+\\cfrac{(2z+2n)(6z+2n)}{5(2z+n)+\\cfrac{(4z+3n)(8z+3n)}{7(2z+n)+\\ddots}}}}\\end{split}$$\n\nCorollaries:\n\nBy taking the limit(which follows after abel's theorem) \\begin{aligned}\\lim_{z\\to0}\\frac{\\displaystyle\\tan\\left(\\frac{z\\pi}{4z+2}\\right)}{2z}=\\frac{\\pi}{4}\\end{aligned}, we recover the well known continued fraction for $$\\pi$$\n\n\\begin{aligned}\\cfrac{4}{1+\\cfrac{1^2}{3+\\cfrac{2^2}{5+\\cfrac{3^2}{7+\\ddots}}}}=\\pi\\end{aligned}\n\nIf we let $$z=1$$ and $$n=2$$,then we have the square root of $$2$$ \\begin{aligned}{1+\\cfrac{1}{2+\\cfrac{1} {2+\\cfrac{1}{2+\\cfrac{1}{2+\\ddots}}}}}=\\sqrt{2}\\end{aligned}\n\nQ: How do we prove rigorously that the conjectured continued fraction is true and converges for all complex numbers $$z$$ with $$x\\gt0$$?\n\nUpdate:I initially defined the continued fraction $$\\displaystyle\\tan\\left(\\frac{z\\pi}{4z+2}\\right)$$ for only natural numbers,but as a matter of fact it holds for all complex numbers $$z$$ with real part greater than zero.Moreover,this continued fraction is a special case of the general continued fraction found in this post.\n\n\u2022 That is impressive. How did you come up with it? \u2013\u00a0marty cohen Oct 13 '15 at 16:20\n\u2022 Really, how did you get this? If you provide some of the methods you used, there is a better chance someone would be able to answer your question. \u2013\u00a0Yuriy S Mar 13 '16 at 19:01\n\u2022 @Nicco, thank you for the link. Sorry, but I don't have anything to contribute so far \u2013\u00a0Yuriy S Apr 7 '16 at 17:04\n\u2022 I have changed the formatting of the title so as to make it take up less vertical space -- this is a policy to ensure that the scarce space on the main page is distributed evenly over the questions. See here for more information. Please take this into consideration for future questions. Thanks in advance. \u2013\u00a0GNUSupporter 8964\u6c11\u4e3b\u5973\u795e \u5730\u4e0b\u6559\u6703 Mar 12 '18 at 17:02\n\nThe proposed continued fraction $$$$\\displaystyle\\tan\\left(\\frac{z\\pi}{4z+2n}\\right)=\\cfrac{2z}{2z+n+\\cfrac{(n)(4z+n)} {3(2z+n)+\\cfrac{(2z+2n)(6z+2n)}{5(2z+n)+\\cfrac{(4z+3n)(8z+3n)}{7(2z+n)+\\ddots}}}}$$$$ can be written as $$$$\\displaystyle\\tan\\left(\\frac{z\\pi}{4z+2n}\\right)=\\cfrac{2z\/\\left( 2z+n \\right)}{1+\\cfrac{(n)\/\\left( 2z+n \\right)\\cdot(4z+n)\/\\left( 2z+n \\right)} {3+\\cfrac{(2z+2n)\/\\left( 2z+n \\right)\\cdot(6z+2n)\/\\left( 2z+n \\right)}{5+\\cfrac{(4z+3n)\/\\left( 2z+n \\right)\\cdot(8z+3n)\/\\left( 2z+n \\right)}{7+\\ddots}}}}$$$$ Denoting $$u=\\cfrac{z}{4z+2n}$$, the factors of the numerators are $$$$\\frac{n}{2z+n}=1-4u\\,;\\quad\\frac{4z+n}{2z+n}=1+4u\\,;\\quad\\frac{2z+2n}{2z+n}=2-4u\\,;\\quad\\frac{6z+2n}{2z+n}=2+4u\\,;\\cdots$$$$ Then, the fraction can be simplified as $$$$\\displaystyle\\tan\\left(\\pi u\\right)=\\cfrac{4u}{1+\\cfrac{\\cfrac{1-16u^2}{1\\cdot3}} {1+\\cfrac{\\cfrac{4-16u^2}{3\\cdot5}}{1+\\cfrac{\\cfrac{9-16u^2}{5\\cdot7}}{1+\\ddots}}}}$$$$ It is thus a special case of the continued fraction found in this answer: $$$$\\tan\\left(\\alpha\\tan^{-1}z\\right)=\\cfrac{\\alpha z}{1+\\cfrac{\\frac{(1^2-\\alpha^2)z^2}{1\\cdot 3}} {1+\\cfrac{\\frac{(2^2-\\alpha^2)z^2}{3\\cdot 5}}{1+\\cfrac{\\frac{(3^2-\\alpha^2)z^2}{5\\cdot 7}}{1+\\ddots}}}}$$$$ here $$z=1$$ and $$\\alpha=4u$$. The brilliant proof is based on a continued fraction due to N\u00f6rlund.\nThe ratio $$\\tan\\dfrac{\\pi z}{4z+2n} = \\dfrac{\\Gamma\\left(\\dfrac{z+n}{4z+2n}\\right)\\Gamma\\left(\\dfrac{3z+n}{4z+2n}\\right)}{\\Gamma\\left(\\dfrac{z}{4z+2n}\\right)\\Gamma\\left(\\dfrac{3z+2n}{4z+2n}\\right)}\\hspace{100mu}\\tag1$$ can be obtained, applying \"real\" identity\n$$\\sin\\pi x = \\dfrac\\pi{\\Gamma(x)\\Gamma(1-x)}\\hspace{100mu}\\tag2$$\nto the expression $$\\tan\\dfrac\\pi2\\dfrac z{2z+n} = \\dfrac{\\sin\\pi\\dfrac z{4z+2n}}{\\sin\\pi\\dfrac{z+n}{4z+2n}},$$ so it looks nice and quite correct.\nContinued fraction can be obtained, using known continued fraction of the tangent function in the form of $$\\tan \\dfrac{\\pi x}4 = \\cfrac x{1+\\operatorname{ \\Large K}\\hspace{-27mu}\\phantom{\\Big|}_{k=1}^{\\large ^{\\,\\infty}}\\cfrac{(2k-1)^2-x^2}2}\\hspace{100mu}\\tag3$$ with $$x=\\dfrac{2z}{2z+n}.$$","date":"2019-08-22 13:14:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 29, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 5, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8786535859107971, \"perplexity\": 619.949298904178}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-35\/segments\/1566027317130.77\/warc\/CC-MAIN-20190822130553-20190822152553-00264.warc.gz\"}"} | null | null |
{"url":"https:\/\/math.stackexchange.com\/questions\/2319562\/determine-the-radius-of-convergence-of-given-power-series","text":"Determine the radius of convergence of given power series.\n\nDetermine the radius of convergence of given power series.\n\n(a) $\\sum_{n=1}^\\infty {(n^{1\/n} - 1)x^n}$\n\n(b) $\\sum_{n=1}^\\infty {{n^c \\over n!}x^n}$ ($c \\in \\mathbb R$)\n\nI tried root test for (a) and got\n\n$$0 \\le |n^{1\/n} - 1|^{1\/n} \\lt |n^{1\/n}|^{1\/n} \\le n^{1\/n}$$\n\nbut I'm not sure if I can use this result.\n\nAnd for (b) I tried ratio test, and found that if $c=1$, the power series clearly converges for all $x$. But I'm stuck in how to solve the rest case of $c$.\n\n(a) Let $f(x)=x^{-x}$. Then $\\lim_{x\\to0}f(x)=1$ and $f'(x)=-x^{-x}\\bigl(1+\\log(x)\\bigr)$. Therefore, $\\lim_{x\\to0}\\frac{f(x)-1}x=\\lim_{x\\to0}f'(x)=+\\infty$. So$$\\lim_{n\\in\\mathbb N}\\frac{\\sqrt[n]n-1}{1\/n}=+\\infty$$and so, if $n\\gg1$, $\\displaystyle\\frac1n<\\sqrt[n]n-1<1$. Since the radius of convergence of both series $\\sum_{n=0}^\\infty\\frac{x^n}n$ and $\\sum_{n=0}^\\infty x^n$ is $1$, the radius of convergence of your series is also $1$.\n(b) $\\displaystyle\\frac{\\frac{(n+1)^c}{(n+1)!}}{\\frac{n^c}{n!}}=\\frac1{n+1}\\left(1+\\frac1n\\right)^c\\to0\\times1=0$. So, the radius of convergence is $+\\infty$.\n\u2022 I suppose in (a) you meant $$\\lim_{x\\to\\color{red}{0^+}}x^{-x}=1\\;\\ldots$$ \u2013\u00a0DonAntonio Jun 12 '17 at 12:16\n\u2022 @DonAntonio Yes. Since $x^x$ is defined only when $x>0$, I did not feel the need of writing $0^+$ instead of simply writing $0$. \u2013\u00a0Jos\u00e9 Carlos Santos Jun 12 '17 at 12:18","date":"2021-08-04 16:38:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.9850659370422363, \"perplexity\": 118.78336552704289}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046154878.27\/warc\/CC-MAIN-20210804142918-20210804172918-00488.warc.gz\"}"} | null | null |
<?php
namespace Ahshok\Tests\Service;
use Ahshok\Service\ProductService;
class ProductServiceTest extends \PHPUnit_Framework_TestCase {
protected $dao;
protected $service;
protected function setUp() {
$this->dao = $this->getMockBuilder('Ahshok\DAO\ProductDAO')
->disableOriginalConstructor()
->getMock();
$this->amazon = $this->getMockBuilder('Ahshok\Service\AmazonService')
->disableOriginalConstructor()
->getMock();
$this->service = new ProductService($this->dao, $this->amazon);
}
protected function tearDown() {
$this->service = null;
}
public function testFindOrCreateWhenProductExists() {
$productData = array(
'asin' => '0135974445',
'title' => 'Agile Software Development'
);
$this->dao->expects($this->once())
->method('find')
->with($productData['asin'])
->will($this->returnValue($productData));
$result = $this->service->findOrCreate($productData['asin']);
$this->assertEquals($productData, $result);
}
public function testFindOrCreateWhenProductDoesNotExist() {
$productData = array(
'asin' => '0321146530',
'title' => 'Test Driven Development'
);
$this->dao->expects($this->once())
->method('find')
->with($productData['asin'])
->will($this->returnValue(false));
$this->amazon->expects($this->once())
->method('lookup')
->with($productData['asin'])
->will($this->returnValue($productData));
$this->dao->expects($this->once())
->method('create')
->with($productData)
->will($this->returnValue($productData));
$result = $this->service->findOrCreate($productData['asin']);
$this->assertEquals($productData, $result);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,948 |
Q: Vertical space between longtable and top of the page With the environment longtable we make a table crossing the page boundary.
If a table has a head (\endhead command), longtable works perfectly: the baseline of the first line in a usual text page and the baseline of the first line of text in the table are coinside.
But, if a table doesn't have a head, the baseline of the first line of text in the table is lower then the baseline of the first line in a usual text page.
How to reduce the space between top of a page and a table without head?
Is it possible to slightly modify the longtable package so that all tables in a document would be fixed?
\documentclass{article}
\usepackage[a5paper]{geometry}
\usepackage{lipsum}
\usepackage{fancyhdr}
\usepackage{longtable}
\pagestyle{fancy}
\rhead{\thepage}
\lhead{\thepage}
\begin{document}
\lipsum[2]
\begin{longtable}{|p{0.4\textwidth}|p{0.4\textwidth}|}
\hline
Header & Header \\
%\endhead %% Uncomment this line for table with head
\hline
\lipsum[2] & \lipsum[2] \\
\hline
\lipsum[2] & \lipsum[2] \\
\hline
\end{longtable}
\end{document}
A: You can use \firsthline from the array package instead of \hline:
\documentclass{book}%
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{array,longtable}
\usepackage{geometry,tikz}
\usepackage{eso-pic}
\geometry{top=2cm}
\AddToShipoutPictureFG{%
\tikz[overlay,remember picture] \draw[red] ([yshift=-2cm-\topskip]current page.north west) -- ++(\textwidth,0);}
\begin{document}
some text
\newpage
\begin{longtable}{l}
\hline
abc
\end{longtable}
\newpage
\begin{longtable}{l}
\firsthline
abc
\end{longtable}
\end{document}
A: The next patch to longtable works for me, assuming that if \hline is present in table, it is present after all \\.
%
% Adjustment of longtable environment
%
\makeatletter
\newskip\ax@LT@topskip
\newif\ifax@LT@hline
% Save original definitions
\let\ax@LT@longtable=\longtable
\let\ax@LT@endlongtable=\endlongtable
\let\ax@LT@output=\LT@output
\let\ax@LT@hline=\LT@hline
% Remember, there was \hline
\def\LT@hline{%
\noalign{\global\ax@LT@hlinetrue}%
\ax@LT@hline
}
% Forget about \hline, store initial value of \topskip
\def\longtable{%
\global\ax@LT@hlinefalse
\ax@LT@topskip=\topskip
\ax@LT@longtable
}
% Adjust \topskip, if there was \hline and wasn't \endhead
\def\LT@output{%
\ifax@LT@hline\ifvoid\LT@head
\global\topskip=\ax@LT@topskip
\global\advance\topskip by -\ht\@arstrutbox
\fi\fi
\ax@LT@output
}
% Restore \topskip
\def\endlongtable{%
\ax@LT@endlongtable
\global\topskip=\ax@LT@topskip
}
\makeatother
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,870 |
package modem
import (
"bytes"
"encoding/hex"
"errors"
"fmt"
"log"
"regexp"
"strconv"
"strings"
"sync"
"time"
"github.com/tarm/serial"
pdu "github.com/xlab/at/pdu"
)
var err error
var lock sync.Mutex
const waitReps int = 5
var m *modem
type Modem interface {
Connect() (err error)
}
type modem struct {
ComPort string
BaudRate int
Port Port
}
type message struct {
Labels string
Sender string
Date time.Time
Body string
Index int
}
type Port interface {
Read(b []byte) (n int, err error)
Write(b []byte) (n int, err error)
Flush() error
Close() (err error)
}
func InitModem(ComPort string, BaudRate int) (err error) {
m = &modem{ComPort: ComPort, BaudRate: BaudRate}
config := &serial.Config{Name: m.ComPort, Baud: m.BaudRate, ReadTimeout: time.Second}
m.Port, err = serial.OpenPort(config)
if err != nil {
return fmt.Errorf("InitModem: Failed to open port. %s", err.Error())
}
return nil
}
func SendCommand(command string, wait bool) (string, error) {
log.Println("SendCommand...", command)
lock.Lock()
m.Port.Flush()
_, err = m.Port.Write([]byte(command))
lock.Unlock()
if err != nil {
return "", fmt.Errorf("SendCommand: Failed to write to port.\n%s", err.Error())
}
var output string
if wait {
output, err = WaitForOutput(waitReps, "OK\r\n")
if err != nil {
return "", fmt.Errorf("SendCommand: Failed to wait for output.\n%s", err.Error())
}
}
return output, nil
}
func WaitForOutput(reps int, suffix string) (string, error) {
log.Printf("WaitForOutput... %d %#v", reps, suffix)
var status string
var buffer bytes.Buffer
buf := make([]byte, 32)
lock.Lock()
defer lock.Unlock()
for i := 1; i < reps+1; {
// ignoring error as EOF raises error on Linux
n, _ := m.Port.Read(buf)
if n > 0 {
buffer.Write(buf[:n])
status = buffer.String()
log.Printf("WaitForOutput: received %d bytes: %#v\n", n, string(buf[:n]))
if strings.HasSuffix(status, suffix) {
return status, nil
} else if regexp.MustCompile(`[A-Z ]*ERROR[0-9A-Za-z ]*`).MatchString(status) {
errorCodes := regexp.MustCompile(`([A-Z ]*)ERROR([0-9A-Za-z :]*)`).FindAllStringSubmatch(status, -1)
if errorCodes[0][1] == "" && errorCodes[0][2] == "" {
return status, fmt.Errorf("WaitForOutput: Found unknown ERROR")
} else {
return status, fmt.Errorf("WaitForOutput: Found %vERROR%v", errorCodes[0][1], errorCodes[0][2])
}
}
} else {
log.Printf("WaitForOutput: No output on %dth iteration", i)
// time.Sleep(time.Millisecond * 500)
i++
}
}
return status, errors.New("WaitForOutput: Timed out.")
}
func GetSignal() (float64, error) {
log.Println("GetSignal...")
status, err := SendCommand("AT+CSQ\r", true)
if err != nil {
return 0.0, err
}
return strconv.ParseFloat(
strings.Replace(
regexp.MustCompile(`\d+,\d+`).FindString(status), ",", ".", 1), 64)
}
func GetCharset() (string, error) {
log.Println("GetCharset...")
status, err := SendCommand("AT+CSCS?\r", true)
if err != nil {
return "", err
}
return regexp.MustCompile(`\"[A-Za-z0-9]+\"`).FindString(status), nil
}
func CheckConnection() error {
log.Println("CheckConnection...")
_, err = SendCommand("AT\r", true)
if err != nil {
return err
}
return nil
}
func Reset() error {
log.Println("Reset...")
InitCommands := []string{
"ATZ\r",
"ATE0\r",
"AT+CFUN=1\r",
"AT+CMEE=1\r",
"AT+COPS=3,0\r",
"AT+CMGF=1\r",
"AT+CSMP=49,167,0,0\r",
"AT+CPMS=\"ME\",\"ME\",\"ME\"\r",
"AT+CNMI=2,1,0,2\r",
"AT+CSCS=\"GSM\"\r",
}
// Send C^Z first
_, err = SendCommand(string(26), false)
for _, c := range InitCommands {
for i := 0; i < 10; i++ {
log.Printf("%v, %#v", i, c)
_, err = SendCommand(c, true)
if err != nil && i < 9 {
log.Println(err)
time.Sleep(time.Millisecond * 500)
} else if err != nil && i == 9 {
return err
} else {
break
}
}
}
return nil
}
func GetBalance(ussdRequest string) (float64, error) {
log.Println("GetBalance...")
//re-set encoding here?
//m.SendCommand("AT+CSCS=\"GSM\"\r", true)
//TODO: Is it necessery to run AT+CMGF=0 ???
SendCommand("AT+CMGF=0\r", true)
SendCommand("AT^USSDMODE=1\r", true)
request := strings.ToUpper(fmt.Sprintf("%x", pdu.Encode7Bit(ussdRequest)))
_, err = SendCommand(fmt.Sprintf("AT+CUSD=1,\"%s\",15\r", request), true)
if err != nil {
return 0.0, err
}
status, err := WaitForOutput(10, "15\r\n")
regex := regexp.MustCompile(`\+CUSD: \d{1},\"([a-zA-Z0-9]*)\",\d*`)
if regex.MatchString(status) {
balanceRaw := regex.FindStringSubmatch(status)[1]
bytesWritten, _ := hex.DecodeString(balanceRaw)
log.Println("Before decode", bytesWritten)
balanceRaw, _ = pdu.Decode7Bit(bytesWritten)
log.Println("After decode", balanceRaw)
balanceParsed := regexp.MustCompile(`\d+\.\d+`).FindString(balanceRaw)
if balanceParsed == "" {
return 0.0, fmt.Errorf("GetBalance: Failed to find balance string in \"%s\"", balanceRaw)
}
balance, err := strconv.ParseFloat(balanceParsed, 64)
if err != nil {
return 0.0, fmt.Errorf("GetBalance: Failed to convert to float64 \"%s\"", balanceRaw)
}
return balance, nil
}
if err != nil {
return 0.0, err
}
return 0.0, errors.New("GetBalace: Failed to get balance.")
}
func SendMessage(mobile string, message string) error {
log.Println("SendMessage...", mobile, message)
// Put Modem in SMS Text Mode
_, err = SendCommand("AT+CMGF=1\r", true)
if err != nil {
return fmt.Errorf("SendMessage: Failed to send command.\n%s", err.Error())
}
// Send message
_, err = SendCommand("AT+CMGS=\""+mobile+"\"\r", false)
if err != nil {
return fmt.Errorf("SendMessage: Failed to send command.\n%s", err.Error())
}
_, err = WaitForOutput(waitReps, "\r\n> ")
if err != nil {
return fmt.Errorf("SendMessage: Failed to wait for output.\n%s", err.Error())
}
// EOM CTRL-Z = 26
_, err = SendCommand(message+string(26), true)
if err != nil {
return fmt.Errorf("SendMessage: Failed to send command.\n%s", err.Error())
}
return nil
}
func DeleteMessage(messageIndex int) error {
log.Println("DeleteMessage...")
// Put Modem in SMS Text Mode
SendCommand("AT+CMGF=1\r", true)
_, err = SendCommand(fmt.Sprintf("AT+CMGD=%d\r", messageIndex), true)
if err != nil {
return fmt.Errorf("DeleteMessage: Failed to send command.\n%s", err.Error())
}
return nil
}
func GetMessage(messageIndex int) (*message, error) {
log.Println("GetMessage...")
status, err := SendCommand(fmt.Sprintf("AT+CMGR=%d\r", messageIndex), true)
if err != nil {
return nil, fmt.Errorf("GetMessage: Failed to send command.\n%s", err.Error())
}
log.Printf("GetMessage: %#v\n", status)
regex := regexp.MustCompile(`(?Us)CMGR: "([A-Z ]*)","([+\d]*)",,"([0-9/,:\+]*)"\r\n(.*)\r\n\r\nOK`)
if regex.MatchString(status) {
msg := regex.FindStringSubmatch(status)
messageLabels := msg[1]
messageSender := msg[2]
messageDate, _ := time.Parse("06/01/02,15:04:05-07", msg[3])
var messageBody string
if regexp.MustCompile(`^[0-9A-F]+$`).MatchString(msg[4]) {
bytesWritten, _ := hex.DecodeString(msg[4])
log.Println("GetMessage: bytesWritten =", bytesWritten)
regex := regexp.MustCompile(`[a-z ]{3}`)
if regex.MatchString(string(bytesWritten)) {
log.Printf("GetMessage: Decoding message #%d using plain text", messageIndex)
messageBody = string(bytesWritten)
} else {
log.Printf("GetMessage: Decoding message #%d using Ucs2", messageIndex)
messageBody, err = pdu.DecodeUcs2(bytesWritten)
if err != nil {
log.Printf("GetMessage: Failed to decode message #%d using Ucs2", messageIndex)
}
}
} else {
messageBody = msg[4]
}
log.Printf("GetMessage: %v %#v %#v %v %#v\n", messageIndex, messageLabels, messageSender, messageDate.Format(time.RFC3339), messageBody)
return &message{
Labels: messageLabels,
Date: messageDate,
Sender: messageSender,
Body: messageBody,
Index: messageIndex,
}, nil
} else {
return nil, fmt.Errorf("GetMessage: Failed to parse response: %v", status)
}
}
func GetMessageIndexes() ([]int, error) {
var messageIndexes []int
log.Println("GetMessageIndexes...")
// Put Modem in SMS Text Mode
SendCommand("AT+CMGF=1\r", true)
// Get message indexes
status, err := SendCommand("AT+CMGD=?\r", true)
if err != nil {
return messageIndexes, err
}
regex := regexp.MustCompile(`\+CMGD: \(([0-9,]*)\)`)
if regex.MatchString(status) {
var messageIndexesRaw []string
statusParsed := regex.FindStringSubmatch(status)[1]
if statusParsed != "" {
messageIndexesRaw = strings.Split(statusParsed, ",")
}
for _, messageIndex := range messageIndexesRaw {
index, err := strconv.Atoi(messageIndex)
if err != nil {
log.Printf("GetMessages: Failed to convert messageIndex=\"%v\" to int", messageIndex)
} else {
messageIndexes = append(messageIndexes, index)
}
}
return messageIndexes, nil
} else {
return nil, errors.New("GetMessageIndexes: Failed to get message indexes")
}
}
func GetMessages() ([]*message, error) {
log.Println("GetMesages...")
var messages []*message
messageIndexes, err := GetMessageIndexes()
if err != nil {
return messages, err
}
log.Println("GetMessages:", messageIndexes)
for _, messageIndex := range messageIndexes {
msg, err := GetMessage(messageIndex)
if err != nil {
return messages, err
} else {
messages = append(messages, msg)
}
}
return messages, nil
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,286 |
import java.util.Scanner;
public class test02 {
public static void main(String[] args) {
// TODO Auto-generated method stub
Scanner scn = new Scanner(System.in);
int n = scn.nextInt();
int a = 1,sum = n;
while(sum<=10000){
sum=sum*n;
a++;
}System.out.println(a);
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,700 |
Devolver Digital delivers another off-the-wall title.
It looks a bit weird, its brightly coloured and apparently its super inventive, Pikuniku has garnered the usual level of attention that Devolver titles get, and it's out today on Nintendo Switch and PC.
As the game's potato-with-legs character players must navigate absurd puzzles, even absurder environments and the colourful cast of people they meet along the way. Help these characters overcome their personal struggles and uncover the seedy conspiracy that lies at the root of the seemingly cheerful world. It's a dystopian adventure with plenty of platforms to get around.
When we reviewed Pikuniku we found it just about as fun as it sounds, although perhaps a little on the short side. To find out more check out the game on Steam, where it is available on PC and Nintendo Switch from today. | {
"redpajama_set_name": "RedPajamaC4"
} | 518 |
Die Liste der Städte in North Carolina nach Einwohnerzahl enthält alle Orte im US-Bundesstaat North Carolina sortiert nach ihrer Einwohnerzahl, die mindestens eine Bevölkerung von 40.000 aufweisen. Hauptstadt des Staates ist Raleigh, die von der Einwohnerzahl her zweitgrößte Stadt North Carolinas.
Stand 1. Juli 2017
Siehe auch
Liste der Städte in North Carolina
Quelle
CityPopulation.de
!
North Carolina, Einwohnerzahl | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,807 |
Improved UX and business results using automation
Tyler Foster is the Senior Vice President of Technology at Evolv, a cloud platform on the bleeding edge of evolutionary algorithms for automation and optimization that vastly outperforms manually implemented A/B testing for e-commerce and other conversion-driven businesses. Tyler and Ledge discuss how optimization and automation engines like Evolv improve user experience and business results, and how to effectively organize multi-disciplinary teams of data scientists and engineers. Tyler shares his hiring philosophy, and how to prioritize a well-rounded team around the individual technology expertise of each team member.
Tyler Foster | Evolv Technologies
Senior Vice President of Technology
Tyler Foster is the Vice President of Engineering for Evolv Technologies. As both a senior individual contributor and executive, Tyler has spent more than 18 years delivering technical solutions to the worlds hardest problems.
Tyler's past-experience includes leading firmware and control system development for subglacial lake exploration ROVs deployed in Antarctica with the MSLED / Wissard project, front-end platform architecture and service design at Apollo Group, one of the world's largest private education companies, and distributed systems deployed by many Fortune 500 companies to solve their most complex data problems at Cloudera. Most recently Tyler led a cloud infrastructure startup with operations in the US, UK, and Asia.
Keep up with Tyler on LinkedIn, Twitter, and GitHub.
DAVID LEDGERWOOD: Tyler, thanks for joining us. It's really cool to have you.
TYLER FOSTER: Thanks, Ledge.
LEDGE: Can you give your two- or three-minute background, story, and history about you and your work just for the listeners?
TYLER: Sure. My background is primarily large-scale distributed systems. I've built platforms at a few different companies including one of the largest education companies in the world as well as Cloudera, a Hadoop distribution company.
Lately, I've been working on evolutionary algorithms and evolutionary computation with Sentient which is the company that I'll be talking to you about today.
We do optimization of stochastic problems and online learning ─
LEDGE: Talk to me about stochastic problems and online learning. What is the particular domain there? I mean, you're talking about some pretty serious technology. Break it down for the business guy.
TYLER: Basically, what our platform does is it takes a series of ideas, potentially millions of permutations of different possible solutions to a problem and then, over time, optimizes it to find early wins and early performance gains but it will also identify the global maxima of the problem space.
What that really means in kind of practical terms is that we're focused on optimizing the user experience of websites where people can put in tens of ideas to how they want to change their website sort of different contents to try, etcetera.
And then, we will collect data about those and identify the optimal combination of those changes at any given time.
LEDGE: And so, business-benefit wise, is it a really predictive engine for the user experience and engagement?
TYLER: Yes, basically, it's an optimization engine. So you turn us on; you put a bunch of ideas in; and if you're an e-commerce site your revenue goes up, basically. It continually improves throughout the time that our system is running. It's a faster automated optimization compared to an A/B or a standard multivariate testing approach.
LEDGE: How does that actually work? What's the stack of technologies and how does that integration work?
TYLER: As far as the stack of technologies go, we're primarily a JavaScript and Python shop. Most of our algorithms were implemented in Python; and then we have a lot of Node and Python services for different aspects.
We are in AWS primarily so we use Lamda Edge quite significantly as well as containers in ECS and then different data-handling systems: Athena, Kinesis, the usual suspects.
So, basically, the algorithms are based around the concept of genetic algorithms where we will introduce a lot of the ideas. We'll create possible solutions, test fitness on those possible solutions; and then, we'll continue to produce variations on those tested solutions to find the local maxima but also to continue to search until we find the global maxima as well.
LEDGE: And how do you know when you've reached the optimal solution? In any given case, that's a broader algorithmic question there. The universe keeps changing.
TYLER: Yes. We kind of adhere to the idea that there isn't a permanent global maxima. We're constantly trying to find the best combination for now. So as your user interaction changes, as the market changes as you run different campaigns, your user behavior will change.
And so, what we're really looking for is the best combination of changes at any given time.
So, in some cases, our customers will choose to find the best combination for now, take it, and implement it based on our Bayesian statistics telling them, "This is the most likely combination to improve the goal achievement of your users." But, in the end, the system can continue to search in the long term.
So the end product is that applied to web and user experience.
LEDGE: I have a question. I've talked to other technology leaders particularly around companies where the end product is very ML- or AI-based and the organization of an engineering team and group of team under that context because it's very scientific and experimental may not, in fact, fit along the lines of normal Scrum and Agile.
I'm curious as to what you, guys, have run into there? Do you treat engineering teams and sort of scientific teams differently? How do you handle the organizational flow there?
TYLER: As far as that goes, we do find that it works sort of well enough but we have the team structured in such a way that we have a couple of data scientists who are permanently exploring new ideas and new improvements to algorithms. Sometimes, that's a simulation task; sometimes, it's productizing an algorithm task.
But it's available. And so, we try to democratize it more than just isolate a team of smart people in a room who spend all of their time researching and not propagating that data and experience to the rest of the engineering team. We try to give all of our engineers the opportunity to do research and to find the right answers because I think that that makes the team better overall.
It makes everyone value the output of the research more significantly if they're involved in producing and if they understand the justifications for it and they help to productize those solutions.
LEDGE: Do you have cross-functional product managers and people who are thinking particularly on the product angle there? It seem like it would be maybe an engineering science and product kind of triangle.
TYLER: Yes. We have an engineer with a math background as our product manager within the company. And so, he's quite cross functional and can speak on both sides. And I tend to bridge the gap so I'm much more technical in my background. We work together and kind of identify the best path to take the technology.
LEDGE: Fantastic! I ask all the guests this because we're in the business of evaluating and vetting and finding the very best engineers and it ends up being less than one percent getting through the net.
So we have a pretty strong sort of proprietary process, if you will, for doing that. And yet, I like to continue to improve that process so I ask all our guests, "How do you find and hire ─ and more on the hire and sort of identify ─ the very best unicorn senior A-plus engineers? What are the heuristics that are most important for these roles in your company?"
TYLER: I think that it really depends on the team. At any given time, a team has its strengths and weaknesses. And we're always trying to be aware of where we're strong and where we're weak.
I'm looking for people who fill the gap. So much about good engineering is context. So does that person have the background to solve the problems we have right now?
As far as the process goes, I find that engineers we've worked with previously within our network, people we have long track records of delivering great products with always end up being the best hires.
And good engineers know good engineers. So it's a good sort of practice to get people who are both good engineers and good people because they generally have a strong network that we can continue to expand the team through.
And then, as far as the evaluation of the individuals, we try and do blind panels where each person will test them on what matters to that individual and review then in that context. And then, we can see where their strengths and weaknesses are and determine maybe that they're great technically, in general, but they're not filling the gaps that we actually have.
Or vice-versa ─ they aren't the most technical candidate but maybe they have exactly the skillset that we need and the skillset where we're weak.
So, overall, I think that teams will evolve. You'll use certain people who have strengths. You'll need to backfill those.
The problem space changes. And so, we adapt our teams relative to the current context.
LEDGE: That's great. I love that it's not just about any particular profile or test or anything; it really depends on the organizational and business problem that you're trying to solve with that collection of skills in context.
TYLER: Definitely!
LEDGE: Of course, now, I want you to develop the AI to just battle test that and figure out how to fit the exact right person at the right time. People have been hit in that problem for a while.
TYLER: Yes. I think some problems are good for AI and some problems are good for traditional intelligence.
LEDGE: We used to joke that we use AI, actual intelligence.
TYLER: I think a lot of it is empathy as well.
LEDGE: Absolutely! And it's just necessary. Good to have you today, man! I appreciate it. Best of luck with the growth of the company!
TYLER: Great! Thank you very much. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 576 |
Losing your business data, your family photos or important files because your hard drive crashed or your RAID failed is extremely stressful. But in most cases, your critical data can be recovered if you find a data recovery company that has the technology and experience to get your data back.
Many residents of Peoria, IL prefer to deal with ACE Data Recovery to recover irreplaceable files from failed hard drives, solid-state drives, all kinds of RAIDs, USB Flash drives, memory cards and tapes. They ship their failed storage media to the company's lab in Chicago because they are sure that ACEs can provide a level of service you can count on.
ACE Data Recovery has been providing A-grade services for more than 35 years and now it has a nationwide reputation for industry leading results, for leading-edge facilities, and a proven commitment to exceptional customer service. ACEs have partnered with leading HDD and computer manufacturers such as Dell and Western Digital.
ACE Data Recovery experienced engineers are hired from all around the world and can successfully recover data that other companies can't - from failed internal and external hard disk drives to SSDs, USB Flash drives, RAID, NAS, SAN, memory cards, and tapes. ACEs always do their recovery work in-house and never outsource. All data recovery operations are performed in the proprietary laboratories with Class 100 clean rooms for HDD recovery. That ensures comprehensive data verification process and quality control.
ACE Data Recovery has helped thousands of small and large businesses, corporations, and individual users recover their data from failed HDDs, SSDs, USB Flash drives, RAID arrays, camera memory cards, and tapes. ACE Data Recovery specialists will welcome the opportunity to help residents of Peoria, IL with their data recovery needs and guarantee fast proven results.
When your data is at risk, don't leave it up to chance. Trust the real expert from ACE Data Recovery to rescue your irreplaceable files. When you need to recover important documents or photos from failed HHD, SSD, USB flash drive, memory card, RAID, NAS, SAN or tape, contact ACE Data Recovery support team and speak with an expert about your case.
It's important that ACE Data Recovery has competitive pricing that includes a free professional diagnostic evaluation for a single media and 'No data - No charge' guarantee. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,043 |
Мину́та дуги́, углова́я мину́та или просто мину́та является единицей измерения углов, равной одной шестидесятой части () от градуса, или () радиан. В свою очередь, секунда дуги равна одной шестидесятой части () от минуты дуги. Эти единицы измерения используются в расчётах с применением СИ.
Поскольку градус определяется как одна триста шестидесятая () часть окружности, минута дуги равна окружности. Минута дуги используется в тех областях, где требуются единицы измерения для малых углов, таких как астрономия, навигация или меткость стрельбы.
Количество квадратных минут дуги в полной сфере равно:
,
или приблизительно квадратной минуты дуги.
Секунда дуги равна от градуса, или от полной окружности, или ≈ радиан.
Чтобы выразить ещё меньшие углы, можно использовать стандартные приставки СИ, например, в астрономии используются миллисекунды, сокращённо mas.
В литературе на русском языке иногда встречаются жаргонные названия «аркминута» для минуты дуги и «арксекунда» для секунды дуги, которые являются транслитерацией английских слов и . Эти названия считаются ошибочными.
Обозначения и аббревиатура
Стандартным символом для обозначения минуты дуги является штрих (′, U+2032), но в тех случаях, когда допускаются только ASCII-символы, применяется символ одиночной кавычки (', U+0027). Таким образом, одна минута угла записывается как1′.
Стандартным символом для обозначения секунды дуги является двойной штрих (″, U+2033), но в тех случаях, когда допускаются только ASCII-символы, применяется символ двойной кавычки (", U+0022). Таким образом, одна секунда дуги записывается как 1″.
В астрономической навигации секунды дуги редко используются в расчетах, предпочтение обычно отдаётся градусам, минутам и десятичным долям минут, например, 42°25′,32 или 42°25′,322. Такая же форма записи была перенесена в морские приемники GPS, в которых широта и долгота по умолчанию обычно отображаются в вышеприведённом формате.
Использование
Огнестрельное оружие
Угловая минута обычно используется в литературе и промышленной документации, относящейся к огнестрельному оружию, в частности, для описания точности стрельбы винтовок. Популярность этой единицы измерения связана с удобством, потому что 1 минута дуги стягивает примерно один дюйм на расстоянии 100 ярдов, традиционной дистанции в тире. Стрелок может легко настроить свой оптический прицел, измеряя расстояние в дюймах от пулевого отверстия на мишени до желаемой точки попадания, при этом величина корректировки прицела в минутах численно равна измеренному расстоянию в дюймах. Большинство прицелов для стрельбы на большие расстояния имеет шкалу регулировки в одну четвёртую () или одну восьмую () минуты. Одна восьмая минуты равна примерно восьмой части дюйма на расстоянии 100 ярдов, или одному дюйму на расстоянии 800 ярдов.
Расчёт физически эквивалентного размера, равного одной угловой минуте, можно сделать с помощью уравнения: эквивалентный размер= tg()× расстояние. В вышеприведённом примере, подставляя 3600 дюймов вместо 100 ярдов: 3600⋅tg() дюймов= 1,047 дюйма, то есть на расстоянии 100 ярдов одной угловой минуте эквивалентно 1,047 дюйма.
В метрических единицах 1 угловая минута на расстоянии 100 метров = 2,908 сантиметра.
Иногда точность огнестрельного оружия измеряется в минутах дуги. Это означает, что в оптимальных условиях (то есть в благоприятных климатических условиях, с качественным матчевым боеприпасом и зажатый в тиски) образец оружия способен произвести серию выстрелов, центры точек попадания которых вписываются в окружность с диаметром, эквивалентным заявленной точности в дуговых минутах. Например, винтовка с точностью 1 минута дуги способна в оптимальных условиях стрельбы обеспечить точность попадания серии выстрелов в окружность диаметром 1 дюйм на расстоянии 100 ярдов, винтовка с точностью 2 минуты дуги — в окружность диаметром 2 дюйма, и т. д. Некоторые производители оружия, такие как «Weatherby» или «Cooper Firearms of Montana», дают реальные гарантии показателей стрельбы своего оружия в дуговых минутах.
Производители винтовок часто пишут в рекламе своей продукции, что их оружие имеет субминутную точность, то есть оно стреляет с точностью менее 1 дуговой минуты. Как правило, проверка делается на одной серии из 3—5 выстрелов на расстоянии 100 ярдов или усреднением при стрельбе несколькими сериями. Если число проб возрастает (то есть больше выстрелов в каждой серии), то количество серий обычно тоже увеличивается.
Например, статистический расчёт даёт следующую зависимость точности от величины боекомплекта для одной и той же винтовки (стандартное отклонение каждого выстрела от центра составляет 1 угловую минуту):
Картография
Угловые минуты и секунды используются также в картографии и навигации. Одна минута угла на уровне моря (по экватору или меридиану) составляет примерно 1,86 километра или одну морскую милю («примерно» потому, что Земля не является идеальным шаром, а слегка сплюснута). Секунда угла равна одной шестидесятой этой величины: около 30 метров или 100 футов.
Традиционно положение объекта задаётся в градусах, минутах и секундах для двух координат: широты, равной углу к северу или к югу от экватора, и долготы, равной углу к востоку или к западу от нулевого меридиана. Используя этот метод, любое положение на Земле или над референц-эллипсоидом Земли может быть задано точно. Однако из-за несколько непривычного шестидесятиричного характера минут и секунд многие люди теперь предпочитают задавать позицию с использованием только градусов, выраженных в десятичной форме, чтобы обеспечить одинаковую точность вычислений. Градусы, заданные с точностью до трех знаков после запятой ( от градуса), имеют точность примерно от выражения в градусах-минутах-секундах ( от градуса), что эквивалентно местоположению в пределах около 120 метров или 400 футов.
Кадастровая съёмка
Относящаяся к картографии геодезическая съёмка территориальных границ с использованием системы межевания использует доли градуса при описании углов линий имущественных владений по отношению к сторонам света. Каждая прямая линия границы каждого участка описывается начальной точкой привязки, двумя направлениями по отношению к сторонам света (север-юг и восток-запад), одним углом по отношению к северу или югу (в зависимости от того, какой угол меньше 90 градусов) и длиной линии. Например, описание «север 65°39′18″, запад 45,67метра» описывает линию, проходящую от точки привязки 45,67метра в направлении к западу и под углом 65°39′18″ (или 65,655°) по отношению к северу.
Астрономия
Угловые минуты и секунды используются также в астрономии. Градусы (и, следовательно, угловые минуты) используются для измерения склонения (то есть углового расстояния на север или юг от небесного экватора). Угловые секунды также часто используются для описания параллакса из-за очень небольших значений углов параллакса для звёзд и крошечного углового диаметра (например, для Венеры он колеблется от 10′′ до 60′′). Параллакс, собственное движение и угловой диаметр звезды может быть записан в угловых миллисекундах (mas) или в тысячных долях секунды. Парсек получил своё название от «параллакса секунд», от тех же угловых секунд.
Астрометрический космический зонд Gaia Европейского космического агентства будет измерять положение звёзд с точностью до 20 угловых микросекунд (μas). В окружности около 1,3 квадриллиона угловых микросекунд. Чтобы получить представление о таких величинах, заметим, что угловой размер в одну угловую микросекунду имеет для земного наблюдателя точка в конце предложения в руководстве по эксплуатации, оставленном на Луне экспедицией Аполлона.
Человеческое зрение
Острота зрения людей позволяет различать пространственные структуры, разделённые углом зрения одна минуты дуги. При проверке зрения с помощью таблицы Снеллена нормальным считается зрение, при котором человек различает буквы в шестой строке с расстояния 6 метров. При этих условиях каждая буква этой строки стягивает дугу размером 5 минут.
Человеко-машинный интерфейс
Согласно эргономическим требованиям к интерфейсам «человек-машина», минимальный элемент значка интерфейса не должен быть меньше 6 угловых минут, размер простых иконок — не менее 20, а сложных — не менее 35 угловых минут. Для оператора, глаза которого находятся в 80 см от монитора, это составит приблизительно 1,4 мм, 4,2 мм и 8,1 мм соответственно.
Технологии
В оптической технике отклонение от параллельности между двумя поверхностями обычно измеряется в минутах или секундах дуги.
Примечания
Тригонометрия
Единицы измерения плоских углов | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/2151432\/finding-the-radius-of-convergence-of-the-complex-power-series","text":"# Finding the Radius of Convergence of the complex power series\n\nThe power series I have is this:\n\n$$\\sum_{n=0}^\\infty a_n z^n ~~~such~that~~ \\sum_{n=0}^\\infty 2^n a_n ~~converges~ while~ \\sum_{n=0}^\\infty (-1)^n2^na_n~~diverges$$\n\nThis is my attempt:\n\nlet $u_n = 2^na_n$, then $limsup \\sqrt[n]{|u_n|} = limsup\\sqrt[n]{|2^na_n|} = 2 limsup\\sqrt[n]{|a_n|} = l_1 < 1$\n\nAlso $limsup\\sqrt[n]{|(-1)^n2^na_n|} = 2 limsup\\sqrt[n]{|a_n|} = l_2 > 1$\n\nThen the radius of convergence of $\\sum a_n z^n$ is at least $\\frac{2}{l_1}$ and at most $\\frac{2}{l_2}$. So the radius is between these two values.\n\nI'm not sure if my approach is right because I'm not getting a definite value.\n\nAny help would be much appreciated.\n\nEdit:\n\nIs it true that $l_1 = l_2$? and therefore in this case the radius = 2\n\n\u2022 Why not just let $a_n=\\frac{(-1)^n}{n2^n}$? \u2013\u00a0Simply Beautiful Art Feb 19 '17 at 14:03\n\u2022 but I am allowed to take a specific value for $a_n$..am I wrong? \u2013\u00a0user368063 Feb 19 '17 at 14:05\n\u2022 If you are doing a problem that asks you to find an example of such a sequence, then sure, you are allowed your own choices. How could it be otherwise? There's an infinity of possible solutions out there! And if that is not the problem statement, well, you haven't told us what is being asked. \u2013\u00a0Harald Hanche-Olsen Feb 19 '17 at 14:42\n\u2022 The $<1$ and $>1$ claims are incorrect. \u2013\u00a0zhw. Feb 19 '17 at 20:12\n\nYou have got $$2 \\lim \\sup \\sqrt[n] {|a_n|} \\le 1 \\Rightarrow 2 \\le \\frac {1}{\\lim \\sup \\sqrt[n] {|a_n|}}=R........(1)$$\nAnd $$2 \\lim \\sup \\sqrt[n] {|a_n|} \\ge 1 \\Rightarrow 2 \\ge \\frac {1}{\\lim \\sup \\sqrt[n] {|a_n|}}=R........(2)$$\nFrom $(1)$ and $(2)$, $R=2$.","date":"2019-07-21 13:37:24","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\": 0.6907240152359009, \"perplexity\": 215.34010756011168}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195527000.10\/warc\/CC-MAIN-20190721123414-20190721145414-00087.warc.gz\"}"} | null | null |
Media releases are provided as is by companies and have not been edited or checked for accuracy. Any queries should be directed to the company itself.
Australia is being left behind in high-technology development
<p>Government and industry are being warned that the lack of an effective high-technology framework is stifling the growth of Australia's IT sector as local innovators head offshore to undertake important research and development. The CEO for Australian software developer QSR International, John Owen, says it is vital that Australia develops a successful high-tech industry which is recognised and exported throughout the world.</p>
<p>"Australia needs to move past the 'digging it out of the ground and growing it mentality' on which we have based our export industry. We have excellent innovation that goes on throughout Australia on a regular basis but without an effective framework for research and development, this innovation is either developed overseas or never sees the light of day," he said.</p>
<p>"A case in point is the creation of the 'black box' used on board aircrafts around the world. Many people wouldn't be aware that the technology was invented in Australia but developed and commercialised overseas. This is but one example of how Australia is particularly poor at nurturing and promoting home grown innovation."</p>
<p>Mr Owen says that many new companies find it difficult to break into mature international IT markets due to a lack of government and industry support when promoting and exporting new technology products.
"At the moment there are many countries in Europe far smaller than Australia which are renowned for their IT prowess, for instance the development of Nokia in Finland or Ericsson in Sweden. However nothing high-tech will come to mind if you ask a European about Australia.</p>
<p>"We need to have a dramatically improved framework to actively encourage and ensure that we can not only develop new technology but that we can commercialise and market it internationally as well. It's unfortunate that a company that's developed something fantastic in Australia would feel the need to list in America, where venture capitalists are more likely to back new technology, because they won't get the support in research and development at home," he said.</p>
<p>QSR International's outstanding achievements in the development, manufacturing and exporting of world's best practice qualitative data analysis software has shown that Australian designed and manufactured IT products can become successful on a global stage. The company exports to 150 countries and is the market-leader in a competitive international field.</p>
<p>In the past 12 months, QSR has been recognised for its achievement at a range of domestic and international awards and has taken out the top honours at both The Age/D&B 2009 Victorian Business of the Year Award and the Dell Small Business Excellence Award in Australia. QSR has also won the prestigious 2009 Market Research Society/Association for Survey Computing Award for Technology Effectiveness which was announced in London this week and is part of the United Kingdom's premier market research awards program. QSR is currently listed for a second international award, being the 2009 Global Dell Small Business Excellence Award, where the company will represent Australia and compete against winners from 12 other countries.</p>
<p>"Businesses which operate in the IT sector must work together and share skills, knowledge and experiences, while government needs to be more proactive in funding research and helping smart industries develop a good international reputation for IT innovation," Mr Owen said.</p>
<p>"In Australia we have the ability to become internationally recognised as a hub for cutting edge technology if an effective framework is put in place. As the development of high technology continues to grow and expand across the world, we need to move beyond the success of our primary industries and expand and diversify Australia's export market into the IT sector."</p>
<p>Media contact:
To interview QSR International CEO John Owen, phone Kate Bright, Sauce Communications on 0427 728 245 or email kate@saucecommunications.com.au
Please note a media backgrounder on QSR International follows.</p>
<p>Media backgrounder:
* QSR International is based in Australia, with offices in Europe and North America.</p>
<p>* From its Melbourne base and with a staff of nearly 65, it has become the world's largest qualitative research software developer.</p>
<p>* QSR's flagship products – NVivo and XSight – are developed end-to-end in Australia and are sold in more than 150 countries. The data analysis software allows users to upload and analyze video, audio, images and text side-by-side. Its powerful analysis tools help users to interrogate their data - testing out theories, identifying trends and cross-examining information.</p>
<p>* QSR was the first in the world to deliver qualitative research software programs in Japanese and Simplified Chinese. The company's NVivo 8 software is also available in German, French, and Spanish.</p>
<p>* QSR customers are drawn from the academic, government and commercial sectors and its largest geographic markets are the United States, Europe and Australasia.</p>
<p>* More than 400,000 customers use QSR software and more than 500 organizations hold site licenses for its products, including the Children's Hospital Boston, the Chronic Poverty Research Centre, sports coach UK, market research giant GfK-NOP, the Victorian CFA and virtually every major university in the United States, Europe and Australasia.</p>
<p>* QSR is the most published software developer in the qualitative research field - its software is cited in textbooks, research literature, journals and blogs worldwide.</p>
<p>* QSR International is the only developer in its field to earn Microsoft Gold Partner status and ISV competency.</p>
<p>* All QSR software developers are Microsoft Certified Professionals and all software testers have International Software Testing Qualifications Board (ISTQB) certification.</p>
<p>* An early adopter of emerging technology, QSR was amongst the first software developers in the world to utilize .Net and Microsoft SQL Server 2005 Express. The company was also the first in its field to develop software using Microsoft XP guidelines and to receive Microsoft's Certified for Vista accreditation.</p>
<p>* QSR won the 2009 Dell Small Business Excellence Award in Australia, and will compete against 12 other countries for the 2009 Global Dell Small Business Excellence Award. QSR was also named as the winner of the IT & Business Services Category in The Age / Dun & Bradstreet 2009 Victorian Business Awards and went on to be named 2009 Victorian Business of the Year in the same awards.</p>
<p>* QSR won the Information and Communications Technology category at the Governor of Victoria Export Awards in 2008, 2006 and 2001. It was highly commended in the same category in 2004 and 2002. The Company has also been named the winner of the 2009 MRS/ASC Technology Effectiveness Awards in the United Kingdom for its software NVivo 8. It was selected as a finalist for its XSight software in 2007 for the same award. In addition, QSR was selected as a finalist in the Applications and Infrastructure Tools category in the 2007 Australian Information Industry Association (AIIA) iAwards.</p>
<p>* While QSR International was formally established in 1995, the company has its origins in 1981 – when the first software product, NUD*IST, was developed.</p>
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Reborn is the tenth solo studio album from Denny Laine. It is his last all-original release until 2001. It was reissued in 2001 as Reborn...Again by The Store for Music with bonus tracks.
Denny became friends with local property developer John Ashworth in Rossendale, Lancashire and they built a recording studio in the basement of John's home in Whalley - Bramley Meade - which was a former maternity hospital.
The cover photo of Reborn is Denny stood in the entrance to Bramley Meade, with John Ashworth's Bentley parked outside.
Denny also wrote the songs for a musical titled Arctic Song which was performed by the students of Stonyhurst College where John Ashworth's two sons attended.
Arctic Song was released on CD in 1998 but wasn't widely publicised. Denny performed all the songs in the CD.
Track listing
All tracks composed by Denny Laine
"In Time" 5.35
"Reborn" 3.59
"Rollin' Tide" 5.18
"Night Walker" 4.22
"Hard Labour" 5.05
"Misty Mountain" 4.52
"Fanfare" 4.52
"Within Walls" 4.29
"Eternal Quest" 5.29
"Phoenix" 5.33
Bonus tracks on Reborn...Again
"Go Now" 3.23
"Time to Hide"
"Again & Again & Again"
"Mull of Kintyre" 4.21
Personnel
Denny Laine
Alan Wormold, Chip Hawkes, Gary Roberts, Jackie Bodimead - backing vocals
Technical
Brian Adams - executive producer
David Simmonds - cover photography
References
Denny Laine albums
1996 albums | {
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\section{Introduction}
\label{intro}
Recent rapid improvement in motion capture (MoCap) sensor accuracy brought affordable technology that can identify walking people. MoCap technology provides video clips of walking individuals containing structural motion data. The format keeps an overall structure of the human body and holds estimated 3D positions of major anatomical landmarks as the person moves. MoCap data can be collected online by a system of multiple cameras (Vicon) or a depth camera (Microsoft Kinect). To visualize motion capture data (see Figure~\ref{f1}), a simplified stick figure representing the human skeleton (graph of joints connected by bones) can be recovered from body point spatial coordinates.
\begin{figure}[h]
\vspace{-6pt}%
\centering
\includegraphics[height=4cm]{f1a.pdf}\qquad
\includegraphics[height=4cm]{f1b.pdf}
\vspace{-6pt}%
\caption{Motion capture data. Skeleton is represented by a stick figure of 31~joints (only 17 are drawn here). Seven selected video frames of a walk sequence contain 3D coordinates of each joint in time. The red and blue lines track trajectories of hands and feet.~\cite{VBZ16}}
\label{f1}
\end{figure}
Recognizing a person by walk involves capturing and normalizing their walk sample, extracting gait features to compose a template, and finally querying a central database for a set of similar templates to report the most likely identity. This work focuses on extracting robust and discriminative gait features from raw MoCap data.
Many geometric gait features have been introduced over the past few years. They are typically combinations of static body parameters (bone lengths, person's height)~\cite{KKMJ14} with dynamic gait features such as step length, walk speed, joint angles and inter-joint distances~\cite{AA15,APG15,KKMJ14,RCA15}, along with various statistics (mean, standard deviation or maximum) of their signals~\cite{AWLSWZ16}. Clearly, these features are schematic and human-interpretable, which is convenient for visualizations and for intuitive understanding, but unnecessary for automatic gait recognition. Instead, this application prefers learning features that maximally separate the identity classes and are not limited by such dispensable factors.
Methods for 2D~gait recognition extensively use machine learning models for extracting gait features, such as principal component analysis and multi-scale shape analysis~\cite{CT15}, genetic algorithms and kernel principal component analysis~\cite{TBLH15}, radial basis function neural networks~\cite{ZW14}, or convolutional neural networks~\cite{CMGP16}. All of those and many other models are reasonable to be utilized also in 3D~gait recognition.
In the video surveillance environment data need to be acquired without walker's consent and new identities can appear on the fly. Here and also in other applications where labels for all encountered people may not always be available, we value features that have a high power in distinguishing all people and not exclusively who they were learned on. We call these walker-independent features. The main idea is to statistically learn what aspects of walk people generally differ in and extract those as gait features. The features are learned in a supervised manner, as described in the following section.
\section{Learning Gait Features}
\label{meth}
In statistical pattern recognition, reducing space dimensionality is a common technique to overcome class estimation problems. Classes are discriminated by projecting high-dimensional input data onto low-dimensional sub-spaces by linear transformations with the goal of maximizing the class separability. We are interested in finding an optimal feature space where a gait template is close to those of the same walker and far from those of different walkers.
Let the model of a human body have $\gJ$ joints and all samples be linearly normalized to their average length~$\gT$. Labeled learning data in the measurement space $\gL{\gG}$ are in the form $\left\{\left(\gGn{n},\gLAMBDAn{n}\right)\right\}_{n=1}^{\gL{\gN}}$ where
\begin{equation}
\gGn{n}=\left[[\gGAMMAjt{1}{1}\,\cdots\,\gGAMMAjt{\gJ}{1}]^\top\,\cdots\,[\gGAMMAjt{1}{\gT}\,\cdots\,\gGAMMAjt{\gJ}{\gT}]^\top\right]^\top
\end{equation}
is a gait sample (one gait cycle) in which $\gGAMMAjt{j}{t}\in\mathbb{R}^3$ are 3D spatial coordinates of a joint $j\in\left\{1,\ldots,\gJ\right\}$ at time $t\in\left\{1,\ldots,\gT\right\}$ normalized with respect to the person's position and walk direction. See that $\gL{\gG}$ has dimensionality $\gD=3\gJ\gT$. Each learning sample falls strictly into one of the learning identity classes $\left\{\gIc{c}\right\}_{c=1}^{\gC}$ determined by $\gLAMBDAn{n}$. A class $\gIc{c}\subseteq\gL{\gG}$ has $\gNc{c}$ samples. The classes are complete and mutually exclusive. We say that learning samples $\left(\gGn{n},\gLAMBDAn{n}\right)$ and $\left(\gGn{n'},\gLAMBDAn{n'}\right)$ share a common walker if and only if they belong to the same class, i.e., $\left(\gGn{n},\gLAMBDAn{n}\right),\left(\gGn{n'},\gLAMBDAn{n'}\right)\in\gIc{c}\Leftrightarrow\gLAMBDAn{n}=\gLAMBDAn{n'}$.
We measure class separability of a given feature space by a representation of the Maximum Margin Criterion (MMC)~\cite{KKS04,LJZ06} used by the Vapnik's Support Vector Machines (SVM)~\cite{V95}
\begin{equation}
\gCRIT=\frac{1}{2}\sum_{c,c'=1}^{\gL{\gC}}\left(\left(\gMc{c}-\gMc{c'}\right)^\top\left(\gMc{c}-\gMc{c'}\right)-\tr\left(\gSIGMAc{c}+\gSIGMAc{c'}\right)\right)
\end{equation}
which is actually a summation of $\frac{1}{2}\gL{\gC}(\gL{\gC}-1)$ between-class margins. The margin is defined as the Euclidean distance of class means minus both individual variances (traces of scatter matrices $\gSIGMAc{c}=\frac{1}{\gNc{c}}\sum_{n=1}^{\gNc{c}}\left(\gGnc{n}{c}-\gMc{c}\right)\left(\gGnc{n}{c}-\gMc{c}\right)^\top$ and similarly for $\gSIGMAc{c'}$). For the whole labeled data, we denote the between- and within-class and total scatter matrices
\begin{equation}
\begin{split}
\gSIGMAb & =\sum_{c=1}^{\gL{\gC}}\left(\gMc{c}-\gM\right)\left(\gMc{c}-\gM\right)^\top\\
\gSIGMAw & =\sum_{c=1}^{\gL{\gC}}\frac{1}{\gNc{c}}\sum_{n=1}^{\gNc{c}}\left(\gGnc{n}{c}-\gMc{c}\right)\left(\gGnc{n}{c}-\gMc{c}\right)^\top\\
\gSIGMAt & =\sum_{c=1}^{\gL{\gC}}\frac{1}{\gNc{c}}\sum_{n=1}^{\gNc{c}}\left(\gGnc{n}{c}-\gM\right)\left(\gGnc{n}{c}-\gM\right)^\top=\gSIGMAb+\gSIGMAw
\end{split}
\end{equation}
where $\gGnc{n}{c}$ denotes the $n$-th sample in class $\gIc{c}$ and $\gMc{c}$ and $\gM$ are sample means for class $\gIc{c}$ and the whole data set, respectively, that is, $\gMc{c}=\frac{1}{\gNc{c}}\sum_{n=1}^{\gNc{c}}\gGnc{n}{c}$ and $\gM=\frac{1}{\gL{\gN}}\sum_{n=1}^{\gL{\gN}}\gGn{n}$. Now we obtain
\begin{equation}
\begin{split}
\gCRIT & =\frac{1}{2}\sum_{c,c'=1}^{\gL{\gC}}\left(\gMc{c}-\gMc{c'}\right)^\top\left(\gMc{c}-\gMc{c'}\right)-\frac{1}{2}\sum_{c,c'=1}^{\gL{\gC}}\tr\left(\gSIGMAc{c}+\gSIGMAc{c'}\right)\\
& =\frac{1}{2}\sum_{c,c'=1}^{\gL{\gC}}\left(\gMc{c}-\gM+\gM-\gMc{c'}\right)^\top\left(\gMc{c}-\gM+\gM-\gMc{c'}\right)-\sum_{c=1}^{\gL{\gC}}\tr\left(\gSIGMAc{c}\right)\\
& =\tr\left(\sum_{c=1}^{\gL{\gC}}\left(\gMc{c}-\gM\right)\left(\gMc{c}-\gM\right)^\top\right)-\tr\left(\sum_{c=1}^{\gL{\gC}}\gSIGMAc{c}\right)\\
& =\tr\left(\gSIGMAb\right)-\tr\left(\gSIGMAw\right)=\tr\left(\gSIGMAb-\gSIGMAw\right).
\end{split}
\end{equation}
Since $\tr\left(\gSIGMAb\right)$ measures the overall variance of the class mean vectors, a large one implies that the class mean vectors scatter in a large space. On the other hand, a small $\tr\left(\gSIGMAw\right)$ implies that classes have a small spread. Thus, a large $\gCRIT$ indicates that samples are close to each other if they share a common walker but are far from each other if they are performed by different walkers. Extracting features, that is, transforming the input data in the measurement space into a feature space of higher $\gCRIT$, can be used to link new observations of walkers more successfully.
\pagebreak
Feature extraction is given by a linear transformation (feature) matrix $\gPHI\in\mathbb{R}^{\gD\times\gH{\gD}}$ from a $\gD$-dimensional measurement space $\gG=\left\{\gGn{n}\right\}_{n=1}^{\gN}$ of not necessarily labeled gait samples to a $\gH{\gD}$-dimensional feature space $\gH{\gG}=\left\{\gGnH{n}\right\}_{n=1}^{\gN}$ of gait templates where $\gH{\gD}<\gD$ and each gait sample $\gGn{n}$ is transformed into a gait template $\gGnH{n}=\gPHI^\top\gGn{n}$. The objective is to learn a transform $\gPHI$ that maximizes MMC in the feature space
\begin{equation}
\gCRITf{\gPHI}=\tr\left(\gPHI^\top\left(\gSIGMAb-\gSIGMAw\right)\gPHI\right).
\label{e2}
\end{equation}
Once the transformation is found, all measured samples are transformed into templates (in the feature space) along with the class means and covariances. The templates are compared by the Mahalanobis distance function
\begin{equation}
\gDELTAccH{\gGnH{n}}{\gGnH{n'}}=\sqrt{\left(\gGnH{n}-\gGnH{n'}\right)^\top\gSIGMAtH^{-1}\left(\gGnH{n}-\gGnH{n'}\right)}.
