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\section{Introduction}
Self-organized criticality (SOC) is a characteristic of systems
that are driven by a slowly acting external force and organize
themselves through energy dissipating avalanches of all sizes.
Although SOC models were first proposed to explain the origin of
the $1/f$ noise, it is recognized that it can be used to explain a
large class of systems such as earthquakes~\cite{sch91},
evolutionary bursts~\cite{baksne93}, forest fires~\cite{drosch92},
rice piles~\cite{fre96}, financial markets~\cite{bar06}, and so
on.
Recently we have raised the issue that, for some critical
organized systems, the size of the largest avalanches can be
reduced by a control intervention heuristics~\cite{cajand10}. That
was just a first effort to investigate if and how the damaging
energy dissipative bursts in SOC systems could be controlled.
Although such systems organize themselves without external
intervention, this reorganization in a lower level of energy is
very costly for society, since it depends on avalanches of all
sizes. Examples of these events of dissipation of energy in nature
and society are avalanches that arise in snow hills, bubbles that
explode in financial markets and earthquakes. Although it is not
possible to intervene in events such as earthquakes, in some sense
we can intervene in the process that generates large snow
avalanches and the explosion of stock bubbles. In the former case,
small avalanches can be triggered in order to avoid larger
ones~\cite{mccsch93}. In the later case, central banks can in some
sense enforce a monetary policy that can avoid the rising of large
bubbles~\cite{greenspan08}. Regarding this aspect, it is not the
purpose of this work to defend this kind of procedure, but we
surely think that it deserves to be studied. Our first
investigation ~\cite{cajand10}, was based on a replica model of
the region of the system to be controlled, a control scheme was
designed to externally trigger small size avalanches in order to
avoid larger ones. Although we have shown that this principle
works for sandpiles in two-dimensional lattices~\cite{cajand10},
we have no information about how far from the optimal choice these
heuristics are. To fill this gap we resume our investigation with
a rather different approach: we develop a dynamical programming
(DP) approach to control the directed abelian Dhar-Ramaswamy
(DADR) model in a two dimensional lattice. Due to the huge number
of possible states that come to play in DP approaches, any
feasible investigation must be restricted to systems of much
smaller size than those one usually considers when performing
numerical integration of the systems. Nevertheless, we can use
this approach to characterize optimally the problem of controlling
SOC systems, as well as to explore the efficiency of other
heuristics built without any kind of optimization law and as a
basis for approximate optimization principles such as the ones
presented in~\cite{bertsi96}.
It is worth mentioning that recent literature has used
optimization principles to understand the structure and dynamics
of several complex systems.
In~\cite{rodrin92,caj05,jacrog05,mottor07,carior08}, it was shown
that complex networks may arise from optimization principles.
Further, analyzes of optimal navigation in complex
networks~\cite{caj09,caj10} have shown that a walker that
minimizes the cost of walking overlaps the random walker and the
directed walker behaviors. In~\cite{cajmal08,ast08} optimization
has been used to study the complex human dynamics of task
execution. Moreover, reinforcement learning has been used to
explore the problem of learning paths in complex
networks~\cite{cajand09}.
\section{The DP approach for controlling the DADR model}
\label{sec:dp} The DADR model~\cite{dha89} is built on a
two-dimensional square lattice of $L\times L$ sites $(i,j)$,
$i,j=1,\cdots,L$. Each site stores a certain amount $z_{ij}$ of
mass units. At each time step, the system is driven by two update
rules: (a) Addition rule: a mass unit is added to a randomly
selected site $(k,\ell)$, so that $z_{k\ell}\rightarrow
z_{k\ell}+1$. (b) Toppling rule: if $z_{ij}>z_{c}=1$, then
$z_{ij}\rightarrow z_{ij}-2$, $z_{i+1,j-1}\rightarrow
z_{i+1,j-1}+1$ and $z_{i+1,j+1}\rightarrow z_{i+1,j+1}+1$. The
model is usually represented after performing a $5\pi/4$ rotation
of the standard square lattice, in such a way the site $(i+1,j+1)$
lies just below the site $(i,j)$, and the $\mathbf{x}$ and
$\mathbf{y}$ directions are at $5\pi/4$ and $7\pi/4$ angles with
the horizontal axis. Therefore, if deposition occurs in site
$(i,j)$, the only sites that may be affected are those located on
the lines $i+\ell, \ell\geq 1$.
Let $\Gamma$ be the finite set of stable states (configurations)
in the phase space of the DADR model, and $N_\Gamma$ the number of
elements of $\Gamma$. As in any DP study, it is necessary to
identify the different actions (or policies) that can be taken
when the system is in any of these states. So let us note such one
policy as $\pi=[u(1),\cdots,u(N_\Gamma)]$, where $u(i)\in U$
refers to the specific control action that $\pi$ undertakes when
the system is in the state $i$. $U$ represents the set of
admissible controls, i.e. those control actions that do not
violate the system dynamics. Note that the number of elements in
the set $\Pi$ of all admissible policies $\pi$ increases faster
than combinatorial when the system size increases. Indeed, this
number depends both on $N_\Gamma$ and on the number of possible
control actions for each state $i$.
To control the avalanches sizes in our approach, it is important
to consider that events occur according to an ordered sequence, as
discussed in ~\cite{cajand10}. If $t$ is a discrete variable
$t=n\Delta t$, and the DADR model is in a given ``stable'' state
$x_t$ we assume that the following events take place within the
time step $\Delta t$. The control scheme triggers (or not) one
avalanche in a specific site $(i,j)$ of the lattice. If this
occurs, the avalanche starts by emptying the site $(i,j)$, what
amounts to topple the single unit mass with $50\%$ of probability
to the site $(i+1,j-1)$ or to the site $(i+1,j+1)$. This control
induced toppling may lead to further toppling until the system
reaches a new stable state $x_{t}^{c}$ due to the control
intervention $u(x_t)$. After this induced avalanche, which is
absent if the used policy indicates to take no action, the usual
deposition process of the model takes place and the system evolves
from either $x_t$ or $x_t^c$ to a new state $x_{t+1}$. For this
process, we only care that control intervention comes before the
deposition process, and that both relaxations occur within the
same time step $\Delta t$.
However, differently from~\cite{cajand10}, we assume here that
there is only possible at most one control intervention per time
step, and the intervention decision is made according to a dynamic
programming approach.
For instance, if $L=2$, then $N_\Gamma=2^{L^2}=2^{2^2}=16$. One
possible state of $\Gamma$ is
\begin{equation}
x=\left
[\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right].\label{aa}
\end{equation}
\noindent Please note that, for the purpose of keeping a simple
diagram, we did not perform the rotation used for the
representation of the system described before. In such matrix-like
notation, the toppling process makes the grain move either one
column to the right or one line downwards. Assume $x$ to be
$x_t$. In this state, we have three admissible controls, namely
the one that triggers no avalanche, and those that trigger an
avalanche in the sites $(1,2)$ and $(2,1)$, respectively. If there
is no intervention, the intermediate state $x_c(t)=x(t)$. If an
avalanche is triggered in the site $(1,2)$, the system goes to
$x_{t}^{c}$ which, with equal probabilities, is one of the states
\[\left[\begin{array}{cc}
0 & 0 \\
1 & 1 \\
\end{array}\right], \left[\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}\right].\]
\noindent (If the site $(2,1)$ had been chosen, the situation
would be quite similar but, for the sake of brevity, we will not
consider this choice here.) In order to differ one state from the
other, we call the one in the left as $x_{t}^{c,L}$ and the one in
the right as $x_{t}^{c,R}$, making reference to the side followed
by the controlled avalanche. Thus, after the deposition process,
the system can have suffered a transition to one of the following
states, in the case of $x_{t}^{c,L}$,
\[\left[\begin{array}{cc}
1 & 0 \\
1 & 1 \\
\end{array}\right], \left[\begin{array}{cc}
0 & 1 \\
1 & 1 \\
\end{array}\right], \left[\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}\right], \left[\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}\right].\]
\noindent or to one of the possible states, if $x_{t}^{c,R}$ was
taken:
\[\left[\begin{array}{cc}
1 & 0 \\
1 & 0 \\
\end{array}\right], \left[\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}\right], \left[\begin{array}{cc}
0 & 0 \\
0 & 1 \\
\end{array}\right], \left[\begin{array}{cc}
0 & 0 \\
1 & 1 \\
\end{array}\right].\]
\noindent Therefore, with equal probabilities, the state $x_{t+1}$
will be represented by one of these eight states. One also must
note that, associated with the control intervention and the
deposition process, we have respectively two classes of
avalanches: {\it controlled avalanches}, that are triggered by
the control scheme in state $x_t$ with sizes $s_{c,x_t\,
x_{t}^{c,L}}$ or $s_{c,x_t \, x_{t}^{c,R}}$ depending on the side
that the controlled avalanche followed; and {\it uncontrolled
avalanches} with sizes $s_{u,x_{t}^{c,L} \, x_{t+1}}$ or
$s_{u,x_{t}^{c,R} \, x_{t+1}}$ that happen when the system goes
from state $x_{t}^{c,L}$ or $x_{t}^{c,R}$ to state $x_{t+1}$ due
to the deposition process.
Following the DP approach, we assume that the control scheme makes
the decision of triggering an avalanche in one site of the system
or doing nothing in a given state $x$ based on the minimization of
the cost function
\begin{equation}J(x)= \min_{\pi\in\Pi}E^{\pi}\left[\sum_{t=0}^{\infty} \gamma^t g(x_t,u(x_t))|x\right]
\label{eq:prob}
\end{equation}
\noindent where the expectation $E^\pi[\cdot|x]$ is conditional on
the policy $\pi$ and the state $x$. The cost per stage
$g(\cdot,\cdot)$ is given by
\begin{equation}
g(x_t,u(x_t))=\sum_{x_{t+1}}p_{x_t x_{t+1}}(u)
\overline{g}(x_t,u(x_t),x_{t+1}), \label{eq:g}
\end{equation}
\noindent where $u(x_t)$ is the control intervention in state
$x_t$, $\overline{g}(x_t,u(x_t),x_{t+1})$ is the cost of using the
control $u$ at state $x_t$ and moving to state $x_{t+1}$, $p_{x_t
x_{t+1}}(u)$ is the transition probability from state $x_t$ to
state $x_{t+1}$ using the control $u$ at state $x_t$. In the
general expression (\ref{eq:g}), the function form of
$\overline{g}$ has not yet made explicit. In this work, we assume
a simple particular form for $\overline{g}$ that depends only on
two parameters and a functional dependence on the avalanche size
$s$. Therefore, we write
\begin{eqnarray}
\lefteqn{g(x_t,u(x_t))= C_I I(x_t)}\\&& +\frac{1}{2}[ C_A
h(s_{c,x_t\, x_{t}^{c,L}}) + \sum_{x_{t+1}}p_{x_{t}^{c,L} x_{t+1}}
C_A h(s_{u,x_{t}^{c,L}\, x_{t+1}})]\nonumber \\&& +\frac{1}{2}[C_A
h(s_{c,x_t\, x_{t}^{c,R}}) + \sum_{x_{t+1}}p_{x_{t}^{c,R} x_{t+1}}
C_A h(s_{u,x_{t}^{c,R}\, x_{t+1}})], \nonumber \label{eq:gb}
\end{eqnarray}
\noindent where $p_{x_{t}^{c,L} x_{t+1}}$ (or $p_{x_{t}^{c,R}
x_{t+1}}$) is the probability of transition from $x_{t}^{c,L}$ (or
$x_{t}^{c,R}$) to $x_{t+1}$. One must also note that, while
$p_{x_t x_{t+1}}(u)$ indicates explicitly that the probability of
the transition from state $x_t$ to state $x_{t+1}$ depends on the
choice of the control $u$ at state $x_t$, $p_{x_{t}^{c,L}
x_{t+1}}$ (or $p_{x_{t}^{c,R} x_{t+1}}$) does not present this
dependence. $I(x_t)$ is an indicator function that assumes the
value 1, when there is an intervention in state $x_t$, and 0
otherwise. Finally, the two parameters $C_I$ and $C_A$ represent
the fixed cost associated with one intervention and avalanche
size. For the sake of simplicity, we assume here that $h(s)=s^2$,
i.e., we penalize larger avalanches in a power law with exponent
equal to 2.
The term $\gamma^t$ in (\ref{eq:prob}) weights differently the
influence of the present and future costs in the decision process.
Although a realistic optimization process must take into account
the intervention cost, it could be expected that avalanches at a
given time step and those in the next future should have
approximately the same weight in the decision process, what
amounts to take the discount factor $\gamma=1$. We call the
attention that this simple choice leads to a technical difficult,
namely, we can not ensure that this problem has a solution that
does not depend on the kind of the controlled Markov process. In
such situations, the method used to solve the problem may depend
on the type of the controlled Markov chain that we are dealing
with and may be difficult to find by simple
algorithms~\cite{put05}. However, due to the Banach Fixed
Theorem~\cite{put05}, this difficult can be circumvented if we
consider $\gamma\rightarrow 1_-$. Finally we should also note that
the choice $\gamma = 1$ is somewhat unrealistic as it does not
consider the cost of the money over time.
It is easy to show that the solution of problem (\ref{eq:prob}) is
given by the Bellman equation~\cite{bel57}
\begin{equation} J(x)=\min_{u\in U(x)}\left[g(x,u)+\sum_{x'}p_{x x'}(u) \gamma J(x')\right].\label{bellman}\end{equation}
In the rest of this paper, we discuss the solution of problem
(\ref{eq:prob}) using numerical solutions of the Bellman equation
(\ref{bellman}) found by means of the value iteration
algorithm~\cite{put05}.
\section{Results} \label{sec:results}
\begin{figure}[t]
\begin{tabular}{cc}
\includegraphics[width=4cm,height=4cm]{figure1a.eps} & \includegraphics[width=4cm,height=4cm]{figure1b.eps} \\
\end{tabular}
\caption{(a) The avalanche average size for several values of the
ratio $C_I/C_A$: controlled avalanches (hollow circles) and
uncontrolled avalanches (solid circles). (b) The average size of
the avalanches for several values of the mass of the state. While
solid symbols represent uncontrolled avalanches, hollow symbols
represent controlled avalanches: $C_I/C_A=0$ (circles) and
$C_I/C_A=10$ (squares).} \label{figura1}
\end{figure}
The main difficulty associated with DP approach is the rapid
increase of $N_\Gamma$ which, for the current study, behaves like
$2^{L^2}$. For practical purposes, it becomes impossible to study
numerically a system larger than $L=4$. As already discussed, the
size of such system in much smaller than lattice sizes actually
used to compute the time evolution of the model. However, we will
show that this approach can be used to characterize the optimal
solution of the problem and be used as a benchmark to validate
other solutions based on ad-hoc chosen heuristics.
For all simulations of DADR's model reported in this paper, the
corresponding solution of Eq. (\ref{bellman}) was obtained for
$L=4$ and $\gamma=0.999$. For this value of $L$, $N_\Gamma=
65536$, the largest avalanche that can take place in such system
is of size $s=16$, the minimal and maximal amount of mass $M$ kept
in the system are, respectively, $M=0$ and $M=16$. For the next
lattice size $L=5$, solving (\ref{bellman}) requires to find the
minimum of $J(x)$ by taking into account all $2^{25}\sim 3.2
\times 10^7$ states for this lattice size.
At a given time $t$, the number of admissible controls depends on
the state $x$. For instance, while in the unique state of the
system with mass 0, there is only one control, which is to do
nothing, in the unique state of the system with mass 16, there are
17 admissible controls.
Figure \ref{figura1}(a) shows the effect of the the cost $C_A$ and
$C_I$ in the solution of the problem presenting the average
controlled and uncontrolled avalanches $\langle s\rangle$ for
different values of $C_I/C_A$. It is shown that, when the cost of
making interventions becomes high, the control scheme waits until
the system has stored a larger amount of mass $M$ to make
interventions. This causes also the size of the uncontrolled
avalanches to increase. For a given threshold value
$(C_I/C_A)_T\sim 40$, intervention cost becomes so large that the
optimal solution corresponds to not intervene in the system
anymore. Correspondingly, when $C_I/C_A$ is close to
$(C_I/C_A)_T$, the number of interventions decreases exponentially
(not shown). Figure 1a also shows that, for $C_I/C_A >
(C_I/C_A)_T$, only avalanches produced by the system dynamics are
observed.
It turns out that $M$ is an interesting metric that can be used to
order the states of the system in terms of danger of larger
avalanches. Figure \ref{figura1}(b) shows the average size of
avalanches for several values of $M$ for the ratios $C_I/C_A=0$
and $C_I/C_A=10$. This figure shows the effect of the increasing
the ratio $C_I/C_A$ for a state of the system characterized by its
mass. For the no cost intervention situation $C_I/C_A=0$, the
control scheme acts for all states of the system but the one with
$M=0$. The same does not happen for $C_I/C_A=10$, when avalanches
are triggered only for states with $M>4$. The consequence of such
behavior is to increase the size of the uncontrolled avalanches
for the states with low values of $M$. Only for large values of
$M$ the average size of the controlled avalanches becomes larger
than that of the uncontrolled ones. Moreover, one may also see
that increasing $C_I/C_A$ has almost no effect on the avalanche
average size when $M$ grows. Finally, Figure \ref{figura1}(b)
suggests that the $C_I/C_A$ plays a role similar to the acceptable
size of an avalanche considered in~\cite{cajand10}, i.e., when the
ratio $C_I/C_A$ is high, it is not worth triggering small
controlled avalanches anymore.
Now, we compare the results provided by DP control with those from
three other heuristic approaches that we identify as maximal ({\it
max}), minimal ({\it min}) and random ({\it ran}), although none
of them is exactly equivalent to the fixed avalanche size
heuristic discussed in \cite{cajand10}. As in the DP case, all of
them make at most one intervention per time step. We call $p_I$
the fraction of time steps where an intervention occurs. Let $T_I$
be a minimal threshold avalanche size that allows the {\it max}
and {\it min} control schemes to intervene, i.e., they do not
trigger avalanches with size less than $T_I$. This parameter plays
a role similar to the acceptable size of an avalanche
in~\cite{cajand10}. The {\it max} approach works as follows. In
each time step $t$ and corresponding state $x_t$, it triggers only
the maximal avalanche with size $s_{\mathrm{max}}$ that may happen
in this state if $s_{\mathrm{max}}\ge T_I$. On the other hand, the
{\it min} approach triggers the minimal avalanche with size
$s_{\mathrm{min}}$ that may happen in the state $x_t$ if
$s_{\mathrm{min}}\ge T_I$. Finally, the {\it ran} approach
triggers avalanches in saturated sites of the system with the same
frequency of intervention $p_I$.
In order to compare the results of the four approaches, we use the
number of interventions as a tune parameter. Therefore, we choose
$T_I$ large enough in order to have the number of interventions of
the {\it max} and {\it min} schemes of the same order of the DP
control. Figure \ref{figura3} compares the DP results low (a) and
high (b) ratios $C_I/C_A$ with the equivalent {\it max}, {\it min}
and {\it ran} controls. There we measure the efficiency of the
control scheme by the ratio $f$ between the number of avalanches
of the controlled to the uncontrolled system for $T_I=1$
(\ref{figura3}a) and $T_I=8$ (\ref{figura3}b). From these
strategies, we see that the {\it max} scheme performs more closely
to the optimal one when the control scheme is allowed to make
almost one intervention per time step and the {\it min} scheme
performs better when the control schemes are allowed only to make
interventions when there is a probability of large avalanches. It
is clear that the cost is always minimal for the DP scheme.
Furthermore, while the {\it max}, {\it min} and {\it ran} controls
have their performances strongly affected by changes in the ratio
$C_I/C_A$, the DP control makes a good work in reducing the size
of avalanches in both situations (this information can also be
seen with the help of Fig. \ref{figura1}(b)). Figure \ref{figura3}
can also help us to choose when to choose the {\it min} scheme and
the {\it max} scheme. The {\it min} scheme should be used when the
size of $T_I$ is larger -- triggering the minimal avalanches, this
system can avoid uncontrolled triggering of large avalanches. On
the other hand, for low values of $T_I$, one should use the {\it
max} scheme. Since in almost every time step avalanches are being
triggered, triggering the largest ones the {\it max} control
avoids uncontrolled triggering of larger avalanches. Note that the
use of the {\it max\;} scheme for the case of large $T_I$ is
dangerous, since the control by itself will trigger large
avalanches. Finally, the choice of the {\it min} control for small
$T_I$ is useless, since it will trigger only small avalanches that
do not help avoid the largest ones.
Figure \ref{figuraNova} reinforces the results of Figure
\ref{figura3} showing simulations of the DADR's model controlled
by the heuristics {\it max}, {\it min} and {\it ran} for a system
with size $L=32$. While in Figure \ref{figuraNova}(a) $T_I=8$
(small value), in figure Figure \ref{figuraNova}(b) $T_I=32$
(large value). One should note that qualitatively the results are
the same. Furthermore, based on the results of Figure
\ref{figura3}, we are able to say that while in the first case
($T_I=8$) the heuristic {\it max} is closest to the optimality, in
the second case ($T_I=32$) the heuristic ({\it min}) is the one
that it is closest.
\begin{figure}[t]
\begin{tabular}{cc}
\includegraphics[width=4.2cm,height=4.2cm]{figure2a.eps} & \includegraphics[width=4.2cm,height=4.2cm]{figure2b.eps} \\
\end{tabular}
\caption{Ratio $f$ between the total number of avalanches in the
controlled and uncontrolled situations, for several control
schemes. Both in panel (a) as in (b), the following notation is
used to identify parameter values and adopted control scheme:
$(p_I;\mathrm{cost\; scheme}/\mathrm{cost\; DP};
\mathrm{scheme})$. In panel (a), $T_I=1$: solid (1; 1; DP),
dashes (1; 1.29; {\it max}), dots (1; 1.46; {\it min}), dot-dashes
(1; 1.31; {\it ran}). In panel (b), $T_I=8$: solid (0.11; 1; DP),
dashes (0.12; 1.21; {\it max}), dots (0.12; 1.14; {\it min}),
dot-dashes (0.1; 1.19; {\it ran}).}\label{figura3}
\end{figure}
\begin{figure}[t]
\begin{tabular}{cc}
\includegraphics[width=4.2cm,height=4.2cm]{figura3a.eps} & \includegraphics[width=4.2cm,height=4.2cm]{figura3b.eps} \\
\end{tabular}
\caption{Ratio $f$ between the total number of avalanches in the
controlled and uncontrolled situations for square lattices with
size $L=32$ and for several control schemes. Both in panel (a) as
in (b), the following notation is used to identify parameter
values and adopted control scheme: $(p_I; \mathrm{scheme})$. In
panel (a), $T_I=8$: dashes (0.92; {\it max}), dots (1; {\it min}),
dot-dashes (1; {\it ran}). In panel (b), $T_I=32$: dashes (0.33;
{\it max}), dots (0.78; {\it min}), dot-dashes (0.50; {\it
ran}).}\label{figuraNova}
\end{figure}
\section{Final remarks} \label{sec:conc}
We have introduced a DP approach to control SOC in the DADR model.
Although this framework cannot be applied to large system, it is
quite useful to characterize the optimal solution and evaluate
optimality of other heuristics. The reduction in the number of
large avalanches shown in Fig. 2 is similar to those obtained
in~\cite{cajand10}, where a fixed heuristics was considered. In
that work, no cost was associated with intervention, so that it is
not possible to directly compare results predicted in Fig. 1a to
larger systems. However, the sudden vanishing of $\langle
s\rangle$ at $(C_I/C_A)_T$ suggests that, for heuristic based
control, a similar behavior would be observed if cost is
introduced in the model. Finally, this approach can be the basis
to study approximate sub-optimal approaches in the line
of~\cite{bertsi96}, using for instance reinforcement learning
techniques.
\section{Acknowledgment} The authors are grateful to the Brazilian
agency CNPQ and the National Institute of Science and Technology
for Complex Systems (Brazil) for financial support.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,030 |
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In This Article: Randy Houser | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,737 |
\section{Introduction}\label{S:introduction}
\subsection{Gonality}
The \defi{gonality} $\gamma_k(X)$ of a curve\footnote{%
All our curves are assumed to be geometrically integral.%
}
$X$ over a field $k$
is the smallest possible degree of a
dominant rational map $X \dashrightarrow {\mathbb P}^1_k$.
For any field extension $L$ of $k$, we define also the \defi{$L$-gonality}
$\gamma_L(X)$ of $X$ as the gonality
of the base extension $X_L:=X \times_k L$.
It is an invariant of the function field $L(X)$ of $X_L$.
\subsection{General facts about gonality}
\label{S:facts}
The following facts are well-known.
\begin{proposition}
\label{P:facts}
Let $X$ be a curve of genus $g$ over a field $k$.
\begin{romanenum}
\item
If $L$ is a field extension of $k$, then $\gamma_L(X) \le \gamma_k(X)$.
\item
If $k$ is algebraically closed, and $L$ is a field extension of $k$,
then $\gamma_L(X) = \gamma_k(X)$.
\item
If $g>1$, then $\gamma_k(X) \le 2g-2$.
For each $g>1$, there exist $k$ and $X$ for which equality holds.
\item
If $X(k) \ne \emptyset$, then $\gamma_k(X) \le g+1$.
If $X(k) \ne \emptyset$ and $g \ge 2$, then $\gamma_k(X) \le g$.
\item
If $k$ is algebraically closed,
then $\gamma_k(X) \le \lfloor \frac{g+3}{2} \rfloor$.
Equality holds for a general curve of genus $g$ over $k$.
\item
If $\pi\colon X \dashrightarrow Y$ is a dominant rational map
of curves over $k$,
then $\gamma_k(X) \le (\deg \pi) \gamma_k(Y)$.
\item
If $\pi\colon X \dashrightarrow Y$ is a dominant rational map
of curves over $k$, then $\gamma_k(Y) \le \gamma_k(X)$.
\end{romanenum}
\end{proposition}
\begin{proof}\hfill
\begin{romanenum}
\item
Trivial.
\item
Given a map over $L$,
standard specialization arguments give a map over $k$
of the same degree.
\item
The canonical linear system $|K|$ has dimension $g-1 \ge 1$
and degree $2g-2$.
So we may use a rational function whose divisor is the difference of
two different canonical divisors.
Equality holds for the general curve
over the function field of the moduli space of genus-$g$ curves
in characteristic $0$,
since its only line sheaves are the powers of the canonical sheaf:
this was the Franchetta conjecture, proved in \cite{Harer1983}
and strengthened in \cite{Mestrano1987}.
\item
Let $P \in X(k)$.
The Riemann-Roch theorem shows that $\dim |(g+1)P| \ge 1$,
and that $\dim |K -(g-2)P| \ge 1$ if $g-2 \ge 0$.
These linear systems have degree $g+1$ and $g$, respectively.
\item
These are consequences of Brill-Noether theory: see (1.1) and (1.5)
of \cite{Arbarello-et-al1985}*{Chapter~V} for an exposition.
The first statement is proved in arbitrary characteristic
in \cites{Kleiman-Laksov1972,Kleiman-Laksov1974}.
The second statement is proved in characteristic $0$
in \cites{Farkas1966,Martens1967,Martens1968},
and can be deduced in characteristic $p$
from the unramified case of \cite{Osserman2005}*{Theorem~1.2}, for instance.
\item
Trivial.
\item
(The ideas in the following argument
go back at least to \cite{Newman1972}*{Theorem~VII.2}.)
Choose $f \in k(X)$ of degree $d:=\gamma_k(X)$.
Let $r=\deg \pi = [k(X):k(Y)]$.
Let $P(T) \in k(Y)[T]$ be the characteristic polynomial of $f$
viewed as an element of the field extension $k(X)$ of $k(Y)$.
For some finite normal extension $M$ of $k(Y)$,
we may write $P(T)=\prod_{i=1}^r (T-f_i)$ for some $f_i \in M$.
As a function in $M$, $f$ has degree $[M:k(X)] d$.
The same is true of each $f_i$, since they are all in the same
$\Aut(M/k(Y))$-orbit.
The polar divisor of a coefficient of $P$ viewed in $M$
is at most the sum of the polar divisors of the $f_i$,
so each coefficient
has degree at most $r [M:k(X)] d = [M:k(Y)] d$ as a function in $M$,
and hence degree at most $d$ as a function in $k(Y)$.
Since $f$ is non-constant, at least one of these coefficients
is non-constant.
Thus $\gamma_k(Y) \le d$.
\end{romanenum}
\end{proof}
\subsection{Modular curves}
Our goal is to obtain lower bounds on the gonality of modular curves.
By Proposition~\ref{P:facts}(ii), it suffices to consider
$k={\mathbb C}$ and $k={\overline{\F}}_p$ (an algebraic closure of ${\mathbb F}_p$) for each prime $p$.
Suppose $N$ is a positive integer not divisible by the characteristic of $k$.
Choose a primitive $N$-th root of unity $\zeta$.
Let $X(N)$ be the smooth projective model of the (possibly coarse)
moduli space parameterizing triples $(E,P,Q)$ where $P,Q \in E$
are a basis for $E[N]$ with Weil pairing $e_N(P,Q)=\zeta$.
For $k={\mathbb C}$, we can describe $X(N)({\mathbb C})$ alternatively as the quotient
of an extended upper half plane by a finite-index subgroup of $\operatorname{PSL}_2({\mathbb Z})$.
More generally, any congruence subgroup $G \le \operatorname{PSL}_2({\mathbb Z})$
gives rise to a curve $X_G$ over ${\mathbb C}$.
We have the following theorem of Abramovich:
\begin{theorem}[\cite{Abramovich1996}]
\label{T:Abramovich}
Let $D=(\operatorname{PSL}_2({\mathbb Z}):G)$.
Then $\gamma_{\mathbb C}(X_G) \ge \frac{7}{800} D$.
\end{theorem}
\begin{remark}
As mentioned in \cite{Abramovich1996},
combining Theorem~\ref{T:Abramovich}
with the genus bound $g-1 \le D/12$ \cite{Shimura1994}*{Proposition~1.40}
yields
\[
\gamma_{\mathbb C}(X_G) \ge \frac{21}{200}(g-1).
\]
The proof of Theorem~\ref{T:Abramovich}
makes use of a lower bound on the leading nontrivial eigenvalue
of the noneuclidean Laplacian; this bound has been improved since 1996,
so the constants $7/800$ and $21/200$ can be improved too.
See also \cite{Baker-et-al2005}*{\S4.3} for some further results.
\end{remark}
In characteristic $p$, one has other kinds of modular curves,
involving level structure where $p$ divides the level.
If $q=p^e$ for some $e \in {\mathbb Z}_{\ge 1}$, the \defi{Igusa curve of level $q$}
is the smooth projective model $\Ig(q)$ of the curve over ${\mathbb F}_p$
parameterizing pairs $(E,R)$ where $E$ is an ordinary elliptic curve,
and $R$ is a generator of the kernel of the degree-$q$ Verschiebung isogeny
$V_q \colon E^{(q)} \to E$, where $E^{(q)}$ is the elliptic curve obtained by
raising all the coefficients of a model of $E$ to the $q$-th power.
Given $N$ not divisible by $p$, and $q=p^e$,
we can define also a hybrid modular curve $X(p^e;N)$ over ${\overline{\F}}_p$
parameterizing $(E,R,P,Q)$, with $R \in \ker V_q$ and $P,Q \in E[N]$
as above.
The group $G_{p^eN}:=({\mathbb Z}/p^e{\mathbb Z})^\times \times \operatorname{SL}_2({\mathbb Z}/N{\mathbb Z})$
acts on $X(p^e;N)$.
The kernel of the action is $\{\pm 1\}$ embedded diagonally in $G_{p^e N}$.
For any subgroup $G \le G_{p^e N}$ containing $\{\pm 1\}$, let $X_G$
be the smooth projective model of the quotient $X(p^e;N)/G$.
The $G_{p^e N}$ form an inverse system with inverse limit
\[
S:={\mathbb Z}_p^\times \times \prod_{\text{prime $\ell \ne p$}} \operatorname{SL}_2({\mathbb Z}_\ell).
\]
The inverse image of $G$ under $S \twoheadrightarrow G_{p^e N}$
is an open subgroup of the profinite group $S$,
and every open subgroup of $S$ containing $\{\pm 1\}$
arises this way for some $p^e N$.
Thus we may define $X_G$ for any open subgroup $G$ of $S$
containing $\{\pm 1\}$.
It seems likely that there is a constant $c>0$ independent
of $p$ and $G$ such that $\gamma_{{\overline{\F}}_p}(X_G) \ge c (S:G)$.
We are unable to prove such a linear lower bound, even for fixed $p$,
but we can show that the gonality goes to infinity for fixed $p$.
Here is our main theorem:
\begin{theorem}
\label{T:main}
Fix a prime $p$.
Let $G_1,G_2,\ldots$ be a sequence of distinct open subgroups of $S$
containing $\{\pm 1\}$.
Then $\gamma_{{\overline{\F}}_p}(X_{G_i}) \to \infty$ as $i \to \infty$.
\end{theorem}
\subsection{Outline of proof of main theorem}
Many of the ideas used in the proof of Theorem~\ref{T:main}
are due to earlier authors,
though we consider a broader class of modular curves
than had been treated earlier.
Section~\ref{S:change} proves
Theorem~\ref{T:change},
an inequality in the direction opposite to Proposition~\ref{P:facts}(i):
the ideas used here and their application to
the classical modular curves $X_0(N)$ can be found in
\cite{Harris-Silverman1991},
\cite{Nguyen-Saito1996preprint},
and \cite{Baker-thesis}*{Chapter 3}.
Theorem \ref{T:change}
reduces the problem to finding lower bounds on gonality over
{\em finite} fields,
and these can be obtained by counting in the spirit of \cite{Ogg1974},
which among other things
determined the $N$ for which $X_0(N)$ is hyperelliptic.
In Section~\ref{S:prime to p} we find that, as in \cite{Ogg1974},
modular curves of level prime to $p$
have too many supersingular points over ${\mathbb F}_{p^2}$
to have small gonality.
In Section~\ref{S:power of p} we cite results
of Schweizer \cite{Schweizer2005}, who obtained lower bounds
on the ${\overline{\F}}_p$-gonality of Igusa curves directly from the geometry
of the curves, instead of first getting lower bounds on ${\mathbb F}_p$-gonality
by counting ${\mathbb F}_p$-points.
Section~\ref{S:group theory} uses Goursat's lemma to study the subgroups
of $S$, so that the prime-to-$p$ and $p$-power cases can be combined
to prove the general case of Theorem~\ref{T:main}
in Section~\ref{S:general case}.
\subsection{Application to the image of Galois}
One application of results like Theorem~\ref{T:main},
noted already by many other authors,
is to the function field analogue of
the strong uniform boundedness theorem for elliptic curves.
By the work of Mazur, Kamienny, and Merel \cite{Merel1996},
for every $d \in {\mathbb Z}_{\ge 1}$,
there exists a constant $N_d$ such that
for any number field $K$ with $[K:{\mathbb Q}] \le d$
and for any elliptic curve $E$ over $K$,
the torsion subgroup $E(K)_{\operatorname{tors}}$ of
the finitely generated abelian group $E(K)$
satisfies $\#E(K)_{\operatorname{tors}} \le N_d$.
In the function field case, we can prove a stronger result,
one which bounds the index of the image of Galois acting on torsion.
If $E$ is an elliptic curve over a field $K$ of characteristic $p \ge 0$,
and $K^s$ is a separable closure of $K$,
there exists a homomorphism
\[
\rho_E\colon \Gal(K^s/K) \to {\mathbb Z}_p^\times \times \prod_{\ell \ne p} \operatorname{GL}_2({\mathbb Z}_\ell)
\]
describing the Galois action on the Tate modules of $E$.
(Of course, if $\Char K=0$, there is no ${\mathbb Z}_p^\times$ factor.)
\begin{theorem}
\label{T:uniform boundedness}
Given $p \ge 0$ and $d \in {\mathbb Z}_{\ge 1}$,
there exists a constant $N_{p,d}$ such that
for any field $k$ of characteristic $p$,
any field $K$ of degree $\le d$ over $k(t)$,
and any elliptic curve $E$ over $K$ with $j(E)$ not algebraic over $k$,
the index $(S:\rho_E(\Gal(K^s/K)) \intersect S)$ is at most $N_{p,d}$.
\end{theorem}
\begin{remark}
In Theorem~\ref{T:uniform boundedness},
we cannot hope to bound the index of $\rho_E(\Gal(K^s/K))$
in ${\mathbb Z}_p^\times \times \prod_{\ell \ne p} \operatorname{GL}_2({\mathbb Z}_\ell)$
(i.e., with $\operatorname{GL}_2$ instead of the $\operatorname{SL}_2$ in the definition of $S$),
since the determinant of the image in $\operatorname{GL}_2({\mathbb Z}_\ell)$
gives the action of $\Gal(K^s/K)$ on roots of unity,
and this is trivial if $k$ is algebraically closed, for example.
\end{remark}
Theorem~\ref{T:uniform boundedness} will be deduced from
Theorem~\ref{T:main}
in Section~\ref{S:uniform boundedness}.
\section{Change in gonality under extension of the ground field}
\label{S:change}
In this section we give an exposition of the ``tower theorem'' of
Nguyen and Saito \cite{Nguyen-Saito1996preprint}*{Theorem 2.1},
and its implication for relating gonalities of a single curve
over different fields.
We will reprove it as our Proposition~\ref{P:tower},
since \cite{Nguyen-Saito1996preprint} remains unpublished after 10 years,
and since we can simplify the proof slightly.
Throughout this section, $k$ is a perfect field.
\begin{proposition}[Castelnuovo-Severi inequality]
\label{P:Castelnuovo-Severi}
Let $F$, $F_1$, $F_2$ be function fields of curves over $k$,
of genera $g$, $g_1$, $g_2$, respectively.
Suppose that $F_i \subseteq F$ for $i=1,2$
and the compositum of $F_1$ and $F_2$ in $F$ equals $F$.
Let $d_i=[F:F_i]$ for $i=1,2$.
Then
\[
g \le d_1 g_1 + d_2 g_2 + (d_1-1)(d_2-1).
\]
\end{proposition}
\begin{proof}
See \cite{Stichtenoth1993}*{III.10.3}.
\end{proof}
Let $X$ be a curve over $k$.
A subfield $F$ of $k(X)$ will be called \defi{$d$-controlled} if
there exists $e \in {\mathbb Z}_{>0}$ such that
$[k(X):F]=d/e$ and the genus of $F$ is $\le (e-1)^2$.
\begin{lemma}
\label{L:d-controlled}
If $F$ is $d$-controlled, and $f \in k(X)$ is a rational function of
degree $d$, then $F(f)$ is $d$-controlled.
\end{lemma}
\begin{proof}
View $F(f)$ as the compositum of $F$ and $k(f)$ in $k(X)$.
Let $a=[F(f):F]$.
Then $[k(X):F(f)] = d/(ae)$, so $[F(f):k(f)]=ae$.
By Proposition~\ref{P:Castelnuovo-Severi}, the genus of $F(f)$ is at most
\[
a (e-1)^2 + 0 + (a-1)(ae-1)
\quad=\quad
(ae-1)^2 - ae (a-1)(e-1)
\quad\le\quad
(ae-1)^2,
\]
since $a,e \ge 1$.
\end{proof}
\begin{corollary}
\label{C:d-controlled}
A subfield of $k(X)$ generated over $k$ by one or more elements of degree $d$
is $d$-controlled.
\end{corollary}
\begin{proof}
Induction on the number of elements:
the case of one element is trivial ($e=1$),
and Lemma~\ref{L:d-controlled} gives the inductive step.
\end{proof}
\begin{proposition}[Tower theorem]
\label{P:tower}
Let $X$ be a curve over a perfect field $k$.
Let $L \supseteq k$ be an algebraic field extension.
Let $d=\gamma_L(X)$.
Then $k(X)$ has a $d$-controlled subfield.
\end{proposition}
\begin{proof}
Enlarging $L$ cannot increase $\gamma_L(X)$, so we may assume $L/k$ is Galois.
Choose $f \in L(X)$ of degree $d$.
Let $F_L$ be the subfield generated over $L$ by the $\Gal(L/k)$-conjugates
of $f$.
By Corollary~\ref{C:d-controlled}, $F_L$ is $d$-controlled
as a subfield of $L(X)$.
The action of $\Gal(L/k)$ on $L(X)$ preserves $F_L$,
and the invariant subfield $F_k:=F_L^{\Gal(L/k)}$
satisfies $[k(X):F_k]=[L(X):F_L]$ and has the same genus as $F_L$.
Thus $F_k$ is a $d$-controlled subfield of $k(X)$.
\end{proof}
\begin{theorem}
\label{T:change}
Let $X$ be a curve over a perfect field $k$.
Let $L \supseteq k$ be an algebraic field extension.
Let $d=\gamma_L(X)$.
Assume that $X(k) \ne \emptyset$.
\begin{romanenum}
\item
If $d \le 2$, then $\gamma_k(X)=d$.
\item \label{I:d>2}
If $d>2$, then $\gamma_k(X) \le (d-1)^2$.
\item \label{I:simple square root}
In any case, $\gamma_L(X) \ge \sqrt{\gamma_k(X)}$.
\end{romanenum}
\end{theorem}
\begin{proof}\hfill
\begin{romanenum}
\item
If $d=1$, then $X \simeq {\mathbb P}^1_k$, so $\gamma_k(X)=1$.
If $d=2$, then $X$ is elliptic or hyperelliptic;
if elliptic, then $\gamma_k(X)=2$;
if hyperelliptic then the canonical map is a degree-$2$ map to a
genus-$0$ curve $Z$ over $k$, and $Z(k) \ne \emptyset$ so $Z \simeq {\mathbb P}^1_k$,
so $\gamma_k(X)=2$.
\item
Now suppose $d>2$.
By Proposition~\ref{P:tower} there exists $e \in {\mathbb Z}_{>0}$ and a
rational map $\pi\colon X \dashrightarrow Y$ of curves over $k$
such that $\deg \pi = d/e$ and the genus $g$ of $Y$ satisfies $g \le (e-1)^2$.
We have $Y(k)\ne\emptyset$.
If $g=0$, then $Y \simeq {\mathbb P}^1$, so
\[
\gamma_k(X) \le d/e \le d < (d-1)^2.
\]
If $g=1$, then $e \ge 2$ and $\gamma_k(Y)=2$, so
\[
\gamma_k(X) \le (d/e) \gamma_k(Y) \le (d/2) 2 = d < (d-1)^2.
\]
If $g \ge 2$, then $\gamma_k(Y) \le g$ by Proposition~\ref{P:facts}(iv), so
\[
\gamma_k(X) \le \frac{d}{e} \gamma_k(Y) \le \frac{d}{e} (e-1)^2.
\]
For $e \in [1,d]$, the function $\frac{d}{e}(e-1)^2$ is maximized at $e=d$,
and the value there is $(d-1)^2$.
\item
This follows directly from the first two parts.
\end{romanenum}
\end{proof}
\begin{remark}
The hypothesis $X(k)\ne \emptyset$ is necessary:
Genus-$1$ curves over ${\mathbb Q}$ have ${\overline{\Q}}$-gonality $2$,
but their ${\mathbb Q}$-gonality can be arbitrarily large.
\end{remark}
\begin{remark}
We do not know whether the $(d-1)^2$ in Theorem~\ref{T:change}\eqref{I:d>2}
can be improved.
\end{remark}
\begin{remark}
For $N=38, 44, 53, 61$, the modular curve $X_0(N)$ is of genus $4$
and has ${\mathbb Q}$-gonality $4$ and ${\overline{\Q}}$-gonality $3$
\cite{Hasegawa-Shimura1999}*{p.~136}.
In particular, Theorem~\ref{T:change}\eqref{I:d>2} is best possible for $d=3$.
\end{remark}
\section{Level prime to $p$}
\label{S:prime to p}
Suppose $p \nmid N$.
We begin by defining a twisted form $X(N)'$ over ${\mathbb F}_{p^2}$ of $X(N)$.
Let $M$ be $({\mathbb Z}/N {\mathbb Z})^2$ made into a
$\Gal({\overline{\F}}_p/{\mathbb F}_{p^2})$-module
by letting the $p^2$-power Frobenius automorphism act as
multiplication by $-p$.
There exists an isomorphism of Galois modules
$\iota\colon \bigwedge^2 M \to \mu_N$;
fix one.
Let $X(N)'$ be the smooth projective model of
the affine curve over ${\mathbb F}_{p^2}$
parameterizing pairs $(E,\phi)$ where $E$ is an elliptic curve
and $\phi$ is an isomorphism $E[N] \to M$
under which the Weil pairing corresponds to $\iota$.
Over ${\overline{\F}}_p$, $X(N)'$ becomes isomorphic to $X(N)$.
The automorphisms of $M$ as an abelian group automatically commute with
the Galois action, so they induce automorphisms of $X(N)'$
defined over ${\mathbb F}_{p^2}$.
Thus we get
$\operatorname{SL}_2({\mathbb Z}/N{\mathbb Z})/\{\pm 1\} \le \operatorname{Aut} X(N)'$.
Moreover, it follows from \cite{Baker-et-al2005}*{Lemma~3.21}
that all the points of $X(N)'$ corresponding to
supersingular elliptic curves are defined over ${\mathbb F}_{p^2}$.
\begin{proposition}
\label{P:special modular}
Let $p$, $N$, and $X(N)'$ be as above.
Let $G$ be a subgroup of $\operatorname{SL}_2({\mathbb Z}/N{\mathbb Z})/\{\pm 1\}$ of index $D$.
Let $X$ be the curve $X(N)'/G$.
Then the ${\mathbb F}_{p^2}$-gonality $\gamma$ of $X$ satisfies
\[
\gamma \ge \frac{p-1}{12(p^2+1)} D.
\]
\end{proposition}
\begin{proof}
This resembles the proof of \cite{Baker-et-al2005}*{Lemma~3.22}.
By \cite{Baker-et-al2005}*{Lemma~3.20},
the number of supersingular points on $X$ is $\ge (p-1)D/12$,
and these are images of supersingular points on $X(N)'$
so they are defined over ${\mathbb F}_{p^2}$;
thus $\#X({\mathbb F}_{p^2}) \ge (p-1)D/12$.
On the other hand,
$\#X({\mathbb F}_{p^2}) \le \gamma \# {\mathbb P}^1({\mathbb F}_{p^2}) = \gamma (p^2+1)$.
Combine the two previous sentences.
\end{proof}
\begin{remark}
Let $g$ be the genus of $X$.
One could also combine Proposition~\ref{P:special modular} with
the bound $g-1 < D/12$ of \cite{Shimura1994}*{Proposition~1.40}
to give a lower bound for $\gamma$ in terms of $g$ instead of $D$.
\end{remark}
We now consider the ${\overline{\F}}_p$-gonality of
all modular curves of level prime to $p$.
\begin{corollary}
\label{C:Phi}
Fix $p$.
Define $\Phi_p(D):=\sqrt{\frac{p-1}{12(p^2+1)} D}$.
If $G$ is the inverse image under
$S \twoheadrightarrow \prod_{\ell \ne p} \operatorname{SL}_2({\mathbb Z}_\ell)$
of an open subgroup of index $D$ in $\prod_{\ell \ne p} \operatorname{SL}_2({\mathbb Z}_\ell)$
containing $\{\pm 1\}$,
then $\gamma_{{\overline{\F}}_p}(X_G) \ge \Phi_p(D)$.
\end{corollary}
\begin{proof}
Combine Proposition~\ref{P:special modular}
and Theorem~\ref{T:change}\eqref{I:simple square root}.
\end{proof}
\begin{remark}
\label{R:Atkin-Lehner}
One could also consider Atkin-Lehner quotients of $X_0(N)$
for $N$ prime to $p$.
These are generally not of the form $X_G$.
Nevertheless,
gonality bounds tending to infinity for fixed $p$ can be obtained:
first apply Proposition~\ref{P:special modular} to get
a lower bound on $\gamma_{{\mathbb F}_{p^2}}(X_0(N))$,
next use Proposition~\ref{P:facts}(vi) to get a lower bound
on the ${\mathbb F}_{p^2}$-gonality of any quotient of $X_0(N)$,
and finally apply Theorem~\ref{T:change}.
This works since the size of the Atkin-Lehner group (some power of $2$)
is asymptotically small compared to the index of
the congruence subgroup $\Gamma_0(N)$ in $\operatorname{PSL}_2({\mathbb Z})$.
\end{remark}
\section{Level a power of $p$}
\label{S:power of p}
The necessary lower bound on the gonality of $\Ig(p^e)$
has been proved already by Schweizer, in a strong form:
\begin{theorem}[\cite{Schweizer2005}*{Lemma~1.5(d,e)}] \hfill
\label{T:Schweizer}
\begin{romanenum}
\item \label{I:1}
If $p\ge 7$, then
$\frac{p+13}{24} \le \gamma_{{\overline{\F}}_p}(\Ig(p)) \le \frac{p-1}{6}$.
\item \label{I:2}
If $e>1$ and $p^e \notin \{25,9,8,4\}$, then
$\gamma_{{\overline{\F}}_p}(\Ig(p^e)) = p \gamma_{{\overline{\F}}_p}(\Ig(p^{e-1}))$.
\end{romanenum}
\end{theorem}
In fact, \cite{Schweizer2005} proves many more results.
The above are more than we need to deduce the following.
\begin{corollary}
\label{C:Psi}
Let $G$ be the inverse image under $S \twoheadrightarrow {\mathbb Z}_p^\times$
of an open subgroup of index $D$ in ${\mathbb Z}_p^\times$ containing $\{\pm 1\}$.
Then $\gamma_{{\overline{\F}}_p}(X_G) > D/12$.
\end{corollary}
\begin{proof}
First suppose that $X_G=\Ig(p^e)$ for some prime power $p^e>2$.
Then $D=p^{e-1}(p-1)/2$.
For $p^e \le 25$, we have $D<12$,
and $\gamma_{{\overline{\F}}_p}(X_G) \ge 1 > D/12$ trivially.
For $p>25$, Theorem~\ref{T:Schweizer}\eqref{I:1} gives
\[
\gamma_{{\overline{\F}}_p}(\Ig(p)) \ge \frac{p+13}{24} > \frac{p-1}{24} = \frac{D}{12}.
\]
For $p^e>25$ with $e>1$, we use induction on $e$,
with Theorem~\ref{T:Schweizer}\eqref{I:2} giving the inductive step.
Any other $X_G$ in Corollary~\ref{C:Psi} is a quotient of $\Ig(p^e)$
for some $p^e>2$, and the inequality for $X_G$ follows from
the inequality for $\Ig(p^e)$,
by Proposition~\ref{P:facts}(vi).
\end{proof}
\begin{remark}
An alternative approach to lower bounds on the gonality of Igusa curves
is to show that they have many points over certain finite fields,
to deduce that the gonality over these finite fields is large,
and then to apply Theorem~\ref{T:change}.
One can no longer use supersingular points, however, since these
are totally ramified in $\Ig(p^e) \stackrel{j}\to {\mathbb P}^1$,
and hence their number does not grow with $e$.
Instead we could use {\em ordinary} points:
it follows from \cite{Pacheco1996}*{Corollary~2.13}
and Hurwitz class number estimates that $\#\Ig(q)({\mathbb F}_q) \ge q^{1/2+o(1)}$
as $q \to \infty$.
Or one could use cusps,
as in the proof of \cite{Schweizer2004}*{Theorem~6.1},
to get lower bounds on $\#\Ig(q)({\mathbb F}_p)$,
since the cusps split completely
in $\Ig(p^e) \stackrel{j}\to {\mathbb P}^1$ \cite{Katz-Mazur1985}*{Corollary~12.7.2}.
But the lower bounds on gonality obtained by these methods
are weaker than the ones we took from \cite{Schweizer2005}.
\end{remark}
\section{Group theory}
\label{S:group theory}
Here we study the open subgroups of $S$.
Let $S_p={\mathbb Z}_p^\times$ and $S_{\ne p} = \prod_{\ell \ne p} \operatorname{SL}_2({\mathbb Z}_\ell)$,
so $S = S_p \times S_{\ne p}$.
Given an open subgroup $G \le S$,
let $G_p$ and $G_{\ne p}$ be the images of $G$ in $S_p$ and $S_{\ne p}$,
respectively.
\begin{lemma}
\label{L:contains SL_2}
Let $B \in {\mathbb Z}_{>0}$.
Any open subgroup $H$ of $S_{\ne p}$ of index $\le B$ contains
\[
\prod_{\ell \le B!, \; \ell \ne p} \{1\} \times \prod_{\ell > B!, \; \ell \ne p} \operatorname{SL}_2({\mathbb Z}_\ell).
\]
\end{lemma}
\begin{proof}
For each $\ell \ne p$, identify $\operatorname{SL}_2({\mathbb Z}_\ell)$ with a
subgroup of $S_{\ne p}$ in the obvious way.
It suffices to show that $H$ contains $\operatorname{SL}_2({\mathbb Z}_\ell)$
for each $\ell>B!$ with $\ell \ne p$.
The kernel of the action of $S_{\ne p}$ on the coset space $S_{\ne p}/H$
is a normal open subgroup $N \trianglelefteq S_{\ne p}$
contained in $H$.
Let $n:=(S_{\ne p}:N)$, so $n \le B!$.
Now $\ell>B! \ge n$, so $1/n \in {\mathbb Z}_\ell$, and
\[
\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}
= \begin{pmatrix} 1 & 1/n \\ 0 & 1 \end{pmatrix}^n \in N,
\]
where the matrices belong to $\operatorname{SL}_2({\mathbb Z}_\ell) \le S_{\ne p}$.
Similarly $\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \in N$.
But these two matrices generate the dense subgroup $\operatorname{SL}_2({\mathbb Z})$
of $\operatorname{SL}_2({\mathbb Z}_\ell)$, so $\operatorname{SL}_2({\mathbb Z}_\ell) \le N \le H$.
\end{proof}
\begin{lemma}
\label{L:open subgroups of S}
For each $B>0$,
there are at most finitely many open subgroups $G$ of $S$
such that $(S_p:G_p)<B$ and $(S_{\ne p}:G_{\ne p})<B$.
\end{lemma}
\begin{proof}
Fix $B$.
By Lemma~\ref{L:contains SL_2}, it suffices to consider instead
the situation in which $S_{\ne p}$ is replaced
by $S_L:=\prod_{\ell \in L} \operatorname{SL}_2({\mathbb Z}_\ell)$
for a {\em finite} set $L$ of primes $\ne p$:
i.e., $G$ is now an open subgroup of $S_p \times S_L$,
$G_L$ is the image of $G$ in $S_L$,
and we are given $(S_p:G_p)<B$ and $(S_L:G_L)<B$.
Since $S_p$ and $S_{\ne p}$ are topologically finitely generated,
there are finitely many possibilities for $G_p$ and $G_L$.
Goursat's Lemma \cite{LangAlgebra}*{p.~75}
states that each possible $G$ is the inverse image under
\[
G_p \times G_L \twoheadrightarrow \frac{G_p}{H_p} \times \frac{G_L}{H_L}
\]
of the graph of an isomorphism
\[
\frac{G_p}{H_p} \to \frac{G_L}{H_L}
\]
for some normal open subgroups
$H_p \trianglelefteq G_p$ and $H_L \trianglelefteq G_L$.
By the finite generation again,
it suffices to bound $(G_p:H_p)=(G_L:H_L)$.
It is bounded by the supernatural number $\gcd(\#S_p,\#S_L)$,
which is finite, since $S_p$ has a pro-$p$ open subgroup,
while $S_L$ has an open subgroup of order prime to $p$.
(See \cite{SerreGaloisCohomology}*{I.\S1.3} for the notion
of supernatural number.)
\end{proof}
\section{The general case of Theorem~\ref{T:main}}
\label{S:general case}
\begin{proof}[Proof of Theorem~\ref{T:main}]
Let $m>0$.
We will show that $\gamma_{{\overline{\F}}_q}(X_{G_i})>m$
for all but finitely many $i$.
Let $\Phi_p$ be as in Corollary~\ref{C:Phi}.
Choose $B_0$ such that $\min\{\Phi_p(B),B/12 \} > m$ for all $B \ge B_0$.
By Lemma~\ref{L:open subgroups of S},
all but finitely many $G_i$ in our sequence
have either $(S_p:(G_i)_p) \ge B$ or $(S_{\ne p}:(G_i)_{\ne p}) \ge B$.
For each such $i$, $X_{G_i}$ dominates an $X_G$
with $G$ as in Corollary \ref{C:Phi} or Corollary \ref{C:Psi};
then $\gamma_{{\overline{\F}}_p}(X_G)$ exceeds either $\Phi_p(B)$ or $B/12$.
By Proposition~\ref{P:facts}(vii),
\[
\gamma_{{\overline{\F}}_p}(X_{G_i}) \ge \gamma_{{\overline{\F}}_p}(X_G)
\ge \min\{ \Phi_p(B), B/12 \} > m.
\]
\end{proof}
\section{Image of Galois}
\label{S:uniform boundedness}
\begin{proof}[Proof of Theorem~\ref{T:uniform boundedness}]
We may assume that $k$ is algebraically closed.
By Theorem~\ref{T:main},
there are only finitely many subgroups $G \le S$
containing $\{\pm1\}$ such that $\gamma_{{\overline{\F}}_p}(X_G) \le d$.
Choose $N_{p,d}$ such that $N_{p,d} \ge 2(S:G)$ for every such $G$.
Let $K$ be a field of degree $\le d$ over $k(t)$,
and let $E$ be an elliptic curve over $K$
with $j(E)$ not algebraic over $k$ (i.e., not in $k$).
Write $K=k(C)$, where $C$ is a curve over $k$ with $\gamma_k(C) \le d$.
Define $H := \rho_E(\Gal(K^s/K))$.
Since $k$ is algebraically closed, $H \subseteq S$.
We want $(S:H) \le N_{p,d}$.
Suppose not.
If $(S:H)$ is infinite,
then since $S/H$ is a profinite group,
we can find a group $H'$ with $H \le H' \le S$
and $N_{p,d} < (S:H') < \infty$.
If $(S:H)$ is finite, let $H'=H$.
In either case, let $H''$ be the group generated by $H'$ and $-1$,
so $N_{p,d}/2 < (S:H'') < \infty$.
By definition of $N_{p,d}$, the group $H''$ does not equal any of the
groups $G$, so $\gamma_{{\overline{\F}}_p}(X_{H''})>d$.
Equivalently, by Proposition~\ref{P:facts}(ii), $\gamma_k(X_{H''})>d$.
The curve $X_{H''}$ is defined as a quotient of some $X(p^e;N)$.
Choosing level structure for $E$ over $K^s$ gives a point
in $X(p^e;N)(K^s)$,
and the action of $\Gal(K^s/K)$ moves this point within the $H''$-orbit,
since $H \subseteq H''$,
so the image point in $X_{H''}(K^s)$ is $K$-rational.
This point in $X_{H''}(K)$ may be viewed as a rational map
$C \dashrightarrow X_{H''}$,
and this map is non-constant since the composition
$C \dashrightarrow X_{H''} \stackrel{j}\to X(1) \simeq {\mathbb P}^1$
corresponds to $j(E) \in K - k$.
Proposition~\ref{P:facts}(vii) implies
$\gamma_k(X_{H''}) \le \gamma_k(C) \le d$,
contradicting the previous paragraph.
\end{proof}
\section*{Acknowledgements}
Most of all I thank Andreas Schweizer for several discussions,
for suggesting several references, and for pointing out that existing
versions of the Castelnuovo-Severi inequality seem to require a
perfect field of constants.
I also thank Matt Baker, from whom I first learned
the idea in \cite{Harris-Silverman1991}
that the Castelnuovo-Severi inequality
could be used to bound change in gonality under algebraic extensions.
I thank Brian Osserman for pointing out that the lower bound
on the geometric gonality of the general curve of genus $g$
in characteristic $p$ could be deduced from \cite{Osserman2005}.
I thank Doug Ulmer for a remark about cusps of Igusa curves.
Finally I thank Alina Cojocaru, Chris Hall, Dinesh Thakur, and Doug Ulmer
for asking questions that inspired this paper.
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\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,828 |
Heat ghee in a kadai. Add cashew nuts, then raisins and fry them til golden brown.
In a mortar pestle, add cloves and cardamom.
Crush them into a powder. Keep it aside.
Now, add water little by little to form a free flowing batter.
To check the consistency of the batter : Fry few balls of the batter with a whisk or spoon. If it has tail ends then the consistency is thick, little more water need to be added. If it is flat then the batter is thin, add little gram flour and mix till well to avoid lumps. The gram flour and some thin batter can be mixed in a bowl to avoid lumps. Repeat the checking process till we get perfect round balls.
Heat oil in a kadai.
Take a ladle full of batter and spread it over a perforated ladle(ladle with holes) in a circular direction as how we do for dosa.
Remove them from oil once cooked. It should not be crispy. Drain the excess oil in a tissue paper.
Wipe off the excess batter in the ladle or wash it and wipe with a dry cloth after each use.
Repeat the process to finish the entire batter.
In a broad vessel, add sugar and water (water level should be just above the sugar).
Heat it up to get a single string consistency.
Once the sugar syrup is warm, add the boondis, fried cashew nuts, raisins, edible camphor, cardamom powder, sugar candy and clove powder.
Mix it well with ladle or hands if the heat is bearable.
Pinch out around 1 tbsp of boondis, shape them into ladoo. The sugar syrup and oil from the boondi will ooze out while making ladoos. Once cooled it will become dry.
Homemade sweet boodi ladoo is ready to taste.
The sugar syrup should be warm while putting the boondis in that.
Make ladoos when the boondi mixture is lukewarm or bearable heat, else the sugar will be crystallised and ladoos can not be formed.
The ladoo ,if handled properly can be stored for 7 days in the room temperature.
Do not touch ladoos with wet hands.
Store the ladoos in an airtight container.
Do not use water ,if you don't get balls.this will make a fungal layer on the ladoo.
If for any reason you didn't able to make balls, just make 2 tbsp of sugar syrup and pour it over the boondis and make ladoos. | {
"redpajama_set_name": "RedPajamaC4"
} | 169 |
{"url":"https:\/\/meangreenmath.com\/category\/precalculus\/","text":"# Left-Hand Rule?\n\nMisleading pictures in math textbooks always send 10,000 volts of electricity down my spine. Thanks to the right-hand rule, the cross product should be pointing down, not up. This comes from the 2007 edition of Glencoe\u2019s \u201cAdvanced Mathematical Topics,\u201d a high-school Precalculus book.\n\nFor what it\u2019s worth, this is the same line of textbooks that, in a supplementary publication, said that the rational numbers are not countable.\n\n# Once upon a time in algebra\u00a0class\u2026\n\nSide note: Yes, there\u2019s only one true exponential curve on the graph. Yes, the spread of COVID-19 is best modeled with a logistic growth curve or an SEIR model. Nevertheless, this comic absolutely rings true.\n\n# Slightly Incorrect Ugly Mathematical Christmas T-Shirts:\u00a0Index\n\nI\u2019m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on slightly incorrect ugly mathematical Christmas T-shirts.\n\nPart 1: Missing digits in the expansion of $\\pi$.\n\nPart 2: Incorrect computation of Pascal\u2019s triangle.\n\nPart 3: Incorrect name of Pascal\u2019s triangle.\n\nI recently read the delightful article \u201cThe IRS Uses Geometric Series?\u201d by Michelle Ghrist in the August\/September 2019 issue of MAA FOCUS. The article concerns a church raffle for a $4000 ATV in which the church would pay for the tax bill of the winner. This turned out to be an unexpected real-world application of an infinite geometric series. A few key quotes: According to the IRS rules at the time, \u2026winnings below a certain level [were] subject to a 25% regular gambling withholding tax\u2026 My initial thought was that the church would need to pay $0.25 \\times \\4000 = \\1000$ to the IRS. However, I then wondered if this extra $\\1000$ payment would then be considered part of the prize and therefore also subject to 25% withholding, requiring the church to give $0.25 \\times \\1000 = \\250$ more to the IRS. But then this $\\250$ would also be part of the prize and subject to withholding, with this process continuing forever. I got quite excited about the possibility of an infinite geometric series being necessary to implement IRS tax code. By my calculations\u2026 [gave] an effective tax rate of 33-1\/3%. I then read more of the instructions, which clarified if the payer pays the withholding tax rate for the payee, \u201cthe withholding is 33.33% of the FMV [Fair Market Value] of the noncash payment minus the amount of the wager.\u201d It was satisfying to discover the behind-the-scenes math leading to that number\u2026 In any event, I am glad to know that the IRS can properly apply geometric series.\u201d Here\u2019s a link to the whole article: http:\/\/digitaleditions.walsworthprintgroup.com\/publication\/?m=7656&l=1#{%22issue_id%22:606088,%22page%22:%2214%22} Note: The authors notes that, in January 2018, the IRS dropped the two above rates to 24% and 31.58%. # Decimal Approximations of Logarithms: Index I\u2019m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series on the decimal expansions of logarithms. Part 1: Pedagogical motivation: how can students develop a better understanding for the apparently random jumble of digits in irrational logarithms? Part 2: Idea: use large powers. Part 3: Further idea: use very large powers. Part 4: Connect to continued fractions and convergents. Part 5: Tips for students to find these very large powers. # Another Poorly Written Word Problem: Index I\u2019m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series poorly written word problem, taken directly from textbooks and other materials from textbook publishers. Part 1: Addition and estimation. Part 2: Estimation and rounding. Part 3: Probability. Part 4: Subtraction and estimation. Part 5: Algebra and inequality. Part 6: Domain and range of a function. Part 7: Algebra and inequality. Part 8: Algebra and inequality. Part 9: Geometric series. Part 10: Currently infeasible track and field problem. Part 11: Another currently infeasible track and field problem. # Engaging students: Geometric sequences In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place. I plan to share some of the best of these ideas on this blog (after asking my students\u2019 permission, of course). This student submission comes from my former student Victor Acevedo. His topic, from Precalculus: geometric sequences. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)? 2048 is a fun game on mobile phones and online that can help introduce the concept of geometric sequences to students. The game is based on the powers of 2 and trying to reach 2^11 (or 2048). Each time two matching tiles are combined it creates the next power of 2. At first glance, it may seem that you are just adding the two tiles, so it doesn\u2019t look like a geometric sequence. The geometric sequence shows up when you look at the terms in the sequence being each new tile that is introduced. For example, the 8 tile comes from two 4 tiles, and each 4 tile comes from two 2 tiles, but the 8 tile is still the third new tile making it the third term in the sequence. There can be a discussion about how many tiles are needed to create the first several terms in the sequence up until 2048. https:\/\/play2048.co\/ What interesting (i.e., uncontrived) word problems using this topic can your students do now? A fun problem that involves geometric sequences is the doubling penny problem. You are asked to decide whether you would rather have lump sum of$1,000,000 given to you upfront or take an offer that involves doubling pennies for the next 30 days. The second offer would involve you taking a single penny on the first day, then doubling that amount each day until the 30th day. At first it seems like a reasonable choice to take the lump sum of $1,000,000, but you have to remember that we are dealing with and exponential or geometric growth in the second offer. On the 30th day you would receive 2^30 pennies which would be$107,374,182.40. That number doesn\u2019t even include the sum of all the other days you were receiving pennies. This would be a great way to explore that difference between linear (or arithmetic) and exponential (or geometric) growth.\n\nHow have different cultures throughout time used this topic in their society?\n\nThe paradox of\u00a0Achilles\u00a0and the\u00a0tortoise is an example where geometric sequences are applied with philosophical thought. Achilles is racing a tortoise. \u00a0Achilles gives the tortoise a lead because he believes that he is much faster than the tortoise. The paradox arises from the fact that Achilles will have to try and close the gap between him and tortoise while the tortoise keeps moving forward. By having to always get to where the tortoise has been, Achilles can\u2019t catch up. A simplified way of seeing this is by imagining the tortoise already being at the finish line and Achilles just having to close the gap in between him and the tortoise. He does so in a way that cuts the distance between him and the tortoise in half every minute. By doing so, Achilles will never actually catch up since there is always more distance to travel. In this case the common ratio for the geometric sequence would \u00bd and the end goal would be 0 but it could never be attained.\n\n# Engaging students: Introducing the number\u00a0e\n\nIn my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.\n\nI plan to share some of the best of these ideas on this blog (after asking my students\u2019 permission, of course).\n\nThis student submission comes from my former student Julie Thompson. Her topic, from Precalculus: introducing the number $e$.\n\nWhat interesting word problems using this topic can your students do now?\n\nI found a very interesting word problem involving the number e and derangements. A derangement is a permutation of a set in which no element is in its original place. The word problem I found is as follows:\n\n\u201cAt the bohemian jazz parties frequented by aficionados of the number e, the espresso flows freely, and at the end of the evening, party-goers are just as likely to go home in someone else\u2019s overcoat as they are in their own. After such a party, what are the chances that\u00a0at least one\u00a0person goes home wearing the right coat?\u201d\n\nTo start off, we need to find out how many permutations, or how many combinations of ways the coats can be put on at random when guests leave the party. The problem asks us to identify the chance that at least one person IS wearing the right coat. In other words, we need to delete all the combinations in which nobody grabbed the correct coat. These are the derangements. Interestingly, when you divide the number of permutations by the number of derangements, you get a number extremely close to the value of e. And the ratio is always so.\n\nLooking at a numerical example with 10 guests, the number of ways 10 people can pick up 10 coats (permutations) is 3628800, and the number of ways nobody would pick up the right coat is 1334961. Dividing, 3628800\/1334961= 2.71828, which is extremely close to e. Therefore, the chance of nobody getting the right coat is about 1 in e. How interesting. I feel like this word problem would really interest students!\n\nThe number e was not discovered as \u2018naturally\u2019 as you may think. Mathematicians came close to discovering e in their calculations many times in the 17th century but thought it was just a random number without any real significance. The first person to calculate e is not documented, but historians believe it to not even be a mathematician, but a banker or trader. Why is this?\n\nThe number e is very fundamental to a financial process that took off in the 17th century. \u201cThe number e\u00a0lies at the foundations of one of the most fundamental processes of finance: compound interest.\u201d Mathematicians, including Jacob Bernoulli, would later go on to define:\n\n. $e = \\displaystyle \\lim_{x \\to \\infty} \\left(1 + \\frac{1}{x} \\right)^x$\n\n\u201cThis is why the number\u00a0e\u00a0appears so often in modeling the exponential growth or decay of everything from bacteria to radioactivity.\u201d This fact was adopted by the mathematical community and many mathematicians started collaborating and making many more discoveries on the number e, such as Euler, who estimated e correctly to 18 decimal places, gave the continued fraction expansion of e, and made a connection between e and the sine and cosine functions. The number e is one of the most beautiful and powerful number in all of mathematics and the use of it was adopted into mathematics most likely by a banker\u2026how interesting.\n\nHow can technology be used to effectively engage students with this topic?\n\nAny graphing technology, such as a TI calculator, Mathematica, MatLab, Desmos, etc., are great tools to use in order to engage students when discovering the number e. For instance, to convince students that the above limit is true,\n\n$e = \\displaystyle \\lim_{x \\to \\infty} \\left(1 + \\frac{1}{x} \\right)^x$,\n\nI can have them graph the function for themselves and actually see that the function approaches the number e as x gets very large. Similarly, I can simulate numbers of e on a computer program with the expansion\u00a0 1 + 1\/1!\u00a0+\u00a01\/2!\u00a0+\u00a01\/3!\u00a0+ \u2026 to show the sum getting closer and closer to the value of e the more terms I add. I believe this will be really engaging because the expansion for the number e and the limit for e look like they have nothing to do with e at first glance. To make the connection between them graphically would be somewhat magical to students and hopefully make them curious for more.\n\nReferences:\n\nhttp:\/\/wmueller.com\/precalculus\/e\/e6.html (this is word problem from A1)\n\nhttps:\/\/brilliant.org\/wiki\/the-discovery-of-the-number-e\/\n\nhttp:\/\/mathworld.wolfram.com\/e.html\n\nhttp:\/\/www-history.mcs.st-and.ac.uk\/HistTopics\/e.html\n\n# Engaging students: Compound\u00a0interest\n\nIn my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.\n\nI plan to share some of the best of these ideas on this blog (after asking my students\u2019 permission, of course).\n\nThis student submission comes from my former student J. R. Calvillo. His topic, from Precalculus: compound interest.\n\nThe mathematical community adopted the concept of compound interest very well. Albert Einstein was one of if not the biggest mathematician who adopted this policy of compound interest very well. His most notable quote on the topic is, \u201cCompound interest is the eighth wonder of the world. He who understand it, earns it\u2026 He who doesn\u2019t\u2026 pays it.\u201d (Albert Einstein) Compound interest has expanded even from the mathematical community and spread to banks, and how they decide to give out interest on deposits or loans. The concept of compound interest has truly been one of the most well received concepts in the mathematical community, so much so that it even spread outside of that community into the business world. Where it then changed how businesses and banks looked at interest.\n\nHow could you as a teacher create an activity or project that involves your topic?\n\nCompound interest is a very important and very relatable topic for teachers to be able to relate real world examples to.\u00a0 With that, I believe it is very important to make compound interest relatable to the real world uses that students\u2019 will one day see when they get older. To begin the activity, each student will receive $2,000. That$2,000 will be put in the bank and the bank has agreed to add interest. The bank decided to give them the option on how they want that interest compounded; daily, monthly, quarterly or yearly. At the end we will group together the students\u2019 who wanted to compound their interest similarly. Each group will get to explain why they chose how often it will be compounded, then will get the opportunity to solve how much it will grow after 2 years, 4 years and 20 years. This will then allow the students to see the differences and similarities between the different options that the bank provided, and which option will earn you the most money.\n\nHow can this topic be used in your students\u2019 future courses in mathematics or science?\n\nCompound interest is a topic that will originally get introduced in a pre-calculus class, however, if any students\u2019 go onto take classes such as statistics or any other business related math, it will contain material on compound interest. It is used as such a big role in the business world that getting a true understanding how it works and the reasoning behind why it works is crucial from the earliest class that we see it in.\u00a0 In later classes it can be touched on more so, especially reaching the ways that it is beneficial to use it or the ways that it may hurt to use it. Either way it is a concept that will come up again whether you see it in the classroom, or in real life. Compound interest is one of if not the most relatable topic to the outside world with all of its applications to loans and how it is used in banks. Getting the fundamental concepts early is a crucial aspect to understanding its deeper usage in other courses.\n\nResources:\n\n# Engaging students: Graphing Sine and Cosine\u00a0Functions\n\nIn my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.\n\nI plan to share some of the best of these ideas on this blog (after asking my students\u2019 permission, of course).\n\nThis student submission comes from my former student Christian Oropeza. His topic, from Precalculus: graphing sine and cosine functions.\n\nHow could you as a teacher create an activity or project that involves your topic?\n\nAn activity for students to understand how to graph sine and cosine could consist the use of website desmos (Reference 1). In this activity students will be in pairs and must complete a worksheet that list different forms of the equation sine and cosine that illustrate some of the transformations for sine and cosine. One student will enter the equation onto desmos and the other student will draw the graph on a separate worksheet. The pair will switch roles after each equation, so both students understand how to interpret and draw a given sine or cosine equation. After all the equations have been graphed and sketched, each pair will move on to the next part of the activity in which they must manipulate the equation asin(bx+c)+d and acos(bx+c)+d on desmos where a,b,c, and d are all\u00a0 numerical sliders that can be adjusted to help students visually interpret what transformation they represent. Finally, to prove that students understood the material, each pair will come up with a sketch of a transformation of sine or cosine and trade with another pair of students, in which they must figure out the corresponding equation that matches the given graph.\n\nHow can this topic be used in your students\u2019 future courses in mathematics or science?\n\nThis topic comes up any subject that has sinusoidal waves, such as physics, calculus, and some engineering classes. For example, in calculus graphing the derivative of sine gives the graph of cosine. This shows students that the slope at any point on the sine curve is the cosine and the slope of any point on the cosine curve is the negative of the sine. The topic of sine and cosine is a crucial component \u00a0in electrical engineering (EE). For EE, there\u2019s a class called, circuit analysis that has a section named \u201cEuler\u2019s Sine Wave\u201d and \u201cEuler\u2019s Cosine Wave\u201d, which incorporates the use of Euler\u2019s formula (Reference 2). Also, in electrical engineering, there\u2019s a machine called a \u201csignal generator\u201d, which sends different types of signals as inputs to circuit. This machine can alter the frequency and amplitude of the signal, where amplitude represents the amount of voltage inputted into the circuit. In math, there\u2019s a topic called \u201cFourier Series\u201d that also incorporates sine and cosine (Reference 3).\n\nHow can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It\u2019s not enough to say \u201csuch-and-such is a great website\u201d; you need to explain in some detail why it\u2019s a great website.\n\nDesmos (Reference 1), can be used to show students the different transformations of both functions. This way students can visually understand what each component is and how each component affects the functions y=asin(bx+c)+d and y=acos(bx+c)+d, where a is the amplitude, b is the period, c is the phase shift, and d is the vertical shift. Vision learning (Reference 4), is also a great website for students when they are introduced to the topic of sine and cosine. This website goes over the history of sine by relating it to waves and circles. The website first goes over how Hipparchus calculated the trigonometric ratios and how that led to the sine function. This website gives students a background on how the functions sine and cosine came to be over time. Also, this website talks about how when the Unit Circle is placed on a Cartesian graph, this illustrates how sine and cosine take over a periodic trend, so students can see why the graphs of sine and cosine are infinite if the domain is all real numbers.\n\nReferences:","date":"2020-05-31 22:44:44","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\": 8, \"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\": 0.5073146820068359, \"perplexity\": 660.6994617793484}, \"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-2020-24\/segments\/1590347413786.46\/warc\/CC-MAIN-20200531213917-20200601003917-00525.warc.gz\"}"} | null | null |
title: Componente React para Autocompletar
components: TextField, Popper, Autocomplete
githubLabel: 'component: Autocomplete'
waiAria: 'https://www.w3.org/TR/wai-aria-practices/#combobox'
---
# Autocompletar
<p class="description">O autocompletar é uma entrada de texto normal aprimorada por um painel de opções sugeridas.</p>
Essa ferramenta é útil para configurar os valores de um campo de texto quando em um dos dois cenários abaixo:
1. O valor para a caixa de texto deve ser escolhido a partir de um conjunto pré-definido de valores permitidos, por exemplo, um campo de localização deve conter um nome de localização válido: [caixa de combinação](#combo-box).
2. A caixa de texto pode conter qualquer valor arbitrário, mas é mais vantajosa, porque pode sugerir possíveis valores para o usuário, por exemplo, um campo de pesquisa que pode sugerir pesquisas anteriores ou semelhantes para economizar o tempo do usuário: [free solo](#free-solo).
A ideia dessa ferramenta é ser uma versão melhorada das bibliotecas "react-select" e "downshift".
{{"component": "modules/components/ComponentLinkHeader.js"}}
## Caixa de combinação
O valor deve ser escolhido a partir de um conjunto predefinido de valores permitidos.
{{"demo": "pages/components/autocomplete/ComboBox.js"}}
### Estrutura das opções
Por padrão, o componente aceita as seguintes estruturas de opções:
```ts
interface AutocompleteOption {
label: string;
}
// ou
type AutocompleteOption = string;
```
por exemplo:
```js
const options = [
{ label: 'The Godfather', id: 1 },
{ label: 'Pulp Fiction', id: 2 },
];
// or
const options = ['The Godfather', 'Pulp Fiction'];
```
No entanto, você pode usar estruturas diferentes fornecendo um prop `getOptionLabel`.
### Área de exemplos
Each of the following examples demonstrates one feature of the Autocomplete component.
{{"demo": "pages/components/autocomplete/Playground.js"}}
### Seleção de países
Escolha um dos 248 países.
{{"demo": "pages/components/autocomplete/CountrySelect.js"}}
### Controlled states
O componente tem dois estados que podem ser controlados:
1. o estado "value" com a combinação das propriedades `value`/`onChange`. Esse estado representa o valor selecionado pelo usuário, por exemplo, quando pressionando <kbd class="key">Enter</kbd>.
2. o estado "input value" com a combinação das propriedades `inputValue`/`onInputChange`. Esse estado representa o valor exibido na caixa de texto.
> ⚠️ These two states are isolated, they should be controlled independently.
{{"demo": "pages/components/autocomplete/ControllableStates.js"}}
## Free solo
Coloque `freeSolo` como true para que o campo de texto contenha qualquer valor aleatório.
### Campo search
A propriedade é projetada para cobrir o principal caso de uso de uma **caixa de pesquisa** com sugestões, por exemplo, pesquisa do Google ou react-autowhatever.
{{"demo": "pages/components/autocomplete/FreeSolo.js"}}
### Creatable
Se você pretende usar este modo para uma [caixa de combinação](#combo-box), por experiência (uma versão aprimorada de um elemento select) recomendamos a configuração:
- `selectOnFocus` para ajudar o usuário a limpar o valor selecionado.
- `clearOnBlur` para ajudar o usuário a digitar um novo valor.
- `handleHomeEndKeys` para mover o foco dentro do popup com as teclas <kbd class="key">Home</kbd> e <kbd class="key">End</kbd>.
- Adicione uma última opção para indicar a possibilidade de adição, por exemplo `Adicionar "SUA PESQUISA"`.
{{"demo": "pages/components/autocomplete/FreeSoloCreateOption.js"}}
Você pode também exibir um diálogo quando o usuário quiser adicionar um novo valor.
{{"demo": "pages/components/autocomplete/FreeSoloCreateOptionDialog.js"}}
## Agrupamento
Você pode agrupar as opções com o prop `groupBy`. Se você fizer isso, certifique-se de que as opções também estejam classificadas com a mesma dimensão que serão agrupadas, caso contrário, você notará cabeçalhos duplicados.
{{"demo": "pages/components/autocomplete/Grouped.js"}}
## Opções desabilitadas
{{"demo": "pages/components/autocomplete/DisabledOptions.js"}}
## `useAutocomplete`
For advanced customization use cases, a headless `useAutocomplete()` hook is exposed. Ele aceita quase as mesmas opções do componente autocompletar exceto todas as propriedades relacionadas a renderização do JSX. The Autocomplete component is built on this hook.
```jsx
import useAutocomplete from '@material-ui/core/useAutocomplete';
```
- 📦 [4.5 kB gzipado](/size-snapshot).
{{"demo": "pages/components/autocomplete/UseAutocomplete.js", "defaultCodeOpen": false}}
### Hook customizado
{{"demo": "pages/components/autocomplete/CustomizedHook.js"}}
Vá para a seção de [customização](#customization) para um exemplo com o componente `Autocomplete` em vez do hook.
## Requisições assíncronas
O componente suporta duas situações de uso assíncronas diferentes:
- [Carregar ao abrir](#load-on-open): espera uma interação com o componente para carregar as opções.
- [Pesquisar enquanto digita](#search-as-you-type): um novo pedido é feito para cada tecla pressionada.
### Carregar ao abrir
Exibe um estado de progresso enquanto a solicitação de rede estiver pendente.
{{"demo": "pages/components/autocomplete/Asynchronous.js"}}
### Pesquisar enquanto digita
Se sua lógica é buscar novas opções a cada tecla pressionada e usando o valor atual da caixa de texto para filtrar no servidor, você pode querer considerar a limitação de requisições.
Além disso, você precisará desabilitar a filtragem integrada do componente `Autocomplete` sobrescrevendo o prop `filterOptions`:
```jsx
<Autocomplete filterOptions={(x) => x} />
```
### Lugares com a API do Google Maps
Uma customização de UI para o autocompletar de lugares do Google Maps.
{{"demo": "pages/components/autocomplete/GoogleMaps.js"}}
Para esse exemplo, nós precisamos carregar a API de Javascript do [Google Maps](https://developers.google.com/maps/documentation/javascript/tutorial).
> ⚠️ Antes de você começar a usar a API JavaScript do Google Maps você precisará estar cadastrado e ter uma conta.
## Múltiplos valores
Também conhecidos como tags, o usuário pode inserir mais de um valor.
{{"demo": "pages/components/autocomplete/Tags.js"}}
### Opções fixas
Em ocasiões que você necessite travar certa tag para que não possa ser removida da interface, você pode defini-la como desabilitada.
{{"demo": "pages/components/autocomplete/FixedTags.js"}}
### Caixas de seleção
{{"demo": "pages/components/autocomplete/CheckboxesTags.js"}}
### Limitar tags
Você pode usar a propriedade `limitTags` para limitrar o número de opções exibidas quando o componente não estiver com o foco.
{{"demo": "pages/components/autocomplete/LimitTags.js"}}
## Tamanhos
Gosta mais de campos de texto menores? Use a propriedade `size`.
{{"demo": "pages/components/autocomplete/Sizes.js"}}
## Customização
### Input customizado
A propriedade `renderInput` permite que você customize o input renderizado. O primeiro argumento desta propriedade de render, contém propriedades que você precisa encaminhar. Preste atenção específicamente nas chaves `ref` e `inputProps`.
{{"demo": "pages/components/autocomplete/CustomInputAutocomplete.js"}}
### Seletor do GitHub
Esta demonstração reproduz o rótulo de seleção do GitHub's:
{{"demo": "pages/components/autocomplete/GitHubLabel.js"}}
Va para a seção [Hook customizado](#customized-hook) para um exemplo com o uso do hook customizado `useAutocomplete` ao invés do componente.
## Realce
A demonstração a seguir dependem do [autosuggest-highlight](https://github.com/moroshko/autosuggest-highlight), um utilitário pequeno (1 kB) para realçar textos nos componentes autosuggest e autocomplete.
{{"demo": "pages/components/autocomplete/Highlights.js"}}
## Filtro customizado
O componente expõe uma fábrica para criar um método de filtro que pode ser fornecido para a propriedade `filterOptions`. Você pode usar ela para modificar o comportamento padrão do filtro.
```js
import { createFilterOptions } from '@material-ui/core/Autocomplete';
```
### `createFilterOptions(config) => filterOptions`
#### Argumentos
1. `config` (_object_ [optional]):
- `config.ignoreAccents` (_bool_ [optional]): Defaults to `true`. Remover sinais diacríticos.
- `config.ignoreCase` (_bool_ [optional]): Defaults to `true`. Minúsculas em tudo.
- `config.limit` (*number* [opcional]): Padrão null. Limitar o número de opções sugeridas a serem exibidas. Por exemplo, se `config.limit` é `100`, somente as primeiras `100` opções correspondentes são exibidas. Isto pode ser útil se um monte corresponderem e a virtualização não estiver configurada.
- `config.matchFrom` (_'any' | 'start'_ [opcional]): Padrão `'any'`.
- `config.stringify` (*func* [opcional]): Controla a forma como a opção é convertida em texto, dessa forma pode ser comparada com qualquer fragmento de texto.
- `config.trim` (_bool_ [optional]): Defaults to `false`. Remover espaços ao fim.
#### Retornos
`filterOptions`: o método de filtro retornado pode ser fornecido diretamente para a propriedade `filterOptions` do componente `Autocomplete` ou para o parâmetro de mesmo nome no hook.
Na demonstração a seguir, as opções necessárias para o filtro ser aplicado no inicio das opções:
```jsx
const filterOptions = createFilterOptions({
matchFrom: 'start',
stringify: (option) => option.title,
});
<Autocomplete filterOptions={filterOptions} />;
```
{{"demo": "pages/components/autocomplete/Filter.js", "defaultCodeOpen": false}}
### Avançado
Para mecanismos de filtragem mais ricos, como correspondência difusa, recomenda-se explorar o [match-sorter](https://github.com/kentcdodds/match-sorter). Por exemplo:
```jsx
import matchSorter from 'match-sorter';
const filterOptions = (options, { inputValue }) => matchSorter(options, inputValue);
<Autocomplete filterOptions={filterOptions} />;
```
## Virtualização
Pesquise dentro de 10.000 opções geradas aleatoriamente. A lista é virtualizada graças a [react-window](https://github.com/bvaughn/react-window).
{{"demo": "pages/components/autocomplete/Virtualize.js"}}
## Eventos
Se você deseja evitar o comportamento padrão do teclado, você pode definir a propriedade do evento `defaultMuiPrevented` para `true`:
```jsx
<Autocomplete
onKeyDown={(event) => {
if (event.key === 'Enter') {
// Previne o comportamento padrão do 'Enter'.
event.defaultMuiPrevented = true;
// your handler code
}
}}
/>
```
## Limitações
### autocomplete/autofill
Browsers have heuristics to help the user fill in form inputs. However, this can harm the UX of the component.
By default, the component disables the input **autocomplete** feature (remembering what the user has typed for a given field in a previous session) with the `autoComplete="off"` attribute. Atualmente, o Google Chrome não suporta essa configuração de atributo ([Issue 587466](https://bugs.chromium.org/p/chromium/issues/detail?id=587466)). Uma solução alternativa possível é remover o `id` para que o componente gere um aleatório.
No entanto, além de relembrar valores fornecidos anteriormente, o navegador também pode propor sugestões de **autofill** (preenchimento automático para informações de login, endereço ou detalhes de pagamento). No caso de você querer evitar o recurso de preenchimento automático, tente o seguinte:
- Nomeie o campo sem fornecer informações para o navegador do que ele representa. `id="field1"` ao invés de `id="country"`. Se você deixar o id do vazio, o componente utiliza um id aleatório.
- Defina `autoComplete="new-password"` (alguns navegadores irão sugerir uma senha forte para entradas com esta configuração de atributo):
```jsx
<TextField
{...params}
inputProps={{
...params.inputProps,
autoComplete: 'new-password',
}}
/>
```
Leia [este guia na MDN](https://developer.mozilla.org/en-US/docs/Web/Security/Securing_your_site/Turning_off_form_autocompletion) para mais detalhes.
### iOS VoiceOver
VoiceOver no Safari do iOS não suporta o atributo `aria-owns` muito bem. Você pode contornar o problema com a propriedade `disablePortal`.
### ListboxComponent
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## Acessibilidade
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Incentivamos a utilização de um rótulo para a caixa de texto. O componente implementa as práticas de autoria da WAI-ARIA.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,264 |
package com.zavtech.morpheus.index;
/**
* A factory class that manufactures Index instances.
*
* <p>This is open source software released under the <a href="http://www.apache.org/licenses/LICENSE-2.0">Apache 2.0 License</a></p>
*
* @author Xavier Witdouck
*/
abstract class IndexFactory {
private static IndexFactory instance;
/**
* Returns the singleton instance of this class
* @return the singleton index factory
*/
public static synchronized IndexFactory getInstance() {
if (instance == null) {
instance = new IndexFactoryDefault();
}
return instance;
}
/**
* Returns a newly created index from the iterable set of keys provided
* @param keys the array to create index from
* @param <K> the index element type
* @return the newly created Index.
*/
public abstract <K> Index<K> create(Iterable<K> keys);
/**
* Returns a newly created index for the type and with initial size
* @param keyType the array type for index
* @param initialSize the initial size of index
* @param <K> the index element type
* @return the newly created Index.
*/
public abstract <K> Index<K> create(Class<K> keyType, int initialSize);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,148 |
\section{Introduction\label{sintro}}
In the present paper we aim to study some particular shape optimization problems in classes of planar domains having a prescribed topology. The quantities we are going to consider for a general bounded open set $\Omega $ are the distributional perimeter $P(\Omega)$ and the torsional rigidity $T(\Omega)$. More precisely, we deal with a scaling free functional $F_q$ which is expressed as the product of the perimeter, and of a suitable powers of the torsional rigidity and of the Lebesgue measure of $\Omega$, depending on a positive parameter $q$.
The restriction to the planar case is essential and is not made here for the sake of simplicity; indeed, in higher dimension stronger topological constraints have to be imposed to make the problems well posed.
In a previous paper \cite{BBP20} we treated the problem above in every space dimension and, after discussing it for general open sets, we focused to the class of convex open sets.
In the following we consider the optimization problems for $F_q$ in the classes $\mathcal{A}_k$ of planar domains having at most $k$ ``holes".
While the maximization problems are always ill posed, even in the class of smooth open sets in $\mathcal{A}_k$, it turns out that the minimizing problems are interesting if $q\le 1/2$ and some regularity constraints are imposed to the sets $\Omega\in\mathcal{A}_k$.
In this case, we provide a explicit lower bound for $F_q$ in the class of Lipschitz sets in $\mathcal{A}_k$, which turns out to be sharp when $k=0,1$ and $q=1/2$ and coincides with the infimum of $F_q$ in the class of convex sets, as pointed out by Polya in \cite{polya60}.
When $q<1/2$ we study the existence of minimizers for $F_q$ and our approach is the one of direct methods of the calculus of variations which consists in the following steps:
\begin{itemize}
\item[-]defining the functional $F_q$ only for Lipschitz domains of the class $\mathcal{A}_k$;
\item[-]relaxing the functional $F_q$ on the whole class $\mathcal{A}_k$, with respect to a suitable topology;
\item[-]showing that the relaxed functional admits an optimal domain in $\mathcal{A}_k$;
\item[-]proving that such a domain is Lipschitz.
\end{itemize}
The relaxation procedure above is necessary to avoid trivial counterexamples due to the fact that the perimeter is Lebesgue measure sensitive, while the torsional rigidity is capacity sensitive.
As in most of the free boundary problems, the last regularity step presents strong difficulties and, even if the regularity of optimal domains could be expected, we are unable to give a full proof of this fact. It would be very interesting to establish if an optimal domain fulfills some kind of regularity, or at least if its perimeter coincides with the Hausdorff measure of the boundary, which amounts to exclude the presence of internal fractures.
This paper is organized as follows.
In Section \ref{spre}, after recalling the definitions of perimeter and torsional rigidity, we summarize the main results of this paper. In Section \ref{sapp} we describe the key tools necessary to apply the so-called method of \textit{interior parallels}, introduced by Makai in \cite{Ma},\cite{Ma59} and by Polya in \cite{polya60}, to our setting. Section \ref{shau} contains a review of some basic facts concerning the complementary Hausdorff convergence, with respect to which we perform the relaxation procedure. Although Sections \ref{sapp} and \ref{shau} may be seen as preliminary, we believe they contain some interesting results that, as far as we know, are new in literature. Finally, in Section \ref{sexis} we discuss the optimization problem: we extend a well known inequality due to Polya (Theorem \ref{theo.Polya} and Remark \ref{rem.polya}), and we prove the main results (Corollary \ref{coro.polya} and Theorem \ref{theo.exis}).
\section{Preliminaries}\label{spre}
The shape functionals we consider in this paper are of the form
\begin{equation}\label{Fq}F_q(\Omega)=\frac{P(\Omega)T^q(\Omega)}{|\Omega|^{2q+1/2}}
\end{equation}
where $q>0$, $\Omega\subset\mathbb{R}^2$ is a general bounded open set and, $|\Omega|$ denotes its Lebesgue measure.
For the reader's convenience, in the following we report the definitions and the basic properties of the perimeter and of the torsional rigidity. According to the De Giorgi formula, the perimeter is given by
$$P(\Omega)=\sup\left\{\int_\Omega\dive\phi\,dx\ :\ \phi\in C^1_c(\mathbb{R}^2;\mathbb{R}^2),\ \|\phi\|_{L^\infty(\mathbb{R}^2)}\le1\right\},$$
and satisfies:
\begin{itemize}
\item[-]the {\it scaling property}
$$P(t\Omega)=tP(\Omega)\qquad\text{for every }t>0;$$
\item[-] the lower semicontinuity with respect to the $L^1$-convergence, that is the convergence of characteristic functions.
\item[-]the {\it isoperimetric inequality}
\begin{equation}\label{isoper}
\frac{P(\Omega)}{|\Omega|^{1/2}}\ge\frac{P(B)}{|B|^{1/2}}
\end{equation}
where $B$ is any disc in $\mathbb{R}^2$. In addition the inequality above becomes an equality if and only if $\Omega$ is a disc (up to sets of Lebesgue measure zero).
\end{itemize}
The torsional rigidity $T(\Omega)$ is defined as
$$T(\Omega)=\int_\Omega u\,dx$$
where $u$ is the unique solution of the PDE
\begin{equation}\label{pdetorsion}\begin{cases}
-\Delta u=1&\text{in }\Omega,\\
u\in H^1_0(\Omega).
\end{cases}
\end{equation}
By means of an integration by parts we can equivalently express the torsional rigidity as
\begin{equation} \label{vartor}
T(\Omega)=\max\Big\{\Big[\int_\Omega u\,dx\Big]^2\Big[\int_\Omega|\nabla u|^2\,dx\Big]^{-1}\ :\ u\in H^1_0(\Omega)\setminus\{0\}\Big\}.
\end{equation}
The main properties we use for the torsional rigidity are:
\begin{itemize}
\item[-]the monotonicity with respect to the set inclusion
$$\Omega_1\subset\Omega_2\Longrightarrow T(\Omega_1)\le T(\Omega_2);$$
\item[-]the additivity on disjoint families of open sets
$$T\Big(\bigcup_n\Omega_n\Big)=\sum_n T(\Omega_n)\qquad\text{whenever $\Omega_n$ are pairwise disjoint;}$$
\item[-]the scaling property
$$T(t\Omega)=t^4T(\Omega),\qquad\text{for every }t>0;$$
\item[-]the relation between torsional rigidity and Lebesgue measure (known as {\it Saint-Venant inequality})
\begin{equation}\label{stven}
\frac{T(\Omega)}{|\Omega|^2}\le\frac{T(B)}{|B|^2}.
\end{equation}
In addition, the inequality above becomes an equality if and only if $\Omega$ is a disc (up to sets of capacity zero).
\end{itemize}
If we denote by $B_1$ the unitary disc of $\mathbb{R}^2$, then the solution of \eqref{pdetorsion}, with $\Omega=B_1$, is
$$u(x)=\frac{1-|x|^2}{4}$$
which provides
$$T(B_1)=\frac{\pi}{8}.$$
Thanks to the scaling properties of the perimeter and of the torsional rigidity, the functional $F_q$ defined by \eqref{Fq} is {\it scaling free} and optimizing it in a suitable class $\mathcal{A}$ is equivalent to optimizing the product $P(\Omega)T^q(\Omega)$ over $\mathcal{A}$ with the additional measure constraint $|\Omega|=m$, for a fixed $m>0$.
In a previous paper \cite{BBP20} we considered the minimum and the maximum problem for $F_q$ (in every space dimension) in the classes
\[\begin{split}
&\mathcal{A}_{all}:=\big\{\Omega\subset\mathbb{R}^d\ :\ \Omega\ne\emptyset\big\}\\
&\mathcal{A}_{convex}:=\big\{\Omega\subset\mathbb{R}^d\ :\ \Omega\ne\emptyset,\ \Omega\text{ convex}\big\}.
\end{split}\]
We summarize here below the results available in the case of dimension 2:
\begin{itemize}
\item[-]for every $q>0$
$$\inf\big\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_{all},\ \Omega\text{ smooth}\big\}=0;$$
\item[-]for every $q>0$
$$\sup\big\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_{all},\ \Omega\text{ smooth}\big\}=+\infty;$$
\item[-]for every $q>1/2$
$$\begin{cases}
\inf\big\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_{convex}\big\}=0\\
\max\big\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_{convex}\big\}\quad\text{is attained};
\end{cases}$$
\item[-]for every $q<1/2$
$$\begin{cases}
\sup\big\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_{convex}\big\}=+\infty\\
\min\big\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_{convex}\big\}\quad\text{is attained};\\
\end{cases}$$
\item[-]for $q=1/2$
$$\begin{cases}
\inf\big\{F_{1/2}(\Omega)\ :\ \Omega\in\mathcal{A}_{convex}\big\}=(1/3)^{1/2}\\
\sup\big\{F_{1/2}(\Omega)\ :\ \Omega\in\mathcal{A}_{convex}\big\}=(2/3)^{1/2},
\end{cases}$$
asymptotically attained, respectively, when $\Omega$ is a long thin rectangle and when $\Omega$ is a long thin triangle.
\end{itemize}
Here we discuss the optimization problems for $F_q$ on the classes of planar domains
$$\mathcal{A}_k:=\big\{\Omega\subset\mathbb{R}^2\ :\ \Omega\ne\emptyset,\ \Omega\text{ bounded, }\#\Omega^c\le k\big\},$$
where, for every set $E$, we denote by $\#E$ the number of bounded connected components of $E$ and $\Omega^c=\mathbb{R}^2\setminus\Omega$. In particular $\mathcal{A}_0$ denotes the class of simply connected domains (not necessarily connected).
From what seen above the only interesting cases to consider are:
$$\begin{cases}
\text{the maximum problem for $F_q$ on $\mathcal{A}_k$ when $q\ge1/2$ ;}\\
\text{the minimum problem for $F_q$ on $\mathcal{A}_k$ when $q\le1/2$.}
\end{cases}$$
We notice that the maximum problem is not well posed, since for every $q>0$ and every $k\ge0$
$$\sup\big\{F_q(\Omega)\ :\ \Omega\text{ smooth},\ \Omega\in\mathcal{A}_k\big\}=+\infty.$$
Indeed, it is enough to take as $\Omega_n$ a smooth perturbation of the unit disc $B_1$ such that
$$B_{1/2}\subset\Omega_n\subset B_2\qquad\text{and}\qquad P(\Omega_n)\to+\infty.$$
All the domains $\Omega_n$ are simply connected, so belong to $\mathcal{A}_k$ for every $k\ge0$, and
$$|\Omega_n|\le|B_2|,\qquad T(\Omega_n)\ge T(B_{1/2}),$$
where we used the monotonicity of the torsional rigidity. Therefore
$$F_q(\Omega_n)\ge\frac{P(\Omega_n)T^q(B_{1/2})}{|B_2|^{2q+1/2}}\to+\infty.$$
Moreover $$\inf\big\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_k\big\}=0,$$ as we can easily see by taking as $\Omega_n$ the unit disk of $\mathbb{R}^2$ where we remove the $n$ segments (in polar coordinates $r,\theta$)
$$S_i=\big\{\theta=2\pi i/n,\ r\in[1/n,1]\big\}\qquad i=1,\dots,n.$$
We have that all the $\Omega_n$ are simply connected, and
$$|\Omega_n|=\pi,\qquad P(\Omega_n)=2\pi,\qquad T(\Omega_n)\to0,$$
providing then $F_q(\Omega_n)\to0$.
Therefore, the problems we study in the sequel are
$$\inf\big\{F_q(\Omega)\ :\ \Omega\in \mathcal{A}_k,\text{ $\Omega$ Lipschitz}\},$$
when $q\le1/2$ and $k\in\mathbb{N}$. Denoting by $m_{q,k}$ the infimum above we summarize here below our main results.
\begin{itemize}
\item[-] For every $q\le1/2$ the values $m_{q,k}$ are decreasing with respect to $k$ and
$$\lim_{k\to\infty}m_{q,k}=0.$$
\item[-]When $k=0,1$ it holds
$$m_{1/2,0}=m_{1/2,1}=3^{-1/2}=\inf\big\{F_{1/2}(\Omega)\ :\ \Omega\text{ convex}\big\};$$
in particular, for $q=1/2$ there is no gap for $\inf F_{1/2}$ between the classes $\mathcal{A}_{convex}$, $\mathcal{A}_0$, $\mathcal{A}_1$, and the infimum is asymptotically reached by a sequence of long and thin rectangles.
\item[-]For every $q\le1/2$ and $k\in\mathbb{N}$, we have
$$m_{q,k}\ge\begin{cases}(8\pi)^{1/2-q}3^{-1/2}&\text{if }k=0,1, \\
(8\pi)^{1/2-q}(3^{1/2}k)^{-1}&\text{if }k>1.
\end{cases}$$
\item[-]For $q<1/2$, we define a relaxed functional $\mathcal{F}_{q,k}$, which coincides with $F_q$ in the class of the sets $\Omega\in\mathcal{A}_k$ satisfying $P(\Omega)=\mathcal{H}^1(\partial\Omega)$, being $\mathcal{H}^1$ the $1$-dimensional Hausdorff measure.
We also prove that $\mathcal{F}_{q,k}$ admits an optimal domain $\Omega^{\star}\in\mathcal{A}_k$ with $\mathcal{H}^{1}(\partial\Omega^\star)<\infty$.
\end{itemize}
\section{ Approximation by interior parallel sets} \label{sapp}
For a given bounded nonempty open set $\Omega$ we denote by $\rho(\Omega)$ its \textit{inradius}, defined as
$$\rho(\Omega):=\sup\big\{d(x,\partial\Omega)\ :\ x\in\Omega\big\},$$
where, as usual,
$d(x,E):=\inf\big\{d(x,y)\ :\ y\in E\big\}$.
For every $t\ge 0$, we denote by $\Omega(t)$ the \textit{interior parallel set} at distance $t$ from $\partial\Omega$, i.e.
$$\Omega(t):=\big\{x\in\Omega\ :\ d(x,\partial\Omega)>t\big\},$$
and by $A(t):=|\Omega(t)|$. Moreover we denote by $L(t)$ the length of the \textit{interior parallel}, that is the set of the points in $\Omega$ whose distance from $\partial\Omega$ is equal to $t$.
More precisely we set
$$L(t):=\mathcal{H}^1 (\{x\in\Omega\ :\ d(x,\partial\Omega)=t \}).$$
Notice that $\partial \Omega(t)\subseteq \{x\in\Omega\ :\ d(x,\partial\Omega)=t \}$.
Using coarea formula (see \cite{EvGa} Theorem 3.13) we can write the following identity:
\begin{equation}\label{eq.Evans}
A(t)=\int_t^{\rho(\Omega)}L(s)\,ds\qquad\forall t\in(0,\rho(\Omega)).
\end{equation} As a consequence it is easy to verify that for a.e. $t\in(0,\rho(\Omega))$ there exists the derivative $A'(t)$ and it coincides with $-L(t)$. The interior parallel sets $\Omega(t)$ belong to $\mathcal{A}_k$ as soon as $\Omega\in\mathcal{A}_k$, as next elementary argument shows.
\begin{lemm}\label{lemm.innerA_k} Let $\Omega\in\mathcal{A}_k$. Then $\Omega(t)\in \mathcal{A}_k$ for every $t\in [0,\rho(\Omega))$.
\end{lemm}
\begin{proof} Let $\alpha:=\#\Omega^c$ ($\le k$), and $C^1,C^2,\cdots C^\alpha$ be the (closed) bounded connected components of $\Omega^c$ and $C^0$ the unbounded one. Define
$$C^i(t):=\big\{x\in\mathbb{R}^2\ : \ d(x,C^i)\le t\big\}.$$
Since $C^i$ is connected, then $C^i(t)$ is connected and the set $\bigcup_{i=0}^\alpha C^i(t)$ has at most $\alpha+1$ connected components. Since we have
$\Omega^c(t)=\bigcup_{i=0}^\alpha C^i(t)$,
the lemma is proved.
\end{proof}
In the planar case, even without any regularity assumptions on $\partial\Omega$, the sets $\Omega(t)$ are a slightly smoothed version of $\Omega$. In particular the following result (see \cite{Fu85}), that we limit to report in the two dimensional case, proves that $\Omega(t)$ has a Lipschitz boundary for a.e. $t\in(0,\rho(\Omega))$.
\begin{theo}[Fu]\label{theo.Fu}
Let $K\subseteq \mathbb{R}^2$ be a compact set. There exists a compact set $C=C(K)\subseteq [0, 3^{-1/ 2} diam(K)]$ such that $|C|=0$ and if $t\notin C$ then the boundary of $\{x\in\mathbb{R}^2\ :\ d(x,K)>t\}$ is a Lipschitz manifold.
\end{theo}
We recall now some general facts of geometric measure theory. Let $E\subset \mathbb{R}^2$, we denote by $E^{(t)}$ the set of the points where the density of $E$ is $t\in [0,1]$, that is
$$E^{(t)}:=\{ x\in\mathbb{R}^2: \lim_{r\to 0^+} (\pi r^2)^{-1}|E\cap B_r(x)|=t\}.$$
It is well known (see \cite{AFP} Theorem 3.61) that if $E$ is a set of finite perimeter, then $P(E)=\mathcal{H}^1(E^{1/2})$ and $E$ has density either $0$ or $1/2$ or $1$ at $\mathcal{H}^1$-a.e $x\in\mathbb{R}^2$. In particular it holds
\begin{equation}\label{eq.decomp}\mathcal{H}^1(\partial E)= \mathcal{H}^1(\partial E\cap E^{(0)})+ \mathcal{H}^1(\partial E\cap E^{(1)})+ P(E),
\end{equation}
which implies
\begin{equation} \label{eq.decomp2}
P(E)+2\mathcal{H}^1(\partial E\cap E^{(1)})\le 2 \mathcal{H}^1(\partial E)-P(E).
\end{equation}
The Minkowski content and the outer Minkowski content of $E$ are, respectively, defined as
$$\mathcal{M}(E):=\lim_{t\to 0}\frac {|\{x\in\mathbb{R}^2\ :\ d(x,E)\le t\}|}{2t},$$
and
$$\mathcal{SM}(E):=\lim_{t\to 0}\frac {|\{x\in\mathbb{R}^2\ :\ d(x,E)\le t\}\setminus E|}{t},$$
whenever the limits above exist.
We say that a compact set $E\subset\mathbb{R}^2$ is $1$-rectifiable if there exists a compact set $K\subset\mathbb{R}$ and a Lipschitz map $f:\mathbb{R}\to \mathbb{R}^2$ such that $f(K)=E$. Any compact connected set of $\mathbb{R}^2$, namely a \textit{continuum}, with finite $\mathcal{H}^1$-measure is $1$-rectifiable (see, for instance, Theorem 4.4 in \cite{AO}). Finally, if $E$ is $1$-rectifiable then
\begin{equation}\label{eq.Amb}
\mathcal{M}(E)=\mathcal{H}^1(E)
\end{equation} (see Theorem $2.106$ in \cite{AFP})
and by Proposition 4.1 of \cite{V}, if $E$ is a Borel set and $\partial E$ is $1$-rectifiable it holds
\begin{equation}\label{eq.Villa}
\mathcal{SM}(E)=P(E)+2\mathcal{H}^1(\partial E\cap E^{(0)}).
\end{equation}
Next two results are easy consequence of \eqref{eq.Amb} and \eqref{eq.Villa}.
\begin{theo}\label{theo.min}
Let $\Omega$ be a bounded open set with $\mathcal{H}^1(\partial \Omega)<\infty$ and $\#\partial\Omega<+\infty$. Then $
\mathcal{M}(\partial\Omega)=\mathcal{H}^{1}(\partial \Omega)$ and $ \mathcal{SM}(\Omega)=P(\Omega)+2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(0)}).$
\end{theo}
\begin{proof}
Since $\mathcal{H}^1(\partial\Omega)<\infty$, each connected component of $\partial\Omega$ is $1$-rectifiable. Being the connected components pairwise disjoint and compact, we easily prove that their finite union is $1$-rectifiable. Then, applying \eqref{eq.Amb} and \eqref{eq.Villa}, we get the thesis.
\end{proof}
\begin{coro}\label{coro.Mink}
Let $\Omega$ be an open set such that $\mathcal{H}^1(\partial\Omega)<\infty$ and $\#\partial\Omega<+\infty$. Then there exists
$$\lim_{r\to 0^+}\frac 1 r\int_{0}^r L(t)dt= P(\Omega)+2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(1)}).$$
\end{coro}
\begin{proof}
We denote by $L^c(t)$ the following quantity
$$L^c(t):=\mathcal{H}^1(\{x\in\Omega^c\ :\ d(x,\partial\Omega)=t\}).$$
By applying coarea formula and Theorem \ref{theo.min}, it holds
\begin{equation}\label{eq.g1}
\lim_{r\to 0^+} \frac{1}{r}\int_{0}^{r}L^c(t)dt=\mathcal{SM}(\Omega)=P(\Omega)+2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(0)}).
\end{equation}
and
\begin{equation}\label{eq.min}
\lim_{r\to 0^+}\frac{1}{r}\int_{0}^{r}\left[L(t)+L^c(t)\right]dt=2\mathcal{M}(\partial\Omega)=2\mathcal{H}^1(\partial \Omega).
\end{equation}
Combining \eqref{eq.decomp}, \eqref{eq.g1} and \eqref{eq.min} we get
$$
\lim_{r\to 0^+}\frac{1}{r}\int_{0}^{r}L(t) dt=
\lim_{r\to 0^+}\left(\frac{1}{r}\int_{0}^{r}L(t) dt+\frac{1}{r}\int_{0}^{r}L^c(t) dt-\frac{1}{r}\int_{0}^{r}L^c(t) dt\right)$$
$$= 2\mathcal{H}^1(\partial \Omega)-P(\Omega)-2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(0)})= P(\Omega)+2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(1)}) $$
and the thesis is achieved.
\end{proof}
Most of the results we present rely on a geometrical theorem proved by Sz. Nagy in \cite{Nagy59}, concerning the behavior of the function $t\to A(t)=|\Omega(t)|$ for a given set $\Omega\in\mathcal{A}_k$.
\begin{theo}[Sz. Nagy]\label{theo.Na}
Let $\Omega\in\mathcal{A}_k$ and let $\alpha:=\#\Omega^c$. Then the function
$$t\mapsto-A(t)-(\alpha-1)\pi t^{2}$$
is concave in $[0,\rho(\Omega))$.
\end{theo}
As a consequence of Corollary \ref{coro.Mink} and Theorem \ref{theo.Na} we have the following result.
\begin{theo}\label{theo.Nareg}
Let $\Omega\in\mathcal{A}_k$ with $\mathcal{H}^1(\partial \Omega)<\infty$ and $\#\Omega<+\infty$. Then, for a.e. $t\in(0,\rho(\Omega))$, it holds:
\begin{align}\label{eq.boundL}
&L(t)\le P(\Omega)+2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(1)}) +2\pi(k-1)t;\\
\label{eq.boundA}
&A(t)\le (P(\Omega)+2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(1)}))(\rho(\Omega)-t)+\pi(k-1)(\rho(\Omega)-t)^2.
\end{align}
In particular $A\in W^{1,\infty}(0,\rho(\Omega))$.
\end{theo}
\begin{proof}
We denote by $g(t)$ the right derivative of the function $t\mapsto -A(t)-(\alpha-1)\pi t^2$ where $\alpha:=\#\Omega^c$ $(\le k)$. By Theorem \ref{theo.Na}, $g$ is a decreasing function in $(0,\rho(\Omega))$ and an easy computation through \eqref{eq.Evans} shows that
\begin{equation}\label{Lg}
g(t)=L(t)-2\pi(\alpha-1)t\qquad\hbox {for a.e. }t\in(0,\rho(\Omega)).
\end{equation}
Thus,
$$\lim_{r\to 0^+}\frac 1 r\int_{0}^r L(t)dt=\lim_{r\to 0^+}\frac 1 r\int_{0}^rg(t)dt=\sup_{(0,\rho(\Omega))}g(t).$$
Since $\Omega\in\mathcal{A}_k$ and $\#\Omega<\infty$ we have also $\#\partial\Omega<\infty$. Hence we can apply Corollary \ref{coro.Mink} to get
\begin{equation}\label{eq.g}
P(\Omega)+2 \mathcal{H}^1(\partial \Omega\cap \Omega^{(1)})=\sup_{(0,\rho(\Omega))}g(t).
\end{equation}
By using \eqref{Lg} and \eqref{eq.g}, inequality \eqref{eq.boundL} easily follows. Finally, by applying \eqref{eq.Evans}, we get both $A\in W^{1,\infty}(0,\rho(\Omega))$ and formula \eqref{eq.boundA}.
\end{proof}
The following lemma can be easily proved by lower semicontinuity property of the perimeter.
\begin{lemm}\label{lem.top1}
Let $\Omega\subset\mathbb{R}^2$ be an open set. Let $(\Omega^i)$ be its connected components and $\Omega_n:=\bigcup_{i=1}^n\Omega^i$. Then we have:
\begin{enumerate}
\item[(i)]$\partial\Omega_n=\bigcup_{i=1}^n\partial\Omega^i\subseteq\partial\Omega$ and $\mathcal{H}^1(\partial\Omega_n)\le \mathcal{H}^1(\partial\Omega)$;
\item [(ii)]$\displaystyle P(\Omega)\le\liminf_{n\to\infty}P(\Omega_n)\le\limsup_{n\to\infty}P(\Omega_n)\le\limsup_{n\to\infty}\mathcal{H}^1(\partial\Omega_n)\le \mathcal{H}^1(\partial\Omega)$.
\end{enumerate}
\end{lemm}
We are now in a position to prove the main results of this section. In Theorem 1.1 of \cite{Sc15} it is shown that, given any set $\Omega$ of finite perimeter satisfying $\mathcal{H}^1(\partial\Omega)=P(\Omega)$, it is possible to approximate $P(\Omega)$ with the perimeters of smooth open sets compactly contained in $\Omega$. Here we show that, if we assume the further hypothesis $\Omega\in\mathcal{A}_k$, then we can construct an approximation sequence made up of Lipschitz sets in $\mathcal{A}_k$.
\begin{theo}\label{theo.approxim}
Let $\Omega\in\mathcal{A}_k$ be a set of finite perimeter. Then there exists an increasing sequence $(A_n)\subset \mathcal{A}_k$ such that:
\begin{enumerate}
\item [(i)] $\overline A_n\subset \Omega$;
\item[(ii)] $\bigcup_{n} A_n=\Omega$;
\item[(iii)] $A_n$ is a Lipschitz set;
\item[(iv)] $\displaystyle P(\Omega)\le\liminf_{n\to\infty}P(A_n)\le\limsup_{n\to\infty}P(A_n)\le2\mathcal{H}^1(\partial\Omega)-P(\Omega)$.
\end{enumerate}
In addition, if $\# \Omega<\infty$, then
$$\lim_{n\to\infty}P(A_n)=P(\Omega)+2 \mathcal{H}^1(\partial\Omega\cap\Omega^{(1)}).$$
\end{theo}
\begin{proof}
Let $\Omega_n$ be defined as in Lemma \ref{lem.top1}. Clearly $\Omega_n\in\mathcal{A}_k$.
Since $\Omega_n(t)$ converges to $\Omega_n$ in $L^1$ when $t\to0^+$, it follows that, for every $n$,
$$\liminf_{t\to 0^+} P(\Omega_n(t))\ge P(\Omega_n).$$
Then there exists $0<\delta_n<1/n\wedge \rho(\Omega_n)$ such that
\begin{equation}\label{primacond}P(\Omega_n(t))\ge P(\Omega_n)-\frac1n\qquad\forall t<\delta_n.\end{equation}
Since $\#\Omega_n\le n$, by applying Theorem \ref{theo.Fu}, Lemma \ref{lemm.innerA_k} and Theorem \ref{theo.Nareg} to the set $\Omega_n$, we can choose a decreasing sequence $(t_n)$ with $0<t_n< \delta_n$ such that the set $A_n:=\Omega_n(t_n)$ is in $\mathcal{A}_k$, has Lipschitz boundary, and
\begin{equation}\label{secondacond}
\mathcal{H}^1(\{x\in\Omega_n\ :\ d(x,\partial\Omega_n)=t_n \})\le P(\Omega_n)+2\mathcal{H}^1(\partial\Omega_n\cap\Omega_n^{(1)})+2\pi(k-1)t_n.
\end{equation}
It is easy to prove that the sequence $(A_n)$ is increasing and satisfies (i) and (ii).
By putting together \eqref{primacond} and \eqref{secondacond}, we get
\begin{equation*}
P(\Omega_n)-\frac1n\le P(A_n)\le P(\Omega_n)+2\mathcal{H}^1(\partial\Omega_n\cap\Omega_n^{(1)})+2\pi(k-1)t_n.
\end{equation*}
By Lemma \ref{lem.top1}, taking also into account \eqref{eq.decomp2}, the previous inequality implies
$$P(\Omega)\le\liminf_n P(A_n)\le\limsup_n P(A_n)\le2\mathcal{H}^1(\partial\Omega)-P(\Omega)$$
which proves $(iv)$.
To conclude consider the case $\#\Omega<+\infty$. We can choose $n$ big enough such that $\Omega_n=\Omega$, $A_n=\Omega(t_n)$ and $\alpha:=\# \Omega^c=\# A_n^c$. For simplicity we denote $\rho_n:=\rho(A_n)$ and $\rho:=\rho(\Omega)$. By applying equality \eqref{eq.g} to the Lipschitz set $A_n$, we get
\begin{equation}\label{terza}
P(A_n)=\sup_{ (0,\rho_n)}g_n(t)
\end{equation}
where $g_n$ is the right derivative of the function $t\mapsto -|A_n(t)|-(\alpha-1)\pi t^2$.
Now, exploiting the equality $A_n(t)=\Omega(t+t_n)$,
we obtain
$$g_n(t)= g(t+t_n)+2\pi(\alpha-1)t_n$$
for all $0<t<(\rho-t_n)\wedge \rho_n$. Thus, as $t\to 0^+$ and applying \eqref{terza}, we can conclude that, for every $n$, it holds
$$\lim_{t\to 0^+}g(t+t_n)+2\pi(\alpha-1)t_n=\sup_{(0,\rho_n)}g_n(t)=P(A_n).$$
Passing to the limit as $n\to\infty$ in the equality above and taking into account \eqref{eq.g} we achieve the thesis.
\end{proof}
\section{Continuity of volume for co-Hausdorff convergence}\label{shau}
The Hausdorff distance between closed sets $C_1, C_2$ of $\mathbb{R}^2$ is defined by
$$d_H(C_1,C_2):=\sup_{x\in C_1}d(x,C_2)\vee\sup_{x\in C_2}d(x,C_1).$$
Through $d_H$ we can define the so called co-Hausdorff distance $d_{H^c}$ between a pair of bounded open subsets $\Omega_1,\Omega_2$ of $\mathbb{R}^2$
$$d_{H^c}(\Omega_1,\Omega_2):=d_H(\Omega_1^c,\Omega_2^c).$$
We say that a sequence of compact sets $(K_n)$ converges in the sense of Hausdorff to some compact set $K$, if $ (d_H(K_n,K))$ converges to zero. In this case we write $K_n\overset{H}{\to}K$. Similarly we say that a sequence of open sets $(\Omega_n)$ converges in the sense of co-Hausdorff to some open set $\Omega$, if $(d_{H^c}(\Omega_n,\Omega))$ converges to zero, and we write $\Omega_n\overset{H^c}{\to}\Omega$. In the rest of the paper we use some elementary properties of Hausdorff distance and co-Hausdorff distance for which we refer to \cite{bubu05} and \cite{He}, (see, for instance, Proposition 4.6.1 of \cite{bubu05}). In particular we recall that if $(\Omega_n)$ is a sequence of equi-bounded sets in $\mathcal{A}_k$ and $\Omega_n\overset{H^c}{\to}\Omega$, then $\Omega$ still belongs to $\mathcal{A}_k$ (see Remark 2.2.20 of \cite{He}).
The introduction of co-Hausdorff convergence is motivated by Sver\'ak's Theorem (see \cite{sv93}) which ensures the continuity of the torsional rigidity in the class $\mathcal{A}_k$. Actually the result is stronger and gives the continuity with respect to the $\gamma$-convergence (we refer to \cite{bubu05} for its precise definition and the related details).
\begin{theo}[Sver\'ak]\label{theo.Sve}
Let $(\Omega_n)\subset\mathcal{A}_k$ be a sequence of equi-bounded open sets. If $\Omega_n\overset{H^c}{\to}\Omega$, then $\Omega_n\to\Omega$ in the $\gamma$-convergence. In particular $T(\Omega_n)\to T(\Omega)$.
\end{theo}
Combining Sver\'ak theorem and Theorem \ref{theo.approxim}, we prove that we can equivalently minimize the functional $F_q$ either in the class of Lipschitz set in $\mathcal{A}_k$ or in the larger class of those sets $\Omega\in\mathcal{A}_k$ satisfying $P(\Omega)=\mathcal{H}^1(\partial\Omega)$.
\begin{prop}
The following identity holds:
$$m_{q,k}=\inf\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_k,\ P(\Omega)=\mathcal{H}^1(\partial\Omega)\}$$
\end{prop}
\begin{proof}
By Theorem \ref{theo.approxim}, for every $\Omega\in\mathcal{A}_k$ such that $P(\Omega)=\mathcal{H}^1(\partial\Omega)<\infty$, there exists a sequence $(A_n)\subset\mathcal{A}_k$ of Lipschitz sets satisfying $\lim_n P(A_n)=P(\Omega)$. By construction $(A_n)$ is an equi-bounded sequence which converges both in the co-Hausdorff and in the $L^1$ sense. By Theorem \ref{theo.Sve} we have
$$\lim_{n\to\infty} F_q(\Omega_n)=F_q(\Omega),$$
so that
$$m_{q,k}\le\inf\{F_q(\Omega)\ :\ \Omega\in\mathcal{A}_k,\ P(\Omega)=\mathcal{H}^1(\partial\Omega)\}.$$
The thesis is then achieved since the opposite inequality is trivial.
\end{proof}
In general the volume is only lower semicontinuous with respect to the $H^c$-convergence as simple counterexamples may show. In this section we prove that $L^1$-convergence is guaranteed in the class $\mathcal{A}_k$ under some further hypotheses, see Theorem \ref{theo.convmeas}. The proof of this result requires several lemma and relies on the classical Go\l ab's semicontinuity theorem, which deals with the lower semicontinuity of the Hausdorff measure $\mathcal{H}^1$ (see, for instance, \cite{AO}, \cite{amti04}).
\begin{theo}[Go\l ab]\label{theo.Golab}
Let $X$ be a complete metric space and $k\in\mathbb{N}$ let
$$\mathcal{C}_k:=\{K\ :\ K\subset X, \ K\text{ is closed},\ \#K\le k\}.$$
Then the function $K\mapsto\mathcal{H}^1(K)$ is lower semicontinuous on $\mathcal{C}_k$ endowed with the Hausdorff distance.
\end{theo}
\begin{lemm}\label{lem.inradcon}
Let $(\Omega_n)$ be a sequence of equi-bounded open sets. If $\Omega_n\overset{H^c}{\to}\Omega$ we have also $\rho(\Omega_n)\to\rho(\Omega)$.
\end{lemm}
\begin{proof}
For simplicity we denote $\rho:=\rho(\Omega)$, and $\rho_n:=\rho(\Omega_n)$. First we show that \begin{equation}\label{basso}\rho\le\liminf_n\rho_n.\end{equation} Indeed, without loss of generality let us assume $\rho>0$. Then for any $0<{\varepsilon}<\rho$, there exists a ball $B_{\varepsilon}$ whose radius is $\rho-{\varepsilon}$ and whose closure is contained in $\Omega$. By elementary properties of co-Hausdorff convergence, there exists $\nu$ such that $B_{\varepsilon}\subset\Omega_n$, for $n>\nu$, which implies $\rho_n\ge\rho-{\varepsilon}$. Since ${\varepsilon}>0$ is arbitrary, we get \eqref{basso}.
In order to prove the upper semicontinuity, assume by contradiction that there exist ${\varepsilon}>0$ and a subsequence $(n_k)$ such that $\rho_{n_k}>\rho+{\varepsilon}$ for every $k\in \mathbb{N}$. Then there exists a sequence of balls $B_{n_k}=B_{ \rho_{n_k} }(x_{n_k})\subseteq\Omega_{n_k}$. Eventually passing to a subsequence, the sequence $(x_{n_k})$ converges to a point $x_{\infty}$ and the sequence of the translated open set $\Omega_{n_k}-x_{n_k}$ converges to $\Omega-x_{\infty}$. Since $B_{r}(0)\subseteq \Omega_{n_k}-x_{n_k} $ for $r=\rho+{\varepsilon}$, it turns out that $B_{r}(0)\subseteq \Omega-x_{\infty}$, i.e. $B_{r}(x_{\infty})\subseteq\Omega$ which leads to a contradiction.
\end{proof}
\begin{lemm}\label{lem.top2}
Let $\Omega$ be a connected bounded open set of $\mathbb{R}^n$. There exists a sequence of connected bounded open sets $(\Omega_n)$ such that $\overline\Omega_n\subset\Omega_{n+1}$ and $\bigcup_n\Omega_n=\Omega$.
\end{lemm}
\begin{proof}
We construct the sequence by induction. First of all we notice that there exists an integer $\nu_1>0$ such that $\Omega(\nu_1^{-1})$ contains at least one connected component of $\Omega$ with Lebesgue measure greater than $\pi\nu_1^{-2}$. Indeed it suffices to choose
$$\nu_1^{-1}\le\min\{ d(y,\partial\Omega)\ :\ y\in\partial B_r(x)\}\wedge r$$
where $B_r(x)$ is any ball with closure contained in $\Omega$. Now let $M$ be the number of connected components of $\Omega(\nu_1^{-1})$ with Lebesgue measure greater than $\pi\nu_1^{-2}$. If $M=1$ we define $ \Omega_{1}:=\Omega(\nu_1^{-1})$. Otherwise, since $\Omega$ is pathwise connected, we can connect the closures of the $M$ connected components with finitely many arcs to define a connected compact set $K\subset\Omega$. Then, we choose $m$ such that $m>\nu_1$ and $m^{-1}<\inf\{d(x,\partial\Omega) : x\in K\}$ and we set
$$\Omega_{1}:=\{ x\in \Omega: \ d(x,K)< (2m)^{-1} \}.$$
In both cases $\Omega_1$ is a connected open set which contains all the connected components of $\Omega(\nu_1^{-1})$ having Lebesgue measure greater then $\pi\nu_1^{-2}$. Moreover by construction there exists $\nu_2>\nu_1$ such that $\overline \Omega_1\subseteq \Omega(\nu^{-1}_2)$.
Replacing $\nu_1$ with $\nu_2$ we can use the previous argument to define $\Omega_2$ such that $\overline\Omega_1\subset \Omega_2$.
Iterating this argument we eventually define an increasing sequence $\nu_n$ and a sequence of connected open sets $(\Omega_n)$ such that
$\overline \Omega_n\subset\Omega_{n+1}\subset\Omega$ and $\Omega_n$ contains all the connected components of $\Omega(\nu_n^{-1})$ of Lebesgue measure greater than $\pi\nu_{n}^{-2}$.
Since for any $x\in\Omega$ there exists $r>0$ such that $\overline{B}_r(x)\subset\Omega$, choosing $\nu_n^{-1}\le \min\{ d(y,\partial\Omega)\ : y\in\partial B_r(x)\}\wedge r$,
it is easy to show that $x\in\Omega_n$. Thus $\bigcup_{n} \Omega_{n}=\Omega$.
\end{proof}
In the following lemma we establish a Bonnesen-type inequality for sets $\Omega\in \mathcal{A}_k$ satisfying $\mathcal{H}^1(\partial \Omega)<\infty$ (see Theorem 2 in \cite{Oss79} when $\Omega$ is a simply connected plane domain bounded by a rectifiable Jordan curve).
\begin{lemm}\label{lem.coarea2}
Let $\Omega\in\mathcal{A}_k$ with $\mathcal{H}^1(\partial \Omega)<\infty$.
Then \begin{equation}\label{eq.coarea}
|\Omega| \le [2\mathcal{H}^1(\partial \Omega)-P(\Omega)+\pi ( k-1) \rho(\Omega)]\rho(\Omega).
\end{equation}
\end{lemm}
\begin{proof}
If $\#\Omega<\infty$, by Theorem \ref{theo.Nareg} and \eqref{eq.Evans},
$$|\Omega|\le \left(P(\Omega)+2\mathcal{H}^1(\partial\Omega \cap \Omega^{(1)})+\pi(k-1)\rho(\Omega)\right)\rho(\Omega),$$
and we conclude by \eqref{eq.decomp2}. To prove the general case we denote by $(\Omega^i)$ the connected components of $\Omega$ and we set $\Omega_n:=\bigcup_{i=1}^{n}\Omega^i$. By the previous step we have
$$|\Omega_n|\le \big(2\mathcal{H}^1(\partial \Omega_n)-P(\Omega_n)+\pi (k-1) \rho(\Omega_n)\big)\rho(\Omega_n).$$
Since $\Omega_n\overset{H^c}{\to}\Omega$ and $\Omega_n\to\Omega$ in the $L^1$-convergence, taking into account Lemma \ref{lem.inradcon} and Lemma \ref{lem.top1}, we can conclude that
\begin{align*}
|\Omega|=\lim_{n\to\infty}|\Omega_n|&\le\big(2\mathcal{H}^1(\partial\Omega)-\limsup_n P(\Omega_n)+\pi(\alpha-1)\rho(\Omega)\big)\rho(\Omega)\\
&\le\big(2 \mathcal{H}^1(\partial\Omega)-P(\Omega)+\pi (k-1)\rho(\Omega)\big)\rho(\Omega),
\end{align*}
from which the thesis is achieved.
\end{proof}
\begin{theo}\label{theo.convmeas}
Let $(\Omega_n)\subset\mathcal{A}_k$ be a sequence of equi-bounded open sets with
$$\sup_n\mathcal{H}^1(\partial\Omega_n)<\infty.$$
If $\Omega_n\overset{H^c}{\to}\Omega$ then $\Omega\in \mathcal{A}_k$ and $\Omega_n\to \Omega$ in the $L^1$-convergence.
If, in addition, either $\sup_n\#\partial\Omega_n< \infty$ or $\#\Omega<\infty$ then
\begin{equation}
\label{eq.golab}
\mathcal{H}^1(\partial\Omega)\le \liminf_n \mathcal{H}^1(\partial\Omega_n).
\end{equation}
\end{theo}
\begin{proof}
We first deal with the case when $\sup_{n}\#\partial\Omega_n<\infty$, already considered in \cite{ChDo} and \cite{bubu05}. By compactness we can suppose that $\partial\Omega_n$ converges to some nonempty compact set $K$ which contains $\partial\Omega$. Then it is easy to show that $\bar\Omega_n\overset{H}{\to}\Omega\cup K$, which implies $\chi_{\Omega_n}\to\chi_\Omega$ pointwise in $\mathbb{R}^2\setminus K$, where $\chi_E$ denotes the characteristic function of a set $E$. By Theorem \ref{theo.Golab} we have also
\begin{equation}\label{eq.golabK}
\mathcal{H}^1(\partial \Omega)\le\mathcal{H}^1(K)\le\liminf_{n\to\infty}\mathcal{H}^1(\partial \Omega_n)<+\infty,
\end{equation}
which implies \eqref{eq.golab}. In particular, we have $|K|=0$, and $\Omega_n\to \Omega$ in the $L^1$ convergence.
We consider now the general case. Let $(\Omega^i)$ be the connected components of $\Omega$ and ${\varepsilon}>0$. There exists an integer $\nu({\varepsilon})$ such that
$$|\Omega|-{\varepsilon}<|\bigcup_{i=1}^{\nu({\varepsilon})}\Omega^i|\le |\Omega|$$
(when $\#\Omega<\infty$ we simply choose $\nu({\varepsilon})=\#\Omega$).
For each $i\le \nu({\varepsilon})$, and for each set $\Omega^i$, we consider the sequence $(\Omega^i_n)$ given by Lemma \ref{lem.top2}. By elementary properties of co-Hausdorff convergence there exists $l:=l(n)$ such that
$$\bigcup_{i}^{\nu({\varepsilon})}\overline{\Omega^i_n}\subset\Omega_{l}.$$
Let's denote by $\widetilde\Omega^i_{l}$ the connected component of $\Omega_{l}$ which contains $\overline{\Omega^i_n}$ (eventually $\widetilde{\Omega}^h_{l}=\widetilde{\Omega}^s_{l}$), and define
$$\widetilde\Omega_{l}:=\bigcup_{i=1}^{\nu({\varepsilon})} \widetilde\Omega^i_{l}.$$
By compactness, passing eventually to a subsequence, there exists $\widetilde{\Omega}\in\mathcal{A}_k$ such that $\widetilde\Omega_{l}\overset{H^c}{\to}\widetilde\Omega$.
Moreover, since $\widetilde\Omega_l\in\mathcal{A}_k$, $\sup_l\#\widetilde\Omega_l\le\nu({\varepsilon})$, and by Lemma \ref{lem.top1} we have
$$\sup_l \mathcal{H}^1( \partial \widetilde \Omega_{l})\le\sup_l \mathcal{H}^1(\partial\Omega_{l})<\infty,$$
we can apply the first part of the proof to conclude that $\widetilde\Omega_l\to \widetilde\Omega$ in the $L^1$-convergence. If $\#\Omega<\infty$ an easy argument shows that $\widetilde\Omega$ must be equal to $\Omega$ and that \eqref{eq.golabK} holds with $K$ the Hausdorff limit of $(\partial\widetilde\Omega_l)$.
In particular \eqref{eq.golab} holds.
Otherwise we consider the set $\Omega^R_l$ of those connected components of $\Omega_{l}$ that have been neglected in the definition of $\widetilde\Omega_l$, that is
$$\Omega^R_{l}:=\Omega_{l}\setminus\widetilde\Omega_{l}.$$
Passing to a subsequence we can suppose that $\Omega^R_l\overset{H^c}{\to}\Omega^R$, for some open set $\Omega^R\in\mathcal{A}_k$. Moreover since $|\widetilde\Omega|>|\Omega|-{\varepsilon}$, $\Omega^R\cap\tilde\Omega=\emptyset$ and $\Omega^R\subset\Omega$ we have also $|\Omega^R|\le{\varepsilon}$. This implies $\rho(\Omega^R)\le\sqrt{\pi^{-1}{\varepsilon}}$ and by Lemma \ref{lem.inradcon},
$$\lim_{l\to\infty}\rho(\Omega^R_{l})\le \sqrt{\pi^{-1}{\varepsilon}}.$$
Finally, by Lemma \ref{lem.coarea2}, we have
\begin{align*}
|\Omega|&\le\liminf_{n\to\infty}|\Omega_{n}|\le \limsup_{l\to\infty} ( |\widetilde\Omega_{l}|+|\Omega^R_{l}|)= |\widetilde \Omega|+\limsup_{l\to\infty}|\Omega^R_{l}|\le |\Omega|+o({\varepsilon}).
\end{align*}
Since ${\varepsilon}$ was arbitrary this shows that
$$
\liminf_{n\to\infty}|\Omega_n|=|\Omega|,
$$
and the thesis is easily achieved.
\end{proof}
As an application of the previous theorem we prove the following fact.
\begin{coro}
Let $\Omega\in\mathcal{A}_k$ with $\mathcal{H}^1(\partial\Omega)<\infty$ and $\#\Omega<\infty$. Then it holds
$$\mathcal{H}^1(\partial\Omega\cap\Omega^{(0)})\le\mathcal{H}^1(\partial\Omega\cap\Omega^{(1)}).$$
\end{coro}
\begin{proof}
By Theorem \ref{theo.approxim} we can consider a sequence $(A_n)\in\mathcal{A}_k$ of Lipschitz sets such that $A_n\overset{H^c}{\to}\Omega$ and $P(A_n)\to P(\Omega)+2\mathcal{H}^1(\partial\Omega\cap \Omega^{(1)})<\infty$. Then, by Theorem \ref{theo.convmeas}, we conclude
$$\mathcal{H}^1(\partial\Omega)\le \lim_{n\to\infty} P(A_n)\le P(\Omega)+2\mathcal{H}^1(\partial\Omega\cap \Omega^{(1)}),$$
which easily implies the thesis, using \eqref{eq.decomp}.
\end{proof}
\begin{rem} We remark the fact that the inequality
$$\lim_{n\to\infty}P(A_n)\ge\mathcal{H}^1(\partial \Omega)$$
is not in general satisfied when $\#\Omega=\infty$, see also Remark \ref{ex.ce}.
\end{rem}
\section{Existence of relaxed solutions}\label{sexis}
Our next result generalizes the estimate
$F_{1/2}(\Omega)\ge3^{-1/2}$, proved in \cite{polya60} for the class $\mathcal{A}_{convex}$, to the class $\mathcal{A}_k$.
\begin{theo}\label{theo.Polya}
For every $\Omega\in\mathcal{A}_k$ set of finite perimeter we have
\begin{equation}\label{eq.polro}
\frac{T^{1/2}(\Omega)}{|\Omega|^{3/2}}\ge\frac{3^{-1/2}}{\left(2\mathcal{H}^1(\partial\Omega)-P(\Omega)+2\pi(k-1)\rho(\Omega)\right)}.
\end{equation}
\end{theo}
\begin{proof}
Without loss of generality we may assume that $\mathcal{H}^1(\partial\Omega)<\infty$ and we set $\rho:=\rho(\Omega)$. First we consider the case $\#\Omega<\infty$. We define
$$G(t):=\int_{0}^{t}\frac{A(t)}{L(t)}dt, \quad u(x):=G(d(x,\partial\Omega)).$$
Notice that, since for any $t\in (0,\rho)$ it holds $L(t)\ge\mathcal{H}^1(\partial \Omega(t))\ge P(\Omega(t))$,
by isoperimetric inequality \eqref{isoper} we have
$$\frac{A(t)}{L(t)}=\frac{|\Omega(t)|^{1/2}}{L(t)}A^{1/2}(t)\le\frac{|\Omega(t)|^{1/2}}{P(\Omega(t))}A^{1/2}(t)\le\frac{|B_1|^{1/2}}{P(B_1)}A^{1/2}(t).$$
In particular, since $A$ is bounded, we get that $L^{-1}A$ is summable on $(0,\rho)$ and $G$ is a Lipschitz function on in the interval $(0,\rho)$. Thus $u\in H^1_{0}(\Omega)$.
Using \eqref{vartor} and \eqref{eq.boundL} we have
\begin{align*}
T(\Omega)&\ge \frac{\left(\int_{\Omega}udx\right)^{2}}{\int_{\Omega}|\nabla u|^{2}dx}\ge\frac{\left(\int_{0}^{\rho}G(t)L(t)dt\right)^2}{\int_0^\rho (G'(t))^{2}L(t)dt}\ge\int_0^\rho\frac{(A(t))^{2}}{L(t)}\,dt=\int_{0}^\rho \frac{A^2(t)L(t)}{L^{2}(t)}dt\\
&\ge\frac{1}{(P(\Omega)+2\mathcal{H}^1(\partial \Omega\cap\Omega^{(1)})+2\pi(k-1)\rho)^2}\int_0^\rho A^2(t)L(t)\,dt.
\end{align*}
Since $A\in W^{1,\infty}(0,\rho(\Omega))$ by Corollary \ref{theo.Nareg} then, set $\psi(s)=s^2,$ we have that the function $\psi\circ A\in W^{1,\infty}(0,\rho(\Omega))$, so that
$$\int_0^\rho A^2(t)L(t)\,dt=-\int_0^{\rho}A^2(t)A'(t)\,dt=-\frac13\left[A^3(t)\right]_0^{\rho(\Omega)}=\frac13|\Omega|^3.$$
Thus
\begin{equation}\label{eq.polro1}
\frac{T(\Omega)}{|\Omega|^3}\ge\frac{1}{3(P(\Omega)+2\mathcal{H}^1(\partial \Omega\cap\Omega^{(1)})+2\pi(k-1)\rho)^2}.
\end{equation}
Taking into account \eqref{eq.decomp2} we get
$$
\frac{T(\Omega)}{|\Omega|^3}\ge\frac{1}{3(2\mathcal{H}^1(\partial\Omega)-P(\Omega)+2\pi(k-1)\rho)^2}.
$$
To prove the general case, let $\Omega_n$ be defined as in Lemma \ref{lem.top1}. Since $\#\Omega_n<\infty$ and $\Omega_n\in \mathcal{A}_k$, by the first part of this proof we have that
$$\frac{T(\Omega_n)}{|\Omega_n|^3}\left(2\mathcal{H}^1(\partial\Omega_n)-P(\Omega_n)+2\pi(k-1)\rho_n\right)^2\ge\frac1{3},$$
where $\rho_n:=\rho(\Omega_n)$.
When $n\to\infty$ we have $|\Omega_n|\to |\Omega|$, $\rho_n\to\rho$ by Lemma \ref{lem.inradcon} and $T(\Omega_n)\to T(\Omega)$ by Theorem \ref{theo.Sve}. Hence, passing to the $\limsup$ in the previous inequality and using Lemma \ref{lem.top1}, we get \eqref{eq.polro}.
\end{proof}
\begin{rem}\label{rem.polya}
Note that, in the special case of $\Omega\in\mathcal{A}_k$ and $\#\Omega<\infty$, we have the improved estimate \eqref{eq.polro1}.
Moreover, if $k=0,1$, \eqref{eq.polro} implies
\begin{equation}\label{eq.polyagen}
F_{1/2}(\Omega)\ge\frac{3^{-1/2}P(\Omega)}{2\mathcal{H}^1(\partial\Omega)-P(\Omega)}\, ,
\end{equation}
while, if $k>1$, we can use the inequality $2\pi\rho(\Omega)\le P(\Omega)$ (which can be easily derived from \eqref{isoper}), to obtain
\begin{equation}\label{eq.polyagenk}
F_{1/2}(\Omega)\ge\frac{3^{-1/2}P(\Omega)}{2\mathcal{H}^1(\partial\Omega)+(k-2)P(\Omega)}\;.
\end{equation}
\end{rem}
As a consequence of Theorem \ref{theo.Polya}, and using the well known fact that for a Lipschitz open set $\Omega$ it holds $P(\Omega)=\mathcal{H}^1(\partial\Omega)$, we have the following main results.
\begin{coro}\label{coro.polya} For every $q\le1/2$ we have
\begin{equation}\label{eq.m01}
m_{1/2,0}=m_{1/2,1}=3^{-1/2}
\end{equation}
and the value $3^{-1/2}$ is asymptotically reached by a sequence of long thin rectangles. More in general, for $k\ge 1$, it holds
\begin{equation}\label{eq.boundkq}
m_{q,k}\ge (8\pi)^{1/2-q}(3^{1/2}k)^{-1}
\end{equation}
and the sequence $(m_{q,k})$ decreases to zero as $k\to \infty$.
\end{coro}
\begin{proof}
By inequality \eqref{eq.polyagen} we have that $m_{1/2,0}, m_{1/2,1}\ge 3^{-1/2}$. Moreover the computations made in \cite{BBP20} show that the value $3^{-1/2}$ is asymptotically reached by a sequence of long thin rectangles, that are clearly in $\mathcal{A}_0$. Thus, being $A_0\subset\mathcal{A}_1$, \eqref{eq.m01} holds. To prove \eqref{eq.boundkq} it is enough to notice that
$$F_q(\Omega)=F_{1/2}(\Omega)\left(\frac{T(\Omega)}{|\Omega|^{2}}\right)^{q-1/2}$$
and apply \eqref{eq.polyagenk} together with the Saint-Venant inequality \eqref{stven}.
Finally to prove that $m_{q,k}\to 0$ as $k\to \infty$, it is enough to consider the sequence $(\Omega_{1,n})$ defined in Theorem 2.1 of \cite{BBP20}, taking into account that $\Omega_{1,n}\in \mathcal{A}_k$ for $k$ big enough.
\end{proof}
We now introduce a relaxed functional $\mathcal{F}_{q,k}$.
More precisely, for $\Omega\in\mathcal{A}_k$ we denote by $\mathcal{O}_k(\Omega)$ the class of equi-bounded sequences of Lipschitz sets in $\mathcal{A}_k$ which converge to $\Omega$ in the sense of co-Hausdorff and we define $\mathcal{F}_{q,k}$ as follows:
$$\mathcal{F}_{q,k}(\Omega):=\inf\left\{\liminf_{n\to\infty} F_q(\Omega_{n}): \ (\Omega_n)\in\mathcal{O}_k(\Omega) \right\}.$$
It is straightforward to verify that $\mathcal{F}_{q,k}$ is translation invariant and scaling free.
As already mentioned in the introduction, when $q<1/2$, we prove the existence of a minimizer for $\mathcal{F}_{q,k}$. We notice this relaxation procedure can be made on the perimeter term only. More precisely, defining
$$\mathcal{P}_k(\Omega):=\inf \left\{\liminf_{n\to\infty} P(\Omega_{n})\ :\ (\Omega_n)\in\mathcal{O}_k(\Omega)\right\},$$
the following proposition holds.
\begin{prop}\label{prop.PPkF}
For every $\Omega\in\mathcal{A}_k$ we have
$$\mathcal{F}_{q,k}(\Omega)=\frac{\mathcal{P}_k(\Omega)T^{q}(\Omega)}{|\Omega|^{2q+1/2}}.$$
\end{prop}
\begin{proof}
Fix ${\varepsilon}>0$. Suppose that $\infty>\mathcal{P}_k(\Omega)+{\varepsilon}\ge\lim_n P(\Omega_n)$, for some $(\Omega_n)\in\mathcal{O}_{k}(\Omega)$. By Theorems \ref{theo.Sve} and \ref{theo.convmeas}, we have
$$
\frac{(\mathcal{P}_k(\Omega)+{\varepsilon})T^{q}(\Omega)}{|\Omega|^{2q+1/2}}\ge\lim_n\left(\frac{P(\Omega_n)T^q(\Omega_n)}{|\Omega_n|^{2q+1/2}}\right)\ge \mathcal{F}_{q,k}(\Omega),
$$
and since ${\varepsilon}$ is arbitrary we obtain the $\le$ inequality.
Similarly, to prove the opposite inequality assume $\lim_n F_q(\Omega_n)\le \mathcal{F}_{q,k}(\Omega)+{\varepsilon}<\infty$, for some sequence $(\Omega_n)\in \mathcal{O}_k(\Omega)$. Let $D$ be a compact set which contains each $\Omega_n$. Thanks to Theorem \ref{theo.Sve}, we have that $T(\Omega_n)\to T(\Omega)$ and, since $P(\Omega_n)=\mathcal{H}^1(\Omega_n)$, we have also
$$\sup_n\mathcal{H}^1(\partial\Omega_n)=\sup_n\left( \frac{F_q(\Omega_n)|\Omega_n|^{2q+1/2}}{\displaystyle{T^q}(\Omega_n)}\right)\le
\sup_n \left(\frac{F_q(\Omega_n)|D|^{2q+1/2}}{\displaystyle{T^q}(\Omega_n)}\right)<+\infty.$$
Applying again Theorem \ref{theo.convmeas} we have $|\Omega_n|\to|\Omega|$ and we can conclude
$$\frac{\mathcal{P}_k(\Omega)T^q(\Omega)}{|\Omega|^{2q+1/2}}\le\lim_n F_q(\Omega_n)\le\mathcal{F}_{q,k}(\Omega)+{\varepsilon},$$
which implies the $\ge$ inequality as ${\varepsilon}\to 0$.
\end{proof}
The perimeter $\mathcal{P}_k$ satisfies the following properties.
\begin{prop}\label{prop.Pk}
For every $\Omega\in\mathcal{A}_k$ of finite perimeter we have
\begin{equation}\label{eq.PPk}
P(\Omega)\le\mathcal{P}_k(\Omega)\le 2\mathcal{H}^1(\partial\Omega)-P(\Omega).
\end{equation}
Moreover if $\#\Omega<\infty$ and $\mathcal{H}^1(\partial\Omega)<+\infty$ it holds
\begin{equation}\label{eq.PPkH1}
\mathcal{H}^1(\partial\Omega)\le\mathcal{P}_k(\Omega)\le P(\Omega)+2\mathcal{H}^1(\partial \Omega \cap \Omega^{(1)})
\end{equation}
and $P(\Omega)=\mathcal{P}_k(\Omega)$ if and only if $P(\Omega)=\mathcal{H}^1(\partial\Omega)$.
\end{prop}
\begin{proof}
Taking into account Theorem \ref{theo.convmeas} and lower semicontinuity of the perimeter with respect to the $L^1$-convergence we have $\mathcal{P}_k(\Omega)\ge P(\Omega)$.
To prove the right-hand inequalities in \eqref{eq.PPk} and \eqref{eq.PPkH1} it is sufficient to take the sequence $(A_n)$ given by Theorem \ref{theo.approxim}. Finally, when $\#\Omega<\infty$, the inequality $\mathcal{H}^1(\partial\Omega)\le \mathcal{P}_k(\Omega)$ follows by Theorem \ref{theo.convmeas}.
\end{proof}
\begin{rem} \label{ex.ce}
If we remove the assumption $\#\Omega<\infty$, then \eqref{eq.PPkH1} is no longer true. For instance, we can slightly modify the Example $3.53$ in \cite{AFP} to define $\Omega\in\ \mathcal{A}_0$ such that $P(\Omega),\mathcal{P}_0(\Omega)<\infty$ while $\mathcal{H}^1(\partial\Omega)=\infty$. More precisely let $(q_n)$ be an enumeration of $\mathbb{Q}^2\cap B_1(0)$ and $(r_n)\subset(0,{\varepsilon})$ be a decreasing sequence such that
$\sum_n 2\pi r_n\le 1$. We recursively define the following sequence of open sets.
Let
$$\Omega_0:=B_{r_0}(q_0),\ \Omega_{n+1}:=\Omega_n\cup B_{s_n}(q_{h_n}),$$
where
$$h_n:=\inf\{k: q_k \in\overline\Omega_n^c\},\quad s_n:=r_{n+1}\wedge\sup\{r_k: B_{r_k}(q_{h_n})\cap \Omega_n=\emptyset\}.$$
Finally let $\Omega=\bigcup_n\Omega_n$. By construction $\Omega_n\overset{H^c}{\to}\Omega$ and since $\Omega_n\in\mathcal{A}_0$ for all $n$, we have also $\Omega\in\mathcal{A}_0$. Moreover we notice that $P(\Omega)\le 1$ and it is easy to verify that the two dimensional Lebesgue measure of $\partial\Omega$ is positive, which implies $\mathcal{H}^1(\partial\Omega)=\infty$. Finally, since the sequence $(\Omega_n)\in \mathcal{O}_0(\Omega)$, we have also $\mathcal{P}_0(\Omega)\le 1$.
\end{rem}
Next we prove that the relaxed functional $\mathcal{F}_{q,k}$ agrees with $F_q$ on the class of Lipschitz open sets in $\mathcal{A}_k$.
\begin{coro}\label{coro.Frel}
For every $\Omega\in\mathcal{A}_k$ we have
\begin{equation}\label{eq.FgF}
\mathcal{F}_{q,k}(\Omega)\ge F_q(\Omega).
\end{equation}
If, in addition, $P(\Omega)=\mathcal{H}^1(\partial\Omega)$ then we have
\begin{equation} \label{eq.FgF1}
F_q(\Omega)=\mathcal{F}_{q,k}(\Omega).
\end{equation}
In particular $\mathcal{F}_{q,k}$ and $F_q$ coincide on the class of Lipschitz sets and it holds
\begin{equation}\label{eq.infrel}
m_{q,k}=\inf\{\mathcal{F}_{q,k}(\Omega)\ :\ \Omega\in\mathcal{A}_k\}.
\end{equation}
\end{coro}
\begin{proof}
The inequalities \eqref{eq.FgF} and \eqref{eq.FgF1} follow by Proposition \ref{prop.PPkF} and \eqref{eq.PPk}. The last part of the theorem follows as a general property of relaxed functionals.
\end{proof}
\begin{lemm}\label{lem.coninf}
For every Lipschitz set $\Omega\in\mathcal{A}_k$, there exists a sequence of connected open sets $(\Omega_n)\subset\mathcal{A}_k$ such that
$$P(\Omega_n)=\mathcal{H}^1(\partial\Omega_n)\qquad\text{and}\qquad\lim_{n\to\infty}F_q(\Omega_n)=F_q(\Omega).$$
\end{lemm}
\begin{proof}
Since $\Omega$ is a bounded Lipschitz set we necessarily have $\#\Omega<\infty$. If $\Omega$ is connected we can take $\Omega_n$ to be constantly equal to $\Omega$. Suppose instead that $\#\Omega=2$ and let $\Omega^1$ and $\Omega^2$ be the connected components of $\Omega$. Since $\Omega$ is Lipschitz there exist $x_1\in\partial \Omega^1, x_2\in\partial\Omega^2$ such that
$$
0<d:=d(x_1,x_2)=\inf\{d(w,v): \ v\in\Omega^1,\ w\in\Omega^2\}.
$$
Define
$$\Omega^{2}_n:=\Omega^2-\left(1-\frac 1 n\right)(x_2-x_1).$$
Clearly we have $\overline{\Omega^2_{n}}\cap \overline{\Omega^1}=\emptyset$ for every $n\ge 1$ and $\Omega^{2}_1=\Omega^2$.
We set $$x_n=x_2-\left(1-\frac 1 n\right)(x_2-x_1).$$
Now we can join $x_1$ and $x_n$ through a segment $\Sigma_n$. By using the fact that the boundary of both $\partial\Omega^1$ and $\partial\Omega^2_n$ are represented as the graph of a Lipschitz functions in a neighborhood of $x_1$ and $x_n$ respectively, then the thin open channel
$$C_{\varepsilon}:=\{ x\in\mathbb{R}^2\setminus\overline\Omega^1\cup\overline\Omega^2_n\ :\ d(x,\Sigma_n)<{\varepsilon}\}$$
of thickness ${\varepsilon}:={\varepsilon}(n)$ is such that the set
$$\Omega_n:=\Omega^1\cup \Omega^2_n\cup C_{{\varepsilon}}$$
belongs to $\mathcal{A}_k$, it is connected and $P(\Omega_n)=\mathcal{H}^1(\partial\Omega_n)$. The following identities are then verified
$$|\Omega_n|\to|\Omega|,\quad T(\Omega_n)\to T(\Omega),\quad P(\Omega_n)\approx P(\Omega^1)+P(\Omega^2)+\frac{2{\varepsilon}}{n},$$
so that $F_q(\Omega_n)\to F_q(\Omega)$ (notice that this does not imply $\Omega_n\to\Omega$). The general case is achieved by induction on $\#\Omega$. More precisely suppose $\#\Omega=N+1$. Let $(\Omega^i)$ be the connected components of $\Omega$. By induction we have
$$F_q(\Omega^1\cup\dots \cup\Omega^N)=\lim_{n\to\infty}F_q(\Omega'_n),$$
for a sequence $(\Omega'_n)\subset\mathcal{A}_k$ of connected open sets satisfying $P(\Omega'_n)=\mathcal{H}^1(\partial\Omega'_n)$.
Using the fact that, being $\Omega$ Lipschitz, the value of $F_q(\Omega)$ do not change if we translate (possibly in different direction and with different magnitude) each connected component of $\Omega$, being careful to avoid intersections, we can suppose $\overline{\Omega}^{N+1}$ to have a positive distance from $\overline{\Omega}'_n$, as $n$ is large enough.
We then apply the previous step to define a sequence of connected open sets $\Omega_{n,m}\in\ A_k$ such that $P(\Omega_{n,m})=\mathcal{H}^1(\partial\Omega_{n,m})$ and
$$F_q(\Omega_{n,m})\to F_q(\Omega'_n\cup \Omega^{N+1}),$$
as $m\to\infty$. Using a diagonal argument we achieve the thesis.
\end{proof}
We finally show the existence of a relaxed solution to the minimization problem of $\mathcal{F}_{q,k}$ in $\mathcal{A}_k$ when $q<1/2$.
\begin{theo}\label{theo.exis}
For $q<1/2$ there exists a nonempty bounded open set $\Omega^{\star}\in\mathcal{A}_k$ minimizing the functional $\mathcal{F}_{q,k}$ such that $\mathcal{H}^1(\partial\Omega^\star)<\infty$.
\end{theo}
\begin{proof}
Let $(\widetilde\Omega_n)\subset\mathcal{A}_k$ be a sequence of Lipschitz sets such that
$$
\lim_{n\to\infty} F_q(\widetilde\Omega_n)=m_{q,k}.
$$
Applying Lemma \ref{lem.coninf} and \eqref{eq.FgF1}, we can easily replace the sequence $(\widetilde\Omega_n)$ with a sequence $(\Omega_n)\subset\mathcal{A}_k$ of connected (not necessarily Lipschitz) open sets, satisfying $\mathcal{H}^1(\Omega_n)=P(\Omega_n)$ and such that
$$\lim_{n\to\infty} F_q(\Omega_n)=\lim_{n\to\infty} F_q(\widetilde\Omega_n)=m_{q,k}.$$
Eventually using the translation invariance of $F_q$ and possibly rescaling the sequence $(\Omega_n)$, we can assume that $(\Omega_n)$ is equi-bounded and
\begin{equation}\label{ipinf}
\mathcal{H}^1(\Omega_n)=P(\Omega_n)=1.
\end{equation}
By compactness, up to subsequences, there exists an open sets $\Omega^\star\in\mathcal{A}_k$ such that $\Omega_n\overset{H^c}{\to}\Omega^\star$.
By \eqref{eq.infrel} we have
$$m_{q,k}\le \mathcal{F}_{q,k}(\Omega^\star).$$
Let us prove the opposite inequality. We notice that, by Theorem \ref{theo.approxim} and \eqref{ipinf}, for every $n$ there exists a sequence $(A_{n,m})_m\subset\mathcal{A}_k$ of Lipschitz sets, such that, as $m\to\infty$,
$$
P(A_{n,m})\to P(\Omega_n)\ \text{and}\ |A_{n,m}|\to |\Omega_n|.
$$
By Theorem \ref{theo.Sve}, we have also $T(A_{n,m})\to T(\Omega_n)$ as $m\to \infty$.
Thus
$$F_q(\Omega_n)=\lim_{m\to\infty}F_q(A_{n,m}).$$
A standard diagonal argument allows us to define a subsequence $A_{n,m_n}\in\mathcal{O}_k(\Omega^\star)$. Then we have
$$\mathcal{F}_{q,k}(\Omega^\star)\le \lim_n F_{q}(A_{n,m_n})=\lim_{n}F_{q}(\Omega_n)= m_{q,k}.
$$
Hence $\Omega^\star$ is a minimum for $\mathcal{F}_{q,k}$.
Moreover, notice that there exists a compact set $K$ containing $\partial\Omega^\star$ such that, up to a subsequence, $\partial\Omega_n\overset{H}{\to} K$.
So, being $\Omega_n$ connected, we have
$$
\sup_n\#\partial\Omega_n<\infty,
$$
and by Theorem \ref{theo.Golab},
$$
\mathcal{H}^1(\partial\Omega^\star)\le \mathcal{H}^1(K)\le \liminf_{n\to\infty}\mathcal{H}^1(\Omega_n)\le 1.
$$
To conclude we have only to show that $\Omega^*$ is nonempty. Notice that for $n$ big enough there exists $C>0$ such that $F_q(\Omega_n)< C$. Thus we have
\begin{equation} \label{eq.final1}
C>F_q(\Omega_n)=\frac{T^q(\Omega_n)}{|\Omega_n|^{2q+1/2}}=\left(\frac{T(\Omega_n)}{|\Omega_n|^{3}}\right)^{q}|\Omega_n|^{q-1/2}\ge \frac{1}{|\Omega_n|^{1/2-q}(\sqrt{3}k)^{2q}}\;,
\end{equation}
where the last inequality follows by \eqref{eq.polyagen}, using \eqref{ipinf}. By \eqref{eq.coarea} we have also
\begin{equation} \label{eq.final2}
|\Omega_n|\le (1+\pi(k-1)\rho(\Omega_n))\rho(\Omega_n).
\end{equation}
Combining \eqref{eq.final1}, \eqref{eq.final2} and the assumption $q<1/2$, we conclude that the sequence of inradius $(\rho(\Omega_n))$ must be bounded from below by some positive constant. By Lemma \ref{lem.inradcon}, $\Omega^{\star}$ is nonempty.
\end{proof}
\section{Conclusions}\label{sconc}
We have seen that in the planar case the topological constraint present in classes $\mathcal{A}_k$ is strong enough to ensure the existence of at least a relaxed optimizer. In higher dimensions this is no longer true and easy examples show that it is possible to construct sequences $(\Omega_n)$ in $\mathcal{A}_k$ with $P(\Omega_n)$ bounded and $T(\Omega_n)\to0$. This suggests that in higher dimensions stronger constraints need to be imposed in order to have well posed optimization problems.