\label{e3}
\end{equation}
We show that solution to the optimization problem in Equation~\eqref{e2} can be obtained by eigendecomposition of the matrix $\gSIGMAb-\gSIGMAw$. An important property to notice about the objective $\gCRITf{\gPHI}$ is that it is invariant w.r.t.\@ rescalings $\gPHI\rightarrow\alpha\gPHI$. Hence, we can always choose $\gPHI=\gPHId{1}\|\cdots\|\gPHId{\gH{\gD}}$ such that $\gPHId{\gH{d}}^\top\gPHId{\gH{d}}=1$, since it is a scalar itself. For this reason we can reduce the problem of maximizing $\gCRITf{\gPHI}$ into the constrained optimization problem
\begin{equation}
\begin{split}
\max & \quad\sum_{\gH{d}=1}^{\gH{\gD}}\gPHId{\gH{d}}^\top\left(\gSIGMAb-\gSIGMAw\right)\gPHId{\gH{d}}\\
\mathrm{subject\,to} & \quad\gPHId{\gH{d}}^\top\gPHId{\gH{d}}-1=0\qquad\forall\gH{d}=1,\ldots,\gH{\gD}.
\end{split}
\end{equation}
To solve the above optimization problem, let us consider the Lagrangian
\begin{equation}
\gLAG{\gPHId{\gH{d}}}{\gLAMBDAd{\gH{d}}}=\sum_{\gH{d}=1}^{\gH{\gD}}\gPHId{\gH{d}}^\top\left(\gSIGMAb-\gSIGMAw\right)\gPHId{\gH{d}}-\gLAMBDAd{\gH{d}}\left(\gPHId{\gH{d}}^\top\gPHId{\gH{d}}-1\right)
\end{equation}
with multipliers $\gLAMBDAd{\gH{d}}$. To find the maximum, we derive it with respect to $\gPHId{\gH{d}}$ and equate to zero
\begin{equation}
\frac{\partial\gLAG{\gPHId{\gH{d}}}{\gLAMBDAd{\gH{d}}}}{\partial\gPHId{\gH{d}}}=\left(\left(\gSIGMAb-\gSIGMAw\right)-\gLAMBDAd{\gH{d}}\gI\right)\gPHId{\gH{d}}=0
\end{equation}
which leads to
\begin{equation}
\left(\gSIGMAb-\gSIGMAw\right)\gPHId{\gH{d}}=\gLAMBDAd{\gH{d}}\gPHId{\gH{d}}
\end{equation}
where $\gLAMBDAd{\gH{d}}$ are the eigenvalues of $\gSIGMAb-\gSIGMAw$ and $\gPHId{\gH{d}}$ are the corresponding eigenvectors. Putting it all together,
\begin{equation}
\left(\gSIGMAb-\gSIGMAw\right)\gPHI=\gLAMBDA\gPHI
\end{equation}
where $\gLAMBDA=\diag\left(\gLAMBDAd{1},\ldots,\gLAMBDAd{\gH{\gD}}\right)$ is the eigenvalue matrix. Therefore,
\begin{equation}
\gCRITf{\gPHI}=\tr\left(\gPHI^\top\left(\gSIGMAb-\gSIGMAw\right)\gPHI\right)=\tr\left(\gPHI^\top\gLAMBDA\gPHI\right)=\tr\left(\gLAMBDA\right)
\end{equation}
is maximized when $\gLAMBDA$ has $\gH{\gD}$ largest eigenvalues and $\gPHI$ contains the corresponding leading eigenvectors.
In the following we discuss how to calculate the eigenvectors of $\gSIGMAb-\gSIGMAw$ and to determine an optimal dimensionality $\gH\gD$ of the feature space. Rewrite $\gSIGMAb-\gSIGMAw=2\gSIGMAb-\gSIGMAt$. Note that the null space of $\gSIGMAt$ is a subspace of that of $\gSIGMAb$ since the null space of $\gSIGMAt$ is the common null space of $\gSIGMAb$ and $\gSIGMAw$. Thus, we can simultaneously diagonalize $\gSIGMAb$ and $\gSIGMAt$ to some $\gDELTA$ and $\gI$
\begin{equation}
\begin{split}
\gPSI^\top\gSIGMAb\gPSI & =\gDELTA\\
\gPSI^\top\gSIGMAt\gPSI & =\gI
\end{split}
\end{equation}
with the $\gD\times\rank\left(\gSIGMAt\right)$ eigenvector matrix
\begin{equation}
\gPSI=\ensuremath{\gB{\Omega}}\ensuremath{\gB{\Theta}}^{-\frac{1}{2}}\ensuremath{\gB{\Xi}}
\end{equation}
where $\ensuremath{\gB{\Omega}}$ and $\ensuremath{\gB{\Theta}}$ are the eigenvector and eigenvalue matrices of $\gSIGMAt$, respectively, and $\ensuremath{\gB{\Xi}}$ is the eigenvector matrix of $\ensuremath{\gB{\Theta}}^{-1/2}\ensuremath{\gB{\Omega}}^\top\gSIGMAb\ensuremath{\gB{\Omega}}\ensuremath{\gB{\Theta}}^{-1/2}$. To calculate $\gPSI$, we use a fast two-step algorithm in virtue of Singular Value Decomposition (SVD). SVD expresses a real $r \times s$ matrix $\gB{A}$ as a product $\gB{A}=\gB{U}\gB{D}\gB{V}^\top$ where $\gB{D}$ is a diagonal matrix with decreasing non-negative entries, and $\gB{U}$ and $\gB{V}$ are $r\times\min\left\{r,s\right\}$ and $s\times\min\left\{r,s\right\}$ eigenvector matrices of $\gB{A}\gB{A}^\top$ and $\gB{A}^\top\gB{A}$, respectively, and the non-vanishing entries of $\gB{D}$ are square roots of the non-zero corresponding eigenvalues of both $\gB{A}\gB{A}^\top$ and $\gB{A}^\top\gB{A}$. See that $\gSIGMAt$ and $\gSIGMAb$ can be expressed in the forms
\vspace{-4pt}%
\begin{equation}
\begin{split}
\gSIGMAt=&\enskip\ensuremath{\gB{X}}\gCHI^\top\enskip\mathrm{where}\enskip\ensuremath{\gB{X}}=\frac{1}{\sqrt{\gL{\gN}}}\left[\left(\gGn{1}-\gM\right)\cdots\left(\gGn{\gL{\gN}}-\gM\right)\right]\enskip\text{and}\\
\gSIGMAb=&\enskip\ensuremath{\gB{\Upsilon}}\gUPSILON^\top\enskip\text{where}\enskip\ensuremath{\gB{\Upsilon}}=\left[\left(\gMc{1}-\gM\right)\cdots\left(\gMc{\gL{\gC}}-\gM\right)\right],
\end{split}
\end{equation}
respectively. Hence, we can obtain the eigenvectors $\ensuremath{\gB{\Omega}}$ and the corresponding eigenvalues $\ensuremath{\gB{\Theta}}$ of $\gSIGMAt$ through the SVD of $\ensuremath{\gB{X}}$ and analogically $\ensuremath{\gB{\Xi}}$ of $\ensuremath{\gB{\Theta}}^{-1/2}\ensuremath{\gB{\Omega}}^\top\gSIGMAb\ensuremath{\gB{\Omega}}\ensuremath{\gB{\Theta}}^{-1/2}$ through the SVD of $\ensuremath{\gB{\Theta}}^{-1/2}\ensuremath{\gB{\Omega}}^\top\ensuremath{\gB{\Upsilon}}$. The columns of $\gPSI$ are clearly the eigenvectors of $2\gSIGMAb-\gSIGMAt$ with the corresponding eigenvalues $2\gDELTA-\gI$. Therefore, to constitute the transform $\gPHI$ by maximizing the MMC, we should choose the eigenvectors in $\gPSI$ that correspond to the eigenvalues of at least $\frac{1}{2}$ in $\gDELTA$. Note that $\gDELTA$ contains at most $\rank\left(\gSIGMAb\right)=\gL{\gC}-1$ positive eigenvalues, which gives an upper bound on the feature space dimensionality~$\gH\gD$. Algorithm~\ref{a1}~\cite{BS16a} provided below is an efficient way of learning the transform $\gPHI$ for MMC on given labeled learning data~$\gL{\gG}$.
\vspace{-20pt}%
\begin{algorithm}[b]
\caption{LearnTransformationMatrixMMC$\left(\gL{\gG}\right)$}
\label{a1}
\begin{algorithmic}[1]
\State split $\gL{\gG}=\left\{\left(\gGn{n},\gLAMBDAn{n}\right)\right\}_{n=1}^{\gL{\gN}}$ into classes $\left\{\gIc{c}\right\}_{c=1}^{\gL{\gC}}$ of $\gNc{c}=\left|\gIc{c}\right|$ samples
\State compute overall mean $\gM=\frac{1}{\gL{\gN}}\sum_{n=1}^{\gL{\gN}}\gGn{n}$ and individual class means $\gMc{c}=\frac{1}{\gNc{c}}\sum_{n=1}^{\gNc{c}}\gGnc{n}{c}$
\State compute $\gSIGMAb=\sum_{c=1}^{\gL{\gC}}\left(\gMc{c}-\gM\right)\left(\gMc{c}-\gM\right)^\top$
\State compute $\ensuremath{\gB{X}}=\frac{1}{\sqrt{\gL{\gN}}}\left[\left(\gGn{1}-\gM\right)\cdots\left(\gGn{\gL{\gN}}-\gM\right)\right]$
\State compute $\ensuremath{\gB{\Upsilon}}=\left[\left(\gMc{1}-\gM\right)\cdots\left(\gMc{\gL{\gC}}-\gM\right)\right]$
\State compute eigenvectors $\ensuremath{\gB{\Omega}}$ and corresponding eigenvalues $\ensuremath{\gB{\Theta}}$ of $\gSIGMAt$ through SVD of $\ensuremath{\gB{X}}$
\State compute eigenvectors $\ensuremath{\gB{\Xi}}$ of $\ensuremath{\gB{\Theta}}^{\nicefrac{-1}{2}}\ensuremath{\gB{\Omega}}^\top\gSIGMAb\ensuremath{\gB{\Omega}}\ensuremath{\gB{\Theta}}^{\nicefrac{-1}{2}}$ through SVD of $\ensuremath{\gB{\Theta}}^{\nicefrac{-1}{2}}\ensuremath{\gB{\Omega}}^\top\ensuremath{\gB{\Upsilon}}$
\State compute eigenvectors $\gPSI=\ensuremath{\gB{\Omega}}\ensuremath{\gB{\Theta}}^{\nicefrac{-1}{2}}\ensuremath{\gB{\Xi}}$
\State compute eigenvalues $\gDELTA=\gPSI^\top\gSIGMAb\gPSI$
\State return transform $\gPHI$ as eigenvectors in $\gPSI$ that correspond to the eigenvalues of at least $\nicefrac{1}{2}$ in $\gDELTA$
\end{algorithmic}
\end{algorithm}
\section{Experiments and Results}
\label{exp}
\subsection{Database}
\label{exp-db}
For the evaluation purposes we have extracted a large number of samples from the general MoCap database from CMU~\cite{CMU03} as a well-known and recognized database of structural human motion data. It contains numerous motion sequences, including a considerable number of gait sequences. Motions are recorded with an optical marker-based Vicon system. People wear a black jumpsuit and have 41~markers taped on. The tracking space of \unit[30]{m$^2$}, surrounded by 12~cameras of sampling rate of \unit[120]{Hz} in the height from 2 to 4~meters above ground, creates a video surveillance environment. Motion videos are triangulated to get highly accurate 3D data in the form of relative body point coordinates (with respect to the root joint) in each video frame and stored in the standard ASF/AMC data format. Each registered participant is assigned with their respective skeleton described in an ASF file. Motions in the AMC files store bone rotational data, which is interpreted as instructions about how the associated skeleton deforms over time.
These MoCap data, however, contain skeleton parameters pre-calibrated by the CMU staff. Skeletons are unique for each walker and even a trivial skeleton check could result in 100\% recognition. In order to use the collected data in a fairly manner, a prototypical skeleton is constructed and used to represent bodies of all subjects, shrouding the unique skeleton parameters of individual walkers. Assuming that all walking subjects are physically identical disables the skeleton check as a potentially unfair classifier. Moreover, this is a skeleton-robust solution as all bone rotational data are linked with a~fixed skeleton. To obtain realistic parameters, it is calculated as the mean of all skeletons in the provided ASF files.
We calculate 3D joint coordinates using bone rotational data and the prototypical skeleton. One cannot directly use raw values of joint coordinates, as they refer to absolute positions in the tracking space, and not all potential methods are invariant to person's position or walk direction. To ensure such invariance, the center of the coordinate system is moved to the position of root joint $\gGAMMAjt{\mathrm{root}}{t}=[0,0,0]^\top$ for each time~$t$ and axes are adjusted to the walker's perspective: the X~axis is from right (negative) to left (positive), the Y~axis is from down (negative) to up (positive), and the Z~axis is from back (negative) to front (positive). In the AMC file structure notation it is achieved by zeroing the root translation and rotation (\texttt{root 0 0 0 0 0 0}) in all frames of all motion sequences.
Since the general motion database contains all motion types, we extracted a number of sub-motions that represent gait cycles. First, an exemplary gait cycle was identified, and clean gait cycles were then filtered out using the DTW distance over bone rotations. The similarity threshold was set high enough so that even the least similar sub-motion still semantically represents a gait cycle. Finally, subjects that contributed with less than 10~samples were excluded. The final database has 54~walking subjects that performed 3{,}843~samples in total, which makes an average of about 71~samples per subject.
\subsection{Evaluation Setups and Metrics}
\label{exp-su}
Learning data $\gL{\gG}=\left\{\left(\gGn{n},\gLAMBDAn{n}\right)\right\}_{n=1}^{\gL{\gN}}$ of $\gL{\gC}$ identities and evaluation data $\gE{\gG}=\left\{\left(\gGn{n},\gLAMBDAn{n}\right)\right\}_{n=1}^{\gE{\gN}}$ of $\gE{\gC}$ identity classes have to be disjunct at all times. Evaluation is performed exclusively on the evaluation part, taking no observations of the learning part into account. In the following we introduce two setups of data separation: homogeneous and heterogeneous. The homogeneous setup learns the transformation matrix on $\nicefrac{1}{3}$ samples of $\gL{\gC}$ identities and is evaluated on templates derived from other $\nicefrac{2}{3}$ samples of the same $\gE{\gC}=\gL{\gC}$ identities. The heterogeneous setup learns the transform on all samples in $\gL{\gC}$ identities and is evaluated on all templates derived from other $\gE{\gC}$ identities. For better clarification we refer to Figure~\ref{f2}. Note that unlike in homogeneous setup, in heterogeneous setup there is no walker identity ever used for both learning and evaluation at the same time.
\begin{figure}[h]
\vspace{-5pt}%
\centering
\includegraphics[width=0.8\textwidth]{f2.pdf}
\vspace{-5pt}%
\caption{Abstraction of data separation for homogeneous setup of $\gL{\gC}=\gE{\gC}=3$ learning-and-evaluation classes (left) and for heterogeneous setup of~$\gL{\gC}=2$ learning classes and $\gE{\gC}=4$ evaluation classes (right). Black square represents a database and ellipses are identity classes.}
\label{f2}
\vspace{-2pt}%
\end{figure}
Homogeneous setup is parametrized by a single number $\gL{\gC}=\gE{\gC}$ of learning-and-evaluation identity classes, whereas the heterogeneous setup has the form $\left(\gL{\gC},\gE{\gC}\right)$ specifying how many learning and how many evaluation identity classes are randomly selected from the database. Evaluation of each setup is repeated 3~times, selecting new random $\gL{\gC}$ and $\gE{\gC}$ identity classes each time and reporting the average result. Please note that in the heterogeneous setup the learning identities are disjunct from the evaluation identities, that is, there is no single identity used for both learning and evaluation.
Correct Classification Rate (CCR) is a standard qualitative measure; however, if a~method has a low CCR, we cannot directly say if the system is failing because of bad features or a bad classifier. Providing an evaluation in terms of class separability of the feature space gives an estimate on the recognition potential of the extracted features and do not reflect eventual combination with an unsuitable classifier. Quality of features extraction algorithms is reflected in the Davies-Bouldin Index (DBI)
\begin{equation}
\mathrm{DBI}=\frac{1}{\gE{\gC}}\sum_{c=1}^{\gE{\gC}}\max\limits_{1 \leq c'\leq\gE{\gC},\,c' \neq c}\frac{\sigma_c+\sigma_{c'}}{\gDELTAccH{\gMcH{c}}{\gMcH{c'}}}
\end{equation}
where $\sigma_c=\frac{1}{\gNc{c}}\sum_{n=1}^{\gNc{c}}\gDELTAccH{\gGnH{n}}{\gMcH{c}}$ is the average distance of all templates in identity class $\gIc{c}$ to its centroid, and similarly for $\sigma_{c'}$. Templates of low intra-class distances and of high inter-class distances have a low DBI. DBI is measured on the full evaluation part, whereas CCR is estimated with 10-fold cross-validation taking one dis-labeled fold as a~testing set and other nine as gallery. Test templates are classified by the winner-takes-all strategy, in which a test template $\gGnH{}^{\mathrm{test}}$ gets assigned with the label $\gLAMBDAn{\argmin_i\gDELTAccH{\gGnH{}^{\mathrm{test}}}{\gGnH{i}^{\mathrm{gallery}}}}$ of the gallery's closest identity class.
Based on Section~\ref{exp-db}, our database has 54~identity classes in total. We performed the series of experiments \exper{A}, \exper{B}, \exper{C}, \exper{D} below. The experiments \exper{A} and \exper{B} are to compare the homogeneous and heterogeneous setup, whereas \exper{C} and \exper{D} examine how performance of the system in the heterogeneous setup improves with increasing number of learning identities. The results are illustrated in Figure~\ref{f3} and in Figure~\ref{f4} in the next section.
\begin{description}
\item[\exper{A}] homogeneous setup with $\gL{\gC}=\gE{\gC}\in\left\{2,\ldots,27\right\}$;
\item[\exper{B}] heterogeneous setup with $\gL{\gC}=\gE{\gC}\in\left\{2,\ldots,27\right\}$;
\item[\exper{C}] heterogeneous setup with $\gL{\gC}\in\left\{2,\ldots,27\right\}$ and $\gE{\gC}=27$;
\item[\exper{D}] heterogeneous setup with $\gL{\gC}\in\left\{2,\ldots,52\right\}$ and $\gE{\gC}=54-\gL{\gC}$.
\end{description}
\subsection{Results}
\label{exp-r}
Experiments \exper{A} and \exper{B} compare homogeneous and heterogeneous setups by measuring the drop in performance on an identical number of learning and evaluation identities ($\gL{\gC}=\gE{\gC}$). Top plot in Figure~\ref{f3} shows the measured values of DBI and CCR metrics in both alternatives, which not only appear comparable but also in some configurations the heterogeneous setup has an even higher CCR. Bottom plot expresses heterogeneous setup as a percentage of the homogeneous setup in each of the particular metrics. Here we see that with raising number of identities the heterogeneous setup approaches 100\% of the fully homogeneous alternative.
\begin{figure}[h]
\includegraphics[height=3.5cm]{f3a.pdf}\\
\includegraphics[height=3.5cm]{f3b.pdf}
\caption{DBI (left vertical axis) and CCR (right vertical axis) for experiments \exper{A} of homogeneous setup and \exper{B} of heterogeneous setup (top) with $\left(\gL{\gC},\gE{\gC}\right)$ configurations (horizontal axes) and their percentages (bottom).}
\label{f3}
\end{figure}
Experiments \exper{C} and \exper{D} investigate on the impact of the number of learning identities in the heterogeneous setup. Observing from the Figure~\ref{f4}, the performance grows quickly on the first configurations with very few learning identities, which we can interpret as an analogy to the Pareto (80--20) principle. Specifically, the results of experiment \exper{C} say that 8~learning identities achieve almost the same performance (66.78~DBI and 0.902~CCR) to as if learned on 27~identities (68.32~DBI and 0.947~CCR). The outcome of experiment~\exper{D} indicates a similar growth of performance and we see that yet 14~identities can be enough to learn the transformation matrix to distinguish 40 completely different people (0.904~CCR).
\begin{figure}[h]
\vspace{-1pt}%
\includegraphics[height=3.5cm]{f4a.pdf}\\
\includegraphics[height=3.5cm]{f4b.pdf}
\caption{DBI (left vertical axes) and CCR (right vertical axes) for experiments \exper{C} (top) and \exper{D} (bottom) on heterogeneous setup with $\left(\gL{\gC},\gE{\gC}\right)$ configurations (horizontal axes).}
\label{f4}
\end{figure}
The proposed method and seven other state-of-the-art methods~\cite{AAS14,AA15,BRRV12,DMG14,KKMJ14,PKWL12,SCB13} have been subjected to extensive simulations on homogeneous setup in our recent research paper~\cite{BS16a}. A variety of class-separability coefficients and classification metrics allows insights from different statistical perspectives. Results indicate that the proposed method is a leading concept for rank-based classifier systems: lowest Davies-Bouldin Index, highest Dunn Index, highest (and exclusively positive) Silhouette Coefficient, second highest Fisher's Discriminant Ratio and, combined with rank-based classifier, the best Cumulative Match Characteristic, False Accept Rate and False Reject Rate trade-off, Receiver Operating Characteristic (ROC) and recall-precision trade-off scores along with Correct Classification Rate, Equal Error Rate, Area Under ROC Curve and Mean Average Precision. We interpret the high scores as a sign of robustness. Apart from performance merits, the MMC method is also efficient: low-dimensional templates ($\gH{\gD}\leq\gL{\gC}-1=\gE{\gC}-1=53$) and Mahalanobis distance ensure fast distance computations and thus contribute to high scalability.
\section{Conclusions}
\label{con}
Despite many advanced optimization techniques used in statistical pattern recognition, a~common practice of state-of-the-art MoCap-based human identification is still to design geometric gait features by hand. As the first contribution of this paper, the proposed method does not involve any ad-hoc features; on the contrary, they are computed from a~much larger space beyond the limits of human interpretability. The features are learned directly from raw joint coordinates by a~modification of the Fisher's LDA with MMC so that the identities are maximally separated. We believe that MMC is a suitable criterion for optimizing gait features; however, our future work will continue with research on further potential optimality criterions and machine learning approaches. Furthermore, we are in the process of developing an evaluation framework with implementation details and source codes of all related methods, data extraction drive from the general CMU MoCap database and the evaluation mechanism to support reproducible research.
Second contribution lies in showing the possibility of building a representation on a problem and using it on another (related) problem. Simulations on the CMU MoCap database show that our approach is able to build robust feature spaces without pre-registering and labeling all potential walkers. In fact, we can take different people (experiments \exper{A} and \exper{B}) and just a~fraction of them (experiments \exper{C} and \exper{D}). We have observed that with an increasing volume of identities the heterogeneous evaluation setup is on par with the homogeneous setup, that is, it does not matter what identities we learn the features on. One does not have to rely on the availability of all walkers for learning. This is particularly important for a system to aid video surveillance applications where encountered walkers never supply labeled data. Multiple occurrences of individual walkers can now be linked together even without knowing their actual identities.
\vspace{-10pt}%
\subsubsection*{Acknowledgments}
Authors thank to the anonymous reviewers for their detailed commentary and suggestions. Data used in this project was created with funding from NSF EIA-0196217 and was obtained from \url{http://mocap.cs.cmu.edu}. Our extracted database is available at \url{https://gait.fi.muni.cz/} to support results reproducibility.
\vspace{-22pt}%
\bibliographystyle{splncs03}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,173 |
\section{Introduction}
\noindent
\setcounter{equation}{0}
The AdS/CFT correspondence \cite{rads}
provides a nonperturbative model of quantum gravity in which black holes
seem to form and evaporate as standard unitary processes in quantum mechanics.
In this construction,
quantum gravity
in a $d$-dimensional asymptotically Anti-de Sitter spacetime
(AdS) of curvature radius $R$ is {\it defined}
in terms
of a conformal field theory (CFT) on a spatial sphere ${\bf S}^{d-2}$ of
radius $R$.
The effective expansion parameter on the gravity side $1/N^2 \sim G_{\rm N}/R^{d-2}$, maps to
an appropriate large $N$ limit of the CFT. For example, for two-dimensional CFT's
$N^2$ is proportional to the central charge. When the
CFT is a
gauge theory and the AdS side is a string theory, $N$
is the rank of the gauge group, and the string perturbative expansion in powers
of $g_s \sim 1/N$
is identified with 't Hooft's $1/N$ expansion in the gauge theory side.
According to this definition, the formation and evaporation of small black holes
with Schwarschild radius $R_S \ll R$,
should be described by a unitary process in terms of
the CFT Hamiltonian. Thus, there is no room for violations of coherence,
independently of the manner in which the process is encoded in the CFT.
Unfortunately, the CFT states corresponding to small black holes
are hard to describe, and it remains a challenge to put the finger on the precise
error in the standard semiclassical analysis \cite{rsdollar} in that case.
\begin{figure}[hp]
\begin{center}
\epsfysize=1.5in
\epsffile{spec.eps}
\end{center}
\caption{
\small
\sl
The energy spectrum of a CFT representing ${\rm AdS}_d$ quantum gravity. The
spectrum is discrete on a sphere of radius $R$, with gap of order $1/R$.
The asymptotic energy band of
very dense ``black hole" states
sets in beyond energies of order $N^2 /R$. The corresponding density of states is
that of a conformal fixed point in $d-1$ spacetime dimensions.}
\label{figure1}
\end{figure}
For large eternal AdS black holes with Schwarschild radius $R_S \gg R$
one may attempt to rise to the challenge, since they
are thermodynamically stable and can exist in equilibrium
with thermal radiation at fixed (high) temperatures $ 1/\beta \gg 1/R$.
Indeed, the corresponding Bekenstein--Hawking entropy
scales like that of $N^2$ conformal degrees of freedom at high energy,
\beq\label{dens}
S \sim N^{2\over d-1} \;(E\,R)^{d-2 \over d-1} \sim N^2 \, (R/\beta)^{d-2} \;.
\eeq
Therefore, large AdS black holes with large Hawking temperature
$\beta^{-1} \gg R$ describe
the leading approximation to the thermodynamical functions of the canonical CFT state
\beq\label{cano}
\rho_\beta =
{e^{-\beta H} \over Z(\beta)}\;, \qquad Z(\beta) = {\rm Tr} \,\exp(-\beta H)\;.
\eeq
This suggests that we can test the semiclassical unitarity argument by careful analysis
of slight departures from thermal
equilibrium, rather than studying a complete evaporation instability
in the vacuum.
Ref. \cite{rmaldas} proposes to look at the very long time structure of
correlators of the form
\beq\label{timec}
G (t) = {\rm Tr} \,\left[\,\rho\,A(t)\,A(0)\,\right]\;,
\eeq
for appropriate Hermitian operators $A$. In this expression, $\rho$ denotes
the density matrix of the initial state.
In the semiclassical approximation, one expects
these correlators to decay as $\exp(-\Gamma \,t)$
with $\Gamma \sim \beta^{-1}$. However,
because the CFT lives in finite volume,
the spectrum is actually discrete (c.f. Fig 1), and the
correlator must show nontrivial long time
structure in the form of Poincar\'e recurrences, in particular it does
not vanish
(see \cite{rsuscumple, rsuspodos}).
This result,
which is straightforward from the boundary theory point of view,
has far reaching
consequences as far as the bulk physics is concerned.
\begin{figure}[hp]
\begin{center}
\epsfysize=2.5in
\epsffile{flucdisp.eps}
\end{center}
\caption{
\small
\sl
A detailed analysis of dissipation of fluctuations in a finite thermal system
can reveal the effect of large quantum fluctuations in which a black hole
turns into thermal radiation and viceversa. In the semiclassical approximation
to quantum gravity, these processes are represented by a coherent sum
over saddle points of different topology. In the case at hand we can use AdS
space as an effective finite-volume box.}
\label{figure2}
\end{figure}
Hence, the non-vanishing of $G(t)$ as $t\rightarrow \infty$ can be used as a criterion
for unitarity preservation
beyond the semiclassical approximation. This argument can be made more
explicit by checking the effect of coherence loss on the long-time behaviour of $G(t)$.
Using the results of \cite{rpesk} one can simulate the required decoherence by coupling
an ordinary quantum mechanical system to a random classical noise. It is then shown in
\cite{rsus} that this random noise forces $G(t)$ to decay exponentially
for large $t$, despite having a discrete energy spectrum. This indicates
that the long-time
behaviour of correlators probes the strict quantum coherence of the bounded system.
At the same time, one would like to identify what kind of systematic corrections to the
leading semiclassical approximation are capable of restoring unitarity. A proposal was made
in \cite{rmaldas} in terms of topology-changing fluctuations of the AdS background. Our
purpose here is to investigate these questions and offer an explicit estimate of the
instanton effects suggested in \cite{rmaldas}. We conclude (c.f. \cite{
rsus}) that large topological fluctuations are unlikely to restore unitarity
in full detail, although they represent a step forward. In particular,
certain coarse-grained properties, such as the time averages of the correlators
(\ref{timec}), are reproduced in order of magnitude. Related
work, especially in the $d=3$ case, can be found in refs. \cite{rsolo, raul,
solo}.
These considerations should also shed light on the recent proposal in
\cite{rhawtalk}, where topological diversity is credited with the restoration of
S-matrix unitarity in
black hole formation and evaporation.
\section{Long-time details of thermal quasi-equilibrium}
\noindent
Poincar\'e recurrences occur in general bounded systems.
Classically they follow from
the compactness of available phase space, plus the preservation of
the phase-space volume in time (Liouville's theorem).
Quantum mechanically, they follow from discreteness of the energy spectrum
(characteristic of spatially bounded systems) and unitarity. The correlator
(\ref{timec})
\beq\label{quasi}
G (t) =
\sum_{j,k} C_{jk}\,\;e^{i(E_j -E_k)t}
\;,\qquad {\rm with}\;\;\;\;C_{jk} = \sum_{i} \rho_{ij} A_{ki} A_{jk}
\;,\eeq
defines a quasiperiodic function of time, provided the matrix elements
$C_{ij}$ are sufficiently bounded so that $G(t)$ is well defined.
After initial dissipation
on a non-universal
time scale $\Gamma^{-1}$, where $\Gamma$ measures the approximate width of
the matrix elements $C_{ij}$
in the energy basis,
the correlator will show large ``resurgences" when most of the relevant phases complete
a whole period (c.f. Fig 3). The associated time scale is
$t_H \equiv 1/\langle \omega \rangle$, with
$\langle \omega \rangle = \langle E_i - E_j \rangle$ an average frequency in
(\ref{quasi}). We can estimate
$\langle \omega \rangle$ as $\Gamma /\Delta n_\Gamma$, where $\Delta n_\Gamma$ is the number of energy
levels in the relevant band of width $\Gamma$. This must be proportional to
the density of levels, or the exponential of the entropy, i.e. we have
\beq\label{thes}
t_H \sim \Gamma^{-1} \;e^{S(\beta)}\,.
\eeq
\begin{figure}
\center
\epsfysize=2.0in
\epsffile{res.eps}
\caption{
\small
\sl
Schematic representation of
the very long time behaviour of the normalized time correlator $L(t)$ in bounded systems.
The initial decay with lifetime of order $ \Gamma^{-1}$
is followed by O(1) ``resurgences" after the Heisenberg time $t_H \sim
\Gamma^{-1}\,\exp(S)$
has elapsed. Poincar\'e recurrence times can be defined by demanding the resurgences to
approach unity with a given {\it a priori} accuracy, and scale like
a double exponential of the entropy.
}
\label{figure3}
\end{figure}
Following \cite{rsrednicki} we call this the Heisenberg time scale.
Poincar\'e times
can be defined in terms of quasiperiodic returns of $G (t)$
with a given {\it a priori} accuracy. In a
sense, the Heisenberg time is the smallest possible Poincar\'e time.
A more quantitative, albeit more inclusive criterion
can be used by defining a normalized positive correlator,
$L(t)$, satisfying $L(0)=1$, and its infinite time average,
\beq\label{eledef}
L(t) \equiv \left|{G(t) \over G(0)}\right|^2, \qquad {\overline L} \equiv \lim_{T\rightarrow
\infty} {1\over T} \int_0^T dt \,L(t)\;.
\eeq
The profile of $L(t)$ is sketched in Fig 3. The time average can be estimated by noticing
that the graph of $L(t)$ features positive ``bumps" of height $\Delta L$ and width $\Gamma$,
separated a time $t_H$, so that
\beq\label{fores}
{\overline L} \sim {\Delta L \over \Gamma \,t_H}\;.
\eeq
For the case at hand $\Delta L \sim 1$, $t_H \sim \Gamma^{-1} \,e^S$,
and we obtain
(c.f. \cite{rsuspodos,rsus})
\beq\label{otr}
{\overline L} \sim \exp\left(-S(\beta)\right) \;.
\eeq
Hence, the ``recurrence index" scales as ${\overline L} \sim \exp(-N^2)$ in the
high-temperature phase. Since $N^2 \sim G_{\rm N}^{-1}$ in the AdS/CFT
dictionary, the index scales as a nonperturbative
effect in the semiclassical approximation.
\section{Absence of recurrences in semiclassical black holes}
\noindent
The previous considerations suggest that recurrences should be invisible in
gravity perturbation theory, i.e. in an expansion in powers of $1/N^2$
around a black hole solution,
and this is
indeed what is found.
The reason is that the relevant eigenfrequencies $\omega$ (the so-called
normal modes of the black hole) form a continuous spectrum
to all orders in the $1/N$ expansion.
For a static metric
of the form
\beq\label{bhk}
ds^2 = -g(r)\,dt^2 + {dr^2 \over g(r)} + r^2 \,d\Omega_{d-2}^2 \;,
\eeq
the normal frequency spectrum follows from the diagonalization of a radial Schr\"odinger operator
\beq\label{veun}
\omega^2 = - {d^2 \over dr_*^2} +
V_{\rm eff} (r_*)
\;,
\eeq
with
\beq\label{vedo}
V_{\rm eff} = {d-2 \over 2}\,g(r)\left({g'(r)\over r} +
{d-4 \over 2r^2} \,g(r) \right) + g(r) \left(-{\nabla^2_\Omega
\over r^2} + m^2 \right)
\;
\eeq
for a scalar field of mass $m$ (analogous potentials can be deduced
for higher spin fields). Here we have defined the
Regge--Wheeler or ``tortoise" coordinate
$ dr_* = dr / g(r)
$.
\begin{figure}
\center
\epsfysize=1.5in
\epsffile{contis.eps}
\caption{
\small
\sl
The effective potential determining the semiclassical
normal frequency modes in a large AdS black hole
background (left).
In Regge--Wheeler coordinates the horizon is at $r_* = -\infty$, whereas
the boundary of AdS is at $r_* = \pi R/2$ (only the region exterior to the horizon
appears). There is a universal exponential behaviour
in the near-horizon (Rindler) region. The effective one-dimensional Schr\"odinger problem
represents a semi-infinite
barrier
with a continuous energy spectrum. This contrasts with
the analogous effective potential in vacuum AdS with global coordinates (right). The
domain of $r_*$ is compact and the spectrum of normal modes is discrete with gap of
order $1/R$.}
\label{figure4}
\end{figure}
We have shown in Fig. 4 the form of the resulting effective potentials for large AdS black
holes, compared with the case of the vacuum AdS manifold.
The vacuum AdS manifold, corresponding to the choice
$g(r) = 1+r^2 /R^2$ in (\ref{bhk}),
behaves like a finite cavity, as expected. The distinguishing feature
of the black-hole horizon is a
a non-degenerate
zero, $g(r_0) =0$, which induces the universal scaling
\beq\label{univs}
V_{\rm eff} (r_*) \;\propto \;\exp(4\pi r_* /\beta) \;\;\;\;{\rm as}\;\;\;
r_*
\rightarrow -\infty\;,
\eeq
with $1/\beta
= g'(r_0) /4\pi$ the Hawking temperature and the horizon $r=r_0$ appearing
at $r_* = -\infty$. Notice that the near-horizon behaviour (\ref{univs}) only
depends on the Hawking temperature, i.e. the curvature at the horizon, and
is independent of other long-distance features of the gravitational background.
The
spectrum is discrete in pure AdS, and continuous in the AdS black hole.
Physically, this just reflects the fact that the horizon is an infinite redshift surface, so
that we can store an arbitrary number of modes with finite total energy, provided they are
sufficiently red-shifted by approaching the horizon \cite{rbrick}.
Since the thermal entropy of perturbative gravity excitations
in the vacuum AdS spacetime scales as
$S(\beta)_{\rm AdS} \sim N^0$, we see that the perturbative Heisenberg time of
the AdS spacetime is of $O(1)$ in the large-$N$ limit, leading to ${\overline L}_{\rm AdS}
= O(1)$. On the other hand, we have ${\overline L}_{\rm bh} =0$ in this approximation.
Although these
results are based on the leading perturbative approximation in the
classical black hole background, it is unlikely that higher-order
perturbative effects will render the frequency spectrum discrete, because
this feature appears as an infrared property of the potentials in
Fig. 4 (c.f. \cite{barbonh}).
Another argument for the robustness of ${\overline L}_{\rm bh}$ in perturbation
theory comes from the
Euclidean formalism, obtained by $t=-i\tau$ in (\ref{bhk}), followed by an identification
$\tau \equiv \tau + \beta$. The resulting metric for the vacuum AdS spacetime
has a non-contractible ${\bf S}^1$ given by the $\tau$ compact direction. We call
$Y$ this Euclidean manifold. On the other hand, the black hole spacetime with
$g(r_0)=0$ is simply connected, since the thermal ${\bf S}^1$ shrinks to zero
size at $r=r_0$. The choice $1/\beta = g'(r_0)/4\pi$ ensures smoothness at
$r=r_0$. We call this Euclidean black hole manifold $X$.
\begin{figure}
\center
\epsfysize=2in
\epsffile{cigar.eps}
\caption{
\small
\sl
The Euclidean black hole manifold $X$ is
simply connected, unlike standard thermal manifolds in quantum field
theory.}
\label{figure6}
\end{figure}
The real-time correlation functions in the black hole background, $G(t)_X$,
follow by analytic continuation from
their Euclidean counterparts. Since $X$ is a completely smooth manifold in the
$1/N$ expansion, so is the Euclidean correlator $G(it)_X$ for $t > 0$.
The continuous
spectrum arising in the spectral decomposition of $G(t)_X$ is a consequence of
the contractible topology of $X$, since the foliation by $\tau={\rm constant}$
hypersurfaces is singular at the horizon
(c.f. \cite{rsus}).
\begin{figure}
\center
\epsfysize=2in
\epsffile{foll.eps}
\caption{
\small
\sl
In the vicinity of $r=r_0$, the
manifold $X$ is well-approximated by the product of a flat disk and the
${\bf S}^{d-2}$ at the horizon.
Equal-time hypersurfaces of
Hamiltonian foliations on $X$ have a fixed point at $r=r_0$.
This fact is responsible for {\rm both} the classical contribution to
the entropy, and the continuous spectrum of normal frequencies.}
\label{figure7}
\end{figure}
Since the continuous spectrum finds its origin in the topological properties
of $X$, this particular fact will not be affected by perturbative corrections.
Incidentally, the same peculiar behaviour with respect to the Hamiltonian
conjugate to $\tau$ is responsible for the existence of a formally classical
entropy. Namely, the Euclidean action
\beq
\label{eac}
I(X)= -{1\over 16\pi G_{\rm N}} \int_X \; {\cal R} -{1\over 8\pi G_{\rm N}}
\oint_{\partial X}
\;\left({\cal K} + {\rm C.T.}\right)
\eeq
with appropriately defined counterterms, ${\rm C.T.}$, is not just given by
$\beta M_{\rm ADM}$,
despite the fact that $\partial_\tau$ is a Killing vector on $X$.
Rather, one finds \cite{rgh}
\beq
\label{een}
I(X)= \beta\,M(X) - S_{\rm bh} (X)
\;,
\eeq
with $S_{\rm bh} = A_{\rm H} /4G_{\rm N}$ the Bekenstein--Hawking entropy.
The microscopic interpretation of this entropy must be referred back to
the dual CFT. This point of view suggests that the information encoded in the
geometry of $X$ is fundamentaly
coarse-grained, so that the continuous spectrum of
frequencies would also
be a reflection of this coarse-graining.
\section{Topological diversity and unitarity}
\noindent
Our discussion in the previous section suggests that improving on the
semiclassical prediction ${\overline L}_{\rm bh} =0$ requires some
sort of topology-change process.
The proposal of \cite{rmaldas}
is
precisely that: instead of evaluating the semiclassical correlators on $X$, one
should
sum coherently the contribution of $X$ and $Y$. Normally one neglects the contribution of
$Y$ on a quantitative basis (at high temperatures $R\gg \beta$).
However, here the contribution of $X$ to ${\overline L}$
vanishes and one is forced to consider the first correction. Since $Y$ has a discrete spectrum
in perturbation theory, the net result for ${\overline L}$ should be non-vanishing in
this approximation.
Physically, this superposition of Euclidean saddle points (or master fields, in the
language of the CFT) corresponds to large-scale fluctuations in which the AdS
black hole is converted into a graviton gas at the same temperature and viceversa.
The resulting time profile in the instanton approximation takes the form
(c.f. \cite{rsus})
\beq
\label{tprof}
L(t)_{\rm inst} = L(t)_X + C\,e^{-2\Delta I} \; L(t)_Y
\;
\eeq
where $C= O(N^0)$, $\Delta I = I_Y - I_X$ and $I=-\log \,Z(\beta)$, calculated in the classical
gravity approximation. Since $I_Y \sim -N^0$ and $I_X \sim -N^2$, the exponential
suppression factor is of order $\exp(-2 |I_X|) \sim \exp(-N^2)$.
The resulting structure is shown in Fig 8. The instanton approximation to the
normalized correlator features the normal dissipation with lifetime $\Gamma^{-1} \sim
\beta$ coming from the contribution of $X$. However, the resurgences are controlled by
$L(t)_Y$,
damped by a factor $\exp(-2\Delta I) \sim \exp(-N^2)$, and separated a time
$t_H (Y) \sim N^0$.
\begin{figure}
\center
\epsfysize=2in
\epsffile{top.eps}
\caption{
\small
\sl
Summing over large-scale fluctuations of the thermal ensemble in which
a black hole spontaneously turns into radiation (and viceversa) is represented
in
the
Euclidean formalism
as the coherent sum of thermal saddle points of different topology.
The ``cigar-like" geometry $X$
represents the black-hole master field (in the CFT
language) and the cylindrical topology $Y$ represents the thermal gas of particles.
}
\label{figure8}
\end{figure}
We can also find the time scale $t_c$ for which the large-scale instantons considered
here are quantitatively important
on the graph of $L(t)$. This is shown in Fig. 8 and
yields $t_c \sim \Delta I /\Gamma \sim N^2$.
We see that the instanton approximation does not reproduce the
expected pattern of recurrences, particularly at high temperatures. It
is interesting to find out how much this depends on the temperature above
the phase transition.
For large AdS black holes, positive specific heat sets in for $r_0 > R \left(
{d-3 \over d-1}\right)^{1/2}$, but these black holes do not dominate the
leading large-$N$ thermodynamics until $r_0 = R$, the location of the
Hawking--Page phase transition \cite{rhpage}. In the immediate vicinity of the transition
the statistical weight of the two backgrounds is approximately the same,
since $\Delta I \approx 0$. However, the entropy increases by a factor
of order $N^2$ across the transition, and we would expect a sharp change
of behaviour of $t_H$ as a function of $\beta$, as well as the long-time
structure of $L(t)$.
On the other hand, in the instanton approximation the resurgences are
controlled by $t_H (Y)$, which is of $O(1)$ in the large-$N$ limit on
both sides of the phase transition. The only difference is that $\Delta I$
starts increasing away from zero as the temperature increases. The bumps,
spaced $t_H (Y)$ appart and initially of height $O(1)$,
decrease accordingly in size. When
$(r_0 -R)/R \sim 1/N^2$, we reach $\Delta I \sim N^2$ and the pattern
in Fig. 8.
\begin{figure}
\center
\epsfysize=1.8in
\epsffile{comrec.eps}
\caption{
\small
\sl
The instanton approximation to the correlator $L(t)_{\rm inst}$ features
the expected exponential decay $\exp(-\Gamma \,t)$ induced by the contribution of t
he
$X$-manifold, whereas the resurgences
are entirely due to the interference with the $Y$-manifold, leading to
small bumps of order $\exp(-2\Delta I) \sim \exp(-N^2)$, separated a time
$t_H (Y) \sim N^0$. These bumps are noticeable against the background of the $X$-ma
nifold
after a time $t_c \sim \Delta I /\Gamma$. In the dashed line we plot the very
different expected behaviour in the exact CFT: large $O(1)$ bumps separated
by time intervals of order $\exp(N^2)$. Despite the gross differences between
both profiles, their time averages coincide in order of magnitude. }
\label{figure5}
\end{figure}
In the limit of very high temperatures, there are some limitations to
be expected. The free energy of the $Y$ manifold scales as that
of a $D$-dimensional thermal gas, with $D=10$ or $D=11$ depending on
the particular model of AdS/CFT duality considered, i.e.
$I(Y) \sim - (R/\beta)^{D-1}$. Hence, at temperatures of order
\beq
\label{vlar}
R/\beta \sim N^{2 \over D-d-3}
\eeq
the $Y$ manifold would dominate again over $X$ (c.f. \cite{rexten}).
However, perturbative
instabilities of $Y$ appear before this threshold. For example, in
the standard case of ${\rm AdS}_5 \times {\bf S}^5$ duality, the
$Y$ manifold reaches the Hagedorn instability at temperatures
$R/\beta \sim (g_s N)^{1/4}$, and the Jeans instability at temperatures
$R/\beta \sim N^{1/5}$.
Despite all these caveats, the instanton approximation
yields an interesting value for
the infinite time average, at {\it all} temperatures.
\beq\label{isntl}
{\overline L}_{\rm inst} \approx C\;e^{-2\Delta I}\;
.
\eeq
Namely,
we have
\beq\label{degg}
{\overline L}_{\rm inst} \sim {\Delta L \over \Gamma \,t_H} \sim {e^{-N^2} \over \Gamma \cdot \Gamma^{-1}}
\sim {1 \over \Gamma \cdot \Gamma^{-1} \,e^{N^2}} \sim {\overline L}_{\rm CFT}\;.
\eeq
The first estimate obtains ${\overline L}_{\rm inst} \sim \exp(-N^2)$ from
the Boltzman suppresion of the $Y$ manifold, despite the fact that $t_H (Y)
\sim O(1)$, whereas the second estimate is based on $O(1)$ recurrences with
very large Heisenberg time. It is important to stress that (\ref{degg}) holds
up to factors of order $\exp(-c N^2)$ with $c=O(1)$, because in general
$S_X \neq -2|I_X|$, even
at high temperatures \cite{rsus}. For large AdS black holes,
the instanton calculation (\ref{degg}) gives a larger value of the index
than the estimate based on the quantum mechanical density of states
(\ref{otr}).
Thus, we find that a coarse-grained question, such as
the infinite-time averages of correlators, is better accounted for
by the semiclassical instanton approximation than a ``detailed" question,
such as the concrete time structure of the correlators. This is
another indication of
the fundamentally thermodynamical features of relativistic horizons.
A more complicated set of Euclidean saddle points can be analyzed for the
three-dimensional case of BTZ black holes. The authors of \cite{raul, solo}
conclude
that resummation of an infinite family of $SL(2,{\bf Z})$ saddle points
is unlikely to alter the conclusions presented here on the basis of the leading
instanton approximation. The authors of
Ref. \cite{raul} also point out that only a finite
set of black-hole saddle points remains under the control of the semiclassical
approximation after a time of order $t_c \sim c$, where $c$ is the central
charge of the CFT.
\begin{figure}
\center
\epsfysize=2in
\epsffile{thermorec.eps}
\caption{
\small
\sl
A pictorial representation of the compact phase space at very large energy.
Poincar\'e recurrences consist on the time development $U_t (W)$ of a
region $W$ intersecting itself after a period larger than $t_H$. We have
separated the dominant black-hole like states from the relatively scarce
thermal gas states in the phase space. The instanton approximation is
only sensitive to the recurrences in the small patch of thermal gas states,
because the spectrum
of black-hole states is effectively treated as continuous.
}
\label{figure9}
\end{figure}
\section{Conclusions}
\noindent
The study of very long time features of correlators in black hole backgrounds is
a potentially important approach towards unraveling the mysteries of black hole
evaporation and the associated physics at the spacelike singularity.
We have seen that large scale topology-changing fluctuations proposed in
\cite{rmaldas}
begin to restore some of the fine structure required by unitarity, but fall short at
the quantitative level. Presumably the appropriate instantons occur on
microscopic
scales and
involve stringy dynamics.
While semiclassical black holes do
faithfully reproduce
``coarse grained" inclusive properties of the system such as the entropy
(c.f. \cite{rgh}), additional dynamical
features of the horizon may be necessary to resolve finer details of the
information
loss problem.
Roughly, one needs a systematic set of corrections that could
generate a ``stretched horizon" of Planckian thickness \cite{rstre, rbhc}.
The crudest model of such
stretched horizon is the brick-wall model of 't Hooft \cite{rbrick}. In this phenomenological
description one replaces the horizon by a reflecting boundary condition at Planck distance
$\epsilon \sim \ell_P$
from the horizon. This defines a ``mutilated" $X_\epsilon$ manifold, of cylindrical
topology,
leading to a discrete spectrum of the right spacing in order of magnitude.
Notice that, in line with our previous discussion, the discrete spectrum on
the effective manifold $X_\epsilon$ is tied with the absence of classical
contribution to the entropy, whose leading contribution is obtained at
one loop order: $S(X_\epsilon) \sim A_{\rm H} / \epsilon^{d-2}$.
We have also seen that the characteristic time for large topological fluctuations to be
important is $t_c \sim O(N^2)$ in the semiclassical approximation. In
\cite{rshenk} it
was argued that semiclassical two-point functions probe the black hole singularity
on much shorter characteristic times, thereby justifying the analysis on the single standard
black hole manifold. However, detailed
unitarity is only restored on
time scales of order $t_H \sim \exp(N^2)$. Thus $t_c \ll t_H$ and we conclude that such
semiclassical analysis of the singularity is bound to be incomplete, as it misses
whatever
microphysics is responsible for the detailed
unitarity restoration in the quantum mechanical
time evolution.
\begin{figure}
\center
\epsfysize=2in
\epsffile{compen.eps}
\caption{
\small
\sl
Different topological contributions to the path integral of the scattering
matrix. We have drawn classical black hole and white hole spacetimes (for
CPT invariance), as well
as a spacetime of trivial topology. According to \cite{rhawtalk},
trivial topology
contributions would be enough to restore unitarity of the S-matrix.}
\label{haw}
\end{figure}
It is natural to ask at this point what possible lessons can be drawn
regarding the related problem of the black hole S-matrix. In particular,
Ref. \cite{rhawtalk} uses the
main idea of \cite{rmaldas}, extrapolating it to
the S-matrix problem, and claiming that trivial-topology spacetimes contributing
to the path integral are enough to restore unitarity in the complete quantum
amplitudes (c.f. Fig. 10).
The standard black hole spacetimes in Fig. 10 violate quantum coherence
because either the {\it In} or {\it Out}
asymptotic surfaces do not support the complete
{\it In} or {
\it Out} Hilbert space, whereas the topologically trivial spacetimes do
have complete {\it In} and {\it Out} asymptotic surfaces. Thus, the claim
would be that asymptotic completeness for off-shell histories is
enough to cure the lack of asymptotic completeness of many approximately on-shell
histories. Since asymptotic completeness is the key property guaranteeing
S-matrix unitarity (c.f. \cite{rlag, rpesk}), it is hard to imagine that
a path integral featuring {\it both} topologically trivial {\it and}
topologically nontrivial spacetimes could be {\it exactly} unitary. A
more natural expectation would be that such topological diversity could
at best achieve some approximate restoration of unitarity.
One obstacle in assesing the proposal of \cite{rhawtalk} is the technical
difficulty in estimating the quantitative effect of trivial topologies
in the complete path integral. For initial conditions that would
classically produce a black hole, the spacetimes of trivial topology
are far from any semiclassical saddle point, and therefore their contribution
is difficult to evaluate with the required precision. In fact, the
existence of a classical saddle point
with trivial topology, the manifold $Y$, was the main reward for
choosing thermal equilibrium states in Ref. \cite{rmaldas}, as opposed to
S-matrix boundary conditions.
The situation with the topologically trivial spacetimes in the S-matrix
is perhaps analogous to that of the $Y$ manifold at temperatures above
the Jeans instability, when it becomes unstable and hence ceases to be a
good saddle point of the path integral.
At sufficiently high temperatures, the Jeans length
$\ell_J \sim \sqrt{\beta^{d}/G_{\rm N}}$ falls below the curvature radius of
AdS, and some thermal fluctuations of wavelength $\lambda > \ell_J$
develop an imaginary effective mass. In real time, this corresponds to
exponential behaviour of linearized perturbations, proportional to
$\exp(\pm t/\ell_J)$, thus entering the non-linear regime beyond the
applicability of
perturbation theory. On physical grounds, we expect that the endpoint
of this ``tachyonic instability" is the large AdS black hole at
the corresponding temperature, thus reverting back to the $X$ manifold
as the unique stable saddle point.
\begin{figure}
\center
\epsfysize=2in
\epsffile{penrr.eps}
\caption{
\small
\sl
Topologically trivial spacetime describing unitary scattering according
to the principle of black hole complementarity. The picture is not
fully semiclassical because the stretched horizon has no known
semiclassical description, except for thermodynamical, coarse-grained
observables. The ``white hole" stretched horizon geometry has been
introduced in order to respect CPT invariance. }
\label{hawt}
\end{figure}
Therefore, if we are to draw inspiration from the study of thermal boundary
conditions, we would suggest that topological diversity {\it per se}
is not enough to restore unitarity of the S-matrix, unless some
coarse-graining is imposed on the S-matrix itself (the analog of
calculating of ${\overline L}$ as opposed to $L(t)$.)
On general grounds, the AdS/CFT correspondence
suggests that spacetime topology can
be unambiguously defined only in the context of the semiclassical
approximation. Perhaps the best we can do in representing geometrically
the black-hole
S-matrix is the effective spacetime suggested by the principle of
black hole complementarity (c.f. \cite{rbhc}), which involves a topologically
trivial spacetime with a stretched horizon that behaves as a
boundary equipped with an effective Hilbert space and effective Hamiltonian
yet to be found (c.f. Fig. 11). In this effective description of the
stretched horizon, all the quantum fluctuations would be included, and
thus no further summation over topological sectors would be needed.
Perhaps the following metaphor will turn out to be a useful guide line.
In massless QCD an infrared scale is required to control the large
perturbative infrared fluctuations. The scale is indeed dynamically generated
in asymptotically free systems. Although gravity is {\it a priori} equipped with an
intrinsic
Planck scale,
in the near-horizon region this scale
is red-shifted away and
the system seems to posses no scale (c.f. the universal potential of
(\ref{univs})). A dynamically produced scale would presumably
allow the formation of a streched horizon and the restoration unitarity.
Some universal features of such stretched horizon were recently pointed out
in \cite{thor}.
\section*{Acknowledgements}
E. R. would like to thank the KITP at Santa Barbara
for hospitality during the completion of this work, under grant of the National Science
Foundation No. PHY99-07949.
The work of J.L.F.B. was partially supported by MCyT
and FEDER under grant
BFM2002-03881 and
the European RTN network
HPRN-CT-2002-00325. The work of E.R. is supported in part by the
BSF-American Israeli Bi-National Science Foundation, The Israel Science
Foundation
and the European RTN network MRTN-CT-2004-512194.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,574 |
\section{Introduction}
\hspace*{10mm}
Progress in the rigorous treatment of the multiple scattering of
nucleons from nuclei has led to the need to study the influence of
the full target nuclear density matrix in the scattering observables.
In first order the spectator expansion of multiple scattering
theory requires a convolution of the fully off-shell nucleon-nucleon
(NN) scattering amplitude with the nuclear wave functions of the target.
This opens the possibility to assess the influence of the
target wave functions on elastic proton and neutron scattering
observables.
\hspace*{10mm}
In its most general form, the first order single scattering optical
potential within the framework of the spectator expansion is given by
the triangle graph shown in Fig.~1. Since there is one loop, the
evaluation of the graph requires a three-dimensional integration
involving the fully-off-shell two-nucleon scattering amplitude and a
realistic nuclear density matrix.
Usually, one makes the assumption that the NN amplitude varies slowly as
a function of its arguments compared to the nuclear density matrix.
This corresponds to the argument that the range of the NN force is small
compared to the size of the nucleus and leads to the approximate
nonrelativistic form $t(q)\rho(q)$ for the first-order nucleon-nucleus
optical potential. Full-folding calculations, avoiding this
approximation, have already been performed by several groups based on
either the KMT approach \cite{FFC,FFO} or a g-matrix approach
\cite{hugo1,hugo2} using various models for the off-shell density matrix as
well as different models for the NN amplitudes.
In general, this work indicates that an improved
treatment of the off-shell structure of the optical potential improves
the description of the observables.
\hspace*{10mm}
Our approach to elastic scattering from nuclei is based on the spectator
expansion of multiple scattering theory \cite{Corr,Sicil,TT}. Here the
first order term involves two-body interactions between the projectile
and one of the target nucleons. Due to the many-body nature of the free
propagator for the projectile+target nucleon system, a theoretical
treatment of this many-body propagator as affected by the residual
target nucleus is included. The calculation of the optical potential
presented in this paper relies on two basic inputs. One is the
fully off-shell NN t-matrix, which represents the current understanding
of the nuclear force, and the other is the nuclear wave function of the
target, representing the best understanding of the ground state of the
target nucleus. To account for the modifications of the free propagator
inside the nucleus, the same mean field potentials are used from which
the ground state wave functions are derived. There are {\it no}
adjustable parameters present in this calculation.
\hspace*{10mm}
The motivation for ongoing work on this topic is twofold. First,
elastic and inelastic nucleon-nucleus scattering provide an
important and sensitive test for theoretical corrections at the
first-order level of the optical potential ({\it e.g.} as
given by possibly genuine modifications
of the NN interaction in the nuclear environment
and off-shell effects). Rigorous microscopic
calculations are required for discerning these effects.
Second, a better understanding of the theoretical details of the
optical potential are needed to construct realistic and physically
sound wave functions representing continuum nucleons in the
interior of the nucleus. These wave functions will become vital
for future theoretical needs in high-energy coincidence experiments
(at TJNAF e.g.), inelastic scattering studies, and for understanding the
reactions in heavy ion experiments
involving nuclei far from the drip lines. Here experiments involving
the scattering of exotic nuclei from single nucleon targets should
should benefit from full-folding type calculations in order to test
the predicted density distributions of halo-like nuclei.
\hspace*{10mm}
The structure of the paper is as follows. In Section~II we review the
relevant formalism for the single-scattering optical potential and
introduce the full-folding procedure as used in our calculations.
In Section~III we discuss the model densities employed and describe the
calculations of the full folding optical potentials. Elastic scattering
results for neutron and proton scattering form a variety of nuclei in the
energy regime between 65 and 400 MeV are discussed in Section~IV. Our
conclusions are presented in Section~V.
\section{The full-folding optical potential}
\hspace*{10mm}
The standard approach to elastic scattering of a strongly interacting
projectile from
a target of $A$ particles is the separation of the Lippmann-Schwinger
equation for the transition amplitude
\begin{equation}
T = V + V G_0(E) T \label{eq:2.1}
\end{equation}
into two parts, namely an integral equation for $T$:
\begin{equation}
T = U + U G_0(E) P T , \label{eq:2.2}
\end{equation}
where $U$ is the optical potential operator and defined by a second
integral equation
\begin{equation}
U = V + V G_0(E) Q U. \label{eq:2.3}
\end{equation}
In the above equations the operator $V$ represents the external
interactions between the projectile and the target nucleons.
Therefore the Hamiltonian for the
$(A+1)$ particle system is given by
\begin{equation}
H=H_{0}+V . \label{eq:2.4}
\end{equation}
The free propagator $G_{0}(E)$ for the projectile-target system is
given by
\begin{equation}
G_{0}(E)=(E-H_{0}+i\epsilon)^{-1}. \label{eq:2.5}
\end{equation}
The potential operator $V={\sum_{i=1}^{A}} v_{0i}$ consists of the
two-body potential $v_{0i}$ acting between the projectile and
the $i$th target nucleon. The operators $P$ and $Q$ are projection
operators, $P+Q=1$, and $P$ is defined such that Eq.~(\ref{eq:2.2}) is
solvable. In this case, $P$ is conventionally taken to project onto
the elastic channel, such that $[G_{0},P]=0$. The free
Hamiltonian is given by
\begin{equation}
H_{0}=h_{0}+H_{A} \label{eq:2.6}
\end{equation}
where $h_{0}$ is the kinetic energy operator for the projectile
and $H_{A}$ stands for the target Hamiltonian. Thus the projector $P$
can be defined as
\begin{equation}
P={|\Phi_{A}\rangle\langle\Phi_{A}|\over{\langle\Phi_{A}|\Phi_{A}
\rangle}} \label{eq:2.7}
\end{equation}
where $|\Phi_{A}\rangle$ corresponds to the ground state of the target
and fulfills
\begin{equation}
H_{A}|\Phi_{A}\rangle= E_{A}|\Phi_{A}\rangle \label{eq:2.8}
\end{equation}
With these definitions the transition operator for elastic scattering
can be defined as ${T_{el}=PTP}$, in which case Eqs.~(\ref{eq:2.2})
becomes
\begin{equation}
T_{el}=PUP + PUPG_{0}(E)T_{el}. \label{eq:2.9}
\end{equation}
\hspace*{10mm}
The fundamental idea of the spectator expansion for the optical potential
is an ordering of the scattering process according to the number
of active target nucleons interacting directly with the projectile.
The first order term involves two-body interactions between the
projectile and one of the target nucleons, i.e.
\begin{equation}
U = \sum_{i=1}^{A}\tau_{i} , \label{eq:2.10}
\end{equation}
where the operator $\tau_{i}$ is derived to be
\begin{eqnarray}
\tau_i &=& v_{0i} + v_{0i} G_0(E) Q \tau_i \nonumber \\
&=& v_{0i} + v_{0i}G_0(E) \tau_i - v_{0i} G_0(E) P \tau_i \label{eq:2.11} \\
&=& \hat{\tau_i} - \hat{\tau_i} G_0(E) P \tau_i . \nonumber
\end{eqnarray}
For elastic scattering only $P\tau_{i}P$, or equivalently
$\langle\Phi_A| \tau_i | \Phi_A\rangle$ need to be considered,
\begin{equation}
\langle\Phi_A| \tau_i | \Phi_A\rangle = \langle\Phi_A| \hat{\tau_i}|
\Phi_A\rangle - \langle\Phi_A| \hat{\tau_i}| \Phi_A\rangle \frac {1}
{(E-E_A) - h_0 + i\varepsilon} \langle\Phi_A| \tau_i | \Phi_A\rangle ,
\label{eq:2.12}
\end{equation}
where $\hat{\tau_i}$ is defined as the solution of
\begin{equation}
\hat{\tau_i} = v_{0i} + v_{0i} G_0(E) \hat{\tau_i}. \label{eq:2.13}
\end{equation}
It should be noted that Eqs.~(\ref{eq:2.3}) to (\ref{eq:2.13}) all
follow in a straightforward derivation and
correspond
to the first order Watson scattering expansion \cite{Watson}. If the
projectile$\,-\,$target nucleon interaction is assumed to be the
same for all target nucleons and if isospin effects are neglected
then the KMT scattering integral equation \cite{KMT} can be
directly derived from
the first order Watson scattering expansion.
\hspace*{10mm}
Since Eq.~(\ref{eq:2.12}) is a simple one-body integral equation,
the principal problem is to find a solution of Eq.~(\ref{eq:2.13}),
which has a many-body character due to
$G_0(E)=(E -h_{0} -H_{A} +i\varepsilon)^{-1}$.
If the propagator $G_{0}(E)$
is expanded in the the spirit of the spectator
expansion within a single particle description,
one obtains in first order \cite{med2,med1}
\begin{equation}
G_i(E) = [(E-E^i) -h_0 -h_i -W_i + i\varepsilon]^{-1}, \label{eq:2.14}
\end{equation}
where $h_i$ is
the kinetic energy of the $i$th target particle and
$W_i=\sum_{j\neq i}v_{ij}$.
The quantity $W_i$ represents the force acting between the struck
target nucleon and the residual (A-1) nucleus.
Then the operator
$ \hat{\tau_i}$ of Eq.(\ref{eq:2.13}) can be written as
\begin{eqnarray}
\hat{\tau_i}&=&v_{0i} + v_{0i} G_i(E)\hat{\tau_i} \nonumber \\
&=& t_{0i} + t_{0i} g_i W_i G_i(E) \hat{\tau_i}. \label{eq:2.15}
\end{eqnarray}
Here the operators $t_{0i}$ and $g_i$ are defined as
\begin{equation}
t_{0i} = v_{0i} + v_{0i} g_i t_{0i} \label{eq:2.17}
\end{equation}
and
\begin{equation}
g_i =[ (E-E^i) - h_0 -h_i + i\varepsilon]^{-1}. \label{eq:2.18}
\end{equation}
The operator $t_{0i}$ can be identified with the free NN t-matrix,
and in lowest order the operator $\hat{\tau_i}$ of Eq.~(\ref{eq:2.15})
is given by $\hat{\tau_i} \approx t_{0i}$. From now on we consider
for clarity in presentation only this case.
\hspace*{10mm}
The matrix element
$\langle\Phi_A|\tau_i | \Phi_A\rangle$ given in
Eq.~(\ref{eq:2.12}) represents the full-folding optical potential and
is given explicitly as
\begin{equation}
\langle{\bf k}' | U |{\bf k}\rangle =
\langle{\bf k}'\Phi_{A} | \sum_{\alpha=p,n} {\tau_{\alpha}}
|{\bf k}\Phi_{A}\rangle , \label{eq:2.19}
\end{equation}
where $\alpha$ represents the sum over the target protons and neutrons.
Since $\langle{\bf k}' | U |{\bf k}\rangle$ is the solution of the sum
of the
one-body integral equations represented by Eq.~(\ref{eq:2.12}),
it is sufficient to consider the driving term
\begin{equation}
\langle{\bf k}' |\hat{U}|{\bf k}\rangle =
\langle{\bf k}'\Phi_{A} | \sum_{\alpha=p,n} \hat{\tau}_{\alpha}
|{\bf k}\Phi_{A}\rangle , \label{eq:2.20}
\end{equation}
where $\hat{\tau}_{\alpha}$ is given by Eq.~(\ref{eq:2.13}).
\hspace*{10mm}
Inserting a complete set of momenta for the struck target nucleon
before and after the collision Eq.~({\ref{eq:2.20}}) reads
\begin{eqnarray}
\hat{U}\left({\bf k},{\bf k^{\prime}}\right)=
\sum_{\alpha=p,n} \int d^{3}{\bf p^{\prime}}
d^{3}{\bf p} \left\langle {{\bf k^{\prime}}{\bf p^{\prime}}}
\mid \hat{\tau}_{\alpha} (\epsilon)
\mid {{\bf k p}} \right\rangle \rho_{\alpha} \left({{\bf p^{\prime}
+\frac{{\bf k'}}{A}}}, {\bf p}+\frac{{\bf k}}{A}\right)
\delta^{3} ({\bf k^{\prime}} + {\bf p^{\prime}} -
{\bf k} - {\bf p}). \label{eq:2.21}
\end{eqnarray}
The momenta ${\bf k'}$ and ${\bf k}$ are the final and initial momenta
of the projectile in the frame of zero total nucleon-nucleus
momentum. The structure of Eq.~({\ref{eq:2.21}}) is represented
graphically by Fig~1, which also illustrates the momenta ${\bf p'}$ and
${\bf p}$. The proton and neutron density matrices are given
by $\rho_{\alpha}$.
By evaluating the $\delta$-function and introducing the
variables
${\bf q}={\bf k'}-{\bf k}$,
${\bf K}={1\over{2}}({{\bf k} + {\bf k^{\prime}}})$ and
${\bf \hat{p}}={1\over{2}}({{{\bf p^{\prime}}}+{{\bf p}}})$ we obtain
\begin{eqnarray}
\hat{U}({\bf q},{\bf K})=\sum_{\alpha=p,n}
\int d^{3}{\bf\hat{p}}\;\left\langle {\bf k'},
{\bf\hat{p}}-{1\over{2}}{{\bf q}}\left|\;\hat{\tau}_{\alpha}
(\epsilon)\;\right|{{\bf k}},{\bf\hat{p}}+{1\over{2}}{{\bf q}}\right\rangle\;
\nonumber \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\rho_{\alpha}\left({{\bf \hat{p}}}-{{A-1}\over{A}}{{{\bf q\over{2}}}}+
{{\bf K}\over{A}},
{{\bf \hat{p}}}+{{A-1}\over{A}}{{{{\bf q}}\over{2}}+{{{{\bf K}}}\over{A}}
}\right) . \label{eq:2.22}
\end{eqnarray}
A change of the integration variable from $\bf\hat{p}$ to ${\bf P}={\bf\hat p}
+ {{{{\bf K}}}\over{A}}$, accounting for the recoil of the nucleus, gives
\cite{pttw}
\begin{eqnarray}
\hat{U}({\bf q},{\bf K})=\sum_{\alpha=p,n}
\int d^{3}{\bf P}\;\left\langle {\bf k'},{\bf P}-{{\bf q}\over{2}}-
{{\bf K}\over{A}}\left|\;\hat{\tau}_{\alpha}
(\epsilon)\;\right|{\bf k},{\bf P}+{{\bf q}\over{2}}-
{{\bf K}\over{A}}\right\rangle\; \nonumber \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\rho_{\alpha}\left({\bf P}-{{A-1}\over{A}}{{{\bf q}\over{2}}},
{\bf P}+{{A-1}\over{A}}{{\bf q}\over{2}}\right). \label{eq:2.23}
\end{eqnarray}
The NN amplitude ${\hat \tau}_{\alpha}$
in Eq.~({\ref{eq:2.23}}) is evaluated in the zero momentum
frame of the nucleon-nucleus system. The relationship to the corresponding
matrix element evaluated in the zero momentum frame of the two nucleons
is given by
\begin{eqnarray}
{\left\langle {\bf {k'}},{\bf P}-{{\bf q}\over{2}}-
{{\bf K}\over{A}}\left|\;\hat{\tau}_{\alpha}
(\epsilon)\;\right|{\bf {k}},{\bf P}+{{\bf q}\over{2}}-
{{\bf K}\over{A}}\right\rangle}_{NA} =
\eta ({\bf P},{\bf q},{\bf K})
{\left\langle {\cal K}',-{\cal K}'|\;\hat{\tau}_{\alpha}(\epsilon)\;
|{\cal K},-{\cal K}\right\rangle}_{NN},
\end{eqnarray}
where ${\bf\cal{K}'}=\frac {1}{2}({\bf k'}-({\bf P}-\frac{{\bf q}}{2}-
\frac{{\bf K}}{A}))$ and ${\bf\cal{K}}=\frac {1}{2}({\bf k}-
({\bf P}+{{\bf q}\over{2}}- {{\bf K}\over{A}}))$
are the nonrelativistic final and initial nuclear momenta in the
zero momentum frame of the NN system.
The factor $\eta({\bf P},{\bf q},{\bf K})$ is the M\o ller factor for the
frame transformation \cite{joachain} and is given by
\begin{equation}
\eta({\bf P},{\bf q},{\bf K})=\left[{{E_N({\cal K}')E_N(-{\cal K}')
E_N({\cal K})E_N(-{\cal K})}\over{E_N(k')E_N({\bf P}-{{\bf q}\over{2}}-
{{\bf K}\over{A}})E_N(k)E_N({\bf P}+{{\bf q}\over{2}}-
{{\bf K}\over{A}})}}\right], \label{eq:2.25}
\end{equation}
where $E_N({\bf k})$
is the relativistic kinetic energy of a nucleon of
momentum $k$. This factor imposes the Lorentz invariance of the flux.
With this frame transformation taken into account, the full-folding
optical potential of Eq.~({\ref{eq:2.23}}) can be written as
\begin{eqnarray}
\hat{U}({\bf q},{\bf K})=\sum_{\alpha=p,n}
\int d^{3}{\bf P}\;
\eta({\bf P},{\bf q},{\bf K})\;
\hat{\tau}_{\alpha}\left({\bf q},{1\over{2}}({{A+1}\over{A}}{\bf K}-{\bf P});
\epsilon\right)\; \nonumber \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\rho_{\alpha}\left({\bf P}-{{A-1}\over{A}}{{{\bf q}\over{2}}},
{\bf P}+{{A-1}\over{A}}{{\bf q}\over{2}}\right). \label{eq:2.26}
\end{eqnarray}
Here the arguments of ${\hat \tau}_{\alpha}$ are ${\bf q}= {\bf k}'
-{\bf k} = {\bf\cal{K}'}-{\bf\cal{K}}$ and $\frac {1}{2}({\bf\cal{K}'}+
{\bf\cal{K}})= \frac {1}{2}(\frac {A+1}{A} {\bf K} -{\bf P})$.
\hspace*{10mm}
The two-nucleon amplitude $\hat{\tau}_{\alpha}$ is calculated from the
free NN t-matrix according to Eqs.~(\ref{eq:2.15}) and (\ref{eq:2.17})
at an appropriate energy $\epsilon$. In principle, this energy should be the
beam energy minus the kinetic energy of the center of mass of the interacting
pair less the binding energy of the struck particle.
Following this argument, $\epsilon$ should be coupled to the integration
variable ${\bf P}$. The full-folding calculations of
Refs.~\cite{hugo1,hugo2} are carried out along this vain. For our
calculations we take a different approach, namely we fix $\epsilon$
at the two-body center-of-mass (c.m.) energy corresponding to free
NN scattering at the beam energy so that the same laboratory energy
applies to the two-body and nuclear scattering.
This approach has been applied in earlier work \cite{FFO} and is also
used in the work of the Surrey Group \cite{FFC}.
\section{Models for the Off-Shell Density}
\hspace*{10mm}
The evaluation of the full-folding optical potential as given in
Eq.~({\ref{eq:2.26}}) requires a nuclear density matrix, which
in a single particle picture is given as
\begin{equation}
\rho_\alpha ({\bf \tilde p'},{\bf \tilde p}) = \sum_i
\Psi_{\alpha,i}^{\dagger} ({\bf \tilde p'}) \Psi_{\alpha,i}
({\bf \tilde p}) \label{eq:3.1}
\end{equation}
Here $\Psi_{\alpha,i}({\bf \tilde p})$ are the wave functions
describing the single particle nuclear ground state. The index $\alpha$
stands for protons and neutrons, respectively,
and the total nuclear ground state
is given by the sum of the two.
In order to achieve consistency with our
formulation of incorporating effects of the `nuclear medium'
on the scattering process we choose as model density matrices the
ones from which the nuclear mean fields $W_i$ are derived (see
e.g. Eq.~({\ref{eq:2.15}})). The models used are a
non-relativistic reduction of a Dirac-Hartree calculation \cite{DH} and a
non-relativistic
Hartree-Fock-Bogolyubov (HFB) structure calculation \cite {HFB,Gogny}.
The Dirac-Hartree calculation is a spherical solution of the
one-body Dirac equation assuming a scalar potential
and the time component of a
vector potential field. The nonrelativistic HFB microscopic nuclear
structure calculation uses the parameterized effective finite-range,
density dependent Gogny D1S effective
NN interaction. The parameter of the Gogny
D1S interaction are fitted to a certain set of stable nuclei. For this
case an axial harmonic oscillator basis is used.
\hspace*{10mm}
The details of the Dirac-Hartree (DH) calculation
leading to the density matrices employed in our calculations are
given below.
The wave function $\Psi_i({\bf r})$ is a solution of the one-body
Dirac-Hartree equation and given by \cite{TIMORA}
\begin {equation}
{\Psi}_{\beta}({\bf r}) \equiv {\Psi}_{n,\nu,m,t_z}({\bf r})
= \left({\begin{array}{cc}
i{[\frac{G_{t_z,n,\nu}(r)}{{r}}}]\;{\bf \phi}_{\nu m} \\
-{[\frac{F_{t_z,n,\nu}(r)}{{r}}}]\;{\bf \phi}_{-\nu m}\;
\end{array}}
\right) \zeta_{t_z} \label{eq:3.3}
\end {equation}
Here $t_z$ stands for the z component of the isospin and $n$
for the principal quantum number. The phase convention is
taken from Ref.~\cite{TIMORA}. During this derivation we prefer
to omit the index $\alpha$.
The spherical harmonics are determined by ${\bf\phi}_{\nu m}$ which is
defined as
\begin{equation}
{\bf \phi}_{\nu m}=\sum_{m_l,m_s}
\langle lm_l{1\over{2}}m_s|
l{1\over{2}}jm\rangle Y_{l}^{m_l}(\Omega )\chi_{m_s}, \label{eq:3.3b}
\end{equation}
where $Y_{l}^{m_l}(\Omega)$ is a spherical harmonic and $\chi_{m_s}$
a Pauli spinor.
The quantum number $\nu$ uniquely defines $j$ and $l$ as
\begin{equation}
j = |\nu|-\frac{1}{2}\;\;\;\;, l =\left\{\begin{array}{cc}
\nu,\;\; \nu>0 \\
-(\nu+1),\;\;\nu<0\end{array}\right\}. \label{3.3b}
\end{equation}
We used the code {\it Timora} \cite{TIMORA}
and the parameter sets given therein to
generate the functions $G$ and $F$ given in Eq.({\ref{eq:3.3}) for
the nuclei studied in this paper.
Under the assumption of orthogonal single particle states the
density matrix is given in coordinate space by
\begin{eqnarray}
& &\rho ({\bf r}',{\bf r}) = \sum_{n\nu m} \Psi^{\dagger}_{n\nu m t_z}
({\bf r}') \Psi_{n\nu m t_z} ({\bf r}) \nonumber \\
&=& \sum_{n\nu}\left[ \frac {G_{t_z,n,\nu}(r')}{r'}
\frac {G_{t_z,n,\nu}(r)}{r} \sum_m \phi_{\nu m}(r') \phi_{\nu m}(r)
+ \frac {F_{t_z,n,\nu}(r')}{r'}\frac {F_{t_z,n,\nu}(r)}{r}
\sum_m \phi_{-\nu m}(r') \phi_{-\nu m}(r) \right]. \label{eq:3.04}
\end{eqnarray}
Here we should point out that in order to obtain a
density matrix which we can apply in our
formulation of the optical potential, we have a {\bf 1}-operator
between the Dirac wave functions $\Psi_{n\nu m t_z}$, and then
treat $\rho ({\bf r}',{\bf r})$ as nonrelativistic density matrix.
The orthogonality of the spin states leads to
$\delta_{m_s,m_s'}$ and thus to $m_l=m_{l'}$.
Taking advantage of the symmetry properties of the Clebsch-Gordon
coefficients leads to
\begin{equation}
\rho ({\bf r}',{\bf r}) = \sum_{n\nu} \left[ \frac
{G_{t_z,n,\nu}(r')}{r'} \frac {G_{t_z,n,\nu}(r)}{r}
+ \frac {F_{t_z,n,\nu}(r')}{r'}\frac {F_{t_z,n,\nu}(r)}{r}
\right] \frac {2j+1}{2l+1} \sum_{m_l} Y_l^{* m_l}(r')
Y_l^{m_l}(r). \label{eq:3.05}
\end{equation}
\vspace{2mm}
\hspace*{10mm}
The calculation of the full-folding optical potential
${\hat U}({\bf q},{\bf K})$ requires the nuclear density
matrix in momentum space. Thus we need to
double Fourier transform $\rho ({\bf r}',{\bf r})$
to obtain the density $\rho({\bf \tilde p'},{\bf \tilde p})$
in the rest frame of the nucleus. This frame is characterized
by the momenta $\bf\tilde{p}$ and $\bf\tilde{p}'$ and the density
matrix is obtained by
\begin{equation}
\rho_{\alpha}({\bf \tilde{p}'},{\bf \tilde{p}}) =
\frac{1}{8\pi^3} \int d^3{\bf r}^{\prime}
e^{-i {\bf r}^{\prime}\cdot{\bf \tilde{p}}
^{\prime}} \int d^3{\bf r} e^{i {\bf r}\cdot{\bf\tilde{p}}}
\rho_\alpha ({\bf r}',{\bf r}), \label{eq:3.4}
\end{equation}
where we again indicate with the index $\alpha$ that we have to
obtain the density matrix for protons as well as neutrons.
Using the standard expansion of a plane wave, the angular integration
in Eq.~(\ref{eq:3.4}) can be easily carried out, and we obtain
for the density matrix
\begin{eqnarray}
\rho_{\alpha}({{\bf \tilde{p}'}},{{\bf \tilde{p}}}) &=& \frac{1}{2\pi^2}
\sum_J (2J+1) \sum_l
P_l(\cos \theta_{\tilde{p},\tilde{p}'}) \nonumber \\
& [ & \int dr' r' \; j_l(\tilde{p}'r')
F_{\alpha,t_z,n,\nu}(r') \;\int dr\; r\; j_l(\tilde{p}r)
F_{\alpha,t_z,n,\nu}(r) + \nonumber \\
& & \int dr' r' \; j_l(\tilde{p}'r') G_{\alpha,t_z,n,\nu}(r')
\int dr\; r\; j_l(\tilde{p}r)
G_{\alpha,t_z,n,\nu}(r) \;\; ]. {\label{eq:3.7}}
\end {eqnarray}
\hspace*{10mm}
The density matrix $\rho_{\alpha}({\bf \tilde{p}'},{\bf \tilde{p}})$
given in Eqs.~(\ref{eq:3.4}) or (\ref{eq:3.7})
is defined in the rest frame of the nucleus. In order to apply
$\rho_{\alpha}({\bf \tilde{p}'},{\bf \tilde{p}})$ in our calculation
of the full-folding
optical potential for nucleon-nucleus scattering, we have to evaluate
the function at the corresponding momenta in the nucleon-nucleus frame.
This is facilitated by variable transformations ${\bf p}=
{\bf \tilde{p}-{k\over{A}}}$ and ${\bf {p}'}=
{\bf \tilde{p}'-{k'\over{A}}}$, which takes into account recoil. As
an aside, not including recoil would mean the transformation
${\bf p'}= {\bf \tilde{p}'-{k\over{A}}}$.
\hspace*{10mm}
For the calculation of the density matrices derived from a
non-relativistic Hartree-Fock Bogolyubov (HFB) calculation based on the
Gogny-D1S NN interaction we employ essentially the same procedure as
described above. The wave functions are created in r space by a code
provided by Berger \cite{HFB} and are represented in a axially
symmetric harmonic oscillator basis. A double Fourier transform is then
performed using the oscillator
basis and summing over all harmonic oscillator quantum numbers.
This choice of basis takes
advantage of the fact that the Fourier transform of a harmonic
oscillator is again a harmonic oscillator. The density matrix is
given by
\begin{equation}
\rho_{\alpha} ({\bf \tilde p'},{\bf \tilde p}) = \sum_{i,i'}
\rho^{i,i'} \varphi_{i'}^{\dagger} ({\bf \tilde p}') \varphi_i
({\bf \tilde p}), \label{eq:3.8}
\end{equation}
where the indices $i,i'$ count the harmonic oscillator basis states and
$\rho^{i,i'}$ is the density matrix in the oscillator basis.
Again, the index $\alpha$ distinguishes between protons and neutrons.
The basis states are explicitly given by
\begin{equation}
\varphi_i({\bf \tilde p})= \sum_{m} A_{i,m_l}(\beta,\gamma)
\;e^{{-\beta p_z}^2} \;H_i ({\beta,{ \tilde p_z}})
\;\;e^{{-\gamma p_r}^2} \;L_i^{|m|} ({{\gamma, \tilde p_r}})
e^{im\theta}.
\label{eq:3.9}
\end{equation}
Here ${\tilde p_z}$ is the projection of the momentum along the z-axis
and $p_r$ the radial momentum, $\beta,\gamma$ are
harmonic oscillator constants.
$H_{i}(\beta,{\tilde p_z})$ are the Hermite polynomials and
$L_i^{|m|}(\gamma,{\tilde p_r})$ the Laguerre polynomials.
The size of the harmonic oscillator basis used depends on the size of the
nucleus, {\it e.g.} the size of the basis for $^{16}$O is 12 shells
whereas for $^{90}$Zr it is 16 shells.
It should be noted that the indices $i$ and $i'$
are not independent. The size of the basis sets needed makes the
calculation of $\rho_{\alpha}({\bf \tilde p'},{\bf \tilde p})$ quite lengthy,
especially for heavier nuclei.
\hspace*{10mm}
In order to calculate the optical potential $\hat U ({\bf q},{\bf K})$
as given in Eq.~(\ref{eq:2.26}), we need the density matrix as function
of the momentum transfer ${\bf q}$ and
${\bf P = \hat p + \frac{K}{A}}$ as indicated in Eq.~(\ref{eq:2.23}).
In these variables
the density matrix is related to the density
profile $\rho_\alpha(q)$ of the nucleus by
\begin{equation}
\rho_\alpha(q)=\int d^3{\bf P} \rho_{\alpha}\left(
{\bf P}-{{A-1}\over{A}}{{{\bf q}\over{2}}},
{\bf P}+{{A-1}\over{A}}{{\bf q}\over{2}}\right). \label{eq:3.10}
\end{equation}
The normalization is chosen such that $\rho_\alpha(q=0)=Z$ or $N$, the number
of protons or neutrons, respectively.
\hspace*{10mm}
In practice we used the relation given in Eq.~({\ref{eq:3.10}})
for testing our numerical integration schemes with the simple
harmonic oscillator density given in Ref. {\cite{FFO}}. In order
to determine how well the two model density matrices
presented here describe the experimentally determined proton
distribution, we calculate the proton density profiles $\rho_p(q)$ for
both the DH and the HFB models for each nucleus we consider.
In the following we want to discuss two cases, namely $^{16}$O
and $^{90}$Zr. In Fig.~2 we compare the density profiles
calculated from the DH and HFB models to the experimental
proton distribution{\cite{edata}}. Overall the DH profile follows
the experimental distribution closer than the HFB profile.
The HFB profile is shifted to larger momenta indicating that the
HFB model slightly underpredicts the radius of the proton
distribution of $^{16}$O. This feature will be visible in the
proton-scattering observables for $^{16}$O calculated with
the HFB model. In the close-up of the minimum of the density
profile it can be seen that both model densities slightly
deviate from the experimental profile.
In Fig.~3 we carry out the same comparison for a heavier nucleus,
$^{90}$Zr. Here both model densities follow the experimental proton
distribution {\cite{edata}} very closely. The close-up of the minimum reveals
that the HFB profile deviates only slightly from the experimental profile.
This is a general trend, the heavier nuclei are better described by the
model profiles. In fact, the proton distribution of
$^{16}$O represents the worst case of disagreement of the model profiles
with the experimental profiles. This is understandable since the HFB
model is known to provide a better representation of the larger nuclei.
\section{Results and Discussion}
\subsection{Details of the Calculation}
\hspace*{10mm}
In this paper the study of the elastic scattering of
neutrons and protons from spin-zero target nuclei at energies that
range from 65 to 400 MeV (incident projectile energy) is strictly
first order in the spectator expansion. Here the connection
to the propagator $G_0(E)$ due to the coupling of the initially
struck target nucleon to the residual target is considered to be
first-order. The full-folding optical potential is calculated as
outlined in Section II, specifically as given in Eq.~({\ref{eq:2.26}}),
using the model densities described in Section III.
The calculations for scattering at energies smaller than 200~MeV
take into account the coupling of the struck target nucleon to the
residual nucleus via the mean field potential $W_i$,
which is chosen to be consistent with the model density employed.
Details of this procedure are given in Refs.~\cite{med2,med1}.
Calculations using the Dirac-Hartree densities (and the corresponding
potential $W_i$) are labeled DH, while
those using the Hartree-Fock-Bogolyubov
densities (and corresponding potentials $W_i$) are labeled HFB.
\hspace*{10mm}
The convolution of the fully off-shell density matrix $\rho_\alpha$
with the fully off-shell NN t-matrix, and the M\o ller frame
transformation factor $\eta({\bf P},{\bf q},{\bf K})$
as given in Eq.~({\ref{eq:2.25}})
is carried out in three dimensions without partial wave
decomposition and the integration is performed using Monte Carlo
integration techniques.
Our algorithm uses Quasi-Random numbers {\cite{sobol}},
together with
importance sampling, which according to our tests has the advantage
of needing significantly
fewer integration points than algorithms based on
conventional `random number' generators or Gauss-Legendre integration
to obtain the same accuracy. Quasi random numbers provide a `uniform'
random distribution over the integration space.
\hspace*{10mm}
Aside from the density matrices, the fully off-shell NN t-matrix is
another crucial ingredient in the calculation of ${\hat U}({\bf q},{\bf K})$.
The calculations presented use the free
NN interaction based upon the full Bonn potential {\cite{Bonn}}.
This interaction includes the effects of
relativistic kinematics, retarded meson propagators as given by
time-ordered perturbation theory, and iterative and crossed meson-exchanges
with $NN$, $N\Delta$, and $\Delta\Delta$ intermediate states.
For the calculations involving projectile energies greater then 300 MeV we
employ an extension of the Bonn model above pion-production threshold
\cite{D52}.
In this model pion production is described through the decay of the
delta isobar with a width obtained consistently from the imaginary part
of the one-pion loop diagram for the delta self-energy.
It is also to be understood that we perform all spin summations in obtaining
${\hat U}({\bf q},{\bf K})$. This reduces the required NN t-matrix elements
to a spin independent component (corresponding to the Wolfenstein amplitude
A) and a spin-orbit component (corresponding to the Wolfenstein amplitude
C). Since we are assuming that we have spin saturated nuclei,
the components of the NN t-matrix depending on the
spin of the struck nucleon vanish.
For the proton nucleus scattering calculations the Coulomb interaction
between
the projectile and the target is included using the exact formulation
described in Ref. {\cite{coul}}.
\hspace*{10mm}
A common approximation to the full-folding expression of Eq.~({\ref{eq:2.26}}),
which still preserves the non-local character of the NN t-matrix, is obtained
as follows. If
one observes that the nuclear size is significantly larger than the range
of the NN interaction, the amplitude $\hat{\tau}_\alpha$ is expected to be the
slower varying quantity in the integral of Eq.~({\ref{eq:2.26}}).
This argues for the method of optimum factorization \cite{pttw,ernst}
which proceeds via an
expansion of $\hat{\tau}_\alpha$ (including the factor $\eta({\bf q,K,P})$)
in ${\bf P}$ about a fixed value ${\bf P_0}$.
The reference momentum ${\bf P_0}$ is determined by requiring that the
contribution of the first derivative term be minimized. In the elastic
scattering case this contribution can be made to vanish if ${\bf P_0}$
is chosen to be zero. For further details we refer to Ref {\cite{pttw}}.
After the integration over the density matrix to produce the
diagonal density profile $\rho_\alpha(q)$ (Eq.~({\ref{eq:3.10}}) the
`optimum factorized' or `off-shell $t\rho$' form of the optical potential
is given by
\begin{equation}
{\hat U}_{fac}({\bf q},{\bf K})=\sum_{\alpha=p,n}
\;\eta({\bf q},{\bf K})\;\hat{\tau}_\alpha
\left({\bf q},{{A+1}\over{2A}}{\bf K}
,\epsilon\right)\; \rho_{\alpha}\left(q \right) \label{eq:4.1}
\end{equation}
Here the non-local character of the optical potential is solely
determined by the off-shell NN t-matrix.
For harmonic oscillator model densities it has been shown
for light nuclei that the optimum factorized form
represents the non-local character of ${\hat U}({\bf q},{\bf K})$
qualitatively {\cite{FFC,FFO}} when applied within the KMT formalism
to first order at intermediate energies.
When comparing elastic scattering observables obtained from full-folding
optical potentials to those obtained
from `off-shell $t\rho$' optical potentials, the scope is two-fold.
First, we employ here realistic models of the nuclear density for light as
well as heavy nuclei. Second we extend this comparison toward energies
below 100 MeV where it could be expected that the nucleon-nucleus scattering
calculation samples the optical potential further off-shell and thus
the optimum factorized form may not be as good an approximation.
\subsection{Elastic Scattering Results}
\hspace*{10mm}
Elastic scattering calculations from several spherical nuclei
are carried out at a variety of energies between 65 and 400 MeV
to allow for comparisons between
results obtained from the full-folding optical potentials with
those arising from the factorized off-shell `$t\rho$' approximation.
\hspace*{10mm}
The scattering observables for elastic proton scattering from
$^{16}$O are displayed in Fig.~4. The solid line represents the
calculation with the full-folding optical potential based on the DH
density and the Bonn t-matrix defined above pion-production threshold,
while the dashed line represents the optimum factorized form
as defined in Eq.~({\ref{eq:4.1}}). Both calculations are
based on the free NN t-matrix.
Since the two calculations give very similar results,
it can be concluded that the bulk of the non-locality of the optical
potential, which affects the elastic scattering observables,
must come from the off-shell structure of the NN t-matrix.
The off-shell structure of the nuclear density matrix plays an
insignificant role
for elastic scattering observables at these high energies. A similar
conclusion was already drawn in Ref. \cite{FFO} and is here confirmed
using realistic densities. In order to illustrate the effect of
the different density profiles for the DH and the HFB models
(as shown in Fig.~2) on the elastic observables, we display
two calculations based on the factorized optical potential for the DH
(solid line) and the HFB model (dashed line) in Fig.~5. As already discussed
in Section III, especially in the case of $^{16}$O, the HFB density
profile is shifted to larger momenta compared to the DH profile.
This translates directly into a slight shift of the first minimum of
the differential cross-section to larger angles and a slightly smaller
angular spacing of the diffraction minima. We carried out similar comparisons
of full-folding and optimum factorized optical potentials
for heavier nuclei, but there the disagreement between the density
profiles of the two models is much smaller then for $^{16}$O
and consequently the prediction of the observables are very similar.
\hspace*{10mm}
Another effect worthwhile to study in this context is the influence of
the M\o ller factor \cite{joachain}, which takes into account the
transformation of the NN t-matrix evaluated in the NN c.m. frame to the
zero momentum frame of the nucleon-nucleus system. This frame transformation
can be viewed as a relativistic effect and its importance should increase
with higher scattering energies. For these reasons we want to consider
it's effect on the elastic scattering observables for proton scattering
from $^{16}$O at 500 MeV (Fig.~6). The M\o ller factor
$\eta({\bf P},{\bf q},{\bf K})$ as given in Eq.~({\ref{eq:2.25}})
is a function of three vector momenta and is part of the full-folding
integral of Eq.~(\ref{eq:2.26}). The solid line in Fig.~6 represents
the calculation of $\hat{U}({\bf q},{\bf K})$ as given in
Eq.~(\ref{eq:2.26}). In the spirit of the optimum factorized
approximation $\eta({\bf P},{\bf q},{\bf K})$ can be expanded around
a fixed value ${\bf P_0}$ (here ${\bf P_0}$=0), thus becoming
independent of the integration variable ${\bf P}$.
This expansion corresponds to considering $\eta({\bf q},{\bf K})$
at a fixed angle between ${\bf q}$ and ${\bf K}$, specifically here
$\Theta=90^o$. The dashed line therefore in Fig.~6 corresponds
to evaluating the optical potential according to
\begin{eqnarray}
\hat{U}({\bf q},{\bf K})=\sum_{\alpha=p,n}
\eta({\bf q},{\bf K})_{\Theta=90^o}\;
\int d^{3}{\bf P}\;
\hat{\tau}_{\alpha}\left({\bf q},{1\over{2}}({{A+1}\over{A}}{\bf K}-{\bf P});\epsilon\right)\; \nonumber \\
\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
\rho_{\alpha}\left({\bf P}-{{A-1}\over{A}}{{{\bf q}\over{2}}},
{\bf P}+{{A-1}\over{A}}{{\bf q}\over{2}}\right) \label{eq:4.2}
\end{eqnarray}
The dashed and solid lines in Fig.~6 are almost indistinguishable.
This infers the conclusion that $\eta({\bf q},{\bf K})_{\Theta =90^o}$
is a very good representation of the exact expression given in
Eq.~(\ref{eq:2.25}). In order to illustrate the total effect due to the
inclusion of the M\o ller factor, the dotted line in Fig.~6 represents
a calculation with $\eta({\bf q},{\bf K})$ set to one in Eq.~(\ref{eq:4.2}).
\hspace*{10mm}
At energies below 200 MeV, calculations of elastic observables not only
incorporate the effects of the nuclear structure models within the
full-folding procedure but also via the mean field force ( given by the
structure model) which couples the struck target nucleon to the residual
nucleus. Thus it is hoped that the influence of different structure
models on the elastic observables is observable.
In Fig.~7 we display the elastic observables for proton scattering from
$^{40}$Ca at 100 MeV laboratory energy employing the DH model for the
density as well as the mean field force $W_i$.
In Fig.~8 the corresponding calculation is done
using the HFB model. In both figures the solid line represents the
full-folding calculation, and the dashed line the factorized
off-shell `$t\rho$' approximation. All calculations
contain the modification due to the mean field $W_i$.
The off-shell structure of the nuclear density matrix in the
full-folding procedure has at this lower energy a slightly larger
effect on the spin observables then at higher energies. In addition,
the angular distribution of the differential cross section diffracts
at slightly larger angles
in the full-folding calculations compared to those based on the
factorized form. This trend is also be observed for the heavier nuclei
$^{90}$Zr and $^{208}$Pb (Figs.~9 and 10).
\hspace*{10mm}
The elastic observables for proton scattering from $^{90}$Zr at
80 MeV are displayed in Fig.~9. Here the difference between the two
model densities employed is almost negligible for $d\sigma\over{d\Omega }$
and $A_y$. Only for the spin rotation function the difference given by
using two different structure models is at higher angles as large
as the effect of using the factorized approximation. This result,
namely that the observables predicted by the two different models
under consideration are so similar, is not surprising in
that both models predict an almost identical density profile
(Fig.~3). The effect of the off-shell structure of the nuclear
density matrix is relatively small as the comparison between the
full-folding (solid) line and corresponding factorized (dashed) calculation
shows. In the case of proton scattering from $^{208}$Pb at 65 MeV
(Fig.~10) a comparison between a full-folding calculation and the
factorized approximation reveals the same trends as the observables
for the lighter nuclei. At large angles the full-folding calculation
falls below the one given by the factorized form, and in this case also
below the data. The inclusion of the off-shell structure of the nuclear
density matrix makes the nucleus appear slightly larger, which
becomes apparent in the shifted diffraction pattern of
$d\sigma\over{d\Omega }$.
\hspace*{10mm}
It is difficult to extract properties of nonlocal potentials
from elastic scattering observables. Nonlocal effects are presumably
more important in inelastic processes which depend on the
nucleon-nucleus interaction such as quasielastic electron scattering
reactions.
In order to gain more insight
into the difference between a full-folding optical potential
and the factorized off-shell `$t\rho$' approximation to this
potential, we plot in Figs. 11 ($^{40}$Ca) and 14 ($^{208}$Pb)
the real and imaginary parts of the on-shell value of
$\hat{U}({\bf q},{\bf K})$ as a function of the orbital angular
momentum L.
We separate the cases $J=L+{1\over{2}}$ and $J=L-{1\over{2}}$ to
isolate the effect of the spin-orbit force. As is seen in both figures,
the full-folding (solid lines) and the factorized (dashed lines)
on-shell values of the imaginary parts of the optical potential are
quite similar. In both cases the real part of the on-shell
values of the optical potentials exhibit an increasing suppression
for smaller L as the nonlocal effects of the density matrix as well
as the NN t-matrix are treated more adequately in the full-folding
procedure. It should be emphasized again that Figs. 11 and 14 only
show the value of $\hat{U}({\bf q},{\bf K})$ fulfilling the
on-shell condition ${\bf q}\cdot{\bf K}=0$ and ${\bf q}^2 +4{\bf K}^2
= 4 {\bf k_0}^2$ with ${\bf k_0}$ being the on-shell relative momentum for
proton-nucleus (NA)
scattering and do not display any off-shell behavior inherent in the
potentials. After iteration to obtain the Watson optical
potential (Eq.~\ref{eq:2.11}) and then in the integral equation
(Eq.~\ref{eq:2.9}), the on-shell elements of the NA t-matrix display
much smaller differences between the full-folding and it's factorized,
on-shell $`t\rho$' approximation. For $^{208}$Pb the differences in
the real parts are nearly insignificant (Fig.~15), whereas for $^{40}$Ca
a slight suppression of the real part for small L remains for the
full-folding calculation compared to the factorized approximation.
However, the differences occur mainly for smaller L where the imaginary
part of the potentials as well as the t-matrices are relatively large.
Because the absorption is significant for these low partial waves,
the elastic observables are not particularly sensitive to the differences
in the real parts as displayed in Figs. 13 and 16.
\hspace*{10mm}
In Fig. 17 total neutron cross section data for $^{12}$C, $^{16}$O,
$^{28}$Si, $^{40}$Ca, $^{90}$Zr, and $^{208}$Pb are shown along
with various calculations of $\sigma_{tot}(E)$ at a number of
energies. Because the data are so extensive, the `usual' procedure
has been reversed in the plotting of these cross sections so that
the data are represented by dotted curves, and the discrete points
correspond to calculated results. The solid diamonds represent the
full-folding calculations as described in Section II. All calculations
are based on a DH model for the nuclear density. For energies
$\leq 200$ MeV the modification of the free propagator through the
DH mean field is included as described in Ref.~\cite{med1}. It has
been shown in Ref.\cite{med2} that for higher energies this
modification of the free propagator becomes negligible. The open
circles represent calculations based on the factorized, off-shell
$`t\rho$' form using the same NN t-matrix. A general trend to
be observed in Fig. 17 is the slightly lower value of $\sigma_{tot}(E)$
obtained from a full-folding calculation compared to the factorized
approximation. This trend is almost independent of the energy and
the nucleus under consideration and is consistent with the observation
that full-folding calculations of the differential cross sections
fell slightly below the values given by a factorized calculation.
\hspace*{10mm}
At this point it is worthwhile to investigate whether the interactions
of the projectile with the target nucleus is uniformly distributed
over the entire nucleus or if specific regions of the nucleus
play a more dominant role in the scattering process at intermediate
energies. For our study we chose neutron scattering at 200 MeV
projectile energy and consider contributions from specific shells
to the total cross sections. We employ the DH model for the density
and remove outer shells of protons as well as neutrons.
Then we recalculate the scattering from the remaining `inner core',
which is chosen to be doubly magic, so that it is bound.
The results of this procedure for $^{16}$O, $^{40}$Ca, and
$^{208}$Pb are given in Table I. As a technical detail, when
calculating the scattering in these tests we treated the
targets as being infinitely heavy to exclude recoil effects, which
would be larger for smaller cores. In order to give an estimate of
the size of the recoil effect on the total cross section we give the
values of $\sigma_{tot}$ calculated with and without recoil in Table I.
The values for the total cross section for neutron scattering
for `inner cores' of 100, 40, 16, and 4 nucleons are given as
entries of the corresponding nuclei. The entries in Table I marked
`n.b.' indicate that e.g. in the case of $^{208}$Pb the DH calculation
with only 8 neutrons and 8 protons did not result in a bound
system using the parameters of $^{208}$Pb given for those 16 nuclei.
The calculated rms radii for the `inner cores' under consideration
for $^{16}$O, $^{40}$Ca, and $^{208}$Pb are listed in Table~II.
This table also contains the rms radii for the proton and neutron
distributions for the above mentioned nuclei as given by the DH model.
Columns 3 and 4 of Table II compare the percentage of the volume filled
by the `inner core' if either the corresponding rms radius is used
(column 3) or the radius is taken to be proportional to $A^{1\over{3}}$
(column 4). The percentage Of the calculated total cross section
contribution from the inner core nucleus is given in column 5.
The numbers suggest that the nucleons in the interior of the
nucleus contribute to the total cross sections with a percentage
slightly larger then the volume they occupy when the volume
is based on the crude estimate $r\sim A^{1\over{3}}.$ This leads
to the conclusion that all nucleons in the nucleus almost equally
contribute to the scattering process. We performed a similar
study at 100 MeV and 500 MeV projectile energy and did not find any
significant deviations from the ratios ${\sigma_{tot}(core)/{\sigma_{tot}}}$
as given in Table II at 200 MeV.
\section{Conclusion}
\hspace*{10mm}
We have calculated the full-folding integral for the first-order
optical potential within the framework of the spectator expansion
of multiple scattering theory. These optical potentials are based
on realistic models for the nuclear density matrix, namely a
Dirac-Hartree and a Hartree-Fock-Bogolyubov model along with the
full Bonn meson exchange model for the NN t-matrix. Recoil and
frame transformation factors are implemented in the calculation in their
complete form.
We calculated elastic scattering observables for a variety of light
and heavy nuclei at projectile energies from 65 to $\sim$400~MeV
laboratory energy. At energies below 200 MeV we included the
modification of the free propagator due to the coupling of the
struck target nucleon to the residual nucleus via the same mean
field used to model the effect of the nuclear medium is
intrinsically consistent with the nuclear structure. The
predictions from these rigorous calculations of elastic nucleon
nucleus observables provide excellent agreement with the experimental
data in this energy regime.
\hspace*{10mm}
We tested the validity of the factorized off-shell `$t\rho$'
approximation in the energy regime between 65 and 400 MeV and found
that this approximation, which only retains the non-locality
given through the NN t-matrix, is even at lower energies a very
good representation of the full-folding calculation as far as the
elastic nucleon-nucleus observables are concerned. Differences between
the factorized approximation and the full calculation of the optical
potential are present predominantly in lower partial waves. However
due to the cumulative effect of many partial waves the elastic
observables do not reflect these differences.
It should be noted that in {\it e.g.} inelastic scattering of nucleons
from nuclei or quasielastic electron scattering those differences
between full-folding calculations and the corresponding factorized
approximation may become more significant.
We also studied the contribution of the interior structure of the nucleus
to the total cross section and find that all nucleons in the nucleus
contribute almost uniformally to the scattering process.
\vfill
\acknowledgments
The authors want to express their gratitude and appreciation
towards R.M. Thaler for the many stimulating, helpful and critical
discussions during the major stages of this project.
This work was performed in part under the auspices of the U.~S.
Department of Energy under contracts No. DE-FG02-93ER40756 with
Ohio University, DE-AC05-84OR21400 with Martin Marietta Energy
Systems, Inc., and DE-FG05-87ER40376 with Vanderbilt University.
We thank the Arctic Region Supercomputing Center (ARSC) and the Ohio
Supercomputer Center (OSC) for the use of their facilities. These
resources were made available through the metacenter regional
alliance
project funded by the Advanced Scientific Computing Program of the
National Science Foundation, Award number ASC-9418357 and Grant
No.~PHS206 from OSC. We also thank the Pittsburgh Supercomputer Center
(PSC) for the use of their facilities under Grant No. PHY950010P
as well as the National Energy Research Supercomputer Center
(NERSC) for the use of their facilities
under the FY1996 Massively Parallel Processing Access Program.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,480 |
Hi Rita, I just wanted to make sure that you saw Linda Halpin's response to your question on Facebook. If not, here it is!
Would like to learn more about quilting.
Where can the Little House fabrics be ordered on line. I live quite a ways from the stores that come up on the store locator.
I grew up reading these and read them to my daughter. She even named her youngest daughter Laura. I would like to make both of them a quilt or throw with these fabrics.
Just found out the Jackman's Fabric stores listed for the St Louis, Mo as sources for the Little House on the Prairie fabric,The only one still open is the Jackman's on N. Lindbergh. The one on Watson is now a PetCo store. The one on Andes is also closed.
Thank you for that information – We will pass it along to Andover Fabrics so they can have their web person update that feature.
I'm not sure about an online retailer but I will try to find out for you.
To order the Andover LHOP Fabrics you can also go to EQuilter.com in Boulder, Colorado, as an online fabric store.
We have a follow up post with Linda scheduled with some tips for beginners who are learning to quilt. We will also be posting a round up of tutorials and DIYs to help beginners get started with simple, very doable projects. Be sure you are subscribed to the newsletter and following us on our social channels so you will see when these posts go live.
Thanks for this information. Yes, I am a very new beginner. I appreciate the help.
My favorite memory is of my Nanny and Aunt Doll making quilts until they died. They would work for hours hand piecing the tops then quilting them. Even after Aunt Doll became bed ridden she never stopped piecing or embroidering tops and then Nanny would quilt them Nanny's quilt rack was hung from the ceiling she would lower while quilting and put back up when done. She would have to watch for anyone coming up so she could roll it back up to get into the house. One Christmas she made every grandchild a quilt. Those two ladies still make me smile to remember them.
I took my daughter Erin to many places where The Ingalls lived. I made her the quilt that was at the museum in Walnut Grove that was Laura's. She now is the proud owner of the quilt I made for her. She has in her home. It is over one of the railing in her home that is an A frame. When she was younger we had a Laura Ingalls Wilder room with all of our collections of souvenirs. We even had the children come our area school to tour our room that we had made. They were studying her books and her family. Laura was a very special author..
Where can we get Ms. Halpin's quilt patter for the quilt she made for the Andover booth in Houston?
So pleased you enjoyed the quilt I made for the Andover booth at Houston. It is a sampler quilt made up of blocks from my book "Quilting with Laura: Patterns Inspired by Laura Ingalls Wilder's Little House on the Prairie Series." Each of the 14 blocks in the book tells a story related to Laura's adventures. I did not create a pattern specifically for the Andover booth quilt, but rather, quilters are encouraged to pick and choose which blocks they like and create their own prairie adventure quilt that tells their story. If you are interested in adding "Quilting with Laura" to your library of books, it may be ordered directly from me on my website http://www.lindahalpin.com. Also, be sure to check out the terrific giveaway that Andover is offering to celebrate the launch of the Little House on the Prairie® fabrics collection. Type 'Andover giveaway' in the Search area of this website (http://littlehouseontheprairie.com/little-house-on-the-prairie-andover-fabrics-giveaway/).
I too have loved Laura Ingalls Wilder and Little House on the Prairie books for as long as I can remember. I have been to some of her homes. I read Little House in the Big Woods to my 3rd graders every year and usually used Little House on the Prairie in our reading curriculum. Many of my 3rd graders learned to love Laura too. I can't wait to see this fabric since I have not yet seen it in our local quilt shop! I will have to get this book too–both to use and to add to my Laura collection!
This is such a great show I watch it with my grandchildren.
Didn't even know this website existed until this morning. I was searching the web for a quilt pattern that would be appropriate for a "Laura" quilt. Thank you so much for your website.
We're so glad you found us! Be sure to subscribe to the newsletter in the sidebar for a monthly email that will keep you informed of any news, giveaways, or special content. Great to have you join us!
I am so happy to see this! The quilts are beautiful!
I was in the 3rd grade when my teacher read the Laura Ingalls Wilder books to us. We also wrote to her and she answered our letters. I grew up in the Midwest prairie area. My grandmother was a school teacher living with different families and teaching in a one-room school house braving the winter blizzards to get to school and start a fire in the stove. So glad to see that the Laura Ingalls Wilder's heritage lives on.
Although I have never learned to quilt I might just have to give it ago with the introduction of the quilt line. This was a lovely article and learned so much. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,476 |
class ConcreteTool
: public ToolBase
{
Q_OBJECT
public:
ConcreteTool(int& argc, char *argv[])
: ToolBase(argc, argv)
{
}
virtual void setup()
{
setDescription("Calculates the differences/overlap between variant lists.");
addInfile("in1", "Input variant list in GSvar format.", false, true);
addInfile("in2", "Input variant list in GSvar format.", false, true);
//optional
addOutfile("out", "Output file. If unset, writes to STDOUT.", true);
addInt("window", "Window to consider around indel positions to compensate for differing alignments.", true, 100);
addFlag("nei", "Allow non-exact indel matches. If set, all indels in the window are considered matches.");
addFlag("sm", "Also show matches. If unset, matching variants are not printed.");
}
int appendStrippedVariants(VariantList& vl, QString filename, QByteArray source)
{
VariantList tmp;
tmp.load(filename);
int idx_q = tmp.annotationIndexByName("quality", true, false);
QList<int> affected_cols = tmp.getSampleHeader().sampleColumns(true);
int idx_g = affected_cols.count()!=1 ? -1 : affected_cols[0];
for (int i=0; i<tmp.count(); ++i)
{
Variant v = tmp[i];
v.annotations().clear();
v.annotations().append(source);
v.annotations().append("");
if (idx_q==-1)
{
v.annotations().append("n/a");
}
else
{
v.annotations().append(tmp[i].annotations()[idx_q]);
}
if (idx_g==-1)
{
v.annotations().append("n/a");
}
else
{
v.annotations().append(tmp[i].annotations()[idx_g]);
}
vl.append(v);
}
return tmp.count();
}
virtual void main()
{
//init
bool sm = getFlag("sm");
bool nei = getFlag("nei");
QString in1 = getInfile("in1");
QString in2 = getInfile("in2");
//merge input files
VariantList vl;
vl.annotations().append(VariantAnnotationHeader("source"));
vl.annotationDescriptions().append(VariantAnnotationDescription("source", "Source sample."));
vl.annotations().append(VariantAnnotationHeader("match"));
vl.annotationDescriptions().append(VariantAnnotationDescription("match", "Match type, exact or fuzzy."));
vl.annotations().append(VariantAnnotationHeader("quality"));
vl.annotationDescriptions().append(VariantAnnotationDescription("quality", "Variant quality (if available)."));
vl.annotations().append(VariantAnnotationHeader("genotype"));
vl.annotationDescriptions().append(VariantAnnotationDescription("genotype", "Genotype (if available)."));
int c1 = appendStrippedVariants(vl, in1, "in1");
int c2 = appendStrippedVariants(vl, in2, "in2");
//sort by chr/start/end/ref/obs/source
vl.sortCustom([](const Variant& a, const Variant& b)
{
if (a.chr()<b.chr()) return true;
if (a.chr()>b.chr()) return false;
if (a.start()<b.start()) return true;
if (a.start()>b.start()) return false;
if (a.end()<b.end()) return true;
if (a.end()>b.end()) return false;
if (a.ref()<b.ref()) return true;
if (a.ref()>b.ref()) return false;
if (a.obs()<b.obs()) return true;
if (a.obs()>b.obs()) return false;
return a.annotations()[0] < b.annotations()[0];
}
);
//flag matches
ChromosomalIndex<VariantList> file_idx(vl);
// check for exact matches
for (int i=0; i<vl.count(); ++i)
{
Variant& v1 = vl[i];
//skip exact matches we already found
if (v1.annotations()[1]=="=") continue;
QVector<int> matches = file_idx.matchingIndices(v1.chr(), v1.start(), v1.end());
foreach(int index, matches)
{
Variant& v2 = vl[index];
//skip exact matches
if (v2.annotations()[1]=="=") continue;
//skip if v1 and v2 are from the same file
if (v2.annotations()[0]==v1.annotations()[0]) continue;
//check if genotypes match
QByteArray geno_match = "=";
if (v2.annotations()[3]!=v1.annotations()[3] && v2.annotations()[3]!="" && v1.annotations()[3]!="") geno_match = "g";
//exact match (SNP and indel)
if (v1.ref()==v2.ref() && v1.obs()==v2.obs())
{
v1.annotations()[1] = geno_match;
v2.annotations()[1] = geno_match;
}
}
}
// check for exact matches in the given window
if (getInt("window") > 0)
{
for (int i=0; i<vl.count(); ++i)
{
Variant& v1 = vl[i];
//skip exact matches we already found
if (v1.annotations()[1]=="=") continue;
if (v1.annotations()[1]=="g") continue;
//skip SNVs
if (v1.isSNV()) continue;
//indel => fuzzy position search
int start = v1.start() - getInt("window");
int end = v1.end() + getInt("window");
QVector<int> matches = file_idx.matchingIndices(v1.chr(), start, end);
foreach(int index, matches)
{
Variant& v2 = vl[index];
//skip exact matches
if (v2.annotations()[1]=="=") continue;
if (v2.annotations()[1]=="g") continue;
//skip if v1 and v2 are from the same file
if (v2.annotations()[0]==v1.annotations()[0]) continue;
//check if genotypes match
QByteArray geno_match = "=";
if (v2.annotations()[3]!=v1.annotations()[3] && v2.annotations()[3]!="" && v1.annotations()[3]!="") geno_match = "g";
//exact match (SNP and indel)
if (v1.ref()==v2.ref() && v1.obs()==v2.obs())
{
v1.annotations()[1] = geno_match;
v2.annotations()[1] = geno_match;
}
}
}
}
if (nei)
{
//non-exact indel mapping
for (int i=0; i<vl.count(); ++i)
{
Variant& v1 = vl[i];
//skip exact matches we already found
if (v1.annotations()[1]=="=") continue;
if (v1.annotations()[1]=="g") continue;
//indel => fuzzy position search
int start = v1.start();
int end = v1.end();
if (!v1.isSNV())
{
start -= getInt("window");
end += getInt("window");
}
QVector<int> matches = file_idx.matchingIndices(v1.chr(), start, end);
foreach(int index, matches)
{
Variant& v2 = vl[index];
//skip exact matches
if (v2.annotations()[1]=="=") continue;
if (v2.annotations()[1]=="g") continue;
//skip if v1 and v2 are from the same file
if (v2.annotations()[0]==v1.annotations()[0]) continue;
if (!v2.isSNV())
{
v1.annotations()[1] = "=";
v2.annotations()[1] = "=";
}
}
}
}
//output
QSharedPointer<QFile> outfile = Helper::openFileForWriting(getOutfile("out"), true);
QTextStream out(outfile.data());
out << "#change\tchr\tstart\tend\tref\tobs\tgenotype\tquality" << endl;
int u1 = 0;
int u2 = 0;
int g = 0;
for (int i=0; i<vl.count(); ++i)
{
Variant& v1 = vl[i];
QString prefix="";
if (v1.annotations()[1]=="=")
{
if (!sm) continue;
}
else if (v1.annotations()[1]=="g")
{
++g;
if (v1.annotations()[0]=="in1")
{
prefix = "-g";
}
else if (v1.annotations()[0]=="in2")
{
prefix = "+g";
}
}
else if (v1.annotations()[0]=="in1")
{
prefix = "-";
++u1;
}
else if (v1.annotations()[0]=="in2")
{
prefix = "+";
++u2;
}
out << prefix << "\t" << v1.chr().str() << "\t" << v1.start() << "\t" << v1.end() << "\t" << v1.ref() << "\t" << v1.obs() << "\t" << v1.annotations()[3] << "\t" << v1.annotations()[2] << endl;
}
//output to console
QTextStream out2(stdout);
out2 <<"#" << endl;
out2 <<"#in1 : " << in1 << endl;
out2 <<"#count : " << QString::number(c1) << endl;
out2 <<"#unique: " << QString::number(u1) << " (" << QString::number(100.0 * u1 / c1, 'f', 2) << "%)" << endl;
out2 <<"#geno : " << QString::number(g/2) << " (" << QString::number(100.0 * g / c1, 'f', 2) << "%)" << endl;
out2 <<"#match : " << QString::number(c1-u1) << " (" << QString::number(100.0 * (c1-u1) / c1, 'f', 2) << "%)" << endl;
out2 <<"#" << endl;
out2 <<"#in2 : " << in2 << endl;
out2 <<"#count : " << QString::number(c2) << endl;
out2 <<"#unique: " << QString::number(u2) << " (" << QString::number(100.0 * u2 / c2, 'f', 2) << "%)" << endl;
out2 <<"#geno : " << QString::number(g/2) << " (" << QString::number(100.0 * g / c2, 'f', 2) << "%)" << endl;
out2 <<"#match : " << QString::number(c2-u2) << " (" << QString::number(100.0 * (c2-u2) / c2, 'f', 2) << "%)" << endl;
//exit with error code
if (u1!=0 || u2!=0 || g!=0)
{
THROW(ToolFailedException, "Detected differences in input files!");
}
}
};
#include "main.moc"
int main(int argc, char *argv[])
{
ConcreteTool tool(argc, argv);
return tool.execute();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,914 |
Q: When is it safe to use -[NSManagedObjectContext lock]? I know that you should abide by CoreData's thread confinement rules in general, but is it ever safe to use -[NSManagedObjectContext lock] and friends? I know that accessing an NSManagedObject property can trigger an implicit NSManagedObjectContext fetch if the NSManagedObject has unloaded properties, so I assume you would have to wrap all NSManagedObject property accesses around -[NSManagedObjectContext lock] and -[NSManagedObjectContext unlock]. I thought this was the only gotcha. Are there others?
In the comments of this answer, Marcus Zarra says that I'm misinterpreting the documentation about -\[NSManagedObjectContext lock\] and friends:
Sending this message to a managed object context helps the framework
to understand the scope of a transaction in a multi-threaded
environment. It is preferable to use the NSManagedObjectContext's
implementation of NSLocking instead using of a separate mutex object.
Also, the above quote implies that you can use other locks to guard NSManagedObjectContext. Is this true?
I'm not worried about parent/child contexts for this question.
A: In an academic setting, can you use locks? yes.
Should you ever use them in production code? no. Why? Because the odds of getting it right the first time are extra-ordinarily high. Getting it right in maintenance mode rapidly approaches zero.
Using locks to access Core Data is just asking for trouble. When you get it wrong you lose/corrupt data. When you get it right you are breaking even with thread confinement. It is a lose/lose gamble with nothing to gain.
The worst part is that there is virtually no way to know if you got it "right" until or unless you lose data. Never worth the risk.
I would also point you to this response from Ben which should give you some nice history on this subject.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,250 |
var AWS = require('aws-sdk');
var DOC = require('dynamodb-doc');
var dynamo = new DOC.DynamoDB();
exports.handler = function(event, context) {
console.log('Received event:', JSON.stringify(event, null, 2));
var item = { client_project:"default",
appointment: event.appointment || {}
};
var cb = function(err, data) {
if(err) {
console.log(err);
context.fail('unable to update appointments at this time');
} else {
console.log(data);
context.succeed(null, data);
}
};
dynamo.putItem({TableName:"sparkl-appointments", Item:item}, cb);
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,392 |
Frederick Shinton (7 March 1883 – 11 April 1923) was an English footballer who played at centre forward or inside right. He scored 103 goals from 163 appearances in the Football League.
Biography
Shinton was born in Wednesbury. He turned professional with West Bromwich Albion in April 1905 and remained with them until November 1907, when he joined Leicester Fosse for £150. In August 1910 he moved to Bolton Wanderers for a £1000 fee, but re-joined Leicester Fosse just five months later, for £750.
Shinton was not offered a contract by Fosse at the end of the 1910–11 season and instead joined Wednesbury Old Athletic, playing for the club in the Birmingham & District League. He featured in the club's Wednesbury Charity Cup winning side v Darlaston in March 1912, but although captaining the side at the start of the following season, he appears to have made no further appearances after October 1912.
He died at home in Wednesbury in April 1923 after a long illness.