Another interesting issue is the analysis of the same kind of questions when the exponent $2$ is replaced by a general $p>1$ in \eqref{vartor}; the torsional rigidity $T(\Omega)$ then becomes the $p$-torsion $T_p(\Omega)$ and it would be interesting to see how our results depend on the exponent $p$ and if in this case the analysis in dimensions higher than two is possible.
Finally, shape functionals $F(\Omega)$ involving quantities other than perimeter and torsional rigidity are interesting to be studied: we point out some recent results in \cite{bbp20},\cite{FtLa} and references therein. However, to our knowledge, the study of these shape functionals under topological constraints as the ones of classes $\mathcal{A}_k$ is still missing.
\bigskip
\noindent{\bf Acknowledgments.} The work of GB is part of the project 2017TEXA3H {\it``Gradient flows, Optimal Transport and Metric Measure Structures''} funded by the Italian Ministry of Research and University. The authors are member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilit\`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 624 |
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Mary Antin, Poster Child for Letting Immigrants Into America
Mary Antin wrote a poem praising George Washington just a few years after she arrived in Boston a poor, 13-year-old Russian Jew who spoke only Yiddish. She published her first book at 18. At 30 she wrote a national best-seller that launched her into political circles that included President Theodore Roosevelt.
Mary Antin became famous as a symbol of the immigrant who achieved the American Dream. But she also discovered the dark side to the dream, and that fame doesn't necessarily bring happiness.
Mary Antin
To some, her rags-to-riches story seemed too pat, too easy. Later in life, Mary Antin hinted at her embarrassment about winning wide acclaim for a slim accomplishment.
"[S]he considered her position a false one and suffered a nervous breakdown as a consequence," wrote Sarah Blacker Cohen.
Maryashe Antin was born June 13, 1881, to Israel Pinchus and Esther (Weltman) Antin in the shtetl of Polotzk in the Pale of Settlement. The Pale, a western region of Imperial Russia, was crowded with impoverished Jews evicted from the cities. Antisemitism made life dangerous for them.
Map, the Pale of Settlement
The Antin family at first prospered in the Pale, with a large house and servants. As a traditional Jewish girl, Mary Antin received an inferior education to her brother. Later, in a short story, she wrote, "What are daughters worth? They're only good to sit in the house, a burden on their parents' neck, until they're married off."
When illness destroyed her father's business, the family ended up living in a room. "We had absolutely no reliable source of income, no settled home, no immediate prospects," she wrote.
Mary Antin and her older sister Fetchke as children
In 1891, her father borrowed money to get to Germany. There an emigrant society helped bring him to Boston. To Mary, America became the Promised Land. Three years after he left, their father sent them a letter saying he'd saved enough money to bring them to America.
Her mother read the letter aloud. "There was an elation, a hint of triumph, such as had never been in my father's letters before," she wrote in her book, The Promised Land. "He saw something — he promised us something. It was this "america." And "America" became my dream."
In 1894, Mary Antin left for America.
Book cover, From Plotzk to Boston
In a book of her letters (now an audio book), she described how her family traveled in packed, airless fourth-class railroad cars. They encountered corrupt crossing guards and German officials who crudely disinfected them. They were locked in quarantine until finally they took the steamer across the Atlantic and reunited the family in Boston.
Boston Slums
Mary's father's attempts to make money failed, and the family moved from slum to slum: Chelsea and Boston's West and South ends. They lived in gloomy tenements filled with "unkempt, half-washed, toiling, unaspiring foreigners, pitiful in the eyes of social missionaries, the despair of boards of health," she wrote.
Boston's South End, from The Promised Land. The original caption read, "Harrison Avenue in the South End ghetto."
But her father had hope for Mary, frail and clearly intelligent. He enrolled her in the local public school in Chelsea, called the Williams School. Said to be the largest school in New England, it had many non-English-speaking immigrant students.
Mary Antin's father.
Because she spoke no English, Mary Antin had to wedge herself into a kindergarten desk. After four months, a composition she wrote called Snow impressed her teacher. The teacher sent Snow to an education journal, Primary Education, which printed it. And so Mary Antin determined to follow a writer's career.
She wrote a poem called "My Country," which described her thrill at sharing citizenship with George Washington. Then she went to Newspaper Row and found an editor willing to publish it.
Uplifting Tale
Mary learned quickly, and her teachers held her up "as an illustration of what the American system of free education and the European immigrant could make of each other." Many people wanted to hear that story, and Mary Antin made a career of telling it.
Her bestselling book, The Promised Land, offered up a romanticized version of the immigrant's assimilation and rise through public education.
In the book, Mary found a way to glorify their South End tenement on Wheeler Street, which ran crookedly between brothels on Corning Street and a saloon on Shawmut Avenue.
"I delighted in the moonlike splendor of the arc lamp just in front of the saloon," she wrote.
She also rhapsodized about the hours she spent reading and dreaming in the Boston Public Library. "That I who was born in the prison of the Pale should roam at will in the land of freedom was a marvel that it did me good to realize. That I who was brought up to my teens almost without a book should be set down in the midst of all the books that ever were written a miracle as great as any on record. That an outcast should become a privileged citizen, that a beggar should dwell in a palace–this was a romance more thrilling than poet ever sung," she wrote.
Her patriotism would later cost her her marriage.
Mary's sister Frieda went to work at 14, which allowed Mary to attend Girls' Latin School, Boston's premier public prep school for girls.
Assembly at the Girls' Latin School
Lina Hecht, a Jewish philanthropist, took notice of Mary Antin. Hecht arranged to have her letters about her journey to America translated from Yiddish and published. So in 1899, 18-year-old Mary published her first book, From Plotzk to Boston. (The printer misspelled the name of her home town, Polotzk.) The precocious scholar-immigrant thus achieved local celebrity.
Mary's life took another turn on a field trip with the Hale House, a South End settlement house sponsored by Edward Everett Hale.
The trip was led by Amadaus Grabau, who studied at Harvard and worked at the Boston Society of Natural History. Their attraction to each other may have seemed unlikely – she, a Russian-Jewish immigrant who wrote poetry; he, an American-born son of a Lutheran minister who studied geology. Not to mention the 11-year difference in their ages.
Amadaus Grabau
They married on Oct. 5, 1901. She was 20, he was 31. Her marriage outside her faith cost her some friendships, though her family stood by her.
Mary had dreamed of going to Radcliffe. But after marrying Grabau she followed him to New York, where he had gotten a job as a professor of geology at Columbia University. They lived in university housing on Morningside Heights and then in Brooklyn, a far cry from the dingy South End tenements she knew in Boston.
Mary attended Columbia Teacher's College and Barnard, but didn't graduate. They had a daughter, Josephine Esther, on Nov. 21, 1907.
At Columbia, Grabau had a reputation as a loner and a workaholic. His marriage to a 20-year-old Jewish immigrant writer and his failure to develop friendships with his colleagues marked him as an outsider.
Mary Antin, Celebrity
Then in 1912, when Mary was 32, she published the wildly successful The Promised Land. It made her a celebrity. Former president Theodore Roosevelt wrote her letters. He credited her — along with Jane Addams and Frances Kellor — with lighting a fire under him to support women's suffrage.
Mary Antin in 1916
Mary sent her daughter to boarding school and embarked on a national lecture tour about immigration. On Dec. 12, 1912, 1,000 people came to listen to her at the Waldorf-Astoria Hotel in New York. Then from 1913 to 1918 she lectured on "The Responsibility of American Citizenship," "How You and I Can Serve Our Country" and "The Public School as a Test of American Faith."
Her goal was to prove that immigrants could become good American citizens during a time of rising anti-immigrant feeling.
By the time her lecture tour ended, World War I broke out – in Europe and in her home.
Despite virulent anti-German sentiment during World War I, Grabau didn't hide his attachment to the German culture. That contributed to the loss of his job at the university and to the end of his marriage.
Their daughter, Josephine, remembered their battles during World War I. "We fought the World War right in our house in Scarsdale. Mother was for the Allies and Father was for the Germans. Mother hung the Allied flag out her study window and Father put the German flag out his study window. They fought the war upstairs and downstairs, into the attic and into the cellar. It was too much for me and I fell apart. They saw what they were doing to me and finally agreed to separate for my sake."
After they separated, he moved to China, where he played a key role in establishing Chinese geology. Along with expatriates and Chinese academics, Grabau recruited and trained Chinese geologists and set up institutions devoted to geology. He never reconciled with his wife, and he only returned to the United States once for the 1933 International Geological Congress.
Their separation marked the beginning of Mary's decline. Her third book didn't have nearly the success of The Promised Land. She wrote no fourth book. She withdrew from her friends and family, and suffered from health problems – both physical and mental.
Mary spent several years at the Austen Riggs Psychiatric Center in Stockbridge, Mass. From 1922 to her death in 1949, she checked in and out of the Gould Farm, a healing community for people "experiencing emotional and psychiatric vulnerabilities" in the Berkshire mountains.
In 1930 she wrote to a friend, "I have so little mastered the art of tranquil living that wherever go I trail storm clouds of drama around me."
Mehar Baba.
She eventually followed of Meher Baba, an Indian mystic who claimed he was the avatar. Baba stopped speaking on July 10, 1925 and communicated using hand gestures or an alphabet board for the rest of his life. He traveled to Hollywood in the 1930s, where he enjoyed the attention of celebrities like Gary Cooper and Tallulah Bankhead.
Antin then followed Rudolf Steiner, an Austrian clairvoyant who founded another esoteric spiritual movement, anthroposophy.
She died in 1949 of cancer. By then, The Promised Land had gone through 34 editions. The book opened with a bold statement: "I was born, I have lived, and I have been made over."
But perhaps Mary Antin had wanted more than to be an assimilated immigrant. "One of the best novels I never wrote is called The Unwilling Celebrity," she wrote in a letter. "It deals with the embarrassment of a woman who never succeeded in reconciling a large measure of public recognition with her insufficient achievement."
With thanks to Sarah Blacker Cohen in Mary Antin's "The Promised Land": A Breach of Promise in Studies in American Jewish Literature, Winter 1977-78.
Also Keren R. McGinity in The Real Mary Antin: Woman on a Mission in the Promised Land, American Jewish History Sept. 1998.
And to Alan Mazur, A ROMANCE IN NATURAL HISTORY; THE LIVES AND WORKS OF AMADEUS GRABAU AND MARY ANTIN.
This story updated in 2021.
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Edward Hammond Clarke and the Chance for Girls
Education was a hot topic in the 1870. How much education should one have and who should get it? Edward Hammond Clarke sought to put the debate into a new perspective – before too much education made everyone sick.
Edward Hammond Clarke (Harvard University Archives)
By the 1870s, there was an established consensus that for boys education should extend well beyond the simple reading and writing demanded in colonial days. Math, science and philosophy were all valued subjects.
For girls, the debate still percolated. Girls had finishing schools and schools that taught home economics. But many schools discouraged girls from studying science and math.
Emma Willard
Connecticut's Emma Willard had been fighting this trend as early as 1819 with publication of her Plan for Improving Female Education. Willard opened a school for girls in Troy, N.Y., the following year. Catherine Beecher opened a school for girls in Hartford, Conn. in 1823. Public high schools opened for girls in Boston and New York in 1826.
Inevitably, greater education led to greater demands for equality in other areas. For example, while Willard never supported the women's suffrage movement, the girls who graduated her school did. As more and more girls received a high school education, more began contemplating college.
Edward Hammond Clarke
Edward Hammond Clarke found that idea dangerous as well as disturbing.
Clarke, a Harvard-trained physician, moved in the right Boston social circles. His patients included abolitionist William Lloyd Garrison. Born in 1820, he received his Harvard degree in 1841. He took his medical training in Philadelphia.
In 1872 the New England Women's Club, an organization of progressive Boston women, invited Clarke to speak. By then Clarke had a position as a professor at the Harvard Medical College and had reached peak of his profession. He had a reputation for treating medical disorders successfully with the latest medications.
New England Women's Club headquarters, ca 1903
Clarke chose as the topic for his speech the appropriate education for girls. The thrust of Clarke's talk was that educating girls was fraught with peril. If girls during the ages of 13 to 17 spent too much time learning, the efforts they put into developing their brains would hinder the needed growth of their ovaries and uterus.
The state of womanhood was dire, Clarke observed in his practice, as disease ran rampant. Food and fashion were in part to blame, he acknowledged. Too much cake and pie contributed their part. Corsets and binding clothes were to blame for some problems, as well. But not all disease could be explained this way.
"Leucorrhoea, amenorrhea, dysmenorrhea, chronic and acute ovaritis, prolapsis utari, hysteria, neuralgia and the like are indirectly affected by food, clothing and exercise; they are directly and largely affected by the causes that will presently be pointed out and which arise from a neglect of the peculiarities of a woman's organization," Clarke said. "The regimen of a college arranged for boys, if imposed on girls, would foster it even more."
Girls vs. Boys
Clarke's theory was straightforward. "Brain work and stomach work interfere with each other if attempted together."
This put girls at a particular disadvantage. Boys, he postulated, entered the world much more fully developed than girls. They would withstand the rigors of school and college and still turn into men with functioning reproductive organs. Girls simply couldn't do both. What's more, too much time in school doomed women to a lifetime of sickliness, or as he put it: "A youth of study and an old age of nerves."
Clarke believed that boys in adolescence would withstand six hours of schooling a day and occasionally as much as eight. Girls should be limited to four and never exceed five.
More time spent studying contributed to the puny, unhealthy population Clarke was treating in his practice. The brain and the body were at war, and the brain was too often winning.
To buttress his observations, Clarke told of a recent trip to Halifax. There he noted that boys of 11 grew as tall and broad-shouldered as a 16-year-old from Massachusetts. And the young girls? "Girls of 10 or 11 were there who looked almost like women – that is, like ideal women – simply because they looked so calm and undisturbed."
At civic functions on his Canadian trip, Clarke noted, he was surrounded by healthy and happy men and women. The reason, he concluded, was that Halifax had no public schools.
Clarke's presentation caused quite an uproar. He would research further and publish his theories in book form in 1873: Sex in Education: Or, A Fair Chance for the Girls. In the book, he added research from Europe that argued girls flourished when they dropped out of school as adolescents and pursued their studies more leisurely at home. Clarke then visited the National Education Association in 1874 to deliver his findings.
Clarke's own life may have driven his thinking. He was a studious but sickly youngster, too sick to attend his graduation exercises. He also suffered from maladies of his digestive system throughout his life. Clarke died in 1877 – just four years after he published his book on education for girls.
Nevertheless, his book had influence that lasted long after his death. The book went through 17 printings. The American Association of University Women, founded in 1883, spent decades funding research to debunk Clarke's theories.
This story about Edward Hammond Clarke was updated in 2023.
Hold Please: George Coy Launched the First Commercial Telephone Exchange
In January of 1878, George Coy of New Haven, Conn., gave birth to a new American tradition – the wrong number.
Butt dialing, prank phone calls and the busy signal all owe their existence to Coy, an enterprising inventor inspired by a lecture by Alexander Graham Bell.
Telephone operator, 1911.
Bell had unleashed his telephone invention on the world in 1876, and for the next couple of years the devices were mainly curiosities. To use them, you had to buy two, then string wires between them. So someone with two buildings might find it helpful to connect them via a telephone. But for most people the invention was not particularly useful.
George Coy
Bell knew this; all his partners and competitors knew this. So he began demonstrating the advantages of a three-way phone network. George Coy attended one of Bell's lectures the Skiff Opera House in New Haven in 1877. There the inventor demonstrated a call connecting Hartford and Middletown.
Impressed, Coy rounded up backers and bought a franchise to license Bell's technology and opened the first commercial telephone exchange in the United States in a storefront in the now-demolished Boardman Building in New Haven.
The exchange itself was not exactly a technological marvel, built from pot lids, carriage bolts and whatever else Coy could find, but it worked. It was capable of handling 64 customers. However, only two calls could take place simultaneously and the operator had to use six switches to put a call through.
Nevertheless, it meant telephone owners could now talk to someone else without buying them a phone and connecting their own wires. In January the exchange opened with 21 subscribers, each paying $1.50 per month. Physicians, businesses and police were early adopters. Within a month, Coy had 50 subscribers and had to publish the first phone directory.
Before long, telephone exchanges were springing up across the country. The New Haven District Telephone Company grew rapidly. By 1882, it was rebranded the Southern New England Telephone Co., with the rights to serve all of Connecticut.
This story last updated in 2022. | {
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<?php
namespace Opl\Dependency\Exception;
/**
* The exception that specifies that the interface contract failed.
*
* @author Tomasz Jędrzejewski
* @copyright Invenzzia Group <http://www.invenzzia.org/> and contributors.
* @license http://www.invenzzia.org/license/new-bsd New BSD License
*/
class ContractException extends LocatorException
{
} // end ContractException; | {
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{"url":"https:\/\/xianblog.wordpress.com\/tag\/g-h-hardy\/","text":"## the Ramanujan machine\n\nPosted in Books, Kids, pictures, University life with tags , , , , , , , , , , , on February 18, 2021 by xi'an\n\nNature of 4 Feb. 2021 offers a rather long (Nature-like) paper on creating Ramanujan-like expressions using an automated process. Associated with a cover in the first pages. The purpose of the AI is to generate conjectures of Ramanujan-like formulas linking famous constants like \u03c0 or e and algebraic formulas like the novel polynomial continued fraction of 8\/\u03c0\u00b2:\n\n$\\frac{8}{{{\\rm{\\pi }}}^{2}}=1-\\frac{2\\times {1}^{4}-{1}^{3}}{7-\\frac{2\\times {2}^{4}-{2}^{3}}{19-\\frac{2\\times {3}^{4}-{3}^{3}}{37-\\frac{2\\times {4}^{4}-{4}^{3}}{\\ldots }}}}$\n\nwhich currently remains unproven. The authors of the \u201cmachine\u201d provide Python code that one can run to try uncover new conjectures, possibly named after the discoverer! The article is spending a large proportion of its contents to justify the appeal of generating such conjectures, with several unsuspected formulas later proven for real, but I remain unconvinced of the deeper appeal of the machine (as well as unhappy about the association of Ramanujan and machine, since S. Ramanujan had a mystical and unexplained relation to numbers, defeating Hardy\u2019s logic,\u00a0 \u201ca mathematician of the highest quality, a man of altogether exceptional originality and power\u201d). The difficulty is in separating worthwhile from anecdotal (true) conjectures, not to mention wrng conjectures. This is certainly of much deeper interest than separating chihuahua faces from blueberry muffins, but does it really \u201chelp to create mathematical knowledge\u201d?","date":"2022-08-16 09:51:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 1, \"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\": 0.8005528450012207, \"perplexity\": 2588.65365442243}, \"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-2022-33\/segments\/1659882572286.44\/warc\/CC-MAIN-20220816090541-20220816120541-00226.warc.gz\"}"} | null | null |
Illinois woman pleads guilty to meth delivery
On behalf of The Law Office of Jessica Koester, LLC | Aug 5, 2014 | Drug Charges
A Griggsville resident entered a plea of guilty to a charge of methamphetamine delivery in Adams County Circuit Court on Aug. 4. The 48-year-old woman is scheduled for sentencing on Sept. 25, and will face up to six years imprisonment. A man charged as a co-conspirator, a 43-year-old Quincy resident, was sentenced to six years imprisonment for an identical charge in May. In exchange for the defendant's guilty plea, prosecutors agreed to drop a charge of aggravated unlawful participation in meth manufacturing; she could have been sentenced to up to 15 years imprisonment had she not agreed to the plea deal.
An investigation into methamphetamine trafficking from Missouri by the West Central Illinois Task Force eventually led law enforcement to search the home where the woman and the other defendant had been staying in the 900 block of North 12th. Police claimed to have discovered 6.7 grams of methamphetamine stashed beneath a sofa seat cushion and also allegedly seized drug paraphernalia, scales and packaging. Police also reported that a child was in the house when the search warrant was executed.
According the assistant state's attorney, the woman admitted to police that she bought, sold and used methamphetamine. The assistant state's attorney also stated that a confidential source had shot a video of the woman with drugs in her possession.
Anyone accused of a drug charge may wish to employ a criminal defense attorney. A criminal defense attorney in a case like this one might attempt to argue that the prosecution's evidence had been obtained illegally. The attorney might also attempt to arrange for a favorable plea bargain, such as the plea deal in this case.
Source: Quincy Herald-Whig, "Griggsville woman admits guilt in meth case", Don O'Brien, August 04, 2014 | {
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Q: Is there any way to lower the fps on a mac? So I play this game on safari (mac). Freeriderhd.com to be exact. I want to know if there is any way to lower the fps on that specific game while I am playing it on safari because it would make it a lot easier. If there is no way to do this, can I use a macro to click the spacebar an infinite amount of times with x amount of time in between each click? If someone can help with any of these questions that would be great. Thanks.
A: I don't really know - sorry - but this seems unlikely. FPS is usually controlled internally by the game software. I have seen a very few games that allow the user to change the frame rate, or at least the requested frame rate. This is usually for the purpose of raising the FPS, not lowering it. Usually the game has an optimum frame rate that it strives for. Especially browser-based ones (which I've written some of).
Your alternative sounds like a means to "bog down the browser" which might peg your processor; not a thing I would tempt the Fates with, personally.
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{"url":"https:\/\/openstax.org\/books\/chemistry-2e\/pages\/12-exercises","text":"Chemistry 2e\n\n# Exercises\n\nChemistry 2eExercises\n\n### 12.1Chemical Reaction Rates\n\n1.\n\nWhat is the difference between average rate, initial rate, and instantaneous rate?\n\n2.\n\nOzone decomposes to oxygen according to the equation $2O3(g)\u27f63O2(g).2O3(g)\u27f63O2(g).$ Write the equation that relates the rate expressions for this reaction in terms of the disappearance of O3 and the formation of oxygen.\n\n3.\n\nIn the nuclear industry, chlorine trifluoride is used to prepare uranium hexafluoride, a volatile compound of uranium used in the separation of uranium isotopes. Chlorine trifluoride is prepared by the reaction $Cl2(g)+3F2(g)\u27f62ClF3(g).Cl2(g)+3F2(g)\u27f62ClF3(g).$ Write the equation that relates the rate expressions for this reaction in terms of the disappearance of Cl2 and F2 and the formation of ClF3.\n\n4.\n\nA study of the rate of dimerization of C4H6 gave the data shown in the table:\n$2C4H6\u27f6C8H122C4H6\u27f6C8H12$\n\n Time (s) 0 1600 3200 4800 6200 [C4H6] (M) 1.00 $\u00d7\u00d7$ 10\u22122 5.04 $\u00d7\u00d7$ 10\u22123 3.37 $\u00d7\u00d7$ 10\u22123 2.53 $\u00d7\u00d7$ 10\u22123 2.08 $\u00d7\u00d7$ 10\u22123\n\n(a) Determine the average rate of dimerization between 0 s and 1600 s, and between 1600 s and 3200 s.\n\n(b) Estimate the instantaneous rate of dimerization at 3200 s from a graph of time versus [C4H6]. What are the units of this rate?\n\n(c) Determine the average rate of formation of C8H12 at 1600 s and the instantaneous rate of formation at 3200 s from the rates found in parts (a) and (b).\n\n5.\n\nA study of the rate of the reaction represented as $2A\u27f6B2A\u27f6B$ gave the following data:\n\n Time (s) 0 5 10 15 20 25 35 [A] (M) 1 0.775 0.625 0.465 0.36 0.285 0.23\n\n(a) Determine the average rate of disappearance of A between 0.0 s and 10.0 s, and between 10.0 s and 20.0 s.\n\n(b) Estimate the instantaneous rate of disappearance of A at 15.0 s from a graph of time versus [A]. What are the units of this rate?\n\n(c) Use the rates found in parts (a) and (b) to determine the average rate of formation of B between 0.00 s and 10.0 s, and the instantaneous rate of formation of B at 15.0 s.\n\n6.\n\nConsider the following reaction in aqueous solution:\n$5Br\u2212(aq)+BrO3\u2212(aq)+6H+(aq)\u27f63Br2(aq)+3H2O(l)5Br\u2212(aq)+BrO3\u2212(aq)+6H+(aq)\u27f63Br2(aq)+3H2O(l)$\n\nIf the rate of disappearance of Br(aq) at a particular moment during the reaction is 3.5 $\u00d7\u00d7$ 10\u22124 mol L\u22121 s\u22121, what is the rate of appearance of Br2(aq) at that moment?\n\n### 12.2Factors Affecting Reaction Rates\n\n7.\n\nDescribe the effect of each of the following on the rate of the reaction of magnesium metal with a solution of hydrochloric acid: the molarity of the hydrochloric acid, the temperature of the solution, and the size of the pieces of magnesium.\n\n8.\n\nExplain why an egg cooks more slowly in boiling water in Denver than in New York City. (Hint: Consider the effect of temperature on reaction rate and the effect of pressure on boiling point.)\n\n9.\n\nGo to the PhET Reactions & Rates interactive. Use the Single Collision tab to represent how the collision between monatomic oxygen (O) and carbon monoxide (CO) results in the breaking of one bond and the formation of another. Pull back on the red plunger to release the atom and observe the results. Then, click on \u201cReload Launcher\u201d and change to \u201cAngled shot\u201d to see the difference.\n\n(a) What happens when the angle of the collision is changed?\n\n(b) Explain how this is relevant to rate of reaction.\n\n10.\n\nIn the PhET Reactions & Rates interactive, use the \u201cMany Collisions\u201d tab to observe how multiple atoms and molecules interact under varying conditions. Select a molecule to pump into the chamber. Set the initial temperature and select the current amounts of each reactant. Select \u201cShow bonds\u201d under Options. How is the rate of the reaction affected by concentration and temperature?\n\n11.\n\nIn the PhET Reactions & Rates interactive, on the Many Collisions tab, set up a simulation with 15 molecules of A and 10 molecules of BC. Select \u201cShow Bonds\u201d under Options.\n\n(a) Leave the Initial Temperature at the default setting. Observe the reaction. Is the rate of reaction fast or slow?\n\n(b) Click \u201cPause\u201d and then \u201cReset All,\u201d and then enter 15 molecules of A and 10 molecules of BC once again. Select \u201cShow Bonds\u201d under Options. This time, increase the initial temperature until, on the graph, the total average energy line is completely above the potential energy curve. Describe what happens to the reaction.\n\n### 12.3Rate Laws\n\n12.\n\nHow do the rate of a reaction and its rate constant differ?\n\n13.\n\nDoubling the concentration of a reactant increases the rate of a reaction four times. With this knowledge, answer the following questions:\n\n(a) What is the order of the reaction with respect to that reactant?\n\n(b) Tripling the concentration of a different reactant increases the rate of a reaction three times. What is the order of the reaction with respect to that reactant?\n\n14.\n\nTripling the concentration of a reactant increases the rate of a reaction nine-fold. With this knowledge, answer the following questions:\n\n(a) What is the order of the reaction with respect to that reactant?\n\n(b) Increasing the concentration of a reactant by a factor of four increases the rate of a reaction four-fold. What is the order of the reaction with respect to that reactant?\n\n15.\n\nHow will the rate of reaction change for the process: $CO(g)+NO2(g)\u27f6CO2(g)+NO(g)CO(g)+NO2(g)\u27f6CO2(g)+NO(g)$ if the rate law for the reaction is $rate=k[NO2]2?rate=k[NO2]2?$\n\n(a) Decreasing the pressure of NO2 from 0.50 atm to 0.250 atm.\n\n(b) Increasing the concentration of CO from 0.01 M to 0.03 M.\n\n16.\n\nHow will each of the following affect the rate of the reaction: $CO(g)+NO2(g)\u27f6CO2(g)+NO(g)CO(g)+NO2(g)\u27f6CO2(g)+NO(g)$ if the rate law for the reaction is $rate=k[NO2][CO]rate=k[NO2][CO]$?\n\n(a) Increasing the pressure of NO2 from 0.1 atm to 0.3 atm\n\n(b) Increasing the concentration of CO from 0.02 M to 0.06 M.\n\n17.\n\nRegular flights of supersonic aircraft in the stratosphere are of concern because such aircraft produce nitric oxide, NO, as a byproduct in the exhaust of their engines. Nitric oxide reacts with ozone, and it has been suggested that this could contribute to depletion of the ozone layer. The reaction $NO+O3\u27f6NO2+O2NO+O3\u27f6NO2+O2$ is first order with respect to both NO and O3 with a rate constant of 2.20 $\u00d7\u00d7$ 107 L\/mol\/s. What is the instantaneous rate of disappearance of NO when [NO] = 3.3 $\u00d7\u00d7$ 10\u22126 M and [O3] = 5.9 $\u00d7\u00d7$ 10\u22127 M?\n\n18.\n\nRadioactive phosphorus is used in the study of biochemical reaction mechanisms because phosphorus atoms are components of many biochemical molecules. The location of the phosphorus (and the location of the molecule it is bound in) can be detected from the electrons (beta particles) it produces:\n$1532P\u27f6 1632S+e\u22121532P\u27f6 1632S+e\u2212$\nrate = 4.85 $\u00d7\u00d7$ 10\u22122 $day\u22121[32P]day\u22121[32P]$\n\nWhat is the instantaneous rate of production of electrons in a sample with a phosphorus concentration of 0.0033 M?\n\n19.\n\nThe rate constant for the radioactive decay of 14C is 1.21 $\u00d7\u00d7$ 10\u22124 year\u22121. The products of the decay are nitrogen atoms and electrons (beta particles):\n$614C\u27f6714N+e\u2212614C\u27f6714N+e\u2212$\n$rate=k[614C]rate=k[614C]$\n\nWhat is the instantaneous rate of production of N atoms in a sample with a carbon-14 content of 6.5 $\u00d7\u00d7$ 10\u22129 M?\n\n20.\n\nThe decomposition of acetaldehyde is a second order reaction with a rate constant of 4.71 $\u00d7\u00d7$ 10\u22128 L mol\u22121 s\u22121. What is the instantaneous rate of decomposition of acetaldehyde in a solution with a concentration of 5.55 $\u00d7\u00d7$ 10\u22124 M?\n\n21.\n\nAlcohol is removed from the bloodstream by a series of metabolic reactions. The first reaction produces acetaldehyde; then other products are formed. The following data have been determined for the rate at which alcohol is removed from the blood of an average male, although individual rates can vary by 25\u201330%. Women metabolize alcohol a little more slowly than men:\n\n [C2H5OH] (M) 4.4 $\u00d7\u00d7$ 10\u22122 3.3 $\u00d7\u00d7$ 10\u22122 2.2 $\u00d7\u00d7$ 10\u22122 Rate (mol L\u22121 h\u22121) 2.0 $\u00d7\u00d7$ 10\u22122 2.0 $\u00d7\u00d7$ 10\u22122 2.0 $\u00d7\u00d7$ 10\u22122\n\nDetermine the rate law, the rate constant, and the overall order for this reaction.\n\n22.\n\nUnder certain conditions the decomposition of ammonia on a metal surface gives the following data:\n\n [NH3] (M) 1.0 $\u00d7\u00d7$ 10\u22123 2.0 $\u00d7\u00d7$ 10\u22123 3.0 $\u00d7\u00d7$ 10\u22123 Rate (mol L\u22121 h\u22121) 1.5 $\u00d7\u00d7$ 10\u22126 1.5 $\u00d7\u00d7$ 10\u22126 1.5 $\u00d7\u00d7$ 10\u22126\n\nDetermine the rate law, the rate constant, and the overall order for this reaction.\n\n23.\n\nNitrosyl chloride, NOCl, decomposes to NO and Cl2.\n$2NOCl(g)\u27f62NO(g)+Cl2(g)2NOCl(g)\u27f62NO(g)+Cl2(g)$\n\nDetermine the rate law, the rate constant, and the overall order for this reaction from the following data:\n\n [NOCl] (M) 0.10 0.20 0.30 Rate (mol L\u22121 h\u22121) 8.0 $\u00d7\u00d7$ 10\u221210 3.2 $\u00d7\u00d7$ 10\u22129 7.2 $\u00d7\u00d7$ 10\u22129\n24.\n\nFrom the following data, determine the rate law, the rate constant, and the order with respect to A for the reaction $A\u27f62C.A\u27f62C.$\n\n [A] (M) 1.33 $\u00d7\u00d7$ 10\u22122 2.66 $\u00d7\u00d7$ 10\u22122 3.99 $\u00d7\u00d7$ 10\u22122 Rate (mol L\u22121 h\u22121) 3.80 $\u00d7\u00d7$ 10\u22127 1.52 $\u00d7\u00d7$ 10\u22126 3.42 $\u00d7\u00d7$ 10\u22126\n25.\n\nNitrogen monoxide reacts with chlorine according to the equation:\n$2NO(g)+Cl2(g)\u27f62NOCl(g)2NO(g)+Cl2(g)\u27f62NOCl(g)$\n\nThe following initial rates of reaction have been observed for certain reactant concentrations:\n\n[NO] (mol\/L) [Cl2] (mol\/L) Rate (mol L\u22121 h\u22121)\n0.50 0.50 1.14\n1.00 0.50 4.56\n1.00 1.00 9.12\n\nWhat is the rate law that describes the rate\u2019s dependence on the concentrations of NO and Cl2? What is the rate constant? What are the orders with respect to each reactant?\n\n26.\n\nHydrogen reacts with nitrogen monoxide to form dinitrogen monoxide (laughing gas) according to the equation: $H2(g)+2NO(g)\u27f6N2O(g)+H2O(g)H2(g)+2NO(g)\u27f6N2O(g)+H2O(g)$\n\nDetermine the rate law, the rate constant, and the orders with respect to each reactant from the following data:\n\n [NO] (M) 0.30 0.60 0.60 [H2] (M) 0.35 0.35 0.70 Rate (mol L\u22121 s\u22121) 2.835 $\u00d7\u00d7$ 10\u22123 1.134 $\u00d7\u00d7$ 10\u22122 2.268 $\u00d7\u00d7$ 10\u22122\n27.\n\nFor the reaction $A\u27f6B+C,A\u27f6B+C,$ the following data were obtained at 30 \u00b0C:\n\n [A] (M) 0.230 0.356 0.557 Rate (mol L\u22121 s\u22121) 4.17 $\u00d7\u00d7$ 10\u22124 9.99 $\u00d7\u00d7$ 10\u22124 2.44 $\u00d7\u00d7$ 10\u22123\n\n(a) What is the order of the reaction with respect to [A], and what is the rate law?\n\n(b) What is the rate constant?\n\n28.\n\nFor the reaction $Q\u27f6W+X,Q\u27f6W+X,$ the following data were obtained at 30 \u00b0C:\n\n [Q]initial (M) 0.170 0.212 0.357 Rate (mol L\u22121 s\u22121) 6.68 $\u00d7\u00d7$ 10\u22123 1.04 $\u00d7\u00d7$ 10\u22122 2.94 $\u00d7\u00d7$ 10\u22122\n\n(a) What is the order of the reaction with respect to [Q], and what is the rate law?\n\n(b) What is the rate constant?\n\n29.\n\nThe rate constant for the first-order decomposition at 45 \u00b0C of dinitrogen pentoxide, N2O5, dissolved in chloroform, CHCl3, is 6.2 $\u00d7\u00d7$ 10\u22124 min\u22121.\n$2N2O5\u27f64NO2+O22N2O5\u27f64NO2+O2$\n\nWhat is the rate of the reaction when [N2O5] = 0.40 M?\n\n30.\n\nThe annual production of HNO3 in 2013 was 60 million metric tons Most of that was prepared by the following sequence of reactions, each run in a separate reaction vessel.\n\n(a) $4NH3(g)+5O2(g)\u27f64NO(g)+6H2O(g)4NH3(g)+5O2(g)\u27f64NO(g)+6H2O(g)$\n\n(b) $2NO(g)+O2(g)\u27f62NO2(g)2NO(g)+O2(g)\u27f62NO2(g)$\n\n(c) $3NO2(g)+H2O(l)\u27f62HNO3(aq)+NO(g)3NO2(g)+H2O(l)\u27f62HNO3(aq)+NO(g)$\n\nThe first reaction is run by burning ammonia in air over a platinum catalyst. This reaction is fast. The reaction in equation (c) is also fast. The second reaction limits the rate at which nitric acid can be prepared from ammonia. If equation (b) is second order in NO and first order in O2, what is the rate of formation of NO2 when the oxygen concentration is 0.50 M and the nitric oxide concentration is 0.75 M? The rate constant for the reaction is 5.8 $\u00d7\u00d7$ 10\u22126 L2 mol\u22122 s\u22121.\n\n31.\n\nThe following data have been determined for the reaction:\n$I\u2212+OCl\u2212\u27f6IO\u2212+Cl\u2212I\u2212+OCl\u2212\u27f6IO\u2212+Cl\u2212$\n\n1 2 3\n$[I\u2212]initial[I\u2212]initial$ (M) 0.10 0.20 0.30\n$[OCl\u2212]initial[OCl\u2212]initial$ (M) 0.050 0.050 0.010\nRate (mol L\u22121 s\u22121) 3.05 $\u00d7\u00d7$ 10\u22124 6.20 $\u00d7\u00d7$ 10\u22124 1.83 $\u00d7\u00d7$ 10\u22124\n\nDetermine the rate law and the rate constant for this reaction.\n\n### 12.4Integrated Rate Laws\n\n32.\n\nDescribe how graphical methods can be used to determine the order of a reaction and its rate constant from a series of data that includes the concentration of A at varying times.\n\n33.\n\nUse the data provided to graphically determine the order and rate constant of the following reaction: $SO2Cl2\u27f6SO2+Cl2SO2Cl2\u27f6SO2+Cl2$\n\n Time (s) 0 5.00 $\u00d7\u00d7$ 103 1.00 $\u00d7\u00d7$ 104 1.50 $\u00d7\u00d7$ 104 [SO2Cl2] (M) 0.100 0.0896 0.0802 0.0719 Time (s) 2.50 $\u00d7\u00d7$ 104 3.00 $\u00d7\u00d7$ 104 4.00 $\u00d7\u00d7$ 104 [SO2Cl2] (M) 0.0577 0.0517 0.0415\n34.\n\nPure ozone decomposes slowly to oxygen, $2O3(g)\u27f63O2(g).2O3(g)\u27f63O2(g).$ Use the data provided in a graphical method and determine the order and rate constant of the reaction.\n\n Time (h) 0 2.0 $\u00d7\u00d7$ 103 7.6 $\u00d7\u00d7$ 103 1.00 $\u00d7\u00d7$ 104 [O3] (M) 1.00 $\u00d7\u00d7$ 10\u22125 4.98 $\u00d7\u00d7$ 10\u22126 2.07 $\u00d7\u00d7$ 10\u22126 1.66 $\u00d7\u00d7$ 10\u22126 Time (h) 1.23 $\u00d7\u00d7$ 104 1.43 $\u00d7\u00d7$ 104 1.70 $\u00d7\u00d7$ 104 [O3] (M) 1.39 $\u00d7\u00d7$ 10\u22126 1.22 $\u00d7\u00d7$ 10\u22126 1.05 $\u00d7\u00d7$ 10\u22126\n35.\n\nFrom the given data, use a graphical method to determine the order and rate constant of the following reaction:\n$2X\u27f6Y+Z2X\u27f6Y+Z$\n\n Time (s) 5 10 15 20 25 30 35 40 [X] (M) 0.099 0.0497 0.0332 0.0249 0.02 0.0166 0.0143 0.0125\n36.\n\nWhat is the half-life for the first-order decay of phosphorus-32? $( 1532P \u27f6 1632S +e\u2212)(1532P\u27f61632S+e\u2212)$ The rate constant for the decay is 4.85 $\u00d7\u00d7$ 10\u22122 day\u22121.\n\n37.\n\nWhat is the half-life for the first-order decay of carbon-14? $( 614C \u27f6 714N +e\u2212)(614C\u27f6714N+e\u2212)$ The rate constant for the decay is 1.21 $\u00d7\u00d7$ 10\u22124 year\u22121.\n\n38.\n\nWhat is the half-life for the decomposition of NOCl when the concentration of NOCl is 0.15 M? The rate constant for this second-order reaction is 8.0 $\u00d7\u00d7$ 10\u22128 L mol\u22121 s\u22121.\n\n39.\n\nWhat is the half-life for the decomposition of O3 when the concentration of O3 is 2.35 $\u00d7\u00d7$ 10\u22126 M? The rate constant for this second-order reaction is 50.4 L mol\u22121 h\u22121.\n\n40.\n\nThe reaction of compound\u00a0\u00a0A to give compounds\u00a0\u00a0C and\u00a0\u00a0D was found to be second-order in\u00a0\u00a0A. The rate constant for the reaction was determined to be 2.42 L mol\u22121 s\u22121. If the initial concentration is 0.500 mol\/L, what is the value of t1\/2?\n\n41.\n\nThe half-life of a reaction of compound A to give compounds D and E is 8.50 min when the initial concentration of A is 0.150 M. How long will it take for the concentration to drop to 0.0300 M if the reaction is (a) first order with respect to A or (b) second order with respect to A?\n\n42.\n\nSome bacteria are resistant to the antibiotic penicillin because they produce penicillinase, an enzyme with a molecular weight of 3 $\u00d7\u00d7$ 104 g\/mol that converts penicillin into inactive molecules. Although the kinetics of enzyme-catalyzed reactions can be complex, at low concentrations this reaction can be described by a rate law that is first order in the catalyst (penicillinase) and that also involves the concentration of penicillin. From the following data: 1.0 L of a solution containing 0.15 \u00b5g (0.15 $\u00d7\u00d7$ 10\u22126 g) of penicillinase, determine the order of the reaction with respect to penicillin and the value of the rate constant.\n\n[Penicillin] (M) Rate (mol L\u22121 min\u22121)\n2.0 $\u00d7\u00d7$ 10\u22126 1.0 $\u00d7\u00d7$ 10\u221210\n3.0 $\u00d7\u00d7$ 10\u22126 1.5 $\u00d7\u00d7$ 10\u221210\n4.0 $\u00d7\u00d7$ 10\u22126 2.0 $\u00d7\u00d7$ 10\u221210\n43.\n\nBoth technetium-99 and thallium-201 are used to image heart muscle in patients with suspected heart problems. The half-lives are 6 h and 73 h, respectively. What percent of the radioactivity would remain for each of the isotopes after 2 days (48 h)?\n\n44.\n\nThere are two molecules with the formula C3H6. Propene, $CH3CH=CH2,CH3CH=CH2,$ is the monomer of the polymer polypropylene, which is used for indoor-outdoor carpets. Cyclopropane is used as an anesthetic:\n\nWhen heated to 499 \u00b0C, cyclopropane rearranges (isomerizes) and forms propene with a rate constant of\n5.95 $\u00d7\u00d7$ 10\u22124 s\u22121. What is the half-life of this reaction? What fraction of the cyclopropane remains after 0.75 h at 499 \u00b0C?\n\n45.\n\nFluorine-18 is a radioactive isotope that decays by positron emission to form oxygen-18 with a half-life of 109.7 min. (A positron is a particle with the mass of an electron and a single unit of positive charge; the equation is $918F \u27f6 818O ++10e)918F\u27f6818O++10e)$ Physicians use 18F to study the brain by injecting a quantity of fluoro-substituted glucose into the blood of a patient. The glucose accumulates in the regions where the brain is active and needs nourishment.\n\n(a) What is the rate constant for the decomposition of fluorine-18?\n\n(b) If a sample of glucose containing radioactive fluorine-18 is injected into the blood, what percent of the radioactivity will remain after 5.59 h?\n\n(c) How long does it take for 99.99% of the 18F to decay?\n\n46.\n\nSuppose that the half-life of steroids taken by an athlete is 42 days. Assuming that the steroids biodegrade by a first-order process, how long would it take for $164164$ of the initial dose to remain in the athlete\u2019s body?\n\n47.\n\nRecently, the skeleton of King Richard III was found under a parking lot in England. If tissue samples from the skeleton contain about 93.79% of the carbon-14 expected in living tissue, what year did King Richard III die? The half-life for carbon-14 is 5730 years.\n\n48.\n\nNitroglycerine is an extremely sensitive explosive. In a series of carefully controlled experiments, samples of the explosive were heated to 160 \u00b0C and their first-order decomposition studied. Determine the average rate constants for each experiment using the following data:\n\n Initial [C3H5N3O9] (M) 4.88 3.52 2.29 1.81 5.33 4.05 2.95 1.72 t (s) 300 300 300 300 180 180 180 180 % Decomposed 52 52.9 53.2 53.9 34.6 35.9 36 35.4\n49.\n\nFor the past 10 years, the unsaturated hydrocarbon 1,3-butadiene $(CH2=CH\u2013CH=CH2)(CH2=CH\u2013CH=CH2)$ has ranked 38th among the top 50 industrial chemicals. It is used primarily for the manufacture of synthetic rubber. An isomer exists also as cyclobutene:\n\nThe isomerization of cyclobutene to butadiene is first-order and the rate constant has been measured as 2.0 $\u00d7\u00d7$ 10\u22124 s\u22121 at 150 \u00b0C in a 0.53-L flask. Determine the partial pressure of cyclobutene and its concentration after 30.0 minutes if an isomerization reaction is carried out at 150 \u00b0C with an initial pressure of 55 torr.\n\n### 12.5Collision Theory\n\n50.\n\nChemical reactions occur when reactants collide. What are two factors that may prevent a collision from producing a chemical reaction?\n\n51.\n\nWhen every collision between reactants leads to a reaction, what determines the rate at which the reaction occurs?\n\n52.\n\nWhat is the activation energy of a reaction, and how is this energy related to the activated complex of the reaction?\n\n53.\n\nAccount for the relationship between the rate of a reaction and its activation energy.\n\n54.\n\nDescribe how graphical methods can be used to determine the activation energy of a reaction from a series of data that includes the rate of reaction at varying temperatures.\n\n55.\n\nHow does an increase in temperature affect rate of reaction? Explain this effect in terms of the collision theory of the reaction rate.\n\n56.\n\nThe rate of a certain reaction doubles for every 10 \u00b0C rise in temperature.\n\n(a) How much faster does the reaction proceed at 45 \u00b0C than at 25 \u00b0C?\n\n(b) How much faster does the reaction proceed at 95 \u00b0C than at 25 \u00b0C?\n\n57.\n\nIn an experiment, a sample of NaClO3 was 90% decomposed in 48 min. Approximately how long would this decomposition have taken if the sample had been heated 20 \u00b0C higher? (Hint: Assume the rate doubles for each 10 \u00b0C rise in temperature.)\n\n58.\n\nThe rate constant at 325 \u00b0C for the decomposition reaction $C4H8\u27f62C2H4C4H8\u27f62C2H4$ is 6.1 $\u00d7\u00d7$ 10\u22128 s\u22121, and the activation energy is 261 kJ per mole of C4H8. Determine the frequency factor for the reaction.\n\n59.\n\nThe rate constant for the decomposition of acetaldehyde, CH3CHO, to methane, CH4, and carbon monoxide, CO, in the gas phase is 1.1 $\u00d7\u00d7$ 10\u22122 L mol\u22121 s\u22121 at 703 K and 4.95 L mol\u22121 s\u22121 at 865 K. Determine the activation energy for this decomposition.\n\n60.\n\nAn elevated level of the enzyme alkaline phosphatase (ALP) in human serum is an indication of possible liver or bone disorder. The level of serum ALP is so low that it is very difficult to measure directly. However, ALP catalyzes a number of reactions, and its relative concentration can be determined by measuring the rate of one of these reactions under controlled conditions. One such reaction is the conversion of p-nitrophenyl phosphate (PNPP) to p-nitrophenoxide ion (PNP) and phosphate ion. Control of temperature during the test is very important; the rate of the reaction increases 1.47 times if the temperature changes from 30 \u00b0C to 37 \u00b0C. What is the activation energy for the ALP\u2013catalyzed conversion of PNPP to PNP and phosphate?\n\n61.\n\nIn terms of collision theory, to which of the following is the rate of a chemical reaction proportional?\n\n(a) the change in free energy per second\n\n(b) the change in temperature per second\n\n(c) the number of collisions per second\n\n(d) the number of product molecules\n\n62.\n\nHydrogen iodide, HI, decomposes in the gas phase to produce hydrogen, H2, and iodine, I2. The value of the rate constant, k, for the reaction was measured at several different temperatures and the data are shown here:\n\nTemperature (K) k (L mol\u22121 s\u22121)\n555 6.23 $\u00d7\u00d7$ 10\u22127\n575 2.42 $\u00d7\u00d7$ 10\u22126\n645 1.44 $\u00d7\u00d7$ 10\u22124\n700 2.01 $\u00d7\u00d7$ 10\u22123\n\nWhat is the value of the activation energy (in kJ\/mol) for this reaction?\n\n63.\n\nThe element Co exists in two oxidation states, Co(II) and Co(III), and the ions form many complexes. The rate at which one of the complexes of Co(III) was reduced by Fe(II) in water was measured. Determine the activation energy of the reaction from the following data:\n\nT (K) k (s\u22121)\n293 0.054\n298 0.100\n64.\n\nThe hydrolysis of the sugar sucrose to the sugars glucose and fructose,\n$C12H22O11+H2O\u27f6C6H12O6+C6H12O6C12H22O11+H2O\u27f6C6H12O6+C6H12O6$\n\nfollows a first-order rate law for the disappearance of sucrose: rate = k[C12H22O11] (The products of the reaction, glucose and fructose, have the same molecular formulas but differ in the arrangement of the atoms in their molecules.)\n\n(a) In neutral solution, k = 2.1 $\u00d7\u00d7$ 10\u221211 s\u22121 at 27 \u00b0C and 8.5 $\u00d7\u00d7$ 10\u221211 s\u22121 at 37 \u00b0C. Determine the activation energy, the frequency factor, and the rate constant for this equation at 47 \u00b0C (assuming the kinetics remain consistent with the Arrhenius equation at this temperature).\n\n(b) When a solution of sucrose with an initial concentration of 0.150 M reaches equilibrium, the concentration of sucrose is 1.65 $\u00d7\u00d7$ 10\u22127 M. How long will it take the solution to reach equilibrium at 27 \u00b0C in the absence of a catalyst? Because the concentration of sucrose at equilibrium is so low, assume that the reaction is irreversible.\n\n(c) Why does assuming that the reaction is irreversible simplify the calculation in part (b)?\n\n65.\n\nUse the PhET Reactions & Rates interactive simulation to simulate a system. On the \u201cSingle collision\u201d tab of the simulation applet, enable the \u201cEnergy view\u201d by clicking the \u201c+\u201d icon. Select the first $A+BC\u27f6AB+CA+BC\u27f6AB+C$ reaction (A is yellow, B is purple, and C is navy blue). Using the \u201cstraight shot\u201d default option, try launching the A atom with varying amounts of energy. What changes when the Total Energy line at launch is below the transition state of the Potential Energy line? Why? What happens when it is above the transition state? Why?\n\n66.\n\nUse the PhET Reactions & Rates interactive simulation to simulate a system. On the \u201cSingle collision\u201d tab of the simulation applet, enable the \u201cEnergy view\u201d by clicking the \u201c+\u201d icon. Select the first $A+BC\u27f6AB+CA+BC\u27f6AB+C$ reaction (A is yellow, B is purple, and C is navy blue). Using the \u201cangled shot\u201d option, try launching the A atom with varying angles, but with more Total energy than the transition state. What happens when the A atom hits the BC molecule from different directions? Why?\n\n### 12.6Reaction Mechanisms\n\n67.\n\nWhy are elementary reactions involving three or more reactants very uncommon?\n\n68.\n\nIn general, can we predict the effect of doubling the concentration of A on the rate of the overall reaction $A+B\u27f6CA+B\u27f6C$? Can we predict the effect if the reaction is known to be an elementary reaction?\n\n69.\n\nDefine these terms:\n\n(a) unimolecular reaction\n\n(b) bimolecular reaction\n\n(c) elementary reaction\n\n(d) overall reaction\n\n70.\n\nWhat is the rate law for the elementary termolecular reaction $A+2B\u27f6products?A+2B\u27f6products?$ For $3A\u27f6products?3A\u27f6products?$\n\n71.\n\nGiven the following reactions and the corresponding rate laws, in which of the reactions might the elementary reaction and the overall reaction be the same?\n\n$(a)Cl2+CO\u27f6Cl2COrate=k[Cl2]3\/2[CO](a)Cl2+CO\u27f6Cl2COrate=k[Cl2]3\/2[CO]$\n\n$(b)PCl3+Cl2\u27f6PCl5rate=k[PCl3][Cl2](b)PCl3+Cl2\u27f6PCl5rate=k[PCl3][Cl2]$\n\n$(c)2NO+H2\u27f6N2+H2O2rate=k[NO][H2](c)2NO+H2\u27f6N2+H2O2rate=k[NO][H2]$\n\n$(d)2NO+O2\u27f62NO2rate=k[NO]2[O2](d)2NO+O2\u27f62NO2rate=k[NO]2[O2]$\n\n$(e)NO+O3\u27f6NO2+O2rate=k[NO][O3](e)NO+O3\u27f6NO2+O2rate=k[NO][O3]$\n\n72.\n\nWrite the rate law for each of the following elementary reactions:\n\n(a) $O3\u2192sunlightO2+OO3\u2192sunlightO2+O$\n\n(b) $O3+Cl\u27f6O2+ClOO3+Cl\u27f6O2+ClO$\n\n(c) $ClO+O\u27f6Cl+O2ClO+O\u27f6Cl+O2$\n\n(d) $O3+NO\u27f6NO2+O2O3+NO\u27f6NO2+O2$\n\n(e) $NO2+O\u27f6NO+O2NO2+O\u27f6NO+O2$\n\n73.\n\nNitrogen monoxide, NO, reacts with hydrogen, H2, according to the following equation:\n$2NO+2H2\u27f6N2+2H2O2NO+2H2\u27f6N2+2H2O$\n\nWhat would the rate law be if the mechanism for this reaction were:\n$2NO+H2\u27f6N2+H2O2(slow)H2O2+H2\u27f62H2O(fast)2NO+H2\u27f6N2+H2O2(slow)H2O2+H2\u27f62H2O(fast)$\n\n74.\n\nExperiments were conducted to study the rate of the reaction represented by this equation.1\n$2NO(g)+2H2(g)\u27f6N2(g)+2H2O(g)2NO(g)+2H2(g)\u27f6N2(g)+2H2O(g)$\n\nInitial concentrations and rates of reaction are given here.\n\nExperiment Initial Concentration [NO] (mol L\u22121) Initial Concentration, [H2] (mol L\u22121 min\u22121) Initial Rate of Formation of N2 (mol L\u22121 min\u22121)\n1 0.0060 0.0010 1.8 $\u00d7\u00d7$ 10\u22124\n2 0.0060 0.0020 3.6 $\u00d7\u00d7$ 10\u22124\n3 0.0010 0.0060 0.30 $\u00d7\u00d7$ 10\u22124\n4 0.0020 0.0060 1.2 $\u00d7\u00d7$ 10\u22124\n\nConsider the following questions:\n\n(a) Determine the order for each of the reactants, NO and H2, from the data given and show your reasoning.\n\n(b) Write the overall rate law for the reaction.\n\n(c) Calculate the value of the rate constant, k, for the reaction. Include units.\n\n(d) For experiment 2, calculate the concentration of NO remaining when exactly one-half of the original amount of H2 had been consumed.\n\n(e) The following sequence of elementary steps is a proposed mechanism for the reaction.\n\nStep 1: $NO+NO\u21ccN2O2NO+NO\u21ccN2O2$\n\nStep 2: $N2O2+H2\u21ccH2O+N2ON2O2+H2\u21ccH2O+N2O$\n\nStep 3: $N2O+H2\u21ccN2+H2ON2O+H2\u21ccN2+H2O$\n\nBased on the data presented, which of these is the rate determining step? Show that the mechanism is consistent with the observed rate law for the reaction and the overall stoichiometry of the reaction.\n\n75.\n\nThe reaction of CO with Cl2 gives phosgene (COCl2), a nerve gas that was used in World War I. Use the mechanism shown here to complete the following exercises:\n$Cl2(g)\u21cc2Cl(g)Cl2(g)\u21cc2Cl(g)$ (fast, k1 represents the forward rate constant, k\u22121 the reverse rate constant)\n$CO(g)+Cl(g)\u27f6COCl(g)CO(g)+Cl(g)\u27f6COCl(g)$ (slow, k2 the rate constant)\n$COCl(g)+Cl(g)\u27f6COCl2(g)COCl(g)+Cl(g)\u27f6COCl2(g)$ (fast, k3 the rate constant)\n\n(a) Write the overall reaction.\n\n(b) Identify all intermediates.\n\n(c) Write the rate law for each elementary reaction.\n\n(d) Write the overall rate law expression.\n\n### 12.7Catalysis\n\n76.\n\nAccount for the increase in reaction rate brought about by a catalyst.\n\n77.\n\nCompare the functions of homogeneous and heterogeneous catalysts.\n\n78.\n\nConsider this scenario and answer the following questions: Chlorine atoms resulting from decomposition of chlorofluoromethanes, such as CCl2F2, catalyze the decomposition of ozone in the atmosphere. One simplified mechanism for the decomposition is:\n$O3\u2192sunlightO2+OO3+Cl\u27f6O2+ClOClO+O\u27f6Cl+O2O3\u2192sunlightO2+OO3+Cl\u27f6O2+ClOClO+O\u27f6Cl+O2$\n\n(a) Explain why chlorine atoms are catalysts in the gas-phase transformation:\n$2O3\u27f63O22O3\u27f63O2$\n\n(b) Nitric oxide is also involved in the decomposition of ozone by the mechanism:\n$O3\u2192sunlightO2+OO3+NO\u27f6NO2+O2NO2+O\u27f6NO+O2O3\u2192sunlightO2+OO3+NO\u27f6NO2+O2NO2+O\u27f6NO+O2$\n\n79.\n\nFor each of the following pairs of reaction diagrams, identify which of the pair is catalyzed:\n\n(a)\n\n(b)\n\n80.\n\nFor each of the following pairs of reaction diagrams, identify which of the pairs is catalyzed:\n\n(a)\n\n(b)\n\n81.\n\nFor each of the following reaction diagrams, estimate the activation energy (Ea) of the reaction:\n\n(a)\n\n(b)\n\n82.\n\nFor each of the following reaction diagrams, estimate the activation energy (Ea) of the reaction:\n\n(a)\n\n(b)\n\n83.\n\nAssuming the diagrams in Exercise 12.81 represent different mechanisms for the same reaction, which of the reactions has the faster rate?\n\n84.\n\nConsider the similarities and differences in the two reaction diagrams shown in Exercise 12.82. Do these diagrams represent two different overall reactions, or do they represent the same overall reaction taking place by two different mechanisms? Explain your answer.\n\n### Footnotes\n\n\u2022 1This question is taken from the Chemistry Advanced Placement Examination and is used with the permission of the Educational Testing Service.\nOrder a print copy\n\nAs an Amazon Associate we earn from qualifying purchases.","date":"2021-03-01 01:46:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 160, \"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\": 0.7551223039627075, \"perplexity\": 1824.3914018389132}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178361808.18\/warc\/CC-MAIN-20210228235852-20210301025852-00321.warc.gz\"}"} | null | null |
JF Auburtin: Born December 2, 1866
Jean Francis Auburtin was a 19th Century Symbolist painter, an heir of Impressionism, influenced by Japonisme, and sometimes referred to as "the Symbolist of the Sea". Born December 2, 1866, Auburtin was apprenticed early to the painter Louis-Theodore Devilly. He then enrolled at the Alsatian School of Paris in 1875 where he met his future wife Marthe Deloy, a sister of one of his classmates. After further education at the Ecole des Beaux-Arts, Auburtin found himself attracted to painting the cliffs and the ever-changing effects of light on the sea and as a result lived in various locations that offered these views. Auburtin mainly painted in gouache and watercolor, depicting the Normandy coastline, the sea, and later in life figures of dancers. At the end of the nineteenth century, Auburtin became interested in Japanese art and began a small collection of prints, some painted by the famous Japanese painter Hokusaï, which influenced his own work in no small measure. Jean Francis Auburtin rose to the rank of Officer of the Legion of Honor. From 1904 onwards he lived in Varengeville; when he died in 1930 he was buried in the cliff-top cemetery of Varengeville-sur-Mer, which is also the final resting place of Georges Braque.
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Produced by Marilynda Fraser-Cunliffe, Sam W. and the
Online Distributed Proofreading Team at http://www.pgdp.net
Transcriber's Note
Illustration captions in {brackets} have been added by the transcriber
for the convenience of the reader.
CURIOUS MYTHS
OF
THE MIDDLE AGES.
BY
S. BARING-GOULD, M.A.
BOSTON:
ROBERTS BROTHERS.
1867.
STEREOTYPED AT THE
BOSTON STEREOTYPE FOUNDRY,
No. 4 Spring Lane.
University Press: Welch, Bigelow, & Co.,
Cambridge.
[Illustration: POPE JOAN.
From Joh. Wolfii Lect. Memorab. (LavingA|, 1600.)]
CONTENTS.
PAGE
The Wandering Jew 1
Prester John 30
The Divining Rod 54
The Seven Sleepers of Ephesus 92
William Tell 110
The Dog Gellert 132
Tailed Men 144
Antichrist and Pope Joan 160
The Man in the Moon 189
The Mountain of Venus 207
Fatality of Numbers 221
The Terrestrial Paradise 242
MEDIA†VAL MYTHS.
The Wandering Jew.
Who, that has looked on Gustave DorA(C)'s marvellous illustrations to
this wild legend, can forget the impression they made upon his
imagination?
I do not refer to the first illustration as striking, where the Jewish
shoemaker is refusing to suffer the cross-laden Savior to rest a
moment on his door-step, and is receiving with scornful lip the
judgment to wander restless till the Second Coming of that same
Redeemer. But I refer rather to the second, which represents the Jew,
after the lapse of ages, bowed beneath the burden of the curse, worn
with unrelieved toil, wearied with ceaseless travelling, trudging
onward at the last lights of evening, when a rayless night of
unabating rain is creeping on, along a sloppy path between dripping
bushes; and suddenly he comes over against a wayside crucifix, on
which the white glare of departing daylight falls, to throw it into
ghastly relief against the pitch-black rain-clouds. For a moment we
see the working of the miserable shoemaker's mind. We feel that he is
recalling the tragedy of the first Good Friday, and his head hangs
heavier on his breast, as he recalls the part he had taken in that
awful catastrophe.
Or, is that other illustration more remarkable, where the wanderer is
amongst the Alps, at the brink of a hideous chasm; and seeing in the
contorted pine-branches the ever-haunting scene of the Via Dolorosa,
he is lured to cast himself into that black gulf in quest of
rest,--when an angel flashes out of the gloom with the sword of flame
turning every way, keeping him back from what would be to him a
Paradise indeed, the repose of Death?
Or, that last scene, when the trumpet sounds and earth is shivering to
its foundations, the fire is bubbling forth through the rents in its
surface, and the dead are coming together flesh to flesh, and bone to
bone, and muscle to muscle--then the weary man sits down and casts off
his shoes! Strange sights are around him, he sees them not; strange
sounds assail his ears, he hears but one--the trumpet-note which gives
the signal for him to stay his wanderings and rest his weary feet.
I can linger over those noble woodcuts, and learn from them something
new each time that I study them; they are picture-poems full of latent
depths of thought. And now let us to the history of this most
thrilling of all mediA|val myths, if a myth.
If a myth, I say, for who can say for certain that it is not true?
"Verily I say unto you, There be some standing here, which shall not
taste of death till they see the Son of Man coming in His kingdom,"[1]
are our Lord's words, which I can hardly think apply to the
destruction of Jerusalem, as commentators explain it to escape the
difficulty. That some should live to see Jerusalem destroyed was not
very surprising, and hardly needed the emphatic Verily which Christ
only used when speaking something of peculiarly solemn or mysterious
import.
Besides, St. Luke's account manifestly refers the coming in the
kingdom to the Judgment, for the saying stands as follows: "Whosoever
shall be ashamed of Me, and of My words, of him shall the Son of Man
be ashamed, when He shall come in His own glory, and in His Father's,
and of the holy angels. But I tell you of a truth, there be some
standing here, which shall not taste of death till they see the
kingdom of God."[2]
There can, I think, be no doubt in the mind of an unprejudiced person
that the words of our Lord do imply that some one or more of those
then living should not die till He came again. I do not mean to insist
on the literal signification, but I plead that there is no
improbability in our Lord's words being fulfilled to the letter. That
the circumstance is unrecorded in the Gospels is no evidence that it
did not take place, for we are expressly told, "Many other signs truly
did Jesus in the presence of His disciples, which are not written in
this book;"[3] and again, "There are also many other things which
Jesus did, the which, if they should be written every one, I suppose
that even the world itself could not contain the books that should be
written."[4]
We may remember also the mysterious witnesses who are to appear in the
last eventful days of the world's history and bear testimony to the
Gospel truth before the antichristian world. One of these has been
often conjectured to be St. John the Evangelist, of whom Christ said
to Peter, "If I will that he tarry till I come, what is that to thee?"
The historical evidence on which the tale rests is, however, too
slender for us to admit for it more than the barest claim to be more
than myth. The names and the circumstances connected with the Jew and
his doom vary in every account, and the only point upon which all
coincide is, that such an individual exists in an undying condition,
wandering over the face of the earth, seeking rest and finding none.
The earliest extant mention of the Wandering Jew is to be found in the
book of the chronicles of the Abbey of St. Albans, which was copied
and continued by Matthew Paris. He records that in the year 1228, "a
certain Archbishop of Armenia the Greater came on a pilgrimage to
England to see the relics of the saints, and visit the sacred places
in the kingdom, as he had done in others; he also produced letters of
recommendation from his Holiness the Pope, to the religious and the
prelates of the churches, in which they were enjoined to receive and
entertain him with due reverence and honor. On his arrival, he came to
St. Albans, where he was received with all respect by the abbot and
the monks; and at this place, being fatigued with his journey, he
remained some days to rest himself and his followers, and a
conversation took place between him and the inhabitants of the
convent, by means of their interpreters, during which he made many
inquiries relating to the religion and religious observances of this
country, and told many strange things concerning the countries of the
East. In the course of conversation he was asked whether he had ever
seen or heard any thing of Joseph, a man of whom there was much talk
in the world, who, when our Lord suffered, was present and spoke to
Him, and who is still alive, in evidence of the Christian faith; in
reply to which, a knight in his retinue, who was his interpreter,
replied, speaking in French, 'My lord well knows that man, and a
little before he took his way to the western countries, the said
Joseph ate at the table of my lord the Archbishop of Armenia, and he
has often seen and conversed with him.'
"He was then asked about what had passed between Christ and the said
Joseph; to which he replied, 'At the time of the passion of Jesus
Christ, He was seized by the Jews, and led into the hall of judgment
before Pilate, the governor, that He might be judged by him on the
accusation of the Jews; and Pilate, finding no fault for which he
might sentence Him to death, said unto them, "Take Him and judge Him
according to your law;" the shouts of the Jews, however, increasing,
he, at their request, released unto them Barabbas, and delivered Jesus
to them to be crucified. When, therefore, the Jews were dragging Jesus
forth, and had reached the door, Cartaphilus, a porter of the hall in
Pilate's service, as Jesus was going out of the door, impiously struck
Him on the back with his hand, and said in mockery, "Go quicker,
Jesus, go quicker; why do you loiter?" and Jesus, looking back on him
with a severe countenance, said to him, "I am going, and you shall
wait till I return." And according as our Lord said, this Cartaphilus
is still awaiting His return. At the time of our Lord's suffering he
was thirty years old, and when he attains the age of a hundred years,
he always returns to the same age as he was when our Lord suffered.
After Christ's death, when the Catholic faith gained ground, this
Cartaphilus was baptized by Ananias (who also baptized the Apostle
Paul), and was called Joseph. He dwells in one or other divisions of
Armenia, and in divers Eastern countries, passing his time amongst the
bishops and other prelates of the Church; he is a man of holy
conversation, and religious; a man of few words, and very circumspect
in his behavior; for he does not speak at all unless when questioned
by the bishops and religious; and then he relates the events of olden
times, and speaks of things which occurred at the suffering and
resurrection of our Lord, and of the witnesses of the resurrection,
namely, of those who rose with Christ, and went into the holy city,
and appeared unto men. He also tells of the creed of the Apostles,
and of their separation and preaching. And all this he relates without
smiling, or levity of conversation, as one who is well practised in
sorrow and the fear of God, always looking forward with dread to the
coming of Jesus Christ, lest at the Last Judgment he should find him
in anger whom, when on his way to death, he had provoked to just
vengeance. Numbers came to him from different parts of the world,
enjoying his society and conversation; and to them, if they are men of
authority, he explains all doubts on the matters on which he is
questioned. He refuses all gifts that are offered him, being content
with slight food and clothing.'"
Much about the same date, Philip Mouskes, afterwards Bishop of
Tournay, wrote his rhymed chronicle (1242), which contains a similar
account of the Jew, derived from the same Armenian prelate:--
"Adonques vint un arceveskes
De ASec.A mer, plains de bonnes tA"ques
Par samblant, et fut d'Armenie,"
and this man, having visited the shrine of "St. Tumas de Kantorbire,"
and then having paid his devotions at "Monsigour St. Jake," he went on
to Cologne to see the heads of the three kings. The version told in
the Netherlands much resembled that related at St. Albans, only that
the Jew, seeing the people dragging Christ to his death, exclaims,--
"AtendA(C)s moi! g'i vois,
S'iert mis le faus profA"te en crois."
Then
"Le vrais Dieux se regarda,
Et li a dit qu'e n'i tarda,
Icist ne t'atenderont pas,
Mais saces, tu m'atenderas."
We hear no more of the wandering Jew till the sixteenth century, when
we hear first of him in a casual manner, as assisting a weaver, Kokot,
at the royal palace in Bohemia (1505), to find a treasure which had
been secreted by the great-grandfather of Kokot, sixty years before,
at which time the Jew was present. He then had the appearance of being
a man of seventy years.[5]
Curiously enough, we next hear of him in the East, where he is
confounded with the prophet Elijah. Early in the century he appeared
to Fadhilah, under peculiar circumstances.
After the Arabs had captured the city of Elvan, Fadhilah, at the head
of three hundred horsemen, pitched his tents, late in the evening,
between two mountains. Fadhilah, having begun his evening prayer with
a loud voice, heard the words "Allah akbar" (God is great) repeated
distinctly, and each word of his prayer was followed in a similar
manner. Fadhilah, not believing this to be the result of an echo, was
much astonished, and cried out, "O thou! whether thou art of the angel
ranks, or whether thou art of some other order of spirits, it is well;
the power of God be with thee; but if thou art a man, then let mine
eyes light upon thee, that I may rejoice in thy presence and society."
Scarcely had he spoken these words, before an aged man, with bald
head, stood before him, holding a staff in his hand, and much
resembling a dervish in appearance. After having courteously saluted
him, Fadhilah asked the old man who he was. Thereupon the stranger
answered, "Bassi Hadhret Issa, I am here by command of the Lord Jesus,
who has left me in this world, that I may live therein until he comes
a second time to earth. I wait for this Lord, who is the Fountain of
Happiness, and in obedience to his command I dwell behind yon
mountain." When Fadhilah heard these words, he asked when the Lord
Jesus would appear; and the old man replied that his appearing would
be at the end of the world, at the Last Judgment. But this only
increased Fadhilah's curiosity, so that he inquired the signs of the
approach of the end of all things, whereupon Zerib Bar Elia gave him
an account of general, social, and moral dissolution, which would be
the climax of this world's history.[6]
In 1547 he was seen in Europe, if we are to believe the following
narration:--
"Paul von Eitzen, doctor of the Holy Scriptures, and Bishop of
Schleswig,[7] related as true for some years past, that when he was
young, having studied at Wittemberg, he returned home to his parents
in Hamburg in the winter of the year 1547, and that on the following
Sunday, in church, he observed a tall man, with his hair hanging over
his shoulders, standing barefoot, during the sermon, over against the
pulpit, listening with deepest attention to the discourse, and,
whenever the name of Jesus was mentioned, bowing himself profoundly
and humbly, with sighs and beating of the breast. He had no other
clothing, in the bitter cold of the winter, except a pair of hose
which were in tatters about his feet, and a coat with a girdle which
reached to his feet; and his general appearance was that of a man of
fifty years. And many people, some of high degree and title, have seen
this same man in England, France, Italy, Hungary, Persia, Spain,
Poland, Moscow, Lapland, Sweden, Denmark, Scotland, and other places.
"Every one wondered over the man. Now, after the sermon, the said
Doctor inquired diligently where the stranger was to be found; and when
he had sought him out, he inquired of him privately whence he came, and
how long that winter he had been in the place. Thereupon he replied,
modestly, that he was a Jew by birth, a native of Jerusalem, by name
Ahasverus, by trade a shoemaker; he had been present at the crucifixion
of Christ, and had lived ever since, travelling through various lands
and cities, the which he substantiated by accounts he gave; he related
also the circumstances of Christ's transference from Pilate to Herod,
and the final crucifixion, together with other details not recorded in
the Evangelists and historians; he gave accounts of the changes of
government in many countries, especially of the East, through several
centuries; and moreover he detailed the labors and deaths of the holy
Apostles of Christ most circumstantially.
"Now when Doctor Paul v. Eitzen heard this with profound astonishment,
on account of its incredible novelty, he inquired further, in order
that he might obtain more accurate information. Then the man answered,
that he had lived in Jerusalem at the time of the crucifixion of
Christ, whom he had regarded as a deceiver of the people, and a
heretic; he had seen Him with his own eyes, and had done his best,
along with others, to bring this deceiver, as he regarded Him, to
justice, and to have Him put out of the way. When the sentence had
been pronounced by Pilate, Christ was about to be dragged past his
house; then he ran home, and called together his household to have a
look at Christ, and see what sort of a person He was.
"This having been done, he had his little child on his arm, and was
standing in his doorway, to have a sight of the Lord Jesus Christ.
"As, then, Christ was led by, bowed under the weight of the heavy
cross, He tried to rest a little, and stood still a moment; but the
shoemaker, in zeal and rage, and for the sake of obtaining credit
among the other Jews, drove the Lord Christ forward, and told Him to
hasten on His way. Jesus, obeying, looked at him, and said, 'I shall
stand and rest, but thou shalt go till the last day.' At these words
the man set down the child; and, unable to remain where he was, he
followed Christ, and saw how cruelly He was crucified, how He
suffered, how He died. As soon as this had taken place, it came upon
him suddenly that he could no more return to Jerusalem, nor see again
his wife and child, but must go forth into foreign lands, one after
another, like a mournful pilgrim. Now, when, years after, he returned
to Jerusalem, he found it ruined and utterly razed, so that not one
stone was left standing on another; and he could not recognize former
localities.
"He believes that it is God's purpose, in thus driving him about in
miserable life, and preserving him undying, to present him before the
Jews at the end, as a living token, so that the godless and
unbelieving may remember the death of Christ, and be turned to
repentance. For his part he would well rejoice were God in heaven to
release him from this vale of tears. After this conversation, Doctor
Paul v. Eitzen, along with the rector of the school of Hamburg, who
was well read in history, and a traveller, questioned him about events
which had taken place in the East since the death of Christ, and he
was able to give them much information on many ancient matters; so
that it was impossible not to be convinced of the truth of his story,
and to see that what seems impossible with men is, after all, possible
with God.
"Since the Jew has had his life extended, he has become silent and
reserved, and only answers direct questions. When invited to become
any one's guest, he eats little, and drinks in great moderation; then
hurries on, never remaining long in one place. When at Hamburg,
Dantzig, and elsewhere, money has been offered him, he never took more
than two skillings (fourpence, one farthing), and at once distributed
it to the poor, as token that he needed no money, for God would
provide for him, as he rued the sins he had committed in ignorance.
"During the period of his stay in Hamburg and Dantzig he was never
seen to laugh. In whatever land he travelled he spoke its language,
and when he spoke Saxon, it was like a native Saxon. Many people came
from different places to Hamburg and Dantzig in order to see and hear
this man, and were convinced that the providence of God was exercised
in this individual in a very remarkable manner. He gladly listened to
God's word, or heard it spoken of always with great gravity and
compunction, and he ever reverenced with sighs the pronunciation of
the name of God, or of Jesus Christ, and could not endure to hear
curses; but whenever he heard any one swear by God's death or pains,
he waxed indignant, and exclaimed, with vehemence and with sighs,
'Wretched man and miserable creature, thus to misuse the name of thy
Lord and God, and His bitter sufferings and passion. Hadst thou seen,
as I have, how heavy and bitter were the pangs and wounds of thy Lord,
endured for thee and for me, thou wouldst rather undergo great pain
thyself than thus take His sacred name in vain!'
"Such is the account given to me by Doctor Paul von Eitzen, with many
circumstantial proofs, and corroborated by certain of my own old
acquaintances who saw this same individual with their own eyes in
Hamburg.
"In the year 1575 the Secretary Christopher Krause, and Master Jacob
von Holstein, legates to the Court of Spain, and afterwards sent into
the Netherlands to pay the soldiers serving his Majesty in that
country, related on their return home to Schleswig, and confirmed with
solemn oaths, that they had come across the same mysterious individual
at Madrid in Spain, in appearance, manner of life, habits, clothing,
just the same as he had appeared in Hamburg. They said that they had
spoken with him, and that many people of all classes had conversed
with him, and found him to speak good Spanish. In the year 1599, in
December, a reliable person wrote from Brunswick to Strasburg that the
same mentioned strange person had been seen alive at Vienna in
Austria, and that he had started for Poland and Dantzig; and that he
purposed going on to Moscow. This Ahasverus was at Lubeck in 1601,
also about the same date in Revel in Livonia, and in Cracow in Poland.
In Moscow he was seen of many and spoken to by many.
"What thoughtful, God-fearing persons are to think of the said
person, is at their option. God's works are wondrous and past finding
out, and are manifested day by day, only to be revealed in full at the
last great day of account.
"Dated, Revel, August 1st, 1613.
"D. W.
"D.
"Chrysostomus DudulA"us,
"Westphalus."
The statement that the Wandering Jew appeared in Lubeck in 1601, does
not tally with the more precise chronicle of Henricus Bangert, which
gives: "Die 14 Januarii Anno MDCIII., adnotatum reliquit LubecA| fuisse
JudA|um illum immortalem, qui se Christi crucifixioni interfuisse
affirmavit."[8]
In 1604 he seems to have appeared in Paris. Rudolph Botoreus says,
under this date, "I fear lest I be accused of giving ear to old wives'
fables, if I insert in these pages what is reported all over Europe of
the Jew, coeval with the Savior Christ; however, nothing is more
common, and our popular histories have not scrupled to assert it.
Following the lead of those who wrote our annals, I may say that he
who appeared not in one century only, in Spain, Italy, and Germany,
was also in this year seen and recognized as the same individual who
had appeared in Hamburg, anno MDLXVI. The common people, bold in
spreading reports, relate many things of him; and this I allude to,
lest anything should be left unsaid."[9]
J. C. Bulenger puts the date of the Hamburg visit earlier. "It was
reported at this time that a Jew of the time of Christ was wandering
without food and drink, having for a thousand and odd years been a
vagabond and outcast, condemned by God to rove, because he, of that
generation of vipers, was the first to cry out for the crucifixion of
Christ and the release of Barabbas; and also because soon after, when
Christ, panting under the burden of the rood, sought to rest before
his workshop (he was a cobbler), the fellow ordered Him off with
acerbity. Thereupon Christ replied, 'Because thou grudgest Me such a
moment of rest, I shall enter into My rest, but thou shalt wander
restless.' At once, frantic and agitated, he fled through the whole
earth, and on the same account to this day he journeys through the
world. It was this person who was seen in Hamburg in MDLXIV. Credat
JudA|us Apella! _I_ did not see him, or hear anything authentic
concerning him, at that time when I was in Paris."[10]
A curious little book,[11] written against the quackery of Paracelsus,
by Leonard Doldius, a NA1/4rnberg physician, and translated into Latin
and augmented, by Andreas Libavius, doctor and physician of Rotenburg,
alludes to the same story, and gives the Jew a new name nowhere else
met with. After having referred to a report that Paracelsus was not
dead, but was seated alive, asleep or napping, in his sepulchre at
Strasburg, preserved from death by some of his specifics, Libavius
declares that he would sooner believe in the old man, the Jew,
Ahasverus, wandering over the world, called by some ButtadA|us, and
otherwise, again, by others.
He is said to have appeared in Naumburg, but the date is not given; he
was noticed in church, listening to the sermon. After the service he
was questioned, and he related his story. On this occasion he
received presents from the burgers.[12] In 1633 he was again in
Hamburg.[13] In the year 1640, two citizens, living in the
Gerberstrasse, in Brussels, were walking in the Sonian wood, when they
encountered an aged man, whose clothes were in tatters and of an
antiquated appearance. They invited him to go with them to a house of
refreshment, and he went with them, but would not seat himself,
remaining on foot to drink. When he came before the doors with the two
burgers, he told them a great deal; but they were mostly stories of
events which had happened many hundred years before. Hence the burgers
gathered that their companion was Isaac Laquedem, the Jew who had
refused to permit our Blessed Lord to rest for a moment at his
door-step, and they left him full of terror. In 1642 he is reported to
have visited Leipzig. On the 22d July, 1721, he appeared at the gates
of the city of Munich.[14] About the end of the seventeenth century or
the beginning of the eighteenth, an impostor, calling himself the
Wandering Jew, attracted attention in England, and was listened to by
the ignorant, and despised by the educated. He, however, managed to
thrust himself into the notice of the nobility, who, half in jest,
half in curiosity, questioned him, and paid him as they might a
juggler. He declared that he had been an officer of the Sanhedrim, and
that he had struck Christ as he left the judgment hall of Pilate. He
remembered all the Apostles, and described their personal appearance,
their clothes, and their peculiarities. He spoke many languages,
claimed the power of healing the sick, and asserted that he had
travelled nearly all over the world. Those who heard him were
perplexed by his familiarity with foreign tongues and places. Oxford
and Cambridge sent professors to question him, and to discover the
imposition, if any. An English nobleman conversed with him in Arabic.
The mysterious stranger told his questioner in that language that
historical works were not to be relied upon. And on being asked his
opinion of Mahomet, he replied that he had been acquainted with the
father of the prophet, and that he dwelt at Ormuz. As for Mahomet, he
believed him to have been a man of intelligence; once when he heard
the prophet deny that Christ was crucified, he answered abruptly by
telling him he was a witness to the truth of that event. He related
also that he was in Rome when Nero set it on fire; he had known
Saladin, Tamerlane, Bajazeth, Eterlane, and could give minute details
of the history of the Crusades.[15]
Whether this wandering Jew was found out in London or not, we cannot
tell, but he shortly after appeared in Denmark, thence travelled into
Sweden, and vanished.
Such are the principal notices of the Wandering Jew which have
appeared. It will be seen at once how wanting they are in all
substantial evidence which could make us regard the story in any other
light than myth.
But no myth is wholly without foundation, and there must be some
substantial verity upon which this vast superstructure of legend has
been raised. What that is I am unable to discover.
It has been suggested by some that the Jew Ahasverus is an
impersonation of that race which wanders, Cain-like, over the earth
with the brand of a brother's blood upon it, and one which is not to
pass away till all be fulfilled, not to be reconciled to its angered
God till the times of the Gentiles are accomplished. And yet, probable
as this supposition may seem at first sight, it is not to be
harmonized with some of the leading features of the story. The
shoemaker becomes a penitent, and earnest Christian, whilst the Jewish
nation has still the veil upon its heart; the wretched wanderer
eschews money, and the avarice of the Israelite is proverbial.
According to local legend, he is identified with the Gypsies, or
rather that strange people are supposed to be living under a curse
somewhat similar to that inflicted on Ahasverus, because they refused
shelter to the Virgin and Child on their flight into Egypt.[16]
Another tradition connects the Jew with the wild huntsman, and there
is a forest at Bretten, in Swabia, which he is said to haunt. Popular
superstition attributes to him there a purse containing a groschen,
which, as often as it is expended, returns to the spender.[17]
In the Harz one form of the Wild Huntsman myth is to this effect:
that he was a Jew who had refused to suffer our Blessed Lord to drink
out of a river, or out of a horse-trough, but had contemptuously
pointed out to Him the hoof-print of a horse, in which a little water
had collected, and had bid Him quench His thirst thence.[18]
As the Wild Huntsman is the personification of the storm, it is
curious to find in parts of France that the sudden roar of a gale at
night is attributed by the vulgar to the passing of the Everlasting
Jew.
A Swiss story is, that he was seen one day standing upon the
Matterberg, which is below the Matterhorn, contemplating the scene
with mingled sorrow and wonder. Once before he stood on that spot, and
then it was the site of a flourishing city; now it is covered with
gentian and wild pinks. Once again will he revisit the hill, and that
will be on the eve of Judgment.
Perhaps, of all the myths which originated in the middle ages, none is
more striking than that we have been considering; indeed, there is
something so calculated to arrest the attention and to excite the
imagination in the outline of the story, that it is remarkable that
we should find an interval of three centuries elapse between its first
introduction into Europe by Matthew Paris and Philip Mouskes, and its
general acceptance in the sixteenth century. As a myth, its roots lie
in that great mystery of human life which is an enigma never solved,
and ever originating speculation.
What was life? Was it of necessity limited to fourscore years, or
could it be extended indefinitely? were questions curious minds never
wearied of asking. And so the mythology of the past teemed with
legends of favored or accursed mortals, who had reached beyond the
term of days set to most men. Some had discovered the water of life,
the fountain of perpetual youth, and were ever renewing their
strength. Others had dared the power of God, and were therefore
sentenced to feel the weight of His displeasure, without tasting the
repose of death.
John the Divine slept at Ephesus, untouched by corruption, with the
ground heaving over his breast as he breathed, waiting the summons to
come forth and witness against Antichrist. The seven sleepers reposed
in a cave, and centuries glided by like a watch in the night. The
monk of Hildesheim, doubting how with God a thousand years could be as
yesterday, listened to the melody of a bird in the green wood during
three minutes, and found that in three minutes three hundred years had
flown. Joseph of ArimathA|a, in the blessed city of Sarras, draws
perpetual life from the Saint Graal; Merlin sleeps and sighs in an old
tree, spell-bound of Vivien. Charlemagne and Barbarossa wait, crowned
and armed, in the heart of the mountain, till the time comes for the
release of Fatherland from despotism. And, on the other hand, the
curse of a deathless life has passed on the Wild Huntsman, because he
desired to chase the red-deer for evermore; on the Captain of the
Phantom Ship, because he vowed he would double the Cape whether God
willed it or not; on the Man in the Moon, because he gathered sticks
during the Sabbath rest; on the dancers of Kolbeck, because they
desired to spend eternity in their mad gambols.
I began this article intending to conclude it with a bibliographical
account of the tracts, letters, essays, and books, written upon the
Wandering Jew; but I relinquish my intention at the sight of the
multitude of works which have issued from the press upon the subject;
and this I do with less compunction as the bibliographer may at little
trouble and expense satisfy himself, by perusing the lists given by
GrA¤sse in his essay on the myth, and those to be found in "Notice
historique et bibliographique sur les Juifs-errants: par O. B."
(Gustave Brunet), Paris, TA(C)chener, 1845; also in the article by M.
Mangin, in "Causeries et MA(C)ditations historiques et littA(C)raires,"
Paris, Duprat, 1843; and, lastly, in the essay by Jacob le Bibliophile
(M. Lacroix) in his "CuriositA(C)s de l'Histoire des Croyances
populaires," Paris, Delahays, 1859.
Of the romances of EugA"ne Sue and Dr. Croly, founded upon the legend,
the less said the better. The original legend is so noble in its
severe simplicity, that none but a master mind could develop it with
any chance of success. Nor have the poetical attempts upon the story
fared better. It was reserved for the pencil of Gustave DorA(C) to treat
it with the originality it merited, and in a series of woodcuts to
produce at once a poem, a romance, and a chef-d'A"uvre of art.
FOOTNOTES:
[1] Matt. xvi. 28. Mark ix. 1.
[2] Luke ix.
[3] John xx. 30.
[4] John xxi. 25.
[5] Gubitz, Gesellsch. 1845, No. 18.
[6] Herbelot, Bibl. Orient, iii. p. 607.
[7] Paul v. Eitzen was born January 25, 1522, at Hamburg; in 1562 he
was appointed chief preacher for Schleswig, and died February 25,
1598. (Greve, Memor. P. ab. Eitzen. Hamb. 1844.)
[8] Henr. Bangert, Comment. de Ortu, Vita, et Excessu Coleri, I. Cti.
Lubec.
[9] R. Botoreus, Comm. Histor. lii. p. 305.
[10] J. C. Bulenger, Historia sui Temporis, p. 357.
[11] Praxis AlchymiA|. Francfurti, MDCIV. 8vo.
[12] Mitternacht, Diss. in Johann. xxi. 19.
[13] Mitternacht, ut supra.
[14] Hormayr, Taschenbuch, 1834, p. 216.
[15] Calmet, Dictionn. de la Bible, t. ii. p. 472.
[16] Aventinus, Bayr. Chronik, viii.
[17] Meier, SchwA¤bischen Sagen, i. 116.
[18] Kuhn u. Schwarz Nordd. Sagen, p. 499.
Prester John.
[Illustration: Arms of the See of Chichester.]
About the middle of the twelfth century, a rumor circulated through
Europe that there reigned in Asia a powerful Christian Emperor,
Presbyter Johannes. In a bloody fight he had broken the power of the
Mussulmans, and was ready to come to the assistance of the Crusaders.
Great was the exultation in Europe, for of late the news from the East
had been gloomy and depressing, the power of the infidel had
increased, overwhelming masses of men had been brought into the field
against the chivalry of Christendom, and it was felt that the cross
must yield before the odious crescent.
The news of the success of the Priest-King opened a door of hope to
the desponding Christian world. Pope Alexander III. determined at
once to effect a union with this mysterious personage, and on the 27th
of September, 1177, wrote him a letter, which he intrusted to his
physician, Philip, to deliver in person.
Philip started on his embassy, but never returned. The conquests of
Tschengis-Khan again attracted the eyes of Christian Europe to the
East. The Mongol hordes were rushing in upon the west with devastating
ferocity; Russia, Poland, Hungary, and the eastern provinces of
Germany, had succumbed, or suffered grievously; and the fears of other
nations were roused lest they too should taste the misery of a
Mongolian invasion. It was Gog and Magog come to slaughter, and the
times of Antichrist were dawning. But the battle of Liegnitz stayed
them in their onward career, and Europe was saved.
Pope Innocent IV. determined to convert these wild hordes of
barbarians, and subject them to the cross of Christ; he therefore sent
among them a number of Dominican and Franciscan missioners, and
embassies of peace passed between the Pope, the King of France, and
the Mogul Khan.
The result of these communications with the East was, that the
travellers learned how false were the prevalent notions of a mighty
Christian empire existing in Central Asia. Vulgar superstition or
conviction is not, however, to be upset by evidence, and the locality
of the monarchy was merely transferred by the people to Africa, and
they fixed upon Abyssinia, with a show of truth, as the seat of the
famous Priest-King. However, still some doubted. John de Plano Carpini
and Marco Polo, though they acknowledged the existence of a Christian
monarch in Abyssinia, yet stoutly maintained as well that the Prester
John of popular belief reigned in splendor somewhere in the dim
Orient.
But before proceeding with the history of this strange fable, it will
be well to extract the different accounts given of the Priest-King and
his realm by early writers; and we shall then be better able to judge
of the influence the myth obtained in Europe.
Otto of Freisingen is the first author to mention the monarchy of
Prester John with whom we are acquainted. Otto wrote a chronicle up to
the date 1156, and he relates that in 1145 the Catholic Bishop of
Cabala visited Europe to lay certain complaints before the Pope. He
mentioned the fall of Edessa, and also "he stated that a few years ago
a certain King and Priest called John, who lives on the farther side
of Persia and Armenia, in the remote East, and who, with all his
people, were Christians, though belonging to the Nestorian Church, had
overcome the royal brothers Samiardi, kings of the Medes and Persians,
and had captured Ecbatana, their capital and residence. The said kings
had met with their Persian, Median, and Assyrian troops, and had
fought for three consecutive days, each side having determined to die
rather than take to flight. Prester John, for so they are wont to call
him, at length routed the Persians, and after a bloody battle,
remained victorious. After which victory the said John was hastening
to the assistance of the Church at Jerusalem, but his host, on
reaching the Tigris, was hindered from passing, through a deficiency
in boats, and he directed his march North, since he had heard that the
river was there covered with ice. In that place he had waited many
years, expecting severe cold; but the winters having proved
unpropitious, and the severity of the climate having carried off many
soldiers, he had been forced to retreat to his own land. This king
belongs to the family of the Magi, mentioned in the Gospel, and he
rules over the very people formerly governed by the Magi; moreover,
his fame and his wealth are so great, that he uses an emerald sceptre
only.
"Excited by the example of his ancestors, who came to worship Christ
in his cradle, he had proposed to go to Jerusalem, but had been
impeded by the above-mentioned causes."[19]
At the same time the story crops up in other quarters; so that we
cannot look upon Otto as the inventor of the myth. The celebrated
Maimonides alludes to it in a passage quoted by Joshua Lorki, a Jewish
physician to Benedict XIII. Maimonides lived from 1135 to 1204. The
passage is as follows: "It is evident both from the letters of Rambam
(Maimonides), whose memory be blessed, and from the narration of
merchants who have visited the ends of the earth, that at this time
the root of our faith is to be found in the lands of Babel and Teman,
where long ago Jerusalem was an exile; not reckoning those who live in
the land of Paras[20] and Madai,[21] of the exiles of Schomrom, the
number of which people is as the sand: of these some are still under
the yoke of Paras, who is called the Great-Chief Sultan by the Arabs;
others live in a place under the yoke of a strange people ... governed
by a Christian chief, Preste-Cuan by name. With him they have made a
compact, and he with them; and this is a matter concerning which there
can be no manner of doubt."
Benjamin of Tudela, another Jew, travelled in the East between the
years 1159 and 1173, the last being the date of his death. He wrote an
account of his travels, and gives in it some information with regard
to a mythical Jew king, who reigned in the utmost splendor over a
realm inhabited by Jews alone, situate somewhere in the midst of a
desert of vast extent. About this period there appeared a document
which produced intense excitement throughout Europe--a letter, yes! a
letter from the mysterious personage himself to Manuel Comnenus,
Emperor of Constantinople (1143-1180). The exact date of this
extraordinary epistle cannot be fixed with any certainty, but it
certainly appeared before 1241, the date of the conclusion of the
chronicle of Albericus Trium Fontium. This Albericus relates that in
the year 1165 "Presbyter Joannes, the Indian king, sent his wonderful
letter to various Christian princes, and especially to Manuel of
Constantinople, and Frederic the Roman Emperor." Similar letters were
sent to Alexander III., to Louis VII. of France, and to the King of
Portugal, which are alluded to in chronicles and romances, and which
were indeed turned into rhyme, and sung all over Europe by minstrels
and trouvA"res. The letter is as follows:--
"John, Priest by the Almighty power of God and the Might of our Lord
Jesus Christ, King of Kings, and Lord of Lords, to his friend Emanuel,
Prince of Constantinople, greeting, wishing him health, prosperity,
and the continuance of Divine favor.
"Our Majesty has been informed that you hold our Excellency in love,
and that the report of our greatness has reached you. Moreover, we
have heard through our treasurer that you have been pleased to send to
us some objects of art and interest, that our Exaltedness might be
gratified thereby.
"Being human, I receive it in good part, and we have ordered our
treasurer to send you some of our articles in return.
"Now we desire to be made certain that you hold the right faith, and
in all things cleave to Jesus Christ, our Lord, for we have heard that
your court regard you as a god, though we know that you are mortal,
and subject to human infirmities.... Should you desire to learn the
greatness and excellency of our Exaltedness and of the land subject to
our sceptre, then hear and believe:--I, Presbyter Johannes, the Lord
of Lords, surpass all under heaven in virtue, in riches, and in power;
seventy-two kings pay us tribute.... In the three Indies our
Magnificence rules, and our land extends beyond India, where rests the
body of the holy Apostle Thomas; it reaches towards the sunrise over
the wastes, and it trends towards deserted Babylon near the tower of
Babel. Seventy-two provinces, of which only a few are Christian, serve
us. Each has its own king, but all are tributary to us.
"Our land is the home of elephants, dromedaries, camels, crocodiles,
meta-collinarum, cametennus, tensevetes, wild asses, white and red
lions, white bears, white merules, crickets, griffins, tigers, lamias,
hyenas, wild horses, wild oxen and wild men, men with horns, one-eyed,
men with eyes before and behind, centaurs, fauns, satyrs, pygmies,
forty-ell-high giants, Cyclopses, and similar women; it is the home,
too, of the phA"nix, and of nearly all living animals. We have some
people subject to us who feed on the flesh of men and of prematurely
born animals, and who never fear death. When any of these people die,
their friends and relations eat him ravenously, for they regard it as
a main duty to munch human flesh. Their names are Gog and Magog, Anie,
Agit, Azenach, Fommeperi, Befari, Conei-Samante, Agrimandri,
Vintefolei, Casbei, Alanei. These and similar nations were shut in
behind lofty mountains by Alexander the Great, towards the North. We
lead them at our pleasure against our foes, and neither man nor beast
is left undevoured, if our Majesty gives the requisite permission. And
when all our foes are eaten, then we return with our hosts home again.
These accursed fifteen nations will burst forth from the four quarters
of the earth at the end of the world, in the times of Antichrist, and
overrun all the abodes of the Saints as well as the great city Rome,
which, by the way, we are prepared to give to our son who will be
born, along with all Italy, Germany, the two Gauls, Britain and
Scotland. We shall also give him Spain and all the land as far as the
icy sea. The nations to which I have alluded, according to the words
of the prophet, shall not stand in the judgment, on account of their
offensive practices, but will be consumed to ashes by a fire which
will fall on them from heaven.
"Our land streams with honey, and is overflowing with milk. In one
region grows no poisonous herb, nor does a querulous frog ever quack
in it; no scorpion exists, nor does the serpent glide amongst the
grass, nor can any poisonous animals exist in it, or injure any one.
"Among the heathen, flows through a certain province the River Indus;
encircling Paradise, it spreads its arms in manifold windings through
the entire province. Here are found the emeralds, sapphires,
carbuncles, topazes, chrysolites, onyxes, beryls, sardius, and other
costly stones. Here grows the plant Assidos, which, when worn by any
one, protects him from the evil spirit, forcing it to state its
business and name; consequently the foul spirits keep out of the way
there. In a certain land subject to us, all kinds of pepper is
gathered, and is exchanged for corn and bread, leather and cloth....
At the foot of Mount Olympus bubbles up a spring which changes its
flavor hour by hour, night and day, and the spring is scarcely three
days' journey from Paradise, out of which Adam was driven. If any one
has tasted thrice of the fountain, from that day he will feel no
fatigue, but will, as long as he lives, be as a man of thirty years.
Here are found the small stones called Nudiosi, which, if borne about
the body, prevent the sight from waxing feeble, and restore it where
it is lost. The more the stone is looked at, the keener becomes the
sight. In our territory is a certain waterless sea, consisting of
tumbling billows of sand never at rest. None have crossed this sea; it
lacks water altogether, yet fish are cast up upon the beach of various
kinds, very tasty, and the like are nowhere else to be seen. Three
days' journey from this sea are mountains from which rolls down a
stony, waterless river, which opens into the sandy sea. As soon as the
stream reaches the sea, its stones vanish in it, and are never seen
again. As long as the river is in motion, it cannot be crossed; only
four days a week is it possible to traverse it. Between the sandy sea
and the said mountains, in a certain plain is a fountain of singular
virtue, which purges Christians and would-be Christians from all
transgressions. The water stands four inches high in a hollow stone
shaped like a mussel-shell. Two saintly old men watch by it, and ask
the comers whether they are Christians, or are about to become
Christians, then whether they desire healing with all their hearts. If
they have answered well, they are bidden to lay aside their clothes,
and to step into the mussel. If what they said be true, then the water
begins to rise and gush over their heads; thrice does the water thus
lift itself, and every one who has entered the mussel leaves it cured
of every complaint.
"Near the wilderness trickles between barren mountains a subterranean
rill, which can only by chance be reached, for only occasionally the
earth gapes, and he who would descend must do it with precipitation,
ere the earth closes again. All that is gathered under the ground
there is gem and precious stone. The brook pours into another river,
and the inhabitants of the neighborhood obtain thence abundance of
precious stones. Yet they never venture to sell them without having
first offered them to us for our private use: should we decline them,
they are at liberty to dispose of them to strangers. Boys there are
trained to remain three or four days under water, diving after the
stones.
"Beyond the stone river are the ten tribes of the Jews, which, though
subject to their own kings, are, for all that, our slaves and
tributary to our Majesty. In one of our lands, hight Zone, are worms
called in our tongue Salamanders. These worms can only live in fire,
and they build cocoons like silk-worms, which are unwound by the
ladies of our palace, and spun into cloth and dresses, which are worn
by our Exaltedness. These dresses, in order to be cleaned and washed,
are cast into flames.... When we go to war, we have fourteen golden
and bejewelled crosses borne before us instead of banners; each of
these crosses is followed by 10,000 horsemen, and 100,000 foot
soldiers fully armed, without reckoning those in charge of the luggage
and provision.
"When we ride abroad plainly, we have a wooden, unadorned cross,
without gold or gem about it, borne before us, in order that we may
meditate on the sufferings of Our Lord Jesus Christ; also a golden
bowl filled with earth, to remind us of that whence we sprung, and
that to which we must return; but besides these there is borne a
silver bowl full of gold, as a token to all that we are the Lord of
Lords.
"All riches, such as are upon the world, our Magnificence possesses in
superabundance. With us no one lies, for he who speaks a lie is
thenceforth regarded as dead; he is no more thought of, or honored by
us. No vice is tolerated by us. Every year we undertake a pilgrimage,
with retinue of war, to the body of the holy prophet Daniel, which is
near the desolated site of Babylon. In our realm fishes are caught,
the blood of which dyes purple. The Amazons and the Brahmins are
subject to us. The palace in which our Supereminency resides, is built
after the pattern of the castle built by the Apostle Thomas for the
Indian king Gundoforus. Ceilings, joists, and architrave are of Sethym
wood, the roof of ebony, which can never catch fire. Over the gable of
the palace are, at the extremities, two golden apples, in each of
which are two carbuncles, so that the gold may shine by day, and the
carbuncles by night. The greater gates of the palace are of sardius,
with the horn of the horned snake inwrought, so that no one can bring
poison within.
"The other portals are of ebony. The windows are of crystal; the
tables are partly of gold, partly of amethyst, and the columns
supporting the tables are partly of ivory, partly of amethyst. The
court in which we watch the jousting is floored with onyx in order to
increase the courage of the combatants. In the palace, at night,
nothing is burned for light but wicks supplied with balsam.... Before
our palace stands a mirror, the ascent to which consists of five and
twenty steps of porphyry and serpentine." After a description of the
gems adorning this mirror, which is guarded night and day by three
thousand armed men, he explains its use: "We look therein and behold
all that is taking place in every province and region subject to our
sceptre.
"Seven kings wait upon us monthly, in turn, with sixty-two dukes, two
hundred and fifty-six counts and marquises: and twelve archbishops
sit at table with us on our right, and twenty bishops on the left,
besides the patriarch of St. Thomas, the Sarmatian Protopope, and the
Archpope of Susa.... Our lord high steward is a primate and king, our
cup-bearer is an archbishop and king, our chamberlain a bishop and
king, our marshal a king and abbot."
I may be spared further extracts from this extraordinary letter, which
proceeds to describe the church in which Prester John worships, by
enumerating the precious stones of which it is constructed, and their
special virtues.
Whether this letter was in circulation before Pope Alexander wrote
his, it is not easy to decide. Alexander does not allude to it, but
speaks of the reports which have reached him of the piety and the
magnificence of the Priest-King. At the same time, there runs a tone
of bitterness through the letter, as though the Pope had been galled
at the pretensions of this mysterious personage, and perhaps winced
under the prospect of the man-eaters overrunning Italy, as suggested
by John the Priest. The papal epistle is an assertion of the claims of
the See of Rome to universal dominion, and it assures the Eastern
Prince-Pope that his Christian professions are worthless, unless he
submits to the successor of Peter. "Not every one that saith unto me,
Lord, Lord," &c., quotes the Pope, and then explains that the will of
God is that every monarch and prelate should eat humble pie to the
Sovereign Pontiff.
Sir John Maundevil gives the origin of the priestly title of the
Eastern despot, in his curious book of travels.
"So it befelle, that this emperour cam, with a Cristene knyght with
him, into a chirche in Egypt: and it was Saterday in Wyttson woke. And
the bishop made orders. And he beheld and listened the servyse fulle
tentyfly: and he asked the Cristene knyght, what men of degree thei
scholden ben, that the prelate had before him. And the knyght
answerede and seyde, that thei scholde ben prestes. And then the
emperour seyde, that he wolde no longer ben clept kyng ne emperour,
but preest: and that he wolde have the name of the first preest, that
wente out of the chirche; and his name was John. And so evere more
sittiens, he is clept Prestre John."
It is probable that the foundation of the whole Prester-John myth lay
in the report which reached Europe of the wonderful successes of
Nestorianism in the East, and there seems reason to believe that the
famous letter given above was a Nestorian fabrication. It certainly
looks un-European; the gorgeous imagery is thoroughly Eastern, and the
disparaging tone in which Rome is spoken of could hardly have been the
expression of Western feelings. The letter has the object in view of
exalting the East in religion and arts to an undue eminence at the
expense of the West, and it manifests some ignorance of European
geography, when it speaks of the land extending from Spain to the
Polar Sea. Moreover, the sites of the patriarchates, and the dignity
conferred on that of St. Thomas, are indications of a Nestorian bias.
A brief glance at the history of this heretical Church may be of value
here, as showing that there really was a foundation for the wild
legends concerning a Christian empire in the East, so prevalent in
Europe. Nestorius, a priest of Antioch and a disciple of St.
Chrysostom, was elevated by the emperor to the patriarchate of
Constantinople, and in the year 428 began to propagate his heresy,
denying the hypostatic union. The Council of Ephesus denounced him,
and, in spite of the emperor and court, Nestorius was anathematized
and driven into exile. His sect spread through the East, and became a
flourishing church. It reached to China, where the emperor was all but
converted; its missionaries traversed the frozen tundras of Siberia,
preaching their maimed Gospel to the wild hordes which haunted those
dreary wastes; it faced Buddhism, and wrestled with it for the
religious supremacy in Thibet; it established churches in Persia and
in Bokhara; it penetrated India; it formed colonies in Ceylon, in
Siam, and in Sumatra; so that the Catholicos or Pope of Bagdad
exercised sway more extensive than that ever obtained by the successor
of St. Peter. The number of Christians belonging to that communion
probably exceeded that of the members of the true Catholic Church in
East and West. But the Nestorian Church was not founded on the Rock;
it rested on Nestorius; and when the rain descended, and the winds
blew, and the floods came, and beat upon that house, it fell, leaving
scarce a fragment behind.
Rubruquis the Franciscan, who in 1253 was sent on a mission into
Tartary, was the first to let in a little light on the fable. He
writes, "The Catai dwelt beyond certain mountains across which I
wandered, and in a plain in the midst of the mountains lived once an
important Nestorian shepherd, who ruled over the Nestorian people,
called Nayman. When Coir-Khan died, the Nestorian people raised this
man to be king, and called him King Johannes, and related of him ten
times as much as the truth. The Nestorians thereabouts have this way
with them, that about nothing they make a great fuss, and thus they
have got it noised abroad that Sartach, Mangu-Khan, and Ken-Khan were
Christians, simply because they treated Christians well, and showed
them more honor than other people. Yet, in fact, they were not
Christians at all. And in like manner the story got about that there
was a great King John. However, I traversed his pastures, and no one
knew anything about him, except a few Nestorians. In his pastures
lives Ken-Khan, at whose court was Brother Andrew, whom I met on my
way back. This Johannes had a brother, a famous shepherd, named Unc,
who lived three weeks' journey beyond the mountains of Caracatais."
This Unk-Khan was a real individual; he lost his life in the year
1203. Kuschhik, prince of the Nayman, and follower of Kor-Khan, fell
in 1218.
Marco Polo, the Venetian traveller (1254-1324), identifies Unk-Khan
with Prester John; he says, "I will now tell you of the deeds of the
Tartars, how they gained the mastery, and spread over the whole earth.
The Tartars dwelt between Georgia and Bargu, where there is a vast
plain and level country, on which are neither cities nor forts, but
capital pasturage and water. They had no chief of their own, but paid
to Prester Johannes tribute. Of the greatness of this Prester
Johannes, who was properly called Un-Khan, the whole world spake; the
Tartars gave him one of every ten head of cattle. When Prester John
noticed that they were increasing, he feared them, and planned how he
could injure them. He determined therefore to scatter them, and he
sent barons to do this. But the Tartars guessed what Prester John
purposed ... and they went away into the wide wastes of the North,
where they might be beyond his reach." He then goes on to relate how
Tschengis-(Jenghiz-)Khan became the head of the Tartars, and how he
fought against Prester John, and, after a desperate fight, overcame
and slew him.
The Syriac Chronicle of the Jacobite Primate, Gregory Bar-HebrA|us
(born 1226, died 1286), also identifies Unk-Khan with Prester John.
"In the year of the Greeks 1514, of the Arabs 599 (A. D. 1202), when
Unk-Khan, who is the Christian King John, ruled over a stock of the
barbarian Hunns, called Kergt, Tschingys-Khan served him with great
zeal. When John observed the superiority and serviceableness of the
other, he envied him, and plotted to seize and murder him. But two
sons of Unk-Khan, having heard this, told it to Tschingys; whereupon
he and his comrades fled by night, and secreted themselves. Next
morning Unk-Khan took possession of the Tartar tents, but found them
empty. Then the party of Tschingys fell upon him, and they met by the
spring called Balschunah, and the side of Tschingys won the day; and
the followers of Unk-Khan were compelled to yield. They met again
several times, till Unk-Khan was utterly discomfited, and was slain
himself, and his wives, sons, and daughters carried into captivity.
Yet we must consider that King John the Kergtajer was not cast down
for nought; nay, rather, because he had turned his heart from the fear
of Christ his Lord, who had exalted him, and had taken a wife of the
Zinish nation, called Quarakhata. Because he forsook the religion of
his ancestors and followed strange gods, therefore God took the
government from him, and gave it to one better than he, and whose
heart was right before God."
Some of the early travellers, such as John de Plano Carpini and Marco
Polo, in disabusing the popular mind of the belief in Prester John as
a mighty Asiatic Christian monarch, unintentionally turned the popular
faith in that individual into a new direction. They spoke of the black
people of Abascia in Ethiopia, which, by the way, they called Middle
India, as a great people subject to a Christian monarch.
Marco Polo says that the true monarch of Abyssinia is Christ; but that
it is governed by six kings, three of whom are Christians and three
Saracens, and that they are in league with the Soudan of Aden.
Bishop Jordanus, in his description of the world, accordingly sets
down Abyssinia as the kingdom of Prester John; and such was the
popular impression, which was confirmed by the appearance at intervals
of ambassadors at European courts from the King of Abyssinia. The
discovery of the Cape of Good Hope was due partly to a desire
manifested in Portugal to open communications with this monarch,[22]
and King John II. sent two men learned in Oriental languages through
Egypt to the court of Abyssinia. The might and dominion of this
prince, who had replaced the Tartar chief in the popular creed as
Prester John, was of course greatly exaggerated, and was supposed to
extend across Arabia and Asia to the wall of China. The spread of
geographical knowledge has contracted the area of his dominions, and a
critical acquaintance with history has exploded the myth which
invested Unk-Khan, the nomad chief, with all the attributes of a
demigod, uniting in one the utmost pretensions of a Pope and the
proudest claims of a monarch.
FOOTNOTES:
[19] Otto, Ep. Frising., lib. vii. c. 33.
[20] Persia.
[21] Media.
[22] Ludolfi Hist. A†thiopica, lib. ii. cap. 1, 2. Petrus, Petri filius
LusitaniA| princeps, M. Pauli Veneti librum (qui de Indorum rebus
multa: speciatim vero de Presbytero Johanne aliqua magnifice scripsit)
Venetiis secum in patriam detulerat, qui (Chronologicis Lusitanorum
testantibus) prA|cipuam Johanni Regi ansam dedit IndicA| navigationis,
quam Henricus Johannis I. filius, patruus ejus, tentaverat,
prosequendA|, &c.
The Divining Rod.
From the remotest period a rod has been regarded as the symbol of
power and authority, and Holy Scripture employs it in the popular
sense. Thus David speaks of "Thy rod and Thy staff comforting me;" and
Moses works his miracles before Pharaoh with the rod as emblem of
Divine commission. It was his rod which became a serpent, which turned
the water of Egypt into blood, which opened the waves of the Red Sea
and restored them to their former level, which "smote the rock of
stone so that the water gushed out abundantly." The rod of Aaron acted
an oracular part in the contest with the princes; laid up before the
ark, it budded and brought forth almonds. In this instance we have it
no longer as a symbol of authority, but as a means of divining the
will of God. And as such it became liable to abuse; thus Hosea rebukes
the chosen people for practising similar divinations. "My people ask
counsel at their stocks, and their staff declareth unto them."[23]
Long before this, Jacob had made a different use of rods, employing
them as a charm to make his father-in-law's sheep bear pied and
spotted lambs.
We find rhabdomancy a popular form of divination among the Greeks, and
also among the Romans. Cicero in his "De Officiis" alludes to it. "If
all that is needful for our nourishment and support arrives to us by
means of some divine rod, as people say, then each of us, free from
all care and trouble, may give himself up to the exclusive pursuit of
study and science."
Probably it is to this rod that the allusion of Ennius, as the agent
in discovering hidden treasures, quoted in the first book of his "De
Divinatione," refers.
According to Vetranius Maurus, Varro left a satire on the "Virgula
divina," which has not been preserved. Tacitus tells us that the
Germans practised some sort of divination by means of rods. "For the
purpose their method is simple. They cut a rod off some fruit-tree
into bits, and after having distinguished them by various marks, they
cast them into a white cloth.... Then the priest thrice draws each
piece, and explains the oracle according to the marks." Ammianus
Marcellinus says that the Alains employed an osier rod.
The fourteenth law of the Frisons ordered that the discovery of
murders should be made by means of divining rods used in Church. These
rods should be laid before the altar, and on the sacred relics, after
which God was to be supplicated to indicate the culprit. This was
called the Lot of Rods, or Tan-teen, the Rod of Rods.
But the middle ages was the date of the full development of the
superstition, and the divining rod was believed to have efficacy in
discovering hidden treasures, veins of precious metal, springs of
water, thefts, and murders. The first notice of its general use among
late writers is in the "Testamentum Novum," lib. i. cap. 25, of Basil
Valentine, a Benedictine monk of the fifteenth century. Basil speaks
of the general faith in and adoption of this valuable instrument for
the discovery of metals, which is carried by workmen in mines, either
in their belts or in their caps. He says that there are seven names by
which this rod is known, and to its excellences under each title he
devotes a chapter of his book. The names are: Divine Rod, Shining Rod,
Leaping Rod, Transcendent Rod, Trembling Rod, Dipping Rod, Superior
Rod. In his admirable treatise on metals, Agricola speaks of the rod
in terms of disparagement; he considers its use as a relic of ancient
magical forms, and he says that it is only irreligious workmen who
employ it in their search after metals. Goclenius, however, in his
treatise on the virtue of plants, stoutly does battle for the
properties of the hazel rod. Whereupon Roberti, a Flemish Jesuit,
falls upon him tooth and nail, disputes his facts, overwhelms him with
abuse, and gibbets him for popular ridicule. Andreas Libavius, a
writer I have already quoted in my article on the Wandering Jew,
undertook a series of experiments upon the hazel divining rod, and
concluded that there was truth in the popular belief. The Jesuit
Kircher also "experimentalized several times on wooden rods which were
declared to be sympathetic with regard to certain metals, by placing
them on delicate pivots in equilibrium; but they never turned on the
approach of metal." (De Arte Magnetica.) However, a similar course of
experiments over water led him to attribute to the rod the power of
indicating subterranean springs and water-courses; "I would not affirm
it," he says, "unless I had established the fact by my own
experience."
Dechales, another Jesuit, author of a treatise on natural springs, and
of a huge tome entitled "Mundus Mathematicus," declared in the latter
work, that no means of discovering sources is equal to the divining
rod; and he quotes a friend of his who, with a hazel rod in his hand,
could discover springs with the utmost precision and facility, and
could trace on the surface of the ground the course of a subterranean
conduit. Another writer, Saint-Romain, in his "Science dA(C)gagA(C)e des
ChimA"res de l'A%cole," exclaims, "Is it not astonishing to see a rod,
which is held firmly in the hands, bow itself and turn visibly in the
direction of water or metal, with more or less promptitude, according
as the metal or the water are near or remote from the surface!"
In 1659 the Jesuit Gaspard Schott writes that the rod is used in every
town of Germany, and that he had frequent opportunity of seeing it
used in the discovery of hidden treasures. "I searched with the
greatest care," he adds, "into the question whether the hazel rod had
any sympathy with gold and silver, and whether any natural property
set it in motion. In like manner I tried whether a ring of metal, held
suspended by a thread in the midst of a tumbler, and which strikes the
hours, is moved by any similar force. I ascertained that these effects
could only have rise from the deception of those holding the rod or
the pendulum, or, may be, from some diabolic impulsion, or, more
likely still, because imagination sets the hand in motion."
The Sieur le Royer, a lawyer of Rouen, in 1674, published his "TraitA(C)
du BActon universel," in which he gives an account of a trial made with
the rod in the presence of Father Jean FranASec.ois, who had ridiculed the
operation in his treatise on the science of waters, published at
Rennes in 1655, and which succeeded in convincing the blasphemer of
the divine Rod. Le Royer denies to it the power of picking out
criminals, which had been popularly attributed to it, and as had been
unhesitatingly claimed for it by Debrio in his "Disquisitio Magica."
And now I am brought to the extraordinary story of Jacques Aymar,
which attracted the attention of Europe to the marvellous properties
of the divining rod. I shall give the history of this man in full, as
such an account is rendered necessary by the mutilated versions I have
seen current in English magazine articles, which follow the lead of
Mrs. Crowe, who narrates the earlier portion of this impostor's
career, but says nothing of his _exposA(C)_ and downfall.
On the 5th July, 1692, at about ten o'clock in the evening, a
wine-seller of Lyons and his wife were assassinated in their cellar,
and their money carried off. On the morrow, the officers of justice
arrived, and examined the premises. Beside the corpses, lay a large
bottle wrapped in straw, and a bloody hedging bill, which undoubtedly
had been the instrument used to accomplish the murder. Not a trace of
those who had committed the horrible deed was to be found, and the
magistrates were quite at fault as to the direction in which they
should turn for a clew to the murderer or murderers.
At this juncture a neighbor reminded the magistrates of an incident
which had taken place four years previous. It was this. In 1688 a
theft of clothes had been made in Grenoble. In the parish of CrA'le
lived a man named Jacques Aymar, supposed to be endowed with the
faculty of using the divining rod. This man was sent for. On reaching
the spot where the theft had been committed, his rod moved in his
hand. He followed the track indicated by the rod, and it continued to
rotate between his fingers as long as he followed a certain direction,
but ceased to turn if he diverged from it in the smallest degree.
Guided by his rod, Aymar went from street to street, till he was
brought to a standstill before the prison gates. These could not be
opened without leave of the magistrate, who hastened to witness the
experiment. The gates were unlocked, and Aymar, under the same
guidance, directed his steps towards four prisoners lately
incarcerated. He ordered the four to be stood in a line, and then he
placed his foot on that of the first. The rod remained immovable. He
passed to the second, and the rod turned at once. Before the third
prisoner there were no signs; the fourth trembled, and begged to be
heard. He owned himself the thief, along with the second, who also
acknowledged the theft, and mentioned the name of the receiver of the
stolen goods. This was a farmer in the neighborhood of Grenoble. The
magistrate and officers visited him and demanded the articles he had
obtained. The farmer denied all knowledge of the theft and all
participation in the booty. Aymar, however, by means of his rod,
discovered the secreted property, and restored it to the persons from
whom it had been stolen.
On another occasion Aymar had been in quest of a spring of water, when
he felt his rod turn sharply in his hand. On digging at the spot,
expecting to discover an abundant source, the body of a murdered woman
was found in a barrel, with a rope twisted round her neck. The poor
creature was recognized as a woman of the neighborhood who had
vanished four months before. Aymar went to the house which the victim
had inhabited, and presented his rod to each member of the household.
It turned upon the husband of the deceased, who at once took to
flight.
The magistrates of Lyons, at their wits' ends how to discover the
perpetrators of the double murder in the wine shop, urged the
Procureur du Roi to make experiment of the powers of Jacques Aymar.
The fellow was sent for, and he boldly asserted his capacity for
detecting criminals, if he were first brought to the spot of the
murder, so as to be put _en rapport_ with the murderers.
He was at once conducted to the scene of the outrage, with the rod in
his hand. This remained stationary as he traversed the cellar, till he
reached the spot where the body of the wine seller had lain; then the
stick became violently agitated, and the man's pulse rose as though he
were in an access of fever. The same motions and symptoms manifested
themselves when he reached the place where the second victim had lain.
Having thus received his _impression_, Aymar left the cellar, and,
guided by his rod, or rather by an internal instinct, he ascended into
the shop, and then stepping into the street, he followed from one to
another, like a hound upon the scent, the track of the murderers. It
conducted him into the court of the archiepiscopal palace, across it,
and down to the gate of the Rhone. It was now evening, and the city
gates being all closed, the quest of blood was relinquished for the
night.
Next morning Aymar returned to the scent. Accompanied by three
officers, he left the gate, and descended the right bank of the Rhone.
The rod gave indications of there having been three involved in the
murder, and he pursued the traces till two of them led to a gardener's
cottage. Into this he entered, and there he asserted with warmth,
against the asseverations of the proprietor to the contrary, that the
fugitives had entered his room, had seated themselves at his table,
and had drunk wine out of one of the bottles which he indicated. Aymar
tested each of the household with his rod, to see if they had been in
contact with the murderers. The rod moved over the two children only,
aged respectively ten and nine years. These little things, on being
questioned, answered, with reluctance, that during their father's
absence on Sunday morning, against his express commands, they had left
the door open, and that two men, whom they described, had come in
suddenly upon them, and had seated themselves and made free with the
wine in the bottle pointed out by the man with the rod. This first
verification of the talents of Jacques Aymar convinced some of the
sceptical, but the Procurateur GA(C)nA(C)ral forbade the prosecution of the
experiment till the man had been further tested.
As already stated, a hedging bill had been discovered, on the scene of
the murder, smeared with blood, and unquestionably the weapon with
which the crime had been committed. Three bills from the same maker,
and of precisely the same description, were obtained, and the four
were taken into a garden, and secretly buried at intervals. Aymar was
then brought, staff in hand, into the garden, and conducted over the
spots where lay the bills. The rod began to vibrate as his feet stood
upon the place where was concealed the bill which had been used by the
assassins, but was motionless elsewhere. Still unsatisfied, the four
bills were exhumed and concealed anew. The comptroller of the province
himself bandaged the sorcerer's eyes, and led him by the hand from
place to place. The divining rod showed no signs of movement till it
approached the blood-stained weapon, when it began to oscillate.
The magistrates were now so far satisfied as to agree that Jacques
Aymar should be authorized to follow the trail of the murderers, and
have a company of archers to follow him.
Guided by his rod, Aymar now recommenced his pursuit. He continued
tracing down the right bank of the Rhone till he came to half a league
from the bridge of Lyons. Here the footprints of three men were
observed in the sand, as though engaged in entering a boat. A rowing
boat was obtained, and Aymar, with his escort, descended the river; he
found some difficulty in following the trail upon water; still he was
able, with a little care, to detect it. It brought him under an arch
of the bridge of Vienne, which boats rarely passed beneath. This
proved that the fugitives were without a guide. The way in which this
curious journey was made was singular. At intervals Aymar was put
ashore to test the banks with his rod, and ascertain whether the
murderers had landed. He discovered the places where they had slept,
and indicated the chairs or benches on which they had sat. In this
manner, by slow degrees, he arrived at the military camp of Sablon,
between Vienne and Saint-Valier. There Aymar felt violent agitation,
his cheeks flushed, and his pulse beat with rapidity. He penetrated
the crowds of soldiers, but did not venture to use his rod, lest the
men should take it ill, and fall upon him. He could not do more
without special authority, and was constrained to return to Lyons. The
magistrates then provided him with the requisite powers, and he went
back to the camp. Now he declared that the murderers were not there.
He recommenced his pursuit, and descended the Rhone again as far as
Beaucaire.
On entering the town he ascertained by means of his rod that those
whom he was pursuing had parted company. He traversed several streets,
then crowded on account of the annual fair, and was brought to a
standstill before the prison doors. One of the murderers was within,
he declared; he would track the others afterwards. Having obtained
permission to enter, he was brought into the presence of fourteen or
fifteen prisoners. Amongst these was a hunchback, who had only an hour
previously been incarcerated on account of a theft he had committed at
the fair. Aymar applied his rod to each of the prisoners in
succession: it turned upon the hunchback. The sorcerer ascertained
that the other two had left the town by a little path leading into the
Nismes road. Instead of following this track, he returned to Lyons
with the hunchback and the guard. At Lyons a triumph awaited him. The
hunchback had hitherto protested his innocence, and declared that he
had never set foot in Lyons. But as he was brought to that town by the
way along which Aymar had ascertained that he had left it, the fellow
was recognized at the different houses where he had lodged the night,
or stopped for food. At the little town of Bagnols, he was confronted
with the host and hostess of a tavern where he and his comrades had
slept, and they swore to his identity, and accurately described his
companions: their description tallied with that given by the children
of the gardener. The wretched man was so confounded by this
recognition, that he avowed having staid there, a few days before,
along with two ProvenASec.als. These men, he said, were the criminals; he
had been their servant, and had only kept guard in the upper room
whilst they committed the murders in the cellar.
On his arrival in Lyons he was committed to prison, and his trial was
decided on. At his first interrogation he told his tale precisely as
he had related it before, with these additions: the murderers spoke
patois, and had purchased two bills. At ten o'clock in the evening all
three had entered the wine shop. The ProvenASec.als had a large bottle
wrapped in straw, and they persuaded the publican and his wife to
descend with them into the cellar to fill it, whilst he, the
hunchback, acted as watch in the shop. The two men murdered the
wine-seller and his wife with their bills, and then mounted to the
shop, where they opened the coffer, and stole from it one hundred and
thirty crowns, eight louis-d'ors, and a silver belt. The crime
accomplished, they took refuge in the court of a large house,--this
was the archbishop's palace, indicated by Aymar,--and passed the night
in it. Next day, early, they left Lyons, and only stopped for a moment
at a gardener's cottage. Some way down the river, they found a boat
moored to the bank. This they loosed from its mooring and entered.
They came ashore at the spot pointed out by the man with the stick.
They staid some days in the camp at Sablon, and then went on to
Beaucaire.
Aymar was now sent in quest of the other murderers. He resumed their
trail at the gate of Beaucaire, and that of one of them, after
considerable _dA(C)tours_, led him to the prison doors of Beaucaire, and
he asked to be allowed to search among the prisoners for his man. This
time he was mistaken. The second fugitive was not within; but the
jailer affirmed that a man whom he described--and his description
tallied with the known appearance of one of the ProvenASec.als--had called
at the gate shortly after the removal of the hunchback to inquire
after him, and on learning of his removal to Lyons, had hurried off
precipitately. Aymar now followed his track from the prison, and this
brought him to that of the third criminal. He pursued the double scent
for some days. But it became evident that the two culprits had been
alarmed at what had transpired in Beaucaire, and were flying from
France. Aymar traced them to the frontier, and then returned to Lyons.
On the 30th of August, 1692, the poor hunchback was, according to
sentence, broken on the wheel, in the Place des Terreaux. On his way
to execution he had to pass the wine shop. There the recorder publicly
read his sentence, which had been delivered by thirty judges. The
criminal knelt and asked pardon of the poor wretches in whose murder
he was involved, after which he continued his course to the place
fixed for his execution.
It may be well here to give an account of the authorities for this
extraordinary story. There are three circumstantial accounts, and
numerous letters written by the magistrate who sat during the trial,
and by an eye-witness of the whole transaction, men honorable and
disinterested, upon whose veracity not a shadow of doubt was supposed
to rest by their contemporaries.
M. Chauvin, Doctor of Medicine, published a "_Lettre A Mme. la
Marquise de Senozan, sur les moyens dont on s'est servi pour dA(C)couvrir
les complices d'un assassinat commis A Lyon, le 5 Juillet, 1692_."
Lyons, 1692. The _procA"s-verbal_ of the Procureur du Roi, M. de
Vanini, is also extant, and published in the _Physique occulte_ of the
AbbA(C) de Vallemont.
Pierre Gamier, Doctor of Medicine of the University of Montpellier,
wrote a _Dissertation physique en forme de lettre, A M. de SA"ve,
seigneur de FlA(C)chA"res_, on Jacques Aymar, printed the same year at
Lyons, and republished in the _Histoire critique des pratiques
superstitieuses du PA"re Lebrun_.
Doctor Chauvin was witness of nearly all the circumstances related, as
was also the AbbA(C) Lagarde, who has written a careful account of the
whole transaction as far as to the execution of the hunchback.
Another eye-witness writes to the AbbA(C) Bignon a letter printed by
Lebrun in his _Histoire critique_ cited above. "The following
circumstance happened to me yesterday evening," he says: "M. le
Procureur du Roi here, who, by the way, is one of the wisest and
cleverest men in the country, sent for me at six o'clock, and had me
conducted to the scene of the murder. We found there M. Grimaut,
director of the customs, whom I knew to be a very upright man, and a
young attorney named Besson, with whom I am not acquainted, but who M.
le Procureur du Roi told me had the power of using the rod as well as
M. Grimaut. We descended into the cellar where the murder had been
committed, and where there were still traces of blood. Each time that
M. Grimaut and the attorney passed the spot where the murder had been
perpetrated, the rods they held in their hands began to turn, but
ceased when they stepped beyond the spot. We tried experiments for
more than an hour, as also with the bill, which M. le Procureur had
brought along with him, and they were satisfactory. I observed several
curious facts in the attorney. The rod in his hands was more violently
moved than in those of M. Grimaut, and when I placed one of my fingers
in each of his hands, whilst the rod turned, I felt the most
extraordinary throbbings of the arteries in his palms. His pulse was
at fever heat. He sweated profusely, and at intervals he was compelled
to go into the court to obtain fresh air."
The Sieur Pauthot, Dean of the College of Medicine at Lyons, gave his
observations to the public as well. Some of them are as follows: "We
began at the cellar in which the murder had been committed; into this
the man with the rod (Aymar) shrank from entering, because he felt
violent agitations which overcame him when he used the stick over the
place where the corpses of those who had been assassinated had lain.
On entering the cellar, the rod was put in my hands, and arranged by
the master as most suitable for operation; I passed and repassed over
the spot where the bodies had been found, but it remained immovable,
and I felt no agitation. A lady of rank and merit, who was with us,
took the rod after me; she felt it begin to move, and was internally
agitated. Then the owner of the rod resumed it, and, passing over the
same places, the stick rotated with such violence that it seemed
easier to break than to stop it. The peasant then quitted our company
to faint away, as was his wont after similar experiments. I followed
him. He turned very pale and broke into a profuse perspiration, whilst
for a quarter of an hour his pulse was violently troubled; indeed, the
faintness was so considerable, that they were obliged to dash water in
his face and give him water to drink in order to bring him round." He
then describes experiments made over the bloody bill and others
similar, which succeeded in the hands of Aymar and the lady, but
failed when he attempted them himself. Pierre Garnier, physician of
the medical college of Montpellier, appointed to that of Lyons, has
also written an account of what he saw, as mentioned above. He gives a
curious proof of Aymar's powers.
"M. le Lieutenant-GA(C)nA(C)ral having been robbed by one of his lackeys,
seven or eight months ago, and having lost by him twenty-five crowns
which had been taken out of one of the cabinets behind his library,
sent for Aymar, and asked him to discover the circumstances. Aymar
went several times round the chamber, rod in hand, placing one foot on
the chairs, on the various articles of furniture, and on two bureaux
which are in the apartment, each of which contains several drawers. He
fixed on the very bureau and the identical drawer out of which the
money had been stolen. M. le Lieutenant-GA(C)nA(C)ral bade him follow the
track of the robber. He did so. With his rod he went out on a new
terrace, upon which the cabinet opens, thence back into the cabinet
and up to the fire, then into the library, and from thence he went
direct up stairs to the lackeys' sleeping apartment, when the rod
guided him to one of the beds, and turned over one side of the bed,
remaining motionless over the other. The lackeys then present cried
out that the thief had slept on the side indicated by the rod, the bed
having been shared with another footman, who occupied the further
side." Garnier gives a lengthy account of various experiments he made
along with the Lieutenant-GA(C)nA(C)ral, the uncle of the same, the AbbA(C) de
St. Remain, and M. de Puget, to detect whether there was imposture in
the man. But all their attempts failed to discover a trace of
deception. He gives a report of a verbal examination of Aymar which is
interesting. The man always replied with candor.
The report of the extraordinary discovery of murder made by the
divining rod at Lyons attracted the attention of Paris, and Aymar was
ordered up to the capital. There, however, his powers left him. The
Prince de CondA(C) submitted him to various tests, and he broke down
under every one. Five holes were dug in the garden. In one was
secreted gold, in another silver, in a third silver and gold, in the
fourth copper, and in the fifth stones. The rod made no signs in
presence of the metals, and at last actually began to move over the
buried pebbles. He was sent to Chantilly to discover the perpetrators
of a theft of trout made in the ponds of the park. He went round the
water, rod in hand, and it turned at spots where he said the fish had
been drawn out. Then, following the track of the thief, it led him to
the cottage of one of the keepers, but did not move over any of the
individuals then in the house. The keeper himself was absent, but
arrived late at night, and, on hearing what was said, he roused Aymar
from his bed, insisting on having his innocence vindicated. The
divining rod, however, pronounced him guilty, and the poor fellow took
to his heels, much upon the principle recommended by Montesquieu a
while after. Said he, "If you are accused of having stolen the towers
of Notre-Dame, bolt at once."
A peasant, taken at haphazard from the street, was brought to the
sorcerer as one suspected. The rod turned slightly, and Aymar declared
that the man did not steal the fish, but ate of them. A boy was then
introduced, who was said to be the keeper's son. The rod rotated
violently at once. This was the finishing stroke, and Aymar was sent
away by the Prince in disgrace. It now transpired that the theft of
fish had taken place seven years before, and the lad was no relation
of the keeper, but a country boy who had only been in Chantilly eight
or ten months. M. Goyonnot, Recorder of the King's Council, broke a
window in his house, and sent for the diviner, to whom he related a
story of his having been robbed of valuables during the night. Aymar
indicated the broken window as the means whereby the thief had entered
the house, and pointed out the window by which he had left it with the
booty. As no such robbery had been committed, Aymar was turned out of
the house as an impostor. A few similar cases brought him into such
disrepute that he was obliged to leave Paris, and return to Grenoble.
Some years after, he was made use of by the MarA(C)chal Montrevel, in his
cruel pursuit of the Camisards.
Was Aymar an impostor from first to last, or did his powers fail him
in Paris? and was it only then that he had recourse to fraud?
Much may be said in favor of either supposition. His _exposA(C)_ at Paris
tells heavily against him, but need not be regarded as conclusive
evidence of imposture throughout his career. If he really did possess
the powers he claimed, it is not to be supposed that these existed in
full vigor under all conditions; and Paris is a place most unsuitable
for testing them, built on artificial soil, and full of disturbing
influences of every description. It has been remarked with others who
used the rod, that their powers languished under excitement, and that
the faculties had to be in repose, the attention to be concentrated on
the subject of inquiry, or the action--nervous, magnetic, or
electrical, or what you will--was impeded.
Now, Paris, visited for the first time by a poor peasant, its
_salons_ open to him, dazzling him with their splendor, and the
novelty of finding himself in the midst of princes, dukes, marquises,
and their families, not only may have agitated the countryman to such
an extent as to deprive him of his peculiar faculty, but may have led
him into simulating what he felt had departed from him, at the moment
when he was under the eyes of the grandees of the Court. We have
analogous cases in Bleton and Angelique Cottin. The former was a
hydroscope, who fell into convulsions whenever he passed over running
water. This peculiarity was noticed in him when a child of seven years
old. When brought to Paris, he failed signally to detect the presence
of water conveyed underground by pipes and conduits, but he pretended
to feel the influence of water where there certainly was none.
Angelique Cottin was a poor girl, highly charged with electricity. Any
one touching her received a violent shock; one medical gentleman,
having seated her on his knee, was knocked clean out of his chair by
the electric fluid, which thus exhibited its sense of propriety. But
the electric condition of Angelique became feebler as she approached
Paris, and failed her altogether in the capital.
I believe that the imagination is the principal motive force in those
who use the divining rod; but whether it is so solely, I am unable to
decide. The powers of nature are so mysterious and inscrutable that we
must be cautious in limiting them, under abnormal conditions, to the
ordinary laws of experience.
[Illustration: {How to hold a divining rod.}]
The manner in which the rod was used by certain persons renders
self-deception possible. The rod is generally of hazel, and is forked
like a Y; the forefingers are placed against the diverging arms of the
rod, and the elbows are brought back against the side; thus the
implement is held in front of the operator, delicately balanced before
the pit of the stomach at a distance of about eight inches. Now, if
the pressure of the balls of the digits be in the least relaxed, the
stalk of the rod will naturally fall. It has been assumed by some,
that a restoration of the pressure will bring the stem up again,
pointing towards the operator, and a little further pressure will
elevate it into a perpendicular position. A relaxation of force will
again lower it, and thus the rotation observed in the rod be
maintained. I confess myself unable to accomplish this. The lowering
of the leg of the rod is easy enough, but no efforts of mine to
produce a revolution on its axis have as yet succeeded. The muscles
which would contract the fingers upon the arms of the stick, pass the
shoulder; and it is worthy of remark that one of the medical men who
witnessed the experiments made on Bleton the hydroscope, expressly
alludes to a slight rising of the shoulders during the rotation of the
divining rod.
But the manner of using the rod was by no means identical in all
cases. If, in all cases, it had simply been balanced between the
fingers, some probability might be given to the suggestion above made,
that the rotation was always effected by the involuntary action of the
muscles.
The usual manner of holding the rod, however, precluded such a
possibility. The most ordinary use consisted in taking a forked stick
in such a manner that the palms were turned upwards, and the fingers
closed upon the branching arms of the rod. Some required the normal
position of the rod to be horizontal, others elevated the point,
others again depressed it.
If the implement were straight, it was held in a similar manner, but
the hands were brought somewhat together, so as to produce a slight
arc in the rod. Some who practised rhabdomancy sustained this species
of rod between their thumbs and forefingers; or else the thumb and
forefingers were closed, and the rod rested on their points; or again
it reposed on the flat of the hand, or on the back, the hand being
held vertically and the rod held in equilibrium.
A third species of divining rod consisted in a straight staff cut in
two: one extremity of the one half was hollowed out, the other half
was sharpened at the end, and this end was inserted in the hollow, and
the pointed stick rotated in the cavity.
[Illustration: POSITIONS OF THE HANDS.
From "Lettres qui dA(C)couvrent l'Illusion des Philosophes sur la
Baguette." Paris, 1693.]
The way in which Bleton used his rod is thus minutely described: "He
does not grasp it, nor warm it in his hands, and he does not regard
with preference a hazel branch lately cut and full of sap. He
places horizontally between his forefingers a rod of any kind given to
him, or picked up in the road, of any sort of wood except elder, fresh
or dry, not always forked, but sometimes merely bent. If it is
straight, it rises slightly at the extremities by little jerks, but
does not turn. If bent, it revolves on its axis with more or less
rapidity, in more or less time, according to the quantity and current
of the water. I counted from thirty to thirty-five revolutions in a
minute, and afterwards as many as eighty. A curious phenomenon is,
that Bleton is able to make the rod turn between another person's
fingers, even without seeing it or touching it, by approaching his
body towards it when his feet stand over a subterranean watercourse.
It is true, however, that the motion is much less strong and less
durable in other fingers than his own. If Bleton stood on his head,
and placed the rod between his feet, though he felt strongly the
peculiar sensations produced in him by flowing water, yet the rod
remained stationary. If he were insulated on glass, silk, or wax, the
sensations were less vivid, and the rotation of the stick ceased."
But this experiment failed in Paris, under circumstances which either
proved that Bleton's imagination produced the movement, or that his
integrity was questionable. It is quite possible that in many
instances the action of the muscles is purely involuntary, and is
attributable to the imagination, so that the operator deceives himself
as well as others.
This is probably the explanation of the story of Mdlle. Olivet, a
young lady of tender conscience, who was a skilful performer with the
divining rod, but shrank from putting her powers in operation, lest
she should be indulging in unlawful acts. She consulted the PA"re
Lebrun, author of a work already referred to in this paper, and he
advised her to ask God to withdraw the power from her, if the exercise
of it was harmful to her spiritual condition. She entered into retreat
for two days, and prayed with fervor. Then she made her communion,
asking God what had been recommended to her at the moment when she
received the Host. In the afternoon of the same day she made
experiment with her rod, and found that it would no longer operate.
The girl had strong faith in it before--a faith coupled with fear; and
as long as that faith was strong in her, the rod moved; now she
believed that the faculty was taken from her; and the power ceased
with the loss of her faith.
If the divining rod is put in motion by any other force except the
involuntary action of the muscles, we must confine its powers to the
property of indicating the presence of flowing water. There are
numerous instances of hydroscopes thus detecting the existence of a
spring, or of a subterranean watercourse; the most remarkably endowed
individuals of this description are Jean-Jacques Parangue, born near
Marseilles, in 1760, who experienced a horror when near water which no
one else perceived. He was endowed with the faculty of seeing water
through the ground, says l'AbbA(C) Sauri, who gives his history. Jenny
Leslie, a Scotch girl, about the same date claimed similar powers. In
1790, Pennet, a native of DauphinA(C), attracted attention in Italy, but
when carefully tested by scientific men in Padua, his attempts to
discover buried metals failed; at Florence he was detected in an
endeavor to find out by night what had been secreted to test his
powers on the morrow. Vincent Amoretti was an Italian, who underwent
peculiar sensations when brought in proximity to water, coal, and
salt; he was skilful in the use of the rod, but made no public
exhibition of his powers.
The rod is still employed, I have heard it asserted, by Cornish
miners; but I have never been able to ascertain that such is really
the case. The mining captains whom I have questioned invariably
repudiated all knowledge of its use.
In Wiltshire, however, it is still employed for the purpose of
detecting water; and the following extract from a letter I have just
received will show that it is still in vogue on the Continent:--
"I believe the use of the divining rod for discovering springs of
water has by no means been confined to mediA|val times; for I was
personally acquainted with a lady, now deceased, who has successfully
practised with it in this way. She was a very clever and accomplished
woman; Scotch by birth and education; by no means credulous; possibly
a little imaginative, for she wrote not unsuccessfully; and of a
remarkably open and straightforward disposition. Captain C----, her
husband, had a large estate in Holstein, near Lubeck, supporting a
considerable population; and whether for the wants of the people or
for the improvement of the land, it now and then happened that an
additional well was needed.
"On one of these occasions a man was sent for who made a regular
profession of finding water by the divining rod; there happened to be
a large party staying at the house, and the whole company turned out
to see the fun. The rod gave indications in the usual way, and water
was ultimately found at the spot. Mrs. C----, utterly sceptical, took
the rod into her own hands to make experiment, believing that she
would prove the man an impostor; and she said afterwards she was never
more frightened in her life than when it began to move, on her walking
over the spring. Several other gentlemen and ladies tried it, but it
was quite inactive in their hands. 'Well,' said the host to his wife,
'we shall have no occasion to send for the man again, as you are such
an adept.'
"Some months after this, water was wanted in another part of the
estate, and it occurred to Mrs. C---- that she would use the rod
again. After some trials, it again gave decided indications, and a
well was begun and carried down a very considerable depth. At last she
began to shrink from incurring more expense, but the laborers had
implicit faith; and begged to be allowed to persevere. Very soon the
water burst up with such force that the men escaped with difficulty;
and this proved afterwards the most unfailing spring for miles round.
"You will take the above for what it is worth; the facts I have given
are undoubtedly true, whatever conclusions may be drawn from them. I
do not propose that you should print my narrative, but I think in
these cases personal testimony, even indirect, is more useful in
forming one's opinion than a hundred old volumes. I did not hear it
from Mrs. C----'s own lips, but I was sufficiently acquainted with her
to form a very tolerable estimate of her character; and my wife, who
has known her intimately from her own childhood, was in her younger
days often staying with her for months together."
I remember having been much perplexed by reading a series of
experiments made with a pendulous ring over metals, by a Mr. Mayo: he
ascertained that it oscillated in various directions under peculiar
circumstances, when suspended by a thread over the ball of the thumb.
I instituted a series of experiments, and was surprised to find the
ring vibrate in an unaccountable manner in opposite directions over
different metals. On consideration, I closed my eyes whilst the ring
was oscillating over gold, and on opening them I found that it had
become stationary. I got a friend to change the metals whilst I was
blindfolded--the ring no longer vibrated. I was thus enabled to judge
of the involuntary action of muscles, quite sufficient to have
deceived an eminent medical man like Mr. Mayo, and to have perplexed
me till I succeeded in solving the mystery.[24]
FOOTNOTES:
[23] Hos. iv. 12.
[24] A similar series of experiments was undertaken, as I learned
afterwards, by M. Chevreuil in Paris, with similar results.
The Seven Sleepers of Ephesus.
One of the most picturesque myths of ancient days is that which forms
the subject of this article. It is thus told by Jacques de Voragine,
in his "Legenda Aurea:"--
"The seven sleepers were natives of Ephesus. The Emperor
Decius, who persecuted the Christians, having come to
Ephesus, ordered the erection of temples in the city, that
all might come and sacrifice before him; and he commanded
that the Christians should be sought out and given their
choice, either to worship the idols, or to die. So great was
the consternation in the city, that the friend denounced his
friend, the father his son, and the son his father.
"Now there were in Ephesus seven Christians, Maximian,
Malchus, Marcian, Dionysius, John, Serapion, and Constantine
by name. These refused to sacrifice to the idols, and
remained in their houses praying and fasting. They were
accused before Decius, and they confessed themselves to be
Christians. However, the emperor gave them a little time to
consider what line they would adopt. They took advantage of
this reprieve to dispense their goods among the poor, and
then they retired, all seven, to Mount Celion, where they
determined to conceal themselves.
"One of their number, Malchus, in the disguise of a
physician, went to the town to obtain victuals. Decius, who
had been absent from Ephesus for a little while, returned,
and gave orders for the seven to be sought. Malchus, having
escaped from the town, fled, full of fear, to his comrades,
and told them of the emperor's fury. They were much alarmed;
and Malchus handed them the loaves he had bought, bidding
them eat, that, fortified by the food, they might have
courage in the time of trial. They ate, and then, as they sat
weeping and speaking to one another, by the will of God they
fell asleep.
"The pagans sought everywhere, but could not find them, and
Decius was greatly irritated at their escape. He had their
parents brought before him, and threatened them with death
if they did not reveal the place of concealment; but they
could only answer that the seven young men had distributed
their goods to the poor, and that they were quite ignorant as
to their whereabouts.
"Decius, thinking it possible that they might be hiding in a
cavern, blocked up the mouth with stones, that they might
perish of hunger.
"Three hundred and sixty years passed, and in the thirtieth
year of the reign of Theodosius, there broke forth a heresy
denying the resurrection of the dead....
"Now, it happened that an Ephesian was building a stable on
the side of Mount Celion, and finding a pile of stones handy,
he took them for his edifice, and thus opened the mouth of
the cave. Then the seven sleepers awoke, and it was to them
as if they had slept but a single night. They began to ask
Malchus what decision Decius had given concerning them.
"'He is going to hunt us down, so as to force us to sacrifice
to the idols,' was his reply. 'God knows,' replied Maximian,
'we shall never do that.' Then exhorting his companions, he
urged Malchus to go back to the town to buy some more bread,
and at the same time to obtain fresh information. Malchus
took five coins and left the cavern. On seeing the stones he
was filled with astonishment; however, he went on towards the
city; but what was his bewilderment, on approaching the gate,
to see over it a cross! He went to another gate, and there he
beheld the same sacred sign; and so he observed it over each
gate of the city. He believed that he was suffering from the
effects of a dream. Then he entered Ephesus, rubbing his
eyes, and he walked to a baker's shop. He heard people using
our Lord's name, and he was the more perplexed. 'Yesterday,
no one dared pronounce the name of Jesus, and now it is on
every one's lips. Wonderful! I can hardly believe myself to
be in Ephesus.' He asked a passer-by the name of the city,
and on being told it was Ephesus, he was thunderstruck. Now
he entered a baker's shop, and laid down his money. The
baker, examining the coin, inquired whether he had found a
treasure, and began to whisper to some others in the shop.
The youth, thinking that he was discovered, and that they
were about to conduct him to the emperor, implored them to
let him alone, offering to leave loaves and money if he might
only be suffered to escape. But the shop-men, seizing him,
said, 'Whoever you are, you have found a treasure; show us
where it is, that we may share it with you, and then we will
hide you.' Malchus was too frightened to answer. So they put
a rope round his neck, and drew him through the streets into
the market-place. The news soon spread that the young man had
discovered a great treasure, and there was presently a vast
crowd about him. He stoutly protested his innocence. No one
recognized him, and his eyes, ranging over the faces which
surrounded him, could not see one which he had known, or
which was in the slightest degree familiar to him.
"St. Martin, the bishop, and Antipater, the governor, having
heard of the excitement, ordered the young man to be brought
before them, along with the bakers.
"The bishop and the governor asked him where he had found the
treasure, and he replied that he had found none, but that the
few coins were from his own purse. He was next asked whence
he came. He replied that he was a native of Ephesus, 'if this
be Ephesus.'
"'Send for your relations--your parents, if they live here,'
ordered the governor.
"'They live here, certainly,' replied the youth; and he
mentioned their names. No such names were known in the town.
Then the governor exclaimed, 'How dare you say that this
money belonged to your parents when it dates back three
hundred and seventy-seven years,[25] and is as old as the
beginning of the reign of Decius, and it is utterly unlike
our modern coinage? Do you think to impose on the old men and
sages of Ephesus? Believe me, I shall make you suffer the
severities of the law till you show where you made the
discovery.'
"'I implore you,' cried Malchus, 'in the name of God, answer
me a few questions, and then I will answer yours. Where is
the Emperor Decius gone to?'
"The bishop answered, 'My son, there is no emperor of that
name; he who was thus called died long ago.'
"Malchus replied, 'All I hear perplexes me more and more.
Follow me, and I will show you my comrades, who fled with me
into a cave of Mount Celion, only yesterday, to escape the
cruelty of Decius. I will lead you to them.'
"The bishop turned to the governor. 'The hand of God is
here,' he said. Then they followed, and a great crowd after
them. And Malchus entered first into the cavern to his
companions, and the bishop after him.... And there they saw
the martyrs seated in the cave, with their faces fresh and
blooming as roses; so all fell down and glorified God. The
bishop and the governor sent notice to Theodosius, and he
hurried to Ephesus. All the inhabitants met him and conducted
him to the cavern. As soon as the saints beheld the emperor,
their faces shone like the sun, and the emperor gave thanks
unto God, and embraced them, and said, 'I see you, as though
I saw the Savior restoring Lazarus.' Maximian replied,
'Believe us! for the faith's sake, God has resuscitated us
before the great resurrection day, in order that you may
believe firmly in the resurrection of the dead. For as the
child is in its mother's womb living and not suffering, so
have we lived without suffering, fast asleep.' And having
thus spoken, they bowed their heads, and their souls
returned to their Maker. The emperor, rising, bent over them
and embraced them weeping. He gave them orders for golden
reliquaries to be made, but that night they appeared to him
in a dream, and said that hitherto they had slept in the
earth, and that in the earth they desired to sleep on till
God should raise them again."
Such is the beautiful story. It seems to have travelled to us from the
East. Jacobus Sarugiensis, a Mesopotamian bishop, in the fifth or
sixth century, is said to have been the first to commit it to writing.
Gregory of Tours (De Glor. Mart. i. 9) was perhaps the first to
introduce it to Europe. Dionysius of Antioch (ninth century) told the
story in Syrian, and Photius of Constantinople reproduced it, with the
remark that Mahomet had adopted it into the Koran. Metaphrastus
alludes to it as well; in the tenth century Eutychius inserted it in
his annals of Arabia; it is found in the Coptic and the Maronite
books, and several early historians, as Paulus Diaconus, Nicephorus,
&c., have inserted it in their works.
A poem on the Seven Sleepers was composed by a trouvA"re named
Chardri, and is mentioned by M. Fr. Michel in his "Rapports Ministre
de l'Instruction Public;" a German poem on the same subject, of the
thirteenth century, in 935 verses, has been published by M. Karajan;
and the Spanish poet, Augustin Morreto, composed a drama on it,
entitled "Los Siete Durmientes," which is inserted in the 19th volume
of the rare work, "Comedias Nuevas Escogidas de los Mejores Ingenios."
Mahomet has somewhat improved on the story. He has made the Sleepers
prophesy his coming, and he has given them a dog named Kratim, or
Kratimir, which sleeps with them, and which is endowed with the gift
of prophecy.
As a special favor this dog is to be one of the ten animals to be
admitted into his paradise, the others being Jonah's whale, Solomon's
ant, Ishmael's ram, Abraham's calf, the Queen of Sheba's ass, the
prophet Salech's camel, Moses' ox, Belkis' cuckoo, and Mahomet's ass.
It was perhaps too much for the Seven Sleepers to ask, that their
bodies should be left to rest in earth. In ages when saintly relics
were valued above gold and precious stones, their request was sure to
be shelved; and so we find that their remains were conveyed to
Marseilles in a large stone sarcophagus, which is still exhibited in
St. Victor's Church. In the MusA|um Victorium at Rome is a curious and
ancient representation of them in a cement of sulphur and plaster.
Their names are engraved beside them, together with certain
attributes. Near Constantine and John are two clubs, near Maximian a
knotty club, near Malchus and Martinian two axes, near Serapion a
burning torch, and near Danesius or Dionysius a great nail, such as
those spoken of by Horace (Lib. 1, Od. 3) and St. Paulinus (Nat. 9, or
Carm. 24) as having been used for torture.
In this group of figures, the seven are represented as young, without
beards, and indeed in ancient martyrologies they are frequently called
boys.
It has been inferred from this curious plaster representation, that
the seven may have suffered under Decius, A. D. 250, and have been
buried in the afore-mentioned cave; whilst the discovery and
translation of their relics under Theodosius, in 479, may have given
rise to the fable. And this I think probable enough. The story of
long sleepers and the number seven connected with it is ancient
enough, and dates from heathen mythology.
Like many another ancient myth, it was laid hold of by Christian hands
and baptized.
Pliny relates the story of Epimenides the epic poet, who, when tending
his sheep one hot day, wearied and oppressed with slumber, retreated
into a cave, where he fell asleep. After fifty-seven years he awoke,
and found every thing changed. His brother, whom he had left a
stripling, was now a hoary man.
Epimenides was reckoned one of the seven sages by those who exclude
Periander. He flourished in the time of Solon. After his death, at the
age of two hundred and eighty-nine, he was revered as a god, and
honored especially by the Athenians.
This story is a version of the older legend of the perpetual sleep of
the shepherd Endymion, who was thus preserved in unfading youth and
beauty by Jupiter.
According to an Arabic legend, St. George thrice rose from his grave,
and was thrice slain.
In Scandinavian mythology we have Siegfrid or Sigurd thus resting,
and awaiting his call to come forth and fight. Charlemagne sleeps in
the Odenberg in Hess, or in the Untersberg near Salzburg, seated on
his throne, with his crown on his head and his sword at his side,
waiting till the times of Antichrist are fulfilled, when he will wake
and burst forth to avenge the blood of the saints. Ogier the Dane, or
Olger Dansk, will in like manner shake off his slumber and come forth
from the dream-land of Avallon to avenge the right--O that he had
shown himself in the Schleswig-Holstein war!
Well do I remember, as a child, contemplating with wondering awe the
great KyffhA¤userberg in Thuringia, for therein, I was told, slept
Frederic Barbarossa and his six knights. A shepherd once penetrated
into the heart of the mountain by a cave, and discovered therein a
hall where sat the emperor at a stone table, and his red beard had
grown through the slab. At the tread of the shepherd Frederic awoke
from his slumber, and asked, "Do the ravens still fly over the
mountains?"
"Sire, they do."
"Then we must sleep another hundred years."
But when his beard has wound itself thrice round the table, then will
the emperor awake with his knights, and rush forth to release Germany
from its bondage, and exalt it to the first place among the kingdoms
of Europe.
In Switzerland slumber three Tells at Rutli, near the
VierwaldstA¤tter-see, waiting for the hour of their country's direst
need. A shepherd crept into the cave where they rest. The third Tell
rose and asked the time. "Noon," replied the shepherd lad. "The time
is not yet come," said Tell, and lay down again.
In Scotland, beneath the Eilden hills, sleeps Thomas of Erceldoune;
the murdered French who fell in the Sicilian Vespers at Palermo are
also slumbering till the time is come when they may wake to avenge
themselves. When Constantinople fell into the hands of the Turks, a
priest was celebrating the sacred mysteries at the great silver altar
of St. Sophia. The celebrant cried to God to protect the sacred host
from profanation. Then the wall opened, and he entered, bearing the
Blessed Sacrament. It closed on him, and there he is sleeping with
his head bowed before the Body of Our Lord, waiting till the Turk is
cast out of Constantinople, and St. Sophia is released from its
profanation. God speed the time!
In Bohemia sleep three miners deep in the heart of the Kuttenberg. In
North America Rip Van Winkle passed twenty years slumbering in the
Katskill mountains. In Portugal it is believed that Sebastian, the
chivalrous young monarch who did his best to ruin his country by his
rash invasion of Morocco, is sleeping somewhere; but he will wake
again to be his country's deliverer in the hour of need. Olaf
Tryggvason is waiting a similar occasion in Norway. Even Napoleon
Bonaparte is believed among some of the French peasantry to be
sleeping on in a like manner.
St. Hippolytus relates that St. John the Divine is slumbering at
Ephesus, and Sir John Mandeville relates the circumstances as follows:
"From Pathmos men gone unto Ephesim a fair citee and nyghe to the see.
And there dyede Seynte Johne, and was buryed behynde the highe
Awtiere, in a toumbe. And there is a faire chirche. For Christene mene
weren wont to holden that place alweyes. And in the tombe of Seynt
John is noughte but manna, that is clept Aungeles mete. For his body
was translated into Paradys. And Turkes holden now alle that place and
the citee and the Chirche. And all Asie the lesse is yclept Turkye.
And ye shalle undrestond, that Seynt Johne bid make his grave there in
his Lyf, and leyd himself there-inne all quyk. And therefore somme men
seyn, that he dyed noughte, but that he resteth there till the Day of
Doom. And forsoothe there is a gret marveule: For men may see there
the erthe of the tombe apertly many tymes steren and moven, as there
weren quykke thinges undre." The connection of this legend of St. John
with Ephesus may have had something to do with turning the seven
martyrs of that city into seven sleepers.
The annals of Iceland relate that, in 1403, a Finn of the name of
Fethmingr, living in Halogaland, in the North of Norway, happening to
enter a cave, fell asleep, and woke not for three whole years, lying
with his bow and arrows at his side, untouched by bird or beast.
There certainly are authentic accounts of persons having slept for an
extraordinary length of time, but I shall not mention any, as I
believe the legend we are considering, not to have been an
exaggeration of facts, but a Christianized myth of paganism. The fact
of the number seven being so prominent in many of the tales, seems to
lead to this conclusion. Barbarossa changes his position every seven
years. Charlemagne starts in his chair at similar intervals. Olger
Dansk stamps his iron mace on the floor once every seven years. Olaf
Redbeard in Sweden uncloses his eyes at precisely the same distances
of time.
I believe that the mythological core of this picturesque legend is the
repose of the earth through the seven winter months. In the North,
Frederic and Charlemagne certainly replace Odin.
The German and Scandinavian still heathen legends represent the heroes
as about to issue forth for the defence of Fatherland in the hour of
direst need. The converted and Christianized tale brings the martyr
youths forth in the hour when a heresy is afflicting the Church, that
they may destroy the heresy by their witness to the truth of the
Resurrection.
If there is something majestic in the heathen myth, there are
singular grace and beauty in the Christian tale, teaching, as it does,
such a glorious doctrine; but it is surpassed in delicacy by the
modern form which the same myth has assumed--a form which is a real
transformation, leaving the doctrine taught the same. It has been made
into a romance by Hoffman, and is versified by Trinius. I may perhaps
be allowed to translate with some freedom the poem of the latter:--
In an ancient shaft of Falun
Year by year a body lay,
God-preserved, as though a treasure,
Kept unto the waking day.
Not the turmoil, nor the passions,
Of the busy world o'erhead,
Sounds of war, or peace rejoicings,
Could disturb the placid dead.
Once a youthful miner, whistling,
Hewed the chamber, now his tomb:
Crash! the rocky fragments tumbled,
Closed him in abysmal gloom.
Sixty years passed by, ere miners
Toiling, hundred fathoms deep,
Broke upon the shaft where rested
That poor miner in his sleep.
As the gold-grains lie untarnished
In the dingy soil and sand,
Till they gleam and flicker, stainless,
In the digger's sifting hand;--
As the gem in virgin brilliance
Rests, till ushered into day;--
So uninjured, uncorrupted,
Fresh and fair the body lay.
And the miners bore it upward,
Laid it in the yellow sun;
Up, from out the neighboring houses,
Fast the curious peasants run.
"Who is he?" with eyes they question;
"Who is he?" they ask aloud;
Hush! a wizened hag comes hobbling,
Panting, through the wondering crowd.
O! the cry,--half joy, half sorrow,--
As she flings her at his side:
"John! the sweetheart of my girlhood,
Here am I, am I, thy bride.
"Time on thee has left no traces,
Death from wear has shielded thee;
I am agA(C)d, worn, and wasted,
O! what life has done to me!"
Then his smooth, unfurrowed forehead
Kissed that ancient withered crone;
And the Death which had divided
Now united them in one.
FOOTNOTE:
[25] This calculation is sadly inaccurate.
William Tell.
I suppose that most people regard William Tell, the hero of
Switzerland, as an historical character, and visit the scenes made
memorable by his exploits, with corresponding interest, when they
undertake the regular Swiss round.
It is one of the painful duties of the antiquarian to dispel many a
popular belief, and to probe the groundlessness of many an historical
statement. The antiquarian is sometimes disposed to ask with Pilate,
"What is truth?" when he finds historical facts crumbling beneath his
touch into mythological fables; and he soon learns to doubt and
question the most emphatic declarations of, and claims to,
reliability.
Sir Walter Raleigh, in his prison, was composing the second volume of
his History of the World. Leaning on the sill of his window, he
meditated on the duties of the historian to mankind, when suddenly
his attention was attracted by a disturbance in the court-yard before
his cell. He saw one man strike another whom he supposed by his dress
to be an officer; the latter at once drew his sword, and ran the
former through the body. The wounded man felled his adversary with a
stick, and then sank upon the pavement. At this juncture the guard
came up, and carried off the officer insensible, and then the corpse
of the man who had been run through.
Next day Raleigh was visited by an intimate friend, to whom he related
the circumstances of the quarrel and its issue. To his astonishment,
his friend unhesitatingly declared that the prisoner had mistaken the
whole series of incidents which had passed before his eyes.
The supposed officer was not an officer at all, but the servant of a
foreign ambassador; it was he who had dealt the first blow; he had not
drawn his sword, but the other had snatched it from his side, and had
run _him_ through the body before any one could interfere; whereupon a
stranger from among the crowd knocked the murderer down with his
stick, and some of the foreigners belonging to the ambassador's
retinue carried off the corpse. The friend of Raleigh added that
government had ordered the arrest and immediate trial of the murderer,
as the man assassinated was one of the principal servants of the
Spanish ambassador.
"Excuse me," said Raleigh, "but I cannot have been deceived as you
suppose, for I was eye-witness to the events which took place under my
own window, and the man fell there on that spot where you see a
paving-stone standing up above the rest."
"My dear Raleigh," replied his friend, "I was sitting on that stone
when the fray took place, and I received this slight scratch on my
cheek in snatching the sword from the murderer; and upon my word of
honor, you have been deceived upon every particular."
Sir Walter, when alone, took up the second volume of his History,
which was in MS., and contemplating it, thought--"If I cannot believe
my own eyes, how can I be assured of the truth of a tithe of the
events which happened ages before I was born?" and he flung the
manuscript into the fire.[26]
Now, I think that I can show that the story of William Tell is as
fabulous as--what shall I say? any other historical event.
It is almost too well known to need repetition.
In the year 1307, Gessler, Vogt of the Emperor Albert of Hapsburg, set
a hat on a pole, as symbol of imperial power, and ordered every one
who passed by to do obeisance towards it. A mountaineer of the name of
Tell boldly traversed the space before it without saluting the
abhorred symbol. By Gessler's command he was at once seized and
brought before him. As Tell was known to be an expert archer, he was
ordered, by way of punishment, to shoot an apple off the head of his
own son. Finding remonstrance vain, he submitted. The apple was placed
on the child's head, Tell bent his bow, the arrow sped, and apple and
arrow fell together to the ground. But the Vogt noticed that Tell,
before shooting, had stuck another arrow into his belt, and he
inquired the reason.
"It was for you," replied the sturdy archer. "Had I shot my child,
know that it would not have missed your heart."
This event, observe, took place in the beginning of the fourteenth
century. But Saxo Grammaticus, a Danish writer of the twelfth century,
tells the story of a hero of his own country, who lived in the tenth
century. He relates the incident in horrible style as follows:--
"Nor ought what follows to be enveloped in silence. Toki, who had for
some time been in the king's service, had, by his deeds, surpassing
those of his comrades, made enemies of his virtues. One day, when he
had drunk too much, he boasted to those who sat at table with him,
that his skill in archery was such, that with the first shot of an
arrow he could hit the smallest apple set on the top of a stick at a
considerable distance. His detractors, hearing this, lost no time in
conveying what he had said to the king (Harald Bluetooth). But the
wickedness of this monarch soon transformed the confidence of the
father to the jeopardy of the son, for he ordered the dearest pledge
of his life to stand in place of the stick, from whom, if the utterer
of the boast did not at his first shot strike down the apple, he
should with his head pay the penalty of having made an idle boast. The
command of the king urged the soldier to do this, which was so much
more than he had undertaken, the detracting artifices of the others
having taken advantage of words spoken when he was hardly sober. As
soon as the boy was led forth, Toki carefully admonished him to
receive the whir of the arrow as calmly as possible, with attentive
ears, and without moving his head, lest by a slight motion of the body
he should frustrate the experience of his well-tried skill. He also
made him stand with his back towards him, lest he should be frightened
at the sight of the arrow. Then he drew three arrows from his quiver,
and the very first he shot struck the proposed mark. Toki being asked
by the king why he had taken so many more arrows out of his quiver,
when he was to make but one trial with his bow, 'That I might avenge
on thee,' he replied, 'the error of the first, by the points of the
others, lest my innocence might happen to be afflicted, and thy
injustice go unpunished.'"
The same incident is told of Egil, brother of the mythical Velundr,
in the Saga of Thidrik.
In Norwegian history also it appears with variations again and again.
It is told of King Olaf the Saint (d. 1030), that, desiring the
conversion of a brave heathen named Eindridi, he competed with him in
various athletic sports; he swam with him, wrestled, and then shot
with him. The king dared Eindridi to strike a writing-tablet from off
his son's head with an arrow. Eindridi prepared to attempt the
difficult shot. The king bade two men bind the eyes of the child and
hold the napkin, so that he might not move when he heard the whistle
of the arrow. The king aimed first, and the arrow grazed the lad's
head. Eindridi then prepared to shoot; but the mother of the boy
interfered, and persuaded the king to abandon this dangerous test of
skill. In this version, also, Eindridi is prepared to revenge himself
on the king, should the child be injured.
But a closer approximation still to the Tell myth is found in the life
of Hemingr, another Norse archer, who was challenged by King Harald,
Sigurd's son (d. 1066). The story is thus told:--
"The island was densely overgrown with wood, and the people went into
the forest. The king took a spear and set it with its point in the
soil, then he laid an arrow on the string and shot up into the air.
The arrow turned in the air and came down upon the spear-shaft and
stood up in it. Hemingr took another arrow and shot up; his was lost
to sight for some while, but it came back and pierced the nick of the
king's arrow.... Then the king took a knife and stuck it into an oak;
he next drew his bow and planted an arrow in the haft of the knife.
Thereupon Hemingr took his arrows. The king stood by him and said,
'They are all inlaid with gold; you are a capital workman.' Hemingr
answered, 'They are not my manufacture, but are presents.' He shot,
and his arrow cleft the haft, and the point entered the socket of the
blade.
"'We must have a keener contest,' said the king, taking an arrow and
flushing with anger; then he laid the arrow on the string and drew his
bow to the farthest, so that the horns were nearly brought to meet.
Away flashed the arrow, and pierced a tender twig. All said that this
was a most astonishing feat of dexterity. But Hemingr shot from a
greater distance, and split a hazel nut. All were astonished to see
this. Then said the king, 'Take a nut and set it on the head of your
brother Bjorn, and aim at it from precisely the same distance. If you
miss the mark, then your life goes.'
"Hemingr answered, 'Sire, my life is at your disposal, but I will not
adventure that shot.' Then out spake Bjorn--'Shoot, brother, rather
than die yourself.' Hemingr said, 'Have you the pluck to stand quite
still without shrinking?' 'I will do my best,' said Bjorn. 'Then let
the king stand by,' said Hemingr, 'and let him see whether I touch the
nut.'
"The king agreed, and bade Oddr Ufeigs' son stand by Bjorn, and see
that the shot was fair. Hemingr then went to the spot fixed for him by
the king, and signed himself with the cross, saying, 'God be my
witness that I had rather die myself than injure my brother Bjorn; let
all the blame rest on King Harald.'
"Then Hemingr flung his spear. The spear went straight to the mark,
and passed between the nut and the crown of the lad, who was not in
the least injured. It flew farther, and stopped not till it fell.
"Then the king came up and asked Oddr what he thought about the
shot."
Years after, this risk was revenged upon the hard-hearted monarch. In
the battle of Stamfordbridge an arrow from a skilled archer penetrated
the windpipe of the king, and it is supposed to have sped, observes
the Saga writer, from the bow of Hemingr, then in the service of the
English monarch.
The story is related somewhat differently in the Faroe Isles, and is
told of Geyti, Aslak's son. The same Harald asks his men if they know
who is his match in strength. "Yes," they reply; "there is a peasant's
son in the uplands, Geyti, son of Aslak, who is the strongest of men."
Forth goes the king, and at last rides up to the house of Aslak. "And
where is your youngest son?"
"Alas! alas! he lies under the green sod of Kolrin kirkgarth." "Come,
then, and show me his corpse, old man, that I may judge whether he was
as stout of limb as men say."
The father puts the king off with the excuse that among so many dead
it would be hard to find his boy. So the king rides away over the
heath. He meets a stately man returning from the chase, with a bow
over his shoulder. "And who art thou, friend?" "Geyti, Aslak's son."
The dead man, in short, alive and well. The king tells him he has
heard of his prowess, and is come to match his strength with him. So
Geyti and the king try a swimming-match.
The king swims well; but Geyti swims better, and in the end gives the
monarch such a ducking, that he is borne to his house devoid of sense
and motion. Harald swallows his anger, as he had swallowed the water,
and bids Geyti shoot a hazel nut from off his brother's head. Aslak's
son consents, and invites the king into the forest to witness his
dexterity.
"On the string the shaft he laid,
And God hath heard his prayer;
He shot the little nut away,
Nor hurt the lad a hair."
Next day the king sends for the skilful bowman:--
"List thee, Geyti, Aslak's son,
And truly tell to me,
Wherefore hadst thou arrows twain
In the wood yestreen with thee?"
The bowman replies,--
"Therefore had I arrows twain
Yestreen in the wood with me,
Had I but hurt my brother dear,
The other had piercA(C)d thee."
A very similar tale is told also in the celebrated Malleus Maleficarum
of a man named Puncher, with this difference, that a coin is placed on
the lad's head instead of an apple or a nut. The person who had dared
Puncher to the test of skill, inquires the use of the second arrow in
his belt, and receives the usual answer, that if the first arrow had
missed the coin, the second would have transfixed a certain heart
which was destitute of natural feeling.
We have, moreover, our English version of the same story in the
venerable ballad of William of Cloudsley.
The Finn ethnologist CastrA(C)n obtained the following tale in the
Finnish village of Uhtuwa:--
A fight took place between some freebooters and the inhabitants of the
village of AlajA¤wi. The robbers plundered every house, and carried off
amongst their captives an old man. As they proceeded with their spoils
along the strand of the lake, a lad of twelve years old appeared from
among the reeds on the opposite bank, armed with a bow, and amply
provided with arrows; he threatened to shoot down the captors unless
the old man, his father, were restored to him. The robbers mockingly
replied that the aged man would be given to him if he could shoot an
apple off his head. The boy accepted the challenge, and on
successfully accomplishing it, the surrender of the venerable captive
was made.
Farid-Uddin A,ttar was a Persian dealer in perfumes, born in the year
1119. He one day was so impressed with the sight of a dervish, that he
sold his possessions, and followed righteousness. He composed the poem
Mantic UttaA-r, or the language of birds. Observe, the Persian A,ttar
lived at the same time as the Danish Saxo, and long before the birth
of Tell. Curiously enough, we find a trace of the Tell myth in the
pages of his poem. According to him, however, the king shoots the
apple from the head of a beloved page, and the lad dies from sheer
fright, though the arrow does not even graze his skin.
The coincidence of finding so many versions of the same story
scattered through countries as remote as Persia and Iceland,
Switzerland and Denmark, proves, I think, that it can in no way be
regarded as history, but is rather one of the numerous household myths
common to the whole stock of Aryan nations. Probably, some one more
acquainted with Sanskrit literature than myself, and with better
access to its unpublished stores of fable and legend, will some day
light on an early Indian tale corresponding to that so prevalent among
other branches of the same family. The coincidence of the Tell myth
being discovered among the Finns is attributable to Russian or Swedish
influence. I do not regard it as a primeval Turanian, but as an Aryan
story, which, like an erratic block, is found deposited on foreign
soil far from the mountain whence it was torn.
German mythologists, I suppose, consider the myth to represent the
manifestation of some natural phenomena, and the individuals of the
story to be impersonifications of natural forces. Most primeval
stories were thus constructed, and their origin is traceable enough.
In Thorn-rose, for instance, who can fail to see the earth goddess
represented by the sleeping beauty in her long winter slumber, only
returning to life when kissed by the golden-haired sun-god PhA"bus
or Baldur? But the Tell myth has not its signification thus painted
on the surface; and those who suppose Gessler or Harald to be the
power of evil and darkness,--the bold archer to be the storm-cloud
with his arrow of lightning and his iris bow, bent against the sun,
which is resting like a coin or a golden apple on the edge of the
horizon, are over-straining their theories, and exacting too much from
our credulity.
In these pages and elsewhere I have shown how some of the ancient
myths related by the whole Aryan family of nations are reducible to
allegorical explanations of certain well-known natural phenomena; but
I must protest against the manner in which our German friends fasten
rapaciously upon every atom of history, sacred and profane, and
demonstrate all heroes to represent the sun; all villains to be the
demons of night or winter; all sticks and spears and arrows to be the
lightning; all cows and sheep and dragons and swans to be clouds.
In a work on the superstition of Werewolves, I have entered into this
subject with some fulness, and am quite prepared to admit the premises
upon which mythologists construct their theories; at the same time I
am not disposed to run to the extravagant lengths reached by some of
the most enthusiastic German scholars. A wholesome warning to these
gentlemen was given some years ago by an ingenious French
ecclesiastic, who wrote the following argument to prove that Napoleon
Bonaparte was a mythological character. Archbishop Whately's "Historic
Doubts" was grounded on a totally different line of argument; I
subjoin the other, as a curiosity and as a caution.
Napoleon is, says the writer, an impersonification of the sun.
1. Between the name Napoleon and Apollo, or Apoleon, the god of the
sun, there is but a trifling difference; indeed, the seeming
difference is lessened, if we take the spelling of his name from the
column of the Place VendA'me, where it stands NA(C)apoleA cubed. But this
syllable _Ne_ prefixed to the name of the sun-god is of importance;
like the rest of the name it is of Greek origin, and is I1/2I. or I1/2I+-I¹,
a particle of affirmation, as though indicating Napoleon as the very
true Apollo, or sun.
His other name, Bonaparte, makes this apparent connection between the
French hero and the luminary of the firmament conclusively certain.
The day has its two parts, the good and luminous portion, and that
which is bad and dark. To the sun belongs the good part, to the moon
and stars belongs the bad portion. It is therefore natural that Apollo
or NA(C)-ApoleA cubedn should receive the surname of _Bonaparte_.
2. Apollo was born in Delos, a Mediterranean island; Napoleon in
Corsica, an island in the same sea. According to Pausanias, Apollo was
an Egyptian deity; and in the mythological history of the fabulous
Napoleon we find the hero in Egypt, regarded by the inhabitants with
veneration, and receiving their homage.
3. The mother of Napoleon was said to be Letitia, which signifies joy,
and is an impersonification of the dawn of light dispensing joy and
gladness to all creation. Letitia is no other than the break of day,
which in a manner brings the sun into the world, and "with rosy
fingers opes the gates of Day." It is significant that the Greek name
for the mother of Apollo was Leto. From this the Romans made the name
Latona, which they gave to his mother. But _LA|to_ is the unused form
of the verb _lA|tor_, and signified to inspire joy; it is from this
unused form that the substantive _Letitia_ is derived. The identity,
then, of the mother of Napoleon with the Greek Leto and the Latin
Latona, is established conclusively.
4. According to the popular story, this son of Letitia had three
sisters; and was it not the same with the Greek deity, who had the
three Graces?
5. The modern Gallic Apollo had four brothers. It is impossible not to
discern here the anthropomorphosis of the four seasons. But, it will
be objected, the seasons should be females. Here the French language
interposes; for in French the seasons are masculine, with the
exception of autumn, upon the gender of which grammarians are
undecided, whilst Autumnus in Latin is not more feminine than the
other seasons. This difficulty is therefore trifling, and what follows
removes all shadow of doubt.
Of the four brothers of Napoleon, three are said to have been kings,
and these of course are, Spring reigning over the flowers, Summer
reigning over the harvest, Autumn holding sway over the fruits. And as
these three seasons owe all to the powerful influence of the Sun, we
are told in the popular myth that the three brothers of Napoleon drew
their authority from him, and received from him their kingdoms. But if
it be added that, of the four brothers of Napoleon, one was not a
king, that was because he is the impersonification of Winter, which
has no reign over anything. If, however, it be asserted, in
contradiction, that the winter has an empire, he will be given the
principality over snows and frosts, which, in the dreary season of the
year, whiten the face of the earth. Well, the fourth brother of
Napoleon is thus invested by popular tradition, commonly called
history, with a vain principality accorded to him _in the decline of
the power of Napoleon_. The principality was that of Canino, a name
derived from _cani_, or the whitened hairs of a frozen old age,--true
emblem of winter. To the eyes of poets, the forests covering the hills
are their hair, and when winter frosts them, they represent the snowy
locks of a decrepit nature in the old age of the year:--
"Cum gelidus crescit _canis_ in montibus humor."
Consequently the Prince of Canino is an impersonification of
winter;--winter whose reign begins when the kingdoms of the three fine
seasons are passed from them, and when the sun is driven from his
power by the children of the North, as the poets call the boreal
winds. This is the origin of the fabulous invasion of France by the
allied armies of the North. The story relates that these invaders--the
northern gales--banished the many- flag, and replaced it by a
white standard. This too is a graceful, but, at the same time, purely
fabulous account of the Northern winds driving all the brilliant
colors from the face of the soil, to replace them by the snowy sheet.
6. Napoleon is said to have had two wives. It is well known that the
classic fable gave two also to Apollo. These two were the moon and the
earth. Plutarch asserts that the Greeks gave the moon to Apollo for
wife, whilst the Egyptians attributed to him the earth. By the moon he
had no posterity, but by the other he had one son only, the little
Horus. This is an Egyptian allegory, representing the fruits of
agriculture produced by the earth fertilized by the Sun. The pretended
son of the fabulous Napoleon is said to have been born on the 20th of
March, the season of the spring equinox, when agriculture is assuming
its greatest period of activity.
7. Napoleon is said to have released France from the devastating
scourge which terrorized over the country, the hydra of the
revolution, as it was popularly called. Who cannot see in this a
Gallic version of the Greek legend of Apollo releasing Hellas from the
terrible Python? The very name _revolution_, derived from the Latin
verb _revolvo_, is indicative of the coils of a serpent like the
Python.
8. The famous hero of the 19th century had, it is asserted, twelve
Marshals at the head of his armies, and four who were stationary and
inactive. The twelve first, as may be seen at once, are the signs of
the zodiac, marching under the orders of the sun Napoleon, and each
commanding a division of the innumerable host of stars, which are
parted into twelve portions, corresponding to the twelve signs. As for
the four stationary officers, immovable in the midst of general
motion, they are the cardinal points.
9. It is currently reported that the chief of these brilliant armies,
after having gloriously traversed the Southern kingdoms, penetrated
North, and was there unable to maintain his sway. This too represents
the course of the Sun, which assumes its greatest power in the South,
but after the spring equinox seeks to reach the North; and after a
_three months'_ march towards the boreal regions, is driven back upon
his traces following the sign of Cancer, a sign given to represent
the retrogression of the sun in that portion of the sphere. It is on
this that the story of the march of Napoleon towards Moscow, and his
humbling retreat, is founded.
10. Finally, the sun rises in the East and sets in the Western sea.
The poets picture him rising out of the waters in the East, and
setting in the ocean after his twelve hours' reign in the sky. Such is
the history of Napoleon, coming from his Mediterranean isle, holding
the reins of government for twelve years, and finally disappearing in
the mysterious regions of the great Atlantic.
To those who see in Samson, the image of the sun, the correlative of
the classic Hercules, this clever skit of the accomplished French AbbA(C)
may prove of value as a caution.
FOOTNOTE:
[26] This anecdote is taken from the _Journal de Paris_, May, 1787;
but whence did the _Journal_ obtain it?
The Dog Gellert.
Having demolished William Tell, I proceed to the destruction of
another article of popular belief.
Who that has visited Snowdon has not seen the grave of Llewellyn's
faithful hound Gellert, and been told by the guide the touching story
of the death of the noble animal? How can we doubt the facts, seeing
that the place, Beth-Gellert, is named after the dog, and that the
grave is still visible? But unfortunately for the truth of the legend,
its pedigree can be traced with the utmost precision.
The story is as follows:--
The Welsh Prince Llewellyn had a noble deerhound, Gellert, whom he
trusted to watch the cradle of his baby son whilst he himself was
absent.
One day, on his return, to his intense horror, he beheld the cradle
empty and upset, the clothes dabbled with blood, and Gellert's mouth
dripping with gore. Concluding hastily that the hound had proved
unfaithful, had fallen on the child and devoured it,--in a paroxysm of
rage the prince drew his sword and slew the dog. Next instant the cry
of the babe from behind the cradle showed him that the child was
uninjured; and, on looking farther, Llewellyn discovered the body of a
huge wolf, which had entered the house to seize and devour the child,
but which had been kept off and killed by the brave dog Gellert.
In his self-reproach and grief, the prince erected a stately monument
to Gellert, and called the place where he was buried after the poor
hound's name.
Now, I find in Russia precisely the same story told, with just the
same appearance of truth, of a Czar Piras. In Germany it appears with
considerable variations. A man determines on slaying his old dog
Sultan, and consults with his wife how this is to be effected. Sultan
overhears the conversation, and complains bitterly to the wolf, who
suggests an ingenious plan by which the master may be induced to spare
his dog. Next day, when the man is going to his work, the wolf
undertakes to carry off the child from its cradle. Sultan is to attack
him and rescue the infant. The plan succeeds admirably, and the dog
spends his remaining years in comfort. (Grimm, K. M. 48.)
But there is a story in closer conformity to that of Gellert among the
French collections of fabliaux made by Le Grand d'Aussy and EdA(C)lA(C)stand
du MA(C)ril. It became popular through the "Gesta Romanorum," a
collection of tales made by the monks for harmless reading, in the
fourteenth century.
In the "Gesta" the tale is told as follows:--
"Folliculus, a knight, was fond of hunting and tournaments. He had an
only son, for whom three nurses were provided. Next to this child, he
loved his falcon and his greyhound. It happened one day that he was
called to a tournament, whither his wife and domestics went also,
leaving the child in the cradle, the greyhound lying by him, and the
falcon on his perch. A serpent that inhabited a hole near the castle,
taking advantage of the profound silence that reigned, crept from his
habitation, and advanced towards the cradle to devour the child. The
falcon, perceiving the danger, fluttered with his wings till he awoke
the dog, who instantly attacked the invader, and after a fierce
conflict, in which he was sorely wounded, killed him. He then lay down
on the ground to lick and heal his wounds. When the nurses returned,
they found the cradle overturned, the child thrown out, and the ground
covered with blood, as was also the dog, who they immediately
concluded had killed the child.
"Terrified at the idea of meeting the anger of the parents, they
determined to escape; but in their flight fell in with their mistress,
to whom they were compelled to relate the supposed murder of the child
by the greyhound. The knight soon arrived to hear the sad story, and,
maddened with fury, rushed forward to the spot. The poor wounded and
faithful animal made an effort to rise and welcome his master with his
accustomed fondness; but the enraged knight received him on the point
of his sword, and he fell lifeless to the ground. On examination of
the cradle, the infant was found alive and unhurt, with the dead
serpent lying by him. The knight now perceived what had happened,
lamented bitterly over his faithful dog, and blamed himself for having
too hastily depended on the words of his wife. Abandoning the
profession of arms, he broke his lance in pieces, and vowed a
pilgrimage to the Holy Land, where he spent the rest of his days in
peace."
The monkish hit at the wife is amusing, and might have been supposed
to have originated with those determined misogynists, as the gallant
Welshmen lay all the blame on the man. But the good compilers of the
"Gesta" wrote little of their own, except moral applications of the
tales they relate, and the story of Folliculus and his dog, like many
others in their collection, is drawn from a foreign source.
It occurs in the Seven Wise Masters, and in the "Calumnia Novercalis"
as well, so that it must have been popular throughout mediA|val Europe.
Now, the tales of the Seven Wise Masters are translations from a
Hebrew work, the Kalilah and Dimnah of Rabbi Joel, composed about
A. D. 1250, or from Simeon Seth's Greek Kylile and Dimne, written in
1080. These Greek and Hebrew works were derived from kindred sources.
That of Rabbi Joel was a translation from an Arabic version made by
Nasr-Allah in the twelfth century, whilst Simeon Seth's was a
translation of the Persian Kalilah and Dimnah. But the Persian
Kalilah and Dimnah was not either an original work; it was in turn a
translation from the Sanskrit Pantschatantra, made about A. D. 540.
In this ancient Indian book the story runs as follows:--
A Brahmin named Devasaman had a wife, who gave birth to a son, and
also to an ichneumon. She loved both her children dearly, giving them
alike the breast, and anointing them alike with salves. But she feared
the ichneumon might not love his brother.
One day, having laid her boy in bed, she took up the water jar, and
said to her husband, "Hear me, master! I am going to the tank to fetch
water. Whilst I am absent, watch the boy, lest he gets injured by the
ichneumon." After she had left the house, the Brahmin went forth
begging, leaving the house empty. In crept a black snake, and
attempted to bite the child; but the ichneumon rushed at it, and tore
it in pieces. Then, proud of its achievement, it sallied forth, all
bloody, to meet its mother. She, seeing the creature stained with
blood, concluded, with feminine precipitance, that it had fallen on
the baby and killed it, and she flung her water jar at it and slew it.
Only on her return home did she ascertain her mistake.
The same story is also told in the Hitopadesa (iv. 13), but the animal
is an otter, not an ichneumon. In the Arabic version a weasel takes
the place of the ichneumon.
The Buddhist missionaries carried the story into Mongolia, and in the
Mongolian Uligerun, which is a translation of the Tibetian Dsanghen,
the story reappears with the pole-cat as the brave and suffering
defender of the child.
Stanislaus Julien, the great Chinese scholar, has discovered the same
tale in the Chinese work entitled "The Forest of Pearls from the
Garden of the Law." This work dates from 668; and in it the creature
is an ichneumon.
In the Persian Sindibad-nAcmeh is the same tale, but the faithful
animal is a cat. In Sandabar and Syntipas it has become a dog. Through
the influence of Sandabar on the Hebrew translation of the Kalilah and
Dimnah, the ichneumon is also replaced by a dog.
Such is the history of the Gellert legend; it is an introduction into
Europe from India, every step of its transmission being clearly
demonstrable. From the Gesta Romanorum it passed into a popular tale
throughout Europe, and in different countries it was, like the Tell
myth, localized and individualized. Many a Welsh story, such as those
contained in the Mabinogion, are as easily traced to an Eastern
origin.
But every story has its root. The root of the Gellert tale is this: A
man forms an alliance of friendship with a beast or bird. The dumb
animal renders him a signal service. He misunderstands the act, and
kills his preserver.
We have tracked this myth under the Gellert form from India to Wales;
but under another form it is the property of the whole Aryan family,
and forms a portion of the traditional lore of all nations sprung from
that stock.
Thence arose the classic fable of the peasant, who, as he slept, was
bitten by a fly. He awoke, and in a rage killed the insect. When too
late, he observed that the little creature had aroused him that he
might avoid a snake which lay coiled up near his pillow.
In the Anvar-i-Suhaili is the following kindred tale. A king had a
falcon. One day, whilst hunting, he filled a goblet with water
dropping from a rock. As he put the vessel to his lips, his falcon
dashed upon it, and upset it with its wings. The king, in a fury, slew
the bird, and then discovered that the water dripped from the jaws of
a serpent of the most poisonous description.
This story, with some variations, occurs in A†sop, A†lian, and
Apthonius. In the Greek fable, a peasant liberates an eagle from the
clutches of a dragon. The dragon spirts poison into the water which
the peasant is about to drink, without observing what the monster had
done. The grateful eagle upsets the goblet with his wings.
The story appears in Egypt under a whimsical form. A Wali once smashed
a pot full of herbs which a cook had prepared. The exasperated cook
thrashed the well-intentioned but unfortunate Wali within an inch of
his life, and when he returned, exhausted with his efforts at
belaboring the man, to examine the broken pot, he discovered amongst
the herbs a poisonous snake.
How many brothers, sisters, uncles, aunts, and cousins of all degrees
a little story has! And how few of the tales we listen to can lay any
claim to originality! There is scarcely a story which I hear which I
cannot connect with some family of myths, and whose pedigree I cannot
ascertain with more or less precision. Shakespeare drew the plots of
his plays from Boccaccio or Straparola; but these Italians did not
invent the tales they lent to the English dramatist. King Lear does
not originate with Geofry of Monmouth, but comes from early Indian
stores of fable, whence also are derived the Merchant of Venice and
the pound of flesh, ay, and the very incident of the three caskets.
But who would credit it, were it not proved by conclusive facts, that
Johnny Sands is the inheritance of the whole Aryan family of nations,
and that Peeping Tom of Coventry peeped in India and on the Tartar
steppes ages before Lady Godiva was born?
If you listen to Traviata at the opera, you have set before you a tale
which has lasted for centuries, and which was perhaps born in India.
If you read in classic fable of Orpheus charming woods and meadows,
beasts and birds, with his magic lyre, you remember to have seen the
same fable related in the Kalewala of the Finnish Wainomainen, and in
the Kaleopoeg of the Esthonian Kalewa.
If you take up English history, and read of William the Conqueror
slipping as he landed on British soil, and kissing the earth, saying
he had come to greet and claim his own, you remember that the same
story is told of Napoleon in Egypt, of King Olaf Harold's son in
Norway, and in classic history of Junius Brutus on his return from the
oracle.
A little while ago I cut out of a Sussex newspaper a story purporting
to be the relation of a fact which had taken place at a fixed date in
Lewes. This was the story. A tyrannical husband locked the door
against his wife, who was out having tea with a neighbor, gossiping
and scandal-mongering; when she applied for admittance, he pretended
not to know her. She threatened to jump into the well unless he opened
the door.
The man, not supposing that she would carry her threat into execution,
declined, alleging that he was in bed, and the night was chilly;
besides which he entirely disclaimed all acquaintance with the lady
who claimed admittance.
The wife then flung a log into a well, and secreted herself behind the
door. The man, hearing the splash, fancied that his good lady was
really in the deeps, and forth he darted in his nocturnal costume,
which was of the lightest, to ascertain whether his deliverance was
complete. At once the lady darted into the house, locked the door,
and, on the husband pleading for admittance, she declared most
solemnly from the window that she did not know _him_.
Now, this story, I can positively assert, unless the events of this
world move in a circle, did not happen in Lewes, or any other Sussex
town.
It was told in the Gesta Romanorum six hundred years ago, and it was
told, may be, as many hundred years before in India, for it is still
to be found in Sanskrit collections of tales.
Tailed Men.
I well remember having it impressed upon me by a Devonshire nurse, as
a little child, that all Cornishmen were born with tails; and it was
long before I could overcome the prejudice thus early implanted in my
breast against my Cornubian neighbors. I looked upon those who dwelt
across the Tamar as "uncanny," as being scarcely to be classed with
Christian people, and certainly not to be freely associated with by
tailless Devonians. I think my eyes were first opened to the fact that
I had been deceived by a worthy bookseller of L----, with whom I had
contracted a warm friendship, he having at sundry times contributed
pictures to my scrapbook. I remember one day resolving to broach the
delicate subject with my tailed friend, whom I liked, notwithstanding
his caudal appendage.
"Mr. X----, is it true that you are a Cornishman?"
"Yes, my little man; born and bred in the West country."
"I like you very much; but--have you really got a tail?"
When the bookseller had recovered from the astonishment which I had
produced by my question, he stoutly repudiated the charge.
"But you are a Cornishman?"
"To be sure I am."
"And all Cornishmen have tails."
I believe I satisfied my own mind that the good man had sat his off,
and my nurse assured me that such was the case with those of sedentary
habits.
It is curious that Devonshire superstition should attribute the tail
to Cornishmen, for it was asserted of certain men of Kent in olden
times, and was referred to Divine vengeance upon them for having
insulted St. Thomas A Becket, if we may believe Polydore Vergil.
"There were some," he says, "to whom it seemed that the king's secret
wish was, that Thomas should be got rid of. He, indeed, as one
accounted to be an enemy of the king's person, was already regarded
with so little respect, nay, was treated with so much contempt, that
when he came to Strood, which village is situated on the Medway, the
river that washes Rochester, the inhabitants of the place, being eager
to show some mark of contumely to the prelate in his disgrace, did not
scruple to cut off the tail of the horse on which he was riding; but
by this profane and inhospitable act they covered themselves with
eternal reproach; for it so happened after this, by the will of God,
that all the offspring born from the men who had done this thing, were
born with tails, like brute animals. But this mark of infamy, which
formerly was everywhere notorious, has disappeared with the extinction
of the race whose fathers perpetrated this deed."
John Bale, the zealous reformer, and Bishop of Ossory in Edward VI.'s
time, refers to this story, and also mentions a variation of the scene
and cause of this ignoble punishment. He writes, quoting his
authorities, "John Capgrave and Alexander of Esseby sayth, that for
castynge of fyshe tayles at thys Augustyne, Dorsettshyre men had
tayles ever after. But Polydorus applieth it unto Kentish men at
Stroud, by Rochester, for cuttinge off Thomas Becket's horse's tail.
Thus hath England in all other land a perpetual infamy of tayles by
theye wrytten legendes of lyes, yet can they not well tell where to
bestowe them truely." Bale, a fierce and unsparing reformer, and one
who stinted not hard words, applying to the inventors of these legends
an epithet more strong than elegant, says, "In the legends of their
sanctified sorcerers they have diffamed the English posterity with
tails, as has been showed afore. That an Englyshman now cannot
travayle in another land by way of marchandyse or any other honest
occupyinge, but it is most contumeliously thrown in his tethe that all
Englyshmen have tails. That uncomely note and report have the nation
gotten, without recover, by these laisy and idle lubbers, the monkes
and the priestes, which could find no matters to advance their
canonized gains by, or their saintes, as they call them, but manifest
lies and knaveries."[27]
Andrew Marvel also makes mention of this strange judgment in his
_Loyal Scot_:--
"But who considers right will find, indeed,
'Tis Holy Island parts us, not the Tweed.
Nothing but clergy could us two seclude,
No Scotch was ever like a bishop's feud.
All Litanys in this have wanted faith,
There's no--_Deliver us from a Bishop's wrath._
Never shall Calvin pardoned be for sales,
Never, for Burnet's sake, the Lauderdales;
For Becket's sake, Kent always shall have tails."
It may be remembered that Lord Monboddo, a Scotch judge of last
century, and a philosopher of some repute, though of great
eccentricity, stoutly maintained the theory that man ought to have a
tail, that the tail is a _desideratum_, and that the abrupt
termination of the spine without caudal elongation is a sad blemish in
the origination of man. The tail, the point in which man is inferior
to the brute, what a delicate index of the mind it is! how it
expresses the passions of love and hate! how nicely it gives token of
the feelings of joy or fear which animate the soul! But Lord Monboddo
did not consider that what the tail is to the brute, that the eye is
to man; the lack of one member is supplied by the other. I can tell a
proud man by his eye just as truly as if he stalked past one with
erect tail; and anger is as plainly depicted in the human eye as in
the bottle-brush tail of a cat. I know a sneak by his cowering glance,
though he has not a tail between his legs; and pleasure is evident in
the laughing eye, without there being any necessity for a wagging
brush to express it.
Dr. Johnson paid a visit to the judge, and knocked on the head his
theory that men ought to have tails, and actually were born with them
occasionally; for said he, "Of a standing fact, sir, there ought to be
no controversy; if there are men with tails, catch a _homo caudatus_."
And, "It is a pity to see Lord Monboddo publish such notions as he has
done--a man of sense, and of so much elegant learning. There would be
little in a fool doing it; we should only laugh; but, when a wise man
does it, we are sorry. Other people have strange notions, but they
conceal them. If they have tails they hide them; but Monboddo is as
jealous of his tail as a squirrel." And yet Johnson seems to have been
tickled with the idea, and to have been amused with the notion of an
appendage like a tail being regarded as the complement of human
perfection. It may be remembered how Johnson made the acquaintance of
the young Laird of Col, during his Highland tour, and how pleased he
was with him. "Col," says he, "is a noble animal. He is as complete an
islander as the mind can figure. He is a farmer, a sailor, a hunter,
a fisher: he will run you down a dog; _if any man has a tail_, it is
Col." And notwithstanding all his aversion to puns, the great Doctor
was fain to yield to human weakness on one occasion, under the
influence of the mirth which Monboddo's name seems to have excited.
Johnson writes to Mrs. Thrale of a party he had met one night, which
he thus enumerates: "There were Smelt, and the Bishop of St. Asaph,
who comes to every place; and Sir Joshua, and Lord Monboddo, and
ladies _out of tale_."
There is a Polish story of a witch who made a girdle of human skin and
laid it across the threshold of a door where a marriage-feast was
being held. On the bridal pair stepping across the girdle they were
transformed into wolves. Three years after the witch sought them out,
and cast over them dresses of fur with the hair turned outward,
whereupon they recovered their human forms, but, unfortunately, the
dress cast over the bridegroom was too scanty, and did not extend over
his tail, so that, when he was restored to his former condition, he
retained his lupine caudal appendage, and this became hereditary in
his family; so that all Poles with tails are lineal descendants of
the ancestor to whom this little misfortune happened. John Struys, a
Dutch traveller, who visited the Isle of Formosa in 1677, gives a
curious story, which is worth transcribing.
"Before I visited this island," he writes, "I had often heard tell
that there were men who had long tails, like brute beasts; but I had
never been able to believe it, and I regarded it as a thing so alien
to our nature, that I should now have difficulty in accepting it, if
my own senses had not removed from me every pretence for doubting the
fact, by the following strange adventure: The inhabitants of Formosa,
being used to see us, were in the habit of receiving us on terms which
left nothing to apprehend on either side; so that, although mere
foreigners, we always believed ourselves in safety, and had grown
familiar enough to ramble at large without an escort, when grave
experience taught us that, in so doing, we were hazarding too much. As
some of our party were one day taking a stroll, one of them had
occasion to withdraw about a stone's throw from the rest, who, being
at the moment engaged in an eager conversation, proceeded without
heeding the disappearance of their companion. After a while, however,
his absence was observed, and the party paused, thinking he would
rejoin them. They waited some time; but at last, tired of the delay,
they returned in the direction of the spot where they remembered to
have seen him last. Arriving there, they were horrified to find his
mangled body lying on the ground, though the nature of the lacerations
showed that he had not had to suffer long ere death released him.
Whilst some remained to watch the dead body, others went off in search
of the murderer; and these had not gone far, when they came upon a man
of peculiar appearance, who, finding himself enclosed by the exploring
party, so as to make escape from them impossible, began to foam with
rage, and by cries and wild gesticulations to intimate that he would
make any one repent the attempt who should venture to meddle with him.
The fierceness of his desperation for a time kept our people at bay;
but as his fury gradually subsided, they gathered more closely round
him, and at length seized him. He then soon made them understand that
it was he who had killed their comrade, but they could not learn from
him any cause for this conduct. As the crime was so atrocious, and, if
allowed to pass with impunity, might entail even more serious
consequences, it was determined to burn the man. He was tied up to a
stake, where he was kept for some hours before the time of execution
arrived. It was then that I beheld what I had never thought to see. He
had a tail more than a foot long, covered with red hair, and very like
that of a cow. When he saw the surprise that this discovery created
among the European spectators, he informed us that his tail was the
effect of climate, for that all the inhabitants of the southern side
of the island, where they then were, were provided with like
appendages."[28]
After Struys, Hornemann reported that, between the Gulf of Benin and
Abyssinia, were tailed anthropophagi, named by the natives
_Niam-niams_; and in 1849, M. Descouret, on his return from Mecca,
affirmed that such was a common report, and added that they had long
arms, low and narrow foreheads, long and erect ears, and slim legs.
Mr. Harrison, in his "Highlands of Ethiopia," alludes to the common
belief among the Abyssinians, in a pygmy race of this nature.
MM. Arnault and VayssiA"re, travellers in the same country, in 1850,
brought the subject before the Academy of Sciences.
In 1851, M. de Castelnau gave additional details relative to an
expedition against these tailed men. "The Niam-niams," he says, "were
sleeping in the sun: the Haoussas approached, and, falling on them,
massacred them to the last man. They had all of them tails forty
centimetres long, and from two to three in diameter. This organ is
smooth. Among the corpses were those of several women, who were
deformed in the same manner. In all other particulars, the men were
precisely like all other <DW64>s. They are of a deep black, their
teeth are polished, their bodies not tattooed. They are armed with
clubs and javelins; in war they utter piercing cries. They cultivate
rice, maize, and other grain. They are fine looking men, and their
hair is not frizzled."
M. d'Abbadie, another Abyssinian traveller, writing in 1852, gives the
following account from the lips of an Abyssinian priest: "At the
distance of fifteen days' journey south of Herrar is a place where all
the men have tails, the length of a palm, covered with hair, and
situated at the extremity of the spine. The females of that country
are very beautiful and are tailless. I have seen some fifteen of these
people at Besberah, and I am positive that the tail is natural."
It will be observed that there is a discrepancy between the accounts
of M. de Castelnau and M. d'Abbadie. The former accords tails to the
ladies, whilst the latter denies it. According to the former, the tail
is smooth; according to the latter, it is covered with hair.
Dr. Wolf has improved on this in his "Travels and Adventures," vol.
ii. 1861. "There are men and women in Abyssinia with tails like dogs
and horses." Wolf heard also from a great many Abyssinians and
Armenians (and Wolf is convinced of the truth of it), that "there are
near Narea, in Abyssinia, people--men and women--with large tails,
with which they are able to knock down a horse; and there are also
such people near China." And in a note, "In the College of Surgeons
at Dublin may still be seen a human skeleton, with a tail seven inches
long! There are many known instances of this elongation of the caudal
vertebra, as in the Poonangs in Borneo."
But the most interesting and circumstantial account of the Niam-niams
is that given by Dr. Hubsch, physician to the hospitals of
Constantinople. "It was in 1852," says he, "that I saw for the first
time a tailed negress. I was struck with this phenomenon, and I
questioned her master, a slave dealer. I learned from him that there
exists a tribe called Niam-niam, occupying the interior of Africa. All
the members of this tribe bear the caudal appendage, and, as Oriental
imagination is given to exaggeration, I was assured that the tails
sometimes attained the length of two feet. That which I observed was
smooth and hairless. It was about two inches long, and terminated in a
point. This woman was as black as ebony, her hair was frizzled, her
teeth white, large, and planted in sockets which inclined considerably
outward; her four canine teeth were filed, her eyes bloodshot. She ate
meat raw, her clothes fidgeted her, her intellect was on a par with
that of others of her condition.
"Her master had been unable, during six months, to sell her,
notwithstanding the low figure at which he would have disposed of her;
the abhorrence with which she was regarded was not attributed to her
tail, but to the partiality, which she was unable to conceal, for
human flesh. Her tribe fed on the flesh of the prisoners taken from
the neighboring tribes, with whom they were constantly at war.
"As soon as one of the tribe dies, his relations, instead of burying
him, cut him up and regale themselves upon his remains; consequently
there are no cemeteries in this land. They do not all of them lead a
wandering life, but many of them construct hovels of the branches of
trees. They make for themselves weapons of war and of agriculture;
they cultivate maize and wheat, and keep cattle. The Niam-niams have a
language of their own, of an entirely primitive character, though
containing an infusion of Arabic words.
"They live in a state of complete nudity, and seek only to satisfy
their brute appetites. There is among them an utter disregard for
morality, incest and adultery being common. The strongest among them
becomes the chief of the tribe; and it is he who apportions the shares
of the booty obtained in war. It is hard to say whether they have any
religion; but in all probability they have none, as they readily adopt
any one which they are taught.
"It is difficult to tame them altogether; their instinct impelling
them constantly to seek for human flesh; and instances are related of
slaves who have massacred and eaten the children confided to their
charge.
"I have seen a man of the same race, who had a tail an inch and a half
long, covered with a few hairs. He appeared to be thirty-five years
old; he was robust, well built, of an ebon blackness, and had the same
peculiar formation of jaw noticed above; that is to say, the tooth
sockets were inclined outwards. Their four canine teeth are filed
down, to diminish their power of mastication.
"I know also, at Constantinople, the son of a physician, aged two
years, who was born with a tail an inch long; he belonged to the white
Caucasian race. One of his grandfathers possessed the same appendage.
This phenomenon is regarded generally in the East as a sign of great
brute force."
About ten years ago, a newspaper paragraph recorded the birth of a
boy at Newcastle-on-Tyne, provided with a tail about an inch and a
quarter long. It was asserted that the child when sucking wagged this
stump as token of pleasure.
Yet, notwithstanding all this testimony in favor of tailed men and
women, it is simply a matter of impossibility for a human being to
have a tail, for the spinal vertebrA| in man do not admit of
elongation, as in many animals; for the spine terminates in the os
sacrum, a large and expanded bone of peculiar character, entirely
precluding all possibility of production to the spine as in caudate
animals.
FOOTNOTES:
[27] "Actes of English Votaries."
[28] "Voyages de Jean Struys," An. 1650.
Antichrist and Pope Joan.
From the earliest ages of the Church, the advent of the Man of Sin has
been looked forward to with terror, and the passages of Scripture
relating to him have been studied with solemn awe, lest that day of
wrath should come upon the Church unawares. As events in the world's
history took place which seemed to be indications of the approach of
Antichrist, a great horror fell upon men's minds, and their
imaginations conjured up myths which flew from mouth to mouth, and
which were implicitly believed.
Before speaking of these strange tales which produced such an effect
on the minds of men in the middle ages, it will be well briefly to
examine the opinions of divines of the early ages on the passages of
Scripture connected with the coming of the last great persecutor of
the Church. Antichrist was believed by most ancient writers to be
destined to arise out of the tribe of Dan, a belief founded on the
prediction of Jacob, "Dan shall be a serpent by the way, an adder in
the path" (conf. Jeremiah viii. 16), and on the exclamation of the
dying patriarch, when looking on his son Dan, "I have waited for Thy
Salvation, O Lord," as though the long-suffering of God had borne long
with that tribe, but in vain, and it was to be extinguished without
hope. This, indeed, is implied in the sealing of the servants of God
in their foreheads (Revelation vii.), when twelve thousand out of
every tribe, except Dan, were seen by St. John to receive the seal of
adoption, whilst of the tribe of Dan _not one_ was sealed, as though
it, to a man, had apostatized.
Opinions as to the nature of Antichrist were divided. Some held that
he was to be a devil in phantom body, and of this number was
Hippolytus. Others, again, believed that he would be an incarnate
demon, true man and true devil; in fearful and diabolical parody of
the Incarnation of our Lord. A third view was, that he would be merely
a desperately wicked man, acting upon diabolical inspirations, just as
the saints act upon divine inspirations. St. John Damascene expressly
asserts that he will not be an incarnate demon, but a devilish man;
for he says, "Not as Christ assumed humanity, so will the devil become
human, but the Man will receive all the inspiration of Satan, and will
suffer the devil to take up his abode within him." In this manner
Antichrist could have many forerunners; and so St. Jerome and St.
Augustine saw an Antichrist in Nero, not _the_ Antichrist, but one of
those of whom the Apostle speaks--"Even now are there many
Antichrists." Thus also every enemy of the faith, such as Diocletian,
Julian, and Mahomet, has been regarded as a precursor of the
Arch-persecutor, who was expected to sum up in himself the cruelty of
a Nero or Diocletian, the show of virtue of a Julian, and the
spiritual pride of a Mahomet.
From infancy the evil one is to take possession of Antichrist, and to
train him for his office, instilling into him cunning, cruelty, and
pride. His doctrine will be--not downright infidelity, but a "show of
godliness," whilst "denying the power thereof;" i. e., the miraculous
origin and divine authority of Christianity. He will sow doubts of our
Lord's manifestation "in the flesh," he will allow Christ to be an
excellent Man, capable of teaching the most exalted truths, and
inculcating the purest morality, yet Himself fallible and carried away
by fanaticism.
In the end, however, Antichrist will "exalt himself to sit as God in
the temple of God," and become "the abomination of desolation standing
in the holy place." At the same time there is to be an awful alliance
struck between himself, the impersonification of the world-power and
the Church of God; some high pontiff of which, or the episcopacy in
general, will enter into league with the unbelieving state to oppress
the very elect. It is a strange instance of religionary virulence
which makes some detect the Pope of Rome in the Man of Sin, the
Harlot, the Beast, and the Priest going before it. The Man of Sin and
the Beast are unmistakably identical, and refer to an Antichristian
world-power; whilst the Harlot and the Priest are symbols of an
apostasy in the Church. There is nothing Roman in this, but something
very much the opposite.
How the Abomination of Desolation can be considered as set up in a
Church where every sanctuary is adorned with all that can draw the
heart to the Crucified, and raise the thoughts to the imposing ritual
of Heaven, is a puzzle to me. To the man uninitiated in the law that
Revelation is to be interpreted by contraries, it would seem more like
the Abomination of Desolation in the Holy Place if he entered a Scotch
Presbyterian, or a Dutch Calvinist, place of worship. Rome does not
fight against the Daily Sacrifice, and endeavor to abolish it; that
has been rather the labor of so-called Church Reformers, who with the
suppression of the doctrine of Eucharistic Sacrifice and Sacramental
Adoration have well nigh obliterated all notion of worship to be
addressed to the God-Man. Rome does not deny the power of the
godliness of which she makes show, but insists on that power with no
broken accents. It is rather in other communities, where authority is
flung aside, and any man is permitted to believe or reject what he
likes, that we must look for the leaven of the Antichristian spirit at
work.
It is evident that this spirit will infect the Church, and especially
those in place of authority therein; so that the elect will have to
wrestle against both "principalities and powers" in the state, and
also "spiritual wickedness in the high places" of the Church. Perhaps
it will be this feeling of antagonism between the inferior orders and
the highest which will throw the Bishops into the arms of the state,
and establish that unholy alliance which will be cemented for the
purpose of oppressing all who hold the truth in sincerity, who are
definite in their dogmatic statements of Christ's having been
manifested in the flesh, who labor to establish the Daily Sacrifice,
and offer in every place the pure offering spoken of by Malachi.
Perhaps it was in anticipation of this, that ancient mystical
interpreters explained the scene at the well in Midian as having
reference to the last times.
The Church, like the daughters of Reuel, comes to the Well of living
waters to water her parched flock; whereupon the shepherds--her chief
pastors--arise and strive with her. "Fear not, O flock, fear not, O
daughter!" exclaims the commentator; "thy true Moses is seated on the
well, and He will arise out of His resting-place, and will with His
own hand smite the shepherds, and water the flock." Let the sheep be
in barren and dry pastures,--so long the shepherds strive not; let the
sheep pant and die,--so long the shepherds show no signs of
irritation; but let the Church approach the limpid well of life, and
at once her prelates will, in the latter days, combine "to strive"
with her, and keep back the flock from the reviving streams.
In the time of Antichrist the Church will be divided: one portion will
hold to the world-power, the other will seek out the old paths, and
cling to the only true Guide. The high places will be filled with
unbelievers in the Incarnation, and the Church will be in a condition
of the utmost spiritual degradation, but enjoying the highest State
patronage. The religion in favor will be one of morality, but not of
dogma; and the Man of Sin will be able to promulgate his doctrine,
according to St. Anselm, through his great eloquence and wisdom, his
vast learning and mightiness in the Holy Scriptures, which he will
wrest to the overthrowing of dogma. He will be liberal in bribes, for
he will be of unbounded wealth; he will be capable of performing great
"signs and wonders," so as "to deceive--the very elect;" and at the
last, he will tear the moral veil from his countenance, and a monster
of impiety and cruelty, he will inaugurate that awful persecution,
which is to last for three years and a half, and to excel in horror
all the persecutions that have gone before.
In that terrible season of confusion faith will be all but
extinguished. "When the Son of Man cometh, shall He find faith on the
earth?" asks our Blessed Lord, as though expecting the answer, No; and
then, says Marchantius, the vessel of the Church will disappear in the
foam of that boiling deep of infidelity, and be hidden in the
blackness of that storm of destruction which sweeps over the earth.
The sun shall "be darkened, and the moon shall not give her light, and
the stars shall fall from heaven;" the sun of faith shall have gone
out; the moon, the Church, shall not give her light, being turned into
blood, through stress of persecution; and the stars, the great
ecclesiastical dignitaries, shall fall into apostasy. But still the
Church will remain unwrecked, she will weather the storm; still will
she come forth "beautiful as the moon, terrible as an army with
banners;" for after the lapse of those three and a half years, Christ
will descend to avenge the blood of the saints, by destroying
Antichrist and the world-power.
Such is a brief sketch of the scriptural doctrine of Antichrist as
held by the early and mediA|val Church. Let us now see to what myths it
gave rise among the vulgar and the imaginative. Rabanus Maurus, in his
work on the life of Antichrist, gives a full account of the miracles
he will perform; he tells us that the Man-fiend will heal the sick,
raise the dead, restore sight to the blind, hearing to the deaf,
speech to the dumb; he will raise storms and calm them, will remove
mountains, make trees flourish or wither at a word. He will rebuild
the temple at Jerusalem, and making the Holy City the great capital of
the world. Popular opinion added that his vast wealth would be
obtained from hidden treasures, which are now being concealed by the
demons for his use. Various possessed persons, when interrogated,
announced that such was the case, and that the amount of buried gold
was vast.
"In the year 1599," says Canon Moreau, a contemporary historian, "a
rumor circulated with prodigious rapidity through Europe, that
Antichrist had been born at Babylon, and that already the Jews of that
part were hurrying to receive and recognize him as their Messiah. The
news came from Italy and Germany, and extended to Spain, England, and
other Western kingdoms, troubling many people, even the most discreet;
however, the learned gave it no credence, saying that the signs
predicted in Scripture to precede that event were not yet
accomplished, and among other that the Roman empire was not yet
abolished.... Others said that, as for the signs, the majority had
already appeared to the best of their knowledge, and with regard to
the rest, they might have taken place in distant regions without their
having been made known to them; that the Roman empire existed but in
name, and that the interpretation of the passage on which its
destruction was predicted, might be incorrect; that for many
centuries, the most learned and pious had believed in the near
approach of Antichrist, some believing that he had already come, on
account of the persecutions which had fallen on the Christians;
others, on account of fires, or eclipses, or earthquakes.... Every
one was in excitement; some declared that the news must be correct,
others believed nothing about it, and the agitation became so
excessive, that Henry IV., who was then on the throne, was compelled
by edict to forbid any mention of the subject."
The report spoken of by Moreau gained additional confirmation from the
announcement made by an exorcised demoniac, that in 1600, the Man of
Sin had been born in the neighborhood of Paris, of a Jewess, named
Blanchefleure, who had conceived by Satan. The child had been baptized
at the Sabbath of Sorcerers; and a witch, under torture, acknowledged
that she had rocked the infant Antichrist on her knees, and she
averred that he had claws on his feet, wore no shoes, and spoke all
languages.
In 1623 appeared the following startling announcement, which obtained
an immense circulation among the lower orders: "We, brothers of the
Order of St. John of Jerusalem, in the Isle of Malta, have received
letters from our spies, who are engaged in our service in the country
of Babylon, now possessed by the Grand Turk; by the which letters we
are advertised, that, on the 1st of May, in the year of our Lord
1623, a child was born in the town of Bourydot, otherwise called
Calka, near Babylon, of the which child the mother is a very aged
woman, of race unknown, called Fort-Juda: of the father nothing is
known. The child is dusky, has pleasant mouth and eyes, teeth pointed
like those of a cat, ears large, stature by no means exceeding that of
other children; the said child, incontinent on his birth, walked and
talked perfectly well. His speech is comprehended by every one,
admonishing the people that he is the true Messiah, and the son of
God, and that in him all must believe. Our spies also swear and
protest that they have seen the said child with their own eyes; and
they add, that, on the occasion of his nativity, there appeared
marvellous signs in heaven, for at full noon the sun lost its
brightness, and was for some time obscured." This is followed by a
list of other signs appearing, the most remarkable being a swarm of
flying serpents, and a shower of precious stones.
According to Sebastian Michaeliz, in his history of the possessed of
Flanders, on the authority of the exorcised demons, we learn that
Antichrist is to be a son of Beelzebub, who will accompany his
offspring under the form of a bird, with four feet and a bull's head;
that he will torture Christians with the same tortures with which the
lost souls are racked; that he will be able to fly, speak all
languages, and will have any number of names.
We find that Antichrist is known to the Mussulmans as well as to
Christians. Lane, in his edition of the "Arabian Nights," gives some
curious details on Moslem ideas regarding him. According to these,
Antichrist will overrun the earth, mounted on an ass, and followed by
40,000 Jews; his empire will last forty days, whereof the first day
will be a year long, the duration of the second will be a month, that
of the third a week, the others being of their usual length. He will
devastate the whole world, leaving Mecca and Medina alone in security,
as these holy cities will be guarded by angelic legions. Christ at
last will descend to earth, and in a great battle will destroy the
Man-devil.
Several writers, of different denominations, no less superstitious
than the common people, connected the apparition of Antichrist with
the fable of Pope Joan, which obtained such general credence at one
time, but which modern criticism has at length succeeded in excluding
from history.
Perhaps the earliest writer to mention Pope Joan is Marianus Scotus,
who in his chronicle inserts the following passage: "A. D. 854,
Lotharii 14, Joanna, a woman, succeeded Leo, and reigned two years,
five months, and four days." Marianus Scotus died A. D. 1086. Sigebert
de Gemblours (d. 5th Oct., 1112) inserts the same story in his
valuable chronicle, copying from an interpolated passage in the work
of Anastasius the librarian. His words are, "It is reported that this
John was a female, and that she conceived by one of her servants. The
Pope, becoming pregnant, gave birth to a child; wherefore some do not
number her among the Pontiffs." Hence the story spread among the
mediA|val chroniclers, who were great plagiarists. Otto of Frisingen
and Gotfrid of Viterbo mention the Lady-Pope in their histories, and
Martin Polonus gives details as follows: "After Leo IV., John Anglus,
a native of Metz, reigned two years, five months, and four days. And
the pontificate was vacant for a month. He died in Rome. He is related
to have been a female, and, when a girl, to have accompanied her
sweetheart in male costume to Athens; there she advanced in various
sciences, and none could be found to equal her. So, after having
studied for three years in Rome, she had great masters for her pupils
and hearers. And when there arose a high opinion in the city of her
virtue and knowledge, she was unanimously elected Pope. But during her
papacy she became in the family way by a familiar. Not knowing the
time of birth, as she was on her way from St. Peter's to the Lateran
she had a painful delivery, between the Coliseum and St. Clement's
Church, in the street. Having died after, it is said that she was
buried on the spot; and therefore the Lord Pope always turns aside
from that way, and it is supposed by some out of detestation for what
happened there. Nor on that account is she placed in the catalogue of
the Holy Pontiffs, not only on account of her sex, but also because of
the horribleness of the circumstance."
Certainly a story at all scandalous _crescit eundo_.
William Ocham alludes to the story, and John Huss, only too happy to
believe it, provides the lady with a name, and asserts that she was
baptized Agnes, or, as he will have it with a strong aspirate, Hagnes.
Others, however, insist upon her name having been Gilberta; and some
stout Germans, not relishing the notion of her being a daughter of
Fatherland, palm her off on England. As soon as we arrive at
Reformation times, the German and French Protestants fasten on the
story with the utmost avidity, and add sweet little touches of their
own, and draw conclusions galling enough to the Roman See,
illustrating their accounts with wood engravings vigorous and graphic,
but hardly decent. One of these represents the event in a peculiarly
startling manner. The procession of bishops, with the Host and tapers,
is sweeping along, when suddenly the cross-bearer before the
triple-crowned and vested Pope starts aside to witness the unexpected
arrival. This engraving, which it is quite impossible for me to
reproduce, is in a curious little book, entitled "Puerperium Johannis
PapA| 8, 1530."
The following jingling record of the event is from the Rhythmical VitA|
Pontificum of Gulielmus Jacobus of Egmonden, a work never printed.
This fragment is preserved in "Wolfii Lectionum Memorabilium
centenarii, XVI.:"--
"PriusquA m reconditur Sergius, vocatur
Ad summam, qui dicitur Johannes, huic addatur
Anglicus, Moguntia iste procreatur.
Qui, ut dat sententia, fA"minis aptatur
Sexu: quod sequentia monstrant, breviatur,
HA|c vox: nam prolixius chronica procedunt.
Ista, de qua brevius dicta minus lA|dunt.
Huic erat amasius, ut scriptores credunt.
Patria relinquitur Moguntia, GrA|corum
StudiosA" petitur schola. PA squaredst doctorum
HA|c doctrix efficitur RomA| legens: horum
HA|c auditu fungitur loquens. Hinc prostrato
Summo hA|c eligitur: sexu exaltato
Quandoque negligitur. Fatur quA squaredd hA|c nato
Per servum conficitur. Tempore gignendi
Ad processum equus scanditur, vice flendi,
Papa cadit, panditur improbis ridendi
Norma, puer nascitur in vico Clementis,
ColossA"um jungitur. Corpus parentis
In eodem traditur sepulturA| gentis,
Faturque scriptoribus, quA squaredd Papa prA|fato,
Vico senioribus transiens amato
Congruo ductoribus sequitur negato
Loco, quo Ecclesia partu denigratur,
Quamvis inter spacia Pontificum ponatur,
Propter sexum."
Stephen Blanch, in his "Urbis RomA| Mirabilia," says that an angel of
heaven appeared to Joan before the event, and asked her to choose
whether she would prefer burning eternally in hell, or having her
confinement in public; with sense which does her credit, she chose the
latter. The Protestant writers were not satisfied that the father of
the unhappy baby should have been a servant: some made him a
Cardinal, and others the devil himself. According to an eminent Dutch
minister, it is immaterial whether the child be fathered on Satan or a
monk; at all events, the former took a lively interest in the youthful
Antichrist, and, on the occasion of his birth, was seen and heard
fluttering overhead, crowing and chanting in an unmusical voice the
Sibylline verses announcing the birth of the Arch-persecutor:--
"Papa pater patrum, PapissA| pandito partum
Et tibi tunc eadem de corpore quando recedam!"
which lines, as being perhaps the only ones known to be of diabolic
composition, are deserving of preservation.
The Reformers, in order to reconcile dates, were put to the somewhat
perplexing necessity of moving Pope Joan to their own times, or else
of giving to the youthful Antichrist an age of seven hundred years.
It must be allowed that the _accouchement_ of a Pope in full
pontificals, during a solemn procession, was a prodigy not likely to
occur more than once in the world's history, and was certain to be of
momentous import.
It will be seen by the curious woodcut reproduced as frontispiece
from Baptista Mantuanus, that he consigned Pope Joan to the jaws of
hell, notwithstanding her choice. The verses accompanying this picture
are:--
"Hic pendebat adhuc sexum mentita virile
FA"mina, cui triplici Phrygiam diademate mitram
Extollebat apex: et pontificalis adulter."
It need hardly be stated that the whole story of Pope Joan is
fabulous, and rests on not the slightest historical foundation. It was
probably a Greek invention to throw discredit on the papal hierarchy,
first circulated more than two hundred years after the date of the
supposed Pope. Even Martin Polonus (A. D. 1282), who is the first to
give the details, does so merely on popular report.
The great champions of the myth were the Protestants of the sixteenth
century, who were thoroughly unscrupulous in distorting history and
suppressing facts, so long as they could make a point. A paper war was
waged upon the subject, and finally the whole story was proved
conclusively to be utterly destitute of historical truth. A melancholy
example of the blindness of party feeling and prejudice is seen in
Mosheim, who assumes the truth of the ridiculous story, and gravely
inserts it in his "Ecclesiastical History." "Between Leo IV., who died
855, and Benedict III., a woman, who concealed her sex and assumed the
name of John, it is said, opened her way to the Pontifical throne by
her learning and genius, and governed the Church for a time. She is
commonly called the Papess Joan. During the five subsequent centuries
the witnesses to this extraordinary event are without number; nor did
any one, prior to the Reformation by Luther, regard the thing as
either incredible or disgraceful to the Church." Such are Mosheim's
words, and I give them as a specimen of the credit which is due to his
opinion. The "Ecclesiastical History" he wrote is full of perversions
of the plainest facts, and that under our notice is but one out of
many. "During the five centuries after her reign," he says, "the
witnesses to the story are innumerable." Now, for two centuries there
is not an allusion to be found to the events. The only passage which
can be found is a universally acknowledged interpolation of the "Lives
of the Popes," by Anastasius Bibliothecarius; and this interpolation
is stated in the first printed edition by BusA|us, Mogunt. 1602, to be
only found in two MS. copies.
From Marianus Scotus or Sigebert de Gemblours the story passed into
other chronicles _totidem verbis_, and generally with hesitation and
an expression of doubt in its accuracy. Martin Polonus is the first to
give the particulars, some four hundred and twenty years after the
reign of the fabulous Pope.
Mosheim is false again in asserting that no one prior to the
Reformation regarded the thing as either incredible or disgraceful.
This is but of a piece with his malignity and disregard for truth,
whenever he can hit the Catholic Church hard. Bart. Platina, in his
"Lives of the Popes," written before Luther was born, after relating
the story, says, "These things which I relate are popular reports, but
derived from uncertain and obscure authors, which I have therefore
inserted briefly and baldly, lest I should seem to omit obstinately
and pertinaciously what most people assert." Thus the facts were
justly doubted by Platina on the legitimate grounds that they rested
on popular gossip, and not on reliable history. Marianus Scotus, the
first to relate the story, died in 1086. He was a monk of St. Martin
of Cologne, then of Fulda, and lastly of St. Alban's, at Metz. How
could he have obtained reliable information, or seen documents upon
which to ground the assertion? Again, his chronicle has suffered
severely from interpolations in numerous places, and there is reason
to believe that the Pope-Joan passage is itself a late interpolation.
If so, we are reduced to Sigebert de Gemblours (d. 1112), placing two
centuries and a half between him and the event he records, and his
chronicle may have been tampered with.
The historical discrepancies are sufficiently glaring to make the
story more than questionable.
Leo IV. died on the 17th July, 855; and Benedict III. was consecrated
on the 1st September in the same year; so that it is impossible to
insert between their pontificates a reign of two years, five months,
and four days. It is, however, true that there was an antipope elected
upon the death of Leo, at the instance of the Emperor Louis; but his
name was Anastasius. This man possessed himself of the palace of the
Popes, and obtained the incarceration of Benedict. However, his
supporters almost immediately deserted him, and Benedict assumed the
pontificate. The reign of Benedict was only for two years and a half,
so that Anastasius cannot be the supposed Joan; nor do we hear of any
charge brought against him to the effect of his being a woman. But the
stout partisans of the Pope-Joan tale assert, on the authority of the
"Annales Augustani,"[29] and some other, but late authorities, that
the female Pope was John VIII., who consecrated Louis II. of France,
and Ethelwolf of England. Here again is confusion. Ethelwolf sent
Alfred to Rome in 853, and the youth received regal unction from the
hands of Leo IV. In 855 Ethelwolf visited Rome, it is true, but was
not consecrated by the existing Pope, whilst Charles the Bald was
anointed by John VIII. in 875. John VIII. was a Roman, son of Gundus,
and an archdeacon of the Eternal City. He assumed the triple crown in
872, and reigned till December 18, 882. John took an active part in
the troubles of the Church under the incursions of the Sarasins, and
325 letters of his are extant, addressed to the princes and prelates
of his day.
Any one desirous of pursuing this examination into the untenable
nature of the story may find an excellent summary of the arguments
used on both sides in Gieseler, "Lehrbuch," &c., Cunningham's trans.,
vol. ii. pp. 20, 21, or in Bayle, "Dictionnaire," tom. iii. art.
Papesse.
The arguments in favor of the myth may be seen in Spanheim, "Exercit.
de Papa FA"mina," Opp. tom. ii. p. 577, or in Lenfant, "Histoire de
la Papesse Jeanne," La Haye, 1736, 2 vols. 12mo.
The arguments on the other side may be had in "Allatii Confutatio
FabulA| de Johanna Papissa," Colon. 1645; in Le Quien, "Oriens
Christianus," tom. iii. p. 777; and in the pages of the Lutheran
Huemann, "Sylloge Diss. Sacras.," tom. i. par. ii. p. 352.
The final development of this extraordinary story, under the delicate
fingers of the German and French Protestant controversialists, may not
prove uninteresting.
Joan was the daughter of an English missionary, who left England to
preach the Gospel to the recently converted Saxons. She was born at
Engelheim, and according to different authors she was christened
Agnes, Gerberta, Joanna, Margaret, Isabel, Dorothy, or Jutt--the last
must have been a nickname surely! She early distinguished herself for
genius and love of letters. A young monk of Fulda having conceived for
her a violent passion, which she returned with ardor, she deserted her
parents, dressed herself in male attire, and in the sacred precincts
of Fulda divided her affections between the youthful monk and the
musty books of the monastic library. Not satisfied with the restraints
of conventual life, nor finding the library sufficiently well provided
with books of abstruse science, she eloped with her young man, and
after visiting England, France, and Italy, she brought him to Athens,
where she addicted herself with unflagging devotion to her literary
pursuits. Wearied out by his journey, the monk expired in the arms of
the blue-stocking who had influenced his life for evil, and the young
lady of so many aliases was for a while inconsolable. She left Athens
and repaired to Rome. There she opened a school and acquired such a
reputation for learning and feigned sanctity, that, on the death of
Leo IV., she was unanimously elected Pope. For two years and five
months, under the name of John VIII., she filled the papal chair with
reputation, no one suspecting her sex. But having taken a fancy to one
of the cardinals, by him she became pregnant. At length arrived the
time of Rogation processions. Whilst passing the street between the
amphitheatre and St. Clement's, she was seized with violent pains,
fell to the ground amidst the crowd, and, whilst her attendants
ministered to her, was delivered of a son. Some say the child and
mother died on the spot, some that she survived but was incarcerated,
some that the child was spirited away to be the Antichrist of the last
days. A marble monument representing the papess with her baby was
erected on the spot, which was declared to be accursed to all ages.
I have little doubt myself that Pope Joan is an impersonification of
the great whore of Revelation, seated on the seven hills, and is the
popular expression of the idea prevalent from the twelfth to the
sixteenth centuries, that the mystery of iniquity was somehow working
in the papal court. The scandal of the Antipopes, the utter
worldliness and pride of others, the spiritual fornication with the
kings of the earth, along with the words of Revelation prophesying the
advent of an adulterous woman who should rule over the imperial city,
and her connection with Antichrist, crystallized into this curious
myth, much as the floating uncertainty as to the signification of our
Lord's words, "There be some standing here which shall not taste of
death till they see the kingdom of God," condensed into the myth of
the Wandering Jew.
The literature connected with Antichrist is voluminous. I need only
specify some of the most curious works which have appeared on the
subject. St. Hippolytus and Rabanus Maurus have been already alluded
to. Commodianus wrote "Carmen Apologeticum adversus Gentes," which has
been published by Dom Pitra in his "Spicilegium Solesmense," with an
introduction containing Jewish and Christian traditions relating to
Antichrist. "De Turpissima Conceptione, Nativitate, et aliis PrA|sagiis
Diaboliciis illius Turpissimi Hominis Antichristi," is the title of a
strange little volume published by Lenoir in A. D. 1500, containing
rude yet characteristic woodcuts, representing the birth, life, and
death of the Man of Sin, each picture accompanied by French verses in
explanation. An equally remarkable illustrated work on Antichrist is
the famous "Liber de Antichristo," a blockbook of an early date. It is
in twenty-seven folios, and is excessively rare. Dibdin has reproduced
three of the plates in his "Bibliotheca Spenseriana," and Falckenstein
has given full details of the work in his "Geschichte der
Buchdruckerkunst."
There is an Easter miracle-play of the twelfth century, still extant,
the subject of which is the "Life and Death of Antichrist." More
curious still is the "Farce de l'AntA(C)christ et de Trois Femmes"--a
composition of the sixteenth century, when that mysterious personage
occupied all brains. The farce consists in a scene at a fish-stall,
with three good ladies quarrelling over some fish. Antichrist steps
in,--for no particular reason that one can see,--upsets fish and
fish-women, sets them fighting, and skips off the stage. The best book
on Antichrist, and that most full of learning and judgment, is
Malvenda's great work in two folio volumes, "De Antichristo, libri
xii." Lyons, 1647.
For the fable of the Pope Joan, see J. Lenfant, "Histoire de la
Papesse Jeanne." La Haye, 1736, 2 vols. 12mo. "Allatii Confutatio
FabulA| de Johanna Papissa." Colon. 1645.
FOOTNOTE:
[29] These Annals were written in 1135.
The Man in the Moon.
[Illustration: From L. Richter.]
Every one knows that the moon is inhabited by a man with a bundle of
sticks on his back, who has been exiled thither for many centuries,
and who is so far off that he is beyond the reach of death.
He has once visited this earth, if the nursery rhyme is to be
credited, when it asserts that--
"The Man in the Moon
Came down too soon,
And asked his way to Norwich;"
but whether he ever reached that city, the same authority does not
state.
The story as told by nurses is, that this man was found by Moses
gathering sticks on a Sabbath, and that, for this crime, he was doomed
to reside in the moon till the end of all things; and they refer to
Numbers xv. 32-36:--
"And while the children of Israel were in the wilderness, they found a
man that gathered sticks upon the Sabbath day. And they that found him
gathering sticks brought him unto Moses and Aaron, and unto all the
congregation. And they put him in ward, because it was not declared
what should be done to him. And the Lord said unto Moses, The man
shall be surely put to death: all the congregation shall stone him
with stones without the camp. And all the congregation brought him
without the camp, and stoned him with stones till he died."
Of course, in the sacred writings there is no allusion to the moon.
The German tale is as follows:--
Ages ago there went one Sunday morning an old man into the wood to hew
sticks. He cut a fagot and slung it on a stout staff, cast it over his
shoulder, and began to trudge home with his burden. On his way he met
a handsome man in Sunday suit, walking towards the Church; this man
stopped and asked the fagot-bearer, "Do you know that this is Sunday
on earth, when all must rest from their labors?"
"Sunday on earth, or Monday in heaven, it is all one to me!" laughed
the wood-cutter.
"Then bear your bundle forever," answered the stranger; "and as you
value not Sunday on earth, yours shall be a perpetual Moon-day in
heaven; and you shall stand for eternity in the moon, a warning to all
Sabbath-breakers." Thereupon the stranger vanished, and the man was
caught up with his stock and his fagot into the moon, where he stands
yet.
The superstition seems to be old in Germany, for the full moon is
spoken of as _wadel_, or _wedel_, a fagot. Tobler relates the story
thus: "An arma mAe ket alawel am Sonnti holz ufglesa. Do hedem der
liebe Gott dwahl gloh, A¶b er lieber wott ider sonn verbrenna oder im
mo verfrura, do willer lieber inn mo ihi. Dromm siedma no jetz an ma
im mo inna, wenns wedel ist. Er hed a pA1/4scheli uffem rogga."[30] That
is to say, he was given the choice of burning in the sun, or of
freezing in the moon; he chose the latter; and now at full moon he is
to be seen seated with his bundle of fagots on his back.
In Schaumburg-Lippe,[31] the story goes, that a man and a woman stand
in the moon, the man because he strewed brambles and thorns on the
church path, so as to hinder people from attending Mass on Sunday
morning; the woman because she made butter on that day. The man
carries his bundle of thorns, the woman her butter-tub. A similar tale
is told in Swabia and in Marken. Fischart[32] says, that there "is to
be seen in the moon a manikin who stole wood;" and PrA|torius, in his
description of the world,[33] that "superstitious people assert that
the black flecks in the moon are a man who gathered wood on a Sabbath,
and is therefore turned into stone."
The Dutch household myth is, that the unhappy man was caught stealing
vegetables. Dante calls him Cain:--
"... Now doth Cain with fork of thorns confine,
On either hemisphere, touching the wave
Beneath the towers of Seville. Yesternight
The moon was round."
_Hell_, cant. xx.
And again,--
"... Tell, I pray thee, whence the gloomy spots
Upon this body, which below on earth
Give rise to talk of Cain in fabling quaint?"
_Paradise_, cant. ii.
Chaucer, in the "Testament of Cresside," adverts to the man in the
moon, and attributes to him the same idea of theft. Of Lady Cynthia,
or the moon, he says,--
"Her gite was gray and full of spottis blake,
And on her brest a chorle painted ful even,
Bering a bush of thornis on his backe,
Whiche for his theft might clime so ner the heaven."
Ritson, among his "Ancient Songs," gives one extracted from a
manuscript of the time of Edward II., on the Man in the Moon, but in
very obscure language. The first verse, altered into more modern
orthography, runs as follows:--
"Man in the Moon stand and stit,
On his bot-fork his burden he beareth,
It is much wonder that he do na doun slit,
For doubt lest he fall he shudd'reth and shivereth.
...
"When the frost freezes must chill he bide,
The thorns be keen his attire so teareth,
Nis no wight in the world there wot when he syt,
Ne bote it by the hedge what weeds he weareth."
Alexander Necham, or Nequam, a writer of the twelfth century, in
commenting on the dispersed shadows in the moon, thus alludes to the
vulgar belief: "Nonne novisti quid vulgus vocet rusticum in luna
portantem spinas? Unde quidam vulgariter loquens ait:--
"Rusticus in Luna,
Quem sarcina deprimit una
Monstrat per opinas
Nulli prodesse rapinas,"
which may be translated thus: "Do you know what they call the rustic
in the moon, who carries the fagot of sticks?" So that one vulgarly
speaking says,--
"See the rustic in the Moon,
How his bundle weighs him down;
Thus his sticks the truth reveal,
It never profits man to steal."
Shakspeare refers to the same individual in his "Midsummer Night's
Dream." Quince the carpenter, giving directions for the performance of
the play of "Pyramus and Thisbe," orders: "One must come in with a
bush of thorns and a lantern, and say he comes in to disfigure, or to
present, the person of Moonshine." And the enacter of this part says,
"All I have to say is, to tell you that the lantern is the moon; I the
man in the moon; this thorn-bush my thorn-bush; and this dog my dog."
Also "Tempest," Act 2, Scene 2:--
"_Cal._ Hast thou not dropt from heaven?
"_Steph._ Out o' th' moon, I do assure thee. I was the man in
th' moon when time was.
"_Cal._ I have seen thee in her; and I do adore thee. My
mistress showed me thee, and thy dog, and thy bush."
The dog I have myself had pointed out to me by an old Devonshire
crone. If popular superstition places a dog in the moon, it puts a
lamb in the sun; for in the same county it is said that those who see
the sun rise on Easter-day, may behold in the orb the lamb and flag.
I believe this idea of locating animals in the two great luminaries of
heaven to be very ancient, and to be a relic of a primeval
superstition of the Aryan race.
There is an ancient pictorial representation of our friend the
Sabbath-breaker in Gyffyn Church, near Conway. The roof of the
chancel is divided into compartments, in four of which are the
Evangelistic symbols, rudely, yet effectively painted. Besides these
symbols is delineated in each compartment an orb of heaven. The sun,
the moon, and two stars, are placed at the feet of the Angel, the
Bull, the Lion, and the Eagle. The representation of the moon is as
below; in the disk is the conventional man with his bundle of sticks,
but without the dog. There is also a curious seal appended to a deed
preserved in the Record Office, dated the 9th year of Edward the Third
(1335), bearing the man in the moon as its device. The deed is one of
conveyance of a messuage, barn, and four acres of ground, in the
parish of Kingston-on-Thames, from Walter de Grendesse, clerk, to
Margaret his mother. On the seal we see the man carrying his sticks,
and the moon surrounds him. There are also a couple of stars added,
perhaps to show that he is in the sky. The legend on the seal reads:--
"Te Waltere docebo
cur spinas phebo
gero,"
which may be translated, "I will teach thee, Walter, why I carry
thorns in the moon."
[Illustration: {Representation of the moon in Gyffyn Church.}]
[Illustration: {The seal with the legend visible.}]
The general superstition with regard to the spots in the moon may
briefly be summed up thus: A man is located in the moon; he is a thief
or Sabbath-breaker;[34] he has a pole over his shoulder, from which
is suspended a bundle of sticks or thorns. In some places a woman is
believed to accompany him, and she has a butter-tub with her; in other
localities she is replaced by a dog.
The belief in the Moon-man seems to exist among the natives of British
Columbia; for I read in one of Mr. Duncan's letters to the Church
Missionary Society, "One very dark night I was told that there was a
moon to see on the beach. On going to see, there was an illuminated
disk, with the figure of a man upon it. The water was then very low,
and one of the conjuring parties had lit up this disk at the water's
edge. They had made it of wax, with great exactness, and presently it
was at full. It was an imposing sight. Nothing could be seen around
it; but the Indians suppose that the medicine party are then holding
converse with the man in the moon.... After a short time the moon
waned away, and the conjuring party returned whooping to their house."
Now let us turn to Scandinavian mythology, and see what we learn from
that source.
MAcni, the moon, stole two children from their parents, and carried
them up to heaven. Their names were Hjuki and Bil. They had been
drawing water from the well Byrgir, in the bucket SA"gr, suspended
from the pole Simul, which they bore upon their shoulders. These
children, pole, and bucket were placed in heaven, "where they could be
seen from earth." This refers undoubtedly to the spots in the moon;
and so the Swedish peasantry explain these spots to this day, as
representing a boy and a girl bearing a pail of water between them.
Are we not reminded at once of our nursery rhyme--
"Jack and Jill went up a hill
To fetch a pail of water;
Jack fell down, and broke his crown,
And Jill came tumbling after"?
This verse, which to us seems at first sight nonsense, I have no
hesitation in saying has a high antiquity, and refers to the Eddaic
Hjuki and Bil. The names indicate as much. Hjuki, in Norse, would be
pronounced Juki, which would readily become Jack; and Bil, for the
sake of euphony, and in order to give a female name to one of the
children, would become Jill.
The fall of Jack, and the subsequent fall of Jill, simply represent
the vanishing of one moon-spot after another, as the moon wanes.
But the old Norse myth had a deeper signification than merely an
explanation of the moon-spots.
Hjuki is derived from the verb jakka, to heap or pile together, to
assemble and increase; and Bil from bila, to break up or dissolve.
Hjuki and Bil, therefore, signify nothing more than the waxing and
waning of the moon, and the water they are represented as bearing
signifies the fact that the rainfall depends on the phases of the
moon. Waxing and waning were individualized, and the meteorological
fact of the connection of the rain with the moon was represented by
the children as water-bearers.
But though Jack and Jill became by degrees dissevered in the popular
mind from the moon, the original myth went through a fresh phase, and
exists still under a new form. The Norse superstition attributed
_theft_ to the moon, and the vulgar soon began to believe that the
figure they saw in the moon was the thief. The lunar specks certainly
may be made to resemble one figure, and only a lively imagination can
discern two. The girl soon dropped out of popular mythology, the boy
oldened into a venerable man, he retained his pole, and the bucket
was transformed into the thing he had stolen--sticks or vegetables.
The theft was in some places exchanged for Sabbath-breaking,
especially among those in Protestant countries who were acquainted
with the Bible story of the stick-gatherer.
The Indian superstition is worth examining, because of the connection
existing between Indian and European mythology, on account of our
belonging to the same Aryan stock.
According to a Buddhist legend, SAckyamunni himself, in one of his
earlier stages of existence, was a hare, and lived in friendship with
a fox and an ape. In order to test the virtue of the Bodhisattwa,
Indra came to the friends, in the form of an old man, asking for food.
Hare, ape, and fox went forth in quest of victuals for their guest.
The two latter returned from their foraging expedition successful, but
the hare had found nothing. Then, rather than that he should treat the
old man with inhospitality, the hare had a fire kindled, and cast
himself into the flames, that he might himself become food for his
guest. In reward for this act of self-sacrifice, Indra carried the
hare to heaven, and placed him in the moon.[35]
Here we have an old man and a hare in connection with the lunar
planet, just as in Shakspeare we have a fagot-bearer and a dog.
The fable rests upon the name of the moon in Sanskrit, ASec.aASec.in, or "that
marked with the hare;" but whether the belief in the spots taking the
shape of a hare gave the name ASec.aASec.in to the moon, or the lunar name
ASec.aASec.in originated the belief, it is impossible for us to say.
Grounded upon this myth is the curious story of "The Hare and the
Elephant," in the "Pantschatantra," an ancient collection of Sanskrit
fables. It will be found as the first tale in the third book. I have
room only for an outline of the story.
THE CRAFTY HARE.
In a certain forest lived a mighty elephant, king of a herd, Toothy by
name. On a certain occasion there was a long drought, so that pools,
tanks, swamps, and lakes were dried up. Then the elephants sent out
exploring parties in search of water. A young one discovered an
extensive lake surrounded with trees, and teeming with water-fowl. It
went by the name of the Moon-lake. The elephants, delighted at the
prospect of having an inexhaustible supply of water, marched off to
the spot, and found their most sanguine hopes realized. Round about
the lake, in the sandy soil, were innumerable hare warrens; and as the
herd of elephants trampled on the ground, the hares were severely
injured, their homes broken down, their heads, legs, and backs crushed
beneath the ponderous feet of the monsters of the forest. As soon as
the herd had withdrawn, the hares assembled, some halting, some
dripping with blood, some bearing the corpses of their cherished
infants, some with piteous tales of ruination in their houses, all
with tears streaming from their eyes, and wailing forth, "Alas, we are
lost! The elephant-herd will return, for there is no water elsewhere,
and that will be the death of all of us."
But the wise and prudent Longear volunteered to drive the herd away;
and he succeeded in this manner: Longear went to the elephants, and
having singled out their king, he addressed him as follows:--
"Ha, ha! bad elephant! what brings you with such thoughtless frivolity
to this strange lake? Back with you at once!"
When the king of the elephants heard this, he asked in astonishment,
"Pray, who are you?"
"I," replied Longear,--"I am Vidschajadatta by name; the hare who
resides in the Moon. Now am I sent by his Excellency the Moon as an
ambassador to you. I speak to you in the name of the Moon."
"Ahem! Hare," said the elephant, somewhat staggered; "and what message
have you brought me from his Excellency the Moon?"
"You have this day injured several hares. Are you not aware that they
are the subjects of me? If you value your life, venture not near the
lake again. Break my command, and I shall withdraw my beams from you
at night, and your bodies will be consumed with perpetual sun."
The elephant, after a short meditation, said, "Friend! it is true that
I have acted against the rights of the excellent Majesty of the Moon.
I should wish to make an apology; how can I do so?"
The hare replied, "Come along with me, and I will show you."
The elephant asked, "Where is his Excellency at present?"
The other replied, "He is now in the lake, hearing the complaints of
the maimed hares."
"If that be the case," said the elephant, humbly, "bring me to my
lord, that I may tender him my submission."
So the hare conducted the king of the elephants to the edge of the
lake, and showed him the reflection of the moon in the water, saying,
"There stands our lord in the midst of the water, plunged in
meditation; reverence him with devotion, and then depart with speed."
Thereupon the elephant poked his proboscis into the water, and
muttered a fervent prayer. By so doing he set the water in agitation,
so that the reflection of the moon was all of a quiver.
"Look!" exclaimed the hare; "his Majesty is trembling with rage at
you!"
"Why is his supreme Excellency enraged with me?" asked the elephant.
"Because you have set the water in motion. Worship him, and then be
off!"
The elephant let his ears droop, bowed his great head to the earth,
and after having expressed in suitable terms his regret for having
annoyed the Moon, and the hare dwelling in it, he vowed never to
trouble the Moon-lake again. Then he departed, and the hares have ever
since lived there unmolested.
FOOTNOTES:
[30] Tobler, Appenz. Sprachsbuch, 20.
[31] Wolf, Zeitschrift fA1/4r Deut. Myth. i. 168.
[32] Fischart, Garg. 130.
[33] PrA|torius, i. 447.
[34] Hebel, in his charming poem on the Man in the Moon, in
"Allemanische Gedichte," makes him both thief and Sabbath-breaker.
[35] "MA(C)moires ... par Hjouen Thsang, traduits du Chinois par
Stanislas Julien," i. 375. Upham, "Sacred Books of Ceylon," iii. 309.
The Mountain of Venus.
Ragged, bald, and desolate, as though a curse rested upon it, rises
the HA¶rselberg out of the rich and populous land between Eisenach and
Gotha, looking, from a distance, like a huge stone sarcophagus--a
sarcophagus in which rests in magical slumber, till the end of all
things, a mysterious world of wonders.
High up on the north-west flank of the mountain, in a precipitous wall
of rock, opens a cavern, called the HA¶rselloch, from the depths of
which issues a muffled roar of water, as though a subterraneous stream
were rushing over rapidly-whirling millwheels. "When I have stood
alone on the ridge of the mountain," says Bechstein, "after having
sought the chasm in vain, I have heard a mighty rush, like that of
falling water, beneath my feet, and after scrambling down the scarp,
have found myself--how, I never knew--in front of the cave."
("Sagenschatz des ThA1/4ringes-landes," 1835.)
In ancient days, according to the ThA1/4ringian Chronicles, bitter cries
and long-drawn moans were heard issuing from this cavern; and at
night, wild shrieks and the burst of diabolical laughter would ring
from it over the vale, and fill the inhabitants with terror. It was
supposed that this hole gave admittance to Purgatory; and the popular
but faulty derivation of HA¶rsel was _HA¶re, die Seele_--Hark, the
Souls!
But another popular belief respecting this mountain was, that in it
Venus, the pagan Goddess of Love, held her court, in all the pomp and
revelry of heathendom; and there were not a few who declared that they
had seen fair forms of female beauty beckoning them from the mouth of
the chasm, and that they had heard dulcet strains of music well up
from the abyss above the thunder of the falling, unseen torrent.
Charmed by the music, and allured by the spectral forms, various
individuals had entered the cave, and none had returned, except the
TanhA¤user, of whom more anon. Still does the HA¶rselberg go by the name
of the Venusberg, a name frequently used in the middle ages, but
without its locality being defined.
"In 1398, at midday, there appeared suddenly three great fires in the
air, which presently ran together into one globe of flame, parted
again, and finally sank into the HA¶rselberg," says the ThA1/4ringian
Chronicle.
And now for the story of TanhA¤user.
A French knight was riding over the beauteous meadows in the HA¶rsel
vale on his way to Wartburg, where the Landgrave Hermann was holding a
gathering of minstrels, who were to contend in song for a prize.
TanhA¤user was a famous minnesinger, and all his lays were of love and
of women, for his heart was full of passion, and that not of the
purest and noblest description.
It was towards dusk that he passed the cliff in which is the
HA¶rselloch, and as he rode by, he saw a white glimmering figure of
matchless beauty standing before him, and beckoning him to her. He
knew her at once, by her attributes and by her superhuman perfection,
to be none other than Venus. As she spake to him, the sweetest strains
of music floated in the air, a soft roseate light glowed around her,
and nymphs of exquisite loveliness scattered roses at her feet. A
thrill of passion ran through the veins of the minnesinger; and,
leaving his horse, he followed the apparition. It led him up the
mountain to the cave, and as it went flowers bloomed upon the soil,
and a radiant track was left for TanhA¤user to follow. He entered the
cavern, and descended to the palace of Venus in the heart of the
mountain.
Seven years of revelry and debauch were passed, and the minstrel's
heart began to feel a strange void. The beauty, the magnificence, the
variety of the scenes in the pagan goddess's home, and all its
heathenish pleasures, palled upon him, and he yearned for the pure
fresh breezes of earth, one look up at the dark night sky spangled
with stars, one glimpse of simple mountain-flowers, one tinkle of
sheep-bells. At the same time his conscience began to reproach him,
and he longed to make his peace with God. In vain did he entreat Venus
to permit him to depart, and it was only when, in the bitterness of
his grief, he called upon the Virgin-Mother, that a rift in the
mountain-side appeared to him, and he stood again above ground.
How sweet was the morning air, balmy with the scent of hay, as it
rolled up the mountain to him, and fanned his haggard cheek! How
delightful to him was the cushion of moss and scanty grass after the
downy couches of the palace of revelry below! He plucked the little
heather-bells, and held them before him; the tears rolled from his
eyes, and moistened his thin and wasted hands. He looked up at the
soft blue sky and the newly-risen sun, and his heart overflowed. What
were the golden, jewel-incrusted, lamp-lit vaults beneath to that pure
dome of God's building!
The chime of a village church struck sweetly on his ear, satiated with
Bacchanalian songs; and he hurried down the mountain to the church
which called him. There he made his confession; but the priest,
horror-struck at his recital, dared not give him absolution, but
passed him on to another. And so he went from one to another, till at
last he was referred to the Pope himself. To the Pope he went. Urban
IV. then occupied the chair of St. Peter. To him TanhA¤user related the
sickening story of his guilt, and prayed for absolution. Urban was a
hard and stern man, and shocked at the immensity of the sin, he thrust
the penitent indignantly from him, exclaiming, "Guilt such as thine
can never, never be remitted. Sooner shall this staff in my hand grow
green and blossom, than that God should pardon thee!"
Then TanhA¤user, full of despair, and with his soul darkened, went
away, and returned to the only asylum open to him, the Venusberg. But
lo! three days after he had gone, Urban discovered that his pastoral
staff had put forth buds, and had burst into flower. Then he sent
messengers after TanhA¤user, and they reached the HA¶rsel vale to hear
that a wayworn man, with haggard brow and bowed head, had just entered
the HA¶rselloch. Since then TanhA¤user has not been seen.
Such is the sad yet beautiful story of TanhA¤user. It is a very ancient
myth Christianized, a wide-spread tradition localized. Originally
heathen, it has been transformed, and has acquired new beauty by an
infusion of Christianity. Scattered over Europe, it exists in various
forms, but in none so graceful as that attached to the HA¶rselberg.
There are, however, other Venusbergs in Germany; as, for instance, in
Swabia, near Waldsee; another near Ufhausen, at no great distance from
Freiburg (the same story is told of this Venusberg as of the
HA¶rselberg); in Saxony there is a Venusberg not far from Wolkenstein.
Paracelsus speaks of a Venusberg in Italy, referring to that in which
A†neas Sylvius (Ep. 16) says Venus or a Sibyl resides, occupying a
cavern, and assuming once a week the form of a serpent. Geiler v.
Keysersperg, a quaint old preacher of the fifteenth century, speaks of
the witches assembling on the Venusberg.
The story, either in prose or verse, has often been printed. Some of
the earliest editions are the following:--
"Das Lied von dem Danhewser." NA1/4rnberg, without date; the same,
NA1/4rnberg, 1515.--"Das Lyedt v. d. Thanheuser." Leyptzk, 1520.--"Das
Lied v. d. DanheA1/4ser," reprinted by Bechstein, 1835.--"Das Lied vom
edlen Tanheuser, Mons Veneris." Frankfort, 1614; Leipzig, 1668.--"Twe
lede volgen Dat erste vain DanhA1/4sser." Without date.--"Van heer
Danielken." Tantwerpen, 1544.--A Danish version in "Nyerup, Danske
Viser," No. VIII.
Let us now see some of the forms which this remarkable myth assumed in
other countries. Every popular tale has its root, a root which may be
traced among different countries, and though the accidents of the
story may vary, yet the substance remains unaltered. It has been said
that the common people never invent new story-radicals any more than
we invent new word-roots; and this is perfectly true. The same
story-root remains, but it is varied according to the temperament of
the narrator or the exigencies of localization. The story-root of the
Venusberg is this:--
The underground folk seek union with human beings.
I+-. A man is enticed into their abode, where he unites
with a woman of the underground race.
I squared. He desires to revisit the earth, and escapes.
I cubed. He returns again to the region below.
Now, there is scarcely a collection of folk-lore which does not
contain a story founded on this root. It appears in every branch of
the Aryan family, and examples might be quoted from Modern Greek,
Albanian, Neapolitan, French, German, Danish, Norwegian and Swedish,
Icelandic, Scotch, Welsh, and other collections of popular tales. I
have only space to mention some.
There is a Norse ThAittr of a certain Helgi Thorir's son, which is, in
its present form, a production of the fourteenth century. Helgi and
his brother Thorstein went on a cruise to Finnmark, or Lapland. They
reached a ness, and found the land covered with forest. Helgi explored
this forest, and lighted suddenly on a party of red-dressed women
riding upon red horses. These ladies were beautiful and of troll race.
One surpassed the others in beauty, and she was their mistress. They
erected a tent and prepared a feast. Helgi observed that all their
vessels were of silver and gold. The lady, who named herself
Ingibjorg, advanced towards the Norseman, and invited him to live with
her. He feasted and lived with the trolls for three days, and then
returned to his ship, bringing with him two chests of silver and gold,
which Ingibjorg had given him. He had been forbidden to mention where
he had been and with whom; so he told no one whence he had obtained
the chests. The ships sailed, and he returned home.
One winter's night Helgi was fetched away from home, in the midst of a
furious storm, by two mysterious horsemen, and no one was able to
ascertain for many years what had become of him, till the prayers of
the king, Olaf, obtained his release, and then he was restored to his
father and brother, but he was thenceforth blind. All the time of his
absence he had been with the red-vested lady in her mysterious abode
of GlA"sisvellir.
The Scotch story of Thomas of Ercildoune is the same story. Thomas met
with a strange lady, of elfin race, beneath Eildon Tree, who led him
into the underground land, where he remained with her for seven years.
He then returned to earth, still, however, remaining bound to come to
his royal mistress whenever she should summon him. Accordingly, while
Thomas was making merry with his friends in the Tower of Ercildoune, a
person came running in, and told, with marks of fear and astonishment,
that a hart and a hind had left the neighboring forest, and were
parading the street of the village. Thomas instantly arose, left his
house, and followed the animals into the forest, from which he never
returned. According to popular belief, he still "drees his weird" in
Fairy Land, and is one day expected to revisit earth. (Scott,
"Minstrelsy of the Scottish Border.") Compare with this the ancient
ballad of Tamlane.
Debes relates that "it happened a good while since, when the burghers
of Bergen had the commerce of the Faroe Isles, that there was a man in
Serraade, called Jonas Soideman, who was kept by the spirits in a
mountain during the space of seven years, and at length came out, but
lived afterwards in great distress and fear, lest they should again
take him away; wherefore people were obliged to watch him in the
night." The same author mentions another young man who had been
carried away, and after his return was removed a second time, upon the
eve of his marriage.
Gervase of Tilbury says that "in Catalonia there is a lofty mountain,
named Cavagum, at the foot of which runs a river with golden sands, in
the vicinity of which there are likewise silver mines. This mountain
is steep, and almost inaccessible. On its top, which is always covered
with ice and snow, is a black and bottomless lake, into which if a
stone be cast, a tempest suddenly arises; and near this lake is the
portal of the palace of demons." He then tells how a young damsel was
spirited in there, and spent seven years with the mountain spirits. On
her return to earth she was thin and withered, with wandering eyes,
and almost bereft of understanding.
A Swedish story is to this effect. A young man was on his way to his
bride, when he was allured into a mountain by a beautiful elfin woman.
With her he lived forty years, which passed as an hour; on his return
to earth all his old friends and relations were dead, or had forgotten
him, and finding no rest there, he returned to his mountain elf-land.
In Pomerania, a laborer's son, Jacob Dietrich of Rambin, was enticed
away in the same manner.
There is a curious story told by Fordun in his "Scotichronicon," which
has some interest in connection with the legend of the TanhA¤user. He
relates that in the year 1050, a youth of noble birth had been married
in Rome, and during the nuptial feast, being engaged in a game of
ball, he took off his wedding-ring, and placed it on the finger of a
statue of Venus. When he wished to resume it, he found that the stony
hand had become clinched, so that it was impossible to remove the
ring. Thenceforth he was haunted by the Goddess Venus, who constantly
whispered in his ear, "Embrace me; I am Venus, whom you have wedded; I
will never restore your ring." However, by the assistance of a
priest, she was at length forced to give it up to its rightful owner.
The classic legend of Ulysses, held captive for eight years by the
nymph Calypso in the Island of Ogygia, and again for one year by the
enchantress Circe, contains the root of the same story of the
TanhA¤user.
What may have been the significance of the primeval story-radical it
is impossible for us now to ascertain; but the legend, as it shaped
itself in the middle ages, is certainly indicative of the struggle
between the new and the old faith.
We see thinly veiled in TanhA¤user the story of a man, Christian in
name, but heathen at heart, allured by the attractions of paganism,
which seems to satisfy his poetic instincts, and which gives full rein
to his passions. But these excesses pall on him after a while, and the
religion of sensuality leaves a great void in his breast.
He turns to Christianity, and at first it seems to promise all that he
requires. But alas! he is repelled by its ministers. On all sides he
is met by practice widely at variance with profession. Pride,
worldliness, want of sympathy exist among those who should be the
foremost to guide, sustain, and receive him. All the warm springs
which gushed up in his broken heart are choked, his softened spirit is
hardened again, and he returns in despair to bury his sorrows and
drown his anxieties in the debauchery of his former creed.
A sad picture, but doubtless one very true.
Fatality of Numbers.
The laws governing numbers are so perplexing to the uncultivated mind,
and the results arrived at by calculation are so astonishing, that it
cannot be matter of surprise if superstition has attached itself to
numbers.
But even to those who are instructed in numeration, there is much that
is mysterious and unaccountable, much that only an advanced
mathematician can explain to his own satisfaction. The neophyte sees
the numbers obedient to certain laws; but _why_ they obey these laws
he cannot understand; and the fact of his not being able so to do,
tends to give to numbers an atmosphere of mystery which impresses him
with awe.
For instance, the property of the number 9, discovered, I believe, by
W. Green, who died in 1794, is inexplicable to any one but a
mathematician. The property to which I allude is this, that when 9 is
multiplied by 2, by 3, by 4, by 5, by 6, &c., it will be found that
the digits composing the product, when added together, give 9. Thus:--
2 A-- 9 = 18, and 1 + 8 = 9
3 A-- 9 = 27, " 2 + 7 = 9
4 A-- 9 = 36, " 3 + 6 = 9
5 A-- 9 = 45, " 4 + 5 = 9
6 A-- 9 = 54, " 5 + 4 = 9
7 A-- 9 = 63, " 6 + 3 = 9
8 A-- 9 = 72, " 7 + 2 = 9
9 A-- 9 = 81, " 8 + 1 = 9
10 A-- 9 = 90, " 9 + 0 = 9
It will be noticed that 9 A-- 11 makes 99, the sum of the digits of
which is 18 and not 9, but the sum of the digits 1 + 8 equals 9.
9 A-- 12 = 108, and 1 + 0 + 8 = 9
9 A-- 13 = 117, " 1 + 1 + 7 = 9
9 A-- 14 = 126, " 1 + 2 + 6 = 9
And so on to any extent.
M. de Maivan discovered another singular property of the same number.
If the order of the digits expressing a number be changed, and this
number be subtracted from the former, the remainder will be 9 or a
multiple of 9, and, being a multiple, the sum of its digits will be 9.
For instance, take the number 21, reverse the digits, and you have
12; subtract 12 from 21, and the remainder is 9. Take 63, reverse the
digits, and subtract 36 from 63; you have 27, a multiple of 9, and 2 +
7 = 9. Once more, the number 13 is the reverse of 31; the difference
between these numbers is 18, or twice 9.
Again, the same property found in two numbers thus changed, is
discovered in the same numbers raised to any power.
Take 21 and 12 again. The square of 21 is 441, and the square of 12 is
144; subtract 144 from 441, and the remainder is 297, a multiple of 9;
besides, the digits expressing these powers added together give 9. The
cube of 21 is 9261, and that of 12 is 1728; their difference is 7533,
also a multiple of 9.
The number 37 has also somewhat remarkable properties; when multiplied
by 3 or a multiple of 3 up to 27, it gives in the product three digits
exactly similar. From the knowledge of this the multiplication of 37
is greatly facilitated, the method to be adopted being to multiply
merely the first cipher of the multiplicand by the first multiplier;
it is then unnecessary to proceed with the multiplication, it being
sufficient to write twice to the right hand the cipher obtained, so
that the same digit will stand in the unit, tens, and hundreds places.
For instance, take the results of the following table:--
37 multiplied by 3 gives 111, and 3 times 1 = 3
37 " 6 " 222, " 3 " 2 = 6
37 " 9 " 333, " 3 " 3 = 9
37 " 12 " 444, " 3 " 4 = 12
37 " 15 " 555, " 3 " 5 = 15
37 " 18 " 666, " 3 " 6 = 18
37 " 21 " 777, " 3 " 7 = 21
37 " 24 " 888, " 3 " 8 = 24
37 " 27 " 999, " 3 " 9 = 27
The singular property of numbers the most different, when added, to
produce the same sum, originated the use of magical squares for
talismans. Although the reason may be accounted for mathematically,
yet numerous authors have written concerning them, as though there
were something "uncanny" about them. But the most remarkable and
exhaustive treatise on the subject is that by a mathematician of
Dijon, which is entitled "TraitA(C) complet des CarrA(C)s magiques, pairs et
impairs, simple et composA(C)s, A Bordures, Compartiments, Croix,
Chassis, A%querres, Bandes dA(C)tachA(C)es, &c.; suivi d'un TraitA(C) des Cubes
magiques et d'un Essai sur les Cercles magiques; par M. Violle,
GA(C)omA"tre, Chevalier de St. Louis, avec Atlas de 54 grandes Feuilles,
comprenant 400 figures." Paris, 1837. 2 vols. 8vo., the first of 593
pages, the second of 616. Price 36 fr.
I give three examples of magical squares:--
2 7 6
9 5 1
4 3 8
These nine ciphers are disposed in three horizontal lines; add the
three ciphers of each line, and the sum is 15; add the three ciphers
in each column, the sum is 15; add the three ciphers forming
diagonals, and the sum is 15.
1 2 3 4 1 7 13 19 25
2 3 2 3 18 24 5 6 12
4 1 4 1 10 11 17 23 4
3 4 1 2 22 3 9 15 16
14 20 21 2 8
The sum is 10. The sum is 65.
But the connection of certain numbers with the dogmas of religion was
sufficient, besides their marvellous properties, to make superstition
attach itself to them. Because there were thirteen at the table when
the Last Supper was celebrated, and one of the number betrayed his
Master, and then hung himself, it is looked upon through Christendom
as unlucky to sit down thirteen at table, the consequence being that
one of the number will die before the year is out. "When I see," said
Vouvenargues, "men of genius not daring to sit down thirteen at table,
there is no error, ancient or modern, which astonishes me."
Nine, having been consecrated by Buddhism, is regarded with great
veneration by the Moguls and Chinese: the latter bow nine times on
entering the presence of their Emperor.
Three is sacred among Brahminical and Christian people, because of the
Trinity of the Godhead.
Pythagoras taught that each number had its own peculiar character,
virtue, and properties.
"The unit, or the monad," he says, "is the principle and the end of
all; it is this sublime knot which binds together the chain of causes;
it is the symbol of identity, of equality, of existence, of
conservation, and of general harmony. Having no parts, the monad
represents Divinity; it announces also order, peace, and tranquillity,
which are founded on unity of sentiments; consequently ONE is a good
principle.
"The number TWO, or the dyad, the origin of contrasts, is the symbol
of diversity, or inequality, of division and of separation. TWO is
accordingly an evil principle, a number of bad augury, characterizing
disorder, confusion, and change.
"THREE, or the triad, is the first of unequals; it is the number
containing the most sublime mysteries, for everything is composed of
three substances; it represents God, the soul of the world, the spirit
of man." This number, which plays so great a part in the traditions of
Asia, and in the Platonic philosophy, is the image of the attributes
of God.
"FOUR, or the tetrad, as the first mathematical power, is also one of
the chief elements; it represents the generating virtue, whence come
all combinations; it is the most perfect of numbers; it is the root of
all things. It is holy by nature, since it constitutes the Divine
essence, by recalling His unity, His power, His goodness, and His
wisdom, the four perfections which especially characterize God.
Consequently, Pythagoricians swear by the quaternary number, which
gives the human soul its eternal nature.
"The number FIVE, or the pentad, has a peculiar force in sacred
expiations; it is everything; it stops the power of poisons, and is
redoubted by evil spirits.
"The number SIX, or the hexad, is a fortunate number, and it derives
its merit from the first sculptors having divided the face into six
portions; but, according to the Chaldeans, the reason is, because God
created the world in six days.
"SEVEN, or the heptad, is a number very powerful for good or for evil.
It belongs especially to sacred things.
"The number EIGHT, or the octad, is the first cube, that is to say,
squared in all senses, as a die, proceeding from its base two, an even
number; so is man four-square, or perfect.
"The number NINE, or the ennead, being the multiple of three, should
be regarded as sacred.
"Finally, TEN, or the decad, is the measure of all, since it contains
all the numeric relations and harmonies. As the reunion of the four
first numbers, it plays an eminent part, since all the branches of
science, all nomenclatures, emanate from, and retire into it."
It is hardly necessary for me here to do more than mention the
peculiar character given to different numbers by Christianity. One is
the numeral indicating the Unity of the Godhead; Two points to the
hypostatic union; Three to the Blessed Trinity; Four to the
Evangelists; Five to the Sacred Wounds; Six is the number of sin;
Seven that of the gifts of the Spirit; Eight, that of the Beatitudes;
Ten is the number of the commandments; Eleven speaks of the Apostles
after the loss of Judas; Twelve, of the complete apostolic college.
I shall now point out certain numbers which have been regarded with
superstition, and certain events connected with numbers which are of
curious interest.
The number 14 has often been observed as having singularly influenced
the life of Henry IV. and other French princes. Let us take the
history of Henry.
On the 14th May, 1029, the first king of France named Henry was
consecrated, and on the 14th May, 1610, the last Henry was
assassinated.
Fourteen letters enter into the composition of the name of Henri de
Bourbon, who was the 14th king bearing the titles of France and
Navarre.
The 14th December, 1553, that is, 14 centuries, 14 decades, and 14
years after the birth of Christ, Henry IV. was born; the ciphers of
the date 1553, when added together, giving the number 14.
The 14th May, 1554, Henry II. ordered the enlargement of the Rue de la
Ferronnerie. The circumstance of this order not having been carried
out, occasioned the murder of Henry IV. in that street, four times 14
years after.
The 14th May, 1552, was the date of the birth of MarguA(C)rite de Valois,
first wife of Henry IV.
On the 14th May, 1588, the Parisians revolted against Henry III., at
the instigation of the Duke of Guise.
On the 14th March, 1590, Henry IV. gained the battle of Ivry.
On the 14th May, 1590, Henry was repulsed from the Fauxbourgs of
Paris.
On the 14th November, 1590, the Sixteen took oath to die rather than
serve Henry.
On the 14th November, 1592, the Parliament registered the Papal Bull
giving power to the legate to nominate a king to the exclusion of
Henry.
On the 14th December, 1599, the Duke of Savoy was reconciled to Henry
IV.
On the 14th September, 1606, the Dauphin, afterwards Louis XIII., was
baptized.
On the 14th May, 1610, the king was stopped in the Rue de la
Ferronnerie, by his carriage becoming locked with a cart, on account
of the narrowness of the street. Ravaillac took advantage of the
occasion for stabbing him.
Henry IV. lived four times 14 years, 14 weeks, and four times 14 days;
that is to say, 56 years and 5 months.
On the 14th May, 1643, died Louis XIII., son of Henry IV.; not only on
the same day of the same month as his father, but the date, 1643, when
its ciphers are added together, gives the number 14, just as the
ciphers of the date of the birth of his father gave 14.
Louis XIV. mounted the throne in 1643: 1 + 6 + 4 + 3 = 14.
He died in the year 1715: 1 + 7 + 1 + 5 = 14.
He lived 77 years, and 7 + 7 = 14.
Louis XV. mounted the throne in the same year; he died in 1774, which
also bears the stamp of 14, the extremes being 14, and the sum of the
means 7 + 7 making 14.
Louis XVI. had reigned 14 years when he convoked the States General,
which was to bring about the Revolution.
The number of years between the assassination of Henry IV. and the
dethronement of Louis XVI. is divisible by 14.
Louis XVII. died in 1794; the extreme digits of the date are 14, and
the first two give his number.
The restoration of the Bourbons took place in 1814, also marked by the
extremes being 14; also by the sum of the ciphers making 14.
The following are other curious calculations made respecting certain
French kings.
Add the ciphers composing the year of the birth or of the death of
some of the kings of the third race, and the result of each sum is
the titular number of each prince. Thus:--
Louis IX. was born in 1215; add the four ciphers of this date, and you
have IX.
Charles VII. was born in 1402; the sum of 1 + 4 + 2 gives VII.
Louis XII. was born in 1461; and 1 + 4 + 6 + 1 = XII.
Henry IV. died in 1610; and 1 + 6 + 1 = twice IV.
Louis XIV. was crowned in 1643; and these four ciphers give XIV. The
same king died in 1715; and this date gives also XIV. He was aged 77
years, and again 7 + 7 = 14.
Louis XVIII. was born in 1755; add the digits, and you have XVIII.
What is remarkable is, that this number 18 is double the number of the
king to whom the law first applies, and is triple the number of the
kings to whom it has applied.
Here is another curious calculation:--
Robespierre fell in 1794;
Napoleon in 1815, and Charles X. in 1830.
Now, the remarkable fact in connection with these dates is, that the
sum of the digits composing them, added to the dates, gives the date
of the fall of the successor. Robespierre fell in 1794; 1 + 7 + 9 + 4
= 21, 1794 + 21 = 1815, the date of the fall of Napoleon; 1 + 8 + 1 +
5 = 15, and 1815 + 15 = 1830, the date of the fall of Charles X.
There is a singular rule which has been supposed to determine the
length of the reigning Pope's life, in the earlier half of a century.
Add his number to that of his predecessor, to that add ten, and the
result gives the year of his death.
Pius VII. succeeded Pius VI.; 6 + 7 = 13; add 10, and the sum is 23.
Pius VII. died in 1823.
Leo XII. succeeded Pius VII.; 12 + 7 + 10 = 29; and Leo XII. died in
1829.
Pius VIII. succeeded Leo XII.; 8 + 12 + 10 = 30; and Pius VIII. died
in 1830.
However, this calculation does not always apply.
Gregory XVI. ought to have died in 1834, but he did not actually
vacate his see till 1846.
It is also well known that an ancient tradition forbids the hope of
any of St. Peter's successors, _pervenire ad annos Petri_; i. e., to
reign 25 years.
Those who sat longest are
Years. Months. Days.
Pius VI., who reigned 24 6 14
Hadrian I. " 23 10 17
Pius VII. " 23 5 6
Alexander III. " 21 11 23
St. Silvester I. " 21 0 4
There is one numerical curiosity of a very remarkable character, which
I must not omit.
The ancient Chamber of Deputies, such as it existed in 1830, was
composed of 402 members, and was divided into two parties. The one,
numbering 221 members, declared itself strongly for the revolution of
July; the other party, numbering 181, did not favor a change. The
result was the constitutional monarchy, which re-established order
after the three memorable days of July. The parties were known by the
following nicknames. The larger was commonly called _La queue de
Robespierre_, and the smaller, _Les honnAªtes gens_. Now, the
remarkable fact is, that if we give to the letters of the alphabet
their numerical values as they stand in their order, as 1 for A, 2 for
B, 3 for C, and so on to Z, which is valued at 25, and then write
vertically on the left hand the words, _La queue de Robespierre_,
with the number equivalent to each letter opposite to it, and on the
right hand, in like manner, _Les honnAªtes gens_, if each column of
numbers be summed up, the result is the number of members who formed
each party.
1 2 3 4 5 6 7 8 9 10 11 12 13
A B C D E F G H I J K L M
14 15 16 17 18 19 20 21 22 23 24 25
N O P Q R S T U V X Y Z
L--12 L--12
A-- 1 E-- 5
S--19
Q--17
U--21 H-- 8
E-- 5 O--15
U-- 5 N--14
E-- 5 N--14
E-- 5
D-- 4 T--20
E-- 5 E-- 5
S--19
R--18
O--15 G-- 7
B-- 2 E-- 5
E-- 5 N--14
S--19 S--19
P--16 -----
I-- 9 181
E-- 5
R--18
R--18
E-- 5
-----
221
Majority 221
Minority 181
----
Total 402
Some coincidences of dates are very remarkable.
On the 25th August, 1569, the Calvinists massacred the Catholic nobles
and priests at BA(C)arn and Navarre.
On the same day of the same month, in 1572, the Calvinists were
massacred in Paris and elsewhere.
On the 25th October, 1615, Louis XIII. married Anne of Austria,
infanta of Spain, whereupon we may remark the following
coincidences:--
The name Loys[36] de Bourbon contains 13 letters; so does the name
Anne d'Austriche.
Louis was 13 years old when this marriage was decided on; Anne was the
same age.
He was the thirteenth king of France bearing the name of Louis, and
she was the thirteenth infanta of the name of Anne of Austria.
On the 23d April, 1616, died Shakspeare: on the same day of the same
month, in the same year, died the great poet Cervantes.
On the 29th May, 1630, King Charles II. was born.
On the 29th May, 1660, he was restored.
On the 29th May, 1672, the fleet was beaten by the Dutch.
On the 29th May, 1679, the rebellion of the Covenanters broke out in
Scotland.
The Emperor Charles V. was born on February 24, 1500; on that day he
won the battle of Pavia, in 1525, and on the same day was crowned in
1530.
On the 29th January, 1697, M. de Broquemar, president of the
Parliament of Paris, died suddenly in that city; next day his brother,
an officer, died suddenly at Bergue, where he was governor. The lives
of these brothers present remarkable coincidences. One day the
officer, being engaged in battle, was wounded in his leg by a
sword-blow. On the same day, at the same moment, the president was
afflicted with acute pain, which attacked him suddenly in the same leg
as that of his brother which had been injured.
John Aubrey mentions the case of a friend of his who was born on the
15th November; his eldest son was born on the 15th November; and his
second son's first son on the same day of the same month.
At the hour of prime, April 6, 1327, Petrarch first saw his mistress
Laura, in the Church of St. Clara in Avignon. In the same city, same
month, same hour, 1348, she died.
The deputation charged with offering the crown of Greece to Prince
Otho, arrived in Munich on the 13th October, 1832; and it was on the
13th October, 1862, that King Otho left Athens, to return to it no
more.
On the 21st April, 1770, Louis XVI. was married at Vienna, by the
sending of the ring.
On the 21st June, in the same year, took place the fatal festivities
of his marriage.
On the 21st January, 1781, was the _fAªte_ at the HA'tel de Ville, for
the birth of the Dauphin.
On the 21st June, 1791, took place the flight to Varennes.
On the 21st January, 1793, he died on the scaffold.
There is said to be a tradition of Norman-monkish origin, that the
number 3 is stamped on the Royal line of England, so that there shall
not be more than three princes in succession without a revolution.
William I., William II., Henry I.; then followed the revolution of
Stephen.
Henry II., Richard I., John; invasion of Louis, Dauphin of France, who
claimed the throne.
Henry III., Edward I., Edward II., who was dethroned and put to death.
Edward III., Richard II., who was dethroned.
Henry IV., Henry V., Henry VI.; the crown passed to the house of York.
Edward IV., Edward V., Richard III.; the crown claimed and won by
Henry Tudor.
Henry VII., Henry VIII., Edward VI.; usurpation of Lady Jane Grey.
Mary I., Elizabeth; the crown passed to the house of Stuart.
James I., Charles I.; Revolution.
Charles II., James II.; invasion of William of Orange.
William of Orange and Mary II., Anne; arrival of the house of
Brunswick.
George I., George II., George III., George IV., William IV., Victoria.
The law has proved faulty in the last case; but certainly there was a
crisis in the reign of George IV.
As I am on the subject of the English princes, I will add another
singular coincidence, though it has nothing to do with the fatality of
numbers.
It is that Saturday has been a day of ill omen to the later kings.
William of Orange died Saturday, 18th March, 1702.
Anne died Saturday, 1st August, 1704.
George I. died Saturday, 10th June, 1727.
George II. died Saturday, 25th October, 1760.
George III. died Saturday, 30th January, 1820.
George IV. died Saturday, 26th June, 1830.
FOOTNOTE:
[36] Up to Louis XIII. all the kings of this name spelled Louis as
Loys.
The Terrestrial Paradise.
The exact position of Eden, and its present condition, do not seem to
have occupied the minds of our Anglo-Saxon ancestors, nor to have
given rise among them to wild speculations.
The map of the tenth century in the British Museum, accompanying the
Periegesis of Priscian, is far more correct than the generality of
maps which we find in MSS. at a later period; and Paradise does not
occupy the place of Cochin China, or the isles of Japan, as it did
later, after that the fabulous voyage of St. Brandan had become
popular in the eleventh century.[37] The site, however, had been
already indicated by Cosmas, who wrote in the seventh century, and had
been specified by him as occupying a continent east of China, beyond
the ocean, and still watered by the four great rivers Pison, Gihon,
Hiddekel, and Euphrates, which sprang from subterranean canals. In a
map of the ninth century, preserved in the Strasbourg library, the
terrestrial Paradise is, however, on the Continent, placed at the
extreme east of Asia; in fact, is situated in the Celestial Empire. It
occupies the same position in a Turin MS., and also in a map
accompanying a commentary on the Apocalypse in the British Museum.
According to the fictitious letter of Prester John to the Emperor
Emanuel Comnenus, Paradise was situated close to--within three days'
journey of--his own territories, but where those territories were, is
not distinctly specified.
"The River Indus, which issues out of Paradise," writes the mythical
king, "flows among the plains, through a certain province, and it
expands, embracing the whole province with its various windings: there
are found emeralds, sapphires, carbuncles, topazes, chrysolites, onyx,
beryl, sardius, and many other precious stones. There too grows the
plant called Asbetos." A wonderful fountain, moreover, breaks out at
the roots of Olympus, a mountain in Prester John's domain, and "from
hour to hour, and day by day, the taste of this fountain varies; and
its source is hardly three days' journey from Paradise, from which
Adam was expelled. If any man drinks thrice of this spring, he will
from that day feel no infirmity, and he will, as long as he lives,
appear of the age of thirty." This Olympus is a corruption of Alumbo,
which is no other than Columbo in Ceylon, as is abundantly evident
from Sir John Mandeville's Travels; though this important fountain has
escaped the observation of Sir Emmerson Tennant.
"Toward the heed of that forest (he writes) is the cytee of Polombe,
and above the cytee is a great mountayne, also clept Polombe. And of
that mount, the Cytee hathe his name. And at the foot of that Mount is
a fayr welle and a gret, that hathe odour and savour of all spices;
and at every hour of the day, he chaungethe his odour and his savour
dyversely. And whoso drynkethe 3 times fasting of that watre of that
welle, he is hool of alle maner sykenesse, that he hathe. And thei
that duellen there and drynken often of that welle, thei nevere han
sykenesse, and thei semen alle weys yonge. I have dronken there of 3
of 4 sithes; and zit, methinkethe, I fare the better. Some men clepen
it the Welle of Youthe: for thei that often drynken thereat, semen
alle weys yongly, and lyven withouten sykenesse. And men seyn, that
that welle comethe out of Paradys: and therefore it is so vertuous."
Gautier de Metz, in his poem on the "Image du Monde," written in the
thirteenth century, places the terrestrial Paradise in an
unapproachable region of Asia, surrounded by flames, and having an
armed angel to guard the only gate.
Lambertus Floridus, in a MS. of the twelfth century, preserved in the
Imperial Library in Paris, describes it as "Paradisus insula in oceano
in oriente:" and in the map accompanying it, Paradise is represented
as an island, a little south-east of Asia, surrounded by rays, and at
some distance from the main land; and in another MS. of the same
library,--a mediA|val encyclopA|dia,--under the word Paradisus is a
passage which states that in the centre of Paradise is a fountain
which waters the garden--that in fact described by Prester John, and
that of which story-telling Sir John Mandeville declared he had
"dronken 3 or 4 sithes." Close to this fountain is the Tree of Life.
The temperature of the country is equable; neither frosts nor burning
heats destroy the vegetation. The four rivers already mentioned rise
in it. Paradise is, however, inaccessible to the traveller on account
of the wall of fire which surrounds it.
Paludanus relates in his "Thesaurus Novus," of course on
incontrovertible authority, that Alexander the Great was full of
desire to see the terrestrial Paradise, and that he undertook his wars
in the East for the express purpose of reaching it, and obtaining
admission into it. He states that on his nearing Eden an old man was
captured in a ravine by some of Alexander's soldiers, and they were
about to conduct him to their monarch, when the venerable man said,
"Go and announce to Alexander that it is in vain he seeks Paradise;
his efforts will be perfectly fruitless; for the way of Paradise is
the way of humility, a way of which he knows nothing. Take this stone
and give it to Alexander, and say to him, 'From this stone learn what
you must think of yourself.'" Now, this stone was of great value and
excessively heavy, outweighing and excelling in value all other gems;
but when reduced to powder, it was as light as a tuft of hay, and as
worthless. By which token the mysterious old man meant, that Alexander
alive was the greatest of monarchs, but Alexander dead would be a
thing of nought.
That strangest of mediA|val preachers, Meffreth, who got into trouble
by denying the Immaculate Conception of the Blessed Virgin, in his
second sermon for the Third Sunday in Advent, discusses the locality
of the terrestrial Paradise, and claims St. Basil and St. Ambrose as
his authorities for stating that it is situated on the top of a very
lofty mountain in Eastern Asia; so lofty indeed is the mountain, that
the waters of the four rivers fall in cascade down to a lake at its
foot, with such a roar that the natives who live on the shores of the
lake are stone-deaf. Meffreth also explains the escape of Paradise
from submergence at the Deluge, on the same grounds as does the Master
of Sentences (lib. 2, dist. 17, c. 5), by the mountain being so very
high that the waters which rose over Ararat were only able to wash the
base of the mountain of Paradise.
The Hereford map of the thirteenth century represents the terrestrial
Paradise as a circular island near India, cut off from the continent
not only by the sea, but also by a battlemented wall, with a gateway
to the west.
Rupert of Duytz regards it as having been situated in Armenia.
Radulphus Highden, in the thirteenth century, relying on the authority
of St. Basil and St. Isidore of Seville, places Eden in an
inaccessible region of Oriental Asia; and this was also the opinion of
Philostorgus. Hugo de St. Victor, in his book "De Situ Terrarum,"
expresses himself thus: "Paradise is a spot in the Orient productive
of all kind of woods and pomiferous trees. It contains the Tree of
Life: there is neither cold nor heat there, but perpetual equable
temperature. It contains a fountain which flows forth in four rivers."
Rabanus Maurus, with more discretion, says, "Many folk want to make
out that the site of Paradise is in the east of the earth, though cut
off by the longest intervening space of ocean or earth from all
regions which man now inhabits. Consequently, the waters of the
Deluge, which covered the highest points of the surface of our orb,
were unable to reach it. However, whether it be there, or whether it
be anywhere else, God knows; but that there _was_ such a spot once,
and that it was on earth, that is certain."
Jacques de Vitry ("Historia Orientalis"), Gervais of Tilbury, in his
"Otia Imperalia," and many others, hold the same views, as to the site
of Paradise, that were entertained by Hugo de St. Victor.
Jourdain de SA"verac, monk and traveller in the beginning of the
fourteenth century, places the terrestrial Paradise in the "Third
India;" that is to say, in trans-Gangic India.
Leonardo Dati, a Florentine poet of the fifteenth century, composed a
geographical treatise in verse, entitled "Della Sfera;" and it is in
Asia that he locates the garden:--
"Asia e le prima parte dove l'huomo
Sendo innocente stava in Paradiso."
But perhaps the most remarkable account of the terrestrial Paradise
ever furnished, is that of the "Eireks Saga VA-dfA¶rla," an Icelandic
narrative of the fourteenth century, giving the adventures of a
certain Norwegian, named Eirek, who had vowed, whilst a heathen, that
he would explore the fabulous Deathless Land of pagan Scandinavian
mythology. The romance is possibly a Christian recension of an ancient
heathen myth; and Paradise has taken the place in it of
GlA"sisvellir.
According to the majority of the MSS. the story purports to be nothing
more than a religious novel; but one audacious copyist has ventured to
assert that it is all fact, and that the details are taken down from
the lips of those who heard them from Eirek himself. The account is
briefly this:--
Eirek was a son of Thrand, king of Drontheim, and having taken upon
him a vow to explore the Deathless Land, he went to Denmark, where he
picked up a friend of the same name as himself. They then went to
Constantinople, and called upon the Emperor, who held a long
conversation with them, which is duly reported, relative to the truths
of Christianity and the site of the Deathless Land, which, he assures
them, is nothing more nor less than Paradise.
"The world," said the monarch, who had not forgotten his geography
since he left school, "is precisely 180,000 stages round (about
1,000,000 English miles), and it is not propped up on posts--not a
bit!--it is supported by the power of God; and the distance between
earth and heaven is 100,045 miles (another MS. reads 9382 miles--the
difference is immaterial); and round about the earth is a big sea
called Ocean." "And what's to the south of the earth?" asked Eirek.
"O! there is the end of the world, and that is India." "And pray where
am I to find the Deathless Land?" "That lies--Paradise, I suppose, you
mean--well, it lies slightly east of India."
Having obtained this information, the two Eireks started, furnished
with letters from the Greek Emperor.
They traversed Syria, and took ship--probably at Balsora; then,
reaching India, they proceeded on their journey on horseback, till
they came to a dense forest, the gloom of which was so great, through
the interlacing of the boughs, that even by day the stars could be
observed twinkling, as though they were seen from the bottom of a
well.
On emerging from the forest, the two Eireks came upon a strait,
separating them from a beautiful land, which was unmistakably
Paradise; and the Danish Eirek, intent on displaying his scriptural
knowledge, pronounced the strait to be the River Pison. This was
crossed by a stone bridge, guarded by a dragon.
The Danish Eirek, deterred by the prospect of an encounter with this
monster, refused to advance, and even endeavored to persuade his
friend to give up the attempt to enter Paradise as hopeless, after
that they had come within sight of the favored land. But the Norseman
deliberately walked, sword in hand, into the maw of the dragon, and
next moment, to his infinite surprise and delight, found himself
liberated from the gloom of the monster's interior, and safely placed
in Paradise.
"The land was most beautiful, and the grass as gorgeous as purple; it
was studded with flowers, and was traversed by honey rills. The land
was extensive and level, so that there was not to be seen mountain or
hill, and the sun shone cloudless, without night and darkness; the
calm of the air was great, and there was but a feeble murmur of wind,
and that which there was, breathed redolent with the odor of
blossoms." After a short walk, Eirek observed what certainly must have
been a remarkable object, namely, a tower or steeple self-suspended in
the air, without any support whatever, though access might be had to
it by means of a slender ladder. By this Eirek ascended into a loft of
the tower, and found there an excellent cold collation prepared for
him. After having partaken of this he went to sleep, and in vision
beheld and conversed with his guardian angel, who promised to conduct
him back to his fatherland, but to come for him again and fetch him
away from it forever at the expiration of the tenth year after his
return to Dronheim.
Eirek then retraced his steps to India, unmolested by the dragon,
which did not affect any surprise at having to disgorge him, and,
indeed, which seems to have been, notwithstanding his looks, but a
harmless and passive dragon.
After a tedious journey of seven years, Eirek reached his native land,
where he related his adventures, to the confusion of the heathen, and
to the delight and edification of the faithful. "And in the tenth
year, and at break of day, as Eirek went to prayer, God's Spirit
caught him away, and he was never seen again in this world: so here
ends all we have to say of him."[38]
The saga, of which I have given the merest outline, is certainly
striking, and contains some beautiful passages. It follows the
commonly-received opinion which identified Paradise with Ceylon; and,
indeed, an earlier Icelandic work, the "Rymbegla," indicates the
locality of the terrestrial Paradise as being near India, for it
speaks of the Ganges as taking its rise in the mountains of Eden. It
is not unlikely that the curious history of Eirek, if not a
Christianized version of a heathen myth, may contain the tradition of
a real expedition to India, by one of the hardy adventurers who
overran Europe, explored the north of Russia, harrowed the shores of
Africa, and discovered America.
Later than the fifteenth century, we find no theories propounded
concerning the terrestrial Paradise, though there are many treatises
on the presumed situation of the ancient Eden. At Madrid was published
a poem on the subject, entitled "Patriana decas," in 1629. In 1662
G. C. Kirchmayer, a Wittemberg professor, composed a thoughtful
dissertation, "De Paradiso," which he inserted in his "DeliciA|
A†stivA|." Fr. Arnoulx wrote a work on Paradise in 1665, full of the
grossest absurdities. In 1666 appeared Carver's "Discourse on the
Terrestrian Paradise." Bochart composed a tract on the subject; Huet
wrote on it also, and his work passed through seven editions, the last
dated from Amsterdam, 1701. The PA"re Hardouin composed a "Nouveau
TraitA(C) de la Situation du Paradis Terrestre," La Haye, 1730. An
Armenian work on the rivers of Paradise was translated by M. Saint
Marten in 1819; and in 1842 Sir W. Ouseley read a paper on the
situation of Eden, before the Literary Society in London.
FOOTNOTES:
[37] St. Brandan was an Irish monk, living at the close of the sixth
century; he founded the Monastery of Clonfert, and is commemorated on
May 16. His voyage seems to be founded on that of Sinbad, and is full
of absurdities. It has been republished by M. Jubinal from MSS. in the
BibliothA"que du Roi, Paris, 8vo. 1836; the earliest printed English
edition is that of Wynkyn de Worde, London, 1516.
[38] Compare with this the death of Sir Galahad in the "Morte
d'Arthur" of Sir Thomas Malory.
THE END.
_The Genius of Solitude._
THE SOLITUDES OF NATURE AND OF MAN; OR, THE LONELINESS OF HUMAN LIFE.
By WM. ROUNSEVILLE ALGER.
CONTENTS.
The Solitudes of Nature.
The Solitudes of Man.
The Morals of Solitude.
Sketches of Lonely Characters: or, Personal Illustrations
of the Good and Evil of Solitude.
Summary of the Subject.
In one handsome volume. 16mo. Cloth. Price $2.00.
"This volume is the result of much investigation, much
meditation, and much experience; and is very comprehensive in
its scope.... The author has shown the influence of solitude
on every grade of mind and character, has discriminated its
beneficent form and its morbid action, and has shown how it
nurtures lofty thoughts as well as how it pampers self-will,
and, in the throng of his personal illustrations, has
indicated its effect on representative men of genius in
almost every department of human effort."--_Boston
Transcript._
"We know of no work like it, and question whether any of its
size has appeared in this generation with an equal amount of
intellectual enrichment and stimulus, moral nutriment, and
invaluable ethical instruction."--_The Liberal Christian._
"This book is a worthy mate to Burton's famous Anatomy of
Melancholy. The fortunate reader may learn from it how to win
the benefits and shun the evils of being alone."--_N. Y.
Express._
"We envy the heart of no one who, unmoved, and with tearless
eye, can read them (The Solitude of the RUIN and the Solitude
of DEATH)."--_West. Missionary._
Mailed, post paid, to any address, on receipt of the price, by the
Publishers,
ROBERTS BROTHERS, Boston.
_Memoirs and Correspondence of Madame RA(C)camier._
Translated and Edited by MISS LUYSTER. 1 vol., 16mo., with a finely
engraved Portrait. Price $2.00.
"The diversified contents of this volume can hardly fail to
gain for it a wide perusal. It has the interest, in a greater
or less degree, of history and romance; of truth stranger
than fiction; of personal sketches; of the curious phases of
an exceptional social life; of singular admixtures of piety
and folly, of greatness and profligacy, fidelity and
intrigue, all mingling or revealed in connection with the
prolonged career of one who was, in certain respects, the
most remarkable woman of her time."--_Boston Transcript._
"With nothing like the talents which immortalized the author
of _Corinne_, Madame RA(C)camier won herself a place of not less
social influence among the men and women of her day. We must
clearly look elsewhere than either to intellect, wealth,
beauty, or all three combined, for the secret of that
witchery which was so distinctive of her. There was
something, we are led to infer, in her constitutional
temperament, which, even beyond her delicate and indefinable
tact, may afford the real clew to much of her mysterious
ascendency. Love seems to have existed in her as a yearning
of the soul almost entirely free from those elements of
passion which are grounded in the difference of the sexes.
There was in it not so much of the desire which centres in a
single object, as of the emotion which seeks to diffuse
itself over the very widest sphere of objects. It could thus
be warm and deep, while pure and inaccessible to evil.
Sainte-Beuve's remark, that she had carried the art of
friendship to perfection, helps us here to give the true key
to her character. A warm and constant friend, she never
admitted, never showed herself, a lover. Satisfied with the
arrangement which gave her from an early age nothing more
than the name and status of a wife, she could let her natural
affection range with freedom and security wherever it met
with a response that left intact her dignity and
self-respect. Such coquetry as she showed arose rather from
an instinctive desire to please and attract, than from
anything approaching to a vicious instinct, or a silly desire
to swell the list of her conquests. What seemed to begin in
flirtation never went to the point of danger, and men who at
first sight loved her passionately usually ended by becoming
her true friends."--_The London Saturday Review._
Mailed, post paid, to any address, by the Publishers,
ROBERTS BROTHERS, Boston.
Transcriber's Note
Archaic spelling is preserved as printed. Variable spelling is also
preserved as printed, where both forms are recognised; for example,
Gervase/Gervais of Tilbury, Sir John Mandeville/Maundevil.
Unk-Khan is given as another name for Prester John. There is one
instance of Un-Khan; however, this is in quoted material, and so is
preserved as printed.
Page 46 includes the phrase, "it was Saterday in Wyttson woke"; the
word 'woke' may be a typographic error for 'weke', but as it cannot
be ascertained for certain, it is preserved as printed.
At page 118, Hemingr is described as throwing a spear rather than
shooting an arrow as challenged. This is presumably an error in the
story, but is preserved as printed.
Page 168 includes "He will rebuild the temple at Jerusalem, and making
the Holy City the great capital of the world." The 'and making' may be
an error for 'and make' or simply 'making'; as it is impossible to be
sure, it is preserved as printed.
Minor punctuation errors have been repaired. Hyphenation and accent
usage have been made consistent.
The following amendments have been made:
Page 21--Labavius amended to Libavius--"... Libavius declares
that he would sooner believe ..."
Page 88--repeated 'a' deleted--"... possibly a little
imaginative, for she wrote not unsuccessfully; ..."
Page 118--it at amended to at it--"... and aim at it from
precisely the same distance."
Page 175--Wolffii amended to Wolfii--"This fragment is
preserved in "Wolfii Lectionum Memorabilium centenarii, XVI.:"
..."
Page 215--omitted word 'on' added--"Helgi and his brother
Thorstein went on a cruise ..."
Page 222--multiplication sign changed to plus--"... but the
sum of the digits 1 + 8 = 9."
The frontispiece illustration has been moved to follow the front
matter. Other illustrations have been moved where necessary so that
they are not in the middle of a paragraph.
Advertising material has been moved from the beginning of the book to
the end.
End of the Project Gutenberg EBook of Curious Myths of the Middle Ages, by
Sabine Baring-Gould
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Q: Storefront UI SfSidebar: Set different visibility for mobile & desktop I'm new to Vue Storefront & Storefront UI.
For my project I have implemented my main menu into an SfSidebar component.
On mobile it should only be visible after clicking on the menu icon.
On desktop it should be visible by default.
This is the beginning of my template for the Sidebar:
<SfSidebar
:visible="isMenuSidebarOpen"
title="Navigation"
class="sf-sidebar--left"
@close="toggleMenuSidebar"
:overlay="toggleOverlay"
:persistent="togglePersistence"
>
As you can see, for visibility it uses "isMenuSidebarOpen" function from useUiState.ts. In useUiState.ts that is set to false by default.
For desktop I want to set it to true by default.
But when I implement something like this into useUiState.ts:
import { reactive, computed } from '@nuxtjs/composition-api';
import {mapMobileObserver} from "@storefront-ui/vue/src/utilities/mobile-observer";
const state = reactive({
isCartSidebarOpen: false,
isWishlistSidebarOpen: false,
isLoginModalOpen: false,
isNewsletterModalOpen: false,
isCategoryGridView: true,
isFilterSidebarOpen: false,
isMobileMenuOpen: false,
isMenuSidebarOpen: false
});
const useUiState = () => {
const isMobile = computed(() => mapMobileObserver().isMobile.get());
const isMobileMenuOpen = computed(() => state.isMobileMenuOpen);
const toggleMobileMenu = () => {
state.isMobileMenuOpen = !state.isMobileMenuOpen;
};
const isMenuSidebarOpen = computed(() => {
if (!isMobile.value) {
return !state.isMenuSidebarOpen
} else {
return state.isMenuSidebarOpen
}
});
I get the following error:
[Vue warn]: You may have an infinite update loop in a component render function.
I understand that this is because I am changing a reactive data property while it is being rendered, so it goes into an infinite loop. But I don't know how else to implement the different visibility-defaults.
Can anyone help me with this?
Thank you!
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{"url":"https:\/\/codereview.stackexchange.com\/questions\/254791\/fizzbuzz-optimisation-modulus-vs-decrement-operator-performance","text":"# FizzBuzz Optimisation | Modulus vs Decrement Operator Performance\n\nThis is not the typical debate in FizzBuzz code about how to handle repeated if statement. I believe that it is a matter of personal preference.\n\nI am here to ask about the choice of repeated mathematical operation.\n\nUsually the solution to a FizzBuzz code is to use the modulus operator on every iteration. However, I'm concerned with its performance impact.\n\nFrom my knowledge, modulus operator is tightly connected with division operator, and has a considerable performance overhead compared to simple increment and decrement operator.\n\nIn addition, I don't believe compiler has the ability to analyse and optimise the use of modulus operator in a loop.\n\nHence, I give the alternative solution that uses only simple decrement operator. The following code are written in C++.\n\nI seek insight on whether my approach will provide performance benefit, or if it is actually a horrible idea that worsen performance.\n\nAnd if there are other ways I can optimize a FizzBuzz code even further. (I think in a modified FizzBuzz puzzle with lots of cases to compare against, a lookup table might be preferable)\n\nTypical FizzBuzz solution.\n\nfor (unsigned int i = 1; i <= 1000000; ++i) {\nif (!(i % 15))\nstd::cout << \"FizzBuzz\\n\";\nelse if (!(i % 3))\nstd::cout << \"Fizz\\n\";\nelse if (!(i % 5))\nstd::cout << \"Buzz\\n\";\nelse\nstd::cout << i << '\\n';\n}\n\n\nMy optimized code.\n\nstatic const unsigned int fa {3};\nstatic const unsigned int fb {5};\n\nunsigned int a {fa};\nunsigned int b {fb};\n\nfor (unsigned int i = 1; i <= 1000000; ++i) {\n--a, --b;\nif (!a && !b) {\na = fa;\nb = fb;\nstd::cout << \"FizzBuzz\\n\";\n} else if (!a) {\na = fa;\nstd::cout << \"Fizz\\n\";\n} else if (!b) {\nb = fb;\nstd::cout << \"Buzz\\n\";\n} else {\nstd::cout << i << '\\n';\n}\n}\n$$$$\n\n\u2022 \"I seek insight on whether my approach will provide performance benefit, or if it is actually a horrible idea that worsen performance.\" You have two horses. Race them. Do you know what a profiler is? \u2013\u00a0Mast Jan 16 at 11:35\n\u2022 @Mast No. What's a profiler? \u2013\u00a0Desmond Rhodes Jan 16 at 11:37\n\u2022 Analysis of a program to find time\/space complexity, frequency and duration of function calls and execution time. Wiki \u2013\u00a0Mast Jan 16 at 11:40\n\u2022 Fully agree with Mast, instead of speculating which is faster, test it! Write both pieces of code run then and measure the time it takes. Optimizing for performance without measuring is like painting your house blind. \u2013\u00a0Emily L. Jan 16 at 13:16\n\nLooks good - you're using only simple integer addition and subtraction.\n\nExcept for the division hidden here:\n\n std::cout << i << '\\n';\n\n\nNote that division by a constant isn't necessarily as expensive as you think: it's always possible for an optimising compiler to implement it in terms of multiplication and bitwise operations. And a test for being an exact multiple is often simpler (think of the well-known algorithms for testing divisibility by 9 or 11 in decimal).\n\nIf you really want to drive up performance, consider storing the integer value as a string (any kind), and implement the ++ operation character by character.\n\nYou may also be interested in High throughput Fizz Buzz over on the Programming Puzzles site.\n\n\u2022 Thanks, the linked thread is crazy, and full of resources to read over. \u2013\u00a0Desmond Rhodes Jan 16 at 11:01\n\u2022 Can you please elaborate on how printing 'i' contain a hidden division? or link me to a resource to read in? Thanks in advance. \u2013\u00a0a.Li Jan 17 at 4:25\n\u2022 @a.Li - I'll turn that around into a challenge for you - convert an integer to a decimal string without library functions. Now do it without any \/ or %`. It's not impossible, but not obvious either. \u2013\u00a0Toby Speight Jan 17 at 9:15","date":"2021-04-18 17:18:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 1, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19225497543811798, \"perplexity\": 2276.1483911780897}, \"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-2021-17\/segments\/1618038507477.62\/warc\/CC-MAIN-20210418163541-20210418193541-00084.warc.gz\"}"} | null | null |
Q: How to initialize model with previous simulation solution? Having run out of ideas again I turn to you.
I'm sure this has to have been answered before though for the life of me I can't find a clear description of it.
Scenario:
You work hard to get a model all set up with initial guesses, etc., and then solve for the 'real' steady state (SS) condition of all parameters in the model. From this solution I want to investigate events effects on the system.
To perform this deviation from SS analysis, it would be expedient to be able to simply load the SS solution .mat file with all the model parameters that is already generated by Dymola. Loading this solution removes any need to first simulate the model to achieve the SS solution before performing the deviations from the SS investigations. I can conceive that a possible step at which this takes place is during the 'Initialization' phase of the simulation.
Given that this would seem to be an important concept, I imagine that this idea would have been considered from a very early stage of Modelica/Dymola development. Manual extraction of each individual parameter does not seem proper given that the size of these models quickly become very large.
Question:
*
*Is it possible to solve for the state of your model (.mat file) and
then load that simulation as the 'initial' values of all variables?
(this would override/replace the previous initial guesses in the
system)
*If it is possible, how is this achieved? (run some script, some
hidden menu option, add some sort of import command in model/simulation?)
I very much appreciate your help.
I am using Dymola 2016.
A: I have found the following answers to my own questions posted above.
Edit: matth also pointed to the following helpful reference on the subject:
http://claytex.com/blog/how-to-restart-a-simulation
These are instructions for Dymola. I cannot verify if they are applicable to other Modelica based programs. I also found some documentation in the Dymola User Manual Volume 1 Section 5.3.3 Simulation > Continue > Import Initial/Continue.
1) Yes it is possible to start your model from the generated .mat file. You can also continue it from the dsfinal.txt file.
2) Below are the steps using the GUI:
*
*Backup the 'YOUR_MODEL.mat' file (e.g. YOUR_MODEL_orig.txt). Once you have run the simulation a new .mat file will be created and you probably don't want to overwrite it by accident.
*Translate your model.
*In the ribbon, go to Simulation > Continue > Import Initial. Select the .mat file from which you want to continue the simulation.
*A prompt will be generated requested a time input with a specified range given. This allows you to continue the simulation at any point in time within the .mat simulation results.
*Go to Simulation Setup and select the new simulation parameters. If you want the solution to register a start time of 0 rather than some large number associated with the Initial SS simulation than update the 'Start time' to 0. This does not affect the initial time used for parameter data from the Model_Sim.mat file.
*Now simulate the model. A new .mat file will be generated starting off from the point you specified.
Alternative method: (an odd thing though with this method no .mat file seems to be generated or updated.)
*
*Backup the 'dsfinal.txt' file (e.g. dsfinal_orig.txt). Once you have run the simulation a new dsfinal.txt file will be created and you probably don't want to overwrite it by accident.
*Translate your model.
*In the ribbon, go to Simulation > Continue > Continue. Simulation will start using whatever dsfinal.txt file is on the path once this option is pressed.
*The simulation will now continue from where the final conditions of the simulation that generated the dsfinal.txt file.
*If it is desired to start at time 0 then the dsfinal.txt file can be modified (line 9, 10, and 12 for me) which corresponds to simulation start (9), stop (10) , and number of timesteps (12). This appears to do the trick.
| {
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} | 3,059 |
xkcd
====
xkcd is a module that exposes the XKCD API to Python.
How to use
----------
Clone this repository inside your working directory and go!
>> import xkcd
>> comic = xkcd.Comic(10)
>> comic.url
https://xkcd.com/10
License
-------
See the LICENSE file.
Copyright (C) 2012 Kostis Karantias <karantiaskostis@gmail.com>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 47 |
Lionel Richie's Latest Instagram Post Talks about His Upcoming Tour
By: Ashmeet Bagga - Published: February 7, 2017 at 10:41 am
Photo: Gary Gershoff / Stringer / Getty Images
Lionel Richie's latest Instagram post has caused excitement among his fans, because in it, he revealed an upcoming tour! Find out more about Lionel Richie's tour in the following article.
Lionel Richie is apparently working extremely hard preparing for his next tour. His latest Instagram photo is proof that the 67-year-old singer is quite excited to perform for fans. Richie captioned the photo of him on stage saying, "see you on the road my friends …"
Lionel Richie is one of the most celebrated artists in the music industry, and has achieved a lot throughout his career. He has sold more than 100 million records worldwide, making him one of the world's best-selling artists of all time. We have Lionel Richie's tour dates for 2017 below. Tickets will sell out fast, so make sure you buy yours soon!
Photo: instagram.com/p/BQM_Fh_DzAZ/
Also Read: Sofia Richie Looks Stunning in Her Latest Instagram Pic
Lionel Richie's Tour Dates
Date Location City
March 15, 2017 Royal Farms Arena Baltimore, MD
March 17, 2017 Prudential Center Newark, NJ
March 18, 2017 Wells Fargo Center Philadelphia, PA
March 21, 2017 PPG Paints Arena Pittsburgh, PA
March 24, 2017 Xcel Energy Center St. Paul, MN
March 25, 2017 United Center Chicago, IL
March 28, 2017 Van Andel Arena Grand Rapids, MI
March 30, 2017 Air Canada Centre Toronto, ON
April 1, 2017 Madison Square Garden New York, NY
April 5, 2017 Quicken Loans Arena Cleveland, OH
April 7, 2017 TD Garden Boston, MA
April 8, 2017 Nassau Coliseum Uniondale, NY
April 11, 2017 The Palace of Auburn Hills Auburn Hills, MI
April 14, 2017 Schottenstein Columbus, OH
April 16, 2017 Sprint Center Kansas City, MO
April 18, 2017 Scottrade Center St. Louis, MO
April 21, 2017 Bok Center Tulsa, OK
April 23, 2017 Pepsi Center Denver, CO
April 25, 2017 Vivint Smart Home Arena Salt Lake City, UT
April 27, 2017 Vancouver, BC Vancouver, BC
April 28, 2017 Key Arena Seattle, WA
April 30, 2017 Oracle Arena Oakland, CA
May 2, 2017 Golden 1 Center Sacramento, CA
May 5, 2017 Hollywood Bowl Los Angeles, CA
May 8, 2017 Viejas Arena San Diego, CA
May 10, 2017 Honda Center Anaheim, CA
May 13, 2017 Toyota Center Houston, TX
May 15, 2017 American Airlines Center Dallas, TX
May 18, 2017 Smoothie King Center New Orleans, LA
May 20, 2017 Bridgestone Arena Nashville, TN
May 21, 2017 Spectrum Center Charlotte, NC
May 24, 2017 Philips Arena Atlanta, GA
May 26, 2017 Amalie Arena Tampa, FL
May 27, 2017 BB&T Center Fort Lauderdale, FL
July 19, 2017 The California Mid-State Fair Paso Robles, CA
If you do not want to miss out on the opportunity to watch one of the greatest singers of all time perform his hits live, make sure you buy tickets for Lionel Richie's tour for a city near you! | {
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Forcepont
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,822 |
Q: Magento 2 category image 404 with store code in url I have a phtml file in my theme that is added to category page via XML updates tab.
I read image in phtml like so:
$this->getUrl('media/catalog/category/').$_category->getImageUrl()
My theme has 2 store view nl-be, fr-be and one of them has store code to url like so:
NL-BE = https://magento2.com/media/catalog/category/image.jpg
NL-FR = https://magento2.com/fr-be/media/catalog/category/image.jpg //THIS IMAGE GOES 404
Anyone any idea?
A: Basically you need to create Symlinks for the same Directory.
If you don't know, how to Symlinks then please refer this link:
*
*For Windows
*For Linux
A: If you have having store codes in URL's then to get the media files correctly working you need to add symlinks in the corresponding directory for your store view for media as follows
ln -s source destination
Example
ln -s ../pub/media media
This if for linux.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,484 |
Forêt Domaniale de Zeralda är en skog i Algeriet. Den ligger i provinsen Tipaza, i den norra delen av landet, km väster om huvudstaden Alger.
Klimatet i området är tempererat. Årsmedeltemperaturen i trakten är °C. Den varmaste månaden är augusti, då medeltemperaturen är °C, och den kallaste är januari, med °C. Genomsnittlig årsnederbörd är millimeter. Den regnigaste månaden är februari, med i genomsnitt mm nederbörd, och den torraste är juli, med mm nederbörd.
Källor
Skogar i Tipaza (provins) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,981 |
Q: Changing variable count on html script Everytime I click a button I try to add 1 to a count variable and that changes the iframe link to its selected number:
const number = document.getElementsByClassName("nextepisode")
const count = 1;
number.addEve = function() {
count+= 2
number.innerHTML = count
}
<iframe id="iframe" src="https://www.2embed.ru/embed/tmdb/tv?id=1421&s=1&e=${count}" width="100%" height="100%" frameborder="0"></iframe>
function prepareFrame () {
const ifrm = document.createElement("iframe")
ifrm.setAttribute("src", `https://www.2embed.ru/embed/tmdb/tv?id=1421&s=1&e=${count}`)
ifrm.style.width = "800"
ifrm.style.height = "600"
ifrm.allowFullscreen = true
ifrm.allow = "autoplay"
ifrm.scrolling = false
ifrm.frameBorder = "0"
document.body.appendChild(ifrm)
}
<body>
<div style="text-align:center">
<button><a class="nextepisode">Next Episode⏭</a></button>
</div>
</body>
</html>
It just shows a blank screen, I'm kind of new to JavaScript so this may be something simple.
A:
const number = document.getElementById("nextepisode")
let count = 1;
const ifrm = document.createElement("iframe")
ifrm.setAttribute("src", `https://www.2embed.ru/embed/tmdb/tv?id=1421&s=1&e=${count}`)
ifrm.style.width = "800"
ifrm.style.height = "600"
ifrm.allowFullscreen = true
ifrm.allow = "autoplay"
ifrm.scrolling = false
ifrm.frameBorder = "0"
document.body.appendChild(ifrm)
function changeNumber() {
count += 2
ifrm.setAttribute("src", `https://www.2embed.ru/embed/tmdb/tv?id=1421&s=1&e=${count}`)
console.log(number)
number.innerHTML = count
}
<!DOCTYPE html>
<html>
<head>
<meta charset='utf-8'>
</head>
<div style="text-align:center">
<button onClick='changeNumber()'>Next <span id="nextepisode"></span></button>
</div>
</html>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 119 |
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"redpajama_set_name": "RedPajamaC4"
} | 1,923 |
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"redpajama_set_name": "RedPajamaC4"
} | 4,771 |
\section{Introduction}
Depth information is essential to numerous computer vision applications, including robotics, mixed reality, and scene understanding. Traditionally, accurate depth measurements are acquired using stereo or multi-view setups~\cite{hartley2003multiple,szeliski2011structure} or active sensors such as ToF cameras, LIDARs. However, deploying such methods for resource-limited devices is costly or may even be infeasible in practice. Considering this, current advances in learning-based monocular depth estimation proffering them as viable alternatives to conventional approaches.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.84\linewidth]{figures/rel_params_chartv2.pdf}
\end{center}
\caption{
Absolute relative error and thresholded accuracy ($\delta_1$) vs.~the number of parameters for recent depth estimation methods on NYU-Depth-v2 (top) and KITTI (bottom) -- the LiDNAS models outperforms the lightweight baselines (black), while using substantially less parameters than the current state-of-the-art methods (blue). Compared to the recent NAS-based depth estimation method (red), LiDNAS improves in both performance and compactness.
} \vspace{-0.45cm}
\label{fig:rel_params_chartv1} %
\end{figure}
Recent deep neural networks (DNN) show compelling results on single image depth estimation by formulating apparent depth cues~\cite{bhat2021adabins,chen2019structure,facil2019cam,Hu2018RevisitingSI,huynh2020guiding,lee2019big,liu2019planercnn,liu2018planenet,ramamonjisoa2019sharpnet,yang2021transformers} or estimating relative depth in unconstrained settings~\cite{chen2016single,lee2019monocular,Ranftl2021}. Moreover, self-supervised methods~\cite{garg2016unsupervised,godard2019digging} offered appealing solutions for single image depth estimation. Nevertheless, most studies focus on increasing accuracy at the expense of model complexity, making them infeasible to devices with limited hardware capabilities.
To tackle this problem, lightweight depth estimation methods~\cite{wofk2019fastdepth} were proposed by utilizing small and straightforward architecture. Usually such simple designs are unreliable and yield low-quality predictions. Other popular strategies include quantizing the weights of a network into low-precision fixed-point operations~\cite{han2015deep} or pruning by directly cut off less important filters~\cite{yang2018netadapt}. That being said, these methods depend on a baseline model, tend to degrade its performance afterward and incapable of exploring new combinations of DNN operations. Moreover, creating a resource-constrained model is a non-trivial task requiring 1) expert knowledge to carefully balance accuracy and resource and 2) plenty of tedious trial-and-error work.
Neural architecture search (NAS), proposed recently~\cite{zoph2016neural,zoph2018learning}, exhibits compelling results, and more importantly, promises to relieve from the manual tweaking of deep neural architectures. Unfortunately, NAS methods mostly obligate thousands of training hours on hundreds of GPUs. To address this, recent NAS studies introduced various efficiency increasing techniques, which include weight sharing~\cite{pham2018efficient}, and network transformation~\cite{elsken2018efficient}. These methods show promising results, but they are still expensive and mainly focus on classification and detection.
This paper introduces LiDNAS, an efficient model compactness-aware NAS framework, with the objective of searching for accurate and lightweight monocular depth estimation architectures.
The approach is based on two main ideas. First, we observe that previous NAS methods essentially search for a few types of cells and then repeatedly accumulate the same cells to build the whole network. Although doing this simplifies the search process, it also restrains layer diversity that is important for computational efficiency. Instead, we construct a pre-defined backbone network that utilizes different layers striving for the right balance between flexibility and search space size. Secondly, we proposed the Assisted Tabu Search (ATS) for efficient neural architecture search. Inspired by the recent NAS study that suggests estimating network performance without training~\cite{mellor2021neural}, we integrate this idea into our multi-objective search function to swiftly evaluate our candidate networks. This, in turn, reduces $\sim 90\%$ search time compared to state-of-the-art NAS-based disparity and depth estimation approaches~\cite{saikia2019autodispnet}.
Figure~\ref{fig:rel_params_chartv1} summarizes a comparison between our LiDNAS models and
other state-of-the-art lightweight approaches. Compared to PyD-Net~\cite{poggi2018towards}, our method improves the REL, RMSE, and thresholded accuracy by $13.6\%$, $8.3\%$, and $3\%$ with similar execution time on the Google Pixel 3a phone (see Table~\ref{tab:runtime_comparison}). Compared to FastDepth~\cite{wofk2019fastdepth} and EDA~\cite{tu2020efficient}, our model achieves higher accuracy with fewer parameters.
To summarize, our work makes the following contributions:
\begin{itemize}[noitemsep,topsep=3pt,parsep=3pt,partopsep=3pt]
\item We propose a multi-objective exploration framework, LiDNAS, searching for accurate and lightweight monocular depth estimation architectures.
\item We introduce a novel scheme called Assisted Tabu Search, enabling fast neural architecture search.
\item We create a well-defined search space that allows computational flexibility and layer diversity.
\item We achieve the state-of-the-art results compared to the lightweight baselines on NYU-Depth-v2, KITTI, and ScanNet while using less parameters.
\end{itemize}
\noindent The implementation of LiDNAS will be made publicly available upon publication of the paper.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.99\linewidth]{figures/search_space_ver2.pdf}
\end{center}
\caption{The search space of our LiDNAS framework. Models are constructed from a pre-defined backbone network containing encoder, decoder, refine, downsample and upsample \textit{blocks} (green). A block is formed by several identical layers (orange) that are generated from a pool of operations and connections. Layers within a block are the same while layers of different blocks can be different.} \vspace{-0.45cm}
\label{fig:search_space_ver2}
\end{figure*}
\begin{figure}[!b]
\begin{center}
\vspace{-0.45cm}
\includegraphics[width=0.99\linewidth]{figures/overview.pdf}
\end{center}
\caption{Overview of the proposed approach.}
\label{fig:overview}
\end{figure}
\section{Related work}
\noindent \textbf{Monocular depth estimation}
Learning-based single image depth estimation was first introduced by Saxena et al.~\cite{saxena2006learning} and Eigen et al.~\cite{eigen2015predicting,eigen2014depth}. Later studies improved accuracy by using large network architectures~\cite{chen2019structure,Hu2018RevisitingSI,laina2016deeper} or integrating semantic information~\cite{jiao2018look} and surface normals~\cite{qi2018geonet}. Fu et al.~\cite{fu2018deep} formulated depth estimation as an ordinal regression problem, while~\cite{chen2016single,lee2019monocular} estimated relative instead of metric depth. Facil et al.~\cite{facil2019cam} proposed to learn camera calibration from the images for depth estimation. Recent approaches further improve the performance by exploiting monocular priors such as planarity constraints~\cite{liu2019planercnn,liu2018planenet,Yin2019enforcing,huynh2020guiding,lee2019big} or occlusion~\cite{ramamonjisoa2019sharpnet}. Gonzalez and Kim~\cite{gonzalezbello2020forget} estimated depth by synthesizing stereo pairs from a single image, while~\cite{yang2021transformers} and~\cite{Ranftl2021} applied vision-transformer for depth prediction. However, these studies mostly focus on increasing accuracy at the cost of model complexity that is infeasible in resource-limited settings.
\noindent \textbf{Lightweight depth estimation architectures} For resource-limited hardware, it is more desirable to not only have a fast but also accurate model. One simple alternative is employing lightweight architectures such as MobileNet~\cite{howard2019searching,howard2017mobilenets,sandler2018mobilenetv2,wofk2019fastdepth}, GhostNet~\cite{han2020ghostnet}, and FBNet~\cite{tu2020efficient}. One popular approach is utilizing network compression techniques, including quantization~\cite{han2015deep}, network pruning~\cite{yang2018netadapt}, and knowledge distillation~\cite{yucel2021real,aleotti2021real}. Other methods employing well-known pyramid networks or dynamic optimization schemes. However, these tasks are tedious, require a lot of trial-and-error, and usually lead to architectures with low accuracy.
\noindent \textbf{Neural Architecture Search} There has been increasing interest in automating network design using neural architecture search. Most of these methods focus on searching high-performance architecture using reinforcement learning~\cite{baker2016designing,liu2018progressive,pham2018efficient,zoph2016neural,zoph2018learning}, evolutionary search~\cite{real2019regularized}, differentiable search~\cite{liu2018darts}, or other learning algorithms~\cite{luo2018neural}. However, these methods are usually very slow and require huge resources for training. Other studies~\cite{dong2018dpp,elsken2018multi,hsu2018monas} also attempt to optimize multiple objectives like model size and accuracy. Nevertheless, their search
process optimizes only on small tasks like CIFAR. In contrast,
our proposed method targets real-world data such as NYU, KITTI and ScanNet.