References
1883 births
1923 deaths
Sportspeople from Wednesbury
English footballers
Association football forwards
West Bromwich Albion F.C. players
Leicester City F.C. players
Bolton Wanderers F.C. players
Wednesbury Old Athletic F.C. players
English Football League players | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,811 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/302518\/how-to-combine-paths-defined-by-functions-to-fill-area","text":"# How to combine paths defined by functions to fill area\n\nInspired by this question I stumbled upon a problem I couldn't solve. Probably somebody can help me out. The actual question was, how to fill an area enclosed by four functions. Taking a sheet of paper its easy to calculate the intersection points of the functions. But however, even though I can define the paths and draw them separately, I'm not able to combine them and fill the area enclosed by them. I found a number of posts handling the question of how to combine paths, but all I found where just working with straight paths defined by single points rather than by functions. So is it possible to fill paths defined by a set of functions and their corresponding domains?\n\n\\documentclass[border=2mm]{standalone}\n\\usepackage{tikz}\n\\usetikzlibrary{intersections}\n%\n\\begin{document}\n\\begin{tikzpicture}\n%\n\\draw[very thin,color=gray] (1,0) grid (3,3);\n%\n% the desired functions plotted\n%\n\\path[draw,color=blue, domain=2:3, samples=100] plot (\\x,{sqrt(\\x^2-4)}) node[right] {$f(x) = \\sqrt{x^2-4}$};\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{sqrt(\\x^2-1)}) node[above right] {$f(x) = \\sqrt{x^2-1}$};\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{1\/\\x}) node[below right] {$f(x) = \\frac{1}{x}$};\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{2\/\\x}) node[above right] {$f(x) = \\frac{2}{x}$};\n%\n% calculated intersection points, just for annotation\n%\n\\filldraw ({sqrt(2+sqrt(5))},{1\/sqrt(2+sqrt(5))}) circle (1pt); %intersection of 1\/x=sqrt(x^2-4)\n\\filldraw ({sqrt(2+sqrt(8))},{2\/sqrt(2+sqrt(8))}) circle (1pt); %intersection of 2\/x=sqrt(x^2-4)\n\\filldraw ({sqrt(1\/2+sqrt(5\/4))},{1\/sqrt(1\/2+sqrt(5\/4))}) circle (1pt); %intersection of 1\/x=sqrt(x^2-1)\n\\filldraw ({sqrt(1\/2+sqrt(17\/4))},{2\/sqrt(1\/2+sqrt(17\/4))}) circle (1pt); %intersection of 2\/x=sqrt(x^2-1)\n%\n% I can define the paths and plot them seperately\n%\n\\draw[red, dashed, domain={sqrt(1\/2+sqrt(5\/4))}:{sqrt(2+sqrt(5))}, samples=100] plot (\\x,{1\/\\x});\n\\draw[red, dashed, domain={sqrt(2+sqrt(5))}:{sqrt(2+sqrt(8))}, samples=100] plot (\\x,{sqrt(\\x^2-4)});\n\\draw[red, dashed, domain={sqrt(2+sqrt(8))}:{sqrt(1\/2+sqrt(17\/4))}, samples=100] plot (\\x,{2\/\\x});\n\\draw[red, dashed, domain={sqrt(1\/2+sqrt(17\/4))}:{sqrt(1\/2+sqrt(5\/4))}, samples=100] plot (\\x,{sqrt(\\x^2-1)});\n%\n% But how to combine and fill the area enclosed by them?\n%\n%\\path[name path=A, domain={sqrt(1\/2+sqrt(5\/4))}:{sqrt(2+sqrt(5))}, samples=100] plot (\\x,{1\/\\x});\n%\\path[name path=B, domain={sqrt(2+sqrt(5))}:{sqrt(2+sqrt(8))}, samples=100] plot (\\x,{sqrt(\\x^2-4)});\n%\\path[name path=C, domain={sqrt(2+sqrt(8))}:{sqrt(1\/2+sqrt(17\/4))}, samples=100] plot (\\x,{2\/\\x});\n%\\path[name path=D, domain={sqrt(1\/2+sqrt(17\/4))}:{sqrt(1\/2+sqrt(5\/4))}, samples=100]\n%\n\\end{tikzpicture}\n\\end{document}\n\n\nThe following example draws the area by clipping. Then it is not necessary to calculate the intersection points. The four functions are paired, that one pair makes the left and right boundary and the other pair the upper and lower boundary of the area:\n\n \\begin{scope}[samples=100]\n\\clip\n(1, 0) -- plot[domain=2:3] (\\x, {sqrt(\\x^2-4)})\n-- plot[domain=3:1] (\\x, {sqrt(\\x^2-1)}) -- cycle;\n\\clip\nplot[domain=1:3] (\\x, 1\/\\x)\n-- plot[domain=3:1] (\\x, 2\/\\x) -- cycle;\n\\fill[yellow] (1, 0) rectangle (3, 3);\n\\end{scope}\n\n\nFull example:\n\ndocumentclass[border=2mm]{standalone}\n\\usepackage{tikz}\n\\usetikzlibrary{intersections}\n%\n\\begin{document}\n\\begin{tikzpicture}\n%\n\\draw[very thin,color=gray] (1,0) grid (3,3);\n%\n% the filled enclosed area above the grid, but below the function drawings\n%\n\\begin{scope}[samples=100]\n\\clip\n(1, 0) -- plot[domain=2:3] (\\x, {sqrt(\\x^2-4)})\n-- plot[domain=3:1] (\\x, {sqrt(\\x^2-1)}) -- cycle;\n\\clip\nplot[domain=1:3] (\\x, 1\/\\x)\n-- plot[domain=3:1] (\\x, 2\/\\x) -- cycle;\n\\fill[yellow] (1, 0) rectangle (3, 3);\n\\end{scope}\n%\n% the desired functions plotted\n%\n\\path[draw,color=blue, domain=2:3, samples=100] plot (\\x,{sqrt(\\x^2-4)}) node[right] {$f(x) = \\sqrt{x^2-4}$};\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{sqrt(\\x^2-1)}) node[above right] {$f(x) = \\sqrt{x^2-1}$};\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{1\/\\x}) node[below right] {$f(x) = \\frac{1}{x}$};\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{2\/\\x}) node[above right] {$f(x) = \\frac{2}{x}$};\n%\n% calculated intersection points, just for annotation\n%\n\\filldraw ({sqrt(2+sqrt(5))},{1\/sqrt(2+sqrt(5))}) circle (1pt); %intersection of 1\/x=sqrt(x^2-4)\n\\filldraw ({sqrt(2+sqrt(8))},{2\/sqrt(2+sqrt(8))}) circle (1pt); %intersection of 2\/x=sqrt(x^2-4)\n\\filldraw ({sqrt(1\/2+sqrt(5\/4))},{1\/sqrt(1\/2+sqrt(5\/4))}) circle (1pt); %intersection of 1\/x=sqrt(x^2-1)\n\\filldraw ({sqrt(1\/2+sqrt(17\/4))},{2\/sqrt(1\/2+sqrt(17\/4))}) circle (1pt); %intersection of 2\/x=sqrt(x^2-1)\n%\n% I can define the paths and plot them seperately\n%\n\\draw[red, dashed, domain={sqrt(1\/2+sqrt(5\/4))}:{sqrt(2+sqrt(5))}, samples=100] plot (\\x,{1\/\\x});\n\\draw[red, dashed, domain={sqrt(2+sqrt(5))}:{sqrt(2+sqrt(8))}, samples=100] plot (\\x,{sqrt(\\x^2-4)});\n\\draw[red, dashed, domain={sqrt(2+sqrt(8))}:{sqrt(1\/2+sqrt(17\/4))}, samples=100] plot (\\x,{2\/\\x});\n\\draw[red, dashed, domain={sqrt(1\/2+sqrt(17\/4))}:{sqrt(1\/2+sqrt(5\/4))}, samples=100] plot (\\x,{sqrt(\\x^2-1)});\n%\n\\end{tikzpicture}\n\\end{document}\n\n\n## Variant with the area as closed path via the intersection points\n\n\\documentclass[border=2mm]{standalone}\n\\usepackage{tikz}\n\\usetikzlibrary{intersections}\n%\n\\begin{document}\n\\begin{tikzpicture}\n%\n\\draw[very thin,color=gray] (1,0) grid (3,3);\n%\n% the desired functions plotted\n%\n\\path[draw,color=blue, domain=2:3, samples=100] plot (\\x,{sqrt(\\x^2-4)}\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{sqrt(\\x^2-1)}\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{1\/\\x}) node[b\n\\path[draw,color=blue, domain=1:3, samples=100] plot (\\x,{2\/\\x}) node[a\n%\n% calculated intersection points, just for annotation\n%\n\\filldraw[\ndraw=red,\nfill=yellow,\nthick,\nsamples=50,\n]\nplot[domain=sqrt(2+sqrt(5)):sqrt(2+sqrt(8))]\n(\\x, {sqrt(\\x*\\x-4)})\n--\nplot[domain=sqrt(2+sqrt(8)):sqrt(1\/2+sqrt(17\/4))]\n(\\x, 2\/\\x)\n--\nplot[domain=sqrt(1\/2+sqrt(17\/4)):sqrt(1\/2+sqrt(5\/4))]\n(\\x, {sqrt(\\x*\\x-1)})\n--\nplot[domain=sqrt(1\/2+sqrt(5\/4)):sqrt(2+sqrt(5))]\n(\\x, 1\/\\x)\n-- cycle\n;\n\\end{tikzpicture}\n\\end{document}\n\n\n\u2022 Thanks for this answer, but actually my question was how to combine the paths rather than how to fill the area. \u2013\u00a0JMP Apr 4 '16 at 21:28\n\u2022 @JMP See updated answer. \u2013\u00a0Heiko Oberdiek Apr 4 '16 at 21:47\n\u2022 Thanks, that was my problem. I gave the options for the plotting domains of the functions at the wrong position. :-\/ \u2013\u00a0JMP Apr 4 '16 at 21:53\n\u2022 Probably you should give this answer to the question mentioned above as well. \u2013\u00a0JMP Apr 4 '16 at 22:02\n\nAs I have learned here (German) it is possible to use the pgfplotslibrary fillbetween with TikZ.\n\n\\documentclass[border=2mm]{standalone}\n\\usepackage{pgfplots}\n\\pgfplotsset{compat=1.13}\n\\usepgfplotslibrary{fillbetween}\n\n%\n\\begin{document}\n\n\\begin{tikzpicture}\n\\draw[very thin,color=gray] (1,0) grid (3,3);\n\\path[name path=A,draw,blue, domain=2:3, samples=100]\nplot (\\x,{sqrt(\\x^2-4)}) node[right] {$f(x) = \\sqrt{x^2-4}$};\n\\path[name path=B,draw,blue, domain=1:3, samples=100]\nplot (\\x,{sqrt(\\x^2-1)}) node[right] {$f(x) = \\sqrt{x^2-1}$};\n\\path[name path=C,draw,blue, domain=1:3, samples=100]\nplot (\\x,{1\/\\x}) node[right,yshift=-.5ex] {$f(x) = \\frac{1}{x}$};\n\\path[name path=D,draw,blue, domain=1:3, samples=100]\nplot (\\x,{2\/\\x}) node[right,yshift=.5ex] {$f(x) = \\frac{2}{x}$};\n\n\\path[%draw,line width=3,orange,\nname path=AandC,\nintersection segments={\nof=A and C,\nsequence={R1 -- L2}\n}\n];\n\\path[%draw,line width=3,purple,\nname path=BandD,\nintersection segments={\nof=B and D,\nsequence={L1 -- R2}\n}\n];\n\n\\path [\ndraw=red,\nfill=yellow,\nintersection segments={\nof=AandC and BandD,\nsequence={L2[reverse] -- R2}\n}\n]--cycle;\n\\end{tikzpicture}\n\\end{document}\n\n\nResult:","date":"2019-08-25 07:59:53","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\": 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\": 0.7921880483627319, \"perplexity\": 13076.287646124632}, \"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\/1566027323221.23\/warc\/CC-MAIN-20190825062944-20190825084944-00229.warc.gz\"}"} | null | null |
{"url":"https:\/\/em.shawnzhong.com\/5.1%20-%20maxwell's%20equations","text":"# Revisiting Ampere's Law\n\n\u2022 Ampere's Law as written allows us to calculate the magnetic field due to an electric current.\n\n\u2022 We also know that a changing electric field produces a magnetic field\n\n\u2022 Combine effects of electric current and changing E field on magnetic field to obtain a more complete version of Ampere's Law\n\n\u2022 Contribution due to the penetrating current is known a conduction current.\n\n\u2022 Contribution due to changing electric field is known as the displacement current\n\n\u2022 Final Ampere's Law","date":"2019-11-20 20:45:43","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8404285311698914, \"perplexity\": 483.4526948006304}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-47\/segments\/1573496670601.75\/warc\/CC-MAIN-20191120185646-20191120213646-00370.warc.gz\"}"} | null | null |
Q: Is there a way to run a function what will self recompile if you make any changes? I know that I can use the below commands:
sf run function start local
and
sf run function start container
But if I am making lots of small changes it is very annoying to keep killing processes and restarting them to get the latest iteration change. Unless they are meant to auto-update at which point I'll stand corrected and ask why mine seem to not.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,065 |
Hipster reformism and the technological fix
Submitted by AWL on 17 July, 2019 - 9:25
Bruce Robinson reviews Aaron Bastani's 'Fully Automated Luxury Communism'
Back in 2013-14 there was a lot of excitement on the left about "left accelerationism" and the prospect of a transition to a "post-capitalism" fuelled by technological advances based on information.
Aaron Bastani coined the meme of "Fully Automated Luxury Communism" (FALC), and it led a fitful life on the Internet. It has now returned in the form of a book which sets out to be a manifesto. Since 2015 Bastani has moved from a politics rooted in "post-workerist" thinkers to become a born-again supporter of Jeremy Corbyn.
The book divides into two parts: the first containing the basis for and outline of FALC as a future communist society near the "end of history", and the second providing a political and economic platform rooted in the present, self-consciously populist and anti-globalist, in which FALC is "a beginning, not a destination".
The basic thesis underlying the book is that we are undergoing a "Third Disruption". The first was agriculture, the second industry and the third is based on information.
"The defining feature is ever-greater abundance in information." As information goods have a cost that declines to almost zero as more are produced, we live on the brink of "extreme supply", a post-scarcity society delivered by courtesy of technological breakthroughs produced by capitalism. Labour is also no longer scarce (there are a number of economic objections to this, and issues of viability, which I will skip over for lack of space).
On this basis, Bastani details a number of technologies that he claims will resolve contemporary crises. Energy scarcity will be overcome by harnessing solar energy on a massive scale. Raw material scarcity will be overcome by mining in space, using asteroids. Problems of an ageing population are solved by gene editing to prevent genetically determined illnesses. The provision of sufficient food is ensured by the creation of synthetic protein that'll taste as good as meat and by the completion of the Green Revolution of the 50s and 60s that introduced higher yielding crops and the use of chemical fertilisers to countries such as India. These measures combined will enable a slowing and eventual end to global warning.
A lot of the book is taken up with advocating these technologies and demonstrating that they are already exist – or are about to – so that in places it reads like a publicity blurb for synthetic hamburgers or reusable rockets.
This is the politics of the technological fix, where social and political problems are taken to have technological solutions. The technologies are assumed to function well and not to have detrimental social, economic and environmental side effects (the Green Revolution is disputed on all three grounds).
If you look closely, Bastani has caveats — not quite there yet, but success is just coming. Those are not allowed to tarnish the overall confidence that the technology developed under capitalism will lead to FALC. This is based on the assertion that " capitalism is incompatible with natural abundance".
"Facing such conditions… production for profit begins to malfunction." FALC is therefore the conclusion of the Third Disruption – capitalism will be driven by its own dynamics to innovate and thus hasten its own demise.
This represents an extreme but not original reading of Marx which takes his words on the development of the productive forces under capitalism (narrowly understood as technology) to imply its transcendence. Productive forces clash with the social relations of production and capitalism cannot survive, in this case because it cannot deal with "extreme supply", even though, as Bastani accepts, today's capitalism is finding ways to circumvent that by controlling and restricting supply through enforcing monopoly rights.
In one of the many absences from the book, the human side of the social relations of production gets little attention, whether in the workplace or society in general.
Both the working class and class more generally are absent as agency and struggle. Class struggle also affects not merely the way in which technologies are developed and implemented under capital but also the content of the technologies themselves. We need a means for the democratic assessment of technologies. Instead here we have uncritical technophilia.
His reading of Marx leads Bastani to conclude that the productive forces needed to support "a post-scarcity, post-work" world were in existence only from the late 60s. To attempt socialism before then was impossible: "You could conceive of it… but you could not create it. This was… simply an inevitability of history."
But it was well within the economic potential of the mid 20th century to provide sufficient housing, healthcare food and education to create a viable socialism, even if not a post-scarcity utopia. It was quite possible to provide a number of the free services that Bastani advocates as transitional measures to FALC.
For Bastani revolutionary socialists in the 20th century were simply before their time and their failure an inevitability. The Russian Revolution was "an anti-liberal coup" (was Kerensky really a liberal?).
The consequence of that reasoning is to airbrush Stalinism as something inevitable and indistinguishable from the revolutionary years of the USSR: "Its [the Soviet Union's] seven decade survival was one of the great political achievements of the last century."
His vision of communism is "a society in which work is eliminated, scarcity replaced by abundance and where labour and leisure blend into one another". He takes up Marx's notion of "free individuality" as the essence of communism, but ignores its grounding in social labour, leaving out the need for collectivity and forms of social solidarity and democratic control that flow from the need to produce.
The realm of necessity – the labour of the associated producers — is not abolished, however many robots there are, but rather diminishes relative to free time.
With social labour deleted, Bastani's communism reduces to individualism. Freedom is "self- authorship… Liberal ends… are impossible without communist means."
FALC is "the politics of the self-help guru – be precisely who you want to be – embedded within a programme for political change."
Looking at "full automation", Bastani argues that, despite the waged working class having grown massively to be the majority on the planet, we have reached "peak labour" and that AI and automation will shrivel the amount of work that needs to be done. Such projections remain speculative.
As Bastani concedes, not all jobs will disappear (he points to health, education, geriatric care and jobs requiring creativity and emotional connection). If social labour continues, then the need continues for decisions about how remaining work is divided up and how a division of labour is put in place that enables needed skills to be developed.
Bastani never considers whether full automation is something desirable from the point of view of a socialism that puts humans and the environment first. Should we just assume there is no alternative to the technological path enabled by capitalism?
For example, the machine learning techniques on which contemporary AI is based are inherently open to bias, false assumptions and false positives. Do we want to live in a machine-run society? Who decides on how technology develops and is implemented? Technocrats or workers?
If the first part of the book might be considered an exercise in utopian thought, the second brings us back to earth with a crash. Purporting to set out the political and economic road from here to FALC, it aims to provide theoretical ballast for Corbynism. In doing so it embraces various classical reformist aims and methods put in a modern context.
The "concrete politics" consist of "a break with neo-liberalism, a shift towards worker-owned production, a state-financed transition to renewable energy and universal services." Bastani's "communist means" are based on "reforging the capitalist state", "demanding that the conscious, intentional planning at the heart of modern capitalism be repurposed to socially useful ends." This rests on "the re-localisation of economies", "socialising finance" and a range of free services that will put much of the economy under public ownership.
Relocalisation is based on the premise that also underlies Bastani's opposition to "globalism": that "locally we can start right away" and "break with neo-liberalism without needing national state power" via "local protectionism" (the Preston model). But, for Bastani, the national state is the best environment for beginning FALC.
This approach, like Brexit, is both regressive and utopian in trying to reverse capital's integration and development across local and national boundaries. Of course useful action can be taken at national or even local levels, but to see the local as the source of spreading worker enterprises that will eventually bring us to FALC is an illusion.
Even if central and local bankers favour worker-owned enterprises (Bastani believes central bankers should become central planners), they still have to compete with much larger capitalist enterprises. The Preston model does not "scale".
As Rosa Luxemburg pointed out in her 1899 reply to Eduard Bernstein's "revisionism" of that era, cooperatives can only survive if protected from the operation of capitalist competition. Rather than being the means to implement new technologies as Bastani argues, small and local firms, even if worker-owned, are less likely to be able to afford and be able to implement the new technology that he sees leading to FALC.
Why are they able to deal with "extreme supply" if large capitalist enterprises can't? A big gap remains between the communist model supposedly just around the corner and Bastani's immediate programme, which essentially gives a contemporary gloss to long established social democratic strategies for improving the capitalist state piecemeal.
Having freed himself from any concept of class, Bastani unashamedly embraces populism. "The people [is] not "a permanent and immutable entity" but has its roots in "certain kinds of assembly, social trait or capacity." He recognises that there is nothing fundamental here to distinguish this from the populism of the right – it just depends who you think the people are and which traits you choose. The book doesn't give a clear answer on Bastani's criteria here. How are the "people" mobilised? Here the Bastani of 2010 who favoured the network organisation of the Internet reappears: "the party form… makes increasingly little sense. The same is true of worker organising, radical or reformist, which are [sic] erroneously premised on the society of work enduring forever." But a few lines later the Bastani of 2019 counters "The role of the labour movement is to liberate the working class... We must build a workers' party against work..."
Bastani here makes increasingly little sense.
This book is notable for a number of absences. There is no conception of working class self- activity either in bringing FALC about or in managing production under it. There is no conception of democratic control in the workplace, in governance of technology or in society more generally. There is no notion of struggle from below to transform economy or society. Those things are presumably out of date.
Instead the book combines a view of a future close by in which technology enables us to forget the collective and focus on self with an immediate platform for Corbynism which repackages some traditional left social democratic policies and ideas about how it might come about. These ideas may become fashionable for a while in the same way as Bastani's original meme.
But, however well- wrapped in the ultra-modernity of new technology in a sort of hipster reformism, they do not offer a road to emancipation from capitalism.
Marx's telescope
Martin Thomas examines Karl Marx's Grundrisse
The future and robots
Charlie Applebaum discusses the fourth industrial revolution
Solidarity with the strikes in France
Labour for a Socialist Europe organised a demonstration at the French Embassy in London on...
"No retreat, no truce!" say French strikers
The government hopes that a few limited or even cosmetic concessions will be enough to...
ReviewsScience and Technology Social and Economic PolicyRosa Luxemburg The Russian Revolution and Its FateSolidarity 513, 17 July 2019 Automation and the working class: Workers' Liberty 3/70Jeremy Corbyn | {
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using BrightData;
namespace BrightWire.ExecutionGraph.WeightInitialisation
{
/// <summary>
/// Identity matrix: https://arxiv.org/abs/1504.00941
/// </summary>
internal class Identity : IWeightInitialisation
{
readonly ILinearAlgebraProvider _lap;
readonly float _value;
public Identity(ILinearAlgebraProvider lap, float value)
{
_lap = lap;
_value = value;
}
public IFloatVector CreateBias(uint size)
{
return _lap.CreateVector(size);
}
public IFloatMatrix CreateWeight(uint rows, uint columns)
{
return _lap.CreateMatrix(rows, columns, (x, y) => x == y ? _value : 0f);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,004 |
That is Joaquin Phoenix's Wife? Plus Much More Details on the Actor's Dating History
Joaquin Phoenix has invested a chunk that is large of life when you look at the limelight, but due to their leading role in Joker, he has got also been up against a lot more questions about their profession along with his personal life.
Even though the 45-year-old star is undoubtedly no complete complete stranger to red carpets and honor shows (this is certainly his 3rd Oscar nom), we really know almost no about their relationships (both past and present). For example, that is Joaquin Phoenix's spouse?
Needless to say, Phoenix is anticipated become in the 2020 Academy Awards, but will he be going to with a spouse? A gf? Here's anything you must know.
1. Who's Joaquin Phoenix's Wife?
Well, he does not get one. But he could be involved ( more about that later). Although Phoenix has dated a number of individuals (both in and away from Hollywood), the actor that is award-winning never ever really been hitched.
It really is well worth noting which he once faked an engagement and pretty much tricked the world that is whole. In 2014, Phoenix showed up from the "Late Show" and told a whole tale about a yoga trainer, whom he later revealed to be their fiancee. "I think she's the main one, " Phoenix told Letterman. Her, and she said yes. " we proposed to" needless to say, the world-wide-web went crazy, simply to find out of the day that is following the complete tale ended up being constructed. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 567 |
Corduroy è l'ottava traccia di Vitalogy, album dei Pearl Jam del 1994. Nonostante non fosse stata pubblicata come singolo, la canzone raggiunse il tredicesimo posto della Billboard Modern Rock Tracks chart. Fu inclusa nella tracklist del greatest hits Rearviewmirror: Greatest Hits 1991-2003.
Il brano
Corduroy comincia con un potente riff suonato come un arpeggio di cui le prime due note sono di un power chord. La canzone decolla, procedendo con una struttura strofa-ritornello-strofa-ritornello-ponte-ritornello-strofa. Sebbene non rivoluzionaria, la struttura della canzone non è completamente normale, infatti quasi nessun verso è ripetuto (anche nel ritornello) e la dissolvenza della canzone comincia dopo una strofa, piuttosto che una tradizionale fine della canzone dopo il terzo ritornello.
La canzone è divenuta una delle più eseguite dal vivo, sebbene viene proposta con un ritmo più veloce. Alcune performance sono precedute da una breve improvvisazione sulla canzone dei Pink Floyd "Interstellar Overdrive". Performance dal vivo di Corduroy sono disponibili su Live on Two Legs e Live at the Gorge 05/06. Disponibile anche su due DVD della band, Touring Band 2000 e Live at the Showbox, oltre che nel DVD Immagine in cornice, uscito nel 2007.
Significato del testo
Il contenuto del testo può essere interpretato in molteplici modi, ma una teoria comune dice che riguardi il peso e le pressioni della popolarità. In una intervista Eddie Vedder dichiarò:
Riguardo al titolo della canzone, Vedder dichiarò:
Note
Collegamenti esterni
Brani musicali dei Pearl Jam | {
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} | 637 |
\subsection*{Acknowledgments}
\begingroup
\footnotesize
The authors thank Matthew Foulkes, David Ceperley, Lucas Wagner, Gareth Conduit, and Ke Liao for helpful discussions. We thank ByteDance AML team specially for their technical and computing support. We also thank ByteDance AI-Lab LIT Group and the rest of ByteDance AI-Lab research team for inspiration and encouragement. This work is directed and supported by Hang Li and ByteDance AI-Lab. J.C. is supported by the National Natural Science Foundation of China under Grant No. 92165101.
\endgroup
\subsection*{Author contributions}
\begingroup
\footnotesize
X.L. and J.C. conceived the study; X.L. developed the method, performed implementations, simulations, and data analyses; Z.L. contributed to the code development and simulation of HEG; J.C. supervised the project. X.L., Z.L., and J.C. wrote the paper.
\endgroup
\end{document}
\section{Hyperparameters for simulations}
The recommended hyperparameters are listed in Supplementary Table~\ref{tab:ge_hyper}. Some employed hyperparameters of the presented results differ from the recommended ones, which are specially given in Supplementary Table~\ref{tab:hyper}.
\begin{table}[htb]
\centering
\begin{tabular}{lclc}
\toprule
Hyperparameter & Value & Hyperparameter & Value \\
\midrule
Pretrain basis & ccpvdz & Pretrain iterations & 1e3\\
Dimension of one electron layer $\mathbf{V}$ & 256 & Dimension of two electron layer $\mathbf{W}$ & 32 \\
Number of layers & 4 & Number of determinants & 8\\
Optimizer & KFAC & Learning rate & 3e-2\\
Damping & 1e-3 & Constrained norm of gradient & 1e-3 \\
Momentum of optimizer & 0.0 & Batch size & \num{4096} \\
Number of training steps & 2e5 & Clipping window of gradient & 5 \\
MCMC burn in & 1e3 & MCMC steps between each iterations & 20 \\
MCMC move width & 2e-2 & Target MCMC acceptance & 55\% \\
Precision & Float64 & Number of inference steps & 5e4 \\
\bottomrule
\end{tabular}
\caption{\textbf{Recommended hyperparameters}
}\label{tab:ge_hyper}
\end{table}
\begin{table}[htb]
\centering
\begin{tabular}{|c|c|c|c|c|c|}\hline
System & Layer dimension & Layer & Determinants & Batch size & Training steps \\\hline
Hydrogen chain & (256, 32) & 3 & 8 & 4096 & 1e5\\
Graphene & (256, 32) & 4 & 8 & 4096 & 3e5\\
$2\times2\times2$ Lithium hydride& (256, 32)& 4& 8 & 4096 & 3e5\\
$3\times3\times3$ Lithium hydride & (256, 32)& 4& 1& 8192 & 4e5\\
Homogeneous electron gas & (256, 32) & 3 & 1 & 4096 & 3e5 \\
\hline
\end{tabular}
\caption{\textbf{Some system dependent hyperparameters}}
\label{tab:hyper}
\end{table}
\section{Hydrogen chain}
\subsection{Training curve}
The training curve of ${\rm H}_{10}$ in PBC is plotted in Supplementary Fig.~\ref{fig:h10_training}.
The correlation error is defined as
\begin{equation}
{\rm Correlation\ error} = \Big(1 - \frac{E_{\rm Net}-E_{\rm HF}}{E_{\rm DMC} - E_{\rm HF}}\Big) \times 100\%\ ,
\end{equation}
where $E_{\rm HF}$ is calculated using the ccpvdz basis set and $E_{\rm DMC}$ is taken from Ref.~\cite{hydrogen_chain}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.9\columnwidth]{hydrogen_train.pdf} \caption{\textbf{Hydrogen chain training curve.} For clarity, at each iteration number, we plot the median correlation error of the last 10$\%$ of the corresponding iteration.}
\label{fig:h10_training}
\end{figure}
\subsection{${\rm H}_{10}$ dissociation curve}
Energy of ${\rm H}_{10}$ chain per atom is given in Supplementary Table~\ref{tab:h10}, LR-DMC and VMC results from Ref.~\citep{hydrogen_chain} are also listed for comparison.
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|}\hline
Bond length (${\rm \AA}$)& Net & LR-DMC(LDA) & VMC(LDA) \\\hline
1.4 & -0.551677(1) & -0.55178(1) & -0.55049(1) \\
1.6 & -0.568740(1) & -0.56881(1) & -0.56752(1) \\
1.8 & -0.572922(1) & -0.57304(1) & -0.57172(1) \\
2.0 & -0.570401(1)& -0.57055(1)& -0.56911(1)\\
2.4 & -0.556861(1) & -0.55703(1)& -0.55522(1)\\
2.8 & -0.540783(1) & -0.54102(1)& -0.53831(1)\\
\hline
\end{tabular}
\caption{\textbf{Energy of ${\rm H}_{10}$ chain}.}
\label{tab:h10}
\end{table}
\subsection{Finite-size error extrapolation}
Energies of different hydrogen chains are given in Supplementary Table~\ref{tab:hn}.
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|}\hline
Size & Net & LR-DMC(LDA) & VMC(LDA) \\\hline
10 & -0.572922(1) & -0.57304(1) & -0.57172(1) \\
18 & -0.567776(1) & -0.56796(1) & -0.56644(1)\\
30 & -0.566114(1) & -0.56627(1) & -0.56478(1)\\
50 & -0.565419(1) & -0.56560(1) & -0.56409(1)\\
\hline
\end{tabular}
\caption{Energies of different hydrogen chains, energies are given in Hartree and the bond length of hydrogen chain is fixed at 1.8 Bohr.}
\label{tab:hn}
\end{table}
\section{Graphene}
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|}\hline
Atom & Position (${\rm \AA}$) & Lattice vector & Position (${\rm \AA}$) \\\hline
C1 & (1.421, 0.0, 0.0) & $\mathbf{a}_1$ & (2.1315, -1.2306, 0.0)\\
C2 & (2.842, 0.0, 0.0) & $\mathbf{a}_2$ & (2.1315, 1.2306, 0.0)\\
& & $\mathbf{a}_3$ & (0, 0, 52.9177)\\
\hline
\end{tabular}
\caption{\textbf{Geometry of Graphene}}
\label{tab:graphene}
\end{table}
\subsection{Geometry}
The primitive cell lattice vectors as well as carbon atom coordinates are given in Supplementary Table~\ref{tab:graphene}. The size of supercell is $2\times 2$.
\subsection{Twist average boundary condition (TABC)}
A $3\times3$ Monkhorst-Pack mesh in the first Brillouin zone of the supercell reciprocal space with $\Gamma$ point centered is used to approximate the twist average integral, which reads
\begin{equation}
\begin{gathered}
E_{\rm TABC}=\frac{\Omega_S}{(2\pi)^3}\int_{\rm 1. B.Z.} d^3\mathbf{k}_{S}~\frac{\Psi^*_{\mathbf{k}_S}\hat{H}_S\Psi_{\mathbf{k}_S}}{\Psi^*_{\mathbf{k}_S}\Psi_{\mathbf{k}_S}} \approx \frac{1}{9} E_{\rm \mathbf{k}_1} + \frac{2}{3}E_{\rm \mathbf{k}_2} + \frac{2}{9}E_{\rm \mathbf{k}_3}, \\
\mathbf{k}_1 = 0,\
\mathbf{k}_2 = \frac{1}{3} \mathbf{b}_1^S + \frac{1}{3} \mathbf{b}_2^S,\
\mathbf{k}_3 = \frac{2}{3} \mathbf{b}_1^S + \frac{1}{3} \mathbf{b}_2^S,
\end{gathered}
\end{equation}
and the weight factors origin from the different number of symmetry equivalent $\mathbf{k}$ points.
\subsection{Training curves}
Training curves at different $\mathbf{k}_S$ are plotted in Supplementary Fig.~\ref{fig:graphene_training}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\columnwidth]{Graphene_training.pdf} \caption{\textbf{$2\times 2$ Graphene training curve.} For clarity, at each iteration number, we show the median energy per primitive cell over the last 10$\%$ of iteration.}
\label{fig:graphene_training}
\end{figure}
The final results are listed in Supplementary Table~\ref{tab:graphene_energy}.
The energy of an isolated carbon atom is taken from Ref.~\citep{FermiNet}, $E=-37.84471$ Hartree.
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|}\hline
& $\mathbf{k}_1$ & $\mathbf{k}_2$ & $\mathbf{k}_3$ \\\hline
Energy (Hartree) & -76.15588(6) & -76.24949(5) & -76.26314(5) \\
\hline
\end{tabular}
\caption{\textbf{Energy of graphene at different twists}}
\label{tab:graphene_energy}
\end{table}
\subsection{Structure factor correction}
TABC technique is usually combined with structure factor corrections \citep{sf_correction}, and the combination is now seen as the standard scheme of applying QMC to solids. Structure factor $S(\mathbf{k})$ is calculated to correct the exchange-correlation part, namely $V_{\rm xc}$, of the total potential energy, which reads
\begin{equation}
\begin{gathered}
\frac{\Delta V_{\rm xc}}{N_{\rm e}}=\frac{2\pi}{\Omega_S}\lim_{\mathbf{k}\rightarrow 0}\frac{S(\mathbf{k})}{\mathbf{k}^2}\ , \\ S(\mathbf{k})=\frac{1}{N_{\rm e}}[\langle\rho(\mathbf{k})\rho^*(\mathbf{k})\rangle-\langle\rho(\mathbf{k})\rangle \langle\rho^*(\mathbf{k})\rangle] \ ,\
\rho(\mathbf{k})=\sum_i \exp(i\mathbf{k}\cdot \mathbf{r}_i)\ ,
\end{gathered}
\end{equation}
where $\mathbf{r}_i$ refers to the coordinate of each electron, and $N_{\rm e}$ denotes the number of electrons in the simulation cell. The calculated $S(\mathbf{k})$ and corresponding $\Delta V_{\rm xc}$ of $\Gamma$ point is plotted in Supplementary Fig.~\ref{fig:graphene_sf}, and corrections of all twists are quite close to each other.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\columnwidth]{sf.pdf} \caption{\textbf{Structure factor correction of Graphene.} The lines are fitted with the formula: $S(\mathbf{k})=1-\exp(-a\cdot \mathbf{k}^2)$.}
\label{fig:graphene_sf}
\end{figure}
The final correction from structure factor is 0.00122 Hartree / atom.
\section{Lithium hydride}
\subsection{Geometry}
Lithium hydride crystal has a rock-salt structure, whose lattice vectors and atom positions are given in Supplementary Table~\ref{tab:lih}.
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|}\hline
Atom & Position & lattice vector & Position \\\hline
Li & (0.0, 0.0, 0.0) & $\mathbf{a}_1$ & (0.0, L/2, L/2)\\
H & (L/2, L/2, L/2) & $\mathbf{a}_2$ & (L/2, 0.0, L/2)\\
& & $\mathbf{a}_3$ & (L/2, L/2, 0.0) \\
\hline
\end{tabular}
\caption{\textbf{Geometry of LiH crystal}}
\label{tab:lih}
\end{table}
\subsection{Training curves}
Training curves of the $2\times2\times2$ LiH crystal is plotted in Supplementary Fig.~\ref{fig:lih_training}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\columnwidth]{lih_training.pdf} \caption{\textbf{$2\times 2\times2$ LiH training curve.} For clarity, at each iteration number, we show the median energy of primitive cell over the last 10$\%$ of iteration.}
\label{fig:lih_training}
\end{figure}
\subsection{Dissociation curve}
The energy of $2\times2\times2$ LiH is listed in Supplementary Table~\ref{tab:lih_energy}.
The energy of an isolated lithium atom is taken from Ref.~\citep{FermiNet}, $E=-7.47798 ~{\rm Hartree}$.
Corresponding Hatree-Fock corrections are calculated with the ccpvdz basis set and the convergence behavior of HF calculation is plotted in Supplementary Fig.~\ref{fig:hf_fn}.
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|c|c|}\hline
L (${\rm \AA}$)& Net & HF correction & L (${\rm \AA}$) & Net & HF correction \\\hline
3.4 & -8.12185(1) & -0.0099 & 4.2 & -8.15112(1) & -0.0004\\
3.6 & -8.13738(1) & -0.0067 & 4.4 & -8.14967(1) & 0.0009\\
3.8 & -8.146147(1) & -0.0042 & 4.6 & -8.14502(1) & 0.0020\\
4.0 & -8.15096(1)& -0.0021& 4.8 & -8.14094(1)& 0.0030\\
\hline
\end{tabular}
\caption{\textbf{Energy of $2\times 2 \times 2$ LiH crystal.} Energies are all given in Hartree.}
\label{tab:lih_energy}
\end{table}
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\columnwidth]{hf_fn.pdf} \caption{\textbf{Hartree-Fock corrections.} The convergence behavior of HF calculations with respect to the number of $\mathbf{k}$ points.}
\label{fig:hf_fn}
\end{figure}
\subsection{Birch-Murnaghan fit}
The third order Birch-Murnaghan equation of state is employed to fit the dissociation curve, which reads
\begin{equation}
E(V) = E_0 + \frac{9V_0B_0}{16}\Big\{\Big[\Big(\frac{V_0}{V}\Big)^{2/3}-1\Big]^3B_0'+\Big[\Big(\frac{V_0}{V}\Big)^{2/3}-1\Big]^2\Big[6-4\Big(\frac{V_0}{V}\Big)^{2/3}\Big]\Big\},
\end{equation}
where $E_0,V_0,B_0,B_0'$ are fitted quantities, their results and corresponding experiment data \citep{qmc_lih} are listed in Supplementary Table~\ref{tab:bh_fit}.
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|c|}\hline
&$a_0$ (\AA) & $B_0$ (GPa) & $E_{\rm coh} $ (eV) \\\hline
Net & 4.022 & 36.89 & -4.757 \\
Exp & 4.061(1) & 33-38 & -4.778,-4.759 \\
\hline
\end{tabular}
\caption{\textbf{Parameters of Birch-Murnaghan equation of state}}
\label{tab:bh_fit}
\end{table}
\subsection{$3\times3\times3$ LiH}
The training curve of the $3\times3\times3$ LiH crystal at its equilibrium lattice constant ${\rm L}=4.061{\rm \AA}$ is plotted in Supplementary Fig.~\ref{fig:lih_333_training}, corresponding Hartree-Fock corrections are also given. The final inference results from neural network are listed in Supplementary Table~\ref{tab:lih_333_energy}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\columnwidth]{lih_333_training.pdf} \caption{\textbf{$3\times 3\times3$ LiH.} Left panel plots the training curve of the $3\times3\times3$ LiH. For clarity, at each iteration number, we show the median energy per unit cell over the last 10$\%$ of iteration. Right panel plots the corresponding Hartree-Fock corrections with the ccpvdz basis set.}
\label{fig:lih_333_training}
\end{figure}
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|}\hline
L (${\rm \AA}$)& Net & HF correction \\\hline
4.061 & -8.16020(2) & 0.0069\\
\hline
\end{tabular}
\caption{\textbf{Energy of the $3\times 3 \times 3$ LiH crystal.} Energies are all given in Hartree.}
\label{tab:lih_333_energy}
\end{table}
\section{Homogeneous electron gas}
\subsection{Training curve}
The training curve of HEG system containing 54 electrons is plotted in Supplementary Fig.~\ref{fig:electron_gas_training}.
$E_{\rm HF}$ and $E_{\rm DMC}$ are taken from Ref. \cite{heg_method_3}.
Final results of neural network, BF-DMC, BF-VMC and DCD \citep{heg_method_3,heg_method_5} are listed in Supplementary Table~\ref{tab:electron_gas}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\columnwidth]{electron_gas_training.pdf} \caption{\textbf{Homogeneous electron gas.} For clarity, at each iteration number, we show the median correlation error over the last 10$\%$ of iteration.}
\label{fig:electron_gas_training}
\end{figure}
\begin{table}[!h]
\centering
\begin{tabular}{|c|c|c|c|c|}\hline
$r_s$ & Net & BF-DMC & BF-VMC & DCD \\\hline
0.5 & 3.221226(2) & 3.22112(4) & 3.22132(7) & 3.22052\\
1 & 0.530019(1) & 0.52989(4) & 0.53009(3) & 0.53001\\
2 & -0.013840(1) & -0.013966(9) & -0.01382(2) & -0.01286 \\
5 & -0.0788354(2) & -0.079036(3) & -0.078961(5) & -0.07655\\
10 & -0.0542785(1) & -0.054443(2) & -0.054389(2)& -0.05157\\
20 & -0.0316886(1) & -0.032047(2) & -0.0319984(8)& -0.02925\\
\hline
\end{tabular}
\caption{\textbf{Energy per electron of HEG at different mean radius of electrons $r_s$.} HEG system contains 54 electrons and energies are all given in Hartree, $r_s$ is given in Bohr.}
\label{tab:electron_gas}
\end{table}
\section{Bader charge analysis}
Detailed Bader charge analysis \cite{bader_charge} is applied to conventional LiH crystal, and the result is plotted in Fig.~\ref{fig:bader}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\columnwidth]{bader.pdf} \caption{\textbf{Bader charge of LiH crystal.} Calculated Bader charge of atoms in LiH crystal, point denotes the number uniformly dividing the crystal.}
\label{fig:bader}
\end{figure}
According to the result, Li and H atoms in LiH become ${\rm Li}^{0.67+}$ and ${\rm H}^{0.67-}$ ions respectively.
\section{Introduction}
\section{Introduction}
Solving the many-body electronic structure of real solids from \textit{ab initio} is one of the grand challenges in condensed matter physics and materials science \citep{KohnNobel}.
More accurate \textit{ab initio} solutions can push the limit of our understanding in many fundamental and mysterious emergent phenomena, such as superconductivity, light-matter interaction, heterogeneous catalysis, to name just a few \citep{martin_2004}.
The current workhorse method
is density functional theory (DFT), whose accuracy
depends quite sensitively on the choice of the so-called exchange-correlation functional and unfortunately there lacks a systematic routine towards the exact \citep{DFT-JONES,kirkpatrick_pushing_2021}.
Other commonly used \textit{ab initio} quantum chemistry methods, such as the coupled-cluster and configuration interaction theories \cite{simons2020}, can provide more accurate solutions for molecules but face severe difficulty when applied to solid systems.
Recently, several breakthroughs have been made in applying these quantum chemistry methods on solids \citep{Booth2013TowardsAE,mihm_shortcut_2021}, driving the study of solid systems towards a new frontier.
Meanwhile, in the last few years, new attempts to tackle the correlated wavefunction problem in molecules or model Hamiltonians using neural network based approaches have been reported by different groups \citep{han_solving_2019,RBM_ising,FermiNet,spencer2020better,PauliNet,ferminetecp, ferminet_dmc}.
The key idea is to use the neural network as the wavefunction ansatz in quantum Monte Carlo (QMC) simulations.
The stochastic nature of QMC enables a considerably economical
\citep{guther_neci_2020,FoulkesReview,ShiAFQMC,Booth2013TowardsAE} time scaling and efficient parallelization.
The universal approximation theorem behind neural network based ansatz significantly improves the accuracy of the traditional QMC method and the strategy has been proved successful in studying small molecules \cite{FermiNet,spencer2020better,PauliNet}.
However, how to apply such neural network ansatz for real solids, i.e. how to apply periodic boundary conditions (PBC) in the neural network, and whether it can describe the long-range electron correlations in extended systems remain as open questions.
Here we propose a powerful periodic neural network ansatz, based on which we develop a highly efficient QMC method for \textit{ab initio} calculation of real solid and general periodic systems with unprecedented accuracy.
We apply our method to periodic hydrogen chains, graphene, lithium hydride (LiH) crystal, and homogeneous electron gas.
These systems cover a wide range of interests, including materials dimension from one to three, electronic structure from metallic to insulating, and bonding type from covalent to ionic.
The calculated dissociation curve, cohesive energy and correlation energy, can be compared satisfactorily with available experimental values and other state-of-the-art computational approaches.
Electron densities of typical systems are further calculated to test our neural network and explore the underlying physics.
All the results demonstrates that our method can achieve accurate electronic structure calculations of real solid/periodic systems.
\section{Results}
\subsection{Neural network for solid system}
\begin{figure*}[t]
\centering
\includegraphics[width=2.0\columnwidth]{figs/net_row.pdf}
\caption{\textbf{Sketch of neural network architecture.} The electron coordinates ${\mathbf{r}_i}$ are passed to two channels. In the first one, they build the periodic distance features $d(\mathbf{r})$ using the periodic metric matrix $\mathbf{M}$ and the lattice vectors ${\mathbf{a}}$, and then $d(\mathbf{r})$ features are fed into two molecular neural networks, that represent separately the real and the imaginary part of wavefunction. In the second channel, ${\mathbf{r}_i}$ constructs the plane-wave phase factors on a selected set of crystal momentum vectors. The total wavefunctions of solids are constructed by the two channels following the expression of Eq.~\eqref{eq:ansatz}.}
\label{fig:net}
\end{figure*}
Periodicity and anti-symmetry are two fundamental properties of the wavefunction of a solid system.
The anti-symmetry can be ensured by the Slater determinant, which has been commonly used as the basic block in molecular neural networks.
We also express the wavefunction by two Slater determinants of one spin-up channel and one spin-down channel,
\begin{equation}
\Psi(\mathbf{r}) = {\rm Det}_\uparrow[e^{i\mathbf{k}\cdot \mathbf{r}_\uparrow}u^\uparrow_{\rm mol}(d)]{\rm Det}_\downarrow[e^{i\mathbf{k}\cdot \mathbf{r}_\downarrow}u_{\rm mol}^\downarrow(d)]\ .
\label{eq:ansatz}
\end{equation}
In this regard, our ansatz resembles the structure of FermiNet \citep{FermiNet,spencer2020better}, whereas other neural network wavefunction ansatz may include extra terms in addition to the Slater determinants \citep{PauliNet}.
Each determinant is then constructed from a set of periodic orbitals, which inherits the physics captured by the Bloch function form by a product of phase factor $e^{i\mathbf{k}\cdot \mathbf{r}}$ and collective molecular orbital $u_{\rm mol}$.
Fig.~\ref{fig:net} displays more details on the structure of our neural network.
Building an efficient and accurate periodic ansatz is the key step in developing \textit{ab initio} methods for solid.
Here we have followed the recently proposed scheme of Whitehead et al. to construct a set of periodic distance features $d(\mathbf{r})$ \citep{PeriodicDis} using lattice vectors in real and reciprocal space $(\mathbf{a}_i,\mathbf{b}_i)$,
\begin{equation}
\begin{gathered}
d(\mathbf{r})=\frac{\sqrt{\mathbf{A}\mathbf{M}\mathbf{A}^T}}{2\pi}~,~
\mathbf{A}=(\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3)\ ,\\
\mathbf{M}_{ij}=f^2(\omega_i)\delta_{ij}+g(\omega_i)g(\omega_j)(1-\delta_{ij})
~,~\omega_i=\mathbf{r}\cdot\mathbf{b}_i\ .
\end{gathered}
\label{eq:periodic_metric}
\end{equation}
The periodic metric matrix $\mathbf{M}$ is constructed by periodic functions $f,g$, which retains ordinary distances at the origin and regulates them to periodic ones at far distances, ensuring asymptotic cusp form, continuity, and periodicity requirement at the same time.
The constructed periodic distance features $d(\mathbf{r})$ can then be fed into molecular neural networks to form collective orbitals $u_{\rm mol}$. Specifically, in this work we represent the molecular networks with FermiNet \citep{FermiNet}, which incorporates the electron-electron interactions.
The inclusion of all-electron features promotes the traditional single-particle orbitals to the collective ones,
and hence the description of wavefunction and correlation effects can be improved while fewer Slater determinants are required.
In addition, the wavefunction of solid systems is necessarily complex-valued,
and we introduce two sets of molecular orbitals
to represent the real and imaginary parts of the solid wavefunction, respectively.
The plane-wave phase factors $e^{{\rm i}\mathbf{k}\cdot\mathbf{r}}$ in Fig.~\ref{fig:net} are used to construct the Bloch function like orbitals, and
the corresponding $\mathbf{k}$ points are selected to minimize the Hartree-Fock (HF) energy.
Based on the variational principle, our neural network is trained using the variational Monte Carlo (VMC) approach. To efficiently optimize the network, a modified Kronecker factored curvature estimator (KFAC) optimizer \citep{kfac} is adopted,
which significantly outperforms traditional energy minimization algorithms.
Calculations are also ensured by efficient and massive parallelization on multiple nodes of high-performance GPUs.
More details on the theories, methods, and computations are included in the Methods section and the supplementary information.
\begin{figure*}[t]
\centering
\includegraphics[width=2.0\columnwidth]{figs/unify_sf.pdf}
\caption{\textbf{Calculated results of neural network}. Our results are all labeled as Net.
${\rm \mathbf{a}}$, ${\rm H}_{10}$ dissociation curve is plotted.
${\rm \mathbf{b}}$, energy of different hydrogen chain sizes N, the bond length of hydrogen chain is fixed at 1.8 Bohr. LR-DMC and VMC both use TZ-LDA basis and AFQMC is pushed to complete basis limit \citep{hydrogen_chain}.
${\rm \mathbf{c}}$, structure of graphene.
${\rm \mathbf{d}}$, the cohesive energy per atom of $\Gamma$ point and finite-size error corrected result is plotted. Experiment cohesive energy is from Ref.~\cite{Graphene_exp}. Graphene is calculated at its equilibrium length 1.421 \AA.
${\rm \mathbf{e}}$, structure of rock-salt lithium hydride crystal.
${\rm \mathbf{f}}$, equation of state of LiH crystal is plotted, fitted Birch-Murnaghan parameters and experimental data are also given. HF corrections are calculated using ccpvdz basis, and $E_\infty^{\rm HF}$ is approximated by $E_{\rm N=8}^{\rm HF}$. The arrows denote the corresponding HF corrections.
${\rm \mathbf{g}}$, plot of homogeneous electron gas system.
${\rm \mathbf{h}}$, correlation error of 54 electrons HEG systems at different $r_s$. DCD and BF-VMC results are displayed for comparison, and BF-DMC data is used as reference \citep{heg_method_3,heg_method_5}.}
\label{fig:unify}
\end{figure*}
\subsection{Ground-state energy}
\subsubsection{Hydrogen chain}
Hydrogen chain is one of the simplest models in condensed matter research. Despite its simplicity, hydrogen chain is a challenging and interesting system, serving as a benchmark system for electronic structure methods and featuring intriguing correlated phenomena \citep{hydrogen_chain}.
The calculated energy of the periodic $\text{H}_{10}$ chain as a function of the bond length is shown in Fig.~\ref{fig:unify}a.
The results from lattice-regularized diffusion Monte Carlo (LR-DMC) and
traditional VMC
are also plotted for comparison \citep{hydrogen_chain}.
We can see that our results nearly coincide with the LR-DMC results and significantly outperform traditional VMC (see Supplementary Table 3).
In Fig.~\ref{fig:unify}b, the energy of hydrogen chains of different atom numbers are calculated for extrapolation to the
thermodynamic limit (TDL).
The shaded bar in Fig.~\ref{fig:unify}b illustrates the extrapolated energy of the periodic hydrogen chain at TDL from auxiliary field quantum Monte Carlo (AFQMC), which is considered as the current state-of-the-art along with LR-DMC.
Our TDL result is comparable with both AFQMC and LR-DMC (see Supplementary Table 4).
\subsubsection{Graphene}
Graphene is arguably the most famous two-dimensional system (Fig.~\ref{fig:unify}c)
receiving broad attention in the past two decades for its mechanical, electronic, and chemical applications \citep{geim_nobel}.
Here we carry out simulations
to estimate its cohesive energy, which measures the strength of C-C chemical bonding and long-range dispersion interactions.
The calculations are performed on a $2\times2$ supercell of graphene using twist average boundary condition (TABC) \citep{twist_average} in conjunction with structure factor $S(\mathbf{k})$ correction \citep{sf_correction} (see Supplementary Fig. 3) to
reduce the finite-size error.
The calculated results are plotted in Fig.~\ref{fig:unify}d along with the experimental value \cite{Graphene_exp}, and it shows that our neural network can deal with graphene very well, producing a cohesive energy of graphene within 0.1 eV/atom to the experimental reference (see Supplementary Table 6).
We also plotted the results with PBC,
namely the $\Gamma$ point only result, which deviates from the experiment data by 1.25 eV/atom.
\subsubsection{Lithium hydride crystal}
For three-dimensional system, we consider the LiH crystal with a rock-salt structure (Fig.~\ref{fig:unify}e),
another benchmark system for accurate \textit{ab initio} methods \citep{Booth2013TowardsAE, scf_lih, qmc_lih}.
Despite consisting of only simple elements, LiH represents typical ionic and covalent bondings upon changing the lattice constants.
Using our neural network, we first simulate the equation of state of LiH on a $2\times2\times2$ supercell, as shown in Fig.~\ref{fig:unify}f.
In addition, we employ a standard finite-size correction based on Hartree-Fock calculations of a large supercell (see Supplementary Fig. 5).
In Fig.~\ref{fig:unify}f we also show the Birch-Murnaghan fitting to the equation of state, based on which we can obtain thermodynamic quantities such as the cohesive energy, the bulk modulus, and the equilibrium lattice constant of LiH.
As shown in the inset, our results on the thermodynamic quantities agree very well with experimental data \citep{scf_lih} (see Supplementary Table 8, 9).
For further validation, we have also computed directly the $3\times3\times3$ supercell of LiH at its equilibrium length $4.061 \AA$, which contains 108 electrons.
To the best of our knowledge, this is the largest electronic system computed using a high-quality neural network ansatz. The $3\times3\times3$ supercell calculation predicts the total energy per unit cell of LiH is $-8.160$ Hartree and the cohesive energy per unit cell is $-4.770~{\rm eV}$ after the finite-size correction (see Supplementary Table 10), which is also very close to the experimental value $-4.759~{\rm eV}$ \citep{scf_lih}.
\subsubsection{Homogeneous electron gas}
In addition to the real solids, our computational framework can also apply straightforwardly to model systems such as homogeneous electron gas (HEG).
HEG has been studied for a long time to understand the fundamental behavior of metals and electronic phase transitions \citep{ElectronGas}.
Several seminal \textit{ab initio} works have reported the energy of HEG at different densities \citep{ElectronGas, heg_method_3,heg_method_5, wapnet, ferminetGas}, and
recently more investigations have been conducted using neural network ansatz \citep{wapnet,ferminetGas}.
Here we broaden our tests to simulate a simple cubic cell containing 54 electrons in closed-shell configuration (Fig.~\ref{fig:unify}g).
Fig.~\ref{fig:unify}h shows the correlation energy error from our neural network calculations on HEG at 6 different densities from $r_s = 0.5 $ Bohr to 20.0 Bohr.
The state-of-the-art results, namely VMC with backflow correlation (BF) \citep{heg_method_3} and distinguishable cluster with double excitations (DCD) \citep{heg_method_5} are also plotted for comparison, and the most accurate BF-DMC result is used as the reference energy of correlation error.
Overall, our neural network performs very well, with an error of less than 1\% in a wide range of density (see Supplementary Table 11).
Generally, the correlation error increases as the density of HEG decreases when the correlation effects become larger, which also appears in DCD calculations.
\subsection{Electron density}
\begin{figure}[t]
\centering
\includegraphics[width=0.70\columnwidth]{figs/h10_order.pdf}
\caption{\textbf{Electron density of $\mathbf{{\rm H}_{10}}$ chains.} Horizontal axis is scaled by the corresponding bond length. Complex polarization modulus ${|Z|}$ as a function of bond length is plotted in the inset. }
\label{fig:h10_density}
\end{figure}
Besides the total energy of solid systems, the electron density is also a key quantity to be calculated.
For example, electron density is crucial for characterizing different solids, such as the difference between insulators and conductors, and the distinct nature of ionic and covalent crystals.
In DFT the one-to-one correspondence between many-body wavefunction and electron density proved by Hohenberg and Kohn in 1964 suggests that electron density is a fundamental quantity of materials.
However,
an interesting survey found that while new functionals in DFT improve the energy calculation the obtained density somehow can deviate from the exact \citep{dft_science}.
Here, with our accurate neural network wavefunction, we can also obtain accurate electron density of solids
and provide a valuable benchmark and guidance for method development.
A conductor features free-moving electrons, which would have macroscopic movements under external electric fields.
In contrast, electrons are localized and constrained in insulators and cause considerable electron resistance.
In Fig.~\ref{fig:h10_density}, as an example, we show the calculated electron density of the hydrogen chain at two different bond lengths.
As we can see, for the compressed hydrogen chain (${\rm L} = 2$ Bohr), the electron density is rather uniform and has small fluctuations.
As the chain is stretched, the electrons are more localized and the density profile has larger variations.
The observation is consistent with the well-known insulator-conductor transition on the hydrogen chain by varying the bond length.
To measure the transition more quantitatively, we further calculate the complex polarization $Z$ as the order parameter for insulator-conductor transition \citep{vmc_complex_polarization}.
A conducting state is characterized by a vanishing complex polarization modulus $|Z|\sim 0$, while an insulating state has finite $|Z|\sim1$. We can see that the conductor-insulator transition bond length of hydrogen chain is around 3 Bohr according to the calculated results, which is also consistent with previous studies \cite{vmc_complex_polarization}.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{figs/density.pdf}
\caption{\textbf{Electron density of solids.} \textbf{a}, structures of solids, where the lattice planes for plotting electron densities are indicated. \textbf{b}, electron density of diamond-structured Si in its ($01\bar{1}$) plane, ccECP[Ne] is employed, and the bond length of Si equals $5.42 \AA$. \textbf{c}, electron density of NaCl crystal in its $xy$-plane, ccECP[Ne] is employed, and the bond length of NaCl equals $5.7 \AA$. \textbf{d}, electron density of LiH crystal in its $xy$-plane, and the bond length of LiH equals $4.0 \AA$.}
\label{fig:density}
\end{figure}
Ionic and covalent bonds are the most fundamental chemical bonds in solids. While the physical pictures of these two types of bonding are very different, they both lie in the behavior of electrons as the "quantum glue" and electron density distribution is a simple way to visualize different bonding types.
Here we choose to calculate the electron density of the diamond-structured Si, rock-salt NaCl and LiH crystals at their equilibrium position. They are representative of covalent and ionic crystals and have also been investigated by other high-level wavefunction methods, e.g. AFQMC \citep{afqmc_density}.
Note that in the calculations of NaCl and Si, correlation-consistent effective core potential (ccECP) is employed to reduce the cost, which removes the inertia core electrons and keeps the behavior of active valence electrons \citep{ccecp, ferminetecp}.
The electron density of diamond-structured Si in its $(01\bar{1})$ plane is plotted in Fig.~\ref{fig:density}b. We can see that valence electrons are shared by nearest Si atoms, forming apparent Si-Si covalent bonds. In contrast, valence electrons are located around atoms in NaCl crystal as Fig.~\ref{fig:density}c shows. All the valence electrons are attracted around Cl atoms, forming effective $\rm{Na^+}$ and $\rm{Cl^-}$ ions in the crystal. Moreover, the electron density of LiH crystal is also calculated and plotted in Fig.~\ref{fig:density}d. LiH crystal is a moderate system between a typical ionic and covalent crystal.
According to the result, the electrons are nearly equally distributed near Li and H atoms for our network. Detailed Bader charge analysis \citep{bader_charge} manifests the atoms in the crystal become ${\rm Li^{0.67+}}$ and ${\rm H^{0.67-}}$ ions respectively (resolution $\sim {\rm 0.015 \AA}$), which slightly deviates from the stable closed-shell configuration (see Supplementary Note 6 for more details).
\section{Conclusion}
The construction of a wave function for solid systems is a crucial but unsolved problem in the neural network community.
The core mechanism of our neural network is the use of the periodic distance feature, which promotes molecule neural networks elegantly to the corresponding periodic ones and avoids time-consuming lattice summation.
Considering the high-accuracy results obtained in this work, our neural network can be further applied to study more delicate physics and materials problems, such as the phase transitions of solids, surfaces, interfaces, and disordered systems, to name just a few.
Our ansatz also offers a flexible extension to other neural networks and an easy integration into traditional computational techniques.
The naturally evolved many-body wavefunction from the neural network may provide more physical and chemical insights to emergent phenomena of complex materials.
\section{Methods}
\paragraph{Supercell approximation.}
Simulating a solid system requires solving the Schr\"{o}dinger equation of many electrons within a large bulk. Supercell approximation is usually adopted to simplify the problem,
considering a finite number of electrons and nuclei with periodic boundary conditions, whose Hamiltonian reads
\begin{equation}
\begin{gathered}
\hat{H}_S=\sum_i -\frac{1}{2}\Delta_i + \frac{1}{2}\sum_{\mathbf{L}_S,i,j}'\frac{1}{|\mathbf{r}_i-\mathbf{r}_j+\mathbf{L}_S|} \\
- \sum_{\mathbf{L}_S,i,I}\frac{Z_I}{|\mathbf{r}_i-\mathbf{R}_I+\mathbf{L}_S|} + \frac{1}{2}\sum_{\mathbf{L}_S,I,J}'\frac{Z_I Z_J}{|\mathbf{R}_I-\mathbf{R}_J+\mathbf{L}_S|}\ ,
\end{gathered}\label{eq:supercell_h}
\end{equation}
where $\mathbf{r}_i$ denotes the spatial position of i-th electron in the supercell. $\mathbf{R}_I,Z_I$ are the spatial position and charge of I-th nucleus and $\{\mathbf{L}_S\}$ is the set of supercell lattice vectors, which is usually a subset of primitive cell lattice vectors $\{\mathbf{L}_p\}$.
In order to simulate the real environments of electrons in solids, the interactions between the particles and their images are also included in $\hat{H}_S$, and the prime symbol in summation means $i=j$ terms are omitted for $\mathbf{L}_S=0$.
Supercell Hamiltonian $\hat{H}_S$ is invariant under translation of any electron by a vector in $\{\mathbf{L}_S\}$ as well as a simultaneous translation of all electrons by a vector in $\{\mathbf{L}_p\}$. As a consequence, two periodic conditions are required for the ground-state wavefunction $\Psi$,
\begin{equation}
\label{eq:periodic_con}
\begin{gathered}
\Psi(\mathbf{r}_1+\mathbf{L}_p,...,\mathbf{r}_N+\mathbf{L}_p)=\exp({i\mathbf{k}_p\cdot\mathbf{L}_p})\Psi(\mathbf{r}_1,...,\mathbf{r}_N)\ , \\
\Psi(\mathbf{r}_1+\mathbf{L}_S,...,\mathbf{r}_N)=\exp({i\mathbf{k}_S\cdot\mathbf{L}_S})\Psi(\mathbf{r}_1,...,\mathbf{r}_N)\ ,
\end{gathered}
\end{equation}
where $\mathbf{k}_S, \mathbf{k}_p$ denote the momentum vectors reduced in the first Brillouin zone of the supercell and the primitive cell, respectively. Eq.~\eqref{eq:periodic_con} and the anti-symmetry condition together form the fundamental requirements for $\Psi$. As the size of supercell increases, simulation results gradually converge to the thermodynamic limit of real solid system.