\section{LiDNAS}
We propose the LiDNAS framework to search for accurate and lightweight monocular depth estimation architectures. The overview the our approach is presented in Figure~\ref{fig:overview}. It takes in a dataset as input to search for the best possible model. This model can be deployed for depth estimation on hardware-limited devices. The first subsection defines the search space while the remaining two describe our multi-objective exploration and search algorithm.
\subsection{Search Space}
Previous neural architecture search (NAS) studies demonstrated the significance of designing a well-defined search space. A common choice of NAS is searching for a small set of complicated cells from a smaller dataset~\cite{zoph2018learning,liu2018progressive,real2019regularized}. These cells are later replicated to construct the entire architecture that hindered layer diversity and suffered from domain differences~\cite{tan2019mnasnet}. On the other hand, unlike classification tasks, dense prediction problems involve mapping a feature representation in the encoder to predictions at larger spatial resolution in the decoder.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.99\linewidth]{figures/naswot_scoring_ver3.pdf}
\end{center}
\caption{Plots of the \textit{score} at initialisation of untrained architectures against evaluation metrics after training:
(a), (b) accuracy ($\delta_{1}$);
(e) mean absolute error of the inverse depth (iMAE);
and (d), (e), (f) absolute relative error (REL). Plots from the first, second and third row are obtained from NYU-Depth-v2, KITTI and ScanNet dataset, respectively.} \vspace{-0.5cm}
\label{fig:naswot_scoring_ver2}
\end{figure}
To this end, we build our search space upon a pre-defined backbone that is shown as the set of green blocks in Figure~\ref{fig:search_space_ver2}. The backbone is divided into multi-scale pyramid networks operating at different spatial resolutions. Each network scale consists of two encoder blocks, two decoder blocks, a refine block, a downsample and a upsample block (except for scale 1). Each block is constructed from a set of identical layers (marked as orange in Figure~\ref{fig:search_space_ver2}). Inspired by~\cite{tan2019mnasnet}, we search for the layer from a pool of operations and connections, including:
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.99\linewidth]{figures/search_algorithm.pdf}
\end{center}
\caption{The flowchart of our architecture search that utilizes the Assisted Tabu Search (ATS) with mutation to search for accurate and lightweight monocular depth estimation networks.} \vspace{-0.45cm}
\label{fig:search_algorithm}
\end{figure*}
\begin{itemize}[noitemsep,topsep=0pt,parsep=1pt,partopsep=1pt]
\item The number of resolution scales $S$.
\item The number of layer for each block $N_{i,j}$.
\item Convolutional operations (ConvOps): vanilla 2D convolution, depthwise convolution, and inverted bottleneck convolution.
\item Convolutional kernel size (KSize): $3 \times 3$, $5 \times 5$.
\item Squeeze and excitation ratio (SER): $0$, $0.25$.
\item Skip connections (SOps): residual or no connection.
\item The number of output channels: $F_{i,j}$.
\end{itemize}
\noindent where $i$ indicates the resolution scale and $j$ is the block index at the same resolution. Internal operations such as ConvOps, KSize, SER, SOps, $F_{i,j}$ are utilized to construct the layer while $N_{i,j}$ determines the number of layer that will be replicated for block$_{i,j}$. In other words, as shown in Figure~\ref{fig:search_space_ver2}, layers within a block (e.g. layers 1 to N$_{1,2}$ of Encoder Block 1,2 are the same) are similar while layers of different blocks (e.g. Layer 1 in Refine Block 1,5 versus Layer 1 in Upsample Block S,7) can be different.
We also perform layer mutation to further diversifying the network structure during the architecture search process. The mutation operations include:
\begin{itemize}[noitemsep,topsep=0pt,parsep=1pt,partopsep=1pt]
\item Swapping operations of two random layers with compatibility check.
\item Modifying a layer with a new valid layer from the predefined operations.
\end{itemize}
Moreover, we also set computational constraints to balance the kernel size with the number of output channels. Therefore, increasing the kernel size of one layer usually results in decreasing output channels of another layer.
Assuming we have a network of $S$ scales, and each block has a sub-search space of size $M$ then our total search space will be $M^{5 + [(S - 1) * 7]}$. Supposedly, a standard case with $M=192$, $S=5$ will result in a search space of size $\sim 2 \times 10^{75}$.
\subsection{Multi-Objective Exploration}
We introduce a multi-objective search paradigm seeking for both accurate and compact architectures. For this purpose, we monitor the \textit{validation grade} $\mathcal{G}$ that formulates both accuracy and the number of parameter of a trained model. It is defined by
\begin{equation}
\label{eq:validation_grade}
\mathcal{G}(m) = \alpha \times A(m) + (1 - \alpha) \times \bigg[\frac{P}{P(m)}\bigg]^{r}
\end{equation}
\noindent where $A(m)$ and $P(m)$ are validation accuracy and the number of parameters of model $m$. $P$ is the target compactness, $\alpha$ is the balance coefficient, and $r$ is an exponent with $r=0$ when $P(m) \leq P$ and otherwise $r=1$. The goal is to search for an architecture $m^{*}$ where $G(m^{*})$ is maximum.
However, computing $G$ requires training for every architecture candidate, resulting in considerable search time. To mitigate this problem, Mellor et al.~\cite{mellor2021neural} suggested to score an architecture at initialisation to predict its performance before training. For a network $f$, the \textit{score(f)} is defined as:
\begin{equation}
\label{eq:jacob_cov_score}
score(f) = log|K_{H}|
\end{equation}
\noindent where $K_{H}$ is the kernel matrix.
Assume the mapping of model $f$ from a batch of data $X = \{x_i\}^{N}_{i=1}$ is $f(x_i)$.
By assigning binary indicators to every activation units in $f$, a linear region $x_i$ of data point $i$ is represented by the binary code $c_i$. The kernel matrix $K_{H}$ is defined as:
\begin{equation}
\label{eq:kernel_matrix}
K_{H} = \begin{pmatrix}
N_{A} - d_{H}(c_1, c_1) & \dots & N_{A} - d_{H}(c_1, c_N) \\
\vdots & \ddots & \vdots \\
N_{A} - d_{H}(c_N, c_1) & \dots & N_{A} - d_{H}(c_N, c_N)
\end{pmatrix}
\end{equation}
\noindent where $N_A$ is the number of activation units, and $d_{H}(c_i, c_j)$ is the Hamming distance between two binary codes. Inspired by this principle, we generate and train a set of different architectures on NYU, KITTI, and ScanNet. We evaluate the performance of these models and visualize the results against the \textit{score} that in our case is the mapping of depth values within image batches. Plots in Figure~\ref{fig:naswot_scoring_ver2} show that models with higher \textit{score} tend to yield better results. Leveraging this observation, we 1) utilize the \textit{score} in our initial network ranking, and 2) define the mutation exploration reward $\mathcal{R}$ as:
\begin{equation}
\label{eq:mutation_exploration_reward}
\mathcal{R}(m_i,m_j) = \alpha \times \frac{score(m_j)}{score(m_i)} + (1 - \alpha) \times \bigg[\frac{P}{P(m_j)}\bigg]^{r}
\end{equation}
\noindent where $m_j$ is a child network that is mutated from $m_i$ architecture.
\subsection{Search Algorithm}
The flowchart of our architecture search is presented in Figure~\ref{fig:search_algorithm}. We first randomly generate $60K$ unique parent models and create the initial network ranking based on the \textit{score} in Eq.~\ref{eq:jacob_cov_score}. We then select \textit{six} architectures in which \textit{three} are the highest-ranked while the other \textit{three} have the highest score of the networks with the size closest to the target compactness.
Starting from these initial networks, we strive for the best possible model utilizing the Assisted Tabu Search (ATS). Tabu search (TS)~\cite{glover1986future} is a high level procedure for solving multicriteria optimization problems. It is an iterative algorithm that starts from some initial feasible solutions and aims to determine better solutions while being designed to avoid traps at local minima.
We propose ATS by applying Eq.~\ref{eq:validation_grade} and~\ref{eq:mutation_exploration_reward} to TS to speed up the searching process. Specifically, we mutate numerous children models ($m_1$, $m_2$, .., $m_n$) from the current architecture ($m_c$). The mutation exploration reward $\mathcal{R}(m_c, m_i)$ is calculated using Eq.~\ref{eq:mutation_exploration_reward}.
ATS then chooses to train the mutation with the highest rewards (e.g. architecture $m_i$ as demonstrated in Figure~\ref{fig:search_algorithm}). The validation grade of this model $\mathcal{G}(m_i)$ is calculated after the training. The performance of the chosen model is assessed by comparing $\mathcal{G}(m_i)$ with $\mathcal{G}(m_c)$. If $\mathcal{G}(m_i)$ is larger than $\mathcal{G}(m_c)$, then $m_i$ is a good mutation, and we opt to build the next generation upon its structure. Otherwise, we swap to use the best option in the tabu list for the next mutation. The process stops when reaching a maximum number of iterations or achieving a terminal condition. The network ranking will be updated, and the search will continue for the remaining parent architectures.
\begin{table*}[t!]
\caption{\label{tab:eval_nyuv2}Evaluation on the NYU-Depth-v2 dataset. Metrics with $\downarrow$ mean lower is better and $\uparrow$ mean higher is better. Type column shows the exploration method used to obtain the model. RL, ATS, and manual, refer to reinforcement learning, assisted tabu search, and manual design, respectively.}
\centering
\small
\begin{tabular}{@{}llrcccccccc@{}}
\hline
\multicolumn{2}{c}{\textbf{Architecture}} & \textbf{\#params} & \textbf{Type} & \textbf{Search Time} & \textbf{REL$\downarrow$} & \textbf{RMSE$\downarrow$} & \(\boldsymbol{\delta_{1}}\)$\uparrow$ & \(\boldsymbol{\delta_{2}}\)$\uparrow$ & \(\boldsymbol{\delta_{3}}\)$\uparrow$ \\ \hline
AutoDepth-BOHB-S & Saikia et al.'19~\cite{saikia2019autodispnet} & 63.0M & RL & 42 GPU days & 0.170 & 0.599 & - & - & - \\ \hline
EDA & Tu et al.'21~\cite{tu2020efficient} & 5.0M & Manual & - & 0.161 & 0.557 & 0.782 & 0.945 & 0.984 \\ \hline
FastDepth & Wofk et al.'19~\cite{wofk2019fastdepth} & 3.9M & Manual & - & 0.155 & 0.599 & 0.778 & 0.944 & 0.981 \\ \hline
SparseSupNet & Yucel et al.'21~\cite{yucel2021real} & 2.6M & Manual & - & 0.153 & 0.561 & 0.792 & 0.949 & 0.985 \\ \hline
Ef+FBNet & Tu \& Wu et al.~\cite{tu2020efficient,wu2019fbnet} & 4.7M & Manual & - & 0.149 & 0.531 & 0.803 & 0.952 & 0.987 \\ \hline
LiDNAS-N & Ours & \textbf{2.1M} & ATS & 4.3 GPU days & \textbf{0.132} & \textbf{0.487} & \textbf{0.845} & \textbf{0.965} & \textbf{0.993} \\ \hline
\end{tabular} \vspace{-0.45cm}
\end{table*}
\section{Performance analysis}
In this section, we evaluate the performance of the proposed method and compare it with several baselines on the NYU-Depth-v2, KITTI, and ScanNet datasets.
\subsection{Datasets} We evaluate the proposed method using NYU-Depth-v2 \cite{silberman2012indoor}, ScanNet~\cite{dai2017scannet} and KITTI~\cite{geiger2013vision} datasets. NYU-Depth-v2 contains $\sim120K$ RGB-D images obtained from 464 indoor scenes. From the entire dataset, we use 50K images for training and the official test set of 654 images for evaluation. The ScanNet dataset comprises of 2.5 million RGB-D images acquired from 1517 scenes. For this dataset, we use the training subset of $\sim20K$ images provided by the Robust Vision Challenge 2018 \cite{robustvision2018} (ROB). In this paper, we report the results on the ScanNet official test set of $5310$ images instead. KITTI is an outdoor driving dataset, where we use the standard Eigen split~\cite{eigen2015predicting,eigen2014depth} for training (39K images) and testing (697 images).
\subsection{Evaluation metrics} The performance is assessed using the standard metrics provided for each dataset. That is, for NYU-Depth-v2 and KITTI we calculate the mean absolute relative error (REL), root mean square error (RMSE), and thresholded accuracy ($\delta_i$). For the ScanNet dataset, we provide the mean absolute relative error (REL), mean square relative error (sqREL), scale-invariant mean squared error (SI), mean absolute error (iMAE), and root mean square error (iRMSE) of the inverse depth values.
\subsection{Implementation Details} \label{implementation_detail}
For searching, we directly perform our architecture exploration on the training samples of the target dataset. We set the target compactness parameter $P$ using the previously published compact models as a guideline. We set the maximum number of exploration iteration to 100 and stop the exploration procedure if a better solution cannot be found after 10 iterations. The total search time required to find optimal architecture is $\sim 4.3$ GPU days.
For training, we use the Adam optimizer~\cite{kingma2014adam} with $(\beta_1, \beta_2, \epsilon) = (0.9, 0.999, 10^{-8})$. The initial learning rate is $7*10^{-4}$, but from epoch 10 the learning is reduced by $5\%$ per $5$ epochs. We use batch size 256 and augment the input RGB and ground truth depth images using random rotation ([-5.0, +5.0] degrees), horizontal flip, rectangular window droppings, and colorization (RGB only).
\subsection{Comparison with state-of-the-art}
\vspace{-5mm}
\noindent \paragraph{NYU-Depth-v2:}
We set the target compactness $P=2M$ with the balance coefficient $\alpha=0.6$ to search for the optimized model on NYU-Depth-v2. We then select the best performance model (LiDNAS-N) and compare its results with lightweight state-of-the-art methods~\cite{tu2020efficient,wofk2019fastdepth,wu2019fbnet,yucel2021real} along with their numbers of parameters. As shown in Table~\ref{tab:eval_nyuv2}, LiDNAS-N outperforms the baseline while containing the least amount of parameters. Comparing with the best-performing approach, the proposed model improves the REL, RMSE, and $\theta_{1}$ by $11.4\%$, $8.2\%$, and $6.8\%$ while compressing the model size by $55\%$. Our method produces high-quality depth maps with sharper details as presented in Figure~\ref{fig:qualitative_nyu}. However, we observe that all methods still struggle in challenging cases, such as the scene containing Lambertian surfaces as illustrated by the example in the third column of Figure~\ref{fig:qualitative_nyu}. Moreover, the proposed method improves REL and RMSE by $22.3\%$ and $18.7\%$ while using only $3\%$ of the model parameters comparing to the state-of-the-art NAS-based disparity and depth estimation approaches~\cite{saikia2019autodispnet}. In addition, our method requires $90\%$ less search time than~\cite{saikia2019autodispnet}.
\begin{table}[b!]
\caption{\label{tab:eval_kitti}Evaluation on the KITTI dataset. Metrics with $\downarrow$ mean lower is better and $\uparrow$ mean higher is better.}
\centering
\small
\adjustbox{width=\columnwidth}{\begin{tabular}{@{}lrccccc@{}}
\hline
\textbf{Method} &\textbf{\#params} & \textbf{REL$\downarrow$} & \textbf{RMSE$\downarrow$} & \(\boldsymbol{\delta_{1}}\)$\uparrow$ & \(\boldsymbol{\delta_{2}}\)$\uparrow$ & \(\boldsymbol{\delta_{3}}\)$\uparrow$ \\ \hline
FastDepth~\cite{wofk2019fastdepth} & 3.93M & 0.156 & 5.628 & 0.801 & 0.930 & 0.971 \\ \hline
PyD-Net~\cite{poggi2018towards} & 1.97M & 0.154 & 5.556 & 0.812 & 0.932 & 0.970 \\ \hline
EQPyD-Net~\cite{cipolletta2021energy} & 1.97M & 0.135 & 5.505 & 0.821 & 0.933 & 0.970 \\ \hline
DSNet~\cite{aleotti2021real} & 1.91M & 0.159 & 5.593 & 0.800 & 0.932 & 0.971 \\ \hline
LiDNAS-K & \textbf{1.78M} & \textbf{0.133} & \textbf{5.157} & \textbf{0.842} & \textbf{0.948} & \textbf{0.980} \\ \hline
\end{tabular}} %
\end{table}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.99\linewidth]{figures/convergence_v2.pdf}
\end{center}
\caption{The progress of different searching scenarios on the NYU-Depth-v2 dataset. From left to right, charts show the accuracy, the number of parameters, and validation grade vs. the number of searching iterations.} \vspace{-0.25cm}
\label{fig:convergence_v2}
\end{figure*}
\vspace{-2mm}
\noindent \paragraph{KITTI:}
In the case of KITTI, we aim at the target compactness of $P=1.5M$ with $\alpha=0.55$. We then train our candidate architectures with the same self-supervised procedure proposed by~\cite{godard2019digging} and adopted by the state-of-the-art approaches~\cite{aleotti2021real,cipolletta2021energy,poggi2018towards,wofk2019fastdepth}. After the search, we pick the best architecture (LiDNAS-K) to compare with the baselines and report the performance figures in Table~\ref{tab:eval_kitti}. The LiDNAS-K model yields competitive results with the baselines while also being the smallest model. We observe that our proposed method provides noticeable improvement from PyD-Net and EQPyD-Net. Examples from Figure~\ref{fig:qualitative_kitti} show that the predicted depth maps from LiDNAS-K are more accurate and contain fewer artifacts.
\noindent \paragraph{ScanNet:}
For ScanNet, we set the target compactness to $4.5M$ with $\alpha=0.57$ for searching. Despite of being compact, our best performance model (LiDNAS-S) produces competitive results compared with state-of-the-art methods, as shown in Table~\ref{tab:eval_scannet}. More specifically, it requires only $20\%$ of the number of parameters in comparison with the best performance baseline.
We also observe that although SARPN~\cite{chen2019structure} and Hu et al.~\cite{Hu2018RevisitingSI} models are multiple times larger than DS-SIDENet~\cite{ren2019deep} or DAV~\cite{huynh2020guiding}, the latter still yield better results, emphasizing the importance of optimal network structure. Furthermore, our model produces comparable depth maps as shown in Figure~\ref{fig:qualitative_scannet}. Details of the generated architectures are provided in the supplementary material.
\begin{figure}[!b]
\begin{center}
\vspace{-2mm}
\includegraphics[width=0.87\linewidth]{figures/qualitative_kitti_v2.pdf}
\end{center}
\caption{Comparison on the Eigen split of KITTI.
(a) input image,
(b) LiDNAS-K,
(c) DSNet~\cite{aleotti2021real},
(d) PyD-Net~\cite{poggi2018towards},
and (e) FastDepth~\cite{wofk2019fastdepth}.
Images in the right column presented zoom-in view for better visualization.}
\label{fig:qualitative_kitti}
\end{figure}
\begin{table}[b!]
\caption{\label{tab:eval_scannet}Evaluation results on ScanNet \cite{dai2017scannet} dataset.}
\centering
\small
\adjustbox{width=\columnwidth}{\begin{tabular}{@{}lrccccc@{}}
\hline
\textbf{Architecture} & \textbf{\#params} & \textbf{REL} & \textbf{sqREL} & \textbf{SI} & \textbf{iMAE} & \textbf{iRMSE} \\ \hline
SARPN~\cite{chen2019structure} & 210.3M & 0.134 & 0.077 & \textbf{0.015} & 0.093 & 0.100 \\ \hline
Hu et al.~\cite{Hu2018RevisitingSI} & 157.0M & 0.139 & 0.081 & 0.016 & 0.100 & 0.105 \\ \hline
DS-SIDENet~\cite{ren2019deep} & 49.8M & 0.133 & \textbf{0.057} & - & - & - \\ \hline
DAV~\cite{huynh2020guiding} & 25.1M & 0.118 & \textbf{0.057} & \textbf{0.015} & \textbf{0.089} & \textbf{0.097} \\ \hline
LiDNAS-S & \textbf{5.2M} & \textbf{0.117} & 0.059 & \textbf{0.015} & 0.090 & \textbf{0.097} \\ \hline
\end{tabular}}
\end{table}
\vspace{-2mm}
\noindent \paragraph{Runtime Measurement:}
We also compare the runtime of our models with state-of-the-art lightweight methods on an Android device using the app from the Mobile AI benchmark developed by Ignatov et al.~\cite{ignatov2021fast}. To this end, we utilize the pre-trained models provided by the authors (Tensorflow~\cite{poggi2018towards}, PyTorch~\cite{wofk2019fastdepth}) and convert them to \textit{tflite}. Unfortunately, we can only report the measurement on the mobile CPU due to the technical issues occurring when converting PyTorch models to TFLite GPU delegate. That being said, the results in Table~\ref{tab:runtime_comparison} suggest that the proposed approaches produce competing performance, with the potential of running real-time on mobile devices with further optimization.
\begin{table}[!b]
\caption{\label{tab:runtime_comparison}Average runtime comparison of the proposed method and other lightweight models. Runtime values are measured using a Pixel 3a phone with input image resolution ($640 \times 480$).}
\centering
\small
\begin{tabular}{lc}
\hline
\textbf{Architecture} & \textbf{CPU(ms)} \\ \hline
FastDepth~\cite{wofk2019fastdepth} & 458 \\ \hline
Ef+FBNet~\cite{tu2020efficient,wu2019fbnet} & 852 \\ \hline
PyD-Net~\cite{poggi2018towards} & 226 \\ \hline
LiDNAS-K & \textbf{205} \\ \hline
LiDNAS-N & 262 \\ \hline
LiDNAS-S & 380 \\ \hline
\end{tabular}
\end{table}
\subsection{Ablation studies}
\noindent \paragraph{Exploration Convergence:}
We experiment with various settings for the multi-objective balance coefficient ($\alpha$) to assess its effect on the performance. For this purpose, we perform the architecture search with $\alpha$ set to $0.0$, $0.4$, $0.5$, $0.6$, and $1.0$ while the target compactness $P=2.0M$. Figure~\ref{fig:convergence_v2} presents the searching progress for accuracy (left), the number of parameters (center), and validation grade (right) from one parent architecture on NYU-Depth-v2. We observe that, scenario with $\alpha=0.0$ quickly becomes saturated as it only gives reward to the smallest model. Searching with $\alpha=0.4$ favors models with compact size but also with limited accuracy. The case with $\alpha=0.5$ provides a more balanced option, but accuracy is hindered due to fluctuation during searching. The exploration with $\alpha=1.0$ seeks for the network with the best accuracy yet producing significantly larger architecture while the case where $\alpha=0.6$ achieves promising accuracy although with slightly bigger model than the target compactness.
\vspace{-2mm}
\noindent \paragraph{Searching Scenarios:}
To further analyze the outcome of different searching scenarios, we perform architecture searches for \textit{six} parent networks in five settings with $\alpha=0.0, 0.4, 0.5, 0.6, 1.0$ and $P=2.0M$ on NYU-Depth-v2. Results in Figure~\ref{fig:searching_scenarios} show that best performance models in case $\alpha=0.5$ are more spread out, while training instances with $\alpha=0.6$ tend to produce both accurate and lightweight architectures. This, in turn, emphasizes the trade-off between validation accuracy and network size.
\begin{figure}[!b]
\begin{center}
\vspace{-0.35cm}
\includegraphics[width=0.65\linewidth]{figures/searching_scenarios.pdf}
\end{center}
\vspace{-0.35cm}
\caption{Trade-off between accuracy vs. the number of parameters of best models trained with different searching scenarios on NYU-Depth-v2 dataset.}
\label{fig:searching_scenarios}
\end{figure}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.935\linewidth]{figures/qualitative_nyu_v1.pdf}
\end{center}
\caption{Comparison on the NYU test set.
(a) input image,
(b) ground truth,
(c) LiDNAS-N,
(d) Ef+FBNet~\cite{tu2020efficient,wu2019fbnet},
and (e) FastDepth~\cite{wofk2019fastdepth}.}
\label{fig:qualitative_nyu}
\end{figure*}
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.935\linewidth]{figures/qualitative_scannet_v1.pdf}
\end{center}
\caption{Comparison on the ScanNet test set.
(a) input image,
(b) ground truth,
(c) LiDNAS-S,
(d) DAV~\cite{huynh2020guiding},
and (e) SARPN~\cite{chen2019structure}.}
\label{fig:qualitative_scannet}
\end{figure*}
\section{Conclusion}
This paper proposed a novel NAS framework to construct lightweight monocular depth estimation architectures using Assisted Tabu Search and employing a well-defined search space for balancing layer diversity and search volume. The proposed method achieves competitive accuracy on three benchmark datasets while running faster on mobile devices and being more compact than state-of-the-art handcrafted and automatically generated models. Our work provides a potential approach towards optimizing the accuracy and the network size for dense depth estimation without the need for manual tweaking of deep neural architectures.
{\small
\bibliographystyle{ieee_fullname}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,536 |
{"url":"https:\/\/electronics.stackexchange.com\/questions\/202302\/measuring-bjt-ebers-moll-parameters","text":"# Measuring BJT (Ebers-Moll) parameters\n\nI am currently attempting to measure the parameters of a \"vintage\" germanium BJT in the hope of creating an Ebers-Moll model from the data. So far I have been unable to find a comprehensive source on the topic as it is slightly out of the region of my expertise.\n\nSo far my primary source of advice has come from \"and yet another Definitive Handbook of Transistor Modeling\" which is fantastic, but has no author, no date, and some of the pages are shuffled.\n\nIn search of a more up to date source I have been using the doctoral thesis of Martin Linder, \"DC Parameter Extraction and Modeling of Bipolar Transistors\". This goes too in depth in some areas and does not cover the basics as they are probably not worthy of writing in a thesis.\n\nThis did however lead me to the original paper by Gummel and Poon, \"An integral charge control model of bipolar transistors\" (ran out of links). This has lots of relevant information but is written like a research paper so not very useful when trying to practically apply it.\n\nHas this information been covered in a text book anywhere? A rigorous method of extracting $I_S,\\; \\beta_F,\\; \\beta_R,\\; N_F,\\; N_R$?\n\n\u2022 nxp.com\/wcm_documents\/models\/bipolar-models\/mextram\/\u2026 That's one of the most sophisticated BJT models, by the way. \u2013\u00a0Fizz Nov 23 '15 at 16:39\n\u2022 Also for GP: ftp.elo.utfsm.cl\/~lsb\/elo102\/ejercicios\/GP_DOCU.pdf People who write such detailed guides don't extract E-M, because it's not used. GP is used in SPICE. And they write the guide assuming you're having some equipment like curve tracers and you can run IC-CAP. If you want to do it the old-fashioned way... you're going to bite the bullet and read old fashioned papers. \u2013\u00a0Fizz Nov 23 '15 at 17:11\n\u2022 Since GP reduces to EM you could extract just what you care about, but I don't think I can find any guide [written in the last 30 years] that does't involve equipment you probably don't have (or you wouldn't be asking this question). Maybe some graduate textbook on semiconductor fabrication would cover the theory of the parameter extraction at a basic\/textbook level. I don't see undergraduate textbooks bothering with this because you buy stuff with datasheet at that level. \u2013\u00a0Fizz Nov 23 '15 at 17:23\n\u2022 This book has chapter on it, but it's somewhat focused on RF. It's wort reading at least that intro page though. \u2013\u00a0Fizz Nov 23 '15 at 17:29\n\u2022 Thanks for all the links @RespawnedFluff that's exactly what I was looking for. I'm using a DAQ from to measure $V_{BE}$ and $V_{CE}$ with known resistances between those voltages and the sources to calculate the currents. The resolution is not perfect but it's enough to get a good approximation. \u2013\u00a0loudnoises Nov 24 '15 at 11:59\n\n# Ebers-Moll Model\n\n\\begin{align} I_{\\mathrm{b}} & = \\frac{I_{\\mathrm{s}}}{\\beta_{\\mathrm{f}}}\\left(\\mathrm{e}^{\\frac{V_{\\mathrm{eb}}}{NV_{\\mathrm{t}}}} - 1\\right) + \\frac{I_{\\mathrm{s}}}{\\beta_{\\mathrm{r}}}\\left(\\mathrm{e}^{\\frac{V_{\\mathrm{eb}} - V_{\\mathrm{ec}}}{NV_{\\mathrm{t}}}} - 1\\right) \\\\[0.9em] I_{\\mathrm{c}} &= I_{\\mathrm{s}}\\left(\\mathrm{e}^{\\frac{V_{\\mathrm{eb}}}{NV_{\\mathrm{t}}}} - 1\\right) - I_{\\mathrm{s}}\\frac{\\beta_{\\mathrm{r}} + 1}{\\beta_{\\mathrm{r}}}\\left(\\mathrm{e}^{\\frac{V_{\\mathrm{eb}} - V_{\\mathrm{ec}}}{NV_{\\mathrm{t}}}} - 1\\right). \\end{align}\n\nModel parameters:\n\n\u2022 Saturation current $I_\\mathrm{s}$\n\u2022 Ideality factor $N$\n\u2022 Thermal voltage $V_\\mathrm{t}$\n\u2022 Forward gain $\\beta_\\mathrm{f}$\n\u2022 Reverse gain $\\beta_\\mathrm{r}$\n\n# Direct parameter extraction\n\nThe most straight-forward way of finding parameter values for the Ebers-Moll model from measurements is using direct extraction.\n\nThe below figure illustrates a Forward Gummel measurement of the 2N3906 BJT, which is when $V_\\mathrm{ec}$ is kept a constant potential and $V_\\mathrm{eb}$ is swept over a range. For this figure $V_\\mathrm{ec} = 0.3\\ \\text{V}$. (As an aside I chose this badly as you want the value of $V_\\mathrm{eb} - V_\\mathrm{ec}$ to be small for the Ebers-Moll model to be a good approximation).\n\nIn the middle of the measurement we see an ideal region which appears linear but is actually exponential as the current is on a logarithmic axis. The Ebers-Moll model only captures the exponential behaviour of the BJT and not the high and low-current regions, so the parts at the top and bottom can be ignored. From the middle of the measurement 3 out of 4 parameters can be determined.\n\nThe gradient of the ideal region controlled by the parameter combination $N V_\\mathrm{t}$. Inspecting the Ebers-Moll equations when biased in the forward-active region (i.e. $V_\\mathrm{eb} - V_\\mathrm{ec}$ is negative) we can see $$I_\\mathrm{c} = I_\\mathrm{s} \\mathrm{e}^\\frac{V_\\mathrm{eb}}{NV_\\mathrm{t}}$$ where $NV_\\mathrm{t}$ is the gradient of the exponential current with respect to $V_\\mathrm{eb}$. The thermal voltage can be found by measuring the junction temperature during the current measurements and applying this formula.\n\nAlso from this equation we can see that when the exponent is 0, $I_\\mathrm{c} = I_\\mathrm{s}$. When $V_\\mathrm{eb}$ approaches 0 however the BJT does not behave ideally, so to find $I_\\mathrm{s}$ we extrapolate from the ideal region.\n\nFinally the gain can be found from the approximation $I_\\mathrm{c} = I_\\mathrm{b}\\beta_\\mathrm{f}$, i.e. by selecting a point in the ideal region and finding the multiplicative difference.\n\nFor reverse measurements $V_\\mathrm{eb}$ is simply made negative which reveals a similar plot but with the reverse gain factor $\\beta_\\mathrm{r}$.\n\n# References\n\nIan Getreu, 'Modeling the Bipolar Transistor', 1976, Tektronix\n\nThe main problem you will find when trying to get the parameters of a BJT is that they are quite dependent on many variable conditions, as temperature. However, there is no problem to get those values in a given time.\n\nThe simplest way to get forward beta is to provide a current smaller than the saturation one in active mode, so measuring the current in base and emitter, you can directly calculate beta:\n\nbeta =Iemitter\/Ibase - 1\n\nThe saturation current is obtained just in the point where the result of the equation isn\u00b4t constant anymore and starts to decrease.\n\nI have never used BJT in reverse mode, but those parameters will probably be calculated in an analog way.\n\n\u2022 I have been attempting to use a DS18B20 to approximate the temperature, which I know fails as it is only the external temperature, but I am hoping it puts it into the right area. The other methods I have been using are extracting gradients from $V_{CE}$ vs. $I_C$ and $V_{BE}$ vs. $I_B$. I was just hoping there was a textbook that might cover the area. \u2013\u00a0loudnoises Nov 23 '15 at 16:26\n\u2022 The book Microelectronic Circuits, from Sedra and Smith, has a whole chapter dedicated to BJTs and thereis explained the Ebers-Moll model \u2013\u00a0Zero point Nov 23 '15 at 16:50","date":"2020-04-04 19:18:14","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 1, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6765692830085754, \"perplexity\": 942.6551245222477}, \"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-2020-16\/segments\/1585370524604.46\/warc\/CC-MAIN-20200404165658-20200404195658-00250.warc.gz\"}"} | null | null |
#ifndef THREAD_H
#define THREAD_H
#include <pthread.h>
#include "mutex.h"
class Thread : public Mutex
{
public:
Thread();
virtual ~Thread();
void start();
void waitForFinish();
static void delay(const unsigned int us);
protected:
virtual void run() = 0;
protected:
pthread_t MyThread;
private:
static void* startRoutine(void* object);
};
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,480 |
My first book of baseball / by Beth Bugler and Mark Bechtel ; illustrations by Bill Hinds.
Bugler, Beth, (author.). Bechtel, Mark, (author.). Hinds, Bill, 1950- (illustrator.).
Physical Description: 1 volume (unpaged) : color illustrations ; 25 x 28 cm.
Publisher: New York, New York : Liberty Street, an imprint of Time Inc. Books, c2016.
Explains baseball concepts to young readers, including common baseball terms such as strikes, outs, steals, foul balls, home runs and more.
Subject: Baseball > Juvenile literature. | {
"redpajama_set_name": "RedPajamaC4"
} | 457 |
{"url":"https:\/\/mailmanbroy.informatik.tu-muenchen.de\/pipermail\/isabelle-dev\/2018-September\/016521.html","text":"[isabelle-dev] Frag \/ Poly_Mapping\n\nAlexander Maletzky alexander.maletzky at risc.jku.at\nMon Sep 24 11:30:56 CEST 2018\n\nSome notions defined in \"Frag.thy\" already exist in \"Poly_Mapping.thy\":\n\"support\" is called \"keys\" there, and I think \"frag_cmul\" could easily\nbe defined in terms of \"map\".\n\n\"frag_extend\" looks like a special case of a more general subsitution\nhomomorphism \"subst\" of type \"('a => 'b => 'c) => ('a =>_0 'b) => 'c\",\ndefined as \"subst f x = (\\Sum i\\in keys x. f i (lookup x i))\", which\ncould indeed be added to \"Poly_Mapping.thy\". The insertion morphism in\n\"MPoly_Type.thy\" could then perhaps be defined in terms of \"subst\" (at\nleast partially; for power-products, the above sum would have to be\nreplaced by a product).\n\nBest regards,\nAlexander\n\nOn 9\/23\/18 20:59, Lawrence Paulson wrote:\n> Attached is a port of the HOL Light \u201cfrag\u201d library (free Abelian groups) built upon Poly_Mapping. It\u2019s a mess, especially with the combination of frag and Poly_Mapping names. Some of it clearly could be added to Poly_Mapping, maybe all of it. But it needs to be rationalised.\n>\n>\n> Larry\n>\n>\n> _______________________________________________\n> isabelle-dev mailing list\n> isabelle-dev at in.tum.de\n> https:\/\/mailmanbroy.informatik.tu-muenchen.de\/mailman\/listinfo\/isabelle-dev\n-------------- next part --------------\nAn HTML attachment was scrubbed...\nURL: <https:\/\/mailman46.in.tum.de\/pipermail\/isabelle-dev\/attachments\/20180924\/8c896426\/attachment-0002.html>","date":"2019-09-18 07:45:38","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\": 0.905832052230835, \"perplexity\": 14163.25411658255}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573258.74\/warc\/CC-MAIN-20190918065330-20190918091330-00128.warc.gz\"}"} | null | null |
package org.nd4j.resources;
import org.apache.commons.io.FileUtils;
import org.apache.commons.io.IOUtils;
import org.apache.commons.io.LineIterator;
import org.junit.Rule;
import org.junit.Test;
import org.junit.rules.TemporaryFolder;
import org.nd4j.config.ND4JSystemProperties;
import org.nd4j.resources.strumpf.ResourceFile;
import org.nd4j.resources.strumpf.StrumpfResolver;
import java.io.BufferedReader;
import java.io.File;
import java.io.FileReader;
import java.io.Reader;
import java.nio.charset.StandardCharsets;
import static org.junit.Assert.assertEquals;
import static org.junit.Assert.assertTrue;
public class TestStrumpf {
@Rule
public TemporaryFolder testDir = new TemporaryFolder();
@Test
public void testResolvingReference() throws Exception {
File f = Resources.asFile("big/raw_sentences.txt");
assertTrue(f.exists());
System.out.println(f.getAbsolutePath());
try(Reader r = new BufferedReader(new FileReader(f))){
LineIterator iter = IOUtils.lineIterator(r);
for( int i=0; i<5 && iter.hasNext(); i++ ){
System.out.println("LINE " + i + ": " + iter.next());
}
}
}
@Test
public void testResolvingActual() throws Exception {
File f = Resources.asFile("data/irisSvmLight.txt");
assertTrue(f.exists());
//System.out.println(f.getAbsolutePath());
int count = 0;
try(Reader r = new BufferedReader(new FileReader(f))){
LineIterator iter = IOUtils.lineIterator(r);
while(iter.hasNext()){
String line = iter.next();
//System.out.println("LINE " + i + ": " + line);
count++;
}
}
assertEquals(12, count); //Iris normally has 150 examples; this is subset with 12
}
@Test
public void testResolveLocal() throws Exception {
File dir = testDir.newFolder();
String content = "test file content";
String path = "myDir/myTestFile.txt";
File testFile = new File(dir, path);
testFile.getParentFile().mkdir();
FileUtils.writeStringToFile(testFile, content, StandardCharsets.UTF_8);
System.setProperty(ND4JSystemProperties.RESOURCES_LOCAL_DIRS, dir.getAbsolutePath());
try{
StrumpfResolver r = new StrumpfResolver();
assertTrue(r.exists(path));
File f = r.asFile(path);
assertTrue(f.exists());
assertEquals(testFile.getAbsolutePath(), f.getAbsolutePath());
String s = FileUtils.readFileToString(f, StandardCharsets.UTF_8);
assertEquals(content, s);
} finally {
System.setProperty(ND4JSystemProperties.RESOURCES_LOCAL_DIRS, "");
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,851 |
{"url":"https:\/\/tex.stackexchange.com\/questions\/251463\/parametric-surfaces-with-dependent-parameters","text":"# Parametric surfaces with dependent parameters\n\nI'm trying to create an image that shows (for example) the portion of a standard cylinder (\\cos(v),\\sin(v),u) that lies between the planes z=0 and z+y=5. Generating the cylinder itself is not an issue, but I would need to set the bounds for u from 0 to 5-\\sin(v), which pstricks seems to hate; it gives me an Invalid argument error. Is there some simple way around this that I'm missing?\n\nEdit: By request, what I'm trying right now thanks to Herbert. At this point I get no errors, but it gets hung up indefinitely in the compile:\n\n\\documentclass[pstricks]{standalone}\n\\usepackage{pst-solides3d}\n\n\\begin{document}\n\\begin{pspicture}(-10,-10)(10,15)\n\\psset{\nunit=1.0,\nDecran=50,\nviewpoint=15 55 35 rtp2xyz,\nlightsrc=20 60 20 rtp2xyz,\nngrid=50 50,\ngrid,\nresolution=720\n}\n\n\\defFunction[algebraic]{cylinder}(u,v){cos(v)}{sin(v)}{u}\n\n\\psSolid[object=surfaceparametree,\nbase=0 2 pi pi neg,\nfunction=cylinder,\nincolor=red!30,\nfillcolor=red!30,\nopacity=0.4,\nplansepare={[0 0 1 -1]},\nname=partiescylindre,\n]\n\n\\axesIIID(0,0,0)(2,2,3)\n\n\\end{pspicture}\n\\end{document}\n\n\u2022 I remember a graphic very similar, that your case, in this page; but using tikz or pgfplots. But I don't remember the link nor tags. :( \u2013\u00a0juanuni Jun 22 '15 at 2:54\n\u2022 give an example what you already tried. \u2013\u00a0user2478 Jun 22 '15 at 5:53\n\u2022 @Herbert: I'm on a mobile right now, but as I said the cylinder draws just fine with u from 0 to 5 and v from pi to pi neg. But if I change the 5 to anything involving a v, I get an Invalid argument error. \u2013\u00a0Paul Jun 22 '15 at 6:03\n\u2022 your example dosn't hang, but it needs in fact of ngrid=50 50, a lot of time for the calculations. However, in the end there is a postscript error. I must have a look at it \u2013\u00a0user2478 Jun 24 '15 at 8:38\n\u2022 @Herbert: You're right, I changed it to ngrid=5 30 and it loads much faster. I must have just been impatient with my slow machine. I see the error now too; it goes away if I comment out the plansepare line. I've seen this error before when fooling around and using a plane that doesn't intersect the surface at all, but that shouldn't be the case this time. \u2013\u00a0Paul Jun 25 '15 at 0:56\n\nDon't know if you are looking for something like this:\n\n\\documentclass[pstricks]{standalone}\n\\usepackage{pst-solides3d}\n\n\\begin{document}\n\\begin{pspicture}(-6,-5)(6,8)\n\\psset{lightsrc=viewpoint,viewpoint=100 30 40 rtp2xyz,Decran=100}\n\\psSolid[object=grille,base=-4 4 -4 4,ngrid=8](0,0,-1)\n\\defFunction[algebraic]{G9}(t){1*cos(t)}{1*sin(t)}{1*sin(5*t)}\n\\psSolid[object=cylindre,range=0 6.28,h=5,function=G9,axe=0 0 1,\nincolor=green!50,fillcolor=yellow!50,linewidth=0.01,ngrid=10 72](0,-3,0)\n\\psSolid[object=cylindre,r=1,h=5,incolor=green!50,fillcolor=yellow!50,\nngrid=5 36](0,3,0)\n\\psSolid[object=plan,definition=equation,opacity=0.5,args={[0 1 1 -5] 180},\nplanmarks,plangrid,base=-2 2 -2 2,showBase]\n\\psSolid[object=cylindre,range=0 6.28,h=5,function=G9,axe=0 0 1,\nincolor=green!50,fillcolor=yellow!50,plansepare={[0 1 1 -5]},\nname=partiescylindre,\nlinewidth=0.01,ngrid=10 72]\n\\end{pspicture}\n\\end{document}\n\n\n\u2022 YES, the plansepare command is exactly what I was looking for. I notice that you use a pre-defined cylinder function. do you know if it will work for general parametric surfaces? I included some code in an edit above; without the plansepare and name parameters it works just fine, but with them it seems to get hung up indefinitely. I get no errors, it just doesn't finish compiling. \u2013\u00a0Paul Jun 23 '15 at 20:58","date":"2019-06-17 17:19:34","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\": 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\": 0.8071516752243042, \"perplexity\": 1014.0556892435482}, \"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-2019-26\/segments\/1560627998513.14\/warc\/CC-MAIN-20190617163111-20190617185111-00483.warc.gz\"}"} | null | null |
using System;
using Newtonsoft.Json;
using VkNet.Utils.JsonConverter;
namespace VkNet.Model.RequestParams;
/// <summary>
/// Список параметров для метода prettyCards.create
/// </summary>
[Serializable]
public class PrettyCardsCreateParams
{
/// <summary>
/// Идентификатор сообщества.
/// </summary>
[JsonProperty("owner_id")]
public long? OwnerId { get; set; }
/// <summary>
/// Фотография карточки.
/// Используйте значение, полученное после загрузки фотографии на сервер.См. метод prettyCards.getUploadURL.
/// Также можно переиспользовать существующую фотографию из другой карточки.
/// Используйте значение поля photo, которое возвращает метод prettyCards.get или prettyCards.getById.
/// </summary>
[JsonProperty("photo")]
public string Photo { get; set; }
/// <summary>
/// Заголовок.
/// </summary>
[JsonProperty("title")]
public string Title { get; set; }
/// <summary>
/// Ссылка.
/// Кроме http(s)-ссылок также допускается указание телефонных номеров в виде tel:+79111234567
/// </summary>
[JsonProperty("link")]
public string Link { get; set; }
/// <summary>
/// Цена.
/// «0» будет отображён как «Бесплатно».
/// Не передавайте этот параметр, чтобы не указывать цену.
/// </summary>
[JsonProperty("price")]
public string Price { get; set; }
/// <summary>
/// Старая цена. Отображается зачёркнутой.
/// «0» будет отображён как «Бесплатно».
/// Не передавайте этот параметр, чтобы не указывать старую цену.
/// </summary>
[JsonProperty("price_old")]
public string PriceOld { get; set; }
/// <summary>
/// Кнопка.
/// Не передавайте этот параметр, чтобы не использовать кнопку.
/// </summary>
[JsonProperty("button")]
[JsonConverter(typeof(SafetyEnumJsonConverter))]
public Enums.SafetyEnums.Button Button { get; set; }
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,147 |
Q: 3.2.0-29.46-generic-pae kernel boot problem After updating my kernel to 3.2.0-27 I had problems getting into the system. It was working with the 3.2.0-26 though. I deleted the new kernel and everything was working fine until I started to have the same problem with the 3.2.0-26 too, after a minor update.
Every time I was getting a black screen and when I was force shutting down the PC the error message Broken pipe could be read for a fraction of a second. Then I thought maybe the new kernel was fixed and searched for an update and I saw the 3.2.0-29, which has the same problem.
I read some other posts with similar problems and, it's not from any nVidia driver nor ATI, I have the default VGA driver. The only option on booting up the system now (with any other kernel) is entering "Recovery mode" and then simply "resume" to normal boot which is unpleasant since I have to stay at the PC until the system boots up.
Edit/Add
Still not working after running boot-repair. This is what I got in the terminal:
"Traceback (most recent call last):
File "/usr/bin/glade2script", line 2339, in set_widget
exec( arg )
File "", line 1, in
File "/usr/lib/python2.7/dist-packages/gi/types.py", line 43, in function
return info.invoke(*args, **kwargs)
gi._glib.GError: Failed to open file 'boot-repair.png': No such file or directory
Error in sys.excepthook:
Traceback (most recent call last):
File "/usr/lib/python2.7/dist-packages/apport_python_hook.py", line 68, in apport_excepthook
binary = os.path.realpath(os.path.join(os.getcwdu(), sys.argv[0]))
OSError: [Errno 2] No such file or directory
Original exception was:
Traceback (most recent call last):
File "/usr/bin/glade2script", line 2339, in set_widget
exec( arg )
File "", line 1, in
File "/usr/lib/python2.7/dist-packages/gi/types.py", line 43, in function
return info.invoke(*args, **kwargs)
gi._glib.GError: Failed to open file 'boot-repair.png': No such file or directory
Traceback (most recent call last):
File "/usr/bin/glade2script", line 2339, in set_widget
exec( arg )
File "", line 1, in
File "/usr/lib/python2.7/dist-packages/gi/types.py", line 43, in function
return info.invoke(*args, **kwargs)
gi._glib.GError: Failed to open file 'boot-repair.png': No such file or directory
Error in sys.excepthook:
Traceback (most recent call last):
File "/usr/lib/python2.7/dist-packages/apport_python_hook.py", line 68, in apport_excepthook
binary = os.path.realpath(os.path.join(os.getcwdu(), sys.argv[0]))
OSError: [Errno 2] No such file or directory
Original exception was:
Traceback (most recent call last):
File "/usr/bin/glade2script", line 2339, in set_widget
exec( arg )
File "", line 1, in
File "/usr/lib/python2.7/dist-packages/gi/types.py", line 43, in function
return info.invoke(*args, **kwargs)
gi._glib.GError: Failed to open file 'boot-repair.png': No such file or directory"
A: Open up ubuntu from rescue mode.Open up the terminal
sudo add-apt-repository ppa:yannubuntu/boot-repair && sudo apt-get update
-Press Enter - Then type:
sudo apt-get install -y boot-repair && boot-repair
Then open boot repair. Perform recommended repairs. If that does not fix it then paste the link here for the boot summary.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,023 |
Cosco Shipping Ports Limited and Volcan Compañía Minera will invest US$ 3,000 million in Chancay Port.
Port Terminals of Chancay will have a strategic partner. After several months of negotiations, Volcan Compañía Minera and the Chinese shipping company Cosco Shipping Ports Limited signed a subscription and investment agreement to build a port terminal in the sea of ??Chancay (north of Lima), which will have an investment of US$ 3,000 millions.
In a first stage, the Chancay Port will be developed in an area of ??141 hectares. Among the investments, a logistical entrance complex, an underground tunnel and the port operational area are included.
In the next few days, the concession contract will be signed for the start of the natural gas massification project in Tumbes. The investment will be US$ 25 million.
The process of awarding the Ancon industrial park will be in charge of Proinversión. Construction is estimated to begin in early 2021 and it will conclude in 2025.
The Ministry of Economy (Ministerio de Economía - MEF) noted that since last year has been announced investments in big project such as, the investment of Toromocho, Quellaveco, Mina Justa, the Port of Salaverry, the destination of the Jorge Chávez International Airport, among others.
New Southern Gas Pipeline would be co-financed with the Government. The tender will be held in 2020 and by March of this year, the report of the Project will be finalited with the final route of the Integrated Gas Transportation System that replaces the Gasoducto Sur Peruano - GSP.
The National Port Authority (Autoridad Portuaria Nacional - APN) indicated that for this year implementation and investment activities are expected in at least seven port terminals in the country, which will mean a joint investment commitment of approximately US$ 1,230 million.
The mayor of Lima, Jorge Muñoz, affirmed that he will promote the construction of two cable cars. The first, will link the districts of San Juan de Lurigancho with Independencia; while the second will serve as transport between the high and low areas of the El Agustino district.
Perupetro will review incentives linked to reducing royalties to boost oil production. The entity said that it is preparing to tender four oil lots for the next month (lot 201 and three others in the north). They will also start the international promotion of another 80 lots.
Hannan Metals, a company with Canadian capitals, announced the submission of an application to access 14,800 hectares of land in the San Martin region, where copper and silver are expected to be present.
The concessionaire Aeropuertos del Perú (Adp) obtained from the Ministry of Transport and Communications (Ministerio de Transportes y Comunicaciones) the approval of the viability of the Chiclayo International Airport modernization project, which will require an investment of more than US$ 300 million. | {
"redpajama_set_name": "RedPajamaC4"
} | 592 |
Rebecca Olson Executive Director of 59 Days of Code
On Episode 414 of The Waves of Tech, we are speaking with Rebecca Olson – Executive Director of 59 Days of Code. 59 Days of Code is based in Fresno, California and envisions a community and city using technology for good by finding solutions to local issues. The nonprofit is creating a community that uses technology to create more sustainable, richer, and efficient ways of conducting business and resolving problems. The main highlight is their expo, dubbed #thecompetition, which throws developers, designers, and entrepreneurs against each other. We dive into the origins, the now, and the future of the nonprofit as they look to expand and create a new environment to live and work. The Waves of Tech is powered by modernlife.network – Modern Issues. Modern Discussions. Enjoy the podcast and continue to ride…The Waves of Tech.
59 Days of Code
We had the privilege of sitting down with Rebecca Olson, Executive Director of 59 Days of Code – a nonprofit for the growth of the technology industry and community in the San Joaquin Valley. Based out of Fresno, California, Rebecca and her team are finding solutions to local community issues by using technology. Essentially, the nonprofit is providing an opportunity to use technology for good.
From their website – "59DaysOfCode is here to lead a new revolution, one where our community takes over the world. Our ragged band of developers, students, companies, and communities will now use our power for good. To create new geeks out of those individuals on the fringes of society. To change the way we do business so that life is more sustainable, richer, efficient. To keep investing in our people and giving power back to you. Because you are the ones who will change the world."
We chat about to origins of the nonprofit and #thecompetition – the largest technology competition in the Central Valley. The expo assimilated over 250 students, created 25 businesses, handed out over $80,000 in prizes, and discovered over 400 developers. It continues to grow year after year. The expo pits competitors against each other and a formal judging panel determines the winner based on four criteria – technology (code), design, business & marketing, and impact.
From their website – "59DaysOfCode's #thecompetition returns to pit the best developers, designers, and entrepreneurs against one another in a giant geek-tastic battle. We start with the Kickoff Event, followed by almost two months of frenzied typing and clicking, and end with the Expo, a huge showcase where the entries are judged, juried, and declared (for those lucky few), winners. And yes, of course there are amazing prizes."
We dive into the catalyst of the nonprofit and the future goals to expand beyond the borders of the city limits. Rebecca shares the positive influence the nonprofit is having in the community, both in terms of reaching students and reaching a new community of supporters and sponsors.
#thecompetition – YouTube Video
Teams that Competed in Expo
Washington Post New Owner and Experiencing Breaking News
Discussion about Jeff Bezos' purchase of The Washington Post and what they may mean for the company, traditional print media, and the digital presence of the paper. Dave retells a personal account of experiencing breaking news and how that event developed through social media and real eyewitnesses.
Google Glass In The NBA and Skype Group Video
Google Glass makes an appearance in the NBA with the Indiana Pacers franchise. Skype opens the digital door to free group video calling! Latest Twitter profile page and Homeland Security's statement regarding Internet Explore.
Back To The Future Day and YouTube Red
We take a deep look into Back to the Future Day and see what predictions made in 1989 became a reality in 2015. Brick-and-mortar stores, such as Target and Lowe's, are incorporating some new app features and in-store tech advancements to improve the customer experience. Also on the agenda, we examine the new subscription-based YouTubeRed announced by the video giant last week and lay out the plan Yahoo has to win you back. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,227 |
\section*{ACKNOWLEDGMENTS}}
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\title{Boolean Hedonic Games\thanks{This paper was presented at the Eleventh Conference on Logic and the Foundations of Game and Decision Theory (LOFT 2014) in Bergen, Norway, July 27-30, 2014.}}
\author[1]{Haris Aziz}
\author[2]{Paul Harrenstein}
\author[3]{J{\'e}r{\^o}me Lang}
\author[2]{Michael Wooldridge}
\affil[1]{\small NICTA and University of New South Wales, Australia}
\affil[2]{\small Department of Computer Science, University of Oxford, UK}
\affil[3]{\small LAMSADE, Universit{\'e} Paris-Dauphine, France}
\begin{document}
\nocomments
\sloppy
\maketitle
\begin{abstract}
\noindent We study \emph{hedonic games with dichotomous
preferences}. Hedonic games are cooperative games in which
players desire to form coalitions, but only care about the makeup
of the coalitions of which they are members; they are indifferent
about the makeup of other coalitions. The assumption of
dichotomous preferences means that, additionally, each player's
preference relation partitions the set of coalitions of which that
player is a member into just two equivalence classes: satisfactory
and unsatisfactory. A player is indifferent between satisfactory
coalitions, and is indifferent between unsatisfactory coalitions,
but strictly prefers any satisfactory coalition over any
unsatisfactory coalition. We develop a succinct representation for
such games, in which each player's preference relation is
represented by a propositional formula. We show how solution
concepts for hedonic games with dichotomous preferences are
characterised by propositional formulas.
\end{abstract}
\section{Introduction}\label{intro}
\hanote{We use set of players, group of players and coalition of players. Are they terms used systematically for different contexts?}
\bphnote{My suggestion: S is a coalition of players if a partition~$\pi$ is given and $S\in\pi$. A group of players I would say is a subset of players that need not be a coalition in a given partition. ``Set of players'' I would only use once, when introducing the framework: ``$N$ is a set of players''.}
Hedonic games are cooperative games in which players desire to form
coalitions, but only care about the makeup of the coalitions of which
they are members; they are indifferent about the makeup of other
coalitions~\citep{dreze:80a,chalkiadakis:2011a}. Because the specification of a
hedonic game requires the expression of each player's ranking over all
sets of players including him, in general, such a specification
requires exponential space -- and, when used by a centralised
mechanism, exponential elicitation time. Such an exponential blow-up
severely limits the practical applicability of hedonic games, and for
this reason researchers have investigated compactly represented
hedonic games. One approach to this problem has been to consider
possible restrictions on the possible preferences that players
have. For example, one may assume that each player specifies only a
ranking over single players, and that her preferences over coalitions
are defined according to the identity of the best (respectively,
worst) element of the coalition~\citep{CeHa04a,Cech08a}. One may also
assume that each player's preferences depend only on the number of
players in her coalition~\citep{bogomolnaia:2002}. These representations come
with a domain restriction, i.e., a loss of expressivity:
\citet{ElWo09a} consider a fully expressive representation for hedonic
games, based on weighted logical formulas.
In the worst case, the representation
of~\citeauthor{ElWo09a}
requires space exponential in the number of players,
but in many cases the space requirement is much smaller.
In this paper, we consider another natural restriction on player
preferences. We consider hedonic games with \emph{dichotomous
preferences}. The assumption of dichotomous preferences means that
each player's preference relation partitions the set of coalitions of
which that player is a member into just two equivalence classes:
satisfactory and unsatisfactory. A player is indifferent between
satisfactory coalitions, and is indifferent between unsatisfactory
coalitions, but strictly prefers any satisfactory coalition over any
unsatisfactory coalition.
While to the best of our knowledge dichotomous preferences have not
been previously studied in the context of hedonic games, they have of
course been studied in other economic settings, such as by
\citet{BMS05a}, \citet{BoMo04a}, and \citet{Bouveret08Jair} in the
context of fair division, by ~\citet{HHMW01a} in the context of Boolean
games, by \citet{KP02} in the context of belief merging, by
\citet{BoMo04a} in the context of matching, and by \citet{BrFi07c} (and
many others) in the context of approval voting.
When the space of all possible alternatives has a combinatorial
structure, propositional formulas are a very natural representation of
dichotomous preferences. In such a representation, variables
correspond to goods (in fair division), outcome variables (Boolean
games), state variables (belief merging), or players (coalition
formation). In the latter case, which we will be concerned with in
the present paper, each player~$i$ can express her preferences over
coalitions containing her by using propositional atoms of the form~$ij$ ($j \neq i$), meaning that~$j$ is in the same coalition as
$i$. Thus, for example, player~1 can express by the formula $(12 \vee
13) \wedge \neg 14$ that he wants to be in a coalition with player~$2$
or with player~$3$, but not with player~$4$.
Our primary aim in this paper is to present such a propositional
framework for specifying hedonic games and computing various solution
concepts. We will first define a propositional logic using atoms of
the form $ij$, together with domain axioms expressing that the output
of the game should be a partition of the set of players. Then we
consider a range of solution concepts, and show that they can be
characterised by some specific classes of (sometimes polysize)
formulas, and solved using propositional satisfiability solvers. The
result is a simple, natural, and compact representation scheme for
expressing preferences, and a machinery based on satisfiability for
computing partitions satisfying some specific stability criteria such
as Nash stability or core stability.
\hanote{Need to clearly show why the new work is different from previous logical approaches to coalition formation. Is it that the framework is old but only characterisations are new?}
\section{Preliminaries}
In this section, we recall some definitions relating to
coalitions, coalition structures (or partitions),
and hedonic games. See, e.g.,~\cite{chalkiadakis:2011a} for an in-depth
discussion of these and related concepts.
\paragraph{Coalitions and Partitions}
\bphnote{Standard notation in the context of hedonic games is: $S,T,\dots$ for coalitions,~$\mathscr N_i$ for the set of coalitions~$i$ belongs to. Let's stick to that, even though~$\mathscr N_i$ is rather wild and nevertheless used quite a lot in what follows.}
We consider a setting in which there is a set~$N$ of~$n$ players with
typical elements~$i,j,k,\dots$. Players can form \emph{coalitions},
which we will denote by~$S,T,\dots$. A coalition is simply a subset of
the players~$N$. One may usefully think of the players as getting together to form
teams that will work together. A \emph{coalition structure} is an
exhaustive partition~$\pi=\set{S_1,\dots,S_m}$ of the players into
disjoint coalitions, i.e.,\xspace $S_1\cup\dots\cup S_m=N$ and $S_i\cap
S_j=\emptyset$ for all~$S_i,S_j\in\pi$ such that~$i \neq j$. {For
technical convenience, we slightly deviate from standard conventions
and require that every coalition structure~$\pi$ contains the empty
set~$\emptyset$.}
We commonly refer to coalition structures simply as \emph{partitions}.
In examples, we also write, e.g.,\xspace
$[\mathwordbox{12}{mi}|\mathwordbox{34}{mi}|\mathwordbox{5}{m}]$
rather than the more cumbersome
$\set{\set{1,2},\set{3,4},\set{5},\emptyset}$.
For each player~$i$ in~$N$,
we let $\mathscr N_i=\set{S\subseteq N\mathrel{:} i\in S}$ denote the set of
coalitions over~$N$ that contain~$i$. If $\pi=\set{S_1,\dots,S_m}$ is
a partition, then~$\pi(i)$ refers to the coalition in~$\pi$ that
player~$i$ is a member of.
The notion of players leaving their own coalition and joining another
lies at the basis of many of the solution concepts that we will come
to consider. We introduce some notation to represent such situations.
For~$T$ a group of players (not necessarily a coalition in~$\pi$), by~$\pi|_T$ we
refer to the partition $\set{S_1\cap T,\dots,S_m\cap T}$ and we
write~$\pi|_{-T}$ for~$\pi|_{N\setminus T}$. Moreover, for~$S$ a
coalition in partition~$\pi|_{-T}$, we use $\pi[T \to S]$ to refer to
the partition that results if the players in~$T$ leave their
respective coalitions in~$\pi$ and join coalition~$S$. We also
allow~$T$ to form a coalition of its own, in which case we write $\pi[T\to\emptyset]$.
Formally, we have, for~$S\in\pi|_{-T}$,
\begin{align*}
\pi[T\to S] & \mathwordbox[c]{=}{==} \set{S_j\in\pi|_{-T}\mathrel{:} S_j\neq S}\cup\set{S\cup T,\emptyset}\text.
\end{align*}
\bphnote{In a later version we might also take into account \emph{simultaneous} deviations of coalitions. This would require slightly more complicated definitions, which are undesirable here.}
If~$T$ is a singleton~$\set i$ we also write~$\pi|_{-i}$ and $\pi[i\to S]$ instead of~$\pi|_{-\set i}$ and $\pi[\set i\to S]$, respectively.
Thus, e.g.,\xspace $S\cup\set i\in\pi[i\to S]$ and $\pi[i\to\pi(i)\setminus\set i]=\pi$.
Finally, define $\pi[i\rightleftarrows j]$ as the partition where $i$ and $j$ exchange their places, i.e.:
\[
\scalebox{1}[1]{$
\pi[i\rightleftarrows j] = (\pi\setminus\set{\pi(i),\pi(j)})\cup\set{(\pi(i)\setminus\set i)\cup\set j,(\pi(j)\setminus\set j)\cup\set i}\text.
$}
\]
Thus, for partition $\pi = [123|45]$, we have $\pi(1) = \pi(2)=\{1,2,3\}$ and $\pi(4)=\set{4,5}$. Furthermore, $\pi|_{\set{1,2,4,5}}=[12|45]$ and $\pi|_{-\set{3,4}}=[12|5]$. Also, $\pi[1\to \{4,5\}] = [23|145]$, $\pi[1\to \emptyset] = [1|23|45]$, and $\pi[3\rightleftarrows 4] = [124|35]$.
%
\noindent\paragraph{Hedonic games}
Hedonic games are the class of coalition formation games in which each
player is only interested in the coalition he is a member of, and is
indifferent as to how the players outside his own coalition are
grouped. Hedonic games were originally introduced by \cite{dreze:80a} and further developed by, e.g., \cite{bogomolnaia:2002}. Also see \cite{hajdukova:2006} for a survey from a more computational point of view. Formally, a \emph{hedonic game} is a
tuple~$(N,R_1,\dots,R_n)$, where~$R_i$ represents~$i$'s transitive,
reflexive, and complete preferences over the set of all coalitions
$\mathscr N_i$ containing $i$. Thus, $S \mathrel{R_i}T$ intuitively
signifies that player~$i$ considers coalition~$S$ at least as
desirable as coalition~$T$, where~$S$ and~$T$ are coalitions
in~$\mathscr N_i$. By~$P_i$ and~$I_i$ we denote the strict and the
indifferent part of~$R_i$, respectively. The preferences~$R_i$ of a
player~$i$ are said to be \emph{dichotomous} whenever $\mathscr N_i$
can be partitioned into two disjoint sets~$\mathscr N_i^+$
and~$\mathscr N_i^-$ such that~$i$ strictly prefers all coalitions
in~$\mathscr N_i^+$ to those in~$\mathscr N_i^-$ and is indifferent
otherwise, i.e.,\xspace $S\mathrel{P_i}T$ if and only if $S\in \mathscr N_i^+$
and $T\in \mathscr N_i^-$. A coalition~$S$ in~$\mathscr N_i$ is \emph{acceptable} to~$i$ if~$i$ (weakly) prefers~$S$ to coalition~$\{i\}$, where he is on his own, i.e.,\xspace if~$S\mathrel{R_i}\set i$. By contrast, we say that a coalition~$S$ is \emph{satisfactory} or \emph{desirable} for~$i$ if $S\in\mathscr N_i^+$. Satisfactory partitions are thus generally acceptable to all players. The implication in the other direction, however, does not hold.
We lift preferences on coalitions to preferences on partitions in a natural way: player~$i$ prefers partition~$\pi$ to partition~$\pi'$ whenever~$i$ prefers coalition~$\pi(i)$ to coalition~$\pi'(i)$. We also extend the concepts of acceptability and desirability of coalitions to partitions.
\begin{example}\label{ex1}
Consider the following Boolean game with four players, $1$, $2$, $3$, and $4$, whose (dichotomous) preferences are as follows. (Indifferences are indicated by commas.)
\begin{align*}
1\colon & \scalebox{1}[1]{$\set{1,2,3},\set{1,2,4},\set{1,3,4},\set{1,2,3,4}$} \mathrel P_1 \scalebox{1}[1]{$\set{1},\set{1,2},\set{1,3},\set{1,4}$}\\
2\colon & \scalebox{1}[1]{$\set{2,1,3},\set{2,1,4},\set{2,3,4}$} \mathrel P_2 \scalebox{1}[1]{$\set{2},\set{2,1},\set{2,3},\set{2,4},\set{2,1,3,4}$}\\
3\colon & \scalebox{1}[1]{$\set{3,1},\set{3,2},\set{3,1,2}$} \mathrel P_3 \scalebox{1}[1]{$\set{3},\set{3,4},\set{3,1,4},\set{3,2,4},\set{3,1,2,4}$}\\
4\colon & \scalebox{1}[1]{$\set{4,1},\set{4,2},\set{4,3},\set{4,1,2},\set{4,1,3},\set 4$} \mathrel P_4 \scalebox{1}[1]{$\set{4,2,3},\set{4,1,2,3}$}
\end{align*}
Thus, player~$1$ wants to be in a coalition of at least three and player~$2$ wishes to be in a coalition of exactly three.
Moreover, player~$3$ wants to be in the same coalition as player~$1$ or as~$2$. He does not want to be in a coalition with player~$4$.
Finally, player~$4$ does not want to be with players~$2$ and~$3$ together. There is exactly one partition that is satisfactory for all four
players, namely $[123\mathwordbox{|}{x} 4]$. For players~$1$,~$2$, and~$3$, all coalitions are acceptable. For player~$4$, however, $\set{4,2,3}$ and $\set{1,2,3,4}$ are unacceptable.
\end{example}
\paragraph{Solution Concepts for Hedonic Games}
A \emph{solution concept} associates with every hedonic game~$(N,R_1,\dots,R_n)$ a (possibly empty) set of partitions of~$N$. Here we review some of the most common solution concepts for hedonic games.
\label{page:stability_concepts}
\begin{itemize}[label={},leftmargin=0em,itemsep=0ex]
\item Individual rationality captures the idea that every player prefers the coalition he is in to being on his own, i.e., that coalitions are acceptable to its members. Thus, formally,~$\pi$ is \emph{individually rational} if, for all players~$i$ in~$N$,
\[
\text{$\pi(i)\mathrel{R_i}\{i\}$.}
\]
This condition is obviously equivalent to $\pi\mathrel{R_i}\pi[i\to\emptyset]$.
\item For dichotomous hedonic games, a partition~$\pi$ is said to be \emph{social welfare optimal} if it maximises the number of players who are in a satisfactory coalition, that is, if~$\pi$ maximises $|\set{i\in N\mathrel{:} \pi(i)\in\mathscr N^+_i}|$. In a similar way, a partition~$\pi$ is \emph{Pareto optimal} if it maximises the set of players being in a satisfactory coalition with respect to set-inclusion, that is, if there is no partition~$\pi'$ with
\[
\set{i\in N\mathrel{:} \pi(i)\in\mathscr N^+_i}\subsetneq
\set{i\in N\mathrel{:} \pi'(i)\in\mathscr N^+_i}\text.
\]
In the extreme case in which every player is in a most preferred coalition,~$\pi$ is said to be \emph{perfect} \citep[cf.,~][]{ABH11c}.
A perfect partition satisfies any other of our stability concepts.
\item A partition is \emph{Nash stable} if no player would like to
unilaterally abandon the coalition he is in and join any other
existing coalition or stay on his own, that is, if, for all~$i\in N$ and all $S\in\pi$,
\[
\text{$\pi(i)\mathrel{R_i}S \cup \{i\}$.
}
\]
Observe that this condition is equivalent to $\pi\mathrel{R_i}\pi[i\to S]$.
\item Core stability concepts consider group deviations instead of individual ones. A group of players, possibly from different coalitions, is said to block a partition if they would all benefit by joining together in a separate coalition. Formally,~$T$ \emph{blocks} (or \emph{is blocking}) partition~$\pi$ if, for all~$i\in T$,
\[
T \mathrel{P_i} \pi(i)\text.
\]
Thus,~$T$ blocks~$\pi$ if and only if $\pi[T\to\emptyset]\mathrel{P_i}\pi$ for all $i\in T$.
A group~$T$ \emph{weakly blocks} (or \emph{is weakly blocking})~$\pi$ if
$T \mathrel{R_i} \pi(i)$ holds for all~$i\in T$ and $T \mathrel{P_i} \pi(i)$ holds for some $i \in T$.
Then, $\pi$ is \emph{core stable} if no group is blocking it
and $\pi$ is \emph{strict core stable} if no group is weakly blocking it.
\item Partition $\pi$ is {envy-free} if no player is envious of another player, that is, if no player~$i$ would prefer to change places with another player~$j$. Formally, partition~$\pi$ is \emph{envy-free} if, for all players~$i$ and~$j$,
\[
\pi\mathrel{R_i}\pi[i\leftrightarrows j].
\]
If $\pi[i\leftrightarrows j]\mathrel{P_i}\pi$ we also say that player~$i$ \emph{envies} player~$j$.
\end{itemize}
{
\setcounter{example}{0}
\begin{example}[continued] In our example, in partition~$[1,2,3\mathwordbox{|}{x} 4]$ each player is in a most preferred coalition. As such~$[1,2,3\mathwordbox{|}{x} 4]$ is perfect as well as social welfare optimal and satisfies all solution concepts mentioned above.
Moreover, all partitions except $[1\mathwordbox{|}{x} 2,3,4]$ and $[1,2,3,4]$ individually rational.
Now, consider partition $\pi=[1\mathwordbox{|}{x} 2,3\mathwordbox{|}{x} 4]$.
Here, player~$2$ does not want to abandon her coalition~$\set{2,3}$ and join another as she prefers none of the following partitions to~$\pi$: $\pi[2\to\set 1]=[1,2\mathwordbox{|}{x} 3\mathwordbox{|}{x} 4]$, $\pi[2\to\set{2,3}]=[1\mathwordbox{|}{x} 2,3\mathwordbox{|}{x} 4]$, $\pi[2\to\set 4]$, and $\pi[2\to\emptyset]=[1\mathwordbox{|}{x} 2\mathwordbox{|}{x} 3\mathwordbox{|}{x} 4]$.
As, however, $\pi[1\to\set{2,3}]=[1,2,3\mathwordbox{|}{x} 4]$ and $[1,2,3\mathwordbox{|}{x} 4]\mathrel P_1\pi$, partition~$\pi$ is not Nash stable.
Also observe that for $\pi=[1\mathwordbox{|}{x} 2,3\mathwordbox{|}{x} 4]$ the group $\set{1,2,3}$ is strongly blocking, as $\pi[\set{1,2,4}\to\emptyset]=[1,2,4\mathwordbox{|}{x} 3]$ and $[1,2,4\mathwordbox{|}{x} 3]\mathrel P_{i}\pi$ for all $i\in\set{1,2,4}$.
Thus, $\pi$ is not core stable. By contrast, $[1,4\mathwordbox{|}{x} 2,3]$ is core stable as only player~$1$ and~$2$ are not satisfied and both of them will only be if they can form a blocking coalition of exactly three.
However, $\set{1,2,4}$ is still weakly blocking, and as such $[1,4\mathwordbox{|}{x} 2,3]$ is not strict core stable.
For envy-freeness, consider partition~$\pi'=[1\mathwordbox{|}{x} 2,4\mathwordbox{|}{x} 3]$. Then, player~$3$ envies player~$4$, as $\pi'[3\leftrightarrows 4]=[1\mathwordbox{|}{x} 2,3\mathwordbox{|}{x} 4]$ and $[1\mathwordbox{|}{x} 2,3\mathwordbox{|}{x} 4]\mathrel{P_3}\pi'$. By contrast, player~$3$ does not envy player~$2$: we have $\pi'[3\leftrightarrows 2]=[1\mathwordbox{|}{x} 2\mathwordbox{|}{x} 3,4]$ but not $[1\mathwordbox{|}{x} 2\mathwordbox{|}{x} 3,4]\mathrel{P_3}\pi'$.
\end{example}
}
\mjwnote{
\paragraph{Strategyproofness} Strategyproofness promises to be very interesting as in hedonic Boolean games and Boolean coalition formation games you can reason about preferences logically.
}
\hanote{I think SP can be easily achieved by translating the setting to a coalition choice setting in which partitions are the alternatives. In that case approval voting is strategyproof for dichotomous preferences. The problem of finding a partition with the maximum number of approvals is at least hard as checking whether a perfect partition exists. Hence the problem is NP-hard.}
\bphnote{This is a very interesting remark and we could insert it somewhere in the text.}
\section{{A Logic for Coalition Structures}}
In this section, we develop a logic for representing coalition
structures. We will then use this logic as a compact specification
language for dichotomous preference relations in hedonic games.
\paragraph{Syntax}
Given a set~$N$ of~$n$ players, we define a propositional language~$L_N$
built from the usual connectives and with for every (unordered)
pair~$\set{i,j}$ of distinct players a propositional variable
$p_{\set{i,j}}$. The set of propositional variables we denote
by~$\ensuremath{\mathit{V}}$. Observe that $|V|=\binom{n}{2}$. \bphnote{I am not overly happy with the notation
$\ensuremath{\mathit{V}}$ for propositional variables/symbols. I would
prefer~$X$, $A$, $Q$, or perhaps even $\Phi$. The reason is that
$\exists \ensuremath{\mathit{V}}\varphi$ looks awkward. I am no great fan of
introducing the notation $\exists i\varphi$ for
$\exists\ensuremath{\mathit{V}}_i\varphi$ either as this may be confused with
$\bigvee_i\varphi$, which we are also using. Moreover, if we choose
for~$X$, we can use~$x,y,z,\dots$ as metavariables over
propositional variables and $\vec x$ for sequences/vectors of
propositional variables. If you agree, please, change the macro
\texttt{$\backslash$vars} in the preamble accordingly.} For
notational convenience we will write~$ij$ for~$p_{\set{i,j}}$. Thus,~$ij$ and~$ji$ refer to the same symbol. The
language is interpreted on coalition structures on~$N$ and the
informal meaning of~$ij$ is ``$i$ and $j$ are in the same coalition''.
Formally, the formulas of the language~$L_N$, with typical
element~$\varphi$ is given by the following grammar
\[
\varphi \mathrel{\Coloneqq} ij
\mathrel{|} \neg\varphi
\mathrel{|} (\varphi\vee\varphi)
\]
where $i,j\in N$ and $i\neq j$. By $|\varphi|$ we denote the \emph{size} of~$\varphi$.
For a given coalition~$S$ of players, we write $\ensuremath{\mathit{V}}_S$ for the propositional variables in which some~$i\in S$ appears, i.e.,
\[
\ensuremath{\mathit{V}}_S = \set{ij\in\ensuremath{\mathit{V}}\mathrel{:} \text{$i\in S$ or $j\in S$}}\text.
\]
Note that for distinct players~$i$ and~$j$ we have~$\ensuremath{\mathit{V}}_i\cap\ensuremath{\mathit{V}}_j=\set{ij}$.
The propositional language over~$\ensuremath{\mathit{V}}_S$ we denote by $L_S$. We write~$\ensuremath{\mathit{V}}_i$ and~$L_i$ for $\ensuremath{\mathit{V}}_{\set i}$ and $L_{\set i}$, respectively.
The remaining classical connectives~$\bot$, $\top$, $\wedge$, $\to$,
and $\leftrightarrow$ are defined in the usual way. Moreover, for
formulas ${\psi_1,\dots,\psi_k}$ of formulas, we have
$\bigwedge_{1\le m\le k}\psi_m$ and $\bigvee_{1\le m\le k}\psi_m$ abbreviate
$\psi_1\wedge\dots\wedge\psi_k$ and $\psi_1\vee\dots\vee\psi_k$,
respectively.
We also make use of the following useful
notational shorthand:
\begin{align*}
i_1 \cdots i_m \overline{i}_{m+1} \cdots \overline{i}_p
& =
\bigwedge_{1 \leq j \leq m}i_1 i_j\wedge \bigwedge_{m < k \leq p}\neg i_1 i_k\text.
\end{align*}
\bphnote{I have changed the definition here, in such a way that $1234=12\wedge13\wedge14$. This is not only shorter, but also helps to guarantee that our examples are still examples of hedonic games. The examples, however, need to be reformulated in such a way that when formulating a player's goal that player should always be mentioned first. $1234$ is \emph{not} the same symbol as, for instance, $3214$. Another option, to get things correct is not requiring that in hedonic games the goals of each player~$i$ is actually from~$L_i$ but \emph{equivalent} to some formula in~$L_i$. This might be more elegant.}
Thus,
$i_1 \cdots i_m \overline{i}_{m+1} \cdots \overline{i}_p$ conveys that $i_1, \dots, i_m$ are in the same coalition and each of them in another coalition than ${i}_{m+1} \cdots {i}_p$. Thus, where $N=\set{1,2,3,4}$,
$
12\overline{3}\overline{4} \vee 13\overline{2}\overline{4} \vee 14\overline{2}\overline{3}
$
abbreviates
$
(12 \wedge \neg 13\wedge \neg 14) \vee (13 \wedge \neg 12\wedge\neg 14) \vee(14\wedge\neg 12\wedge \neg 13)
$
and signifies that player~$1$ is in a coalition of two players.
\paragraph{Semantics}
We interpret the formulas of $L_N$ on partitions~$\pi$ as follows.
\[
\begin{array}{lcl}
\pi \models ij& \mbox{if and only if} & \pi(i)=\pi(j)\\
\pi \models \neg\varphi & \mbox{if and only if} & \pi\not\models\varphi\\
\pi \models \varphi\to\psi & \mbox{if and only if} & \mbox{$\pi\not\models\varphi$ or $\pi\models\psi$}
\end{array}
\]
For~$\Psi\subseteq L_N$, we have $\Psi\models\varphi$ if $\pi\models\psi$ for all $\psi\in\Psi$ implies $\pi\models\varphi$. If $\Psi=\emptyset$, we write $\models\varphi$ and say that~$\varphi$ is \emph{valid}.
Notice that partitions play a dual role in our framework: both
their initial role as coalition structures, and the role of models
in our logic. This dual role is key to using formulas of our
propositional language as a specification language for preference
relations. Thus, e.g., partition~$[1|2|345]$ satisfies
the following formulas of $L_N$: $345$, $3\overline{1}$, $345\overline{1}\overline{2}$, $\neg 12 \wedge (23 \vee 34)$, and $12 \leftrightarrow 23$.
\paragraph{Axiomatisation} We have the following axiom schemes
for mutually distinct players~$i$,~$j$, and~$k$,
\begin{enumerate}[label=\ensuremath{(\mathrm A\arabic*)},leftmargin=3em]
\setcounter{enumi}{-1}
\item\label{axiom:taut} all propositional tautologies
\item\label{axiom:trans} $ij\wedge jk \to ik$ \hfill (\emph{transitivity})
\end{enumerate}
as well as \emph{modus ponens} as the only rule of the system:
\begin{enumerate}[label=\ensuremath{(\mathrm{MP})},leftmargin=3em]
\item \text{from \wordbox{$\varphi$}{ $\varphi$ } and \wordbox{$\varphi\to\psi$}{ $\varphi\to\psi$ } infer \wordbox{$\psi$.}{ $\psi.$ }}
\hfill (\text{\emph{modus ponens}})
\end{enumerate}
The resulting logic we refer to as \ensuremath{\mathbf{P}}\ and write $\Psi\vdash_{\ensuremath{\mathbf{P}}}\varphi$ if there is a derivation of~$\varphi$ from~$\Psi$, \ref{axiom:taut}, and \ref{axiom:trans}, using modus ponens.
\begin{theorem}[Completeness]\label{thm:completeness} Let $\Psi\cup\set\varphi\subseteq L_N$. Then,
\[
\text{$\Psi\vdash_\ensuremath{\mathbf{P}}\varphi$ \wordbox{ if and only if }{ iif and only iff } $\Psi\models\varphi$.}
\]
\end{theorem}
\begin{proof}[(sketch)]
Soundness is straightforward. For completeness a standard Lindenbaum construction can be used. To this end, assume $\Psi\not\vdash_{\ensuremath{\mathbf{P}}}\varphi$. Then, $\Psi\cup\set{\neg\varphi}$ is consistent and can as such be extended to a maximal consistent theory $\Psi^*$. Define a relation~$\sim_{\Psi^*}$ such that for all $i,j\in N$,
\[
\text{$i\sim_{\Psi^*}j$ \wordbox{ if and only if }{ iif and only iff } $ij\in\Psi^*$.}
\]
The axiom schemes~\ref{axiom:taut} and~\ref{axiom:trans} ensure that~$\sim_{\Psi^*}$ is a well-defined equivalence relation. Let $[\mathwordbox{i}{n}]_{\sim_{\Psi^*}}=\set{j\in N\mathrel{:} i\sim_{\Psi^*}j}$ be the equivalence class under~$\sim_{\Psi^*}$ to which player~$i$ belongs. Then define the partition $\pi_{\Psi^*}=\set{[\mathwordbox{i}{n}]_{\sim_{\Psi^*}}\mathrel{:} i\in N}$. By a straightforward structural induction, it can then be shown that for all $\psi\in L_N$,
\[
\text{$\pi_{\Psi^*}\models\psi$ \wordbox{ if and only if }{ iif and only iff } $\psi\in\Psi^*$.}
\]
It follows that $\pi_{\Psi^*}\models\Psi$ and $\pi_{\Psi^*}\not\models\varphi$. Hence, $\Psi\not\models\varphi$.
\end{proof}
Alternatively, one can reason with coalition structures in standard
propositional logic, by writing the transitivity axiom directly as a
propositional logic formula. Let
\[
\mathit{trans} = \bigwedge_{i,j,k\in N} (ij \wedge jk \rightarrow ik)\text.
\]
Then, for any propositional formulas~$\varphi$ and~$\psi$ of $L_N$,
\[
\varphi \vdash_\ensuremath{\mathbf{P}} \psi \text{\wordbox{ if and only if }{ iif and only iff }} \varphi \wedge \mathit{trans} \vdash \psi
\]
that is, checking whether a formula~$\varphi$ implies another formula~$\psi$ in~$\ensuremath{\mathbf{P}}$ is equivalent to saying that~$\varphi$ together with the transitivity constraint implies~$\psi$. This means that reasoning tasks in~$\ensuremath{\mathbf{P}}$ can be done with a classical propositional theorem prover. In what follows we say that two formulas~$\varphi$ and~$\psi$ are $\ensuremath{\mathbf{P}}$-equivalent whenever their equivalence can be proven in~$\ensuremath{\mathbf{P}}$, i.e.,\xspace $\vdash_\ensuremath{\mathbf{P}}\varphi\leftrightarrow\psi$.
\section{Boolean Hedonic Games}
The denotation of a formula $\varphi$ of our propositional language is
a set of coalition structures, and we can naturally interpret these as
being the desirable or satisfactory coalition structures for a particular player.
Thus, instead of writing a hedonic game with dichotomous preferences
as a structure $(N,R_1, \ldots, R_n)$, in which we explicitly
enumerate preference relations $R_i$, we can instead write
$(N,\gamma_1,\dots,\gamma_n)$, where $\gamma_i$ is a formula of our
propositional language that acts as a specification of the preference
relation $R_i$. Intuitively,~$\gamma_i$ represents player~$i$'s
`goal' and player~$i$ is satisfied if his goal is achieved and
unsatisfied if he is not. We refer to a structure $(N,\gamma_1,
\ldots, \gamma_n)$ as a \emph{Boolean hedonic game}. Thus, a Boolean
hedonic game $(N,\gamma_1,\dots,\gamma_{n})$ represents the
(standard) hedonic game $(N,R_1,\dots,R_n)$ with for each~$i$,
\[
\text{$\pi(i)\mathrel{R_i}\pi'(i)$ \wordbox{if and only if}{xif and only ifx} $\pi\models\gamma_i$ implies $\pi'\models\gamma_i$.}
\]
Observe that, defined thus, the preferences of each player in a
hedonic Boolean game are dichotomous.
It should be clear that every dichotomous preference relation~$R_i$ can be
specified by a propositional formula~$\gamma_i$, and hence our
propositional language forms a fully expressive representation
scheme for Boolean hedonic games.%
\footnote{Let~$i$ be a player with dichotomous preferences~$R_i$ and let~$X_i$ be the set of coalitions
most preferred by~$i$, i.e.,
$S \in X_i$ if and only if $S \mathrel{R_i}S'$ for all coalitions~$S$ and~$S'$ containing~$i$.
Then,~$R_i$ is represented by following formula of~$L_i$ in disjunctive normal form:
\[
\bigvee_{S \in X_i}\Big(\bigwedge_{j\in S}ij\wedge\bigwedge_{k\notin S}\neg ik\Big)\text.
\]
} In fact, formulas in~$L_N$ are strictly more expressive in the sense that they can represent \emph{any} dichotomous preference relation over partitions rather than just preference relations over partitions as induced by a preference relation~$R_i$ for a player~$i$ over \emph{coalitions} in~$\mathscr N_i$. We find, however, that every Boolean hedonic game~$(N,\gamma_1,\dots,\gamma_n)$ represents a hedonic game with dichotomous preferences provided that every player's goal~$\gamma_i$ is equivalent to a formula in the language~$L_i$, the sublanguage of~$L_N$ in which only variables in $V_{i}=\set{ij\mathrel{:} j\in N\setminus\set i}$ occur. Intuitively, formulas in~$L_i$ only convey information about the coalitions player~$i$ is in or she is not in.
\begin{proposition}\label{proposition:characterisation_hedonic_aspect}
If a Boolean hedonic game $(N,\gamma_1,\dots,\gamma_n)$ represents a hedonic game with dichotomous preferences, then for every player~$i$ there is a formula~$\varphi_i\in L_i$ that is $\ensuremath{\mathbf{P}}$-equivalent to~$\gamma_i$. Moreover, if for every player~$i$ there is a formula~$\varphi_i\in L_i$ that is $\ensuremath{\mathbf{P}}$-equivalent to~$\gamma_i$, then $(N,\gamma_1,\dots,\gamma_n)$ represents a hedonic game with dichotomous preferences.
\end{proposition}
\begin{proof}[(sketch)]
For a player~$i$ and $\varphi$ a formula in~$L_i$, a straightforward inductive argument shows that
\[
\text{$\pi\models\varphi$ \wordbox{if and only if}{iiif and only ifff} $\pi'\models\varphi$ for all $\pi'$ with $\pi'(i)=\pi(i)$.}
\]
Then, the result follows as a corollary.
\end{proof}
Often, the use of propositional
formulas $\gamma_i$ gives a `concise' representation of the
preference relation $R_i$, although of course in the worst case the
shortest formula $\gamma_i$ representing $R_i$ may be of size
exponential in the number of players. In what follows, we will write
$(N,\gamma_1, \ldots, \gamma_n)$, understanding that we are
referring to the game $(N,R_1, \ldots, R_n)$ corresponding to this
specification.
{
\setcounter{example}{0}
\begin{example}[continued]
The hedonic game with dichotomous preferences in Example~\ref{ex1} is represented by the Boolean hedonic game $(N,\gamma_1,\gamma_2,\gamma_3,\gamma_4)$ with $N=\set{1,2,3,4}$ and the players' goals given by:
\begin{align*}
\gamma_1 & = (123 \vee 124 \vee 134) &
\gamma_2 & = (213\overline{4} \vee 214\overline{3} \vee 234\overline{1}) \\
\gamma_3 & = (31\vee 32)\wedge\neg34 &
\gamma_4 & = \neg 423\text.
\end{align*}
For each player~$i$ we then have that $\pi\models\gamma_i$ if and only if $\pi\in\mathscr N_i^+$.
\end{example}
}
\bphnote{Would this be a good place to present the (easy) characterisation of hedonic games? ($(N,\gamma_1,\dots,\gamma_n)$ is a hedonic game if and only if for each player~$i$ there is a formula $\gamma_i$ in the language~$L_i$ on $\ensuremath{\mathit{V}}_i$ such that $\gamma_i\equiv_{P} \gamma_i$)}
\section{Substitution and Deviation}\label{section:substition_and_deviation}
\bphnote{It might be better to have this section \emph{after} Boolean hedonic games are being introduced.}
We establish a formal link between substitution in formulas of our language and
the possibility of players deviating from their respective coalition in a
given partition and joining other coalitions.
\paragraph{Substitution}
We first introduce some formal notation and terminology with respect to substitution of formulas for variables in our logic.
For~$ij$ a propositional variable in~$\ensuremath{\mathit{V}}_N$
and~$\varphi$ and~$\psi$ formulas of~$L_N$, we denote by $\varphi_{ij \leftarrow \psi}$ the \emph{uniform
substitution} of variable~$ij$ by~$\psi$ in $\varphi$. If
$\vec{\imath\jmath}=i_1j_1,\dots,i_kj_k$ is a sequence of~$k$ distinct variables in~$\ensuremath{\mathit{V}}$
and $\vec\psi=\psi_1,\dots,\psi_k$ a sequence of~$k$ formulas,
\begin{align*}
\varphi_{\vec{\imath\jmath}\leftarrow\vec\psi}
& \mathwordbox{=}{===} \varphi_{i_1j_1,\dots,i_kj_k\leftarrow \psi_1,\dots,\psi_k}
\end{align*}
denotes the \emph{simultaneous substitution} of each $i_mj_m$ by $\psi_m$ ($1\le m\le k$).
Thus, e.g., $(ij\vee\neg jk)_{ij,jk\leftarrow jk,ik}=jk\vee\neg ik$.
A special case, which recurs frequently in what follows, is if every~$\psi_i$ is a Boolean, i.e., if $\psi_1,\dots,\psi_k\in\set{\top,\bot}$.
Sequences $\vec b=b_1,\dots,b_k$ where $b_1,\dots,b_k\in\set{\top,\bot}$ we will also refer to as \emph{Boolean vectors of length~$k$}.
Thus, e.g., $\top,\bot$ is a Boolean vector of length~$2$ and $(ij\wedge jk\to ki)_{ij,ki\leftarrow\top,\bot}=\top\wedge jk\to\bot$.
\paragraph{Characterising individual deviations}\label{subsection:characterising_individual_diviations}
Some of the stability concepts for Boolean hedonic games we consider
in this paper, e.g., Nash stability, are based on which coalitions an individual player~$i$ can join given a partition~$\pi$. Recall that these coalitions are given by $\pi|_{-i}$. Of course, not all groups of agents are included in~$\pi_{-i}$. For instance, let partition~$\pi$ be given by $[\,12\sep34\mathwordbox{|}{x} 5\,]$. Then, player~$1$ can join coalition~$\set{3,4}$ but cannot form a coalition with players~$4$ and~$5$ by unilaterally deviating from~$\pi$. We find that the set~$\pi|_{-i}$ can be characterised in our logic. This furthermore yields a logical characterisation of when a player~$i$ can unilaterally break loose from his coalition, join another one and thereby guarantee that a given formula~$\varphi$ will be satisfied. A particularly interesting case is if~$\varphi$ implies the respective player's goal. We thus gain expressive power with respect to whether a player can \emph{beneficially} deviate from a given partition, a crucial concept.
\bphnote{The proofs of the following lemmas are included. Presumably, it would be better to leave them out from the submission.}
\begin{lemma}\label{lemma:forgetting1}
Let~$\pi$ be a partition,~$i$ a player, $B$~a group of players in $N\setminus\set i$. Let furthermore~$\vec b=b_1,\dots,b_{n-1}$ be a Boolean vector of length~$n-1$ and $i\vec\jmath=ij_1,\dots,ij_{n-1}$ an enumeration of~$\ensuremath{\mathit{V}}_i$ such that $B=\set{j\mathrel{:} ij_{i\vec\jmath\leftarrow \vec b} =\top}$.
Then,
\begin{enumerate}[label={\ensuremath{(\roman*)}},leftmargin=2.25em]
\item\label{item:forgettting1i} $B\in\pi|_{-i}$ \wordbox{iff}{iifff} $\pi\models\mathit{trans}_{i\vec\jmath\leftarrow \vec b} $,
\item\label{item:forgettting1ii} $B\in\pi|_{-i}$ and $\pi[i\to B]\models\varphi$ \wordbox{iff}{iifff} $\pi\models(\varphi\wedge\mathit{trans})_{i\vec\jmath\leftarrow \vec b}$.
\end{enumerate}
\end{lemma}
\begin{proof}
{
\renewcommand{_{i\vec\jmath\leftarrow \vec b}}{'}
We prove~\ref{item:forgettting1i}; the proof for~\ref{item:forgettting1ii} is by structural induction on~$\varphi$ and relies on similar principles as~\ref{item:forgettting1i}.
As~$\vec b$ and $i\vec\jmath$ are fixed throughout the proof,
for better readability, we write~$\varphi'$ for~$\varphi_{i\vec\jmath\leftarrow\vec b}$.
For the ``only if''-direction, assume that $B\in \pi_{-i}$ as well as $\pi\not\models\mathit{trans}_{i\vec\jmath\leftarrow \vec b}$. Observe that
$
\mathit{trans}_{i\vec\jmath\leftarrow \vec b} \mathwordbox{=}{==}
\bigwedge_{k,l,m}\big(kl_{i\vec\jmath\leftarrow \vec b} \wedge lm_{i\vec\jmath\leftarrow \vec b} \rightarrow km_{i\vec\jmath\leftarrow \vec b}\big)\text.
$
Accordingly, there are some (mutually distinct)~$k$, $l$,~and~$m$ such that $\pi\not\models kl_{i\vec\jmath\leftarrow \vec b} \wedge lm_{i\vec\jmath\leftarrow \vec b} \rightarrow km_{i\vec\jmath\leftarrow \vec b}$.
It suffices to consider the following three cases.
\begin{align*}
(a)&\quad i\notin\set{k,l,m}\text,&
(b)&\quad i=k\text{,} &
(c)&\quad i=l\text.
\end{align*}
Case~$(a)$ cannot occur as we would have $kl_{i\vec\jmath\leftarrow \vec b}=kl$, $lm_{i\vec\jmath\leftarrow \vec b}=lm$, $km_{i\vec\jmath\leftarrow \vec b}=km$, and $kl \wedge lm \rightarrow km$ is a theorem of the system.
If~$(b)$, then
$
\pi\not\models il_{i\vec\jmath\leftarrow \vec b} \wedge lm_{i\vec\jmath\leftarrow \vec b} \rightarrow im_{i\vec\jmath\leftarrow \vec b}\text.
$
It follows that $\pi\models il_{i\vec\jmath\leftarrow \vec b}$, $\pi\models lm_{i\vec\jmath\leftarrow \vec b}$, and $\pi\not\models im_{i\vec\jmath\leftarrow \vec b}$. Observe that in this case $lm_{i\vec\jmath\leftarrow \vec b}=lm$. Hence, $\pi(l)=\pi(m)$. Also notice that $il_{i\vec\jmath\leftarrow \vec b},im_{i\vec\jmath\leftarrow \vec b}\in\set{\top,\bot}$ and, thus, $im_{i\vec\jmath\leftarrow \vec b}=\bot$ and $il_{i\vec\jmath\leftarrow \vec b}=\top$.
Accordingly, $l\in B$ but $m\notin B$. As $i\neq m$ and having assumed $B\in\pi|_{-i}$, a contradiction follows:
\[
\pi(m)\neq\pi(i)=\pi(l)=\pi(m)\text.
\]
If~$(c)$, we have
$
\pi\not\models ik_{i\vec\jmath\leftarrow \vec b} \wedge im_{i\vec\jmath\leftarrow \vec b} \rightarrow km_{i\vec\jmath\leftarrow \vec b}\text.
$
Thus, $\pi\models ik_{i\vec\jmath\leftarrow \vec b}$, $\pi\models im_{i\vec\jmath\leftarrow \vec b}$, and $\pi\not\models km_{i\vec\jmath\leftarrow \vec b}$. Observe that $km_{i\vec\jmath\leftarrow \vec b}=km$. Hence, $\pi(k)\neq\pi(m)$. Moreover, $ik_{i\vec\jmath\leftarrow \vec b},im_{i\vec\jmath\leftarrow \vec b}\in\set{\top,\bot}$, from which follows that
$ik_{i\vec\jmath\leftarrow \vec b}=\top$ and $im_{i\vec\jmath\leftarrow \vec b}=\top$.
Accordingly, both $k,m\in B$. With $B\in\pi|_{-i}$, we obtain that $\pi(k)=\pi(m)$, a contradiction.
For the ``if''-direction, assume $B\notin\pi|_{-i}$ and~$B\neq\emptyset$. Because of the latter, there is some $j\in B$. Accordingly, $ij_{i\vec\jmath\leftarrow \vec b}=\top$.
As $B\notin\pi|_{-i}$, and thus in particular $B\neq \pi(j)\setminus\set{i}$, there are two possibilities:
\begin{enumerate}[label=\ensuremath{(\arabic*)},leftmargin=*,itemsep=.4ex]
\item there is some~$k\neq i$ with $k\in\pi(j)$ and $k\notin B$, or
\item there is some~$k\neq i$ with $k\notin\pi(j)$ and $k\in B$.
\end{enumerate}
If~$(1)$, we have $\pi(j)=\pi(k)$ as well as $ik_{i\vec\jmath\leftarrow \vec b}=\bot$. As $jk_{i\vec\jmath\leftarrow \vec b}=jk$, it holds that $\pi\models ij_{i\vec\jmath\leftarrow \vec b}\wedge jk_{i\vec\jmath\leftarrow \vec b}$ but
$\pi\not\models ik_{i\vec\jmath\leftarrow \vec b}$. If~$(2)$, however, we have $\pi(j)\neq\pi(k)$ and $ik_{i\vec\jmath\leftarrow \vec b}=\top$. As $jk_{i\vec\jmath\leftarrow \vec b}=jk$, it holds that $\pi\models ij_{i\vec\jmath\leftarrow \vec b}\wedge ik_{i\vec\jmath\leftarrow \vec b}$ but
$\pi\not\models jk_{i\vec\jmath\leftarrow \vec b}$. In either case it follows that $\pi\not\models\mathit{trans}_{i\vec\jmath\leftarrow \vec b}$.
}
\end{proof}
%
The following example illustrates Lemma~\ref{lemma:forgetting1}.
\begin{example}
Consider the partition $\pi=[12\sep34\mathwordbox{|}{x} 5]$. Then, $\pi|_{-1}=\set{\set 2,\set{34},\set 5,\emptyset}$.
Let $1\vec\jmath=12,13,14,15$ be a fixed enumeration of $\ensuremath{\mathit{V}}_1$. Also let $\vec b_1=\bot,\top,\top,\bot$ and $\vec b_2=\bot,\top,\bot,\top$ be Boolean vectors (of length~$4$). Then,
\[
[\,12\sep34\mathwordbox{|}{x} 5\,]\models\mathit{trans}_{12,13,14,15\leftarrow\bot,\top,\top,\bot}\text.
\]
(This may be established, somewhat tediously, by painstakingly
checking all~$30$ conjuncts of the form $(kl\wedge lm)\to km$ of
$\mathit{trans}$.) Now, observe that $\set{j\mathrel{:} 1j_{1\vec j\leftarrow\vec
b_1}}=\set{3,4}$ and that $\set{3,4}\in\pi_{-1}$. On the other
hand, observe that $(13\wedge 15 \to 35)_{1\vec\jmath\leftarrow\vec
b_2}=(\top\wedge\top)\to 35$. It is easily established, however,
that $ [12|34|5]$ does not satisfy $(\top\wedge\top)\to
35$ and, hence, neither $\mathit{trans}_{1\vec\jmath\leftarrow\vec
b_2}$. Finally, observe that $\set{j\mathrel{:} 1j_{1\vec
\jmath\leftarrow\vec b_1}}=\set{3,5}$ and that~$\set{3,5}$ is not
in~$\pi|_{-1}$.
\end{example}
\bphnote{Lemma~\ref{lemma:forgetting2} is methinks the crucial for the characterisation of the stability concept. Silly me, only formulated Lemmas~\ref{lemma:forgetting1} and~\ref{lemma:forgetting2} for individual deviations, believing that similar results for ``group deviations'' would straightforwardly follow from them. It appears, however, we need ``group deviation'' variants so as also have (uniform) characterisations of blocking coalitions (which I believe can then be achieved in polynomial space) and core (which cannot). Obviously, these stronger versions would subsume Lemmas~\ref{lemma:forgetting1} and~\ref{lemma:forgetting2} and we do not need both.}
\bphnote{For the moment the quantifiers $\hat\exists i$ and $\hat\forall i$ only play a role in the characterisation of Nash stability. }
We now introduce the following abbreviation, where $i\vec\jmath=ij_1,\dots,ij_{n-1}$ is assumed to be a fixed enumeration of~$\ensuremath{\mathit{V}}_i$.
\begin{align*}
\hat\exists i\,\varphi\; & = \scalebox{1}[1]{$\displaystyle \bigvee_{\vec b\in\set{\bot,\top}^{n-1}} (\varphi\wedge\mathit{trans})_{i\vec\jmath\leftarrow \vec b}$}
\end{align*}
Thus, $\hat\exists i$ can be understood as the operation
of forgetting everything about player $i$ (in the sense of
\cite{lin1994forget}) while taking the transitivity constraint into
account. Intuitively, $\hat\exists i\,\varphi$ signifies that given partition~$\pi$ player~$i$ can deviate to some coalition such that that~$\varphi$ is satisfied.
%
%
\begin{proposition}\label{proposition:quantification}
Let~$\pi$ be a partition,~$i$ a player, and~$\varphi$ a formula of~$L_N$. Then,
\[
\text{$\pi\models\hat\exists i\,\varphi $ \text{\wordbox{iff}{iiiifffff} $\pi[i\to S]\models\varphi$ \;\wordbox[l]{for some $S\in \pi|_{-i}$,}{\quad for some $S\in \pi|_{-i}$}}}
\]
\end{proposition}
\begin{proof}
First assume $\pi\models\hat\exists i\,\varphi$. Then, $\pi\models(\varphi\wedge\mathit{trans})_{i\vec\jmath\leftarrow \vec b}$\quad for some $\vec b\in\set{\bot,\top}^{n-1}$.
Define $S=\set{j\mathrel{:} ij_{i\vec\jmath\leftarrow \vec b} =\top}$. By Lemma~\ref{lemma:forgetting1}\ref{item:forgettting1ii}, we then obtain $\pi[i\to S]\models\varphi$.
For the opposite direction, assume that $\pi[i\to S]\models\varphi$ for some $S\in\pi|_{-i}$. Define $\vec b=b_1,\dots,b_{n-1}$ as the Boolean vector of length~$n-1$ such that
for every $1\le k\le n-1$,
\[
b_k
\mathwordbox{=}{==}
\begin{cases}
\top & \text{if $j\in S\cup\set i$}\\
\bot & \text{otherwise.}
\end{cases}
\]
Then, clearly, $S=\set{j\mathrel{:} ij_{i\vec\jmath\leftarrow \vec b}=\top}$. By Lemma~\ref{lemma:forgetting1}\ref{item:forgettting1ii}, it follows that $\pi\models\varphi_{i\vec\jmath\leftarrow \vec b}$. We may conclude that
$\pi\models\hat\exists i\,\varphi$.
\end{proof}
It is important to note, however, that the number of Boolean vectors of length~$k$ is exponential in~$k$. Accordingly, $\hat\exists i\,\varphi$ abbreviates a formula whose size is exponential in the size of~$\varphi$.
\paragraph{Characterising group deviations}\label{subsection:characterising_group_diviations}
\bphnote{The following lemma is useful for characterising both individual rationality and blocking coalitions. Additional text and an example should still be supplied.}
Besides a single player deviating from its coalition and joining another, multiple players (from possibly different coalitions) could also deviate together and form a coalition of their own. This concept lies at the basis of, e.g., the core stability concept. We establish a formal connection between substitution and group deviations.
Let~$T=\set{i_1,\dots,i_t}$ be a group of players.
Observe that $|\ensuremath{\mathit{V}}_T|=\binom{n}{2}-\binom{n-t}{2}$ and let $\vec{\imath\jmath}_T$ be a fixed enumeration of $\ensuremath{\mathit{V}}_T$. By the \emph{$T$-separating Boolean vector} (given~$\vec{\imath\jmath}_T$) we define as the unique Boolean vector~$\vec b_T$ of length~$\binom{n}{2}-\binom{n-t}{2}$ such that for all~$i\in T$ and all~$j\in N$,
\[
ij_{\vec{\imath\jmath}_T\leftarrow\vec b_T} \mathwordbox{=}{==}
\begin{cases}
\top & \text{if $j\in T$,}\\
\bot & \text{otherwise.}
\end{cases}
\]
Intuitively,~$\vec b_T$ represents the choice of group~$T$ to form a coalition of their own.
Whenever~$T$ is clear from the context we omit the subscript in $\vec b_T$ and $\vec{\imath\jmath}_T$. The following characterisation now holds.
\begin{lemma}\label{lemma:group_deviation}
Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game,~$T$ a group of players,~$\pi$ a partition,~$\vec{\imath\jmath}$ a fixed enumeration of~$\ensuremath{\mathit{V}}_T$, and~$\vec b_T$ the corresponding $T$-separating Boolean vector. Then, for every formula~$\varphi\in L_N$,
\[
\text{$\pi\models\varphi_{\vec{\imath\jmath}\leftarrow\vec b_T} $ \wordbox{ if and only if }{ iif and only iff } $\pi[T\to\emptyset]\models\varphi$.}
\]
\end{lemma}
\section{{Characterising Solutions}}
Our task in this section is to show how the various solution
concepts we introduced above can be \emph{characterised} as formulas
of our propositional language. Let~$f$ be a function mapping each Boolean hedonic game~$G$ for~$N$ to a formula~$f(G)$ of~$L_N$. Given a solution concept~$\theta$,
we say that~$f$ is a {\em
characterisation} of~$\theta$ if for every Boolean hedonic game~$G$ on~$N$
and every partition $\pi$, we have that~$\pi$ is a solution
according to~$\theta$ for game~$G$ if and only if $\pi \models
f(G)$. If, furthermore, there exists a polynomial $p$ such that
$|f(G)| \leq p(|N|)$, then $f$ is a {\em polynomial
characterisation} of $\theta$.
\hanote{This definitions above can feature more prominently perhaps in definition environment.}
Once we have a characterisation of $\theta$, we know that there is a one-to-one correspondence between the partitions of $N$ satisfying $\theta$ and the models of $f(G)$. Therefore, given a Boolean hedonic game~$G$:
\begin{itemize}
\item checking whether there exists a partition satisfying $\theta$ in~$G$ amounts to checking whether~$f(G)$ is satisfiable;
\item computing a partition satisfying~$\theta$ in~$G$ amounts to finding a model of~$f(G)$;
\item computing all partitions satisfying~$\theta$ in~$G$ amounts to finding all models of~$f(G)$.
\end{itemize}
Thus, once we have a characterisation of a solution concept, one can
use a SAT solver to find (some or all) or to check the existence of partitions that satisfy it.
This carries over to {\em conjunctions} of solution
concepts. For instance, if individual rationality is characterised by
$f_{\mathit{IR}}$ and envy-freeness by $f_{\mathit{EF}}$, the there is
a one-to-one correspondence between the individual rational envy-free
partitions for~$G$ and the models of $f_{\mathit{IR}}(G) \wedge
f_{\mathit{EF}}(G)$. More generally, these techniques can be used for finding or checking partitions satisfying~$\theta$ that also have certain other properties expressible in~$L_N$.
In the remainder of the section we focus on how a number of classical solution concepts, and see how they can be characterised in our logic.
\paragraph{Individual rationality, perfection, and optimality}\label{ir}
Recall that a partition is individually rational if any player is at
least as happy in her coalition as being alone, that is, no player
would prefer to leave her coalition to form a singleton
coalition. Now we have the following characterisation of individual rationality in our logic.
%
%
%
%
%
\bphnote{The definition of the characterising formula could perhaps be presented in the main text, rendering the formulation of the proposition shorter and clearer. \textbf{The important thing is the definition of the single Boolean vector $\vec b=(\bot,\dots,\bot)$}.}
\begin{proposition}\label{proposition:individual_rationality}
Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game, let~$i$ be a player with goal~$\gamma$, and let $\pi$ be a partition. Let, furthermore,
$i\vec\jmath$ be a fixed enumeration of~$\ensuremath{\mathit{V}}_i$ and let
$\vec b=\bot,\dots,\bot$ be the Boolean vector of length~$n-1$ only containing~$\bot$. Then,
\begin{enumerate}[label=\ensuremath{(\roman*)},leftmargin=2.5em,itemsep=.5ex]
\item
\text{$\pi$ is acceptable to~$i$}
iff
$\pi\models(\gamma_i)_{i\vec\jmath\leftarrow \vec b}\to \gamma_i$,
\item
\scalebox{1}[1]{$\pi$ is individually rational iff $\displaystyle\pi\models\bigwedge_{i\in N}\big((\gamma_i)_{i\vec\jmath\leftarrow \vec b}\to \gamma_i\big)$.}
\end{enumerate}
\end{proposition}
\begin{proof}
We only give the proof for~$(i)$, as~$(ii)$ follows as an immediate consequence.
{
For~$(i)$, merely consider the following equivalences, of which the third one follows from Lemma~\ref{lemma:forgetting1}\ref{item:forgettting1ii}.
\[
\renewcommand{\arraystretch}{1.3}
\begin{array}{lll}
\multicolumn{1}{l}{\text{$\pi$ is acceptable to~$i$}}
& \text{iff} & \text{$\pi\mathrel{R_i}\pi[i\to\emptyset]$}\\
& \text{iff} & \text{$\pi[i\to\emptyset]\models\gamma_i$ implies $\pi\models\gamma_i$}\\
& \text{iff} & \text{$\pi\models(\gamma_i)_{i\vec\jmath\leftarrow \vec b}$ implies $\pi\models\gamma_i$}\\
& \text{iff} & \text{$\pi\models(\gamma_i)_{i\vec\jmath\leftarrow \vec b}\to \gamma_i$.}
\end{array}
\]
This concludes the proof.
}
\end{proof}
To illustrate Proposition~\ref{proposition:individual_rationality} we consider again \exref{ex1}.
{
\setcounter{example}{0}
\begin{example}[continued]
In the game of our example, all partitions are acceptable to player~$1$, whose goal is given by $\gamma_1=123 \vee 124 \vee 134$.
Let~$\ensuremath{\mathit{V}}_1$ be enumerated by $1\vec\jmath=12,13,14$ and let $\vec b=\bot,\bot,\bot$. Then, $(\gamma_2)_{12,13,14\leftarrow\bot,\bot,\bot}$ is $\ensuremath{\mathbf{P}}$-equivalent to~$\bot$ and, hence, $\pi\models(\gamma_2)_{12,13,14\leftarrow\bot,\bot,\bot}\to\gamma_1$ for all partitions~$\pi$.
According to Proposition~\ref{proposition:individual_rationality} this signifies that to player~$1$ every partition is acceptable.
Now consider player~$4$, whose goal is given by $\neg423$, that is, by $\neg(42\wedge 43)$. Let~$\ensuremath{\mathit{V}}_4$ be enumerated by $41,42,43$ and let $\vec b=\bot,\bot,\bot$.
Then, $\neg(42\wedge 43)_{41,42,43\leftarrow\bot,\bot,\bot}=\neg(\bot\wedge\bot)$, which is obviously $\ensuremath{\mathbf{P}}$-equivalent to~$\top$.
Hence,
\[
\text{$\pi\models \neg(42\wedge 43)_{41,42,43\leftarrow\bot,\bot,\bot}$ \wordbox{if and only if}{if and only if} $\pi\models \neg(42\wedge 43)$,}
\]
meaning that a partition~$\pi$ is acceptable to player~$4$ if and only if~$\pi$ satisfies his goal.
\end{example}
}
The logical characterisation of perfect perfect partition is immediate, as witnessed by the following proposition.
\begin{proposition} Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game. Then,
a partition~$\pi$ is perfect \wordbox{if and only if}{ if and only if } $\displaystyle\pi \models \bigwedge_{i\in N} \gamma_i\text.$
\end{proposition}
%
As a consequence, a perfect partition exists if and
only if the formula $\mathit{trans}\wedge \bigwedge_{i\in N} \gamma_i$ is
satisfiable. Moreover, finding a social welfare maximising partition reduces to
finding valuation satisfying a maximum number of formulas~$\gamma_i\wedge\mathit{trans}$, that is,
to solving a {\sc maxsat} problem.
Leveraging the same idea of iteratively checking whether a perfect partition can be found for a subset of agents, one can compute Pareto optimal solutions for a given game. A subset~$\Psi$ of formulas is said to be a \emph{maximal trans-consistent} if both
\begin{enumerate}[label=$(\roman*)$]
\item $\Psi\cup\set{\mathit{trans}}$ is consistent, and
\item $\Psi'\cup\set{\mathit{trans}}$ is inconsistent for all sets of formulas~$\Psi'$ with~$\Psi\subsetneq\Psi'$.
\end{enumerate}
We now have the following proposition.
\begin{proposition}
A partition~$\pi$ of a Boolean hedonic game is Pareto optimal if and only if $\set{\gamma_i \mathrel{:} \pi \models \gamma_i}$ is a maximal $trans$-consistent subset of $\{\gamma_1, \ldots, \gamma_n\}$
\end{proposition}
Algorithms for computing maximal consistent subsets are well-known and could thus be exploited for the computation of Pareto optimal partitions.
%
\bphnote{Social welfare maximisation in the context of dichotomous or Boolean games seems to be equivalent to Pareto optimality. It seems to me that it is hard to characterise Pareto efficiency in a general way without ``quantifying'' over all partitions.}
\jlnote{Yes, it is possible in polynomial space using cardinality formulas (which requires additional propositional symbols in the language). Probably we should not do this here, but in the future long version of the paper.}
\paragraph{Nash stability}
Recall that a partition~$\pi$ is Nash stable, if no player~$i$ wishes to leave his coalition~$\pi(i)$ and join another (possibly empty) coalition so as to satisfy his goal. Leveraging our results from Section~\ref{section:substition_and_deviation}, we obtain the following characterisation of this fundamental solution concept.
\begin{proposition}
Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game and~$\pi$ a partition. Then,
\[
\text{$\pi$ is Nash stable}
\wordbox{ if and only if }{ iif and only iff }
\text{$\pi\models\bigwedge_{i\in N}\big((\hat\exists i\,\gamma_i)\to\gamma_i\big)$.}
\]
\end{proposition}
\begin{proof}
Consider an arbitrary player~$i$ and observe that following equivalences hold. The fourth equivalence holds in virtue of Proposition~\ref{proposition:quantification}. The third one is a standard law of logic: merely observe that $\pi\models\gamma_i$ is not dependent on~$S$.
\[
\renewcommand{\arraystretch}{1.3}
\begin{array}{l@{\;\;}l}
\multicolumn{2}{l}{\text{$\pi$ is Nash stable}} \\
\text{iff} & \text{for all $i\in N$ and $S\in\pi|_{-i}$: $\pi\mathrel{R_i}\pi[i\to S]$}\\
\text{iff} & \text{for all $i\in N$ and $S\in\pi|_{-i}$: if $\pi[i\to S]\models\gamma_i$ then $\pi\models\gamma_i$} \\
\text{iff} & \scalebox{1}[1]{\text{for all $i\in N$: if $\pi[i\to S]\models\gamma_i$ for some $S\in\pi|_{-i}$ then $\pi\models\gamma_i$}} \\
\text{iff} & \text{for all $i\in N$: if $\pi\models\hat\exists i\,\gamma_i$ then $\pi\models\gamma_i$}\\
\text{iff} & \text{for all $i\in N$: $\pi\models(\hat\exists i\,\gamma_i)\to\gamma_i$} \\
\text{iff} & \text{$\pi\models\bigwedge_{i\in N}\big((\hat\exists i\,\gamma_i)\to\gamma_i\big)$}
\end{array}
\]
This concludes the proof.
\end{proof}
Our running example illustrates this result.
{
\setcounter{example}{0}
\begin{example}[continued]
Consider again the game of Example~\ref{ex1}. Partition~$[123|4]$ satisfies each player's goal and, consequently, is Nash stable. We also have that
$
[123|4]\models\gamma_1\wedge\gamma_2\wedge\gamma_3\wedge\gamma_4
$
and, thus,
\[
[123|4]\models\bigwedge_{i\in N}\big((\hat\exists i\,\gamma_i)\to\gamma_i\big)\text.
\]
Now recall that for partition~$\pi=[1|23|4]$ player~$2$'s goal is not satisfied and that she cannot deviate and join another coalition to make this happen.
In this case, $\pi|_{-2}=\set{\set 1,\set 3,\set 4}$. Moreover, $\pi[2\to\set 1]=[12|3|4]$, $\pi[2\to\set 3]=[1|23|4]$, and $\pi[2\to\set 4]=[1|3|24]$.
Since,
$
[12|3|4] \not\models\gamma_2$,
$ [1|23|4] \not\models\gamma_2$, and
$ [1|3|24] \not\models\gamma_2$,
it follows that $\pi\not\models\hat\exists 2\,\gamma_2$. Hence, $\pi\models(\hat\exists 2\,\gamma_2)\to\gamma_2$.
Player~$1$, however, could deviate from~$\pi_2$ and join~$\set{2,3}$ and thus have his goal satisfied. Thus, $\pi$ is not Nash stable.
Now observe that $\set{2,3}\in\pi|_{-1}$ and that $\pi[1\to\set{2,3}]=[123|4]$. Moreover,
$[123|4]\models\gamma_1$. As thus $\pi\models\hat\exists1\,\gamma_1$, also $\pi\not\models(\hat\exists1\,\gamma_1)\to\gamma_1$. We may
conclude that
\[
[1|23|4]\not\models\bigwedge_{i\in N}\big((\hat\exists i\,\gamma_i)\to\gamma_i\big)\text.
\]
\end{example}
}
Nash stable partitions are not guaranteed to exist in Boolean hedonic games. The two-player game $(\set{1,2},12,\neg 21)$ witnesses this fact, as can easily be appreciated. The translation into a SAT instance gives us a way to compute all Nash stable partitions of a given Boolean hedonic game. Recall, however, that the size of~$\hat\exists i\,\gamma_i$ is generally exponential in the size of~$\gamma_i$.
\paragraph{Core and strict core stability}
\bphnote{The above Lemma and Proposition seem to have been commented out...}
Core and strict core stability relate to group deviations much in the same way as Nash stability relates to individual deviations.
Group deviations we characterised in Section~\ref{subsection:characterising_group_diviations}.
We thus find that Lemma~\ref{lemma:group_deviation} yields a straightforward characterisation in our logic of a specific group blocking or weakly blocking a given partition.
\begin{proposition}\label{proposition:blocking_coalition}
Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game and~$T$ a group of players, and $\pi$ be a partition. Let, furthermore,
$\vec{\imath\jmath}$ a fixed enumeration of~$\ensuremath{\mathit{V}}_T$ and
$\vec b$ \emph{the corresponding $T$-separating Boolean vector}. Then,
\begin{enumerate}[label=\ensuremath{(\roman*)},leftmargin=2.5em,itemsep=.5ex]
\item \text{
\text{$T$ blocks~$\pi$}
\wordbox{ if and only if }{ iif and only iff }
$\pi\models\displaystyle\bigwedge_{i\in T}\big(\neg\gamma_i\wedge(\gamma_i)_{\vec{\imath\jmath}\leftarrow\vec b}\big)$,
}
\item $T$ weakly blocks~$\pi$ if and only if
\[
\pi\models\bigwedge_{j\in T}\big(\gamma_j\to(\gamma_j)_{\vec{\imath\jmath}\leftarrow\vec b}\big)\wedge\bigvee_{i\in T}\big(\neg\gamma_i\wedge(\gamma_i)_{\vec{\imath\jmath}\leftarrow\vec b}\big)\text.
\]
\end{enumerate}
\end{proposition}
\begin{proof} We give the proof for~$(i)$, as the one for~$(ii)$ runs along analogous lines.
Consider the following equivalences, of which the third one follows immediately from Lemma~\ref{lemma:group_deviation}.
\[
\renewcommand{\arraystretch}{1.3}
\begin{array}{lll}
\multicolumn{1}{l}{\text{$T$ blocks $\pi$}}
& \text{iff} & \text{for all $i\in T$: $\pi[T\to\emptyset]\mathrel{P_i}\pi$}\\
& \text{iff} & \text{for all $i\in T$: $\pi[T\to\emptyset]\models\gamma_i$ and $\pi\not\models\gamma_i$}\\
& \text{iff} & \text{for all $i\in T$: $\pi\models(\gamma_i)_{\vec{\imath\jmath}\leftarrow\vec b}$ and $\pi\not\models\gamma_i$}\\
& \text{iff} & \text{$\pi\models\displaystyle\bigwedge_{i\in T}\big(\neg\gamma_i\wedge(\gamma_i)_{\vec{\imath\jmath}\leftarrow\vec b}\big)$.}\\
\end{array}
\]
This concludes the proof.
\end{proof}
Observe that the size of $\bigwedge_{i\in T}\big(\neg\gamma_i\wedge(\gamma_i)_{\vec{\imath\jmath}\leftarrow\vec b}\big)$ is obviously polynomial in $\sum_{i\in T}|\gamma_i|$ and, hence, a partition~$\pi$ being blocking by particular group~$T$ of players can be polynomially characterised.
It might also be worth observing that this characterisation is reminiscent of that for individual rationality and, surprisingly, much more so than of the one for Nash stability.
As a corollary of Proposition~\ref{proposition:blocking_coalition} and de Morgan laws, we obtain the following characterisations of a partition being core stable and of a partition being strict core stable. The characterisations, however, involve a conjunctions over all groups of players and as such is not polynomial.
\begin{corollary}
Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game and~$\pi$ be a partition. Let for each coalition~$T$, $\vec{\imath\jmath}$ be an enumeration of~$\ensuremath{\mathit{V}}_T$ and~$\vec b$ the corresponding $T$-separating Boolean vector. Then,
\begin{enumerate}[label=\ensuremath{(\roman*)},leftmargin=2.5em,itemsep=.5ex]
\item
$\pi$ is core stable if and only if
\scalebox{1}[1]{$
\displaystyle\pi\models \bigwedge_{T\subseteq N}\bigvee_{i\in T}\big((\gamma_i)_{\vec{\imath\jmath}\leftarrow\vec b}\to \gamma_i\big)
$,}
\item Then, $\pi$ is strict core stable if and only if
\[
\pi\models \bigwedge_{T\subseteq N}\big(\bigvee_{j\in T}(\gamma_j\wedge\neg(\gamma_j)_{\vec{\imath\jmath}\leftarrow\vec b})\vee\bigwedge_{i\in T}((\gamma_i)_{\vec{\imath\jmath}\leftarrow\vec b}\to\gamma_i)\big)\text.
\]
\end{enumerate}
\end{corollary}
%
%
Although core stable coalition structure are not guaranteed to exist in general hedonic games, the restriction to dichotomous preferences allows us to derive this positive result.
\begin{proposition}
For every Boolean hedonic game, a core stable coalition structure is guaranteed to exist.
\end{proposition}
\begin{proof}
We initialise $N'$ to $N$ and partition $\pi$ to $\{\emptyset\}$. We find a maximal subset of $S\subset N'$ for which all players are in an approved coalition that satisfies their formulas. We modify $\pi$ to $\pi\cup \{S\}$ and $N'$ to $N'\setminus S$. The procedure is repeated until no such maximal subset $S$ exists. If $N'\neq \emptyset$, then $\pi$ is set to $\pi\cup \{\{i\}\mathrel{:} i\in N'\}$.
We now argue that $\pi$ is core stable.
We note that each player who was in some subset $S$ will never be part of a blocking coalition. If $N'$ was non-empty in the last iteration, then no subset of players in $N'$ can form a deviating coalition among themselves.
\end{proof}
By contrast, a strict core stable partition is not guaranteed to exist. To see this consider the three-player Boolean hedonic game $(\set{1,2,3},12,21\vee 23,32)$. It is not hard to see that each of the five possible partitions is weakly blocked by either $\set{1,2}$ or $\set{2,3}$.
\paragraph{Envy-freeness}
Recall that a partition is envy-free if no player would strictly prefer to exchange places with another player. Observe that for the trivial partitions
$\pi^0=[1\mathwordbox{|}{x}\cdots\mathwordbox{|}{x} n]$ and
$\pi^1=[1,\dots,n]$,
we have $\pi^0[i\leftrightarrows j]=\pi^0$ and $\pi^1[i\leftrightarrows j]=\pi^1$ for all players~$i$ and~$j$. Accordingly~$\pi^0$ and~$\pi^1$ are envy-free. Envy-free partitions are thus guaranteed to exist in our setting. The following lemma allows us to derive a polynomial characterisation of envy-freeness.
\bphnote{What follows implies a \textbf{polynomial} characterisation of envy-freeness. It is important to observe that the characterisation depends on the hedonic aspect, that is, we are making essential use of the fact that each player~$i$'s goal is a formulas of~$L_i$ rather than~$L_N$.}
\bphnote{How to get rid of the bloody case distinctions in the following proof?!}
\newcommand{_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}}{_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}}
\begin{lemma}\label{lemma:swapsub}
Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game and~$i$ and~$j$ players in~$N$, and $\varphi$ a formula in~$L_N$. Fix, furthermore, an enumeration
$k_1,\dots,k_{n-2}$ of $N\setminus\set{i,j}$ and let ${i\vec k}=ik_1,\dots,ik_{n-2}$ and $j\vec k=jk_1,\dots,jk_{n-2}$ enumerate $\ensuremath{\mathit{V}}_i\setminus\set{ij}$ and $\ensuremath{\mathit{V}}_j\setminus\set{ji}$, respectively. Then,
\[
\text{
$\pi\models\varphi_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$
\wordbox{ if and only if }{ if and only if }
$\pi[i\leftrightarrows j]\models\varphi$.
}
\]
\end{lemma}
\begin{proof}
With~$i\vec k$ and~$j\vec k$ being fixed we write~$\varphi'$ for~$\varphi_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$. The proof is then by induction on~$\varphi$.
{
\renewcommand{_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}}{'}
For the basis, let $\varphi=lm$. There are three possibilities:
\[
\scalebox{1}[1]{
\begin{enumerate*}[label=$(\alph{*})$,leftmargin=2.25em]
\item $lm=ij$,
\item $lm\in (V_i\cup V_j)\setminus\set{ij}$, and
\item $lm\notin V_i\cup V_j$.
\end{enumerate*}}
\]
If~$(a)$, we have that $lm_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}=ij_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}=ij=lm$.
Now, either $\pi(i)=\pi(j)$ or $\pi(i)\neq\pi(j)$. If the former, $\pi[i\leftrightarrows j]=\pi$ as well as both $\pi\models ij_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$ and $\pi[i\leftrightarrows j]\models ij$. If the latter, however, it can easily be seen that both $\pi\not\models ij_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$ and $\pi[i\leftrightarrows j]\not\models ij$.
For case~$(b)$, we may assume without loss of generality that $lm=ik$ for some $k\neq j$. Then, $ik_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k} = jk$. In case $\pi(i)=\pi(j)$, obviously, $\pi=\pi[i\leftrightarrows j]$ as well as $k\in \pi(i)$ if and only if $k\in\pi(j)$. Hence, $\pi\models ik_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$ if and only if $\pi[i
\leftrightarrows j]\models ik$. So, assume $\pi(i)\neq \pi(j)$. Now, either
\begin{enumerate*}[label=$(\roman{*})$,leftmargin=2.25em]
\item $k\in\pi(i)$ and $k\notin\pi(j)$,
\item $k\notin\pi(k)$ and $k\in\pi(j)$, or
\item $k\notin\pi(i)$ and $k\notin\pi(j)$.
\end{enumerate*}
If~$(i)$, $\pi\models ik_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$ as well as $\pi[i\leftrightarrows j]\models jk$. In cases~$(ii)$ and~$(iii)$, we have $\pi\not\models ik_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$ and $\pi[i\leftrightarrows j]\not\models jk$.
Finally, if~$(c)$, we have $lm_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}=lm$. As $l,m\notin\set{i,j}$, it can then easily be seen that $\pi\models lm_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$ if and only if $\pi[i\leftrightarrows j]\models lm$.
The cases $\varphi=\neg\psi$ and $\varphi=\psi\to\chi$ follow by induction.
}
\end{proof}
%
We are now in a position to state the following result.
\begin{proposition}\label{proposition:envyfree}
Let $(N,\gamma_1,\dots,\gamma_n)$ be a Boolean hedonic game. Furthermore, for every two players,~$i$ and~$j$, and enumeration
$k_1,\dots,k_{n-2}$ of $N\setminus\set{i,j}$, let ${i\vec k}=ik_1,\dots,ik_{n-2}$ and $j\vec k=jk_1,\dots,jk_{n-2}$ enumerate $\ensuremath{\mathit{V}}_i\setminus\set{ij}$ and $\ensuremath{\mathit{V}}_j\setminus\set{ij}$, respectively.
Then,
\[
\scalebox{1}[1]{
$\pi$ is envy-free \wordbox{if and only if}{ if and only if } $\displaystyle\pi\models\bigwedge_{i,j\in N}\big((\gamma_i)_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}\to\gamma_i\big)$.
}
\]
\end{proposition}
\begin{proof}
By virtue of Lemma~\ref{lemma:swapsub}, the following equivalences hold:
\[
\renewcommand{\arraystretch}{1.3}
\begin{array}{ll}
\multicolumn{2}{l}{\text{$\pi$ is envy-free}}\\
\text{iff} & \text{for all $i,j\in N$: $\pi\mathrel{R_i}\pi[i\leftrightarrows j]$}\\
\text{iff} & \text{for all $i,j\in N$: $\pi[i\leftrightarrows j]\models\gamma_i$ implies $\pi\models\gamma_i$}\\
\text{iff} & \text{for all $i,j\in N$: $\pi\models(\gamma_i)_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}$ implies $\pi\models\gamma_i$}\\
\text{iff} & \text{for all $i,j\in N$: $\pi\models(\gamma_i)_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}\to\gamma_i$}\\
\text{iff} & \text{$\pi\models\displaystyle\bigwedge_{i,j\in N}\big((\gamma_i)_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}\to\gamma_i\big)$}
\end{array}
\]
This concludes the proof.
\end{proof}
Observe that the size of $\bigwedge_{i,j\in N}\big((\gamma_i)_{i\vec k,j\vec k\leftarrow j\vec k,i\vec k}\to\gamma_i\big)$ is clearly polynomial in $\sum_{i\in T}|\gamma_i|$. Hence, a partition~$\pi$ being envy-free can be polynomially characterised.
{
\setcounter{example}{0}
\begin{example}[continued]
Recall that $\gamma_3=(31\vee 32)\wedge\neg 34$ and that player~$3$ envies player~$4$ if partition~$\pi'=[1|24|3]$ obtains.
To see how this is reflected by Proposition~\ref{proposition:envyfree}, let~$31,32$ and~$41,42$ enumerate $V_3\setminus\set{34}$ and $V_4\setminus\set{43}$, respectively. Then,
\[
((31\vee 32)\wedge\neg 34)_{31,32,41,42\leftarrow 41,42,31,32}
=
(41\vee 42)\wedge\neg 34\text.
\]
Now, both $\pi'\models (41\vee 42)\wedge\neg 34$ and $\pi'\not\models(31\vee 32)\wedge\neg 34$, and, hence,
$
\pi'\not\models(\gamma_3)_{34,31,32\leftarrow43,41,42}\to\gamma_3
$.
\end{example}
}
\mjwnote{
\section{The Scope of Boolean Hedonic Games}
\subsection{Boolean Hedonic Games Unleashed}
Some informal remarks concerning
\begin{itemize}
\item the possibility of extending the framework with general preferences---there are several possibilities,
\item the possibility of extending the framework to general coalition formation games,
\item the completeness of Boolean hedonic games, in the sense that every dichotomous hedonic game can be represented by a Boolean hedonic game.
\end{itemize}
\subsection{Applications}
\bphnote{J\'er\^ome has some ideas for this section.}
\begin{itemize}
\item Kidney exchange
\item Marriage markets
\end{itemize}
}
\section{Related Work and Conclusions}
Our motivation and approach is strongly reminiscent of the setting of
Boolean games in the context of non-cooperative game
theory~\citep{HHMW01a}. A major difference with Boolean games and
propositional hedonic games is that in Boolean games, players have
preferences over outcomes, where an outcome is a truth assignment to
outcome variables, and each outcome variable is controlled by a
specific player. This control assignment function, which is a central
notion in Boolean games, has no counterpart here, where the outcome is
a partition of the players. However, there are technical
similarities with and conceptual connections to Boolean games, especially when characterising solution
concepts. For instance, the characterisation of Nash stable partitions
by propositional formulas (Section 4) is similar to the
characterisation of Nash equilibria by propositional formulas in
Boolean games as by \citet{bonzon:2009a}. The basic Boolean games model
of~\citet{HHMW01a} was adapted to the setting of cooperative games
by \citet{dunne:2008a}. However, the logic used to specific player's
goals in the work of~\citeauthor{dunne:2008a} was not intended for
specifying desirable coalition structures, as we have done in the
present paper.
Our work also shares some common ground with the work
of \citet{bonzon2012effectivity}, who study the formation of efficient
coalitions in Boolean games, that is, coalitions whose joint abilities
allow their members to jointly achieve their goals. Our work also
bears some resemblance to the work of \citet{ElWo09a}, who were
interested in using logic as a foundation upon which to build a
compact representation scheme for hedonic games; more precisely, their
work made use of weighted Boolean formulas, and was inspired by the
\emph{marginal contribution nets} representation for cooperative games
in characteristic function form
proposed by\citet{ieong:2005a}. The focus of~\citet{ElWo09a}, however, was more
on complexity issues than in finding exact characterisations for
solution concepts.
Finally, our work
contributes to the extensive literature on compact
representations for cooperative games, which has expanded rapidly over
the past decade~\citep{chalkiadakis:2011a}.
Our characterisations of solution concepts enable to compute, using an off-the-shelf
SAT solver, a partition or all partitions satisfying a solution concept or a logical combination of solution concepts.
Of course, this translation is interesting only when we cannot do better. For instance, for solution concepts
leading to a polynomial characterisation, we cannot do better if and only if the corresponding decision problem
is {\sf NP}-complete. Identifying the complexity of finding partitions satisfying solution concepts for Boolean hedonic games
is therefore the most immediate direction of further research.
\balance
There are at least three more directions in which our work might be further developed.
First, we could think of relaxing our restriction to
dichotomous preferences and study more general hedonic games with
compact logical representations and derive exact
characterisations of solution concepts. There are several ways in which more general preferences can be incorporated in our logical framework for hedonic games. For instance, instead of a single goal, we could associate with each player a prioritised set of goals. The different possibilities in this respect, however, vary in their level of sophistication. For some of the cruder extensions our results extend naturally and straightforwardly. For the more sophisticated settings more research seems to be required, which falls beyond the scope of this paper.
Second, our restriction to
{\em hedonic} preferences can also be relaxed, so that players may
have preferences that do depend not only on on the coalition to which
they belong. This would also pave the way to a more general {\em logic
of coalition structures}. Solution concepts, once generalised, can
hopefully be characterised. (We have positive preliminary results that
go into this direction).
A third topic of future research would be the characterisation of classes of hedonic and coalition formation games in our logic. As mentioned above, various classes of hedonic games that allow for a concise representation have been proposed in the literature. It would be interesting to see whether these classes can also be polynomially characterised in our logic.
\paragraph{Acknowledgments}
The authors would like to thank the anonymous referees of LOFT 2014 for their constructive comments.
Haris Aziz has been supported by NICTA which is funded by the Australian
Government as represented by the Department of Broadband,
Communications and the Digital Economy and the Australian Research
Council through the ICT Centre of Excellence program.
J{\'e}r{\^o}me Lang has been supported by the ANR project CoCoRICo-CoDec.
Paul Harrenstein and Michael Wooldridge have been supported by the ERC under Advanced Grant 291528 (``RACE'').
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,332 |
namespace arrow {
namespace internal {
template <typename c_int>
static std::vector<std::string> MakeIntStrings(int32_t num_items) {
using c_int_limits = std::numeric_limits<c_int>;
std::vector<std::string> base_strings = {"0",
"5",
c_int_limits::is_signed ? "-12" : "12",
"34",
"99",
c_int_limits::is_signed ? "-111" : "111",
std::to_string(c_int_limits::min()),
std::to_string(c_int_limits::max())};
std::vector<std::string> strings;
for (int32_t i = 0; i < num_items; ++i) {
strings.push_back(base_strings[i % base_strings.size()]);
}
return strings;
}
static std::vector<std::string> MakeFloatStrings(int32_t num_items) {
std::vector<std::string> base_strings = {"0.0", "5", "-12.3",
"98765430000", "3456.789", "0.0012345",
"2.34567e8", "-5.67e-8"};
std::vector<std::string> strings;
for (int32_t i = 0; i < num_items; ++i) {
strings.push_back(base_strings[i % base_strings.size()]);
}
return strings;
}
static std::vector<std::string> MakeTimestampStrings(int32_t num_items) {
std::vector<std::string> base_strings = {"2018-11-13 17:11:10", "2018-11-13 11:22:33",
"2016-02-29 11:22:33"};
std::vector<std::string> strings;
for (int32_t i = 0; i < num_items; ++i) {
strings.push_back(base_strings[i % base_strings.size()]);
}
return strings;
}
template <typename ARROW_TYPE, typename C_TYPE = typename ARROW_TYPE::c_type>
static void IntegerParsing(benchmark::State& state) { // NOLINT non-const reference
auto strings = MakeIntStrings<C_TYPE>(1000);
StringConverter<ARROW_TYPE> converter;
while (state.KeepRunning()) {
C_TYPE total = 0;
for (const auto& s : strings) {
C_TYPE value;
if (!converter(s.data(), s.length(), &value)) {
std::cerr << "Conversion failed for '" << s << "'";
std::abort();
}
total = static_cast<C_TYPE>(total + value);
}
benchmark::DoNotOptimize(total);
}
state.SetItemsProcessed(state.iterations() * strings.size());
}
template <typename ARROW_TYPE, typename C_TYPE = typename ARROW_TYPE::c_type>
static void FloatParsing(benchmark::State& state) { // NOLINT non-const reference
auto strings = MakeFloatStrings(1000);
StringConverter<ARROW_TYPE> converter;
while (state.KeepRunning()) {
C_TYPE total = 0;
for (const auto& s : strings) {
C_TYPE value;
if (!converter(s.data(), s.length(), &value)) {
std::cerr << "Conversion failed for '" << s << "'";
std::abort();
}
total += value;
}
benchmark::DoNotOptimize(total);
}
state.SetItemsProcessed(state.iterations() * strings.size());
}
template <TimeUnit::type UNIT>
static void TimestampParsing(benchmark::State& state) { // NOLINT non-const reference
using c_type = TimestampType::c_type;
auto strings = MakeTimestampStrings(1000);
auto type = timestamp(UNIT);
StringConverter<TimestampType> converter(type);
while (state.KeepRunning()) {
c_type total = 0;
for (const auto& s : strings) {
c_type value;
if (!converter(s.data(), s.length(), &value)) {
std::cerr << "Conversion failed for '" << s << "'";
std::abort();
}
total += value;
}
benchmark::DoNotOptimize(total);
}
state.SetItemsProcessed(state.iterations() * strings.size());
}
BENCHMARK_TEMPLATE(IntegerParsing, Int8Type);
BENCHMARK_TEMPLATE(IntegerParsing, Int16Type);
BENCHMARK_TEMPLATE(IntegerParsing, Int32Type);
BENCHMARK_TEMPLATE(IntegerParsing, Int64Type);
BENCHMARK_TEMPLATE(IntegerParsing, UInt8Type);
BENCHMARK_TEMPLATE(IntegerParsing, UInt16Type);
BENCHMARK_TEMPLATE(IntegerParsing, UInt32Type);
BENCHMARK_TEMPLATE(IntegerParsing, UInt64Type);
BENCHMARK_TEMPLATE(FloatParsing, FloatType);
BENCHMARK_TEMPLATE(FloatParsing, DoubleType);
BENCHMARK_TEMPLATE(TimestampParsing, TimeUnit::SECOND);
BENCHMARK_TEMPLATE(TimestampParsing, TimeUnit::MILLI);
BENCHMARK_TEMPLATE(TimestampParsing, TimeUnit::MICRO);
BENCHMARK_TEMPLATE(TimestampParsing, TimeUnit::NANO);
} // namespace internal
} // namespace arrow
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,574 |
\section{Introduction}
Given a collection of $n$ independent
uniformly distributed random points in a $d$-dimensional
cube of volume $n$
(the so-called \textit{binomial point process}),
let $V$ denote the (random) total volume
of the union of interpenetrating balls of fixed radius
$\rho$ centered at these points,
and let $S$ denote the number of balls of radius $\rho/2$
(centered at the same set of points)
which are singletons,
that is, do not overlap any other such ball.
These variables are fundamental topics of interest
in the stochastic geometry of coverage processes
and random geometric graphs \cite{Hallbk,Molchanov,Pe,SKM}.
As $n \to\infty$ with $\rho$ fixed
(the so-called thermodynamic limit), both $V$ and $S$
are known to satisfy a central limit theorem (CLT)
\cite{Moran2,Pe,PY1}.
In the present work we provide
associated Berry--Esseen
type
results; that is, we show under periodic boundary conditions
that the cumulative distribution
functions converge to that of the normal at
the same $O(n^{-1/2})$ rate as for a sum of $n$ independent identically
distributed variables, and provide bounds on
the quality of the normal approximation for finite $n$.
Were we to consider instead a
Poisson-distributed number of points,
that is, a~Poisson point process instead of a binomial one,
both of our variables of interest could be expressed as sums
of locally dependent random variables, and thereby
Berry--Esseen
type bounds could be (and have been) obtained
by known methods \cite{AB,GHH,PR,PYStein}. But with
a nonrandom number of points, the local dependence is lost
and the de-Poissonization arguments in \cite{Pe,PY1} do
not provide error bounds for the de-Poissonized CLTs.
The early work of Moran \cite{Moran1,Moran2} on $V$ was
in response to queries in the
statistical physics literature (including the well-known
paper of Widom and Rowlinson \cite{WR}) which specifically addressed
normal approximation of $V$ for nonrandom $n$, and
in general, it seems worthwhile to study the
de-Poissonized setting since in practice one might well
observe the actual number of points, in which case
the conditional distribution of
any test statistic, based on what is observed,
will be over a binomial rather than a Poisson point process.
The variables $V$ and $S$
are just two of a large class
of variables of interest that can be expressed
as a sum, over the $n$ points, of terms that depend only
on the configuration of nearby points in some sense.
General CLTs have been developed
for such variables
\cite{PY1,PeEJP}
and general Berry--Esseen
type results are available in the Poissonized setting
\cite{GHH,PR,PYStein},
but it remains open to provide a generally applicable
Berry--Esseen
type result for such sums when $n$ is nonrandom
(see however \cite{Chat}, which is discussed further in Section
\ref{secgr}).
However, there seem to be good prospects of adapting
the approach of the present paper (which is new in
the geometrical setting) to
a wider class of geometrical sums.
Our approach to normal approximation is based on
Stein's method via size-biased couplings.
Given a nonnegative random variable $Y$ with
positive finite mean $\mu= \mathbb EY$, we say $Y'$ has the $Y$ size-biased
distribution if $P[Y' \in dy] = (y/\mu) P[Y \in dy]$, or more formally,
if
\begin{eqnarray}\label{formalsb}
\mathbb E[Yg(Y)]=\mu\mathbb Eg(Y')\qquad
\mbox{for bounded continuous functions $f$.}
\end{eqnarray}
The method of size-biased couplings was introduced
by Baldi, Rinott and Stein \cite{BRS}, who used it to develop bounds of order
$\sigma^{-1/2}$ to the normal approximation to the number of local
maxima $Y$ of a random function on a graph, where
$\sigma^2=\mbox{Var}(Y)$. Goldstein and Rinott \cite{GR} extended the
technique to multivariate normal approximations, and improved the
rate to $\sigma^{-1}$ for the expectation of smooth functions of a
vector $\mathbf{Y}$ recording the number of edges with certain fixed
degrees in a random graph.
In \cite{Gold},
the method is used to
give bounds of order $\sigma^{-1}$
for various functions on graphs and permutations.
Here we shall use Lemma \ref{LGthm} below, which
improves the constant
in a more general result from
\cite{Gold}.
Loosely speaking, this result says that given any coupling of $Y$ and
$Y'$ on a common space, an upper bound on the distance between the
distribution of $Y$ and the normal can be found which involves
functions of the joint distribution of $Y,Y'$ in terms of (i) the
uniform distance between $Y$ and $Y'$, that is, the $L^\infty$ norm of
$Y-Y'$, and (ii) the
variance of $\mathbb E[Y-Y'|Y]$.
In Section \ref{couplsec} we show how to find a coupled realization of
$Y'$ that is uniformly close to $Y$, for those $Y$ under consideration
here. To do this we show that here the size-biasing amounts to
conditioning the (binomial) number of points falling in a certain
(randomly located) $\rho$-ball to be nonzero, and can be achieved by
modifying at most a single point location to obtain $Y'$ from $Y$, so
that $\| Y'-Y\|_\infty$ is bounded.
This construction may be of independent interest, along with Lemma
\ref{condsblem} (a general result on how to size-bias a conditional
probability) and Lemma \ref{LGthm}.
\section{Results}
\label{secgr}
Let $d \geq1$ and $n \geq4$ be integers. Suppose $U_1, \ldots, U_n$ are
independent random $d$-vectors, uniformly distributed over the cube
$C_n := [0,n^{1/d})^d$ (we write $U_{i}$ rather than $U_{n,i}$ because
the value of $n$ should be clear from the context). Write $\mathcaligr
U_n$ for the point set $\{U_1,\ldots,U_n\}$. For $x, y$ in the cube
$C_n$, let $D(x,y)$ denote the distance between $x$ and $y$ under the
Euclidean toroidal metric on $C_n$. For $x \in C_n$ and $r >0$ let
$B_r(x)$ denote the ball $\{y \in C_n\dvtx D(x,y) \leq r\}$. Let
$B_{i,r}$ denote the ball $B_r(U_i)$. Given $r$, the collection of
balls $B_{i,r}$ form a coverage process (also known as a germ-grain
model) in $C_n$; see \cite{Hallbk,SKM}. Let $\rho> 0$, and define
\begin{eqnarray}
\label{Vndef}
V &:=& \operatorname{Volume} \Biggl( \bigcup_{i=1}^n
B_{i,\rho} \Biggr) ;
\\
\label{Yndef}
S &:=& \sum_{i=1}^n \mathbf{1} \bigl\{
\mathcaligr{U}_n
\cap
B_{i,\rho}
=
\{U_i\}
\bigr\}.
\end{eqnarray}
Then $V$ is the total covered volume for the coverage process
with $r = \rho$, while $ S$ is the number of
singletons (isolated balls) in the case $r= \rho/2$, and may also be
viewed as the number of isolated points in the geometric graph
on vertex set $\mathcaligr{U}_n$ with distance parameter $\rho$ \cite{Pe}.
Let $Z$ denote a standard normal random variable.
Given a random variable $X$ with $\operatorname{SD}(X) := \sqrt
{\operatorname{Var}(X)} \in(0,
\infty)$,
let $D_X$ denote
the Kolmogorov distance between
the distribution of $X$ (scaled and centered) and that of $Z$, that is,
\[
D_X:= \sup_{t \in\mathbb{R}} \biggl| P \biggl[\frac{ X- \mathbb
EX}{\operatorname{SD}(X) }
\leq t \biggr] - P[Z \leq t] \biggr|.
\]
Our main results provide bounds in the normal approximation
for $V$ and $S$; if $\rho$ is fixed then as $n \to\infty$,
\begin{eqnarray}
\label{main02}
D_V
= O(n^{-1/2}) ;\qquad
D_S
= \Theta(n^{-1/2}).
\end{eqnarray}
Recall that $a_n = \Theta(b_n)$ means that $a_n = O(b_n)$
and $b_n = O(a_n)$.
We conjecture that the first bound in
(\ref{main02}) can
be improved to $\Theta(n^{-1/2})$.
To state our results more precisely, we need further notation.
Set $\pi_d$ to be the volume of the unit ball in $d$ dimensions,
that is, $\pi_d: = \pi^{d/2} / \Gamma(1 +d/2)$,
and $\phi: = \pi_d\rho^d$.
We say two unit balls \textit{touch} if their closures intersect, but
their interiors do not.
Let $\kappa_d$ (respectively, $\kappa_d^*$)
denote the maximum number of closed unit balls in
$d$ dimensions that can be packed so they all intersect
(respectively, touch)
a~closed unit ball at the origin, but are
disjoint from each other (respectively, have disjoint interiors).
Then $\kappa^*_d$ is
the so-called \textit{kissing number} in $d$ dimensions, which has been studied
for centuries (see \cite{Conway,Zong}).
It is not hard to see
$\kappa_d^*$ is an upper bound for $\kappa_d$,
and in most dimensions it seems likely that $\kappa_d = \kappa_d^*$,
but $\kappa_2 =5$ whereas $\kappa_2^*=6$.
It is known that
$\kappa_3 =
\kappa^*_3 = 12$.
Set $\kappa_d^+: = 1 + \kappa_d$.
Set $\mu_V := \mathbb E[V]$,
$\mu_S := \mathbb E[S]$,
$\sigma_V := \operatorname{SD}(V)$,
and $\sigma_S := \operatorname{SD}(S)$.
It is straightforward to write down formulae for
$\mu_V$, $\mu_S$, $\sigma_V^2$ and $\sigma_S^2$; see
(\ref{0522a}),
(\ref{varV}) and
(\ref{0521a}).
Our first two main results provide nonasymptotic upper bounds
on the Kolmorogorov distance.
\begin{theo}
\label{thm2}
If $ n > 6^d \phi$, then
\begin{eqnarray*}
D_V
\leq
\frac{ \mu_V}{5 \sigma_V^2}
\Biggl(
\sqrt{ \frac{11 \phi^2}{\sigma_V} + \frac{5 \sqrt{\eta_V(n,\rho
)}}{\sqrt{n}}
}
+ \frac{2\phi}{\sqrt{\sigma_V}}
\Biggr)^2
\end{eqnarray*}
with
\begin{eqnarray}\label{etaVdef}\qquad
\eta_V(n,\rho) &:=&
2 \phi^2 \bigl((3^{d} +1)\phi+1\bigr)^2
\nonumber\\
&&{}\times
\biggl( 1 +
(2^d +1) 6^d \phi+
\biggl( \frac{2 n - 6^d \phi}{
n - 6^d \phi} \biggr)
6^{2d} \phi^2
\biggr)
\\
&&{} + 2\phi^4
\biggl(
3 (4^d + 2^d) \phi+
3(4^d) \phi^{2} \biggl(
\frac{2n- 3(2^d) \phi}
{n- 3(2^d) \phi} \biggr)
+ 4 + \frac{2}{n} \biggr).\nonumber
\end{eqnarray}
\end{theo}
\begin{theo}
\label{thm1}
If
$n > \max( 3^d, 2^{d+1} +1) \phi$, then
\begin{eqnarray*}
D_S
\leq
\frac{n- \mu_S}{ 5 \sigma_S^2}
\Biggl( \sqrt{ \frac{ 11 (\kappa_d^+)^2}{ \sigma_S}
+ \frac{5 \sqrt{\eta_S(n,\rho)}}{\sqrt{n}} }
+ \frac{2 \kappa_d^+}{\sqrt{\sigma_S}}
\Biggr)^{2}
\end{eqnarray*}
with
\begin{eqnarray}
\label{etaYdef}\hspace*{30pt}
\eta_S(n,\rho) &:=&
2(1+ 2 \kappa_d)^2 \biggl( 1 + (2^d+1) 3^d \phi+
\biggl( \frac{2n- 3^d \phi}{
n - 3^d \phi} \biggr) 9^{d} \phi^2 \biggr)
\nonumber\\
&&{} + \frac{
( \kappa_d^+)^2
}{2} \biggl(
\bigl(2^d + 2(3^d) +3\bigr) \phi\\
&&\hspace*{47pt}{} +
(2^{d+1}+1)\biggl( \frac{2n - (2^{d+1} +1) \phi
}{n - (2^{d+1} +1) \phi}
\biggr)\phi^2
+ \frac{4n -2}{n-1}
\biggr).
\nonumber
\end{eqnarray}
\end{theo}
By using
the inequality $(x+y)^2 \leq2(x^2+ y^2)$, the
bounds in Theorems \ref{thm2} and~\ref{thm1}
can replaced by bounds which are simpler, though
less sharp.\vspace*{1pt}
The next result confirms that
for large $n$, all of
$\mu_V$, $\sigma_V^2$, $\mu_S$ and $\sigma_S^2$
are $\Theta(n)$, so that (\ref{main02})
follows from Theorems \ref{thm2} and \ref{thm1}.
To provide details we require further notation.
For $0 \leq r \leq2$, write
$\omega_d(r)$ for the volume of the union of two unit
balls in $\mathbb{R}^d$ with
centers distant $r$ apart (see (\ref{omMoran})
for a formula).
Define the integral
\begin{equation}\label{Jdef}
J_{r,d}(\rho) := d \pi_d\int_0^r \exp( - \rho^d \omega_d(t) ) t^{d-1}\,dt
\end{equation}
and the functions
\begin{eqnarray}\qquad
\label{gVdef}
g_V(\rho) & := & \rho^d J_{2,d}(\rho) -
(2^d \phi+ \phi^2)
e^{-2\phi} ;
\\
\label{gWdef}
g_S(\rho) & := &
e^{-\phi} - \bigl(1 + (2^d-2) \phi+ \phi^2\bigr)
e^{-2 \phi} + \rho^d \bigl(J_{2,d}(\rho) -J_{1,d}(\rho)\bigr).
\end{eqnarray}
Also, define
$ \eta_V(\rho) : = \lim_{n \to\infty} \eta_V(n,\rho) $
and
$ \eta_S(\rho)
: = \lim_{n \to\infty} \eta_S(n,\rho)$.
Formulae for
these limits
are immediate from the definitions
(\ref{etaVdef}) and (\ref{etaYdef}).
\begin{theo}
\label{thmlimsup}
If $\rho$ is fixed then as $n \to\infty$,
\begin{eqnarray}
\label{meanlim}
\lim_{n \to\infty} \bigl( 1-n^{-1} \mu_V(\rho) \bigr) & = &
\lim_{n \to\infty} ( n^{-1} \mu_S(\rho)) =
e^{-\phi};
\\
\label{varVlim}
\lim_{n \to\infty} ( n^{-1} \sigma_V^2 ) & = & g_V(\rho) > 0;
\\
\label{varWlim}
\lim_{n \to\infty} ( n^{-1} \sigma_S^2 ) & = & g_S(\rho) > 0
\end{eqnarray}
and
\begin{eqnarray}\qquad
\label{main2}
\limsup_{n\to\infty}
(n^{1/2}
D_V
)
& \leq&
\frac{1 - e^{-\phi}}{5 g_V(\rho)}
\Biggl(
\sqrt{
\frac{11\phi^2}{g_V(\rho)^{1/2}} + 5
\eta_V^{1/2}
}
+ \frac{2 \phi}{g_V(\rho)^{1/4}
}
\Biggr)^2;
\\
\label{main1}
\limsup_{n\to\infty} (n^{1/2}
D_S
)
& \leq&
\frac{1 - e^{-\phi}}{5 g_S(\rho)}
\Biggl(
\sqrt{
\frac{11(\kappa_d^+)^2}{g_S(\rho)^{1/2}} + 5
\eta_S^{1/2}
}
+ \frac{2 \kappa_d^+}{g_S(\rho)^{1/4}}
\Biggr)^2;
\\
\label{SLB}
\liminf_{n\to\infty} (n^{1/2}
D_S
)
& \geq& (8 \pi g_S(\rho))^{-1/2} .
\end{eqnarray}
\end{theo}
Theorems \ref{thm2} and \ref{thm1}
are proved in Sections \ref{proof-2}
and \ref{proof-1}, respectively.
Theorem \ref{thmlimsup} is proved in Section \ref{secvar},
where we also derive numerical values for
the asymptotic
upper bounds in Theorem \ref{thmlimsup}, for some particular cases.
\subsection*{Remarks}
The limiting variances in (\ref{varVlim}),
respectively (\ref{varWlim}),
are consistent with those
given by Moran \cite{Moran1,Moran2},
respectively,
Penrose (\cite{Pe}, Theorem 4.14).
Moran and Penrose do not
explicitly rule out the possibility that these limiting variances might be
zero, as we do here.
Clearly (\ref{main2}) and (\ref{main1}) imply central limit theorems whereby
both $(V- \mu_V)/\sigma_V$ and $(S- \mu_S)/\sigma_S$
converge in distribution to the standard normal,
thereby providing an alternative to existing
proofs of these central limit theorems \mbox{\cite{Moran2,PY1,Pe}}.
In the Poissonized setting, nonasymptotic
bounds analogous to those in Theorems \ref{thm2} and \ref{thm1}
are given in \cite{PR}
and imply $O(n^{-1/2})$
bounds analogous to
(\ref{main2}) and (\ref{main1}).
In the de-Poissonized setting
considered here,
Chatterjee \cite{Chat} provides
bounds similar to those in (\ref{main2}) and (\ref{main1}),
which hold for general metric spaces,
but using the Kantorovich--Wasserstein distance, rather than
the Kolmogorov distance considered here, and without
providing any explicit constants.
As stated in~\cite{Chat}, ``obtaining optimal rates for the Kolmogorov
distance requires extra work and new ideas.''
Generalizations of our results
should be possible
in many directions.
These include:
\subsection*{More general germ-grain models}
Replace the balls of fixed radius in the description of
$V$ and $S$ by
(independent identically distributed)
balls of
random radius, or more generally, random shapes.
\subsection*{Random measures}
Consider the random measure associated
with $V$ (the Lebesgue measure on the covered region)
or with $S$ (a sum of Dirac measures at
the isolated points), and look at normal approximation for
the random variable
given by the integral of a test function $f$ on $C_n$ with respect
to that measure.
\subsection*{Euclidean distance}
Suppose in the definition of $V$ and $S$, that
the periodic boundary conditions on $C_n$ are dropped, that is, the toroidal
distance $D$ is replaced by the ordinary Euclidean distance.
\subsection*{Nonuniform points}
Consider a sequence of
independent random points $(\mathbf{X}_n)_{n \geq1}$ with a common
density function
$\nu\dvtx \mathbb{R}^d \to\mathbb{R}$. Placing balls of radius $r_n$
around each point of $\mathcaligr{X}_n := \{ \mathbf{X}_1,\ldots
,\mathbf{X}_n\}$,
for some specified sequence $r_n$ tending to zero,
one may define quantities analogous to $V$ and $S$.
When $r_n \propto n^{-1/d}$ this is a rescaling of our
model but allows for nonuniform $\nu$. Our approach
might also provide information
about other asymptotic regimes.
\subsection*{$k$-nearest neighbors}
Let $k \in\mathbb{N}$ and consider the
number of points $U_i$ whose $k$th nearest neighbor in the point
set $\mathcaligr{U}_n \setminus\{U_i\}$ lies at a distance greater
than $\rho$.
The case $k=1$ reduces to $S$.
These extensions generally seem to be nontrivial, and worthy of further
study.
\section{Lemmas}
The proof of (\ref{main2}) and (\ref{main1}) is based on
the following result. This result
improves the constant which would be obtained by applying
the more general
Theorem 1.2 of
\cite{Gold} to the particular case of Kolmogorov distance.
\begin{lemm}
\label{LGthm}
Let $Y \ge0$ be a random variable with mean $\mu$ and variance
$\sigma^2 \in(0,\infty)$, and let $Y^s$ be defined on the same
probability
space, with the $Y$-size biased distribution. If $P[|Y^s-Y| \le B] =1$
for some constant $B >0$,
then
\begin{equation}\label{090520a}
D_Y
\le
\frac{\mu}{5 \sigma^2} \Biggl(\sqrt{\frac{11B^2}{\sigma} + 5 \Delta} + \frac{2 B}{
\sqrt{\sigma}}\Biggr)^2,
\end{equation}
where
$\Delta: =
\sqrt{\operatorname{Var}(\mathbb E[Y^s-Y|Y])}$.
\end{lemm}
\begin{pf}
Given $z \in\mathbb{R}$ and $\varepsilon>0$, define real-valued functions
$h_z$ and $h_{z,\varepsilon}$
by
\[
h_z(x)= \mathbf{1}_{(-\infty,z]}(x),\qquad
h_{z,\varepsilon}(x)=\varepsilon^{-1}\int_0^\varepsilon
h_{z}(x-t)\,dt,\qquad
z \in\mathbb{R}.
\]
Then with $W:=(Y-\mu)/\sigma$ and
$Z$
denoting a standard normal,
by definition
\begin{equation}\label{090528a}
D_Y =\sup\{|\mathbb Eh_z(W) - \mathbb Eh_z( Z)|\dvtx z \in\mathbb{R}\}.
\end{equation}
For $\varepsilon>0$,
set
\begin{equation}\label{090528b}
D_Y^\varepsilon:=\sup\{ |\mathbb Eh_{z,\varepsilon}(W) - \mathbb
Eh_{z,\varepsilon}(Z)|\dvtx
z \in\mathbb{R}\}.
\end{equation}
Fix $z$ and $\varepsilon$,
and let $f$ be the unique bounded solution of the Stein equation
\[
f'(w)-wf(w)=h_{z,\varepsilon}(w)-\mathbb Eh_{z,\varepsilon}(Z)
\]
for $h_{z,\varepsilon}$; see \cite{CS}.
With some abuse of notation, let $W^s=(Y^s-\mu)/\sigma$.
Then
\begin{eqnarray}
\label{L-BD:KSepsm}
&&\mathbb E [ h_{z,\varepsilon}(W)-\mathbb Eh_{z,\varepsilon}(Z)
] \nonumber\\
&&\qquad= \mathbb E [ f'(W)-Wf(W) ] \nonumber\\
&&\qquad= \mathbb E \biggl[ f'(W)- \frac{\mu}{\sigma}\bigl(f(W^s)-f(W)\bigr)
\biggr]\\
&&\qquad=\mathbb E \biggl[f'(W)\biggl(1-\frac{\mu}{\sigma}(W^s-W)\biggr)\nonumber\\
&&\qquad\quad\hspace*{10.8pt}{} - \frac{\mu
}{\sigma
}\int_0^{W^s-W}\bigl(f'(W+t)-f'(W)\bigr)\,dt \biggr] \nonumber.
\end{eqnarray}
The following bounds on the solution $f$ can be found in \cite{CS}:
\begin{equation}
\label{L-BD:ChSh04bd}
|f'(w)| \le 1
\end{equation}
and
\begin{equation}
\label{090615a}
|f'(w+t)-f'(w)|
\le (|w|+1)|t|+ \varepsilon^{-1} \int_{t \wedge0}^{t \vee0}
\mathbf{1}_{[z ,z+\varepsilon]} (w+u) \,du.
\end{equation}
Noting that $\mathbb EY^s=\mathbb EY^2/\mu$ by (\ref{formalsb}) with
$g(y)=y$, we find
that
\[
\frac{\mu}{\sigma}\mathbb E[W^s-W]=\frac{\mu}{\sigma^2} \biggl(
\frac
{\mathbb EY^2}{\mu}-\mu\biggr)=1,
\]
and therefore, taking expectation by conditioning, and then
using
(\ref{L-BD:ChSh04bd}), we have
\[
\biggl\vert\mathbb E \biggl\{ f'(W) \mathbb E \biggl[ 1 - \frac{ \mu}{\sigma}
(W^s-W)\Big|W \biggr]
\biggr\}\biggr\vert
\le\frac{ \mu}{\sigma} \sqrt{\mbox{Var}(\mathbb E
[W^s-W|W])}= \frac{ \mu}{\sigma^2}\Delta.
\]
Now, using
(\ref{L-BD:KSepsm}) and
(\ref{090615a})
yields
\begin{eqnarray}\label{L-BD:sbbound}\qquad
&&|\mathbb E [ h_{z,\varepsilon}(W)- \mathbb Eh_{z,\varepsilon}
(Z) ]|
\nonumber\\
&&\qquad\leq\frac{\mu}{\sigma^2}\Delta
+ \frac{\mu}{\sigma}\mathbb E
\biggl[\int_{(W^s-W)\wedge0}^{(W^s-W)\vee0}
(|W|+1)|t| \,dt
\nonumber\\
&&\qquad\quad\hspace*{57.2pt}{} +
\int_{-B/\sigma}^{B/\sigma}
\varepsilon^{-1}\int_{t \wedge0}^{t \vee0}
\mathbf{1}_{[z , z+\varepsilon]}(W+u) \,du\,dt \biggr]\\
&&\qquad\le\frac{\mu}{\sigma^2}\Delta+
\frac{\mu B^2}{2 \sigma^3}(\mathbb E|W|+1) +\frac{\mu}{\sigma
}\varepsilon
^{-1}\int_{-B/\sigma}^{B/\sigma} \int_{t \wedge0}^{t \vee
0}(0.4\varepsilon+2 D_Y)\,du\,dt\nonumber\\
&&\qquad\le
\frac{\mu}{\sigma^2}\Delta+ 1.4\frac{\mu}{\sigma^3}B^2 +\frac
{2\mu}{\sigma^3}B^2\varepsilon^{-1}D_Y,\nonumber
\end{eqnarray}
where in the second-to-last inequality above we have used the fact that
\[
P [\alpha\le W \le\beta] \le(\beta-\alpha)/\sqrt{2 \pi} + 2 D_Y,
\]
and in the last, the fact that $\mathbb E|W| \le1$.
By (\ref{090528a})
we see that $D_Y^\varepsilon$ is bounded by (\ref{L-BD:sbbound}),
and since
(\ref{090528a}) and (\ref{090528b}) imply
$
D_Y \le0.4\varepsilon+ D_Y^\varepsilon$,
substitution yields
\[
D_Y \le\gamma(\varepsilon) :=\frac{a\varepsilon
+b}{1-c/\varepsilon},
\]
where
\[
a:=\frac{2}{5},\qquad
b:= \frac{\mu}{\sigma^2}\Delta+\frac{7}{5}\frac{\mu}{\sigma
^3}B^2\quad \mbox{and}\quad c:= \frac{2\mu B^2}{\sigma^3}.
\]
The optimum bound on $D_Y$ is at the positive root of
$\gamma'(\varepsilon)=0$, namely
$\varepsilon=c + r $ where
$r :=\sqrt{c^2+cb/a}$.
We wish to calculate $\gamma(c+r)$.
The denominator equals
\[
1-\frac{c}{c+r}=1-\frac{c(c-r)}{c^2-r^2}
=1+\frac{a(c-r)}{b}=\frac{b+a(c-r)}{b},
\]
and therefore
\begin{eqnarray*}
\gamma(c+r)
&=&
b \biggl( \frac{a(c+r)+b}{a(c-r)+b} \biggr)
= b \biggl( \frac{c+b/a+r}{c+b/a-r} \biggr)
\\ &=&
b \biggl( \frac{(c+b/a+r)^2}{(c+b/a)^2-r^2} \biggr)
= b \biggl( \frac{(c+b/a+r)^2}{cb/a+(b/a)^2} \biggr)\\
&=& a \biggl( \frac{(c+b/a+r)^2}{c+(b/a)} \biggr)
=
a \biggl( \frac{(c+b/a+\sqrt{c}\sqrt{c+b/a})^2}{c+(b/a)} \biggr)
\\
&= & a \bigl( \sqrt{c+b/a}+\sqrt{c} \bigr)^2
= \frac{2}{5} \Biggl(\sqrt{\frac{11}{2}\frac{\mu B^2}{\sigma
^3}+\frac{5}{2}\frac{\mu}{\sigma^2}\Delta} + \sqrt{\frac{2\mu
B^2}{\sigma^3}} \Biggr)^2,
\end{eqnarray*}
and this bound on $D_Y$
yields (\ref{090520a}).
\end{pf}
Let $\operatorname{Bin}(n,p)$ denote the binomial distribution with
parameters $n \in\mathbb{N}$ and $p \in(0,1)$.
Our next two lemmas are concerned with binomial
and conditioned binomial distributions.
Lemma \ref{binlem} is used to prove Lemma
\ref{bincouplem}.
\begin{lemm}
\label{binlem}
Let $m \in\mathbb{N}$ and $p \in(0,1)$.
Suppose $N \sim\operatorname{Bin}(m,p)$,
and $\mathcaligr{L}(N') = \mathcaligr{L}(N | N >0)$,
$N''-1 \sim\operatorname{Bin}(m-1,p)$. Then
for all $k \in\mathbb{N}$,
\begin{equation}\label{binlemeq}
P[ N \geq k ] \leq P [ N' \geq k ] \leq P[ N'' \geq k].
\end{equation}
\end{lemm}
\begin{pf}
The first inequality in (\ref{binlemeq}) is easy since
for $k \geq1$, by definition
$P[N' \geq k]
= P[N \geq k]/P[N \geq1]$.
It remains to prove the second inequality.
Suppose $\xi_1,\xi_2,\ldots$ are independent Bernoulli random
variables with parameter $p$. Let $M= \min\{ i\dvtx \xi_i =1\}$ and
$\tilde{N}'' := \sum_{i=M}^{M+m-1} \xi_i$.
Then $M$ and $\tilde{N}''$
are independent and $\mathcaligr{L}(\tilde{N}'') = \mathcaligr{L}(N'')$.
Define the random variables
\[
J:= \biggl\lceil\frac{M}{m} \biggr\rceil\quad\mbox{and}\quad
\tilde{N}' := \sum_{i=m(J-1)+1}^{mJ} \xi_{i}.
\]
In other words,
split the sequence of Bernoulli trials into disjoint intervals
of length~$m$, and let
$\tilde{N}'$ denote the number
of successful Bernoulli trials in the
first such interval that contains at least one successful trial.
Then $\tilde{N}'$ has the distribution of $N'$, and by construction
$ \tilde{N}' \leq\tilde{N}''$ almost surely.
Since $\tilde{N}''$ has the distribution of $N''$, this shows that
$N'$ is stochastically
dominated by $N''$, that is, the second inequality in (\ref{binlemeq}) holds.
\end{pf}
Our next lemma demonstrates the existence of a ``uniformly close
coupling'' of
random variables with a binomial distribution, and with
the same distribution conditioned to be nonzero (denoted,
respectively, $N$ and $M$ in the lemma).
This result will be used in Section \ref{couplsec} to
provide a uniformly close coupling of
$V$ [given by (\ref{Yndef})] and its size
biased version,
and likewise for
$S$ (in fact, for $n -S$).
\begin{lemm}
\label{bincouplem} Let $m \in\mathbb{N}$ and $p \in(0,1)$. Suppose $N
\sim\operatorname{Bin}(m,p)$, with
$N= \sum_{i=1}^m \xi_i$ where $\xi_i$ are independent
Bernoulli variables with parameter $p$.
Defining $\pi_k$
by
\begin{equation}\label{pikdef}
\pi_k :=
\cases{
\dfrac{P[N > k |N >0]
- P[N > k] }{P[N=k](1 - (k/m))},
&\quad if $0 \leq k \leq m-1$,
\cr
0, &\quad if $k= m$,}
\end{equation}
we then have $0 \leq\pi_k \leq1$ for all $k \in\{0,\ldots,m\}$.
Suppose also that $\mathcaligr{B}$ is a further Bernoulli variable with
$P(\mathcaligr{B}=1|\xi_1,\ldots,\xi_m)=\pi_N$, and suppose $I$ is
an independent
discrete uniform random variable over $\{1,2,\ldots,m\}$.
Set $ M := N + (1-\xi_I)\mathcaligr{B}$,
that is, let $M$ be given by the same sum as $N$ except that
if $\mathcaligr{B}=1$ the $I$th term is set to 1.
Then
\begin{equation}\label{1210a}
\mathcaligr{L}(M)=\mathcaligr{L}(N|N>0).
\end{equation}
\end{lemm}
\begin{pf}
Lemma \ref{binlem} shows $\pi_k \ge0$.
For the upper bound,
set $N'' = 1 + \sum_{i=2}^m \xi_i$.
Then $N'' - 1 \sim\operatorname{Bin}(m-1,p)$ and
$N'' $ is equal either to $N$ or to $N+1$, with
$P[N''=k+1 |N=k] = 1-(k/m)$
for $0 \leq k \leq m$.
Hence for all $k$,
by Lem\-ma~\ref{binlem},
\[
P[N > k] + P[N=k](1 -k/m) = P[N'' > k] \geq P[N > k |N > 0]
\]
so $\pi_k \leq1$.
Also, assertion (\ref{1210a})
follows
by (\ref{pikdef}) and the fact that
\[
\{M > k \} = \{N > k\} \cup\{N=k, \mathcaligr{B}=1,
\xi_I=0\}.
\]
\upqed\end{pf}
Our next result refers to measurable real-valued functions
$\psi$ defined on all pairs $(x,\mathcaligr{X})$ such that
$\mathcaligr{X}$ is a finite subset of $C_n$ and $x \in\mathcaligr{X}$.
We say that such a functional $\psi$
is \textit{translation-invariant} if
$\psi(x,\mathcaligr{X}) = \psi(y+x,y+\mathcaligr{X})$
for all $x, \mathcaligr{X}$ and all $y \in C_n$ (here
addition is in the torus $C_n$, and $y + \mathcaligr{X}:= \{y+w \dvtx w
\in
\mathcaligr{X}\}$). For $ r >0$, we say that $\psi$ has \textit{radius
$r$} if
$\psi(x,\mathcaligr{X})$ is unaffected by the addition of points to,
or removal of
points from, the point set $\mathcaligr{X}$ at a distance more than
$r$ from $x$,
that is, if for all $(x,\mathcaligr{X})$ we have $\psi(x,\mathcaligr
{X}) = \psi(x,\mathcaligr{X}\cap B_r(x))$.
The notion of radius is the same as that of \textit{range of interaction}
used in \cite{PR}; see
also the notion of \textit{radius of stabilization}, in
\cite{PYStein,PR} and elsewhere. We also define
\begin{eqnarray*}
\| \psi\| &: =& \mathop{\operatorname{ess}\operatorname{sup}}_{x,\mathcaligr
{X}} \{ |\psi(x,\mathcaligr{X}) |\} ; \\
\operatorname{rng}( \psi) &: =&
\mathop{\operatorname{ess}\operatorname{sup}}_{x,\mathcaligr{X}} \{ \psi
(x,\mathcaligr{X}) \}
- \mathop{\operatorname{ess}\operatorname{inf}}_{x,\mathcaligr{X}} \{ \psi
(x,\mathcaligr{X}) \}.
\end{eqnarray*}
Recall that $\mathcaligr{U}_n := \{U_1,\ldots,U_n\}$
denotes a collection of $n$ independent
uniformly distributed points in $C_n$, and $\pi_d$ is
the volume of the unit $d$-ball.
\begin{lemm}
\label{momlem}
Let $n \in\mathbb{N}$ and $k \in\mathbb{N}$ with $2 \leq k \leq n$.
Suppose that for $i=1,\ldots, k$, $\psi_{i}$
is a measurable real-valued function
defined on all pairs $(x,\mathcaligr{X})$ with
$\mathcaligr{X}$ a finite set in $C_n$ and $x \in\mathcaligr{X}$.
Suppose for each $i$ that $\psi_{i}$ is translation-invariant
and has radius $r_i$ for some $r_i \in(0,\infty)$, and that
$\| \psi_{i} \| < \infty$,
and $\mathbb E[\psi_{1}(U_1,\mathcaligr{U}_n) ]=0$.
With $\phi_i := \pi_dr_i^d$,
suppose also that
$\phi_2 + \cdots+ \phi_k <n$.
Then
\begin{eqnarray*}
\Biggl| \mathbb E \Biggl[ \prod_{i=1}^k \psi_{i}(U_i,\mathcaligr
{U}_n) \Biggr] \Biggr|
&\leq&
\Biggl( n^{-1} \prod_{i=2}^k \|\psi_{i} \| \Biggr)
\operatorname{rng}(\psi_1)
\\
&&{}\times\Biggl( \pi_d
\Biggl( \sum_{i=2}^k (r_1 + r_i)^d \Biggr)
\\
&&\hspace*{17.8pt}{}
+ \phi_1 \Biggl( k-1 + \Biggl(\sum_{i=2}^k \phi_i \Biggr)
\Biggl(\frac{ 2n - \sum_{i=2}^k \phi_i }{n - \sum_{i=2}^k \phi_i }\Biggr)
\Biggr)\Biggr).
\end{eqnarray*}
\end{lemm}
\begin{pf}
Given $\mathbf{x}= (x_1,\ldots,x_k) \in C_n^k$, define the set of
points
\[
\mathcaligr{U}_n^\mathbf{x}:= \{x_1,\ldots,x_k, U_{k+1},\ldots, U_n
\}.
\]
Let $F_n$ be the set of $\mathbf{x}= (x_1,\ldots,x_k) \in C_n^k$
such that
$D(x_1,x_i) > r_1 + r_i $
for $i \in\{2,\ldots,k\}$, and let $F_n^c:= C_n^k \setminus F_n$.
Then by the law of total probability,
\begin{eqnarray*}
\mathbb E \Biggl[
\prod_{i=1}^k \psi_{i}(U_i,\mathcaligr{U}_n)
\Biggr]
&=& n^{-k} \int_{F_n} \mathbb E\prod_{i=1}^k \psi_{i}(x_i,\mathcaligr
{U}_n^\mathbf{x}) \,d\mathbf{x}
\\
&&{}
+ n^{-k} \int_{F_n^c} \mathbb E\prod_{i=1}^k \psi
_{i}(x_i,\mathcaligr{U}_n^\mathbf{x}) \,d\mathbf{x}.
\end{eqnarray*}
Since
$\mathbb E[\psi_1(U_1,\mathcaligr{U}_n)]=0$ it follows that
$\| \psi_1\| \leq\operatorname{rng}(\psi_1)$, so that
\begin{eqnarray}\label{0422c}\qquad\quad
\Biggl|n^{-k} \int_{F_n^c} \mathbb E\prod_{i=1}^k \psi_{i}(x_i,\mathcaligr
{U}_n^\mathbf{x}) \,d\mathbf{x}\Biggr|
& \leq&
\Biggl( \prod_{i=1}^k \|\psi_{i} \| \Biggr)
P [(U_1,\ldots,U_k) \in F_n^c]
\nonumber\\[-8pt]\\[-8pt]
& \leq&
\operatorname{rng}(\psi_1)
\Biggl( \prod_{i=2}^k \|\psi_{i} \| \Biggr) \sum_{ i =2}^k
\pi_d(r_1+r_i)^d/n.\nonumber
\end{eqnarray}
Fix $\mathbf{x}= (x_1,\ldots,x_k) \in F_n$.
For $m \in\mathbb{Z}_+$,
let $h_1(m):= \mathbb E\psi_{1}(x_1,\{x_1\} \cup\mathcaligr{Y}_m)$, where
$\mathcaligr{Y}_m$ denotes
a collection of $m$ uniformly distributed points in $B_{r_1}(x_1)$.
Let
$h_2(m):= \mathbb E\prod_{i=2}^k \psi_{i}(x_i,\{x_2,\ldots,x_k\}
\cup\mathcaligr{Y}'_m)$,
where
$\mathcaligr{Y}'_m$ denotes
a collection of $m$ uniformly distributed points in
$\bigcup_{i=2}^k
B_{r_i}(x_i)$.
If $N_1$ and $N_2$ denote the number of points of
$\{U_{k+1},\ldots,U_n\}$ in
$B_{r_1}(x_1)$ and in $\bigcup_{i=2}^k B_{r_i}(x_i)$,
respectively,
then the values of
$\psi_{1}(x_1,\mathcaligr{U}^\mathbf{x}_n) $ and of
$ \prod_{i=2}^k \psi_i(x_i, \mathcaligr{U}^\mathbf{x}_n)$
are conditionally
independent, given $(N_1,N_2)$, because the regions $B_{r_1}(x_1)$ and
$\bigcup_{i=2}^k B_{r_i}(x_i)$ are disjoint since we assume $\mathbf
{x}\in F_n$.
Hence, we assert that
\begin{equation}\label{0422d}
\mathbb E \Biggl[ \prod_{i=1}^k \psi_{i}(x_i,\mathcaligr{U}^\mathbf{x}_n)\Biggr]
= \mathbb E [ h_{1}(N_1) h_2(N_2) ],
\end{equation}
where $(N_1,N_2,N_3)$ have the multinomial distribution
\begin{equation}\label{mul}
( N_1,N_2,N_3 ) \sim\operatorname{Mult}
\biggl( n-k;\frac{a_1}{n},
\frac{a_2}{n},
\frac{a_3}{n}\biggr)
\end{equation}
with $a_1$
denoting the volume of a ball of radius $r_1$ in $C_n$
[so that $a_1 \leq\phi_1 = \pi_dr_1^d$ with equality if
$r_1 \leq(1/2) n^{1/d}$]
while $a_2$ is the volume of
$\bigcup_{i=2}^k B_{r_i}(x_i)$
in $C_n$ and $a_3 := n-a_1 -a_2$.
To verify (\ref{0422d}),
use the law of total probability to decompose
the left-hand side as a sum over possible values
of $(N_1,N_2)$.
Also, if $\tilde{N}_1 \sim\operatorname{Bin}(n-1,\frac{a_1}{n})$
then $\mathbb E[ h_{1}(\tilde{N}_1)] = 0$,
because of the assumption that
$\mathbb E[\psi_{1}(U_1,\mathcaligr{U}_n) ]=0$,
along with translation invariance; the
value of $\mathbb E[ h_{1}(\tilde{N}_1)] $ does not depend on $x_1$.
We give a coupling of $N_1$ to another random variable
$N'_1$ with the same distribution as $\tilde{N}_1$
that is independent of $N_2$,
for which we can give a useful bound on
$P[N_1 \neq N'_1]$.
Consider throwing a series of colored balls
so each ball can land in one of three urns, where the probability
of landing in urn $i$ is $a_i/n$ for $1 \leq i \leq3$.
First, throw $n-k$ white balls and let $N_1^*, N_2,N^*_3$ denote
the number of
white balls in urn $i$ for $i = 1,2,3$, respectively, that is, let
$(N^*_1,N_2,N^*_3)$ have
the $\operatorname{Mult}(n-k;\frac{a_1}{n},\frac{a_2}{n},\frac{a_3}{n})$
distribution. Now pick out
the $n-k- N_2 $ balls in urns 1 and 3, paint them red,
and throw them again; that is,
given the values of $N^*_1,N_2,N^*_3$ let
$N_1^r,N_2^r,N_3^r$ count the number of red balls in urns
$1,2,3$, respectively,
and so be nonnegative integer valued
variables such that
\[
\mathcaligr{L} ( ( N_1^r,N_2^r,N_3^r ) |N_1^*,N_2)
= \operatorname{Mult}
\biggl( n- k - N_2 ;\frac{a_1}{n}, \frac{a_2}{n} , \frac{a_3}{n}\biggr).
\]
Now take the $N_2^r$ red balls in urn $2$,
paint them blue, and throw them again but condition them to land
in urns 1 and 3 (or equivalently, throw each blue ball
again and again until
it avoids urn 2), so that
\[
\mathcaligr{L} ( ( N_1^b,N_3^b
)|N^*_1,N_2,N_1^r,N_2^r )
=\operatorname{Mult} \biggl(N_2^r ; \frac{a_1}{a_1+a_3},\frac{
a_3}{a_1+a_3} \biggr).
\]
Finally, throw $k-1+N_2$ green balls, making the total number
of green, red and blue balls
$n-1$, and record how many land in urn 1, so
\[
\mathcaligr{L}
(N_1^g
| N^*_1,N_2,N_1^r,N_2^r,N_1^b ) =
\operatorname{Bin}
\biggl( k-1 + N_2; \frac{a_1}{n} \biggr).
\]
Now set
\[
N_1 = N_1^r + N_1^b ,\qquad
N_3 = N_3^r + N_3^b\quad \mbox{and}\quad N'_1 = N_1^r + N_1^g.
\]
Then $(N_1,N_2,N_3) $ have the multinomial distribution given
by (\ref{mul}).
Also, $N'_1 \sim\operatorname{Bin}(n-1,\frac{a_1}{n})$
and $N'_1$ is independent of $N_2$.
Since $N'_1 = N_1 - N_1^b +N_1^g$,
we have that
\begin{eqnarray*}
P[ N_1 \neq N'_1] & \leq&
\mathbb E[N_1^g]
+
\mathbb E[N_1^b]
\leq
\frac{a_1}{n}
(k-1 + \mathbb EN_2 )
+ \biggl(
\frac{a_1}{a_1 +a_3} \biggr) \mathbb E[N_2^r]
\\
& \leq&
\frac{a_1}{n}
( k-1 + a_2 )
+
\biggl( \frac{a_1}{n - a_2} \biggr)
a_2
\end{eqnarray*}
so that
\[
\bigl|
\mathbb E\bigl[ h_{2}(N_2) \bigl( h_{1}(N_1) -
h_1(N'_1) \bigr) \bigr]
\bigr|
\leq\frac{a_1}{n} \biggl(
k-1 + a_2
+
\biggl(
\frac{
n a_2
}{n - a_2}
\biggr)
\biggr) \operatorname{rng}(\psi_1) \prod_{i=2}^k \|\psi_{i}\|
\]
and since $N'_1$ is independent of $N_2$ with
$N'_1 \sim\operatorname{Bin}(n-1,\frac{a_1}{n})$,
\[
\mathbb E[ h_{1}(N'_1) h_{2}(N_2) ] = 0,
\]
so by (\ref{0422d})
and the fact that $a_1 \leq\phi_1$ and $a_2 \leq\sum_{i=2}^k \phi
_i $
and the assumption that $ \sum_{i=2}^k \phi_i < n$,
\begin{eqnarray*}
&&\Biggl|
\mathbb E \Biggl[ \prod_{i=1}^k \psi_{i}(x_i,\mathcaligr{U}_n^{\mathbf{x}})
\Biggr] \Biggr|
\\
&&\qquad\leq\frac{a_1}{n} \biggl(
k-1+ a_2
\biggl( \frac{ 2n - a_2 }{ n - a_2}
\biggr) \biggr) \operatorname{rng}(\psi_1) \prod_{i=2}^k \|\psi
_{i}\|
\\
&&\qquad\leq\frac{\phi_1}{n} \Biggl(
k-1+
\Biggl(
\sum_{i=2}^k \phi_i
\Biggr)
\biggl( \frac{ 2n -
\sum_{i=2}^k \phi_i
}{n -
\sum_{i=2}^k \phi_i
}
\biggr) \Biggr) \operatorname{rng}(\psi_1) \prod_{i=2}^k \|\psi
_{i}\|.
\end{eqnarray*}
The preceding bound holds uniformly over all possible values
of ${\mathbf x}= (x_1,\ldots,x_k) \in F_n$. Combined with (\ref{0422c}),
this shows that the asserted bound
holds.
\end{pf}
\begin{lemm}
\label{varbdlem}
Suppose $ \psi_1$ is as defined in Lemma \ref{momlem}.
Then with notation from that result, if $\phi_1 < n$ then
\begin{eqnarray*}
&&\operatorname{Var} \Biggl[ \frac{1}{n} \sum_{i=1}^n \psi
_1(U_i,\mathcaligr{U}_n) \Biggr]
\\
&&\qquad\leq
\frac{ \| \psi_1\|^2 }{n}
( 1 +
2^d
\phi_1
)
+
\frac{\|\psi_1\|
}{n}
\biggl(\phi_1+\phi^2_1\biggl( \frac{2 n -\phi_1 }{n -\phi_1} \biggr)
\biggr) \operatorname{rng}(\psi_1).
\end{eqnarray*}
\end{lemm}
\begin{pf}
By the case $k=2$ of Lemma \ref{momlem},
\begin{eqnarray*}
&&\operatorname{Cov}( \psi_1(U_1,\mathcaligr{U}_n),\psi
_1(U_2,\mathcaligr{U}_n))
\\
&&\qquad
= \mathbb E[ \psi_1(U_1,\mathcaligr{U}_n)
\psi_1(U_2,\mathcaligr{U}_n) ]
\\
&&\qquad
\leq\frac{\|\psi_1\|}{n}\biggl(2^d \phi_1\|\psi_1\|+\phi_1
\biggl(1 + \phi_1\biggl( \frac{2n- \phi_1}{n - \phi_1} \biggr)\biggr)
\operatorname{rng}(\psi_1)\biggr)
\end{eqnarray*}
and since
\begin{eqnarray*}
\operatorname{Var} \Biggl[ \frac{1}{n} \sum_{i=1}^n \psi
_1(U_i,\mathcaligr{U}_n) \Biggr]
&=&
n^{-1}
\operatorname{Var}[ \psi_1(U_1,\mathcaligr{U}_n) ]
\\
&&{}+ \frac{n-1}{n}
\operatorname{Cov}( \psi_1(U_1,\mathcaligr{U}_n),\psi
_1(U_2,\mathcaligr{U}_n)),
\end{eqnarray*}
the result follows.
\end{pf}
\section{Size-biased coupling constructions}
\label{couplsec}
We now give a
simple
lemma which shows how to size-bias a random variable
that can be expressed as a conditional probability
of an event arising from some further randomization.
\begin{lemm}
\label{condsblem}
Suppose $Y$ is a random variable given by
$Y= a P[A|\mathcaligr{F}]$, where $\mathcaligr{F}$ is some $\sigma$-algebra,
$a>0$ is a constant,
and
$A$ is an event with $0<P[A]< 1$.
Then
$Y'$ has the $Y$ size biased distribution
if
\begin{equation}\label{newsbeq}
\mathcaligr{L}( Y') = \mathcaligr{L}(Y|A).
\end{equation}
\end{lemm}
\begin{pf}
With $\mathcaligr{L}(Y')$ defined by (\ref{newsbeq}),
we must show for all bounded and continuous $g\dvtx\mathbb{R}\to\mathbb{R}$,
that $\mathbb E[ g(Y') ] = \mathbb E[Y g(Y)]/ \mathbb E[Y]$
[see (\ref{formalsb})]. But
\begin{eqnarray*}
\mathbb E[g(Y')] & = &
\mathbb E[g(Y)|A] = \mathbb E[ g(Y)\mathbf{1}_A]/P[A]
\\
& = & \mathbb E[ g(Y) P[A|\mathcaligr{F}] ] / P[A],
\end{eqnarray*}
where the last equality follows because $g(Y)$ is $\mathcaligr{F}$-measurable.
The last expression equals $\mathbb E[Y g(Y)]/\mathbb E[Y]$, as
required.
\end{pf}
Let $V$ and $S$ be given by (\ref{Vndef}), (\ref{Yndef}), respectively.
Set $W=n-S$ (the number of nonsingletons).
We assert that either $V$ or $W$
can be expressed as $n$ times the
conditional probability of some event $A$, given
the locations of the points of $\mathcaligr{U}_n$, so that
Lemma~\ref{condsblem} is applicable.
For $V$, take $A= A_V$ to be the event that
an additional uniformly distributed random point $U_0$ in $C_n$ lies
in the covered region $\bigcup_{i=1}^n B_{i,\rho}$.
For $W$, take $A = A_W$
to be the event that
an element of $\mathcaligr{U}_n$, selected uniformly at random,
is
nonisolated.
Event $A_V$ can be written as
the event that $N_V > 0$, where
$N_V$ denotes the
number of points of $\mathcaligr{U}_n$ in
$B_\rho(U_0)$, and
$N_V \sim\operatorname{Bin}(n,\phi/n)$ (recall $\phi:= \pi_d\rho^d$
and $C_n$ has volume $n$).
A point set
(denoted $\mathcaligr{U}_V$)
with the
conditional distribution of $\mathcaligr{U}_n$ given $N_V$
can be obtained as follows:
\begin{enumerate}[III.]
\item[I.]
Sample a uniform random point in $C_n$, denoted $U_0$.
\item[II.]
Set $m=n$.
Sample $N=N_V$ independent uniform random points in $B_\rho(U_0)$,
and $m-N$ independent uniform random points in $C_n \setminus B_\rho(U_0)$.
\item[III.]
Let $\mathcaligr{U}_V$ be the union of the two samples of uniform points.
\end{enumerate}
Therefore, coupled realizations of $\mathcaligr{U}_V$ and $\mathcaligr{U}'_V$
(having, respectively, the distribution of $\mathcaligr{U}_n$ and
the conditional distribution of $\mathcaligr{U}_n$ given $N_V>0$),
and hence coupled realizations of $V$ and $V'$,
can be obtained
as follows.
\begin{enumerate}
\item Set $m=n$.
\item
Sample $U_0$ uniformly at random over $C_n$.
\item
Sample $m$ random $d$-vectors
independently and uniformly over
$C_n$,
and denote this
point set by $\mathcaligr{U}_{m,1}$.
\item Let $N$ denote the number of points of $\mathcaligr{U}_{m,1}$ in
$B_\rho(U_0)$.
\item
Sample a Bernoulli random variable
$\mathcaligr{B}$ with $P[\mathcaligr{B}=1] = \pi_N$,
where $(\pi_k, k \geq0)$ is given by (\ref{pikdef}).
\item Sample a random $d$-vector $U$ which is uniform over
$B_\rho(U_0)$.
\item
If $\mathcaligr{B}=1$, then select one of the points of
$\mathcaligr{U}_{m,1} $
uniformly at random, and move it to $U$. Denote the resulting
modification of
$\mathcaligr{U}_{m,1} $
by $\mathcaligr{U}_{m,2}$.
If $\mathcaligr{B}=0$ then set $\mathcaligr{U}_{m,2} : = \mathcaligr
{U}_{m,1} $.
\item
Set $\mathcaligr{U}_V := \mathcaligr{U}_{m,1} $ and
$\mathcaligr{U}'_V := \mathcaligr{U}_{m,2}$.
Set $V := g_V(\mathcaligr{U}_V)$ and
$V':=g_V(\mathcaligr{U}'_V)$, where $g_V(\mathcaligr{U}):=
\operatorname{Vol}( \bigcup_{x \in\mathcaligr{U}} B_{\rho}(x))$.
\end{enumerate}
By Lemma \ref{bincouplem},
the number of points of
$\mathcaligr{U}_{m,2}$ in $\mathcaligr{B}_\rho(U_0)$ has the distribution
$\mathcaligr{L}(N_V|N_V>0)$, and hence $\mathcaligr{L}(\mathcaligr
{U}'_V) = \mathcaligr{L}(\mathcaligr{U}_V|N_V >0)$.
So by Lemma \ref{condsblem}, $V'$ has the $V$ size biased distribution.
In the case of $W$, $A_W$ is the event that $N_W>0$,
where $N_W$ denotes the number of points of $\mathcaligr{U}_n
\setminus\{U_0\}$
in $B_\rho(U_0)$, and now $U_0$ denotes a
point of $\mathcaligr{U}_n$ selected uniformly at random.
So $N_W \sim\operatorname{Bin}(n-1,\phi/n)$. We can obtain
a point set (denoted $\mathcaligr{U}_W$)
with the conditional distribution of $\mathcaligr{U}_n$ given $N_W$
by the same steps as for $\mathcaligr{U}_V$ except that now in Step II
we put
$m=n-1$ and $N=N_W$, and in Step III, $\mathcaligr{U}_W$ is the union
of the two
samples of uniform random points with an added point at $U_0$.
Hence,
we can obtain coupled realizations
of
$W$ and $W'$
by the same
sequence of steps as described above for $(V,V')$, except
that the following steps are modified:
\begin{itemize}
\item In Step 1, set $m= n-1$ (this affects Steps 3 and 5.)
\item
In Step 8,
set $\mathcaligr{U}_W := \mathcaligr{U}_{m,1} \cup\{U_0\}$,
and $\mathcaligr{U}'_W := \mathcaligr{U}_{m,2} \cup\{U_0\}$.
Set $W := g_W(\mathcaligr{U}_W)$ and $W':= g_W(\mathcaligr{U}'_W)$
with $g_W(\mathcaligr{U}) := \sum_{x \in\mathcaligr{U}} \mathbf{1}
\{\mathcaligr{U}\cap B_{\rho}(x) \neq\{x\} \}$.
\end{itemize}
By a similar argument to the $V$ case,
$W'$ has the $W$ size biased distribution.
\section{Proof of Theorem \protect\ref{thm2}}
\label{proof-2}
We couple $V'$ to
$V$ as described in Section \ref{couplsec}.
Since $V'$ differs from $V$ through the moving of at most a
single point, clearly $|V'-V| \leq\pi_d\rho^d := \phi$.
Hence, by Lemma \ref{LGthm} with $B = \phi$, to prove Theorem \ref
{thm2} it suffices
to prove the following.
\begin{prop}
\label{convar2prop}
Under the assumptions of Theorem \ref{thm2},
$\operatorname{Var}(\mathbb E[V'-V|V]) \leq n^{-1}\eta_V(n,\rho)$,
where $\eta_V(n,\rho)$ is given by (\ref{etaVdef}).
\end{prop}
\begin{pf}
Let $\mathcaligr{G}$
be the $\sigma$-algebra generated by the point set
$\mathcaligr{U}_V$.
List the points of $\mathcaligr{U}_V$, in an order chosen uniformly at random,
as $U_1,\ldots,U_n$, and set $\mathbf{U}:= (U_1,\ldots,U_{n})$.
Then $V$ is $\mathcaligr{G}$-measurable.
The conditional variance
formula, with $X = \mathbb E[V'-V|\mathcaligr{G}]$, yields
\[
\operatorname{Var}(\mathbb E[V'-V|V]) = \operatorname{Var}( \mathbb
E[X|V]) \leq\operatorname{Var}(X) ,
\]
so it suffices to prove
\begin{equation}\label{a0422a}
\operatorname{Var}(\mathbb E[V'-V|\mathcaligr{G}])
\leq n^{-1} \eta_V(n,\rho).
\end{equation}
For $x \in C_n$,
let $\xi_x$ denote the probability that $\mathcaligr{B}=1$, given
$\mathcaligr{U}_n$ and given that $U_0=x$,
that is, $\xi_x=\pi_{N_x}$, where $N_x$ denotes the number
of points of $\mathcaligr{U}_V$ in $B_\rho(x)$.
Let $R_{xj}$ denote the expectation (over $U$) of the increment in
the covered volume
if $U_j$ is moved to a uniform
randomly selected location $U$ in $B_\rho(x)$.
Note that
for $x$ and $j$ fixed, $R_{xj}$ is determined by
$\mathbf{U}$.
Then, since both $U_0$ and $I$ are independent of $\mathcaligr{G}$,
\[
\mathbb E[V'-V|\mathcaligr{G}] = \frac{1}{n} \int_{C_n} \xi_x
\Biggl( \frac{1}{n} \sum_{j=1}^n R_{xj} \Biggr)\,
dx,
\]
where the first factor of $1/n$ comes from the probability density
of $U_0$, and the second arises as the probability that $I$ takes the
value $j$.
Let $H_x$ be the expectation (over $U$) of the increment in the
covered volume when a point is inserted into $\mathcaligr{U}_V$ at a uniform
random location $U\in B_\rho(x)$, and let $T_j$ be the increment in
the covered volume when point $U_j$ is removed from $\mathcaligr{U}$
(for fixed
$x$ and $j$, both $H_x$ and $T_j$ are determined by $\mathbf{U}$).
If $U_j$ is far distant from $x$ then $R_{xj}= H_x + T_j$.
Set $ Q_{xj} := R_{xj} - H_x - T_j $, which is in fact
the expectation (over $U$) of the total volume
of the otherwise uncovered regions lying within
distance $\rho$ both of $U$ and of $U_j$
(such regions contribute to $T_j$ but not to $H_x$ or $R_{xj})$.
Then
\begin{eqnarray}\label{a0422b}\quad
\mathbb E[V'-V|\mathcaligr{G}]
& = & \frac{1}{n^2} \int_{C_n}
\sum_{j =1}^n \xi_x(H_x+T_j +Q_{xj})\, dx
\nonumber\\[-8pt]\\[-8pt]
& = & \frac{1}{n} \int_{C_n} \xi_x \Biggl(H_x + \frac{1}{n} \sum
_{j =1
}^n Q_{xj} \Biggr)\,dx + \frac{1}{n^2} \int_{C_n} \sum_{j=1}^n \xi_x
T_j \,dx .\nonumber
\end{eqnarray}
Set $\phi:= \pi_d\rho^d$.
We have that
$0 \leq H_x \leq\phi$, $ 0 \geq T_j \geq- \phi$ and
if $D(x,U_j) > 3 \rho$ then
$Q_{xj}=0$.
Moreover,
if $D(x,U_j) > 3 \rho$ for all $j \in\{1,\ldots,n\}$, then
$ H_x = \phi$ and
if $D(x,U_j) > \rho$ for all $j \in\{1,\ldots,n\}$, then
$\xi_x =1$.
Finally, $Q_{xj} \geq0$ and
\[
0 \le H_x + \frac{1}{n} \sum_{j=1}^n Q_{xj} \leq
H_x + \sum_{j=1}^n Q_{xj} \leq
\phi.
\]
Hence setting
\[
\tau_x := \xi_x \Biggl(
H_x + \frac{1}{n} \sum_{j=1}^n Q_{xj} \Biggr) - \phi,
\]
we have that $ -\phi\leq\tau_x \leq0$,
and
$\tau_x$ is determined by the collection of points
of $\mathcaligr{U}_n$ within distance $3\rho$ of $x$, and $\tau_x
=0$ if there
are no
such points of $\mathcaligr{U}_n$. We can rewrite (\ref{a0422b}) as
\[
\mathbb E[V'-V|\mathcaligr{G}] = \phi+
\frac{1}{n} \int_{C_n} \tau_x \,dx + \frac{1}{n} \Biggl( \sum_{j=1}^n
T_j \Biggr)
+ \frac{1}{n^2} \int_{C_n} \sum_{j=1}^n ( \xi_x -1) T_j \,dx.
\]
Recalling that $B_r(x): = \{ y \in C_n \dvtx D(x,y) \leq r\}$, let
$\Gamma_{i,r}$ be the set of points
$y \in B_{r}(U_i)$
such that $D(y,U_i) < D(y,U_j)$ for all $j \in
\{1,\ldots,n\}
\setminus\{i\}$ (i.e., the intersection of the $r$-ball around
$U_i$ and the Voronoi cell of $U_i$ relative to $\mathcaligr{U}_n$).
Set
\[
S'_i := \int_{\Gamma_{i,3\rho}} \tau_x\,dx,\qquad S''_i := \int
_{\Gamma_{i,\rho}}
(\xi_x -1)\,dx.
\]
Then
\begin{eqnarray}\label{0423c}
\mathbb E[V'-V|\mathcaligr{G}] &=& \phi+ \Biggl( \frac{1}{n}
\sum_{i=1}^n S'_i \Biggr) + \Biggl( \frac{1}{n} \sum_{j=1}^n T_j
\Biggr) + \Biggl( \frac{1}{n^2} \sum_{i=1}^n S''_iT_i \Biggr)
\nonumber\\[-8pt]\\[-8pt]
&&{} + \biggl( \frac{1}{n^2} \sum_{(i,j)\dvtx i \neq j} S''_iT_j
\biggr),\nonumber
\end{eqnarray}
and if we put
$b=\mathbb ET_i$ (which
does not depend on $i$), we have
\[
\frac{1}{n^2} \sum_{(i,j)\dvtx i \neq j} S''_i T_j
=
\frac{1}{n^2} \biggl( \sum_{(i,j)\dvtx i \neq j} S''_i ( T_j -b) \biggr)
+ \frac{b(n-1)}{n^2} \Biggl( \sum_{i=1}^n S''_i \Biggr),
\]
so by (\ref{0423c}),
\begin{eqnarray*}
&&\mathbb E[V'-V|\mathcaligr{G}] \\
&&\qquad = \phi
+ \frac{1}{n^2} \biggl( \sum_{(i,j)\dvtx i \neq j} S''_i ( T_j -b) \biggr)
\\
&&\qquad\quad{} + \frac{1}{n}\sum_{i=1}^n \bigl( S'_i
+ T_i + \bigl( n^{-1} T_i + (1 -n^{-1})b \bigr) S''_i \bigr).
\end{eqnarray*}
Since $(x+y)^2 \leq2(x^2 +y^2)$ for any real $x,y$,
\begin{eqnarray}\label{0525}\qquad
&&\operatorname{Var}(\mathbb E[V'-V|\mathcaligr{G}])
\nonumber\\
&&\qquad\leq
2\operatorname{Var} \Biggl( \frac{1}{n}\sum_{i=1}^n \bigl( S'_i
+ T_i + \bigl(n^{-1} T_i + (1 -n^{-1}) b\bigr) S''_i \bigr) \Biggr)
\\
&&\qquad\quad{} + 2 \operatorname{Var} \biggl(\frac{1}{n^2} \biggl(
\sum_{(i,j)\dvtx i \neq j} S''_i ( T_j -b) \biggr)\biggr).\nonumber
\end{eqnarray}
Table \ref{table1} summarizes upper and lower bounds and the radius of
the relevant variables, where the \textit{radius} of a variable indexed by $i$ is
the smallest distance from $U_i$ one needs to look to establish its value
(as with the functionals considered in Lemma \ref{momlem}).
\begin{table}[b]
\caption{Radii and bounds for covered volume}
\label{table1}
\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccccc@{}}
\hline
\textbf{Variable} & $\bolds{\tau_x}$ & $\bolds{T_i}$ & $\bolds{S''_i}$
& $\bolds{S'_i}$ & $\bolds{(n^{-1}T_i + (1-n^{-1})b)S''_i}$\\
\hline
Radius & $3 \rho$ & $2 \rho$ & $2\rho$ & $6 \rho$ & $2\rho$\\
Lower bound & $-\phi$ & $-\phi$ & $-\phi$ & $-3^d \phi^2$ & 0\\
Upper bound & 0 & 0 & 0 & 0 & $\phi^2$\\
\hline
\end{tabular*}
\legend{Note: The last two columns are deduced from the previous columns.}
\end{table}
Hence, the variable
\[
S'_i +
T_i + \bigl(n^{-1} T_i + (1 -n^{-1}) b\bigr) S''_i
\]
has radius $6\rho$ relative to $U_i$ and
lies between $-\phi- 3^d \phi^2$ and
$\phi^2$,
so that
its centered value is bounded
in absolute value by $(3^d+1)\phi^2 + \phi
$, and this also bounds its range of possible values.
So by Lemma \ref{varbdlem} and the assumption that $6^d \phi< n$,
\begin{eqnarray}\label{0527c}\quad
&&\operatorname{Var} \Biggl( \frac{1}{n} \sum_{i=1}^n \bigl( S'_i + T_i + (n^{-1}
T_i + (1 -n^{-1}) b) S''_i \bigr) \Biggr)
\nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq\frac{\phi^2 ( (3^{d} +1)\phi+ 1 )^2}{n} \biggl( 1 + (2^d +1) 6^d
\phi + \biggl( \frac{2 n - 6^d \phi }{ n - 6^d \phi } \biggr) 6^{2d} \phi^2 \biggr)
.\nonumber
\end{eqnarray}
Now consider the last term in the right-hand side of (\ref{0525}).
Set
$\bar{T}_j:= T_j - b$. Then
\begin{eqnarray}\label{0527a}\quad
&&\operatorname{Var} \biggl( \sum_{(i,j)\dvtx i \neq j} S''_i \bar{T}_j\biggr)
\nonumber\\
&&\qquad= n(n-1)(n-2)(n-3) \operatorname{Cov}( S''_1 \bar{T}_2,
S''_3\bar{T}_4)
\nonumber\\
&&\qquad\quad{} + n(n-1)(n-2) \bigl( \operatorname{Cov}(S''_1 \bar{T}_2,
S''_1 \bar{T}_3)
\\
&&\qquad\quad\hspace*{86.6pt}{}+ \operatorname{Cov}(S''_2 \bar{T}_1, S''_3 \bar{T}_1)
+ 2\operatorname{Cov}(S''_1 \bar{T}_2, S''_3 \bar{T}_1)\bigr)
\nonumber\\
&&\qquad\quad{} + n(n-1) \bigl(\operatorname{Var}( S''_1 \bar{T}_2 ) +
\operatorname{Cov}(S''_1 \bar{T}_2, S''_2 \bar{T}_1 )\bigr).\nonumber
\end{eqnarray}
It follows from the case $k=4$ of Lemma \ref{momlem} and
the assumption $6^d \phi< n$ (which implies $3(2^d \phi) < n$)
that
\begin{eqnarray*}
\operatorname{Cov}( S''_1 \bar{T_2}, S''_3 \bar{T_4} )
& = &
\mathbb E[S''_1 \bar{T_2}S''_3 \bar{T}_4 ] - (\mathbb E[S''_1 \bar{T}_2 ])^2
\leq \mathbb E[ S''_1 \bar{T_2}S''_3 \bar{T}_4 ]
\\
& \leq& \frac{\phi^4}{n} \biggl( 3\pi_d(4\rho)^d + 2^d \phi\biggl(3 + 3 (2^d) \phi
\biggl( \frac{2n- 3(2^d) \phi}{n- 3(2^d) \phi} \biggr) \biggr) \biggr)
\\
& = & \frac{3 \phi^4}{n} \biggl( (4^d + 2^d) \phi+ 4^d \phi^{2} \biggl(\frac{2n-
3(2^d) \phi}{n- 3(2^d) \phi}\biggr)\biggr).
\end{eqnarray*}
Since we can always bound
$\operatorname{Cov}(S''_i\bar{T}_j,S''_{i'} \bar{T}_{j'})$
above
by $\phi^4$, we have from (\ref{0527a}) that
\begin{eqnarray}\label{0527b}
&& \operatorname{Var} \biggl( \frac{1}{n^2}
\biggl( \sum_{(i,j)\dvtx i \neq j} S''_i ( T_j -b) \biggr) \biggr)
\nonumber\\[-8pt]\\[-8pt]
&&\qquad \leq \frac{\phi^4}{n} \biggl( 3 (4^d + 2^d) \phi+ 3 (4^d) \phi^{2} \biggl(
\frac{2n- 3(2^d) \phi} {n- 3(2^d) \phi} \biggr) + 4 + \frac{2}{n}
\biggr).\nonumber
\end{eqnarray}
By (\ref{0525}), (\ref{0527c}) and (\ref{0527b}) we have that
\begin{eqnarray*}
&& (n/2) \operatorname{Var} ( \mathbb E[V'-V|\mathcaligr{G}] )
\\
&&\qquad \leq
\phi^2 \bigl( (3^{d} +1)\phi+ 1\bigr)^2 \biggl( 1 + (2^d +1) 6^d \phi+ \biggl( \frac{2 n -
6^d \phi}{ n - 6^d \phi} \biggr) 6^{2d} \phi^2\biggr)
\\
&&\qquad\quad{} + \phi^4\biggl(3(4^d + 2^d) \phi+ 3(4^d) \phi^{2} \biggl( \frac{2n- 3(2^d) \phi}
{n- 3(2^d) \phi} \biggr) + 4 + \frac{2}{n} \biggr).
\end{eqnarray*}
This completes the proof of
Proposition \ref{convarprop}, and hence of Theorem \ref{thm2}.
\end{pf}
\section{Proof of Theorem \protect\ref{thm1}}
\label{proof-1}
We couple $W'$ to
$W$ as described in Section \ref{couplsec}.
Thus $W= g_W(\mathcaligr{U}_W)$ and $W'= g_W(\mathcaligr{U}'_W)$, where
$\mathcaligr{U}'_W$ is obtained from
$\mathcaligr{U}_W$ by moving at most a single
randomly selected point of $\mathcaligr{U}_W \setminus\{U_{0} \} $ to
a (uniform random) location in $B_\rho(U_0)$, if $\mathcaligr{B}=1$,
and leaving $\mathcaligr{U}_W$ unchanged if $\mathcaligr{B}=0$.
The number of points that can be made isolated by removing a single
point from $\mathcaligr{U}_W$ is almost surely bounded by
$\kappa_d$. Moreover, the number of points that can be made
nonisolated by inserting a point (including the inserted
point itself) is almost surely bounded by $\kappa_d + 1$.
Hence $|W-W'| \leq\kappa_d + 1$, so we may
take $B=\kappa_d + 1$.
By the symmetry of the normal distribution,
$D_{-S} = D_{S}$ and hence $D_W = D_S$.
Thus,
Theorem \ref{thm1}
follows from Lemma \ref{LGthm}
along with the following:
\begin{prop}
\label{convarprop}
Under the assumptions of Theorem \ref{thm1},
$\operatorname{Var}(\mathbb E[W'-W|W]) \leq
n^{-1}
\eta_S(n,\rho)$, where $\eta_S(n,\rho)$ is
given by (\ref{etaYdef}).
\end{prop}
\begin{pf}
Here we let $\mathcaligr{G}$ denote
the $\sigma$-algebra generated by the unlabelled point set
$\mathcaligr{U}:= \mathcaligr{U}_W$.
Then $W'$ is $\mathcaligr{G}$-measurable,
and by the conditional variance formula
(as in the proof of Proposition \ref{convar2prop}),
it suffices to prove that
\begin{equation}\label{0422a}
\operatorname{Var}(\mathbb E[W'-W|\mathcaligr{G}]) \leq n^{-1} \eta
_S(n,\rho).
\end{equation}
Label the points of $\mathcaligr{U}$, in an order chosen uniformly at random,
as $U_1,\ldots,U_n$, and set $\mathbf{U}:= (U_1,\ldots,U_n)$.
$\xi_i=\pi_{N_i}$, where
$N_i$ denotes the number of points of $\mathcaligr{U}\setminus\{U_i\}
$ in
$B_\rho(U_i)$. Let $R_{ij}$ denote the
expectation (over $U$) of the increment in the number of
nonisolated points when $U_j$ is moved to a uniform
randomly selected location $U$ in $B_\rho(U_i)$.
Then
\[
\mathbb E[W'-W|\mathcaligr{G}] = \frac{1}{n(n-1)} \sum_{(i,j)\dvtx i \neq
j} \xi_i R_{ij} ,
\]
where $\sum_{(i,j)\dvtx i \neq j}$ denotes summation over
pairs of distinct integers $i,j$ in $[1,n]$.
Now let $H_i$ be the expectation (over $U$)
of the increment in
the number of isolated points
when a point is inserted into $\mathcaligr{U}$ at a uniform random location
$U\in B_\rho(U_i)$, and let $T_j$ be the increment in
the number of isolated points
when point $U_j$ is removed from $\mathcaligr{U}$ (both $H_i$ and
$T_j$ are
determined by $\mathbf{U}$).
If $U_j$ is far distant from $U_i$ then $R_{ij}= - H_i - T_j$.
In fact, setting $ Q_{ij} := R_{ij} + H_i + T_j $,
we have that $Q_{ij}$ is
the expectation (over $U$) of the
number of otherwise isolated points of $\mathcaligr{U}$
within distance $\rho$ both of
$U$ and of $U_j$
(such points contribute to $T_j$ but not to $H_i$ or $R_{ij})$.
Then
\begin{eqnarray}\label{0422b}
\mathbb E[W'-W|\mathcaligr{G}]
&=& \frac{1}{n(n-1)} \sum_{(i,j)\dvtx i \neq j} \xi_i(-H_i-T_j +Q_{ij})
\nonumber\\[-8pt]\\[-8pt]
&=& \frac{1}{n} \sum_{i=1}^n \xi_i \tau_i
- \frac{1}{n(n-1)} \sum_{(i,j)\dvtx i \neq j} \xi_i T_j,\nonumber
\end{eqnarray}
where we set
\begin{equation}\label{taudef}
\tau_i := -H_i + \frac{1}{n-1} \sum_{j \dvtx j\neq i} Q_{ij}.
\end{equation}
Put $a := \mathbb E[\xi_i]$
(given $n$, this expectation does not
depend on $i$)
and put $b := (\kappa_d -1)/2$. Then
\begin{eqnarray*}
\frac{1}{n(n-1)} \sum_{(i,j)\dvtx i \neq j} \xi_i T_j &=&
\frac{1}{n(n-1)} \biggl( \sum_{(i,j)\dvtx i \neq j} (\xi_i -a) ( T_j - b)\biggr)
\\
&&{} + \frac{a}{n} \Biggl( \sum_{j=1}^n
T_j \Biggr) + \frac{b}{n} \Biggl( \sum_{i=1}^n ( \xi_i -a)\Biggr).
\end{eqnarray*}
Hence we can rewrite (\ref{0422b}) as
\[
\mathbb E[ W'-W|\mathcaligr{G}] = \frac{1}{n} \sum_{i=1}^n \bigl( \xi_i
\tau_i - aT_i - b (\xi_i-a) \bigr)- \frac{1}{n(n-1)} \sum_{(i,j)\dvtx i
\neq j} (\xi_i-a) (T_j-b) .
\]
Since $(x+y)^2 \leq2(x^2 + y^2)$ for any real $x,y$,
it follows that
\begin{eqnarray}\label{0513c}\quad
\operatorname{Var} ( \mathbb E[W'-W|\mathcaligr{G}] ) &\leq& 2
\operatorname{Var} \Biggl( \frac{1}{n} \sum_{i=1}^n \bigl( \xi_i (\tau_i -b) +
a(b-T_i) \bigr) \Biggr)
\nonumber\\[-8pt]\\[-8pt]
&&{} + 2 \operatorname{Var} \biggl( \frac{1}{n(n-1)} \sum_{(i,j)\dvtx i \neq
j} (\xi_i-a) ( T_j -b)\biggr).\nonumber
\end{eqnarray}
We have that $-\kappa_d \leq H_i \leq0$, $ -1 \leq T_j \leq\kappa_d$,
and $Q_{ij} \geq0$. Also,
\[
0 \le-H_i + \sum_{j\dvtx j \neq i} Q_{ij} \leq\kappa_d,
\]
and if
$D(U_i,U_j) >3\rho$ then
$Q_{ij}=0$.
\begin{table}[b]
\caption{Radii and bounds for singletons}
\label{table2}
\tabcolsep=0pt
\noindent\begin{tabular*}{\tablewidth}{@{\extracolsep{\fill}}lccccccc@{}}
\hline
\textbf{Variable} & $\bolds{H_i}$ & $\bolds{T_i}$ & $\bolds{\xi_i}$
& $\bolds{\tau_i}$ & $\bolds{\xi_i(\tau_i-b)}$ &
$\bolds{a(b- T_i)}$ & $\bolds{\xi_i( \tau_i-b) + a(b- T_i)}$ \\
\hline
Radius & $3 \rho$ & $2\rho$ & $\rho$ & $3\rho$ & $3\rho$ &
$2 \rho$ & $3 \rho$\\
$\operatorname{ess}\operatorname{inf}$ & $-\kappa_d$ & $-1$ & 0 & 0
&
$(1 - \kappa_d)/2$ & $- ( \kappa_d +1 )/2$ & $- \kappa_d$\\
$\operatorname{ess}\operatorname{sup}$ & 0 & $\kappa_d$ & 1 & $\kappa_d$
& $( \kappa_d +1)/2$ & $( \kappa_d +1)/2$ & $\kappa_d + 1$\\
\hline
\end{tabular*}
\legend{Note: The last three columns are deduced from
the preceding columns and the definitions of $a,b$.}
\end{table}
Hence,
$ 0 \leq\tau_i \leq\kappa_d$, and
$\tau_i$ is determined by the collection of points
of $\mathcaligr{U}$ within distance $3 \rho$ of $U_i$.
Table \ref{table2} summarizes this discussion; recall from Table \ref{table1}
the notion of radius.
From the last column in this table, we see that after centering,
the terms in first sum in the right-hand side of (\ref{0513c})
have radius $3\rho$ and
absolute values bounded by
$ 1 + 2 \kappa_d $.
Moreover, even after centering each of these terms
has range (i.e., essential supremum minus essential infimum)
which is also bounded by $1 + 2 \kappa_d$ (this range is
unaffected by the centering).
Hence with $\phi:= \pi_d\rho^d$, Lemma \ref{varbdlem},
using the assumption
$3^d \phi< n$, yields
\begin{eqnarray}\label{0519b}
&&\operatorname{Var} \Biggl( \frac{1}{n} \sum_{i=1}^n \bigl( \xi_i (\tau_i -b) + a(b
- T_i) \bigr) \Biggr)
\nonumber\\[-8pt]\\[-8pt]
&&\qquad\leq\frac{(1 + 2 \kappa_d)^2}{n} \biggl(1 + (2^d+1) 3^d \phi + \biggl( \frac{2n-
3^d \phi}{ n - 3^d \phi} \biggr) (3^d\phi)^{2}\biggr).\nonumber
\end{eqnarray}
Now consider the second sum in the right-hand side of (\ref{0513c}).
Set $\bar{\xi}_i := \xi_i -a$
and $\bar{T}_j:= T_j - b$.
Then
\begin{eqnarray}\label{0519a}
&&\operatorname{Var} \biggl( \sum_{(i,j)\dvtx i \neq j} (\xi_i -a)( T_j -b)\biggr)
\nonumber\\
&&\qquad = n(n-1)(n-2)(n-3) \operatorname{Cov}( \bar{\xi}_1 \bar{T}_2,
\bar{\xi}_3\bar{T}_4)
\nonumber\\
&&\qquad\quad{} + n(n-1)(n-2) \bigl(
\operatorname{Cov}(\bar{\xi}_1 \bar{T}_2, \bar{\xi}_1 \bar{T}_3)\\
&&\qquad\quad\hspace*{86.06pt}{}
+ \operatorname{Cov}(\bar{\xi}_2 \bar{T}_1, \bar{\xi}_3 \bar{T}_1)
+ 2\operatorname{Cov}(\bar{\xi}_1 \bar{T}_2, \bar{\xi}_3 \bar{T}_1)\bigr)
\nonumber\\
&&\qquad\quad{} + n(n-1) \bigl(\operatorname{Var}( \bar{\xi}_1 \bar{T}_2 )
+ \operatorname{Cov}(\bar{\xi}_1 \bar{T}_2,
\bar{\xi}_2 \bar{T}_1)\bigr).\nonumber
\end{eqnarray}
Note that $\bar{\xi}_i$ has absolute value bounded by 1, and range
of possible values also bounded by 1, and mean zero.
Also, $\bar{T}_j$ has absolute value almost surely bounded by
$(\kappa_d + 1)/2$ (its mean
might not be zero). Hence,
the case $k=4$ of
Lemma \ref{momlem} [taking $r_1=r_2= \rho$
and $r_3=r_4 = 2 \rho$ so that $\phi_2+ \phi_3
+ \phi_4 = (2^{d+1} +1)\phi$] yields
\begin{eqnarray*}
&&\operatorname{Cov}( \bar{\xi}_1 \bar{T}_2, \bar{\xi}_3 \bar{T}_4 ) \\
&&\qquad=
\mathbb E[ \bar{\xi}_1 \bar{T}_2\bar{\xi}_3 \bar{T}_4 ] - (\mathbb
E[\bar{\xi_1} \bar{T}_2 ])^2 \leq \mathbb E[ \bar{\xi}_1
\bar{T}_2\bar{\xi}_3 \bar{T}_4 ]
\\
&&\qquad\leq\frac{ (\kappa_d + 1)^2 }{4 n} \biggl( \phi\bigl( 2(3^d) + 2^d\bigr) + 3 \phi+
(2^{d+1}+1) \phi^2 \biggl( \frac{2n - (2^{d+1} +1) \phi}{n - (2^{d+1} +1)
\phi} \biggr) \biggr),
\end{eqnarray*}
where we have also used the assumption
that $(2^{d+1} +1) \phi< n$.
Since we can always bound
$\operatorname{Cov}(\bar{\xi}_i\bar{T}_j,\bar{\xi}_{i'} \bar{T}_{j'})$
by $((\kappa_d +1)/2)^2$, we have from (\ref{0519a}) that
\begin{eqnarray}\label{0519d}
&&\operatorname{Var} \biggl( \frac{1}{n(n-1)} \biggl(\sum_{(i,j)\dvtx i
\neq j} (\xi_i -a) ( T_j - b) \biggr) \biggr)
\nonumber\\
&&\qquad\leq\frac{
(\kappa_d +1)^2 }{4 n}
\biggl(\bigl( 2(3^d) + 2^d +3\bigr) \phi\nonumber\\[-8pt]\\[-8pt]
&&\hspace*{83pt}{} + (2^{d+1}+1) \phi^2 \biggl(
\frac{2n - (2^{d+1} +1) \phi }{n - (2^{d+1} +1) \phi} \biggr) \biggr)
\nonumber\\
&&\qquad\quad{} + \biggl(\frac{ \kappa_d +1}{2} \biggr)^2
\biggl(\frac{4}{n} + \frac{2}{n(n-1)} \biggr).\nonumber
\end{eqnarray}
By (\ref{0513c}), (\ref{0519b}) and (\ref{0519d}),
we have that
\begin{eqnarray}
&& n \operatorname{Var} ( \mathbb E[W'-W|\mathcaligr{G}] )
\nonumber\\
&&\qquad\leq 2 (1+ 2\kappa_d)^2 \biggl( 1 + (2^d+1) 3^d \phi+ \biggl(
\frac{2n- 3^d \phi}{ n - 3^d \phi} \biggr) 9^d\phi^2 \biggr)
\nonumber\\
&&\qquad\quad{} + \frac{(\kappa_d +1)^2}{2} \biggl( \bigl( 2(3^d) + 2^d
+3\bigr) \phi + (2^{d+1}+1) \biggl( \frac{2n - (2^{d+1} +1) \phi }{n -
(2^{d+1} +1) \phi} \biggr) \phi^{2} \biggr)
\nonumber\\
&&\qquad\quad{} + \frac{ (\kappa_d +1)^2}{2} \biggl( 4 + \frac{2}{n-1}
\biggr). \nonumber
\end{eqnarray}
This completes the proof of
Proposition \ref{convarprop}, and hence of Theorem \ref{thm1}.
\end{pf}
\section{Proof of Theorem \protect\ref{thmlimsup} and numerics}
\label{secvar}
Again set $\phi:= \pi_d\rho^d$.
It is easy to see that provided $2 \rho< n^{1/d}$,
\begin{equation}\label{0522a}
\mathbb E[V] = n \bigl(1-(1 - \phi/n)^{n}\bigr);\qquad
\mathbb E[S] = n (1 - \phi/n)^{n-1} ,
\end{equation}
and (\ref{meanlim}) follows from this.
Write $|\cdot|$ for the Euclidean norm and
recall that $\omega_d(|x|)$ denotes the volume of the union of unit
balls centered at the origin $\mathbf{0}$ and at $x$.
If $I_x$ denotes the indicator of the event that
$x$ is not contained in any of the balls $B_{\rho,i}$,
then
provided $4\rho< n^{1/d}$
we have the exact formula
\begin{eqnarray}\label{varV}
\operatorname{Var}(V) &=& \operatorname{Var}(n-V)\nonumber\\
&=& \operatorname
{Var}\int_{C_n} I_x \,dx
\nonumber\\
&=& \int_{C_n} \int_{C_n} \mathbb E[I_x I_y] \,dx \,dy
- \bigl(n ( 1 - \phi/n)^n \bigr)^2
\\
&=& n\int_{B_{2\rho}(\mathbf{0})}
\biggl( 1 - \frac{\rho^d \omega_d(|y|/\rho) }{n} \biggr)^n \, dy
\nonumber\\
&&{}
+ n(n- 2^d\phi) \biggl( 1 - \frac{ 2\phi}{n} \biggr)^n
- n^2 ( 1 - \phi/n)^{2n}.\nonumber
\end{eqnarray}
\begin{pf*}{Proof of (\protect\ref{varVlim})}
For asymptotics as $n \to\infty$ with $\rho$ fixed,
use the MacLaurin expansion of $\log(1-x)$
to obtain
\begin{eqnarray*}
\biggl(1 - \frac{2\phi}{n} \biggr)^n &=& e^{-2 \phi} \exp\biggl( -
\frac{2 \phi^2}{n} + O( n^{-2} ) \biggr);
\\
\biggl(1 - \frac{\phi}{n} \biggr)^{2n} &=& e^{-2 \phi} \exp\biggl( -
\frac{\phi^2}{n} + O(n^{-2}) \biggr)
\end{eqnarray*}
so that
\begin{eqnarray*}
n^{-1} \operatorname{Var}( V ) & = & \int_{B_{2\rho}(\mathbf{0})}
\biggl( 1 - \frac{\rho^d \omega_d(|y|/\rho) }{n} \biggr)^n \, dy
\\
&&{} + ne^{-2\phi} \biggl( \biggl(1 - \frac{ 2^d \phi}{n} \biggr)
\exp\biggl( - \frac{2\phi^2}{n} \biggr)
- \exp\biggl( - \frac{\phi^2}{n} \biggr) + O(n^{-2}) \biggr)
\\
&\to& \biggl( \int_{B_{2\rho}(\mathbf{0})}
\exp\bigl( - \rho^d \omega_d(|y|/\rho) \bigr) \,dy \biggr) - e^{-2 \phi}
( 2^d \phi+ \phi^2)
\end{eqnarray*}
and this limit is equal to $g_V(\rho)$ as defined by
(\ref{gVdef}), so the first part of (\ref{varVlim}) is proven.
It remains to show that $g_V(\rho)>0$.
This can be done either by
using
the last part of Theorem 2.1 of
\cite{PY1}, or directly. We leave it
to the reader to check that the conditions
of
the last part of Theorem 2.1 are satisfied here, or to look
up the direct argument which is in the first version of this
paper
(\href{http://www.arxiv.org/abs/0812.3084}{arXiv:0812.3084}).
Thus (\ref{varVlim}) holds in its entirety.
\end{pf*}
The computations for $S$ are somewhat similar.
With $X_i$ denoting the indicator of the event that
$U_i$ is isolated,
\begin{eqnarray*}
\operatorname{Var}(S)
& = &
n \operatorname{Var}(X_1) + n(n-1) \operatorname{Cov}(X_1,X_2)
\\
& = & n (1 - \phi/n)^{n-1} \bigl(1 - (1 - \phi/n)^{n-1}\bigr)
+ n (n-1) \operatorname{Cov}(X_1,X_2).
\end{eqnarray*}
Since
$\operatorname{Cov}(X_1,X_2) = \mathbb E[X_1X_2] - \mathbb E[X_1]^2$,
provided $4\rho< n^{1/d}$ we can write
\begin{eqnarray}\label{0521a}\quad
\operatorname{Var}(S)
& = & n (1 - \phi/n)^{n-1} \bigl(1 - (1 - \phi/n)^{n-1}\bigr)
\nonumber\\
& &{} + (n-1) \int_{B_{2 \rho} (\mathbf{0}) \setminus B_\rho(\mathbf{0})}
\biggl( 1- \frac{\rho^d \omega_d(|y|/\rho)}{n} \biggr)^{n-2} \, dy
\\
& &{} + n(n-1) \biggl( \biggl(1 - \frac{2^d \phi}{n} \biggr) \biggl(1 -
\frac{2\phi}{n} \biggr)^{n-2} - \biggl(1 - \frac{\phi}{n}
\biggr)^{2n-2}\biggr).\nonumber
\end{eqnarray}
\begin{pf*}{Proof of (\protect\ref{varWlim})}
For asymptotics as $n \to\infty$ with $\rho$ fixed,
by again using the MacLaurin expansion of $\log(1-x)$ we obtain
\begin{eqnarray*}
\biggl(1-\frac{2\phi}{n} \biggr)^{n-2} & = &
\exp\biggl((n-2) \biggl(-\frac{2\phi}{n}-\frac{2\phi^2}{n^2} +
O(n^{-3})\biggr) \biggr)
\\
& = &\exp\biggl(-2\phi+ \frac{ 4 \phi- 2 \phi^2}{n} + O(n^{-2})\biggr)
\end{eqnarray*}
and
\begin{eqnarray*}
\biggl(1-\frac{\phi}{n}
\biggr)^{2n-2}
& = &\exp\biggl((2n-2)\biggl(-\frac{\phi}{n}-\frac{\phi^2}{2n^2} +
O(n^{-3})\biggr) \biggr)
\\
& = &
\exp\biggl(-2\phi+ \frac{ 2 \phi- \phi^2}{n} +
O(n^{-2}) \biggr)
\end{eqnarray*}
and hence the last term in the right-hand side of (\ref{0521a})
is equal to
\begin{eqnarray*}
&&
n(n-1)
\exp(-2\phi)
\\
&&\quad{} \times\biggl( \biggl(1 - \frac{2^d \phi}{n}
\biggr)\exp\biggl(\frac{4 \phi- 2\phi^2}{n}\biggr)
- \exp\biggl(\frac{2\phi- \phi^2}{n} \biggr)
+ O(n^{-2}) \biggr)\\
&&\qquad= n(n-1) \exp(-2\phi) \biggl(- \frac{2^d \phi}{n} + \frac
{2 \phi}{n} -
\frac{\phi^2}{n} + O(n^{-2}) \biggr),
\end{eqnarray*}
so that
\begin{eqnarray*}
&& \lim_{n \to\infty} n^{-1}
\operatorname{Var}(S)
\\
&&\qquad = e^{-\phi} (1 - e^{-\phi})
- e^{-2 \phi}\bigl( (2^d -2) \phi+ \phi^2\bigr)
+ \int_{B_{2\rho}(\mathbf{0}) \setminus B_\rho(\mathbf{0})}
e^{- \rho^d \omega_d(|y|/\rho)} \,dy
\\
&&\qquad = e^{-\phi} - \bigl(1 + (2^d-2) \phi+ \phi^2\bigr)
e^{-2 \phi} + \rho^d \int_{B_{2}(\mathbf{0}) \setminus B_1(\mathbf{0})}
e^{- \rho^d \omega_d(|u|)} \,du,
\end{eqnarray*}
and since this limit is equal to $g_S(\rho)$ as
defined by (\ref{gWdef}),
we have proved the first
part of (\ref{varWlim}), namely, convergence to $g_S(\rho)$.
To complete the proof of (\ref{varWlim}), we need to
show that $g_S(\rho) > 0$. This can be done by
the same arguments as for the
proof of (\ref{varVlim}).
Hence,
(\ref{varWlim}) holds in its entirety.
\end{pf*}
\begin{pf*}{Proof of Theorem \protect\ref{thmlimsup}}
It remains only to prove
(\ref{main2}),
(\ref{main1})
and (\ref{SLB}).
By definition
$ \eta_V(\rho)
= \lim_{n \to\infty} \eta_V(n,\rho) $
and
$ \eta_S(\rho)
= \lim_{n \to\infty} \eta_S(n,\rho)$.
Then
(\ref{main2})
follows at once from Theorem \ref{thm2}, along with
(\ref{meanlim}) and (\ref{varVlim}).
Similarly,
(\ref{main1})
follows at once from Theorem \ref{thm1} along with (\ref{meanlim}),
and
(\ref{varWlim}).
Finally, we demonstrate the asymptotic lower bound (\ref{SLB}).
For any random variable $X$,
let $F_X$ denote its cumulative distribution function
and
let
$f_X$ denote its probability density function (if it has one).
Let $\varepsilon\in(0,1)$.
Set
\[
t_1 : = \frac{ [\mu_S] -\mu_S }{\sigma_S} ;\qquad
t_2 := \frac{ [\mu_S] -\mu_S +1 - \varepsilon}{\sigma_S} .
\]
Here $[\cdot]$ denotes integer part, so that $|t_i| \leq\sigma_S^{-1}
$ for $i=1,2$.
By the unimodality of the standard normal density,
\begin{eqnarray}\label{1031a}
F_Z(t_2 ) - F_Z(t_1) & \geq&
(t_2-t_1) \min(f_Z(t_1),f_Z(t_2)) \nonumber\\[-8pt]\\[-8pt]
& \geq&
(1 - \varepsilon) \sigma_S^{-1} f_Z(\sigma_S^{-1}).\nonumber
\end{eqnarray}
On the other hand, since $S$ is integer-valued, $F_{(S-\mu_S)/\sigma_S}(t_1) $
is equal to\break
$F_{(S-\mu_S)/\sigma_S}(t_2)$,
so that by (\ref{1031a})
\[
D_S
\geq(1/2)
(1 - \varepsilon) \sigma_S^{-1} f_Z(\sigma_S^{-1}).
\]
Scaling by $n^{1/2}$,
letting $n \to\infty$, using (\ref{varWlim}) and
letting $\varepsilon\to0$
yields (\ref{SLB}).
\end{pf*}
To conclude,
we compute some numerical values for
the asymptotic upper bounds
appearing in
(\ref{main2}) and (\ref{main1}).
For this we need
to compute $J_{r,d}(\rho)$ defined by (\ref{Jdef}) (for $r=1$ and $r=2$),
and for this in turn,
we need to compute $\omega_d(u) $, the volume of the
union of two unit balls in $d$-space whose centers are at points
($x,x'$ say) distance $u$ apart ($u \leq2 $).
Clearly,
$ \omega_1(u) = 2 + u$, and
generalizing (6) of \cite{Moran1}
to arbitrary $d \geq2$, we have
\begin{equation}\label{omMoran}
\omega_d(u) = \pi_d + \pi_{d-1} \int_0^u \bigl( 1- (t/2)^2 \bigr)^{(d-1)/2}
\,dt,\qquad
d \geq2.
\end{equation}
Using the preceding formulae,
we have computed numerical values for the asymptotic
upper bounds in Theorem \ref{thmlimsup},
for the cases with $\rho=1$ and $d \leq3$. These are as follows to
five significant figures, where $\delta_V(\rho)$ denotes the right-hand
side of (\ref{main2}) and $\delta_S(\rho)$ denotes the right-hand
side of (\ref{main1}):
\begin{eqnarray*}
\delta_V (1) &=& \cases{
6.4252 \times10^3, &\quad if $d=1$, \cr
8.6212 \times10^5, &\quad if $d=2$, \cr
1.4451 \times10^8, &\quad if $d=3$,}
\\
\delta_S (1) &=& \cases{
2.1024 \times10^3, &\quad if $d=1$, \cr
4.6833 \times10^4, &\quad if $d=2$, \cr
1.0578 \times10^6, &\quad if $d=3$.}
\end{eqnarray*}
\section*{Acknowledgments}
This work was partly done at meetings in
the Institute for Mathematical Sciences at the National University
of Singapore, whose
support we gratefully acknowledge.
We also thank Joseph Yukich for conversations which stimulated
our initial interest in this topic.
We are indebted to the referee for the idea for Lemma \ref{LGthm},
and other helpful suggestions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,693 |
We all had those days that nothing goes right. When your bad day comes from homeschooling your child in math you want to reflect on what happened so the next day will be great. Let me give you hope when you have a bad day in math- it's ONE DAY not 1 year. Of course if every day is a bad day that's another topic but let's just stick to dealing with days that nothing goes well.
I'm sure you could add many more reasons to my list but here's my point: try to determine the main reason your day in learning math did not work.
The key to improving anything is to know what the issue is so focus your energy on lessons learned after a bad day of math. Some of the adjustments you will need to make maybe small like changing the time you teach math so your child has more energy or you may have to make bigger changes if you have many bad days with math. These "big changes" could include switching to another curriculum or moving your child to another level. However before you take a big step in your child's education you really need to pin point the root causes of why things are not working.
Listen we all have bad days it's going to happen just remember these bad days can be a "bump" in the road or a warning sign to consider significant change. As long as you are paying attention to the causes and effects in your child's learning these "bad days" can be valuable indicators that help us improve. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,788 |
Замков — имя собственное; распространено в виде фамилий, имеет женскую форму Замкова.
Замков, Алексей Андреевич (1883—1942) — русский, советский врач, хирург, терапевт, уролог, создатель препарата «Гравидан».
Замков, Василий Михайлович (1936—2013) — участник войны во Вьетнаме, Герой Народных Вооружённых сил Вьетнама.
Замков, Владимир Константинович (1925—1998) — советский художник-монументалист, член-корреспондент АХ СССР.
Замкова
Смирнова-Замкова, Александра Ивановна (1880—1962) — русский, советский и украинский патологоанатом, академик АН УССР (1951-62).
См. также
Замок | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,115 |
{"url":"https:\/\/gmatclub.com\/forum\/the-mean-of-fourteen-consecutive-integers-is-27-1-2-if-the-integers-a-328940.html","text":"GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video\n\n It is currently 03 Aug 2020, 10:09\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\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# The mean of fourteen consecutive integers is 27 1\/2. If the integers a\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\nAuthor Message\nTAGS:\n\n### Hide Tags\n\nMath Expert\nJoined: 02 Sep 2009\nPosts: 65764\nThe mean of fourteen consecutive integers is 27 1\/2. If the integers a\u00a0 [#permalink]\n\n### Show Tags\n\n10 Jul 2020, 03:41\n00:00\n\nDifficulty:\n\n35% (medium)\n\nQuestion Stats:\n\n72% (02:25) correct 28% (02:08) wrong based on 36 sessions\n\n### HideShow timer Statistics\n\nThe mean of fourteen consecutive integers is $$27 \\frac{1}{2}$$. If the integers are arranged in increasing order. what is the mean of the largest seven integers?\n\nA. 24\nB. 25\nC. 31\nD. 34\nE. 35\n\n_________________\nManager\nJoined: 05 Jan 2020\nPosts: 142\nThe mean of fourteen consecutive integers is 27 1\/2. If the integers a\u00a0 [#permalink]\n\n### Show Tags\n\nUpdated on: 10 Jul 2020, 04:29\nMean = 27.5\nSince number of elements is even and elements are consecutive integers, the mean will the average of 7th and 8th integers => 7th element = 27\n=> first element = 21\n=> 4th element = 24\n\nEdited: mean of largest 7 numbers will be the 11th integer\n11th integer = 31\nAns: C\n\nOriginally posted by Lipun on 10 Jul 2020, 03:45.\nLast edited by Lipun on 10 Jul 2020, 04:29, edited 1 time in total.\nSenior Manager\nJoined: 16 May 2011\nPosts: 283\nConcentration: Finance, Real Estate\nGMAT Date: 12-27-2011\nWE: Law (Law)\nRe: The mean of fourteen consecutive integers is 27 1\/2. If the integers a\u00a0 [#permalink]\n\n### Show Tags\n\n10 Jul 2020, 03:47\nbecause they are Even numbers the mean is between 7th and 8th items so 27 an 28\n\n28 is the smallest of the largest seven numbers and 34 is the largest, so 28+34\/2= 31\nIntern\nJoined: 21 Aug 2015\nPosts: 16\nLocation: India\nConcentration: Strategy, Technology\nGMAT 1: 680 Q49 V32\nGPA: 3.4\nWE: Information Technology (Consulting)\nRe: The mean of fourteen consecutive integers is 27 1\/2. If the integers a\u00a0 [#permalink]\n\n### Show Tags\n\n10 Jul 2020, 03:51\n(C). mean = 55\/2. Since there are 14 consecutive numbers, mean 55\/2 is also mean of 7th and 8th numbers. Therefore 7th number is 27 and 8th is 28.\nSo, largest 7 numbers are - 28, 29, 30, 31, 32, 33, 34. And mean of these 7 numbers would be 4th number - 31.\nManager\nJoined: 11 Mar 2012\nPosts: 178\nLocation: India\nConcentration: General Management, Marketing\nGMAT 1: 600 Q50 V22\nGPA: 3.48\nWE: Project Management (Real Estate)\nRe: The mean of fourteen consecutive integers is 27 1\/2. If the integers a\u00a0 [#permalink]\n\n### Show Tags\n\n10 Jul 2020, 05:50\nBunuel wrote:\nThe mean of fourteen consecutive integers is $$27 \\frac{1}{2}$$. If the integers are arranged in increasing order. what is the mean of the largest seven integers?\n\nA. 24\nB. 25\nC. 31\nD. 34\nE. 35\n\nAs it is mentioned that the integers are consecutive\n\nLet the first integer be a, the 14th integer will be a+13d, where d = 1\n\nAverage of the 14 integers = {[a+(a+13)]}\/2 = 27.5\n\n=> a+6.5 = 27.5\n\n=> a = 21\n\nFinally, 8th integer = a+7d = 28\nAnd, 14th integer = a+13d = 34\n\nMean of last 7 integers = (28+34)\/2 = 31\n\nOA: C\n\n_________________\nTop Scorer of SP Jain Global's SPJAT with above 99 percentile score\n\nAiming for a real big thing through GMAT\nRe: The mean of fourteen consecutive integers is 27 1\/2. If the integers a \u00a0 [#permalink] 10 Jul 2020, 05:50\n\n# The mean of fourteen consecutive integers is 27 1\/2. If the integers a\n\n new topic post reply Question banks Downloads My Bookmarks Reviews Important topics\n\n Powered by phpBB \u00a9 phpBB Group | Emoji artwork provided by EmojiOne","date":"2020-08-03 18:09:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6886599063873291, \"perplexity\": 7730.623055508611}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439735823.29\/warc\/CC-MAIN-20200803170210-20200803200210-00276.warc.gz\"}"} | null | null |
Q: How to create nested ViewComponents in Monorail and NVelocity? I have been asked to update the menu on a website we maintain. The website uses Castle Windors Monorail and NVelocity as the template. The menu is currently rendered using custom made subclasses of ViewComponent, which render li elements. At the moment there is only one (horizontal) level, so the current mechanism is fine.
I have been asked to add drop down menus to some of the existing menus. As this is the first time I have seen Monorail and NVelocity, I'm a little lost.
What currently exists:
<ul>
#component(MenuComponent with "title=Home" "hover=autoselect" "link=/")
#component(MenuComponent with "title=Videos" "hover=autoselect")
#component(MenuComponent with "title=VPS" "hover=autoselect" "link=/vps")
#component(MenuComponent with "title=Add-Ons" "hover=autoselect" "link=/addons")
#component(MenuComponent with "title=Hosting" "hover=autoselect" "link=/hosting")
#component(MenuComponent with "title=Support" "hover=autoselect" "link=/support")
#component(MenuComponent with "title=News" "hover=autoselect" "link=/news")
#component(MenuComponent with "title=Contact Us" "hover=autoselect" "link=/contact-us")
</ul>
Is it possible to have nested MenuComponents (or a new SubMenuComponent) something like:
<ul>
#component(MenuComponent with "title=Home" "hover=autoselect" "link=/")
#component(MenuComponent with "title=Videos" "hover=autoselect")
#blockcomponent(MenuComponent with "title=VPS" "hover=autoselect" "link=/vps")
#component(SubMenuComponent with "title="Plans" "hover=autoselect" "link=/vps/plans")
#component(SubMenuComponent with "title="Operating Systems" "hover=autoselect" "link=/vps/os")
#component(SubMenuComponent with "title="Supported Applications" "hover=autoselect" "link=/vps/apps")
#end
#component(MenuComponent with "title=Add-Ons" "hover=autoselect" "link=/addons")
#component(MenuComponent with "title=Hosting" "hover=autoselect" "link=/hosting")
#component(MenuComponent with "title=Support" "hover=autoselect" "link=/support")
#component(MenuComponent with "title=News" "hover=autoselect" "link=/news")
#component(MenuComponent with "title=Contact Us" "hover=autoselect" "link=/contact-us")
</ul>
I need to draw the sub menu (ul and li elements) inside the overridden Render method on MenuComponent, so using nested ViewComponent derivatives may not work. I would like a method keep the basically declarative method for creating menus, if at all possible.
edit: I can use Context.RenderBody() to render the nested ViewComponent derivitives, but they're being rendered before the parent. I guess the rending of the sub menus need to somehow hook into the same output as the parent?
A: My original render method looked like
public override void Render()
{
var buffer = new StringBuilder();
var extraClass = _hoverState == MenuHoverState.Selected ? "class=\"Selected\"" : "";
// Menu Item Start
buffer.Append("<li><a href=\"" + ComponentParams["link"] + "\"" + extraClass + ">");
// Menu Text
buffer.Append(ComponentParams["title"]);
// Menu Item End
buffer.Append("</a></li>");
RenderText(buffer.ToString());
}
I needed to hook into the Context.Writer:
public override void Render()
{
var buffer = new StringBuilder();
var extraClass = _hoverState == MenuHoverState.Selected ? "class=\"Selected\"" : "";
// Menu Item Start
buffer.Append("<li><a href=\"" + ComponentParams["link"] + "\"" + extraClass + ">");
// Menu Text
buffer.Append(ComponentParams["title"]);
// Menu Item End
buffer.Append("</a><ul class=\"subMenu\" style=\"display:none;\">");
Context.Writer(buffer.ToString());
Context.RenderBody(Context.Writer);
Contet.Writer("</ul></li>");
}
A:
I can use Context.RenderBody() to render the nested ViewComponent derivitives
in your Render method override, you could use something like
RenderView("header");
RenderBody();
RenderView("footer");
and maybe use RenderSection could be useful to be able to override some parts from the template you use the component
if(HasSection("header")){
RenderSection("header");
} else {
RenderView("header");
}
it is also possible to itterate and alter the context:
for(var item in this.SubItems){
PropertyBag["item"] = item;
if(HasSection("item")){
RenderSection("item");
} else {
RenderView("item");
}
}
all these solution are fancy, but I generaly prefer having a viewcomponent which takes a purpose specific viewmodel (say HierarchicalMenuViewModel) as a parameter and keep the templating logic simple, it's more easy to use, and output customization happen
at least for simple controls (that would sometime only deserve a macro or partial depending on the viewengine).
In the end, viewcomponent concepts illustrated above are still nice to have when doing control that need more customization. An advice is to take care of documenting the rendering logic or keeping it simple (<= 10 lines in render method)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,291 |
{"url":"https:\/\/discourse.pymc.io\/t\/speeding-up-inference-on-large-datasets\/7450","text":"# Speeding up inference on large datasets\n\nI started learning about bayesian methods, becuase I am trying to solve a problem where I would like to modify a time series\u2019 values to correct for a bias factor that is known to artificially inflate or deflate the value of the time series.\n\nI am using a model that inspired by HMMs but is continuous.\n\nObserved values are in green and latent variables are in red.\n\nI have specified the variables as follows:\nTrue\\_Value_t \\sim \\mathcal{N}(True\\_Value_{t-1}, \\sigma_t)\nTrue\\_Value_0 \\sim \\mathcal{N}(0, \\sigma_t)\nBias_t \\sim \\mathcal{N}(0, \\sigma_b)\nBias\\_Coeff \\sim \\mathcal{N}(0, 1)\nMeasurement_t \\sim \\mathcal{N}(True\\_Value_t + Bias\\_Coeff \\cdot Bias_t, \\sigma_m)\n\nUsing this model I can infer the most likley sequence of True\\_Value, to \u2018correct\u2019 the series of measurements accounting for the bias, using find_MAP.\n\nThis works well but I want to perform this analysis for a large dataset, and use the result of this inference for a downstream ML task.\n\nI have 4k of these sequences with an average length of 20 meaning I will be running inference roughly 80k times!\n\nAs i want to use the smoothed series for a downstream ML task I will need to calculate a smoothed series for each step to prevent dataleakage into my downstream model.\n\nNpVector = np.ndarray\ndef create_model(measurements: NpVector, bias_values: NpVector):\nn = len(measurements)\ntrue_value = pm.distributions.timeseries.GaussianRandomWalk('true_value', sigma=1, shape=n)\nbias = pm.Normal('bias', mu=0, sigma=1, shape=n, observed=bias_values)\nbias_coeff = pm.Normal('bias_coeff', mu=0, sigma=1)\nmeasurement = pm.Normal('measurement', mu=true_value + bias_coeff*bias, sigma=1, shape=n, observed=measurements)\n\n\nAt the moment I am now doing inference in a for loop:\n\nfor measurements, biases in data:\nwith pm.Model as model:\ncreate_model(measurements, biases)\nMAP = pm.find_MAP()\n\n\nI have tried to use multiprocessing's pool to parallelise this for loop, but am not sure how this will play with pymc3\u2019s implementation of find_MAP.\n\nI was wondering if there is some way for me to do inferences on chains of the same length fitted with different independent sets of observations?\n\nOr if there would be a way to do some sort of online learning as the length of the sequence grows?\n\n1 Like","date":"2022-07-04 15:13:38","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\": 0.7610816359519958, \"perplexity\": 2438.022351784886}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104432674.76\/warc\/CC-MAIN-20220704141714-20220704171714-00554.warc.gz\"}"} | null | null |
import { DataEntity, pDelay } from '@terascope/utils';
import { DelayConfig } from './interfaces';
import { BatchProcessor } from '../../operations';
export default class Delay extends BatchProcessor<DelayConfig> {
async onBatch(data: DataEntity[]): Promise<DataEntity[]> {
await pDelay(this.opConfig.ms);
return data;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,640 |
{"url":"https:\/\/mathoverflow.net\/questions\/297486\/how-many-conjugacy-classes-of-elementary-abelian-subgroups-of-rank-2-does-gl","text":"# How many conjugacy classes of elementary abelian subgroups of rank $2$ does $GL_{n}(Z \/ pZ)$ have?\n\nLet $p$ be a prime number and $G=GL_n ( \\mathbb{Z} \/ p \\mathbb{Z} )$. Consider the set $U$ of upper-triangular matrices of $G$ having entries of $1$ on the diagonal. The cardinality of $U$ is $p^{\\frac {n(n-1)} 2}$ and $U$ is a subgroup of $G$, in particular $U$ is a Sylow $p$-subgroup of $G$. It is well known that the Sylow $p$-subgroups of a group $G$ are conjugate, and every $p$-subgroup $H$ of $G$ is contained in some Sylow $p$-subgroup of $G$. Then there exists $g\\in G$ such that $H\\leq gUg^{-1}$, which allows us to compute the number of conjugacy classes of elementary abelian subgroups of rank $2$ ($H=( \\mathbb{Z} \/ p \\mathbb{Z} ) ^2$) in the Sylow $p$-subgroup $U$. Any help would be appreciated so much. Thank you all.","date":"2019-03-26 02:59:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7841154932975769, \"perplexity\": 23.81549132852966}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912204768.52\/warc\/CC-MAIN-20190326014605-20190326040605-00269.warc.gz\"}"} | null | null |
#import "TiModule.h"
#ifdef USE_TI_MAP
@interface MapModule : TiModule {
}
@property(nonatomic,readonly) NSNumber *STANDARD_TYPE;
@property(nonatomic,readonly) NSNumber *SATELLITE_TYPE;
@property(nonatomic,readonly) NSNumber *HYBRID_TYPE;
@property(nonatomic,readonly) NSNumber *ANNOTATION_RED;
@property(nonatomic,readonly) NSNumber *ANNOTATION_GREEN;
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\section{Introduction}
This paper is devoted to the introduction of a representation theoretic approach to the study of the topology of moduli spaces and stacks of (semi)stable Higgs bundles and Higgs sheaves, or of other moduli spaces of vector bundles with extra structure, on a fixed smooth projective curve over a field $k$.
Let us recall that a Higgs bundle on a complex Riemann surface $X$ of arbitrary genus is a pair $(\mathcal{F}, \phi\colon \mathcal{F} \to \mathcal{F}\otimes \omega_X)$ consisting of a vector bundle $\mathcal{F}$ and a morphism $\phi$ called a \emph{Higgs field}; here $\omega_X$ is the canonical line bundle of $X$. Moduli spaces of \emph{stable} Higgs bundles of fixed rank and degree over $X$ were introduced by Hitchin in the late 80's \cite{art:hitchin1987, art:hitchin1987-II} --- see, e.g., \cite[Appendix]{book:wells2008} and \cite{art:casalainawise2017}, respectively, for a differential-geometric point of view and for an algebro-geometric one to the Hitchin moduli spaces. These moduli spaces have a rich geometry: for example, they are smooth quasi-projective varieties and, from a differential point of view, they are endowed with a complete hyperk\"ahler metric. In addition, the map which associates with any stable Higgs bundle $(\mathcal{F}, \phi)$ the characteristic polynomial of $\phi$ defines a complete integrable system, called the \emph{Hitchin fibration}. The preimage with respect to zero of the Hitchin fibration is the so-called \emph{global nilpotent cone}, which parametrizes stable Higgs bundles $(\mathcal{F}, \phi)$ with nilpotent Higgs field $\phi$. Since its introduction, the Hitchin moduli space has played a preeminent role in the theory of moduli spaces, integrable systems, mirror symmetry, number theory, and string and gauge theories.
In the well-documented analogy between (smooth, projective) curves and quivers the role of the Hitchin moduli stack is played by the \emph{preprojective stack}, and the analog of the global nilpotent cone is the \emph{Lusztig nilpotent stack}. As for the moduli space of (stable) Higgs bundles, the closest analog is another family of non-compact hyperk\"ahler manifolds which share similar geometric properties, the \emph{Nakajima quiver varieties} --- introduced in \cite{art:nakajima:1994-3}. They admit as well a canonical projective morphism to an affine variety, which is the affinization map --- such a morphism plays the role of the Hitchin fibration; the global nilpotent cone is replaced by the \emph{Lagrangian Nakajima quiver variety}. See \cite{art:schiffmann2006-II, art:ginzburg2012} for an introduction to the theory of quiver varieties, and \cite{art:bozec2016} for more details on the case of quivers with edge loops.
As illustrated by the classical results of Nakajima and others, the (co)homology (or K-theory) of quiver varieties is extremely rich from the point of view of representation theory, and many of its topological invariants have representation-theoretic meanings. For instance, the computation of the Poincar\'e polynomials of Nakajima quiver varieties associated with an arbitrary quiver was done by Hausel in \cite{art:hausel2010}, where he showed that such a polynomial is related to the \emph{Kac's A-polynomial} of the quiver. Recall that, as proved by Kac and Stanley \cite{art:kac1982, art:kac1983} the number of geometrically indecomposable $\mathbb{F}_q$-representations of a quiver $Q$ of given dimension $\mathbf{d}$ is given by a polynomial $A_{Q, \mathbf{d}}(q)$ in $q$, called Kac's $A$-polynomial. Another geometric interpretation of the Kac's A-polynomial is the one in terms of the Poincar\'e polynomial of the preprojective stack, i.e., the stack of representations of the preprojective algebra $\Pi_Q$ associated with $Q$ \footnote{One has also a nilpotent version of such a relation, by considering from the algebraic side nilpotent versions of the Kac's A-polynomial and from the geometric side the generalizations of Lusztig's nilpotent variety introduced in \cite{art:bozecschiffmannvasserot2017,art:schiffmannvasserot2017} (see also \cite{art:bozec2015, art:bozec2016}).}. Much more recently, this relation between a polynomial of geometric nature, such as the Poincar\'e polynomials of Nakajima quiver variety associated with $Q$ and of the stack of representations of $\Pi_Q$, and a polynomial of representation-theoretic nature, such as the Kac's A-polynomial $A_{Q, \mathbf{d}}(q)$, has been ``categorified" in the following way (cf.\ \cite{art:schiffmannvasserot2017}): there exists an associative algebra structure on the Borel-Moore homology of the stack of representations of $\Pi_Q$ --- the so-called \emph{cohomological Hall algebra} --- whose Hilbert series is given by the Kac's A-polynomial of $Q$. Moreover, such an algebra is conjecturally\footnote{The conjecture is true for finite and affine quivers. At the moment there is only a partial result for general quivers: see \cite{art:schiffmannvasserot2017-II}.} isomorphic to the positive part of the Yangian algebra $\mathsf{Y}(\mathfrak{g}_Q)$ of the Maulik-Okounkov graded Lie algebra $\mathfrak{g}_Q$ (cf.\ \cite{art:maulikokounkov2012, art:schiffmannvasserot2017-II} for a definition of $\mathfrak{g}_Q$). What's more, this algebra acts on the Borel-Moore homology of Nakajima quiver varieties associated with the same quiver, and such an action extends to a larger\footnote{At least when the quiver is not of finite type.} algebra of symmetries Nakajima's construction of representations of $\mathsf{U}(\mathfrak{g}_Q)$ on the Borel-Moore homology of Nakajima quiver varieties.
Let us return to the curve case, for which the situation is (from that point of view) much less developed. The Poincar\'e polynomial of the moduli stack of Higgs bundles is ill-defined (i.e. the Betti numbers are in general infinite), and the Betti numbers of the moduli spaces of stable Higgs bundles on curve $X$, for coprime rank and degree, were only recently computed in terms of the \emph{Kac polynomial} of $X$ in \cite{art:schiffmann2016}. We refer to \emph{loc. cit.} for a precise definition of these Kac polynomials $A_{r,\, d,\, g}(z_1, \ldots, z_{2g})$, which depend on the rank $r$, the degree $d$ and the genus $g$ of the curve\footnote{The independence on the degree $d$ was proved in \cite{art:mellit2017}.}, and whose evaluation at the Weil numbers $(\sigma_1, \ldots, \sigma_{2g})$ of the curve is equal to the number of geometrically indecomposable vector bundles of rank $r$, degree $d$ on the curve $X$ defined over $\mathbb{F}_q$.
It is natural to wonder if in the curve case also there is a deeper representation theoretic result behind such an enumerative relation, which would allow us to bring representation theoretic tools to the study the topology of moduli spaces of stable Higgs bundles, or similar moduli spaces. The aim of the present paper is to perform the first step of this program, namely to construct the \emph{cohomological Hall algebra} attached to the stacks of Higgs sheaves over a smooth projective curve $X$ of genus $g$. Note that we consider here the entire stack $\mathbf{Higgs}(X)\coloneqq\bigsqcup_{r, \, d} \mathbf{Higgs}_{r, \, d} $ and not only of its stable part. Although our main potential applications in mind are in the context of Borel-Moore homology or K-theory, we develop the theory of these cohomological Hall algebra for an arbitrary free oriented Borel-Moore (OBM) homology theory (as is done in \cite{art:yangzhao2014} in the context of quivers).
The construction and detailed study of the cohomological Hall algebra for the stack of Higgs \emph{torsion} sheaves is the subject of the recent work by Minets in \cite{art:minets2018}. Our first main result extends the construction to the higher rank case.
\begin{theorem}[Theorem \ref{theorem:defproduct}]
Let $X$ be an irreducible smooth projective curve\footnote{We do not see any major difficulty to assume more generally that $X$ is a Gorenstein factorial projective (stacky) curve, but also no crucial benefit, and so we leave the details to the interested reader.} over a field $k$. Let $A$ be either the Borel-Moore homology or an arbitrary free oriented Borel-Moore homology theory\footnote{Since we are dealing with algebraic stacks with infinitely many irreducible components, we consider rather a subgroup $A^0 \subseteq A$ of classes satisfying some support condition, see Section \ref{sec:BM}.}. Then there is a canonical graded associative algebra structure on
\begin{align}
\mathbf{AHA}_{\mathsf{Higgs}(X)}\coloneqq\bigoplus_{r,\, d} A_\ast(\mathbf{Higgs}_{r,\, d}) \ .
\end{align}
\end{theorem}
There are some natural variants of this algebra, in which we replace the stacks $\mathbf{Higgs}_{r, \, d}$ by the global nilpotent cones $\mathbf{\Lambda}_{r, \, d}$ or the stacks of semistable Higgs bundles $\mathbf{Higgs}_{r, \, d}^{\mathsf{ss},\, \nu}$, of a fixed slope $\nu$. We can as well introduce an equivariant parameter coming from the action of $T\coloneqq\mathbb{G}_m$ by dilations on the Higgs field.
\begin{corollary}[Corollaries~\ref{cor:Higgsvariantnilp}, \ref{cor:Higgsvariantss}]
There are canonical graded associative algebra structures on
\begin{align}
\mathbf{AHA}_{\mathbf{\Lambda}}\coloneqq\bigoplus_{r, \, d} A_\ast(\mathbf{\Lambda}_{r, \, d})\ , \qquad \mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}\coloneqq\bigoplus_{\substack{d/r=\nu}} A_\ast(\mathbf{Higgs}^{\mathsf{ss}}_{r, \, d}) \quad \mbox{for all } \nu \in \mathbb{Q} \cup \{\infty\}
\end{align}
and on their $T$-equivariant cousins $\mathbf{AHA}^T_{\mathbf{\Lambda}}$, $\mathbf{AHA}^T_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}$.
\end{corollary}
There are some strong relations between these variants and the original cohomological Hall algebra of $\mathbf{Higgs}(X)$. For instance, the proper pushforward map $\mathbf{AHA}_{\mathbf{\Lambda}} \to \mathbf{AHA}_{\mathsf{Higgs}(X)}$ is an algebra homomorphism. Likewise, the open restriction map $\mathbf{AHA}_{\mathsf{Higgs}^\nu(X)} \to \mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}$ is an algebra homomorphism, where
\begin{align}
\mathbf{AHA}_{\mathsf{Higgs}^{\nu}(X)}\coloneqq\bigoplus_{d/r=\nu} A_\ast(\mathbf{Higgs}_{r, \, d})\ .
\end{align}
Moreover, the proper pushforward induces an isomorphism of localized algebras
\begin{align}\label{eqintro:1}
\mathbf{AHA}^T_{\mathbf{\Lambda}} \otimes_{A_T(\mathsf{pt})} \mathsf{Frac}(A_T(\mathsf{pt})) \stackrel{\sim}{\longrightarrow} \mathbf{AHA}^T_{\mathsf{Higgs}(X)} \otimes_{A_T(\mathsf{pt})} \mathsf{Frac}(A_T(\mathsf{pt})).
\end{align}
(see Proposition~\ref{prop:localizationT}).
Although the definition of cohomological Hall algebras can be given for an arbitrary free OBM theory in a very uniform fashion, the properties of these algebras strongly depend on the choice of the OBM theory. Our results concerning the structure of $\mathbf{AHA}_{\mathsf{Higgs}(X)}$ are for the moment restricted to the cases of usual Borel-Moore homology (or Chow groups). So \emph{we assume until the end of this introduction that $A=H_\ast$ and we restrict ourselves to $A^0$}. The cohomology ring of the stack $\mathbf{Coh}(X)$ of coherent sheaves on $X$ acts on $\mathbf{AHA}_{\mathsf{Higgs}(X)}$ by pullback to $\mathbf{Higgs}(X)$ and cap product. By Heinloth's generalization of the Atiyah-Bott theorem (see Theorem \ref{T:Heinloth}), this ring is (freely\footnote{Only in the positive rank case.}) generated by tautological classes, and we can define a universal ring (in fact, (co)commutative Hopf algebra) $\mathbb{H}$ which acts on $\mathbf{AHA}_{\mathsf{Higgs}(X)}$ (and on all its cousins).
The second main result of the paper concerns torsion-freeness. It can be seen as a key technical step to embed our algebra in a bigger shuffle-type algebra (as done in the rank zero case in \cite[Section~3]{art:minets2018}).
\begin{theorem}[Theorem \ref{T:torsionfree}]
Let $\alpha\in (\mathbb{Z}^2)^+$. Then $H^T_\ast(\mathbf{\Lambda}_{\alpha})$ is a torsion-free $H^\ast(\bCoh_\alpha)\otimes \mathbb{Q}[t]$-module.
\end{theorem}
From the analogy with the case of quivers, it is natural to expect that in fact $\mathbf{AHA}^T_{\Lambda}$ is of generic rank one (not free!), but we do not prove this here.
Our final main result, in a spirit similar to \cite[Theorem~B (e)]{art:schiffmannvasserot2017}, provides a family of generators for $\mathbf{AHA}_{\Lambda}$.
\begin{theorem}[Theorem \ref{T:gen}, Corollary \ref{cor:gen2}]
The $\mathbb{H}$-algebra $\mathbf{AHA}_{\Lambda}$ is generated by the collection of fundamental classes $\{[\mathbf{\Lambda}_{(r,\, d)}]\}_{r, \, d}$ of the zero sections of the projections $\mathbf{Higgs}_{r, \, d} \to \mathbf{Coh}_{r, \, d}$.
\end{theorem}
Of course, using Formula \eqref{eqintro:1} we may deduce similar results for $\mathbf{AHA}_{\mathsf{Higgs}(X)}$.
Let us conclude this introduction with some heuristics and speculations concerning the structure and representation theory of $\mathbf{AHA}_{\mathsf{Higgs}(X)}$.
First, we expect that our cohomological Hall algebra acts on the (oriented) Borel-Moore homology of moduli spaces of stable Higgs bundles (of fixed slope) and of Minets' generalization of Nakajima quiver varieties\footnote{These moduli spaces parametrize stable point in the cotangent stack of the stack of \emph{coframed pairs}, i.e., pairs $(\mathcal{F}, \mathcal{E}\to \mathcal{F})$ where $\mathcal{E}, \mathcal{F}$ are coherent sheaves on $X$, and $\mathcal{E}$ is fixed, see \cite{phdthesis:minets2018-II}.}. Slightly more generally, it is natural to expect that it will also act on suitable moduli spaces of stable and framed sheaves on smooth (stacky) surfaces containing the curve $X$ as an embedded divisor of self-intersection $2(1-g)$\footnote{More general embedded curves would require constructing a cohomological Hall algebra for \emph{arbitrarily} twisted Higgs bundles.}. If $X=\mathbb{P}^1$, examples of such surfaces are those described in \cite[Section~2, Remarks (ii)]{art:ginzburgkapranovvasserot1995} and the stacky surfaces defined in \cite{art:bruzzopedrinisalaszabo2016}.
Next, by analogy with the case of quivers, one can hope for a strong relation between the algebra $\mathbf{AHA}_{\mathsf{Higgs}(X)}$ and the (usual) Hall algebra of curves of genus $g$ over finite fields. Very slightly more precisely, one would expect the existence of a graded Lie algebra $\mathfrak{g}_{g}$ whose Hilbert series is given by the Kac polynomials $A_{g,\, r,\, d}$, which would be a 'generic' form of the Hall-Lie algebra of curves of genus $g$ on the one hand (cf.\ \cite[Section~8.3]{art:schiffmann2016} for the definition of such a Lie algebra), and whose affinization (or Yangian) would be isomorphic to $\mathbf{AHA}_{\mathsf{Higgs}(X)}$.
Finally, by the nonabelian Hodge correspondence \cite{art:simpson1994-II}, moduli spaces of stable rank $r$ Higgs bundles on $X$ are diffeomorphic to (twisted) character varieties of $X$ for the group $\mathsf{GL}(r)$. The topology of the latter moduli space has been extensively studied by Hausel, Letellier and Rodriguez-Villegas (cf.\ \cite{art:hauselletellierrodriguezvillegas2013} and the conjectures stated therein). To better understanding \emph{loc.cit.} from a representation theoretic point of view, Davison and Meinhardt \cite{art:davisonmeinhardt2016} have introduced a cohomological Hall algebra à la Kontsevich-Soibelman associated with the untwisted character stack for $\mathsf{GL}(r)$. We propose the following conjecture, which can be seen as a representation theoretical version of the nonabelian Hodge correspondence. The following conjecture was also made by B. Davison (see e.g. \cite{art:davison2016}).
\begin{conjecture}
The algebra $\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},0}(X)}$ is isomorphic to the Davison-Meinhard Cohomological Hall algebra of the genus $g$ untwisted character variety.
\end{conjecture}
This paper is organized as follows. Section \ref{sec:CohHiggs} provides notations and serves as a reminder concerning stacks of coherent and Higgs sheaves on smooth projective curves. The cohomological Hall algebras (or $A$-homological Hall algebras) are defined in Section \ref{sec:COHAHiggs}. From this point on, we restrict ourselves to the context of Borel-Moore homology. In Section \ref{sec:torsionfreeness} we introduce the universal cohomology ring of the stacks of coherent sheaves on curves of a fixed genus, and prove the torsion-freeness result. Section \ref{sec:generation} is devoted to the generation theorem.
\subsection*{Acknowledgements}
The first seed of the present paper can be traced back to some discussions during two visits of the first-named author to Paris: the first one was under the umbrella of the Research in Paris program supported by the Institut Henri Poincaré, while the host of the second visit was the Université of Paris-Sud. The first-named author thanks both istitutions for the hospitality and support. In addition, some of the results of the present paper were presented during the workshop on ``Hitchin systems in Mathematics and Physics" (February 2017, Perimeter Institute, Canada) and the workshop on ``Geometric Representation Theory" (July 2017, University of Glasgow, UK). The first-named author thanks the organizers and the participants of both workshops for interesting discussions. Finally, we thank B. Davison, D. E. Diaconescu, S. Meinhardt, A. Minets, A. Negut, Y. Soibelman and É. Vasserot for interesting discussions and comments.
\bigskip\section{Stacks of coherent and Higgs sheaves on a curve}\label{sec:CohHiggs}
In this section we introduce the stacks of coherent and Higgs sheaves on smooth projective curves, and recall some of their key properties. Because our construction of the multiplication in the cohomological Hall algebras uses the local charts defined in terms of Quot schemes, we go into some depth in describing the latter explicitly.
\subsection{The curve}
Let $X$ be an irreducible smooth projective curve of genus $g$ over a field $k$, and $\omega_X$ its canonical line bundle. As usual, we denote by $\operatorname{rk}(\mathcal{F}), \deg(\mathcal{F})$ the rank and degree of a coherent sheaf $\mathcal{F}$ on $X$ and by
\begin{align}
\mu(\mathcal{F}) =\frac{\deg(\mathcal{F})}{\operatorname{rk}(\mathcal{F})}\in \mathbb{Q} \cup \{ \infty\}
\end{align}
its slope. Denote by $\mathsf{Coh}(X)$ the category of coherent sheaves on $X$. It is an abelian category of homological dimension one. Denote by $\mathsf{K}(X)$ the \emph{Grothendieck group} of $X$ and by $\left[\mathcal{F}\right]$ the class of a coherent sheaf $\mathcal{F}$. Let $\mathsf{K}(X)^+$ be the semigroup of $\mathsf{K}(X)$ consisting of classes of the form $\left[\mathcal{F}\right]$, for a coherent sheaf $\mathcal{F}$ on $X$. There are natural maps
\begin{align}
\operatorname{rk}\colon \mathsf{K}(X)\to \mathbb{Z}_{\geq 0}\qquad \mbox{and} \qquad\deg\colon \mathsf{K}(X)\to \mathbb{Z}
\end{align}
assigning to $[\mathcal{F}]$ the rank and degree of $\mathcal{F}$ respectively. This yields a projection $\mathsf{K}(X)\to \mathsf{K}^{\mathsf{num}}(X)$, where $\mathsf{K}^{\mathsf{num}}(X)\coloneqq\mathbb{Z}^2$ is the \emph{numerical Grothendieck group} of $X$. We define the (numerical) class of a coherent sheaf $\mathcal{F}$ as the pair $\overline{\mathcal{F}}\coloneqq\big(\operatorname{rk}(\mathcal{F}), \deg(\mathcal{F})\big)$. We accordingly set
\begin{align}
\Ksf^{\mathsf{num},+}(X)=\{(r,d)\in \mathbb{Z}^2\;\vert \; r >0, d \in \mathbb{Z}\;\mbox{or}\; r=0, d \geq 0\}\eqqcolon (\mathbb{Z}^2)^+\ .
\end{align}
Finally, recall that the Euler form on $\mathsf{K}(X)$, which descends to $\mathsf{K}^{\mathsf{num}}(X)$, is explicitly given by the following formula:
\begin{multline}
\langle\, \overline{\mathcal{E}}, \overline{\mathcal{F}}\,\rangle\coloneqq\dim \mathsf{Hom}(\mathcal{E},\mathcal{F})-\dim \mathsf{Ext}^1(\mathcal{E}, \mathcal{F})\\
=(1-g)\operatorname{rk}(\mathcal{E})\operatorname{rk}(\mathcal{F})+(\operatorname{rk}(\mathcal{E})\deg(\mathcal{F})-\operatorname{rk}(\mathcal{F})\deg(\mathcal{E}))\ .
\end{multline}
\subsection{Stack of coherent sheaves}\label{sec:stackcoherentsheaves}
For $\alpha\in \Ksf^{\mathsf{num},+}(X)$, let $\bCoh_\alpha$ be the stack parameterizing coherent sheaves on $X$ of class $\alpha$. It is a smooth algebraic stack, locally of finite type over $\mathsf{Spec}(k)$, and irreducible of dimension $-\langle \alpha, \alpha \rangle$; in addition, $\bCoh_\alpha$ is equipped with a \emph{tautological sheaf} $\mathfrak{E}_\alpha\in \mathsf{Coh}\big(\bCoh_\alpha\times X)$ (see \cite[Théorème~4.6.2.1]{book:laumonmoretbailly2000}; the smoothness follows, e.g., from the description of an atlas of $\bCoh_\alpha$ given below). Since $\bCoh_\alpha$ is smooth, the cotangent complex\footnote{The theory of cotangent complexes for algebraic stacks is developed in \cite[Chapter~16]{book:laumonmoretbailly2000} and \cite{art:olsson2007}.} $\mathbb{L}_{\bCoha}$ of $\bCoh_\alpha$ is perfect (hence dualizable) of Tor-amplitude [0, 1] (cf.\ \cite[Proposition~17.10]{book:laumonmoretbailly2000}); the dual complex, the tangent complex $\mathbb{T}_{\bCoha}$, can be described explicitly as (cf.\ \cite{art:toenvaquie2007})
\begin{align}\label{eq:tangentcomplex}
\mathbb{T}_{\bCoha} =\mathbb{R} p_\ast\, \mathbb{R} \mathcal{H} om(\mathfrak{E}_\alpha, \mathfrak{E}_\alpha)[1]\ ,
\end{align}
where $p\colon \bCoh_\alpha\times X\to \bCoh_\alpha$ is the projection.
For later purposes, let us give an atlas for $\bCoh_\alpha$; this will be used in Section~3 for the definition of the cohomological Hall algebra associated with the moduli stacks of Higgs sheaves. Let us fix a line bundle $\mathcal{L}$ on $X$. We will say that a coherent sheaf $\mathcal{F}$ is \emph{strongly generated by} $\mathcal{L}$ if the canonical morphism $\mathsf{Hom}(\mathcal{L},\mathcal{F}) \otimes \mathcal{L} \to \mathcal{F}$ is surjective and $\mathsf{Ext}^1(\mathcal{L},\mathcal{F})=\{0\}$. We denote by $\Coh^{> \Lcal}(X)\subset \mathsf{Coh}(X)$ the full subcategory of coherent sheaves on $X$ which are strongly generated by $\mathcal{L}$. Note that $\Coh^{> \Lcal}(X)$ is stable under quotients and extensions and that
\begin{align}
\dim \mathsf{Hom}(\mathcal{L},\mathcal{F})=\langle \mathcal{L},\mathcal{F}\rangle
\end{align}
for all $\mathcal{F} \in \Coh^{> \Lcal}(X)$. Let $u_\alpha^\mathcal{L}\colon \bCoh_\alpha^{> \Lcal}\hookrightarrow \bCoh_\alpha$ be the open substack of $\bCoh_\alpha$ parameterizing sheaves strongly generated by $\mathcal{L}$ and of class $\alpha$. We call $\bCoh_\alpha^{> \Lcal}$ a \emph{local chart} of $\bCoh_\alpha$. The stack $\bCoh_\alpha^{> \Lcal}$ can be realized as a global quotient stack as follows. Let $\mathsf{Quot}_\alpha^\mathcal{L}\coloneqq\mathsf{Quot}_{X/k}\big(\mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}}, \alpha\rangle}, \alpha\big)$ be the Quot scheme parameterizing isomorphism classes of quotients $\phi \colon \mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}}, \alpha\rangle}\twoheadrightarrow \mathcal{F}$ such that $\overline{\mathcal{F}}=\alpha$ (see \cite[Section~2.2]{book:huybrecthslehn2010} for an introduction to the theory of Quot schemes). This is a projective $k$-scheme, which is singular in general, of finite type and carries a canonical $\mathsf{G}_\alpha^\Lcal\coloneqq\mathsf{GL}(k,\langle \overline{\mathcal{L}},\alpha\rangle)$-action defined by $g \cdot \phi\coloneqq\phi \circ (\mathsf{id}_\mathcal{L} \otimes g^{-1})$. Its Zariski tangent space at a point $\big[\phi \colon \mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}\twoheadrightarrow \mathcal{F}\big]$ is $\mathsf{Hom}(\ker(\phi), \mathcal{F})$, while the obstruction to the smoothness lies in $\mathsf{Ext}^1(\ker(\phi),\mathcal{F})$. Consider the open subscheme $\mathsf{Q}_\alpha^\Lcal \subset \mathsf{Quot}_\alpha^\mathcal{L}$ be the open subscheme whose $k$-points are
\begin{align}
\mathsf{Q}_\alpha^\Lcal(k)\coloneqq\{\big[\phi\colon\mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}\twoheadrightarrow \mathcal{F}\big] \in \mathsf{Quot}_\alpha^\mathcal{L}(k)\;\vert\; \phi_\ast\colon k^{\,\langle \overline{\mathcal{L}},\alpha\rangle} \stackrel{\sim}{\longrightarrow} \mathsf{Hom}(\mathcal{L},\mathcal{F})\}\ .
\end{align}
\begin{proposition}
The following hold:
\begin{itemize}
\item[(i)] The scheme $\mathsf{Q}_\alpha^\Lcal$ is $\mathsf{G}_\alpha^\Lcal$-invariant and there is a canonical isomorphism of algebraic stacks
\begin{align}
\bCoh_\alpha^{> \Lcal} \simeq \big[ \mathsf{Q}_\alpha^\Lcal/\mathsf{G}_\alpha^\Lcal\big]\ .
\end{align}
\item[(ii)] $\mathsf{Q}_\alpha^\Lcal$ is smooth and reduced.
\end{itemize}
\end{proposition}
Statement (i) is shown for example in the proof of \cite[Théorème~4.6.2.1]{book:laumonmoretbailly2000}, while (ii) follows from the vanishing of $\mathsf{Ext}^1(\ker(\phi),\mathcal{F})$ for any point $\big[\phi \colon \mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}\twoheadrightarrow \mathcal{F}\big]\in \mathsf{Q}_\alpha^\Lcal$ and \cite[Theorem~5.3]{book:newstead1978}. We may think of $\mathsf{Q}_\alpha^\Lcal$ as the fine moduli space parameterizing pairs $(\mathcal{F},u)$ where $\mathcal{F} \in \Coh^{> \Lcal}(X)$ is of class $\alpha$ and $u$ is a trivialization $k^{\, \langle \overline{\mathcal{L}},\alpha\rangle} \stackrel{\sim}{\longrightarrow} \mathsf{Hom}(\mathcal{L},\mathcal{F})$.
Now, we shall provide an explicit description of $(u_\alpha^\mathcal{L})^\ast \mathbb{T}_{\bCoha}$ and $(u_\alpha^\mathcal{L})^\ast \mathbb{L}_{\bCoha}$, which will be useful later on. On $\bCoh_\alpha^{> \Lcal} \simeq \big[ \mathsf{Q}_\alpha^\Lcal/\mathsf{G}_\alpha^\Lcal\big]$, the tautological sheaf $\mathfrak{E}_\alpha$ fits into a short exact sequence of tautological $\mathsf{G}_\alpha^\Lcal$-equivariant sheaves on $\mathsf{Q}_\alpha^\Lcal \times X$
\begin{align}
0\to \mathfrak{K}_\alpha^\mathcal{L} \to \mathcal{O}_{\mathsf{Q}_\alpha^\Lcal}^{\oplus\, \langle \overline{\mathcal{L}},\alpha\rangle}\boxtimes \mathcal{L} \to \mathfrak{E}_\alpha^\mathcal{L}=(u_\alpha^\mathcal{L})^\ast\mathfrak{E}_\alpha\to 0
\end{align}
such that the fibers over $\big\{\big[\phi\colon \mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}\twoheadrightarrow \mathcal{F}\big]\big\}\times X$ are
\begin{align}
\mathfrak{K}_\alpha^\mathcal{L}\vert_{\{[\phi]\}\times X}=\ker \phi \quad\mbox{and}\quad \mathfrak{E}_\alpha^\mathcal{L} \vert_{\{[\phi]\}\times X}=\mathcal{F}=\mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}/\ker \phi\ .
\end{align}
Let $p_\alpha^\mathcal{L}\colon \mathsf{Q}_\alpha^\Lcal \times X \to \mathsf{Q}_\alpha^\Lcal$ denote the projection. Since $\mathsf{Ext}^1(\mathfrak{K}_\alpha^\mathcal{L}\vert_{\{[\phi]\}\times X}, \mathfrak{E}_\alpha^\mathcal{L}\vert_{\{[\phi]\}\times X})=\{0\}$ for any $[\phi]\in \mathsf{Q}_\alpha^\Lcal$, the complex $\mathbb{R}(p_\alpha^\mathcal{L})_\ast((\mathfrak{K}_\alpha^\mathcal{L})^\vee \otimes \mathfrak{E}_\alpha^\mathcal{L})$ is a locally free sheaf on $\mathsf{Q}_\alpha^\Lcal$ of rank $\dim \mathsf{Q}_\alpha^\Lcal$, which we will simply denote by $\mathsf{Hom}(\mathfrak{K}_\alpha^\mathcal{L},\mathfrak{E}_\alpha^\mathcal{L})$. Such a locally free sheaf coincides with the tangent bundle $\mathcal{T}_{\mathsf{Q}_\alpha^\Lcal}$ of $\mathsf{Q}_\alpha^\Lcal$. Likewise, the fiber of $\mathbb{R}(p_\alpha^\mathcal{L})_\ast((\mathcal{O}_{\mathsf{Q}_\alpha^\Lcal}^{\oplus\, \langle \overline{\mathcal{L}},\alpha\rangle}\boxtimes \mathcal{L}^\vee) \otimes \mathfrak{E}_\alpha^\mathcal{L})$ over $\big\{\big[\phi\colon \mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}\twoheadrightarrow \mathcal{F}\big]\big\}\times X$ is identified, via $\phi_\alpha$, with $\mathfrak{g}_\alpha^\Lcal\coloneqq \mathfrak{gl}(k,\langle \overline{\mathcal{L}},\alpha\rangle)$, the Lie algebra of $\mathsf{G}_\alpha^\Lcal$, hence $\mathbb{R}(p_\alpha^\mathcal{L})_\ast((\mathcal{O}_{\mathsf{Q}_\alpha^\Lcal}^{\oplus\, \langle \overline{\mathcal{L}},\alpha\rangle}\boxtimes \mathcal{L}^\vee) \otimes \mathfrak{E}_\alpha^\mathcal{L})\simeq \mathfrak{g}_\alpha^\Lcal \otimes \mathcal{O}_{\mathsf{Q}_\alpha^\Lcal}$. Collecting the above, from Formula \eqref{eq:tangentcomplex} we get that
\begin{align}
(u_\alpha^\mathcal{L})^\ast \mathbb{T}_{\bCoha}\simeq \big[\mathfrak{g}_\alpha^\Lcal \otimes \mathcal{O}_{\mathsf{Q}_\alpha^\Lcal} \stackrel{\delta_\alpha^\mathcal{L}}{\longrightarrow}\mathcal{T}_{\mathsf{Q}_\alpha^\Lcal}\big]\ ,
\end{align}
where the complex on the right-hand-side is concentrated in degree [-1, 0]. Thus,
\begin{align}\label{eq:cotancomplex}
(u_\alpha^\mathcal{L})^\ast \mathbb{L}_{\bCoha}\simeq \big[\mathcal{T}_{\mathsf{Q}_\alpha^\Lcal}^\ast \stackrel{\tilde \mu_\alpha^\mathcal{L}}{\longrightarrow} (\mathfrak{g}_\alpha^\Lcal)^\ast \otimes \mathcal{O}_{\mathsf{Q}_\alpha^\Lcal}\big]\ ,
\end{align}
where $\tilde \mu_\alpha^\mathcal{L}$ (the moment map) is obtained by dualizing the canonical restriction morphism $\delta_\alpha^\mathcal{L}$ defined, at the level of points $\big[\phi \colon \mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}\twoheadrightarrow \mathcal{F}\big]$, as $\delta_\alpha^\mathcal{L}(u)=( \phi \circ u)\vert_{\ker \phi}$ for $u \in \mathfrak{g}_\alpha^\Lcal = \operatorname{End}(\mathcal{L} \otimes k^{\langle \mathcal{L}, \alpha\rangle})$.
Next, let us realize $\bCoh_\alpha$ as an \emph{ind-algebraic stack}\footnote{We consider ind-algebraic stacks in a very broad sense, as stated in \cite[Definition~4.2.1]{art:emertongee2015}.}. As a first thing, let us make explicit the inductive system of $\bCoh_\alpha^{> \Lcal}$'s. Let $\mathfrak{Pic}(X)$ be the groupoid formed by all line bundles on $X$ with their isomorphisms. We define the following preorder $\prec$ on (the set of objects of) $\mathfrak{Pic}(X)$ such that it is endowed with the structure of a directed groupoid: we say that $\mathcal{L} \prec \mathcal{L}'$, for two line bundles $\mathcal{L}$ and $\mathcal{L}'$, if $\mathcal{L}'$ is strongly generated by $\mathcal{L}$. In that situation, any coherent sheaf $\mathcal{F}$, which is strongly generated by $\mathcal{L}'$, is also strongly generated by $\mathcal{L}$. Hence, we have a chain of open embeddings
\begin{align}
\bCoh_\alpha^{> \Lcal'} \subseteq \bCoh_\alpha^{> \Lcal}\subseteq \bCoh_\alpha
\end{align}
coming from the inclusions of full subcategories $\Coh^{> \Lcal'}(X) \subset \Coh^{> \Lcal}(X)$. We will describe these embeddings explicitly in local atlases, namely we will
construct the map $j_{\mathcal{L},\mathcal{L}',\alpha}\colon \big[ \mathsf{Q}_\alpha^{\Lcal'}/\mathsf{G}_\alpha^{\Lcal'}\big] \hookrightarrow \big[\mathsf{Q}_\alpha^\Lcal/\mathsf{G}_\alpha^\Lcal\big]$.
To define $j_{\mathcal{L},\mathcal{L}',\alpha}$, we shall provide another equivalent description of $\bCoh_\alpha^{> \Lcal'}$ as a global quotient stack. For this, consider the open subscheme $\mathsf{Q}_\alpha^{\Lcal, \Lcal'} \subset \mathsf{Q}_\alpha^\Lcal$ consisting of all points $\big[\phi\colon \mathcal{L} \otimes k^{\, \langle \overline{\mathcal{L}}, \alpha\rangle} \twoheadrightarrow \mathcal{F}\big]$ for which $\mathcal{F} \in \Coh^{> \Lcal'}(X)$. Then $\bCoh_\alpha^{> \Lcal'}\simeq \big[ \mathsf{Q}_\alpha^{\Lcal, \Lcal'}/\mathsf{G}_\alpha^\Lcal\big]$ and we have a canonical open embedding $\big[ \mathsf{Q}_\alpha^{\Lcal, \Lcal'}/\mathsf{G}_\alpha^\Lcal\big]\hookrightarrow \big[\mathsf{Q}_\alpha^\Lcal/\mathsf{G}_\alpha^\Lcal\big]=\bCoh_\alpha^{> \Lcal}$. Now, we need to compare the two realizations $\big[\mathsf{Q}_\alpha^{\Lcal, \Lcal'}/\mathsf{G}_\alpha^\Lcal\big]$ and $ \big[ \mathsf{Q}_\alpha^{\Lcal'}/\mathsf{G}_\alpha^{\Lcal'}\big]$ by providing a canonical explicit isomorphism between them. Let $p_X\colon \mathsf{Q}_\alpha^{\Lcal'}\times X\to X$ be the projection. Consider the $\mathsf{G}_\alpha^{\Lcal'}$-equivariant sheaf $\mathsf{Hom}(p_X^\ast\mathcal{L}, \mathfrak{E}_\alpha^{\mathcal{L}'})$ over $\mathsf{Q}_\alpha^{\Lcal'}$, which by the same reasoning as above is locally free and of rank $\langle \mathcal{L}, \alpha\rangle$. Let $\mathsf{R}_\alpha^{\Lcal, \Lcal'}$ be the total space of the associated $\mathsf{G}_\alpha^\Lcal$-bundle. Therefore $\mathsf{R}_\alpha^{\Lcal, \Lcal'}$ carries an action of $\mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}$ and
\begin{align}
\mathsf{R}_\alpha^{\Lcal, \Lcal'} / \mathsf{G}_\alpha^\Lcal \simeq \mathsf{Q}_\alpha^{\Lcal'}, \qquad
\big[ \mathsf{R}_\alpha^{\Lcal, \Lcal'} / \mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}\big] \simeq \bCoh_\alpha^{> \Lcal'} \ ,
\end{align}
the first isomorphism being in the category of $\mathsf{G}_\alpha^{\Lcal'}$-schemes\footnote{Here and in the following, we call \emph{$G$-scheme} a scheme endowed with an action of an algebraic group $G$.}. Likewise, let $\mathsf{R}_\alpha^{\Lcal', \Lcal}$ be the total space of the $\mathsf{G}_\alpha^\Lcal$-equivariant $\mathsf{G}_\alpha^{\Lcal'}$-bundle
$\mathsf{Hom}(p_X^\ast \mathcal{L}', \mathfrak{E}_\alpha^\mathcal{L})$ over $\mathsf{Q}_\alpha^{\Lcal, \Lcal'}$, so that
\begin{align}
\mathsf{R}_\alpha^{\Lcal', \Lcal} / \mathsf{G}_\alpha^{\Lcal'} \simeq \mathsf{Q}_\alpha^{\Lcal, \Lcal'}, \qquad
\big[ \mathsf{R}_\alpha^{\Lcal', \Lcal} / \mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}\big] \simeq \bCoh_\alpha^{> \Lcal'}\ ,
\end{align}
the first isomorphism being in the category of $\mathsf{G}_\alpha^\Lcal$-schemes.
\begin{lemma}\label{L:rall}
There is a canonical isomorphism $\mathsf{R}_\alpha^{\Lcal, \Lcal'} \simeq \mathsf{R}_\alpha^{\Lcal', \Lcal}$ in the category of $\mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}$-schemes.
\end{lemma}
\proof
By construction, $\mathsf{R}_\alpha^{\Lcal, \Lcal'}$ represents the contravariant functor $\mathsf{Aff}/k \to (\mathsf{Sets})$ which assigns to an affine $k$-variety $S$ the set of pairs $([\phi],u)$ where $\big[\phi\colon \mathcal{L}' \boxtimes \mathcal{O}_S^{\oplus\, \langle \overline{\mathcal{L}'}, \alpha\rangle} \twoheadrightarrow \mathcal{F}\big]$ belongs to $\mathsf{Q}_\alpha^{\Lcal'}(S)$ and $u$ is a trivialization $\mathcal{O}_S^{\oplus \langle \mathcal{L}, \alpha\rangle} \simeq \mathsf{Hom}(\mathcal{L} \boxtimes \mathcal{O}_S, \mathcal{F})$. Similarly, $\mathsf{R}_\alpha^{\Lcal', \Lcal}$ represents the contravariant functor $\mathsf{Aff}/k \to (\mathsf{Sets})$ which assigns to an affine $k$-variety $S$ the set of pairs $([\psi],v)$ where
$\big[\psi\colon \mathcal{L} \boxtimes \mathcal{O}_S^{\oplus\, \langle \overline{\mathcal{L}}, \alpha\rangle} \twoheadrightarrow \mathcal{F}\big]$ belongs to $\mathsf{Q}_\alpha^{\Lcal, \Lcal'}(S)$ and $v$ is a trivialization $\mathcal{O}_S^{\oplus \, \langle \overline{\mathcal{L}'}, \alpha\rangle} \simeq \mathsf{Hom}(\mathcal{L}' \boxtimes \mathcal{O}_S, \mathcal{F})$. The isomorphism between the two functors is given by the assignment $(\phi,u) \mapsto (\overline{u},\overline{\phi})$ where $\overline{u}: \mathcal{L} \boxtimes \mathcal{O}_S^{\oplus \langle \mathcal{L},\alpha\rangle} \twoheadrightarrow \mathcal{F}$ and $\overline{\phi}: \mathcal{O}_S^{\oplus \langle \mathcal{L}', \alpha\rangle} \simeq \mathsf{Hom}_{\mathcal{O}_X}(\mathcal{L}' \boxtimes \mathcal{O}_S, \mathcal{F})$ are canonically associated to $u$ and $\phi$ respectively.
\endproof
To sum up, $\mathsf{R}_\alpha^{\Lcal, \Lcal'}$ is a smooth $\mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}$-scheme such that
\begin{align}\label{eq:RLLp}
\mathsf{R}_\alpha^{\Lcal, \Lcal'} / \mathsf{G}_\alpha^\Lcal \simeq \mathsf{Q}_\alpha^{\Lcal'}\ , \qquad \mathsf{R}_\alpha^{\Lcal, \Lcal'} / \mathsf{G}_\alpha^{\Lcal'} \simeq \mathsf{Q}_\alpha^{\Lcal, \Lcal'}\ .
\end{align}
Note that $\mathsf{R}_\alpha^{\Lcal, \Lcal'}$ is nothing but the fiber product of stacks
\begin{align}
\mathsf{R}_\alpha^{\Lcal, \Lcal'}=\mathsf{Q}_\alpha^{\Lcal'} \underset{\bCoh_\alpha^{> \Lcal'}}{\times} \mathsf{Q}_\alpha^{\Lcal, \Lcal'}\ ,
\end{align}
and can be thought of as the fine moduli space parameterizing triples $(\mathcal{F},u,v)$ where $\mathcal{F}$ is a coherent sheaf on $X$ of class $\alpha$ which is strongly generated by $\mathcal{L}'$ and $u,v$ a pair of trivializations of $\mathsf{Hom}(\mathcal{L}, \mathcal{F}), \mathsf{Hom}(\mathcal{L}',\mathcal{F})$ respectively. The open embedding $j_{\mathcal{L},\mathcal{L}',\alpha}$ is now given by the composition
\begin{align}\label{eq:transmapscoh}
\bCoh_\alpha^{> \Lcal'}=\big[\mathsf{Q}_\alpha^{\Lcal'}/\mathsf{G}_\alpha^{\Lcal'}\big]\simeq [\mathsf{R}_\alpha^{\Lcal, \Lcal'}/\mathsf{G}_\alpha^\Lcal\times \mathsf{G}_\alpha^{\Lcal'}]\simeq [\mathsf{R}_\alpha^{\Lcal', \Lcal}/\mathsf{G}_\alpha^\Lcal\times \mathsf{G}_\alpha^{\Lcal'}]\simeq \big[\mathsf{Q}_\alpha^{\Lcal, \Lcal'}/\mathsf{G}_\alpha^\Lcal\big]\hookrightarrow \bCoh_\alpha^{> \Lcal}\ .
\end{align}
Thus we have a direct system $\langle \bCoh_\alpha^{> \Lcal}, j_{\mathcal{L}, \mathcal{L}', \alpha}\rangle$ and thanks to Serre's theorem we get
\begin{align}
\bCoh_\alpha \simeq \lim_{\genfrac{}{}{0pt}{}{\to}{\mathcal{L}}} \, \bCoh_\alpha^{> \Lcal} \coloneqq\lim_{\genfrac{}{}{0pt}{}{\to}{\mathfrak{Pic}(X)}} \, \bCoh_\alpha^{> \Lcal} \ .
\end{align}
This provides the desired description of $\bCoh_\alpha$ as an ind-algebraic stack.
Let us finish this section by recalling the structure of the (singular) cohomology ring of the stacks $\bCoh_\alpha$. Here we assume that $k=\mathbb{C}$ and simply write $H^\ast(\bullet)$ for $H^\ast(\bullet, \mathbb{Q})$. Let us fix a basis $\Pi=\{1, \pi_1, \ldots, \pi_{2g}, \varpi\}$ of $H^\ast(X)$, with $1 \in H^0(X), \pi_1, \ldots, \pi_{2g} \in H^1(X)$ and $\varpi \in H^2(X)$. For $i \in \mathbb{N}$, let
\begin{align}\label{eq:chernclasses}
c_i(\mathfrak{E}_\alpha)=\sum_{\pi \in \Pi} c_{i,\pi}(\mathfrak{E}_\alpha) \otimes \pi \in H^\ast(\bCoh_\alpha) \otimes H^\ast(X)
\end{align}
be the K\"unneth decomposition of the $i$-th Chern class of the tautological sheaf $\mathfrak{E}_\alpha$.
\begin{theorem}[Heinloth, \cite{art:heinloth2012}]\label{T:Heinloth}
The rational cohomology ring $H^\ast(\bCoh_\alpha)$ is described as follows:
\begin{enumerate}\itemsep0.2cm
\item[(a)] If $\alpha=(0,d)$ then $H^\ast(\bCoh_\alpha) \simeq S^d(H^\ast(X)[z])$,
\item[(b)] If $\alpha=(r,d)$ with $r >0$ then $H^\ast(\bCoh_\alpha) \simeq \mathbb{Q}[c_{i,\pi}(\mathfrak{E}_\alpha)]_{i,\pi}$ is freely generated (as a supercommutative algebra)
by the classes $c_{i,\pi}(\mathfrak{E}_\alpha)$ for $i \geq 2, \pi \in \Pi$ and $i=1, \pi \in \Pi \setminus \{\varpi\}$.
\end{enumerate}
Moreover, the stack $\bCoh_\alpha$ is cohomologically pure\footnote{The Hodge theory of algebraic stacks, locally of finite type, has been introduced for example in \cite[Section~2]{art:dhillon2006}.} for any $\alpha$.
\end{theorem}
By Poincar\'e duality, the assignment $c \mapsto c \cap [\bCoh_\alpha]$ identifies $H^i(\bCoh_\alpha)$ with $H_{-\langle \alpha,\alpha\rangle-i}(\bCoh_\alpha)$, where $H_\ast(\bullet)$ stands for the Borel-Moore homology with rational coefficients. Hence the above theorem also yields a description of the Borel-Moore homology groups $H_\ast(\bCoh_\alpha)$.
\subsection{Higgs sheaves}
Recall that a \emph{Higgs sheaf} on $X$ is a pair $(\mathcal{F},\theta)$ with $\mathcal{F} \in \mathsf{Coh}(X)$ and $\theta \in \mathsf{Hom}(\mathcal{F},\mathcal{F} \otimes \omega_X)$. We say that $(\mathcal{F},\theta)$ is of numerical class $\alpha=(r,d)$ if $\mathcal{F}$ is. Higgs sheaves form the object of a Calabi-Yau two-dimensional abelian category $\mathsf{Higgs}(X)$, in which the Euler form and Serre duality take the following form (see e.g. \cite{art:gothenking2005}). Define, for $\underline{\mathcal{F}}\coloneqq(\mathcal{F}, \theta_\mathcal{F})$, $\underline{\mathcal{G}}\coloneqq(\mathcal{G}, \theta_\mathcal{G}) \in \mathsf{Higgs}(X)$
\begin{align}
\langle \underline{\mathcal{F}},\underline{\mathcal{G}}\rangle = \dim \mathsf{Hom}(\underline{\mathcal{F}},\underline{\mathcal{G}}) -\dim \mathsf{Ext}^1(\underline{\mathcal{F}},\underline{\mathcal{G}}) + \dim \mathsf{Ext}^2(\underline{\mathcal{F}},\underline{\mathcal{G}})\ .
\end{align}
Then
\begin{align}\label{eq:eulerformhiggs}
\langle \underline{\mathcal{F}},\underline{\mathcal{G}} \rangle=\langle \mathcal{F}, \mathcal{G} \rangle - \langle \mathcal{F}, \mathcal{G}\otimes \omega_X\rangle = \langle \mathcal{F}, \mathcal{G} \rangle +\langle\mathcal{G}, \mathcal{F} \rangle =2(1-g)\operatorname{rk}(\mathcal{F})\operatorname{rk}(\mathcal{G})\ .
\end{align}
Moreover, Serre's duality holds:
\begin{align}
\mathsf{Ext}^i(\underline{\mathcal{F}},\underline{\mathcal{G}}) \simeq \mathsf{Ext}^{2-i}(\underline{\mathcal{G}},\underline{\mathcal{F}})^\ast
\end{align}
for all $i=0,1,2$. The slope of a Higgs sheaf is the slope of its underlying coherent sheaf, i.e.,
\begin{align}
\mu(\underline{\mathcal{F}})=\mu(\mathcal{F})=\frac{\deg(\mathcal{F})}{\operatorname{rk}(\mathcal{F})} \ .
\end{align}
A Higgs sheaf is \emph{semistable} if $\mu(\underline{\mathcal{G}}) \leq \mu(\underline{\mathcal{F}})$ for any Higgs subsheaf $\underline{\mathcal{G}} \subset \underline{\mathcal{F}}$, i.e. if $\mu(\mathcal{G}) \leq \mu(\mathcal{F})$ for any subsheaf $\mathcal{G}$ of $\mathcal{F}$ such that $\theta_\mathcal{F}(\mathcal{G}) \subset \mathcal{G} \otimes \omega_X$. Semistable Higgs sheaves of fixed slope $\nu \in \mathbb{Q} \cup \{\infty\}$ form an abelian subcategory $\mathsf{Higgs}^{\mathsf{ss},\nu}(X)$ of $\mathsf{Higgs}(X)$, which is stable under extensions.
For a Higgs sheaf $(\mathcal{F}, \theta)$, we denote by $\theta^{\, k}$ the composition
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=3.5,yscale=-1]
\node (A0_0) at (0.3, 0) {$\theta^{\, k}\colon \mathcal{F}$};
\node (A1_0) at (1, 0) {$ \mathcal{F}\otimes \omega_X$};
\node (A2_0) at (1.8, 0) {$\cdots$};
\node (A3_0) at (2.8, 0) {$\mathcal{F}\otimes \omega_X^{\otimes\, k}$};
\path (A0_0) edge [->]node [auto] {$\scriptstyle{\theta}$} (A1_0);
\path (A1_0) edge [->]node [auto] {$\scriptstyle{\theta\otimes\mathsf{id}_{\omega_X}}$} (A2_0);
\path (A2_0) edge [->]node [auto] {$\scriptstyle{\theta^{\otimes\, (k-1)}\otimes\mathsf{id}_{\omega_X}}$} (A3_0);
\end{tikzpicture}
\end{aligned}
\end{align}
$(\mathcal{F}, \theta)$ is called \emph{nilpotent} if there exists $s >0$ such that $\theta^{\, s}$ vanishes; we call $s$ the \emph{nilpotency index of $\theta$}. Nilpotent Higgs sheaves form an abelian subcategory $\mathsf{Higgs}^{\mathsf{nilp}}(X)$ of $\mathsf{Higgs}(X)$ which is closed under extensions, subobjects and quotients.
Let $(\mathcal{F}, \theta)$ be a nilpotent Higgs sheaf on $X$ with nilpotency index $s$. For $k\geq 1$, define $\mathcal{F}_k\coloneqq\mathsf{Im}\big(\theta^{\, k}\otimes \mathsf{id}_{\omega_X^{\, \otimes -k}}\big)$. Then $\mathcal{F}_k$ is a subsheaf of $\mathcal{F}$. Finally, set $\mathcal{F}_0\coloneqq\mathcal{F}$. Then there are chains respectively of inclusions and of epimorphisms
\begin{align}
\{0\}=\mathcal{F}_s\subset \mathcal{F}_{s-1}\subset \cdots \mathcal{F}_1\subset \mathcal{F}_0=\mathcal{F} \quad\mbox{and}\quad \mathcal{F}=\mathcal{F}_0 \twoheadrightarrow \mathcal{F}_1\otimes \omega_X\twoheadrightarrow \cdots \twoheadrightarrow \mathcal{F}_s\otimes \omega_X^{\otimes\, s}=0 \ .
\end{align}
Define for $k\geq 0$
\begin{align}
\mathcal{F}_k'\coloneqq\ker\big(\mathcal{F}_k \twoheadrightarrow \mathcal{F}_{k+1}\otimes \omega_X\big) \quad\mbox{and}\quad \mathcal{F}_k''\coloneqq\mathcal{F}_k/\mathcal{F}_{k+1} \ .
\end{align}
Therefore we have chains respectively of inclusions and of epimorphisms
\begin{align}
\{0\}=\mathcal{F}_s'\subset\mathcal{F}_{s-1}' \subset \cdots \mathcal{F}_1'\subset \mathcal{F}_0'\quad\mbox{and}\quad \mathcal{F}_0'' \twoheadrightarrow \mathcal{F}_1''\otimes \omega_X\twoheadrightarrow \cdots \twoheadrightarrow \mathcal{F}_m''\otimes \omega_X^{\otimes\, m}=0 \ .
\end{align}
For $k\geq 1$, set
\begin{align}
\alpha_k\coloneqq\overline{\ker\big(\mathcal{F}_{k-1}''\otimes \omega_X^{\otimes\, k-1}\to \mathcal{F}_{k}''\otimes \omega_X^{\otimes\, k}\big)}\ .
\end{align}
One has
\begin{align}\label{eq:kernelk}
\overline{\ker\big(\theta^{\, k}\big)}=\sum_{h=1}^k\,\sum_{j=h}^s\, \alpha_j((h-j)\,\ell)
\end{align}
for any $k\geq 1$.
This computation justifies the following definition. Given $\alpha \in (\mathbb{Z}^2)^+$, a finite sequence $\underline{\alpha}=(\alpha_1, \ldots, \alpha_s)$ of elements of $(\mathbb{Z}^2)^+$ is called a \emph{Jordan type of class $\alpha$} if satisfies
\begin{align}
\alpha=\sum_{i=1}^s \sum_{k=0}^{i-1} \alpha_i(-k\ell)\ ,
\end{align}
where for any $\beta=(r, d)\in (\mathbb{Z}^2)^+$ and $n\in \mathbb{Z}$, we set $\beta(n)\coloneqq(r, d+nr)$. Here $\ell=\deg(\omega_X)$. We call $s$ the \emph{length $\ell(\underline{\alpha})$} of $\underline{\alpha}$. Note that unless $\operatorname{rk}(\alpha)=0$ there are countably many Jordan types of class $\alpha$. Let $J_\alpha$ be the set of all Jordan types of class $\alpha$.
We may helpfully represent a Jordan type by its associated colored Young diagram as follows (here with $s=4$)
\begin{align}\label{diag:Young}
\begin{aligned}
\ytableausetup {mathmode, boxsize=3.1em,centertableaux}
\begin{ytableau}
\scriptstyle{\alpha_4} \\
\scriptstyle\alpha_4(-\ell) & \scriptstyle\alpha_3 \\
\scriptstyle{\alpha_4(-2\ell)} &\scriptstyle \alpha_3(-\ell) & \scriptstyle\alpha_2\\
\scriptstyle{\alpha_4(-3\ell)} &\scriptstyle \alpha_3(-2\ell) & \scriptstyle\alpha_2(-\ell)&\scriptstyle\alpha_1
\end{ytableau}
\end{aligned}
\end{align}
Given a nilpotent Higgs sheaf $(\mathcal{F}, \theta)$ of class $\alpha$, we will say that $(\mathcal{F}, \theta)$ is \emph{of Jordan type $\underline{\alpha}$} if for all $k \geq 1$, the sheaf $\ker\big(\theta^{\, k}\big)$ satisfies \eqref{eq:kernelk}. In the pictorial description of $\underline{\alpha}$, this corresponds to the bottom $k$ rows of the Young tableaux; thus one can think of the Higgs field $\theta$ as the composition of 'going down one box' and 'tensoring by $\omega_X$').
\subsection{Stacks of Higgs sheaves}
Let us denote by $\bHiggs_\alpha$ the stack parameterizing Higgs sheaves over $X$ of class $\alpha$. Similarly, let $\mathbf{\Lambda}_{\alpha}$ and $\mathbf{Higgs}^{\mathsf{ss}}_{\alpha}$ stand for the respectively closed and open substacks of $\bHiggs_\alpha$ parametrizing respectively nilpotent and semistable Higgs sheaves\footnote{For a derived point of view to the stack of (semistable) Higgs bundles, see e.g. \cite{art:halpernleistner2016,art:ginzburgrozenblyum2017} and references therein. For the GIT approach to the construction of moduli spaces of semistable Higgs bundles, see \cite{art:nitsure1991, art:simpson1994, art:simpson1994-II}.}. The following is well-known, see e.g. \cite{art:ginzburg2001} and \cite[Section~7]{art:casalainawise2017}.
\begin{theorem}\label{T:StackHiggs}
The following hold:
\begin{enumerate}\itemsep0.2cm
\item[(a)] The stack $\bHiggs_\alpha$ is locally of finite type, of dimension $-2\langle \alpha,\alpha\rangle$ and is canonically isomorphic to the underived cotangent stack $T^\ast\bCoh_\alpha$ of $\bCoh_\alpha$:
\begin{align}
\bHiggs_\alpha \simeq T^\ast \bCoh_\alpha\coloneqq\mathsf{Spec}\,\mathsf{Sym}\mathcal{H}^0(\mathbb{T}_{\bCoha})\ .
\end{align}
\item[(b)] The stack $\mathbf{\Lambda}_{\alpha}$ is a Lagrangian substack of $\bHiggs_\alpha$.
\item[(c)] The stack $\bHiggs_\alpha^{\mathsf{ss}}$ is a global quotient stack, and it is smooth if $\alpha=(r,d)$ with $\mathsf{gcd}(r,d)=1$.
\end{enumerate}
\end{theorem}
Let us denote by $r_\alpha\colon \bHiggs_\alpha \to \bCoh_\alpha$ the projection map, forgetting the Higgs field.
There exists an action of the multiplicative group $T=\mathbb{G}_m$ on $\bHiggs_\alpha$, which at level of families reads as $z \cdot (\mathcal{F},\theta)\coloneqq(\mathcal{F},z\theta)$ for $z\in T$ and $(\mathcal{F}, \theta)$ a flat family of Higgs sheaves. Such an action is simply the action scaling the fibers of $r_\alpha\colon T^\ast \bCoh_\alpha\to \bCoh_\alpha$. Indeed, $r_\alpha$ is $T$-equivariant with respect to the trivial action of $T$ on $\bCoh_\alpha$.
It is known that, for $g>1$, the preimage under the projection map $r_\alpha$ of the open substack of vector bundles $\bBun_\alpha\subset \bCoh_\alpha$ is irreducible (cf.\ \cite[Section~2.10]{art:beilinsondrinfeld1991}), and that the irreducible components of $\bHiggs_\alpha$ are in fact given by the Zariski closures of the substacks $r_\alpha^{-1}(\bCoh_\alpha^{\mathsf{tor=d}})$ for $d \geq 0$, where $\bCoh_\alpha^{\mathsf{tor=d}}$ stands for the substack of $\bCoh_\alpha$ parametrizing sheaves whose torsion part is of degree $d$. We thank Jochen Heinloth for explanations concerning these facts. We will not consider these irreducible components, and instead focus on the irreducible components of $\mathbf{\Lambda}_{\alpha}$, which we now describe explicitly following \cite{art:bozec2017}.
There is a partition $\mathbf{\Lambda}_{\alpha}=\bigsqcup_{\underline{\alpha} \in J_\alpha} \mathbf{\Lambda}_{\underline{\alpha}}$ where $\mathbf{\Lambda}_{\underline{\alpha}}$ is the locally closed substack of $\mathbf{\Lambda}_{\alpha}$ parametrizing nilpotent Higgs sheaves of Jordan type $\underline{\alpha}$.
As shown in the proof of \cite[Proposition~5.2]{art:mozgovoyschiffmann2017}, we have the following.
\begin{proposition}\label{prop:MS5.2}
For any $\alpha$ and any $\underline{\alpha}=(\alpha_1, \ldots, \alpha_s) \in J_{\alpha}$ the morphism
\begin{align}
\pi_{\underline{\alpha}}\colon \mathbf{\Lambda}_{\underline{\alpha}} \to& \prod_{i=1}^s \mathbf{Coh}_{\alpha_i}\\
(\mathcal{F}, \theta) \mapsto & \Big(\ker\big(\mathcal{F}_{k-1}''\otimes \omega_X^{\otimes\, k-1}\to \mathcal{F}_{k}''\otimes \omega_X^{\otimes\, k}\big)\Big)_i
\end{align}
is an iterated vector bundle stack morphism\footnote{See \cite[Section~3.1]{art:garcia-pradaheinlothschmitt2014} for the definition of vector bundle stack morphisms.}.
\end{proposition}
\begin{corollary}[{cf.\ \cite[Proposition~2.3 and Corollary~2.4]{art:bozec2017}}]\label{T:Bozec}
For any $\alpha \in (\mathbb{Z}^2)^+$, the irreducible components of $\mathbf{\Lambda}_{\alpha}$ are the Zariski closures $\overline{\mathbf{\Lambda}_{\underline{\alpha}}}$ for $\underline{\alpha} \in J_{\alpha}$.
\end{corollary}
Define the following partial order $\preceq$ on $J_{\alpha}$. For $\underline{\alpha}=(\alpha_1, \ldots, \alpha_s), \underline{\beta}=(\beta_1, \ldots, \beta_t)\in J_{\underline{\alpha}}$, we have $ \underline{\beta}\preceq\underline{\alpha}$ if and only if for any $k\geq 1$ the following inequality holds:
\begin{align}\label{eq:preceq}
\sum_{h=1}^k\,\sum_{j=h}^s\, \alpha_j((h-j)\,\ell)\leq \sum_{h=1}^k\,\sum_{j=h}^t\, \beta_j((h-j)\,\ell)\ ,
\end{align}
where the partial order $\leq$ on $(\mathbb{Z}^2)^+$ is the ``standard" order: for $\underline{\alpha}, \underline{\beta}\in (\mathbb{Z}^2)^+$, we have $\underline{\beta}\leq \underline{\alpha}$ if and only if $\underline{\alpha}-\underline{\beta}\in (\mathbb{Z}^2)^+$. Note that in particular $\underline{\beta}\preceq \underline{\alpha}$ implies $\ell(\underline{\beta})\leq \ell(\underline{\alpha})$. The minimal element in $J_{\alpha}$ for this order is $(\alpha)$.
By Formula \eqref{eq:kernelk}, one can reinterpret the inequality \eqref{eq:preceq} as an inequality for the classes of kernels of subsequent powers of Higgs fields associated with $\underline{\alpha}$ and $\underline{\beta}$. This observation together with the semicontinuity of the rank and the dimensions of cohomology groups of a coherent sheaf (cf.\ \cite[Example~12.7.2 and Theorem~12.8]{book:hartshorne1977}), imply the following result.
\begin{proposition}\label{prop:preceq}
Let $\alpha \in (\mathbb{Z}^2)^+$. For any $\underline{\alpha}\in J_{\alpha}$,
\begin{align}
\mathbf{\Lambda}_{\preceq \underline{\alpha}} \coloneqq \bigsqcup_{\underline{\beta}\preceq \underline{\alpha}} \mathbf{\Lambda}_{\underline{\beta}}
\end{align}
is a closed algebraic substack.
\end{proposition}
An important example of an irreducible component of $\mathbf{\Lambda}_{\alpha}$ is the \emph{zero section} $\mathbf{\Lambda}_{(\alpha)}= \bCoh_\alpha \subset \mathbf{\Lambda}_{\alpha}$, obtained for the unique Jordan type of length $s=1$. It is a closed substack by the previous proposition.
\begin{remark}
By Proposition~\ref{prop:MS5.2} and Theorem~\ref{T:Heinloth}, each strata $\mathbf{\Lambda}_{\underline{\alpha}}$ is cohomologically pure (indeed, this property is preserved under vector bundle stack morphisms). But then $\mathbf{\Lambda}$ is itself pure since it has a locally finite partition into pure strata. Define
\begin{align}
\mathbf{\Lambda}_{\prec \underline{\alpha}} \coloneqq \bigsqcup_{\underline{\beta}\prec \underline{\alpha}} \mathbf{\Lambda}_{\underline{\beta}} \ .
\end{align}
Then $\mathbf{\Lambda}_{\preceq \underline{\alpha}}=\mathbf{\Lambda}_{\prec \underline{\alpha}}\sqcup \mathbf{\Lambda}_{\underline{\alpha}}$. Thanks to the purity of $\mathbf{\Lambda}_{\underline{\alpha}}$, one can show that also $\mathbf{\Lambda}_{\preceq \underline{\alpha}}$ and $\mathbf{\Lambda}_{\prec \underline{\alpha}}$ are pure. In addition, there are short exact sequences
\begin{align}\label{eq:filtration}
0\to H_k(\mathbf{\Lambda}_{\prec \underline{\alpha}})\to H_k(\mathbf{\Lambda}_{\preceq \underline{\alpha}}) \to H_k(\mathbf{\Lambda}_{\underline{\alpha}})\to 0\ .
\end{align}
Thus there is an induced filtration of $H_\ast(\mathbf{\Lambda}_{\alpha})$, whose associated graded is
\begin{align}\label{eq:gradedcoh}
\mathsf{gr}(H_\ast(\mathbf{\Lambda}_{\alpha}))=\bigoplus_{\underline{\alpha} \in J_{\alpha}} H_\ast(\mathbf{\Lambda}_{\underline{\alpha}}) \simeq \bigoplus_{\underline{\alpha} \in J_{\alpha}} H_\ast(\mathbf{Coh}_{\alpha_1}) \otimes \cdots \otimes H_\ast(\mathbf{Coh}_{\alpha_{\ell(\underline{\alpha})}})\ .
\end{align}
Such results hold in the $T$-equivariant setting as well.
\end{remark}
\subsection{Local charts of the stack of Higgs sheaves}\label{sec:statificationHiggs}
Let us now proceed with the description of the \emph{local charts} of the stacks $\bHiggs_\alpha$. Let $\mathcal{L}$ be a line bundle on $X$. By \cite[Proposition~14.2.4]{book:laumonmoretbailly2000}, we have a cartesian diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2.3,yscale=-1]
\node (A0_0) at (0, 0) {$\mathsf{Spec}\,\mathsf{Sym}\mathcal{H}^0(\mathbb{T}_{\bCohaL})$};
\node (A2_0) at (2, 0) {$\bHiggs_\alpha$};
\node (A0_2) at (0, 2) {$\bCoh_\alpha^{> \Lcal}$};
\node (A2_2) at (2, 2) {$\bCoh_\alpha$};
\node(A1_1) at (1,1) {$\square$};
\path (A0_0) edge [->]node [auto] {$\scriptstyle{v_\alpha^\mathcal{L}}$} (A2_0);
\path (A0_0) edge [->]node [auto] {$\scriptstyle{r_\alpha^\mathcal{L}}$} (A0_2);
\path (A2_0) edge [->]node [auto] {$\scriptstyle{r_\alpha}$} (A2_2);
\path (A0_2) edge [->]node [auto] {$\scriptstyle{u_\alpha^\mathcal{L}}$} (A2_2);
\end{tikzpicture}
\end{aligned}
\end{align}
where the map $u_\alpha^\mathcal{L}$ is an open embedding, and hence also $v_\alpha^\mathcal{L}$. Define
\begin{align}
\bHiggs_\alpha^{> \Lcal} \coloneqq\mathsf{Spec}\,\mathsf{Sym}\mathcal{H}^0(\mathbb{T}_{\bCohaL})\ .
\end{align}
Thus $\bHiggs_\alpha^{> \Lcal}$ is the algebraic stack parameterizing Higgs sheaves on $X$ of class $\alpha$ such that the underlying coherent sheaf is strongly generated by $\mathcal{L}$. Such a stack can be realized as a global quotient stack. Indeed, by the explicit description \eqref{eq:cotancomplex} of $\mathbb{T}_{\bCohaL}$, we get
\begin{align}
\bHiggs_\alpha^{> \Lcal}\simeq [T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal/\mathsf{G}_\alpha^\Lcal]\ ,
\end{align}
where $T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal\coloneqq\big(\mu_\alpha^\Lcal\big)^{-1}(0)$ and $\mu_\alpha^\Lcal$ is the composition
\begin{align}
\mathsf{Spec}\, \mathsf{Sym} \mathcal{T}_{\mathsf{Q}_\alpha^\Lcal}=T^\ast \mathsf{Q}_\alpha^\Lcal\to \mathsf{Spec}\, \mathsf{Sym}\, (\mathfrak{g}_\alpha^\Lcal\otimes \mathcal{O}_{\mathsf{Q}_\alpha^\Lcal}) = \big(\mathfrak{g}_\alpha^\Lcal\big)^\ast\times \mathsf{Q}_\alpha^\Lcal \to \big(\mathfrak{g}_\alpha^\Lcal\big)^\ast \ .