\paragraph{Wavefunction ansatz.}
In conventional QMC simulation of solids, Hartree-Fock type wavefunction anzatz composed of Bloch functions is often used, which reads
\begin{equation}
\begin{gathered}
\label{eq:hf}
\Psi^{\rm HF}_{\mathbf{k}_S,\mathbf{k}_p}(\mathbf{r})={\rm Det}\left|
\begin{matrix}
e^{i\mathbf{k}_1\cdot\mathbf{r}_1}u_{\mathbf{k}_1}(\mathbf{r}_1) & \cdots & e^{i\mathbf{k}_N\cdot\mathbf{r}_1}u_{\mathbf{k}_N}(\mathbf{r}_1) \\
\cdot & & \cdot\\
\cdot & & \cdot\\
\cdot & & \cdot\\
e^{i\mathbf{k}_1\cdot\mathbf{r}_N}u_{\mathbf{k}_1}(\mathbf{r}_N) & \cdots & e^{i\mathbf{k}_N\cdot\mathbf{r}_N}u_{\mathbf{k}_N}(\mathbf{r}_N) \\
\end{matrix}
\right|\ .
\end{gathered}
\end{equation}
In order to satisfy Eq.~\eqref{eq:periodic_con}, $\mathbf{k}_i$ in the determinant should lie on the grid of supercell reciprocal lattice vectors $\{\mathbf{G}_S\}$ offset by $\mathbf{k}_S$ within the first Brillouin zone of primitive cell. Moreover, $u_{\mathbf{k}}$ functions in Eq.~\eqref{eq:hf} should satisfy the translation invariant condition by the primitive cell lattice vectors,
\begin{equation}
\begin{gathered}
\label{eq:uk_con}
u_{\mathbf{k}}(\mathbf{r}+\mathbf{L}_p)=u_{\mathbf{k}}(\mathbf{r})\ .
\end{gathered}
\end{equation}
Following the strategy of FermiNet \citep{FermiNet}, Bloch functions in Eq.~\eqref{eq:hf} can be promoted with collective distances,
\begin{equation}
\begin{gathered}
\label{eq:promotion}
e^{i\mathbf{k}\cdot\mathbf{r}_i}u_{\mathbf{k}}(\mathbf{r}_i)\rightarrow e^{i\mathbf{k}\cdot\mathbf{r}_i}u_\mathbf{k}(\mathbf{r}_i;\mathbf{r}_{\neq i})\ ,
\end{gathered}
\end{equation}
where $\mathbf{r}_{\neq i}$ denotes all the electron coordinates except $\mathbf{r}_i$.
These collective orbitals are constructed to achieve the equivalence of electron permutations $P$,
\begin{equation}
P_{i,j}u_{\mathbf{k}_i}(\mathbf{r}_j;\mathbf{r}_{\neq j})=u_{\mathbf{k}_j}(\mathbf{r}_i;\mathbf{r}_{\neq i})\ ,
\label{eq:permutation}
\end{equation}
which combined with the Slater determinant ensure the anti-symmetry nature of electron.
Moreover, we use the periodic distance features $d(\mathbf{r})$ in Eq.~\eqref{eq:periodic_metric} to substitute ordinary $|\mathbf{r}|$ in the molecular neural network.
The periodic functions $f,g$ used in Eq.~\eqref{eq:periodic_metric} read
\begin{equation}
\begin{gathered}
\label{eq:fg}
f(\omega)=|\omega|~(1-\frac{|\omega/\pi|^3}{4})\ ,\\
g(\omega)=\omega~(1-\frac{3}{2}|\omega/\pi|+\frac{1}{2}|\omega/\pi|^2)\ ,
\end{gathered}
\end{equation}
and their arguments $\omega$ are reduced into $[-\pi,\pi]$.
Eq.~\eqref{eq:uk_con} can then be satisfied without causing discontinuity \citep{PeriodicDis}. For an overall sketch of the neural network, see Algorithm~\ref{alg:solidnet}. Note that the distance between electrons and nuclei is omitted for HEG system since it does not contain any nucleus. Specific hyperparameters of each system are listed in Supplementary Note 1.
\begin{algorithm}[h]
\caption{\textbf{Pseudocode of network}}
\label{alg:solidnet}
\begin{algorithmic}[1]
\Require electron positions $\{\mathbf{r}_1^\uparrow, \cdots, \mathbf{r}_{n^\uparrow}^\uparrow, \mathbf{r}_1^\downarrow, \cdots, \mathbf{r}_{n^\downarrow}^\downarrow\}$
\Require nuclear positions $\{\mathbf{R}_I\}$ in the primitive cell
\Require lattice vector $\{\mathbf{a}^{p,S}_1,\mathbf{a}^{p,S}_2,\mathbf{a}^{p,S}_3\}$ of primitive cell and supercell
\Require reciprocal lattice vector $\{\mathbf{b}^{p,S}_1,\mathbf{b}^{p,S}_2,\mathbf{b}^{p,S}_3\}$ of primitive cell and supercell
\Require occupied $\{\mathbf{k_i}\}$ points offered by Hartree-Fock method
\For{each electron e, atom I}
\State $\omega_{e,I}=(\mathbf{r}_e-\mathbf{R}_I)\cdot\{\mathbf{b}^{p}_1,\mathbf{b}^{p}_2,\mathbf{b}^{p}_3\}$
\State $\omega_{e,e'}=(\mathbf{r}_e-\mathbf{r}_{e'})\cdot\{\mathbf{b}^{S}_1,\mathbf{b}^{S}_2,\mathbf{b}^{S}_3\}$
\EndFor
\For {each electron e}
\State $\mathbf{h}_e=\{\Sigma_{i=1}^3g(\omega_{e,I}^i)\ \mathbf{a}^p_i, d(\omega_{e,I})\}$
\State $\mathbf{h}_{e,e'}=\{\Sigma_{i=1}^3g(\omega_{e,e'}^i)\ \mathbf{a}^S_i, d(\omega_{e,e'})\}$
\EndFor
\For{each layer l}
\State $\mathbf{g}^{l,\uparrow}=\frac{1}{n^\uparrow}\sum_e\mathbf{h}_{e}^{l,\uparrow}$
\State $\mathbf{g}^{l,\downarrow}=\frac{1}{n^\downarrow}\sum_e\mathbf{h}_{e}^{l,\downarrow}$
\For{each electron e, spin $\alpha$}
\State $\mathbf{g}^{l,\alpha,\uparrow}_e=\frac{1}{n^\uparrow}\sum_{e'}\mathbf{h}_{e,e'}^{l,\alpha,\uparrow}$
\State $\mathbf{g}^{l,\alpha,\downarrow}_e=\frac{1}{n^\downarrow}\sum_{e'}\mathbf{h}_{e,e'}^{l,\alpha,\downarrow}$
\State $\mathbf{f}^{l,\alpha}_e={\rm cat}(\mathbf{h}_e^{l,\alpha},\mathbf{g}^{l,\uparrow},
\mathbf{g}^{l,\downarrow}, \mathbf{g}_e^{l,\alpha,\uparrow},
\mathbf{g}_e^{l,\alpha,\downarrow})$
\State $\mathbf{h}_e^{l+1,\alpha}={\rm tanh}(\mathbf{V}^l\cdot \mathbf{f}_e^{l,\alpha}+\mathbf{b}^l)+\mathbf{h}_e^{l,\alpha}$
\State $\mathbf{h}_{e,e'}^{l+1,\alpha,\beta}={\rm tanh}(\mathbf{W}^l\cdot \mathbf{h}_{e,e'}^{l,\alpha,\beta}+\mathbf{c}^l)+\mathbf{h}_{e,e'}^{l,\alpha,\beta}$
\EndFor
\EndFor
\For {each orbital i}
\For{each electron e, spin $\alpha$}
\State $u_{i,e}^\alpha={\rm Orb}_{i,\alpha}^{\rm Re}\cdot \mathbf{h}_e^L + \mathbf{i}\times{\rm Orb}_{i,\alpha}^{\rm Im}\cdot\mathbf{h}_e^L$
\State $p_{i,e}^\alpha = \exp(\mathbf{i}\mathbf{k}_i\cdot\mathbf{r}_e^\alpha)$
\State ${\rm enve}_{i,e}^\alpha=\sum_I \pi_{i}^{I,\alpha}\exp(-\sigma_{i}^{I,\alpha} d(\omega_{e,I}))$
\State $\phi_{i,e}^\alpha=p_{i,e}^\alpha u_{i,e}^\alpha {\rm enve}^\alpha_{i,e}$
\EndFor
\EndFor
\State $\Psi={\rm Det}[\phi^{\uparrow}]{\rm Det}[\phi^{\downarrow}]$
\end{algorithmic}
\end{algorithm}
\paragraph{Neural network optimization.}
Parameters $\theta$ within the neural network can be optimized to minimize the energy expectation value $\langle E_l \rangle$, and the gradient $\nabla_\theta\langle E_l\rangle$ reads
\begin{equation}
\label{eq:energy_grad}
\begin{gathered}
\nabla_\theta\langle E_l\rangle = {\rm Re}[\langle E_l\nabla_\theta\ln\Psi^*\rangle - \langle E_l\rangle\langle\nabla_\theta\ln\Psi^*\rangle]\ , \\
E_l = \Psi^{-1}\hat{H}_S\Psi\ ,
\end{gathered}
\end{equation}
where $E_l$ denotes the local energy of neural network ansatz $\Psi$. Besides energy minimization, stochastic reconfiguration optimization \citep{SR} has also been widely adopted and proved to be much more efficient, whose gradient reads
\begin{equation}
\label{eq:sr}
\begin{gathered}
{\rm Grad} = F^{-1}\nabla_\theta\langle E_l\rangle\ ,\\
F_{ij} = {\rm Re}\Big[\langle\frac{\partial\ln\Psi^*}{\partial \theta_i}\frac{\partial\ln\Psi}{\partial \theta_j}\rangle-\langle\frac{\partial\ln\Psi^*}{\partial\theta_i}\rangle\langle\frac{\partial\ln\Psi}{\partial\theta_j}\rangle\Big]\ .
\end{gathered}
\end{equation}
In this work, we adopt a modified KFAC optimizer, which approximates $F$ as
\begin{equation}
\label{eq:kfac}
\begin{aligned}
F& ={\rm Re}\Big[\langle\frac{\partial\ln\Psi^*}{\partial {\rm vec}(W_l)}\frac{\partial\ln\Psi^T}{\partial {\rm vec}(W_l)}\rangle-\langle\frac{\partial\ln\Psi^*}{\partial{\rm vec}(W_l)}\rangle\langle\frac{\partial\ln\Psi^T}{\partial{\rm vec}(W_l)}\rangle\Big] \\
&= {\rm Re}\Big[\langle(a_l\otimes e_l^*)(a_l\otimes e_l)^T\rangle-\langle(a_l\otimes e_l^*)\rangle\langle(a_l\otimes e_l)\rangle^T\Big] \\
&\approx {\rm Re}\Big[\langle a_la_l^T\rangle \otimes \langle e_l^*e_l^T\rangle\Big]\ ,
\end{aligned}
\end{equation}
where $W_l$ denotes the weight parameters of layer $l$, and vec means vectorized form. $a_l,e_l$ denote the activation and sensitivity of layer $l$ respectively. Note that activation $a_l$ is always real-valued, which explains the absence of conjugation of $a_l$ in the second line. The first term in the bracket of Eq.~\eqref{eq:kfac} is approximated as the Kronecker product of the expectation values, and the second term is omitted for simplification.
\paragraph{Twist average boundary condition.}
TABC is a conventional technique to reduce the finite-size error due to the constrained size of supercell \citep{twist_average}. It averages the contributions from different periodic images of the supercell and improve the convergence on the total energy. The formula reads
\begin{equation}
\begin{gathered}
E_{\rm TABC}=\frac{\Omega_S}{(2\pi)^3}\int_{\rm 1. B.Z.} d^3\mathbf{k}_{S}~\frac{\Psi^*_{\mathbf{k}_S}\hat{H}_S\Psi_{\mathbf{k}_S}}{\Psi^*_{\mathbf{k}_S}\Psi_{\mathbf{k}_S}}\ ,
\end{gathered}
\end{equation}
where ${\rm 1. B. Z.}$ denotes the first Brillouin zone of supercell and the integral is practically approximated by a discrete sum of a Monkhorst-Pack mesh (see Supplementary Note 3.3 for more details).
\paragraph{Structure factor correction} Finite-size error can be further reduced via the structure factor $S(\mathbf{k})$ correction \citep{sf_correction}, which is usually calculated to correct the exchange-correlation potential $V_{\rm xc}$ and the formula reads
\begin{equation}
\begin{gathered}
\frac{\Delta V_{\rm xc}}{N_{\rm e}}=\frac{2\pi}{\Omega_S}\lim_{\mathbf{k}\rightarrow 0}\frac{S(\mathbf{k})}{\mathbf{k}^2}, \\ S(\mathbf{k})=\frac{1}{N_{\rm e}}[\langle\rho(\mathbf{k})\rho^*(\mathbf{k})\rangle-\langle\rho(\mathbf{k})\rangle \langle\rho^*(\mathbf{k})\rangle] \ ,\
\end{gathered}
\end{equation}
where $\lim_{\mathbf{k}\rightarrow 0}$ is practically estimated via interpolation (see Supplementary Note 3.4 for more details).
\paragraph{Empirical correction formula.}
Empirical formulas are also commonly employed to reduce the finite-size error \citep{FoulkesReview}, one of which reads
\begin{equation}
\begin{gathered}
E_\infty=E_{\rm N}^{\rm Net}+(E_{\infty}^{\rm HF}-E_{\rm N}^{\rm HF})\ .
\label{eq:fneq}
\end{gathered}
\end{equation}
The simulation size of high-accuracy methods is usually limited due to high computational costs. Hence methods with much more practical time scaling, such as HF, is usually used to give a posterior estimation of the finite-size error. All the results of LiH are corrected using this empirical formula with HF results in ccpvdz basis (see Supplementary Note 4.3 for more details).
\paragraph{Electron density analysis}
Electron density $\rho(\mathbf{r})$ is defined as
\begin{equation}
\rho(\mathbf{r})=N\int d^3\mathbf{r}_2\cdots d^3\mathbf{r}_N \ |\Psi(\mathbf{r},\mathbf{r}_2,\cdots,\mathbf{r}_N)|^2,
\label{eq:density}
\end{equation}
and it's practically evaluated by accumulating Monte Carlo samples of electrons on a uniform grid over the simulation cell. As for the complex polarization $Z$, it is defined as \citep{vmc_complex_polarization}
\begin{equation}
Z = \langle \exp(i\sum_i\frac{2\pi}{L}\mathbf{r}_i^{\parallel})\rangle,
\end{equation}
where $\mathbf{r}^\parallel$ denotes the projection of electron coordinate along the chain direction. Moreover, Bader charge is employed to estimate the charge partition on each atom \citep{bader_charge}.
The convergence test of Bader charge is shown in the Supplementary Fig.8.
\paragraph{Workflow and computational details.}
This work is developed upon open-source FermiNet and PyQMC on Github, deep learning framework JAX \citep{jax2018github} is used which supports flexible and powerful complex number calculation.
Ground-state energy calculations are performed with all-electrons. Diamond-structured Si and NaCl crystal are simulated with ccECP[Ne] \citep{ccecp}.
The neural network is pretrained by Hartree-Fock ansatz, obtained with
PySCF software \citep{pyscf}.
All the expectation values for distribution $|\Psi|^2$ are evaluated via the Monte Carlo approach, and then the energy and wavefunction is optimized using the modified KFAC optimizer (see Supplementary Fig.1, 2, 4, 6, 7).
The Ewald summation technique is implemented for the lattice summation in energy calculation.
After training is converged, energy is calculated in a separate inference phase. Concrete code of this work is developed on Github at \url{https://github.com/bytedance/DeepSolid}.
| {
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} | 2,754 |
namespace blackhole {
inline namespace v1 {
namespace stdext {
namespace {
TEST(string_view, CompareWithString) {
EXPECT_EQ(string_view("message"), std::string("message"));
EXPECT_EQ(std::string("message"), string_view("message"));
}
} // namespace
} // namespace stdext
} // namespace v1
} // namespace blackhole
| {
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} | 4,535 |
Event by admin
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He's Todd Rundgren, and he's returning to Australia for the first time in 5 years.
And to coincide with his 70th birthday and in conjunction with the tour, dedicated Australian and US fans are holding Todd-A-Roo, a special 3-day birthday celebration with the man himself, on Cockatoo Island, Sydney, from October 22nd- 24th October. Details at https://www.toddstock-2018.com/todd-a-roo
Backing Todd are Davey Lane's Drunken Blue Roosters. Lane (solo artist, member of You Am I and collaborator of Jimmy Barnes, Crowded House, Robyn Hitchcock, The Saints, Masters' Apprentices among others) heads the Roosters – Brett Wolfenden (drums), Tony Featherstone (keys) and Luke Hodgson (bass) – to bring Todd's vision of mind-bending psychedelia, perfect power pop, lilting piano balladry and flat-stick rock 'n roll to the stage for 5 very special shows in intimate venues across Australia's East. Promised is a show traversing the absolute greatest of his 5 decades in music along with songs he has never performed live on stage before.+ Google Calendar+ iCal Export
20 Wonderful Hotel Gearin Katoomba Features. Number 15 is Absolutely Stunning | {
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} | 500 |
Пенчо Николов Райков е български химик, работил главно в областта на органичната химия. Провежда изследвания върху окислителните процеси, механизма на асимилация, създава нови лабораторни прибори. Работи и в областта на аналитичната химия.
Биография
Пенчо Райков е роден през 1864 г. в Трявна. Средно образование завършва в пансиона на Тодор Минков в град Николаев. През 1888 г. завършва химия в Лайпцигския университет. Негова статия, публикувана през 1886 г. в "Хемише Централблат", е смятана за първата научна публикация на български автор в областта на химията.
След завръщането си в България Райков е учител в Педагогическото училище в Казанлък, а от 1890 г. – в Първа софийска мъжка гимназия, като започва да преподава и във Висшето училище (днес Софийски университет "Свети Климент Охридски"). От 1892 г. е редовен преподавател в университета, като от 1894 г. до пенсионирането си през 1935 г. оглавява катедрата по органична химия. През 1900 г. става действителен член на Българското книжовно дружество, днес Българска академия на науките. През 1908 – 1909 г. е ректор на университета.
Баща е на химичката Теодора Райкова, родена през 1893 г. Става негова асистентка в Софийския университет през 1918 година.
Пенчо Райков умира през 1940 г.
Библиография
"По въпроса за химическите процеси, по които става фабричното редуциране на ароматните нитросъединения в аминокиселините" (1914)
"По въпроса за асимилирането на CO2 от растенията" (1914)
"Нова теория за структурата на водородния прекис и за механизма на реакциите при неговите превръщания" (1928, на немски език)
"Органическа химия. Част 1. Мастни Съединения" (1935)
Източници
Български химици
Преподаватели в Софийския университет
Академици на БАН
Ректори на Софийския университет
Родени в Трявна
Погребани в Централните софийски гробища | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,173 |
Q: Access to polygon matrix from json files I load polygons from json file using loadGeoJson. In json file I have properties name and id. All Polygons loads correctly on the map.
How can I get matrix of all polygons? I need to check that my point containsLocation of one from this polygons. TIA 4 answer
A: I use :
var woj =[];
map.data.addListener('addfeature', function(event) {
var adm=event.feature.getProperty('id');
woj.push(adm);
});
That solves my problem.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,691 |
Israel set to approve over 900 new settlement homes in West Bank
International, Religion & Spirituality
Civil Administration is expected to okay 732 units in Modi'in Ilit; some isolated settlements to be retroactively approved after construction stopped by a High Court of Justice injunction.
By Chaim Levinson
The Civil Administration's Supreme Planning Council is to meet Wednesday morning to approve the construction of over 900 new homes in the West Bank.
The largest plan is for 732 units in Modi'in Ilit, a plan that was okayed by former Defense Minister Ehud Barak in January. This plan is at an advanced stage of authorization, so if the defense minister permits, construction could start shortly.
The adjacent Palestinian village of Dir Qadis objected to the plan, claiming that some of the construction was slated to take place on its lands. But last month, the planning council rejected the village's objections, saying there was no private land involved.
Other, smaller plans are being approved for isolated settlements. Shiloh will be getting 17 homes, whose construction had previously been stopped by a High Court of Justice injunction.
This retroactive approval is essentially a gift from the state to Ze'ev Hever, secretary general of Amana, the movement that started building the homes without a permit. The construction itself is the subject of an ongoing police investigation.
But it isn't clear whether any police representatives will come to the council hearing to find out who takes responsibility for the illegal construction.
Thirty-eight units in Kokhav Yaakov, near Ramallah, will also be getting retroactive approval. The settlement is home to Avi Roeh, chairman of the Yesha Council of settlements.
"After years in which the plan, and the change in parcellation, have been stuck in the Civil Administration, the time has come for the kibbutz to continue its trend of absorption and development and add families and returning children to the valley," said David Elhayani, chairman of the Jordan Valley Regional Council. "This is welcome news that will strengthen the valley."
Video: Israel slams EU ban on business with settlers
Video: Israel PM criticises EU funding ban
Video: EU laments leaking of Israeli settlement report
EU takes tougher stance on Israeli settlements
'Earthquake' directive will prohibit EU states from signing deals with Israel unless settlement exclusion clause is included
Harriet Sherwood in Jerusalem
The European Union has dealt a harsh blow to the Israeli settlement enterprise in a directive that insists all future agreements between the EU and Israel must explicitly exclude Jewish colonies in the West Bank or East Jerusalem.
The move, described by an Israeli official as an "earthquake", prompted furious criticism from the Israeli prime minister over "external diktats".
But it was hailed by Palestinians and their supporters as a significant political and economic sanction against settlements.
The EU guidelines will prohibit the issuing of grants, funding, prizes or scholarships unless a settlement exclusion clause is included. Israeli institutions and bodies situated across the pre-1967 Green Line – including the Golan Heights, occupied by Israel in 1967 and later annexed — will be automatically ineligible.
In order to secure agreements with the EU in the future, the Israeli government will be required to concede in writing that settlements in the West Bank and East Jerusalem are outside the state of Israel.
The directive, part of the 2014-20 financial framework, covers all areas of co-operation between the EU and Israel, including economics, science, culture, sports and academia.
In a broadcast statementon Tuesday evening , Binyamin Netanyahu said: "As prime minister of Israel, I will not allow the hundreds of thousands of Israelis who live in the West Bank, Golan Heights and our united capital Jerusalem to be harmed. We will not accept any external diktats about our borders. This matter will only be settled in direct negotiations between the parties."
The move was seen in Israel as a penalty that could in future extend to settlement produce and goods destined for European markets.
Israel has become increasingly concerned about the EU adopting a more robust stance against settlements. Some member states are pressing for an EU-wide policy of labelling produce and goods originating in settlements to allow consumers to make informed choices on purchases.
An EU statement said the guidelines "set out the territorial limitations under which the commission will award EU support to Israeli entities … concern has been expressed in Europe that Israeli entities in the occupied territories could benefit from EU support. The purpose of these guidelines is to make a distinction between the state of Israel and the occupied territories when it comes to EU support."
The directive follows a decision by EU foreign ministers last December that "all agreements between the state of Israel and the EU must unequivocally and explicitly indicate their inapplicability to the territories occupied by Israel in 1967". All Israeli settlements are illegal under international law.
Other Israeli government ministers weighed in with criticism. The centrist finance minister, Yair Lapid, said the EU move "pushes peace further away instead of bringing it closer" and was "another in a long line of decisions that isolate Israel". He urged a return to peace talks, adding: "This is a miserable decision, which was made in very bad timing and thus sabotages the efforts that US secretary of state John Kerry is putting into bringing the sides back to the negotiation table."
Ze'ev Elkin, Israel's rightwing deputy foreign minister who lives in a West Bank settlement, told Army Radio the directive was a "big mistake", saying: "This is more fuel for Palestinian rejectionism."
Another minister, Silvan Shalom, said: "Once again, Europe has demonstrated just how detached it is, how it can't really be a full partner to the negotiations."
A senior Israeli official, who declined to be identified, told the Guardian: "Israel will have to explicitly express in writing the EU's position. We don't believe the EU's position should be forced down our throats like geese." He said it was impossible for Israel to agree to such a demand.
The directive would affect "all realms of co-operation", he added, and would result in "rising tension and increased friction" and "create a lot of bad blood".
Another Israeli official told Haaretz, which disclosed the new guidelines, the move was an "earthquake" that unprecedentedly turns "understandings and quiet agreements that the [EU] does not work beyond the Green Line [into] formal, binding policy".
Hanan Ashrawi, a senior Palestinian official, welcomed the guidelines. "The EU has moved from the level of statements, declarations and denunciations to effective policy decisions and concrete steps, which constitute a qualitative shift that will have a positive impact on the chances of peace," she said.
"The Israeli occupation must be held to account, and Israel must comply with international and humanitarian law and the requirements for justice and peace."
The new requirements would affect the EuroMed Youth agreement, under negotiation, which involves joint youth projects and exchanges, said Haaretz.
Another example would be applications from Israel to the EU's research and technical development programme, an EU source told the Guardian.
The directive emerged as Kerry arrived in the region on his sixth visit in a drive to restart peace negotiations between Israel and the Palestinians. He is expected to meet the Palestinian president, Mahmoud Abbas, in Amman on Tuesday.
Unusually, the secretary of state is not scheduled to visit Jerusalem or meet the Israeli prime minister. Some analysts have suggested this is because Israel has signed up to Kerry's parameters for a resumption of talks, but he still needs agreement from the Palestinian side.
However, an unnamed Israeli minister was reported by Israel Radio as saying that Netanyahu's primary objective was merely to show willingness to negotiate and that he did not intend to engage in a far-reaching peace process.
http://www.guardian.co.uk/world/2013/jul/16/eu-israel-settlement-exclusion-clause
EU's new policy on Israeli settlements: The full guidelines
Guidelines state that any private Israeli entity that wants to receive funding from EU must demonstrate that it has no links to West Bank, East Jerusalem, or Golan Heights.
By Barak Ravid
Any Israeli entity seeking funding from or cooperation with the European Union will have to submit a declaration stating that the entity has no direct or indirect links to the West Bank, East Jerusalem or the Golan Heights, according to the new EU guidelines.
The guidelines, which condition all future agreements on Jerusalem's acknowledgement that its occupied territories are not part of Israel, have strained relations with the EU to unprecedented level.
The full text of the guidelines obtained by Haaretz states that any Israeli entity that wants to receive funding, participate in a project or apply for grants or prizes from EU foundations or institutions will have to submit such a declaration.
The guidelines also state that "only Israeli entities having their place of establishment within Israel's pre-1967 borders will be considered eligible" for consideration."
The place of establishment "is understood to be the legal address where the entity is registered, as confirmed by a precise postal address corresponding to a concrete physical location. The use of a post office box is not allowed."
The guidelines, which were drawn up by the European Commission, the EU's executive arm are expected to be officially released on Friday. The document was circulated among all the EU institutions, foundations, investment funds and aid organizations two weeks ago, as well as to all 28 EU member states. They go into effect on January 1.
A senior Israeli official said that according to the Foreign Ministry's preliminary legal opinion, the new guidelines are meant to apply to all the bodies and institutions of the EU itself.
With regard to each of its 28 member states, the Foreign Ministry believes the guidelines are not obligatory but are considered policy recommendations. Nevertheless, the ministry believes that many EU members will adopt the guidelines as their policy.
The document states that the European Commission, "will also endeavor to have the content of these guidelines reflected in international agreements or protocols thereto or Memoranda of Understanding with Israeli counterparts or with other parties."
The guidelines apply to "Israeli regional or local authorities and other public bodies, public or private companies or corporations and other private [entities] including non-governmental not-for-profit organizations."
The new guidelines will not apply to human rights organizations operating in the territories, the Golan Heights or East Jerusalem (like B'Tselem), or to NGOs that work toward promoting peace and operate in the territories, such as the Geneva Initiative or Peace Now.
The guidelines also do not apply to Israeli government ministries or national agencies or to private individuals.
The document makes it clear that the guidelines do not apply to any Palestinian body, governmental or private, in the West Bank or East Jerusalem, and that they do not affect agreements between the EU and the PLO or the Palestinian Authority.
With regard to research grants or scholarship funds, the guidelines are to be imposed on the initial recipient of the funds. However, the EU official who reviews the application has the authority to apply the guidelines to secondary recipients of the funds, as well. Thus for example, if an Israeli company competes for a research grant, it could be asked to commit itself not to transfer any of the funds to a laboratory located over the Green Line.
With regard to investment funds, the guidelines apply to the primary recipient as well to any secondary recipients. Thus, to obtain EU investment funds, a high-tech firm in Tel Aviv will have to declare that it will not use subcontractors located in the settlements. The guidelines will apply to contenders for or winners of prizes or awards from an EU institution.
EU's new policy on Israeli settlement
See link below. Click your back arrow to return to Dilemma X
Har Homa, Jerusalem
Har Homa is an Israeli settlement in southern East Jerusalem. It is constructed on land annexed to the Jerusalem municipality by Israel after the 1967 Six-Day War, it is considered by much of the world an illegal Israeli settlement, although Israel disputes that case.
← Two NFL franchises don't make the Forbes list "The World's 50 Most Valuable Sports Teams 2013"
Nelson Mandela International Day- Happy Birthday 18 July -Madiba → | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,723 |
The Pinker Tones ist eine spanische Band, die 2001 in Barcelona gegründet wurde. Sie besteht aus Mister Furia, Professor Manso und DJ Niño. Ihr Stil wird häufig der Popmusik zugeordnet, allerdings mit Hang zu Electropop und Alternative-Ansätzen. 2014 gaben sie ihre Auflösung bekannt. Dennoch veröffentlichen sie in regelmäßigen Abständen weiterhin Hörbücher, Kollaborationsalben und Soundtracks.
Stil
Die Musikrichtung der Pinker Tones setzt sich vor allem zusammen aus Pop, Funk, Soul, Swing und Lounge. Dabei vereinen sie vereinzelt sogar mehrere musikalische Elemente innerhalb eines Liedes und dies in bis zu vier verschiedenen Sprachen, darunter Spanisch, Französisch, Englisch, aber auch Deutsch. Aufgrund dieser Beschaffenheiten, waren sie vielmals auch international mit ihrer Musik und auf Konzerten vertreten. Sie spielten in weit mehr als 300 Shows in mehr als 40 Ländern weltweit.
Geschichte
Vorgeschichte und Gründung
Professor Manso und Mister Furia, die bereits gemeinsam die Universität besuchten und gleichzeitig dort studierten, sich jedoch lange nicht mehr gesehen hatten, trafen sich immer wieder zufällig in der Stadt, bis sie schließlich herausfanden, dass sie lediglich 500 Meter voneinander entfernt wohnten. Nachdem sie sich das vierte Mal getroffen hatten, bat Furia Manso um Hilfe bei einem Projekt, bei dem er die Hintergrundmusik für die TV-Serie Once Upon a Time in Europe komponieren sollte. Da Manso hervorragende Fähigkeiten im Umgang mit digitalen Produktionsmedien hatte, beschlossen sie darauf in Zukunft öfter gemeinsam zu arbeiten.
Erstes Album: The Pink Connection
Da die Zusammenarbeit so von Erfolg gekrönt war, verabredeten sie sich öfter und bald wurde klar, dass sie ein paar Lieder für ein Debütalbum zusammenstellen müssten, da beider Traum ein durch ihre Musik finanziertes Leben war. Nachdem sie ein paar arbeitsintensive Sessions hinter sich hatten, veröffentlichten sie The Pink Connection über ein kleines lokales Label. Entgegen den Erwartungen erlangte das Album schnell Bekanntheit und verkaufte sich nun auch international gut, sodass sie mit ihm sogar in die MTV Dance Charts einstiegen und schlussendlich mit Hilfe eines mehr bekannten britischen Labels internationale Auftritte bekamen. Hierdurch stand ihnen die Weltbühne nahezu komplett offen.
Weitere Erfolge und internationaler Durchbruch
Um an die Erfolge der ersten drei Alben, die allesamt in den Jahren 2003 bis 2004 veröffentlicht wurden, anzuknüpfen und nicht in Vergessenheit zu geraten, fingen sie schnell an, an vielen weiteren Projekten zu arbeiten, darunter Produktionen für andere Künstler, Soundtracks zu Filmen und Remixe. Außerdem begannen sie die Arbeit an ihrem vierten Album The Million Colour Revolution, welches nach der Veröffentlichung allen Erwartungen gerecht wurde und sich durch die Zusammenarbeit des britischen Labels Outstanding Records und ihrem eigens gegründeten Pinkerland Records auch international in so vielen Ländern, wie nie eines ihrer vorigen Alben verkaufen konnte. Sie beschlossen, dass sie wegen der stetig steigenden Anzahl der Fans eine Live-Show brauchen und engagierten deshalb DJ Niño, dank dem sie nun die Möglichkeiten hatten, Musik in Richtung Rock, als auch in Richtung Electronic, zu machen und dies sogar live.
Weltbühne
Ihre Beliebtheit zeigte sich nunmehr, nachdem sie durch verschiedene Kontinente der Welt tourten und Auftritte in Ländern, wie z. B. China, Südafrika und auch Russland hatten. Sie spielten Lieder für Computerspiele, beispielsweise FIFA und Forza Motorsport, für Hollywood-Produktionen, z. B. Ugly Betty, und Filme ein. Weltweit wurden sie hervorragend rezensiert, auch die New York Times und die Washington Post lobten ihre Werke und sie belegten Platz sechs der laut Amazon.com besten Alben 2006 mit The Million Colour Revolution. Sie ruhten sich jedoch nicht auf ihren Lorbeeren aus und veröffentlichten nur ein Jahr später das Album More Colours!, 2008 Wild Animals, 2010 Modular und im Mai 2012 Life in Stereo.
Auflösung
Nachdem sie sich im Jahr 2012 insbesondere auf ihr Buch Rolf & Flor fokussierten, gaben die Pinker Tones im April 2014 bekannt, dass sie sich zurückziehen möchten, um sich wieder musikfernen Aktivitäten zu widmen.
Diskografie
Alben
The Pink Connection (2003/2004)
Mission Pink (2003/2004)
The BCN Connection (2004)
The Million Colour Revolution (2006)
More Colours! (2007)
Wild Animals (April 2008)
Modular (Juni 2010)
Life in Stereo (Mai 2012)
Leon (2019)
Hörbücher
Rolf & Flor (2012)
Flor & Rolf (2012)
Flor & Rolf (2013)
Rolf & Flor en el Círculo Polar (2014)
Rolf & Flor en Londres (2015)
Soundtracks
El Pregón (Original Motion Picture Sound Track) (2016)
Schriften
Rolf & Flor. Alba Editorial, 2012, ISBN 978-8484289104.
Weblinks
Einzelnachweise
Spanische Band
Popband
Musik (Katalonien) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,754 |
Q: Calculate "camera" projection limit points I need to calculate the eight points which define the limits of a orthographic perspective. I am using OpenGL deprecated functions:
glMatrixMode(GL_PROJECTION)
glLoadIdentity()
# self.zoom is a integer used for zooming in and out the camera
# The near and far values are hardcoded to +-5000
glOrtho(self.zoom, -self.zoom, -self.zoom, self.zoom, -5000, 5000)
up = 1
if self.theta == 360:
up = -1
# self.x self.y self.z are the coords of the looked point
# self.deviation_x/y/z are the coords of the observer
# There is a strange bug, when self.theta = 360 the camera flips, it gets fixed flipping again the camera
gluLookAt(self.x, self.y, self.z, self.deviation_x, self.deviation_y, self.deviation_z, 0, up, 0)
glMatrixMode(GL_MODELVIEW)
# Drawing stuff
I need to get the coordinates of the following points:
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,475 |
Q: Point IntelliJ to installed Scala I'm new to Scala and IntelliJ.
I have already installed Scala on my Ubuntu PC:
printdoc@print$ scala
Welcome to Scala version 2.11.6 (Java HotSpot(TM) 64-Bit Server VM, Java 1.8.0_181).
Type in expressions to have them evaluated.
Type :help for more information.
scala>
Then, I install IntelliJ. I know that we need install Scala plugin to use Scala with IntelliJ. But I wonder if it will be duplicated because I already have Scala in my computer?
If so, how can I point the Scala location for IntelliJ, so that the IDE can re-use scala, instead of download a new one.
Which is a better practice, between installing Scala plugin and reusing installed Scala?
The same question for sbt with IntelliJ.
A: When not using sbt with IntelliJ, you can manually choose to use the Scala jars installed on your system, but I recommend letting IntelliJ handle it. That makes it easier to select or upgrade the Scala SDK used with your project. The files may be duplicated, but that is an insignificant amount of storage.
But to answer your question, to use an installed Scala library:
*
*open Project structure menu / Project Settings / Modules
*click the +
*select Library...
*select New Library / Scala SDK
*To use a specific version, select it if available, or choose Download. OR click Browse and navigate to your scala installation and choose the relevant jars.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,598 |
<div align="center">
<img src="wireit.svg" height="125" alt="wireit"/>
_Wireit upgrades your npm scripts to make them smarter and more efficient._
[](https://www.npmjs.com/package/wireit)
[](https://github.com/google/wireit/actions/workflows/tests.yml)
</div>
## Features
- 🙂 Use the `npm run` commands you already know
- ⛓️ Automatically run dependencies between npm scripts in parallel
- 👀 Watch any script and continuously re-run on changes
- 🥬 Skip scripts that are already fresh
- ♻️ Cache output locally and remotely on GitHub Actions for free
- 🛠️ Works with single packages, npm workspaces, and other monorepos
## Contents
- [Features](#features)
- [Install](#install)
- [Setup](#setup)
- [VSCode Extension](#vscode-extension)
- [Dependencies](#dependencies)
- [Vanilla scripts](#vanilla-scripts)
- [Cross-package dependencies](#cross-package-dependencies)
- [Parallelism](#parallelism)
- [Extra arguments](#extra-arguments)
- [Input and output files](#input-and-output-files)
- [Incremental build](#incremental-build)
- [Caching](#caching)
- [Local caching](#local-caching)
- [GitHub Actions caching](#github-actions-caching)
- [Cleaning output](#cleaning-output)
- [Watch mode](#watch-mode)
- [Services](#services)
- [Execution cascade](#execution-cascade)
- [Failures and errors](#failures-and-errors)
- [Package locks](#package-locks)
- [Recipes](#recipes)
- [TypeScript](#typescript)
- [ESLint](#eslint)
- [Reference](#reference)
- [Configuration](#configuration)
- [Dependency syntax](#dependency-syntax)
- [Environment variables](#environment-variables)
- [Glob patterns](#glob-patterns)
- [Fingerprint](#fingerprint)
- [Requirements](#requirements)
- [Related tools](#related-tools)
- [Contributing](#contributing)
## Install
```sh
npm i -D wireit
```
## Setup
Wireit works _with_ `npm run`, it doesn't replace it. To configure an NPM script
for Wireit, move the command into a new `wireit` section of your `package.json`,
and replace the original script with the `wireit` command.
<table>
<tr>
<th>Before</th>
<th>After</th>
</tr>
<tr>
<td>
<pre lang="json">
{
"scripts": {
"build": "tsc"
}
}
</pre>
</td>
<td>
<pre lang="json">
{
"scripts": {
"build": "wireit"
},
"wireit": {
"build": {
"command": "tsc"
}
}
}
</pre>
</td>
</tr>
</table>
Now when you run `npm run build`, Wireit upgrades the script to be smarter and
more efficient. Wireit works with [yarn](https://yarnpkg.com/)
(both 1.X "[Classic](https://classic.yarnpkg.com/)" and its successor "Berry")
and [pnpm](https://pnpm.io/), too.
You should also add `.wireit` to your `.gitignore` file. Wireit uses the
`.wireit` directory to store caches and other data for your scripts.
```sh
echo .wireit >> .gitignore
```
## VSCode Extension
If you use VSCode, consider installing the `google.wireit` extension. It adds documentation on hover, autocomplete, can diagnose a number of common mistakes, and even suggest a refactoring to convert an npm script to use wireit.
Install it [from the marketplace](https://marketplace.visualstudio.com/items?itemName=google.wireit) or on the command line like:
```
code --install-extension google.wireit
```
## Dependencies
To declare a dependency between two scripts, edit the
`wireit.<script>.dependencies` list:
```json
{
"scripts": {
"build": "wireit",
"bundle": "wireit"
},
"wireit": {
"build": {
"command": "tsc"
},
"bundle": {
"command": "rollup -c",
"dependencies": ["build"]
}
}
}
```
Now when you run `npm run bundle`, the `build` script will automatically run
first.
### Vanilla scripts
The scripts you depend on don't need to be configured for Wireit, they can be
vanilla `npm` scripts. This lets you only use Wireit for some of your scripts,
or to upgrade incrementally. Scripts that haven't been configured for Wireit are
always safe to use as dependencies; they just won't be fully optimized.
### Cross-package dependencies
Dependencies can refer to scripts in other npm packages by using a relative path
with the syntax `<relative-path>:<script-name>`. All cross-package dependencies
should start with a `"."`. Cross-package dependencies work well for npm
workspaces, as well as in other kinds of monorepos.
```json
{
"scripts": {
"build": "wireit"
},
"wireit": {
"build": {
"command": "tsc",
"dependencies": ["../other-package:build"]
}
}
}
```
## Parallelism
Wireit will run scripts in parallel whenever it is safe to do so according to
the dependency graph.
For example, in this diagram, the `B` and `C` scripts will run in parallel,
while the `A` script won't start until both `B` and `C` finish.
```mermaid
graph TD
A-->B;
A-->C;
subgraph parallel
B;
C;
end
```
By default, Wireit will run up to 2 scripts in parallel for every logical CPU
core detected on your system. To change this default, set the `WIREIT_PARALLEL`
[environment variable](#environment-variables) to a positive integer, or
`infinity` to run without a limit. You may want to lower this number if you
experience resource starvation in large builds. For example, to run only one
script at a time:
```sh
export WIREIT_PARALLEL=1
npm run build
```
If two or more separate `npm run` commands are run for the same Wireit script
simultaneously, then only one instance will be allowed to run at a time, while
the others wait their turn. This prevents coordination problems that can result
in incorrect output files being produced. If `output` is set to an empty array,
then this restriction is removed.
## Extra arguments
As with plain npm scripts, you can pass extra arguments to a Wireit script by
placing a `--` double-dash argument in front of them. Any arguments after a `--`
are sent to the underlying command, instead of being interpreted as arguments to
npm or Wireit:
```sh
npm run build -- --verbose
```
## Input and output files
The `files` and `output` properties of `wireit.<script>` tell Wireit what your
script's input and output files are, respectively. They should be arrays of
[glob patterns](#glob-patterns), where paths are interpreted relative to the
package directory. They can be set on some, all, or none of your scripts.
Setting these properties allow you to use more features of Wireit:
| | Requires<br>`files` | Requires<br>`output` |
| ------------------------------------------: | :-----------------: | :------------------: |
| [**Dependency graph**](#dependencies) | - | - |
| [**Watch mode**](#watch-mode) | ☑️ | - |
| [**Clean build**](#cleaning-output) | - | ☑️ |
| [**Incremental build**](#incremental-build) | ☑️ | ☑️ |
| [**Caching**](#caching) | ☑️ | ☑️ |
#### Example configuration
```json
{
"scripts": {
"build": "wireit",
"bundle": "wireit"
},
"wireit": {
"build": {
"command": "tsc",
"files": ["src/**/*.ts", "tsconfig.json"],
"output": ["lib/**"]
},
"bundle": {
"command": "rollup -c",
"dependencies": ["build"],
"files": ["rollup.config.json"],
"output": ["dist/bundle.js"]
}
}
}
```
#### Default excluded paths
By default, the following folders are excluded from the `files` and `output`
arrays:
- `.git/`
- `.hg/`
- `.svn/`
- `.wireit/`
- `CVS/`
- `node_modules/`
In the highly unusual case that you need to reference a file in one of those
folders, set `allowUsuallyExcludedPaths: true` to remove all default excludes.
## Incremental build
Wireit can automatically skip execution of a script if nothing has changed that
would cause it to produce different output since the last time it ran. This is
called _incremental build_.
To enable incremental build, configure the input and output files for each
script by specifying [glob patterns](#glob-patterns) in the
`wireit.<script>.files` and `wireit.<script>.output` arrays.
> ℹ️ If a script doesn't have a `files` or `output` list defined at all, then it
> will _always_ run, because Wireit doesn't know which files to check for
> changes. To tell Wireit it is safe to skip execution of a script that
> definitely has no input and/or files, set `files` and/or `output` to an empty
> array (`files: [], output: []`).
## Caching
If a script has previously succeeded with the same configuration and input
files, then Wireit can copy the output from a cache, instead of running the
command. This can significantly improve build and test time.
To enable caching for a script, ensure you have defined both the [`files` and
`output`](#input-and-output-files) arrays.
> ℹ️ If a script doesn't produce any output files, it can still be cached by
> setting `output` to an empty array (`"output": []`). Empty output is common for
> tests, and is useful because it allows you to skip running tests if they
> previously passed with the exact same inputs.
### Local caching
In _local_ mode, Wireit caches `output` files to the `.wireit` folder inside
each of your packages.
Local caching is enabled by default, unless the
[`CI=true`](https://docs.github.com/en/enterprise-cloud@latest/actions/learn-github-actions/environment-variables#default-environment-variables)
environment variable is detected. To force local caching, set
`WIREIT_CACHE=local`. To disable local caching, set `WIREIT_CACHE=none`.
> ⚠️ Wireit does not currently limit the size of local caches. To free up this
> space, use `rm -rf .wireit/*/cache`. Automatic cache size limits will be added
> in an upcoming release, tracked at
> [wireit#71](https://github.com/google/wireit/issues/71).
### GitHub Actions caching
In _[GitHub Actions](https://github.com/features/actions)_ mode, Wireit caches
`output` files to the [GitHub Actions
cache](https://docs.github.com/en/actions/using-workflows/caching-dependencies-to-speed-up-workflows)
service. This service is available whenever running in GitHub Actions, and is
free for all GitHub users.
> ℹ️ GitHub Actions cache entries are automatically deleted after 7 days, or if
> total usage exceeds 10 GB (the least recently used cache entry is deleted
> first). See the [GitHub Actions
> documentation](https://docs.github.com/en/actions/using-workflows/caching-dependencies-to-speed-up-workflows#usage-limits-and-eviction-policy)
> for more details.
To enable caching on GitHub Actions, add the following
[`uses`](https://docs.github.com/en/actions/using-workflows/workflow-syntax-for-github-actions#jobsjob_idstepsuses)
clause to your workflow. It can appear anywhere before the first `npm run` or
`npm test` command:
```yaml
- uses: google/wireit@setup-github-actions-caching/v1
```
#### Example workflow
```yaml
# File: .github/workflows/tests.yml
name: Tests
on: [push, pull_request]
jobs:
tests:
os: ubuntu-20.04
steps:
- uses: actions/checkout@v3
- uses: actions/setup-node@v3
with:
node-version: 16
cache: true
# Set up GitHub Actions caching for Wireit.
- uses: google/wireit@setup-github-actions-caching/v1
# Install npm dependencies.
- run: npm ci
# Run tests. Wireit will automatically use
# the GitHub Actions cache whenever possible.
- run: npm test
```
## Cleaning output
Wireit can automatically delete output files from previous runs before executing
a script. This is helpful for ensuring that every build is clean and free from
outdated files created in previous runs from source files that have since been
removed.
Cleaning is enabled by default as long as the
[`output`](#input-and-output-files) array is defined. To change this behavior,
set the `wireit.<script>.clean` property to one of these values:
| Setting | Description |
| ------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `true` | Clean before every run (the default). |
| `"if-file-deleted"` | Clean only if an input file has been deleted since the last run.<br><br>Use this option for tools that have incremental build support, but do not clean up outdated output when a source file has been deleted, such as `tsc --build` (see [TypeScript](#typescript) for more on this example.) |
| `false` | Do not clean.<br><br>Only use this option if you are certain that the script command itself already takes care of removing outdated files from previous runs. |
## Watch mode
In _watch_ mode, Wireit monitors all `files` of a script, and all `files` of its
transitive dependencies, and when there is a change, it re-runs only the
affected scripts. To enable watch mode, ensure that the
[`files`](#input-and-output-files) array is defined, and add the `--watch`
flag:
```sh
npm run <script> --watch
```
The benefit of Wireit's watch mode over built-in watch modes are:
- Wireit watches the entire dependency graph, so a single watch command replaces
many built-in ones.
- It prevents problems that can occur when running many separate watch commands
simultaneously, such as build steps being triggered before all preceding steps
have finished.
## Services
By default, Wireit assumes that your scripts will eventually exit by themselves.
This is well suited for build and test scripts, but not for long-running
processes like servers. To tell Wireit that a process is long-running and not
expected to exit by itself, set `"service": true`.
```json
{
"scripts": {
"start": "wireit",
"build:server": "wireit"
},
"wireit": {
"start": {
"command": "node my-server.js",
"service": true,
"files": ["my-server.js"],
"dependencies": [
"build:server",
{
"script": "../assets:build",
"cascade": false
}
]
},
"build:server": {
...
}
}
}
```
### Service lifetime
If a service is run _directly_ (e.g. `npm run serve`), then it will stay running
until the user kills Wireit (e.g. `Ctrl-C`).
If a service is a _dependency_ of one or more other scripts, then it will start
up before any depending script runs, and will shut down after all depending
scripts finish.
### Service readiness
By default, a service is considered _ready_ as soon as its process spawns,
allowing any scripts that depend on that service to start.
However, often times a service needs to perform certain actions before it is
safe for dependents to interact with it, such as starting a server and listening
on a network interface.
Use `service.readyWhen.lineMatches` to tell Wireit to monitor the `stdout` and
`stderr` of the service and defer readiness until a line is printed that matches
the given regular expression.
```json
{
"command": "node my-server.js",
"service": {
"readyWhen": {
"lineMatches": "Server listening on port \\d+"
}
}
}
```
### Service restarts
In watch mode, a service will be restarted whenever one of its input files or
dependencies change, except for dependencies with
[`cascade`](#execution-cascade) set to `false`.
### Service output
Services cannot have `output` files, because there is no way for Wireit to know
when a service has finished writing its output.
If you have a service that produces output, you should define a _non-service_
script that depends on it, and which exits when the service's output is
complete.
## Execution cascade
By default, a script always needs to run (or restart in the case of
[`services`](#services)) if any of its dependencies needed to run, _regardless
of whether the dependency produced new or relevant output_.
This automatic _cascade_ of script execution is the default behavior because it
ensures that any _possible_ output produced by a dependent script propagates to all other
scripts that might depend on it. In other words, Wireit does not assume that the
`files` array completely describes the inputs to a script with dependencies.
### Disabling cascade
This execution cascade behavior can be disabled by expanding a dependency into
an object, and setting the `cascade` property to `false`:
> **Note**
> What really happens under the hood is that the `cascade` property simply controls
> whether the [fingerprint](#fingerprint) of a script _includes the fingerprints
> of its dependencies_, which in turn determines whether a script needs to run or restart.
```json
{
"dependencies": [
{
"script": "foo",
"cascade": false
}
]
}
```
### Reasons to disable cascade
There are two main reasons you might want to set `cascade` to `false`:
1. **Your script only consumes a subset of a dependency's output.**
For example, `tsc` produces both `.js` files and `.d.ts` files, but only the
`.js` files might be consumed by `rollup`. There is no need to re-bundle
when a typings-only changed occured.
> **Note**
> In addition to setting `cascade` to `false`, the subset of output that
> _does_ matter (`lib/**/*.js`) has been added to the `files` array.
```json
{
"scripts": {
"build": "wireit",
"bundle": "wireit"
},
"wireit": {
"build": {
"command": "tsc",
"files": ["src/**/*.ts", "tsconfig.json"],
"output": ["lib/**"]
},
"bundle": {
"command": "rollup -c",
"dependencies": [
{
"script": "build",
"cascade": false
}
],
"files": ["rollup.config.json", "lib/**/*.js"],
"output": ["dist/bundle.js"]
}
}
}
```
2. **Your server doesn't need to restart for certain changes.**
For example, a web server depends on some static assets, but the server
reads those assets from disk dynamically on each request. In [`watch`](#watch-mode) mode,
there is no need to restart the server when the assets change.
> **Note**
> The `build:server` dependency uses the default `cascade` behavior
> (`true`), because changing the implementation of the server itself _does_
> require the server to be restarted.
```json
{
"scripts": {
"start": "wireit",
"build:server": "wireit"
},
"wireit": {
"start": {
"command": "node lib/server.js",
"service": true,
"dependencies": [
"build:server",
{
"script": "../assets:build",
"cascade": false
}
],
"files": ["lib/**/*.js"]
},
"build:server": {
"command": "tsc",
"files": ["src/**/*.ts", "tsconfig.json"],
"output": ["lib/**"]
}
}
}
```
## Failures and errors
By default, when a script fails (meaning it returned with a non-zero exit code),
all scripts that are already running are allowed to finish, but new scripts are
not started.
In some situations a different behavior may be better suited. There are 2
additional modes, which you can set with the `WIREIT_FAILURES` environment
variable. Note that Wireit always ultimately exits with a non-zero exit code if
there was a failure, regardless of the mode.
### Continue
When a failure occurs in `continue` mode, running scripts continue, and new
scripts are started as long as the failure did not affect their dependencies.
This mode is useful if you want a complete picture of which scripts are
succeeding and which are failing.
```bash
WIREIT_FAILURES=continue
```
### Kill
When a failure occurs in `kill` mode, running scripts are immediately killed,
and new scripts are not started. This mode is useful if you want to be notified
as soon as possible about any failures.
```bash
WIREIT_FAILURES=kill
```
## Package locks
By default, Wireit automatically treats
[`package-lock.json`](https://docs.npmjs.com/cli/v8/configuring-npm/package-lock-json)
files in the package directory, plus all parent directories, as input files.
This is useful because installing or upgrading your dependencies can affect the
behavior of your scripts, so it's important to re-run them whenever your
dependencies change.
If you are using an alternative package manager instead of npm, then your
package lock files might be named something else. Some examples are:
- Yarn: [`yarn.lock`](https://yarnpkg.com/configuration/yarnrc#lockfileFilename) (configurable)
- pnpm: [`pnpm-lock.yaml`](https://pnpm.io/git#lockfiles)
To change the name of the package lock files Wireit should look for, specify it
in the `wireit.<script>.packageLocks` array. Wireit will look for the given
filenames in the script's directory, as well as in all of its parent
directories. You can specify multiple filenames here, if needed.
```json
{
"scripts": {
"build": "wireit"
},
"wireit": {
"build": {
"command": "tsc",
"files": ["src/**/*.ts", "tsconfig.json"],
"output": ["lib/**"],
"packageLocks": ["yarn.lock"]
}
}
}
```
If you're sure that a script isn't affected by dependencies at all, you can turn
off this behavior entirely to improve your cache hit rate by setting
`wireit.<script>.packageLocks` to `[]`.
## Recipes
This section contains advice about integrating specific build tools with Wireit.
### TypeScript
```json
{
"scripts": {
"ts": "wireit"
},
"wireit": {
"ts": {
"command": "tsc --build --pretty",
"clean": "if-file-deleted",
"files": ["src/**/*.ts", "tsconfig.json"],
"output": ["lib/**", ".tsbuildinfo"]
}
}
}
```
- Set [`"incremental": true`](https://www.typescriptlang.org/tsconfig#incremental) and use
[`--build`](https://www.typescriptlang.org/docs/handbook/project-references.html#build-mode-for-typescript)
to enable incremental compilation, which significantly improves performance.
- Include
[`.tsbuildinfo`](https://www.typescriptlang.org/tsconfig#tsBuildInfoFile) in
`output` so that it is reset on clean builds. Otherwise `tsc` will get out of
sync and produce incorrect output.
- Set [`"clean": "if-file-deleted"`](#cleaning-output) so that you get fast
incremental compilation when sources are changed/added, but also stale outputs
are cleaned up when a source is deleted (`tsc` does not clean up stale outputs
by itself).
- Include `tsconfig.json` in `files` so that changing your configuration re-runs
`tsc`.
- Use [`--pretty`](https://www.typescriptlang.org/tsconfig#pretty) to get
colorful output despite not being attached to a TTY.
### ESLint
```json
{
"scripts": {
"lint": "wireit"
},
"wireit": {
"lint": {
"command": "eslint --color --cache --cache-location .eslintcache .",
"files": ["src/**/*.ts", ".eslintignore", ".eslintrc.cjs"],
"output": []
}
}
}
```
- Use
[`--cache`](https://eslint.org/docs/user-guide/command-line-interface#caching)
so that `eslint` only lints the files that were added or changed since the
last run, which significantly improves performance.
- Use
[`--color`](https://eslint.org/docs/user-guide/command-line-interface#--color---no-color)
to get colorful output despite not being attached to a TTY.
- Include config and ignore files in `files` so that changing your configuration
re-runs `eslint`.
## Reference
### Configuration
The following properties can be set inside `wireit.<script>` objects in
`package.json` files:
| Property | Type | Default | Description |
| ------------------------- | ------------------------------ | ----------------------- | ----------------------------------------------------------------------------------------------------------- |
| `command` | `string` | `undefined` | The shell command to run. |
| `dependencies` | `string[] \| object[]` | `[]` | [Scripts that must run before this one](#dependencies). |
| `dependencies[i].script` | `string` | `undefined` | [The name of the script, when the dependency is an object.](#dependencies). |
| `dependencies[i].cascade` | `boolean` | `true` | [Whether this dependency always causes this script to re-execute](#execution-cascade). |
| `files` | `string[]` | `undefined` | Input file [glob patterns](#glob-patterns), used to determine the [fingerprint](#fingerprint). |
| `output` | `string[]` | `undefined` | Output file [glob patterns](#glob-patterns), used for [caching](#caching) and [cleaning](#cleaning-output). |
| `clean` | `boolean \| "if-file-deleted"` | `true` | [Delete output files before running](#cleaning-output). |
| `service` | `boolean` | `false` | [Whether this script is long-running, e.g. a server](#cleaning-output). |
| `packageLocks` | `string[]` | `['package-lock.json']` | [Names of package lock files](#package-locks). |
### Dependency syntax
The following syntaxes can be used in the `wireit.<script>.dependencies` array:
| Example | Description |
| ------------ | ----------------------------------------------------------------------------------------------- |
| `foo` | Script named `"foo"` in the same package. |
| `../foo:bar` | Script named `"bar"` in the package found at `../foo` ([details](#cross-package-dependencies)). |
### Environment variables
The following environment variables affect the behavior of Wireit:
| Variable | Description |
| ----------------- | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ |
| `WIREIT_FAILURES` | [How to handle script failures](#failures-and-errors).<br><br>Options:<br><ul><li>[`no-new`](#failures-and-errors) (default): Allow running scripts to finish, but don't start new ones.</li><li>[`continue`](#continue): Allow running scripts to continue, and start new ones unless any of their dependencies failed.</li><li>[`kill`](#kill): Immediately kill running scripts, and don't start new ones.</li></ul> |
| `WIREIT_PARALLEL` | [Maximum number of scripts to run at one time](#parallelism).<br><br>Defaults to 2×logical CPU cores.<br><br>Must be a positive integer or `infinity`. |
| `WIREIT_CACHE` | [Caching mode](#caching).<br><br>Defaults to `local` unless `CI` is `true`, in which case defaults to `none`.<br><br>Automatically set to `github` by the [`google/wireit@setup-github-actions-caching/v1`](#github-actions-caching) action.<br><br>Options:<ul><li>[`local`](#local-caching): Cache to local disk.</li><li>[`github`](#github-actions-caching): Cache to GitHub Actions.</li><li>`none`: Disable caching.</li></ul> |
| `CI` | Affects the default value of `WIREIT_CACHE`.<br><br>Automatically set to `true` by [GitHub Actions](https://docs.github.com/en/actions/learn-github-actions/environment-variables#default-environment-variables) and most other CI (continuous integration) services.<br><br>Must be exactly `true`. If unset or any other value, interpreted as `false`. |
### Glob patterns
The following glob syntaxes are supported in the `files` and `output` arrays:
| Example | Description |
| --------------- | ---------------------------------------------------------------------------------------- |
| `foo` | The file named `foo`, or if `foo` is a directory, all recursive children of `foo`. |
| `foo/*.js` | All files directly in the `foo/` directory which end in `.js`. |
| `foo/**/*.js` | All files in the `foo/` directory, and all recursive subdirectories, which end in `.js`. |
| `foo.{html,js}` | Files named `foo.html` or `foo.js`. |
| `!foo` | Exclude the file or directory `foo` from previous matches. |
Also note these details:
- Paths should always use `/` (forward-slash) delimiters, even on Windows.
- Paths are interpreted relative to the current package even if there is a
leading `/` (e.g. `/foo` is the same as `foo`).
- Whenever a directory is matched, all recursive children of that directory are
included.
- `files` are allowed to reach outside of the current package using e.g.
`../foo`. `output` files cannot reference files outside of the current
package.
- Symlinks in input `files` are followed, so that they are identified by their content.
- Symlinks in `output` files are cached as symlinks, so that restoring from
cache doesn't create unnecessary copies.
- The order of `!exclude` patterns is significant.
- Hidden/dot files are matched by `*` and `**`.
- Patterns are case-sensitive (if supported by the filesystem).
### Fingerprint
The following inputs determine the _fingerprint_ for a script. This value is
used to determine whether a script can be skipped for [incremental
build](#incremental-build), and whether its output can be [restored from
cache](#caching).
- The `command` setting.
- The [extra arguments](#extra-arguments) set on the command-line.
- The `clean` setting.
- The `output` glob patterns.
- The SHA256 content hashes of all files matching `files`.
- The SHA256 content hashes of all files matching `packageLocks` in the current
package and all parent directories.
- The system platform (e.g. `linux`, `win32`).
- The system CPU architecture (e.g. `x64`).
- The system Node version (e.g. `16.7.0`).
- The fingerprint of all transitive dependencies, unless `cascade` is set to
`false`.
When using [GitHub Actions caching](#github-actions-caching), the following
input also affects the fingerprint:
- The `ImageOS` environment variable (e.g. `ubuntu20`, `macos11`).
## Requirements
Wireit is supported on Linux, macOS, and Windows.
Wireit is supported on Node Current (19), LTS (18), and Maintenance (16 and 14).
See [Node releases](https://nodejs.org/en/about/releases/) for the schedule.
> **Warning**
> Wireit will no longer work with Node 14 when it reaches end-of-life on
> 2023-04-30. We recommend upgrading to Node 18 as soon as possible.
Wireit is supported on the npm versions that ship with the latest versions of
the above supported Node versions (6 and 8), Yarn Classic (1), Yarn Berry (3),
and pnpm (7).