\end{align}
Next, let us realize $\bHiggs_\alpha$ as an ind-algebraic stack. By construction (cf.\ proof of Lemma \ref{L:rall}), there are dual pairs of short exact sequences
\begin{align}
0 \to & \mathfrak{g}_\alpha^{\Lcal'} \to T_{([\phi],u)}\mathsf{R}_\alpha^{\Lcal, \Lcal'} \to \mathsf{Hom}(\ker(\phi),\mathcal{F}) \to 0\ ,\\[2pt]
0 \to & \mathfrak{g}_\alpha^\Lcal \to T_{([\psi],v)}\mathsf{R}_\alpha^{\Lcal', \Lcal} \to \mathsf{Hom}(\ker(\psi),\mathcal{F}) \to 0 \ ,
\end{align}
and
\begin{align}
0 \to & \mathsf{Hom}(\ker(\phi),\mathcal{F})^\ast \to T^\ast_{([\phi],u)}\mathsf{R}_\alpha^{\Lcal, \Lcal'} \stackrel{\mu'}{\longrightarrow} (\mathfrak{g}_\alpha^{\Lcal'})^\ast \to 0\ , \\[2pt]
0 \to & \mathsf{Hom}(\ker(\psi),\mathcal{F})^\ast \to T^\ast_{([\psi],v)}\mathsf{R}_\alpha^{\Lcal', \Lcal} \stackrel{\mu}{\longrightarrow} (\mathfrak{g}_\alpha^\Lcal)^\ast \to 0
\end{align}
for any $([\phi],u) \in \mathsf{R}_\alpha^{\Lcal, \Lcal'}, ([\psi],v) \in \mathsf{R}_\alpha^{\Lcal', \Lcal}$. Recall that by Lemma~\ref{L:rall} we have a canonical isomorphism $\mathsf{R}_\alpha^{\Lcal, \Lcal'} \simeq \mathsf{R}_\alpha^{\Lcal', \Lcal}$ as $\mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}$-schemes, which sends $([\phi], u)$ to $([\overline{u}], \overline{\phi})$. The moment map relative to the Hamiltonian $\mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}$-action on $T^\ast\mathsf{R}_\alpha^{\Lcal, \Lcal'}$ is given by
\begin{align}
\mu_\alpha^{\mathcal{L},\mathcal{L}'}\coloneqq\mu'\oplus \mu \colon T^\ast_{(\phi,u)}\mathsf{R}_\alpha^{\Lcal, \Lcal'} \to (\mathfrak{g}_\alpha^{\Lcal'})^\ast \oplus (\mathfrak{g}_\alpha^\Lcal)^\ast\ .
\end{align}
The above complex is quasi-isomorphic to both
\begin{align}
\big[T^\ast_{[\psi]}\mathsf{Q}_\alpha^{\Lcal, \Lcal'} \stackrel{\mu_\alpha^\mathcal{L}}{\to} (\mathfrak{g}_\alpha^\Lcal)^\ast\big] \simeq \big[\mathsf{Hom}(\ker(\phi),\mathcal{F}) \to (\mathfrak{g}_\alpha^\Lcal)^\ast\big]\ ,\\[2pt]
\big[T^\ast_{[\phi]}\mathsf{Q}_\alpha^{\Lcal'} \stackrel{\mu_\alpha^{\mathcal{L}'}}{\to} (\mathfrak{g}_\alpha^{\Lcal'})^\ast\big] \simeq \big[\mathsf{Hom}(\ker(\psi),\mathcal{F}) \to (\mathfrak{g}_\alpha^{\Lcal'})^\ast\big]\ .
\end{align}
It follows that the projection maps $T^\ast_{\mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}}\mathsf{R}_\alpha^{\Lcal, \Lcal'} \to T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^{\Lcal, \Lcal'}$ and $T^\ast_{\mathsf{G}_\alpha^\Lcal \times \mathsf{G}_\alpha^{\Lcal'}}\mathsf{R}_\alpha^{\Lcal, \Lcal'} \to T^\ast_{\mathsf{G}_\alpha^{\Lcal'}}\mathsf{Q}_\alpha^{\Lcal'}$ are respectively principal $\mathsf{G}_\alpha^{\Lcal'}$ and $\mathsf{G}_\alpha^\Lcal$-bundles. Hence the open embedding $j_{\mathcal{L}, \mathcal{L}', \alpha}\colon \bCoh_\alpha^{> \Lcal'} \hookrightarrow \bCoh_\alpha^{> \Lcal}$ lifts to an open embedding $h_{\mathcal{L}, \mathcal{L}', \alpha}\colon \bHiggs_\alpha^{> \Lcal'}\hookrightarrow \bHiggs_\alpha^{> \Lcal}$ obtained as a composition
\begin{align}\label{eq:haLLp}
\begin{aligned}
\bHiggs_\alpha^{> \Lcal'}=&[T^\ast_{\mathsf{G}_\alpha^{\Lcal'}}\mathsf{Q}_\alpha^{\Lcal'}/\mathsf{G}_\alpha^{\Lcal'}]\simeq [T^\ast_{\mathsf{G}_\alpha^\Lcal\times \mathsf{G}_\alpha^{\Lcal'}} \mathsf{R}_\alpha^{\Lcal, \Lcal'}/\mathsf{G}_\alpha^\Lcal\times \mathsf{G}_\alpha^{\Lcal'}]\\[2pt]
&\simeq [T^\ast_{\mathsf{G}_\alpha^\Lcal\times \mathsf{G}_\alpha^{\Lcal'}} \mathsf{R}_\alpha^{\Lcal', \Lcal}/\mathsf{G}_\alpha^\Lcal\times \mathsf{G}_\alpha^{\Lcal'}]\simeq [T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^{\Lcal, \Lcal'}/\mathsf{G}_\alpha^\Lcal]\hookrightarrow \bHiggs_\alpha^{> \Lcal} \ ,
\end{aligned}
\end{align}
where the last morphism is obtained by applying base change with respect to $j_{\mathcal{L}, \mathcal{L}', \alpha}$ and \cite[Proposition~14.2.4]{book:laumonmoretbailly2000}. Thus, we obtain a directed system $\langle \bHiggs_\alpha^{> \Lcal}, h_{\mathcal{L}, \mathcal{L}', \alpha}\rangle$ and thus we get
\begin{align}
\bHiggs_\alpha \simeq \lim_{\genfrac{}{}{0pt}{}{\to}{\mathcal{L}}} \, \bHiggs_\alpha^{> \Lcal} \coloneqq\lim_{\genfrac{}{}{0pt}{}{\to}{\mathfrak{Pic}(X)}} \, \bHiggs_\alpha^{> \Lcal} \ .
\end{align}
\bigskip\section{Definition of the cohomological Hall algebras}\label{sec:COHAHiggs}
The present section is devoted to the construction of $A$-homological Hall algebras associated with the 2-Calabi-Yau category $\mathsf{Higgs}(X)$, and their variants for the category of nilpotent Higgs sheaves and the category of semistable Higgs bundles of fixed slope.
\subsection{Borel-Moore homology theories}\label{sec:BM}
Although most of our results here concern the case of the cohomological Hall algebra for Borel-Moore homology or Chow groups, our constructions make sense for an arbitrary oriented Borel-Moore homology theory (OBM). Let $\mathsf{Sch}/ k$ be the category of separated $k$-schemes of finite type. Recall that an OBM theory on $\mathsf{Sch}/ k$ is the data of
\begin{enumerate}\itemsep0.2cm
\item[(a)] for every $k$-scheme $X$, a graded abelian group $A_\ast(X)$;
\item[(b)] for every projective morphism $f\colon X \to Y$, a homomorphism $f_\ast\colon A_\ast(X) \to A_\ast(Y)$;
\item[(c)] for every locally complete intersection (l.c.i.) morphism $g\colon X \to Y$ of relative dimension $d$, a homomorphism $g^\ast\colon A_\ast(Y) \to A_{\ast+d}(X)$;
\item[(d)] an element $\mathbf{1} \in A_0(\mathsf{pt})$ and for every pair $(X,Y)$ of $k$-schemes, a bilinear pairing $\times \colon A_\ast(X) \otimes A_\ast(Y) \to A_\ast(X \times Y)$ which is associative, commutative and for which $\mathbf{1}$ is a unit,
\end{enumerate}
satisfying a certain number of natural axioms, see \cite{book:levinemorel2007}. Of particular importance to us will be the existence of \emph{refined Gysin pullback morphisms} in the following context: let $f\colon Y \to X$ be an l.c.i morphism and let $g \colon Z \to X$ be an arbitrary morphism; then there exists a pullback morphism
\begin{align}
f^!=f^!_g\colon A_\ast(Z) \to A_\ast(Z \times_X Y)
\end{align}
which coincides with the usual pullback morphism $f^\ast$ if $Z=X$ and $g=\mathsf{id}_X$.
In the following, we shall restrict ourselves to those OBMs which are \emph{free}, see e.g. \cite[Chapter~4]{book:levinemorel2007}. Examples of free OBM theories include K-theory and Chow groups. Although usual Borel-Moore homology is not \emph{per se} an OBM theory because of the presence of odd degree classes, it satisfies all the important properties and all of our constructions will be valid in that situation as well. We refer to the papers \cite[Appendix~A]{art:minets2018} and \cite[Section~1.1]{art:yangzhao2014} which contain all the properties of OBM theories which we will need here.
Let $G$ be a reductive algebraic group, a $G$-equivariant version of OBMs has been defined in \cite{art:hellermalagonlopez2013}, and therefore there exists a theory of OBMs for global quotient stacks. In \cite{art:kresch1999}, Kresch define a theory of Chow groups for algebraic stacks, locally of finite type (and stratified by global quotient stacks).
Since, we are considering $\bHiggs_\alpha$ as an ind-algebraic stack, we give here the following definition. Let $\mathscr{X}$ be an algebraic stack such that $\displaystyle \mathscr{X}\simeq \lim_{\to}\, \mathscr{U}_i$, where the limit is taken with respect to the directed system $(\mathscr{U}_i, \jmath_{i\leq i'}\colon \mathscr{U}_i\to \mathscr{U}_{i'})$, formed by open substacks $\mathscr{U}_i$ of $\mathscr{X}$ and open immersions $\jmath_{i\leq i'}$. We define
\begin{align}
A_\ast(\mathscr{X}):=\lim_{\genfrac{}{}{0pt}{}{\longleftarrow}{\mathscr{U}_i}}\, A_\ast(\mathscr{U}_i)\ .
\end{align}
This graded abelian group tends to be very large and untractable. We also define a smaller graded abelian group $A^0_\ast(\mathscr{X}) \subset A_\ast(\mathscr{X})$ as follows. First, we say that an algebraic stack $\mathscr{Z}$, such that $\displaystyle \mathscr{Z}\simeq \lim_{\to}\, \mathscr{W}_i$, is \emph{admissible} if $\mathsf{codim}(\mathscr{Z}\setminus \mathscr{W}_i)\to \infty$ if $i\to \infty$. Let $A^0_\ast(\mathscr{X})$ be the subgroup of $A_\ast(\mathscr{X})$ consisting of classes supported on an admissible closed substack, i.e., lying in the image of the direct image morphism $i_\ast\colon A_\ast(\mathscr{Z}) \to A_\ast(\mathscr{X})$ for $i\colon \mathscr{Z}\hookrightarrow \mathscr{X}$ an admissible closed substack. The subgroups $A^0_\ast$ are preserved under the standard operations $(-)_\ast,(-)^\ast, (-)^!$ with respect to morphisms which are of finite relative dimension.
\subsection{Cohomological Hall algebra of the stacks of coherent sheaves}\label{sec:COHAcoh}
Let us first define the cohomological Hall algebra attached to the simpler $1$-dimensional category $\mathsf{Coh}(X)$. For this, we need to introduce stacks classifying extensions between coherent sheaves. For $\alpha,\beta \in (\mathbb{Z}^2)^+$, let $\widetilde{\bCoh_{\alpha, \beta}}$ be the stack parameterizing inclusions $\mathcal{G}\subset \mathcal{F}$, where $\mathcal{G}$ is a flat family of coherent sheaves of class $\beta$ and $\mathcal{F}$ is a flat family of coherent sheaves of class $\alpha+\beta$. It is a smooth irreducible algebraic stack, locally of finite type, of dimension $-\langle \alpha,\alpha\rangle -\langle \beta,\beta\rangle -\langle \alpha,\beta\rangle$, equipped with a pair of morphisms
\begin{align}\label{eq:convoldiagramcoh}
\begin{aligned}
\begin{tikzpicture}[xscale=3,yscale=-1]
\node (A0_0) at (0, 0) {$\bCoh_\alpha \times\bCoh_\beta$};
\node (A1_0) at (1, 0) {$\widetilde{\bCoh_{\alpha, \beta}}$};
\node (A2_0) at (2, 0) {$\bCoh_{\alpha+\beta}$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{q_{\alpha, \beta}}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{p_{\alpha,\beta}}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
defined at the level of flat families by $q_{\alpha,\beta}(\mathcal{G}\subset\mathcal{F})=(\mathcal{F}/\mathcal{G},\mathcal{G})$ and $p_{\alpha,\beta}(\mathcal{G}\subset\mathcal{F})=\mathcal{F}$. The map
$q_{\alpha,\beta}$ is a vector bundle stack morphism, while $p_{\alpha,\beta}$ is a proper representable morphism (see e.g. \cite[Section~3.1]{art:garcia-pradaheinlothschmitt2014}). The stack $\widetilde{\bCoh_{\alpha, \beta}}$ is equipped with a short exact sequence of \emph{tautological sheaves}
\begin{align}
0\to \widetilde \mathfrak{E}_\beta\to \widetilde \mathfrak{E}_{\alpha+\beta}\to \widetilde \mathfrak{E}_\alpha \to 0 \ ,
\end{align}
where $\widetilde \mathfrak{E}_\alpha, \widetilde \mathfrak{E}_{\alpha+\beta}, \widetilde \mathfrak{E}_\beta\in \mathsf{Coh}\big(\widetilde{\bCoh_{\alpha, \beta}}\times X
\big)$. Moreover, the following relations hold between tautological sheaves
\begin{align}
\big(q_{\alpha, \beta}\big)^\ast \mathsf{pr}_\alpha^\ast\mathfrak{E}_\alpha \simeq \widetilde \mathfrak{E}_\alpha \ , \quad \big(q_{\alpha, \beta}\big)^\ast \mathsf{pr}_\alpha^\ast \mathfrak{E}_\beta \simeq \widetilde \mathfrak{E}_\beta \quad\mbox{and}\quad \big(p_{\alpha,\beta}\big)^\ast \mathfrak{E}_{\alpha+\beta} = \widetilde \mathfrak{E}_{\alpha+\beta}\ ,
\end{align}
where $\mathsf{pr}_\alpha^\ast, \mathsf{pr}_\beta^\ast$ are the projections from $\bCoh_\alpha\times \bCoh_\beta$ to the two factors respectively.
\begin{definition}
Let $A$ be either Borel-Moore homology or a free oriented Borel-Moore homology theory. The \emph{$A$-homological Hall algebra} of the category $\mathsf{Coh}(X)$
is the abelian group
\begin{align}
\mathbf{AHA}_{\mathsf{Coh}(X)}\coloneqq\bigoplus_{\alpha \in (\mathbb{Z}^2)^+} A_\ast(\bCoh_\alpha)
\end{align}
equipped with the multiplication
\begin{align}
A_\ast(\bCoh_\alpha) \otimes A_\ast(\bCoh_\beta) \to A_*(\bCoh_{\alpha+\beta})\ , \quad c_1 \otimes c_2 \mapsto (p_{\alpha,\beta})_\ast (q_{\alpha,\beta})^\ast(c_1 \boxtimes c_2)\ .
\end{align}
\end{definition}
Since $\bCoh_\alpha, \bCoh_\beta$ and $\widetilde{\bCoh_{\alpha, \beta}}$ are smooth, it is easy to see that $\mathbf{AHA}_{\mathsf{Coh}(X)}$ is a graded associative algebra (see \cite{art:schiffmannvasserot2018} where the case of $A=H_\ast$ is considered). Although this is already a very interesting and still mysterious algebra, the aim of this paper is to study its \emph{two}-dimensional counterpart, defined using the moduli stacks of Higgs sheaves on $X$.
\subsubsection{Local presentation of the diagram \eqref{eq:convoldiagramcoh}}
To unburden the notation, let us set $\mathbb{V}_{\mathcal{L},\gamma}=k^{\, \langle \mathcal{L}, \gamma \rangle}$ for any $\mathcal{L}$ and any $\gamma \in (\mathbb{Z}^2)^+$. Let us also fix an isomorphism $\mathbb{V}_{\mathcal{L},\,\alpha+\beta} \simeq \mathbb{V}_{\mathcal{L},\,\alpha} \oplus \mathbb{V}_{\mathcal{L},\, \beta}$.
\begin{lemma}\label{lem:standard}
Let $\mathcal{L}'$ be a line bundle on $X$ and let $\alpha,\beta \in (\mathbb{Z}^2)^+$. Then there exists $N \ll 0$ such that, for any line bundle $\mathcal{L}$ with $\deg(\mathcal{L}) \leq N$, for any scheme $S$, and any $(\mathcal{G}\subset \mathcal{F}) \in \widetilde{\bCoh_{\alpha, \beta}}(S)$ with $\mathcal{F} \in \bCoh_{\alpha+\beta}^{> \Lcal'}(S)$ we have $(\mathcal{F}/\mathcal{G}, \mathcal{G}) \in \bCoh_\alpha^{> \Lcal}(S)\times \bCoh_\beta^{> \Lcal}(S)$.
\end{lemma}
\begin{proof}
This comes from the fact that $\mathbf{Coh}^{>\mathcal{L}'}_{\alpha+\beta}$ is a global quotient stack, in particular of finite type, and $\displaystyle\mathbf{Coh}_{\gamma}\simeq \lim_{\genfrac{}{}{0pt}{}{\to}{\mathcal{L}}}\, \mathbf{Coh}^{>\mathcal{L}}_{\gamma}$ for any $\gamma\in (\mathbb{Z}^2)^+$. See e.g. \cite[Lemma~2.3]{art:schiffmann2004} for details.
\end{proof}
As a corollary, we see that for any $\alpha,\beta,\mathcal{L}'$ as above, and any $\mathcal{L}$ of sufficiently negative degree we have
\begin{align}
p_{\alpha,\beta}^{-1}(\bCoh_{\alpha+\beta}^{> \Lcal'}) \subset q_{\alpha,\beta}^{-1}(\bCoh_\alpha^{> \Lcal} \times \bCoh_\beta^{> \Lcal})\ .
\end{align}
Now we provide a presentation of $\widetilde{\bCoh_{\alpha, \beta}}$ as an ind-algebraic stack. Define the subscheme of $\widetilde{\mathsf{Q}}^{\Lcal}_{\alpha, \beta}$ consisting of points $\big[\phi \colon \mathcal{L}\otimes\mathbb{V}_{\mathcal{L},\, \alpha+ \beta}\twoheadrightarrow \mathcal{F}\big]$ for which $\phi(\mathcal{L} \otimes \mathbb{V}_{\mathcal{L},\,\beta})$ is strongly generated by $\mathcal{L}$, and
\begin{align}
\widetilde{\mathsf{Q}}^{\Lcal, \Lcal'}_{\alpha, \beta}\coloneqq\widetilde{\mathsf{Q}}^{\Lcal}_{\alpha, \beta} \cap \mathsf{Q}_{\alpha+\beta}^{\Lcal, \Lcal'} \subset \mathsf{Q}_{\alpha+\beta}^\Lcal\ .
\end{align}
Note that $\widetilde{\mathsf{Q}}^{\Lcal}_{\alpha, \beta}$ may not be closed in $\mathsf{Q}_{\alpha+\beta}^\Lcal$ (because of being strongly generated is an open condition), but by Lemma \ref{lem:standard} $\widetilde{\mathsf{Q}}^{\Lcal, \Lcal'}_{\alpha, \beta}$ is closed in $\mathsf{Q}_{\alpha+\beta}^{\Lcal, \Lcal'}$. Let $\mathsf{P}_{\alpha,\beta}^{\Lcal}\subset\mathsf{G}_{\alpha+\beta}^\Lcal$ be the group consisting of $g\in \mathsf{G}_{\alpha+\beta}^\Lcal$ such that $g(\mathbb{V}_{\mathcal{L}, \, \beta})=\mathbb{V}_{\mathcal{L}, \, \beta}$. Define the global quotient stack
\begin{align}
\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}\simeq \big[\widetilde{\mathsf{Q}}^{\Lcal, \Lcal'}_{\alpha, \beta}/\mathsf{P}_{\alpha,\beta}^{\Lcal}\big]\ .
\end{align}
It is the algebraic stack parameterizing extensions $0\to \mathcal{G}\to \mathcal{F}\to \mathcal{E} \to 0$ of coherent sheaves on $X$ with $\overline{\mathcal{G}}=\beta$, $\overline{\mathcal{E}}=\alpha$ and $\mathcal{F}$ strongly generated by $\mathcal{L}'$, while the sheaves $\mathcal{H}$ and $\mathcal{G}$ are strongly generated by $\mathcal{L}$. Moreover, we have an open embedding $u_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}\colon \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}\to \widetilde{\bCoh_{\alpha, \beta}}$.
Let $\alpha, \beta, \mathcal{L}'$ as above and consider two line bundles $\mathcal{L}_1, \mathcal{L}_2$ of degrees less or equal to $N$. By Lemma \ref{lem:standard}, we have an isomorphism $\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal_{1}, > \Lcal'}\simeq \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal_{2}, > \Lcal'}$. An explicit realization of such an isomorphism can be given by following the same reasoning as in Section \ref{sec:stackcoherentsheaves}. Let $\widetilde{\mathsf{R}}^{(\Lcal_1,\Lcal_2),\Lcal'}_{\alpha,\beta}$ be the scheme representing the contravariant functor $\mathsf{Aff}/k \to (\mathsf{Sets})$ which assigns to an affine $k$-variety $S$ the set of pairs $([\phi], u)$, where $[\phi]\in \widetilde{\mathsf{Q}}^{\Lcal_1, \Lcal'}_{\alpha, \beta}(S)$, and $u\colon \mathbb{V}_{\mathcal{L}_2,\,\alpha+\beta}\boxtimes \mathcal{O}_S\simeq \mathsf{Hom}(\mathcal{L}_2\boxtimes\mathcal{O}_S,\mathcal{F})$ is an isomorphism such that $u(\mathbb{V}_{\mathcal{L}_2, \,\beta}\boxtimes\mathcal{O}_S)=\mathsf{Hom}(\mathcal{L}_2\boxtimes\mathcal{O}_S,\mathcal{G})$. One can show that $\widetilde{\mathsf{R}}^{(\Lcal_1,\Lcal_2),\Lcal'}_{\alpha,\beta}$ is a $\mathsf{P}_{\alpha,\beta}^{\Lcal_2}$-principal bundle over $\widetilde{\mathsf{Q}}^{\Lcal_1, \Lcal'}_{\alpha, \beta}$ and a $\mathsf{P}_{\alpha,\beta}^{\Lcal_1}$-principal bundle over $\widetilde{\mathsf{Q}}^{\Lcal_2, \Lcal'}_{\alpha, \beta}$. Therefore, at the level of global quotient stacks we have
\begin{align}
\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal_{1}, > \Lcal'}=[\widetilde{\mathsf{Q}}^{\Lcal_1, \Lcal'}_{\alpha, \beta}/\mathsf{P}_{\alpha,\beta}^{\Lcal_1}]\simeq [\widetilde{\mathsf{R}}^{(\Lcal_1,\Lcal_2),\Lcal'}_{\alpha,\beta}/\mathsf{P}_{\alpha,\beta}^{\Lcal_1}\times \mathsf{P}_{\alpha,\beta}^{\Lcal_2}]\simeq [\widetilde{\mathsf{Q}}^{\Lcal_2, \Lcal'}_{\alpha, \beta}/\mathsf{P}_{\alpha,\beta}^{\Lcal_2}]=\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal_{2}, > \Lcal'}\ .
\end{align}
On the other hand, given two line bundles $\mathcal{L}_1$ and $\mathcal{L}_2$ such that $\mathcal{L}_1\preceq \mathcal{L}_2$ and given $\mathcal{L}$ of sufficiently negative degree (less than the $N$'s of Lemma \ref{lem:standard} both for the triple $\alpha, \beta, \mathcal{L}_1$ and the triple $\alpha, \beta, \mathcal{L}_2$), we have an open embedding $\jmath_{\alpha,\beta}^{\mathcal{L}, (\mathcal{L}_1, \mathcal{L}_2)}\colon \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal_1}\to \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal_2}$. We get a direct system $\langle \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}, \jmath_{\alpha,\beta}^{\mathcal{L}, (\mathcal{L}_1, \mathcal{L}_2)}\rangle$ and therefore
\begin{align}
\widetilde{\bCoh_{\alpha, \beta}} \simeq \lim_{\genfrac{}{}{0pt}{}{\to}{\mathcal{L}'}} \, \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'} :=\lim_{\genfrac{}{}{0pt}{}{\to}{\widetilde{\mathfrak{Pic}}(X)}} \, \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'} \ .
\end{align}
There are maps
\begin{align}\label{eq:overlinemaps}
\overline{p}_{\alpha,\beta}^{\, \mathcal{L}, \mathcal{L}'}\colon \widetilde{\mathsf{Q}}^{\Lcal, \Lcal'}_{\alpha, \beta} \hookrightarrow \mathsf{Q}_{\alpha+\beta}^{\Lcal, \Lcal'}\qquad\mbox{and}\qquad\overline{q}_{\alpha,\beta}^{\, \mathcal{L}, \mathcal{L}'}\colon\widetilde{\mathsf{Q}}^{\Lcal}_{\alpha, \beta} \to \mathsf{Q}_\alpha^\Lcal \times \mathsf{Q}_\beta^\Lcal\ ,
\end{align}
which are respectively a proper morphism and an affine fibration of rank $\langle \mathcal{L},\beta\rangle \langle \mathcal{L},\alpha\rangle-\langle \beta,\alpha\rangle$. Moreover, $H_{\alpha, \beta}^{\mathcal{L}}\coloneqq\mathsf{G}_\alpha^\Lcal \times \mathsf{G}^{\Lcal}_\beta$ acts on $\mathsf{Q}_\alpha^\Lcal \times \mathsf{Q}_\beta^\Lcal$, $\mathsf{P}_{\alpha,\beta}^{\Lcal}$ acts on $\widetilde{\mathsf{Q}}^{\Lcal, \Lcal'}_{\alpha, \beta}$ while
$\mathsf{G}_{\alpha+\beta}^\Lcal$ acts on $\mathsf{Q}_{\alpha+\beta}^{\Lcal, \Lcal'}$. Note that $H_{\alpha, \beta}^{\mathcal{L}} \subset \mathsf{P}_{\alpha,\beta}^{\Lcal} \subset \mathsf{G}_{\alpha+\beta}^\Lcal$ is the inclusion of a Levi factor of a parabolic subgroup of $\mathsf{G}_{\alpha+\beta}^\Lcal$, and that $\overline{p}_{\alpha,\beta}^{\, \mathcal{L}, \mathcal{L}'}, \overline{q}_{\alpha,\beta}^{\, \mathcal{L}, \mathcal{L}'}$ are $\mathsf{P}_{\alpha,\beta}^{\Lcal}$-equivariant (with respect to the canonical maps $\mathsf{P}_{\alpha,\beta}^{\Lcal} \to H_{\alpha, \beta}^{\mathcal{L}}$ and $\mathsf{P}_{\alpha,\beta}^{\Lcal} \to \mathsf{G}_{\alpha+\beta}^\Lcal$). Thus, we have induced morphisms at the stacky level
\begin{align}
p_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}\colon \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'} \to \bCoh_{\alpha+\beta}^{> \Lcal'} \quad\mbox{and}\quad q_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}\colon \widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'} \to \bCoh_\alpha^{> \Lcal} \times \bCoh_\beta^{> \Lcal}\ ,
\end{align}
which fit into, respectively, the cartesian diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2.3,yscale=-1]
\node (A0_0) at (0, 0) {$\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}$};
\node (A2_0) at (2, 0) {$\bCoh_{\alpha+\beta}^{> \Lcal'}$};
\node (A0_2) at (0, 2) {$\widetilde{\bCoh_{\alpha, \beta}}$};
\node (A2_2) at (2, 2) {$\bCoh_{\alpha+\beta}$};
\node(A1_1) at (1,1) {$$};
\path (A0_0) edge [->]node [auto] {$\scriptstyle{p_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}}$} (A2_0);
\path (A0_0) edge [->]node [auto] {$ $} (A0_2);
\path (A2_0) edge [->]node [auto] {$ $} (A2_2);
\path (A0_2) edge [->]node [auto] {$\scriptstyle{p_{\alpha,\beta}}$} (A2_2);
\end{tikzpicture}
\end{aligned}
\end{align}
and the commutative diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2.3,yscale=-1]
\node (A0_0) at (0, 0) {$\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}$};
\node (A2_0) at (2, 0) {$\bCoh_\alpha^{> \Lcal}\times \bCoh_\beta^{> \Lcal}$};
\node (A0_2) at (0, 2) {$\widetilde{\bCoh_{\alpha, \beta}}$};
\node (A2_2) at (2, 2) {$\bCoh_\alpha\times\bCoh_\beta$};
\node(A1_1) at (1,1) {$$};
\path (A0_0) edge [->]node [auto] {$\scriptstyle{q_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}}$} (A2_0);
\path (A0_0) edge [->]node [auto] {$ $} (A0_2);
\path (A2_0) edge [->]node [auto] {$ $} (A2_2);
\path (A0_2) edge [->]node [auto] {$\scriptstyle{q_{\alpha,\beta}}$} (A2_2);
\end{tikzpicture}
\end{aligned}
\end{align}
\subsection{Cohomological Hall algebra of the stack of Higgs sheaves}
We now turn to the more involved construction in the $2$-dimensional case of the category $\mathsf{Higgs}(X)$. Analogously to the case of the stacks $\bCoh_\alpha$, we may consider the stack $\widetilde{\bHiggs_{\alpha, \beta}}$ parameterizing inclusions of Higgs sheaves $\underline{\mathcal{G}} \subset \underline{\mathcal{F}}$ with $\underline{\mathcal{G}} \in \bHiggs_\beta$ and $\underline{\mathcal{F}} \in \bHiggs_{\alpha+\beta}$. One can realize $\widetilde{\bHiggs_{\alpha, \beta}}$ as
\begin{align}
\widetilde{\bHiggs_{\alpha, \beta}} \simeq \mathsf{Spec}\, \mathsf{Sym} \mathcal{H}^0\big(\mathcal{N}_{\alpha, \beta}^\vee\big)\ ,
\end{align}
where $\mathcal{N}_{\alpha, \beta}$ is the \emph{conormal complex} associated with the morphism $(q_{\alpha, \beta}, p_{\alpha, \beta})\colon \widetilde{\bCoh_{\alpha, \beta}}\to \bCoh_\alpha\times \bCoh_\beta\times \bCoh_{\alpha+\beta}$, i.e., $\mathcal{N}_{\alpha, \beta}\coloneqq\mathsf{Cone}\big(\mathbb{L} (q_{\alpha, \beta}, p_{\alpha, \beta})^\ast (\mathbb{L}_{\bCoh_\alpha\times \bCoh_\beta}\oplus \mathbb{L}_{\bCoh_{\alpha+\beta}})\to \mathbb{L}_{\widetilde{\bCoh_{\alpha, \beta}}}\big)[-1]$.
There are again maps
\begin{align}\label{eq:convoldiagramhiggs}
\begin{aligned}
\begin{tikzpicture}[xscale=3,yscale=-1]
\node (A0_0) at (0, 0) {$\bHiggs_\alpha \times\bHiggs_\beta$};
\node (A1_0) at (1, 0) {$\widetilde{\bHiggs_{\alpha, \beta}}$};
\node (A2_0) at (2, 0) {$\bHiggs_{\alpha+\beta}$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{q_{\alpha, \beta}}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{p_{\alpha,\beta}}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
defined by $q_{\alpha,\beta}(\underline{\mathcal{G}} \subset \underline{\mathcal{F}})=(\underline{\mathcal{F}}/\underline{\mathcal{G}},\underline{\mathcal{G}})$ and $p_{\alpha,\beta}(\underline{\mathcal{G}} \subset \underline{\mathcal{F}})=\underline{\mathcal{F}}$. We use the same notation as in Section \ref{sec:COHAcoh}, hoping no confusion may arise.
The map $p_{\alpha,\beta}$ is still a proper representable morphism, but the map $q_{\alpha,\beta}$ is very far from being a vector bundle stack morphism, or even an l.c.i. morphism hence it is not possible to define directly a pullback morphism $q_{\alpha,\beta}^\ast\colon A_\ast(\bHiggs_\alpha \times \bHiggs_\beta) \to A_\ast(\widetilde{\bHiggs_{\alpha, \beta}})$. In order to circumvent this difficulty, we follow \cite[Section~4]{art:schiffmannvasserot2013-II} (see also \cite{art:yangzhao2014} for the case of arbitrary Borel-Moore homology theories) and embed the convolution diagram \eqref{eq:convoldiagramhiggs} into a convolution diagram of smooth varieties and use refined Gysin pullbacks. One caveat of this approach is that we only manage to construct this embedding locally, and hence work with local atlases. The case of rank zero Higgs stacks is studied in details in \cite{art:minets2018}: since rank zero Higgs stacks are global quotient stacks, the author applies directly the machinery of \cite{art:schiffmannvasserot2013-II,art:yangzhao2014}. We shall follow closely \cite{art:minets2018} in some of the arguments here.
\subsubsection{Local charts of the stack $\widetilde{\bHiggs_{\alpha, \beta}}$}\label{sec:localdescriptionHiggstilde}
Let $\alpha, \beta \in (\mathbb{Z}^2)^+$ and $\mathcal{L}'$ be a line bundle on $X$. Let $\mathcal{L}$ be a line bundle of degree less or equal that the $N$, depending on $\alpha, \beta, \mathcal{L}'$ of Lemma \ref{lem:standard}. The diagram \ref{eq:convoldiagramcoh} reduces locally to
\begin{align}\label{eq:convoldiagramcohLLp}
\begin{aligned}
\begin{tikzpicture}[xscale=3.8,yscale=-1]
\node (A0_0) at (0, 0) {$\bCoh_\alpha^{> \Lcal} \times\bCoh_\beta^{> \Lcal}$};
\node (A1_0) at (1, 0) {$\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}$};
\node (A2_0) at (1.8, 0) {$\bCoh_{\alpha+\beta}^{> \Lcal'}$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{q_{\alpha, \beta}^{\mathcal{L}, \mathcal{L}'}}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{p_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
Such a diagram can be realized at the level of atlases in the following way. To unburden the notation, set
\begin{align}
H\coloneqq H_{\alpha, \beta}^{\mathcal{L}} \ ,\ P\coloneqq \mathsf{P}_{\alpha,\beta}^{\Lcal}\ ,\ G\coloneqq \mathsf{G}_{\alpha+\beta}^\Lcal\ ,\ V\coloneqq\widetilde{\mathsf{Q}}^{\Lcal}_{\alpha, \beta} \ , \ V^\circ\coloneqq \widetilde{\mathsf{Q}}^{\Lcal, \Lcal'}_{\alpha, \beta}\ , \ Y\coloneqq \mathsf{Q}_\alpha^\Lcal\times \mathsf{Q}_\beta^\Lcal\ , \ X'\coloneqq \mathsf{Q}_{\alpha+\beta}^{\Lcal, \Lcal'}
\end{align}
and
\begin{align}
W\coloneqq G\underset{P}{\times} V\ , \quad W^{\circ}\coloneqq G\underset{P}{\times} V^{\circ}\ , \quad X\coloneqq G\underset{P}{\times} Y\ .
\end{align}
Since $\bCoh_\alpha^{> \Lcal}\times \bCoh_\beta^{> \Lcal}\simeq [X/ G]$, $\bCoh_{\alpha+\beta}^{> \Lcal'}\simeq [X'/G]$ and $\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}\simeq [W^\circ/G]$, the diagram \eqref{eq:convoldiagramcohLLp} corresponds to the following diagram of $G$-varieties
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=1.8,yscale=-1.3]
\node (A0_0) at (0.2, 1) {$X$};
\node (A1_0) at (1, 0) {$W$};
\node (A2_0) at (2, 0) {$W^\circ$};
\node (A3_0) at (2.8, 1) {$X'$};
\path (A2_0) edge [->]node [above] {$\scriptstyle{i}$} (A1_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{f}$} (A0_0);
\path (A2_0) edge [->]node [above] {$\scriptstyle{g}$} (A3_0);
\end{tikzpicture}
\end{aligned}
\end{align}
where $i\colon W^{\circ} \to W$ is the open immersion, and where $f,g$ are defined respectively by
\begin{align}
f\colon (h,v)\bmod{P} \mapsto (h,\overline{q}_{\alpha,\beta}^{\,\mathcal{L}, \mathcal{L}'}(v))\bmod{P}\quad\mbox{and}\quad g\colon (h,v)\bmod{P} \mapsto h \cdot \overline{p}_{\alpha,\beta}^{\,\mathcal{L}, \mathcal{L}'}(v)\ .
\end{align}
Since $(f\circ i,g)$ is a regular embedding, we can identify $W^\circ$ with a smooth subvariety of $X \times X'$. Put $Z^\circ\coloneqq T^\ast_{W^\circ}(X \times X')$. Denoting by $\Phi$ and $\Psi$ the projections on factors, we obtain a diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2,yscale=-1.3]
\node (A0_0) at (0.2, 1) {$T^\ast X$};
\node (A1_0) at (1, 0) {$Z^\circ $};
\node (A2_0) at (1.8, 1) {$T^\ast X'$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{\Phi}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{\Psi}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
Note that $\Psi$ is proper since $\overline{p}_{\alpha,\beta}^{\,\mathcal{L}, \mathcal{L}'}$ is a closed embedding (see e.g. \cite[Lemma~2.3]{art:schiffmannvasserot2012}), while $\Phi$ is a regular morphism as both $Z^\circ$ and $T^\ast X$ are smooth. Next, set $Z^\circ_G\coloneqq Z^\circ \cap (T^\ast_GX \times T^\ast_GX')$. Then by \emph{loc. cit.} $\Phi^{-1}(T^\ast_G X) = Z^\circ_G$ and $\Psi(Z^\circ_G) \subseteq T^\ast_GX'$. We arrive at the following diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2,yscale=-2]
\node (A0_1) at (0.2, 1) {$T^\ast_G X$};
\node (A1_0) at (1, 0) {$Z^\circ_G$};
\node (A2_1) at (1.8, 1) {$T^\ast_G X'$};
\node (A0_2) at (0.2, 2) {$T^\ast X$};
\node (A1_1) at (1, 1) {$Z^\circ $};
\node (A2_2) at (1.8, 2) {$T^\ast X'$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{}$} (A1_1);
\path (A0_1) edge [->]node [above] {$\scriptstyle{}$} (A0_2);
\path (A2_1) edge [->]node [above] {$\scriptstyle{}$} (A2_2);
\path (A1_0) edge [->]node [above=0.7em, left=0.2em] {$\scriptstyle{\Phi_G}$} (A0_1);
\path (A1_0) edge [->]node [above=0.7em, right=0.2em] {$\scriptstyle{\Psi_G}$} (A2_1);
\path (A1_1) edge [->]node [above=0.7em, left=0.2em] {$\scriptstyle{\Phi}$} (A0_2);
\path (A1_1) edge [->]node [above=0.7em, right=0.2em] {$\scriptstyle{\Psi}$} (A2_2);
\end{tikzpicture}
\end{aligned}
\end{align}
in which the left square is cartesian. By Section \ref{sec:statificationHiggs}, we get $\bHiggs_\alpha^{> \Lcal}\times \bHiggs_\beta^{> \Lcal}\simeq [T^\ast_G X/G]$ and $\bHiggs_{\alpha+\beta}^{> \Lcal'}\simeq [T^\ast_G X'/G]$. Note that the global quotient stack $[Z^\circ_G/G]$ is isomorphic to the stack $\mathsf{Spec}\, \mathsf{Sym} \mathcal{H}^0\big(\mathcal{N}_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}\big)$, where
\begin{align}
\mathcal{N}_{\alpha, \beta}^{\mathcal{L}, \mathcal{L}'}\coloneqq\mathsf{Cone}\Big(\mathbb{L} (q_{\alpha, \beta}^{\mathcal{L}, \mathcal{L}'}, p_{\alpha, \beta}^{\mathcal{L}, \mathcal{L}'})^\ast (\mathbb{L}_{\bCoh_\alpha^{> \Lcal}\times \bCoh_\beta^{> \Lcal}}\oplus \mathbb{L}_{\bCoh_{\alpha+\beta}^{> \Lcal'}})\to \mathbb{L}_{\widetilde{\bCoh_{\alpha, \beta}}^{> \Lcal, > \Lcal'}}\Big)[-1]\ .
\end{align}
In addition, by following some of the arguments in the proof of \cite[Lemma~2.1]{art:minets2018} one can show that $[Z^\circ_G/G]$ is the stack $\widetilde{\bHiggs_{\alpha, \beta}}^{> \Lcal, > \Lcal'}$ parameterizing inclusions of Higgs sheaves $\underline{\mathcal{G}} \subset \underline{\mathcal{F}}$ with $\underline{\mathcal{G}} \in \bHiggs_\beta^{> \Lcal}$ and $\underline{\mathcal{F}} \in \bHiggs_{\alpha+\beta}^{> \Lcal'}$. Therefore, we obtain a diagram
\begin{align}\label{eq:convoldiagramhiggsLLp}
\begin{aligned}
\begin{tikzpicture}[xscale=4,yscale=-1]
\node (A0_0) at (0, 0) {$\bHiggs_\alpha^{> \Lcal} \times\bHiggs_\beta^{> \Lcal}$};
\node (A1_0) at (1, 0) {$\widetilde{\bHiggs_{\alpha, \beta}}^{> \Lcal, > \Lcal'}$};
\node (A2_0) at (1.8, 0) {$\bHiggs_{\alpha+\beta}^{> \Lcal'}$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{q_{\alpha, \beta}^{\mathcal{L}, \mathcal{L}'}}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{p_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
which is the local version of the diagram \eqref{eq:convoldiagramhiggs}.
\subsubsection{Definition of the multiplication}
Now we shall define for any free OBM theory $A_\ast$ (or for the usual Borel-Moore homology $H_\ast$) a map
\begin{align}\label{eq:defproduct}
m_{\alpha,\beta}^{\mathcal{L},\mathcal{L}'}\colon A (\bHiggs_\alpha^{> \Lcal})\otimes A(\bHiggs_\beta^{> \Lcal})\to A(\bHiggs_{\alpha+\beta}^{> \Lcal'})\ .
\end{align}
In the above, we have suppressed the grading index $\ast$ in $A_\ast$ for extra readability.
We shall continue to use the notation of the previous section. Because $\Phi$ is regular (in particular, l.c.i.) there is a refined Gysin pullback morphism
\begin{align}\label{eq:refGysin1}
\Phi^!\colon A^G(T^\ast_G X) \to A^G(Z^\circ_G)\ ,
\end{align}
and a pushforward morphism
\begin{align}\label{eq:refpush}
\Psi_{G, \ast}\colon A^G(Z^\circ_G) \to A^G(T^\ast_GX')\ .
\end{align}
Now, from the isomorphisms \eqref{eq:RLLp} we get the chain of isomorphisms
\begin{align}
A^G(T^\ast_G X')=A^{\mathsf{G}_{\alpha+\beta}^\Lcal}\big(T^\ast_{\mathsf{G}_{\alpha+\beta}^\Lcal} \mathsf{Q}_{\alpha+\beta}^{\Lcal, \Lcal'}\big)\simeq A^{\mathsf{G}_{\alpha+\beta}^\Lcal\times \mathsf{G}_{\alpha+\beta}^{\Lcal'}}\big(T^\ast_{\mathsf{G}_{\alpha+\beta}^\Lcal\times \mathsf{G}_{\alpha+\beta}^{\Lcal'}} \mathsf{R}_{\alpha+\beta}^{\Lcal, \Lcal'}\big)\simeq A^{\mathsf{G}_{\alpha+\beta}^{\Lcal'}}\big(T^\ast_{\mathsf{G}_{\alpha+\beta}^{\Lcal'}} \mathsf{Q}^{\Lcal'}_{\alpha+\beta}\big)\ .
\end{align}
In addition, since $T^\ast_G X =G \underset{P}{\times} T^\ast_H Y$, by \cite[Proposition~A.6]{art:minets2018} we have
\begin{align}
A^{\mathsf{G}_\alpha^\Lcal \times \mathsf{G}^{\Lcal}_\beta}(T^\ast_{\mathsf{G}_\alpha^\Lcal \times \mathsf{G}^{\Lcal}_\beta}(\mathsf{Q}_\alpha^\Lcal \times \mathsf{Q}_\beta^\Lcal))=A^H(T^\ast_H Y)\simeq A^G(T^\ast_G X) \ .
\end{align}
By compositing all these maps together, we get
\begin{align}
A^{\mathsf{G}_\alpha^\Lcal \times \mathsf{G}^{\Lcal}_\beta}(T^\ast_{\mathsf{G}_\alpha^\Lcal \times \mathsf{G}^{\Lcal}_\beta}(\mathsf{Q}_\alpha^\Lcal \times \mathsf{Q}_\beta^\Lcal))=&A^H(T^\ast_H Y)\simeq A^G(T^\ast_G X)\stackrel{\Phi^!}{\longrightarrow} A^G(Z^\circ_G)\\
&\stackrel{\Psi_{G, \ast}}{\longrightarrow} A^G(T^\ast_G X')\simeq A^{\mathsf{G}_{\alpha+\beta}^{\Lcal'}}\big(T^\ast_{\mathsf{G}_{\alpha+\beta}^{\Lcal'}} \mathsf{Q}^{\Lcal'}_{\alpha+\beta}\big)\ ,
\end{align}
whose restriction
\begin{align}\label{eq:defproduct2}
m_{\alpha,\beta}^{\mathcal{L},\mathcal{L}'}\colon A^{\mathsf{G}_\alpha^\Lcal}(T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal) \otimes A^{\mathsf{G}^{\Lcal}_\beta}(T^\ast_{\mathsf{G}^{\Lcal}_\beta}( \mathsf{Q}_\beta^\Lcal)) \to A^{\mathsf{G}_{\alpha+\beta}^{\Lcal'}}(T^\ast_{\mathsf{G}_{\alpha+\beta}^{\Lcal'}}\mathsf{Q}^{\Lcal'}_{\alpha+\beta})
\end{align}
gives \eqref{eq:defproduct}.
\subsection{Main theorem}
\begin{theorem}\label{theorem:defproduct}
The collection of morphisms $m_{\alpha,\beta}^{\mathcal{L},\mathcal{L}'}$ give rise to a canonically defined morphism
\begin{align}
m_{\alpha,\beta}\colon A(\bHiggs_\alpha) \otimes A(\bHiggs_\beta) \to A(\bHiggs_{\alpha+\beta})\ .
\end{align}
Equipped with these morphisms,
\begin{align}
\mathbf{AHA}'_{\mathsf{Higgs}(X)}\coloneqq\bigoplus_{\alpha \in (\mathbb{Z}^2)^+} A(\bHiggs_\alpha)
\end{align}
is an associative algebra.
\end{theorem}
\begin{proof}
The first statement boils down to the following. Let $\alpha,\beta$ be fixed, and let $\mathcal{L}_i$ for $i=1,2,3$ be line bundles on $X$. Assume that $\mathcal{L}_3$ is strongly generated by $\mathcal{L}_2$, itself strongly generated by $\mathcal{L}_1$. Assume in addition that the conclusion of Lemma \ref{lem:standard} holds for the pairs $(\mathcal{L}_1,\mathcal{L}_2)$ and $(\mathcal{L}_2,\mathcal{L}_3)$. Then, denoting by $\mathsf{res}_{\gamma}^{\mathcal{L},\mathcal{L}'} \colon A(\mathbf{Higgs}_{\gamma}^{>\mathcal{L}}) \to A(\mathbf{Higgs}_{\gamma}^{>\mathcal{L}'})$ the pullback induced by the open embedding $h_{\mathcal{L}, \mathcal{L}', \gamma}$ introduced in \eqref{eq:haLLp} for any $\gamma\in (\mathbb{Z}^2)^+$, we have to show that
\begin{align}\label{eq:proof1}
m_{\alpha,\beta}^{\mathcal{L}_1,\mathcal{L}_3} &= m_{\alpha,\beta}^{\mathcal{L}_2,\mathcal{L}_3} \circ (\mathsf{res}^{\mathcal{L}_1,\mathcal{L}_2}_{\alpha} \otimes \mathsf{res}^{\mathcal{L}_1,\mathcal{L}_2}_{\beta}) \ , \\ \label{eq:proof2}
m_{\alpha,\beta}^{\mathcal{L}_1,\mathcal{L}_3} & = \mathsf{res}^{\mathcal{L}_2, \mathcal{L}_3}_{\alpha+\beta} \circ m_{\alpha,\beta}^{\mathcal{L}_1,\mathcal{L}_2}\ ,
\end{align}
as morphisms $A(\bHiggs_\alpha^{> \Lcal_1}) \otimes A(\bHiggs_\beta^{> \Lcal_1}) \to A(\bHiggs_{\alpha+\beta}^{> \Lcal_3})$. We will prove \eqref{eq:proof1} in details and leave the (easier) \eqref{eq:proof2} to the reader. We begin by observing that by Lemma \ref{lem:standard} there is a factorization
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=3.5,yscale=-1.5]
\node (A0_0) at (0, 0) {$\mathsf{Q}_\alpha^{\Lcal_1, \Lcal_2}\times \mathsf{Q}_\beta^{\Lcal_1, \Lcal_2}$};
\node (A1_0) at (1, 0) {$\widetilde{\mathsf{Q}}^{\Lcal_1, \Lcal_3}_{\alpha, \beta}$};
\node (A0_1) at (0, 1) {$\mathsf{Q}_\alpha^{\Lcal_1}\times\mathsf{Q}_\beta^{\Lcal_1}$};
\path (A0_0) edge [->]node [left] {$\scriptstyle{j}$} (A0_1);
\path (A1_0) edge [->]node [above] {$\scriptstyle{\overline{q}_{\alpha, \beta}^{(\mathcal{L}_1, \mathcal{L}_2), \mathcal{L}_3}}$} (A0_0);
\path (A1_0) edge [->]node [auto] {$\scriptstyle{\overline{q}_{\alpha, \beta}^{\mathcal{L}_1, \mathcal{L}_3}}$} (A0_1);
\end{tikzpicture}
\end{aligned}
\end{align}
where $j$ denote the canonical open immersion. Here for any pair of line bundles $\mathcal{L}, \mathcal{L}'$, by abuse of notation we have denoted by $\overline{q}_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}$ the composition of the open embedding $\widetilde{\mathsf{Q}}^{\Lcal, \Lcal'}_{\alpha, \beta}\hookrightarrow \widetilde{\mathsf{Q}}^{\Lcal}_{\alpha, \beta}$ with the map $\overline{q}_{\alpha,\beta}^{\mathcal{L}, \mathcal{L}'}$ introduced in \eqref{eq:overlinemaps}.
Keeping the notations used in Section \ref{sec:localdescriptionHiggstilde}, set $G\coloneqq \mathsf{G}_{\alpha+\beta}^{\mathcal{L}_1}$, $P\coloneqq \mathsf{P}_{\alpha,\beta}^{\Lcal_1}$ and $X_{\mathcal{L}_2}\coloneqq G \underset{P}{\times} (\mathsf{Q}_\alpha^{\Lcal_1, \Lcal_2}\times \mathsf{Q}_\beta^{\Lcal_1, \Lcal_2})$. Then there is also a factorization
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2.5,yscale=-1]
\node (A0_0) at (0, 0) {$T^\ast_G X$};
\node (A2_0) at (2, 0) {$Z^\circ_G$};
\node (A1_1) at (1, 1) {$T^\ast_G X_{\mathcal{L}_2}$};
\node (A0_2) at (0, 2) {$T^\ast X$};
\node (A2_2) at (2, 2) {$Z^\circ $};
\node (A1_3) at (1, 3) {$T^\ast X_{\mathcal{L}_2}$};
\path (A0_0) edge [->]node [above] {$\scriptstyle{}$} (A0_2);
\path (A2_0) edge [->]node [above] {$\scriptstyle{}$} (A2_2);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A1_3);
\path (A2_0) edge [->]node [right=2em, above] {$\scriptstyle{\Phi_G}$} (A0_0);
\path (A2_0) edge [->]node [auto] {$\scriptstyle{\Phi_G'}$} (A1_1);
\path (A2_2) edge [->]node [right=2em, above] {$\scriptstyle{\Phi}$} (A0_2);
\path (A2_2) edge [->]node [auto] {$\scriptstyle{\Phi'}$} (A1_3);
\path (A1_1) edge [->]node [auto] {$\scriptstyle{j_G}$} (A0_0);
\path (A1_3) edge [->]node [auto] {$\scriptstyle{j}$} (A0_2);
\end{tikzpicture}
\end{aligned}
\end{align}
Because Gysin pullbacks commute with the restriction to open subsets, we have $\Phi^!=(\Phi')^! \circ j^\ast = (\Phi')^! \circ (\mathsf{res}^{\mathcal{L}_1,\mathcal{L}_2}_{\alpha} \otimes \mathsf{res}^{\mathcal{L}_1,\mathcal{L}_2}_{\beta})$. In order to conclude, we have to identify $(\Phi')^!$ with the Gysin pullback $\Phi_2^!$ coming from the cartesian square
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=1.5,yscale=-1]
\node (A0_0) at (0, 0) {$T^\ast_{G_2} X_2$};
\node (A2_0) at (2, 0) {$Z^\circ_{G_2}$};
\node (A0_2) at (0, 2) {$T^\ast X_2$};
\node (A2_2) at (2, 2) {$Z^\circ_2 $};
\path (A0_0) edge [->]node [above] {$\scriptstyle{}$} (A0_2);
\path (A2_0) edge [->]node [above] {$\scriptstyle{}$} (A2_2);
\path (A2_0) edge [->]node [above] {$\scriptstyle{\Phi_{2, G_2}}$} (A0_0);
\path (A2_2) edge [->]node [above] {$\scriptstyle{\Phi_2}$} (A0_2);
\end{tikzpicture}
\end{aligned}
\end{align}
itself built from the diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2,yscale=-1]
\node (A0_0) at (0, 0) {$X_2$};
\node (A1_0) at (1, 0) {$W_2^\circ$};
\node (A2_0) at (1.8, 0) {$X_2'$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{\overline{q}_2}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{\overline{p}_2}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
where we set
\begin{align}
G_2\coloneqq\mathsf{G}^{\mathcal{L}_2}_{\alpha+\beta}\ ,\quad P_2\coloneqq \mathsf{P}_{\alpha,\beta}^{\mathcal{L}_2}\ ,\quad X'_2\coloneqq {\mathsf{Q}}^{\mathcal{L}_2,\mathcal{L}_3}_{\alpha+\beta}\ , \quad W^{\circ}_2\coloneqq G_2 \underset{P_2}{\times} \widetilde{\mathsf{Q}}^{\mathcal{L}_2,\mathcal{L}_3}_{\alpha,\beta}\ , \quad X_2\coloneqq G_2 \underset{P_2}{\times}(\mathsf{Q}^{\mathcal{L}_2}_{\alpha} \times \mathsf{Q}^{\mathcal{L}_2}_{\beta})\ .
\end{align}
Consider the commutative diagram
\begin{align}\label{eq:diagram-proof1}
\begin{aligned}
\begin{tikzpicture}[xscale=3.5,yscale=-2]
\node (A0_0) at (0, 0) {$\mathsf{Q}^{\mathcal{L}_2}_{\alpha} \times \mathsf{Q}^{\mathcal{L}_2}_{\beta}$};
\node (A1_0) at (1, 0) {$\widetilde{\mathsf{Q}}^{\mathcal{L}_2,\mathcal{L}_3}_{\alpha,\beta}$};
\node (A2_0) at (2, 0) {$\mathsf{Q}^{\mathcal{L}_2,\mathcal{L}_3}_{\alpha+\beta}$};
\node (A0_1) at (0, 1) {$\mathsf{R}^{\mathcal{L}_1,\mathcal{L}_2}_{\alpha} \times \mathsf{R}^{\mathcal{L}_1,\mathcal{L}_2}_{\beta}$};
\node (A1_1) at (1, 1) {$\widetilde{\mathsf{R}}^{(\mathcal{L}_1,\mathcal{L}_2),\mathcal{L}_3}_{\alpha,\beta}$};
\node (A2_1) at (2, 1) {$\mathsf{R}^{(\mathcal{L}_1,\mathcal{L}_2),\mathcal{L}_3}_{\alpha+\beta}$};
\node (A0_2) at (0, 2) {$\mathsf{Q}^{\mathcal{L}_1,\mathcal{L}_2}_{\alpha} \times \mathsf{Q}^{\mathcal{L}_1,\mathcal{L}_2}_{\beta}$};
\node (A1_2) at (1, 2) {$\widetilde{\mathsf{Q}}^{\mathcal{L}_1,\mathcal{L}_3}_{\alpha,\beta}$};
\node (A2_2) at (2, 2) {$\mathsf{Q}^{\mathcal{L}_1,\mathcal{L}_3}_{\alpha+\beta}$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{}$} (A2_0);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A0_1);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A2_1);
\path (A1_2) edge [->]node [above] {$\scriptstyle{}$} (A0_2);
\path (A1_2) edge [->]node [above] {$\scriptstyle{}$} (A2_2);
\path (A0_1) edge [->]node [above] {$\scriptstyle{}$} (A0_0);
\path (A0_1) edge [->]node [above] {$\scriptstyle{}$} (A0_2);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A1_0);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A1_2);
\path (A2_1) edge [->]node [above] {$\scriptstyle{}$} (A2_0);
\path (A2_1) edge [->]node [above] {$\scriptstyle{}$} (A2_2);
\end{tikzpicture}
\end{aligned}
\end{align}
where ${\mathsf{R}}^{(\mathcal{L}_1,\mathcal{L}_2),\mathcal{L}_3}_{\alpha,\beta}$ is the scheme representing the contravariant functor $\mathsf{Aff}/k \to (\mathsf{Sets})$ which assigns to an affine $k$-variety $S$ the set of pairs $([\phi], u)$, where $[\phi]\in \mathsf{Q}^{\mathcal{L}_1,\mathcal{L}_3}_{\alpha+\beta}(S)$, and $u\colon \mathbb{V}_{\mathcal{L}_2,\,\alpha+\beta}\boxtimes \mathcal{O}_S\simeq \mathsf{Hom}(\mathcal{L}_2\boxtimes\mathcal{O}_S,\mathcal{F})$ is an isomorphism.
The downwards pointing vertical arrows in \eqref{eq:diagram-proof1} are, respectively, a $\mathsf{G}^{\mathcal{L}_2}_{\alpha} \times \mathsf{G}^{\mathcal{L}_2}_{\beta}$, a $P_{\alpha, \beta}^{\mathcal{L}_2}$ and a $\mathsf{G}^{\mathcal{L}_2}_{\alpha+\beta}$-principal bundle, where $P_{\alpha, \beta}^{\mathcal{L}_2}\subset \mathsf{G}^{\mathcal{L}_2}_{\alpha+\beta}$ is a parabolic subgroup with Levi factor $\mathsf{G}^{\mathcal{L}_2}_{\alpha} \times \mathsf{G}^{\mathcal{L}_2}_{\beta}$. Similarly, the upwards pointing vertical arrows in \eqref{eq:diagram-proof1} are respectively a $\mathsf{G}^{\mathcal{L}_1}_{\alpha} \times \mathsf{G}^{\mathcal{L}_1}_{\beta}$, a $P_{\alpha,\beta}^{\mathcal{L}_1}$ and a $\mathsf{G}^{\mathcal{L}_1}_{\alpha+\beta}$-principal bundle, where $P_{\alpha,\beta}^{\mathcal{L}_1} \subset \mathsf{G}^{\mathcal{L}_1}_{\alpha+\beta}$ is a parabolic subgroup with Levi factor $\mathsf{G}^{\mathcal{L}_1}_{\alpha} \times \mathsf{G}^{\mathcal{L}_1}_{\beta}$.
After passing to the cotangent spaces, the Gysin pullback $(\Phi')^!$ comes from the bottom row of \eqref{eq:diagram-proof1} while $\Phi_2^!$ comes from the
top row of \eqref{eq:diagram-proof1}. We are in the following general situation. Let $H \subset P \subset G$, $H'\subset P'\subset G'$ be a pair of Levi factors inclusions of parabolic subgroups of some reductive groups $G,G'$. Set $\overline{G}\coloneqq G \times G', \overline{P}=P \times P', \overline{H}=H \times H'$. Let $Y,V,X'$ be a triple of smooth varieties equipped with respective actions of $H,P$ and $G$ along with $P$-equivariant maps
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=1.5,yscale=-1]
\node (A0_0) at (0, 0) {$Y$};
\node (A1_0) at (1, 0) {$V$};
\node (A2_0) at (2, 0) {$X'$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{q}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{p}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
We further assume that $q$ is smooth and $p$ is a closed embedding. Let $(\overline{Y},\overline{V}, \overline{X'}, \overline{q},\overline{p})$ be similar data for $\overline{H}, \overline{P}, \overline{G}$ and suppose that we have a commuting diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=1.5,yscale=-1]
\node (A0_0) at (0, 0) {$\overline{Y}$};
\node (A1_0) at (1, 0) {$\overline{V}$};
\node (A2_0) at (2, 0) {$\overline{X'}$};
\node (A0_1) at (0, 1) {$Y$};
\node (A1_1) at (1, 1) {$V$};
\node (A2_1) at (2, 1) {$X'$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{\overline{q}}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{\overline{p}}$} (A2_0);
\path (A1_1) edge [->]node [above] {$\scriptstyle{q}$} (A0_1);
\path (A1_1) edge [->]node [above] {$\scriptstyle{p}$} (A2_1);
\path (A0_0) edge [->]node [above] {$\scriptstyle{}$} (A0_1);
\path (A1_0) edge [->]node [above] {$\scriptstyle{}$} (A1_1);
\path (A2_0) edge [->]node [above] {$\scriptstyle{}$} (A2_1);
\end{tikzpicture}
\end{aligned}
\end{align}
in which the vertical arrows are respectively a $H', P'$ and $G'$-principal bundles. Forming the fiber products
\begin{align}
X\coloneqq G \underset{P}{\times}Y\ , \qquad W\coloneqq G \underset{P}{\times} V\ , \qquad \overline{X}\coloneqq \overline{G} \underset{\overline{P}}{\times}\overline{Y}\ , \qquad \overline{W}\coloneqq\overline{G} \underset{\overline{P}}{\times} \overline{V}
\end{align}
yields a commuting diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=1.5,yscale=-1]
\node (A0_0) at (0, 0) {$\overline{X}$};
\node (A1_0) at (1, 0) {$\overline{W}$};
\node (A2_0) at (2, 0) {$\overline{X'}$};
\node (A0_1) at (0, 1) {$X$};
\node (A1_1) at (1, 1) {$W$};
\node (A2_1) at (2, 1) {$X'$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{}$} (A2_0);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A0_1);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A2_1);
\path (A0_0) edge [->]node [above] {$\scriptstyle{}$} (A0_1);
\path (A1_0) edge [->]node [above] {$\scriptstyle{}$} (A1_1);
\path (A2_0) edge [->]node [above] {$\scriptstyle{}$} (A2_1);
\end{tikzpicture}
\end{aligned}
\end{align}
in which all vertical arrows are $G'$-principal bundles. Next, we put
\begin{align}
Z\coloneqq T^\ast_W(X \times X')\ , \ \overline{Z}\coloneqq T^\ast_{\overline{W}}( \overline{X} \times \overline{X'})\ , \ Z_G\coloneqq Z \cap (T^\ast_GX \times T^\ast_GX')\ , \ \overline{Z}_{\overline{G}}\coloneqq \overline{Z} \cap (T^\ast_{\overline{G}}\overline{X} \times T^\ast_{\overline{G}}\overline{X'})\ .
\end{align}
Observe that $T^\ast_{\overline{G}}\overline{X}$ and $\overline{Z}_{\overline{G}}$ are both $G'$-principal bundles over $T^\ast_GX$ and $Z_G$ respectively. In addition, there is a commutative diagram
\begin{align}
\begin{aligned}
\begin{tikzpicture}[xscale=2.5,yscale=-1]
\node (A0_0) at (0, 0) {$T^\ast_{\overline{G}}\overline{X}$};
\node (A2_0) at (2, 0) {$\overline{Z}_{\overline{G}}$};
\node (A1_1) at (1, 1) {$T^\ast_G X$};
\node (A3_1) at (3, 1) {$Z_G$};
\node (A0_2) at (0, 2) {$T^\ast_{{G}'}\overline{X}$};
\node (A2_2) at (2, 2) {$\overline{Z}_{G'}$};
\node (A1_3) at (1, 3) {$T^\ast X$};
\node (A3_3) at (3, 3) {$Z$};
\node (A0_4) at (0, 4) {$T^\ast\overline{X}$};
\node (A2_4) at (2, 4) {$\overline{Z}$};
\path (A0_0) edge [->]node [above] {$\scriptstyle{}$} (A1_1);
\path (A2_0) edge [->]node [above] {$\scriptstyle{}$} (A3_1);
\path (A0_2) edge [->]node [above] {$\scriptstyle{}$} (A1_3);
\path (A2_0) edge [->]node [above] {$\scriptstyle{}$} (A3_1);
\path (A2_2) edge [->]node [above] {$\scriptstyle{}$} (A3_3);
\path (A2_0) edge [->]node [right=2em, above] {$\scriptstyle{\overline{\Phi}_{\overline{G}}}$} (A0_0);
\path (A3_1) edge [->]node [right=2em, above] {$\scriptstyle{\Phi_{G}}$} (A1_1);
\path (A2_2) edge [->]node [right=2em, above] {$\scriptstyle{\overline{\Phi}_{G'}}$} (A0_2);
\path (A3_3) edge [->]node [right=2em, above] {$\scriptstyle{\Phi}$} (A1_3);
\path (A2_4) edge [->]node [right=2em, above] {$\scriptstyle{\overline{\Phi}}$} (A0_4);
\path (A0_0) edge [->]node [above] {$\scriptstyle{}$} (A0_2);
\path (A0_2) edge [->]node [above] {$\scriptstyle{}$} (A0_4);
\path (A2_0) edge [->]node [above] {$\scriptstyle{}$} (A2_2);
\path (A2_2) edge [->]node [above] {$\scriptstyle{}$} (A2_4);
\path (A1_1) edge [->]node [above] {$\scriptstyle{}$} (A1_3);
\path (A3_1) edge [->]node [above] {$\scriptstyle{}$} (A3_3);
\end{tikzpicture}
\end{aligned}
\end{align}
in which all diagonal arrows are $G'$-principal bundles. Note that $\overline{\Phi}_{G'}$ is regular since $\Phi$ is, hence by \cite[Theorem~6.2 (c)]{book:fulton1998} it follows that $\overline{\Phi}^!=\overline{\Phi}_{G'}^!$. Thanks to the identifications $A^{\overline{G}}(T^\ast_{\overline{G}}\overline{X}) \simeq A^G(T^\ast_GX)$ and $A^{\overline{G}}(\overline{Z}_{\overline{G}}) \simeq A^G(Z_G)$, we have that $\Phi^!=\overline{\Phi}^!$.
To conclude it is enough to apply the previous argument first for the pair of reductive groups $G=\mathsf{G}^{\mathcal{L}_1}_{\alpha+\beta}, G'=\mathsf{G}^{\mathcal{L}_2}_{\alpha+\beta}$ and the diagram in \eqref{eq:diagram-proof1} consisting of the central and bottom vertical arrows, later for the pair of reductive groups $G'=\mathsf{G}^{\mathcal{L}_1}_{\alpha+\beta}, G=\mathsf{G}^{\mathcal{L}_2}_{\alpha+\beta}$ and the diagram in \eqref{eq:diagram-proof1} consisting of the central and top vertical arrows. We obtain $\Phi_2^!= \overline{\Phi}=(\Phi')^!$ and this completes the proof of \eqref{eq:proof1}.
The proof of associativity of the multiplication can be made in locally and uses the same arguments as in the proof of \cite[Theorem~2.2]{art:minets2018}.
\end{proof}
\begin{definition} Let $A$ be either a free OBM theory or $A=H_\ast$. We call the algebra $\mathbf{AHA}'_{\mathsf{Higgs}(X)}$ the \emph{unrestricted} $A$-\emph{homological Hall algebra of the category $\mathsf{Higgs}(X)$}.
\end{definition}
It is easy to see from the construction that the subgroup
\begin{align}
\mathbf{AHA}_{\mathsf{Higgs}(X)}\coloneqq \bigoplus_{\alpha \in (\mathbb{Z}^2)^+} A^0(\bHiggs_\alpha)
\end{align}
is a subalgebra, which we call the $A$-\emph{homological Hall algebra of the category $\mathsf{Higgs}(X)$}.
\subsection{Cohomological Hall algebra of nilpotent, semistable and equivariant Higgs sheaves}
The full abelian subcategory $\mathsf{Higgs}^{\mathsf{nilp}}(X)$ of $\mathsf{Higgs}(X)$ is stable under extension, and the same holds for $\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)$ for each fixed slope $\nu$. We may repeat the above constructions \emph{verbatim} in these contexts (using refined Gysin pullback maps on each local charts\footnote{The stack of semistable Higgs bundles is a global quotient stack (see for example \cite[Section~7.7.1]{art:casalainawise2017}), hence we do not need to restrict ourselves to local charts.} obtained by embedding in the \emph{same} l.c.i. morphism). The compatibility of Gysin pullbacks with direct images by proper morphism and and pullback by open immersions (see \cite[Theorem~6.2 (a), (b)]{book:fulton1998}) imply the following.
\begin{corollary}\label{cor:Higgsvariantnilp}
There is a natural associative algebra structure on the group
\begin{align}
\mathbf{AHA}'_{\mathsf{Higgs}^{\mathsf{nilp}}(X)}\coloneqq\bigoplus_{\alpha \in (\mathbb{Z}^2)^+} A(\mathbf{\Lambda}_{\alpha})\ .
\end{align}
Furthermore, the proper pushforward map $\mathbf{AHA}'_{\mathsf{Higgs}^{\mathsf{nilp}}(X)} \to \mathbf{AHA}'_{\mathsf{Higgs}(X)}$ is an algebra homomorphism. Similarly, there is a natural associative algebra structure on the group
\begin{align}
\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{nilp}}(X)}\coloneqq\bigoplus_{\alpha \in (\mathbb{Z}^2)^+} A^0(\mathbf{\Lambda}_{\alpha})\ ,
\end{align}
and the proper pushforward map $\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{nilp}}(X)} \to \mathbf{AHA}_{\mathsf{Higgs}(X)}$ is an algebra homomorphism.
\end{corollary}
\begin{corollary}\label{cor:Higgsvariantss}
For any fixed slope $\nu$, there is a natural associative algebra structure on the group
\begin{align}
\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}\coloneqq\bigoplus_{\genfrac{}{}{0pt}{}{\alpha \in (\mathbb{Z}^2)^+}{\mu(\alpha)=\nu}} A^0(\mathbf{Higgs}^{\mathsf{ss}}_{\alpha})\ .
\end{align}
Furthermore, the open restriction map $\mathbf{AHA}_{\mathsf{Higgs}^\nu(X)} \to \mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}$ is an algebra homomorphism, where
\begin{align}
\mathbf{AHA}_{\mathsf{Higgs}^{\nu}(X)}\coloneqq\bigoplus_{\substack{ \alpha \in (\mathbb{Z}^2)^+ \\ \mu(\alpha)=\nu}} A^0(\mathbf{Higgs}_{\alpha})\ .
\end{align}
\end{corollary}
One may likewise consider equivariant versions of all the above, with respect to the action of the multiplcative group $T=\mathbb{G}_m$ on $\bHiggs_\alpha$ by $z \cdot (\mathcal{F},\theta)\coloneqq(\mathcal{F},z\,\theta)$, and get in this fashion \emph{equivariant} $A$-homological Hall algebras
\begin{align}
\mathbf{AHA}'^{\,T}_{\mathsf{Higgs}(X)}\ , \qquad \mathbf{AHA}'^{\, T}_{\mathsf{Higgs}^{\mathsf{nilp}}(X)}\ , \quad \mathbf{AHA}^T_{\mathsf{Higgs}(X)}\ , \quad \mathbf{AHA}^T_{\mathsf{Higgs}^{\mathsf{nilp}}(X)}\ , \quad \mathbf{AHA}^T_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}\ .
\end{align}
These are, by construction, modules over the ring $A_T^\ast(\mathsf{pt}) \simeq A^\ast(\mathsf{pt})[[c_1(t)]]$, where $c_1(t)$ is the first Chern class of the tautological character of $T$, see e.g. \cite[Appendix~A]{art:minets2018}.
\begin{proposition}\label{prop:localizationT}
The direct image morphism is an isomorphism of \emph{localized} algebras
\begin{align}
\mathbf{AHA}^T_{\mathsf{Higgs}^{\mathsf{nilp}}(X)} \otimes_{A_T(\mathsf{pt})} \mathsf{Frac}(A_T(\mathsf{pt})) \stackrel{\sim}{\to} \mathbf{AHA}^T_{\mathsf{Higgs}(X)} \otimes_{A_T(\mathsf{pt})} Frac(A_T(\mathsf{pt}))\ .
\end{align}
\end{proposition}
\begin{proof}
Fix a line bundle $\mathcal{L}$ and $\alpha \in (\mathbb{Z}^2)^+$. Let $(T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)^{\mathsf{nilp}} \subset T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal$ be the closed $\mathsf{G}_\alpha^\Lcal$-subvariety such that
\begin{align}
\mathbf{\Lambda}_{\alpha}^{>\mathcal{L}}\coloneqq\big[(T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)^{\mathsf{nilp}} / \mathsf{G}_\alpha^\Lcal\big]\simeq \bHiggs_\alpha^{>\mathcal{L}} \underset{\bHiggs_\alpha}{\times} \mathbf{\Lambda}_{\alpha}\ .
\end{align}
It is enough to show that the direct image morphism
\begin{align}
A^{T \times \mathsf{G}_\alpha^\Lcal}((T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)^{\mathsf{nilp}})\otimes_{A_T(\mathsf{pt})} \mathsf{Frac}(A_T(\mathsf{pt})) \to A_{T \times \mathsf{G}_\alpha^\Lcal}(T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)\otimes_{A_T(\mathsf{pt})} \mathsf{Frac}(A^T(\mathsf{pt}))
\end{align}
is an isomorphism. This follows from the same argument as in \cite[Corollary~6.3]{art:minets2018}.
\end{proof}
The above proposition allows one to deduce certain properties of $\mathbf{AHA}^T_{\mathsf{Higgs}(X)}$ from the geometry of $\displaystyle\mathbf{\Lambda}\coloneqq\bigsqcup_{\alpha} \mathbf{\Lambda}_{\alpha}$, which is sometimes more agreable than that of $\displaystyle\mathbf{Higgs}\coloneqq\bigsqcup_{\alpha} \bHiggs_\alpha$. The results in the next two sections provide an illustration of this principle.
\bigskip\section{Torsion-freeness}\label{sec:torsionfreeness}
In this section and in the next, we take $A=H_\ast$, the usual Borel-Moore homology with rational coefficients (most results also hold for Chow groups with appropriate modifications). In addition, we will focus on the cohomological Hall algebras $\mathbf{AHA}^T_{\mathsf{Higgs}^{\mathsf{nilp}}(X)}$ and $\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{nilp}}(X)}$, which we denote simply by $\mathbf{AHA}^T_{\mathbf{\Lambda}}$ and $\mathbf{AHA}_{\mathbf{\Lambda}}$.
\subsection{The universal cohomology ring of $\mathbf{Coh}$}
For any $\alpha \in (\mathbb{Z}^2)^+$ there is an action of $H^\ast(\bCoh_\alpha)$ on $H_\ast(\bHiggs_\alpha)$ defined by $c \cdot u\coloneqq r_{\alpha}^\ast(c) \cap u$, where $r_{\alpha}\colon \bHiggs_\alpha \to \bCoh_\alpha$ is the canonical projection. The ring $H^\ast(\bCoh_\alpha)$ being freely generated by tautological classes (see Theorem \ref{T:Heinloth}), it is independent of $\alpha$ (strictly speaking, this is true only for $\operatorname{rk}(\alpha) >0$). In this paragraph, we consider a universal version $\mathbb{H}$ of this ring, and endow the algebras $\mathbf{AHA}_{\mathbf{\Lambda}}, \mathbf{AHA}_{\mathsf{Higgs}(X)}$ and $\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}$ with $\mathbb{H}$-module algebra structures.
Let us fix a basis $\Pi=\{1, \pi_1, \ldots, \pi_{2g}, \varpi\}$ of $H^\ast(X)$, with $1 \in H^0(X), \pi_1, \ldots, \pi_{2g} \in H^1(X)$ and $\varpi \in H^2(X)$. Let $\mathbb{H}\coloneqq\mathbb{Q}[c_{i,\pi}]_{i,\pi}$ be the graded free supercommutative algebra generated by elements $c_{i,\pi}$ with $i \geq 2, \pi \in \Pi$ or $i=1, \pi \in \Pi \backslash \{\omega\}$. the degree of $c_{i,\pi}$ is defined to be $\deg(c_{i,\pi})=2i-\deg(\pi)$. For any $\alpha$, there is a surjective morphism $\pi_\alpha\colon \mathbb{H} \to H^\ast(\bCoh_\alpha)$ defined by $\pi_\alpha(c_{i,\pi})=c_{i,\pi}(\mathfrak{E}_\alpha)$, where the classes $c_{i,\pi}(\mathfrak{E}_\alpha)$ are defined in \eqref{eq:chernclasses}. The maps $\pi_\alpha$ is an isomorphism whenever $\operatorname{rk}(\alpha)>0$. Via the map $\pi_\alpha$, the ring $\mathbb{H}$ acts on $\mathbf{AHA}_{\mathsf{Higgs}(X)}$, preserving the class $\alpha$ (but shifting the cohomological degree). This action factors through to an action on $\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}$, and there is a compatible action on $\mathbf{AHA}_{\mathbf{\Lambda}}$.
Let $\alpha_1, \alpha_2 \in (\mathbb{Z}^2)^+$ and set $\alpha=\sum_i \alpha_i$. Define a morphism
\begin{align}
\Delta_{\alpha_1,\, \alpha_2}\colon H^\ast(\bCoh_\alpha) \to H^\ast(\mathbf{Coh}_{\alpha_1}) \otimes H^\ast(\mathbf{Coh}_{\alpha_2})
\end{align}
as the pullback by the direct sum morphism
\begin{align}
\bigoplus_{\alpha_1,\, \alpha_2}\colon \mathbf{Coh}_{\alpha_1} \times \mathbf{Coh}_{\alpha_2} \to \mathbf{Coh}_{\alpha}\ ,\quad (\mathcal{F}_1, \mathcal{F}_2) \mapsto \mathcal{F}_1 \oplus \mathcal{F}_2\ .
\end{align}
The map $\Delta_{\alpha_1,\alpha_2}$ is very easy to express in terms of tautological classes as
\begin{align}
{\bigoplus_{\alpha_1,\, \alpha_2}}^{\!\!\ast}(\mathfrak{E}_\alpha)\simeq \mathsf{pr}_1^\ast(\mathfrak{E}_{\alpha_1}) \oplus
\mathsf{pr}_2^\ast(\mathfrak{E}_{\alpha_2})\ ,
\end{align}
where $\mathsf{pr}_i\colon \mathbf{Coh}_{\alpha_1} \times \mathbf{Coh}_{\alpha_2} \to \mathbf{Coh}_{\alpha_i}$ is the projection for $i=1, 2$. For instance, $\Delta_{\alpha_1,\, \alpha_2}(c_{1,\pi}(\mathfrak{E}_\alpha))=c_{1,\pi}(\mathfrak{E}_{\alpha_1}) \otimes 1 + 1 \otimes c_{1,\pi}(\mathfrak{E}_{\alpha_2})$. We won't need the explicit form of $\Delta_{\alpha_1,\alpha_2}$ here; the following result, whose proof we leave to the reader, will suffice:
\begin{lemma}
There exists a (unique) coassociative coproduct $\Delta\colon \mathbb{H} \to \mathbb{H} \otimes \mathbb{H}$ such that, for any $\alpha_1, \alpha_2$ we have
\begin{align}
(\pi_{\alpha_1} \otimes \pi_{\alpha_2} ) \circ \Delta= \Delta_{\alpha_1,\, \alpha_2} \circ \pi_{\alpha_1+\alpha_2} \ .
\end{align}
Equipped with this coproduct, $\mathbb{H}$ is a graded commutative and cocommutative Hopf algebra.
\end{lemma}
We will say that an $\mathbb{H}$-module $\mathbf{M}$ is an $\mathbb{H}$-module algebra if $\mathbf{M}$ is equipped with an associative algebra structure $m\colon \mathbf{M} \otimes \mathbf{M} \to \mathbf{M}$ such that for any $h \in \mathbb{H}$, $x,y \in \mathbf{M}$ it holds $h \cdot m(x\otimes y)=m(\Delta(h) \cdot (x \otimes y))$.
\begin{proposition}\label{prop:Hmodule}
The algebras $\mathbf{AHA}_{\mathsf{Higgs}(X)}, \mathbf{AHA}_{\mathbf{\Lambda}}$ and $\mathbf{AHA}_{\mathsf{Higgs}^{\mathsf{ss},\, \nu}(X)}$ are $\mathbb{H}$-module algebras.
\end{proposition}
\begin{proof}
Fix some $\alpha_1, \alpha_2$ and set $\alpha=\alpha_1+\alpha_2$. Let $\gamma \in H^\ast(\bCoh_\alpha)$ and $u_i \in H^\ast(\mathbf{Higgs}_{\alpha_i})$ for $i=1,2$. Keep the notation of Section \ref{sec:COHAcoh}. By definition, $u_1 \cdot u_2= (p_{\alpha_1,\, \alpha_2})_\ast\Phi^!(u_1 \otimes u_2)$. Set $v\coloneqq\Phi^!(u_1 \otimes u_2)$. By the projection formula, we have
\begin{align}
\gamma \cdot (p_{\alpha_1,\alpha_2})_\ast(v)= (p_{\alpha_1,\,\alpha_2})_\ast((p_{\alpha_1,\,\alpha_2})^\ast r_{\alpha}^\ast(\gamma) \cap v)\ .
\end{align}
Note that
\begin{align}
(p_{\alpha_1,\,\alpha_2})^\ast r_{\alpha}^\ast(\gamma)=\Phi_G^\ast (r_{\alpha_1}^\ast \otimes r_{\alpha_2}^\ast) (\Delta_{\alpha_1,\,\alpha_2}(\gamma))\ ,
\end{align}
while by multiplicativity of the Gysin pullback
\begin{align}
\Phi_G^\ast (r_{\alpha_1}^\ast \otimes r_{\alpha_2}^\ast) (\Delta_{\alpha_1,\, \alpha_2}(\gamma)) \cap \Phi^!(u_1 \otimes u_2)=\Phi^!((r_{\alpha_1}^\ast \otimes r_{\alpha_2}^\ast) (\Delta_{\alpha_1,\, \alpha_2}(\gamma)) \cap (u_1 \otimes u_2))\ .
\end{align}
This yields the desired equality. We are done.
\end{proof}
Proposition \ref{prop:Hmodule} has an obvious equivariant avatar. Note that in that case, $\mathbb{H}$ is replaced by $\mathbb{H}\otimes \mathbb{Q}[t]$.
\subsection{Torsion-freeness}
In the context of quivers, see \cite[Section~4.4]{art:schiffmannvasserot2017} (or rank zero Higgs sheaves, see \cite[Section~6]{art:minets2018}) the following technical result is crucial in describing the cohomological Hall algebras as \emph{shuffle algebras}. Although we do not give such a realization here, we nevertheless state the following theorem.
\begin{theorem}\label{T:torsionfree}
Let $\alpha\in (\mathbb{Z}^2)^+$. Then $H^{0, T}_\ast(\mathbf{\Lambda}_{\alpha})$ is a torsion-free $H^\ast(\bCoh_\alpha)\otimes \mathbb{Q}[t]$-module.
\end{theorem}
\begin{proof}
Our approach will bear some similarity with those followed in the proofs of \cite[Proposition~4.6]{art:schiffmannvasserot2017} and \cite[Theorem~6.4]{art:minets2018}. To simplify the notation, we shall drop $0$ from $H^{0, T}_\ast$.
Let $\mathbb{H}^+ \subset H^\ast(\bCoh_\alpha)$ be the graded augmentation ideal of $H^\ast(\bCoh_\alpha)$ and set $I=\mathbb{H}^+ \otimes \mathbb{Q}[t] \subset H^\ast(\bCoh_\alpha)\otimes \mathbb{Q}[t]$. For any $H^\ast(\bCoh_\alpha)\otimes \mathbb{Q}[t]$-module $M$ we denote by $M_{\mathsf{loc},I}$ the localization of $M$ with respect to the
ideal $I$. We will prove the following two statements:
\begin{enumerate}\itemsep0.2cm
\item[(a)] the natural map $H^T_\ast(\mathbf{\Lambda}_{\alpha}) \to H^T_\ast(\mathbf{\Lambda}_{\alpha})_{\mathsf{loc},I}$ is injective, i.e $H^T_\ast(\mathbf{\Lambda}_{\alpha})$ is $I$-torsion-free,
\item[(b)] the direct image morphism $H^T_\ast(\bCoh_\alpha)_{\mathsf{loc},I} \to H^T_\ast(\mathbf{\Lambda}_{\alpha})_{\mathsf{loc},I}$ is an isomorphism.
\end{enumerate}
The theorem will follow since $H_\ast^T(\bCoh_\alpha)$ is evidently torsion free (in fact, free) as a $H^\ast(\bCoh_\alpha)\otimes \mathbb{Q}[t]$-module (note that $T$ acts trivially on $\bCoh_\alpha$).
We begin with (a). Recall that we are only considering admissible classes in $H^{T}_\ast(\mathbf{\Lambda}_{\alpha})$, i.e., classes supported on finitely many irreducible components. Moreover, (see \eqref{eq:filtration} and \eqref{eq:gradedcoh}), there is a $H^\ast(\bCoh_\alpha)\otimes \mathbb{Q}[t]$-invariant filtration of $H^T_\ast(\mathbf{\Lambda}_{\alpha})$ by $H^T_\ast(\mathbf{\Lambda}_{\preceq \underline{\alpha}})$, whose associated graded is
\begin{align}
\mathsf{gr}(H^{T}_\ast(\mathbf{\Lambda}_{\alpha}))=\bigoplus_{\underline{\alpha} \in J_{\alpha}} H^{T}_\ast(\mathbf{\Lambda}_{\underline{\alpha}})\ , \quad H^{T}_\ast(\mathbf{\Lambda}_{\underline{\alpha}}) \simeq \bigotimes_i\, H^\ast_T(\mathbf{Coh}_{\alpha_i})\ .
\end{align}
In particular, each $H^T_\ast(\mathbf{\Lambda}_{\underline{\alpha}})$ is a \emph{free} graded $\mathbb{Q}[t]$-module, and hence also $I$-torsion-free. Therefore $H^T_\ast(\mathbf{\Lambda}_{\alpha})$ is also $I$-torsion-free.
We will check statement (b) locally, by showing that for any line bundle $\mathcal{L}$, the direct image morphism
\begin{align}\label{eq:prooftortwo}
H^T_\ast(\bCoh_\alpha^{> \Lcal})_{\mathsf{loc},I} \to H^T_\ast(\mathbf{\Lambda}_{\alpha}^{>\mathcal{L}})_{\mathsf{loc},I}
\end{align}
which respect to the closed embedding $\bCoh_\alpha^{> \Lcal}\simeq \mathbf{\Lambda}_{(\alpha)}^{>\mathcal{L}} \hookrightarrow \mathbf{\Lambda}_{\alpha}^{\mathcal{L}}$, is an isomorphism. The isomorphism is preserved by taking the limit with respect to $\mathcal{L}$, indeed we may again consider the $H^\ast(\bCoh_\alpha)\otimes \mathbb{Q}[t]$-invariant filtration as in case (a) above and argue on each $\mathbf{\Lambda}_{\preceq \underline{\alpha}}$ (which is an admissible stack). So let us fix a line bundle $\mathcal{L}$. The open substack $\bCoh_\alpha^{> \Lcal}$ is isomorphic to the quotient $[\mathsf{Q}_\alpha^\Lcal/\mathsf{G}_\alpha^\Lcal]$, while $\mathbf{\Lambda}_{\alpha}^{>\mathcal{L}}=[(T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)^{\mathsf{nilp}} / \mathsf{G}_\alpha^\Lcal]$. Hence $H_\ast(\bCoh_\alpha^{> \Lcal}) \simeq H_\ast^{\mathsf{G}_\alpha^\Lcal}(\mathsf{Q}_\alpha^\Lcal)$ and $H_\ast(\mathbf{\Lambda}_{\alpha}^{>\mathcal{L}})\simeq H_\ast^{\mathsf{G}_\alpha^\Lcal}((T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)^{\mathsf{nilp}})$. In particular, $H_\ast(\bCoh_\alpha^{> \Lcal})$ carries an action of the equivariant cohomology ring $H^\ast_{\mathsf{G}_\alpha^\Lcal}(\mathsf{pt})\eqqcolon \mathbf{R}_{\mathsf{G}_\alpha^\Lcal}$. There is a similar action of $\mathbf{R}_{\mathsf{G}_\alpha^\Lcal} $ on $H_\ast(\bHiggs_\alpha^{> \Lcal})$ and on $H_\ast(\mathbf{\Lambda}_{\alpha}^{>\mathcal{L}})$.
We shall need the following result.
\begin{lemma}\label{L:prooftor1}
For any $\alpha, \mathcal{L}$ there is a surjective algebra morphism $s_{\mathcal{L},\alpha} \colon H^\ast(\bCoh_\alpha) \to \mathbf{R}_{\mathsf{G}_\alpha^\Lcal}$ such that for any $\gamma \in H^\ast(\bCoh_\alpha)$ and any $c \in H_\ast(\bHiggs_\alpha)$ (resp.\ $H_\ast(\mathbf{\Lambda}_{\alpha})$) we have
\begin{align}
(\gamma \cap c)\vert_{\bHiggs_\alpha^{> \Lcal}}& =s_{\mathcal{L},\alpha}(\gamma) \cap c\vert_{\bHiggs_\alpha^{> \Lcal}}\ .\\[4pt]
\mbox{(resp.\ }\ (\gamma \cap c)\vert_{\mathbf{\Lambda}_{\alpha}^{>\mathcal{L}}}& =s_{\mathcal{L},\alpha}(\gamma) \cap c\vert_{\mathbf{\Lambda}_{\alpha}^{>\mathcal{L}}}\ .\mbox{ )}
\end{align}
\end{lemma}
\begin{proof}
Since the $H^\ast(\bCoh_\alpha)$-action is given via pullback with respect to the morphism $r_\alpha\colon \bHiggs_\alpha\to \bCoh_\alpha$, it suffices to prove the statement of the lemma for $\bCoh_\alpha$ in place of $\bHiggs_\alpha$ or $\mathbf{\Lambda}_{\alpha}$.
Let $p_\alpha^\mathcal{L}\colon \mathsf{Q}_\alpha^\Lcal \times X \to \mathsf{Q}_\alpha^\Lcal$ and $p_X\colon \mathsf{Q}_\alpha^\Lcal\times X\to X$ denote the two projections. The action of $\mathbf{R}_{\mathsf{G}_\alpha^\Lcal}$ on $H^{\mathsf{G}_\alpha^\Lcal}_\ast(\mathsf{Q}_\alpha^\Lcal)$ is given by cap product by the Chern classes of the tautological $\mathsf{G}_\alpha^\Lcal$-equivariant vector bundle $\mathbb{V}\coloneqq\mathbb{R}(p_\alpha^\mathcal{L})_\ast(p_X^\ast\mathcal{L} \otimes \mathfrak{E}_\alpha^\mathcal{L})$, whose fiber over a point $\big[\phi\colon \mathcal{L}\otimes k^{\, \langle \overline{\mathcal{L}},\alpha\rangle}\twoheadrightarrow \mathcal{F}\big]$ is $\mathsf{Hom}(\mathcal{L}, \mathcal{F})$. On the other hand, the action of $H^\ast(\bCoh_\alpha)$ is given by cap product with the K\"unneth components of the Chern classes of the tautological sheaf $\mathfrak{E}_\alpha$. Applying the Grothendieck-Riemann-Roch formula \cite[Theorem~15.2]{book:fulton1998} to the morphism $p_\alpha^\mathcal{L}$ and $p_X^\ast\mathcal{L}^\vee \otimes \mathfrak{E}_\alpha^\mathcal{L}$ yields an expression for $\mathsf{ch}(\mathbb{V})$ in terms of $\mathsf{ch}(\mathfrak{E}_\alpha^\mathcal{L})$ as wanted. Note that $(p_\alpha^\mathcal{L})_\ast p_X^\ast \mathcal{L}$ is a trivial $\mathsf{G}_\alpha^\Lcal$-equivariant bundle on $\mathsf{Q}_\alpha^\Lcal$.
\end{proof}
Put $I_{\mathsf{G}_\alpha^\Lcal}\coloneqq\mathbf{R}_{\mathsf{G}_\alpha^\Lcal}^+ \otimes \mathbb{Q}[t]$, where $\mathbf{R}_{\mathsf{G}_\alpha^\Lcal}^+$ is the graded augmentation ideal of $\mathbf{R}_{\mathsf{G}_\alpha^\Lcal}$. By the relative form of the localization theorem (see e.g. \cite[Proposition~A.13, Theorem~A.14]{art:minets2018}) the localized pushforward map
\begin{align}
H_\ast^{\mathsf{G}_\alpha^\Lcal \times T}(\mathsf{Q}_\alpha^\Lcal)_{\mathsf{loc},I_{\mathsf{G}_\alpha^\Lcal}} = H_\ast^{\mathsf{G}_\alpha^\Lcal \times T}((T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)^T)_{\mathsf{loc},I_{\mathsf{G}_\alpha^\Lcal}} \to H_\ast^{\mathsf{G}_\alpha^\Lcal \times T}((T^\ast_{\mathsf{G}_\alpha^\Lcal}\mathsf{Q}_\alpha^\Lcal)^{\mathsf{nilp}})_{\mathsf{loc}, I_{\mathsf{G}_\alpha^\Lcal}}
\end{align}
is an isomorphism. By Lemma \ref{L:prooftor1} we have $s_{\mathcal{L},\alpha}(I)=I_{\mathsf{G}_\alpha^\Lcal}$. This implies \eqref{eq:prooftortwo} and concludes the proof Theorem \ref{T:torsionfree}.
\end{proof}
\bigskip\section{Generation theorem}\label{sec:generation}
In this section, we take again $A=H_\ast$. For any $\alpha \in (\mathbb{Z}^2)^+$ there is a distinguished irreducible component $\mathbf{\Lambda}_{(\alpha)}$ of $\mathbf{\Lambda}_{\alpha}$, namely the zero section of the projection $r_{\alpha}\colon \bHiggs_\alpha \to \bCoh_\alpha$. Thus $\mathbf{\Lambda}_{(\alpha)} \simeq \bCoh_\alpha$. In particular, by Poincar\'e duality,
\begin{align}
H_*(\mathbf{\Lambda}_{(\alpha)}) \simeq \left\{
\begin{array}{ll}
\mathbb{Q}[c_{i,\pi}(\mathfrak{E}_{\alpha}]_{i,\pi} & \mbox{if }\operatorname{rk}(\alpha)>0\ ,\\[4pt]
S^d(H^\ast(X)[z]) & \mbox{if } \alpha=(0,d)\ .
\end{array}
\right.
\end{align}
The following is an analog of \cite[Theorem~B (e)]{art:schiffmannvasserot2017}. It is interesting that the proof, though similar, is simpler in the curve case than in the quiver case as the structure of $\mathbf{\Lambda}_{\alpha}$ is simpler than that of the Lusztig nilpotent stack.
\begin{theorem}\label{T:gen}
For $A=H_\ast$, the algebra $\mathbf{AHA}_{\mathbf{\Lambda}}$ is generated by the collection of subspaces $H_\ast(\mathbf{\Lambda}_{(\alpha)})$ for $\alpha \in (\mathbb{Z}^2)^+$.
\end{theorem}
\begin{proof}
Let us denote by $\mathbf{B}$ the subalgebra of $\mathbf{AHA}_{\mathbf{\Lambda}}$ generated by the collection of subspaces $H_\ast(\mathbf{\Lambda}_{(\alpha)})$ for $\alpha \in (\mathbb{Z}^2)^+$. By definition of $A_\ast^0$, every class in $c \in H_\ast^0(\mathbf{\Lambda}_{\alpha})$ is supported on a finite number of irreducible components, i.e., there exists a finite subset of Jordan types $I_c \subset J_{\alpha}$ such that
\begin{align}
c \in \mathsf{Im}\bigg( H_\ast\Big(\bigsqcup_{\underline{\alpha} \in I_c} \mathbf{\Lambda}_{\underline{\alpha}}\Big) \to H^0_\ast(\mathbf{\Lambda}_{\alpha})\bigg) \ .
\end{align}
Recall the partial order $\prec$ on $J_{\alpha}$ (see \eqref{eq:preceq} and Proposition \ref{prop:preceq}) as well as the induced filtration \eqref{eq:filtration} on $H^0_\ast(\mathbf{\Lambda}_{\alpha})$. We will prove by induction on $\underline{\alpha}$ with respect to $\prec$ that
\begin{align}\label{eq:proofgen1}
\mathsf{Im}\big( H_\ast(\mathbf{\Lambda}_{\preceq \underline{\alpha}}) \to H^0_\ast(\mathbf{\Lambda}_{\alpha})\big) \subset \mathbf{B}\ .
\end{align}
So let us fix $\underline{\alpha} \in J_{\alpha}$ and assume that \eqref{eq:proofgen1} holds for all $\underline{\beta}$ with $\underline{\beta} \prec \underline{\alpha}$. If $\underline{\alpha}=(\alpha)$ then \eqref{eq:proofgen1} holds by definition. Otherwise, let us write $\underline{\alpha}=(\alpha_1, \ldots, \alpha_s)$ and put $\gamma_i\coloneqq\sum_{j \geq i} \alpha_i((i-j)\deg(\omega_X))$; this is the total class of the $i$th row of the colored Young diagram associated with $\underline{\alpha}$, see \eqref{diag:Young}. Consider the (iterated) convolution diagram for Higgs stacks
\begin{align}\label{eq:proofgen2}
\begin{aligned}
\begin{tikzpicture}[xscale=4,yscale=-1]
\node (A0_0) at (0, 0) {$\prod_i \mathbf{Higgs}_{\gamma_i}$};
\node (A1_0) at (1, 0) {$\widetilde{\mathbf{Higgs}}_{\gamma_1, \ldots, \gamma_s}$};
\node (A2_0) at (1.8, 0) {$\mathbf{Higgs}_\alpha$};
\path (A1_0) edge [->]node [above] {$\scriptstyle{q_{\underline{\gamma}}}$} (A0_0);
\path (A1_0) edge [->]node [above] {$\scriptstyle{p_{\underline{\gamma}}}$} (A2_0);
\end{tikzpicture}
\end{aligned}
\end{align}
Using \eqref{eq:eulerformhiggs} and Theore~\ref{T:StackHiggs} (a) we compute
\begin{align}\label{E:proofgen3}
\dim(q_{\underline{\gamma}})=\dim(\widetilde{\mathbf{Higgs}}_{\gamma_1, \ldots, \gamma_s})-\sum_i \dim( \mathbf{Higgs}_{\gamma_i})=-2\sum_{i \neq j} \langle \gamma_i,\gamma_j\rangle\ .
\end{align}
We will use the following three observations:
\begin{enumerate}\itemsep0.2em
\item[(a)] $p_{\underline{\gamma}} \circ q_{\underline{\gamma}}^{-1}(\prod_i \mathbf{\Lambda}_{(\gamma_i)}) \subseteq \mathbf{\Lambda}_{\preceq \underline{\alpha}}$,
\item[(b)] $p_{\underline{\gamma}}\colon p_{\underline{\gamma}}^{-1}(\mathbf{\Lambda}_{\underline{\alpha}}) \to \mathbf{\Lambda}_{\underline{\alpha}}$ is an isomorphism,
\item[(c)] there exists an open subset $\mathscr{U}$ of $\prod_i \mathbf{\Lambda}_{(\gamma_i)}$ over which $q_{\underline{\gamma}}$ is smooth with connected fibers of dimension $-\sum_{i\neq j}\langle \gamma_i,\gamma_j\rangle$ and $q_{\underline{\gamma}}^{-1}(\mathscr{U}) \underset{\widetilde{\mathbf{Higgs}}_{\gamma_1, \ldots, \gamma_s}}{\times} p_{\underline{\gamma}}^{-1}(\mathbf{\Lambda}_{\underline{\alpha}})$ is open in $ p_{\underline{\gamma}}^{-1}(\mathbf{\Lambda}_{\underline{\alpha}})$.
\end{enumerate}
Statement (a) is easy, while (b) comes from the unicity of the iterated kernel filtration $\ker(\theta) \subseteq \ker(\theta^2) \subseteq \cdots \subseteq \mathcal{F}$ for any $\underline{\mathcal{F}} \in \mathbf{\Lambda}_{\underline{\alpha}}$. Statement (c) is proved as \cite[Proposition~1.6, Theorem~1.4]{art:bozec2016}, see also \cite[Lemma~3.19]{art:schiffmannvasserot2017}. Note that
\begin{align}
\dim(\mathbf{\Lambda}_{(\alpha)})-\sum_i \dim(\mathbf{\Lambda}_{(\gamma_i)})=-\sum_{i \neq j} \langle \gamma_i,\gamma_j\rangle\ .
\end{align}
From (a), (b) and (c), using the local construction of the multiplication map, the base change property of refined Gysin pullbacks \cite[Theorem 6.2 (b)]{book:fulton1998} we deduce that
\begin{align}\label{eq:proofgen4}
\Big( [\mathbf{\Lambda}_{(\gamma_s)}] \star[\mathbf{\Lambda}_{(\gamma_{s-1})}] \star \cdots \star [\mathbf{\Lambda}_{(\gamma_1)}]\Big)\vert_{\mathbf{\Lambda}_{\underline{\alpha}}}=[\mathbf{\Lambda}_{\underline{\alpha}}]
\end{align}
while $\mathsf{supp}\Big( [\mathbf{\Lambda}_{(\gamma_s)}] \star [\mathbf{\Lambda}_{(\gamma_{s-1})}] \star \cdots \star [\mathbf{\Lambda}_{(\gamma_1)}]\Big) \subseteq \mathbf{\Lambda}_{\preceq \underline{\alpha}}$. More generally, from the compatibility of refined Gysin morphisms with respect to cap product with Chern classes \cite[Proposition~6.3]{book:fulton1998}, we deduce that for any polynomials $P_1, \ldots, P_s$ in the (K\"unneth components of the) Chern classes of the tautological sheaves $\mathfrak{E}_{\gamma_1}, \ldots, \mathfrak{E}_{\gamma_s}$ on $\mathbf{\Lambda}_{(\gamma_1)} \simeq \mathbf{Coh}_{\gamma_1}, \ldots, \mathbf{\Lambda}_{(\gamma_s)} \simeq \mathbf{Coh}_{\gamma_s}$ respectively, we have
\begin{multline}\label{eq:proofgen5}
\Big(P_s(c_{i,\pi}(\mathfrak{E}_{\gamma_s}) \cap [\mathbf{\Lambda}_{(\gamma_s)}])\Big) \star\cdots \star \Big(P_1(c_{i,\pi}(\mathcal{E}_{\gamma_1}) \cap [\mathbf{\Lambda}_{(\gamma_1)}])\Big)\vert_{\mathbf{\Lambda}_{\underline{\alpha}}}\\
=\Big(P_1(c_{i,\pi}(\mathfrak{E}_1)) \cdots P_s(c_{i,\pi}(\mathfrak{E}_{s})\Big) \cap [\mathbf{\Lambda}_{\underline{\alpha}}]\ ,
\end{multline}
where $\mathfrak{E}_1, \ldots, \mathfrak{E}_s$ are the tautological sheaves over $\mathbf{\Lambda}_{\underline{\alpha}} \times X$ defined as $\mathfrak{E}_i = \ker(\theta^i)/\ker(\theta^{i-1})$, where $\theta$ is the Higgs field on the universal sheaf $\mathfrak{E}_\alpha$. We claim that $H^\ast(\mathbf{\Lambda}_{\underline{\alpha}})$ is generated by the Chern classes $c_{i,\pi}(\mathfrak{E}_j)$ for $j=1, \ldots, s$. Indeed, by Proposition \ref{prop:MS5.2}, $H^\ast(\mathbf{\Lambda}_{\underline{\alpha}})$ is generated by the Chern classes $c_{i,\pi}(\mathfrak{E}_{\alpha_j})$ for $j=1, \ldots, s$, where $[\mathfrak{E}_{\alpha_j}]=[\mathfrak{E}_{i+1}\otimes \omega_X]-[\mathfrak{E}_i]$ in $K_0(\mathbf{\Lambda}_{\underline{\alpha}})$. Since the K\"unneth components $c_{i,\pi}(\mathfrak{E}_{i+1}\otimes \omega_X)$ obviously generate the same algebra as the K\"unneth components $c_{i,\pi}(\mathfrak{E}_{i+1})$, the claim follows.
From all this, we deduce that
\begin{align}
H_\ast(\mathbf{\Lambda}_{\underline{\alpha}}) \subseteq \mathsf{gr}\big( H_\ast(\mathbf{\Lambda}_{(\gamma_s)}) \star \cdots \star H_\ast(\mathbf{\Lambda}_{(\gamma_1)})\big) \subseteq \bigoplus_{\underline{\beta} \preceq \underline{\alpha}} H_\ast(\mathbf{\Lambda}_{\underline{\beta}})\ .
\end{align}
Using the induction hypothesis \eqref{eq:proofgen1} we get that
\begin{align}
\mathsf{Im}\big( H_\ast(\mathbf{\Lambda}_{\preceq \underline{\alpha}}) \to H^0_\ast(\mathbf{\Lambda}_{\alpha})\big) \subset \mathbf{B}
\end{align}
as wanted. Theorem \ref{T:gen} is proved.
\end{proof}
\begin{corollary}\label{cor:gen1}
For $A=H_\ast$, the algebra $\mathbf{AHA}^T_{\mathsf{Higgs}(X)}\otimes \mathbb{Q}(t)$ is generated over $\mathbb{Q}(t)$ by the collection of subspaces $H_\ast(\mathbf{\Lambda}_{(\alpha)})$ for $\alpha \in (\mathbb{Z}^2)^+$.
\end{corollary}
\begin{corollary}\label{cor:gen2}
For $A=H_\ast$, $\mathbf{AHA}_{\mathbf{\Lambda}}$ is generated as an $\mathbb{H}$-module algebra by the collection of elements $[\mathbf{\Lambda}_{(\alpha)}]$ for $\alpha \in (\mathbb{Z}^2)^+$.
\end{corollary}
\begin{proof}
It suffices to observe that, by Poincar\'e duality for the stack $\bCoh_\alpha$, $\mathbb{H} \cdot [\mathbf{\Lambda}_{(\alpha)}]=H_\ast(\mathbf{\Lambda}_{(\alpha)})$.
\end{proof}
\bigskip
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,190 |
Q: Storing music tags in an SQLite database I'm looking for an efficient way to store music tags and comments in an SQLite database running on Android API 8 (Froyo 2.2). They should be in a similar format to Vorbis comments.
So far I've considered:
A JSON Object stored in a column named 'fields' formatted similarly:
+-----+----------------------------------------------------------------------+
| _id | fields |
+-----+----------------------------------------------------------------------+
| 0 | {"artist":"Artist", "title":"Track", "album":"Album 123", "track":3} |
+-----+----------------------------------------------------------------------+
Individual columns for each field. This is the way google.android.music stores fields, but I was worried about new tracks adding too many columns. Source:
On the other hand, many experienced database designers will argue that a well-normalized database will never need more than 100 columns in a table.
http://www.sqlite.org/limits.html
+-----+-------------+----------------+------------+
| _id | artist | album | title |
+-----+-------------+----------------+------------+
| 0 | Test Artist | Test Album 123 | Test Title |
| 1 | An Artist | This Album | This Title |
+-----+-------------+----------------+------------+
Originally I just put them in a
Map<String, String> and invoked toString(), adding it to 'fields' as follows:
+-----+--------------------------------------------------------------+
| _id | fields |
+-----+--------------------------------------------------------------+
| 0 | {title=Example Title, album=Example Album, artist=An Artist} |
+-----+--------------------------------------------------------------+
This is similar to a JSON Object (the first example), but with added difficulties, such as parsing it back into a Java object.
I would prefer having the option of an arbitrary number of fields, so I'm tending towards a JSON Object. Please tell me any alternative options, or what you'd do in a similar situation. Thanks!
I created the ASCII tables with http://www.sensefulsolutions.com/2010/10/format-text-as-table.html
A: I would strongly suggest normalizing your data - don't store JSON, etc, because it drastically reduces your ability to later query that data in any meaningful way (and believe me, NOT normalizing data seems to be the most sure-fire way to guarantee that you will need to query that data in the future, even if you don't now).
Go with your second example of multiple columns, you will be glad you did.
A: Start with individual columns for each field. If you have a need later for several user defined fields, you can always add a json object later, but my guess is you won't need it.
Individual fields have many, many advantages, including the ability to sort and filter on any field. Most sql database abilities are based on having individual fields, so it will be much harder to do anything if you combine the fields.
A: Have you heard of NoSQl databases? It will be a perfect fit for your needs since it stores data in a JSON Model, so each data item can have arbitrary fields. Check out TouchDB-Android.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,206 |
Borough Park is a football stadium in Workington, Cumbria, England. The home ground of Workington A.F.C., it has a capacity of 3,101, of which 500 is seated.
History
Borough Park was built with the assistance of the local council, and opened in 1937, with Workington moving from their previous Lonsdale Park ground, which was next to Borough Park. The ground initially consisted of a 1,000 seat main stand on the western touchline, and banking around the remainder of the pitch, but by 1951 the embankments had been converted to terracing, and two more stands erected in the north-west and south-west corners of the ground.
Workington were elected to the Football League in 1951, and the first League match at Borough Park saw them defeat Chesterfield 3–1 in front of 11,000 spectators. The ground underwent further expansion during the 1950s as the main stand was extended and the terracing on the eastern side of the pitch was roofed in 1956. The record attendance of 21,000 was set on 4 January 1958 for an FA Cup match against Manchester United. The League record attendance of 18,628 was set for a local derby against Carlisle United on 26 December 1963.
Workington were voted out of the League in 1977 and replaced by Wimbledon F.C. The main stand was closed and the roof removed in 1988.
Borough Park was also the home of the town's rugby league club Workington Town from when they formed after the Second World War in 1945 until they moved out in 1956 and into their own new ground nearby, Derwent Park, where they remain today.
Workington Stadium plans
In February 2019, a plan for a new stadium for Workington was announced. This would have involved the demolition of Borough Park and Derwent Park.
In June 2019, it was announced by the new leadership of Allerdale Borough Council that the new sports stadium would not be built.
References
Workington
Workington A.F.C.
Football venues in England
Sports venues completed in 1937
Sports venues in Cumbria
English Football League venues | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,677 |
{"url":"https:\/\/www.aimsciences.org\/article\/doi\/10.3934\/proc.2009.2009.857","text":"Article Contents\nArticle Contents\n\n# Asymptotical dynamics of the modified Schnackenberg equations\n\n\u2022 The existence of a global attractor in the $L^2$ product phase space for the solution semiflow of the modified Schnackenberg equations with the Dirichlet boundary condition on a bounded domain of space dimension $n\\le 3$ is proved. This reaction-diffusion system features two pairs of oppositely-signed nonlinear terms so that the dissipative sign-condition is not satisfied. The proof features two types of rescaling and grouping estimation in showing the absorbing property and the uniform smallness in proving the asymptotical compactness by the approach of a new decomposition.\nMathematics Subject Classification: 37L30, 35B40, 35B41, 35K55, 35K57, 35Q80.\n\n Citation:\n\nOpen Access Under a Creative Commons license","date":"2023-03-21 13:30:59","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 1, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19889770448207855, \"perplexity\": 730.5490880629467}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296943698.79\/warc\/CC-MAIN-20230321131205-20230321161205-00113.warc.gz\"}"} | null | null |
package com.twitter.elephantbird.hive.serde;
import com.twitter.elephantbird.mapreduce.io.ThriftWritable;
import org.apache.hadoop.conf.Configuration;
import org.apache.hadoop.hive.serde.Constants;
import org.apache.hadoop.hive.serde2.SerDe;
import org.apache.hadoop.hive.serde2.SerDeException;
import org.apache.hadoop.hive.serde2.SerDeStats;
import org.apache.hadoop.hive.serde2.objectinspector.ObjectInspector;
import org.apache.hadoop.hive.serde2.objectinspector.ObjectInspectorFactory;
import org.apache.hadoop.io.Writable;
import java.util.Properties;
/**
* SerDe for working with {@link ThriftWritable} records.
* This pairs well with {@link com.twitter.elephantbird.mapred.input.HiveMultiInputFormat}.
*/
public class ThriftSerDe implements SerDe {
private ObjectInspector inspector;
@Override
public void initialize(Configuration conf, Properties properties) throws SerDeException {
String thriftClassName = properties.getProperty(Constants.SERIALIZATION_CLASS, null);
if (thriftClassName == null) {
throw new SerDeException("Required property " + Constants.SERIALIZATION_CLASS + " is null.");
}
Class thriftClass;
try {
thriftClass = conf.getClassByName(thriftClassName);
} catch (ClassNotFoundException e) {
throw new SerDeException("Failed getting class for " + thriftClassName);
}
inspector = ObjectInspectorFactory.getReflectionObjectInspector(
thriftClass, ObjectInspectorFactory.ObjectInspectorOptions.THRIFT);
}
@Override
public Class<? extends Writable> getSerializedClass() {
return null;
}
@Override
public Writable serialize(Object o, ObjectInspector objectInspector) throws SerDeException {
return null;
}
/**
* @param writable the {@link ThriftWritable} to deserialize
* @return the actual thrift object
* @throws SerDeException
*/
@Override
public Object deserialize(Writable writable) throws SerDeException {
if (!(writable instanceof ThriftWritable)) {
throw new SerDeException("Not an instance of ThriftWritable: " + writable);
}
return ((ThriftWritable) writable).get();
}
@Override
public ObjectInspector getObjectInspector() throws SerDeException {
return inspector;
}
@Override
public SerDeStats getSerDeStats() {
return null;
}
}
| {
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{"url":"https:\/\/mattermodeling.stackexchange.com\/questions\/751\/simulating-breaking-bonds-in-molecular-dynamics\/763","text":"# Simulating breaking bonds in molecular dynamics\n\nHow does one introduce the possibility of breaking a bond in a molecule during MD simulation?\n\nI only found the cases when we just introduce the harmonic potential $$U_{ij} = \\frac{1}{2}k(r_i - r_j)^2$$ for atoms in a molecule or introduce an additional equation restricting the length of a bond during simulaton.\n\nNevertheless, it is still not clear for me how we can take into account the breaking bond result?\n\n\u2022 You cannot break bonds in classical MD. See ReaxFF. \u2013\u00a0B. Kelly May 17 '20 at 10:22\n\u2022 Ab-initio molecular dynamics (AIMD), ie using a quantum chemical\/quantum mechanical description of the ensemble does it \u2013\u00a0Greg Nov 23 '20 at 23:06\n\nThe quadratic potential is the simplest possible model for a bond. You can derive it by considering the Taylor expansion of the potential around the natural bond length\n\n$$V(r - r_0) = V(r_0) + \\frac{d V(r_0)}{d r} (r - r_0) + \\frac{1}{2} \\frac{d^2 V}{dr^2} (r - r_0)^2$$\n\nThe constant term can be set to 0 since it does not contribute to the force and just sets the 0 point of your energy scale. The linear term is 0, because you are at a stationary point [1].\n\nThat leaves you with $$V(r - r_0) = \\frac{1}{2} \\frac{d^2 V}{dr^2} (r - r_0)^2$$\n\nIf we require the second derivative to be constant, we have recovered the harmonic potential. As @Charlie Crown illustrated, this force resulting from this potential does not go to 0 at infinity, while the Morse potential does. You can of course take a polynomial of higher order than two, but not every order is suitable. A third order polynomial results in a potential that (typically) goes to negative infinity at large $$r$$, so instead a quartic potential is some times used. It has the advantage of being slightly \"wider\" than the quadratic one. That said, a completely unrelated reason why none of these can simulate bond breaking\/formation is that the implementation requires explicit declaration of which atoms should interact via the stretching potential.\n\nStill, at large $$r$$ both differ significantly from the Morse potential. Why then is the Morse potential not used? The restoring force for large $$r$$ is very low in case of the Morse potential, hence it takes longer for the bond length to return to the equilibrium position. The quadratic potential describes the potential well for displacements close to equilibrium and for moderate temperatures, this is the part of the potential you care about.\n\nObviously that still leaves the question of how to simulate bond breaking in a force field. ReaxFF assumes that the bond order of a pair of atoms can be determined from the interatomic distance alone.\n\n(qualitative recreation from [2])\n\nThe sigma, pi and double pi bonds contribute increasingly to the overall bond order (max individual bond order is 1) as the atoms get closer together. For simplicity I am leaving out the corrections made to the overall bond order necessitated by overcoordination. The bond stretching potential takes the form of a modified Morse potential\n\n$$E_{Bond} = -D_e \\cdot BO_{ij} \\cdot \\exp(p \\cdot (1 - BO_{ij}^p))$$\n\nwhere $$p$$ is a bond specific parameter[2].\n\nReferences:\n\n[1]: Frank Jensen, Introduction to Computational Chemistry Chap. 2\n\n[2]: J. Phys. Chem. A 2001, 105, 9396-9409\n\n\u2022 +1. Great effort went into this. Now if only there's a way to \"tag\" Stack Exchange user Frank Jensen to show him that you cited his book! \u2013\u00a0Nike Dattani May 17 '20 at 21:32\n\n## Very short answer: No, classical molecular dynamics cannot break bonds.\n\nThe potential you showed is the most common form of bond, the harmonic potential a.k.a. Hookes law.\n\nIf you have ever broken a bond in QM (calculated a dissociation curve), you know it is a bit tricky, you need to use \"unrestricted\" settings, meaning, that a given pair of electrons does not need share the same orbital. As an added point for accuracy at the risk of distraction... saying you need \"QM\" is a bit hand wavey, although, not wrong. For real accuracy in QM, you need more than one Slater determinant to approximate the wave function. Now this is somewhat besides the point, but it lets me show a picture hastily drawn in powerpoint, so don't judge...\n\nAs you can see, the potential has a minimum, and at that minimum, and for a small ways out from it, as you stretch or compress the bond from equilibrium (the minimum is the equilibrium energy\/bond length) the form of the potential is pretty close to quadratic. SPOILER ALERT: this is why the potential you showed in your question is used. It is quadratic, and describes the bond energy fairly well, BUT, only close to the minimum. As you get further away the bond energy clearly is not well modeled by a harmonic potential (quadratic)\n\nThus, if you try to break a bond with in classical MD, you will do a terrible job of accurately modelling it. QM can't even do it well without using un-Restricted or Restricted open shell approaches!\n\n## There is also the subtle point that the quadratic we use for bonds will never actually break. The energy will increase to infinity without \"breaking\".\n\nAs I alluded to in my comment \"non-classical\" methods can be used, for instance ab-initio MD which incorporated QM, thus allowing the bond to break. Also, ReaxFF which is closer to a classical FF uses bond-orders rather than actual bonds to describe molecules. I am not experienced with either of these, so I simply mention them and supply these two links for your further reading if you are interested:\n\nab-initio ab-initio Wiki\n\nreaxFF ReaxFF Wiki\n\n\u2022 That's a REALLY good picture, especially considering it was done in PowerPoint! I didn't even know PowerPoint could make such decent figures! \u2013\u00a0Nike Dattani May 17 '20 at 19:15\n\u2022 Being a poor grad student for several years looks to have paid off :) \u2013\u00a0B. Kelly May 17 '20 at 19:18\n\u2022 One quibble - it's not simply a matter with QM of open shell calculations. To really do bond breaking, you need more than one Slater determinant, e.g. multi-reference methods, etc. \u2013\u00a0Geoff Hutchison May 18 '20 at 0:50\n\u2022 @JamesFlash One thing neither of us discussed, is that bond breaking does not effect only the bond energy\/forces. As you break a bond, the angles and torsions those two atoms are invovled in need to be accounted for (eventually turned off). Also, you have to start switching to applying non-bonded electrostatic and vdW forces between them. This also means, how do you figure out the partial charges on each atoms? reaxFF addresses all of these issues, often with empirical parameters. It is good to keep all these other issues in mind though. \u2013\u00a0B. Kelly May 18 '20 at 22:44\n\u2022 Thanks @CharlieCrown, your comments advanced my understaning of the process \u2013\u00a0James Flash May 19 '20 at 9:36\n\nYou can do something with Lennard-Jones potentials instead of harmonic ones. I know this has been done for Coarse-Grained MD simulations using Go-Martini models as in https:\/\/pubs.acs.org\/doi\/abs\/10.1021\/acs.jctc.6b00986\n\nNow this needs more testing, I have done some force-probe simulations with it but nothing too serious.\n\n\u2022 Lennard-Jones potentials describe only van der Waals effects, i.e. not really bond breaking. \u2013\u00a0Susi Lehtola May 23 '20 at 9:47","date":"2021-07-25 22:36:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 8, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5895300507545471, \"perplexity\": 897.4650834701409}, \"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-2021-31\/segments\/1627046151866.98\/warc\/CC-MAIN-20210725205752-20210725235752-00288.warc.gz\"}"} | null | null |
Delivery Day | AirSprint Accepts Its 9th Cessna Citation CJ3+ Aircraft
Posted on Wednesday, 15 June 2022
PRESS RELEASE: TORONTO, ONTARIO - JUNE 15, 2022
AirSprint Inc., the Canadian authority in Fractional Jet Ownership, announces its 9th Cessna Citation CJ3+ has arrived to join its fleet. With this new acceptance, the AirSprint Citation fleet has grown to 15 jets, bringing the total number of aircraft in the AirSprint jet collection to 25. AirSprint now operates with 10 Embraer Praetor 500/Legacy 450/500, 9 Cessna Citation CJ3+, and 6 Cessna Citation CJ2+ aircraft.
"With this new CJ3+ joining the fleet and the jets slotted to arrive later this year, AirSprint is well-positioned to continue to support our current and future Fractional Owners in Canada," said Scott Wenz, AirSprint's Vice President of Sales & Marketing. "The CJ3+ aircraft has always generated a lot of interest as it is the perfect combination of range, speed and efficiency—and has the economics for ownership to make sense."
Individuals and corporations choose Fractional Jet Ownership with AirSprint not only for a cost-effective way to maximize their time but for its expertly selected collection of jets. The Cessna Citation CJ3+ aircraft is essential to the AirSprint fleet, which from an operational standpoint performs remarkably. It has light-jet efficiency and flexibility combined with mid-size jet performance, range, and comfort.
"Our fleet of exceptional aircraft is growing, as is our amazing team. We have introduced new software and advanced training that supports our team's expansion as we continue to elevate the private travel experience of our Fractional Owners," said Jared Williams, AirSprint's Vice President, Operations & COO. "At AirSprint, the acceptance of a new jet involves every department in the company, from sales to finance and flight operations to maintenance. It has been a busy and exciting day."
About AirSprint
AirSprint Private Aviation is a privately-held company with offices in Toronto, Montréal and Calgary. AirSprint maintains the largest fractional fleet of private aircraft in Canada, a jet collection of Embraer Praetor 500s, Embraer Legacy 450/500s, Cessna Citations CJ3+ and Cessna Citations CJ2+. AirSprint proudly flies Canadians from coast-to-coast, including service from Vancouver, Calgary, Edmonton, Winnipeg, Toronto, Ottawa, Montréal and the Maritimes. AirSprint provides discerning Canadians with a better choice for optimizing their time by enhancing the private jet ownership experience with industry-leading safety standards, exceptional turn-key service and increased flexibility; everything personalized for the Owners' individual travel needs. All at a fraction of the cost. AirSprint.com | {
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Academic Understanding
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Richard Green Rate My Professor
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Effective immediately and continuing through approximately May 2019, the 8th Street Bridge in Passaic/Wallington will be closed for reconstruction work. As a result, NJ TRANSIT Bus Route No. 707 (Paterson-Paramus) will operate on a detour.
Should expect service impacts within the vicinity of the construction work including Passaic, Wallington and Garfield.
Customers are advised to plan accordingly and to allow for extra travel time.
The following charts indicate the bus stopsthat WILL NOT be served during this detour as well as the alternate bus stops. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,567 |
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The LA Coliseum track has drawn the most comparisons to the ancient Bowman Gray Stadium in Winston-Salem, N.C. | {
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Albert Watson ist der Name folgender Personen:
* Albert Watson II (1909–1993), US-amerikanischer Generalleutnant, Stadtkommandant von Berlin
Albert Watson (Fotograf) (* 1942), britischer Fotograf
Albert William Watson (1922–1994), amerikanischer Politiker | {
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{"url":"http:\/\/mathhelpforum.com\/statistics\/216097-probability-dice.html","text":"# Math Help - Probability with a dice\n\n1. ## Probability with a dice\n\nI throw a dice 6 times. What are the odds of getting higher than 4 on at least 5 throws?\n\nCan I write this as: $\\left(\\frac{2}{6}\\right)^5+\\frac{2}{6}$ ?\n\nIn other words..The odds of getting a 5 or 6, 5 times in a row..Plus the probability of getting it in the 6th throw?\n\n2. ## Re: Probability with a dice\n\nSo getting higher than 4 on 5 or 6 throws? Yes, the probability of getting 5 or 6 on one through is 2\/6= 1\/3 so the probability of throwing 5 or 6 6 consecutive times is $(1\/3)^6$. The probabilty of get 5 or 6 exactly 5 times is $6(1\/3)^5(2\/3)$. The probabilty of either of those happening is the sum, $(1\/3)^6+ 6(1\/3)^5$.\n\nNotice the \"6\" in that? That is because there are 6 different orders: YYYYYN, YYYYNY, YYYNYY, YYNYYY, YNYYYY, and NYYYY where \"Y\" means a 5 or 6 and \"N\" means not a 5 or 6.\n\n(And, by the way, the word \"dices\" is a verb meaning \"cuts something into small cubes\". The word you want is \"dice\" which is itself the plural of the word \"die\".)\n\n3. ## Re: Probability with a dice\n\nHi:\n\nYou want to use the binomial distribution for this problem. Are you familiar with that distribution?\n\nHoward Heller\nInteractiveMathTutor.com\n\n4. ## Re: Probability with a dice\n\nOriginally Posted by Paze\nI throw a dice 6 times. What are the odds of getting higher than 4 on at least 5 throws?\nCan I write this as: $\\left(\\frac{2}{6}\\right)^5+\\frac{2}{6}$ ?\nIn other words..The odds of getting a 5 or 6, 5 times in a row..Plus the probability of getting it in the 6th throw?\nI think that there is a language(translation) difficulty here.\n\nFirst I do not like the word odds and never use it. So I change it to probability.\n\nThus \"What are the probability of getting higher than 4 on at least 5 throws?\"\nThat means \"getting a five or six at least five times out of six throws\".\n\nIf you agree that is what it means, then the answer is:\n$\\binom{6}{5}\\left( {\\frac{2}{6}} \\right)^5}\\left( {\\frac{4}{6}} \\right) + {\\left( {\\frac{2}{6}} \\right)^6$\n\n5. ## Re: Probability with a dice\n\nThat is correct now.\n\nHoward Heller\nInteractiveMathTutor.com\n\n6. ## Re: Probability with a dice\n\nOriginally Posted by IMTinstructor\nThat is correct now.\n\nHoward Heller\nInteractiveMathTutor.com\nSomeone of your low IQ would not know if that is correct or not.\nAnyone stupid enough to fall for you scam deserves the outcome.\nI can find no entry for you at MathGenealogy Project\n\nSo what equips you to tutor?\n\n7. ## Re: Probability with a dice\n\nMath major, math expert, actuarial science major, MBA\n\n8. ## Re: Probability with a dice\n\nOriginally Posted by IMTinstructor\nMath major, math expert, actuarial science major, MBA\nOh my goodness. Do you really think that has anything to do with mathematics?\nWhat does \"Math major\" mean? \"actuarial science major\" has very little to do with mathematics.\n\nI live in a state in which it is illegal to call oneself a \"blank\" if one does not have a PhD in \"blank\".\nI can thank my brother for that, \"educational psychologist is not a psychologist\".\n\nDo you have a PhD?\n\n9. ## Re: Probability with a dice\n\nThanks guys. This helped me understand my problem! On to the next...\n\n10. ## Re: Probability with a dice\n\nOriginally Posted by Plato\nI think that there is a language(translation) difficulty here.\n\nFirst I do not like the word odds and never use it. So I change it to probability.\n\nThus \"What are the probability of getting higher than 4 on at least 5 throws?\"\nThat means \"getting a five or six at least five times out of six throws\".\n\nIf you agree that is what it means, then the answer is:\n$\\binom{6}{5}\\left( {\\frac{2}{6}} \\right)^5}\\left( {\\frac{4}{6}} \\right) + {\\left( {\\frac{2}{6}} \\right)^6$\nOr wait, hold on. Your answers differs from Halls's answer, doesn't it? I got the same answer as HallsOfIvy. Where does the 4\/6 come from?","date":"2014-12-18 00:28:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 7, \"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\": 0.7712875008583069, \"perplexity\": 1256.4676377895307}, \"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-2014-52\/segments\/1418802765002.8\/warc\/CC-MAIN-20141217075245-00043-ip-10-231-17-201.ec2.internal.warc.gz\"}"} | null | null |
Сіверя́ни () — союз східнослов'янських племен, які жили в басейні річки Десни та над течіями річок Сейму, Сули, Псла і Ворскли. Локалізовані на лівому березі Дніпра в нинішній Чернігівщині, Сумщині й Полтавщині, Брянщині та Курщині. На правому березі Десни вони мешкали до річки Снов, а далі вже були землі радимичів.
Головні міста сіверян: Чернігів, Новгород-Сіверський, Брянськ, Стародуб, Глухів, Путивль, Курськ, Рильськ, Любеч, Переяслав. З заходу сіверяни межували з полянами і дреговичами, з півночі з радимичами, кривичами і в'ятичами, на півдні з уличами.
Етимологія
«Повість временних літ» зафіксувала не один, а кілька споріднених етнонімів: «сѣверъ», «сѣверо», «сѣверы», «сѣверѧне»; останній найчастіше використовувався у фаховій літературі та був загальновизнаним наприкінці 1990-х-2000-х.
Згідно версії В. Седова формування історичних сіверян відбулося в процесі взаємодії прийшлого ранньослов'янського населення з автохтонами Дніпровського Лівобережжя — балто- й іраномовними племенами. В зв'язку з цим виникла гіпотеза, що спочатку назву «сѣверъ» мала локальна група населення, яка згодом була асимільована слов'янами, котрі перейняли й вихідний етнонім. Їй суперечить той факт, що етнонім «сѣверъ» відомий і на нижньому Дунаї, в Мезії. Місто Севеж () столиця Севежського князівства відоме в СілезіЇ. Окрім того, в топонімії Польщі представлені Siewiersk, Siewierska Góra, Małe/Wielkie Siewieruszki. У сучасній лінгвістиці вважається що термін «сѣверъ», як і низка інших племінних назв «Повісті временних літ» («поля», «дерева» тощо), є одним з найбільш архаїчних слов'янських етнонімів, котрі сформувались у період, що передував розпаду праслов'янської спільноти (VI ст. н. е.) та початку міграцій представників окремих груп слов'янства.
Історія
Згідно літопису локалізуються «по Десне и по Семи, по Суле». Це підтверджується даними археології: окреслена область в основному збігається з ареалом роменської культури (VIII—XII ст.), слов'янська атрибуція якої не викликає сумнівів у фахівців.
Під час слов'янської колонізації розселялися із заходу на схід, через Понемання, потім Подвіння, ймовірно, разом з кривичами, досягли верхів'їв Двіни, Волги, Дніпра, звідти вийшли на постійний притулок на Десні, Сеймі, Сулі.
У IX ст.-на поч. X ст платили данину хозарам, від 884 східна частина керувалася з Києва: «иде Олегъ на сѣверяне, и побѣди сѣверяны, и възложи на нь дань легьку». 907 року, за князя Олега (882–912 рр.), були повністю приєднані до Київської Русі. У IX — поч. X ст. брали участь у поході на греків.
Княжий стіл на цих землях виник опісля конфлікту між синами Володимира Святославича — Ярославом, який князював у Києві, й Мстиславом Тмутороканським, котрий 1024 р. захопив Чернігів і переміг брата в битві під Лиственом.
Опісля 1024 р. назва «сіверяни» зникає з історичних джерел; вони влилися до складу українського народу, однак залишилася назва Сіверського князівства.
Востаннє сіверяни згадуються літописом під таким племінним ім'ям у 1183 році.
У джерелах XVI—XVII ст. в басейні р. Сейму згадується етнографічна група севрюків, яких дехто вважає безпосередніми нащадками сіверян VIII-ХІІ ст. Від цього ж етноніму походять деякі географічні назви Лівобережної України: Сіверія, Новгород-Сіверський, Сіверський Донець, Сіверська земля, Сіверщина тощо.
Археологія
Археологи вирізняють їхні кургани за спіралевидними скроневими кільцями. Разом з тим, вони багато в чому нагадують поховання Правобережжя Дніпра (простота костюму, півтораобертові перстеневі кільця). Археологічним відповідником сіверян є волинцівська з черняхівською культурою, роменська археологічні культури лісостепового Лівобережжя Дніпра VIII-ХІІ ст.. Певний вплив на сіверян справила алано-болгарська салтово-маяцька культура басейну Сіверського Дінця (історичні хозари).
Суспільство
Головним заняттям сіверян було хліборобство, скотарство, мисливство і рибальство.
Примітки
Джерела та література
Плахонін А.Г. Сіверяни //
«Літопис Руський», м. Київ, вид. «Дніпро», 1989 р., 591 с. — ISBN 5-308-00052-2
Стецюк Валентин. «Дослідження передісторичних етногенетичних процесів у Східній Європі». Розділ IX. Східні слов'яни.
Горленко В. П. Литвини півночі України — ймовірний уламок нащадків племені літописних сіверян
Голубовський П. В. Сиверцы — Сивер — Северяне
Посилання
Східнослов'янські племена
Середньовічні слов'яни України
Сіверщина
Волинцівська культура
Роменська культура | {
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Q: RecyclerView и Button проблема есть xml разметка . в RecyclerViewя загружаю данные с апи(список товаров)
проблема в том что после того как данные подгружаються то кнопка "Next>" пропадает куда то. не могу понть где ошибка и в чем проблема
<LinearLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical">
<LinearLayout
android:layout_width="match_parent"
android:layout_height="wrap_content"
android:orientation="horizontal">
<EditText
android:id="@+id/search_edit_text"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="4.25" />
<Spinner
android:id="@+id/category_spinner"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="1" />
<Button
android:id="@+id/submit_button"
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="1"
android:text="Submit" />
</LinearLayout>
<android.support.v7.widget.RecyclerView
android:id="@+id/recycler_conteiner"
android:layout_width="match_parent"
android:layout_height="wrap_content" />
<Button
android:id="@+id/next_page"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:text="Next >" />
</LinearLayout>
A: Кнопка у вас не помещается на экран. Установите высоту для RecyclerView так, чтобы он занимал всё место между верхними и нижними кнопками. Например так, через вес:
<android.support.v7.widget.RecyclerView
android:id="@+id/recycler_conteiner"
android:layout_width="match_parent"
android:layout_weight="1"
android:layout_height="0dp" />
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Doty ready to return for Missouri women's basketball
Missouri guard Lianna Doty dribbles past a defender during the game against Western Illinois on Dec. 22,2013, at Mizzou Arena.
Quint Smith
Joyce Peng
COLUMBIA — Last season, Lianna Doty didn't contribute on the court. She didn't play a single minute. She didn't give out assists or passes.
The redshirt junior suffered a Lisfranc injury, a fracture in the middle of the foot, during a 2014 preseason practice, which forced her to miss the entire 2014-15 season. As the team's starting point guard for the 2013-14 campaign, Doty averaged 5.1 points per game and 5.6 assists per game.
But even though she sat out, she gave to her team in a different way. Doty gave her teammates confidence.
Because of Doty's injury, the point guard position went to several other players instead, including Lindsey Cunningham, who saw the most minutes in the position.
Cunningham said Doty often reminded Cunningham of her strengths on the floor. Cunningham, who had limited college playing experience prior to Doty's injury, appreciated the encouragement coming from a fellow point guard.
"It just makes you feel good about yourself and makes you go out there and play a little bit more free," she said.
Doty's compliments boosted Cunningham's confidence, reassured her that her actions on the court were right and soothed her nerves before games.
Doty knows how to uplift people with her words.
She would smile at guard Sierra Michaelis, who always got down on herself, and say, "'We're playing basketball. That's what we love to do.'"
When foreward Michelle Hudyn bemoaned making mistakes during games, Doty would encourage her.
"Next play, next play," Doty told her.
During practice, Doty offered to help Hudyn on ball handling, an area in which the forward strives to improve. She also gave Hudyn specific pointers on how to improve her game, such as reminding her to come to the basket for a dump-off pass for the layup instead of fading away from the basket, as Hudyn liked to do.
Doty was able to learn about the game even when she was unable to play it.
She saw how often players got emotionally overwhelmed on the court, thus hampering their performance.
"Just being able to really step back and take a deep breath," Doty said as a way to calm down. She adds that it's crucial to be in a position where one isn't exhausted and can see the game with a clear mind.
Doty said that the foot injury was an opportunity to improve in different ways. She said she hopes to be, among other things, a better floor general, facilitator, shooter and leader.
Doty received a medical redshirt last season because of her injury and spent about four months on crutches. During rehabilitation, she had to learn how to walk, jog and run again. Doty said this week that she felt good on the court.
"I'm feeling fantastic," she said. "Moving faster than ever, I'm stronger than ever."
Despite starting as point guard in the 2013-14 season, Doty might or might not start this season.
She said being injured was frustrating and hard on her but added that it's sweet to be back.
"It's more of a joy being on the court right now than it ever has been in my life," she said.
Supervising editor is Christian Clark.
From the court to the stage and back: The journey of Michael Porter Sr.