## Related tools
Wireit shares a number of features with these other great tools, and we highly
recommend you check them out too:
- [Nx](https://nx.dev/)
- [Turborepo](https://turborepo.org/)
- [Chomp](https://chompbuild.com/)
- [Bazel](https://bazel.build/)
Here are some things you might especially like about Wireit:
- **Feels like npm**. When you use Wireit, you'll continue typing the same npm
commands you already use, like `npm run build` and `npm test`. There are no
new command-line tools to learn, and there's only one way to run each script.
Your script config stays in your `package.json`, too. Wireit is designed to be
the minimal addition to npm needed to get script dependencies and incremental
build.
- **Caching with GitHub Actions**. Wireit supports caching build artifacts and
test results directly through GitHub Actions, without any extra third-party
services. Just add a single `uses:` line to your workflows.
- **Watch any script**. Want to automatically re-run your build and tests
whenever you make a change? Type `npm test --watch`. Any script you've
configured using Wireit can be watched by typing `--watch` after it.
- **Great for single packages and monorepos**. Wireit has no opinion about how
your packages are arranged. It works great with single packages, because you
can link together scripts within the same package. It also works great with
any kind of monorepo, because you can link together scripts across different
packages using relative paths.
- **Complements npm workspaces**. We think Wireit could be the missing tool that
unlocks the potential for [npm
workspaces](https://docs.npmjs.com/cli/v8/using-npm/workspaces) to become the
best way to set up monorepos. To use Wireit with npm workspaces, you'll just
use standard npm workspace commands like `npm run build -ws`.
- **Adopt incrementally**. Wireit scripts can depend on plain npm scripts, so
they can be freely mixed. This means you can use Wireit only for the parts of
your build that need it most, or you can try it out on a script-by-script
basis without changing too much at the same time.
## Contributing
See [CONTRIBUTING.md](./CONTRIBUTING.md)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,603 |
was formed in 2012 by the merger of the old Nippon Steel and Sumitomo Metal. was established in 1970 by the merger of Fuji Iron & Steel and Yawata Iron & Steel. Nippon Steel is the world's third largest steel producer by volume as of 2019.
History
Early years
Nippon Steel was created by the merger of two giants, Yawata Iron & Steel (八幡製鉄 Yawata Seitetsu) and Fuji Iron & Steel (富士製鉄 Fuji Seitetsu). Beginning in early 1981, however, the company cut production and saw a sharp decline in profit that fiscal year. Forced to close furnaces, the company exhibited a typical Japanese economic aversion to layoffs, opting instead to offer standard early retirement enticements but also less conventional schemes such as a mushroom cultivation venture that used the surplus heat created by steel furnaces to temperature control a fecund fungi complex.
Troubled times
Attributing the drop to higher material costs, the company entered into another troubled year. In 1983, the company reported the end of the fiscal year (March 31) would reveal Nippon Steel was in an even more beleaguered situation. A fall in demand brought about a 39 percent tumble in profits from an already weak previous year. During this time the entire Japanese steel industry struggled in a period of turmoil as other nations such as South Korea, with only a fraction of labor costs, won over business. The company announced a loss in 1986, prompting a determined effort to diversify away from the moribund "smokestack" industrial sector and to provide new work for thousands of employees that would be transferred from closing furnaces.
Diversification
Nippon Steel expanded or further established itself in semiconductors, electronics, a theme park called Space World, software, and even human resources products. Most notable was Librex Computer Systems, Nippon Steel's attempt to sell notebook computers abroad that lasted from 1990 to 1993. The company bucked seven struggling but profitable years when it returned to loss in 1993. Again, thousands of employees would be transferred to new operations. Due to cost-cutting, the company returned to health in 1995. However, Nippon Steel reported earnings in 1999 suffered from an overwhelming charge needed to cover pension costs, a problem not uncommon for shrinking industrial giants. 2002 and 2003 would be back-to-back loss years, but robust demand for steel in the People's Republic of China returned the company to profitability. (However, Nippon Steel had an operating profit for 2002 and 2003. The losses were made of extraordinary losses because of reevaluation of real estate and securities of the company among others.) Following a triple merger of Sumitomo Corporation, Kinzoku Steel Corporation (Sumikin Bussan), and the existing Nippon Steel, NSSC was formed as these companies' conglomerate Stainless Steel division.
Merger
In early 2011, Nippon Steel announced plans to merge with Sumitomo Metal Industries. With Nippon Steel producing ~26.5 million tonnes of steel per year and Sumitomo making ~11 million tonnes, the merged entity would produce close to 37 million tonnes of crude steel per year. This volume of steel output would make Nippon Steel the second largest steelmaker in the world, putting it well ahead of Baosteel – the current number two (making ~31 mt steel / year) – although still well behind ArcelorMittal (who produced 77.5 mt crude steel in 2010).
On October 1, 2012, Nippon Steel formally merged with Sumitomo Metal Industries at a ratio of 0.735 Nippon Steel shares per Sumitomo Metal share. The merged stock is listed (under number 5401, the old Nippon Steel number) as Nippon Steel & Sumitomo Metal Corp. The logistics branches of both companies are announced to be merged on April 1, 2013, under the name "Nippon Steel & Sumikin Logistics Co., Ltd.", wholly owned by Nippon Steel & Sumitomo Metal Corporation. The merged company planned to publish a common fact book in the summer of 2013.
On April 1, 2019, the Japanese name of the company was changed from Nippon Steel & Sumitomo Metal Corporation to Nippon Steel Corporation.
Present
By May 2020, Nippon Steel has announced that it would suspend operations of four furnaces, of which one for permanently, as it booked an annual loss in FY 2019.
Major plant locations
Muroran, Hokkaido
Kamaishi, Iwate
Kimitsu, Chiba: Kimitsu Steel Works
Tokyo
Tōkai, Aichi (Nagoya)
Sakai, Osaka
Himeji, Hyogo (Hirohata)
Hikari, Yamaguchi – steel piping
Kitakyushu, Fukuoka (Yahata)
Oita, Oita
Added after Sumitomo merger
Kashima, Ibaraki
Jōetsu, Niigata (Naoetsu)
Amagasaki, Hyōgo
Osaka
Wakayama
Kitakyushu, Fukuoka (Kokura)
Joint ventures
New Carlisle, Indiana, USA (built 1991)
AM/NS Calvert. Formerly named ThyssenKrupp Steel USA and located in Calvert, Alabama, the facility was purchased from ThyssenKrupp through a 50/50 joint partnership with ArcelorMittal in February 2014 for $1.5 billion and renamed AM/NS Calvert. A greenfield construction project which began in 2007, the facility began operation in 2010 and has a production capacity of 5.3 million tons and includes a hot strip mill, cold roll mill and 4 coating lines. Products from the facility are marketed in the NAFTA region through managing partner ArcelorMittal.
Nippon Steel Trading Co., Ltd., has set up a joint venture with three Indonesian local companies to produce 120,000 tons of sheet steel for the automotive industry. Nippon Steel would control a 30 percent share of the joint venture, PT IndoJapan Steel Center. It is located in the Mitra Karawang Industrial Estate, West Java in a 4.8-hectare area with total investment for first phase $38 million and was expected to start operating in January 2013.
POSCO-Nippon Steel RHF Joint Venture, Co., Ltd., located in Pohang, South Korea. Using rotary hearth furnace technology, the company recycles sludge and dust coming out from the POSCO plants.
Nippon and POSCO reached an agreement with Teck Resources Limited (TSX: TECK.A and TECK.B, NYSE: TECK) to exchange minority interests into some Teck operations for a new interest into Elk Valley Resources Ltd ("EVR") that was spin ned-off on Feb 21st, 2023.
Controversies
On October 30, 2018, the South Korean Supreme Court rejected appeals to overturn a 2013 order requiring Nippon to pay compensation to four South Korean workers who underwent forced labor which occurred during World War II and ordered Nippon to pay each of the workers an individual sum of 100 million won (US$87,700). The four surviving steel workers, who were victims of forced labor which was supervised by Sumitomo, originally filed suit in 2005. A Nippon spokesman called the decision "deeply regretful," while also promising a review of the ruling. The Japanese Minister of Foreign Affairs Taro Kono maintained that the matter "has been resolved following the Treaty on Basic Relations between Japan and the Republic of Korea".
The asset seizure ordered by the Korean supreme court involves Nippon's stake in PNR, the POSCO-Nippon joint venture.
Environmental record
In 2005 the Nippon Steel corporation made a plan to step up its capacity for recycling waste plastics into coke by 30%. Coke is a main resource in steel production. To manage the load they have invested ¥4 billion (about $38.2 million) to install equipment at Oita Mill and set up a second furnace at Kyushu facility.
In 2006 Nippon Steel and Mitsubishi Heavy Industries, Ltd. (MHI) jointly created a high tensile strength steel. The first application this steel was used for was the hulls of container ships. This steel allows the ships to be just as strong without the thick steel that it was requiring for them to grow in size. The smaller thickness allows the ships to attain a greater fuel-efficiency, cutting down on the environmental load of the ships.
Nippon Steel announced a pilot project to process waste food into ethanol in 2006. They have tasked Kitakyushu City with collecting and sorting the food waste and Nishihara Co., a waste management company, with developing new technologies to implement the sorted collecting system. To minimize costs they will use waste heat from an existing incineration facility that had not been effectively utilized, and the residue left after ethanol recovery will be burned in this incinerator.
Nippon Steel has been addressing environmental issues in an integrated manner as part of the overall management since the establishment of the company, aiming at realizing a sustainable society. In 2011, the company was awarded with the Fray International Sustainability Award in Mexico, for its approach in achieving Eco-processes, Eco-products, and Eco-solutions.
Carbon footprint
Nippon Steel reported Total CO2e emissions (Direct + Indirect) for the twelve months ending 31 March 2020 at 73,706 Kt (-16,556 /-18.3% y-o-y). Reported emissions have been on a downward trend since 2016.
See also
History of Nippon Steel Corporation
Anti-Japanese sentiment in Korea
NS Solutions
Kashima Antlers
Nippon Steel Yawata S.C., former company football club, based near the Yawata plant and originally owned by it before the Fuji Steel merger
References
Manufacturing companies based in Tokyo
Anti-Japanese sentiment in Korea
Chemical companies based in Tokyo
Steel companies of Japan
Defense companies of Japan
Japanese brands
Companies listed on the Tokyo Stock Exchange | {
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Following on from our October 2017 article, where we speculated that we would see an increase in the Bank of England base rate from 0.25% to 0.50%, we saw, at the beginning of August, a further increase to the Bank of England base rates from 0.50% to 0.75% – the highest it has been for almost a decade. But what does this mean to your mortgage?
The biggest impact will be seen by those of you who are currently on standard variable rate mortgages. With no certainty in your mortgage payments, you will likely see a small increase in your monthly outgoings, however this small increase could end up costing large amounts over the term of your mortgage if you stay where you are and make no changes to the deal you currently have. For example, if you currently have an outstanding mortgage of £100,000.00, you could potentially see an increase in total interest cost over the full term of around £3,600.00.
If you're on a fixed term mortgage then you may think you don't have anything to worry about – and though this might be true for the short term, have you considered your long term repayment strategy? As there is a great chance the interest rate rise will have an impact on this.
Whether you're on a variable rate mortgage or a fixed rate mortgage, there has never been a better time to review your mortgage options. By sitting down and reviewing what is on offer there is a chance you could save money over the long term, so what is stopping you?
If you would like to review your current mortgage arrangement then please contact the office to arrange a FREE initial discussion with one of our specialist mortgage advisers. To be in with the chance to win fee FREE mortgage advice then please visit our Facebook Page – www.facebook.com/LibertyPartnershipLtd – and enter our competition. | {
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Beer Review: Lagunitas Brewing Co's Imperial Red is like a liquid hop! HOW??
We occasionally get Dogfish Head beers in CA, and they're always a treat. Recently, a friend of mine brought a pack each of the 60 minute and 90 minute IPA's, and they really are stellar. This is a great write-up of the 120-minute flavor, and you should definitely read it! Then drink it. Then drink it some more.
Hello, Congregants. Welcome to the Pastoral Counseling feature of Ale Evangelism. In this feature, we will attempt to answer the questions asked by friends and readers to bring us all to a better understanding of beer appreciation.
Today's question comes from several people over the years who wonder if the substitution of the word "hoppy" for "bitter" is just an attempt to make beer consumption seem more pretentious, or whether there's a good reason for this word choice. | {
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Belgium outrage over woman punched in burger ad
Brussels (AFP)
Belgium's advertising regulator said Wednesday it has clocked up hundreds of complaints over a viral hamburger ad showing a comic-book version of a man punching a woman for handing him the wrong sandwich.
Consumers and politicians called the online ad "sickening" and "irresponsible", saying it went directly against public campaigns denouncing domestic violence.
The ad, published on Facebook, aimed to publicise a Belgian hamburger brand, Bicky Burger.
Its cartoon style borrowed from 1950s and 1960s American pop-art made famous by artist Roy Lichtenstein and by a panel from a 1965 comic showing Batman slapping his sidekick Robin that has been reworked into a long-running internet meme.
The Bicky Burger ad depicts a man in a suit swinging his fist to violently hit a buxom blonde woman carrying a fast-food container as he exclaims: "Seriously, a fake Bicky?"
The head of Belgium's regulatory Advertising Council, Sandrine Sepul, said the agency had received 300 complaints from the public in 24 hours.
She told AFP that a Dutch food company, Izico, commissioned the advertisement and her agency had transmitted the complaints to its counterpart in the Netherlands.
Izico was ordered to explain itself, she said, adding that possible punishment could be decided "in a few weeks".
Belgian media ran indignant reports on the controversy, which led to the ad being pulled from Facebook -- but it was relayed thousands of times in individual social media posts.
Le Soir newspaper asked how such an advertising campaign could be created in 2019 when, in Belgium, "every 10 days a man kills his wife or his ex. And every day a number of women are humiliated, raped, beaten."
Two regional politicians in charge of gender equality for Brussels and for the Wallonia part of Belgium said they had also called on the country's Advertising Council to act.
One of them, Nawal Ben Hamou, said on her Facebook page she found the ad "sickening and totally irresponsible".
The other, Christie Morreale, echoed that in a tweet saying that "using violence towards women in an ad is irresponsible". | {
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Provisional designation in astronomy is the naming convention applied to astronomical objects immediately following their discovery. The provisional designation is usually superseded by a permanent designation once a reliable orbit has been calculated. Approximately 47% of the more than 1,100,000 known minor planets remain provisionally designated, as hundreds of thousands have been discovered in the last two decades.
Minor planets
The current system of provisional designation of minor planets (asteroids, centaurs and trans-Neptunian objects) has been in place since 1925. It superseded several previous conventions, each of which was in turn rendered obsolete by the increasing numbers of minor planet discoveries. A modern or new-style provisional designation consists of the year of discovery, followed by two letters and, possibly, a suffixed number.
New-style provisional designation
For example, the provisional designation stands for the 3910th body identified during 1–15 March 2016:
2016 – the first element indicates the year of discovery.
E – the first letter indicates the half-month of the object's discovery within that year and ranges from A (first half of January) to Y (second half of December), while the letters I and Z are not used (see table below). The first half is always the 1st through to the 15th of the month, regardless of the numbers of days in the second "half". Thus, E indicates the period from March 1 to 15.
K156 – the second letter and a numerical suffix indicate the order of discovery within that half-month. The first 25 discoveries of the half-month only receive a letter (A to Z) without a suffix, while the letter I is not used (to avoid potential confusions with the digit 1). Because modern techniques typically yield hundreds if not thousands of discoveries per half-month, the subscript number is appended to indicate the number of times that the letters from A to Z have cycled through. The suffix 156 indicates 156 completed cycles (156 cycles × 25 letters = 3900), while K is the 10th position in the current cycle. Thus, K156 stands for the 3910th minor planet discovered in a half-month.
The packed form of is written as .
This scheme is now also used retrospectively for pre-1925 discoveries. For these, the first digit of the year is replaced by an A. For example, A801 AA indicates the first object discovered in the first half of January 1801 (1 Ceres).
Further explanations
During the first half-month of January 2014, the first minor planet identification was assigned the provisional designation . Then the assignment continued to the end of the cycle at , which was in turn followed by the first identification of the second cycle, . The assignment in this second cycle continued with , , ... until , and then was continued with the first item in the third cycle. With the beginning of a new half-month on 16 January 2014, the first letter changed to "B", and the series started with .
An idiosyncrasy of this system is that the second letter is listed before the number, even though the second letter is considered "least-significant". This is in contrast to most of the world's numbering systems. This idiosyncrasy is not seen, however, in the so-called packed form (packed designation).
A packed designation has no spaces. It may also use letters to codify for the designation's year and subscript number. It is frequently used in online and electronic documents. For example, the provisional designation is written as K07Tf8A in the packed form, where "K07" stands for the year 2007, and "f8" for the subscript number 418.
90377 Sedna, a large trans-Neptunian object, had the provisional designation , meaning it was identified in the first half of November 2003 (as indicated by the letter "V"), and that it was the 302nd object identified during that time, as 12 cycles of 25 letters give 300, and the letter "B" is the second position in the current cycle.
Survey designations do not follow the rules for new-style provisional designations.
For technical reasons, such as ASCII limitations, the numerical suffix is not always subscripted, but sometimes "flattened out", so that can also be written as .
A very busy half month was the second half of January 2015 (letter "B"), which saw a total of 14,208 new minor planet identifications . One of the last assignments in this period was and corresponds to the 14,208th position in the sequence.
Survey designations
Minor planets discovered during the Palomar–Leiden survey including three subsequent Trojan-campaigns, which altogether discovered more than 4,000 asteroids and Jupiter trojans between 1960 and 1977, have custom designations that consist of a number (order in the survey) followed by a space and one of the following identifiers:
P-L Palomar–Leiden survey (1960–1970)
T-1 Palomar–Leiden Trojan survey (1971)
T-2 Palomar–Leiden Trojan survey (1973)
T-3 Palomar–Leiden Trojan survey (1977)
For example, the asteroid 6344 P-L is the 6344th minor planet in the original Palomar–Leiden survey, while the asteroid 4835 T-1 was discovered during the first Trojan-campaign. The majority of these bodies have since been assigned a number and many are already named.
Historical designations
The first four minor planets were discovered in the early 19th century, after which there was a lengthy gap before the discovery of the fifth. Astronomers initially had no reason to believe that there would be countless thousands of minor planets, and strove to assign a symbol to each new discovery, in the tradition of the symbols used for the major planets. For example, 1 Ceres was assigned a stylized sickle (⚳), 2 Pallas a stylized lance or spear (⚴), 3 Juno a scepter (⚵), and 4 Vesta an altar with a sacred fire (). All had various graphic forms, some of considerable complexity.
It soon became apparent, though, that continuing to assign symbols was impractical and provided no assistance when the number of known minor planets was in the dozens. Johann Franz Encke introduced a new system in the Berliner Astronomisches Jahrbuch (BAJ) for 1854, published in 1851, in which he used encircled numbers instead of symbols. Encke's system began the numbering with Astrea which was given the number (1) and went through (11) Eunomia, while Ceres, Pallas, Juno and Vesta continued to be denoted by symbols, but in the following year's BAJ, the numbering was changed so that Astraea was number (5).
The new system found popularity among astronomers, and since then, the final designation of a minor planet is a number indicating its order of discovery followed by a name. Even after the adoption of this system, though, several more minor planets received symbols, including 28 Bellona the morning star and lance of Mars's martial sister, 35 Leukothea an ancient lighthouse and 37 Fides a Latin cross (). According to Webster's A Dictionary of the English Language, four more minor planets were also given symbols: 16 Psyche, 17 Thetis, 26 Proserpina, and 29 Amphitrite. However, there is no evidence that these symbols were ever used outside of their initial publication in the Astronomische Nachrichten.
134340 Pluto is an exception: it is a high-numbered minor planet that received a graphical symbol with significant astronomical use (♇), because it was considered a major planet on its discovery, and did not receive a minor planet number until 2006.
Graphical symbols continue to be used for some minor planets, and assigned for some recently discovered larger ones, mostly by astrologers (see astronomical symbol and astrological symbol). Three centaurs – 2060 Chiron, 5145 Pholus, and 7066 Nessus – and the other seven large trans-Neptunian dwarf planets – 50000 Quaoar, 90377 Sedna, 90482 Orcus, 136108 Haumea, 136199 Eris, 136472 Makemake, and 225088 Gonggong – have relatively standard symbols among astrologers: the symbols for Haumea, Makemake, and Eris have even been occasionally used in astronomy. However, such symbols are generally not in use among astronomers.
Genesis of the current system
Several different notation and symbolic schemes were used during the latter half of the nineteenth century, but the present form first appeared in the journal Astronomische Nachrichten (AN) in 1892. New numbers were assigned by the AN on receipt of a discovery announcement, and a permanent designation was then assigned once an orbit had been calculated for the new object.
At first, the provisional designation consisted of the year of discovery followed by a letter indicating the sequence of the discovery, but omitting the letter I (historically, sometimes J was omitted instead). Under this scheme, 333 Badenia was initially designated , 163 Erigone was , etc. In 1893, though, increasing numbers of discoveries forced the revision of the system to use double letters instead, in the sequence AA, AB... AZ, BA and so on. The sequence of double letters was not restarted each year, so that followed and so on. In 1916, the letters reached ZZ and, rather than starting a series of triple-letter designations, the double-letter series was restarted with .
Because a considerable amount of time could sometimes elapse between exposing the photographic plates of an astronomical survey and actually spotting a small Solar System object on them (witness the story of Phoebe's discovery), or even between the actual discovery and the delivery of the message (from some far-flung observatory) to the central authority, it became necessary to retrofit discoveries into the sequence — to this day, discoveries are still dated based on when the images were taken, and not on when a human realised they were looking at something new. In the double-letter scheme, this was not generally possible once designations had been assigned in a subsequent year. The scheme used to get round this problem was rather clumsy and used a designation consisting of the year and a lower-case letter in a manner similar to the old provisional-designation scheme for comets. For example, (note that there is a space between the year and the letter to distinguish this designation from the old-style comet designation 1915a, Mellish's first comet of 1915), 1917 b. In 1914 designations of the form year plus Greek letter were used in addition.
Temporary minor planet designations
Temporary designations are custom designation given by an observer or discovering observatory prior to the assignment of a provisional designation by the MPC. These intricate designations were used prior to the Digital Age, when communication was slow or even impossible (e.g. during WWI). The listed temporary designations by observatory/observer use uppercase and lowercase letters (LETTER, letter), digits, numbers and years, as well Roman numerals (ROM) and Greek letters (greek).
Comets
The system used for comets was complex previous to 1995. Originally, the year was followed by a space and then a Roman numeral (indicating the sequence of discovery) in most cases, but difficulties always arose when an object needed to be placed between previous discoveries. For example, after Comet 1881 III and Comet 1881 IV might be reported, an object discovered in between the discovery dates but reported much later couldn't be designated "Comet 1881 III½". More commonly comets were known by the discoverer's name and the year. An alternate scheme also listed comets in order of time of perihelion passage, using lower-case letters; thus "Comet Faye" (modern designation 4P/Faye) was both Comet 1881 I (first comet to pass perihelion in 1881) and Comet 1880c (third comet to be discovered in 1880).
The system since 1995 is similar to the provisional designation of minor planets. For comets, the provisional designation consists of the year of discovery, a space, one letter (unlike the minor planets with two) indicating the half-month of discovery within that year (A=first half of January, B=second half of January, etc. skipping I (to avoid confusion with the number 1 or the numeral I) and not reaching Z), and finally a number (not subscripted as with minor planets), indicating the sequence of discovery within the half-month. Thus, the eighth comet discovered in the second half of March 2006 would be given the provisional designation 2006 F8, whilst the tenth comet of late March would be 2006 F10.
If a comet splits, its segments are given the same provisional designation with a suffixed letter A, B, C, ..., Z, AA, AB, AC...
If an object is originally found asteroidal, and later develops a cometary tail, it retains its asteroidal designation. For example, minor planet 1954 PC turned out to be Comet Faye, and we thus have "4P/1954 PC" as one of the designations of said comet. Similarly, minor planet was reclassified as a comet, and because it was discovered by LINEAR, it is now known as 176P/LINEAR (LINEAR 52) and (118401) LINEAR.
Provisional designations for comets are given condensed or "packed form" in the same manner as minor planets. 2006 F8, if a periodic comet, would be listed in the IAU Minor Planet Database as PK06F080. The last character is purposely a zero, as that allows comet and minor planet designations not to overlap.
Periodic comets
Comets are assigned one of four possible prefixes as a rough classification. The prefix "P" (as in, for example, P/1997 C1, a.k.a. Comet Gehrels 4) designates a "periodic comet", one which has an orbital period of less than 200 years or which has been observed during more than a single perihelion passage (e.g. 153P/Ikeya-Zhang, whose period is 367 years). They receive a permanent number prefix after their second observed perihelion passage (see List of periodic comets).
Non-periodic comets
Comets which do not fulfill the "periodic" requirements receive the "C" prefix (e.g. C/2006 P1, the Great Comet of 2007). Comets initially labeled as "non-periodic" may, however, switch to "P" if they later fulfill the requirements.
Comets which have been lost or have disintegrated are prefixed "D" (e.g. D/1993 F2, Comet Shoemaker-Levy 9).
Finally, comets for which no reliable orbit could be calculated, but are known from historical records, are prefixed "X" as in, for example, X/1106 C1. (Also see List of non-periodic comets and List of hyperbolic comets.)
Satellites and rings of planets
When satellites or rings are first discovered, they are given provisional designations such as "" (the 11th new satellite of Jupiter discovered in 2000), "" (the first new satellite of Pluto discovered in 2005), or "" (the second new ring of Saturn discovered in 2004). The initial "S/" or "R/" stands for "satellite" or "ring", respectively, distinguishing the designation from the prefixes "C/", "D/", "P/", and "X/" used for comets. These designations are sometimes written as "", dropping the second space.
The prefix "S/" indicates a natural satellite, and is followed by a year (using the year when the discovery image was acquired, not necessarily the date of discovery). A one-letter code written in upper case identifies the planet such as J and S for Jupiter and Saturn, respectively (see list of one-letter abbreviations), and then a number identifies sequentially the observation. For example, Naiad, the innermost moon of Neptune, was at first designated "". Later, once its existence and orbit were confirmed, it received its full designation, "".
The Roman numbering system arose with the very first discovery of natural satellites other than Earth's Moon: Galileo referred to the Galilean moons as I through IV (counting from Jupiter outward), in part to spite his rival Simon Marius, who had proposed the names now adopted. Similar numbering schemes naturally arose with the discovery of moons around Saturn and Uranus. Although the numbers initially designated the moons in orbital sequence, new discoveries soon failed to conform with this scheme (e.g. "" is Amalthea, which orbits closer to Jupiter than does Io). The unstated convention then became, at the close of the 19th century, that the numbers more or less reflected the order of discovery, except for prior historical exceptions (see the Timeline of discovery of Solar System planets and their natural satellites). The convention has been extended to natural satellites of minor planets, such as "".
Moons of minor planets
The provisional designation system for minor planet satellites, such as asteroid moons, follows that established for the satellites of the major planets. With minor planets, the planet letter code is replaced by the minor planet number in parentheses. Thus, the first observed moon of 87 Sylvia, discovered in 2001, was at first designated S/2001 (87) 1, later receiving its permanent designation of (87) Sylvia I Romulus. Where more than one moon has been discovered, Roman numerals specify the discovery sequence, so that Sylvia's second moon is designated (87) Sylvia II Remus.
Since Pluto was reclassified in 2006, discoveries of Plutonian moons since then follow the minor-planet system: thus Nix and Hydra, discovered in 2005, were S/2005 P 2 and S/2005 P 1, but Kerberos and Styx, discovered in 2011 and 2012 respectively, were S/2011 (134340) 1 and S/2012 (134340) 1. That said, there has been some unofficial use of the formats "S/2011 P 1" and "S/2012 P 1".
Packed designation
Packed designations are used in online and electronic documents as well as databases.
Packed minor planet designation
The Orbit Database (MPCORB) of the Minor Planet Center (MPC) uses the "packed form" to refer to all provisionally designated minor planets. The idiosyncrasy found in the new-style provisional designations, no longer exists in this packed-notation system, as the second letter is now listed after the subscript number, or its equivalent 2-digit code. For an introduction on provisional minor planet designations in the "un-packed" form, see .
Provisional packed designations
The system of packed provisional minor planet designations:
uses exactly 7 characters with no spaces for all designations
compacts 4 digit years to a 3-character code, e.g. 2014 is written as K14 (see tables below)
converts all subscript numbers to a 2-character code (00 is used when there is no following subscript, 99 is used for subscript 99, A0 is used for subscript 100, and A1 is used for 101)
the packed 2 character subscript code is placed between the half-month letter and the second (discovery order) letter (e.g. has discovery order K so the last three characters for its packed form are A2K)
Contrary to the new-style system, the letter "i" is used in the packed form both for the year and the numeric suffix. The compacting system provides upper and lowercase letters to encode up to 619 "cycles". This means that 15,500 designations () within a half-month can be packed, which is a few times more than the designations assigned monthly in recent years.
Examples
is written as J95X00A
is written as J95X01L
is written as K16EF6K
is written as K07Tf8A
Description
The year 1995 is compacted to J95. As it has no subscript number, 00 is used as placeholder instead, and directly placed after the half-month letter "X".
The year 1995 is compacted to J95. Subscript number "1" is padded to 01 to maintain the length of 7 characters, and placed after the first letter.
The year 2016 is compacted to K16. The subscript number "156" exceeds 2 digits and is converted to F6, (see table below)
The year 2007 is compacted to K07. The subscript number "418" exceeds 2 digits and is converted to f8, (see table below)
Conversion tables
Comets follow the minor-planet scheme for their first four characters. The fifth and sixth characters encode the number. The seventh character is usually 0, unless it is a component of a split comet, in which case it encodes in lowercase the letter of the fragment.
Examples
1995 A1 is written as J95A010
1995 P1-B is written as J95P01b (i.e. fragment B of comet )
2088 A103 is written as K88AA30 (as the subscript number exceeds two digits and is converted according to the above table).
There is also an extended form that adds five characters to the front. The fifth character is one of "C", "D", "P", or "X", according to the status of the comet. If the comet is periodic, then the first four characters are the periodic-comet number (padded to the left with zeroes); otherwise, they are blank.
Natural satellites use the format for comets, except that the last column is always 0.
Packed survey designations
Survey designations used during the Palomar–Leiden Survey (PLS) have a simpler packed form, as for example:
is written as PLS6344
is written as T1S4835
is written as T2S1010
is written as T3S4101
Note that the survey designations are distinguished from provisional designations by having the letter S in the third character, which contains a decimal digit in provisional designations and permanent numbers.
Permanent packed designations
A packed form for permanent designations also exists (these are numbered minor planets, with or without a name). In this case, only the designation's number is used and converted to a 5-character string. The rest of the permanent designation is ignored. Minor planet numbers below 100,000 are simply zero-padded to 5 digits from the left side. For minor planets above 100,000, a single letter (A–Z and a–z) is used, similar as for the provisional subscript number (also see table above):
A covers the number range 100,000–109,999
B covers the number range 110,000–119,999
a covers the number range 360,000–369,999
z covers the number range 610,000–619,999
Examples
00001 encodes 1 Ceres
99999 encodes
A0000 encodes 100000 Astronautica, ()
A9999 encodes ()
B0000 encodes ()
G3693 encodes 163693 Atira ()
Y2843 encodes 342843 Davidbowie ()
g0356 encodes 420356 Praamzius ()
This system permits compression of numbers up to 619,999 (z9999). the list of minor planets already contains 619,150 objects. For minor planets numbered 620,000 or higher, a tilde "~" will be used as the first character. The subsequent 4 characters encoded in Base62 (using 0–9, then A–Z, and a–z, in this specific order) are used to store the difference of the object's number minus 620,000. This extended system will allow for the encoding of more than 15 million minor planet numbers. For example:
(620000) will be represented as ~0000 (
(620061) will be represented as ~000z (
(3140113) will be represented as ~AZaz (
(15396335) will be represented as ~zzzz (
For comets, permanent designations only apply to periodic comets that are seen to return. The first four characters are the number of the comet (left-padded with zeroes). The fifth character is "P", unless the periodic comet is lost or defunct, in which case it is "D".
For natural satellites, permanent packed designations take the form of the planet letter, then three digits containing the converted Roman numeral (left-padded with zeroes), and finally an "S". For example, Jupiter XIII Leda is J013S, and Neptune II Nereid is N002S.
See also
Minor-planet designation
Naming of moons
References
External links
New- And Old-Style Minor Planet Designations (Minor Planet Center)
Astronomical nomenclature
Comets
Minor planets
Moons | {
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You are here: Home » Scholarship Years » 2016 Winner & Finalists for the Tenth Scholarship
2016 Winner & Finalists for the Tenth Scholarship
Non-Commissioned Officer – Australian Defence Force
Andrew is excited at the opportunity to attend the world class MBA at Cranfield, where he can develop his business and managerial skills. He aims to work with the government and senior business leaders in addressing the complex issues faced by veterans making the transition from the ADF to fulfilling civilian careers.
Andrew will be making the move to Cranfield with his wife Caris and three children, Emily, Ella and William.
Clare Donovan
Finalist. Awarded a 50th Anniversary Scholarship by Cranfield School of Management
Sustainability Coordinator, NSW Office of Environment and Heritage
Clare has been working in social and environmental sustainability management roles for over 15 years in the public, private and not-for-profit sectors. Clare is currently managing the Office of Environment and Heritage's internal corporate sustainability program. The program covers over 3,000 staff and 400 sites including offices, science labs and national parks in NSW.
Prior to the environment sector, Clare worked in Mozambique, Africa for several years doing humanitarian work. Clare is concerned that developing countries will be the most affected by climate change, although they have the least resources and capacity to manage change.
Clare would like to strengthen her sustainability leadership potential with the Cranfield MBA to enable her to challenge the status quo and inspire the re-imagination of our future; to be the leader who speaks to influential Australian leaders, finding common ground and sharing her vision of an alternative, better future for Australia.
Clare is an active person who recently hiked the Thorsborne Trail in Queensland and Jatbula Trail in Northern Territory. Clare's love of the outdoors fuels her passion for sustainability and inspiration to protect nature.
2017 Winner and Finalists for the Eleventh Scholarship
2015 Winner & Finalists for the Ninth Scholarship
2015 Scholarship Award events
2014 Winner and Finalists for the Eighth Scholarship
2013 Winner and finalists for the Seventh Scholarship
2012 Scholarship winner and finalists
2011 Award Event Gallery
2007 Winner and Finalists | {
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Couple mauled to death by rabid sloth bear after visiting temple in Indian forest
Bethany Dawson
A sloth bear
Prisma by Dukas/Universal Images Group via Getty Images
A sloth bear has mauled a couple to death in Madhya Pradesh, central India.
An autopsy confirmed the bear had contracted rabies but had not fed on the corpses.
Dependents of the deceased couple will be given 400,000 Indian Rupees ($5,118) in compensation.
A rabid sloth bear has mauled a couple to death in Madhya Pradesh, central India, and played with their remains, Indian news outlets report.
The Times of India said that the couple had gone to pray at a nearby temple when they encountered the sloth bear on June 5.
The newspaper identified the deceased as 43-year-old Mukesh Rai and his wife, 39-year-old Gudiya Rai. The husband was killed when he tried to rescue his wife from the bear's clutches, according to CBS News.
Witness reports from earlier in the week that the bear ate parts of the couple after it had killed them, have been refuted, CBS News reported. The bear died two hours after it was captured, Divisional Forest Officer Gaurav Sharma told CBS News.
An autopsy confirmed the bear had contracted rabies but had not fed on the corpses, Sharma confirmed, said the news outlet.
The Times reports that the dependents of the deceased couple will be given 400,000 Indian Rupees ($5,118) in compensation.
According to the Smithsonian National Zoo, sloth bears are similar in size to the American back bear and grow up to six feet long, weighing, on average, 200 to 300 pounds. They have a shaggy, dusty appearance and have large canines for defense. About 20,000 or fewer sloth bears remain in the wild.
A study from HNG University, Gujarat, said sloth bears are "known for their aggressive and unpredictable behavior." Attacks are increasing as human activity increasingly impinges on the bears' home territories and 13.7% of sloth bear attacks result in death, said the study.
Conservation biologist Neha Sinha told CBS News that conflicts between bears and humans at this time of year are often caused by the Mahua tree flowering. Bears go to feed on the flower whilst humans harvest them to sell, and the bears can become territorial.
A study published in March found that in India, Sloth bear attacks are most common in the Madhya Pradesh region.
bear Bear Attack India | {
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} | 1,501 |
Q: Who was "The Pan" in Pan's Labyrinth? In Spanish, faun is fauno. Why did the English version insist on making the faun into Pan himself? Who was Pan? Why include the god of forests and nature?
A: According to Guillermo del Toro, the use of the word Pan in the non-Spanish versions was just a translation issue. It's not just the English version that has this mistake; the French and German translations also make it. Since he wrote his own subtitles, it was del Toro himself that chose to use the word Pan, but he's has admitted that the name is not accurate.
He likely chose Pan because he is the most well-known faun-like creature in classical mythology (even though it's the wrong mythology), so a name that viewers would identify with. But, The Faun is not Pan:
And the character of the faun is essentially the trickster. He is a character that is neither good nor bad. He's a character ... that's why I chose a faun. Not "Pan." Pan is just the translation, which is not accurate. It's a faun, because the faun in classical mythology was at the same time a creature of destruction, and a creature of nurturing and life. src
| {
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} | 5,991 |
{"url":"https:\/\/studyadda.com\/sample-papers\/jee-main-sample-paper-43_q89\/297\/302860","text":"\u2022 # question_answer Direction: For the following questions, choose the correct answer from the codes [a], [b], [c] and [d] defined as follows. Let us define the function as ${{\\cos }^{-1}}\\,(\\cos \\theta )=\\theta$and$2{{\\tan }^{-1}}x=\\frac{2x}{1-{{x}^{2}}}$. Statement I If $\\sin \\,[2\\,{{\\cos }^{-1}}\\{\\cot \\,(2\\,{{\\tan }^{-1}}x)\\}]=0,$ then $x=\\pm 1,\\,\\,\\pm \\,(1\\pm \\sqrt{2})$ Statement II $\\cot \\,\\,(2\\,{{\\tan }^{-1}}x)=\\frac{1-{{x}^{2}}}{2x}$ A) \u00a0Statement I is true. Statement II is also true and Statement II is the correct explanation of Statement I. B) \u00a0Statement I is true. Statement II is also true and Statement II is not the correct explanation of Statement I. C) \u00a0Statement I is true. Statement II is false. D) \u00a0Statement I is false. Statement II is true.\n\n$\\sin \\,\\theta =0\\,\\,\\,\\Rightarrow \\,\\,\\,\\theta n\\pi$ $\\Rightarrow$ \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 $2\\,{{\\cos }^{-1}}[\\cot (2\\,{{\\tan }^{-1}}x)]\\,=n\\pi$ $\\Rightarrow$\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 ${{\\cos }^{-1}}\\,[\\cot \\,(2\\,{{\\tan }^{-1}}x)]=\\frac{n\\pi }{2}$ $\\Rightarrow$\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 $\\cot \\,(2\\,{{\\tan }^{-1}}x)=\\cos \\,\\frac{n\\pi }{2}$ $\\Rightarrow$\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 $\\frac{1}{\\tan \\,\\left( {{\\tan }^{-1}}\\frac{2x}{1-{{x}^{2}}} \\right)}=\\cos \\,\\frac{n\\pi }{2}$ $\\Rightarrow$\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 $\\frac{1-{{x}^{2}}}{2x}=\\cos \\,\\frac{n\\pi }{2}$ Now, $\\cos \\,\\frac{n\\pi }{2}=0,$ if $n=1,\\,\\,3,\\,5,...$ $\\therefore$\u00a0\u00a0\u00a0 $\\frac{1-{{x}^{2}}}{2x}=0\\,\\,\\Rightarrow \\,\\,x=\\pm 1$ Now, $\\cos \\,\\frac{n\\pi }{2}=1,$ if $n=0,4,...$ $\\therefore$\u00a0\u00a0\u00a0 $\\frac{1-{{x}^{2}}}{2x}=1\\Rightarrow \\,x=-1\\,\\pm \\,\\sqrt{2}$ Now, $\\cos \\,\\frac{n\\pi }{2}=-1,$ if $n=2,\\,\\,6,...$ $\\therefore$\u00a0\u00a0\u00a0 $\\frac{1-{{x}^{2}}}{2x}=-1\\,\\,\\,\\Rightarrow \\,\\,x=1\\pm \\sqrt{2}$ Hence, $x=\\pm 1,\\,\\,\\pm 1\\pm \\sqrt{2}$","date":"2022-01-17 10:07:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7001170516014099, \"perplexity\": 2360.2976085313176}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-05\/segments\/1642320300533.72\/warc\/CC-MAIN-20220117091246-20220117121246-00542.warc.gz\"}"} | null | null |
OK 2019-04-20 07:24:03 0d 2h 25m 46s 1/3 Automatic updates functioning normally.
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OK 2019-04-20 07:21:24 1d 14h 18m 25s 1/3 Automatic updates functioning normally. | {
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MicroStrategy Acquires Extra 7,002 Bitcoins, Bringing its Portfolio to over 121K BTC
Brian Njuguna Nov 30, 2021 03:50 2 Min Read
Ever since institutional investments started trickling into the Bitcoin market, the leading cryptocurrency has experienced exponential growth, with its market capitalization topping $1 trillion.
One of the notable institutional investors includes American business intelligence firm MicroStrategy that has continuously accumulated Bitcoin to 121,044 BTC over the past year. The firm's CEO Michael Saylor confirmed the purchase of an additional over 7,000 Bitcoins and said:
"MicroStrategy has purchased an additional 7,002 Bitcoins for $414.4 million in cash at an average price of ~$59,187 per Bitcoin. As of 11/29/21 we hodl 121,044 Bitcoins acquired for $3.57 billion at an average price of $29,534 per Bitcoin."
Microstrategy bought 21,454 Bitcoins worth $250 million for the first time in August 2020, following a new capital allocation strategy to maximize its shareholders' long-term value.
Saylor believes that Bitcoin becomes the principal asset in the firm's treasury reserves because it showcases itself as a legitimate investment vehicle superior to cash. Since then, Microstrategy has not relented in its quest to accumulate more BTC because it usually takes advantage of buying the dip.
Saylor's Bitcoin advocacy has also not gone unnoticed, as he previously stated:
"We find the global acceptance, brand recognition, ecosystem vitality, network dominance, architectural resilience, technical utility, and community ethos of Bitcoin to be persuasive evidence of its superiority as an asset class for those seeking a long-term store of value."
Meanwhile, Thailand seeks to revamp its ailing tourism sector by targeting crypto investors. This is based on plans by the Tourism Authority of Thailand to collaborate with a local crypto exchange and lawmakers to accept crypto payments for travel.
Therefore, this approach intends to breathe life into the nation's tourism industry that has lost approximately $80 billion through the Covid-19 pandemic.
MicroStrategy Further Accumulates Bitcoin Holdings to 114,042 after additional 5,050 Purchase
MICHAEL SAYLOR | {
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} | 6,580 |
Cooking turkey for the holidays is easy when you have the right tools, and we're your one stop shop for everything other than the bird. Whether you need a roasting pan, baster, brining bucket, cutting board, cheesecloth, or any other kitchen accessory we've got you covered. We've collected some of the more popular items and listed them below.
Looking for a great turkey recipe? Our favorite is Alton Brown's recipe posted on the Food Network website. Combine this with this make-ahead gravy recipe and you're guaranteed to have delicious turkey and gravy for your holiday guests. Gobble gobble! | {
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} | 6,593 |
Q: Celery with Flask and UWSGI I have a simple application flask with a simple celery task:
from flask import Flask
from celery import Celery
app = Flask(__name__)
app.config['CELERY_BROKER_URL'] = 'redis://localhost:6379'
app.config['CELERY_RESULT_BACKEND'] = 'redis://localhost:6379'
celery = Celery(app.name, broker=app.config['CELERY_BROKER_URL'])
celery.conf.update(app.config)
@celery.task
def add(x, y):
return x + y
@app.route('/', methods=['GET'])
def test_func():
res = add.delay(4,5)
while not res.ready():
pass
data = res.get()
return str(data)
if __name__ == '__main__':
app.run(host='0.0.0.0',debug=True)
When I try to use your exemple with uwsgi but I encounter almost the same error than you.
First, I run it simply with python:
python app.py
and the broker with the following command:
celery -A app.celery worker -l info
Everything is working perfectly. Now I try to launch the flask application with uwsgi.
[uwsgi]
mount = /=/home/admin/flask-celery/app.py
callable = app
virtualenv = /home/admin/flask-celery/.venv
socket = :3031
master = true
processes = 2
threads = 4
http = :9000
But when I go on my route, I encounter the following error:
celery.exceptions.NotRegistered: 'uwsgi_file__home_admin_flask-celery_app.add'
A: Pleas take a look at this:https://uwsgi-docs.readthedocs.io/en/latest/AttachingDaemons.html
Although the following configuration does not exactly match your environment, I hope this could help.
[uwsgi]
base = /home/project
chdir = %(base)
module = app
pythonpath = %(base)
virtualenv = %(base)/venv
wsgi-file = %(base)/app.py
master = true
smart-attach-daemon = %(base)/tmp/celery.pid %(virtualenv)/bin/celery -A %(module).celery worker --pidfile=%(base)/tmp/celery.pid
socket = %(base)/socket.sock
chmod-socket = 777
processes = 4
threads = 4
logto = %(base)/log/%n.log
stats = 127.0.0.1:9191
A: Try specifying task name explicitly, for example like this:
@celery.task(name='app.celery.add')
def add(x, y):
return x + y
To ensure changes takes effect, restart celery worker after you made changes and cleanup all *.pyc files with find . -name '*.pyc' -delete
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,798 |
Do People All Throughout the World Wear Wedding Bands?
Wooden Earth Home Info Do People All Throughout the World Wear Wedding Bands?
Jan, 26
If you want to buy a ring but aren't sure which one is ideal, this article may help. The history, cultural significance, and demographics of engagement ring wear will all be discussed. You may get a wide range of styles, from classic to contemporary, at a wide range of price points.
Traditional wedding bands have been used for hundreds of years or more. Natural materials like hemp and ivory were commonly used to craft rings in ancient times. Diamonds and other stones were used to embellish these rings. Rings throughout the Middle Ages were often made of gold or silver and often had jewels for decoration.
In the past, people looked at wedding bands as a symbol of a happy marriage. They were initially used ceremonially by the ancient Egyptians. They also made the first recorded mention of them in print.
They're still used nowadays, but in a more casual context. The wedding ring has several forms throughout different civilizations.
Customary Turkish Engagement Rites
Throughout the years, a variety of distinct customs have emerged in Turkey around the engagement process. There is a great deal of cultural weight attached to each one.
The soz kesmek, or "promise ceremony," is one of the earliest and most significant rituals. As part of a coming-of-age ceremony, the father of the prospective groom will give his son a blessing and a word of purpose.
Making coffee is another custom that is meant to demonstrate a woman's potential suitability for marriage. The mother of the groom lavishes her son-in-law-to-be with luxury gifts including shoes, jewelry, and cosmetics.
Representative of Irish Pride
One of the most recognizable symbols of Ireland is the Claddagh ring. It's also a traditional choice for wedding bands. The ring has a special significance as a timeless token of affection, companionship, and devotion.
Irish fisherman Richard Joyce crafted one of the oldest known examples of the Claddagh ring in the 17th century. They were both clearly smitten with one another after he presented her with the ring. After then, the ring became increasingly popular and is today commonly recognized as a symbol of Irish heritage.
The original location of the Claddagh ring is unknown, however it is said to have originated in the Galway County fishing community of the same name. A lovely and self-sufficient fishing community.
Choices That Deviate from the Norm
It's important to get a wedding band that complements your own style. Fortunately, the market offers a plethora of alternatives. You must also choose between the tried-and-true and a more experimental approach.
Alternative wedding rings are widely available. Sizes and forms are not restricted. You may also discover one-of-a-kind rings fashioned from nontraditional materials like oak, silicone, and glow-in-the-dark resin.
The colorful diamond ring is one of the most well-liked alternatives to the conventional solitaire setting. This ring is a vibrant and striking way to declare your love for your special someone.
Gen Y shells out 157.9 percent more than typical Baby Boomers do on jewelry.
Jewelry is a popular purchase among Millennials. Because this age range includes students, this is to be expected. This demographic has more disposable income than previous generations and can thus afford to make substantial expenditures.
Most Millennials say they have a positive spending momentum, notwithstanding the extent of this spending disparity. They also said they have increased their spending on necessities like food, utilities, and transportation.
The generation known as Millennials shop for jewelry in a variety of ways. There is a significant market for internet shopping. One in five customers is in a committed relationship. An important demographic for the jewelry industry is the single population.
The most expensive engagement ring ever sold was an emerald-cut diamond set on a platinum split-shank band.
The most expensive engagement ring in the world consists of a dazzling emerald-cut diamond mounted on a platinum split-band setting. The cost of an emerald-cut diamond set in platinum has increased by approximately $9 million since the recession of 2008.
Several well-known celebrities who can afford to flaunt their expensive engagement rings are included below. That platinum ring Mariah Carey wore was reportedly worth $10 million. Beyonce was proposed to by Jay-Z with an 18-carat emerald cut diamond.
Even though Priyanka Chopra's ring looks to be merely five carats, analysts believe that it might cost as much as $150,000. The five-carat diamond in the core of Carrie Underwood's ring is responsible for the rumored price tag of $150,000.
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Tweets by @woodenearth | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,889 |
Gold and Silver prices are updated regularly. The following tables shows the latest gold price in Algeria calculated in Algerian Dinar (DZD) and updated regularly. The prices are sometimes updated more frequently at times of strong price moves based on live spot gold price (Bid Price).
Know the latest Silver price in Algeria on the largest website for Silver rates.
algeria 1904 in stock and ready to ship right now online. Locate Algeria 1904 on sale today online.
On Algeria Gold Rate page, user can check Live Gold rate in Algeria. They can find the latest and gold current price in Algeria . Gold price in Algeria is available in various weights like Grams, Carat, and Ounce (example 10 grams, 100 grams, 500 grams amp; 1 Kg).
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Gold price per gram today is DZD 4,673.08 in the Algeria gold market and the gold rate in Algeria today per ounce is DZD 145,349.18. On 22 11 2018, a kilogram (1000 Grams) of gold in Algeria was DZD 4,673,084.72 as per the global market.
Jan 17, 20170183;32;Lead Sand, Lead Sand Suppliers and Manufacturers at gold mineral processing equipment gold concentrator for algeria. polyacrylamide/ PAM in Gold/Silver/ Coal/ Copper/ Zinc/Lead/ .. Lead Silver Ore Zinc Sf Flotation Concentrator Separation Process Machine Used For Mining.
Silver Price Today in Algeria in Algerian Dinar (DZD) This page shows the current today's silver prices in Algeria in Algerian Dinar (DZD) according to the local timezone of Algiers in addition to the last price of yesterday with calculation of the change percent.
Updated Gold Prices In Algeria. Gold rates change almost every day worldwide including Algeria. Get the updates about today gold price in Algeria, and get rates of 18 karat, 20 karat, 21 karat and 22 karat gold.
Unlike the 2017 edition, the 2018 Silver Cup will be split into two groups of 3 teams each. By beating Zambia 31 0, Algeria won the competition and will automatically join the Gold Cup alongside Zambia.
Gold and Silver Price Today in Algeria in Algerian Dinar . The latest gold prices and silver prices by different currencies in different countries in the world. Gold and silver prices are updated daily.
Algeria is the second largest country in Africa (by area) with a population of approximately 34 million. when entering into a formal association with an Algerian state enterprise to exploit strategic deposits such as gold. Total Silver Production ENOR. oz. 4,756. 6,409. 3,748. High Grade Ore Tonnes ex Mine. mt. 36,410. 60,720.
Find great deals on eBay for algeria coins. Shop with confidence.
1 kg pure silver bar price in Algeria is 51,082.38 Algerian Dinar and 10 tola pure silver biscuit rate in Algeria is 5,958.15 Algerian Dinar. 1 oz silver bar price in Algeria is 1,588.84 Algerian Dinar and 1 sovereign silver coin rate in Algeria is 408.66 Algerian Dinar.
Jan 19, 20160183;32;VIDEO FINANCIAL REPORTING Why invest in is the first financial video platform where you can easily search through thousands of videos describing global securities.