Missouri women's basketball picked No. 7 in SEC media poll
Missouri women's basketball Unleashed at homecoming parade
Pingeton signs new five-year contract with Missouri
Mizzou's Pingeton wants 5,000 fans at each home game
Lianna Doty
Missouri Women's Basketball
Lindsey Cunningham
Michelle Hudyn
Sierra Michaelis
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Missourian women's basketball beat reporter, fall/winter 2015, spring 2016 Studying sports journalism Reach me at jpckf@mail.missouri.edu or in the newsroom at 882-5720
Pete Bland
blandp@missouri.edu
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#backend a[href^="#"]:after {
content: "";
}
#backend pre,
#backend blockquote {
border: 1px solid #999;
page-break-inside: avoid;
}
#backend thead {
display: table-header-group;
}
#backend tr,
#backend img {
page-break-inside: avoid;
}
#backend img {
max-width: 100% !important;
}
@page {
margin: 0.5cm;
}
#backend p,
#backend h2,
#backend h3 {
orphans: 3;
widows: 3;
}
#backend h2,
#backend h3 {
page-break-after: avoid;
}
}
#backend body {
margin: 0;
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
font-size: 13px;
line-height: 18px;
color: #333333;
background-color: #ffffff;
}
#backend a {
color: #0088cc;
text-decoration: none;
}
#backend a:hover,
#backend a:focus {
color: #005580;
text-decoration: underline;
}
#backend .img-rounded {
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
}
#backend .img-polaroid {
padding: 4px;
background-color: #fff;
border: 1px solid #ccc;
border: 1px solid rgba(0, 0, 0, 0.2);
-webkit-box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1);
-moz-box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1);
box-shadow: 0 1px 3px rgba(0, 0, 0, 0.1);
}
#backend .img-circle {
-webkit-border-radius: 500px;
-moz-border-radius: 500px;
border-radius: 500px;
}
#backend .row {
margin-left: -20px;
*zoom: 1;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend [class*="span"] {
float: left;
min-height: 1px;
margin-left: 20px;
}
#backend .container,
#backend .navbar-static-top .container,
#backend .navbar-fixed-top .container,
#backend .navbar-fixed-bottom .container {
width: 940px;
}
#backend .span12 {
width: 940px;
}
#backend .span11 {
width: 860px;
}
#backend .span10 {
width: 780px;
}
#backend .span9 {
width: 700px;
}
#backend .span8 {
width: 620px;
}
#backend .span7 {
width: 540px;
}
#backend .span6 {
width: 460px;
}
#backend .span5 {
width: 380px;
}
#backend .span4 {
width: 300px;
}
#backend .span3 {
width: 220px;
}
#backend .span2 {
width: 140px;
}
#backend .span1 {
width: 60px;
}
#backend .offset12 {
margin-left: 980px;
}
#backend .offset11 {
margin-left: 900px;
}
#backend .offset10 {
margin-left: 820px;
}
#backend .offset9 {
margin-left: 740px;
}
#backend .offset8 {
margin-left: 660px;
}
#backend .offset7 {
margin-left: 580px;
}
#backend .offset6 {
margin-left: 500px;
}
#backend .offset5 {
margin-left: 420px;
}
#backend .offset4 {
margin-left: 340px;
}
#backend .offset3 {
margin-left: 260px;
}
#backend .offset2 {
margin-left: 180px;
}
#backend .offset1 {
margin-left: 100px;
}
#backend .row {
margin-left: -20px;
*zoom: 1;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend [class*="span"] {
float: left;
min-height: 1px;
margin-left: 20px;
}
#backend .container,
#backend .navbar-static-top .container,
#backend .navbar-fixed-top .container,
#backend .navbar-fixed-bottom .container {
width: 940px;
}
#backend .span12 {
width: 940px;
}
#backend .span11 {
width: 860px;
}
#backend .span10 {
width: 780px;
}
#backend .span9 {
width: 700px;
}
#backend .span8 {
width: 620px;
}
#backend .span7 {
width: 540px;
}
#backend .span6 {
width: 460px;
}
#backend .span5 {
width: 380px;
}
#backend .span4 {
width: 300px;
}
#backend .span3 {
width: 220px;
}
#backend .span2 {
width: 140px;
}
#backend .span1 {
width: 60px;
}
#backend .offset12 {
margin-left: 980px;
}
#backend .offset11 {
margin-left: 900px;
}
#backend .offset10 {
margin-left: 820px;
}
#backend .offset9 {
margin-left: 740px;
}
#backend .offset8 {
margin-left: 660px;
}
#backend .offset7 {
margin-left: 580px;
}
#backend .offset6 {
margin-left: 500px;
}
#backend .offset5 {
margin-left: 420px;
}
#backend .offset4 {
margin-left: 340px;
}
#backend .offset3 {
margin-left: 260px;
}
#backend .offset2 {
margin-left: 180px;
}
#backend .offset1 {
margin-left: 100px;
}
#backend .row-fluid {
width: 100%;
*zoom: 1;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid [class*="span"] {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
float: left;
margin-left: 2.127659574%;
*margin-left: 2.0744680846382977%;
}
#backend .row-fluid [class*="span"]:first-child {
margin-left: 0;
}
#backend .row-fluid .controls-row [class*="span"] + [class*="span"] {
margin-left: 2.127659574%;
}
#backend .row-fluid .span12 {
width: 99.99999998999999%;
*width: 99.94680850063828%;
}
#backend .row-fluid .span11 {
width: 91.489361693%;
*width: 91.4361702036383%;
}
#backend .row-fluid .span10 {
width: 82.97872339599999%;
*width: 82.92553190663828%;
}
#backend .row-fluid .span9 {
width: 74.468085099%;
*width: 74.4148936096383%;
}
#backend .row-fluid .span8 {
width: 65.95744680199999%;
*width: 65.90425531263828%;
}
#backend .row-fluid .span7 {
width: 57.446808505%;
*width: 57.3936170156383%;
}
#backend .row-fluid .span6 {
width: 48.93617020799999%;
*width: 48.88297871863829%;
}
#backend .row-fluid .span5 {
width: 40.425531911%;
*width: 40.3723404216383%;
}
#backend .row-fluid .span4 {
width: 31.914893614%;
*width: 31.8617021246383%;
}
#backend .row-fluid .span3 {
width: 23.404255317%;
*width: 23.3510638276383%;
}
#backend .row-fluid .span2 {
width: 14.89361702%;
*width: 14.8404255306383%;
}
#backend .row-fluid .span1 {
width: 6.382978723%;
*width: 6.329787233638298%;
}
#backend .row-fluid .offset12 {
margin-left: 104.25531913799999%;
*margin-left: 104.14893615927657%;
}
#backend .row-fluid .offset12:first-child {
margin-left: 102.127659564%;
*margin-left: 102.02127658527658%;
}
#backend .row-fluid .offset11 {
margin-left: 95.744680841%;
*margin-left: 95.63829786227659%;
}
#backend .row-fluid .offset11:first-child {
margin-left: 93.61702126700001%;
*margin-left: 93.5106382882766%;
}
#backend .row-fluid .offset10 {
margin-left: 87.23404254399999%;
*margin-left: 87.12765956527657%;
}
#backend .row-fluid .offset10:first-child {
margin-left: 85.10638297%;
*margin-left: 84.99999999127658%;
}
#backend .row-fluid .offset9 {
margin-left: 78.723404247%;
*margin-left: 78.61702126827659%;
}
#backend .row-fluid .offset9:first-child {
margin-left: 76.59574467300001%;
*margin-left: 76.4893616942766%;
}
#backend .row-fluid .offset8 {
margin-left: 70.21276594999999%;
*margin-left: 70.10638297127657%;
}
#backend .row-fluid .offset8:first-child {
margin-left: 68.085106376%;
*margin-left: 67.97872339727658%;
}
#backend .row-fluid .offset7 {
margin-left: 61.702127653%;
*margin-left: 61.595744674276595%;
}
#backend .row-fluid .offset7:first-child {
margin-left: 59.574468079%;
*margin-left: 59.468085100276596%;
}
#backend .row-fluid .offset6 {
margin-left: 53.19148935599999%;
*margin-left: 53.08510637727659%;
}
#backend .row-fluid .offset6:first-child {
margin-left: 51.06382978199999%;
*margin-left: 50.95744680327659%;
}
#backend .row-fluid .offset5 {
margin-left: 44.680851059%;
*margin-left: 44.574468080276596%;
}
#backend .row-fluid .offset5:first-child {
margin-left: 42.553191485%;
*margin-left: 42.4468085062766%;
}
#backend .row-fluid .offset4 {
margin-left: 36.170212762%;
*margin-left: 36.063829783276596%;
}
#backend .row-fluid .offset4:first-child {
margin-left: 34.042553188%;
*margin-left: 33.9361702092766%;
}
#backend .row-fluid .offset3 {
margin-left: 27.659574465%;
*margin-left: 27.553191486276596%;
}
#backend .row-fluid .offset3:first-child {
margin-left: 25.531914891%;
*margin-left: 25.425531912276597%;
}
#backend .row-fluid .offset2 {
margin-left: 19.148936168%;
*margin-left: 19.042553189276596%;
}
#backend .row-fluid .offset2:first-child {
margin-left: 17.021276594%;
*margin-left: 16.914893615276597%;
}
#backend .row-fluid .offset1 {
margin-left: 10.638297870999999%;
*margin-left: 10.531914892276596%;
}
#backend .row-fluid .offset1:first-child {
margin-left: 8.510638297%;
*margin-left: 8.404255318276597%;
}
#backend .row-fluid {
width: 100%;
*zoom: 1;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid [class*="span"] {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
float: left;
margin-left: 2.127659574%;
*margin-left: 2.0744680846382977%;
}
#backend .row-fluid [class*="span"]:first-child {
margin-left: 0;
}
#backend .row-fluid .controls-row [class*="span"] + [class*="span"] {
margin-left: 2.127659574%;
}
#backend .row-fluid .span12 {
width: 99.99999998999999%;
*width: 99.94680850063828%;
}
#backend .row-fluid .span11 {
width: 91.489361693%;
*width: 91.4361702036383%;
}
#backend .row-fluid .span10 {
width: 82.97872339599999%;
*width: 82.92553190663828%;
}
#backend .row-fluid .span9 {
width: 74.468085099%;
*width: 74.4148936096383%;
}
#backend .row-fluid .span8 {
width: 65.95744680199999%;
*width: 65.90425531263828%;
}
#backend .row-fluid .span7 {
width: 57.446808505%;
*width: 57.3936170156383%;
}
#backend .row-fluid .span6 {
width: 48.93617020799999%;
*width: 48.88297871863829%;
}
#backend .row-fluid .span5 {
width: 40.425531911%;
*width: 40.3723404216383%;
}
#backend .row-fluid .span4 {
width: 31.914893614%;
*width: 31.8617021246383%;
}
#backend .row-fluid .span3 {
width: 23.404255317%;
*width: 23.3510638276383%;
}
#backend .row-fluid .span2 {
width: 14.89361702%;
*width: 14.8404255306383%;
}
#backend .row-fluid .span1 {
width: 6.382978723%;
*width: 6.329787233638298%;
}
#backend .row-fluid .offset12 {
margin-left: 104.25531913799999%;
*margin-left: 104.14893615927657%;
}
#backend .row-fluid .offset12:first-child {
margin-left: 102.127659564%;
*margin-left: 102.02127658527658%;
}
#backend .row-fluid .offset11 {
margin-left: 95.744680841%;
*margin-left: 95.63829786227659%;
}
#backend .row-fluid .offset11:first-child {
margin-left: 93.61702126700001%;
*margin-left: 93.5106382882766%;
}
#backend .row-fluid .offset10 {
margin-left: 87.23404254399999%;
*margin-left: 87.12765956527657%;
}
#backend .row-fluid .offset10:first-child {
margin-left: 85.10638297%;
*margin-left: 84.99999999127658%;
}
#backend .row-fluid .offset9 {
margin-left: 78.723404247%;
*margin-left: 78.61702126827659%;
}
#backend .row-fluid .offset9:first-child {
margin-left: 76.59574467300001%;
*margin-left: 76.4893616942766%;
}
#backend .row-fluid .offset8 {
margin-left: 70.21276594999999%;
*margin-left: 70.10638297127657%;
}
#backend .row-fluid .offset8:first-child {
margin-left: 68.085106376%;
*margin-left: 67.97872339727658%;
}
#backend .row-fluid .offset7 {
margin-left: 61.702127653%;
*margin-left: 61.595744674276595%;
}
#backend .row-fluid .offset7:first-child {
margin-left: 59.574468079%;
*margin-left: 59.468085100276596%;
}
#backend .row-fluid .offset6 {
margin-left: 53.19148935599999%;
*margin-left: 53.08510637727659%;
}
#backend .row-fluid .offset6:first-child {
margin-left: 51.06382978199999%;
*margin-left: 50.95744680327659%;
}
#backend .row-fluid .offset5 {
margin-left: 44.680851059%;
*margin-left: 44.574468080276596%;
}
#backend .row-fluid .offset5:first-child {
margin-left: 42.553191485%;
*margin-left: 42.4468085062766%;
}
#backend .row-fluid .offset4 {
margin-left: 36.170212762%;
*margin-left: 36.063829783276596%;
}
#backend .row-fluid .offset4:first-child {
margin-left: 34.042553188%;
*margin-left: 33.9361702092766%;
}
#backend .row-fluid .offset3 {
margin-left: 27.659574465%;
*margin-left: 27.553191486276596%;
}
#backend .row-fluid .offset3:first-child {
margin-left: 25.531914891%;
*margin-left: 25.425531912276597%;
}
#backend .row-fluid .offset2 {
margin-left: 19.148936168%;
*margin-left: 19.042553189276596%;
}
#backend .row-fluid .offset2:first-child {
margin-left: 17.021276594%;
*margin-left: 16.914893615276597%;
}
#backend .row-fluid .offset1 {
margin-left: 10.638297870999999%;
*margin-left: 10.531914892276596%;
}
#backend .row-fluid .offset1:first-child {
margin-left: 8.510638297%;
*margin-left: 8.404255318276597%;
}
#backend [class*="span"].hide,
#backend .row-fluid [class*="span"].hide {
display: none;
}
#backend [class*="span"].pull-right,
#backend .row-fluid [class*="span"].pull-right {
float: right;
}
#backend .container {
margin-right: auto;
margin-left: auto;
*zoom: 1;
}
#backend .container:before,
#backend .container:after {
display: table;
content: "";
line-height: 0;
}
#backend .container:after {
clear: both;
}
#backend .container:before,
#backend .container:after {
display: table;
content: "";
line-height: 0;
}
#backend .container:after {
clear: both;
}
#backend .container:before,
#backend .container:after {
display: table;
content: "";
line-height: 0;
}
#backend .container:after {
clear: both;
}
#backend .container:before,
#backend .container:after {
display: table;
content: "";
line-height: 0;
}
#backend .container:after {
clear: both;
}
#backend .container-fluid {
padding-right: 20px;
padding-left: 20px;
*zoom: 1;
}
#backend .container-fluid:before,
#backend .container-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .container-fluid:after {
clear: both;
}
#backend .container-fluid:before,
#backend .container-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .container-fluid:after {
clear: both;
}
#backend p {
margin: 0 0 9px;
}
#backend .lead {
margin-bottom: 18px;
font-size: 19.5px;
font-weight: 200;
line-height: 27px;
}
#backend small {
font-size: 85%;
}
#backend strong {
font-weight: bold;
}
#backend em {
font-style: italic;
}
#backend cite {
font-style: normal;
}
#backend .muted {
color: #999999;
}
#backend a.muted:hover,
#backend a.muted:focus {
color: #808080;
}
#backend .text-warning {
color: #c09853;
}
#backend a.text-warning:hover,
#backend a.text-warning:focus {
color: #a47e3c;
}
#backend .text-error {
color: #b94a48;
}
#backend a.text-error:hover,
#backend a.text-error:focus {
color: #953b39;
}
#backend .text-info {
color: #3a87ad;
}
#backend a.text-info:hover,
#backend a.text-info:focus {
color: #2d6987;
}
#backend .text-success {
color: #468847;
}
#backend a.text-success:hover,
#backend a.text-success:focus {
color: #356635;
}
#backend .text-left {
text-align: left;
}
#backend .text-right {
text-align: right;
}
#backend .text-center {
text-align: center;
}
#backend h1,
#backend h2,
#backend h3,
#backend h4,
#backend h5,
#backend h6 {
margin: 9px 0;
font-family: inherit;
font-weight: bold;
line-height: 18px;
color: inherit;
text-rendering: optimizelegibility;
}
#backend h1 small,
#backend h2 small,
#backend h3 small,
#backend h4 small,
#backend h5 small,
#backend h6 small {
font-weight: normal;
line-height: 1;
color: #999999;
}
#backend h1,
#backend h2,
#backend h3 {
line-height: 36px;
}
#backend h1 {
font-size: 35.75px;
}
#backend h2 {
font-size: 29.25px;
}
#backend h3 {
font-size: 22.75px;
}
#backend h4 {
font-size: 16.25px;
}
#backend h5 {
font-size: 13px;
}
#backend h6 {
font-size: 11.049999999999999px;
}
#backend h1 small {
font-size: 22.75px;
}
#backend h2 small {
font-size: 16.25px;
}
#backend h3 small {
font-size: 13px;
}
#backend h4 small {
font-size: 13px;
}
#backend .page-header {
padding-bottom: 8px;
margin: 18px 0 27px;
border-bottom: 1px solid #eeeeee;
}
#backend ul,
#backend ol {
padding: 0;
margin: 0 0 9px 25px;
}
#backend ul ul,
#backend ul ol,
#backend ol ol,
#backend ol ul {
margin-bottom: 0;
}
#backend li {
line-height: 18px;
}
#backend ul.unstyled,
#backend ol.unstyled {
margin-left: 0;
list-style: none;
}
#backend ul.inline,
#backend ol.inline {
margin-left: 0;
list-style: none;
}
#backend ul.inline > li,
#backend ol.inline > li {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
padding-left: 5px;
padding-right: 5px;
}
#backend dl {
margin-bottom: 18px;
}
#backend dt,
#backend dd {
line-height: 18px;
}
#backend dt {
font-weight: bold;
}
#backend dd {
margin-left: 9px;
}
#backend .dl-horizontal {
*zoom: 1;
}
#backend .dl-horizontal:before,
#backend .dl-horizontal:after {
display: table;
content: "";
line-height: 0;
}
#backend .dl-horizontal:after {
clear: both;
}
#backend .dl-horizontal:before,
#backend .dl-horizontal:after {
display: table;
content: "";
line-height: 0;
}
#backend .dl-horizontal:after {
clear: both;
}
#backend .dl-horizontal dt {
float: left;
width: 160px;
clear: left;
text-align: right;
overflow: hidden;
text-overflow: ellipsis;
white-space: nowrap;
}
#backend .dl-horizontal dd {
margin-left: 180px;
}
#backend hr {
margin: 18px 0;
border: 0;
border-top: 1px solid #eeeeee;
border-bottom: 1px solid #ffffff;
}
#backend abbr[title],
#backend abbr[data-original-title] {
cursor: help;
border-bottom: 1px dotted #999999;
}
#backend abbr.initialism {
font-size: 90%;
text-transform: uppercase;
}
#backend blockquote {
padding: 0 0 0 15px;
margin: 0 0 18px;
border-left: 5px solid #eeeeee;
}
#backend blockquote p {
margin-bottom: 0;
font-size: 16.25px;
font-weight: 300;
line-height: 1.25;
}
#backend blockquote small {
display: block;
line-height: 18px;
color: #999999;
}
#backend blockquote small:before {
content: '\2014 \00A0';
}
#backend blockquote.pull-right {
float: right;
padding-right: 15px;
padding-left: 0;
border-right: 5px solid #eeeeee;
border-left: 0;
}
#backend blockquote.pull-right p,
#backend blockquote.pull-right small {
text-align: right;
}
#backend blockquote.pull-right small:before {
content: '';
}
#backend blockquote.pull-right small:after {
content: '\00A0 \2014';
}
#backend q:before,
#backend q:after,
#backend blockquote:before,
#backend blockquote:after {
content: "";
}
#backend address {
display: block;
margin-bottom: 18px;
font-style: normal;
line-height: 18px;
}
#backend code,
#backend pre {
padding: 0 3px 2px;
font-family: Menlo, Monaco, Consolas, "Courier New", monospace;
font-size: 11px;
color: #333333;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
}
#backend code {
padding: 2px 4px;
color: #d14;
background-color: #f7f7f9;
border: 1px solid #e1e1e8;
white-space: nowrap;
}
#backend pre {
display: block;
padding: 8.5px;
margin: 0 0 9px;
font-size: 12px;
line-height: 18px;
word-break: break-all;
word-wrap: break-word;
white-space: pre;
white-space: pre-wrap;
background-color: #f5f5f5;
border: 1px solid #ccc;
border: 1px solid rgba(0, 0, 0, 0.15);
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend pre.prettyprint {
margin-bottom: 18px;
}
#backend pre code {
padding: 0;
color: inherit;
white-space: pre;
white-space: pre-wrap;
background-color: transparent;
border: 0;
}
#backend .pre-scrollable {
max-height: 340px;
overflow-y: scroll;
}
#backend form {
margin: 0 0 18px;
}
#backend fieldset {
padding: 0;
margin: 0;
border: 0;
}
#backend legend {
display: block;
width: 100%;
padding: 0;
margin-bottom: 18px;
font-size: 19.5px;
line-height: 36px;
color: #333333;
border: 0;
border-bottom: 1px solid #e5e5e5;
}
#backend legend small {
font-size: 13.5px;
color: #999999;
}
#backend label,
#backend input,
#backend button,
#backend select,
#backend textarea {
font-size: 13px;
font-weight: normal;
line-height: 18px;
}
#backend input,
#backend button,
#backend select,
#backend textarea {
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
}
#backend label {
display: block;
margin-bottom: 5px;
}
#backend select,
#backend textarea,
#backend input[type="text"],
#backend input[type="password"],
#backend input[type="datetime"],
#backend input[type="datetime-local"],
#backend input[type="date"],
#backend input[type="month"],
#backend input[type="time"],
#backend input[type="week"],
#backend input[type="number"],
#backend input[type="email"],
#backend input[type="url"],
#backend input[type="search"],
#backend input[type="tel"],
#backend input[type="color"],
#backend .uneditable-input {
display: inline-block;
height: 18px;
padding: 4px 6px;
margin-bottom: 9px;
font-size: 13px;
line-height: 18px;
color: #555555;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
vertical-align: middle;
}
#backend input,
#backend textarea,
#backend .uneditable-input {
width: 206px;
}
#backend textarea {
height: auto;
}
#backend textarea,
#backend input[type="text"],
#backend input[type="password"],
#backend input[type="datetime"],
#backend input[type="datetime-local"],
#backend input[type="date"],
#backend input[type="month"],
#backend input[type="time"],
#backend input[type="week"],
#backend input[type="number"],
#backend input[type="email"],
#backend input[type="url"],
#backend input[type="search"],
#backend input[type="tel"],
#backend input[type="color"],
#backend .uneditable-input {
background-color: #ffffff;
border: 1px solid #cccccc;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-webkit-transition: border linear .2s, box-shadow linear .2s;
-moz-transition: border linear .2s, box-shadow linear .2s;
-o-transition: border linear .2s, box-shadow linear .2s;
transition: border linear .2s, box-shadow linear .2s;
}
#backend textarea:focus,
#backend input[type="text"]:focus,
#backend input[type="password"]:focus,
#backend input[type="datetime"]:focus,
#backend input[type="datetime-local"]:focus,
#backend input[type="date"]:focus,
#backend input[type="month"]:focus,
#backend input[type="time"]:focus,
#backend input[type="week"]:focus,
#backend input[type="number"]:focus,
#backend input[type="email"]:focus,
#backend input[type="url"]:focus,
#backend input[type="search"]:focus,
#backend input[type="tel"]:focus,
#backend input[type="color"]:focus,
#backend .uneditable-input:focus {
border-color: rgba(82, 168, 236, 0.8);
outline: 0;
outline: thin dotted \9;
/* IE6-9 */
-webkit-box-shadow: inset 0 1px 1px rgba(0,0,0,.075), 0 0 8px rgba(82,168,236,.6);
-moz-box-shadow: inset 0 1px 1px rgba(0,0,0,.075), 0 0 8px rgba(82,168,236,.6);
box-shadow: inset 0 1px 1px rgba(0,0,0,.075), 0 0 8px rgba(82,168,236,.6);
}
#backend input[type="radio"],
#backend input[type="checkbox"] {
margin: 4px 0 0;
*margin-top: 0;
/* IE7 */
margin-top: 1px \9;
/* IE8-9 */
line-height: normal;
}
#backend input[type="file"],
#backend input[type="image"],
#backend input[type="submit"],
#backend input[type="reset"],
#backend input[type="button"],
#backend input[type="radio"],
#backend input[type="checkbox"] {
width: auto;
}
#backend select,
#backend input[type="file"] {
height: 28px;
/* In IE7, the height of the select element cannot be changed by height, only font-size */
*margin-top: 4px;
/* For IE7, add top margin to align select with labels */
line-height: 28px;
}
#backend select {
width: 220px;
border: 1px solid #cccccc;
background-color: #ffffff;
}
#backend select[multiple],
#backend select[size] {
height: auto;
}
#backend select:focus,
#backend input[type="file"]:focus,
#backend input[type="radio"]:focus,
#backend input[type="checkbox"]:focus {
outline: thin dotted #333;
outline: 5px auto -webkit-focus-ring-color;
outline-offset: -2px;
}
#backend .uneditable-input,
#backend .uneditable-textarea {
color: #999999;
background-color: #fcfcfc;
border-color: #cccccc;
-webkit-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.025);
-moz-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.025);
box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.025);
cursor: not-allowed;
}
#backend .uneditable-input {
overflow: hidden;
white-space: nowrap;
}
#backend .uneditable-textarea {
width: auto;
height: auto;
}
#backend input:-moz-placeholder,
#backend textarea:-moz-placeholder {
color: #999999;
}
#backend input:-ms-input-placeholder,
#backend textarea:-ms-input-placeholder {
color: #999999;
}
#backend input::-webkit-input-placeholder,
#backend textarea::-webkit-input-placeholder {
color: #999999;
}
#backend input:-moz-placeholder,
#backend textarea:-moz-placeholder {
color: #999999;
}
#backend input:-ms-input-placeholder,
#backend textarea:-ms-input-placeholder {
color: #999999;
}
#backend input::-webkit-input-placeholder,
#backend textarea::-webkit-input-placeholder {
color: #999999;
}
#backend .radio,
#backend .checkbox {
min-height: 18px;
padding-left: 20px;
}
#backend .radio input[type="radio"],
#backend .checkbox input[type="checkbox"] {
float: left;
margin-left: -20px;
}
#backend .controls > .radio:first-child,
#backend .controls > .checkbox:first-child {
padding-top: 5px;
}
#backend .radio.inline,
#backend .checkbox.inline {
display: inline-block;
padding-top: 5px;
margin-bottom: 0;
vertical-align: middle;
}
#backend .radio.inline + .radio.inline,
#backend .checkbox.inline + .checkbox.inline {
margin-left: 10px;
}
#backend .input-mini {
width: 60px;
}
#backend .input-small {
width: 90px;
}
#backend .input-medium {
width: 150px;
}
#backend .input-large {
width: 210px;
}
#backend .input-xlarge {
width: 270px;
}
#backend .input-xxlarge {
width: 530px;
}
#backend input[class*="span"],
#backend select[class*="span"],
#backend textarea[class*="span"],
#backend .uneditable-input[class*="span"],
#backend .row-fluid input[class*="span"],
#backend .row-fluid select[class*="span"],
#backend .row-fluid textarea[class*="span"],
#backend .row-fluid .uneditable-input[class*="span"] {
float: none;
margin-left: 0;
}
#backend .input-append input[class*="span"],
#backend .input-append .uneditable-input[class*="span"],
#backend .input-prepend input[class*="span"],
#backend .input-prepend .uneditable-input[class*="span"],
#backend .row-fluid input[class*="span"],
#backend .row-fluid select[class*="span"],
#backend .row-fluid textarea[class*="span"],
#backend .row-fluid .uneditable-input[class*="span"],
#backend .row-fluid .input-prepend [class*="span"],
#backend .row-fluid .input-append [class*="span"] {
display: inline-block;
}
#backend input,
#backend textarea,
#backend .uneditable-input {
margin-left: 0;
}
#backend .controls-row [class*="span"] + [class*="span"] {
margin-left: 20px;
}
#backend input.span12,
#backend textarea.span12,
#backend .uneditable-input.span12 {
width: 926px;
}
#backend input.span11,
#backend textarea.span11,
#backend .uneditable-input.span11 {
width: 846px;
}
#backend input.span10,
#backend textarea.span10,
#backend .uneditable-input.span10 {
width: 766px;
}
#backend input.span9,
#backend textarea.span9,
#backend .uneditable-input.span9 {
width: 686px;
}
#backend input.span8,
#backend textarea.span8,
#backend .uneditable-input.span8 {
width: 606px;
}
#backend input.span7,
#backend textarea.span7,
#backend .uneditable-input.span7 {
width: 526px;
}
#backend input.span6,
#backend textarea.span6,
#backend .uneditable-input.span6 {
width: 446px;
}
#backend input.span5,
#backend textarea.span5,
#backend .uneditable-input.span5 {
width: 366px;
}
#backend input.span4,
#backend textarea.span4,
#backend .uneditable-input.span4 {
width: 286px;
}
#backend input.span3,
#backend textarea.span3,
#backend .uneditable-input.span3 {
width: 206px;
}
#backend input.span2,
#backend textarea.span2,
#backend .uneditable-input.span2 {
width: 126px;
}
#backend input.span1,
#backend textarea.span1,
#backend .uneditable-input.span1 {
width: 46px;
}
#backend input,
#backend textarea,
#backend .uneditable-input {
margin-left: 0;
}
#backend .controls-row [class*="span"] + [class*="span"] {
margin-left: 20px;
}
#backend input.span12,
#backend textarea.span12,
#backend .uneditable-input.span12 {
width: 926px;
}
#backend input.span11,
#backend textarea.span11,
#backend .uneditable-input.span11 {
width: 846px;
}
#backend input.span10,
#backend textarea.span10,
#backend .uneditable-input.span10 {
width: 766px;
}
#backend input.span9,
#backend textarea.span9,
#backend .uneditable-input.span9 {
width: 686px;
}
#backend input.span8,
#backend textarea.span8,
#backend .uneditable-input.span8 {
width: 606px;
}
#backend input.span7,
#backend textarea.span7,
#backend .uneditable-input.span7 {
width: 526px;
}
#backend input.span6,
#backend textarea.span6,
#backend .uneditable-input.span6 {
width: 446px;
}
#backend input.span5,
#backend textarea.span5,
#backend .uneditable-input.span5 {
width: 366px;
}
#backend input.span4,
#backend textarea.span4,
#backend .uneditable-input.span4 {
width: 286px;
}
#backend input.span3,
#backend textarea.span3,
#backend .uneditable-input.span3 {
width: 206px;
}
#backend input.span2,
#backend textarea.span2,
#backend .uneditable-input.span2 {
width: 126px;
}
#backend input.span1,
#backend textarea.span1,
#backend .uneditable-input.span1 {
width: 46px;
}
#backend .controls-row {
*zoom: 1;
}
#backend .controls-row:before,
#backend .controls-row:after {
display: table;
content: "";
line-height: 0;
}
#backend .controls-row:after {
clear: both;
}
#backend .controls-row:before,
#backend .controls-row:after {
display: table;
content: "";
line-height: 0;
}
#backend .controls-row:after {
clear: both;
}
#backend .controls-row [class*="span"],
#backend .row-fluid .controls-row [class*="span"] {
float: left;
}
#backend .controls-row .checkbox[class*="span"],
#backend .controls-row .radio[class*="span"] {
padding-top: 5px;
}
#backend input[disabled],
#backend select[disabled],
#backend textarea[disabled],
#backend input[readonly],
#backend select[readonly],
#backend textarea[readonly] {
cursor: not-allowed;
background-color: #eeeeee;
}
#backend input[type="radio"][disabled],
#backend input[type="checkbox"][disabled],
#backend input[type="radio"][readonly],
#backend input[type="checkbox"][readonly] {
background-color: transparent;
}
#backend .control-group.warning .control-label,
#backend .control-group.warning .help-block,
#backend .control-group.warning .help-inline {
color: #c09853;
}
#backend .control-group.warning .checkbox,
#backend .control-group.warning .radio,
#backend .control-group.warning input,
#backend .control-group.warning select,
#backend .control-group.warning textarea {
color: #c09853;
}
#backend .control-group.warning input,
#backend .control-group.warning select,
#backend .control-group.warning textarea {
border-color: #c09853;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.warning input:focus,
#backend .control-group.warning select:focus,
#backend .control-group.warning textarea:focus {
border-color: #a47e3c;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #dbc59e;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #dbc59e;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #dbc59e;
}
#backend .control-group.warning .input-prepend .add-on,
#backend .control-group.warning .input-append .add-on {
color: #c09853;
background-color: #fcf8e3;
border-color: #c09853;
}
#backend .control-group.warning .control-label,
#backend .control-group.warning .help-block,
#backend .control-group.warning .help-inline {
color: #c09853;
}
#backend .control-group.warning .checkbox,
#backend .control-group.warning .radio,
#backend .control-group.warning input,
#backend .control-group.warning select,
#backend .control-group.warning textarea {
color: #c09853;
}
#backend .control-group.warning input,
#backend .control-group.warning select,
#backend .control-group.warning textarea {
border-color: #c09853;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.warning input:focus,
#backend .control-group.warning select:focus,
#backend .control-group.warning textarea:focus {
border-color: #a47e3c;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #dbc59e;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #dbc59e;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #dbc59e;
}
#backend .control-group.warning .input-prepend .add-on,
#backend .control-group.warning .input-append .add-on {
color: #c09853;
background-color: #fcf8e3;
border-color: #c09853;
}
#backend .control-group.error .control-label,
#backend .control-group.error .help-block,
#backend .control-group.error .help-inline {
color: #b94a48;
}
#backend .control-group.error .checkbox,
#backend .control-group.error .radio,
#backend .control-group.error input,
#backend .control-group.error select,
#backend .control-group.error textarea {
color: #b94a48;
}
#backend .control-group.error input,
#backend .control-group.error select,
#backend .control-group.error textarea {
border-color: #b94a48;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.error input:focus,
#backend .control-group.error select:focus,
#backend .control-group.error textarea:focus {
border-color: #953b39;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #d59392;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #d59392;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #d59392;
}
#backend .control-group.error .input-prepend .add-on,
#backend .control-group.error .input-append .add-on {
color: #b94a48;
background-color: #f2dede;
border-color: #b94a48;
}
#backend .control-group.error .control-label,
#backend .control-group.error .help-block,
#backend .control-group.error .help-inline {
color: #b94a48;
}
#backend .control-group.error .checkbox,
#backend .control-group.error .radio,
#backend .control-group.error input,
#backend .control-group.error select,
#backend .control-group.error textarea {
color: #b94a48;
}
#backend .control-group.error input,
#backend .control-group.error select,
#backend .control-group.error textarea {
border-color: #b94a48;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.error input:focus,
#backend .control-group.error select:focus,
#backend .control-group.error textarea:focus {
border-color: #953b39;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #d59392;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #d59392;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #d59392;
}
#backend .control-group.error .input-prepend .add-on,
#backend .control-group.error .input-append .add-on {
color: #b94a48;
background-color: #f2dede;
border-color: #b94a48;
}
#backend .control-group.success .control-label,
#backend .control-group.success .help-block,
#backend .control-group.success .help-inline {
color: #468847;
}
#backend .control-group.success .checkbox,
#backend .control-group.success .radio,
#backend .control-group.success input,
#backend .control-group.success select,
#backend .control-group.success textarea {
color: #468847;
}
#backend .control-group.success input,
#backend .control-group.success select,
#backend .control-group.success textarea {
border-color: #468847;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.success input:focus,
#backend .control-group.success select:focus,
#backend .control-group.success textarea:focus {
border-color: #356635;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7aba7b;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7aba7b;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7aba7b;
}
#backend .control-group.success .input-prepend .add-on,
#backend .control-group.success .input-append .add-on {
color: #468847;
background-color: #dff0d8;
border-color: #468847;
}
#backend .control-group.success .control-label,
#backend .control-group.success .help-block,
#backend .control-group.success .help-inline {
color: #468847;
}
#backend .control-group.success .checkbox,
#backend .control-group.success .radio,
#backend .control-group.success input,
#backend .control-group.success select,
#backend .control-group.success textarea {
color: #468847;
}
#backend .control-group.success input,
#backend .control-group.success select,
#backend .control-group.success textarea {
border-color: #468847;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.success input:focus,
#backend .control-group.success select:focus,
#backend .control-group.success textarea:focus {
border-color: #356635;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7aba7b;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7aba7b;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7aba7b;
}
#backend .control-group.success .input-prepend .add-on,
#backend .control-group.success .input-append .add-on {
color: #468847;
background-color: #dff0d8;
border-color: #468847;
}
#backend .control-group.info .control-label,
#backend .control-group.info .help-block,
#backend .control-group.info .help-inline {
color: #3a87ad;
}
#backend .control-group.info .checkbox,
#backend .control-group.info .radio,
#backend .control-group.info input,
#backend .control-group.info select,
#backend .control-group.info textarea {
color: #3a87ad;
}
#backend .control-group.info input,
#backend .control-group.info select,
#backend .control-group.info textarea {
border-color: #3a87ad;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.info input:focus,
#backend .control-group.info select:focus,
#backend .control-group.info textarea:focus {
border-color: #2d6987;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7ab5d3;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7ab5d3;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7ab5d3;
}
#backend .control-group.info .input-prepend .add-on,
#backend .control-group.info .input-append .add-on {
color: #3a87ad;
background-color: #d9edf7;
border-color: #3a87ad;
}
#backend .control-group.info .control-label,
#backend .control-group.info .help-block,
#backend .control-group.info .help-inline {
color: #3a87ad;
}
#backend .control-group.info .checkbox,
#backend .control-group.info .radio,
#backend .control-group.info input,
#backend .control-group.info select,
#backend .control-group.info textarea {
color: #3a87ad;
}
#backend .control-group.info input,
#backend .control-group.info select,
#backend .control-group.info textarea {
border-color: #3a87ad;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075);
}
#backend .control-group.info input:focus,
#backend .control-group.info select:focus,
#backend .control-group.info textarea:focus {
border-color: #2d6987;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7ab5d3;
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7ab5d3;
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.075), 0 0 6px #7ab5d3;
}
#backend .control-group.info .input-prepend .add-on,
#backend .control-group.info .input-append .add-on {
color: #3a87ad;
background-color: #d9edf7;
border-color: #3a87ad;
}
#backend input:focus:invalid,
#backend textarea:focus:invalid,
#backend select:focus:invalid {
color: #b94a48;
border-color: #ee5f5b;
}
#backend input:focus:invalid:focus,
#backend textarea:focus:invalid:focus,
#backend select:focus:invalid:focus {
border-color: #e9322d;
-webkit-box-shadow: 0 0 6px #f8b9b7;
-moz-box-shadow: 0 0 6px #f8b9b7;
box-shadow: 0 0 6px #f8b9b7;
}
#backend .form-actions {
padding: 17px 20px 18px;
margin-top: 18px;
margin-bottom: 18px;
background-color: #f5f5f5;
border-top: 1px solid #e5e5e5;
*zoom: 1;
}
#backend .form-actions:before,
#backend .form-actions:after {
display: table;
content: "";
line-height: 0;
}
#backend .form-actions:after {
clear: both;
}
#backend .form-actions:before,
#backend .form-actions:after {
display: table;
content: "";
line-height: 0;
}
#backend .form-actions:after {
clear: both;
}
#backend .help-block,
#backend .help-inline {
color: #595959;
}
#backend .help-block {
display: block;
margin-bottom: 9px;
}
#backend .help-inline {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
vertical-align: middle;
padding-left: 5px;
}
#backend .input-append,
#backend .input-prepend {
display: inline-block;
margin-bottom: 9px;
vertical-align: middle;
font-size: 0;
white-space: nowrap;
}
#backend .input-append input,
#backend .input-prepend input,
#backend .input-append select,
#backend .input-prepend select,
#backend .input-append .uneditable-input,
#backend .input-prepend .uneditable-input,
#backend .input-append .dropdown-menu,
#backend .input-prepend .dropdown-menu,
#backend .input-append .popover,
#backend .input-prepend .popover {
font-size: 13px;
}
#backend .input-append input,
#backend .input-prepend input,
#backend .input-append select,
#backend .input-prepend select,
#backend .input-append .uneditable-input,
#backend .input-prepend .uneditable-input {
position: relative;
margin-bottom: 0;
*margin-left: 0;
vertical-align: top;
-webkit-border-radius: 0 3px 3px 0;
-moz-border-radius: 0 3px 3px 0;
border-radius: 0 3px 3px 0;
}
#backend .input-append input:focus,
#backend .input-prepend input:focus,
#backend .input-append select:focus,
#backend .input-prepend select:focus,
#backend .input-append .uneditable-input:focus,
#backend .input-prepend .uneditable-input:focus {
z-index: 2;
}
#backend .input-append .add-on,
#backend .input-prepend .add-on {
display: inline-block;
width: auto;
height: 18px;
min-width: 16px;
padding: 4px 5px;
font-size: 13px;
font-weight: normal;
line-height: 18px;
text-align: center;
text-shadow: 0 1px 0 #ffffff;
background-color: #eeeeee;
border: 1px solid #ccc;
}
#backend .input-append .add-on,
#backend .input-prepend .add-on,
#backend .input-append .btn,
#backend .input-prepend .btn,
#backend .input-append .btn-group > .dropdown-toggle,
#backend .input-prepend .btn-group > .dropdown-toggle {
vertical-align: top;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .input-append .active,
#backend .input-prepend .active {
background-color: #a9dba9;
border-color: #46a546;
}
#backend .input-prepend .add-on,
#backend .input-prepend .btn {
margin-right: -1px;
}
#backend .input-prepend .add-on:first-child,
#backend .input-prepend .btn:first-child {
-webkit-border-radius: 3px 0 0 3px;
-moz-border-radius: 3px 0 0 3px;
border-radius: 3px 0 0 3px;
}
#backend .input-append input,
#backend .input-append select,
#backend .input-append .uneditable-input {
-webkit-border-radius: 3px 0 0 3px;
-moz-border-radius: 3px 0 0 3px;
border-radius: 3px 0 0 3px;
}
#backend .input-append input + .btn-group .btn:last-child,
#backend .input-append select + .btn-group .btn:last-child,
#backend .input-append .uneditable-input + .btn-group .btn:last-child {
-webkit-border-radius: 0 3px 3px 0;
-moz-border-radius: 0 3px 3px 0;
border-radius: 0 3px 3px 0;
}
#backend .input-append .add-on,
#backend .input-append .btn,
#backend .input-append .btn-group {
margin-left: -1px;
}
#backend .input-append .add-on:last-child,
#backend .input-append .btn:last-child,
#backend .input-append .btn-group:last-child > .dropdown-toggle {
-webkit-border-radius: 0 3px 3px 0;
-moz-border-radius: 0 3px 3px 0;
border-radius: 0 3px 3px 0;
}
#backend .input-prepend.input-append input,
#backend .input-prepend.input-append select,
#backend .input-prepend.input-append .uneditable-input {
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .input-prepend.input-append input + .btn-group .btn,
#backend .input-prepend.input-append select + .btn-group .btn,
#backend .input-prepend.input-append .uneditable-input + .btn-group .btn {
-webkit-border-radius: 0 3px 3px 0;
-moz-border-radius: 0 3px 3px 0;
border-radius: 0 3px 3px 0;
}
#backend .input-prepend.input-append .add-on:first-child,
#backend .input-prepend.input-append .btn:first-child {
margin-right: -1px;
-webkit-border-radius: 3px 0 0 3px;
-moz-border-radius: 3px 0 0 3px;
border-radius: 3px 0 0 3px;
}
#backend .input-prepend.input-append .add-on:last-child,
#backend .input-prepend.input-append .btn:last-child {
margin-left: -1px;
-webkit-border-radius: 0 3px 3px 0;
-moz-border-radius: 0 3px 3px 0;
border-radius: 0 3px 3px 0;
}
#backend .input-prepend.input-append .btn-group:first-child {
margin-left: 0;
}
#backend input.search-query {
padding-right: 14px;
padding-right: 4px \9;
padding-left: 14px;
padding-left: 4px \9;
/* IE7-8 doesn't have border-radius, so don't indent the padding */
margin-bottom: 0;
-webkit-border-radius: 15px;
-moz-border-radius: 15px;
border-radius: 15px;
}
#backend .form-search .input-append .search-query,
#backend .form-search .input-prepend .search-query {
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .form-search .input-append .search-query {
-webkit-border-radius: 14px 0 0 14px;
-moz-border-radius: 14px 0 0 14px;
border-radius: 14px 0 0 14px;
}
#backend .form-search .input-append .btn {
-webkit-border-radius: 0 14px 14px 0;
-moz-border-radius: 0 14px 14px 0;
border-radius: 0 14px 14px 0;
}
#backend .form-search .input-prepend .search-query {
-webkit-border-radius: 0 14px 14px 0;
-moz-border-radius: 0 14px 14px 0;
border-radius: 0 14px 14px 0;
}
#backend .form-search .input-prepend .btn {
-webkit-border-radius: 14px 0 0 14px;
-moz-border-radius: 14px 0 0 14px;
border-radius: 14px 0 0 14px;
}
#backend .form-search input,
#backend .form-inline input,
#backend .form-horizontal input,
#backend .form-search textarea,
#backend .form-inline textarea,
#backend .form-horizontal textarea,
#backend .form-search select,
#backend .form-inline select,
#backend .form-horizontal select,
#backend .form-search .help-inline,
#backend .form-inline .help-inline,
#backend .form-horizontal .help-inline,
#backend .form-search .uneditable-input,
#backend .form-inline .uneditable-input,
#backend .form-horizontal .uneditable-input,
#backend .form-search .input-prepend,
#backend .form-inline .input-prepend,
#backend .form-horizontal .input-prepend,
#backend .form-search .input-append,
#backend .form-inline .input-append,
#backend .form-horizontal .input-append {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
margin-bottom: 0;
vertical-align: middle;
}
#backend .form-search .hide,
#backend .form-inline .hide,
#backend .form-horizontal .hide {
display: none;
}
#backend .form-search label,
#backend .form-inline label,
#backend .form-search .btn-group,
#backend .form-inline .btn-group {
display: inline-block;
}
#backend .form-search .input-append,
#backend .form-inline .input-append,
#backend .form-search .input-prepend,
#backend .form-inline .input-prepend {
margin-bottom: 0;
}
#backend .form-search .radio,
#backend .form-search .checkbox,
#backend .form-inline .radio,
#backend .form-inline .checkbox {
padding-left: 0;
margin-bottom: 0;
vertical-align: middle;
}
#backend .form-search .radio input[type="radio"],
#backend .form-search .checkbox input[type="checkbox"],
#backend .form-inline .radio input[type="radio"],
#backend .form-inline .checkbox input[type="checkbox"] {
float: left;
margin-right: 3px;
margin-left: 0;
}
#backend .control-group {
margin-bottom: 9px;
}
#backend legend + .control-group {
margin-top: 18px;
-webkit-margin-top-collapse: separate;
}
#backend .form-horizontal .control-group {
margin-bottom: 18px;
*zoom: 1;
}
#backend .form-horizontal .control-group:before,
#backend .form-horizontal .control-group:after {
display: table;
content: "";
line-height: 0;
}
#backend .form-horizontal .control-group:after {
clear: both;
}
#backend .form-horizontal .control-group:before,
#backend .form-horizontal .control-group:after {
display: table;
content: "";
line-height: 0;
}
#backend .form-horizontal .control-group:after {
clear: both;
}
#backend .form-horizontal .control-label {
float: left;
width: 160px;
padding-top: 5px;
text-align: right;
}
#backend .form-horizontal .controls {
*display: inline-block;
*padding-left: 20px;
margin-left: 180px;
*margin-left: 0;
}
#backend .form-horizontal .controls:first-child {
*padding-left: 180px;
}
#backend .form-horizontal .help-block {
margin-bottom: 0;
}
#backend .form-horizontal input + .help-block,
#backend .form-horizontal select + .help-block,
#backend .form-horizontal textarea + .help-block,
#backend .form-horizontal .uneditable-input + .help-block,
#backend .form-horizontal .input-prepend + .help-block,
#backend .form-horizontal .input-append + .help-block {
margin-top: 9px;
}
#backend .form-horizontal .form-actions {
padding-left: 180px;
}
#backend table {
max-width: 100%;
background-color: transparent;
border-collapse: collapse;
border-spacing: 0;
}
#backend .table {
width: 100%;
margin-bottom: 18px;
}
#backend .table th,
#backend .table td {
padding: 8px;
line-height: 18px;
text-align: left;
vertical-align: top;
border-top: 1px solid #dddddd;
}
#backend .table th {
font-weight: bold;
}
#backend .table thead th {
vertical-align: bottom;
}
#backend .table caption + thead tr:first-child th,
#backend .table caption + thead tr:first-child td,
#backend .table colgroup + thead tr:first-child th,
#backend .table colgroup + thead tr:first-child td,
#backend .table thead:first-child tr:first-child th,
#backend .table thead:first-child tr:first-child td {
border-top: 0;
}
#backend .table tbody + tbody {
border-top: 2px solid #dddddd;
}
#backend .table .table {
background-color: #ffffff;
}
#backend .table-condensed th,
#backend .table-condensed td {
padding: 4px 5px;
}
#backend .table-bordered {
border: 1px solid #dddddd;
border-collapse: separate;
*border-collapse: collapse;
border-left: 0;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .table-bordered th,
#backend .table-bordered td {
border-left: 1px solid #dddddd;
}
#backend .table-bordered caption + thead tr:first-child th,
#backend .table-bordered caption + tbody tr:first-child th,
#backend .table-bordered caption + tbody tr:first-child td,
#backend .table-bordered colgroup + thead tr:first-child th,
#backend .table-bordered colgroup + tbody tr:first-child th,
#backend .table-bordered colgroup + tbody tr:first-child td,
#backend .table-bordered thead:first-child tr:first-child th,
#backend .table-bordered tbody:first-child tr:first-child th,
#backend .table-bordered tbody:first-child tr:first-child td {
border-top: 0;
}
#backend .table-bordered thead:first-child tr:first-child > th:first-child,
#backend .table-bordered tbody:first-child tr:first-child > td:first-child,
#backend .table-bordered tbody:first-child tr:first-child > th:first-child {
-webkit-border-top-left-radius: 4px;
-moz-border-radius-topleft: 4px;
border-top-left-radius: 4px;
}
#backend .table-bordered thead:first-child tr:first-child > th:last-child,
#backend .table-bordered tbody:first-child tr:first-child > td:last-child,
#backend .table-bordered tbody:first-child tr:first-child > th:last-child {
-webkit-border-top-right-radius: 4px;
-moz-border-radius-topright: 4px;
border-top-right-radius: 4px;
}
#backend .table-bordered thead:last-child tr:last-child > th:first-child,
#backend .table-bordered tbody:last-child tr:last-child > td:first-child,
#backend .table-bordered tbody:last-child tr:last-child > th:first-child,
#backend .table-bordered tfoot:last-child tr:last-child > td:first-child,
#backend .table-bordered tfoot:last-child tr:last-child > th:first-child {
-webkit-border-bottom-left-radius: 4px;
-moz-border-radius-bottomleft: 4px;
border-bottom-left-radius: 4px;
}
#backend .table-bordered thead:last-child tr:last-child > th:last-child,
#backend .table-bordered tbody:last-child tr:last-child > td:last-child,
#backend .table-bordered tbody:last-child tr:last-child > th:last-child,
#backend .table-bordered tfoot:last-child tr:last-child > td:last-child,
#backend .table-bordered tfoot:last-child tr:last-child > th:last-child {
-webkit-border-bottom-right-radius: 4px;
-moz-border-radius-bottomright: 4px;
border-bottom-right-radius: 4px;
}
#backend .table-bordered tfoot + tbody:last-child tr:last-child td:first-child {
-webkit-border-bottom-left-radius: 0;
-moz-border-radius-bottomleft: 0;
border-bottom-left-radius: 0;
}
#backend .table-bordered tfoot + tbody:last-child tr:last-child td:last-child {
-webkit-border-bottom-right-radius: 0;
-moz-border-radius-bottomright: 0;
border-bottom-right-radius: 0;
}
#backend .table-bordered caption + thead tr:first-child th:first-child,
#backend .table-bordered caption + tbody tr:first-child td:first-child,
#backend .table-bordered colgroup + thead tr:first-child th:first-child,
#backend .table-bordered colgroup + tbody tr:first-child td:first-child {
-webkit-border-top-left-radius: 4px;
-moz-border-radius-topleft: 4px;
border-top-left-radius: 4px;
}
#backend .table-bordered caption + thead tr:first-child th:last-child,
#backend .table-bordered caption + tbody tr:first-child td:last-child,
#backend .table-bordered colgroup + thead tr:first-child th:last-child,
#backend .table-bordered colgroup + tbody tr:first-child td:last-child {
-webkit-border-top-right-radius: 4px;
-moz-border-radius-topright: 4px;
border-top-right-radius: 4px;
}
#backend .table-striped tbody > tr:nth-child(odd) > td,
#backend .table-striped tbody > tr:nth-child(odd) > th {
background-color: #f9f9f9;
}
#backend .table-hover tbody tr:hover > td,
#backend .table-hover tbody tr:hover > th {
background-color: #f5f5f5;
}
#backend table td[class*="span"],
#backend table th[class*="span"],
#backend .row-fluid table td[class*="span"],
#backend .row-fluid table th[class*="span"] {
display: table-cell;
float: none;
margin-left: 0;
}
#backend .table td.span1,
#backend .table th.span1 {
float: none;
width: 44px;
margin-left: 0;
}
#backend .table td.span2,
#backend .table th.span2 {
float: none;
width: 124px;
margin-left: 0;
}
#backend .table td.span3,
#backend .table th.span3 {
float: none;
width: 204px;
margin-left: 0;
}
#backend .table td.span4,
#backend .table th.span4 {
float: none;
width: 284px;
margin-left: 0;
}
#backend .table td.span5,
#backend .table th.span5 {
float: none;
width: 364px;
margin-left: 0;
}
#backend .table td.span6,
#backend .table th.span6 {
float: none;
width: 444px;
margin-left: 0;
}
#backend .table td.span7,
#backend .table th.span7 {
float: none;
width: 524px;
margin-left: 0;
}
#backend .table td.span8,
#backend .table th.span8 {
float: none;
width: 604px;
margin-left: 0;
}
#backend .table td.span9,
#backend .table th.span9 {
float: none;
width: 684px;
margin-left: 0;
}
#backend .table td.span10,
#backend .table th.span10 {
float: none;
width: 764px;
margin-left: 0;
}
#backend .table td.span11,
#backend .table th.span11 {
float: none;
width: 844px;
margin-left: 0;
}
#backend .table td.span12,
#backend .table th.span12 {
float: none;
width: 924px;
margin-left: 0;
}
#backend .table tbody tr.success > td {
background-color: #dff0d8;
}
#backend .table tbody tr.error > td {
background-color: #f2dede;
}
#backend .table tbody tr.warning > td {
background-color: #fcf8e3;
}
#backend .table tbody tr.info > td {
background-color: #d9edf7;
}
#backend .table-hover tbody tr.success:hover > td {
background-color: #d0e9c6;
}
#backend .table-hover tbody tr.error:hover > td {
background-color: #ebcccc;
}
#backend .table-hover tbody tr.warning:hover > td {
background-color: #faf2cc;
}
#backend .table-hover tbody tr.info:hover > td {
background-color: #c4e3f3;
}
#backend [class^="icon-"],
#backend [class*=" icon-"] {
display: inline-block;
width: 14px;
height: 14px;
*margin-right: .3em;
line-height: 14px;
vertical-align: text-top;
background-image: url("img/glyphicons-halflings.png");
background-position: 14px 14px;
background-repeat: no-repeat;
margin-top: 1px;
}
#backend .icon-white,
#backend .nav-pills > .active > a > [class^="icon-"],
#backend .nav-pills > .active > a > [class*=" icon-"],
#backend .nav-list > .active > a > [class^="icon-"],
#backend .nav-list > .active > a > [class*=" icon-"],
#backend .navbar-inverse .nav > .active > a > [class^="icon-"],
#backend .navbar-inverse .nav > .active > a > [class*=" icon-"],
#backend .dropdown-menu > li > a:hover > [class^="icon-"],
#backend .dropdown-menu > li > a:focus > [class^="icon-"],
#backend .dropdown-menu > li > a:hover > [class*=" icon-"],
#backend .dropdown-menu > li > a:focus > [class*=" icon-"],
#backend .dropdown-menu > .active > a > [class^="icon-"],
#backend .dropdown-menu > .active > a > [class*=" icon-"],
#backend .dropdown-submenu:hover > a > [class^="icon-"],
#backend .dropdown-submenu:focus > a > [class^="icon-"],
#backend .dropdown-submenu:hover > a > [class*=" icon-"],
#backend .dropdown-submenu:focus > a > [class*=" icon-"] {
background-image: url("img/glyphicons-halflings-white.png");
}
#backend .icon-glass {
background-position: 0 0;
}
#backend .icon-music {
background-position: -24px 0;
}
#backend .icon-search {
background-position: -48px 0;
}
#backend .icon-envelope {
background-position: -72px 0;
}
#backend .icon-heart {
background-position: -96px 0;
}
#backend .icon-star {
background-position: -120px 0;
}
#backend .icon-star-empty {
background-position: -144px 0;
}
#backend .icon-user {
background-position: -168px 0;
}
#backend .icon-film {
background-position: -192px 0;
}
#backend .icon-th-large {
background-position: -216px 0;
}
#backend .icon-th {
background-position: -240px 0;
}
#backend .icon-th-list {
background-position: -264px 0;
}
#backend .icon-ok {
background-position: -288px 0;
}
#backend .icon-remove {
background-position: -312px 0;
}
#backend .icon-zoom-in {
background-position: -336px 0;
}
#backend .icon-zoom-out {
background-position: -360px 0;
}
#backend .icon-off {
background-position: -384px 0;
}
#backend .icon-signal {
background-position: -408px 0;
}
#backend .icon-cog {
background-position: -432px 0;
}
#backend .icon-trash {
background-position: -456px 0;
}
#backend .icon-home {
background-position: 0 -24px;
}
#backend .icon-file {
background-position: -24px -24px;
}
#backend .icon-time {
background-position: -48px -24px;
}
#backend .icon-road {
background-position: -72px -24px;
}
#backend .icon-download-alt {
background-position: -96px -24px;
}
#backend .icon-download {
background-position: -120px -24px;
}
#backend .icon-upload {
background-position: -144px -24px;
}
#backend .icon-inbox {
background-position: -168px -24px;
}
#backend .icon-play-circle {
background-position: -192px -24px;
}
#backend .icon-repeat {
background-position: -216px -24px;
}
#backend .icon-refresh {
background-position: -240px -24px;
}
#backend .icon-list-alt {
background-position: -264px -24px;
}
#backend .icon-lock {
background-position: -287px -24px;
}
#backend .icon-flag {
background-position: -312px -24px;
}
#backend .icon-headphones {
background-position: -336px -24px;
}
#backend .icon-volume-off {
background-position: -360px -24px;
}
#backend .icon-volume-down {
background-position: -384px -24px;
}
#backend .icon-volume-up {
background-position: -408px -24px;
}
#backend .icon-qrcode {
background-position: -432px -24px;
}
#backend .icon-barcode {
background-position: -456px -24px;
}
#backend .icon-tag {
background-position: 0 -48px;
}
#backend .icon-tags {
background-position: -25px -48px;
}
#backend .icon-book {
background-position: -48px -48px;
}
#backend .icon-bookmark {
background-position: -72px -48px;
}
#backend .icon-print {
background-position: -96px -48px;
}
#backend .icon-camera {
background-position: -120px -48px;
}
#backend .icon-font {
background-position: -144px -48px;
}
#backend .icon-bold {
background-position: -167px -48px;
}
#backend .icon-italic {
background-position: -192px -48px;
}
#backend .icon-text-height {
background-position: -216px -48px;
}
#backend .icon-text-width {
background-position: -240px -48px;
}
#backend .icon-align-left {
background-position: -264px -48px;
}
#backend .icon-align-center {
background-position: -288px -48px;
}
#backend .icon-align-right {
background-position: -312px -48px;
}
#backend .icon-align-justify {
background-position: -336px -48px;
}
#backend .icon-list {
background-position: -360px -48px;
}
#backend .icon-indent-left {
background-position: -384px -48px;
}
#backend .icon-indent-right {
background-position: -408px -48px;
}
#backend .icon-facetime-video {
background-position: -432px -48px;
}
#backend .icon-picture {
background-position: -456px -48px;
}
#backend .icon-pencil {
background-position: 0 -72px;
}
#backend .icon-map-marker {
background-position: -24px -72px;
}
#backend .icon-adjust {
background-position: -48px -72px;
}
#backend .icon-tint {
background-position: -72px -72px;
}
#backend .icon-edit {
background-position: -96px -72px;
}
#backend .icon-share {
background-position: -120px -72px;
}
#backend .icon-check {
background-position: -144px -72px;
}
#backend .icon-move {
background-position: -168px -72px;
}
#backend .icon-step-backward {
background-position: -192px -72px;
}
#backend .icon-fast-backward {
background-position: -216px -72px;
}
#backend .icon-backward {
background-position: -240px -72px;
}
#backend .icon-play {
background-position: -264px -72px;
}
#backend .icon-pause {
background-position: -288px -72px;
}
#backend .icon-stop {
background-position: -312px -72px;
}
#backend .icon-forward {
background-position: -336px -72px;
}
#backend .icon-fast-forward {
background-position: -360px -72px;
}
#backend .icon-step-forward {
background-position: -384px -72px;
}
#backend .icon-eject {
background-position: -408px -72px;
}
#backend .icon-chevron-left {
background-position: -432px -72px;
}
#backend .icon-chevron-right {
background-position: -456px -72px;
}
#backend .icon-plus-sign {
background-position: 0 -96px;
}
#backend .icon-minus-sign {
background-position: -24px -96px;
}
#backend .icon-remove-sign {
background-position: -48px -96px;
}
#backend .icon-ok-sign {
background-position: -72px -96px;
}
#backend .icon-question-sign {
background-position: -96px -96px;
}
#backend .icon-info-sign {
background-position: -120px -96px;
}
#backend .icon-screenshot {
background-position: -144px -96px;
}
#backend .icon-remove-circle {
background-position: -168px -96px;
}
#backend .icon-ok-circle {
background-position: -192px -96px;
}
#backend .icon-ban-circle {
background-position: -216px -96px;
}
#backend .icon-arrow-left {
background-position: -240px -96px;
}
#backend .icon-arrow-right {
background-position: -264px -96px;
}
#backend .icon-arrow-up {
background-position: -289px -96px;
}
#backend .icon-arrow-down {
background-position: -312px -96px;
}
#backend .icon-share-alt {
background-position: -336px -96px;
}
#backend .icon-resize-full {
background-position: -360px -96px;
}
#backend .icon-resize-small {
background-position: -384px -96px;
}
#backend .icon-plus {
background-position: -408px -96px;
}
#backend .icon-minus {
background-position: -433px -96px;
}
#backend .icon-asterisk {
background-position: -456px -96px;
}
#backend .icon-exclamation-sign {
background-position: 0 -120px;
}
#backend .icon-gift {
background-position: -24px -120px;
}
#backend .icon-leaf {
background-position: -48px -120px;
}
#backend .icon-fire {
background-position: -72px -120px;
}
#backend .icon-eye-open {
background-position: -96px -120px;
}
#backend .icon-eye-close {
background-position: -120px -120px;
}
#backend .icon-warning-sign {
background-position: -144px -120px;
}
#backend .icon-plane {
background-position: -168px -120px;
}
#backend .icon-calendar {
background-position: -192px -120px;
}
#backend .icon-random {
background-position: -216px -120px;
width: 16px;
}
#backend .icon-comment {
background-position: -240px -120px;
}
#backend .icon-magnet {
background-position: -264px -120px;
}
#backend .icon-chevron-up {
background-position: -288px -120px;
}
#backend .icon-chevron-down {
background-position: -313px -119px;
}
#backend .icon-retweet {
background-position: -336px -120px;
}
#backend .icon-shopping-cart {
background-position: -360px -120px;
}
#backend .icon-folder-close {
background-position: -384px -120px;
width: 16px;
}
#backend .icon-folder-open {
background-position: -408px -120px;
width: 16px;
}
#backend .icon-resize-vertical {
background-position: -432px -119px;
}
#backend .icon-resize-horizontal {
background-position: -456px -118px;
}
#backend .icon-hdd {
background-position: 0 -144px;
}
#backend .icon-bullhorn {
background-position: -24px -144px;
}
#backend .icon-bell {
background-position: -48px -144px;
}
#backend .icon-certificate {
background-position: -72px -144px;
}
#backend .icon-thumbs-up {
background-position: -96px -144px;
}
#backend .icon-thumbs-down {
background-position: -120px -144px;
}
#backend .icon-hand-right {
background-position: -144px -144px;
}
#backend .icon-hand-left {
background-position: -168px -144px;
}
#backend .icon-hand-up {
background-position: -192px -144px;
}
#backend .icon-hand-down {
background-position: -216px -144px;
}
#backend .icon-circle-arrow-right {
background-position: -240px -144px;
}
#backend .icon-circle-arrow-left {
background-position: -264px -144px;
}
#backend .icon-circle-arrow-up {
background-position: -288px -144px;
}
#backend .icon-circle-arrow-down {
background-position: -312px -144px;
}
#backend .icon-globe {
background-position: -336px -144px;
}
#backend .icon-wrench {
background-position: -360px -144px;
}
#backend .icon-tasks {
background-position: -384px -144px;
}
#backend .icon-filter {
background-position: -408px -144px;
}
#backend .icon-briefcase {
background-position: -432px -144px;
}
#backend .icon-fullscreen {
background-position: -456px -144px;
}
#backend .dropup,
#backend .dropdown {
position: relative;
}
#backend .dropdown-toggle {
*margin-bottom: -3px;
}
#backend .dropdown-toggle:active,
#backend .open .dropdown-toggle {
outline: 0;
}
#backend .caret {
display: inline-block;
width: 0;
height: 0;
vertical-align: top;
border-top: 4px solid #000000;
border-right: 4px solid transparent;
border-left: 4px solid transparent;
content: "";
}
#backend .dropdown .caret {
margin-top: 8px;
margin-left: 2px;
}
#backend .dropdown-menu {
position: absolute;
top: 100%;
left: 0;
z-index: 1000;
display: none;
float: left;
min-width: 160px;
padding: 5px 0;
margin: 2px 0 0;
list-style: none;
background-color: #ffffff;
border: 1px solid #ccc;
border: 1px solid rgba(0, 0, 0, 0.2);
*border-right-width: 2px;
*border-bottom-width: 2px;
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
-webkit-box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2);
-moz-box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2);
box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2);
-webkit-background-clip: padding-box;
-moz-background-clip: padding;
background-clip: padding-box;
}
#backend .dropdown-menu.pull-right {
right: 0;
left: auto;
}
#backend .dropdown-menu .divider {
*width: 100%;
height: 1px;
margin: 8px 1px;
*margin: -5px 0 5px;
overflow: hidden;
background-color: #e5e5e5;
border-bottom: 1px solid #ffffff;
}
#backend .dropdown-menu > li > a {
display: block;
padding: 3px 20px;
clear: both;
font-weight: normal;
line-height: 18px;
color: #333333;
white-space: nowrap;
}
#backend .dropdown-menu > li > a:hover,
#backend .dropdown-menu > li > a:focus,
#backend .dropdown-submenu:hover > a,
#backend .dropdown-submenu:focus > a {
text-decoration: none;
color: #ffffff;
background-color: #0081c2;
background-image: -moz-linear-gradient(top, #0088cc, #0077b3);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#0088cc), to(#0077b3));
background-image: -webkit-linear-gradient(top, #0088cc, #0077b3);
background-image: -o-linear-gradient(top, #0088cc, #0077b3);
background-image: linear-gradient(to bottom, #0088cc, #0077b3);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff0088cc', endColorstr='#ff0077b3', GradientType=0);
}
#backend .dropdown-menu > .active > a,
#backend .dropdown-menu > .active > a:hover,
#backend .dropdown-menu > .active > a:focus {
color: #ffffff;
text-decoration: none;
outline: 0;
background-color: #0081c2;
background-image: -moz-linear-gradient(top, #0088cc, #0077b3);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#0088cc), to(#0077b3));
background-image: -webkit-linear-gradient(top, #0088cc, #0077b3);
background-image: -o-linear-gradient(top, #0088cc, #0077b3);
background-image: linear-gradient(to bottom, #0088cc, #0077b3);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff0088cc', endColorstr='#ff0077b3', GradientType=0);
}
#backend .dropdown-menu > .disabled > a,
#backend .dropdown-menu > .disabled > a:hover,
#backend .dropdown-menu > .disabled > a:focus {
color: #999999;
}
#backend .dropdown-menu > .disabled > a:hover,
#backend .dropdown-menu > .disabled > a:focus {
text-decoration: none;
background-color: transparent;
background-image: none;
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
cursor: default;
}
#backend .open {
*z-index: 1000;
}
#backend .open > .dropdown-menu {
display: block;
}
#backend .dropdown-backdrop {
position: fixed;
left: 0;
right: 0;
bottom: 0;
top: 0;
z-index: 990;
}
#backend .pull-right > .dropdown-menu {
right: 0;
left: auto;
}
#backend .dropup .caret,
#backend .navbar-fixed-bottom .dropdown .caret {
border-top: 0;
border-bottom: 4px solid #000000;
content: "";
}
#backend .dropup .dropdown-menu,
#backend .navbar-fixed-bottom .dropdown .dropdown-menu {
top: auto;
bottom: 100%;
margin-bottom: 1px;
}
#backend .dropdown-submenu {
position: relative;
}
#backend .dropdown-submenu > .dropdown-menu {
top: 0;
left: 100%;
margin-top: -6px;
margin-left: -1px;
-webkit-border-radius: 0 6px 6px 6px;
-moz-border-radius: 0 6px 6px 6px;
border-radius: 0 6px 6px 6px;
}
#backend .dropdown-submenu:hover > .dropdown-menu {
display: block;
}
#backend .dropup .dropdown-submenu > .dropdown-menu {
top: auto;
bottom: 0;
margin-top: 0;
margin-bottom: -2px;
-webkit-border-radius: 5px 5px 5px 0;
-moz-border-radius: 5px 5px 5px 0;
border-radius: 5px 5px 5px 0;
}
#backend .dropdown-submenu > a:after {
display: block;
content: " ";
float: right;
width: 0;
height: 0;
border-color: transparent;
border-style: solid;
border-width: 5px 0 5px 5px;
border-left-color: #cccccc;
margin-top: 5px;
margin-right: -10px;
}
#backend .dropdown-submenu:hover > a:after {
border-left-color: #ffffff;
}
#backend .dropdown-submenu.pull-left {
float: none;
}
#backend .dropdown-submenu.pull-left > .dropdown-menu {
left: -100%;
margin-left: 10px;
-webkit-border-radius: 6px 0 6px 6px;
-moz-border-radius: 6px 0 6px 6px;
border-radius: 6px 0 6px 6px;
}
#backend .dropdown .dropdown-menu .nav-header {
padding-left: 20px;
padding-right: 20px;
}
#backend .typeahead {
z-index: 1051;
margin-top: 2px;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .well {
min-height: 20px;
padding: 19px;
margin-bottom: 20px;
background-color: #f5f5f5;
border: 1px solid #e3e3e3;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.05);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.05);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.05);
}
#backend .well blockquote {
border-color: #ddd;
border-color: rgba(0, 0, 0, 0.15);
}
#backend .well-large {
padding: 24px;
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
}
#backend .well-small {
padding: 9px;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
}
#backend .fade {
opacity: 0;
-webkit-transition: opacity 0.15s linear;
-moz-transition: opacity 0.15s linear;
-o-transition: opacity 0.15s linear;
transition: opacity 0.15s linear;
}
#backend .fade.in {
opacity: 1;
}
#backend .collapse {
position: relative;
height: 0;
overflow: hidden;
-webkit-transition: height 0.35s ease;
-moz-transition: height 0.35s ease;
-o-transition: height 0.35s ease;
transition: height 0.35s ease;
}
#backend .collapse.in {
height: auto;
}
#backend .close {
float: right;
font-size: 20px;
font-weight: bold;
line-height: 18px;
color: #000000;
text-shadow: 0 1px 0 #ffffff;
opacity: 0.2;
filter: alpha(opacity=20);
}
#backend .close:hover,
#backend .close:focus {
color: #000000;
text-decoration: none;
cursor: pointer;
opacity: 0.4;
filter: alpha(opacity=40);
}
#backend button.close {
padding: 0;
cursor: pointer;
background: transparent;
border: 0;
-webkit-appearance: none;
}
#backend .btn {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
padding: 4px 12px;
margin-bottom: 0;
font-size: 13px;
line-height: 18px;
text-align: center;
vertical-align: middle;
cursor: pointer;
color: #333333;
text-shadow: 0 1px 1px rgba(255, 255, 255, 0.75);
background-color: #f5f5f5;
background-image: -moz-linear-gradient(top, #ffffff, #e6e6e6);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#ffffff), to(#e6e6e6));
background-image: -webkit-linear-gradient(top, #ffffff, #e6e6e6);
background-image: -o-linear-gradient(top, #ffffff, #e6e6e6);
background-image: linear-gradient(to bottom, #ffffff, #e6e6e6);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffffffff', endColorstr='#ffe6e6e6', GradientType=0);
border-color: #e6e6e6 #e6e6e6 #bfbfbf;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #e6e6e6;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
border: 1px solid #cccccc;
*border: 0;
border-bottom-color: #b3b3b3;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
*margin-left: .3em;
-webkit-box-shadow: inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
-moz-box-shadow: inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
box-shadow: inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
}
#backend .btn:hover,
#backend .btn:focus,
#backend .btn:active,
#backend .btn.active,
#backend .btn.disabled,
#backend .btn[disabled] {
color: #333333;
background-color: #e6e6e6;
*background-color: #d9d9d9;
}
#backend .btn:active,
#backend .btn.active {
background-color: #cccccc \9;
}
#backend .btn:hover,
#backend .btn:focus,
#backend .btn:active,
#backend .btn.active,
#backend .btn.disabled,
#backend .btn[disabled] {
color: #333333;
background-color: #e6e6e6;
*background-color: #d9d9d9;
}
#backend .btn:active,
#backend .btn.active {
background-color: #cccccc \9;
}
#backend .btn:first-child {
*margin-left: 0;
}
#backend .btn:first-child {
*margin-left: 0;
}
#backend .btn:hover,
#backend .btn:focus {
color: #333333;
text-decoration: none;
background-position: 0 -15px;
-webkit-transition: background-position 0.1s linear;
-moz-transition: background-position 0.1s linear;
-o-transition: background-position 0.1s linear;
transition: background-position 0.1s linear;
}
#backend .btn:focus {
outline: thin dotted #333;
outline: 5px auto -webkit-focus-ring-color;
outline-offset: -2px;
}
#backend .btn.active,
#backend .btn:active {
background-image: none;
outline: 0;
-webkit-box-shadow: inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);
-moz-box-shadow: inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);
box-shadow: inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);
}
#backend .btn.disabled,
#backend .btn[disabled] {
cursor: default;
background-image: none;
opacity: 0.65;
filter: alpha(opacity=65);
-webkit-box-shadow: none;
-moz-box-shadow: none;
box-shadow: none;
}
#backend .btn-large {
padding: 11px 19px;
font-size: 16.25px;
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
}
#backend .btn-large [class^="icon-"],
#backend .btn-large [class*=" icon-"] {
margin-top: 4px;
}
#backend .btn-small {
padding: 2px 10px;
font-size: 11.049999999999999px;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
}
#backend .btn-small [class^="icon-"],
#backend .btn-small [class*=" icon-"] {
margin-top: 0;
}
#backend .btn-mini [class^="icon-"],
#backend .btn-mini [class*=" icon-"] {
margin-top: -1px;
}
#backend .btn-mini {
padding: 0 6px;
font-size: 9.75px;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
}
#backend .btn-block {
display: block;
width: 100%;
padding-left: 0;
padding-right: 0;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
}
#backend .btn-block + .btn-block {
margin-top: 5px;
}
#backend input[type="submit"].btn-block,
#backend input[type="reset"].btn-block,
#backend input[type="button"].btn-block {
width: 100%;
}
#backend .btn-primary.active,
#backend .btn-warning.active,
#backend .btn-danger.active,
#backend .btn-success.active,
#backend .btn-info.active,
#backend .btn-inverse.active {
color: rgba(255, 255, 255, 0.75);
}
#backend .btn-primary {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #0074cc;
background-image: -moz-linear-gradient(top, #0088cc, #0055cc);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#0088cc), to(#0055cc));
background-image: -webkit-linear-gradient(top, #0088cc, #0055cc);
background-image: -o-linear-gradient(top, #0088cc, #0055cc);
background-image: linear-gradient(to bottom, #0088cc, #0055cc);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff0088cc', endColorstr='#ff0055cc', GradientType=0);
border-color: #0055cc #0055cc #003580;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #0055cc;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
}
#backend .btn-primary:hover,
#backend .btn-primary:focus,
#backend .btn-primary:active,
#backend .btn-primary.active,
#backend .btn-primary.disabled,
#backend .btn-primary[disabled] {
color: #ffffff;
background-color: #0055cc;
*background-color: #004ab3;
}
#backend .btn-primary:active,
#backend .btn-primary.active {
background-color: #004099 \9;
}
#backend .btn-primary:hover,
#backend .btn-primary:focus,
#backend .btn-primary:active,
#backend .btn-primary.active,
#backend .btn-primary.disabled,
#backend .btn-primary[disabled] {
color: #ffffff;
background-color: #0055cc;
*background-color: #004ab3;
}
#backend .btn-primary:active,
#backend .btn-primary.active {
background-color: #004099 \9;
}
#backend .btn-warning {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #faa732;
background-image: -moz-linear-gradient(top, #fbb450, #f89406);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#fbb450), to(#f89406));
background-image: -webkit-linear-gradient(top, #fbb450, #f89406);
background-image: -o-linear-gradient(top, #fbb450, #f89406);
background-image: linear-gradient(to bottom, #fbb450, #f89406);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fffbb450', endColorstr='#fff89406', GradientType=0);
border-color: #f89406 #f89406 #ad6704;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #f89406;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
}
#backend .btn-warning:hover,
#backend .btn-warning:focus,
#backend .btn-warning:active,
#backend .btn-warning.active,
#backend .btn-warning.disabled,
#backend .btn-warning[disabled] {
color: #ffffff;
background-color: #f89406;
*background-color: #df8505;
}
#backend .btn-warning:active,
#backend .btn-warning.active {
background-color: #c67605 \9;
}
#backend .btn-warning:hover,
#backend .btn-warning:focus,
#backend .btn-warning:active,
#backend .btn-warning.active,
#backend .btn-warning.disabled,
#backend .btn-warning[disabled] {
color: #ffffff;
background-color: #f89406;
*background-color: #df8505;
}
#backend .btn-warning:active,
#backend .btn-warning.active {
background-color: #c67605 \9;
}
#backend .btn-danger {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #da4f49;
background-image: -moz-linear-gradient(top, #ee5f5b, #bd362f);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#ee5f5b), to(#bd362f));
background-image: -webkit-linear-gradient(top, #ee5f5b, #bd362f);
background-image: -o-linear-gradient(top, #ee5f5b, #bd362f);
background-image: linear-gradient(to bottom, #ee5f5b, #bd362f);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffee5f5b', endColorstr='#ffbd362f', GradientType=0);
border-color: #bd362f #bd362f #802420;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #bd362f;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
}
#backend .btn-danger:hover,
#backend .btn-danger:focus,
#backend .btn-danger:active,
#backend .btn-danger.active,
#backend .btn-danger.disabled,
#backend .btn-danger[disabled] {
color: #ffffff;
background-color: #bd362f;
*background-color: #a9302a;
}
#backend .btn-danger:active,
#backend .btn-danger.active {
background-color: #942a25 \9;
}
#backend .btn-danger:hover,
#backend .btn-danger:focus,
#backend .btn-danger:active,
#backend .btn-danger.active,
#backend .btn-danger.disabled,
#backend .btn-danger[disabled] {
color: #ffffff;
background-color: #bd362f;
*background-color: #a9302a;
}
#backend .btn-danger:active,
#backend .btn-danger.active {
background-color: #942a25 \9;
}
#backend .btn-success {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #5bb75b;
background-image: -moz-linear-gradient(top, #62c462, #51a351);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#62c462), to(#51a351));
background-image: -webkit-linear-gradient(top, #62c462, #51a351);
background-image: -o-linear-gradient(top, #62c462, #51a351);
background-image: linear-gradient(to bottom, #62c462, #51a351);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff62c462', endColorstr='#ff51a351', GradientType=0);
border-color: #51a351 #51a351 #387038;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #51a351;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
}
#backend .btn-success:hover,
#backend .btn-success:focus,
#backend .btn-success:active,
#backend .btn-success.active,
#backend .btn-success.disabled,
#backend .btn-success[disabled] {
color: #ffffff;
background-color: #51a351;
*background-color: #499249;
}
#backend .btn-success:active,
#backend .btn-success.active {
background-color: #408140 \9;
}
#backend .btn-success:hover,
#backend .btn-success:focus,
#backend .btn-success:active,
#backend .btn-success.active,
#backend .btn-success.disabled,
#backend .btn-success[disabled] {
color: #ffffff;
background-color: #51a351;
*background-color: #499249;
}
#backend .btn-success:active,
#backend .btn-success.active {
background-color: #408140 \9;
}
#backend .btn-info {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #49afcd;
background-image: -moz-linear-gradient(top, #5bc0de, #2f96b4);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#5bc0de), to(#2f96b4));
background-image: -webkit-linear-gradient(top, #5bc0de, #2f96b4);
background-image: -o-linear-gradient(top, #5bc0de, #2f96b4);
background-image: linear-gradient(to bottom, #5bc0de, #2f96b4);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff5bc0de', endColorstr='#ff2f96b4', GradientType=0);
border-color: #2f96b4 #2f96b4 #1f6377;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #2f96b4;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
}
#backend .btn-info:hover,
#backend .btn-info:focus,
#backend .btn-info:active,
#backend .btn-info.active,
#backend .btn-info.disabled,
#backend .btn-info[disabled] {
color: #ffffff;
background-color: #2f96b4;
*background-color: #2a85a0;
}
#backend .btn-info:active,
#backend .btn-info.active {
background-color: #24748c \9;
}
#backend .btn-info:hover,
#backend .btn-info:focus,
#backend .btn-info:active,
#backend .btn-info.active,
#backend .btn-info.disabled,
#backend .btn-info[disabled] {
color: #ffffff;
background-color: #2f96b4;
*background-color: #2a85a0;
}
#backend .btn-info:active,
#backend .btn-info.active {
background-color: #24748c \9;
}
#backend .btn-inverse {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #414141;
background-image: -moz-linear-gradient(top, #555555, #222222);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#555555), to(#222222));
background-image: -webkit-linear-gradient(top, #555555, #222222);
background-image: -o-linear-gradient(top, #555555, #222222);
background-image: linear-gradient(to bottom, #555555, #222222);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff555555', endColorstr='#ff222222', GradientType=0);
border-color: #222222 #222222 #000000;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #222222;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
}
#backend .btn-inverse:hover,
#backend .btn-inverse:focus,
#backend .btn-inverse:active,
#backend .btn-inverse.active,
#backend .btn-inverse.disabled,
#backend .btn-inverse[disabled] {
color: #ffffff;
background-color: #222222;
*background-color: #151515;
}
#backend .btn-inverse:active,
#backend .btn-inverse.active {
background-color: #080808 \9;
}
#backend .btn-inverse:hover,
#backend .btn-inverse:focus,
#backend .btn-inverse:active,
#backend .btn-inverse.active,
#backend .btn-inverse.disabled,
#backend .btn-inverse[disabled] {
color: #ffffff;
background-color: #222222;
*background-color: #151515;
}
#backend .btn-inverse:active,
#backend .btn-inverse.active {
background-color: #080808 \9;
}
#backend button.btn,
#backend input[type="submit"].btn {
*padding-top: 3px;
*padding-bottom: 3px;
}
#backend button.btn::-moz-focus-inner,
#backend input[type="submit"].btn::-moz-focus-inner {
padding: 0;
border: 0;
}
#backend button.btn.btn-large,
#backend input[type="submit"].btn.btn-large {
*padding-top: 7px;
*padding-bottom: 7px;
}
#backend button.btn.btn-small,
#backend input[type="submit"].btn.btn-small {
*padding-top: 3px;
*padding-bottom: 3px;
}
#backend button.btn.btn-mini,
#backend input[type="submit"].btn.btn-mini {
*padding-top: 1px;
*padding-bottom: 1px;
}
#backend .btn-link,
#backend .btn-link:active,
#backend .btn-link[disabled] {
background-color: transparent;
background-image: none;
-webkit-box-shadow: none;
-moz-box-shadow: none;
box-shadow: none;
}
#backend .btn-link {
border-color: transparent;
cursor: pointer;
color: #0088cc;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .btn-link:hover,
#backend .btn-link:focus {
color: #005580;
text-decoration: underline;
background-color: transparent;
}
#backend .btn-link[disabled]:hover,
#backend .btn-link[disabled]:focus {
color: #333333;
text-decoration: none;
}
#backend .btn-group {
position: relative;
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
font-size: 0;
vertical-align: middle;
white-space: nowrap;
*margin-left: .3em;
}
#backend .btn-group:first-child {
*margin-left: 0;
}
#backend .btn-group:first-child {
*margin-left: 0;
}
#backend .btn-group + .btn-group {
margin-left: 5px;
}
#backend .btn-toolbar {
font-size: 0;
margin-top: 9px;
margin-bottom: 9px;
}
#backend .btn-toolbar > .btn + .btn,
#backend .btn-toolbar > .btn-group + .btn,
#backend .btn-toolbar > .btn + .btn-group {
margin-left: 5px;
}
#backend .btn-group > .btn {
position: relative;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .btn-group > .btn + .btn {
margin-left: -1px;
}
#backend .btn-group > .btn,
#backend .btn-group > .dropdown-menu,
#backend .btn-group > .popover {
font-size: 13px;
}
#backend .btn-group > .btn-mini {
font-size: 9.75px;
}
#backend .btn-group > .btn-small {
font-size: 11.049999999999999px;
}
#backend .btn-group > .btn-large {
font-size: 16.25px;
}
#backend .btn-group > .btn:first-child {
margin-left: 0;
-webkit-border-top-left-radius: 4px;
-moz-border-radius-topleft: 4px;
border-top-left-radius: 4px;
-webkit-border-bottom-left-radius: 4px;
-moz-border-radius-bottomleft: 4px;
border-bottom-left-radius: 4px;
}
#backend .btn-group > .btn:last-child,
#backend .btn-group > .dropdown-toggle {
-webkit-border-top-right-radius: 4px;
-moz-border-radius-topright: 4px;
border-top-right-radius: 4px;
-webkit-border-bottom-right-radius: 4px;
-moz-border-radius-bottomright: 4px;
border-bottom-right-radius: 4px;
}
#backend .btn-group > .btn.large:first-child {
margin-left: 0;
-webkit-border-top-left-radius: 6px;
-moz-border-radius-topleft: 6px;
border-top-left-radius: 6px;
-webkit-border-bottom-left-radius: 6px;
-moz-border-radius-bottomleft: 6px;
border-bottom-left-radius: 6px;
}
#backend .btn-group > .btn.large:last-child,
#backend .btn-group > .large.dropdown-toggle {
-webkit-border-top-right-radius: 6px;
-moz-border-radius-topright: 6px;
border-top-right-radius: 6px;
-webkit-border-bottom-right-radius: 6px;
-moz-border-radius-bottomright: 6px;
border-bottom-right-radius: 6px;
}
#backend .btn-group > .btn:hover,
#backend .btn-group > .btn:focus,
#backend .btn-group > .btn:active,
#backend .btn-group > .btn.active {
z-index: 2;
}
#backend .btn-group .dropdown-toggle:active,
#backend .btn-group.open .dropdown-toggle {
outline: 0;
}
#backend .btn-group > .btn + .dropdown-toggle {
padding-left: 8px;
padding-right: 8px;
-webkit-box-shadow: inset 1px 0 0 rgba(255,255,255,.125), inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
-moz-box-shadow: inset 1px 0 0 rgba(255,255,255,.125), inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
box-shadow: inset 1px 0 0 rgba(255,255,255,.125), inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
*padding-top: 5px;
*padding-bottom: 5px;
}
#backend .btn-group > .btn-mini + .dropdown-toggle {
padding-left: 5px;
padding-right: 5px;
*padding-top: 2px;
*padding-bottom: 2px;
}
#backend .btn-group > .btn-small + .dropdown-toggle {
*padding-top: 5px;
*padding-bottom: 4px;
}
#backend .btn-group > .btn-large + .dropdown-toggle {
padding-left: 12px;
padding-right: 12px;
*padding-top: 7px;
*padding-bottom: 7px;
}
#backend .btn-group.open .dropdown-toggle {
background-image: none;
-webkit-box-shadow: inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);
-moz-box-shadow: inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);
box-shadow: inset 0 2px 4px rgba(0,0,0,.15), 0 1px 2px rgba(0,0,0,.05);
}
#backend .btn-group.open .btn.dropdown-toggle {
background-color: #e6e6e6;
}
#backend .btn-group.open .btn-primary.dropdown-toggle {
background-color: #0055cc;
}
#backend .btn-group.open .btn-warning.dropdown-toggle {
background-color: #f89406;
}
#backend .btn-group.open .btn-danger.dropdown-toggle {
background-color: #bd362f;
}
#backend .btn-group.open .btn-success.dropdown-toggle {
background-color: #51a351;
}
#backend .btn-group.open .btn-info.dropdown-toggle {
background-color: #2f96b4;
}
#backend .btn-group.open .btn-inverse.dropdown-toggle {
background-color: #222222;
}
#backend .btn .caret {
margin-top: 8px;
margin-left: 0;
}
#backend .btn-large .caret {
margin-top: 6px;
}
#backend .btn-large .caret {
border-left-width: 5px;
border-right-width: 5px;
border-top-width: 5px;
}
#backend .btn-mini .caret,
#backend .btn-small .caret {
margin-top: 8px;
}
#backend .dropup .btn-large .caret {
border-bottom-width: 5px;
}
#backend .btn-primary .caret,
#backend .btn-warning .caret,
#backend .btn-danger .caret,
#backend .btn-info .caret,
#backend .btn-success .caret,
#backend .btn-inverse .caret {
border-top-color: #ffffff;
border-bottom-color: #ffffff;
}
#backend .btn-group-vertical {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
}
#backend .btn-group-vertical > .btn {
display: block;
float: none;
max-width: 100%;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .btn-group-vertical > .btn + .btn {
margin-left: 0;
margin-top: -1px;
}
#backend .btn-group-vertical > .btn:first-child {
-webkit-border-radius: 4px 4px 0 0;
-moz-border-radius: 4px 4px 0 0;
border-radius: 4px 4px 0 0;
}
#backend .btn-group-vertical > .btn:last-child {
-webkit-border-radius: 0 0 4px 4px;
-moz-border-radius: 0 0 4px 4px;
border-radius: 0 0 4px 4px;
}
#backend .btn-group-vertical > .btn-large:first-child {
-webkit-border-radius: 6px 6px 0 0;
-moz-border-radius: 6px 6px 0 0;
border-radius: 6px 6px 0 0;
}
#backend .btn-group-vertical > .btn-large:last-child {
-webkit-border-radius: 0 0 6px 6px;
-moz-border-radius: 0 0 6px 6px;
border-radius: 0 0 6px 6px;
}
#backend .alert {
padding: 8px 35px 8px 14px;
margin-bottom: 18px;
text-shadow: 0 1px 0 rgba(255, 255, 255, 0.5);
background-color: #fcf8e3;
border: 1px solid #fbeed5;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .alert,
#backend .alert h4 {
color: #c09853;
}
#backend .alert h4 {
margin: 0;
}
#backend .alert .close {
position: relative;
top: -2px;
right: -21px;
line-height: 18px;
}
#backend .alert-success {
background-color: #dff0d8;
border-color: #d6e9c6;
color: #468847;
}
#backend .alert-success h4 {
color: #468847;
}
#backend .alert-danger,
#backend .alert-error {
background-color: #f2dede;
border-color: #eed3d7;
color: #b94a48;
}
#backend .alert-danger h4,
#backend .alert-error h4 {
color: #b94a48;
}
#backend .alert-info {
background-color: #d9edf7;
border-color: #bce8f1;
color: #3a87ad;
}
#backend .alert-info h4 {
color: #3a87ad;
}
#backend .alert-block {
padding-top: 14px;
padding-bottom: 14px;
}
#backend .alert-block > p,
#backend .alert-block > ul {
margin-bottom: 0;
}
#backend .alert-block p + p {
margin-top: 5px;
}
#backend .nav {
margin-left: 0;
margin-bottom: 18px;
list-style: none;
}
#backend .nav > li > a {
display: block;
}
#backend .nav > li > a:hover,
#backend .nav > li > a:focus {
text-decoration: none;
background-color: #eeeeee;
}
#backend .nav > li > a > img {
max-width: none;
}
#backend .nav > .pull-right {
float: right;
}
#backend .nav-header {
display: block;
padding: 3px 15px;
font-size: 11px;
font-weight: bold;
line-height: 18px;
color: #999999;
text-shadow: 0 1px 0 rgba(255, 255, 255, 0.5);
text-transform: uppercase;
}
#backend .nav li + .nav-header {
margin-top: 9px;
}
#backend .nav-list {
padding-left: 15px;
padding-right: 15px;
margin-bottom: 0;
}
#backend .nav-list > li > a,
#backend .nav-list .nav-header {
margin-left: -15px;
margin-right: -15px;
text-shadow: 0 1px 0 rgba(255, 255, 255, 0.5);
}
#backend .nav-list > li > a {
padding: 3px 15px;
}
#backend .nav-list > .active > a,
#backend .nav-list > .active > a:hover,
#backend .nav-list > .active > a:focus {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.2);
background-color: #0088cc;
}
#backend .nav-list [class^="icon-"],
#backend .nav-list [class*=" icon-"] {
margin-right: 2px;
}
#backend .nav-list .divider {
*width: 100%;
height: 1px;
margin: 8px 1px;
*margin: -5px 0 5px;
overflow: hidden;
background-color: #e5e5e5;
border-bottom: 1px solid #ffffff;
}
#backend .nav-tabs,
#backend .nav-pills {
*zoom: 1;
}
#backend .nav-tabs:before,
#backend .nav-pills:before,
#backend .nav-tabs:after,
#backend .nav-pills:after {
display: table;
content: "";
line-height: 0;
}
#backend .nav-tabs:after,
#backend .nav-pills:after {
clear: both;
}
#backend .nav-tabs:before,
#backend .nav-pills:before,
#backend .nav-tabs:after,
#backend .nav-pills:after {
display: table;
content: "";
line-height: 0;
}
#backend .nav-tabs:after,
#backend .nav-pills:after {
clear: both;
}
#backend .nav-tabs > li,
#backend .nav-pills > li {
float: left;
}
#backend .nav-tabs > li > a,
#backend .nav-pills > li > a {
padding-right: 12px;
padding-left: 12px;
margin-right: 2px;
line-height: 14px;
}
#backend .nav-tabs {
border-bottom: 1px solid #ddd;
}
#backend .nav-tabs > li {
margin-bottom: -1px;
}
#backend .nav-tabs > li > a {
padding-top: 8px;
padding-bottom: 8px;
line-height: 18px;
border: 1px solid transparent;
-webkit-border-radius: 4px 4px 0 0;
-moz-border-radius: 4px 4px 0 0;
border-radius: 4px 4px 0 0;
}
#backend .nav-tabs > li > a:hover,
#backend .nav-tabs > li > a:focus {
border-color: #eeeeee #eeeeee #dddddd;
}
#backend .nav-tabs > .active > a,
#backend .nav-tabs > .active > a:hover,
#backend .nav-tabs > .active > a:focus {
color: #555555;
background-color: #ffffff;
border: 1px solid #ddd;
border-bottom-color: transparent;
cursor: default;
}
#backend .nav-pills > li > a {
padding-top: 8px;
padding-bottom: 8px;
margin-top: 2px;
margin-bottom: 2px;
-webkit-border-radius: 5px;
-moz-border-radius: 5px;
border-radius: 5px;
}
#backend .nav-pills > .active > a,
#backend .nav-pills > .active > a:hover,
#backend .nav-pills > .active > a:focus {
color: #ffffff;
background-color: #0088cc;
}
#backend .nav-stacked > li {
float: none;
}
#backend .nav-stacked > li > a {
margin-right: 0;
}
#backend .nav-tabs.nav-stacked {
border-bottom: 0;
}
#backend .nav-tabs.nav-stacked > li > a {
border: 1px solid #ddd;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .nav-tabs.nav-stacked > li:first-child > a {
-webkit-border-top-right-radius: 4px;
-moz-border-radius-topright: 4px;
border-top-right-radius: 4px;
-webkit-border-top-left-radius: 4px;
-moz-border-radius-topleft: 4px;
border-top-left-radius: 4px;
}
#backend .nav-tabs.nav-stacked > li:last-child > a {
-webkit-border-bottom-right-radius: 4px;
-moz-border-radius-bottomright: 4px;
border-bottom-right-radius: 4px;
-webkit-border-bottom-left-radius: 4px;
-moz-border-radius-bottomleft: 4px;
border-bottom-left-radius: 4px;
}
#backend .nav-tabs.nav-stacked > li > a:hover,
#backend .nav-tabs.nav-stacked > li > a:focus {
border-color: #ddd;
z-index: 2;
}
#backend .nav-pills.nav-stacked > li > a {
margin-bottom: 3px;
}
#backend .nav-pills.nav-stacked > li:last-child > a {
margin-bottom: 1px;
}
#backend .nav-tabs .dropdown-menu {
-webkit-border-radius: 0 0 6px 6px;
-moz-border-radius: 0 0 6px 6px;
border-radius: 0 0 6px 6px;
}
#backend .nav-pills .dropdown-menu {
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
}
#backend .nav .dropdown-toggle .caret {
border-top-color: #0088cc;
border-bottom-color: #0088cc;
margin-top: 6px;
}
#backend .nav .dropdown-toggle:hover .caret,
#backend .nav .dropdown-toggle:focus .caret {
border-top-color: #005580;
border-bottom-color: #005580;
}
#backend .nav-tabs .dropdown-toggle .caret {
margin-top: 8px;
}
#backend .nav .active .dropdown-toggle .caret {
border-top-color: #fff;
border-bottom-color: #fff;
}
#backend .nav-tabs .active .dropdown-toggle .caret {
border-top-color: #555555;
border-bottom-color: #555555;
}
#backend .nav > .dropdown.active > a:hover,
#backend .nav > .dropdown.active > a:focus {
cursor: pointer;
}
#backend .nav-tabs .open .dropdown-toggle,
#backend .nav-pills .open .dropdown-toggle,
#backend .nav > li.dropdown.open.active > a:hover,
#backend .nav > li.dropdown.open.active > a:focus {
color: #ffffff;
background-color: #999999;
border-color: #999999;
}
#backend .nav li.dropdown.open .caret,
#backend .nav li.dropdown.open.active .caret,
#backend .nav li.dropdown.open a:hover .caret,
#backend .nav li.dropdown.open a:focus .caret {
border-top-color: #ffffff;
border-bottom-color: #ffffff;
opacity: 1;
filter: alpha(opacity=100);
}
#backend .tabs-stacked .open > a:hover,
#backend .tabs-stacked .open > a:focus {
border-color: #999999;
}
#backend .tabbable {
*zoom: 1;
}
#backend .tabbable:before,
#backend .tabbable:after {
display: table;
content: "";
line-height: 0;
}
#backend .tabbable:after {
clear: both;
}
#backend .tabbable:before,
#backend .tabbable:after {
display: table;
content: "";
line-height: 0;
}
#backend .tabbable:after {
clear: both;
}
#backend .tab-content {
overflow: auto;
}
#backend .tabs-below > .nav-tabs,
#backend .tabs-right > .nav-tabs,
#backend .tabs-left > .nav-tabs {
border-bottom: 0;
}
#backend .tab-content > .tab-pane,
#backend .pill-content > .pill-pane {
display: none;
}
#backend .tab-content > .active,
#backend .pill-content > .active {
display: block;
}
#backend .tabs-below > .nav-tabs {
border-top: 1px solid #ddd;
}
#backend .tabs-below > .nav-tabs > li {
margin-top: -1px;
margin-bottom: 0;
}
#backend .tabs-below > .nav-tabs > li > a {
-webkit-border-radius: 0 0 4px 4px;
-moz-border-radius: 0 0 4px 4px;
border-radius: 0 0 4px 4px;
}
#backend .tabs-below > .nav-tabs > li > a:hover,
#backend .tabs-below > .nav-tabs > li > a:focus {
border-bottom-color: transparent;
border-top-color: #ddd;
}
#backend .tabs-below > .nav-tabs > .active > a,
#backend .tabs-below > .nav-tabs > .active > a:hover,
#backend .tabs-below > .nav-tabs > .active > a:focus {
border-color: transparent #ddd #ddd #ddd;
}
#backend .tabs-left > .nav-tabs > li,
#backend .tabs-right > .nav-tabs > li {
float: none;
}
#backend .tabs-left > .nav-tabs > li > a,
#backend .tabs-right > .nav-tabs > li > a {
min-width: 74px;
margin-right: 0;
margin-bottom: 3px;
}
#backend .tabs-left > .nav-tabs {
float: left;
margin-right: 19px;
border-right: 1px solid #ddd;
}
#backend .tabs-left > .nav-tabs > li > a {
margin-right: -1px;
-webkit-border-radius: 4px 0 0 4px;
-moz-border-radius: 4px 0 0 4px;
border-radius: 4px 0 0 4px;
}
#backend .tabs-left > .nav-tabs > li > a:hover,
#backend .tabs-left > .nav-tabs > li > a:focus {
border-color: #eeeeee #dddddd #eeeeee #eeeeee;
}
#backend .tabs-left > .nav-tabs .active > a,
#backend .tabs-left > .nav-tabs .active > a:hover,
#backend .tabs-left > .nav-tabs .active > a:focus {
border-color: #ddd transparent #ddd #ddd;
*border-right-color: #ffffff;
}
#backend .tabs-right > .nav-tabs {
float: right;
margin-left: 19px;
border-left: 1px solid #ddd;
}
#backend .tabs-right > .nav-tabs > li > a {
margin-left: -1px;
-webkit-border-radius: 0 4px 4px 0;
-moz-border-radius: 0 4px 4px 0;
border-radius: 0 4px 4px 0;
}
#backend .tabs-right > .nav-tabs > li > a:hover,
#backend .tabs-right > .nav-tabs > li > a:focus {
border-color: #eeeeee #eeeeee #eeeeee #dddddd;
}
#backend .tabs-right > .nav-tabs .active > a,
#backend .tabs-right > .nav-tabs .active > a:hover,
#backend .tabs-right > .nav-tabs .active > a:focus {
border-color: #ddd #ddd #ddd transparent;
*border-left-color: #ffffff;
}
#backend .nav > .disabled > a {
color: #999999;
}
#backend .nav > .disabled > a:hover,
#backend .nav > .disabled > a:focus {
text-decoration: none;
background-color: transparent;
cursor: default;
}
#backend .navbar {
overflow: visible;
margin-bottom: 18px;
*position: relative;
*z-index: 2;
}
#backend .navbar-inner {
min-height: 40px;
padding-left: 20px;
padding-right: 20px;
background-color: #2c2c2c;
background-image: -moz-linear-gradient(top, #333333, #222222);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#333333), to(#222222));
background-image: -webkit-linear-gradient(top, #333333, #222222);
background-image: -o-linear-gradient(top, #333333, #222222);
background-image: linear-gradient(to bottom, #333333, #222222);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff333333', endColorstr='#ff222222', GradientType=0);
border: 1px solid #030303;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
-webkit-box-shadow: 0 1px 4px rgba(0, 0, 0, 0.065);
-moz-box-shadow: 0 1px 4px rgba(0, 0, 0, 0.065);
box-shadow: 0 1px 4px rgba(0, 0, 0, 0.065);
*zoom: 1;
}
#backend .navbar-inner:before,
#backend .navbar-inner:after {
display: table;
content: "";
line-height: 0;
}
#backend .navbar-inner:after {
clear: both;
}
#backend .navbar-inner:before,
#backend .navbar-inner:after {
display: table;
content: "";
line-height: 0;
}
#backend .navbar-inner:after {
clear: both;
}
#backend .navbar .container {
width: auto;
}
#backend .nav-collapse.collapse {
height: auto;
overflow: visible;
}
#backend .navbar .brand {
float: left;
display: block;
padding: 11px 20px 11px;
margin-left: -20px;
font-size: 20px;
font-weight: 200;
color: #999999;
text-shadow: 0 1px 0 #333333;
}
#backend .navbar .brand:hover,
#backend .navbar .brand:focus {
text-decoration: none;
}
#backend .navbar-text {
margin-bottom: 0;
line-height: 40px;
color: #999999;
}
#backend .navbar-link {
color: #999999;
}
#backend .navbar-link:hover,
#backend .navbar-link:focus {
color: #ffffff;
}
#backend .navbar .divider-vertical {
height: 40px;
margin: 0 9px;
border-left: 1px solid #222222;
border-right: 1px solid #333333;
}
#backend .navbar .btn,
#backend .navbar .btn-group {
margin-top: 5px;
}
#backend .navbar .btn-group .btn,
#backend .navbar .input-prepend .btn,
#backend .navbar .input-append .btn,
#backend .navbar .input-prepend .btn-group,
#backend .navbar .input-append .btn-group {
margin-top: 0;
}
#backend .navbar-form {
margin-bottom: 0;
*zoom: 1;
}
#backend .navbar-form:before,
#backend .navbar-form:after {
display: table;
content: "";
line-height: 0;
}
#backend .navbar-form:after {
clear: both;
}
#backend .navbar-form:before,
#backend .navbar-form:after {
display: table;
content: "";
line-height: 0;
}
#backend .navbar-form:after {
clear: both;
}
#backend .navbar-form input,
#backend .navbar-form select,
#backend .navbar-form .radio,
#backend .navbar-form .checkbox {
margin-top: 5px;
}
#backend .navbar-form input,
#backend .navbar-form select,
#backend .navbar-form .btn {
display: inline-block;
margin-bottom: 0;
}
#backend .navbar-form input[type="image"],
#backend .navbar-form input[type="checkbox"],
#backend .navbar-form input[type="radio"] {
margin-top: 3px;
}
#backend .navbar-form .input-append,
#backend .navbar-form .input-prepend {
margin-top: 5px;
white-space: nowrap;
}
#backend .navbar-form .input-append input,
#backend .navbar-form .input-prepend input {
margin-top: 0;
}
#backend .navbar-search {
position: relative;
float: left;
margin-top: 5px;
margin-bottom: 0;
}
#backend .navbar-search .search-query {
margin-bottom: 0;
padding: 4px 14px;
font-family: "Helvetica Neue", Helvetica, Arial, sans-serif;
font-size: 13px;
font-weight: normal;
line-height: 1;
-webkit-border-radius: 15px;
-moz-border-radius: 15px;
border-radius: 15px;
}
#backend .navbar-static-top {
position: static;
margin-bottom: 0;
}
#backend .navbar-static-top .navbar-inner {
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .navbar-fixed-top,
#backend .navbar-fixed-bottom {
position: fixed;
right: 0;
left: 0;
z-index: 1030;
margin-bottom: 0;
}
#backend .navbar-fixed-top .navbar-inner,
#backend .navbar-static-top .navbar-inner {
border-width: 0 0 1px;
}
#backend .navbar-fixed-bottom .navbar-inner {
border-width: 1px 0 0;
}
#backend .navbar-fixed-top .navbar-inner,
#backend .navbar-fixed-bottom .navbar-inner {
padding-left: 0;
padding-right: 0;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
}
#backend .navbar-static-top .container,
#backend .navbar-fixed-top .container,
#backend .navbar-fixed-bottom .container {
width: 940px;
}
#backend .navbar-fixed-top {
top: 0;
}
#backend .navbar-fixed-top .navbar-inner,
#backend .navbar-static-top .navbar-inner {
-webkit-box-shadow: 0 1px 10px rgba(0,0,0,.1);
-moz-box-shadow: 0 1px 10px rgba(0,0,0,.1);
box-shadow: 0 1px 10px rgba(0,0,0,.1);
}
#backend .navbar-fixed-bottom {
bottom: 0;
}
#backend .navbar-fixed-bottom .navbar-inner {
-webkit-box-shadow: 0 -1px 10px rgba(0,0,0,.1);
-moz-box-shadow: 0 -1px 10px rgba(0,0,0,.1);
box-shadow: 0 -1px 10px rgba(0,0,0,.1);
}
#backend .navbar .nav {
position: relative;
left: 0;
display: block;
float: left;
margin: 0 10px 0 0;
}
#backend .navbar .nav.pull-right {
float: right;
margin-right: 0;
}
#backend .navbar .nav > li {
float: left;
}
#backend .navbar .nav > li > a {
float: none;
padding: 11px 15px 11px;
color: #999999;
text-decoration: none;
text-shadow: 0 1px 0 #333333;
}
#backend .navbar .nav .dropdown-toggle .caret {
margin-top: 8px;
}
#backend .navbar .nav > li > a:focus,
#backend .navbar .nav > li > a:hover {
background-color: transparent;
color: #ffffff;
text-decoration: none;
}
#backend .navbar .nav > .active > a,
#backend .navbar .nav > .active > a:hover,
#backend .navbar .nav > .active > a:focus {
color: #ffffff;
text-decoration: none;
background-color: #222222;
-webkit-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125);
-moz-box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125);
box-shadow: inset 0 3px 8px rgba(0, 0, 0, 0.125);
}
#backend .navbar .btn-navbar {
display: none;
float: right;
padding: 7px 10px;
margin-left: 5px;
margin-right: 5px;
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #1f1f1f;
background-image: -moz-linear-gradient(top, #262626, #151515);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#262626), to(#151515));
background-image: -webkit-linear-gradient(top, #262626, #151515);
background-image: -o-linear-gradient(top, #262626, #151515);
background-image: linear-gradient(to bottom, #262626, #151515);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff262626', endColorstr='#ff151515', GradientType=0);
border-color: #151515 #151515 #000000;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #151515;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
-webkit-box-shadow: inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.075);
-moz-box-shadow: inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.075);
box-shadow: inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.075);
}
#backend .navbar .btn-navbar:hover,
#backend .navbar .btn-navbar:focus,
#backend .navbar .btn-navbar:active,
#backend .navbar .btn-navbar.active,
#backend .navbar .btn-navbar.disabled,
#backend .navbar .btn-navbar[disabled] {
color: #ffffff;
background-color: #151515;
*background-color: #080808;
}
#backend .navbar .btn-navbar:active,
#backend .navbar .btn-navbar.active {
background-color: #000000 \9;
}
#backend .navbar .btn-navbar:hover,
#backend .navbar .btn-navbar:focus,
#backend .navbar .btn-navbar:active,
#backend .navbar .btn-navbar.active,
#backend .navbar .btn-navbar.disabled,
#backend .navbar .btn-navbar[disabled] {
color: #ffffff;
background-color: #151515;
*background-color: #080808;
}
#backend .navbar .btn-navbar:active,
#backend .navbar .btn-navbar.active {
background-color: #000000 \9;
}
#backend .navbar .btn-navbar .icon-bar {
display: block;
width: 18px;
height: 2px;
background-color: #f5f5f5;
-webkit-border-radius: 1px;
-moz-border-radius: 1px;
border-radius: 1px;
-webkit-box-shadow: 0 1px 0 rgba(0, 0, 0, 0.25);
-moz-box-shadow: 0 1px 0 rgba(0, 0, 0, 0.25);
box-shadow: 0 1px 0 rgba(0, 0, 0, 0.25);
}
#backend .btn-navbar .icon-bar + .icon-bar {
margin-top: 3px;
}
#backend .navbar .nav > li > .dropdown-menu:before {
content: '';
display: inline-block;
border-left: 7px solid transparent;
border-right: 7px solid transparent;
border-bottom: 7px solid #ccc;
border-bottom-color: rgba(0, 0, 0, 0.2);
position: absolute;
top: -7px;
left: 9px;
}
#backend .navbar .nav > li > .dropdown-menu:after {
content: '';
display: inline-block;
border-left: 6px solid transparent;
border-right: 6px solid transparent;
border-bottom: 6px solid #ffffff;
position: absolute;
top: -6px;
left: 10px;
}
#backend .navbar-fixed-bottom .nav > li > .dropdown-menu:before {
border-top: 7px solid #ccc;
border-top-color: rgba(0, 0, 0, 0.2);
border-bottom: 0;
bottom: -7px;
top: auto;
}
#backend .navbar-fixed-bottom .nav > li > .dropdown-menu:after {
border-top: 6px solid #ffffff;
border-bottom: 0;
bottom: -6px;
top: auto;
}
#backend .navbar .nav li.dropdown > a:hover .caret,
#backend .navbar .nav li.dropdown > a:focus .caret {
border-top-color: #ffffff;
border-bottom-color: #ffffff;
}
#backend .navbar .nav li.dropdown.open > .dropdown-toggle,
#backend .navbar .nav li.dropdown.active > .dropdown-toggle,
#backend .navbar .nav li.dropdown.open.active > .dropdown-toggle {
background-color: #222222;
color: #ffffff;
}
#backend .navbar .nav li.dropdown > .dropdown-toggle .caret {
border-top-color: #999999;
border-bottom-color: #999999;
}
#backend .navbar .nav li.dropdown.open > .dropdown-toggle .caret,
#backend .navbar .nav li.dropdown.active > .dropdown-toggle .caret,
#backend .navbar .nav li.dropdown.open.active > .dropdown-toggle .caret {
border-top-color: #ffffff;
border-bottom-color: #ffffff;
}
#backend .navbar .pull-right > li > .dropdown-menu,
#backend .navbar .nav > li > .dropdown-menu.pull-right {
left: auto;
right: 0;
}
#backend .navbar .pull-right > li > .dropdown-menu:before,
#backend .navbar .nav > li > .dropdown-menu.pull-right:before {
left: auto;
right: 12px;
}
#backend .navbar .pull-right > li > .dropdown-menu:after,
#backend .navbar .nav > li > .dropdown-menu.pull-right:after {
left: auto;
right: 13px;
}
#backend .navbar .pull-right > li > .dropdown-menu .dropdown-menu,
#backend .navbar .nav > li > .dropdown-menu.pull-right .dropdown-menu {
left: auto;
right: 100%;
margin-left: 0;
margin-right: -1px;
-webkit-border-radius: 6px 0 6px 6px;
-moz-border-radius: 6px 0 6px 6px;
border-radius: 6px 0 6px 6px;
}
#backend .navbar-inverse .navbar-inner {
background-color: #1b1b1b;
background-image: -moz-linear-gradient(top, #222222, #111111);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#222222), to(#111111));
background-image: -webkit-linear-gradient(top, #222222, #111111);
background-image: -o-linear-gradient(top, #222222, #111111);
background-image: linear-gradient(to bottom, #222222, #111111);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff222222', endColorstr='#ff111111', GradientType=0);
border-color: #252525;
}
#backend .navbar-inverse .brand,
#backend .navbar-inverse .nav > li > a {
color: #999999;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
}
#backend .navbar-inverse .brand:hover,
#backend .navbar-inverse .nav > li > a:hover,
#backend .navbar-inverse .brand:focus,
#backend .navbar-inverse .nav > li > a:focus {
color: #ffffff;
}
#backend .navbar-inverse .brand {
color: #999999;
}
#backend .navbar-inverse .navbar-text {
color: #999999;
}
#backend .navbar-inverse .nav > li > a:focus,
#backend .navbar-inverse .nav > li > a:hover {
background-color: transparent;
color: #ffffff;
}
#backend .navbar-inverse .nav .active > a,
#backend .navbar-inverse .nav .active > a:hover,
#backend .navbar-inverse .nav .active > a:focus {
color: #ffffff;
background-color: #111111;
}
#backend .navbar-inverse .navbar-link {
color: #999999;
}
#backend .navbar-inverse .navbar-link:hover,
#backend .navbar-inverse .navbar-link:focus {
color: #ffffff;
}
#backend .navbar-inverse .divider-vertical {
border-left-color: #111111;
border-right-color: #222222;
}
#backend .navbar-inverse .nav li.dropdown.open > .dropdown-toggle,
#backend .navbar-inverse .nav li.dropdown.active > .dropdown-toggle,
#backend .navbar-inverse .nav li.dropdown.open.active > .dropdown-toggle {
background-color: #111111;
color: #ffffff;
}
#backend .navbar-inverse .nav li.dropdown > a:hover .caret,
#backend .navbar-inverse .nav li.dropdown > a:focus .caret {
border-top-color: #ffffff;
border-bottom-color: #ffffff;
}
#backend .navbar-inverse .nav li.dropdown > .dropdown-toggle .caret {
border-top-color: #999999;
border-bottom-color: #999999;
}
#backend .navbar-inverse .nav li.dropdown.open > .dropdown-toggle .caret,
#backend .navbar-inverse .nav li.dropdown.active > .dropdown-toggle .caret,
#backend .navbar-inverse .nav li.dropdown.open.active > .dropdown-toggle .caret {
border-top-color: #ffffff;
border-bottom-color: #ffffff;
}
#backend .navbar-inverse .navbar-search .search-query {
color: #ffffff;
background-color: #515151;
border-color: #111111;
-webkit-box-shadow: inset 0 1px 2px rgba(0,0,0,.1), 0 1px 0 rgba(255,255,255,.15);
-moz-box-shadow: inset 0 1px 2px rgba(0,0,0,.1), 0 1px 0 rgba(255,255,255,.15);
box-shadow: inset 0 1px 2px rgba(0,0,0,.1), 0 1px 0 rgba(255,255,255,.15);
-webkit-transition: none;
-moz-transition: none;
-o-transition: none;
transition: none;
}
#backend .navbar-inverse .navbar-search .search-query:-moz-placeholder {
color: #cccccc;
}
#backend .navbar-inverse .navbar-search .search-query:-ms-input-placeholder {
color: #cccccc;
}
#backend .navbar-inverse .navbar-search .search-query::-webkit-input-placeholder {
color: #cccccc;
}
#backend .navbar-inverse .navbar-search .search-query:-moz-placeholder {
color: #cccccc;
}
#backend .navbar-inverse .navbar-search .search-query:-ms-input-placeholder {
color: #cccccc;
}
#backend .navbar-inverse .navbar-search .search-query::-webkit-input-placeholder {
color: #cccccc;
}
#backend .navbar-inverse .navbar-search .search-query:focus,
#backend .navbar-inverse .navbar-search .search-query.focused {
padding: 5px 15px;
color: #333333;
text-shadow: 0 1px 0 #ffffff;
background-color: #ffffff;
border: 0;
-webkit-box-shadow: 0 0 3px rgba(0, 0, 0, 0.15);
-moz-box-shadow: 0 0 3px rgba(0, 0, 0, 0.15);
box-shadow: 0 0 3px rgba(0, 0, 0, 0.15);
outline: 0;
}
#backend .navbar-inverse .btn-navbar {
color: #ffffff;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #0e0e0e;
background-image: -moz-linear-gradient(top, #151515, #040404);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#151515), to(#040404));
background-image: -webkit-linear-gradient(top, #151515, #040404);
background-image: -o-linear-gradient(top, #151515, #040404);
background-image: linear-gradient(to bottom, #151515, #040404);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff151515', endColorstr='#ff040404', GradientType=0);
border-color: #040404 #040404 #000000;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #040404;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
}
#backend .navbar-inverse .btn-navbar:hover,
#backend .navbar-inverse .btn-navbar:focus,
#backend .navbar-inverse .btn-navbar:active,
#backend .navbar-inverse .btn-navbar.active,
#backend .navbar-inverse .btn-navbar.disabled,
#backend .navbar-inverse .btn-navbar[disabled] {
color: #ffffff;
background-color: #040404;
*background-color: #000000;
}
#backend .navbar-inverse .btn-navbar:active,
#backend .navbar-inverse .btn-navbar.active {
background-color: #000000 \9;
}
#backend .navbar-inverse .btn-navbar:hover,
#backend .navbar-inverse .btn-navbar:focus,
#backend .navbar-inverse .btn-navbar:active,
#backend .navbar-inverse .btn-navbar.active,
#backend .navbar-inverse .btn-navbar.disabled,
#backend .navbar-inverse .btn-navbar[disabled] {
color: #ffffff;
background-color: #040404;
*background-color: #000000;
}
#backend .navbar-inverse .btn-navbar:active,
#backend .navbar-inverse .btn-navbar.active {
background-color: #000000 \9;
}
#backend .breadcrumb {
padding: 8px 15px;
margin: 0 0 18px;
list-style: none;
background-color: #f5f5f5;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .breadcrumb > li {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
text-shadow: 0 1px 0 #ffffff;
}
#backend .breadcrumb > li > .divider {
padding: 0 5px;
color: #ccc;
}
#backend .breadcrumb > .active {
color: #999999;
}
#backend .pagination {
margin: 18px 0;
}
#backend .pagination ul {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
margin-left: 0;
margin-bottom: 0;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
-webkit-box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05);
-moz-box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05);
box-shadow: 0 1px 2px rgba(0, 0, 0, 0.05);
}
#backend .pagination ul > li {
display: inline;
}
#backend .pagination ul > li > a,
#backend .pagination ul > li > span {
float: left;
padding: 4px 12px;
line-height: 18px;
text-decoration: none;
background-color: #ffffff;
border: 1px solid #dddddd;
border-left-width: 0;
}
#backend .pagination ul > li > a:hover,
#backend .pagination ul > li > a:focus,
#backend .pagination ul > .active > a,
#backend .pagination ul > .active > span {
background-color: #f5f5f5;
}
#backend .pagination ul > .active > a,
#backend .pagination ul > .active > span {
color: #999999;
cursor: default;
}
#backend .pagination ul > .disabled > span,
#backend .pagination ul > .disabled > a,
#backend .pagination ul > .disabled > a:hover,
#backend .pagination ul > .disabled > a:focus {
color: #999999;
background-color: transparent;
cursor: default;
}
#backend .pagination ul > li:first-child > a,
#backend .pagination ul > li:first-child > span {
border-left-width: 1px;
-webkit-border-top-left-radius: 4px;
-moz-border-radius-topleft: 4px;
border-top-left-radius: 4px;
-webkit-border-bottom-left-radius: 4px;
-moz-border-radius-bottomleft: 4px;
border-bottom-left-radius: 4px;
}
#backend .pagination ul > li:last-child > a,
#backend .pagination ul > li:last-child > span {
-webkit-border-top-right-radius: 4px;
-moz-border-radius-topright: 4px;
border-top-right-radius: 4px;
-webkit-border-bottom-right-radius: 4px;
-moz-border-radius-bottomright: 4px;
border-bottom-right-radius: 4px;
}
#backend .pagination-centered {
text-align: center;
}
#backend .pagination-right {
text-align: right;
}
#backend .pagination-large ul > li > a,
#backend .pagination-large ul > li > span {
padding: 11px 19px;
font-size: 16.25px;
}
#backend .pagination-large ul > li:first-child > a,
#backend .pagination-large ul > li:first-child > span {
-webkit-border-top-left-radius: 6px;
-moz-border-radius-topleft: 6px;
border-top-left-radius: 6px;
-webkit-border-bottom-left-radius: 6px;
-moz-border-radius-bottomleft: 6px;
border-bottom-left-radius: 6px;
}
#backend .pagination-large ul > li:last-child > a,
#backend .pagination-large ul > li:last-child > span {
-webkit-border-top-right-radius: 6px;
-moz-border-radius-topright: 6px;
border-top-right-radius: 6px;
-webkit-border-bottom-right-radius: 6px;
-moz-border-radius-bottomright: 6px;
border-bottom-right-radius: 6px;
}
#backend .pagination-mini ul > li:first-child > a,
#backend .pagination-small ul > li:first-child > a,
#backend .pagination-mini ul > li:first-child > span,
#backend .pagination-small ul > li:first-child > span {
-webkit-border-top-left-radius: 3px;
-moz-border-radius-topleft: 3px;
border-top-left-radius: 3px;
-webkit-border-bottom-left-radius: 3px;
-moz-border-radius-bottomleft: 3px;
border-bottom-left-radius: 3px;
}
#backend .pagination-mini ul > li:last-child > a,
#backend .pagination-small ul > li:last-child > a,
#backend .pagination-mini ul > li:last-child > span,
#backend .pagination-small ul > li:last-child > span {
-webkit-border-top-right-radius: 3px;
-moz-border-radius-topright: 3px;
border-top-right-radius: 3px;
-webkit-border-bottom-right-radius: 3px;
-moz-border-radius-bottomright: 3px;
border-bottom-right-radius: 3px;
}
#backend .pagination-small ul > li > a,
#backend .pagination-small ul > li > span {
padding: 2px 10px;
font-size: 11.049999999999999px;
}
#backend .pagination-mini ul > li > a,
#backend .pagination-mini ul > li > span {
padding: 0 6px;
font-size: 9.75px;
}
#backend .pager {
margin: 18px 0;
list-style: none;
text-align: center;
*zoom: 1;
}
#backend .pager:before,
#backend .pager:after {
display: table;
content: "";
line-height: 0;
}
#backend .pager:after {
clear: both;
}
#backend .pager:before,
#backend .pager:after {
display: table;
content: "";
line-height: 0;
}
#backend .pager:after {
clear: both;
}
#backend .pager li {
display: inline;
}
#backend .pager li > a,
#backend .pager li > span {
display: inline-block;
padding: 5px 14px;
background-color: #fff;
border: 1px solid #ddd;
-webkit-border-radius: 15px;
-moz-border-radius: 15px;
border-radius: 15px;
}
#backend .pager li > a:hover,
#backend .pager li > a:focus {
text-decoration: none;
background-color: #f5f5f5;
}
#backend .pager .next > a,
#backend .pager .next > span {
float: right;
}
#backend .pager .previous > a,
#backend .pager .previous > span {
float: left;
}
#backend .pager .disabled > a,
#backend .pager .disabled > a:hover,
#backend .pager .disabled > a:focus,
#backend .pager .disabled > span {
color: #999999;
background-color: #fff;
cursor: default;
}
#backend .modal-backdrop {
position: fixed;
top: 0;
right: 0;
bottom: 0;
left: 0;
z-index: 1040;
background-color: #000000;
}
#backend .modal-backdrop.fade {
opacity: 0;
}
#backend .modal-backdrop,
#backend .modal-backdrop.fade.in {
opacity: 0.8;
filter: alpha(opacity=80);
}
#backend .modal {
position: fixed;
top: 10%;
left: 50%;
z-index: 1050;
width: 560px;
margin-left: -280px;
background-color: #ffffff;
border: 1px solid #999;
border: 1px solid rgba(0, 0, 0, 0.3);
*border: 1px solid #999;
/* IE6-7 */
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
-webkit-box-shadow: 0 3px 7px rgba(0, 0, 0, 0.3);
-moz-box-shadow: 0 3px 7px rgba(0, 0, 0, 0.3);
box-shadow: 0 3px 7px rgba(0, 0, 0, 0.3);
-webkit-background-clip: padding-box;
-moz-background-clip: padding-box;
background-clip: padding-box;
outline: none;
}
#backend .modal.fade {
-webkit-transition: opacity .3s linear, top .3s ease-out;
-moz-transition: opacity .3s linear, top .3s ease-out;
-o-transition: opacity .3s linear, top .3s ease-out;
transition: opacity .3s linear, top .3s ease-out;
top: -25%;
}
#backend .modal.fade.in {
top: 10%;
}
#backend .modal-header {
padding: 9px 15px;
border-bottom: 1px solid #eee;
}
#backend .modal-header .close {
margin-top: 2px;
}
#backend .modal-header h3 {
margin: 0;
line-height: 30px;
}
#backend .modal-body {
position: relative;
overflow-y: auto;
max-height: 400px;
padding: 15px;
}
#backend .modal-form {
margin-bottom: 0;
}
#backend .modal-footer {
padding: 14px 15px 15px;
margin-bottom: 0;
text-align: right;
background-color: #f5f5f5;
border-top: 1px solid #ddd;
-webkit-border-radius: 0 0 6px 6px;
-moz-border-radius: 0 0 6px 6px;
border-radius: 0 0 6px 6px;
-webkit-box-shadow: inset 0 1px 0 #ffffff;
-moz-box-shadow: inset 0 1px 0 #ffffff;
box-shadow: inset 0 1px 0 #ffffff;
*zoom: 1;
}
#backend .modal-footer:before,
#backend .modal-footer:after {
display: table;
content: "";
line-height: 0;
}
#backend .modal-footer:after {
clear: both;
}
#backend .modal-footer:before,
#backend .modal-footer:after {
display: table;
content: "";
line-height: 0;
}
#backend .modal-footer:after {
clear: both;
}
#backend .modal-footer .btn + .btn {
margin-left: 5px;
margin-bottom: 0;
}
#backend .modal-footer .btn-group .btn + .btn {
margin-left: -1px;
}
#backend .modal-footer .btn-block + .btn-block {
margin-left: 0;
}
#backend .tooltip {
position: absolute;
z-index: 1020;
display: block;
visibility: visible;
font-size: 11px;
line-height: 1.4;
opacity: 0;
filter: alpha(opacity=0);
}
#backend .tooltip.in {
opacity: 0.8;
filter: alpha(opacity=80);
}
#backend .tooltip.top {
margin-top: -3px;
padding: 5px 0;
}
#backend .tooltip.right {
margin-left: 3px;
padding: 0 5px;
}
#backend .tooltip.bottom {
margin-top: 3px;
padding: 5px 0;
}
#backend .tooltip.left {
margin-left: -3px;
padding: 0 5px;
}
#backend .tooltip-inner {
max-width: 200px;
padding: 8px;
color: #ffffff;
text-align: center;
text-decoration: none;
background-color: #000000;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .tooltip-arrow {
position: absolute;
width: 0;
height: 0;
border-color: transparent;
border-style: solid;
}
#backend .tooltip.top .tooltip-arrow {
bottom: 0;
left: 50%;
margin-left: -5px;
border-width: 5px 5px 0;
border-top-color: #000000;
}
#backend .tooltip.right .tooltip-arrow {
top: 50%;
left: 0;
margin-top: -5px;
border-width: 5px 5px 5px 0;
border-right-color: #000000;
}
#backend .tooltip.left .tooltip-arrow {
top: 50%;
right: 0;
margin-top: -5px;
border-width: 5px 0 5px 5px;
border-left-color: #000000;
}
#backend .tooltip.bottom .tooltip-arrow {
top: 0;
left: 50%;
margin-left: -5px;
border-width: 0 5px 5px;
border-bottom-color: #000000;
}
#backend .popover {
position: absolute;
top: 0;
left: 0;
z-index: 1010;
display: none;
max-width: 276px;
padding: 1px;
text-align: left;
background-color: #ffffff;
-webkit-background-clip: padding-box;
-moz-background-clip: padding;
background-clip: padding-box;
border: 1px solid #ccc;
border: 1px solid rgba(0, 0, 0, 0.2);
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
-webkit-box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2);
-moz-box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2);
box-shadow: 0 5px 10px rgba(0, 0, 0, 0.2);
white-space: normal;
}
#backend .popover.top {
margin-top: -10px;
}
#backend .popover.right {
margin-left: 10px;
}
#backend .popover.bottom {
margin-top: 10px;
}
#backend .popover.left {
margin-left: -10px;
}
#backend .popover-title {
margin: 0;
padding: 8px 14px;
font-size: 14px;
font-weight: normal;
line-height: 18px;
background-color: #f7f7f7;
border-bottom: 1px solid #ebebeb;
-webkit-border-radius: 5px 5px 0 0;
-moz-border-radius: 5px 5px 0 0;
border-radius: 5px 5px 0 0;
}
#backend .popover-title:empty {
display: none;
}
#backend .popover-content {
padding: 9px 14px;
}
#backend .popover .arrow,
#backend .popover .arrow:after {
position: absolute;
display: block;
width: 0;
height: 0;
border-color: transparent;
border-style: solid;
}
#backend .popover .arrow {
border-width: 11px;
}
#backend .popover .arrow:after {
border-width: 10px;
content: "";
}
#backend .popover.top .arrow {
left: 50%;
margin-left: -11px;
border-bottom-width: 0;
border-top-color: #999;
border-top-color: rgba(0, 0, 0, 0.25);
bottom: -11px;
}
#backend .popover.top .arrow:after {
bottom: 1px;
margin-left: -10px;
border-bottom-width: 0;
border-top-color: #ffffff;
}
#backend .popover.right .arrow {
top: 50%;
left: -11px;
margin-top: -11px;
border-left-width: 0;
border-right-color: #999;
border-right-color: rgba(0, 0, 0, 0.25);
}
#backend .popover.right .arrow:after {
left: 1px;
bottom: -10px;
border-left-width: 0;
border-right-color: #ffffff;
}
#backend .popover.bottom .arrow {
left: 50%;
margin-left: -11px;
border-top-width: 0;
border-bottom-color: #999;
border-bottom-color: rgba(0, 0, 0, 0.25);
top: -11px;
}
#backend .popover.bottom .arrow:after {
top: 1px;
margin-left: -10px;
border-top-width: 0;
border-bottom-color: #ffffff;
}
#backend .popover.left .arrow {
top: 50%;
right: -11px;
margin-top: -11px;
border-right-width: 0;
border-left-color: #999;
border-left-color: rgba(0, 0, 0, 0.25);
}
#backend .popover.left .arrow:after {
right: 1px;
border-right-width: 0;
border-left-color: #ffffff;
bottom: -10px;
}
#backend .thumbnails {
margin-left: -20px;
list-style: none;
*zoom: 1;
}
#backend .thumbnails:before,
#backend .thumbnails:after {
display: table;
content: "";
line-height: 0;
}
#backend .thumbnails:after {
clear: both;
}
#backend .thumbnails:before,
#backend .thumbnails:after {
display: table;
content: "";
line-height: 0;
}
#backend .thumbnails:after {
clear: both;
}
#backend .row-fluid .thumbnails {
margin-left: 0;
}
#backend .thumbnails > li {
float: left;
margin-bottom: 18px;
margin-left: 20px;
}
#backend .thumbnail {
display: block;
padding: 4px;
line-height: 18px;
border: 1px solid #ddd;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
-webkit-box-shadow: 0 1px 3px rgba(0, 0, 0, 0.055);
-moz-box-shadow: 0 1px 3px rgba(0, 0, 0, 0.055);
box-shadow: 0 1px 3px rgba(0, 0, 0, 0.055);
-webkit-transition: all 0.2s ease-in-out;
-moz-transition: all 0.2s ease-in-out;
-o-transition: all 0.2s ease-in-out;
transition: all 0.2s ease-in-out;
}
#backend a.thumbnail:hover,
#backend a.thumbnail:focus {
border-color: #0088cc;
-webkit-box-shadow: 0 1px 4px rgba(0, 105, 214, 0.25);
-moz-box-shadow: 0 1px 4px rgba(0, 105, 214, 0.25);
box-shadow: 0 1px 4px rgba(0, 105, 214, 0.25);
}
#backend .thumbnail > img {
display: block;
max-width: 100%;
margin-left: auto;
margin-right: auto;
}
#backend .thumbnail .caption {
padding: 9px;
color: #555555;
}
#backend .media,
#backend .media-body {
overflow: hidden;
*overflow: visible;
zoom: 1;
}
#backend .media,
#backend .media .media {
margin-top: 15px;
}
#backend .media:first-child {
margin-top: 0;
}
#backend .media-object {
display: block;
}
#backend .media-heading {
margin: 0 0 5px;
}
#backend .media > .pull-left {
margin-right: 10px;
}
#backend .media > .pull-right {
margin-left: 10px;
}
#backend .media-list {
margin-left: 0;
list-style: none;
}
#backend .label,
#backend .badge {
display: inline-block;
padding: 2px 4px;
font-size: 10.998px;
font-weight: bold;
line-height: 14px;
color: #ffffff;
vertical-align: baseline;
white-space: nowrap;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #999999;
}
#backend .label {
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
}
#backend .badge {
padding-left: 9px;
padding-right: 9px;
-webkit-border-radius: 9px;
-moz-border-radius: 9px;
border-radius: 9px;
}
#backend .label:empty,
#backend .badge:empty {
display: none;
}
#backend a.label:hover,
#backend a.label:focus,
#backend a.badge:hover,
#backend a.badge:focus {
color: #ffffff;
text-decoration: none;
cursor: pointer;
}
#backend .label-important,
#backend .badge-important {
background-color: #b94a48;
}
#backend .label-important[href],
#backend .badge-important[href] {
background-color: #953b39;
}
#backend .label-warning,
#backend .badge-warning {
background-color: #f89406;
}
#backend .label-warning[href],
#backend .badge-warning[href] {
background-color: #c67605;
}
#backend .label-success,
#backend .badge-success {
background-color: #468847;
}
#backend .label-success[href],
#backend .badge-success[href] {
background-color: #356635;
}
#backend .label-info,
#backend .badge-info {
background-color: #3a87ad;
}
#backend .label-info[href],
#backend .badge-info[href] {
background-color: #2d6987;
}
#backend .label-inverse,
#backend .badge-inverse {
background-color: #333333;
}
#backend .label-inverse[href],
#backend .badge-inverse[href] {
background-color: #1a1a1a;
}
#backend .btn .label,
#backend .btn .badge {
position: relative;
top: -1px;
}
#backend .btn-mini .label,
#backend .btn-mini .badge {
top: 0;
}
@-webkit-keyframes progress-bar-stripes {
from {
background-position: 40px 0;
}
to {
background-position: 0 0;
}
}
@-moz-keyframes progress-bar-stripes {
from {
background-position: 40px 0;
}
to {
background-position: 0 0;
}
}
@-ms-keyframes progress-bar-stripes {
from {
background-position: 40px 0;
}
to {
background-position: 0 0;
}
}
@-o-keyframes progress-bar-stripes {
from {
background-position: 0 0;
}
to {
background-position: 40px 0;
}
}
@keyframes progress-bar-stripes {
from {
background-position: 40px 0;
}
to {
background-position: 0 0;
}
}
#backend .progress {
overflow: hidden;
height: 18px;
margin-bottom: 18px;
background-color: #f7f7f7;
background-image: -moz-linear-gradient(top, #f5f5f5, #f9f9f9);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#f5f5f5), to(#f9f9f9));
background-image: -webkit-linear-gradient(top, #f5f5f5, #f9f9f9);
background-image: -o-linear-gradient(top, #f5f5f5, #f9f9f9);
background-image: linear-gradient(to bottom, #f5f5f5, #f9f9f9);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fff5f5f5', endColorstr='#fff9f9f9', GradientType=0);
-webkit-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.1);
-moz-box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.1);
box-shadow: inset 0 1px 2px rgba(0, 0, 0, 0.1);
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .progress .bar {
width: 0%;
height: 100%;
color: #ffffff;
float: left;
font-size: 12px;
text-align: center;
text-shadow: 0 -1px 0 rgba(0, 0, 0, 0.25);
background-color: #0e90d2;
background-image: -moz-linear-gradient(top, #149bdf, #0480be);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#149bdf), to(#0480be));
background-image: -webkit-linear-gradient(top, #149bdf, #0480be);
background-image: -o-linear-gradient(top, #149bdf, #0480be);
background-image: linear-gradient(to bottom, #149bdf, #0480be);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff149bdf', endColorstr='#ff0480be', GradientType=0);
-webkit-box-shadow: inset 0 -1px 0 rgba(0, 0, 0, 0.15);
-moz-box-shadow: inset 0 -1px 0 rgba(0, 0, 0, 0.15);
box-shadow: inset 0 -1px 0 rgba(0, 0, 0, 0.15);
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
-webkit-transition: width 0.6s ease;
-moz-transition: width 0.6s ease;
-o-transition: width 0.6s ease;
transition: width 0.6s ease;
}
#backend .progress .bar + .bar {
-webkit-box-shadow: inset 1px 0 0 rgba(0,0,0,.15), inset 0 -1px 0 rgba(0,0,0,.15);
-moz-box-shadow: inset 1px 0 0 rgba(0,0,0,.15), inset 0 -1px 0 rgba(0,0,0,.15);
box-shadow: inset 1px 0 0 rgba(0,0,0,.15), inset 0 -1px 0 rgba(0,0,0,.15);
}
#backend .progress-striped .bar {
background-color: #149bdf;
background-image: -webkit-gradient(linear, 0 100%, 100% 0, color-stop(0.25, rgba(255, 255, 255, 0.15)), color-stop(0.25, transparent), color-stop(0.5, transparent), color-stop(0.5, rgba(255, 255, 255, 0.15)), color-stop(0.75, rgba(255, 255, 255, 0.15)), color-stop(0.75, transparent), to(transparent));
background-image: -webkit-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -moz-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -o-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
-webkit-background-size: 40px 40px;
-moz-background-size: 40px 40px;
-o-background-size: 40px 40px;
background-size: 40px 40px;
}
#backend .progress.active .bar {
-webkit-animation: progress-bar-stripes 2s linear infinite;
-moz-animation: progress-bar-stripes 2s linear infinite;
-ms-animation: progress-bar-stripes 2s linear infinite;
-o-animation: progress-bar-stripes 2s linear infinite;
animation: progress-bar-stripes 2s linear infinite;
}
#backend .progress-danger .bar,
#backend .progress .bar-danger {
background-color: #dd514c;
background-image: -moz-linear-gradient(top, #ee5f5b, #c43c35);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#ee5f5b), to(#c43c35));
background-image: -webkit-linear-gradient(top, #ee5f5b, #c43c35);
background-image: -o-linear-gradient(top, #ee5f5b, #c43c35);
background-image: linear-gradient(to bottom, #ee5f5b, #c43c35);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffee5f5b', endColorstr='#ffc43c35', GradientType=0);
}
#backend .progress-danger.progress-striped .bar,
#backend .progress-striped .bar-danger {
background-color: #ee5f5b;
background-image: -webkit-gradient(linear, 0 100%, 100% 0, color-stop(0.25, rgba(255, 255, 255, 0.15)), color-stop(0.25, transparent), color-stop(0.5, transparent), color-stop(0.5, rgba(255, 255, 255, 0.15)), color-stop(0.75, rgba(255, 255, 255, 0.15)), color-stop(0.75, transparent), to(transparent));
background-image: -webkit-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -moz-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -o-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
}
#backend .progress-success .bar,
#backend .progress .bar-success {
background-color: #5eb95e;
background-image: -moz-linear-gradient(top, #62c462, #57a957);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#62c462), to(#57a957));
background-image: -webkit-linear-gradient(top, #62c462, #57a957);
background-image: -o-linear-gradient(top, #62c462, #57a957);
background-image: linear-gradient(to bottom, #62c462, #57a957);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff62c462', endColorstr='#ff57a957', GradientType=0);
}
#backend .progress-success.progress-striped .bar,
#backend .progress-striped .bar-success {
background-color: #62c462;
background-image: -webkit-gradient(linear, 0 100%, 100% 0, color-stop(0.25, rgba(255, 255, 255, 0.15)), color-stop(0.25, transparent), color-stop(0.5, transparent), color-stop(0.5, rgba(255, 255, 255, 0.15)), color-stop(0.75, rgba(255, 255, 255, 0.15)), color-stop(0.75, transparent), to(transparent));
background-image: -webkit-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -moz-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -o-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
}
#backend .progress-info .bar,
#backend .progress .bar-info {
background-color: #4bb1cf;
background-image: -moz-linear-gradient(top, #5bc0de, #339bb9);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#5bc0de), to(#339bb9));
background-image: -webkit-linear-gradient(top, #5bc0de, #339bb9);
background-image: -o-linear-gradient(top, #5bc0de, #339bb9);
background-image: linear-gradient(to bottom, #5bc0de, #339bb9);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ff5bc0de', endColorstr='#ff339bb9', GradientType=0);
}
#backend .progress-info.progress-striped .bar,
#backend .progress-striped .bar-info {
background-color: #5bc0de;
background-image: -webkit-gradient(linear, 0 100%, 100% 0, color-stop(0.25, rgba(255, 255, 255, 0.15)), color-stop(0.25, transparent), color-stop(0.5, transparent), color-stop(0.5, rgba(255, 255, 255, 0.15)), color-stop(0.75, rgba(255, 255, 255, 0.15)), color-stop(0.75, transparent), to(transparent));
background-image: -webkit-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -moz-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -o-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
}
#backend .progress-warning .bar,
#backend .progress .bar-warning {
background-color: #faa732;
background-image: -moz-linear-gradient(top, #fbb450, #f89406);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#fbb450), to(#f89406));
background-image: -webkit-linear-gradient(top, #fbb450, #f89406);
background-image: -o-linear-gradient(top, #fbb450, #f89406);
background-image: linear-gradient(to bottom, #fbb450, #f89406);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#fffbb450', endColorstr='#fff89406', GradientType=0);
}
#backend .progress-warning.progress-striped .bar,
#backend .progress-striped .bar-warning {
background-color: #fbb450;
background-image: -webkit-gradient(linear, 0 100%, 100% 0, color-stop(0.25, rgba(255, 255, 255, 0.15)), color-stop(0.25, transparent), color-stop(0.5, transparent), color-stop(0.5, rgba(255, 255, 255, 0.15)), color-stop(0.75, rgba(255, 255, 255, 0.15)), color-stop(0.75, transparent), to(transparent));
background-image: -webkit-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -moz-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: -o-linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
background-image: linear-gradient(45deg, rgba(255, 255, 255, 0.15) 25%, transparent 25%, transparent 50%, rgba(255, 255, 255, 0.15) 50%, rgba(255, 255, 255, 0.15) 75%, transparent 75%, transparent);
}
#backend .accordion {
margin-bottom: 18px;
}
#backend .accordion-group {
margin-bottom: 2px;
border: 1px solid #e5e5e5;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .accordion-heading {
border-bottom: 0;
}
#backend .accordion-heading .accordion-toggle {
display: block;
padding: 8px 15px;
}
#backend .accordion-toggle {
cursor: pointer;
}
#backend .accordion-inner {
padding: 9px 15px;
border-top: 1px solid #e5e5e5;
}
#backend .carousel {
position: relative;
margin-bottom: 18px;
line-height: 1;
}
#backend .carousel-inner {
overflow: hidden;
width: 100%;
position: relative;
}
#backend .carousel-inner > .item {
display: none;
position: relative;
-webkit-transition: 0.6s ease-in-out left;
-moz-transition: 0.6s ease-in-out left;
-o-transition: 0.6s ease-in-out left;
transition: 0.6s ease-in-out left;
}
#backend .carousel-inner > .item > img,
#backend .carousel-inner > .item > a > img {
display: block;
line-height: 1;
}
#backend .carousel-inner > .active,
#backend .carousel-inner > .next,
#backend .carousel-inner > .prev {
display: block;
}
#backend .carousel-inner > .active {
left: 0;
}
#backend .carousel-inner > .next,
#backend .carousel-inner > .prev {
position: absolute;
top: 0;
width: 100%;
}
#backend .carousel-inner > .next {
left: 100%;
}
#backend .carousel-inner > .prev {
left: -100%;
}
#backend .carousel-inner > .next.left,
#backend .carousel-inner > .prev.right {
left: 0;
}
#backend .carousel-inner > .active.left {
left: -100%;
}
#backend .carousel-inner > .active.right {
left: 100%;
}
#backend .carousel-control {
position: absolute;
top: 40%;
left: 15px;
width: 40px;
height: 40px;
margin-top: -20px;
font-size: 60px;
font-weight: 100;
line-height: 30px;
color: #ffffff;
text-align: center;
background: #222222;
border: 3px solid #ffffff;
-webkit-border-radius: 23px;
-moz-border-radius: 23px;
border-radius: 23px;
opacity: 0.5;
filter: alpha(opacity=50);
}
#backend .carousel-control.right {
left: auto;
right: 15px;
}
#backend .carousel-control:hover,
#backend .carousel-control:focus {
color: #ffffff;
text-decoration: none;
opacity: 0.9;
filter: alpha(opacity=90);
}
#backend .carousel-indicators {
position: absolute;
top: 15px;
right: 15px;
z-index: 5;
margin: 0;
list-style: none;
}
#backend .carousel-indicators li {
display: block;
float: left;
width: 10px;
height: 10px;
margin-left: 5px;
text-indent: -999px;
background-color: #ccc;
background-color: rgba(255, 255, 255, 0.25);
border-radius: 5px;
}
#backend .carousel-indicators .active {
background-color: #fff;
}
#backend .carousel-caption {
position: absolute;
left: 0;
right: 0;
bottom: 0;
padding: 15px;
background: #333333;
background: rgba(0, 0, 0, 0.75);
}
#backend .carousel-caption h4,
#backend .carousel-caption p {
color: #ffffff;
line-height: 18px;
}
#backend .carousel-caption h4 {
margin: 0 0 5px;
}
#backend .carousel-caption p {
margin-bottom: 0;
}
#backend .hero-unit {
padding: 60px;
margin-bottom: 30px;
font-size: 18px;
font-weight: 200;
line-height: 27px;
color: inherit;
background-color: #eeeeee;
-webkit-border-radius: 6px;
-moz-border-radius: 6px;
border-radius: 6px;
}
#backend .hero-unit h1 {
margin-bottom: 0;
font-size: 60px;
line-height: 1;
color: inherit;
letter-spacing: -1px;
}
#backend .hero-unit li {
line-height: 27px;
}
#backend .pull-right {
float: right;
}
#backend .pull-left {
float: left;
}
#backend .hide {
display: none;
}
#backend .show {
display: block;
}
#backend .invisible {
visibility: hidden;
}
#backend .affix {
position: fixed;
}
#backend .clearfix {
*zoom: 1;
}
#backend .clearfix:before,
#backend .clearfix:after {
display: table;
content: "";
line-height: 0;
}
#backend .clearfix:after {
clear: both;
}
#backend .hide-text {
font: 0/0 a;
color: transparent;
text-shadow: none;
background-color: transparent;
border: 0;
}
#backend .input-block-level {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
}
@-ms-viewport {
width: device-width;
}
#backend .hidden {
display: none;
visibility: hidden;
}
#backend .visible-phone {
display: none !important;
}
#backend .visible-tablet {
display: none !important;
}
#backend .hidden-desktop {
display: none !important;
}
#backend .visible-desktop {
display: inherit !important;
}
@media (min-width: 768px) and (max-width: 979px) {
#backend .hidden-desktop {
display: inherit !important;
}
#backend .visible-desktop {
display: none !important ;
}
#backend .visible-tablet {
display: inherit !important;
}
#backend .hidden-tablet {
display: none !important;
}
}
@media (max-width: 767px) {
#backend .hidden-desktop {
display: inherit !important;
}
#backend .visible-desktop {
display: none !important;
}
#backend .visible-phone {
display: inherit !important;
}
#backend .hidden-phone {
display: none !important;
}
}
#backend .visible-print {
display: none !important;
}
@media print {
#backend .visible-print {
display: inherit !important;
}
#backend .hidden-print {
display: none !important;
}
}
@media (min-width: 1200px) {
#backend .row {
margin-left: -30px;
*zoom: 1;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend [class*="span"] {
float: left;
min-height: 1px;
margin-left: 30px;
}
#backend .container,
#backend .navbar-static-top .container,
#backend .navbar-fixed-top .container,
#backend .navbar-fixed-bottom .container {
width: 1170px;
}
#backend .span12 {
width: 1170px;
}
#backend .span11 {
width: 1070px;
}
#backend .span10 {
width: 970px;
}
#backend .span9 {
width: 870px;
}
#backend .span8 {
width: 770px;
}
#backend .span7 {
width: 670px;
}
#backend .span6 {
width: 570px;
}
#backend .span5 {
width: 470px;
}
#backend .span4 {
width: 370px;
}
#backend .span3 {
width: 270px;
}
#backend .span2 {
width: 170px;
}
#backend .span1 {
width: 70px;
}
#backend .offset12 {
margin-left: 1230px;
}
#backend .offset11 {
margin-left: 1130px;
}
#backend .offset10 {
margin-left: 1030px;
}
#backend .offset9 {
margin-left: 930px;
}
#backend .offset8 {
margin-left: 830px;
}
#backend .offset7 {
margin-left: 730px;
}
#backend .offset6 {
margin-left: 630px;
}
#backend .offset5 {
margin-left: 530px;
}
#backend .offset4 {
margin-left: 430px;
}
#backend .offset3 {
margin-left: 330px;
}
#backend .offset2 {
margin-left: 230px;
}
#backend .offset1 {
margin-left: 130px;
}
#backend .row {
margin-left: -30px;
*zoom: 1;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend [class*="span"] {
float: left;
min-height: 1px;
margin-left: 30px;
}
#backend .container,
#backend .navbar-static-top .container,
#backend .navbar-fixed-top .container,
#backend .navbar-fixed-bottom .container {
width: 1170px;
}
#backend .span12 {
width: 1170px;
}
#backend .span11 {
width: 1070px;
}
#backend .span10 {
width: 970px;
}
#backend .span9 {
width: 870px;
}
#backend .span8 {
width: 770px;
}
#backend .span7 {
width: 670px;
}
#backend .span6 {
width: 570px;
}
#backend .span5 {
width: 470px;
}
#backend .span4 {
width: 370px;
}
#backend .span3 {
width: 270px;
}
#backend .span2 {
width: 170px;
}
#backend .span1 {
width: 70px;
}
#backend .offset12 {
margin-left: 1230px;
}
#backend .offset11 {
margin-left: 1130px;
}
#backend .offset10 {
margin-left: 1030px;
}
#backend .offset9 {
margin-left: 930px;
}
#backend .offset8 {
margin-left: 830px;
}
#backend .offset7 {
margin-left: 730px;
}
#backend .offset6 {
margin-left: 630px;
}
#backend .offset5 {
margin-left: 530px;
}
#backend .offset4 {
margin-left: 430px;
}
#backend .offset3 {
margin-left: 330px;
}
#backend .offset2 {
margin-left: 230px;
}
#backend .offset1 {
margin-left: 130px;
}
#backend .row-fluid {
width: 100%;
*zoom: 1;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid [class*="span"] {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
float: left;
margin-left: 2.564102564102564%;
*margin-left: 2.5109110747408616%;
}
#backend .row-fluid [class*="span"]:first-child {
margin-left: 0;
}
#backend .row-fluid .controls-row [class*="span"] + [class*="span"] {
margin-left: 2.564102564102564%;
}
#backend .row-fluid .span12 {
width: 100%;
*width: 99.94680851063829%;
}
#backend .row-fluid .span11 {
width: 91.45299145299145%;
*width: 91.39979996362975%;
}
#backend .row-fluid .span10 {
width: 82.90598290598291%;
*width: 82.8527914166212%;
}
#backend .row-fluid .span9 {
width: 74.35897435897436%;
*width: 74.30578286961266%;
}
#backend .row-fluid .span8 {
width: 65.81196581196582%;
*width: 65.75877432260411%;
}
#backend .row-fluid .span7 {
width: 57.26495726495726%;
*width: 57.21176577559556%;
}
#backend .row-fluid .span6 {
width: 48.717948717948715%;
*width: 48.664757228587014%;
}
#backend .row-fluid .span5 {
width: 40.17094017094017%;
*width: 40.11774868157847%;
}
#backend .row-fluid .span4 {
width: 31.623931623931625%;
*width: 31.570740134569924%;
}
#backend .row-fluid .span3 {
width: 23.076923076923077%;
*width: 23.023731587561375%;
}
#backend .row-fluid .span2 {
width: 14.52991452991453%;
*width: 14.476723040552828%;
}
#backend .row-fluid .span1 {
width: 5.982905982905983%;
*width: 5.929714493544281%;
}
#backend .row-fluid .offset12 {
margin-left: 105.12820512820512%;
*margin-left: 105.02182214948171%;
}
#backend .row-fluid .offset12:first-child {
margin-left: 102.56410256410257%;
*margin-left: 102.45771958537915%;
}
#backend .row-fluid .offset11 {
margin-left: 96.58119658119658%;
*margin-left: 96.47481360247316%;
}
#backend .row-fluid .offset11:first-child {
margin-left: 94.01709401709402%;
*margin-left: 93.91071103837061%;
}
#backend .row-fluid .offset10 {
margin-left: 88.03418803418803%;
*margin-left: 87.92780505546462%;
}
#backend .row-fluid .offset10:first-child {
margin-left: 85.47008547008548%;
*margin-left: 85.36370249136206%;
}
#backend .row-fluid .offset9 {
margin-left: 79.48717948717949%;
*margin-left: 79.38079650845607%;
}
#backend .row-fluid .offset9:first-child {
margin-left: 76.92307692307693%;
*margin-left: 76.81669394435352%;
}
#backend .row-fluid .offset8 {
margin-left: 70.94017094017094%;
*margin-left: 70.83378796144753%;
}
#backend .row-fluid .offset8:first-child {
margin-left: 68.37606837606839%;
*margin-left: 68.26968539734497%;
}
#backend .row-fluid .offset7 {
margin-left: 62.393162393162385%;
*margin-left: 62.28677941443899%;
}
#backend .row-fluid .offset7:first-child {
margin-left: 59.82905982905982%;
*margin-left: 59.72267685033642%;
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#backend .row-fluid .offset6 {
margin-left: 53.84615384615384%;
*margin-left: 53.739770867430444%;
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#backend .row-fluid .offset6:first-child {
margin-left: 51.28205128205128%;
*margin-left: 51.175668303327875%;
}
#backend .row-fluid .offset5 {
margin-left: 45.299145299145295%;
*margin-left: 45.1927623204219%;
}
#backend .row-fluid .offset5:first-child {
margin-left: 42.73504273504273%;
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#backend .row-fluid .offset4 {
margin-left: 36.75213675213675%;
*margin-left: 36.645753773413354%;
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#backend .row-fluid .offset4:first-child {
margin-left: 34.18803418803419%;
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#backend .row-fluid .offset3 {
margin-left: 28.205128205128204%;
*margin-left: 28.0987452264048%;
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#backend .row-fluid .offset3:first-child {
margin-left: 25.641025641025642%;
*margin-left: 25.53464266230224%;
}
#backend .row-fluid .offset2 {
margin-left: 19.65811965811966%;
*margin-left: 19.551736679396257%;
}
#backend .row-fluid .offset2:first-child {
margin-left: 17.094017094017094%;
*margin-left: 16.98763411529369%;
}
#backend .row-fluid .offset1 {
margin-left: 11.11111111111111%;
*margin-left: 11.004728132387708%;
}
#backend .row-fluid .offset1:first-child {
margin-left: 8.547008547008547%;
*margin-left: 8.440625568285142%;
}
#backend .row-fluid {
width: 100%;
*zoom: 1;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid [class*="span"] {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
float: left;
margin-left: 2.564102564102564%;
*margin-left: 2.5109110747408616%;
}
#backend .row-fluid [class*="span"]:first-child {
margin-left: 0;
}
#backend .row-fluid .controls-row [class*="span"] + [class*="span"] {
margin-left: 2.564102564102564%;
}
#backend .row-fluid .span12 {
width: 100%;
*width: 99.94680851063829%;
}
#backend .row-fluid .span11 {
width: 91.45299145299145%;
*width: 91.39979996362975%;
}
#backend .row-fluid .span10 {
width: 82.90598290598291%;
*width: 82.8527914166212%;
}
#backend .row-fluid .span9 {
width: 74.35897435897436%;
*width: 74.30578286961266%;
}
#backend .row-fluid .span8 {
width: 65.81196581196582%;
*width: 65.75877432260411%;
}
#backend .row-fluid .span7 {
width: 57.26495726495726%;
*width: 57.21176577559556%;
}
#backend .row-fluid .span6 {
width: 48.717948717948715%;
*width: 48.664757228587014%;
}
#backend .row-fluid .span5 {
width: 40.17094017094017%;
*width: 40.11774868157847%;
}
#backend .row-fluid .span4 {
width: 31.623931623931625%;
*width: 31.570740134569924%;
}
#backend .row-fluid .span3 {
width: 23.076923076923077%;
*width: 23.023731587561375%;
}
#backend .row-fluid .span2 {
width: 14.52991452991453%;
*width: 14.476723040552828%;
}
#backend .row-fluid .span1 {
width: 5.982905982905983%;
*width: 5.929714493544281%;
}
#backend .row-fluid .offset12 {
margin-left: 105.12820512820512%;
*margin-left: 105.02182214948171%;
}
#backend .row-fluid .offset12:first-child {
margin-left: 102.56410256410257%;
*margin-left: 102.45771958537915%;
}
#backend .row-fluid .offset11 {
margin-left: 96.58119658119658%;
*margin-left: 96.47481360247316%;
}
#backend .row-fluid .offset11:first-child {
margin-left: 94.01709401709402%;
*margin-left: 93.91071103837061%;
}
#backend .row-fluid .offset10 {
margin-left: 88.03418803418803%;
*margin-left: 87.92780505546462%;
}
#backend .row-fluid .offset10:first-child {
margin-left: 85.47008547008548%;
*margin-left: 85.36370249136206%;
}
#backend .row-fluid .offset9 {
margin-left: 79.48717948717949%;
*margin-left: 79.38079650845607%;
}
#backend .row-fluid .offset9:first-child {
margin-left: 76.92307692307693%;
*margin-left: 76.81669394435352%;
}
#backend .row-fluid .offset8 {
margin-left: 70.94017094017094%;
*margin-left: 70.83378796144753%;
}
#backend .row-fluid .offset8:first-child {
margin-left: 68.37606837606839%;
*margin-left: 68.26968539734497%;
}
#backend .row-fluid .offset7 {
margin-left: 62.393162393162385%;
*margin-left: 62.28677941443899%;
}
#backend .row-fluid .offset7:first-child {
margin-left: 59.82905982905982%;
*margin-left: 59.72267685033642%;
}
#backend .row-fluid .offset6 {
margin-left: 53.84615384615384%;
*margin-left: 53.739770867430444%;
}
#backend .row-fluid .offset6:first-child {
margin-left: 51.28205128205128%;
*margin-left: 51.175668303327875%;
}
#backend .row-fluid .offset5 {
margin-left: 45.299145299145295%;
*margin-left: 45.1927623204219%;
}
#backend .row-fluid .offset5:first-child {
margin-left: 42.73504273504273%;
*margin-left: 42.62865975631933%;
}
#backend .row-fluid .offset4 {
margin-left: 36.75213675213675%;
*margin-left: 36.645753773413354%;
}
#backend .row-fluid .offset4:first-child {
margin-left: 34.18803418803419%;
*margin-left: 34.081651209310785%;
}
#backend .row-fluid .offset3 {
margin-left: 28.205128205128204%;
*margin-left: 28.0987452264048%;
}
#backend .row-fluid .offset3:first-child {
margin-left: 25.641025641025642%;
*margin-left: 25.53464266230224%;
}
#backend .row-fluid .offset2 {
margin-left: 19.65811965811966%;
*margin-left: 19.551736679396257%;
}
#backend .row-fluid .offset2:first-child {
margin-left: 17.094017094017094%;
*margin-left: 16.98763411529369%;
}
#backend .row-fluid .offset1 {
margin-left: 11.11111111111111%;
*margin-left: 11.004728132387708%;
}
#backend .row-fluid .offset1:first-child {
margin-left: 8.547008547008547%;
*margin-left: 8.440625568285142%;
}
#backend input,
#backend textarea,
#backend .uneditable-input {
margin-left: 0;
}
#backend .controls-row [class*="span"] + [class*="span"] {
margin-left: 30px;
}
#backend input.span12,
#backend textarea.span12,
#backend .uneditable-input.span12 {
width: 1156px;
}
#backend input.span11,
#backend textarea.span11,
#backend .uneditable-input.span11 {
width: 1056px;
}
#backend input.span10,
#backend textarea.span10,
#backend .uneditable-input.span10 {
width: 956px;
}
#backend input.span9,
#backend textarea.span9,
#backend .uneditable-input.span9 {
width: 856px;
}
#backend input.span8,
#backend textarea.span8,
#backend .uneditable-input.span8 {
width: 756px;
}
#backend input.span7,
#backend textarea.span7,
#backend .uneditable-input.span7 {
width: 656px;
}
#backend input.span6,
#backend textarea.span6,
#backend .uneditable-input.span6 {
width: 556px;
}
#backend input.span5,
#backend textarea.span5,
#backend .uneditable-input.span5 {
width: 456px;
}
#backend input.span4,
#backend textarea.span4,
#backend .uneditable-input.span4 {
width: 356px;
}
#backend input.span3,
#backend textarea.span3,
#backend .uneditable-input.span3 {
width: 256px;
}
#backend input.span2,
#backend textarea.span2,
#backend .uneditable-input.span2 {
width: 156px;
}
#backend input.span1,
#backend textarea.span1,
#backend .uneditable-input.span1 {
width: 56px;
}
#backend input,
#backend textarea,
#backend .uneditable-input {
margin-left: 0;
}
#backend .controls-row [class*="span"] + [class*="span"] {
margin-left: 30px;
}
#backend input.span12,
#backend textarea.span12,
#backend .uneditable-input.span12 {
width: 1156px;
}
#backend input.span11,
#backend textarea.span11,
#backend .uneditable-input.span11 {
width: 1056px;
}
#backend input.span10,
#backend textarea.span10,
#backend .uneditable-input.span10 {
width: 956px;
}
#backend input.span9,
#backend textarea.span9,
#backend .uneditable-input.span9 {
width: 856px;
}
#backend input.span8,
#backend textarea.span8,
#backend .uneditable-input.span8 {
width: 756px;
}
#backend input.span7,
#backend textarea.span7,
#backend .uneditable-input.span7 {
width: 656px;
}
#backend input.span6,
#backend textarea.span6,
#backend .uneditable-input.span6 {
width: 556px;
}
#backend input.span5,
#backend textarea.span5,
#backend .uneditable-input.span5 {
width: 456px;
}
#backend input.span4,
#backend textarea.span4,
#backend .uneditable-input.span4 {
width: 356px;
}
#backend input.span3,
#backend textarea.span3,
#backend .uneditable-input.span3 {
width: 256px;
}
#backend input.span2,
#backend textarea.span2,
#backend .uneditable-input.span2 {
width: 156px;
}
#backend input.span1,
#backend textarea.span1,
#backend .uneditable-input.span1 {
width: 56px;
}
#backend .thumbnails {
margin-left: -30px;
}
#backend .thumbnails > li {
margin-left: 30px;
}
#backend .row-fluid .thumbnails {
margin-left: 0;
}
}
@media (min-width: 768px) and (max-width: 979px) {
#backend .row {
margin-left: -20px;
*zoom: 1;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend [class*="span"] {
float: left;
min-height: 1px;
margin-left: 20px;
}
#backend .container,
#backend .navbar-static-top .container,
#backend .navbar-fixed-top .container,
#backend .navbar-fixed-bottom .container {
width: 724px;
}
#backend .span12 {
width: 724px;
}
#backend .span11 {
width: 662px;
}
#backend .span10 {
width: 600px;
}
#backend .span9 {
width: 538px;
}
#backend .span8 {
width: 476px;
}
#backend .span7 {
width: 414px;
}
#backend .span6 {
width: 352px;
}
#backend .span5 {
width: 290px;
}
#backend .span4 {
width: 228px;
}
#backend .span3 {
width: 166px;
}
#backend .span2 {
width: 104px;
}
#backend .span1 {
width: 42px;
}
#backend .offset12 {
margin-left: 764px;
}
#backend .offset11 {
margin-left: 702px;
}
#backend .offset10 {
margin-left: 640px;
}
#backend .offset9 {
margin-left: 578px;
}
#backend .offset8 {
margin-left: 516px;
}
#backend .offset7 {
margin-left: 454px;
}
#backend .offset6 {
margin-left: 392px;
}
#backend .offset5 {
margin-left: 330px;
}
#backend .offset4 {
margin-left: 268px;
}
#backend .offset3 {
margin-left: 206px;
}
#backend .offset2 {
margin-left: 144px;
}
#backend .offset1 {
margin-left: 82px;
}
#backend .row {
margin-left: -20px;
*zoom: 1;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend .row:before,
#backend .row:after {
display: table;
content: "";
line-height: 0;
}
#backend .row:after {
clear: both;
}
#backend [class*="span"] {
float: left;
min-height: 1px;
margin-left: 20px;
}
#backend .container,
#backend .navbar-static-top .container,
#backend .navbar-fixed-top .container,
#backend .navbar-fixed-bottom .container {
width: 724px;
}
#backend .span12 {
width: 724px;
}
#backend .span11 {
width: 662px;
}
#backend .span10 {
width: 600px;
}
#backend .span9 {
width: 538px;
}
#backend .span8 {
width: 476px;
}
#backend .span7 {
width: 414px;
}
#backend .span6 {
width: 352px;
}
#backend .span5 {
width: 290px;
}
#backend .span4 {
width: 228px;
}
#backend .span3 {
width: 166px;
}
#backend .span2 {
width: 104px;
}
#backend .span1 {
width: 42px;
}
#backend .offset12 {
margin-left: 764px;
}
#backend .offset11 {
margin-left: 702px;
}
#backend .offset10 {
margin-left: 640px;
}
#backend .offset9 {
margin-left: 578px;
}
#backend .offset8 {
margin-left: 516px;
}
#backend .offset7 {
margin-left: 454px;
}
#backend .offset6 {
margin-left: 392px;
}
#backend .offset5 {
margin-left: 330px;
}
#backend .offset4 {
margin-left: 268px;
}
#backend .offset3 {
margin-left: 206px;
}
#backend .offset2 {
margin-left: 144px;
}
#backend .offset1 {
margin-left: 82px;
}
#backend .row-fluid {
width: 100%;
*zoom: 1;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid [class*="span"] {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
float: left;
margin-left: 2.7624309392265194%;
*margin-left: 2.709239449864817%;
}
#backend .row-fluid [class*="span"]:first-child {
margin-left: 0;
}
#backend .row-fluid .controls-row [class*="span"] + [class*="span"] {
margin-left: 2.7624309392265194%;
}
#backend .row-fluid .span12 {
width: 100%;
*width: 99.94680851063829%;
}
#backend .row-fluid .span11 {
width: 91.43646408839778%;
*width: 91.38327259903608%;
}
#backend .row-fluid .span10 {
width: 82.87292817679558%;
*width: 82.81973668743387%;
}
#backend .row-fluid .span9 {
width: 74.30939226519337%;
*width: 74.25620077583166%;
}
#backend .row-fluid .span8 {
width: 65.74585635359117%;
*width: 65.69266486422946%;
}
#backend .row-fluid .span7 {
width: 57.18232044198895%;
*width: 57.12912895262725%;
}
#backend .row-fluid .span6 {
width: 48.61878453038674%;
*width: 48.56559304102504%;
}
#backend .row-fluid .span5 {
width: 40.05524861878453%;
*width: 40.00205712942283%;
}
#backend .row-fluid .span4 {
width: 31.491712707182323%;
*width: 31.43852121782062%;
}
#backend .row-fluid .span3 {
width: 22.92817679558011%;
*width: 22.87498530621841%;
}
#backend .row-fluid .span2 {
width: 14.3646408839779%;
*width: 14.311449394616199%;
}
#backend .row-fluid .span1 {
width: 5.801104972375691%;
*width: 5.747913483013988%;
}
#backend .row-fluid .offset12 {
margin-left: 105.52486187845304%;
*margin-left: 105.41847889972962%;
}
#backend .row-fluid .offset12:first-child {
margin-left: 102.76243093922652%;
*margin-left: 102.6560479605031%;
}
#backend .row-fluid .offset11 {
margin-left: 96.96132596685082%;
*margin-left: 96.8549429881274%;
}
#backend .row-fluid .offset11:first-child {
margin-left: 94.1988950276243%;
*margin-left: 94.09251204890089%;
}
#backend .row-fluid .offset10 {
margin-left: 88.39779005524862%;
*margin-left: 88.2914070765252%;
}
#backend .row-fluid .offset10:first-child {
margin-left: 85.6353591160221%;
*margin-left: 85.52897613729868%;
}
#backend .row-fluid .offset9 {
margin-left: 79.8342541436464%;
*margin-left: 79.72787116492299%;
}
#backend .row-fluid .offset9:first-child {
margin-left: 77.07182320441989%;
*margin-left: 76.96544022569647%;
}
#backend .row-fluid .offset8 {
margin-left: 71.2707182320442%;
*margin-left: 71.16433525332079%;
}
#backend .row-fluid .offset8:first-child {
margin-left: 68.50828729281768%;
*margin-left: 68.40190431409427%;
}
#backend .row-fluid .offset7 {
margin-left: 62.70718232044199%;
*margin-left: 62.600799341718584%;
}
#backend .row-fluid .offset7:first-child {
margin-left: 59.94475138121547%;
*margin-left: 59.838368402492065%;
}
#backend .row-fluid .offset6 {
margin-left: 54.14364640883978%;
*margin-left: 54.037263430116376%;
}
#backend .row-fluid .offset6:first-child {
margin-left: 51.38121546961326%;
*margin-left: 51.27483249088986%;
}
#backend .row-fluid .offset5 {
margin-left: 45.58011049723757%;
*margin-left: 45.47372751851417%;
}
#backend .row-fluid .offset5:first-child {
margin-left: 42.81767955801105%;
*margin-left: 42.71129657928765%;
}
#backend .row-fluid .offset4 {
margin-left: 37.01657458563536%;
*margin-left: 36.91019160691196%;
}
#backend .row-fluid .offset4:first-child {
margin-left: 34.25414364640884%;
*margin-left: 34.14776066768544%;
}
#backend .row-fluid .offset3 {
margin-left: 28.45303867403315%;
*margin-left: 28.346655695309746%;
}
#backend .row-fluid .offset3:first-child {
margin-left: 25.69060773480663%;
*margin-left: 25.584224756083227%;
}
#backend .row-fluid .offset2 {
margin-left: 19.88950276243094%;
*margin-left: 19.783119783707537%;
}
#backend .row-fluid .offset2:first-child {
margin-left: 17.12707182320442%;
*margin-left: 17.02068884448102%;
}
#backend .row-fluid .offset1 {
margin-left: 11.32596685082873%;
*margin-left: 11.219583872105325%;
}
#backend .row-fluid .offset1:first-child {
margin-left: 8.56353591160221%;
*margin-left: 8.457152932878806%;
}
#backend .row-fluid {
width: 100%;
*zoom: 1;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid:before,
#backend .row-fluid:after {
display: table;
content: "";
line-height: 0;
}
#backend .row-fluid:after {
clear: both;
}
#backend .row-fluid [class*="span"] {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
float: left;
margin-left: 2.7624309392265194%;
*margin-left: 2.709239449864817%;
}
#backend .row-fluid [class*="span"]:first-child {
margin-left: 0;
}
#backend .row-fluid .controls-row [class*="span"] + [class*="span"] {
margin-left: 2.7624309392265194%;
}
#backend .row-fluid .span12 {
width: 100%;
*width: 99.94680851063829%;
}
#backend .row-fluid .span11 {
width: 91.43646408839778%;
*width: 91.38327259903608%;
}
#backend .row-fluid .span10 {
width: 82.87292817679558%;
*width: 82.81973668743387%;
}
#backend .row-fluid .span9 {
width: 74.30939226519337%;
*width: 74.25620077583166%;
}
#backend .row-fluid .span8 {
width: 65.74585635359117%;
*width: 65.69266486422946%;
}
#backend .row-fluid .span7 {
width: 57.18232044198895%;
*width: 57.12912895262725%;
}
#backend .row-fluid .span6 {
width: 48.61878453038674%;
*width: 48.56559304102504%;
}
#backend .row-fluid .span5 {
width: 40.05524861878453%;
*width: 40.00205712942283%;
}
#backend .row-fluid .span4 {
width: 31.491712707182323%;
*width: 31.43852121782062%;
}
#backend .row-fluid .span3 {
width: 22.92817679558011%;
*width: 22.87498530621841%;
}
#backend .row-fluid .span2 {
width: 14.3646408839779%;
*width: 14.311449394616199%;
}
#backend .row-fluid .span1 {
width: 5.801104972375691%;
*width: 5.747913483013988%;
}
#backend .row-fluid .offset12 {
margin-left: 105.52486187845304%;
*margin-left: 105.41847889972962%;
}
#backend .row-fluid .offset12:first-child {
margin-left: 102.76243093922652%;
*margin-left: 102.6560479605031%;
}
#backend .row-fluid .offset11 {
margin-left: 96.96132596685082%;
*margin-left: 96.8549429881274%;
}
#backend .row-fluid .offset11:first-child {
margin-left: 94.1988950276243%;
*margin-left: 94.09251204890089%;
}
#backend .row-fluid .offset10 {
margin-left: 88.39779005524862%;
*margin-left: 88.2914070765252%;
}
#backend .row-fluid .offset10:first-child {
margin-left: 85.6353591160221%;
*margin-left: 85.52897613729868%;
}
#backend .row-fluid .offset9 {
margin-left: 79.8342541436464%;
*margin-left: 79.72787116492299%;
}
#backend .row-fluid .offset9:first-child {
margin-left: 77.07182320441989%;
*margin-left: 76.96544022569647%;
}
#backend .row-fluid .offset8 {
margin-left: 71.2707182320442%;
*margin-left: 71.16433525332079%;
}
#backend .row-fluid .offset8:first-child {
margin-left: 68.50828729281768%;
*margin-left: 68.40190431409427%;
}
#backend .row-fluid .offset7 {
margin-left: 62.70718232044199%;
*margin-left: 62.600799341718584%;
}
#backend .row-fluid .offset7:first-child {
margin-left: 59.94475138121547%;
*margin-left: 59.838368402492065%;
}
#backend .row-fluid .offset6 {
margin-left: 54.14364640883978%;
*margin-left: 54.037263430116376%;
}
#backend .row-fluid .offset6:first-child {
margin-left: 51.38121546961326%;
*margin-left: 51.27483249088986%;
}
#backend .row-fluid .offset5 {
margin-left: 45.58011049723757%;
*margin-left: 45.47372751851417%;
}
#backend .row-fluid .offset5:first-child {
margin-left: 42.81767955801105%;
*margin-left: 42.71129657928765%;
}
#backend .row-fluid .offset4 {
margin-left: 37.01657458563536%;
*margin-left: 36.91019160691196%;
}
#backend .row-fluid .offset4:first-child {
margin-left: 34.25414364640884%;
*margin-left: 34.14776066768544%;
}
#backend .row-fluid .offset3 {
margin-left: 28.45303867403315%;
*margin-left: 28.346655695309746%;
}
#backend .row-fluid .offset3:first-child {
margin-left: 25.69060773480663%;
*margin-left: 25.584224756083227%;
}
#backend .row-fluid .offset2 {
margin-left: 19.88950276243094%;
*margin-left: 19.783119783707537%;
}
#backend .row-fluid .offset2:first-child {
margin-left: 17.12707182320442%;
*margin-left: 17.02068884448102%;
}
#backend .row-fluid .offset1 {
margin-left: 11.32596685082873%;
*margin-left: 11.219583872105325%;
}
#backend .row-fluid .offset1:first-child {
margin-left: 8.56353591160221%;
*margin-left: 8.457152932878806%;
}
#backend input,
#backend textarea,
#backend .uneditable-input {
margin-left: 0;
}
#backend .controls-row [class*="span"] + [class*="span"] {
margin-left: 20px;
}
#backend input.span12,
#backend textarea.span12,
#backend .uneditable-input.span12 {
width: 710px;
}
#backend input.span11,
#backend textarea.span11,
#backend .uneditable-input.span11 {
width: 648px;
}
#backend input.span10,
#backend textarea.span10,
#backend .uneditable-input.span10 {
width: 586px;
}
#backend input.span9,
#backend textarea.span9,
#backend .uneditable-input.span9 {
width: 524px;
}
#backend input.span8,
#backend textarea.span8,
#backend .uneditable-input.span8 {
width: 462px;
}
#backend input.span7,
#backend textarea.span7,
#backend .uneditable-input.span7 {
width: 400px;
}
#backend input.span6,
#backend textarea.span6,
#backend .uneditable-input.span6 {
width: 338px;
}
#backend input.span5,
#backend textarea.span5,
#backend .uneditable-input.span5 {
width: 276px;
}
#backend input.span4,
#backend textarea.span4,
#backend .uneditable-input.span4 {
width: 214px;
}
#backend input.span3,
#backend textarea.span3,
#backend .uneditable-input.span3 {
width: 152px;
}
#backend input.span2,
#backend textarea.span2,
#backend .uneditable-input.span2 {
width: 90px;
}
#backend input.span1,
#backend textarea.span1,
#backend .uneditable-input.span1 {
width: 28px;
}
#backend input,
#backend textarea,
#backend .uneditable-input {
margin-left: 0;
}
#backend .controls-row [class*="span"] + [class*="span"] {
margin-left: 20px;
}
#backend input.span12,
#backend textarea.span12,
#backend .uneditable-input.span12 {
width: 710px;
}
#backend input.span11,
#backend textarea.span11,
#backend .uneditable-input.span11 {
width: 648px;
}
#backend input.span10,
#backend textarea.span10,
#backend .uneditable-input.span10 {
width: 586px;
}
#backend input.span9,
#backend textarea.span9,
#backend .uneditable-input.span9 {
width: 524px;
}
#backend input.span8,
#backend textarea.span8,
#backend .uneditable-input.span8 {
width: 462px;
}
#backend input.span7,
#backend textarea.span7,
#backend .uneditable-input.span7 {
width: 400px;
}
#backend input.span6,
#backend textarea.span6,
#backend .uneditable-input.span6 {
width: 338px;
}
#backend input.span5,
#backend textarea.span5,
#backend .uneditable-input.span5 {
width: 276px;
}
#backend input.span4,
#backend textarea.span4,
#backend .uneditable-input.span4 {
width: 214px;
}
#backend input.span3,
#backend textarea.span3,
#backend .uneditable-input.span3 {
width: 152px;
}
#backend input.span2,
#backend textarea.span2,
#backend .uneditable-input.span2 {
width: 90px;
}
#backend input.span1,
#backend textarea.span1,
#backend .uneditable-input.span1 {
width: 28px;
}
}
@media (max-width: 767px) {
#backend body {
padding-left: 20px;
padding-right: 20px;
}
#backend .navbar-fixed-top,
#backend .navbar-fixed-bottom,
#backend .navbar-static-top {
margin-left: -20px;
margin-right: -20px;
}
#backend .container-fluid {
padding: 0;
}
#backend .dl-horizontal dt {
float: none;
clear: none;
width: auto;
text-align: left;
}
#backend .dl-horizontal dd {
margin-left: 0;
}
#backend .container {
width: auto;
}
#backend .row-fluid {
width: 100%;
}
#backend .row,
#backend .thumbnails {
margin-left: 0;
}
#backend .thumbnails > li {
float: none;
margin-left: 0;
}
#backend [class*="span"],
#backend .uneditable-input[class*="span"],
#backend .row-fluid [class*="span"] {
float: none;
display: block;
width: 100%;
margin-left: 0;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
}
#backend .span12,
#backend .row-fluid .span12 {
width: 100%;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
}
#backend .row-fluid [class*="offset"]:first-child {
margin-left: 0;
}
#backend .input-large,
#backend .input-xlarge,
#backend .input-xxlarge,
#backend input[class*="span"],
#backend select[class*="span"],
#backend textarea[class*="span"],
#backend .uneditable-input {
display: block;
width: 100%;
min-height: 28px;
-webkit-box-sizing: border-box;
-moz-box-sizing: border-box;
box-sizing: border-box;
}
#backend .input-prepend input,
#backend .input-append input,
#backend .input-prepend input[class*="span"],
#backend .input-append input[class*="span"] {
display: inline-block;
width: auto;
}
#backend .controls-row [class*="span"] + [class*="span"] {
margin-left: 0;
}
#backend .modal {
position: fixed;
top: 20px;
left: 20px;
right: 20px;
width: auto;
margin: 0;
}
#backend .modal.fade {
top: -100px;
}
#backend .modal.fade.in {
top: 20px;
}
}
@media (max-width: 480px) {
#backend .nav-collapse {
-webkit-transform: translate3d(0, 0, 0);
}
#backend .page-header h1 small {
display: block;
line-height: 18px;
}
#backend input[type="checkbox"],
#backend input[type="radio"] {
border: 1px solid #ccc;
}
#backend .form-horizontal .control-label {
float: none;
width: auto;
padding-top: 0;
text-align: left;
}
#backend .form-horizontal .controls {
margin-left: 0;
}
#backend .form-horizontal .control-list {
padding-top: 0;
}
#backend .form-horizontal .form-actions {
padding-left: 10px;
padding-right: 10px;
}
#backend .media .pull-left,
#backend .media .pull-right {
float: none;
display: block;
margin-bottom: 10px;
}
#backend .media-object {
margin-right: 0;
margin-left: 0;
}
#backend .modal {
top: 10px;
left: 10px;
right: 10px;
}
#backend .modal-header .close {
padding: 10px;
margin: -10px;
}
#backend .carousel-caption {
position: static;
}
}
@media (max-width: 979px) {
#backend body {
padding-top: 0;
}
#backend .navbar-fixed-top,
#backend .navbar-fixed-bottom {
position: static;
}
#backend .navbar-fixed-top {
margin-bottom: 18px;
}
#backend .navbar-fixed-bottom {
margin-top: 18px;
}
#backend .navbar-fixed-top .navbar-inner,
#backend .navbar-fixed-bottom .navbar-inner {
padding: 5px;
}
#backend .navbar .container {
width: auto;
padding: 0;
}
#backend .navbar .brand {
padding-left: 10px;
padding-right: 10px;
margin: 0 0 0 -5px;
}
#backend .nav-collapse {
clear: both;
}
#backend .nav-collapse .nav {
float: none;
margin: 0 0 9px;
}
#backend .nav-collapse .nav > li {
float: none;
}
#backend .nav-collapse .nav > li > a {
margin-bottom: 2px;
}
#backend .nav-collapse .nav > .divider-vertical {
display: none;
}
#backend .nav-collapse .nav .nav-header {
color: #999999;
text-shadow: none;
}
#backend .nav-collapse .nav > li > a,
#backend .nav-collapse .dropdown-menu a {
padding: 9px 15px;
font-weight: bold;
color: #999999;
-webkit-border-radius: 3px;
-moz-border-radius: 3px;
border-radius: 3px;
}
#backend .nav-collapse .btn {
padding: 4px 10px 4px;
font-weight: normal;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
}
#backend .nav-collapse .dropdown-menu li + li a {
margin-bottom: 2px;
}
#backend .nav-collapse .nav > li > a:hover,
#backend .nav-collapse .nav > li > a:focus,
#backend .nav-collapse .dropdown-menu a:hover,
#backend .nav-collapse .dropdown-menu a:focus {
background-color: #222222;
}
#backend .navbar-inverse .nav-collapse .nav > li > a,
#backend .navbar-inverse .nav-collapse .dropdown-menu a {
color: #999999;
}
#backend .navbar-inverse .nav-collapse .nav > li > a:hover,
#backend .navbar-inverse .nav-collapse .nav > li > a:focus,
#backend .navbar-inverse .nav-collapse .dropdown-menu a:hover,
#backend .navbar-inverse .nav-collapse .dropdown-menu a:focus {
background-color: #111111;
}
#backend .nav-collapse.in .btn-group {
margin-top: 5px;
padding: 0;
}
#backend .nav-collapse .dropdown-menu {
position: static;
top: auto;
left: auto;
float: none;
display: none;
max-width: none;
margin: 0 15px;
padding: 0;
background-color: transparent;
border: none;
-webkit-border-radius: 0;
-moz-border-radius: 0;
border-radius: 0;
-webkit-box-shadow: none;
-moz-box-shadow: none;
box-shadow: none;
}
#backend .nav-collapse .open > .dropdown-menu {
display: block;
}
#backend .nav-collapse .dropdown-menu:before,
#backend .nav-collapse .dropdown-menu:after {
display: none;
}
#backend .nav-collapse .dropdown-menu .divider {
display: none;
}
#backend .nav-collapse .nav > li > .dropdown-menu:before,
#backend .nav-collapse .nav > li > .dropdown-menu:after {
display: none;
}
#backend .nav-collapse .navbar-form,
#backend .nav-collapse .navbar-search {
float: none;
padding: 9px 15px;
margin: 9px 0;
border-top: 1px solid #222222;
border-bottom: 1px solid #222222;
-webkit-box-shadow: inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.1);
-moz-box-shadow: inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.1);
box-shadow: inset 0 1px 0 rgba(255,255,255,.1), 0 1px 0 rgba(255,255,255,.1);
}
#backend .navbar-inverse .nav-collapse .navbar-form,
#backend .navbar-inverse .nav-collapse .navbar-search {
border-top-color: #111111;
border-bottom-color: #111111;
}
#backend .navbar .nav-collapse .nav.pull-right {
float: none;
margin-left: 0;
}
#backend .nav-collapse,
#backend .nav-collapse.collapse {
overflow: hidden;
height: 0;
}
#backend .navbar .btn-navbar {
display: block;
}
#backend .navbar-static .navbar-inner {
padding-left: 10px;
padding-right: 10px;
}
}
@media (min-width: 980px) {
#backend .nav-collapse.collapse {
height: auto !important;
overflow: visible !important;
}
}
#backend #content {
/*.row;
.span12;*/
}
#backend [class*="span"] [class*="span-"] {
margin-left: 0px;
}
#backend .flash-notice {
padding: 8px 35px 8px 14px;
margin-bottom: 18px;
text-shadow: 0 1px 0 rgba(255, 255, 255, 0.5);
background-color: #fcf8e3;
border: 1px solid #fbeed5;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
color: #c09853;
}
#backend .notice {
padding: 8px 35px 8px 14px;
margin-bottom: 18px;
text-shadow: 0 1px 0 rgba(255, 255, 255, 0.5);
background-color: #fcf8e3;
border: 1px solid #fbeed5;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
color: #c09853;
}
#backend .flash-warning {
padding: 8px 35px 8px 14px;
margin-bottom: 18px;
text-shadow: 0 1px 0 rgba(255, 255, 255, 0.5);
background-color: #fcf8e3;
border: 1px solid #fbeed5;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
color: #c09853;
background-color: #f2dede;
border-color: #eed3d7;
color: #b94a48;
}
#backend .warning {
padding: 8px 35px 8px 14px;
margin-bottom: 18px;
text-shadow: 0 1px 0 rgba(255, 255, 255, 0.5);
background-color: #fcf8e3;
border: 1px solid #fbeed5;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
color: #c09853;
background-color: #f2dede;
border-color: #eed3d7;
color: #b94a48;
}
#backend div.row [class*="span-"] {
/*margin-left: 0px;*/
}
#backend table[class*="span-"] {
float: none;
}
#backend div.form {
min-height: 20px;
padding: 19px;
margin-bottom: 20px;
background-color: #f5f5f5;
border: 1px solid #e3e3e3;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
-webkit-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.05);
-moz-box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.05);
box-shadow: inset 0 1px 1px rgba(0, 0, 0, 0.05);
}
#backend div.form blockquote {
border-color: #ddd;
border-color: rgba(0, 0, 0, 0.15);
}
#backend div.form .row {
margin-bottom: 18px;
*zoom: 1;
margin-left: 0px;
}
#backend div.form .row:before,
#backend div.form .row:after {
display: table;
content: "";
line-height: 0;
}
#backend div.form .row:after {
clear: both;
}
#backend div.form .row:before,
#backend div.form .row:after {
display: table;
content: "";
line-height: 0;
}
#backend div.form .row:after {
clear: both;
}
#backend div.form BUTTON,
#backend div.form INPUT[type="button"],
#backend div.form INPUT[type="submit"] {
display: inline-block;
*display: inline;
/* IE7 inline-block hack */
*zoom: 1;
padding: 4px 12px;
margin-bottom: 0;
font-size: 13px;
line-height: 18px;
text-align: center;
vertical-align: middle;
cursor: pointer;
color: #333333;
text-shadow: 0 1px 1px rgba(255, 255, 255, 0.75);
background-color: #f5f5f5;
background-image: -moz-linear-gradient(top, #ffffff, #e6e6e6);
background-image: -webkit-gradient(linear, 0 0, 0 100%, from(#ffffff), to(#e6e6e6));
background-image: -webkit-linear-gradient(top, #ffffff, #e6e6e6);
background-image: -o-linear-gradient(top, #ffffff, #e6e6e6);
background-image: linear-gradient(to bottom, #ffffff, #e6e6e6);
background-repeat: repeat-x;
filter: progid:DXImageTransform.Microsoft.gradient(startColorstr='#ffffffff', endColorstr='#ffe6e6e6', GradientType=0);
border-color: #e6e6e6 #e6e6e6 #bfbfbf;
border-color: rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.1) rgba(0, 0, 0, 0.25);
*background-color: #e6e6e6;
/* Darken IE7 buttons by default so they stand out more given they won't have borders */
filter: progid:DXImageTransform.Microsoft.gradient(enabled = false);
border: 1px solid #cccccc;
*border: 0;
border-bottom-color: #b3b3b3;
-webkit-border-radius: 4px;
-moz-border-radius: 4px;
border-radius: 4px;
*margin-left: .3em;
-webkit-box-shadow: inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
-moz-box-shadow: inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
box-shadow: inset 0 1px 0 rgba(255,255,255,.2), 0 1px 2px rgba(0,0,0,.05);
}
#backend div.form BUTTON:hover,
#backend div.form INPUT[type="button"]:hover,
#backend div.form INPUT[type="submit"]:hover,
#backend div.form BUTTON:focus,
#backend div.form INPUT[type="button"]:focus,
#backend div.form INPUT[type="submit"]:focus,
#backend div.form BUTTON:active,
#backend div.form INPUT[type="button"]:active,
#backend div.form INPUT[type="submit"]:active,
#backend div.form BUTTON.active,
#backend div.form INPUT[type="button"].active,
#backend div.form INPUT[type="submit"].active,
#backend div.form BUTTON.disabled,
#backend div.form INPUT[type="button"].disabled,
#backend div.form INPUT[type="submit"].disabled,
#backend div.form BUTTON[disabled],
#backend div.form INPUT[type="button"][disabled],
#backend div.form INPUT[type="submit"][disabled] {
color: #333333;
background-color: #e6e6e6;
*background-color: #d9d9d9;
}
#backend div.form BUTTON:active,
#backend div.form INPUT[type="button"]:active,
#backend div.form INPUT[type="submit"]:active,
#backend div.form BUTTON.active,
#backend div.form INPUT[type="button"].active,
#backend div.form INPUT[type="submit"].active {
background-color: #cccccc \9;
}
#backend div.form BUTTON:hover,
#backend div.form INPUT[type="button"]:hover,
#backend div.form INPUT[type="submit"]:hover,
#backend div.form BUTTON:focus,
#backend div.form INPUT[type="button"]:focus,
#backend div.form INPUT[type="submit"]:focus,
#backend div.form BUTTON:active,
#backend div.form INPUT[type="button"]:active,
#backend div.form INPUT[type="submit"]:active,
#backend div.form BUTTON.active,
#backend div.form INPUT[type="button"].active,
#backend div.form INPUT[type="submit"].active,
#backend div.form BUTTON.disabled,
#backend div.form INPUT[type="button"].disabled,
#backend div.form INPUT[type="submit"].disabled,
#backend div.form BUTTON[disabled],
#backend div.form INPUT[type="button"][disabled],
#backend div.form INPUT[type="submit"][disabled] {
color: #333333;
background-color: #e6e6e6;
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,771 |
In some pastures around Sanpete, there is a particularly hearty grass growing anew. It grows in isolated clumps among the rest of the grasses. If left alone by hungry livestock, it will grow six inches long and more and stay green all summer. In some pastures where no animals have grazed for a while, it grows roughly as big as a tire and in patches only two feet apart, more or less. It's greener and more prolific than the rest of the grass. I call it 'buffalo grass,' and like to watch it grow through the spring and summer here and there.