Alibaba offers 71 algeria gold jewellery products. About 25% of these are zinc alloy jewelry, 23% are jewelry sets, and 15% are bracelets amp; bangles. A wide variety of algeria gold jewellery options are available to you, such as anniversary, engagement, and gift. | {
"redpajama_set_name": "RedPajamaC4"
} | 764 |
{"url":"https:\/\/za.limehousetownhall.org.uk\/11003-open-a-osmpbf-file-with-fiona-in-python.html","text":"Open a .osm.pbf file with fiona in python\n\nI would like to open a .osm.pbf using fiona in Python. I can't find much documentation on this. How can I does one do this?\n\nI have done it using ogr2ogr.\n\nFiona is by design restricted to the conventional record model of data, i.e. all records (features) have the same fields associated with them. This means that Fiona reads shapefiles, but does not read more flexible formats such as the OSM PBF format.\n\nYou can check which drivers are supported in Fiona with:\n\nimport fiona list(fiona.drivers)\n\nYou have two options then: use the OGR Python drivers to read the data, or to useogr2ogrto convert the data to a format that Fiona can read. I think the second option is your best bet as I find Fiona much easier to use.\n\nAdding a background map to plots\u00b6\n\nThis example shows how you can add a background basemap to plots created with the geopandas .plot() method. This makes use of the contextily package to retrieve web map tiles from several sources (OpenStreetMap, Stamen). Also have a look at contextily\u2019s introduction guide for possible new features not covered here.\n\nLet\u2019s use the NYC borough boundary data that is available in geopandas datasets. Plotting this gives the following result:\n\n15 Python Libraries for GIS and Mapping\n\nPython libraries are the ultimate extension in GIS because it allows you to boost its core functionality.\n\nBy using Python libraries, you can break out of the mold that is GIS and dive into some serious data science.\n\nThere are 200+ standard libraries in Python. But there are thousands of third-party libraries too. So, it\u2019s endless how far you can take it.\n\nToday, it\u2019s all about Python libraries in GIS. Specifically, what are the most popular Python packages that GIS professionals use today? Let\u2019s get started.\n\nFirst, why even use Python libraries for GIS?\n\nHave you ever noticed how GIS is missing that one capability you need it to do? Because no GIS software can do it all, Python libraries can add that extra functionality you need.\n\nPut simply, a Python library is code someone else has written to make life easier for the rest of us. Developers have written open libraries for machine learning, reporting, graphing, and almost everything in Python.\n\nIf you want this extra functionality, you can leverage those libraries by importing them into your Python script. From here, you can call functions that aren\u2019t natively part of your core GIS software.\n\nPRO TIP: Use pip to install and manage your packages in Python\n\nPython Libraries for GIS\n\nIf you\u2019re going to build an all-star team for GIS Python libraries, this would be it. They all help you go beyond the typical managing, analyzing, and visualizing of spatial data. That is the true definition of a geographic information system.\n\n1 . Arcpy\n\nIf you use Esri ArcGIS, then you\u2019re probably familiar with the ArcPy library. ArcPy is meant for geoprocessing operations. But it\u2019s not only for spatial analysis, it\u2019s also for data conversion, management, and map production with Esri ArcGIS.\n\n2 . Geopandas\n\nGeopandas is like pandas meet GIS. But instead of straight-forward tabular analysis, the geopandas library adds a geographic component. For overlay operations, Geopandas uses Fiona and Shapely, which are Python libraries of their own.\n\n3 . GDAL\/OGR\n\nThe GDAL\/OGR library is used for translating between GIS formats and extensions. QGIS, ArcGIS, ERDAS, ENVI, and GRASS GIS and almost all GIS software use it for translation in some way. At this time, GDAL\/OGR supports 97 vector and 162 raster drivers.\n\n4 . RSGISLib\n\nThe RSGISLib library is a set of remote sensing tools for raster processing and analysis. To name a few, it classifies, filters, and performs statistics on imagery. My personal favorite is the module for object-based segmentation and classification (GEOBIA).\n\n5 . PyProj\n\nThe main purpose of the PyProj library is how it works with spatial referencing systems. It can project and transform coordinates with a range of geographic reference systems. PyProj can also perform geodetic calculations and distances for any given datum.\n\nPython Libraries for Data Science\n\nData science extracts insights from data. It takes data and tries to make sense of it, such as by plotting it graphically or using machine learning. This list of Python libraries can do exactly this for you.\n\n6 . NumPy\n\nNumerical Python (NumPy library) takes your attribute table and puts it in a structured array. Once it\u2019s in a structured array, it\u2019s much faster for any scientific computing. One of the best things about it is how you can work with other Python libraries like SciPy for heavy statistical operations.\n\n7 . Pandas\n\nThe Pandas library is immensely popular for data wrangling. It\u2019s not only for statisticians. But it\u2019s incredibly useful in GIS too. Computational performance is key for pandas. The success of Pandas lies in its data frame. Data frames are optimized to work with big data. They\u2019re optimized to such a point that it\u2019s something that Microsoft Excel wouldn\u2019t even be able to handle.\n\n8 . Matplotlib\n\nWhen you\u2019re working with thousands of data points, sometimes the best thing to do is plot it all out. Enter matplotlib. Statisticians use the matplotlib library for visual display. Matplotlib does it all. It plots graphs, charts, and maps. Even with big data, it\u2019s decent at crunching numbers.\n\n9 . Scikit\n\nLately, machine learning has been all the buzz. And with good reason. Scikit is a Python library that enables machine learning. It\u2019s built into NumPy, SciPy, and Matplotlib. So, if you want to do any data mining, classification or ML prediction, the Scikit library is a decent choice.\n\n10 . Re (regular expressions)\n\nRegular expressions (Re) are the ultimate filtering tool. When there\u2019s a specific string you want to hunt down in a table, this is your go-to library. But you can take it a bit further like detecting, extracting and replacing with pattern matching.\n\n11 . ReportLab\n\nReportLab is one of the most satisfying libraries on this list. I say this because GIS often lacks sufficient reporting capabilities. Especially, if you want to create a report template, this is a fabulous option. I don\u2019t know why the ReportLab library falls a bit off the radar because it shouldn\u2019t.\n\n12 . ipyleaflet\n\nIf you want to create interactive maps, ipyleaflet is a fusion of Jupyter notebook and Leaflet. You can control an assortment of customizations like loading basemaps, geojson, and widgets. It also gives a wide range of map types to pick from including choropleth, velocity data, and side-by-side views.\n\n13 . Folium\n\nJust like ipyleaflet, Folium allows you to leverage leaflet to build interactive web maps. It gives you the power to manipulate your data in Python, then you can visualize it with the leading open-source JavaScript library.\n\n14 . Geemap\n\nGeemap is intended more for science and data analysis using Google Earth Engine (GEE). Although anyone can use this Python library, scientists and researchers specifically use it to explore the multi-petabyte catalog of satellite imagery in GEE for their specific applications and uses with remote sensing data.\n\n15 . LiDAR\n\nSimply named the LiDAR Python Package, the purpose is to process and visualize Light Detection and Ranging (LiDAR) data. For example, it includes tools to smooth, filter, and extract topological properties from digital elevation models (DEMs) data. Althought I don\u2019t see integration with raw LAS files, it serves its purpose for terrain and hydrological analysis.\n\nPRO TIP: If you need a quick and dirty list of functions for Python libraries, check out DataCamp\u2019s Cheat Sheets.\n\nThe Python Libraries All-Star Team\n\nThese are the Python libraries we thought were stand-outs for GIS and data science.\n\nNow, it\u2019s time to turn it on to you.\n\nIf you could build an all-star team of Python libraries, who would you put on your team?\n\nContents\n\nTo communicate information clearly and efficiently, data visualization uses statistical graphics, plots, information graphics and other tools. Numerical data may be encoded using dots, lines, or bars, to visually communicate a quantitative message. [6] Effective visualization helps users analyze and reason about data and evidence. It makes complex data more accessible, understandable, and usable. Users may have particular analytical tasks, such as making comparisons or understanding causality, and the design principle of the graphic (i.e., showing comparisons or showing causality) follows the task. Tables are generally used where users will look up a specific measurement, while charts of various types are used to show patterns or relationships in the data for one or more variables.\n\nData visualization refers to the techniques used to communicate data or information by encoding it as visual objects (e.g., points, lines, or bars) contained in graphics. The goal is to communicate information clearly and efficiently to users. It is one of the steps in data analysis or data science. According to Vitaly Friedman (2008) the \"main goal of data visualization is to communicate information clearly and effectively through graphical means. It doesn't mean that data visualization needs to look boring to be functional or extremely sophisticated to look beautiful. To convey ideas effectively, both aesthetic form and functionality need to go hand in hand, providing insights into a rather sparse and complex data set by communicating its key aspects in a more intuitive way. Yet designers often fail to achieve a balance between form and function, creating gorgeous data visualizations which fail to serve their main purpose \u2014 to communicate information\". [7]\n\nIndeed, Fernanda Viegas and Martin M. Wattenberg suggested that an ideal visualization should not only communicate clearly, but stimulate viewer engagement and attention. [8]\n\nData visualization is closely related to information graphics, information visualization, scientific visualization, exploratory data analysis and statistical graphics. In the new millennium, data visualization has become an active area of research, teaching and development. According to Post et al. (2002), it has united scientific and information visualization. [9]\n\nIn the commercial environment data visualization is often referred to as dashboards. Infographics are another very common form of data visualization.\n\nCharacteristics of effective graphical displays Edit\n\nProfessor Edward Tufte explained that users of information displays are executing particular analytical tasks such as making comparisons. The design principle of the information graphic should support the analytical task. [11] As William Cleveland and Robert McGill show, different graphical elements accomplish this more or less effectively. For example, dot plots and bar charts outperform pie charts. [12]\n\nIn his 1983 book The Visual Display of Quantitative Information, Edward Tufte defines 'graphical displays' and principles for effective graphical display in the following passage: \"Excellence in statistical graphics consists of complex ideas communicated with clarity, precision, and efficiency. Graphical displays should:\n\n\u2022 show the data\n\u2022 induce the viewer to think about the substance rather than about methodology, graphic design, the technology of graphic production, or something else\n\u2022 avoid distorting what the data has to say\n\u2022 present many numbers in a small space\n\u2022 make large data sets coherent\n\u2022 encourage the eye to compare different pieces of data\n\u2022 reveal the data at several levels of detail, from a broad overview to the fine structure\n\u2022 serve a reasonably clear purpose: description, exploration, tabulation, or decoration\n\u2022 be closely integrated with the statistical and verbal descriptions of a data set.\n\nGraphics reveal data. Indeed graphics can be more precise and revealing than conventional statistical computations.\" [13]\n\nFor example, the Minard diagram shows the losses suffered by Napoleon's army in the 1812\u20131813 period. Six variables are plotted: the size of the army, its location on a two-dimensional surface (x and y), time, the direction of movement, and temperature. The line width illustrates a comparison (size of the army at points in time), while the temperature axis suggests a cause of the change in army size. This multivariate display on a two-dimensional surface tells a story that can be grasped immediately while identifying the source data to build credibility. Tufte wrote in 1983 that: \"It may well be the best statistical graphic ever drawn.\" [13]\n\nNot applying these principles may result in misleading graphs, distorting the message, or supporting an erroneous conclusion. According to Tufte, chartjunk refers to the extraneous interior decoration of the graphic that does not enhance the message or gratuitous three-dimensional or perspective effects. Needlessly separating the explanatory key from the image itself, requiring the eye to travel back and forth from the image to the key, is a form of \"administrative debris.\" The ratio of \"data to ink\" should be maximized, erasing non-data ink where feasible. [13]\n\nThe Congressional Budget Office summarized several best practices for graphical displays in a June 2014 presentation. These included: a) Knowing your audience b) Designing graphics that can stand alone outside the report's context and c) Designing graphics that communicate the key messages in the report. [14]\n\nQuantitative messages Edit\n\nAuthor Stephen Few described eight types of quantitative messages that users may attempt to understand or communicate from a set of data and the associated graphs used to help communicate the message:\n\n1. Time-series: A single variable is captured over a period of time, such as the unemployment rate over a 10-year period. A line chart may be used to demonstrate the trend.\n2. Ranking: Categorical subdivisions are ranked in ascending or descending order, such as a ranking of sales performance (the measure) by sales persons (the category, with each sales person a categorical subdivision) during a single period. A bar chart may be used to show the comparison across the sales persons.\n3. Part-to-whole: Categorical subdivisions are measured as a ratio to the whole (i.e., a percentage out of 100%). A pie chart or bar chart can show the comparison of ratios, such as the market share represented by competitors in a market.\n4. Deviation: Categorical subdivisions are compared against a reference, such as a comparison of actual vs. budget expenses for several departments of a business for a given time period. A bar chart can show comparison of the actual versus the reference amount.\n5. Frequency distribution: Shows the number of observations of a particular variable for given interval, such as the number of years in which the stock market return is between intervals such as 0-10%, 11-20%, etc. A histogram, a type of bar chart, may be used for this analysis. A boxplot helps visualize key statistics about the distribution, such as median, quartiles, outliers, etc.\n6. Correlation: Comparison between observations represented by two variables (X,Y) to determine if they tend to move in the same or opposite directions. For example, plotting unemployment (X) and inflation (Y) for a sample of months. A scatter plot is typically used for this message.\n7. Nominal comparison: Comparing categorical subdivisions in no particular order, such as the sales volume by product code. A bar chart may be used for this comparison. or geospatial: Comparison of a variable across a map or layout, such as the unemployment rate by state or the number of persons on the various floors of a building. A cartogram is a typical graphic used. [6][15]\n\nAnalysts reviewing a set of data may consider whether some or all of the messages and graphic types above are applicable to their task and audience. The process of trial and error to identify meaningful relationships and messages in the data is part of exploratory data analysis.\n\nVisual perception and data visualization Edit\n\nA human can distinguish differences in line length, shape, orientation, distances, and color (hue) readily without significant processing effort these are referred to as \"pre-attentive attributes\". For example, it may require significant time and effort (\"attentive processing\") to identify the number of times the digit \"5\" appears in a series of numbers but if that digit is different in size, orientation, or color, instances of the digit can be noted quickly through pre-attentive processing. [16]\n\nCompelling graphics take advantage of pre-attentive processing and attributes and the relative strength of these attributes. For example, since humans can more easily process differences in line length than surface area, it may be more effective to use a bar chart (which takes advantage of line length to show comparison) rather than pie charts (which use surface area to show comparison). [16]\n\nHuman perception\/cognition and data visualization Edit\n\nAlmost all data visualizations are created for human consumption. Knowledge of human perception and cognition is necessary when designing intuitive visualizations. [17] Cognition refers to processes in human beings like perception, attention, learning, memory, thought, concept formation, reading, and problem solving. [18] Human visual processing is efficient in detecting changes and making comparisons between quantities, sizes, shapes and variations in lightness. When properties of symbolic data are mapped to visual properties, humans can browse through large amounts of data efficiently. It is estimated that 2\/3 of the brain's neurons can be involved in visual processing. Proper visualization provides a different approach to show potential connections, relationships, etc. which are not as obvious in non-visualized quantitative data. Visualization can become a means of data exploration.\n\nStudies have shown individuals used on average 19% less cognitive resources, and 4.5% better able to recall details when comparing data visualization with text. [19]\n\nThere is no comprehensive 'history' of data visualization. There are no accounts that span the entire development of visual thinking and the visual representation of data, and which collate the contributions of disparate disciplines. [20] Michael Friendly and Daniel J Denis of York University are engaged in a project that attempts to provide a comprehensive history of visualization. Contrary to general belief, data visualization is not a modern development. Since prehistory, stellar data, or information such as location of stars were visualized on the walls of caves (such as those found in Lascaux Cave in Southern France) since the Pleistocene era. [21] Physical artefacts such as Mesopotamian clay tokens (5500 BC), Inca quipus (2600 BC) and Marshall Islands stick charts (n.d.) can also be considered as visualizing quantitative information. [22] [23]\n\nThe first documented data visualization can be tracked back to 1160 B.C. with Turin Papyrus Map which accurately illustrates the distribution of geological resources and provides information about quarrying of those resources. [24] Such maps can be categorized as thematic cartography, which is a type of data visualization that presents and communicates specific data and information through a geographical illustration designed to show a particular theme connected with a specific geographic area. Earliest documented forms of data visualization were various thematic maps from different cultures and ideograms and hieroglyphs that provided and allowed interpretation of information illustrated. For example, Linear B tablets of Mycenae provided a visualization of information regarding Late Bronze Age era trades in the Mediterranean. The idea of coordinates was used by ancient Egyptian surveyors in laying out towns, earthly and heavenly positions were located by something akin to latitude and longitude at least by 200 BC, and the map projection of a spherical earth into latitude and longitude by Claudius Ptolemy [c.85\u2013c. 165] in Alexandria would serve as reference standards until the 14th century. [24]\n\nThe invention of paper and parchment allowed further development of visualizations throughout history. Figure shows a graph from the 10th or possibly 11th century that is intended to be an illustration of the planetary movement, used in an appendix of a textbook in monastery schools. [25] The graph apparently was meant to represent a plot of the inclinations of the planetary orbits as a function of the time. For this purpose, the zone of the zodiac was represented on a plane with a horizontal line divided into thirty parts as the time or longitudinal axis. The vertical axis designates the width of the zodiac. The horizontal scale appears to have been chosen for each planet individually for the periods cannot be reconciled. The accompanying text refers only to the amplitudes. The curves are apparently not related in time.\n\nBy the 16th century, techniques and instruments for precise observation and measurement of physical quantities, and geographic and celestial position were well-developed (for example, a \u201cwall quadrant\u201d constructed by Tycho Brahe [1546\u20131601], covering an entire wall in his observatory). Particularly important were the development of triangulation and other methods to determine mapping locations accurately. [20] Very early, the measure of time led scholars to develop innovative way of visualizing the data (e.g. Lorenz Codomann in 1596, Johannes Temporarius in 1596 [26] ).\n\nFrench philosopher and mathematician Ren\u00e9 Descartes and Pierre de Fermat developed analytic geometry and two-dimensional coordinate system which heavily influenced the practical methods of displaying and calculating values. Fermat and Blaise Pascal's work on statistics and probability theory laid the groundwork for what we now conceptualize as data. [20] According to the Interaction Design Foundation, these developments allowed and helped William Playfair, who saw potential for graphical communication of quantitative data, to generate and develop graphical methods of statistics. [17]\n\nIn the second half of the 20th century, Jacques Bertin used quantitative graphs to represent information \"intuitively, clearly, accurately, and efficiently\". [17]\n\nJohn Tukey and Edward Tufte pushed the bounds of data visualization Tukey with his new statistical approach of exploratory data analysis and Tufte with his book \"The Visual Display of Quantitative Information\" paved the way for refining data visualization techniques for more than statisticians. With the progression of technology came the progression of data visualization starting with hand-drawn visualizations and evolving into more technical applications \u2013 including interactive designs leading to software visualization. [27]\n\nPrograms like SAS, SOFA, R, Minitab, Cornerstone and more allow for data visualization in the field of statistics. Other data visualization applications, more focused and unique to individuals, programming languages such as D3, Python and JavaScript help to make the visualization of quantitative data a possibility. Private schools have also developed programs to meet the demand for learning data visualization and associated programming libraries, including free programs like The Data Incubator or paid programs like General Assembly. [28]\n\nBeginning with the symposium \"Data to Discovery\" in 2013, ArtCenter College of Design, Caltech and JPL in Pasadena have run an annual program on interactive data visualization. [29] The program asks: How can interactive data visualization help scientists and engineers explore their data more effectively? How can computing, design, and design thinking help maximize research results? What methodologies are most effective for leveraging knowledge from these fields? By encoding relational information with appropriate visual and interactive characteristics to help interrogate, and ultimately gain new insight into data, the program develops new interdisciplinary approaches to complex science problems, combining design thinking and the latest methods from computing, user-centered design, interaction design and 3D graphics.\n\nData visualization involves specific terminology, some of which is derived from statistics. For example, author Stephen Few defines two types of data, which are used in combination to support a meaningful analysis or visualization:\n\n\u2022 Categorical: Represent groups of objects with a particular characteristic. Categorical variables can either be nominal or ordinal. Nominal variables for example gender have no order between them and are thus nominal. Ordinal variables are categories with an order, for sample recording the age group someone falls into. [30]\n\u2022 Quantitative: Represent measurements, such as the height of a person or the temperature of an environment. Quantitative variables can either be continuous or discrete. Continuous variables capture the idea that measurements can always be made more precisely. While discrete variables have only a finite number of possibilities, such as a count of some outcomes or an age measured in whole years. [30]\n\nThe distinction between quantitative and categorical variables is important because the two types require different methods of visualization.\n\nTwo primary types of information displays are tables and graphs.\n\n\u2022 A table contains quantitative data organized into rows and columns with categorical labels. It is primarily used to look up specific values. In the example above, the table might have categorical column labels representing the name (a qualitative variable) and age (a quantitative variable), with each row of data representing one person (the sampled experimental unit or category subdivision).\n\u2022 A graph is primarily used to show relationships among data and portrays values encoded as visual objects (e.g., lines, bars, or points). Numerical values are displayed within an area delineated by one or more axes. These axes provide scales (quantitative and categorical) used to label and assign values to the visual objects. Many graphs are also referred to as charts. [31]\n\nEppler and Lengler have developed the \"Periodic Table of Visualization Methods,\" an interactive chart displaying various data visualization methods. It includes six types of data visualization methods: data, information, concept, strategy, metaphor and compound. [32]\n\n\u2022 length\/count\n\u2022 category\n\u2022 color\n\u2022 Presents categorical data with rectangular bars with heights or lengths proportional to the values that they represent. The bars can be plotted vertically or horizontally.\n\u2022 A bar graph shows comparisons among discretecategories. One axis of the chart shows the specific categories being compared, and the other axis represents a measured value.\n\u2022 Some bar graphs present bars clustered in groups of more than one, showing the values of more than one measured variable. These clustered groups can be differentiated using color.\n\u2022 For example comparison of values, such as sales performance for several persons or businesses in a single time period.\n\nVariable-width (\"variwide\") bar chart\n\n\u2022 category (size\/count\/extent in first dimension)\n\u2022 size\/count\/extent in second dimension\n\u2022 size\/count\/extent as area of bar\n\u2022 color\n\u2022 Includes most features of basic bar chart, above\n\u2022 Area of non-uniform-width bar explicitly conveys information of a third quantity that is implicitly related to first and second quantities from horizontal and vertical axes\n\u2022 bin limits\n\u2022 count\/length\n\u2022 color\n\u2022 An approximate representation of the distribution of numerical data. Divide the entire range of values into a series of intervals and then count how many values fall into each interval this is called binning. The bins are usually specified as consecutive, non-overlapping intervals of a variable. The bins (intervals) must be adjacent, and are often (but not required to be) of equal size.\n\u2022 For example, determining frequency of annual stock market percentage returns within particular ranges (bins) such as 0-10%, 11-20%, etc. The height of the bar represents the number of observations (years) with a return % in the range represented by the respective bin.\n\u2022 x position\n\u2022 y position\n\u2022 symbol\/glyph\n\u2022 color\n\u2022 size\n\u2022 Uses Cartesian coordinates to display values for typically two variables for a set of data.\n\u2022 Points can be coded via color, shape and\/or size to display additional variables.\n\u2022 Each point on the plot has an associated x and y term that determines its location on the cartesian plane.\n\u2022 Scatter plots are often used to highlight the correlation between variables (x and y).\n\u2022 position x\n\u2022 position y\n\u2022 position z\n\u2022 color\n\u2022 symbol\n\u2022 size\n\u2022 Similar to the 2-dimensional scatter plot above, the 3-dimensional scatter plot visualizes the relationship between typically 3 variables from a set of data.\n\u2022 Again point can be coded via color, shape and\/or size to display additional variables\n\u2022 nodes size\n\u2022 nodes color\n\u2022 ties thickness\n\u2022 ties color\n\u2022 Finding clusters in the network (e.g. grouping Facebook friends into different clusters).\n\u2022 Discovering bridges (information brokers or boundary spanners) between clusters in the network\n\u2022 Determining the most influential nodes in the network (e.g. A company wants to target a small group of people on Twitter for a marketing campaign).\n\u2022 Finding outlier actors who do not fit into any cluster or are in the periphery of a network.\n\u2022 color\n\u2022 Represents one categorical variable which is divided into slices to illustrate numerical proportion. In a pie chart, the arc length of each slice (and consequently its central angle and area), is proportional to the quantity it represents.\n\u2022 For example, as shown in the graph to the right, the proportion of English native speakers worldwide\n\u2022 x position\n\u2022 y position\n\u2022 symbol\/glyph\n\u2022 color\n\u2022 size\n\u2022 Represents information as a series of data points called 'markers' connected by straight line segments.\n\u2022 Similar to a scatter plot except that the measurement points are ordered (typically by their x-axis value) and joined with straight line segments.\n\u2022 Often used to visualize a trend in data over intervals of time \u2013 a time series \u2013 thus the line is often drawn chronologically.\n\u2022 width\n\u2022 color\n\u2022 time (flow)\n\u2022 A type of stacked area graph which is displaced around a central axis, resulting in a flowing shape.\n\u2022 Unlike a traditional stacked area graph in which the layers are stacked on top of an axis, in a streamgraph the layers are positioned to minimize their \"wiggle\".\n\u2022 Streamgraphs display data with only positive values, and are not able to represent both negative and positive values.\n\u2022 For example, the right visual shows the music listened to by a user over the start of the year 2012\n\u2022 size\n\u2022 color\n\u2022 Is a method for displaying hierarchical data using nested figures, usually rectangles.\n\u2022 For example disk space by location \/ file type\n\u2022 color\n\u2022 time (flow)\n\u2022 Type of bar chart that illustrates a project schedule\n\u2022 Modern Gantt charts also show the dependency relationships between activities and current schedule status.\n\u2022 For example used in project planning\n\u2022 color\n\u2022 categorical variable\n\u2022 Represents the magnitude of a phenomenon as color in two dimensions.\n\u2022 There are two categories of heat maps:\n\u2022 cluster heat map: where magnitudes are laid out into a matrix of fixed cell size whose rows and columns are categorical data. For example, the graph to the right.\n\u2022 spatial heat map: where no matrix of fixed cell size for example a heat-map. For example, a heat map showing population densities displayed on a geographical map\n\u2022 x position\n\u2022 color\n\u2022 Uses a series of colored stripes chronologically ordered to visually portray long-term temperature trends.\n\u2022 Portrays a single variable\u2014prototypically temperature over time to portray global warming\n\u2022 Deliberately minimalist\u2014with no technical indicia\u2014to communicate intuitively with non-scientists [33]\n\u2022 Can be \"stacked\" to represent plural series (example)\n\u2022 rotating angle (cycling through months)\n\u2022 color (passing years)\n\u2022 Portrays a single dependent variable\u2014prototypically temperature over time to portray global warming\n\u2022 Dependent variable is progressively plotted along a continuous \"spiral\" determined as a function of (a) constantly rotating angle (twelve months per revolution) and (b) evolving color (color changes over passing years) [34]\n\u2022 x axis\n\u2022 y axis\n\u2022 A method for graphically depicting groups of numerical data through their quartiles.\n\u2022 Box plots may also have lines extending from the boxes (whiskers) indicating variability outside the upper and lower quartiles. may be plotted as individual points.\n\u2022 The two boxes graphed on top of each other represent the middle 50% of the data,, with the line separating the two boxes identifying the median data value and the top and bottom edges of the boxes represent the 75th and 25th percentile data points respectively.\n\u2022 Box plots are non-parametric: they display variation in samples of a statistical population without making any assumptions of the underlying statistical distribution, thus are useful for getting an initial understanding of a data set. For example, comparing the distribution of ages between a group of people (e.g. male and females).\nor process\n\u2022 Represents a workflow, process or a step-by-step approach to solving a task.\n\u2022 The flowchart shows the steps as boxes of various kinds, and their order by connecting the boxes with arrows.\n\u2022 For example, outlying the actions to undertake if a lamp is not working, as shown in the diagram to the right.\n\u2022 attributes\n\u2022 value assigned to attributes\n\u2022 Displays multivariatedata in the form of a two-dimensional chart of three or more quantitative variables represented on axes starting from the same point.\n\u2022 The relative position and angle of the axes is typically uninformative, but various heuristics, such as algorithms that plot data as the maximal total area, can be applied to sort the variables (axes) into relative positions that reveal distinct correlations, trade-offs, and a multitude of other comparative measures.\n\u2022 For example, comparing attributes\/skills (e.g. communication, analytical, IT skills) learnt across different a university degrees (e.g. mathematics, economics, psychology)\n\u2022 all possible logical relations between a finite collection of different sets.\n\u2022 Shows all possible logical relations between a finite collection of different sets.\n\u2022 These diagrams depict elements as points in the plane, and sets as regions inside closed curves.\n\u2022 A Venn diagram consists of multiple overlapping closed curves, usually circles, each representing a set.\n\u2022 The points inside a curve labelled S represent elements of the set S, while points outside the boundary represent elements not in the set S. This lends itself to intuitive visualizations for example, the set of all elements that are members of both sets S and T, denoted ST and read \"the intersection of S and T\", is represented visually by the area of overlap of the regions S and T. In Venn diagrams, the curves are overlapped in every possible way, showing all possible relations between the sets.\n\nInteractive data visualization enables direct actions on a graphical plot to change elements and link between multiple plots. [35]\n\nInteractive data visualization has been a pursuit of statisticians since the late 1960s. Examples of the developments can be found on the American Statistical Association video lending library. [36]\n\nCommon interactions include:\n\n\u2022 Brushing: works by using the mouse to control a paintbrush, directly changing the color or glyph of elements of a plot. The paintbrush is sometimes a pointer and sometimes works by drawing an outline of sorts around points the outline is sometimes irregularly shaped, like a lasso. Brushing is most commonly used when multiple plots are visible and some linking mechanism exists between the plots. There are several different conceptual models for brushing and a number of common linking mechanisms. Brushing scatterplots can be a transient operation in which points in the active plot only retain their new characteristics. At the same time, they are enclosed or intersected by the brush, or it can be a persistent operation, so that points retain their new appearance after the brush has been moved away. Transient brushing is usually chosen for linked brushing, as we have just described.\n\u2022 Painting: Persistent brushing is useful when we want to group the points into clusters and then proceed to use other operations, such as the tour, to compare the groups. It is becoming common terminology to call the persistent operation painting,\n\u2022 Identification: which could also be called labeling or label brushing, is another plot manipulation that can be linked. Bringing the cursor near a point or edge in a scatterplot, or a bar in a barchart, causes a label to appear that identifies the plot element. It is widely available in many interactive graphics, and is sometimes called mouseover.\n\u2022 Scaling: maps the data onto the window, and changes in the area of the. mapping function help us learn different things from the same plot. Scaling is commonly used to zoom in on crowded regions of a scatterplot, and it can also be used to change the aspect ratio of a plot, to reveal different features of the data.\n\u2022 Linking: connects elements selected in one plot with elements in another plot. The simplest kind of linking, one-to-one, where both plots show different projections of the same data, and a point in one plot corresponds to exactly one point in the other. When using area plots, brushing any part of an area has the same effect as brushing it all and is equivalent to selecting all cases in the corresponding category. Even when some plot elements represent more than one case, the underlying linking rule still links one case in one plot to the same case in other plots. Linking can also be by categorical variable, such as by a subject id, so that all data values corresponding to that subject are highlighted, in all the visible plots.\n\nThere are different approaches on the scope of data visualization. One common focus is on information presentation, such as Friedman (2008). Friendly (2008) presumes two main parts of data visualization: statistical graphics, and thematic cartography. [37] In this line the \"Data Visualization: Modern Approaches\" (2007) article gives an overview of seven subjects of data visualization: [38]\n\n& resources\n\u2022 Displaying connections\n\u2022 Displaying data\n\u2022 Displaying news\n\u2022 Displaying websites\n\u2022 Tools and services\n\nAll these subjects are closely related to graphic design and information representation.\n\nOn the other hand, from a computer science perspective, Frits H. Post in 2002 categorized the field into sub-fields: [9] [39]\n\nWithin The Harvard Business Review, Scott Berinato developed a framework to approach data visualisation. [40] To start thinking visually, users must consider two questions 1) What you have and 2) what you\u2019re doing. The first step is identifying what data you want visualised. It is data-driven like profit over the past ten years or a conceptual idea like how a specific organisation is structured. Once this question is answered one can then focus on whether they are trying to communicate information (declarative visualisation) or trying to figure something out (exploratory visualisation). Scott Berinato combines these questions to give four types of visual communication that each have their own goals. [40]\n\nThese four types of visual communication are as follows\n\n\u2022 idea illustration (conceptual & declarative). [40]\n\u2022 Used to teach, explain and\/or simply concepts. For example, organisation charts and decision trees.\n\u2022 Used to discover, innovate and solve problems. For example, a whiteboard after a brainstorming session.\n\u2022 Used to spot trends and make sense of data. This type of visual is more common with large and complex data where the dataset is somewhat unknown and the task is open-ended.\n\u2022 The most common and simple type of visualisation used for affirming and setting context. For example, a line graph of GDP over time.\n\nData presentation architecture (DPA) is a skill-set that seeks to identify, locate, manipulate, format and present data in such a way as to optimally communicate meaning and proper knowledge.\n\nHistorically, the term data presentation architecture is attributed to Kelly Lautt: [a] \"Data Presentation Architecture (DPA) is a rarely applied skill set critical for the success and value of Business Intelligence. Data presentation architecture weds the science of numbers, data and statistics in discovering valuable information from data and making it usable, relevant and actionable with the arts of data visualization, communications, organizational psychology and change management in order to provide business intelligence solutions with the data scope, delivery timing, format and visualizations that will most effectively support and drive operational, tactical and strategic behaviour toward understood business (or organizational) goals. DPA is neither an IT nor a business skill set but exists as a separate field of expertise. Often confused with data visualization, data presentation architecture is a much broader skill set that includes determining what data on what schedule and in what exact format is to be presented, not just the best way to present data that has already been chosen. Data visualization skills are one element of DPA.\"\n\nObjectives Edit\n\nDPA has two main objectives:\n\n\u2022 To use data to provide knowledge in the most efficient manner possible (minimize noise, complexity, and unnecessary data or detail given each audience's needs and roles)\n\u2022 To use data to provide knowledge in the most effective manner possible (provide relevant, timely and complete data to each audience member in a clear and understandable manner that conveys important meaning, is actionable and can affect understanding, behavior and decisions)\n\nScope Edit\n\nWith the above objectives in mind, the actual work of data presentation architecture consists of:\n\nTraceback (most recent call last): File \"C:UsersmeAppDataRoamingBlender FoundationBlender2.81scriptsaddonsBlenderGIS-masteroperatorsio_export_shp.py\", line 162, in execute if v.lstrip(\"-+\").isdigit(): AttributeError: 'IDPropertyGroup' object has no attribute 'lstrip'\n\nBasemaps - NaN cast error\n\nThe error that goes with this\n\nOriginally posted by @MikeDabrowski in https:\/\/github.com\/domlysz\/BlenderGIS\/issues\/186#issuecomment-565720570\n\nAdding Mac OSX & Linux support in documentation\n\nI saw your great Wiki and the installation for gdal python bindings inside blender ont this page : https:\/\/github.com\/domlysz\/BlenderGIS\/wiki\/How-to-install-GDAL It's very helpfull but there is windows only installation. I don't use Windows. According to you, I would want to add other plateform to your wiki.\n\nMac Osx Tested on Yosemite 10.10 and Blender 2.74 1) Install Xcode and Macports from this link : https:\/\/www.macports.org\/install.php\n\n2) Install gdal and gdal python bindings Open a terminal from spotlight or from Applications => Utilities => Terminal Then type with administratives rights :\n\nsudo port install gdal py34-gdal\n\n3) Copy osgeo folder from python bindings to blender\n\ncp -rf \/opt\/local\/Library\/Frameworks\/Python.framework\/Versions\/3.4\/lib\/python3.4\/site-packages\/osgeo \/where_you_put_blender_on_your_mac\/Blender\/blender.app\/Contents\/Resources\/2.74\/scripts\/modules\/\n\nReplace where_you_put_blender_on_your_mac with the path where you run or install Blender\n\nTest it in Blender Python console like windows installation.\n\nI think there is a mistake in the wiki with this : Finally, to get GDAL working in Blender, just copy osgeo folder in Python tree folder of Blender (C:Program FilesBlender FoundationBlender2.70pythonlibsite-packages). If I put the osgeo folder in the same path like you recommand (python\/lib\/site-packages), I'm not able to launch gdal from blender. When I put osgeo in blender's module folder, It works !\n\nSorry for my poor english, I'm french .\n\nNo imageIO module\n\nNo imaging library available. ImageIO module was not correctly installed. Please reinstall it or try to install Python GDAL or Pillow module\n\nthis is my problem,when i start BlenderGIS reinstall it 3 times thank you\n\nGaps between DEM's when trying to achieve tiled terrain project\n\nI'm working to take assets imported using BlenderGIS and then work on them with Armory, so I can interact with the terrain and fly through it. Due to the size of some of the rasters 20000+ pixels I am hitting WebGL limitations within Armory exports.\n\nTo resolve this, I attempted to cut my Heightmap up in QGIS and load individual tiles with BlenderGIS, but I got the following gaps between DEM's which were impossible to join:\n\nI spotted this had been mentioned before in following posts: https:\/\/github.com\/domlysz\/BlenderGIS\/issues\/24 https:\/\/github.com\/domlysz\/BlenderGIS\/issues\/98\n\nSo I switched between pxLoc='CENTER' and Loc='CORNER' in operators\/io_import_georaster.py but neither made a difference.\n\nSo I took your advice in one of the posts and just imported the whole heightmap and looked for another route to tile. Having found this script I was able to slice up the mesh into 16 tiles (seperate objects):\n\nAnd started importing sat images that I had already split into tiles within QGIS, this appeared to look nice and worked well:\n\nHowever when I zoomed into the edges I had a similar gap issue:\n\nI feel like I'm getting closer but would appreciate a little help trying to reduce the gap issues.\n\nThe entire sat image in this test is 10000 x 10000 and each tile is 2500 x 2500.\n\nI'm using the following python extract the square HM from the source asc:\n\nAnd the following to generate tiled sat images:\n\nI'm running Blender 2.8 with your latest BlenderGIS build. Projection on the project is QGS 84 \/ UTM zone58S\n\n.blend file for reference: https:\/\/1drv.ms\/u\/s!AjCedBZJ5Eh4i3-eifFqf19IZefa\n\nA couple of the Sat tiles: https:\/\/1drv.ms\/u\/s!AjCedBZJ5Eh4jADXJQmz8O3lyLja\n\nPlace the Georef Cam higher\n\nSometimes I get black holes when I render the image:\n\nThis happens when there is a single peak that is higher than surrounding terrain. Is it possible to have the camera higher by default, so it's above all parts of the DEM?\n\nGet SRTM TimeoutError: [WinError 10060]\n\nI got this error loading the SRTM file, i try another locations but is the same error.\n\nTimeoutError: [WinError 10060]An error occurred during the connection attempt since the connected party did not respond properly after a period of time, or an error occurred in the established connection since the connected host could not respond.","date":"2021-10-16 21:21:08","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\": 0.33017146587371826, \"perplexity\": 2417.522116313046}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585025.23\/warc\/CC-MAIN-20211016200444-20211016230444-00102.warc.gz\"}"} | null | null |
Preamps are designed to enhance the original sound signal of the guitar before it reaches the main amplifier. In using a preamp, guitarists can create a bit of a distortion effect, before it hits the amp, providing more distortion at lower volume levels. Shop our wide selection of preamps today. | {
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Ingeborg Švédská (1277 – 15. srpna 1319) byla jako manželka dánského krále Erika Menveda dánská královna v letech 1296–1319.
Biografie
Původ, mládí
Narodila se jako nejstarší potomek z pěti dětí švédského krále Magnuse Ladulåse a jeho manželky královny Helvig Holštýnské. Písemné prameny vyzdvihují její krásu, půvab a něžnost.
Manželství a potomci
V devatenácti letech, roce 1296 se v Helsingborgu provdala za dánského krále Erika Menveda, s nímž byla z dynastických důvodů zasnoubena již v dětství; v roce 1298 se její nejstarší bratr, švédský král Birger Magnusson oženil se sestrou jejího manžela, dánskou princeznou Martkétou (Märtou). Dispens potřebný k uzavření manželství však byl udělen až v roce 1297 kvůli konfliktu mezi jejím manželem a biskupem Jensem Grandem. Jinak královna Ingeborg nesehrála žádnou významnou politickou roli. Dánský královský pár však byl v bratrovražedných bojích o švédský trůn spojencem švédského krále.
Z manželství se narodilo 8 dětí (některé prameny uvádějí 14). Většinou však šlo o potraty a ostatní děti zemřely buď záhy po porodu nebo zemřely v útlém věku.
Informace se zachovaly pouze o následujících potomcích:
Erik (zemřel jako dítě);
Magnus (zemřel jako dítě);
Valdemar (zemřel jako dítě);
Konec života, smrt
V roce 1318 přivedla Ingeborg na svět své poslední dítě – syna, který se narodil živý a životaschopný. Ukazovala ho veřejnosti jedouc na voze, dítě jí však vyklouzlo z náručí, při pádu si zlomilo vaz a zemřelo. Poté se žalem zdrcená Ingeborg uchýlila do kláštera klarisek v Roskilde; není jasné, zda to bylo její vlastní rozhodnutí nebo příkaz králův, který jí kladl za vinu synovu smrt. Některé prameny považují vedle této tragické události za konečný impuls smrt jejích bratrů Erika a Valdemara (byli v průběhu občanské války ve Švédsku z příkazu svého nejstaršího bratra krále Birgera zajati, uvězněni a ve vězení buď umořeni hladem, nebo zavražděni), jiné spekulují o zapuzení manželem z důvodu psychické choroby, jež jí měla znemožňovat plnění oficiálních povinností. V klášteře měla královna vidění a měla dokonce předpovědět datum vlastní smrti. Zemřela rok po svém vstupu do kláštera, 15. srpna 1319. Pochována v kostele sv. Benedikta v Ringstedu, kam byl tři měsíce po ní pohřben i její manžel, který zemřel 13. listopadu roku 1319.
Vývod z předků
Literatura
Henning Dehn-Nielsen: Kings and Queens of Denmark, Kopenhaga 2007,
Kay Nielsen, Ib Askholm: Danmarks kongelige familier i 1000 år, 2007,
Rikke Agnete Olsen: Kongerækken, Kopenhaga 2005,
Externí odkazy
http://runeberg.org/dbl/8/0280.html
http://www.thepeerage.com/p10551.htm#i105509+
Folkungové
Dánské královny
Švédské princezny
Úmrtí v Roskilde
Pohřbení v klášteře Ringsted
Narození ve 13. století
Úmrtí 15. srpna
Úmrtí v roce 1319
Ženy | {
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Q: How do I collect Tracing information in ASP.NET Web API? I'm trying to log every request for my WEB API.
I followed the instruction in the link below to have tracing information in my output. I wonder what will be a good way to collect those information? For example, I want to store the information into a file.
https://learn.microsoft.com/en-us/aspnet/web-api/overview/testing-and-debugging/tracing-in-aspnet-web-api
A: For my part, I use an ETW for tracing information, it is a very powerful component of Microsoft that provides multiple sink that you can use to store your event, whether in a file or a database .
Documentation : https://learn.microsoft.com/en-us/windows/desktop/etw/about-event-tracing
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,482 |
The Starboard will host its world-famous closing party weekend from Friday to Sunday, Nov. 2 to 4. But as one season draws to a close, another is just beginning, and Dewey Beach is a great hot spot in the cold months.
Steve "Monty" Montgomery, owner of The Starboard, is stoked about the shoulder season and ready to welcome everyone to the Shark Tank. He said, "Winter in Dewey is great because it's a smaller, local crowd, and the Shark Tank is open, Starboard RAW is open all winter, as well as lots of other businesses in Dewey, which are now open year-round." Montgomery said it's the perfect time to come out and a share great food with friends, while the crowds have subsided.
Winter at the beach has come a long way from just 10 years ago when whole towns along the coast shut down for the winter months. While tourism slows considerably, the local population comes out enough to warrant businesses keeping their doors open. Montgomery and many of his fellow restaurant owners in Dewey know fostering that local connection is a cornerstone to their success.
Dewey Beer Company, Woody's Bar & Grill, Gary's Dewey Beach Grill, Sunrise Restaurant, Hammerheads, Dewey Beach Country Club, Sirveza and MezCali are all staying open this winter in Dewey Beach to offer locals the same great food and service they boast all year long.
The Shark Tank will be open Thursday through Sunday all winter long. Come out for bingo every Thursday when the Starboard staff dresses up and keeps the comedy coming all night while auctioning off random prizes. Starboard RAW will be open every day all winter serving amazing craft cocktails and an outstanding menu.
For more information, go to www.thestarboard.com. | {
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} | 8,652 |
Q: Java subclass must be in same block its is subclassing? By subclassing in java, does that mean a subclass must be within the same code block as the class the subclass is subclassing? Or can it be within its own class block?
Example:
public void MyClass(){
private class MySubClass(){
}
}
Compared to:
myClass.java:
public void MyClass(){
}
mySubclass.java:
private class MySubClass(){
}
A: You are thinking of an inner class, which is defined within the scope of a parent class. A subclass simply extends a parent class:
public class Parent { }
public class Child extends Parent { }
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,181 |
We can fix flat tires in a jiffy! There's nothing more inconvenient than a flat tire. Although there isn't much you can do to prevent a flat tire, you can always help minimize the damage to your tire by knowing what to look out for. If you have any questions, talk to one of our highly experienced technicians.
Tires should always be selected by the correct size for the vehicle and purchased in sets of four. If only two tires can be installed, they should be matching sets on the same axle. For some tires, the direction in which they are installed can also be an important factor.
For long tire-life, rotating is essential. The rotation pattern can differ depending on the vehicle and the type of tires, but generally, rotating means switching the front tires and back tires with each other. This ensures that the tires will wear evenly.
Balancing your tires takes only a couple minutes, but it is very important. Unbalanced tires can lead to uneven wear, vibration, and even unsafe driving conditions. Based on mileage and the amount of wear, our technicians can easily determine whether your tired needs to be rotated and balanced.
We stock hundreds of quality, brand-name tires. We carry all sizes of tires from car, to truck, to SUV. Our used tires are thoroughly inspected for defects and can be bought in sets, pairs, and singles. Regardless of your tire situation, we have the solution!
The worst time to think about replacing your spare tire is when you need it most. A flat tire is never convenient, so don't be unprepared. Purchasing a used, quality tire as a spare is an economical investment that you should consider. | {
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Q: VBA Error 1004 Object Range failed for _Global I need to make a little VBA Application for a School project.
I recorded a Macro which Resize all Cells and then make them green. After that I select specific Cells and recolor them in white. So the result should be the Excel logo. However when I run the code there is an Error 1004 Range Object failed for _Global.
Code:
Sub Resize()
Columns("A:BZ").ColumnWidth = 2.71
Rows("1:1000").RowHeight = 15
Cells.Select
With Selection.Interior
.Pattern = xlSolid
.PatternColorIndex = xlAutomatic
.Color = 4485149
.TintAndShade = 0
.PatternTintAndShade = 0
End With
Union(Range( _
"O14:P15,L7:T7,S5:T6,P6:Q6,E38:F47,G46:J47,G42:J43,G38:J39,N38:O39,O40:R41,R38:S39,P42:Q43,O44:R45,N46:O47,R46:S47,W38:X47,Y46:AB47,Y38:AB39,AF38:AK39,AF40:AG47,AH46:AK47,AH42:AK43,AO38:AP47,AQ46:AT47,R6,Y7:AP9,AN10:AP31,Y29:AM31,AF10:AF28,AG24:AM24,Y24:AE24" _
), Range( _
"AG19:AM19,AG14:AM14,Y14:AE14,V4:X33,U5:U32,T12:T25,S14:S23,R16:R21,Q18:Q19,Q28:T32,M26:R27,N24:Q25,O22:P23,L28:P31,H28:K30,F9:G29,H8:J27,K12:K25,L14:L23,M16:M21,N18:N19,K8:T9,M10:R11,N12:Q13" _
)).Select
Cells.Select
With Selection.Interior
.Pattern = xlSolid
.PatternColorIndex = xlAutomatic
.ThemeColor = xlThemeColorDark1
.TintAndShade = 0
.PatternTintAndShade = 0
End With
End Sub
A: the string fed to Range must be less then 256 characters, while your first range has just 257... so just shift some characters to the 2nd range
furthermore you're selecting all cells instead of wanted ones
see code:
Option Explicit
Sub Resize()
With Range("A1:BZ1000")
.ColumnWidth = 2.71
.RowHeight = 15
With .Interior
.Pattern = xlSolid
.PatternColorIndex = xlAutomatic
.Color = 4485149
.TintAndShade = 0
.PatternTintAndShade = 0
End With
End With
With Union(Range( _
"O14:P15,L7:T7,S5:T6,P6:Q6,E38:F47,G46:J47,G42:J43,G38:J39,N38:O39,O40:R41,R38:S39,P42:Q43,O44:R45,N46:O47,R46:S47,W38:X47,Y46:AB47,Y38:AB39,AF38:AK39,AF40:AG47,AH46:AK47,AH42:AK43,AO38:AP47,AQ46:AT47,R6,Y7:AP9,AN10:AP31,Y29:AM31" _
), Range( _
"AF10:AF28,AG24:AM24,Y24:AE24,AG19:AM19,AG14:AM14,Y14:AE14,V4:X33,U5:U32,T12:T25,S14:S23,R16:R21,Q18:Q19,Q28:T32,M26:R27,N24:Q25,O22:P23,L28:P31,H28:K30,F9:G29,H8:J27,K12:K25,L14:L23,M16:M21,N18:N19,K8:T9,M10:R11,N12:Q13" _
)).Interior
.Pattern = xlSolid
.PatternColorIndex = xlAutomatic
.ThemeColor = xlThemeColorDark1
.TintAndShade = 0
.PatternTintAndShade = 0
End With
End Sub
| {
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\section{Introduction}
\label{sec:1}
Tornadoes are considered among the most extreme and violent weather phenomena on Earth. They can occur under appropriate circumstances in all continents expect Antarctic and can be hazardous causing loss of human lives and extensive properties damages.
Meteorologists define as a tornado a rapidly rotating mass of air that extends downward from a cumuliform cloud, i.e. a cloud formed due to vertical motion of air parcels to the ground. There exists several types of tornadoes, such as landspouts and waterspouts. The majority of the most destructive tornadoes are known as supercell since they are generated within supercell thunderstorms \cite{MR10}, \cite{MR14}.
Due to the complexity of tornadoes, the current knowledge about them comes mainly from laboratory experiments and numerical models of idealized supercell thunderstorms, as Rotunno (2013) stated in \cite{Rot13}. In 1972, Ward \cite{Ward} conducted a pioneering laboratory experiment reproducing a tornado-like flow using a simplified model for a steady flow and a fluid with constant density. Based on this work, several experimental and numerical simulations have taken place and provided important information in the field of fluid dynamics of tornadoes, \cite{Rot13}. Furthermore, various attempts have been made to analytically model a tornado-like flow. Assuming that a vortex line resembles the tornado core, these models are derived using the basic motion of equations of fluid dynamics for an axisymmetric flow, i.e. the axisymmetric Euler and Navier - Stokes equations, for incompressible homogeneous fluids. A detailed presentation can be found in \cite{Kim17} and \cite{Gillmeier-p} and in references therein.
Motivated by the aforementioned vortex models, a different, theoretical approach was introduced by Long (1958, 1961) \cite{Long58}, \cite{Long61}. Considering the existence of an infinite vortex line in a fluid interacting with a plane boundary surface, he presented the reduction of incompressible axisymmetric Navier-Stokes equations to a system of differential equations. Independently, Goldshtik (1960) showed that a similar reduction of incompressible axisymmetric Navier-Stokes equations to a system of differential equations leads to a class of exact self-similar solutions, \cite{Gold60}. Serrin (1972) broadened this class of solutions and described the existence of three different solution profiles depending on an arbitrary parameter and the kinematic viscosity, \cite{Serrin}. There are several studies of mathematical aspects of the aforementioned system of differential equations under other types of boundary conditions, \cite{GS89}, \cite{GS90}, \cite{Gold90}, and also studies of the related subject of conical flows, \cite{SH1999}, \cite{FFA00}, \cite{Shtern12}.
Here, we first develop a class of exact stationary solutions for Euler and equations. Afterwards, we consider the problem of whether such solutions can be connected with slip-type discontinuities. If this was the case, it would provide a relation with "two-cell" solutions of Serrin, \cite{Serrin}. We show that they do not exist for the given set of boundary conditions. The same holds true for conical flows. This manuscript is an extract of the work presented in \cite{KMTz} where the connection of such Euler and Navier-Stokes solutions is examined using boundary layer analysis.
\section{Cylindrical Axisymmetric Navier-Stokes Equations}
\subsection{Introduction}
We consider the system of Navier-Stokes equations for an incompressible homogeneous fluid formulated as follows:
\begin{subequations}
\label{NS}
\begin{align}
\vec{u}_t + (\vec{u} \cdot \nabla) \vec{u} &= - \nabla p + \nu \, \Delta \vec{u} , \\
\nabla\cdot \vec{u} & = 0,
\end{align}
\end{subequations}
where $\vec{u} : \mathbb{R}^3\times \mathbb{R}_+ \to \mathbb{R}^3$ is the velocity vector of the fluid, $p : \mathbb{R}^3 \times \mathbb{R}_+\to \mathbb{R}$ is pressure and $\nu \ge 0$ is the coefficient of kinematic viscosity. Motivated by the shape of a tornado, we introduce cylindrical coordinates $(r,\theta,z)$
\begin{align*}
x_1 = r \,\cos\theta, \quad x_2 = r\,\sin\theta, \quad x_3 = z,
\end{align*}
and focus on axisymmetric flows, i.e. a flow where the velocity vector $\vec{u} = (u,v,w)$ does not depend on azimuth angle $\theta$. The axisymmetric Navier-Stokes equations take the form
\begin{subequations}
\begin{align}
\label{tNS1}
\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial r} + w \frac{\partial u}{\partial z} - \frac{v^2}{r} & = \nu \Big[\frac{1}{r}\frac{\partial }{\partial r} \Big(r \frac{\partial u}{\partial r}\Big)+ \frac{\partial^2 u}{\partial z^2} - \frac{u}{r^2} \Big] - \frac{\partial p}{\partial r} \\
\label{tNS2}
\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial r} + w \frac{\partial v}{\partial z} + \frac{uv}{r} & = \nu \Big[\frac{1}{r}\frac{\partial }{\partial r} \Big(r \frac{\partial v}{\partial r}\Big)+ \frac{\partial^2 v}{\partial z^2} - \frac{v}{r^2} \Big] \\
\label{tNS3}
\frac{\partial w}{\partial t} + u \frac{\partial w}{\partial r} + w \frac{\partial w}{\partial z} \qquad & = \nu \Big[\frac{1}{r}\frac{\partial }{\partial r} \Big(r \frac{\partial w}{\partial r}\Big)+ \frac{\partial^2 w}{\partial z^2} \qquad \Big] - \frac{\partial p}{\partial z} \\
\label{tNS4}
\frac{1}{r} \frac{\partial }{\partial r} (ru) + \frac{\partial w}{\partial z} & = 0
\end{align}
\end{subequations}
\subsection{Self-Similar Formulation}
The Navier-Stokes equations remain invariant under scaling
\begin{equation*}
\vec{u}_\lambda(t,r,z) = \lambda \vec{u}(\lambda^2 t,\lambda r, \lambda z) \quad \textrm{and} \quad p_\lambda(t,r,z) = \lambda^2 p(\lambda^2 t,\lambda r, \lambda z).
\end{equation*} Looking for self-similar solutions and focusing only on stationary flows, we establish the ansatz
\begin{equation*}
\label{ansatz}
u(r,z) = \frac{1}{r} U(\xi), \quad v(r,z) = \frac{1}{r} V(\xi), \quad w(r,z) =\frac{1}{r} W(\xi) \quad \textrm{and} \quad p(r,z) =\frac{1}{r^2} P(\xi).
\vspace{7pt}
\end{equation*}
Such an ansatz induces a singularity at $r=0$ which in the applied math literature is considered as the line vortex resembling the tornado core. For convenience, we also introduce a new variable $\theta(\xi)$, namely we set $\theta(\xi) = W -\xi U$, which coincides with the self-similar form of the stream function. After a lengthy calculation, we obtain a system of ordinary differential equations
\begin{subequations}
\label{ssform}
\begin{align}
\bigg[\frac{\theta^2}{2} + (1+\xi^2)P \bigg]' &= \nu \bigg[\xi\theta - (1+\xi^2) \theta' \bigg]' -\xi V^2 \\
V' \theta &= \nu \Big[3\xi V' + (1+\xi^2) V''\Big] \\
\bigg[\theta^2 - \xi \Big(\frac{\theta^2}{2}\Big)' + P \bigg]' &= \nu \Big[\xi\theta - \xi^2 \theta' - \xi (1+\xi^2) \theta'' \Big]' \\
\theta' &= - U
\end{align}
\end{subequations}
This is viewed as a coupled system of $\theta(\xi)$, $V(\xi)$ and $P(\xi)$ where $U(\xi) = - \theta'$ and $W(\xi) = \theta - \xi \theta'$. After imposing boundary conditions, the problem can be reformulated as
\begin{subequations}
\label{Th-V-eq}
\begin{align}
\frac{\theta^2}{2} - \nu \bigg[(1+\xi^2) \theta' + \xi \theta \bigg] &= G(\xi) + \,\, {E_0} \big(\xi\sqrt{1+\xi^2} - \xi^2 \big) \\
\nu V'' + \frac{3\nu\xi - \theta}{1+\xi^2} V' &= 0
\end{align}
\end{subequations}
where
\vspace{-5pt}
\begin{align*}
G(\xi) = \xi\sqrt{1+\xi^2} \int_{\xi}^{\infty} \bigg[\frac{1}{\zeta^2(1+\zeta^2)^\frac{3}{2}} \int_{0}^{\zeta} s V^2(s) \,ds \bigg] d\zeta.
\end{align*}
Here we consider no-slip conditions on r-axis, i.e. ${\vec{u} = 0}$ at ${\xi=0}$, and no-penetration condition on z-axis, i.e. ${\vec{u} \cdot \vec{n} = 0}$ as $\mathbf{\xi \to \infty}$. A restriction on swirl $V$ is also added to close the system. Namely, we take $V \to V_\infty$, as $\xi \to \infty$. System \eqref{Th-V-eq} can now be solved numerically. After multiple numerical experiments, we observe that under certain combinations of parameters $\nu, V_\infty, E_0$ there exist three different profiles of solution. In the first case, the flow is directed outward near the plane $z = 0$ and downward near the vortex line. In the second case it is inward near the plane $z = 0$ and upward near the vortex line. For the last case, the flow is directed inward near the plane $z = 0$ and downward near the vortex line. These are in agreement with results presented in \cite{Serrin}. Under a suitable change of variables, i.e. setting $x = \frac{\xi}{\sqrt{1+ \xi^2}}$ and $\bar{\Theta} (x) = - \sqrt{1+ \xi^2} \, \theta(\xi)$, $\bar{V} (x) = V(\xi)$, system \eqref{Th-V-eq} takes a similar form as Serrin presented in \cite{Serrin} and thus, his results also hold for \eqref{Th-V-eq}.
\section{Stationary Euler Equations}
\label{inviscid}
\subsection{Continuous Solution}
Let us consider the case of inviscid Navier-Stokes system, i.e. the case where kinematic viscosity is equal to zero. Therefore, setting $\nu = 0$ into \eqref{ssform}, the system becomes
\begin{subequations}
\begin{align}
\label{theta-eq}
\bigg[\frac{\theta^2}{2} + (1+\xi^2)P \bigg]' &= -\xi V^2 \\
\label{v-eq}
V' \theta &= 0 \\
\label{p-eq}
\bigg[\theta^2 - \xi \Big(\frac{\theta^2}{2}\Big)' + P \bigg]' &= 0
\end{align}
\end{subequations}
Equation $\eqref{v-eq}$ implies that either $\theta(\xi)$ is equal to zero or $V(\xi)$ is a constant function. Supposed that $\theta \neq 0$ and thus $V(\xi)$ is continuous, we have
\begin{equation*}
V \equiv V_{0},
\end{equation*}
where $V_{0}$ is a given constant. This yields to a simple system of differential equation which can be solved analytically. In order to define the constants arising after integration, boundary conditions are imposed. Motivated by the structure of the problem, we consider no-penetration boundary conditions on both axes, i.e. $\vec{u} \cdot \vec{n} = 0$. In other words, we require that the orthogonal component of the velocity vector is equal to zero on the axes, which implies that $W = 0$ at $\xi = 0$ and $U \to 0$ as $\xi \to \infty$. Consequently, an explicit family of solutions that depends on parameters $V_0 = V(0)$ and $E_0 = P(0)$ is derived as follows
\begin{subequations}
\begin{align*}
\theta^2 (\xi) &= 2 {k_0} \ \phi(\xi) \quad \textrm{ and }
\quad V (\xi) = V_{0}
\end{align*}
\end{subequations}
where $\phi(\xi) = \xi \sqrt{1+\xi^2} - \xi^2$ and $k_0 = E_0 + \frac{V_0^2}{2}$ must be a positive constant. Expressions for $U, W$ and $P$ can easily be calculated using the definition of $\theta(\xi)$.
It is worth mentioning that if $\theta$ is positive, then the flow is directed inward near the plane $z = 0$ and upward near the vortex line. Conversely, if $\theta$ is negative, the flow has the reverse direction, i.e it is directed outward near the plane $z = 0$ and downward near the vortex line, see Fig.1. Such behaviors also occur when solving Navier - Stokes equations, \cite{Serrin}.
\begin{figure}[htbp]
\label{fig:th}
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{euler-sol-th_pos}
\end{minipage}\hfill
\begin{minipage}{0.5\textwidth}
\centering
\includegraphics[width=.9\linewidth]{euler-sol-th_neg}
\end{minipage}
\caption{Velocity vector field $(u,w)$ in $(r,z)$ plane for $V_0 = 1$ and $E_0 = 1$. Left, $\theta>0$; right, $\theta<0$}
\vspace{-15pt}
\end{figure}
\subsection{Discontinuous Solutions}
Although the flow patterns described in the previous section coincide with flows derived using the stationary Navier-Stokes equations, the interesting case where $\theta$ changes sign and thus flow changes direction is not observed. To examine whether this phenomenon is feasible, we assume that a velocity solution of $ \eqref{theta-eq} - \eqref{p-eq}$ has a discontinuity at some point $\xi = \sigma$, for $\sigma \in (0,\infty)$. Hence, we introduce an ansatz
\begin{subequations}
\begin{align*}
\theta =
\left\{
\begin{aligned}
& \theta_{-}\,\, , \quad &&\xi \in (0,\sigma) \\
&\theta_{+} \,\, , \quad &&\xi \in (\sigma,\infty)
\end{aligned}
\right.
\quad \text{and} \quad
V =
\left\{
\begin{aligned}
& V_- \,\, , \quad &&\xi \in (0,\sigma) \\
& V_+ \,\, , \quad &&\xi \in (\sigma,\infty)
\end{aligned}
\right.
\end{align*}
\end{subequations}
and seek for solutions in each domain independently. Under the restriction of continuity of $\theta$ at $\xi=\sigma$, i.e. ${\theta}_+(\sigma) = {\theta}_-(\sigma) = 0$, and no-penetration boundary conditions on the axes, i.e. $W_-(0) = 0,\, U_+(\xi) \to 0 \textrm{ as }\xi \to \infty$, the discontinuous solution becomes
\begin{equation}
\label{th-disc}
\frac{\theta^2}{2} =
\left\{
\begin{aligned}
& {k_-} \Bigg[\phi(\xi) -\phi(\sigma) - \frac{ \phi(\sigma)}{\sigma^2} (\xi^2 - \sigma^2)\Bigg] \,\, , \quad &&\xi \in (0,\sigma) \\
& {k_+} \bigg[ \phi(\xi) - \phi(\sigma) \bigg] \,\, , \quad &&\xi \in (\sigma,\infty)
\end{aligned}
\right.
\end{equation}
where $k_+,k_-$ are constants.
\subsubsection{Weak Formulation}
Let $(U,V,W,P)$ be a (generally weak) self-similar solution of Euler equations which satisfies the system of ordinary differential equations \eqref{theta-eq} - \eqref{p-eq} in the sense of distributions. Under a suitable choice of test function, the weak form of the system can be defined over a closed interval $[a,b] \subset \mathbb{R}_{+}$ and we obtain
\begin{subequations}
\label{weak2}
\begin{align}
\bigg(\frac{\theta^2(\xi)}{2} + (1+\xi^2)P(\xi) \bigg)\Bigg|_a^b &= - \int_a^b \xi V^2(\xi) d\xi \\
\bigg(\theta(\xi) V(\xi)\bigg) \Bigg|_a^b &= - \int_a^b U(\xi) V(\xi) d\xi \ \\
\bigg(\theta^2 - \xi \Big(\frac{\theta^2}{2}\Big)' + P(\xi) \bigg)\Bigg|_a^b &= 0 \\
\label{weak-theta}
\theta(\xi) \Big|_a^b &= - \int_a^b U(\xi) d\xi
\end{align}
\end{subequations}
From \eqref{weak-theta}, we infer that $\theta'(\xi) = U$ exists and is locally integrable. Recalling \eqref{th-disc}, $U$ is indeed integrable. This implies that $\theta(\xi)$ and $\theta^2(\xi)$ are absolutely continuous on $[a,b]$. Using this observation, we can easily conclude that \eqref{weak2} is a good definition of weak solution of \eqref{theta-eq} - \eqref{p-eq} in the class of functions of bounded variation. Therefore, there exists a countable set $S \subset (0,\infty)$ consisting of the points of jump discontinuity and the right and left limits of the solution exist at any $\xi \in S$. In addition, the jump conditions
\begin{subequations}
\begin{align*}
\frac{1}{2}\Big(\theta_+^2 - \theta_-^2\Big) + (1+\xi^2)\Big(P_+ - P_-\Big) &= 0 \\
\theta_+ V_+ - \theta_- V_- &= 0 \ \\
\bigg(\theta_+^2 - \xi \Big(\frac{\theta_+^2}{2}\Big)' - \Big(\theta_-^2 - \xi \Big(\frac{\theta_-^2}{2}\Big)'\Big) \bigg) + \bigg(P_+ - P_- \bigg) &= 0 \\
\theta_+ - \theta_- &= 0
\end{align*}
\end{subequations}
hold for any $\xi \in S$. Here, we denote the one-sided limits as $(\theta\pm,U\pm,V\pm,W\pm,P\pm)$. The last equation implies that $\theta$ is continuous for any $\xi \in (0, \infty)$. Hence, the jump conditions reduce to
\begin{subequations}
\label{jump}
\begin{align}
\Big[P \Big] = 0 \\
\Big[\theta^2 - \xi \theta \, \theta' \Big] = 0
\end{align}
\end{subequations}
If $\theta(\xi)$ is non-zero, the prior identities are satisfied when all functions $(\theta,U,V,W,P)$ are continuous for all $\xi \in (0, \infty)$. Therefore, it is sufficient to consider that there exists a point of jump discontinuity $\sigma \in S$ such that ${\theta}_+(\sigma) = {\theta}_-(\sigma) = 0$. This implies $P(\xi)$ is continuous for any $\xi$ while $V(\xi)$ and $\theta'$, and thus $U(\xi)$ and $W(\xi)$, have a jump discontinuity at $\xi = \sigma$.
\begin{proposition}[Nonexistence of solutions]
Although from the prospective of regularity it could be a weak solution, the class of discontinuous solutions \eqref{th-disc} does not exist.
\end{proposition}
\begin{proof}
Suppose $\theta$ is given in form \eqref{th-disc}. From jump condition \eqref{jump}, we have
\begin{equation}
\label{j2}
\big({k_-}-k_+\big) \phi'(\sigma) = 2{k_-}\frac{ \phi(\sigma)}{\sigma}\quad \Rightarrow \quad \frac{k_+}{k_-}= 1 - 2\frac{\phi(\sigma)}{\sigma \, \phi'(\sigma)}
\end{equation}
which provides an additional relation for constants $k_+,k_-$, with the right hand-side to be negative. We want to check if this relation is compatible with sign restrictions for constants $k_+,k_-$. By construction $k_+$ is always positive since $\phi(\xi)$ is a non-negative function. Therefore, it is sufficient to examine the sign of $k_-$ by finding the sign of $\theta_-^2$. For $\xi\in(0,\sigma)$, set
\begin{equation*}
J(\xi) = \phi(\xi) - \phi(\sigma) - \frac{\phi(\sigma)}{\sigma^2} (\xi^2 - \sigma^2)
\end{equation*}
We observe that $J(0)= J(\sigma) = 0$, $J'(0) = \phi'(0)>0$ and $J''<0$. This implies that $J(\xi)>0 \forall \xi\in(0,\sigma)$, and thus $k_-$ is also positive. So, we get a contradiction.
\end{proof}
\section{Conical Flows}
Motivated by the study of Euler equations presented in the previous section, we are interested in extending it for the case of axisymmetric conical flows, i.e. for flows in a cone-shaped domain. Suppose there exists $\xi_0 \in \mathbb{R}$, we seek solutions of $\eqref{theta-eq} - \eqref{p-eq}$ defined over the interval $[\xi_0, \infty)$.
\begin{wrapfigure}{r}{0.35\textwidth}
\centering
\begin{tikzpicture}[xscale=3, yscale=10]
\draw [<-] (0,0.2) -- (0,0) ;
\draw [dashed,->] (0,0) -- (1.1,0.0);
\node [left] at (0,0.2) {$z$};
\node [below] at (1.1,0) {$r$};
\draw (0,0) -- (1,0.18);
\node [below] at (1,0.15) {$\xi = \xi_0>0$};
\draw [blue] (0,0) -- (0.7,0.2);
\node [below] at (0.8,0.21) {$\xi = \sigma$};
\draw (0,0) -- (1,-0.12);
\node [below] at (1,-0.125) {$\xi = \xi_0<0$};
\end{tikzpicture}
\caption{Conical shaped domain}
\vspace{-20pt}
\end{wrapfigure}
\subsection{Continuous Solutions}
Let us begin with the case where solutions are continuous. As before, we assume $\theta \neq 0$ and $V_0 = V(\xi_0)$. If no-penetration boundary conditions are imposed on both ends of the domain $[\xi_0, \infty)$, we get the conditions:
\begin{align*}
W(\xi_0) - \xi_0 U(\xi_0) = 0& \text{ at } \xi = \xi_0, \\
U(\xi) \to 0& \text{ as } \xi \to \infty
\end{align*}
Therefore, solutions of $\eqref{theta-eq} - \eqref{p-eq}$ for a conical domain $\xi \in [\xi_0, \infty)$ become
\begin{subequations}
\begin{align*}
\frac{\theta^2}{2} &= \bigg(\frac{V_0^2}{2} + A_0\bigg) \ \frac{\phi(\xi) - \phi(\xi_0)}{1-\phi(\xi_0)} \\
V &= V_{0} \\
\end{align*}
\end{subequations}
\subsection{Discontinuous Solutions}
To investigate now the existence of discontinuous solutions, we consider a solution of $ \eqref{theta-eq} - \eqref{p-eq}$ with a discontinuity at some point $\xi = \sigma$, for $\sigma \in (\xi_0,\infty)$. Under the restriction of continuity of $\theta$ at $\xi=\sigma$, i.e. ${\theta}_+(\sigma) = {\theta}_-(\sigma) = 0$, and no-penetration boundary conditions, the discontinuous solution takes the form
\begin{equation}
\label{th-con}
\frac{\theta^2}{2} =
\left\{
\begin{aligned}
& {k_-} \Bigg[\bigg(\phi(\xi) -\phi(\sigma)\bigg) - \frac{\phi(\xi_0) - \phi(\sigma)}{\xi_0^2-\sigma^2} \,\, (\xi^2 - \sigma^2)\Bigg] \,\, , \quad &&\xi \in (\xi_0,\sigma) \\
& {k_+} \bigg[ \phi(\xi) - \phi(\sigma) \bigg] \,\, , \quad &&\xi \in (\sigma,\infty)
\end{aligned}
\right.
\end{equation}
where $k_+$, $k_-$ are constants.
\begin{proposition}[Nonexistence of solutions]
Although from the prospective of regularity it could be a weak solution, the class of discontinuous solutions \eqref{th-con} does not exist.
\end{proposition}
\begin{proof}
Suppose there exists $\theta$ expressed as \eqref{th-con}. Because of jump conditions \eqref{jump}, we request
\begin{equation}
\label{jump-con}
\frac{k_+}{k_-}= 1 - 2 \, \frac{\phi(\xi_0) - \phi(\sigma)}{\xi_0^2-\sigma^2} \,\, \sigma \quad \Rightarrow \quad \frac{\xi_0+\sigma}{2\sigma} \bigg(\frac{k_+}{k_-} \bigg)= \frac{\xi_0+\sigma}{2\sigma} - \frac{1}{\phi(\sigma)} \frac{\phi(\xi_0) - \phi(\sigma)}{\xi_0-\sigma}
\end{equation}
As before, it is sufficient to check if this relation is compatible with sign restrictions for constants $k_+,k_-$. Since $\phi$ is decreasing, it is clear that $k_+$ is positive for all $\xi \in (\sigma,\infty)$.
\begin{itemize}
\item {\underline{Case 1:} $\xi_0>0$} \\
If $\xi_0 < \sigma$, we get that the right hand-side of \eqref{jump-con} is negative. Therefore, it is satisfied if $k_-$ is also negative. To find this, we check the sign of $\theta_-^2$. Set
\begin{align*}
J_{con}(\xi) &= \phi(\xi) -\phi(\sigma) - \frac{\phi(\xi_0) - \phi(\sigma)}{\xi_0^2-\sigma^2} \,\, (\xi^2 - \sigma^2) = (\xi^2 - \sigma^2) \bigg(F(\xi) - F(\xi_0)\bigg)
\end{align*}
where $F(\xi) = \frac{\phi(\xi) - \phi(\sigma)}{\xi^2-\sigma^2}$. Using that $F(\xi)$ is a decreasing function and $\xi_0 < \sigma$, we get that $J(\xi)$ is positive. This implies that $k_-$ is positive and leads to contradiction.
\item {\underline{Case 2:} $\xi_0<0$} \\
We consider first the instance where $|\xi_0|<\sigma$. This is equivalent to case $1$ described above. So, let us move to the instance where $|\xi_0|>\sigma$. From \eqref{jump-con}, we have that the right hand-side of the above relation is negative. Since $\frac{\xi_0+\sigma}{2\sigma}<0$, \eqref{jump-con} is satisfied if $k_-$ is positive. To find this, we check again the sign of $\theta_-^2$. It is clear that $(\xi^2 - \sigma^2)>0$ for $\xi \in (-|\xi_0|,\sigma)$. Since $F(\xi)$ is a decreasing function, we conclude that $J_{con}(\xi)$ is negative and as consequence $k_-$ is also negative. This also leads to contradiction.
\end{itemize}
\end{proof}
\input{references}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,568 |
\section{Introduction and summary}
The Novikov-Thorne geometrically thin and optically thick disk model (\cite{25334}) and its non-relativistic limit (\cite{1973A&A....24..337S}) are benchmarks of black hole accretion theory (see also \cite{25341,1975ApJ...200..187E,25363,1997ApJ...479..179A,2002ApJ...567..463L} for important corrections and refinements; for reviews, see \cite{Blaes:2002wz,2007A&ARv..15....1D,lrr-2013-1,Blaes:2013toa,2014ARA&A..52..529Y,2016ASSL..440....1L}). The model is analytical, based on falsifiable assumptions whose range of validity can be tested, and it only depends on four free parameters that include the phenomenological Shakura-Sunyaev $\alpha$ ``viscosity''. In particular, it predicts a blackbody thermal spectrum in the range $10^4-10^7$K which makes it suitable as a rough model of several classes of black hole binaries and luminous active galactic nuclei (see e.g. \cite{1999PASP..111....1K,McClintock:2013vwa}).
One important boundary condition imposed in the original Novikov-Thorne model is that the torque vanishes at the ISCO, which leads to an inconsistency of the model (the fluid velocity diverges at the ISCO even though observables remain finite). This hypothesis has also been challenged by external considerations (\cite{1999ApJ...515L..73K,1999ApJ...522L..57G,2000ApJ...533L.115L,2010ApJ...711..959N}). Modelling physical boundary conditions at the ISCO is crucial in particular for highly spinning black holes in order to accurately calibrate their spin estimate (\cite{Li:2004aq}). In \cite{Penna:2011rw}, a self-consistent boundary condition hereafter called \emph{``sonic-ISCO''} was proposed, which consists in equating the radial comoving fluid velocity at the ISCO with the sound speed. Two arguments were presented in its favour. First, in slim disk models around the Schwarzschild black hole analysed in \cite{Abramowicz:2010nk}, the sonic point asymptotes to the ISCO in the thin disk regime (where the accretion rate is sub-Eddington), independently of $\alpha$. Secondly, the boundary condition implies that advection terms are negligible with respect to the stresses at the ISCO in the energy balance, in the limit where the disk height is negligible with respect to $\alpha$. We will check that the latter assumption is self-consistent for typical stellar-mass black holes and luminous AGN parameters. A crucial consistency condition of thin disk is therefore satisfied. As we will discuss, the boundary condition also implies that the specific internal energy is negligible, which automatically enforces another hypothesis of the Novikov-Thorne model. In addition, we will show that the sonic-ISCO boundary condition allows for a well-defined near-horizon near-extremal scaling behavior, in contrast to the no-torque boundary condition. Our first objective is to derive the complete thin disk model with the sonic-ISCO boundary condition from first principles. We took particular care to correct various algebraic errors and typos found in the literature.
A new feature emerges from our analysis. For highly spinning stellar-mass black holes, gas pressure at the ISCO becomes negligible with respect to radiation pressure. The sonic-ISCO boundary condition then implies that the accretion rate is not a free parameter of the model. Instead, the disk height at the ISCO is the fourth independent parameter. It allows us to infer a best-fitting value for the Shakura-Sunyaev parameter $\alpha$ in terms of the total luminosity, radiative efficiency and spin of the source. For example in the case of the X-ray binary source GRS 1915+105, recent estimates of the mass and spin, using the Very Long Baseline Array, give $M \sim 12 \pm 2$ M$_{\astrosun}$ and $J/M^2=0.98 \pm 0.01$ (\cite{Reid:2014ywa}). Also, the best-fitting model of the \emph{NuSTAR} observation of GRS 1915+105 in the plateau state gives a disk luminosity at $23\% \pm 4 \%$ of the Eddington rate (\cite{2041-8205-775-2-L45}). Taking into account the torque contribution to the radiative efficiency of thin accretion disks (\cite{2002ApJ...567..463L}), we will infer that the best-fitting Novikov-Thorne model is a disk dominated by radiation around the ISCO with $\alpha =0.43$ and $\dot M =1.6 \times 10^{18} g\ sec^{-1}$.
It is well established that radiation-dominated phases in thin disks are both thermally and viscously unstable (\cite{1974ApJ...187L...1L,1976MNRAS.175..613S}). Without a stabilization mechanism, such disk models are not expected to occur as steady flows in nature or be visible in numerical simulations (see the thermal instability in \cite{Mishra:2016yyp}). Instead, the radiation pressure instability has been argued to drive the variability of radiation-dominated objects such as GRS 1915+105 (\cite{1538-4357-542-1-L33}). The thermal and viscous instabilities are the main open issues to be understood in the Novikov-Thorne model. In this paper, we will further explore the radiation-dominated phase with the sonic-ISCO boundary condition in the extremely high spin regime, where new features arise.
According to the cosmic censorship conjecture, the spin of the Kerr black hole is bounded by its value at extremality. Extreme spinning black holes admit a near-horizon region with global conformal symmetry $SO(2,1)$ (\cite{Bardeen:1999px}) and probe fields can therefore be described using the language of conformal representation theory (\cite{Porfyriadis:2014fja,Lupsasca:2014pfa,Zhang:2014pla,Lupsasca:2014hua,Li:2014bta,Compere:2015pja}) or critical phenomena (\cite{Gralla:2016jfc}). Approaching the extremality bound by realistic accreting processes is, however, limited by the absorption cross-section of retrograde photons as shown by \cite{1974ApJ...191..507T}, leading to the bound $J/M^2 <99.8\%$. This bound is commonly accepted (see e.g. recent work by \cite{Kesden:2009ds}), even though it can be lowered by magnetic fields (\cite{Gammie:2003qi})
or instead challenged (\cite{2011A&A...532A..41S}). More fundamentally, for accretion disk models where the inner edge approaches the marginally bound orbit instead of the innermost stable orbit, the capture of retrograde photons asymptotes to zero, which allows spinning black hole to further approach extremality (\cite{1980AcA....30...35A}). Such scenarios have been shown to occur in slim disk models (\cite{2011A&A...532A..41S}). It is therefore worthwhile to explore which specific signatures may arise in the extremely high spin regime, where conformal symmetry is only slightly broken. This programme has already been carried out for gravitational wave signatures (\cite{Hadar:2014dpa,Hadar:2015xpa,Gralla:2016qfw}) and we initiate this programme for accretion disks in the simplest case of thin disk models. (for earlier related work see also \cite{2009arXiv0909.2041G}).
The paper is organized as follows. In Sec. \ref{sec:cr}, we first show how the emergent conformal symmetry constraints finite observables in the near-extremal near-horizon limit of Kerr spacetime, which completes the analysis of \cite{Gralla:2016jfc}.
Sec. \ref{sec:disk} is devoted to revisit the thin disk equations and the main assumptions of the Novikov-Thorne model. We impose the sonic-ISCO boundary condition (\cite{Penna:2011rw}), and we explain its physical implications. In Sec. \ref{sec:features}, we analyse the features of the Novikov-Thorne model and we construct piece-wisely the global solution. We focus on parameters for two notable cases of interest: stellar-mass and supermassive black holes. Moreover, we study the phase diagram of thin accretion disks in either cases in terms of the spin parameter. Remarkably, thin accretion disks reach a critical spin beyond which a phase transition occurs, close to the ISCO, from gas pressure dominated to radiation pressure dominated. This happens in the high spin regime. As an example, we consider the Novikov-Thorne model parameters to be those of the best-fitting values of GRS1915+105. Finally, in Sec. \ref{NHEK sol}, we present a first explicit example of critical (self-similar) disk in the near-horizon region of near-extremal Kerr. In Appendix \ref{appKerr}, the reader can find all the necessary properties of the Kerr spacetime and the main definitions we used throughout the paper. Appendix \ref{Solutions} contains the local solutions to the thin disk equations.
\section{Critical phenomena around near-extremal black holes}
\label{sec:cr}
It was originally discovered by \cite{Bardeen:1999px} that in the extreme spinning limit
\begin{eqnarray}
\frac{J}{M^2}=\sqrt{1-\sigma^2},\quad \sigma \ll 1,
\end{eqnarray}
a scale invariant spacetime region emerges in the near-horizon region of the Kerr black hole. More precisely, starting with the extreme ($\sigma = 0$) Kerr black hole in Boyer-Linquist (BL) coordinates $(t,r,\theta,\phi)$, one can take the near-horizon corotating limit by defining new coordinates
$(T, R, \theta, \Phi)$ as
\begin{equation}\label{NHEKs}
T = \frac{\lambda t}{2 M}, \qquad R = \frac{r - M}{\lambda M}, \qquad \Phi = \phi - \frac{t}{2M},
\end{equation}
and let $\lambda \rightarrow 0$ with $T$, $R$ and $\Phi$ fixed. The asymptotically flat region then completely decouples and the resulting near-horizon extreme Kerr (NHEK) geometry is geodesically complete and given by
\begin{equation} \label{NHEK}
ds^2 = 2M^2 \Gamma\left[-R^2dT^2 +\frac{dR^2}{R^2} + d\theta^2 + \gamma^{2}\left(d\Phi + RdT\right)^2\right],
\end{equation}
where $\Gamma(\theta) = \left(1+\cos^2(\theta)\right)/2$ and $\gamma(\theta) = \sin(\theta)/\Gamma(\theta)$.
Since it is obtained as a scaling limit of Kerr, it obeys the vacuum Einstein's equations and it admits enhanced scaling symmetry along the Killing vector $H_0 \equiv T \partial_T - R \partial_R$. It turns out that the exact symmetry is further enhanced to the conformal group times the abelian group of azimuthal rotations, $SO(2,1)\times U(1)$.
Conformal symmetry also appears for infinitesimal deviations from extremality (\cite{Amsel:2009ev,Bredberg:2009pv}), which implies that near-extremal black holes admit critical phenomena in their near-horizon region. In order to make that statement precise in the presence of matter accretion, it is fundamental to control the location of the ISCO.\footnote{Local conformal symmetry also arises (\cite{Guica:2008mu}) and plays an important role in understanding the quantum physics of black holes (see the reviews \cite{Bredberg:2011hp,2012arXiv1203.3561C}). Conformal symmetry also appears for specific probes away from extremality (\cite{Castro:2010fd}). We will, however, not use these additional structures for the present considerations.}
It has been already noticed in the original paper of \cite{Bardeen:1972fi} (see their Fig. 2) that the BL coordinate radius of the ISCO ($r_{0}$), as well as photon and marginally bound orbits, asymptotically merges with the BL coordinate radius of the horizon $r_{+}$ towards extremality,
\begin{equation}
\frac{r_{+}}{M} = 1 + \sigma, \qquad \frac{r_{0}}{M} = 1 + 2^{1/3}\sigma^{2/3}+\mathcal{O}\left(\sigma^{4/3}\right) .
\end{equation}
Therefore, in order to obtain a finite coordinate location for the ISCO in the near-horizon limit, it is necessary to scale the near-extremality parameter as $\sigma = \bar\sigma \lambda^{3/2}$, where $\bar{\sigma}$ is fixed, as described in \cite{Gralla:2015rpa}. The $\lambda \rightarrow 0$ limit still gives the NHEK spacetime in Eq. \eqref{NHEK} and the ISCO is now located at the finite radial position $R=R_{0} \equiv 2^{1/3} \bar{\sigma}^{2/3}$.
The near-horizon near-extremal limit above is invariant under the scaling $\lambda \rightarrow c \lambda$ for arbitrary $c$. Therefore, all physical quantities, which are finite in this limit, will be invariant under a simultaneous rescaling of $R \rightarrow c R$, $T \rightarrow T/c$ and $R_{0} \rightarrow c R_{0}$. In other words, finite observables in the above limit are zero eigenfunctions of the total dilatation operator $T \partial_T - R \partial_R - R_{0}\partial_{R_{0}}$. In particular, stationary observables $O$ ought to be of the form $\left(R_{0}/R\right)^{{\mathfrak h}}$, where ${\mathfrak h}$ is the critical exponent, which equals the conformal weight of the observable in the near-horizon region
\begin{eqnarray}
\mathcal L_{H_0} O = {\mathfrak h}\, O.
\end{eqnarray}
Thanks to an additional length-scale, namely $R_0$, near extremal configurations allow a richer variety of critical exponents as compared with the extremal case discussed in \cite{Gralla:2016jfc}.
In Sec. \ref{NHEK sol}, we will provide an explicit example of such a critical behavior within the Novikov-Thorne accretion disk model with critical exponents
\begin{eqnarray}\label{cw}
{}[ u^R]=[h]= [F]=2,\qquad [T]=[p]=0,\qquad [ \rho ]=-4.
\end{eqnarray}
Here, $u^R$ is the near-horizon radial velocity, $h$ the disk opening angle, $F$ the vertical energy flux, $T$ the temperature, $p$ the pressure and $\rho$ the rest-mass density.
\section{Thin disk equations} \label{sec:disk}
In this section, we revisit the accretion thin disk model around Kerr spacetime, originally presented in \cite{25334} and \cite{25341}, but instead of imposing the \emph{``no-torque''} boundary condition at the ISCO, we will impose the \emph{``sonic-ISCO''} boundary condition introduced in \cite{Penna:2011rw}. The familiar reader might skip sub-sections \ref{sec:1}-\ref{sec:3} and directly reach sub-section \ref{ISCOBC}, where new features arise.
\subsection{Fundamental equations}\label{sec:1}
We assume a single component relativistic viscous fluid, flowing along the four-velocity $u^{\mu}$ in the fixed Kerr background.
The stress-tensor can then be split uniquely with respect to $u^\mu$ as
\begin{equation}
T^{\mu\nu} = \rho (1+\Pi) u^{\mu}u^{\nu} + p h^{\mu\nu} + S^{\mu\nu} + u^{\mu}q^{\nu} + u^{\nu}q^{\mu},
\end{equation}
where $\rho$ is the rest-mass density, $\Pi$ the specific internal energy, $p$ the total isotropic pressure, $h_{\mu\nu}= u_{\mu}u_{\nu} + g_{\mu\nu}$ the projector into the local rest frame (LRF), $S^{\mu\nu}$ the symmetric, transverse and traceless anisotropic stress-tensor that takes into account viscous stresses, and $q^{\mu}$ the transverse energy flux. All these quantities are relative to the LRF and all indices are raised with $g^{\mu\nu}$.
The disk is assumed not to self-irradiate. Magnetic fields are ignored except for their contribution to the viscous stresses. Neutrinos and dark matter are ignored.
The fundamental equations governing the dynamics of the accretion disk are
\begin{equation} \label{fundamental eqs}
0 = \left(\rho u^{\mu}\right)_{; \mu}, \qquad 0 = h_{\mu\sigma}T^{\sigma\nu}_{\;\;\;\;\; ; \nu}, \qquad
0 = u_{\mu}T^{\mu\nu}_{\;\;\;\; ; \nu},
\end{equation}
respectively, the rest-mass conservation law, the relativistic Navier-Stokes equations and the energy conservation equation. These fundamental equations must be supplemented by the equation of state, the radiative energy transport law and prescriptions about the nature of the viscous effects and opacity. Conservation of the rest-mass along the fluid four-velocity is valid for energies much below $2m_b$, where $m_b$ is the rest-mass of the baryon species, or equivalently, for temperatures much below $10^{10}$ K for electrons and $10^{13}$ K for protons/neutrons. We will see that the last hypothesis is self-consistent after presenting the solution to the model.
Assuming thermal equilibrium and the existence of an equation of state of the form $\Pi=\Pi(p,v)$ where $v=1/\rho$ is the specific volume of the fluid, one can define the temperature $T$ and specific entropy $s$ by the first law $T ds = d\Pi +p dv$. The energy conservation equation might then be written in terms of local entropy production (\cite{Ellis:1971pg})
\begin{equation}
\rho T u^{\mu}s_{,\mu} = - \left( S^{\mu\nu}\sigma_{\mu\nu} + q^{\mu}a_{\mu} + q^{\mu}_{\;\; ;\mu}\right),
\end{equation}
where $\sigma_{\mu\nu} = h^{\alpha}_{\;\; \mu}h^{\beta}_{\;\; \nu}u_{(\alpha ; \beta)} - (1/3)h_{\mu\nu}u^{\alpha}_{\;\; ;\alpha}$ is the shear tensor and $a^{\mu} = u^{\nu}u^{\mu}_{\;\; ;\nu}$ is the acceleration. The four-entropy flux is defined as $S^{\mu} = s \rho u^{\mu} + q^{\mu}/T$ and the local form of the second law of thermodynamics $S^{\mu}_{\;\; ;\mu} \geq 0$ implies the two constitutive equations
\begin{equation}\label{consrel}
S^{\mu\nu} = - \eta \sigma^{\mu\nu}, \qquad q^{\mu} = -\lambda h^{\mu\nu}\left(T_{,\nu} + T a_{\nu}\right)
\end{equation}
with $\eta, \lambda \geq 0$ being the viscosity and heat conduction coefficients, respectively.
\subsection{Thin disk approximation}\label{sec:2}
The fundamental equations \eqref{fundamental eqs} require additional simplifying assumptions in order to construct analytical accretion disk models. The original thin disk model provides a set of working assumptions, carefully listed in \cite{25341}, to transform the full system of partial differential equations to an algebraic non-linear system of equations and obtain local solutions in analytical form.
In what follows, we will use cylindrical BL coordinates $(t,r,z,\phi)$ where $z=r \cos(\theta)$ (see Appendix \ref{appKerr} for details and notation). We assume stationarity and axisymmetry. The half-thickness $H$ of the disk is defined as the vertical distance between its upper surface and the equatorial plane at $z=0$. A geometrical thin disk has an opening angle $h(r) \equiv H/r \ll 1$. We suppose that all radiation is vertical, $q^\mu \sim \delta^\mu_z$. We are only interested in vertically integrated quantities between $z=-H$ and $z=+H$. For example the surface density of the disk is defined as
\begin{equation} \label{surface density}
\Sigma(r) \equiv \int_{-H}^{H} \rho(r, z) dz = 2 \rho H.
\end{equation}
In order to directly obtain the equations for vertically integrated quantities from Eqs. \eqref{fundamental eqs} as the leading order of a Taylor expansion at the equator, one can simply ignore all $z$ dependence of all physical quantities, except for the pressure $p$ and radiation flux $q^z$, which can be assumed to be (see discussion in \cite{1997ApJ...479..179A}):
\begin{eqnarray}
p(r,z) =p(r) \left(1-\frac{z^2}{H^2}\right),\qquad q^z(r,z) = F(r) \frac{z}{H} \;\;\; (z \leq H).
\end{eqnarray}
The function $F$ is then the radiation flux emitted from the upper or lower side of the disk. The orbital motion of the fluid is taken to follow nearly circular equatorial geodesics with a small radial (non-geodesic) component $u^r$ produced by viscous stresses and responsible for accretion on to the black hole.
Since the shear tensor $\sigma_{\hat \mu \hat \nu}$ only admits the non-vanishing component $\sigma_{\hat r\hat \phi} = \sigma_{\hat \phi \hat r} < 0$ in the local rest frame (see Eq. \eqref{shear tensor}), and given the first constitutive equation \eqref{consrel}, one can define the vertically integrated shear tensor as
\begin{equation} \label{int shear tensor}
W(r) \equiv \int_{-H}^{H} S_{\hat{r}\hat{\phi}}(r, z) dz = 2 S_{\hat{r}\hat{\phi}} H.
\end{equation}
We assume the $\alpha$-viscosity prescription of \cite{1973A&A....24..337S}
\begin{eqnarray}
S_{\hat{r}\hat{\phi}} = \alpha p,
\end{eqnarray}
where $\alpha$ is a free parameter and $p$ is total pressure. \footnote{Some specific MHD turbulent disks might be modelled by this prescription (\cite{1999ApJ...521..650B}). The modification of $\alpha p$ to $\alpha p_{gas}$ was argued in \cite{1974ApJ...187L...1L} and further developed in \cite{1977A&A....59..111B} and \cite{1981ApJ...247...19S}. Such models, later called $\beta p$ models, do not suffer from thermal instabilities. However, MHD simulations do not conclude on their validity (\cite{Hirose:2008hi,2009PASJ...61L...7O,Ross:2015gga}). More elaborated prescriptions have been developed by \cite{2003MNRAS.340..969O} and \cite{PhysRevLett.97.221103}.}
Two further assumptions were used in the original analysis of \cite{25334}, namely negligible specific internal energy density $\Pi = 0$ and negligible advection. We will see that these hypotheses are obeyed as a result of the sonic-ISCO boundary conditions and $ h \ll \alpha $, which will be obeyed in turn by explicit check on the solutions.
\subsection{Thin disk equations}\label{sec:3}
The dynamics in the radial direction is governed by the rest-mass, energy and angular momentum conservation laws. The radial Navier-Stokes equation is trivially satisfied with the additional assumptions that velocity and pressure gradients are negligible. Standard manipulations of Eqs. \eqref{fundamental eqs} lead to
\begin{subequations}
\begin{align}
\dot{M} &= -2 \pi r \Sigma u^{r}, \label{re1} \\
F &= - \sigma_{\hat{r}\hat{\phi}} W, \label{re2} \\
-4 \pi r \frac{(E - \Omega L)^2}{\Omega_{,r}} \frac{F}{\dot{M}} &= \int_{r_0}^{r}(E - \Omega L)L_{,r^{\prime}} dr^{\prime} + M \mathcal P_0. \label{re3}
\end{align}
\end{subequations}
Equation \eqref{re1} is the conservation of rest mass, Eq. \eqref{re2} conservation of energy and Eq. \eqref{re3} is a combination of energy and angular momentum conservation laws.
The constant of integration $\dot{M}$ is the accretion rate, while $E$, $L$, $\Omega$ and $\sigma_{\hat{r}\hat{\phi}}$ are kinematic quantities of circular equatorial geodesics (see Appendix \ref{CEGs} for their expressions).
The right-hand side of Eq. \eqref{re3} is nothing else than $M \mathcal{P}$ defined in \eqref{P}.
The dimensionless integration constant $\mathcal P_0$ is fixed by the boundary conditions, which are discussed in sub-section \ref{ISCOBC}.
The vertical Navier-Stokes equation describes the pressure balance along the cylindrical vertical coordinate $z$ and reads as
\begin{equation} \label{vertical equation1}
\frac{2p}{\rho} = h^2 \frac{\mathcal{L}_{\star}^2}{r^2}
\end{equation}
as derived in \cite{1997ApJ...479..179A}\footnote{Note the factor 2 typo in Eq. (B12) of \cite{Penna:2011rw}.}. The total pressure is the sum of the radiation pressure and the gas pressure
\begin{subequations}\label{EOS}
\begin{align}
&p = p^{(gas)} + p^{(rad)} ,\\
&p^{(gas)} = \frac{k_B \rho}{m_p} T,\\
&p^{(rad)} =\frac{1}{3} b T^4,
\end{align}
\end{subequations}
where $b = 4\sigma_{SB}/c$ is the radiation constant density, $k_B$ Boltzmann's constant, $\sigma_{SB}$ Stefan-Boltzmann's constant and $m_p$ the rest-mass of the proton.
We impose the energy transport law
\begin{equation} \label{vertical equation2}
b T^4 =\bar{\kappa} \Sigma F
\end{equation}
where $\bar\kappa$ is the optical opacity of the disk
\begin{subequations} \label{opacity prescriptions}
\begin{align}
\bar{\kappa} &= \bar{\kappa}_{ff} + \bar{\kappa}_{es},\\
\bar{\kappa}_{ff} &= \left(0.64 \times 10^{23} cm^2 g^{-1}\right)\left(\frac{\rho}{g/cm^3}\right)\left(\frac{T}{K}\right)^{-7/2},\\
\bar{\kappa}_{es} &= 0.40 \; cm^2 g^{-1} ,
\end{align}
\end{subequations}
originating from free-free (ff) absorption and electron scattering (es).
\subsection{Sonic-ISCO boundary condition}
\label{ISCOBC}
A physical system is determined by its equations of motion and its boundary conditions.
In the original analysis of \cite{25334}, it was assumed that there is no torque at the ISCO (located at $r=r_0$), which is equivalent to assume that there is no radiation at that point,
\begin{equation}
F(r_0) = 0,
\end{equation}
which is also equivalent to fixing the integration constant $\mathcal P_0 =0$ in Eq. \eqref{re3}.
Following \cite{Penna:2011rw}, we impose instead that the (purely radial) fluid velocity in the frame corotating with the geodesic flow equals (minus) the sound speed at the ISCO,
\begin{equation}
c_s (r_0) =-u^{\hat r}(r_0). \label{sound}
\end{equation}
This boundary condition has three important advantages that we will describe in what follows.
So far, we can rewrite the energy conservation equation as
\begin{equation}
Q_{diss} = Q_{cool}+Q_{adv},
\end{equation}
where $Q_{diss} = - S^{\mu\nu} \sigma_{\mu\nu}$ is the dissipation function, $Q_{cool} = q^{\mu}_{\;\; ;\mu}$ is the cooling function and $Q_{adv} = \rho Tu^{\mu}s_{,\mu}$ is the advection function that takes into account the rate of change of the specific entropy along the four-velocity. Now, as shown in \cite{Penna:2011rw}, the boundary condition \eqref{sound} implies the following scaling relations: $Q_{adv} \sim h^2$, $Q_{cool}\sim Q_{diss} \sim \alpha h$; so if $h \ll \alpha$, advection can indeed be neglected.
Thin disk models usually assume that the specific internal energy density is negligible $\Pi = 0$. This hypothesis is justified if the sound speed is nonrelativistic, $c_s \ll 1$. Due to the gravitational potential, the sound speed is highest in the near-horizon region of the disk, where we will impose the boundary condition \eqref{sound}. The hypothesis $c_s \ll 1$ will therefore be obeyed as long as $|u^{\hat r}| \ll 1$ at the ISCO, which is already part of the hypotheses since we assumed that the fluid follows nearly circular geodesics. We will check that the solutions indeed obey $|u^{\hat r}| \ll 1$ and $c_s \ll 1$.
Finally, with the no-torque boundary condition, the disk model has no regular limit at the ISCO as we will review below, while the boundary condition \eqref{sound} regularizes the model.
Let us now fix the remaining integration constant $\mathcal P_0$ for the sonic-ISCO boundary condition. It will be fixed as a function of the disk height at the ISCO, $h_0$, as follows. The integral in the right-hand side of \eqref{re3} is zero when evaluated at the ISCO. Thus, Eq. \eqref{re3} reads as
\begin{equation} \label{c1}
\left(\frac{F}{\dot{M}}\right)_0 = - \frac{M \Omega_{,r}|_0 \mathcal P_0}{4\pi r_0(E_0 - \Omega_0 L_0)^2}.
\end{equation}
On the other hand, dividing the first two equations \eqref{re1} and \eqref{re2}, we get
\begin{equation} \label{c2}
\left(\frac{F}{\dot{M}}\right)_0 = \left(\frac{\sigma_{\hat{r}\hat{\phi}} W}{2 \pi r \Sigma u^{r}}\right)_0 = - \frac{1}{\sqrt{2}} \frac{A_0}{4\pi r^{4}_0 \Delta_0^{1/2}}\alpha h_0\gamma^2_0\mathcal{L}_{\star,0}\Omega_{,r}|_0.
\end{equation}
In the second step, we have expressed the radial velocity component in the LRF, $u^{r} = (\Delta^{1/2}/ r) u^{\hat{r}}$, we have substituted the definitions of $\Sigma$ and $W$, we have used the Shakura-Sunyaev prescription and assumed the sonic-ISCO boundary condition \eqref{sound}, $u^{\hat{r}}_{0} = -\sqrt{p_0 /\rho_0}$. Equating \eqref{c1} and \eqref{c2}, we find
\begin{align} \label{C}
\mathcal P_0 &= \frac{1}{\sqrt{2}} \frac{\alpha h_0}{M} \frac{A_0}{r^3_0 \Delta^{1/2}_0} \gamma^2_0 (E_0 - \Omega_0 L_0)^2 \mathcal{L}_{\star,0} \nonumber\\
& = \frac{1}{\sqrt{2}} \alpha h_0 x_0 \mathcal{D}^{1/2}_{0} \mathcal{R}^{1/2}_{0},
\end{align}
after some algebra involving the functions defined in Appendix \ref{appKerr}. Our final formula for $M \mathcal P_0$ disagrees with the constant $C$ derived in \cite{Penna:2011rw}, which is easily seen to be incorrect since it has the wrong dimension of length.
The physical meaning of the integration constant $\mathcal{P}_{0}$ is to introduce a torque at the ISCO. More precisely, the torque might be derived by comparison of Eq. \eqref{re3} and Eq. (12) of \cite{2002ApJ...567..463L}. The torque $g_{0}$ is then
\begin{equation} \label{torque}
g_{0} = \frac{M \dot{M} \mathcal{P}_{0}}{E_{0} - \Omega_{0}L_{0}}.
\end{equation}
The total energy radiated per unit time as measured by an observer at infinity is therefore given by both accretion and torque contribution (\cite{2002ApJ...567..463L})
\begin{equation}
\mathcal{L}_{tot} = \eta_{0} \dot{M} + g_{0} \Omega_{0} \equiv \eta \dot{M},
\end{equation}
where $\eta_{0} = 1 - E_{0} $ is the specific conserved energy of a particle orbiting along the ISCO \eqref{kin quantities1} and $\eta$ is the radiative efficiency of the disk
\begin{equation} \label{radiative efficiency}
\eta = \eta_{0} + \frac{g_{0}\Omega_{0}}{\dot{M}} = \eta_0\left(\frac{a}{M}\right) + \alpha h_0 g\left(\frac{a}{M}\right),
\end{equation}
where $g\left(\frac{a}{M}\right)=2^{-1/2}M\Omega_0 (E_{0} - \Omega_{0}L_{0})^{-1}x_0 \sqrt{\mathcal D_0 \mathcal R_0} = 2^{-1/2}x_{0}^{-2}\mathcal{C}_{0}^{-1/2}\sqrt{\mathcal{D}_0 \mathcal{R}_0}$. Here, we are ignoring the capture of radiation from the hole, which decreases the efficiency for high spin (\cite{1974ApJ...191..507T}).
\section{Features of the general solution} \label{sec:features}
We introduce the dimensionless mass and mass accretion rate
\begin{equation}
M_{\star} \equiv \frac{M}{3 \mbox{M}_{\astrosun}}, \qquad \dot{M}_\star \equiv \frac{\dot{M}}{10^{17} g\; sec^{-1}}
\end{equation}
where M$_{\astrosun}$ is the mass of the Sun.
The global solution can be approximated by a piece-wise construction of three local solutions described in Appendix \ref{Solutions}, which are patched according to their range of validity. The qualitative features of the global solution depend upon the region that dominates at the ISCO. There are three possibilities depending which of the three relevant local solution is valid around the ISCO.
\subsection{Gas-pressure-dominated ISCO}
In the ``standard'' first two cases, the ISCO lies in the region dominated by gas pressure, either Region [Gas-es] or Region [Gas-ff] of Appendix \ref{Solutions}. The four free parameters of the model are $(M,a,\dot M,\alpha)$ and the disk height at the ISCO, $h_0$, is fixed. Indeed, if the ISCO lies in Region [Gas-es], we evaluate \eqref{h gases} at the ISCO using Eq. \eqref{C}. The result is
\begin{equation} \label{h0 gases}
h_0 = \left(1.8 \times 10^{-3}\right)\left( \alpha^{1/8} M_{\star}^{-3/8} \dot{M}_{\star}^{1/4}\right) x_0^{1/8}\mathcal{C}_0^{-1/8}\mathcal{R}_0^{-1/2}.
\end{equation}
If the ISCO lies in Region [Gas-ff] instead, we evaluate \eqref{hB3} at the ISCO using Eq. \eqref{C}. The result is
\begin{equation} \label{h0B3}
h_0 = \left(1.3 \times 10^{-3}\right)\left( \alpha^{1/17} M_{\star}^{-5/17} \dot{M}_{\star}^{3/17}\right) x_0^{5/17}\mathcal{C}_0^{-1/17}\mathcal{D}_0^{-1/34}\mathcal{R}_0^{-8/17}.
\end{equation}
We checked that the hypothesis $h_0 \ll \alpha$ is obeyed at the ISCO in both cases \eqref{h0 gases} and \eqref{h0B3} for the range $\alpha \sim 0.01-1$ and $a/M \sim 0 - 0.999$ and either $(M_{\star},\dot{M}_\star)\sim (1, 1)$ or $(M_{\star},\dot{M}_\star)\sim (10^{7}, 10^{5})$.
As explicit examples, the disk regions and their transitions are plotted in Figs \ref{regions} and \ref{regions2} for the spin range $0 \leq a \leq 0.999 M$ assuming $\alpha = 0.2$ for either $(M_{\star},\dot{M}_\star)=(1, 1)$ (modelling a stellar-mass black hole) and $(M_{\star},\dot{M}_\star)= (10^{7}, 10^{5})$ (modelling an AGN).
\begin{figure}
\centering
\includegraphics[width=8.5cm]{regions}
\caption{Disk regions for stellar-mass black holes with $(M_{\star},\dot{M}_\star)= (1, 1)$ and $\alpha = 0.2$ in the spin range $0 \leq a \leq 0.999 M$.}
\label{regions}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{regions2}
\caption{Disk regions for supermassive black holes with $(M_{\star},\dot{M}_\star)= (10^{7}, 10^{5})$ and $\alpha = 0.2$ in the spin range $0 \leq a \leq 0.999 M$.}
\label{regions2}
\end{figure}
In Fig. \ref{regions}, the first region in which the ISCO is located is called the \emph{edge region}. There, the gas pressure overwhelms the radiation pressure and the opacity due to electron scattering is dominant over the free-free absorption. The disk height is given by Eq. \eqref{h0 gases}. The transition to the \emph{inner region} occurs when radiation pressure starts becoming predominant over the gas pressure. A new transition occurs when the gas pressure becomes dominant again over radiation pressure and the resulting region is called the \emph{middle region}. For very low spins, the inner region is absent and the edge and middle regions merge. The \emph{outer region} is still gas-pressure-dominated, but the main mechanism responsible for the opacity is the free-free absorption. We checked that the height satisfies $h \ll \alpha = 0.2$ for $r \ll 10^{13}M$ where the model breaks down for other reasons (because self-gravitation is not negligible). The temperature of the accretion disk is hotter in the region near the ISCO, and in general the temperature is higher for faster spin. By evaluating the temperature at the ISCO for $a=0.999M$, we found $T \sim 10^7 $K. This implies that the assumption of conservation of mass is valid. We also checked that the sound speed is negligible with respect to light speed, $c_s =\sqrt{p/\rho}\ll c$, in the entire disk.
The original Novikov-Thorne model did not contain the edge region, but it should contain it by consistency: for standard spins, the radiation pressure in the inner region goes to zero at the ISCO and therefore the gas pressure needs to dominate at low enough radius. Yet, if one assumes the no-torque boundary condition, the Novikov-Thorne model is singular at the ISCO in the edge region. The reason is readily seen because in the [Gas-es] solution \eqref{sol gases} the function $\mathcal{P}$ defined in \eqref{P} vanishes at the ISCO if $\mathcal P_0=0$, and the radial velocity is then divergent at the ISCO. The non-zero torque introduced by the sonic-ISCO boundary condition allows us to regulate the model as claimed earlier.
In Fig. \ref{regions2}, the disk is always dominated by gas pressure, but the dominant contribution to opacity varies with radius. In the \emph{edge region}, where the ISCO lies, the free-free absorptions are dominant. The disk height is therefore given by Eq. \eqref{h0B3}. The \emph{middle region} is dominated by electron scattering. The \emph{outer region} is again dominated by free-free absorption. The assumption $h \ll \alpha = 0.2$ is obeyed for $r \ll 10^{21} M$ where the model breaks down. The accretion disk for supermassive black holes is colder with respect to the stellar-mass black holes; it never exceeds $T \sim 10^4$K and the conservation of mass is obeyed. We also checked that the sound speed is negligible with respect to light speed, $c_s \ll c$, in the entire disk.
\subsection{Radiation-pressure-dominated ISCO}
Let us now discuss the configurations where the ISCO lies in the region dominated by radiation pressure and electron scattering (Region [Rad-es] of Appendix \ref{Solutions}). This scenario happens for very high spins as we will discuss below. A new feature arises as a result of the sonic-ISCO boundary condition: a constraint relates the accretion rate $\dot{M}$, the mass, spin and $\alpha$ parameter, while the opening angle at the ISCO is unconstrained.
Indeed, the disk height \eqref{h rades} in geometric units is given by
\begin{equation}
h = \frac{\bar{\kappa}_{es}}{2\pi}\frac{\dot{M}}{M} x^{-3}\mathcal{C}^{-1}\mathcal{R}^{-1}\mathcal{P}.
\end{equation}
At the ISCO, $\mathcal{P}$ is given by $\mathcal P_0$ (see Eq. \eqref{P}) that can be evaluated using the expression for the sonic-ISCO boundary condition in Eq. \eqref{C}. Therefore, $h_0$ appears linearly in both sides and we are left with a constraint among the parameters of the model given by
\begin{equation} \label{constraint rades}
\frac{\alpha \bar{\kappa}_{es}}{4\pi} \frac{\dot{M}}{M} = \frac{1}{\sqrt{2}} x_0^{2} \mathcal{C}_0 \mathcal{D}_0^{-1/2} \mathcal{R}_0^{1/2} \equiv f \left(\frac{a}{M}\right)
\end{equation}
The function $f\left(\frac{a}{M}\right)$ monotonically decreases and vanishes at extremality.
It is instructive to compare the accretion rate in Eq. \eqref{constraint rades} with the Eddington accretion rate
$ \dot M_{Edd} = 4\pi M / (\bar \kappa_{es} \eta)$, where $\eta$ is the radiative efficiency defined in Eq. \eqref{radiative efficiency}. We find the reduced accretion rate
\begin{equation}\label{MMEdd}
\dot m \equiv \frac{\dot M}{\dot M_{Edd}} = \frac{\eta}{\alpha} f \left(\frac{a}{M}\right).
\end{equation}
Since the $\alpha$ parameter is usually hard to estimate, it is useful to solve the relation \eqref{MMEdd} for $\alpha$ in terms of $\dot m$ using Eq. \eqref{radiative efficiency}. The accretion rate is then determined from Eq. \eqref{constraint rades} and we obtain
\begin{equation}\label{solalpha}
\alpha = \frac{\eta_0 f }{\dot m - h_0 f g } ,\qquad \dot M = \frac{4\pi M }{\eta_0 \bar \kappa_{es}}( \dot m - h_0 fg ).
\end{equation}
The free parameters of the model where the ISCO lies in a radiation-dominated region can be finally taken to be $(M,a,\dot m,h_0)$.
In the phase diagrams of both typical stellar-mass and supermassive black holes displayed in Figs \ref{regions} and \ref{regions2}, the gas pressure dominates at the ISCO for all standard spins. However, for sufficiently high spins, radiation pressure dominates as we will now show. Let us first discuss configurations where the ISCO lies in the Region [Gas-es] for standard spins such as the case studied in Fig.~1. If one (wrongly) assumes that the ISCO lies in Region [Gas-es] for very high spins, one deduces from Eqs. \eqref{consistency gases1} and \eqref{h0 gases} that the ratio of pressures at the ISCO is
\begin{eqnarray}
\frac{p_{rad}}{p_{gas}}\Bigg|_{0} = \frac{\alpha \kappa_{es}}{2\sqrt{2}\pi} \frac{\dot M}{M} \frac{\sqrt{\mathcal D_0}}{\mathcal C_0 \sqrt{\mathcal R}_0 x_0^2} = 0.27 \frac{\alpha \dot M_{\star}}{M_\star} \sigma^{-2/3}+\mathcal{O}(\sigma),
\end{eqnarray}
where in the last step we took the near-extremal scaling $a/M=\sqrt{1-\sigma^2}$. For $\sigma \ll 1$, one obtains that radiation pressure will instead dominate. In the example of $M_\star = \dot M_\star = 1$ and $\alpha = 0.2$, the transition occurs (in the sense that $p_{rad}=p_{gas}$) at $a/M = 0.99996$ which is much above the Thorne bound of $0.998$ (\cite{1974ApJ...191..507T}) and therefore much probably unrealistic.
If instead the ISCO lies in the Region [Gas-ff] for standard spins such as the case studied in Fig. \ref{regions2}, one deduces from Eqs. \eqref{consistency gasff1} and \eqref{h0B3} that the ratio of pressures at the ISCO is
\begin{equation}
\frac{p_{rad}}{p_{gas}}\Bigg|_0 = 0.02 \frac{\alpha^{8/17}\dot{M}_\star^{7/17}}{M_\star^{6/17}} \sigma^{-14/51}+\mathcal{O}\left(\sigma^{19/51}\right),
\end{equation}
which is $\gg 1$ for $\sigma \ll 1$. Again, radiation pressure dominates for sufficiently high spins and a new \emph{near-ISCO region} opens up. However, for typical parameters $M_\star = 10^7$, $\dot M_\star = 10^5$ and $\alpha=0.2$, the transition to the near region occurs at $a/M=1-10^{-18}$ which is unreasonably high to be realistic.
However, there are more interesting parameters to consider. Let us take an accretion rate at $23 \%$ Eddington ($\dot m = 0.23$) as a model for the plateau state of GRS 1915+105 (\cite{2041-8205-775-2-L45}) with the spin estimate $J/M^2=0.98$ (\cite{Reid:2014ywa}). Assuming a radiation-dominated ISCO, we can derive the $\alpha$ parameter using Eq. \eqref{solalpha} after fixing an estimate for $h_0$. We checked that for any value $0 < h_0< 0.01$, the resulting values of $\alpha=0.43$ and $\dot{M}_{\star} = 16.5$ differ by $1\%$ or less. The continuous transition between the gas-pressure- and radiation-dominated phases occurs at $a/M=0.980 \pm 0.001$. For definiteness, we choose $h_{0}=0.002$, so that the transition exactly occurs at $a/M=0.98$.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{regions3}
\caption{Disk regions for fastly accreting highly spinning stellar-mass black hole with $M_\star = 4$, accretion rate at $23\%$ of the Eddington limit and disk height at the ISCO $h_0 = 0.002$. The radiation-dominated region extends from the ISCO up to $r \sim 150 M$.}
\label{regions3}
\end{figure}
We conclude that GRS 1915+105 could be modelled by a high spin $a/M > 0.98$ and fastly accreting thin disk, which is radiation-dominated at the ISCO. As an example, we depict in Fig.~\ref{regions3} the phase diagram for the parameters $M_\star = 4$, $\dot m =0.23$, $h_0 =0.002$ and spins $0.98 < a/M < 1$. The \emph{inner region} is radiation-dominated. At finite radius away from the ISCO, there is a transition to a \emph{middle region}, which is gas pressure-dominated, but whose opacity is still dominated by electron scattering. There is also an \emph{outer region} further away, where the main mechanism for opacity is free-free absorption.
\section{Near-horizon near-extremal solution}
\label{NHEK sol}
In the near-extremal regime, a new spacetime region, the NHEK region, opens up as explained in Section \ref{sec:cr}, which is characterized by conformal symmetry. Since the radiation-dominated solution is the only one relevant in the limit of extremely high spins, we will perform the near-extremal near-horizon limit of the solution \eqref{sol rades} in order to exhibit its conformal properties.
We perform the scaling \eqref{NHEKs} with $a/M=\sqrt{1-\bar\sigma^2 \lambda^3}$ and let $\lambda \rightarrow 0$. The ISCO is located at $R_{0}=2^{1/3} \bar{\sigma}^{2/3}$. The accretion rate $\dot M$ (in terms of asymptotic time) is constrained as in Eq. \eqref{constraint rades} which implies the scaling $\dot M \sim \lambda$. In terms of near-horizon time $T$, the accretion rate $M^{\prime}$ is finite,
\begin{equation} \label{constraint rades NHEK}
M^{\prime} = \frac{\partial t}{\partial T} \dot{M} = \frac{2M}{\lambda}\dot{M} = \pi \sqrt{6(7-\sqrt{3})}\frac{M^2}{\alpha \bar{\kappa}_{es}} R_{0}.
\end{equation}
Therefore, contrary to the asymptotically flat observer, the NHEK observer measures a finite non-zero accretion rate $M^{\prime}$. The ratio $M^{\prime}/R_0$ is fixed by the parameters of the model.
The near-horizon behaviour of the solution \eqref{sol rades} can be obtained by performing the near-extremal near-horizon scaling and trading $\dot M$ for $M^\prime$. We get the following expressions at leading order ($O(\lambda^0)$):
\begin{subequations} \label{NHEK sol rades}
\begin{align}
F &= \frac{7-\sqrt{3}}{4} \frac{h_0}{M \bar{\kappa}_{es}} \left(\frac{R_{0}}{R}\right)^2 \label{F scaling}\\
&= \left(2.0 \times 10^{26} erg/(cm^2 sec)\right)\left(h_0 M_{\star}^{-1}\right)\left(\frac{R_{0}}{R}\right)^2, \nonumber\\
\Sigma &= \frac{3}{2}\frac{1}{\alpha h_0 \bar{\kappa}_{es}} \left(\frac{R}{R_{0}}\right)^2 \\
&= \left(3.75 g/cm^2\right)\left(\alpha^{-1} h_{0}^{-1}\right) \left(\frac{R}{R_{0}}\right)^2, \nonumber\\
h &=h_{0} \left(\frac{R_{0}}{R}\right)^2, \\
u^{R} &= -\sqrt{\frac{7 - \sqrt{3}}{6}} \frac{h_0 R_0}{M} \left(\frac{R_{0}}{R}\right)^2 \\
&= \left(-6.3 \times 10^{4} /sec\right)\left(h_0 R_{0} M_{\star}^{-1}\right) \left(\frac{R_{0}}{R}\right)^2, \nonumber \\
p &= \frac{7 - \sqrt{3}}{8} \frac{1}{\alpha M \bar{\kappa}_{es}} \\
&= \left(3.4 \times 10^{15} dyn/cm^2 \right)\left(\alpha^{-1} M_{\star}^{-1}\right), \nonumber\\
\rho &= \frac{3}{4}\frac{1}{M \alpha h_0^2 \bar{\kappa}_{es}} \left(\frac{R}{R_0}\right)^4 \\
&= \left(4.21 \times 10^{-6} g/cm^3 \right)\left(\alpha^{-1} h_{0}^{-2}M_{\star}^{-1}\right) \left(\frac{R}{R_0}\right)^4, \nonumber\\
T &= \left(\frac{3(7 - \sqrt{3})}{8\alpha b M \bar{\kappa}_{es}}\right)^{1/4} = \left(3.39 \times 10^{7} K \right)\left(\alpha^{-1/4} M_{\star}^{-1/4}\right). \label{Tnear}
\end{align}
\end{subequations}
All these quantities are defined for $R \geq R_0$. Note that all quantities, except $p$ and $T$, depend on the ratio
\begin{equation}
\frac{R_{0}}{R} = \frac{\left[2\left(1-(a/M)^2\right)\right]^{1/3}}{x^2-1},
\end{equation}
which is independent of the choice of the constant $\bar{\sigma}$. The only exception is the radial component of the four-velocity, which has an additional power of $R_0$, and therefore depends on the position of the ISCO. The radial scaling exponents reproduce the table of conformal weights in Eq. \eqref{cw}, announced in Section \ref{sec:cr}. Note one unusual property of the self-similar solution: starting from $h_0$ at the ISCO the disk height $h$ \emph{decreases} with the radius; it increases again outside of the range of validity of the self-similar solution.
The self-similar solution is an approximate solution of the disk around the ISCO. It is important to discuss its range of validity. The critical temperature does not depend upon the accretion rate $\dot M$ or the height of the disk $h_0$. Since $\alpha$ or $M$ are overall factors of the temperature profile \eqref{T inner}, the relative error between the actual temperature profile and the constant critical temperature only depends upon the radius and the spin. We find that for near-extremal spins $0.96 \leq a/M \leq 1$, the actual temperature profile deviates from the critical temperature by less than $25\%$ only in the range $r_{0} \leq r \leq 2.1-2.2 M$, where the upper bound is nearly independent of the spin. This very limited \emph{near-ISCO region} is the region where the disk is approximately described by the self-similar solution. The region is biggest when the ISCO approaches $M$, which occurs closest to extremality. If a higher precision is required, the region of validity shrinks accordingly. We plot in Fig. \ref{Prec1} the range of validity of the temperature of the self-similar solution with 25$\%$, $15\%$ and $10\%$ relative precision.
Other physical quantities can be analysed similarly. Unfortunately, the relative precision of the pressure requires a spin higher than the Thorne bound $a/M=0.998$ and a narrower region around the ISCO, as plotted in Fig. \ref{Prec2}. Another important physical quantity is the radiation flux $F$. Fig. \ref{Prec3} shows that the range of validity of the self-similar solution is limited to a narrow region around the ISCO. The physical relevance of the self-similar solution \eqref{NHEK sol rades} is therefore uncertain.
In conclusion, we found a first example of critical accretion around a near-extremal black hole, which admits a scale invariance in a region close to the ISCO. More elaborated models are required to find more realistic solutions.
\begin{figure}
\centering
\includegraphics[width=8.5cm]{Prec1}
\caption{Relative precision of the self-similar critical temperature with respect to the actual temperature profile in the radiation-dominated region around the ISCO as a function of the spin. The relative precision is independent of other parameters of the model.}
\label{Prec1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{Prec2}
\caption{Relative precision of the self-similar critical pressure with respect to the actual pressure profile in the radiation-dominated region around the ISCO as a function of the spin. The relative precision is independent of other parameters of the model.}
\label{Prec2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{Prec3}
\caption{Relative precision of the self-similar critical radiation flux with respect to the actual radiation flux profile in the radiation-dominated region around the ISCO as a function of the spin for $h_{0}= 0.002$, $\dot m =0.23$. The auxiliary parameter $\alpha$ is fixed through Eq. \eqref{solalpha}. The relative precision is independent of the mass, because it factors out upon substituting the accretion rate \eqref{solalpha} into Eq. \eqref{F rades} and taking the ratio with Eq. \eqref{F scaling}.}
\label{Prec3}
\end{figure}
\vspace{-0.2cm}
\section*{Acknowledgements}
We are grateful to A. Lupsasca and A. Strominger for their comments and especially A. Lupsasca for pointing out several typos. We thank our referee, M. Abramowicz, for constructive remarks. G.C. and R.O. acknowledge the current support of the ERC Starting Grant No. 335146 ``HoloBHC''. G.C. is a Research Associate of the Fonds de la Recherche Scientifique F.R.S.-FNRS (Belgium).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,976 |
package com.jian.zero.interation.androidinterationzero;
public class AppEnvironment {
public static final boolean ENABLE_LOG = false;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 735 |
That time of the year when I decided, on a whim, to attempt the triple chilli challenge at Meat Mission, and didn't even finish my plate, let alone do it within 10 minutes. Yorkie Josh is clearly a freak of nature (he's done it in not much above 2 minutes!). I could barely walk for the rest of the night, such was the bowling ball in my stomach thanks to my lack of adequate preparation (ie, maybe don't eat 2 hours before attempting this challenge).
Its that time when burger outlets nationwide offer 20% off their wares. Simply go here, choose your restaurant, fill in the form for the voucher, print out the voucher from the email you get, go stuff your face.
Its that time when grown men starve themselves all day so as to eat a minimum of 6 of the burgers below at Hawker House.
Its that time of the year where burger restaurants out-do each other with weird and wonderful one-day-only creations to moisten the palate and excite the loins of red-blooded men (and women, lets not be sexist here) across the land.
If you see someone at Hawker House with this t-shirt on, come and say hello, as it's me! I may not be able to reply, as it's likely I will have a burger in my fat face.
It's still Monday somewhere! I've been a bit sloppy with my Monday posts, but this week I'm just outside the right timezone for this to be legit – my excuse, it was my birthday today!
Anyway, to the music. Marcus Marr is someone I've long been a fan of, thanks to his superb disco-tinged productions, especially this cracker.
The mix here is superb. I have this habit of screencapping mixes while I'm listening to them so that a couple of times a week I can go and buy the tracks that stood out for me in the music I've been listening to. Most mixes you might get 1 or 2, but this mix clocked in at a hefty 7 screencaps! So get your ear'oles round this bad boy, it really is excellent.
This week I bring something a little different. As with the Leftfield episode a while ago, it's a video of a live performance. But in this case it's even further removed from being a mixtape, instead being a recording of DJ Craze's 2 rounds from the 1998 ITF scratch battle against DJ First Rate (then of the Scratch Perverts). This video here only shows Craze's sections – now, while First Rate is an absolute badman on the cut, I don't think I'm being unfair to say that this battle was a massacre!
Craze has a good claim to being the greatest of all time, which really shouldn't be a shock when you know he won 3 straight DMC world titles (arguably only missing out the year before that winning streak because of a terribly unlucky needle mishap in the USA final that knocked him out of contention). The ITFs were often considered to be a bit more of a "purists" competition, with the battles broken down into technical sections for scratching, juggling, teams etc. In this battle he followed First Rate, with each having 2 x 3 minute sections to demonstrate their scratching prowess.
For me, what always made him stand out, alongside his unquestionable technical skills and prodigiously funky cuts, was his transitions in his routines, getting from this bit to that bit to the next bit. Many scratch sets were just a hodge-podge of as many short routines as the DJ could cram into their allotted time, with little or no thought given to the journey between these landmarks. Craze basically managed to make his 3 or 6 minute sets into tiny mixtapes with their own internal narrative and logic, seamlessly flowing between styles. It's no wonder, therefore, that Red Bull Thre3style brought him on board to be part of their team, with him performing and judging the inaugural world final in Paris (that I was lucky enough to be competing in /brag).
These 2 short sets absolutely blitzed this contest, and included 3 of the absolute best disses I've ever seen in a battle context – the "you're going too fast…" bit at the start of the 2nd routine only really makes full sense in the context of the approach First Rate had taken. Watch, and enjoy an absolute master at work.
For my money, nobody does mixtapes quite like J-Rocc does mixtapes. While this isn't exactly a mixtape, and rather a live recording, it is probably my personal favourite. Indeed, the fact that it is done live just makes the technical prowess on display all the more jaw-dropping.
Probably best known as a key member of the Beat Junkies, J-Rocc is also the DJ for Madlib's live shows and was the 3rd member of Jaylib (Madlib and J Dilla's project) when they performed live. He's also a totally great guy, I had the pleasure of hosting an event with him and Shortkut performing, at Rescue Rooms in Nottingham way back in 2003!
Enjoy this superb selection of hip hop, funk and reggae, and please go and explore his other mixes. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,321 |
Recently, a close friend of mine asked me to do an appraisal on a round brilliant cut diamond that she inherited. The diamond had been her grandmother's and was passed down to her mom and then to her. According to her family, the diamond was of very high quality (F-G color with VVS range clarity), although not certified by any gemological laboratory. She was under the impression it was a very valuable diamond.
she purchased from a local retail store. After wearing the necklace for some time, she decided that she would like to have insurance on it and so I was asked to examine the necklace for an insurance replacement value appraisal. When I looked at the diamond under magnification I was shocked to find that the stone was clearly chipped…. and of course, I had to be the one to break the disappointing news.
To my friend's dismay, I was immediately concerned that the diamond might have been damaged somehow during the time of her wearing it. I asked if she was aware that the diamond had a chip…she replied that she did not. I asked if the previous jeweler who set the diamond had disclosed with her that there had been any damage to the stone when setting it, she also replied no. As a gemologist, I knew any reputable jeweler would have performed a formal "check in" screening of the diamond before ever tampering with it. This is standard procedure at deBebians. The fact that the jeweler had not disclosed that the diamond was chipped either before or after setting it in the pendant was extremely unsettling for me. Although the chip was not huge, it was enough to knock the diamond out of a VVS-VS clarity range for it was visible to the naked eye. What could have been evaluated in an insurance replacement value appraisal for around $18,000 (based upon the initial information provided) was really only valued at around $8,000.
As the story turns out, my friend went home and did a little more research on the stone and discovered that her family was well aware that the diamond was chipped. They never knew a chip qualified as a clarity characteristic so they never thought to say anything. I am still shocked that a jewelry store would set a diamond without disclosing that the diamond was chipped. Fortunately the diamond was otherwise okay, and in the end there was no further harm done.
It is my recommendation to have a jewelry appraisal done on valuable jewelry as soon as possible once the item has been purchased and received. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,355 |
Fire Emblem Engage's difficulty cracked my brain open like an egg
January 5, 2023 by Riya Chakraborty
My army marches through an over-the-top anime castle. In the shadow of its sandstone walls, the forces of good make their stand against the machinations of the evil Fel Dragon. Our mission is simple: protect the gate. I've got Permadeath enabled, so every move counts in this turn-based tactical RPG. I agonize over every decision, trying to predict my enemies' moves. My Pegasus Rider stays away from enemy archers lest he be shot. Wander, my tough-as-nails Great Knight, holds the vanguard while my archers and mages take refuge behind a line of unbreakable steel.
Despite my solid formation, the enemy is clever and maneuvers to edge me. I fully delve into Fire Emblem: Engage's elegant combat system and customize it as best I can. My slow, steady and strategic approach works as we take down the enemy leader, finally completing the mission. However, at this point, I realized that something was not quite right. I watch from my Nintendo Switch. It's 1 o'clock.
No Fire Emblem has been as effective at triggering my voracious "one more twist" appetite since Fire Emblem 7, the first title in the series to hit west shores back in 2003. In those days, grand sprites fought on 2D grounds, and permadeath was mandatory. In almost every aspect, Engage is a powerful and conscious attempt to evoke the nail-biting tactical demands of "classic" Fire Emblem.
This contrasts with the gentler approach of its predecessor: Fire Emblem Three Houses, which combined the series' turn-based combat with a relaxed anime-academy simulation experience. Fire Emblem Engage returns to the grit and ruthlessness at the heart of the tactical series.
Help features old and new to reinforce Fire Emblem's challenging turn-based battles. The weapon triangle returns, taking on the game's weapons in a rock, paper, scissors-style relationship — swords meet against axes, axes against axes, and spears against swords. But developer Intelligent Systems has added welcome twists to the mechanic.
The new "break" system means that if you hit an opponent with a weapon they are weak against, they will drop their weapon, disabling their counterattack. This can be a great way to take out difficult boss enemies, while removing the threat of their (often fatal) counterattack. However, you will need to be careful as your units are just as vulnerable to "breaking through" as your opponents. Putting your swordsman in range of a lancer's weapon is just asking for trouble.
With the "engage" command, your fighter can merge with the ghost, unlocking a bunch more skills and single-use ultimate moves
The big new system is the one that gives Fire Emblem Engage its name. Throughout your campaign, your warriors will find ghost rings from the past of Fire Emblem heroes. Wearing just one insignia ring gives the character access to additional skills and abilities reminiscent of the hero in question. However, with the "engage" command, your fighter can merge with the ghost, unlocking a bunch more skills and single-use ultimate moves. Although you usually only have three-turns of the clock to use these powers, they can radically change the course of battle.
It also allows for some wild combinations. Give your slow, stoic knight Sigurd a ring, and he'll be able to zip across the battlefield at an uncanny pace. Characters can also gain abilities from their symbol rings, making these changes permanent. The scope for versatility and inventiveness is staggering; Emblem rings are far from an empty gimmick.
I'm playing the game on Hard difficulty with Permadeath enabled, and I need every advantage I can get to overpower my enemies. While there's no right or wrong way to play Fire Emblem Engage, I've always seen the threat of character death as a hallmark of the series. The threat of losing one of your dear anime friends adds considerable weight to the fight – and it's something that goes all the way back Turn-based RPG like Tactics Ogre,
You can rewind time if you mess things up, but get-out-of-jail cards have limited charges, so you can only bail out so many times. Even with the rewinding feel softened, I've found Classic mode to be the best kind of uncompromising puzzle box. The punishment for a bad tactical move is lethal, so I go through every troop order thinking through how I'm opening myself up to the enemy, but I make mistakes nonetheless. In some of the more challenging battles, 10 rewinds feels like a drop in the ocean.
When it comes to strategy and the classic "just one more turn" itch, Engage works in spades
The puzzle goodness cuts through each of Engage's modes and difficulty levels, so you shouldn't have to ratchet things up to see what Fire Emblem Engage has to offer. No matter your preferred difficulty, it offers a deeply satisfying and well-realized combat experience. As a somewhat long-time Fire Emblem fan myself, I'm exactly the kind of person Engage is trying to appeal to. When it comes to strategy and scratching the classic "just one more turn" itch, Engage works in spades, delivering one of the most fleshed-out and satisfying tactical combat experiences in Fire Emblem history.
However, as a fan of the three Houses' nuanced storytelling and organic friendship-building, I found Engage's out-of-combat experience somewhat lackluster. The complex and fascinating web of relationships that defined The Three Houses has been discarded in favor of more streamlined, but less deep character interactions. Despite this, I thoroughly enjoyed my time with Engage. The nuanced challenge of its well-crafted battles is enough to keep me coming back for more.
Prince Harry accuses William of physical assault in new book
Asphalt Hamlin update: As Bills player remains in critical condition, Colts' Rodney Thomas II details his bedside hospital visit | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,711 |
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