Last year the kids cut our few wee patches down to feed to the horses next door. It was fun for all, except me, who felt like a friend had been shorn of her lovely tresses and left to cry. Sentimental attachment to an unidentified species of grass. I must truly be a city-slicker at heart. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,009 |
\section{Introduction}
Spoken language understanding (SLU) is a key component of task-oriented dialogue systems, which assist user to complete tasks such as booking flight tickets. SLU parses user utterances into semantic frames, including intents, slots, and user dialogue acts~\cite{tur2011spoken}. The semantic frame for a restaurant reservation query is shown in Figure~\ref{fig:example}. Both intents and user dialogue acts represent the user's intentions, but intents and user acts could be defined with different granularities. In this work, we model both intent and user act classification when both of them are available in the dialogue corpora.
Previous research in SLU has significantly focused on single-turn SLU, that is, understanding the current user utterance. However, completing a task usually necessitates multiple turns of back-and-forth conversations between the user and the system. Multi-turn SLU imposes different challenges from single-turn SLU, for example, entities introduced earlier in conversation may be referred later by the user and the system, information mentioned earlier may be skipped later, causing ambiguities, as shown in Figure~\ref{fig:example}. Incorporating contextual information has been shown useful for multi-turn SLU~\cite{DBLP:conf/icassp/BhargavaCHS13,DBLP:conf/icassp/XuS14,DBLP:conf/icmi/ChenSRG15,DBLP:conf/iui/SunCR16,DBLP:conf/icassp/ShiYCPHP15,DBLP:conf/interspeech/GuptaRH18}. Information from previous intra-session utterances was explored by applying SVM-HMMs to sequence tagging for SLU~\cite{DBLP:conf/icassp/BhargavaCHS13}. Contextual information was incorporated into the recurrent neural network (RNN) structure~\cite{DBLP:conf/icassp/XuS14,DBLP:conf/icassp/ShiYCPHP15}. Chen et al.~\cite{DBLP:conf/interspeech/ChenHTGD16} proposed a memory network based approach for multi-turn SLU by encoding history utterances and leveraging the memory embeddings through attention. Bapna et al.~\cite{DBLP:conf/sigdial/BapnaTHH17} enhanced the memory network architecture by adding a BiRNN session encoder temporally combining the current utterance encoding and the memory vectors. Su et al.~\cite{DBLP:conf/naacl/SuYC18} investigated different time-decay attention mechanisms. Gupta el al.~\cite{DBLP:conf/interspeech/GuptaRH18} proposed an approach to encode system dialogue acts for SLU, substituting the use of system utterances. Also, various models have been proposed for jointly modeling intent and slot predictions and achieved significant performance improvement over models that model these predictions independently~\cite{DBLP:conf/asru/XuS13,DBLP:conf/interspeech/Hakkani-TurTCCG16,DBLP:conf/ijcai/ZhangW16a,DBLP:conf/interspeech/LiuL16,DBLP:conf/naacl/GooGHHCHC18,DBLP:journals/corr/abs-1812-09471}. In this work, we also follow the joint learning paradigm.
\begin{figure}[htb]
\centering
\includegraphics[width=0.3\textwidth]{example}
\caption{An example user query in a multi-turn conversation and its semantic frame with slot, intent and user dialogue act annotations. For user query ``5 at sakoon'', ``5'' could indicate date, time, number of people, etc; yet with the context, it is most likely resolved as the number of people. }
\label{fig:example}
\end{figure}
However, lack of human-labeled data for SLU
results in poor generalization capability.
A variety of transfer learning (TL) techniques were proposed for addressing the data sparsity challenge. One category of TL approaches includes training general purpose language representation models using a large amount of unlabeled text, such as ELMo~\cite{DBLP:conf/naacl/PetersNIGCLZ18}, GPT~\cite{DBLP:techreport/ge1ne8r}, and BERT~\cite{DBLP:journals/corr/abs-1810-04805}. Pre-trained models can be fine-tuned on NLP tasks and have achieved significant improvement over training on the task-specific annotated data. Bapna et al.~\cite{DBLP:journals/corr/BapnaTHH17aa} leveraged slot name and description encodings within a multi-task model for domain adaptation. Lee et al.~\cite{DBLP:journals/aaai/Lee19} proposed zero-shot adaptive transfer for slot tagging by embedding the slot descriptions and fine-tuning a pre-trained model on the target domain. Siddhant et al.~\cite{DBLP:journals/aaai/Siddhant19} used a light-weight ELMo model for pre-training and unsupervised and supervised transfer.
Our contribution in this paper is threefold: \textbf{First}, we propose a Context Encoding Language Transformer (CELT) model for context-aware SLU. Different from previous work of exploring various encoding schemes and attention mechanisms to encode context for multi-turn SLU, CELT facilitates encoding various context information in the dialogue history for SLU, such as user and system utterances, speaker information, system acts, and utilizing these information in a unified framework through a multi-head self-attention mechanism. The context information that CELT can exploit is extensible. For example, for a conversational system that facilitates the use of a screen for multi-modal interactions, screen-displayed information can be treated similarly as context in CELT and help understand user query. \textbf{Second}, we develop a multi-step TL approach on CELT, namely, unsupervised pre-training to exploit large-scale general purpose unlabeled text, unsupervised adaptive training and supervised adaptive training to exploit other in-domain and out-of-domain dialogue corpora. To our knowledge, the CELT model and the multi-step TL approach on CELT are first proposed in this work for multi-turn SLU. \textbf{Third}, we systematically evaluate the efficacy on SLU from various context information and TL approaches. The proposed CELT model together with the proposed TL approaches significantly outperform the state-of-the-art performance on two large-scale single-turn dialogue benchmarks and one large-scale multi-turn dialogue benchmark.
\section{Proposed Approach}
Figure~\ref{fig:model} provides a high-level illustration of CELT, which consists of the input embedding layer, the encoder representation layer, and the final classifier layer.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{model}
\caption{A high-level view of the CELT model. It consists of the input embedding layer, the encoder representation layer, and the final classifier layer. Details of the input embedding layer are illustrated in Figure~\ref{fig:input_embedding}. ``Trm'' denotes Transformer blocks. }
\label{fig:model}
\end{figure}
\subsection{Input Embedding Layer}
Given the current query token sequence ${\vect q=q_1,\dots, q_T}$
at turn ${t}$, and previous turns in the dialogue session, i.e., user turns ${\vect u}^i, i=[1, \dots, t-1]$, and system turns ${\vect s}^i, i=[1, \dots, t-1]$, the target is to predict the semantic frame, including intent, user acts, and slots, for ${\vect q}$. We concatenate all the previous turns chronologically in a dialogue session and the current query as the input text ${\vect x}$, i.e., ${\vect x} = ({\vect u}^1,{\vect s}^1,\dots, {\vect u}^{t-1}, {\vect s}^{t-1}, {\vect q})$. The first token of every input text is always the special classification embedding ([CLS]) which is used to predict the intent and user acts. Each utterance in the previous turns is inserted an end-of-utterance ([EOU]) token. The previous utterances and the current turn are separated by a special token ([SEP]).
For a token in the input text, its input embedding is an element-wise sum of token embeddings, position embeddings, segment embeddings, and embeddings of other context information. In this work, we add speaker embeddings and system act embeddings into the sum to obtain the final input embedding. Figure~\ref{fig:input_embedding} illustrates the input embedding layer.
The learned \textit{WordPiece embeddings}~\cite{DBLP:journals/corr/WuSCLNMKCGMKSJL16} are used to alleviate the out-of-vocabulary (OOV) problem. The learned \textit{position embeddings} are used to capture the sequence order information. The learned \textit{segment embeddings} are used to distinguish the previous turns and the current query, hence all previous turns have the same segment embeddings. \textit{Speaker embeddings} are used to distinguish the user's turns or the system's turns, considering that speaker role information has been shown useful for SLU in complex dialogues~\cite{DBLP:conf/naacl/SuYC18}. \textit{System act embeddings} encode the system act information. Each system act contains an act type and optional slot and value parameters. The acts are categorized into two broad types: acts with an associated slot (i.e. request(date), select(time=7pm))
and acts without associated slots (e.g. negate). We keep the slot type, ignore the slot values, and convert the system acts into embeddings just like word embeddings. We define an n-hot binary vector ${\vect a}$ to represent system acts of the previous system turn in the system act vocabulary $A$\footnote{We only use the system acts from the previous system turn instead of using all system acts in the dialogue history, in order to keep the same setting as Gupta et al.~\cite{DBLP:conf/interspeech/GuptaRH18}.}, and convert the n-hot binary vector to a fixed-sized vector by multiplying it with a system act embedding matrix ${\mat E}^a$, that is, ${\vect e}^a = {\mat E}^a {\vect a}$.
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{input_embedding}
\caption{The input embedding layer of CELT.}
\label{fig:input_embedding}
\end{figure}
\subsection{Encoder Representation Layer}
The encoder representation layer is a multi-layer Transformer~\cite{DBLP:conf/nips/VaswaniSPUJGKP17} consisting of multi-head self-attention sub-layer and feed-forward sub-layer in each layer. The multi-head self-attention mechanism builds upon scaled dot-product attention, operating on query $Q$, key $K$, and value $V$~\cite{DBLP:conf/nips/VaswaniSPUJGKP17}:
\begin{align}
\mathrm{Attention}(Q,K,V) = \mathrm{softmax}(\frac{Q K^T}{\sqrt{d_k}})V \,,
\label{equ:attention}
\end{align}
\noindent where $d_k$ is the dimension of the keys. Multi-head attention mechanisms obtain $h$ different representations of $(Q, K, V)$, compute scaled dot-product attention for each representation, and concatenate the results. This can be expressed in the same notation as Equation (\ref{equ:attention}):
\begin{align}
\mathrm{head}_i = \mathrm{Attention}(QW_i^Q,KW_i^K,VW_i^V) \,, \\
\mathrm{MultiHead(Q,K,V)} = \mathrm{Concat}_i(\mathrm{head}_i)W^O \,,
\end{align}
\noindent where $W_i^Q,W_i^K,W_i^V,W^O$ are projection matrices~\cite{DBLP:conf/nips/VaswaniSPUJGKP17}.
The concatenation is projected with the feed-forward neural network (FFN) sub-layer. We propose using a two-layered network with GELU~\cite{DBLP:journals/corr/HendrycksG16} activation. Given trainable weights $W_1$, $W_2$, $b_1$, $b_2$, this sub-layer is defined as:
\begin{align}
\mathrm{FFN}(x) = W_2 \mathrm{GELU}(W_1x + b_1) + b_2 \,,
\end{align}
After feeding the input embedding sequence into the encoder representation layer,
the output hidden states are ${\mat H} = ({\vect h}_1,\dots, {\vect h}_T)$, where ${\vect h}_1$ corresponds to [CLS].
\subsection{Final Classifier Layer}
\subsubsection{Intent and user act classification}
We assume that each user query contains a single intent and multiple user dialogue acts. Intent classification (IC) predicts the intent probability distribution $p^i$, using Equation~(\ref{equ:intent}). User act classification (UAC) is defined as a multi-label binary classification problem.
The probability of the presence of the $k$-th user act in the user query, $p^a(k)$, is calculated by Equation~(\ref{equ:act}).
\begin{align}
\label{equ:intent}
p^i = \mathrm{softmax}({\mat W}_i F_i({\vect h}_{[CLS]}) + {\vect b}_i) \,, \\
\label{equ:act}
p^a = \mathrm{sigmoid}({\mat W}_a F_a({\vect h}_{[CLS]}) + {\vect b}_a) \,,
\end{align}
\noindent where $F_i$ and $F_a$ are non-linear feed-forward layers with tanh activation. During inference, the intent label is predicted by $\mathrm{argmax}(p^i)$, and user acts are predicted when the probability $p^a(k)$ is greater than the threshold $t_u$, where $t_u$ is a hyperparameter tuned on the validation set.
\subsubsection{Slot filling}
Slot filling (SF) identifies the values for different slots present in the user utterance. We use the BIO (begin-inside-outside) tagging scheme to assign a label to each token. We feed the final hidden states $\vect{h}_2,\dots,\vect{h}_T$ into a softmax layer to classify over the SF labels. To make this procedure compatible with the WordPiece tokenization, we feed each tokenized input word into a WordPiece tokenizer and use the hidden state corresponding to the first sub-token as the input to the softmax classifier.
\begin{align}
p^s_i = \mathrm{softmax}({\mat W}_s F_s({\vect h}_i) + {\vect b}_s) \,,\
\end{align}
\noindent where \({\vect h}_i\) is the hidden state corresponding to the first sub-token of word \(q_i\) in the current query ${\vect q}$, and $F_s$ is a non-linear feed-forward layer with GELU activation.
For joint learning, the objective is to minimize the sum of the softmax cross-entropy losses of IC and SF and the sigmoid cross-entropy loss of UAC. Previous work has shown that an additional CRF layer on top of a BiLSTM can improve the performance for sequence tagging~\cite{DBLP:conf/acl/ZhouX15,DBLP:journals/corr/HuangXY15}. Hence we investigate the efficacy of adding a CRF layer for modeling slot label dependencies, on top of the Encoder Representation Layer, similar to~\cite{DBLP:conf/icml/LaffertyMP01}.
\subsection{Transfer Learning}
To improve SLU on the target domains and reduce dependency on data collection and annotation, we explore TL based on the CELT model, by leveraging large-scale unlabeled text and other multi-turn dialogue corpora either unlabeled or labeled with different intent/slot/dialog act labels, from the same or different domains w.r.t. the target domains. We develop a multi-step transfer learning approach. In the \textit{first} step, to exploit large-scale unlabeled text, we use unsupervised pre-training based on the BERT model with two tasks trained together, i.e., masked language model (MLM) and next sentence prediction (NSP)~\cite{DBLP:journals/corr/abs-1810-04805}. The resulting model is denoted \(\theta_A\). Next, to exploit other dialogue corpora, we propose two TL methods, namely, unsupervised adaptive training and supervised adaptive training. In the \textit{second} step, the unsupervised adaptive training approach trains \(\theta_A\) on other unlabeled dialogue corpora, using the MLM and NSP losses. The resulting model is denoted \(\theta_B\). In the \textit{third} step, given other labeled dialogue corpora, the proposed supervised adaptive training approach fine-tunes \(\theta_B\) on the labeled data, based on the combined loss of IC and SF\footnote{Note that in the combined loss, when samples have multiple intent labels, sigmoid cross-entropy loss is used for IC instead of softmax cross-entropy loss.}. The resulting weights for the input embedding layer and the encoder representation layer of CELT are then used to initialize the new CELT model. This model is denoted \(\theta_C\) for the next step fine-tuning for the target domain SLU. This way, our supervised adaptive training can exploit labeled data with intent/slot labels different from labels used for the target domains. In the \textit{fourth} step, \(\theta_C\) is fine-tuned on the target domain labeled data based on the combined loss of IC, SF, and UAC.
\section{Experiments and Analysis}
\subsection{Data}
We conduct two sets of experiments. For the first set of experiments, we evaluate the efficacy of BERT pre-train and joint modeling of IC and SF in CELT on the single-turn ATIS~\cite{DBLP:conf/slt/TurHH10} and Snips~\cite{DBLP:journals/corr/abs-1805-10190} dialogue corpora. ATIS includes audio recordings of people making flight reservations. Snips is collected from the Snips personal voice assistant. We use the same data division as~\cite{DBLP:conf/naacl/GooGHHCHC18} for both datasets. The data statistics are summarized in Table~\ref{tab:statistics}. For these two datasets, system act embeddings and user act classifier are not used, because system and user dialogue acts are not annotated. Speaker embeddings are not used since there is only the current query in single-turn dialogues.
\begin{table}[ht!]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{l | r r r}
\hline
\textbf{Dataset} & \textbf{Snips} &\textbf{ATIS} & \textbf{GSD} \\
\hline
Intents & 7 & 21 & 3* \\
Slots & 72 & 120 & 21 \\
User Act & -& - & 22 \\
Training samples & 13,084 & 4,478 & 8,148\\
Validation samples & 700 & 500 & 2,116 \\
Test samples & 700 & 893 & 4,800 \\
\hline
\end{tabular}
}
\end{center}
\caption{Statistics for the Snips, ATIS and GSD data sets, including the number of intent types, slot labels (after applying the BIO scheme on the original slots, including O) and user acts for the training set, the number of samples in the training, validation, and test sets, respectively. *: note that for GSD, intents are quite high-level while the user acts have the same level of granularity as intents for Snips and ATIS.}
\label{tab:statistics}
\end{table}
For the second set of experiments, we evaluate the proposed model and TL approaches on the multi-turn Google Simulated Dialogues (GSD)\footnote{https://github.com/google-research-datasets/simulated-dialogue}~\cite{DBLP:conf/interspeech/GuptaRH18}. We explore Microsoft Dialogue Challenge (MDC)\footnote{https://github.com/xiul-msr/e2e\_dialog\_challenge}~\cite{DBLP:journals/corr/abs-1807-11125} and MultiWOZ 2.0 (WOZ)\footnote{https://www.repository.cam.ac.uk/handle/1810/280608. Note that WOZ does not have user act annotations so cannot be directly used for SLU.}~\cite{DBLP:conf/emnlp/BudzianowskiWTC18} datasets as other dialogue corpora for evaluating the proposed TL approaches.
We use the same data division as~\cite{DBLP:conf/interspeech/GuptaRH18}.
The GSD dataset covers restaurant (GSD-Resturant) and movie (GSD-Moive) domains. The entire GSD dataset (GSD-Overall) consists of 3 intents, 12 slot types, and 22 user dialogue act types. The data statistics are summarized in Table~\ref{tab:statistics}. Note that the 3 intents (``BUY\_MOVIE\_TICKETS",
``FIND\_RESTAURANT", ``RESERVE\_RESTAURANT") are quite high-level; instead, the 22 user dialog act types provide the user intent information for the SLU task.
The MDC dataset covers restaurant, movie, and taxi domains, with 4103, 2890, and 3094 training dialogues, 11 intents, and 30, 29, and 19 slots for the three domains, respectively. The WOZ dataset consists of human-human written conversations spanning 7 domains and 10,438 dialogues in total.
\subsection{Training Details}
The Transformer block in CELT has 12 layers, 768 hidden states, 3072 feed-forward size, and 12 self-attention heads. The size of hidden states in the final classifier layer is 768.
For pre-training, we use the English uncased BERT-Base model\footnote{https://github.com/google-research/bert}, pre-trained on the BooksCorpus~\cite{DBLP:conf/iccv/ZhuKZSUTF15} and English Wikipedia.
For unsupervised/supervised adaptive training on MDC and WOZ and fine-tuning on the GSD-overall dataset, all hyper-parameters are tuned on the GSD-overall validation set. For the first set of experiments on ATIS and Snips, the maximum sequence length is 50, the batch size is 128, and the number of training epochs is 30. For the second set of experiments on multi-turn dialogues, the maximum sequence length is 128 and the batch size is 32. Adam~\cite{DBLP:journals/corr/KingmaB14} is used for optimization. The initial learning rate is 5e-5 for the supervised adaptive training and fine-tuning (in both sets of experiments), and 2e-5 for the unsupervised adaptive training. The dropout probability is 0.1.
The mask probability of the MLM task is 15\% for the unsupervised adaptive training. We compare using different numbers of previous user and system turns in the dialogue session and observe the best SLU performance from using all previous turns.
The threshold $t_u$ for user act classification is selected from $[0.3, 0.4, 0.5]$ and tuned on the validation set.
\subsection{Results and Discussion}
\label{sec:results}
\subsubsection{Single-Turn SLU}
\label{subsubsec:single-turn-slu}
\begin{table*}[htb]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{l |c c c |c c c}
\hline
\multirow{2}{*}{\textbf{Models}} & \multicolumn{3}{c|}{\textbf{Snips}} & \multicolumn{3}{c}{\textbf{ATIS}} \\
& \textbf{Intent} & \textbf{Slot} & \textbf{Frame} & \textbf{Intent} & \textbf{Slot} & \textbf{Frame} \\
& \textbf{(Acc)} & \textbf{(F1)} & \textbf{(Acc)} & \textbf{(Acc)} & \textbf{(F1)} & \textbf{(Acc)} \\
\hline
RNN-LSTM~\cite{DBLP:conf/interspeech/Hakkani-TurTCCG16} & 96.9 & 87.3 & 73.2 & 92.6 & 94.3 & 80.7 \\
Atten.-BiRNN~\cite{DBLP:conf/interspeech/LiuL16} & 96.7 & 87.8 & 74.1 & 91.1 & 94.2 & 78.9 \\
Slot-Gated~\cite{DBLP:conf/naacl/GooGHHCHC18} & 97.0 & 88.8 & 75.5 & 94.1 & 95.2 & 82.6 \\
Capsule Neural Networks~\cite{DBLP:journals/corr/abs-1812-09471} & 97.3 & 91.8 & 80.9 & 95.0 & 95.2 & 83.4\\
\hline
(1) CELT w/o BERT pre-train & 97.8$\pm$0.2 & 90.0$\pm$0.6 & 79.3$\pm$1.4 & 96.9$\pm$0.1 & 92.7$\pm$0.1 & 80.5$\pm$0.4\\
(2) CELT w/o BERT pre-train + CRF & 97.9$\pm$0.3 & 90.8$\pm$0.2 & 80.9$\pm$0.5 & 97.0$\pm$0.3 & 93.1$\pm$0.2 & 81.6$\pm$0.4 \\
(3) (1) w/ BERT pre-train & \textbf{98.3}$\pm$0.3 & 96.4$\pm$0.2 & \textbf{91.9}$\pm$0.2 & {97.4}$\pm$0.4 & \textbf{95.9}$\pm$0.1 & \textbf{87.9}$\pm$0.4 \\
(4) (2) w/ BERT pre-train & \textbf{98.3}$\pm$0.1 & \textbf{96.5}$\pm$0.2 & 91.8$\pm$0.5 & \textbf{97.6}$\pm$0.1 & {95.7}$\pm$0.1 & 87.6$\pm$0.2\\
\hline
\end{tabular}
}
\end{center}
\caption{SLU performance on the single-turn Snips and ATIS testsets. Note that since Snips and ATIS are single-turn dialogues, all models in this table do not use context information. All models are trained and tested on the same training and test partitions of Snips and ATIS, respectively (no transfer learning is applied). The mean and standard deviation of SLU results from CELT w/o and with BERT pre-train, w/o and with replacing the softmax layer with a CRF layer, from 5 different models with different random initialization are given here. The metrics are the intent classification accuracy, slot filling F1, and sentence-level semantic frame accuracy. The results for the first group of models are cited from~\cite{DBLP:conf/naacl/GooGHHCHC18,DBLP:journals/corr/abs-1812-09471}.}
\label{tab:single-turn:result}
\end{table*}
Table~\ref{tab:single-turn:result} shows the SLU performance as SF F1, IC accuracy, and sentence-level semantic frame accuracy on the Snips and ATIS datasets. The first group of models is considered as the baselines and it consists of the state-of-the-art joint IC and SF models: the sequence-based joint model using BiLSTM~\cite{DBLP:conf/interspeech/Hakkani-TurTCCG16}, the attention-based model~\cite{DBLP:conf/interspeech/LiuL16}, the slot-gated model~\cite{DBLP:conf/naacl/GooGHHCHC18}, and the capsule neural network based model~\cite{DBLP:journals/corr/abs-1812-09471}.
The second group of models in Table~\ref{tab:single-turn:result} includes the proposed CELT models. CELT with BERT pre-train significantly outperforms the baselines on both datasets. Compared to ATIS, Snips includes multiple domains and has a larger vocabulary. For the more complex Snips dataset, CELT with BERT pre-train achieves intent accuracy of 98.3\% (from 97.3\%), slot F1 of 96.4\% (from 91.8\%), and sentence accuracy of 91.9\% (from 80.9\%). On ATIS, CELT achieves intent accuracy of 97.4\% (from 95.0\%), slot F1 of 95.9\% (from 95.2\%), and sentence accuracy of 87.9\% (from 83.4\%). The gain from CELT with BERT pre-train on Snips over the baselines is much more significant on slot F1 and sentence frame accuracy than intent accuracy. Further analysis shows that 53.4\% of slots in the Snips test set can be found in Wikipedia. Since BERT pre-training data includes English Wikipedia, the model may have encoded the knowledge in representations and hence improves slot F1 and sentence accuracy.
Without BERT pre-train, the SLU performance degrades drastically on both datasets. These results demonstrate the strong generalization and semantic representation capability of the BERT pre-train model, considering that it is pre-trained on large-scale text from mismatched domains and genres (books and Wikipedia). Without BERT pre-train, replacing the softmax layer with CRF consistently improves the sentence accuracy (3\% and 2.4\% relative gains for Snips and ATIS, respectively); whereas, adding CRF for CELT with BERT pre-train performs comparably. Hence, the second set of experiments uses CELT without CRF.
Ablation analysis on Snips shows that when fine-tuning the BERT pre-train model separately for IC and SF, intent accuracy drops to $97.8 \pm 0.4$\% (from $98.3 \pm 0.3$\%), and slot F1 drops to $96.3 \pm 0.1$\% (from $96.4 \pm 0.2$\%). These results demonstrate that joint modeling in CELT improves the performance for both tasks. We compare CELT models with different fine-tuning epochs. The CELT model fine-tuned with only 1 epoch already outperforms the baselines in Table~\ref{tab:single-turn:result}.
\begin{table*}[ht]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{l | c c c c | c c c c | c c c c}
\hline
\multirow{3}{*}{\textbf{Models}} & \multicolumn{4}{c|}{\textbf{GSD-Resturant}} & \multicolumn{4}{c|}{\textbf{GSD-Movie}} & \multicolumn{4}{c}{\textbf{GSD-Overall}} \\
& \textbf{Intent} & \textbf{Act} & \textbf{Slot} & \textbf{Frame} & \textbf{Intent} & \textbf{Act} & \textbf{Slot} & \textbf{Frame} & \textbf{Intent} & \textbf{Act} & \textbf{Slot} & \textbf{Frame} \\
& \textbf{(Acc)} & \textbf{(F1)} & \textbf{(F1)} & \textbf{(Acc)} & \textbf{(Acc)} & \textbf{(F1)} & \textbf{(F1)} & \textbf{(Acc)} & \textbf{(Acc)} & \textbf{(F1)} & \textbf{(F1)} & \textbf{(Acc)} \\
\hline
RNN-NoContext~\cite{DBLP:conf/interspeech/GuptaRH18} & 83.61 & 87.13 & 94.24 & 65.51 & 88.51 & 93.49 & 86.91 & 62.17 & 84.76 & 89.03 & 92.01 & 64.56 \\
RNN-PreviousTurn~\cite{DBLP:conf/interspeech/GuptaRH18} & 99.37 & 90.10 & 94.96 & 86.93 & 99.12 & 93.58 & 88.63 & 77.27 & 99.31 & 91.13 & 93.06 & 84.19 \\
MemNet-20~\cite{DBLP:conf/interspeech/ChenHTGD16} & 99.67 & 95.67 & 94.28 & 89.52 & 98.76 & 96.25 & 90.70 & 80.35 & 99.29 & 95.85 & 93.21 & 86.92 \\
SDEN-20~\cite{DBLP:conf/sigdial/BapnaTHH17} & 99.84 & 94.43 & 94.81 & 89.46 & 99.60 & 97.56 & 90.93 & 82.55 & 99.81 & 95.38 & 93.65 & 87.50 \\
HRNN-SystemAct~\cite{DBLP:conf/interspeech/GuptaRH18} & \textbf{99.98} & 95.42 & 95.38 & 89.26 & 99.71 & 96.35 & 91.58 & 83.36 & \textbf{99.92} & 95.70 & 94.22 & 87.58 \\
\hline
CELT & 99.88 & \textbf{98.47} & \textbf{97.12} & \textbf{95.63} & \textbf{99.71} & \textbf{98.45} & \textbf{95.74} & \textbf{93.48} & 99.83 & \textbf{98.47} & \textbf{96.70} & \textbf{95.02} \\
\hline
\end{tabular}
}
\end{center}
\caption{SLU performance on different test sets of the multi-turn GSD dialogue corpus, from baselines and our proposed CELT model, when trained on the GSD-overall training set. The results for the first group of models are cited from~\cite{DBLP:conf/interspeech/GuptaRH18}. MemNet-20 and SDEN-20 denote models with memory size 20.}
\label{tab:result}
\end{table*}
\begin{table}[ht]
\renewcommand{\arraystretch}{0.9}
\begin{center}
\scalebox{0.9}{
\begin{tabular}{l | c c c c}
\hline
\multirow{2}{*}{\textbf{Model}} & \textbf{Intent} & \textbf{Act} & \textbf{Slot} & \textbf{Frame} \\
& \textbf{(Acc)} & \textbf{(F1)} & \textbf{(F1)} & \textbf{(Acc)} \\
\hline
a. CELT & 99.83 & 98.47 & 96.70 & \textbf{95.02} \\ \hline
b. a-UA-SA & 99.62 & 98.29 & 94.99 & 93.00 \\ \hline \hline
c. b+UA(ID) & 99.88 & 98.03 & 96.64 & 94.40 \\
d. b+UA(OOD) & 99.92 & 98.18 & 95.07 &93.35 \\
e. b+UA(ID)+SA(ID) & 99.90 & 98.43 & 96.17 & 94.60 \\
f. b+UA(ID)+SA(ID+OOD) &99.83 & 98.47 & 96.70 & \textbf{95.02} \\
g. b+UA(ID+OOD) & 99.90 & 98.24 & 95.88 & 94.25 \\
h. b+UA(ID+OOD) &\multirow{2}{*}{99.92} &\multirow{2}{*}{96.58} &\multirow{2}{*}{96.58} & \multirow{2}{*}{94.81} \\
~~~~+SA(ID+OOD) & & & & \\
i. b+UA(ID+OOD+WOZ) & 99.92 & 98.52 & 95.02 & 93.71 \\
\hline \hline
j. b-BERT pre-train & 99.92 & 92.44 & 91.12 & 86.54 \\
k. j-speaker embeddings & 99.96 & 92.22 & 90.43 & 86.15 \\
l. k-context utterances & 94.52 & 93.55 & 91.61 & 82.35 \\
m. l-system act embeddings & 77.33 & 89.11 & 90.93 & 66.88 \\
\hline
\end{tabular}
}
\end{center}
\caption{Ablation Analysis on the GSD-overall test set. UA and SA denote unsupervised and supervised adaptive training, respectively. ID and OOD denote in-domain and out-of-domain dialogues in MDC w.r.t. the GSD-overall test set, i.e., MDC restaurant and movie domain dialogues are ID data, MDC taxi domain dialogues are OOD data. +WOZ denotes using MDC+WOZ data for adaptive training.}
\label{tab:ablation}
\end{table}
\subsubsection{Multi-Turn SLU and Transfer Learning}
\label{subsec:multi-turn-slu}
Table~\ref{tab:result} shows the IC accuracy, UAC F1, SF F1, and sentence frame accuracy on the GSD test sets.
The first group of models includes the baselines. \textit{RNN-NoContext}~\cite{DBLP:conf/interspeech/GuptaRH18} uses two-layer stacked BiRNN with GRU and LSTM cells respectively, and no context information is used. \textit{RNN-PreviousTurn}~\cite{DBLP:conf/interspeech/GuptaRH18} is similar to the \textit{RNN-NoContext} model, with a different BiGRU layer encoding the previous system turn for slot tagging.
\textit{MemNet}~\cite{DBLP:conf/interspeech/ChenHTGD16} uses memory network to encode the dialogue history utterances from both user and system. \textit{SDEN}~\cite{DBLP:conf/sigdial/BapnaTHH17} uses the dialogue history utterances from both user and system through a BiGRU for combining memory embeddings. \textit{HRNN-SystemAct}~\cite{DBLP:conf/interspeech/GuptaRH18} is the previous state-of-the-art (SOTA) system, using a hierarchical RNN to encode the dialogues acts of the previous system turn as the context information.
The second group of models includes the proposed CELT model after applying the proposed TL approaches. CELT achieves new SOTA results and the absolute gains from CELT over the previous SOTA user act F1 and sentence frame accuracy are 2.8\% and 6.11\% on the GSD-Restaurant testset, 0.89\% and 10.12\% on the GSD-Movie testset, and 2.62\% and 7.44\% on the GSD-Overall testset.
Table~\ref{tab:ablation} shows the ablation analysis on the GSD-Overall test set. Removing all unsupervised and supervised adaptive training for CELT (CELT-UA-SA) degrades the sentence accuracy from 95.02\% to 93.00\%. Further removing the unsupervised BERT pre-train step degrades sentence accuracy to 86.54\%. Particularly, user act F1 decreases from 98.29\% to 92.44\%, and slot F1 decreases from 94.99\% to 91.12\%. These results demonstrate that the contextual representations learned from the large-scale general purpose unlabeled text significantly help improve user act classification and slot filling. After further removing the speaker embeddings, slot F1 drops from 91.12\% to 90.43\% and sentence accuracy drops from 86.54\% to 86.15\%, suggesting that CELT is capable of exploiting the additional discriminative information provided by the speaker embeddings. After further removing the context utterances, intent accuracy drops from 99.96\% to 94.52\% and sentence accuracy drops from 86.15\% to 82.35\%, indicating that the context utterances play a key role in intent prediction. After further removing system act embeddings, that is, no context is used, intent accuracy drops from 94.52\% to 77.33\%, user act F1 drops from 93.55\% to 89.11\% and sentence frame accuracy drops from 82.35\% to 66.88\%. These results show that using no context information degrades the SLU performance significantly. It is noticeable that although intent accuracy and slot F1 of CELT-NoContext (the last row in Table~\ref{tab:ablation}) are both lower than those of RNN-NoContext (the first row in Table~\ref{tab:result}), CELT-NoContxt achieves a better sentence accuracy (66.88\%) than RNN-NoContext (64.56\%), demonstrating the strength of CELT to enforce intent and slot coherence.
We further analyze the efficacy of using in-domain (ID) and out-of-domain (OOD) data for unsupervised adaptive (UA) and supervised adaptive (SA) training. As shown in Table~\ref{tab:ablation}, using OOD data for UA (model d) achieves small SLU improvement over model b. UA(ID) (model c) yields a significantly larger gain over model b compared to model d.
However, adding OOD data to ID data for UA (model g) degrades the performance slightly compared to model c. Adding WOZ to MDC ID+OOD data for UA (model i) further degrades the performance over model g. In contrast, after applying UA(ID) (model c), adding OOD to ID for SA (model f) outperforms SA(ID) (model e), achieving 95.02\% sentence frame accuracy. These results suggest that SA can benefit from both ID and OOD data, probably due to the combined loss of IC and SF. We will explore different losses for UA other than MLM and NSP losses in order to benefit from both ID and OOD data.
We also observe fast convergence speed of both CELT-UA-SA and CELT models on the GSD-overall testset, consistent with previous observations on models using BERT pre-train. For CELT-UA-SA, the user act F1 and sentence frame accuracy increase from 87.50 and 52.52 (epoch 1) to 97.19 and 90.31 (epoch 5), and keep improving until epoch 20 (98.29 and 93.00) but degrade from epoch 20 to 40. For CELT, these two results increase from 92.34 and 62.60 (epoch 1) to 97.17 and 93.79 (epoch 5), and keep improving from epoch 5 to 40 (98.47 and 95.02).
\section{Conclusions}
We propose Context Encoding Language Transformer for SLU facilitating exploiting various context information and different transfer learning approaches for leveraging external resources. Experimental results demonstrate that CELT with TL achieves new SOTA SLU performance on two large-scale single-turn dialogue benchmarks and one multi-turn dialogue benchmark. Future work includes improving supervised and unsupervised TL and exploring TL on knowledge bases.
\bibliographystyle{IEEEbib}
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NFL Week 1 Game Picks & Predictions (2019)
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Hey, what is going on? Everybody welcome in back to the sports that she aren't sportscaster, and we got the NFL season kicking off on Thursday night, the Bears versus the Packers in the NFL 100 season to open it up.
But before I do so getting into all the game picks year, I like to let you guys know that I'll be doing the NFL game picks throughout the season.
Here on the sports dish on sportscaster.
I'll be doing game picks each week on Tuesday night for you guys from my predictions for every single NFL game. Drop the season.
So here we are, weak one of the NFL season up and coming with my game picks and predictions as we take a look towards the NFL opening night and I will take place on Thursday, September 5th, with the Green Bay Packers taking on the Chicago Bears and Action.
That's game will be 8:20 p.m.
Eastern time on NBC.
And honestly, this game could go either way between both of these teams.
Just like last season's matchup toe open up the season, I feel that this game will be a lower scoring game just like last season with the Eagles versus Falcons to kick things off.
But towards the end here I got the Chicago Bears defense.
I think that they'll just be too good for Aaron Rodgers and company in action for this game.
As far as what to expect.
Coolio Mac is a force with that pass rush and stopping the run as well.
He'll be the key factor for the Chicago Bears as we look towards the Green Bay Packers and they're rushing offense, led by Aaron Jones, but also the passing game with arid Rogers.
It's gonna be an interesting match about their toe.
Chicago Bear Secondary's really, really stacked.
Kyle Fuller will lead the NFL interceptions last season with seven I and teases.
This should be a really close game to open up the NFL season.
I've got the Chicago Bears, and that's 1 17 14 pulling out that w over to Green Bay Packers.
So that's my Thursday Night football protection now moving on to the sleigh of Sunday afternoon.
Games will go over the first three games here on 1 p.m.
Eastern time on Sunday.
We'll start out with this one, the Tennessee Titans taking on the Cleveland Browns.
This game at Cleveland, 1 p.m.
Eastern time on CBS, and I've got the Cleveland Browns winning this game $34.
13 will be the final form.
My game pick of this.
When the Cleveland Browns, I'm going to say, Well, just have too much firepower.
Where the addition of Odell Beckham Jr on the roster on M v P.
Quarterback candidate possible for this season.
Baker. Mayfield.
I just think that the Cleveland Browns walked too much fire power in this game with Baker and O.
B. J. Also Jarvis Landry.
And there's a second wide out behind the O. B. J.
Don't forget about the tight and David the Joker as well for the Cleveland Browns and the running back Nick Shoved trying to build off his second season here in the NFL.
I feel like the Browns are just gonna have too much firepower in that game.
I feel like they're gonna get the W 34 to 13 Tennessee.
Offensively, I feel like they just won't have enough to keep up.
Keep up with the Cleveland Browns in this game.
Tennessee rank towards the bottom of the NFL last season in passing offense.
Wide receiver Kori Davidson, A J.
Brown if they wanted, when they really need to step up their game and receive some passes and gain some separation that get open and score touchdown receptions and catches around the field.
And Delaney Walker is back.
That's gonna help out the Tennessee Titans is he got injured in the first game of last season in the season opener against Miami, but still, I've got the Cleveland Browns all over Tennessee and this 1 34 to 13.
Hey, what's good? We have a six d A back in here, just going over my week.
One NFL game picks.
I'll be doing this every Tuesday night for you guys. My Week.
One NFL game picks as we're on the first way.
The games.
I just went over the Browns versus Packers a moment ago, but I'll recap towards the end of the show that scores for you guys as well.
I had Bears 17 14 over the Packers, moving on to the next game.
We're only at the one o'clock time slot right now.
The Baltimore Ravens take on the Miami Dolphins for this one.
I've got the Ravens all over the direction.
This game.
One PM on CBS Nothing really looks promising at all for this Dolphins team for this upcoming season.
They're basically rebuilding, that's for sure.
With Brian Floor is coming in as the new head coach, heading over from the Patriots, a defense of coordinator to now coaching for the Dolphins.
But the Ravens will have a second season in a row that they open up with the easiest game on the schedule last season that crush Buffalo 47 to 3.
This season, they should be on their route to another a f C East, routing this one.
They should crush the Miami Dolphins.
I see the score there, 31 23 So possible turnovers by the Dolphins could be likely in this game.
And they're likely that happened, for sure, as Baltimore has had a great defense in the past, they did lose a couple of their players over the free agency Eric Weddle, CJ Mosley.
But still, I think the Baltimore Ravens, they're going across crush the Miami Dolphins in action in this game 31 23 I have is the final prediction in the next game.
Honestly, I have the Atlanta Falcons taking on the Minnesota Vikings.
This game will be on Fox.
m. Eastern time will be the start of this one.
Honestly, same thing as the Packers vs Bears, This game could go either way.
I have this as the game of the week.
This game, in my opinion, will be the best game of the week.
Could go down to the wire.
Could be a walkoff touchdown.
Or so that's how close this game will be.
But I'm taking a lancer here on the road at the Vikings toe win this one.
So I'm taking the Falcons in this game, and a few guys tuned into my NFL standings predictions.
I was actually really high in the Falcons this season.
I think that their defense will be back, Their defense will be healthy, and I think Matt Ryan will have three solid wide receivers that throw two in the line of Julio Jones, Calvin Ridley and also Mohammad Sanu said this Atlanta Falcons offense and defense.
If it both clicks, this team could be reminiscent of what we saw back in 2016 where they want all the way to the Super Bowl.
But the defense is back and healthy.
That's the thing that watch out now for if you're taking on the Atlanta Falcons last season, the Vikings were able to win the big games when it mattered against really good teams above 500 records.
I think that the Viking start out with a disappointment, losing a close one at home against the Atlanta Falcons in this game.
So it's gonna be a close one, I think.
But I got Atlanta and in action here, winning the match up.
We'll move on towards the next late the next two games of the one o'clock time slot here, as we have, the Buffalo bill is going up against the New York Jets.
This one will kick off 1 p.
Eastern time on CBS, and it's gonna be a battle for sure.
I'm thinking this game will be close, but I like Sam Darnold over Josh.
Alan, That's quarterback in this game.
I'm seeing Sam Darnold as the number two quarterback this season in the A F C East.
Of course, the goat the greatest of all time.
Tom already will be your number one again at each 42.
But as far as the number two Q B and A A F C East.
I've got Sam Darnold over Josh.
Alan in this game.
Will show for sure is that Jets will win this game at home at the Meadowlands by six against the Buffalo Bills.
Lavia Bell will be a factor.
He'll be back.
Last season, the running game wasn't there for the New York Jets Day at Trenton.
Cannon and Elijah McGuire did two running backs last season for the New York Jets.
Getting Levy on Bill will be huge towards the New York Jets that help them out so much.
He was the number one player in all fantasy football back in 2017.
He missed all last season because of his hold out.
But now on the New York Jets, and he has this team on peace.
That go, I'm guessing I'd say like nine and seven for the record, if I was to predict the New York Jets record.
But I say they're going to get this game during the Buffalo Bills as the New York Jets will win by six as faras Buffalo the opposite for their running game.
How Levy on Bella's a strong running back for the Jets.
The opposite can be said for Buffalo is they do not have a strong running game.
It's led by running back past his prime 15 season in the NFL.
Frank Gore, who had his best days playing in the late to thousands and mid tooth that his best days are yet behind them, even with all his Pro Bowl appearances.
But Devon Singletary will hopefully emerge sometime later this season for the Buffalo Bills, but I just don't see it as early as weak one as they'll give the keys sometime to Devon Singletary.
But in this game, I see the New York Jets winning this one by 6 22 14 the final and now moving on to the NFC match of one o'clock Eastern time on Fox's this matchup between the Washington Redskins in the Philadelphia Eagles.
Now I've got the Eagles and this one just crushing the Redskins 37 that sent the final here.
I said that Washington stands no chance against Philadelphia.
The Redskins didn't name Case Keenum, their starting quarterback for Week one over Dwayne Haskins.
So Dwayne Haskins, the rookie pick of number 20th in the draft by Washington, not enough to earn a starting job, though case Keenum will get that for Week one.
We all know how last season want one case. Keenum played for Denver.
He struggled early in the season, that is for sure, as I think that this filly defense will just pick apart the Redskins offense in this game and Seamus the offense for Philadelphia.
Don't pick apart the Redskins defense Carson Wentz in the Zach Ertz Connection that will be there throughout the game.
I'm seeing Seamus all Sean, Jeffrey Hell be a viable targets in the in the game as well with Carson once when they take on Washington in the first game.
It's at home, by the way, at Philadelphia.
Very tough place to play.
I say Philadelphia is gonna win single handedly by three scores, arm or in action here.
I'm saying 37 a 10. The final.
Philadelphia's defense looks like one of the best.
If you guys you fantasy football.
Philly's defense this week looks like one of the best plug in please for this upcoming week in fantasy football, Of course, the Washington Redskins haven't had a strong offense for the past couple years or so, and they don't look like they're gonna be strong on the offensive end.
Really? Weakened pleaded on their wide receiving corps.
I've got the Philadelphia Eagles here crushing the Redskins, 30 7 to 10.
Now we move on to our next play, the games.
On the map here, 1 p.m.
Eastern time will move on to the Los Angeles Rams, kicking off against the Carolina Panthers this game.
I have the Rams winning by eight points, 35 to 27 so even were being held at three points in that Super Bowl.
I still think that the L.
Rams will still be able to put up a ton of points in the Sean McVeigh style type of offense here for the L.
A Rams, of course, they use so many things so well with their wide receivers running batches, integrating them all into the game. For the L.
A Rams weather.
It's the end around.
Jets sweep that they do, faking that and around Jet sweeping, getting a wide receiver open down the field.
Whether it's Robert Woods, Cooper Cup or somebody else in the roster, there seems Brandin cooks with a deep bog.
There's just so many things that the L. A.
Rams did well besides when they put in that Super Bowl last year.
But the Rams have so much to look forward to with their offensive end once again with head coach Sean McVeigh.
I think that, though, win this game 35 27 I'm thinking they'll be up 35 to 20 against Carolina.
And then Carolina will probably get a garbage time touched down to put this game a whole lot closer, 35 27.
The final as far as the offense for Carolina there wide receiving corps is not as strong at all compared to the L A Rams.
I like the Rams in this one D.
Moore is the number one wide receiver for the Carolina Panthers. He's on.
Lee caught two career reception touchdowns in its time for Carolina.
Greg Olsen is back after 16 games in the past two seasons with foot injuries up looks like Carolina Panthers could be banged up for Week one.
Also, Cam Newton status as a quarterback for Week one is also up in the air of Cam.
Cannot start that score for Carolina will lower significantly.
Ham does not start week one.
I predict the score cam doesn't start Week one.
The score be different.
It looked like 35 to 14 ourselves.
Right now, I'm guessing that Cam will start this game.
Oh, be a little bit closer.
I have the score.
The next game on the slate here.
Eastern time on Sunday, September 8th is a game on CBS between the Kansas City Chiefs and the Jacksonville Jaguars.
This game I have that.
She's winning by two touchdowns, 34 to 22 finalists.
Casey often just so difficult to stop her there.
Tyreek Hill Speediest player.
All of the NFL.
Patrick Mahomes can throw it better than anybody.
How he slings the ball up in the air.
It's incredible what the phenomenon do out there on the field, but just too many weapons.
Travis Kelsey In this Kansas City offense, Jacksonville isn't really known for their offense.
Their defense, though, is pretty good. But their offense.
It's gonna be interesting to see what Jacksonville can do without offense that try to keep up with Kansas City.
Of course, Nick folks will go in with his first ever NFL career start for the Jacksonville Jaguars Is this upcoming Sunday LB at home at Jacksonville, But still, I like the Kansas City Chiefs, even though they're not an arrowhead for this game.
I like the Chiefs on the road, winning by two touchdowns against Jacksonville, and that brings me to my next slate of games here.
Will look towards the late afternoon, the 405 start times and then the 4 25 star times.
We have two games starting at 405 We'll look at the 1st 1 the Indianapolis Colts taking on the Los Angeles Chargers in action.
This game. I had a whole lot closer.
I have the courts.
Two and 1/2 weeks ago, when Andrew Luck was still in the NFL, I had the Colts winning this game by a close margin.
Believe their night.
Think the score was 24 21 but since now Andrew Luck's retired from the NFL, I've got the Chargers rolling in this 1 28 to 6, just two little time.
That put something together for Jacoby percent and company for this Indianapolis Colts offense.
Chances are slim when you're starting quarterback retires, and he was the 2018 comeback player of the year.
Andrew Luck.
Its chances are slim to win this game when the best player on the field for your team retires two weeks before the season opener, Jacoby percent now will step in as starting quarterback for the Indianapolis Colts.
Look how it went to two years ago in 2017.
They lost that game, 46 to 9 against the L A Rams in Week one, but two that are on Sunday doesn't look too promising.
I'd predict for them 28 to 6.
I have the L. A Chargers.
Winning this game, even without Melvin Gordon in the lineup, is the holdout continues on most, predicting their Melvin Gordon is seeking trade options as well as we're gonna look towards that and the L.
A Chargers right now, I think that they'll win that game 28 to 6, moving on to the next prediction.
Here I have the score production.
I think that the Seattle Seahawks will win the Bangles and that Game 20.
The six will be the final on protecting injuries in just a terrible defense last season, where the big time bring down in the bane of the bangles team last season.
Andy Dalton got injured around Week 11 or so.
I think that Andy Dalton will have a terrible week playing against the Seattle defense.
If you guys do fantasy football, please do not stream Andy Dalton.
And this is not a week that do so at Seattle against a very tough defense.
I don't even think they score a touchdown in this game.
The only future that the Bangles basically have is their running back, Joe Mixon.
But rush carries and yards could be tough to come by in this game as well.
For the Bangles, they most likely will probably beat down all game at Seattle A.
Green not helping out this team as well as he's still recovering from an ankle injury.
He will not be able to play in the week one season opener at Seattle, just to add another punch there for the Bangles to recover from.
And it's gonna be tough for the Bengals that get the W here.
I'm thinking Seattle will be all over Cincinnati and expect Chris Carson and Rashawn Penny, both of the running backs for the Seahawks, have big games to run the ball all over Cincinnati in this game for the Seahawks.
I'm taking the Seahawks, and that's 1 20 to 6.
Moving out to my next prediction.
This game for 25 p.
Eastern time on Fox. The Giants versus the Cowboys.
I'm picking the Giants here.
Set the Cowboys year, 17 13 over the Giants.
Now the it's gonna be a close game feeling.
Both of these teams have played close games in the past.
Giants and Cowboys and Opening Day.
Just don't expect too much from the Giants offense, I'd say with their wide receivers sake, want Barkley the running back? We'll pick up that load for the Giants here in Week one.
He might be a possible M V P player this season, depending on his numbers.
For sure, he should definitely go over 1000 rushing yards sake.
Want Barkley shut? But I like the Dallas Cowboys defense much better than the Giants in action here.
As far as the Giants wide receivers, they're banged up.
I know Sterling Shepard is recovering from a thumb injury.
He's hoping to play in the season opener, but other than that, so the other wide receivers on this team golden teeth the number two wide out from the New York Giants, is suspended for the 1st 4 games for violating the NFL substance policy.
So it's looking like the Cowboys for this one on predicting I've got Dallas winning this 1 17 to 13 now moving on to the next batch of games.
Well, go over in this batch.
We have the Detroit Lions taking on the Arizona Cardinals here in this portion of these games, these air still the late Sunday afternoon games right now.
So I have the Cardinals winning by four against the lines.
I think that, honestly, Cuyler Murray can shock people here when its first ever game in the NFL.
I am a believer of Kyler.
I don't believe that they will be able to get to the playoffs, but I believe that those six in 10 this season projecting their record.
But Skylar murdered Murray Aiken see improving doubters, winning his first ever NFL career start here.
Just it will be interesting to see Cuyler how on the field ends.
I'm saying Detroit Survivor Steven Core.
It's really not that strong.
Kenny Golladay is one of the leading by receiver.
Seems Marvin Jones Jr.
The number three White House Danny Amendola.
But other than that, the whole other lines receiving corps is really weak after those three in the lineup and then a bunch of basically no need wide receivers that you ever heard before.
So Arizona's rushing attack, I think, is gonna be the big difference in this game will be led by David Johnson, who was a fantasy stud the Ree years ago or so back in 2016 when he was one of the top scores on all fantasy football deejay.
Hopefully, he'll be back and emerge again this season.
There was a slow season tow the star of last season for David Johnson, but he got the wheels going later in the season.
But I think that David Johnson will get a great start here at home for Arizona with a victory, 27 2 23 over the lines.
Moving on to the next game on predicting I'm predicting 40 Niners vs Buccaneers.
I've got this game.
A 10 point victory by the 40 Niners.
Now Jimmy Garoppolo is very lucky.
He gets up fairly easy.
Buccaneers defense to start out the season.
Last season, the Buccaneers defense was horrendous.
In the secondary. They were giving up passing touchdown.
After passing touchdown, they gave up six passing touchdown.
Some Mitch Mitchell Travis Key of the Chicago Bears last season.
Now I have Jimmy Garoppolo throwing 33 touchdown passes and weak one.
After Week one, everybody will be so high on Jimmy Garoppolo.
I field.
I'll be like you. Oh, Jimmy G is back.
This guy is awesome.
San Francisco 40 Niners going to the playoffs year.
This guy is great, and then it's just because they point Tampa Bay.
So don't hold out for a minute.
Don't expect those numbers that continue for Jimmy Garoppolo throughout the season.
They'll get the victory, but it's because they played against Tampa Bay in the poor secondary that they have.
The 40 Niners will get the victory 30 to 20 and this one that I'm predicting.
Moving on to the next way. The games.
Now, since we have the Sunday Night Football one, I cannot wait for this game to get into action.
Sunday Night Football Raising the banner.
Gillette Stadium here with the Super Bowl, 53 banners.
We got the New England Patriots out home actual at taking on the Pittsburgh Steelers, 8:20 p.
Eastern time on NBC will be the kickoff time.
I've got the Patriots by seven in this 1 24 to 17 a final, and this should be a great game for sure.
So Patriots by a touchdown I feel like they're just gonna be way too talented on both sides of the ball, especially that secondary.
They're fun.
Gilmore, the McCourty Brothers, J. C.
Jackson, Jonathan Jones, Diran.
Harming inside the secondary.
And also the pass rush is something that we haven't seen from the past with the Patriots the past few years and actually the preseason we saw that nine sacks and Week one of the preseason opener against the Detroit Lions.
Have you seen a good amount of sex from that passion? What Rush? One of the big players is from the left side of the defense of EJ Chase window.
Vich um, hoping that he could be a factor there week one that got him round up round number three in the draft from the University of Michigan.
But I've got the Patriots year, I think, just two talented on both sides of the ball.
Josh Gordon is back now in the NFL once again for the Patriots.
He's gonna be a major downfield threat in this one.
Julian Edelman, of course, Inside the slot for the Patriots, as well as the other players.
Key breakout preseason player Jacoby Meyers as a wide receiver.
And also look towards a lot of running plays for the Patriots, led by Sony.
Michelle, the running back in the past, catching back as well, James White as they do it on all sides of the football.
I've got the Patriots year 24 to 17 the final up against the Pittsburgh Steelers, but the Steelers? I feel like we'll be a really good team this season.
There will be back in the playoffs is a wild card team led by three players who two seasons ago it was Rock less Burger, Antonio Brown and Levy on Bell.
Now the whole new trio of the Pittsburgh Steelers goes like this.
Ben Roethlisberger, juju, Smith, Schuster and James Connor are the whole new three, but I still get the Patriots up by seven, and this one I have them predicted to win that game.
Now we move on towards the lastly, the game's year Monday Night Football on September 9th.
We got some good games on the slate for this one.
The Houston Texans will go that take on the New Orleans Saints, and it should be a good one at the Dome here.
But I get the Saints winning this game.
The game will kick off 7:10 p.
m. Eastern time on ESPN.
I had Houston scoring 27 points, but I dropped it a 23 because of the Lamar Miller injury before they lost Lamar Miller to the tourney a C offered this season.
Still, the offensive line for the Texas looks shaky, That is for sure.
Now the New Orleans Saints were a top five defense against the run this past season.
They were really quiet, but really good this past season in the NFL against the run.
I expect that to continue here for the New Orleans Saints.
Interesting thing to look out for is this question.
How well Sean Payton integrate tastes on the Hill into the game plan here.
Now we've seen the Oakland Athletics talking about baseball here in the wild card game last season, introducing bullpen ing to Major League Baseball, where they had a pitcher pitching inning here, another pitcher coming in something.
The pigeon inning Here.
Now in football, this will be interesting.
Which John Peyton? How he integrates Taysom Hill into the lineup.
Will it be like one out of every five snaps? Taysom Hill comes in one out of every 10 snaps to come in and fill in for Drew Brees.
That's quarterback.
It'll be interesting.
I know when they go for for 1/4 down conversion, expect taste and hell to be out there.
He is somebody who absolutely does everything on the field, from quarterback to running back to special teams.
Taysom Hill does it all for the Saints.
It'll be interesting to see El Sean Payton integrates him into the game plan this season to see if he does take off the load a little bit off Drew Brees.
As far as that's concerned, though, I still have the Saints with too much firepower.
Behind Michael Thomas is why receiver? They also have a new tight end in town.
Jared Krupp coming from the Raiders over to the Saints.
I believe that he's an upgrade that tight ends from Ben Watson last season and as well as the running back.
Watch out, guys.
Alvin, CA.
Mera will have huge season in the NFL.
I'm predicting.
And now by Alvin Kamara will be the main bell cow this upcoming season for the New Orleans Saints since Mark Ingraham is now playing for Baltimore.
He's gonna have a huge season this season in the NFL, and I think that Ingram playing alongside Lamar Jackson, We'll have a pretty big season as well.
Both of those players could be pro Bowlers for separate squads.
The FC in NFC.
So now it just moves on to one more game of the week here.
Monday, September 9th.
The late game.
This one will kick off at 10:20 p.m.
Eastern time.
It'll be between the Denver Broncos and the Oakland Raiders, and this will be the lowest scoring game of the week.
And I have a surprise here.
I got I have the Oakland Raiders.
Winning this game is a shocker here. 14.
That 10 will be the final.
This will be the lowest scoring game of the week on predicting defensively the Broncos still look good in the defense of end.
They should be a top eight defense for this season.
But as far as the offense, I feel like the Denver Broncos will be the weakest offense for this upcoming season in the NFL.
They haven't really done much to address the offense in the off season there, so running back Philip Lindsay is one of the breaks Spots in the offense seem as the wide receiver Emanuel Sanders.
But if Sanders ever goes down there, Lindsay, this team is in big trouble.
Why receiver to at this point is Courtland sucked in because of the tree that they did? Tamir.
Any pleas for the Patriots at the tread trade deadline last season? But now they have a new quarterback in town.
Joe Flacco, now the quarterback for the Denver Broncos, is.
He was benched last season with seven games left to go in the season from Lamar Jackson and the Ravens.
Here they have.
Joe Flacco is not known to be a deep threat pastor to pass the ball downfield that often more like a game manager like Blake Bortles or so.
So I think that the Broncos will struggle to score points throughout the season unless they address that category and make some time throughout the season.
a bigtime trade for a really good Why receiver? Something like that.
But as far as this game, I actually have the Raiders winning this 1 14 to 10 over Denver.
I believe that this is how the season will go for Denver this upcoming season.
They're gonna play a lot of close games.
The defense will keep a minute, but it's just offense.
They're going to struggle to score points 10 13 16 17 points.
But they'll give up just a tad morn and what they've got on the board.
So it's gonna be interesting.
They're really interested to see Antonio Brown for the first time, suiting up for the Oakland Raiders.
If he does, so, that bolsters Derek Carr's chances that throw the ball more to him that target him her.
So So it's gonna be interesting.
The CEO Antonio Brown shows up this season.
He led the NFL last season and reception touchdown with 15 receiving touchdowns, but that is a wrap here Thio the score predictions A week one of the NFL.
Before you go, I'm just gonna go through once again, every single score here of the NFL for you guys.
In case you missed it.
So here's the NFL opening night Thursday, September 5th.
I got the Chicago Bears by three over the Green Bay Packers.
Honestly, I feel like this game could go either way, but I've got the Packers.
Their defense will just be too strong on the field.
Coolio, Mac, Akeem, Hex, Prince Takemura, Kyle Fuller.
That defense will just be too strong for the Chicago Bears going up against one of the greatest quarterbacks in the NFL right now.
Aaron Rodgers.
Aaron Rodgers had an M V P type season last season.
25 touchdown passes toe only two interceptions, but they finished off the season with a disappointing six and nine record with a tie in there.
Basically, because I had coach Mike McCarthy and how that whole offense for the Packers hadn't jelled throughout the season, they couldn't get the running game going throughout the first, like 75 percent of this season.
But that is the production to the opening night game and then moving on to the 1st 3 games here. The 1 p.m.
Times Live Baker, Mayfield an o b J all over that Tennessee Titans in this game.
Cleveland Browns gonna win this 1 34 to 13 on predicting I've got the Ravens all over the Dolphins, 31 23 the final and then a close game game of the week right here.
Atlanta Falcons squeaking out a victory on the root against the minister.
The Vikings, 28 2 27 will be a close game, but I get the Falcons and that one in case you missed them, moving on to the next batch of games here. 1 p.m.
time slot on Sunday.
When I do my game picks year predicting this one.
The New York Jets at home, Sam Darnold and Levy on Bill will get the W with the metal lands against the Buffalo Bills by 6 22 14 will be the final in predicting Eagles just all over the Redskins man, 37 to 10.
On predicting this final score to be, I think that the Eagles would just beat down all day gets Washington.
This game will not be closed from the start, moving on to the next batch of games that I'm predicting here with my game picks.
I've got the Rams over the Panthers, 35 to 27 now keep an eye on Cam Newton status.
If Cam Newton cannot sit up for a week one and he's still injured without sprained ankle, that score for the Panthers will drop.
So that will be something more like 35 14 of Cam.
Newton does not play weak one.
But I haven't rained out playing for Week one, and I have them in the predictions right now, 35 to 27 on predicting that game in Kansas City.
Just two top offensively to stop here, going up against the Jacksonville Jaguars.
I have the Kansas City Chiefs winning this game by two touchdowns, 34 to 20 moving on to the four o'clock time games year in a 405 games on the time slot.
In the 4 25 games, first two games are four or five matchups.
Indianapolis Colts up against the Los Angeles Chargers.
I've got the Chargers here 28 to 6, the final over Indy.
Just too much to ask for, with Jacoby percent coming in two weeks before the start of the season, when Andrew Luck retired here to take over his quarterback for India, thinking chargers all the way in this 1 28 The sex as far as the Bangles go, is gonna be tough playing at Seattle, that's for sure.
A tough defense, tough stadium, the plan as well.
I've got the Seahawks in this game for Sher, 20 to 6, and they're running backs.
Will run a right all over the Bengals defense.
Watch out for both.
They're running us.
Chris Carson and Rashawn Penny toe have huge games in action in this one.
The Bengals offense will struggle mightily against the Seahawks, and in this game, I have a close one.
Dallas Cowboys winning 17 13 against the New York Giants.
It'll be close, but I have Dallas getting the W in action.
Look for Ezekiel Elliott Elliott.
The finally play his first game of the season as hell, hopefully report later this week.
That deal is almost done, basically, should be done by tomorrow, Most likely for Zeke to be back onto the field this upcoming season for the Cowboys.
Next slate of games coming up here in the 405 R 4 25 times lots.
I've got Kyle or Murray, with his first ever NFL career, start, getting the W here against Matt.
Patricia and the Detroit lines.
This game is at Arizona, 27 23.
The final.
I've got Kyla Murray getting the wind in action.
Jimmy Garoppolo will get his first start of the seasonal wind as well over the Tampa Bay Buccaneers of the Pirate ship. 30.
That's when he will be the final.
This is a make it or break a year for Jamie's Winston here.
If he doesn't perform well up to exact expectations, he's gonna be done.
It's quarterback for the bucking year, so this is it.
This is the last season for Jameis Winston if he doesn't perform two caliber, but I've got Jimmy Garoppolo in the 40 Niners year.
The 20 bucks expect Jimmy Garoppolo toe Probably dropped down after this game.
Help post the great game against, Ah, pretty poor secondary there with the Tampa Bay Buccaneers.
They were terrible last season with their defense up against the past, but it's just I think, that Jimmy G will throw over the Buccaneers, but don't expect that trend that continue for Jimmy G throughout the season.
That's just going up against ah Buccaneers defense.
That really struggles as far as the Sunday night football game.
I'm predicting this to be the final score.
Patriots over the Steelers at Gillette raising the banner there.
Gillette Stadium 24 to 17 of final for the Patriots.
Getting the W in the money.
No football games.
Year before I sign off on the video, I've got the New Orleans Saints winning their game 36 to 23.
I did my NFL predictions earlier today.
I had the Saints with the best record in the NFL.
I've got them winning by 13 against the Texans.
And then in the last game, I've got the Oakland Raiders shocking the Denver Broncos 14 the 10 and this one.
I feel like Denver will have trouble scoring points in this game.
I like the Raiders to win this game. 14. Attend.
So that is a wrap here to my video of you doing game picks every single week here to the NFL every single Tuesday nights or so.
So feel free to join me here on sportscaster on the sports dish on sportscaster.
I'll be doing my game pics of the NFL each Tuesday night year on the sports dish on sportscaster.
If you haven't done so already, feel free to follow me here on the sports dish on sportscaster on more streams throughout the week.
Also, I'll be doing play by play.
Chicago Bears hosting the Green Bay Packers all be doing that game for you guys.
That will be on Thursday nights 8:20 p.m.
All have the call for you guys doing the play by play.
But thank you so much, everybody.
This has been my NFL Week one game picks and predictions more to come next week, as I'll do Week two game picks and predictions for this upcoming Tuesday, a week from today.
Thanks so much for watching this video is this Is that sports this year on sportscaster?
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{"url":"https:\/\/www.imrpress.com\/journal\/JOMH\/18\/6\/10.31083\/j.jomh1806132\/htm","text":"NULL\nCountries | Regions\nCountries | Regions\nArticle Types\nArticle Types\nYear\nVolume\nIssue\nPages\nIMR Press \/ JOMH \/ Volume 18 \/ Issue 6 \/ DOI: 10.31083\/j.jomh1806132\nOpen Access Review\nEvidence-based circumcision policy for Australia\nShow Less\n1 School of Medical Sciences, University of Sydney, Sydney, NSW 2006,\u00a0Australia\n2 Department of Urology, St George Hospital, Sydney, NSW 2217,\u00a0Australia\n3 Department of Obstetrics and Gynaecology, SAN Clinic, Wahroonga, NSW 2076,\u00a0Australia\n4 Victoria Circumcision Clinic, The Regent Medical Group, Preston, VIC 3072,\u00a0Australia\n5 Mulgoa Medical Centre, Mulgoa, NSW 2745,\u00a0Australia\n6 Department of Medicine, Royal North Shore Hospital, St Leonards, NSW 2065,\u00a0Australia\n7 School of Biomedical Sciences and Pharmacy, University of Newcastle, Pregnancy and Reproduction Program, Hunter Medical Research Institute, New Lambton Heights; Priority Research Centre for Reproductive Sciences, University of Newcastle, Callaghan, NSW 2308,\u00a0Australia\n8 St Vincent's Hospital, Australian Tobacco Harm Reduction Association and Australia21, Darlinghurst, NSW 2010,\u00a0Australia\n9 Katelaris Urology, North Shore Private Hospital, St Leonards, NSW 2065,\u00a0Australia\n*Correspondence: brian.morris@sydney.edu.au (Brian J. Morris)\nJ. Mens. Health 2022 , 18(6), 132; https:\/\/doi.org\/10.31083\/j.jomh1806132\nSubmitted: 22 November 2021 | Revised: 24 January 2022 | Accepted: 9 February 2022 | Published: 30 May 2022\nThis is an open access article under the CC BY 4.0 license.\nAbstract\n\nThe aim was (1) to perform an up-to-date systematic review of the male circumcision (MC) literature and (2) to determine the number of adverse medical conditions prevented by early MC in Australia. Searches of PubMed using \u201ccircumcision\u201d with 39 keywords and bibliography searches yielded 278 publications meeting our inclusion criteria. Early MC provides immediate and lifetime benefits, including protection against: urinary tract infections, phimosis, inflammatory skin conditions, inferior penile hygiene, candidiasis, various STIs, and penile and prostate cancer. In female partners MC reduces risk of STIs and cervical cancer. A risk-benefit analysis found benefits exceeded procedural risks, which are predominantly minor, by approximately 200 to 1. It was estimated that more than 1 in 2 uncircumcised males will experience an adverse foreskin-related medical condition over their lifetime. An increase in early MC in Australia to mid-1950s prevalence of 85% from the current level of 18.75% would avoid 77,000 cases of infections and other adverse medical conditions over the lifetime for each annual birth cohort. Survey data, physiological measurements, and the anatomical location of penile sensory receptors responsible for sexual sensation indicate that MC has no detrimental effect on sexual function, sensitivity or pleasure. US studies found that early infant MC is cost saving. Evidence-based reviews by the AAP and CDC support early MC as a desirable public health measure. Although MC can be performed at any age, early MC maximizes benefits and minimises procedural risks. Parents should routinely be provided with accurate, up-to-date evidence-based information in an unbiased manner early in a pregnancy so that they have time to weigh benefits and risks of early MC and make an informed decision should they have a son. Parental choice should be respected. A well-trained competent practitioner is essential and local anaesthesia should be routinely used. Third party coverage of costs is advocated.\n\nKeywords\ncircumcision male\npolicy\nurinary tract infection\nsexually transmitted infections\ninflammatory conditions\npenile cancer\nprostate cancer\nsexual function\ncomplications\nrisk benefit\ncost benefit\n1. Introduction\n\nCircumcision of males (MC) involves removal of the foreskin. It has been practiced for thousands of years by diverse cultural groups globally [1]. In Victorian times medical circumcison became popular to prevent syphilis, phimosis, penile cancer and inferior hygiene [1]. In the 21st century it was approved for protection against HIV in epidemic settings [2, 3, 4, 5]. MC is currently the world\u2019s most widely performed surgical procedure, prevalence globally being 37\u201339% [6]. In Australia, a 2010 telephone survey found that 33% of Australian men under 30 years of age were circumcised [7]. Recent data show a reduction in early MC prevalence from a peak of 85% in the 1950s to 18.75% in 2019 [8]. In the US, Centers for Disease Control and Prevention (CDC) estimates show an increasing trend in MC prevalence to 92% in white, 76% in black and 44% of Hispanic males aged 14\u201359 years [9].\n\nEvidence-based reviews of MC were published by the American Academy of Pediatrics (AAP) in 2012 [10, 11], the CDC in 2018 [12, 13], and the Circumcision Academy of Australia in 2012 [14]. (See Supplementary Material 1 for summaries of available circumcision policy statements.) The aim of the present study was to conduct a systematic review of the literature and use this to develop an up-to-date MC policy tailored to the setting of Australia. Data for Australia were used when possible, otherwise information was from mostly from the comparable setting of the US, which is the richest data source.\n\n2. Literature searches\n\nArticles were retrieved from PubMed using the keyword \u201dcircumcision\u201d together with one of 39 other relevant keywords (Supplementary Material 2), leading to 253 that were included. Additional publications (10 articles and 15 Internet publications) were identified in bibliographies of these. In total 278 publications meeting the inclusion criteria were obtained. Particular priority was given to randomized controlled trials, systematic reviews and meta-analyses. Studies were rated Level 1+, 1++, 1\u2013, 2++, 2+, 2\u2013 and 3 by the Scottish Intercollegiate Guidelines Network (SIGN) system [15] (Supplementary Material 3).\n\n3. Phimosis\n\nIn newborn males the inner surface of the foreskin adheres lightly to the underlying glans making foreskin retraction difficult, a condition termed phimosis [16]. During childhood the foreskin gradually separates from the glans. By age five most boys are able to retract their foreskin partially, with some adhesions usually remaining. By puberty full retraction is generally possible [17]. Forceable retraction can be painful, and could injure the foreskin, leading to scarring and persistence of the phimosis [16]. Gentle manipulation during bathing is helpful.\n\nA recent systematic review of phimosis prevalence at all ages found that the condition remained in 3.4% (range 0.5\u201313%) of uncircumcised males aged $\\geq$18 years [18] (SIGN rating: Level 2++). Phimosis can result in pain, especially during erections, sexual dysfunction, increased risk of penile inflammatory conditions such as balanitis and penile cancer. Lichen sclerosus (next section) is usually accompanied by secondary phimosis. Steroid creams can be used [16], but are not always successful (see below), and circumcision is the definitive option [16] (Level 2+). Paraphimosis is an even more serious condition and involves failure of the foreskin to return after retraction. Constriction of the the glans leads to oedema, and in some cases ischaemia with a risk of progression to gangrene. Pararaphimosis is a urological emergency which may require immediate surgery, particularly if not detected in a timely fashion. Adolescent and young adult males may not know that they have phimosis and could suffer in silence.\n\n4. Penile inflammation\n\nThe most common forms are balanitis and balanoposthitis that affect the glans penis and foreskin of uncircumcised boys [19]. In a meta-analysis, reported prevalence was 68% lower in uncircumcised males [20] (Level 1+). Circumcision was a common treatment for penile inflammation (as well as phimosis), but in recent years steroid creams have become more common [19, 21] (Level 1+). A recent meta-analysis of the devastating penile inflammatory condition lichen sclerosus (old term: balanitis xerotica obliterans) in boys aged 1 month to 15 years, found that steroid treatment for an average of 4 months (range 6 weeks to 5 years) avoided circumcision in just 35% of cases [21] (Level 1+). A commitment to regular application is required, which may limit compliance to prescribed treatment protocols, and there is a risk of side effects from long-term usage. In contrast, circumcision is ~100% effective and protection is lifelong [22].\n\n5. Candidiasis\n\nIn a large Australian survey, this fungal infection of the penis (commonly known in Australia as thrush), was reported by 7.7% of uncircumcised vs. 4.9% of circumcised men [7] (Level 2+). In boys of mean age 6.4 years (range 8 months to 18 years), prevalence was 18% in those who were circumcised vs. 44% in the uncircumcised [23]. Of interest, cases of phimosis, balanitis and candidiasis can occur in isolation or simultaneously.\n\n6. Urinary tract infections\n\nUrinary tract infection (UTI) is more common among uncircumcised boys, especially those with underlying renal tract anomalies [16, 24, 25, 26]. UTIs are common in infancy and often present with the infant febrile, distressed and in pain. In infancy, the prevalence of febrile UTIs is highest (8.7%) in those aged $<$3 months, 3.3% in those aged 3\u20136 months and 1.7% in the 6\u201312 month old age group [16, 27]. Pediatric UTI can lead to significant short and long term morbidity [28]. The younger the infant, the higher the likelihood of progression to sepsis, and greater risk of fatality [29]. A survey in Sydney, Australia, found that by age 7 years, 2.1% of boys have had at least one UTI and another 4.8% have probably had one [30]. The fact that the infant kidney is still growing means greater susceptibility to renal injury and scarring [31, 32], so exposing half to serious, life-threatening conditions later in life [33].\n\nThe acute febrile illness results in 25% of boys with UTI being hospitalised and receiving a period of parenteral antibiotics [34, 35]. Older children are more likely to be able to be managed with oral antibiotics on an outpatient basis. Oral administration in infants is difficult and absorption is low, requiring hospitalisation to enable intravenous antibiotic administration [36, 37]. Emergence of resistance to most or all antibiotics, including methicillin, will make treatment of UTI more challenging [38, 39]. Maternal antibiotic use during pregnancy also increases the risk of resistant pathogens during neonatal UTI [40].\n\nPyelonephritis develops in ~80% of febrile infants and young boys diagnosed with UTI [41, 42]. In the US ~20,000 annual cases of acute pyelonephritis in infancy were prevented by MC [43]. Pyelonephritis leads to renal scarring in 36\u201352% of cases [41, 44]. Nuclear imaging studies have confirmed that renal scarring occurs following pyelonephritis even in the absence of vesicoureteric reflux (VUR) [45]. In boys without VUR, ~36% have recurrent UTI [46]. The reason is that the aetiology of renal scarring from pyelonephritis is parenchymal infection and inflammation rather than VUR [44, 45]. Permanent kidney damage is seen in 10\u201315% of boys with high grade VUR [47].\n\nThere are strong biological reasons why MC can prevent UTI [16] (Level 2+). Concentration of uropathogenic organisms near the urethral meatus is much higher in uncircumcised male infants than circumcised male infants in the highest risk period of 6 months post-birth [48]. The bacteria adhere to the foreskin\u2019s mucosal surface and readily colonize it [49]. Since uropathogens are substantially lower by 3 weeks after circumcision of boys, it was suggested that by removing the foreskin MC eliminates the haven for organisms responsible for ascending UTI by changing it into an external skin surface [50, 51, 52]. For boys with hydronephrosis, MC is strongly recommended [53, 54].\n\nA systematic review and meta-analysis that included data for 296,837 circumcised and 111,065 uncircumcised males (from 1 randomized controlled trial, 6 cohort studies, 11 case-control studies, 2 cross-sectional studies, 1 retrospective cross-sectional study and 1 retrospective analysis) found that for uncircumcised vs. circumcised boys, relative risk (RR) of UTI was 9.91-fold higher for age 0\u20131 year, 6.56-fold higher for age 1\u201316 years, and 3.41-fold higher for males aged over 16 years of age [55] (Level 1++). It calculated that, over the lifetime, 32.1% of uncircumcised males vs. 8.8% circumcised males develop UTI (RR = 3.65). Value for number needed to treat (NNT) was 4.29. Data from bag specimens or clean-catch urine samples were similar to those for studies in which most samples were from suprapubic aspiration or bladder catheterization. Risk reduction from being circumcised was, in older meta-analyses, 10\u201312 fold in infants [27, 56] (Level 2++), and 8-fold in a study combining infants and older males [57] (Level 2\u2013). The latter reported a cumulative incidence of UTI of 1.1% in uncircumcised infant boys [57]. In boys aged under 5 years of age in Western Sydney, UTI was diagnosed in 6% of those uncircumcised and 1% (n = 2) of the circumcised [58]. Prevalence by age 2 years was 2.2% in a Swedish study [59] and was 3.6% to age 16 in a UK study [60]. Recurrence of UTI was seen in 35% of boys diagnosed with UTI in the first year of life [57]. Most (up to 12%) of recurrence occurs after the age of 12 months. Boys with more than 2 recurrent UTIs often have urinary tract abnormalities. For those with high grade VUR, NNT by circumcision is low [57] (Level 2+). In uncircumcised boys with recurrent UTI MC should be advised for treatment. A past chair of the AAP Task Force on infant MC strongly recommended early MC to avoid risk of renal damage in immature kidneys and of VUR from pyelonephritis [29]. He compared postponing MC to postponing vaccinations. The level of protection that newborn MC affords against UTIs is comparable to that of many vaccines given to children to prevent other infections and diseases [61], an example being vaccination against influenza [62, 63].\n\n7. Sexually transmitted infections\n7.1 Human immunodeficiency virus (HIV)\n\nRandomized controlled trials (RCTs) in Africa found MC was protective against HIV transmission from infected women [64, 65, 66] (Level 1++). Overall efficacy was ~60% [67]. A Cochrane committee meta-analysis found high consistency of the trial results [68] (Level 1++). The World Health Organization (WHO) and the Joint United Nations Programme on HIV\/AIDS (UNAIDS) then recommended adoption of voluntary medical MC (VMMC) for reduction in HIV prevalence in epidemic settings in Africa [2, 3] (Level 1+). Roll-out has resulted in over 20 million procedures in high-priority African countries [4], and has reduced HIV infections by up to 50% [5]. Levels of protection found in recent meta-analyses were 70% [69] and 72% [70] (Level 1++). Meta-analyses found risk compensation after VMMC, such as not using condoms, was negligible [71] (Level 1++).\n\nThe CDC [72, 73] has endorsed MC as a means of protection against HIV during heterosexual intercourse, as confirmed in US studies [74, 75] (Level 2+). In the US most men are circumcised in infancy. In the Netherlands and France, where MC prevalence is low, but sexual behaviour indices are comparable, heterosexually-acquired HIV diagnoses were 6 times higher in men and 10 times higher in women than in Israel, where infant MC prevalence is very high [76]. A systematic review of contrary arguments by MC opponents found their statements misrepresent good studies, selectively cite references containing fallacious information, draw erroneous conclusions, and are contradicted by evidence from high-quality studies [77, 78] (Level 2++). The late David Cooper, when director of the Kirby Institute at UNSW, argued in favour of infant MC for HIV prevention in Australia [79]. Similarly, the Canadian Urological Association states that circumcision \u201cis one of several partially effective risk-reduction alternatives for heterosexual men that should be used in combination with other measures\u201d [16].\n\nThe biology of the foreskin makes it vulnerable to HIV infection [80, 81, 82, 83, 84, 85]. Inflammatory conditions and ulcerative STIs increase risk [86, 87, 88, 89, 90], as do coital injuries, which uncircumcised men are prone to [91, 92, 93], and risk is higher when foreskin size is large [94]. Some protection against low levels of HIV is afforded by langerin, which is produced by the inner foreskin mucosal epithelium [95]. Langerin becomes overwhelmed, however, at high HIV loads [95, 96].\n\n7.2 Human papillomavirus (HPV)\n\nHPV prevalence in developed countries is ~75% [16]. High-risk (oncogenic) HPV genotypes mostly infect the foreskin and underlying glans [97]. Meta-analyses found MC to be associated with 32\u201365% reduction in genital HPV prevalence [98, 99, 100] (Level 1+). Reduction averged 40% in data from the African RCTs [101, 102, 103, 104, 105, 106] (Level 1+). In one of these studies, flat penile lesions (an indicator of high-risk HPV) were 98% less common in circumcised men [101] (Level 1+). A study involving 1913 couples in 5 European, Asian and South American settings found penile HPV prevalence of 5.5% in circumcised and 19.6% in uncircumcised men [107] (Level 2++). After adjustment for potential confounding factors, HPV infection risk in circumcised men was 63% lower than in uncircumcised men. A large survey in the UK found 86% lower prevalence of high-risk HPV genotypes in circumcised men [108]. Low-risk HPV genotypes responsible for genital warts infect the anogenital region more broadly and therefore MC is less effective in prevention of these genotypes [97]. In a RCT, duration of infection of the glans\/coronal sulcus by high-risk HPV was shorter for circumcised men [109] (Level 1+), but circumcision status did not affect duration of infection in the penile shaft, scrotum or all genital sites combined. Thus, clearance is greatest in the glans, the area of the penis exposed by circumcision. In confirmation, a US study found 2.7-fold greater likelihood of clearance of any HPV infection, a 3.2-fold increased clearance of oncogenic HPV infection, but no difference in clearance of non-oncogenic HPV infection in circumcised vs. uncircumcised men [110] (Level 2++). Men with phimosis have higher prevalence of HPV infection of their foreskin [111].\n\n7.3 Other STIs\n\nGenital herpes simplex virus-2 (HSV-2) prevalence was 45%, 30%, and 28% lower in circumcised men in the RCTs in Uganda, South Africa and Kenya, respectively [86, 112, 113, 114, 115]. Protection was ~50% against Trichomonas vaginalis [116], ~40% against Mycoplasma genitalium [117], 33\u201350% against Treponema pallidum (syphilis) [118, 119, 120], ~50% against chancroid [118], and ~50% against genital ulcer disease [86, 121, 122], as found in RCTs (Level 1++) and observational studies (Level 2++ and 2+). Data from a RCT noted that MC reduces total prevalence and load of anaerobic bacteria as well as microbiota biodiversity [123]. RCT data [124] (Level 1+) and a meta-analysis [125] (Level 1+) found that MC does not protect men against sexually transmitted urethritis (gonorrhoea, chlamydia and nonspecific urethritis). For more on the role of MC in protection against STIs in men see reviews [126, 127, 128, 129].\n\n7.4 STIs in women\n\nRecent systematic reviews of RCTs and numerous observational studies found that MC was associated with reduced risk of infection by HSV-2, chlamydia, syphilis, high- and low-risk HPV genotypes, genital warts, Mycoplasma genitalium, candidiasis, dysuria, and possibly bacterial vaginosis, HIV, non-specific genital ulcers, trichomoniasis and vaginal discharge [130, 131] (see also an editorial [132]). HIV prevalence in South African women who only had circumcised male partners was significantly lower by 78% [133]. Meta-analyses of all studies, however, found non-significantly lower HIV risk reduction of 20% [134] and 32% [69]. In one trial, disobeying medical advice to abstain from sexual intercourse for 6 weeks after MC was responsible for slightly higher HIV infection in female partners [135].\n\n7.5 STIs in men who have sex with men (MSM)\n\nA recent study from Melbourne, noted a reduction of barrier contraception use and an increase in casual sex, HIV, syphilis and gonorrhoea over the past decade in MSM [136]. For MSM who adopt the insertive role during anal intercourse, a Cochrane meta-analysis found MC was associated with 73% lower HIV infection risk [137]. As expected, for men who adopt the receptive role there was no significant protection. Another meta-analysis found MC was associated with a significant 23% reduction in overall risk of HIV infection [138]. The findings led to a call for action [139]. Each of these studies noted the highly significant 89% risk reduction amongst circumcised insertive MSM in Sydney [140].\n\nFor HPV, MC afforded 57% protection against the most common genotype, HPV16, in MSM who practiced predominantly insertive anal intercourse, but there was no protection in the receptive partner [141]. HIV-infected MSM who were circumcised had 29% lower HPV in a 2019 meta-analysis [138] (Level 1+). HSV-2 infection was found to be 16% lower in circumcised MSM overall in this meta-analysis. In Sydney, a 65% lower prevalence of incident syphilis, was found amongst circumcised MSM and was 90% lower in the one-third who engaged predominantly in insertive anal intercourse [142] (Level 2+). The finding for incident but not prevalent syphilis in that study was because MSM who initiated sexual activity during the late 1980s and 1990s when syphilis prevalence was low would have been at very low risk of acquiring syphilis irrespective of their MC status, whereas only since 2001 has syphilis seen a re-emergence amongst Australian MSM [142].\n\nIt has been emphasized that bisexual men pose a particular risk for STI transmission to women [136, 140].\n\n7.6 Condoms\n\nCondoms provide 80% [143] to 71\u201377% [144] protection against HIV infection, but only if used consistently and correctly [143, 145]. Condoms may break or slip off. A Cochrane systematic review and meta-analysis of RCTs of condom use (2 in the US, one in England and 4 in Africa) found, \u201clittle clinical evidence of effectiveness\u201d and no, \u201cfavourable results\u201d for HIV prevention [146]. Condoms were, however, 42% effective in prevention of syphilis [146].\n\nIt should be noted, moreover, that condoms must be used at each sexual encounter, whereas MC is a one-off procedure that is always in place. MC and condom use each provide a reasonable degree of protection against STIs. When both are in place protection is higher [77]. Vaccination too can be compared with behavioural and barrier protections against infectious agents, but the only STI for which a vaccine offering reasonable, but not complete, protection is directed at common anogenital HPV genotypes. HPV vaccination is available early in high school, with parent approval required.\n\n8. Genital cancers\n8.1 Penile cancer\n\nCancer of the penis has a lifetime risk in uncircumcised men of ~1 in 1000 [147], making it uncommon, but not rare. It is rare in circumcised men, prevalence being 0.00008\u20130.02 in 1,000 [148, 149]. Consistent with a role for MC in prevention, annual incidence was highest in England and Wales (1.44 per 100,000), lower in Australia (0.80 per 100,000) and lowest in the US (0.66 per 100,000) [150], commensurate with MC prevalence in each country. A study in California found risk was 22-fold higher in uncircumcised men [151]. The disease is debilitating. It results in substantial functional impairment and devastating psychological effects [152, 153]. Recurrence is 28% following penile preserving therapies and 5-year mortality is 90%, whereas recurrence is 5.3% after ablation of all or part of the penis [152].\n\nFactors associated with increased risk of penile cancer were shown in meta-analyses to include phimosis (12.1-fold), balanitis (3.8-fold) and smegma (3.0-fold) [154] (Level 1+). Another meta-analysis found an average of 47% of penile cancers contain high-risk HPV genotypes [98] (Level 1+). Genital warts, smoking, STI history, extramarital relationships, multiple sexual partners, inferior genital hygiene, previous genital conditions, protracted penile rash, and penile tear are also risk factors. If the quadrivalent HPV vaccine was fully implemented in the target population of boys early in high school, population prevalence of the most common oncogenic HPV genotypes (16 and 18) could be reduced by ~70%. Consequently, HPV vaccination could reduce penile cancer prevalence by 47 $\\times{}$ 0.7 = 33% [155]. Vaccines are ineffective for non-HPV related causes [156]. The overall level of effectiveness of vaccination is similar to effectiveness of MC found in a meta-analysis [100] and RCTs [101, 102, 103, 104, 105, 106].\n\n8.2 Prostate cancer\n\nLifetime risk of prostate cancer is $\\geq$10%. One in 6 men in Australia are at risk of developing the condition by the age of 85 years [157], making it the most common male cancer. In 2020 there were 17,000 new cases and 3200 deaths, representing 12% of cancer deaths in Australian males [157]. Globally, there is an inverse correlation between prostate cancer incidence and MC prevalence [158] (Level 2++). After correction for potential confounding factors, countries with high MC prevalence have lower prostate cancer-related mortality, which is a harder end-point than prevalence [159]. Meta-analyses found prostate cancer risk is ~10% lower in circumcised men [160, 161, 162] (Level 1+). Risk reduction was 12% lower (p = 0.01) in the post-PSA testing era, 16% lower in population-based studies (p = 0.05), 17% lower in studies that collected data by personal interview (p = 0.03), 41% lower in studies of black race (p = 0.02) [160] (36% in US [163] and 60% in Canadian [164] studies), and 16% lower for more aggressive prostate cancer (p = 0.02) [161]. Thus, risk reduction associated with MC is on a par with other factors associated with reduced risk of prostate cancer [165, 166].\n\n8.3 Cervical cancer in women\n\nOncogenic HPV genotypes are responsible for 99% of cervical cancers. Since women may have a history of circumcised and uncircumcised sexual partners, a large multinational study focused on women who had had only one sexual partner. Monogamous women whose male partner had a high sexual-behaviour risk index ($\\geq$6 sexual partners and first intercourse prior to 17 years of age; n = 1420) were 82% less likely to have had a cervical cancer diagnosis if their male partner was circumcised [107] (Level 2++). Monogamous women whose male partner had an intermediate risk index and was circumcised were 50% less likely to be diagnosed with cervical cancer than if their male partner was uncircumcised. Cervical cancer incidence was 35 per 100,000 women per year in 51 countries in which MC prevalence was low ($<$20%) but was 20 per 100,000 in 52 countries with high ($>$80%) MC prevalence (p $<$ 0.001) [167]. The study examined many factors and being uncircumcised was the strongest risk factor for cervical cancer. In Israel, low cervical cancer prevalence compared with the 11.7% global prevalence [168] was attributed in part to MC [169]. In Kuwait, where males are circumcised prior to puberty, HPV prevalence is 2.3%, one of the lowest in the world [170]. In a Danish study, the 5-fold lower HPV prevalence in circumcised men was implicated in lower cervical cancer prevalence in their female partners [171]. Women in Myanmar with circumcised husbands had significanty lower cervical cancer prevalence [172]. In Seoul, South Korea, 53% lower risk of invasive cervical cancer was seen in women with circumcised male sexual partners [173]. Amongst 3261 women in Spain, HPV infection risk was 40% lower in those with $\\geq$2 lifetime sexual partners who were circumcised [174]. There were similar HPV findings in Ghana [175] and in a Nigerian study, which also found a 14-fold difference in cytological abnormalities (5% vs. 63%) in women with a circumcised vs. uncircumcised male partner [176].\n\nA meta-analysis of 2 studies in Australia, 5 in the US, 2 in Mexico, and one each in South Korea, Denmark, England, Kenya and the multinational study in Brazil, Colombia, Spain, Thailand and The Philippines [107] found cervical cancer to be less common in women whose male partner was circumcised (OR = 0.75 overall, and 0.18 for those whose husband had a high sexual behaviour risk index) [177] (Level 1+). (See also systematic reviews [130, 131]).\n\nVaccination against up to 9 anogenital HPV genotypes early in high school should help reduce cervical cancer. But vaccines are not directed at all of the $>$14 mucosotropic HPV genotypes. Overall vaccine uptake in the 10\u201320 year old age group in high income countries is only 33.6% [178]. In Australia, however, full vaccination by age 15 was 78.6% in girls and 72.9% in boys [179]. Ideally, if vaccine coverage in school children were universal and if the nonavalent HPV vaccine were effective, total HPV infections could be reduced by 93%. A systematic review of HPV vaccination experience revealed effectiveness was suboptimal (see Fig. 3C of that publication) [180]. In Australia, HPV 6, 11, 16 and 18 targeted by the quadrivalent vaccine were reduced by 86% (not 100%) [180]. While prevalence of high-risk HPV 16 and 18 has declined, replacement by HPV genotypes not included in vaccines used has been seen [181].\n\nWhile HPV vaccination against a subset of HPV genotypes in early adolescence should help mitigate cervical cancer risk, uptake is not widespread in all settings and durability of effectiveness is not assured. Adoption of multiple effective preventive measures in usual for public health recommendations. Thus, early MC plus vaccination should have a greater impact than vaccination or MC alone. More accurate screening by the advent of PCR-based detection of HPV [182, 183] should further reduce cervical cancer prevalence.\n\n9. Trends\n\nA 2021 study of Medical Benefits Scheme (MBS) claims found that early MC in Australia declined from a peak of ~85% in the 1950s to 18.75% in 2019 [8]. The authors concluded that \u201cMedical and surgical authorities may have played an important role in the gradual reduction of procedures over the last decade\u201d. In particular, negative policies instituted in the 1970s following the appointment of paediatricians from the UK to Chairs of pediatrics contributed to these [184]. In the UK, MC is a \u201cmark\u201d of the upper classes. An overall decline in early MC of boys in the UK occurred after 1949 following the withdrawal of coverage by the National Health Service (NHS).\n\nA study examining the US Pediatric Health Information System database of MC prevalence at different ages in US hospitals found an increase in the rate of neonatal MC ensued in response to the AAP\u2019s 2012 affirmative policy in which a literature review led the AAP to conclude that the benefits of MC during the neonatal period outweigh the risks and recommended various means to increase rates, partly because \u201ccircumcision during the birth hospitalization in the neonatal period is more resource-effective than postponing until later in infancy\u201d [185]. In the US, up until 2012 there had been a downturn in neonatal MC prevalence. This was attributed to weak paediatric policy statements prior to 2012, increased immigration from countries, particularly Hispanic, in which MC is uncommon, a dimunition in access and affordability owing to non-coverage in some states by Medicaid, and lobbying by MC opponents [186].\n\n10. Sexual function and pleasure\n\nRCT findings [187, 188] (Level 1+), a large UK survey [189] (Level 2++), 4 systematic reviews [190, 191, 192, 193] (Level 2++) and 2 Meta-analyses [191, 192] (Level 1+) showed that MC has no adverse effect on sexual function, penile sensitivity, nor sexual sensation, arousal, or pleasure. The most recent meta-analysis found 64% of circumcised vs. uncircumcised men experienced less pain during intercourse, 28% had lower ejaculation latency time, and 58% had less erectile dysfunction [192]. An Australian study found sexual experience in homosexual men circumcised early was unaffected [194] (Level 2++). However, homosexual men circumcised later in life for medical reasons were more likely to report sexual problems. A systematic review critically comparing high quality evidence with evidence of sexual harms from infant MC strongly favoured the former over the latter [78] (Level 2++). A study involving only men who believed their sex life had been diminished by their early MC [195] (Level 2\u2013) was critically evaluated and shown to be flawed owing to recruitment bias, none of the self-selected participants claimed problems having been confirmed by a medical practitioner, \u201cloaded\u201d and subjective questions and exaggerated responses, \u201ccherry-picked\u201d information that contradicted high-quality evidence, and confirmation bias [196].\n\nQuantitative sensory testing found no difference in penile sensitivity between circumcised and uncircumcised men [197]. Using thermal imaging, another study found basal temperature of the penis of circumcised men was higher, and in response to an erotic video, temperature during erection more rapidly reached the same plateau as uncircumcised men, and a greater proportion of circumcised men reported being sexually aroused whereas a greater proportion of uncircumcised men reported being unaffected [198] (Level 2+). Such methods, moreover, revealed the foreskin is not involved in sexual sensitivity, sensation or pleasure [198, 199] (Level 2++). The neuroreceptors responsible are genital corpuscles located in the glans and underside of the distal shaft, thus further ruling out the foreskin as a location of pleasure response [200] (Level 2++). Tugging the foreskin could, via the frenulum, stimulate genital corpuscles in the shaft. Less pain and better erectile function in circumcised men were found in a large Australian survey [201].\n\nWomen\u2019s experiences of circumcised vs. uncircumcised male sexual partners were found in systematic reviews to favour the circumcised penis [202, 203] (Level 2++). The reasons were esthetics, ease of vaginal penetration, less dyspareunia, better hygiene, and reduced risk of infection [202, 203].\n\n11. Benefit to risk ratio\n\nConsidering data relevant to an Australian context, a risk-benefit analysis found that based on data for level of protection and prevalence of conditions for which early MC provides protection and the frequency of procedural complications benefits were calculated to exceed risk by approximately 200 to 1 (Table 1, Ref. [7, 15, 18, 20, 55, 70, 98, 112, 113, 114, 115, 116, 117, 118, 121, 122, 151, 162, 190, 191, 200, 204, 205, 206, 207, 208, 209, 210, 211]). Furthermore, over their lifetime an estimated 80% of uncircumcised males would likely suffer an adverse medical condition attributable to their foreskin.\n\nTable 1.Risk-benefit analysis for newborn male circumcision in Australia.\n (A) Medical conditions, risk reduction and number of cases prevented Condition Decrease in risk${}^{a}$ Approximate % affected${}^{b}$ Study type [Ref] Quality score${}^{c}$ Approximate number of cases Urinary tract infections (lifetime) 72% 27 Meta-analysis [55] 1+ 30,300 Phimosis persistence at age $\\geq$18 years 97% 3 Systematic review [18] 2+ 3400 Balanitis 68% 10 Meta-analysis [20] 1+ 11,000 Candidiasis (thrush) 60% 10 Original study [7] 2+ 11,000 High-risk HPV infection 60% 10 Meta-analysis [98] 1++ 11,000 HIV (acquired heterosexually) 72% 0.1 Meta-analysis [70] 1++ 100 Genital ulcer disease 50% 1 Original study [121, 122, 204] 2+ 1100 Syphilis 47% 1 Meta-analysis [118] 1+ 1100 Trichomonas vaginalis 50% 1 RCT [116] 1+ 1100 Mycoplasma genitalium 40% 0.5 RCT [117] 1+ 500 Herpes simplex virus type 2 30% 4 RCTs [112, 113, 114, 115] 1++ 4500 Chancroid 50% 1 Meta [118] 1+ 1000 Penile cancer (lifetime) 95% 0.1 Original study [151, 205, 206] 2+ 100 Prostate cancer: population-based 10% 2.1 Meta-analysis [162] 1+ 1100 Totals 80 \u2013 \u2013 77,300 Total percentage of uncircumcised males affected = approximately 80% (B) Risks posed by infant MC and percent affected Condition \u2013 Approximate % affected Study type [Ref] Quality score \u2013 Excessive minor bleeding \u2013 0.1\u20130.2 Original study [208, 207] 2++ \u2013 Infection, local \u2013 0.06 Original study [208, 207] 2++ \u2013 Infection, systemic \u2013 0.03 Original study [208] 2++ \u2013 Need for repeat surgery \u2013 0.08 Original study [208] 2++ \u2013 Meatal stenosis \u2013 0.007 Original study [208, 209, 210, 211] 2++ \u2013 Partial loss of penis \u2013 0.0002 Original study [208] 2++ \u2013 Death \u2013 $<$0.000001 Original study [206] 2++ \u2013 Reduced penile function, sensitivity, sexual pleasure \u2013 0 Systematic review [190, 191, 200] 2++ \u2013 Reduced penile function \u2013 0 Meta-analysis [191] 1+ \u2013 Risk:benefit Thus, over the lifetime, the risk to an uncircumcised male of developing a foreskin-related condition requiring medical attention may be up to 80%. In comparison the procedural risk during infant MC of experiencing an easily treatable condition is approximately 1 in 250. The risk of a moderate or serious complication is approximately 1 in 3000. Thus benefit to risk = 1:200. ${}^{a}$Based on data for circumcised vs. uncircumcised males. ${}^{b}$The percentage of males who will be affected as a result of the single risk factor of retention of the foreskin. Data for STIs were estimated after taking into account the external factor of heterosexual exposure, which is dependent on population prevalence of each STI in Australia and risk reduction conferred by MC. ${}^{c}$Quality rating was based on an international grading system [15] (Supplementary Material 3). Rating was 1++ or 1+ for well-conducted meta-analysis and RCTs, was 2++ for well-conducted systematic reviews, and was 2++ or 2+ for the original studies cited.\n12. Procedures used for neonatal circumcison\n\nThe Plastibell, Gomco and Mogen devices are commonly used for neonatal MC, the Plastibell being particularly common in Australia. For a detailed description of the technique involved in each of these see: [212]. Circumcision should be indicated for most male neonates. The practioner needs to be aware, however, that there are several contraindications (Table 2).\n\nTable 2.Contraindications to infant circumcision.*\n Anatomical (1) Congenital abnormality of penile curvature (cordee). (2) Concealed or buried penis, including from large suprapublic fat pade. (3) Congenital megaprepuce. This is a specific form of buried penis characterized by extensive redundancy and balloming of the inner foreskin as a result of foreskin stenosis and phimosis, resulting in voiding difficulties. (4) Micropenis. (5) Epispadias. This is a rare congenital abnormality in which the urethra opens on the upper surface of the penis rather than the distal end. The space between the opening and the tip of the penis has the appearance of a gutter. (6) Hypospadias. This condition involves the urethra opening on the ventral shaft rather than the tip, causing downward curvature of the penis and spraying of urine during urination. (7) Penile torsion. This presents as a rotation of the penis or a corkscrew-like appearance of the penis and affects approximatey 1 in 80 male neonates. It is mostly seen in uncircumcised boys. (8) Penoscrotal webbing in when the skin of the scrotum is attached to the underside of the shaft. Apart from abnormal cosmetic appearance it does not cause functional problems. (9) Posthitis: substantial inflammation of the penis or foreskin presenting as a red, tender, sensitive rash and oedema. Medical (1) Unstable or premature infant admitted to the neonatal ICU. (2) Neonatal age less than 12 hours. (3) Bleeding diathesis, an unusual susceptibility to haemorrhage, mostly due to hypocoagulability. (4) Curremt illness. (5) Jaundice. (6) Vitamin K not yet administered or parental refusal. *See Supplementary Material 4 for glossary of terms used.\n\nRisk of an adverse event from MC is $\\leq$0.5% during infancy [11, 16, 207, 208, 213]. Most adverse events are minor, and can be immediately and easily treated, with complete resolution, but some very rare complications can be severe [16]. In older boys and men complications are 10\u201320 times higher [207, 208] (Level 2++). Traditional\/ritual MC presents a higher risk than medical MC by a competent practitioner [214]. Provider training is essential to reduce risk of complications [215]. A New Zealand birth cohort study found neonatally circumcised males followed from infancy had fewer penile problems than the uncircumcised [216], and no differences in breastfeeding outcomes, health in infancy nor cognitive ability in later childhood [217] (Level 2++). US findings were similar [218, 219].\n\nRisk of post-MC meatal stenosis was low (0.66%) in a recent Meta-analysis [220] (Level 1++). Its diagnosis by visual inspection is subjective, leading to over-estimation of prevalence. Most cases were asymptotic with no obstructive uropathy. An appearance of meatal stenosis at age 3\u20138 years in boys circumcised neonatally may be an illusion arising from a ventral \u201cmeatal web\u201d [221] (Level 2+). Monitoring for meatal stenosis onset by repeated visual inspection found that most cases developed on average 2\u20134 weeks after neonatal MC and 95% were asymptomatic [222] (Level 2+). This challenges the idea that meatal stenosis is a long-term complication of MC. Diagnosis should only be made on the basis of urine flow rate, evidence of urinary tract blockage, or testing of kidney function.\n\nMeatal examination in the circumcised male is trivial. In uncircumcised infants only 54% had a visible meatus [17], as did 47% of uncircumcised boys aged $<$3 years [223]. The reason is because non-retractile foreskins are common [224], so impeding visual inspection. Data from a Danish study of meatal stenosis [225] (Level 2+), when examined in detail by others, revealed overall prevalence of 0.12% in uncircumcised males and 0.099% in circumcised males [226]. Prevalence of meatal stenosis increases with age, a major cause being from penile inflammation secondary to lichen sclerosis, a condition much more common in uncircumcised than in circumcised males [19], as was apparent in the Danish study [226].\n\n14. Anaesthesia\n\nCircumcision must be performed using adequate anaesthesia and analgesia [16]. For a comprehensive review see [16]. Local anaesthesia is recommended for neonatal MC. After the infant becomes mobile general anaesthesia may be required.\n\nBoys circumcised neonatally without anaesthetic exhibited greater pain and crying response during routine immunisation at age 4\u20136 months compared with uncircumcised boys and boys who had received topical anaesthesia during their circumcision [227, 228] (Level 2+). A systematic review found there was little effect on breastfeeding or cognitive ability, and that low quality studies reporting associations with sudden infant death syndrome, autism, alexithymia, impaired sexual experience and socio-affective processing contained flaws in study design, statistical analysis, sample size and other factors rendering them unreliable [78] (Level 2++).\n\nThe AAP and Canadian Paediatric Society issued joint guidelines in 2000 for prevention and management of pain and stress in the neonate [229] (Level 2++). Anaesthetic techniques were reviewed in the AAP\u2019s 2012 policy statement [11]. Topical administration of eutectic mixture of local anaesthetics (EMLA 5%, an emulsion containing 2.5% lidocaine and 2.5% prilocaine), when applied 60 to 80 minutes before the procedure, was superior to placebo in attenuating MC pain measured by heart rate, oxygen saturation, facial responses, as well as period and characteristics of crying [230, 231] (Level 2+). LMX4 lidocaine 4% is a more recent local anaesthetic cream. Methods more effective than topical creams include dorsal penile nerve block (DPNB) and subcutaneous ring block [232, 233] (Level 1+). Each require training in application and avoidance of complications [234, 235, 236]. In its 2012 policy review, the AAP [11] referred to a landmark ultrasound guided technique developed by Sydney paediatric anaesthetists for correct needle placement during DPNB in children under general anaesthesia [235, 237]. This resulted in lower pain scores in the first postoperative hour and a longer interval should rescue analgesia be required. When the infant is younger than 6 months, general anaesthesia for MC should be avoided [238]. General anaesthesia has inherent risks, albiet low. Local anaesthesia is much cheaper, especially as it does not require the services of an anaesthetist [239]. Another technique is caudal epidural block, which can be used during MC of older children [240] (Level 1+).\n\nThe 2012 AAP policy statement mentioned the possible risk of methaemoglobinemia with lidocaine-prilocaine [11], but noted that when methaemoglobin has been measured after lidocaine-prilocaine application, the level, although elevated, was not clinically significant [231]. The AAP nevertheless noted isolated case reports of clinically significant methaemoglobinemia, but those involved prolonged application time or its use in premature infants [11].\n\n15. Cost benefit\n\nThe reasons for the decline in early MC in the US has included cessation of Medicaid coverage for the procedure in 18 States. Any such decline was deemed, in the long-term, to result in substantially higher costs because of: (1) the need for more expensive MC to treat medical conditions that could have been prevented had MC been performed shortly after birth [75, 241, 242, 243, 244], (2) the fact that later MC is associated with a 10\u201320 fold higher risk of complications [208], and (3) treatment required for the wide array of adverse medical conditions that would have likely been prevented or reduced in frequency had the boy been circumcised early [75, 155, 241, 242, 243, 244, 245, 246]. It was estimated by researchers at Johns Hopkins University that if MC declined from the high US levels to a level of 10%, direct costs for treatment of UTIs and STIs would rise to US$4.4 billion for 10 annual birth cohorts [241] (Level 2++). The increase in expenditure was said to be on average US$313 per foregone MC. Indirect costs for just HIV may be more than 4 times the direct medical costs [247]. The CDC reported that in the US MC was cost-saving for HIV prevention in black and Hispanic males in whom HIV prevalence is highest [75]. If one took into account the other conditions prevented by MC, direct and indirect costs would be even higher. For prostate cancer, without MC there would be 24\u201340% more cases in the US and US$0.8\u20131.1 billion extra in costs for treatment and terminal care per year [165]. Several US states do not provide Medicaid coverage for elective MC, so making MC unaffordable for poor families. As a result, the decrease in infant MC in the poor has resulted in over 100 additional HIV cases and US$30M in medical treatment costs annually [242]. The MC cost in the birth cohort was US$4,856,000, which was found to be 6% of the cost just for treatment of HIV. In Louisiana [243] and Florida [244], cost savings initially generated by not allowing Medicaid to cover elective infant MC were mitigated by increases in rate and expense of medically indicated MC required later to treat various conditions. Since the Louisiana study only considered costs of later MC of boys aged 0\u20135 years, lifetime costs would likely be far greater, impacting healthcare systems. Medicaid defunding in Florida was shown to result in a 6-fold increase in publicly-funded MC and to cost US$112M [244]. Florida responded by restoring Medicaid coverage for elective MC [248]. In Australia and New Zealand, the lack of government coverage for non-therapeutic MC in public health systems would similarly be having cost impacts for treatment of medical conditions protected against by neonatal MC. An increase in early MC in Australia to 85% from the current level of 18.75% [8] would avoid 77,000 cases of infections and other adverse medical conditions over the lifetime for each annual birth cohort (Table 1).\n\n16. Legality of circumcision of boys\n\nCircumcision of males is a legal procedure in virtually all countries worldwide, including Australia, New Zealand, the UK, the USA and Canada. In Australia and New Zealand legality is based on well-established rights of parents to make decisions about medical care for their children. Generally, both parents should agree. Australia has ratified Article 24(3) of the United Nations Convention on the Rights of the Child [249]. Consistent with Australian legislation, Article 24(3) requires that the best interests of the child shall be the primary consideration.\n\nDespite attempts to legislate against circumcision of male minors in Scandinavian countries, circumcision of boys remains legal. A controversial case in Cologne in 2012 concerning a bleeding complication in a Muslim boy circumcised by a Muslim doctor was misconstrued by news media and others as Germany having banned MC, whereas that regional court had ruled the illegality of MC of boys to be among the \u201cundecided questions of law,\u201d concluding that the defendant was not guilty of a criminal, act and was acquitted, with costs ordered to be paid from public funds [250]. An appeal failed. The German Parliament then enacted legislation upholding the legal right of parents to choose MC for their sons, providing that it was performed by a trained professional in a safe environment [251].\n\nAn attempt to have infant MC banned in San Francisco was challenged in court and a bill was subsequently passed unanimously by both houses of the California legislature to prevent any future municipal initiatives to ban MC and other medical procedures [252]. Arguments supporting the legality of infant MC were presented by a member of the AAP\u2019s 2012 Task Force on MC [253]. Arguments challenging the legality of MC of minors in the US were considered by legal, bioethics and medical academics to depend on speculative claims, obfuscation of scientific data, failure to appreciate benefits or the higher risks and barriers to later MC, to be inconsistent with evidence that parent-approved MC is legal, ethical (see next section), is in the best interests of the health of the male child, and consistent with the Hippocratic Oath which contains the statement \u201cI will prevent disease whenever I can, for prevention is preferable to cure\u201d [254, 255, 256]. The oft quoted \u201cFirst do no harm\u201d (Latin: \u201cprimum non nocere\u201d) is a mistranslation of the Greek text \u201c\u1f60$\\varphi{}$$\\epsilon{}$$\\lambda{}$$\\acute{\\epsilon}$$\\epsilon{}$$\\iota{}$$\\nu{}$$\\acute{\\eta}$$\\mu{}$$\\beta{}$$\\lambda{}$$\u0301\\alpha$$\\pi{}$$\\tau{}$$\\epsilon{}$$\\iota{}$$\\nu{}$\u201d, the English translation of which is \u201cfor better or for worse\u201d or \u201cfor good or ill\u201d.\n\nDecisions by legislative and judicial bodies in Australia upholding the legality of MC appear in a review by a lawyer and medical experts [257]. That review found a Tasmanian report recommending prohibition [258] to be illogical, dangerous, unworkable, and that doctors should have guaranteed protection in performing medical procedures based on sound evidence of effectiveness and safety [257]. The report has never been presented to the Tasmanian Parliament.\n\n17. Ethics\n\nParents\u2019 reasons for choosing circumcision for a son include better health, hygiene, appearance, culture and religion [259]. Scholarly assessments support circumcision of male minors as being ethical [253, 257, 260, 261, 262, 263, 264, 265, 266]. When considering the wide-ranging protection that MC affords against an array of adverse medical conditions and infections in infancy and childhood, and STIs in adolescent boys who become sexually active, there are cogent arguments as to why it would be unethical to leave boys uncircumcised [257, 263]. Ethicists and others have interpreted Article 24(3) of the United Nations International Convention of the Rights of the Child as mandating MC, since not doing so would be prejudicial to male health [263]. Nevertheless, in line with views published by AAP Task Force member and professor of bioethics Douglas Diekema [267], the AAP\u2019s 2012 infant MC policy states, \u201cparents should weigh health benefits and risks in light of their own religious, cultural, and personal preferences, as the medical benefits alone may not outweigh these other considerations for individual families\u201d [11]. Medical practitioners with a conscientious objection to performing the procedure should refer parents to another doctor.\n\nAccurate information on benefits and risks should be provided to all parents in an unbiassed manner, ideally early in a pregnancy should they be having a son.Parents should be informed that the option of delaying MC beyond early infancy, or leaving it to the boy to decide, will mean missing out on benefits early in life and pose substantial obstacles that may ultimately mean it will not happen, so diminishing the health and other benefits and increasing the risk of adverse medical conditions over his lifespan (Table 3). While some males may resent their parents\u2019 decision to have them circumcised as a baby, others who were not circumcised in infancy may resent their parents\u2019 decision not to have them circumcised, especially if suffering from infections and other medical conditions that may have been avoided by being circumcised.\n\nTable 3.Issues to consider for time of male circumcision: neonatal vs. later.\n Neonatal circumcision Circumcision of older boys and men \u2022 Simple \u2022 More complex \u2022 Quick (takes several minutes) \u2022 Half an hour or more to perform \u2022 Cost is lower \u2022 Much more expensive (often unaffordable) \u2022 Low risk (adverse events 0.4%) \u2022 Moderate risk (adverse events 4\u20138%) \u2022 Bleeding (uncommon) is minimal and easily stopped \u2022 Bleeding more common, requiring cautery or other interventions \u2022 Sutures not needed \u2022 Sutures or tissue glue needed \u2022 Convenient for patient (sleeps mostly) \u2022 Inconvenient (time off school or work) \u2022 Local anaesthesia for age $<$2 months \u2022 General anaesthesia for age $>$2 months to 9 years. Local anaesthesia for men, although general anaesthesia often preferred by surgeon \u2022 Healing is fast ($<$2 weeks) \u2022 Healing takes 6 weeks or more \u2022 Cosmetic outcome usually good \u2022 If stitches used then stitch marks may be seen \u2022 No long-term memory of the procedure \u2022 Fear of undergoing an operation \u2022 Does not disrupt feeding or other day-to-day activities \u2022 Abstinence from sexual intercourse during the 6-week healing period\n\nOpponents of boyhood circumcision have used ethical arguments in support of their cause. A consortium of mostly Northern Europeans alleged that the AAP\u2019s 2012 infant MC policy was culturally biased, arguing that the only relevant benefit was protection against UTI, extent of complications was unknown, and that there \u201care no compelling reasons for surgery before boys are old enough to decide for themselves\u201d [268]. In response, the AAP Task Force on infant MC found the opinions expressed were \u201cnot comprehensive, systematic, or unbiased,\u201d instead containing false and one-sided information, suggested that the \u201cobvious\u201d cultural bias referred to stemmed from \u201cthe normality of non-therapeutic MC in the US,\u201d arguing that because \u201capproximately half of US males are circumcised, and half are not,\u201d any bias \u201cis more likely likely to be neutral\u2026 so predisposing the AAP Task Force to a more dispassionate analysis of the scientific literature than a culture with a bias that is either strongly opposed to circumcision or strongly in favor of it\u201d [269]. Arguments that the AAP\u2019s policy was unethical and unlawful [35] were shown by academics with expertise in medicine, ethics and law to lack merit, because arguments against MC involve \u201cpoor understanding of epidemiology, erroneous interpretation of the evidence, selective citation of the literature, statistical manipulation of data, and circular reasoning\u201d [254, 270]. Similarly, such experts repudiated [255, 271] criticisms of the CDC\u2019s 2014 draft recommendations [272, 273] by pointing out that the strong medical evidence would make it unethical to withhold information about the risks and benefits of MC from parents of boys. They quoted the following from Article 24(1) of the United Nations International Convention on the Rights of the Child: \u201cStates Parties recognize the right of the child to the enjoyment of the highest attainable standard of health\u201d and \u201cshall strive to ensure that no child is deprived of his or her right of access to health care services.\u201d A recent systematic review [78] has provided a detailed evaluation of the contrasting arguments and counter-arguments published by MC opponents and proponents.\n\n18. Conclusions\n\nThis review finds that circumcision of boys early in infancy is a low risk procedure providing a lifetime of benefits by protecting against infection and disease. Medical practitioners, nurses and other health professions in Australia have an ethical duty to present clear and unbiased information to parents of boys and to men regarding the range of benefits from MC, the net level of lifetime protection against disease, the low prevalence of procedural risks, that MC is generally performed using local anaesthesia in neonates, and, if need be, to direct parents to competent operators when they choose to proceed. The Circumcision Academy of Australia\u2019s policy recommendations appear in Table 4.\n\nTable 4.Recommendations by the Circumcision Academy of Australia.\n (1) Circumcision must be performed by a well-trained competent practitioner under sterile conditions using appropriate anaesthesia for pain management according to the age of the patient. (2) Parents should routinely be informed accurately early in a pregnancy in an unbiased manner about (i) the range of health benefits conferred by neonatal circumcision, (ii) the low risk of complications and that if any occur most are minor and easily treated with complete resolution, severe complications being rare, (iii) when performed in older boys complications are more common and the procedure is more expensive, (iv) circumcision is a well tolerated, minor procedure, and (v) pain will be managed. (3) The benefits of circumcision compared with the low risk in newborn boys are sufficient to justify nation-wide access to the procedure. (4) Third-party payment of costs by the federal government under Medicare and private health insurance is warranted. (5) After being fully informed, it is up to the parents to decide whether their boy should receive circumcision. In so doing, they will need to weigh up the medical information in the context of their own beliefs, be they cultural or religious practices or ethical views. The parents\u2019 decision should be respected.\nAbbreviations\n\nAAP, American Academy of Pediatrics; CDC, Centers for Disease Control and Prevention; HPV, human papillomavirus; LMX4, a topical anesthetic cream containing 4% lidocaine; MC, male circumcision; MSM, men who have sex with men; RCT, randomized controlled trial; RR, relative risk; SIGN, Scottish Intercollegiate Guidelines Network; STIs, sexually transmitted infections; UNAIDS, Joint United Nations Programme on HIV\/AIDS; UTI, urinary tract infection; VMMC, voluntary medical MC; VUR, vesicoureteric reflux.\n\nAuthor contributions\n\nBJM conceived, designed the study, performed literature searches, and prepared the initial draft of the manuscript; AK, NJB, MH, ACS, LS, ERL, ADW and PK provided input to successive drafts.\n\nEthics approval and consent to participate\n\nNot applicable.\n\nAcknowledgment\n\nThanks to all the peer reviewers for their opinions and suggestions.\n\nFunding\n\nThis research received no external funding.\n\nConflict of interest\n\nBJM is a Member of the Editorial Board of Journal of Men\u2019s Health. MH is medical director of Quick Medical Pty Ltd, a company that markets medical devices, including circumcision devices, in Australia. All authors are Members of the Circumcision Academy of Australia, a not-for-profit, government registered, medical society that provides accurate, evidence-based information on male circumcision to parents, practitioners and others, as well as contact details of doctors who perform the procedure in Australia and New Zealand; PK is President, BJM is Secretary, NB is Treasurer and MH is Surgical Training Co-ordinator of this organization.\n\nPublisher\u2019s Note: IMR Press stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.\nShare","date":"2022-07-03 11:07:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 39, \"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\": 0.2721402049064636, \"perplexity\": 12412.54285116502}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104240553.67\/warc\/CC-MAIN-20220703104037-20220703134037-00778.warc.gz\"}"} | null | null |
Trimalaconothrus repetitus är en kvalsterart som beskrevs av Subías 2004. Trimalaconothrus repetitus ingår i släktet Trimalaconothrus och familjen Malaconothridae. Inga underarter finns listade i Catalogue of Life.
Källor
Spindeldjur
repetitus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,237 |
Q: How to update a texture from array in Kivy? I'm new to Kivy, but have watched the tutorials. I want to have a widget containing a texture or image generated from an array, which will change at each frame. See below for what I currently have. Current behaviour is wrong when I resize the window - I think that the old Rectangle is never being deleted, but I can't see how to do that. It also shows the same image in a default (100,100) view at the bottom left of the main window. What do I need to change to achieve the desired behaviour, and not get artifacts when resizing the window?
from kivy.app import App
from kivy.properties import ObjectProperty
from kivy.uix.boxlayout import BoxLayout
from kivy.uix.layout import Layout
from kivy.graphics import Rectangle
from kivy.graphics.texture import Texture
from kivy.clock import Clock
import numpy as np
import random
class MainDisplay(Layout):
tex = ObjectProperty(None)
def __init__(self, **kwargs):
super(MainDisplay, self).__init__(**kwargs)
Clock.schedule_once(self.texture_init, 0)
def texture_init(self, instance):
self.tex = Texture.create()
def update(self, dt):
size = 64 * 64 * 3
buf = np.array([int(random.random() * x * 255 / size) for x in range(size)])
print('update', max(buf), min(buf), np.mean(buf))
# then blit the buffer
self.tex.blit_buffer(buf.tostring(), colorfmt='bgr', bufferfmt='ubyte')
print('end update')
print(self.canvas)
print(self.size, self.pos, self, self.parent)
with self.canvas:
Rectangle(texture=self.tex, size=(self.width / 2, self.height / 2), pos=(self.center_x / 2, self.center_y / 2))
class MainWindow(BoxLayout):
md = ObjectProperty(None)
def __init__(self, **kwargs):
super(MainWindow, self).__init__(**kwargs)
def update(self, dt):
self.md.update(dt)
class ProtoApp(App):
def build(self):
mainWindow = MainWindow()
Clock.schedule_interval(mainWindow.update, 1.0/10.0)
return mainWindow
if __name__ == "__main__":
ProtoApp().run()
with the proto.kv file:
<MainWindow>:
md: md
MainDisplay:
id: md
size_hint: (0.5, 0.5)
Thanks in advance for your help!
A: Problem
Whenever the window is resized, it is creating new rectangle and leaving traces of the previous one.
Solution
Use the canvas's built-in function, clear()
Snippets
def update(self, dt):
size = 64 * 64 * 3
buf = np.array([int(random.random() * x * 255 / size) for x in range(size)])
# then blit the buffer
self.tex.blit_buffer(buf.tostring(), colorfmt='bgr', bufferfmt='ubyte')
with self.canvas:
self.rect = Rectangle(texture=self.tex, size=(self.width / 2, self.height / 2),
pos=(self.center_x / 2, self.center_y / 2))
self.bind(pos=self.update_rect, size=self.update_rect)
def update_rect(self, *args):
self.canvas.clear()
self.rect.pos = self.pos
self.rect.size = self.size
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,284 |
Q: What is the most acceptable way to let a visitor know they are on an unsupoorted browser? I'm developing a site that uses JavaScript, HTML5 and CSS3 heavily. I have no desire to support last generations browsers (IE8 and under for example.) What is the most acceptable way to let the visitor know they are on an old browser?
Redirect to a place letting them know? Should I let them in anyway after I let them know even though a large portion of the site wont work? Should I recommend a browser to use, or list all functional browsers?
A: You're on a dangerous road trying to come up with a "list of functional browsers". I'd strongly reconsider that entire approach if I were you. There are so many great ways to support "older" browsers. It's not nearly as hard as it used to be.
Start by reading: http://www.alistapart.com/articles/understandingprogressiveenhancement/
Then look into using jQuery, Modernizr, and similar tools to help with your cross-browser/cross version woes.
A: Its not suggested to allow users to use your page if you are confident that the experience will be a disaster. If you know that 80% of the page won't work, you should advice your users to change the browser or don't use the page at all. If you don't do it, your credibility could be affected...a user could think the page is broken, or not professional.
The best way is to redirect to a page explaining the problem and suggesting A LIST of possible browsers, never a single one. If you suggest a single one, you could be seen as propaganda of this browser.
Good luck!
A: As a user, nothing is more irritating than being shut out of something because my browser has been rejected.
I would suggest notifying the visitor that their browser is unlikely to deliver a satisfying experience on your site (and explain that you are using the latest standards). Provide links to perhaps three acceptable browsers. Then let them in anyway. You never know when someone may show up with something that works but wasn't on your list of approved browsers.
Ideally, have your site degrade gracefully in older browsers by writing semantically-correct HTML that still renders even if some of the CSS and HTML features you're using are not supported.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,012 |
Biografia
Figlio di Luigi si trasferì ancora bambino a Firenze. Lavorò come carbonaio e falegname ed entrò in contatto con i fotografi fiorentini Alinari e Brogi. La sua attività di fotografo iniziò a Prato intorno al 1902. Quando i Salvi cessaroro l'attività, nello stabilimento fotografico, Coppi subentrò e si dichiarò "successore di G. Salvi". Dopo il 1905 si trasferì nei locali di piazza Buonamici. La nuova attività "Fotografia Coppi" venne inaugurata nell'aprile 1905 ed fu destinata a segnare la storia della fotografia a Prato.
Membro del circolo dei Misoduli, Coppi si inserì negli spazi lasciati dai Salvi e puntando sulla novità di un moderno studio fotografico, secondo quanto accadeva da tempo nelle grandi città. L'attività fotografica divenne attrasse i pratesi facoltosi, intenzionati ad avere un ritratto per sé o per la propria famiglia. Coppi eseguì non solo ritratti in studio ma anche fotografie all'aperto, soprattutto nella campagna di Schignano dove possedeva una casa. Fotografò mutilati e invalidi di guerra in gita a Trieste e Venezia, documentando una mostra di tessuti pratesi in Tripolitania. Si dedicò anche ai collage fotografici, ai fotomontaggi e alle stampe eseguite con diversi procedimenti.
Nel 1929 cessò l'attività nello studio di piazza Buonamici a causa di problemi di salute e forse anche dalla concorrenza.
Domenico Coppi ebbe come allievi i futuri fotografi professionisti: Carlo Silli, Adolfo Massai e Alfredo Ranfagni, al quale insegnò il mestiere e al quale la moglie lasciò l'archivio dei negativi, acquisito successivamente dall'Archivio Fotografico Toscano.
Note
Collegamenti esterni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,577 |
Eurepa är ett släkte av insekter. Eurepa ingår i familjen syrsor.
Kladogram enligt Catalogue of Life:
Källor
Syrsor
Eurepa | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,338 |
Q: Ordering columns according to columns values in Python SO I have the below
Name Tues Mon Tues Mon
col 0 0 1 1 <-
bill 2 1 2 1
jon 4 3 4 3
and i want to order the dataframe columns according to the "col" row to group 0's and 1's
in order but also in order according the days of the week so below is the result.
Name Mon Tues Mon Tues
col 0 0 1 1
bill 1 2 1 2
jon 3 4 3 4
A: *
*It's easier to sort by rows, so you can use .T to transpose the dataframe and then use .T to transpose it back after running some operations.
*The first thing it looks like you need to do is sort by the day of the week? You can crate a new column that replaces the partial Weekday strings to numbers, so you can sort in order by day of week, and then drop that column after sorting.
df = df.set_index('Name').T.reset_index()
df['day'] = df['index'].replace(['Su.*', 'M.*', 'Tu.*', 'W.*', 'Th.*', 'F.*', 'Sa.*'],
[1,2,3,4,5,6,7], regex=True)
df = df.sort_values(['col', 'day']).drop('day', axis=1).set_index('index').T.reset_index()
df
Out[1]:
index Name Mon Tues Mon.1 Tues.1
0 col 0 0 1 1
1 bill 1 2 1 2
2 jon 3 4 3 4
You can chnage the column names with:
df.columns = [col.split('.')[0] for col in df.columns]
Name Mon Tues Mon Tues
0 col 0 0 1 1
1 bill 1 2 1 2
2 jon 3 4 3 4
A: You can sort df by the following simple self-explained line
df_sorted = df.T.sort_values(['col', 'bill']).T
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,976 |
Плен-Вілем () — округ Маврикію, розташований в центральній частині країни і не має виходу до океану. Згідно перепису 2010 року, чисельність населення становить 385 034 осіб. Район займає площу в 203,3 км, щільність населення — 1893,92 чол./км². Округ ділитися на верхній Плен-Вілем з адміністративним центром Кьюрпайп і нижній з адміністративним центром Роз-Гілл.
Примітки
Адміністративний поділ Маврикію | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,692 |
Q: Plotting the deformed shape of a rectangle I want to plot a rectangle using graphics :
rc = Graphics[{Blue, Rectangle[{0, 0}, {1, 1}]}];
and add a deformation function to it:
$$\chi(\mathbf X)=X_1(1+X_2)\hat{\mathbf e}_1+X_2(1+3X_1)\hat{\mathbf e}_2$$
This deformation function specifies new coordinates as functions of old ones. How can I plot the new shape?
A: As J. M. suggested in comments, ParametricPlot[] can be used to show this deformation:
{ParametricPlot[{u, v}, {u, 0, 1}, {v, 0, 1}],
ParametricPlot[{u (1 + v), v (1 + 3 u)}, {u, 0, 1}, {v, 0,1}]} // GraphicsRow
A: Another way of approaching this is to map the image of the rectangle rather than the geometric object itself.
img = Image[Graphics[{Blue, Rectangle[{0, 0}, {1, 1}]}]]
This turns the graphic into an image which can then be mapped using the specified function:
ImageForwardTransformation[img, {#[[1]] (1 + #[[2]])/4, #[[2]] (1 + 3 #[[1]])/4} &]
The only wrinkle here is that the range of the image needs to be in {0,1} so I scaled both entries by the largest value (4) so that would show the whole image of the distorted rectangle.
Of course, this method maps not only the rectangle itself, but also the contents of the rectangle (if any). So for example, the same mapping can be applied to an image.
img = Import["http://i.stack.imgur.com/6zVEI.png"];
ImageForwardTransformation[img, {#[[1]] (#[[2]] + 1)/4,
(1 + 3 #[[1]]) #[[2]]/4} &, Background -> White]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,021 |
Me and a colleague were missing some features in mysql-query-browser and am trying to have a look at them, since no one at MySQL AB is very interested in supporting it. So I thought I could have a look at it. System is Fedora 10 (still), and I use it mainly because it's small, simple to use, and it's GTK!
You'll need a patch from Oden Eriksson attached to Bug #32184, or you can use the one from the RPM - otherwise you'll get the error error: 'SigC' has not been declared found on that bug report. I had to cut it for building from the SVN tree, and patched mysql-gui-common and mysql-query-browser independently (split the patch).
So it's easy to spot the needed --with switch. I had to apply several other patches that I just took source RPM. Most of them were applied with -p2.
Hunk #1 succeeded at 119 (offset 17 lines).
And that should be it - actually there was a path concatenation issue (looking for ...fake/usr/local/share...) which I quickly fixed with symlinks. After that, we should be ready to rock.
One of the features I miss most is the number of affected rows of some DML commands, such as UPDATE and INSERT. This was not easy to do in five minutes because of the UI split: mysql_affected_rows() doesn't seem to reach the GUI. So I've made a simple test, and succeeded.
This looks promising. I just set a global var, which will do for now. I still have to check for potential race conditions, but expect the polished patch, along with a new RPM for Fedora 10, at least, in the near future. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,223 |
Battle comes home, with clarinet
By Carol Simmons
Yellow Springs native Mark Battle will play his clarinet with pianist George Lopez on Tuesday, March 18, at 5 p.m. at a free concert. (Submitted Photo)
Yellow Springs native and one-time Yellow Springs News paper carrier Mark Battle will return to town with his clarinet in tow next week to perform a house concert with colleague and friend, pianist George Lopez.
The now 48-year-old Battle, who has a degree in clarinet performance from the New England Conservatory of Music, is a faculty member in the physics and astronomy department at Bowdoin College, in Maine, where pianist Lopez is an artist in residence.
The son of longtime villagers Esther and David Battle, and a 1983 graduate of Yellow Springs High School, Battle pursued undergraduate music studies at the conservatory in Boston as part of a five-year dual degree program with nearby Tufts University, where he studied physics. He received separate degrees from both institutions in 1988, and then continued his education at the University of Rochester, earning a masters and a Ph.D. in physics. He has been at Bowdoin for 14 years and lives in Brunswick, Maine, with his wife, Kathy Thorson, and their two children, Zoë and Felix. His research focuses on climate-related issues.
In a phone call last week, he described his life as busy and full, but without a lot of opportunities in recent years for clarinet performance.
The addition of George Lopez to the Bowdoin community, however, has helped the professional physicist expand his music-making efforts, Battle said. He began playing with Lopez regularly after first collaborating with him on a February 2013 recital at the college. He said Lopez, who has a concert this weekend in Hilliard, Ohio, suggested the duo performance in Yellow Springs when he realized he would be traveling near his friend's hometown.
The 5 p.m. performance on Tuesday, March 18, which is being presented as a gift to the community, is literally a community event. It will take place in the Phillips Street home of Jane Baker, a Battle family friend. And Mark Battle's parents will serve as co-hosts. Admission is free, but contributions to Chamber Music Yellow Springs will be accepted.
Battle said he is happy not only to be able to play for the town that nurtured him as a youth, but also to introduce Lopez's artistry to the community. Lopez has performed around the world, including Paris, London, Cologne, New York's Weill Recital Hall at Carnegie Hall, and in Los Angeles, where a Los Angeles Times critic hailed his "musical perspective, continuity and kaleidoscopic colors."
Although Battle said his own playing had been "languishing" before its recent renewal with Lopez, music has been a passion his whole life. And that passion was fed by opportunities he had growing up in Yellow Springs, where he went through Mills Lawn and then the middle and high schools. Mills Lawn didn't have an instrumental program during his years there, but there were many other resources in town.
He began taking lessons when he was in fourth-grade, he said. "I wanted to play sooner, but my fingers were too small to cover the holes."
Initially attracted to the saxophone — "I liked how shiny it was and how the keys worked" — his parents suggested he start with the clarinet. And once he started, he was hooked.
His first teacher was a high school student, as a part of a successful student-mentor program that earned widespread recognition. "Shirley Mullins (the district's long-time orchestra teacher) wrote an article about the program" that introduced it to other school districts, he said.
After a couple of years under the tutelage of an older student, Battle progressed to private lessons with Richard York, the principal clarinetist for the Springfield Symphony Orchestra. He continued his studies with faculty members at Wright State University, while also playing in regional and state groups, including the Dayton and Springfield youth orchestras, with additional lessons in Columbus and Cincinnati, before heading off to college.
The dual-degree program in music and physics allowed him to pursue his "two passions," he said. A shared love of science and the arts, particularly music, seems natural, he said.
"There are many people in the sciences and mathematics who have a strong interest in music. Anecdotally, it seems that a brain wired for math and physics is also wired for music."
In putting together the program for Tuesday's performance, Battle said he "wanted to play really good music" that also offered a variety of time periods and styles. "It's good to bring less similar works to light as well," he added. He and Lopez plan to perform works by Johannes Brahms, Bohuslav Martinu and Witold Lutoslawski. In addition, Lopez will play the Argentine Dances by Alberto Ginastera and three sonatas by Domenico Scarlatti.
Seating is limited for the performance, so those interested in attending are asked to make a reservation by calling Jane Baker at 767-7129.
Aaron's Lens - Snow
Merry marchers, friendly fire on the Fourth
Brash brass
First day, second grade
A Tour of Yellow Springs Murals
Block by block, summer's last hurrah plays out
Climb for a cause
YS Arts Council Yellow Springs School Board Bulldogs boys soccer Village Council Miami Township Fire-Rescue Bulldogs swimming Mckinney Middle School CBE Antioch College alumni summer youth baseball Yellow Springs Schools Bulldogs girls basketball Village of Yellow Springs First Presbyterian Church WYSO Tecumseh Land Trust Glen Helen Yellow Springs Arts Council Bulldogs boys cross country Bulldogs basketball YS School Board Home Inc. Bulldogs cross country Bulldogs football Friends Care Community Antioch College Bulldogs volleyball Bulldogs golf Perry League t-ball Yellow Springs Police Department Little Art Theatre Yellow Springs High School Mills Lawn School Bulldog girls soccer Bulldogs girls cross country
Street Fair YSHS spring musical New Year's Eve gay pride march Spring Street Fair Yellow Springs Experience art opening farmers' market holiday Summer in the Springs MLK march Simply Women Antioch College reunion Easter Egg Hunt protest DeWine briefings Christmas FMC benefit concert open gym basketball elections 2017 elections Art On the Lawn Challenger Sports British Soccer Camp 2018 Election Community Dance Concert holiday in the village SpringsFest AACW Blues Fest Fourth of July Earth Day YSHS graduation Election 2016 business opening fundraiser PBL Exhibition Night
local food local business coronavirus APRIL FOOLS! African-American culture and history First Lines energy efficiency Antioch College revival Obituary racism land use alternative energy Glen Helen economic development theater affordable housing Yellow Springs Schools community support bulldog sports COVID-19 antioch college YS COVID news fundraising Village water project-based learning retirement property tax levy ys covid-19 news environmental sustainability Village Manager solar energy protest village council music food and drink | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,087 |
Mahonia trifoliolata, branch in fruit. Family Berberidaceae, Subclass Magnoliidae. Origin: Cultivated.
James R. Manhart (2011). Mahonia trifoliolata (Cultivated). Available electronically from http : / /hdl .handle .net /1969 .1 /107279. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,071 |
package org.apache.eagle.app.environment.builder;
import backtype.storm.task.IOutputCollector;
import backtype.storm.task.OutputCollector;
import backtype.storm.tuple.Tuple;
import org.apache.eagle.app.utils.ClockWithOffset;
import org.apache.eagle.app.utils.ManualClock;
import org.junit.Assert;
import org.junit.Test;
import java.lang.reflect.Field;
import java.lang.reflect.InvocationTargetException;
import java.lang.reflect.Method;
import java.util.*;
import java.util.concurrent.TimeUnit;
public class CounterToRateFunctionTest {
private Map mkCountTypeEvent(long ts, double value) {
Map event = new HashMap();
event.put("timestamp", ts);
event.put("metric", "hadoop.hbase.regionserver.server.totalrequestcount");
event.put("component", "hbasemaster");
event.put("site", "sandbox");
event.put("value", value);
event.put("host", "xxx-xxx.int.xxx.com");
return event;
}
private Map mkCountTypeEventWithMetricName(long ts, double value, String metric) {
Map event = new HashMap();
event.put("timestamp", ts);
event.put("metric", metric);
event.put("component", "hbasemaster");
event.put("site", "sandbox");
event.put("value", value);
event.put("host", "xxx-xxx.int.xxx.com");
return event;
}
private Map mkOtherTypeEvent(long ts, double value) {
Map event = new HashMap();
event.put("timestamp", ts);
event.put("metric", "hadoop.memory.heapmemoryusage.used");
event.put("component", "hbasemaster");
event.put("site", "sandbox");
event.put("value", value);
event.put("host", "xxx-xxx.int.xxx.com");
return event;
}
@Test
public void testToMetricAndCounterValue() throws NoSuchMethodException, InvocationTargetException, IllegalAccessException {
long baseTime = System.currentTimeMillis() + 100000L;
MetricDescriptor metricDefinition = MetricDescriptor
.metricGroupByField("group")
.siteAs("siteId")
.namedByField("metric")
.eventTimeByField("timestamp")
.dimensionFields("host", "component", "site")
.granularity(Calendar.MINUTE)
.valueField("value");
CounterToRateFunction counterToRateFunction = new CounterToRateFunction(metricDefinition, 3, TimeUnit.MINUTES, ClockWithOffset.INSTANCE);
Map event = mkCountTypeEvent((baseTime + 0), 374042741.0);
Method toMetricMethod = counterToRateFunction.getClass().getDeclaredMethod("toMetric", Map.class);
toMetricMethod.setAccessible(true);
CounterToRateFunction.Metric metric = (CounterToRateFunction.Metric) toMetricMethod.invoke(counterToRateFunction, event);
Assert.assertEquals("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.totalrequestcount", metric.getMetricName());
Assert.assertEquals(374042741.0, Double.valueOf(metric.getValue().toString()), 0.00001);
Assert.assertEquals(374042741.0, metric.getNumberValue().doubleValue(), 0.00001);
Assert.assertTrue(metric.isCounter());
event = mkOtherTypeEvent((baseTime + 0), 100);
metric = (CounterToRateFunction.Metric) toMetricMethod.invoke(counterToRateFunction, event);
Assert.assertEquals("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.memory.heapmemoryusage.used", metric.getMetricName());
Assert.assertEquals(100, Double.valueOf(metric.getValue().toString()), 0.00001);
Assert.assertEquals(100, metric.getNumberValue().doubleValue(), 0.00001);
Assert.assertTrue(!metric.isCounter());
}
@Test
public void testTransformToRate() throws NoSuchFieldException, IllegalAccessException {
List<Map> result = new ArrayList<>();
OutputCollector collector = new OutputCollector(new IOutputCollector() {
@Override
public List<Integer> emit(String streamId, Collection<Tuple> anchors, List<Object> tuple) {
result.add((Map) tuple.get(1));
return null;
}
@Override
public void emitDirect(int taskId, String streamId, Collection<Tuple> anchors, List<Object> tuple) {
}
@Override
public void ack(Tuple input) {
}
@Override
public void fail(Tuple input) {
}
@Override
public void reportError(Throwable error) {
}
});
MetricDescriptor metricDefinition = MetricDescriptor
.metricGroupByField("group")
.siteAs("siteId")
.namedByField("metric")
.eventTimeByField("timestamp")
.dimensionFields("host", "component", "site")
.granularity(Calendar.MINUTE)
.valueField("value");
CounterToRateFunction counterToRateFunction = new CounterToRateFunction(metricDefinition, 3, TimeUnit.MINUTES, ClockWithOffset.INSTANCE);
counterToRateFunction.open(new StormOutputCollector(collector));
long baseTime = System.currentTimeMillis() + 100000L;
//put first count sample
Map event = mkCountTypeEvent((baseTime + 0), 374042741.0);
counterToRateFunction.transform(event);
Assert.assertTrue(result.isEmpty());
Field cacheField = counterToRateFunction.getClass().getDeclaredField("cache");
cacheField.setAccessible(true);
Map<String, CounterToRateFunction.CounterValue> cache = (Map<String, CounterToRateFunction.CounterValue>) cacheField.get(counterToRateFunction);
Assert.assertTrue(cache.size() == 1);
CounterToRateFunction.CounterValue counterValue = cache.get("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.totalrequestcount");
Assert.assertEquals((long) event.get("timestamp"), counterValue.getTimestamp());
Field valueField = counterValue.getClass().getDeclaredField("value");
valueField.setAccessible(true);
double value = (double) valueField.get(counterValue);
Assert.assertEquals(374042741.0, value, 0.00001);
result.clear();
//put not count sample
event = mkOtherTypeEvent((baseTime + 0), 100);
counterToRateFunction.transform(event);
Assert.assertTrue(result.size() == 1);
Assert.assertTrue(cache.size() == 1);
Assert.assertEquals(baseTime + 0, counterValue.getTimestamp());
Assert.assertEquals(374042741.0, value, 0.00001);
Assert.assertEquals("hadoop.memory.heapmemoryusage.used", event.get("metric"));
Assert.assertEquals(100, (Double) event.get("value"), 0.00001);
result.clear();
//delta of 10 in 5 seconds
event = mkCountTypeEvent((baseTime + 5000), 374042751.0);
counterToRateFunction.transform(event);
Assert.assertTrue(result.size() == 1);
Map transedEvent = result.get(0);
Assert.assertEquals(baseTime + 5000, transedEvent.get("timestamp"));
Assert.assertEquals(2.0, (double) transedEvent.get("value"), 0.00001);
Assert.assertEquals(baseTime + 5000, counterValue.getTimestamp());
value = (double) valueField.get(counterValue);
Assert.assertEquals(374042751.0, value, 0.00001);
result.clear();
//delta of 15 in 5 seconds
event = mkCountTypeEvent((baseTime + 10000), 374042766.0);
counterToRateFunction.transform(event);
Assert.assertTrue(result.size() == 1);
transedEvent = result.get(0);
Assert.assertEquals(baseTime + 10000, transedEvent.get("timestamp"));
Assert.assertEquals(3.0, (double) transedEvent.get("value"), 0.00001);
Assert.assertEquals(baseTime + 10000, counterValue.getTimestamp());
value = (double) valueField.get(counterValue);
Assert.assertEquals(374042766.0, value, 0.00001);
result.clear();
//No change from previous sample
event = mkCountTypeEvent((baseTime + 15000), 374042766.0);
counterToRateFunction.transform(event);
Assert.assertTrue(result.size() == 1);
transedEvent = result.get(0);
Assert.assertEquals(baseTime + 15000, transedEvent.get("timestamp"));
Assert.assertEquals(0.0, (double) transedEvent.get("value"), 0.00001);
Assert.assertEquals(baseTime + 15000, counterValue.getTimestamp());
value = (double) valueField.get(counterValue);
Assert.assertEquals(374042766.0, value, 0.00001);
result.clear();
//Decrease from previous sample
event = mkCountTypeEvent((baseTime + 20000), 1.0);
counterToRateFunction.transform(event);
Assert.assertTrue(result.size() == 1);
transedEvent = result.get(0);
Assert.assertEquals(baseTime + 20000, transedEvent.get("timestamp"));
Assert.assertEquals(0.0, (double) transedEvent.get("value"), 0.00001);
Assert.assertEquals(baseTime + 20000, counterValue.getTimestamp());
value = (double) valueField.get(counterValue);
Assert.assertEquals(1.0, value, 0.00001);
result.clear();
}
@Test
public void testTransformToRateWithExpiration() throws NoSuchFieldException, IllegalAccessException {
MetricDescriptor metricDefinition = MetricDescriptor
.metricGroupByField("group")
.siteAs("siteId")
.namedByField("metric")
.eventTimeByField("timestamp")
.dimensionFields("host", "component", "site")
.granularity(Calendar.MINUTE)
.valueField("value");
List<Map> result = new ArrayList<>();
OutputCollector collector = new OutputCollector(new IOutputCollector() {
@Override
public List<Integer> emit(String streamId, Collection<Tuple> anchors, List<Object> tuple) {
result.add((Map) tuple.get(1));
return null;
}
@Override
public void emitDirect(int taskId, String streamId, Collection<Tuple> anchors, List<Object> tuple) {
}
@Override
public void ack(Tuple input) {
}
@Override
public void fail(Tuple input) {
}
@Override
public void reportError(Throwable error) {
}
});
ManualClock manualClock = new ManualClock(0);
manualClock.set(30000L);
CounterToRateFunction counterToRateFunction = new CounterToRateFunction(metricDefinition, 60, TimeUnit.SECONDS, manualClock);
counterToRateFunction.open(new StormOutputCollector(collector));
Map event = mkCountTypeEventWithMetricName(manualClock.now(), 110, "hadoop.hbase.regionserver.server.totalrequestcount");
counterToRateFunction.transform(event);
Field cacheField = counterToRateFunction.getClass().getDeclaredField("cache");
cacheField.setAccessible(true);
Map<String, CounterToRateFunction.CounterValue> cache = (Map<String, CounterToRateFunction.CounterValue>) cacheField.get(counterToRateFunction);
Assert.assertTrue(cache.size() == 1);
manualClock.set(50000L);
event = mkCountTypeEventWithMetricName(manualClock.now(), 130, "hadoop.hbase.regionserver.server.readerrequestcount");
counterToRateFunction.transform(event);
cache = (Map<String, CounterToRateFunction.CounterValue>) cacheField.get(counterToRateFunction);
Assert.assertEquals(2, cache.size());
Assert.assertEquals("CounterValue{timestamp=30000, value=110.0}", cache.get("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.totalrequestcount").toString());
Assert.assertEquals("CounterValue{timestamp=50000, value=130.0}", cache.get("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.readerrequestcount").toString());
manualClock.set(100000L);
event = mkCountTypeEventWithMetricName(manualClock.now(), 120, "hadoop.hbase.regionserver.server.totalrequestcount");
counterToRateFunction.transform(event);
cache = (Map<String, CounterToRateFunction.CounterValue>) cacheField.get(counterToRateFunction);
Assert.assertEquals(2, cache.size());
Assert.assertEquals("CounterValue{timestamp=100000, value=120.0}", cache.get("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.totalrequestcount").toString());
Assert.assertEquals("CounterValue{timestamp=50000, value=130.0}", cache.get("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.readerrequestcount").toString());
manualClock.set(160001L);
event = mkCountTypeEventWithMetricName(manualClock.now(), 10, "hadoop.hbase.regionserver.server.writerrequestcount");
counterToRateFunction.transform(event);
Assert.assertEquals(2, cache.size());
Assert.assertEquals("CounterValue{timestamp=160001, value=10.0}", cache.get("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.writerrequestcount").toString());
Assert.assertEquals("CounterValue{timestamp=50000, value=130.0}", cache.get("xxx-xxx.int.xxx.com-hbasemaster-sandbox-hadoop.hbase.regionserver.server.readerrequestcount").toString());
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 4,977 |
import gym
import random
import numpy as np
import tflearn
from tflearn.layers.core import input_data, dropout, fully_connected
from tflearn.layers.estimator import regression
from statistics import median, mean
from collections import Counter
import tensorflow as tf
import sys
tf.logging.set_verbosity(tf.logging.FATAL)
LR = 1e-3
env = gym.make("CartPole-v1")
env.reset()
DROPOUT_RATE = 0.3
goal_steps = 500
score_requirement = 50
initial_games = 100000
n_nodes_hl1 = 128
n_nodes_hl2 = 256
n_nodes_hl3 = 512
n_nodes_hl4 = 256
n_nodes_hl5 = 128
n_classes = 2
epochN=15
def initial_population():
"""
Extracts good runs from random games. Code from sentdex
:return training_data:
"""
# [OBS, MOVES]
training_data = []
# all scores:
scores = []
# just the scores that met our threshold:
accepted_scores = []
# iterate through however many games we want:
for _ in range(initial_games):
score = 0
# moves specifically from this environment:
game_memory = []
# previous observation that we saw
prev_observation = []
# for each frame in 200
for _ in range(goal_steps):
# choose random action (0 or 1)
action = random.randrange(0, 2)
# do it!
observation, reward, done, info = env.step(action)
# notice that the observation is returned FROM the action
# so we'll store the previous observation here, pairing
# the prev observation to the action we'll take.
if len(prev_observation) > 0:
game_memory.append([prev_observation, action])
prev_observation = observation
score += reward
if done: break
# IF our score is higher than our threshold, we'd like to save
# every move we made
# NOTE the reinforcement methodology here.
# all we're doing is reinforcing the score, we're not trying
# to influence the machine in any way as to HOW that score is
# reached.
if score >= score_requirement:
accepted_scores.append(score)
for data in game_memory:
# convert to one-hot (this is the output layer for our neural network)
if data[1] == 1:
output = [0, 1]
elif data[1] == 0:
output = [1, 0]
# saving our training data
training_data.append([data[0], output])
# reset env to play again
env.reset()
# save overall scores
scores.append(score)
# some stats here, to further illustrate the neural network magic!
print('Average accepted score:', mean(accepted_scores))
print('Median score for accepted scores:', median(accepted_scores))
print(Counter(accepted_scores))
return training_data
def neural_network_modelv2():
x = tf.placeholder(tf.float32, shape=(None, 4), name='x')
# # #(input_data * weights) +biases
#
# hidden_1_layer = {'weights': tf.Variable(tf.random_normal([4, n_nodes_hl1])),
# 'biases': tf.Variable(tf.random_normal([n_nodes_hl1]))}
#
# hidden_2_layer = {'weights': tf.Variable(tf.random_normal([n_nodes_hl1, n_nodes_hl2])),
# 'biases': tf.Variable(tf.random_normal([n_nodes_hl2]))}
#
# hidden_3_layer = {'weights': tf.Variable(tf.random_normal([n_nodes_hl2, n_nodes_hl3])),
# 'biases': tf.Variable(tf.random_normal([n_nodes_hl3]))}
#
# hidden_4_layer = {'weights': tf.Variable(tf.random_normal([n_nodes_hl3, n_nodes_hl4])),
# 'biases': tf.Variable(tf.random_normal([n_nodes_hl4]))}
#
# hidden_5_layer = {'weights': tf.Variable(tf.random_normal([n_nodes_hl4, n_nodes_hl5])),
# 'biases': tf.Variable(tf.random_normal([n_nodes_hl5]))}
#
# output_layer = {'weights': tf.Variable(tf.random_normal([n_nodes_hl5, n_classes])),
# 'biases': tf.Variable(tf.random_normal([n_classes])), }
#
# l1 = tf.add(tf.matmul(x, hidden_1_layer['weights']), hidden_1_layer['biases'])
# l1 = tf.nn.relu(l1)
# #l1 = tf.nn.dropout(l1,0.2)
#
# l2 = tf.add(tf.matmul(l1, hidden_2_layer['weights']), hidden_2_layer['biases'])
# l2 = tf.nn.relu(l2)
# #l2 = tf.nn.dropout(l2,0.2)
#
# l3 = tf.add(tf.matmul(l2, hidden_3_layer['weights']), hidden_3_layer['biases'])
# l3 = tf.nn.relu(l3)
# #l3 = tf.nn.dropout(l3,0.2)
#
# l4 = tf.add(tf.matmul(l3, hidden_4_layer['weights']), hidden_4_layer['biases'])
# l4 = tf.nn.relu(l4)
# #l4 = tf.nn.dropout(l4,0.2)
#
# l5 = tf.add(tf.matmul(l4, hidden_5_layer['weights']), hidden_5_layer['biases'])
# l5 = tf.nn.relu(l5)
# #l5 = tf.nn.dropout(l5,0.2)
#
# output_layer = tf.matmul(l5, output_layer['weights']) + output_layer['biases']
# output_layer = tf.nn.softmax(output_layer)
#
# return output_layer
network = tf.contrib.layers.relu(x, 128)
network = tf.layers.dropout(network, rate=DROPOUT_RATE, training=dropout)
network = tf.contrib.layers.relu(network, 256)
network = tf.layers.dropout(network, rate=DROPOUT_RATE, training=dropout)
network = tf.contrib.layers.relu(network, 512)
network = tf.layers.dropout(network, rate=DROPOUT_RATE, training=dropout)
network = tf.contrib.layers.relu(network, 512)
network = tf.layers.dropout(network, rate=DROPOUT_RATE, training=dropout)
network = tf.contrib.layers.relu(network, 256)
network = tf.layers.dropout(network, rate=DROPOUT_RATE, training=dropout)
network = tf.contrib.layers.relu(network, 128)
network = tf.layers.dropout(network, rate=DROPOUT_RATE, training=dropout)
output = tf.contrib.layers.fully_connected(network, 2, activation_fn=tf.nn.softmax)
return output
dropout = tf.placeholder(tf.bool,shape=None,name ='dropout')
def playthegame(training_data):
y=tf.placeholder(tf.float32,shape=(None,2), name='y')
trainingX = np.array([i[0] for i in training_data]).reshape(-1, len(training_data[0][0]))
trainingY = [i[1] for i in training_data]
nn = neural_network_modelv2()
cost = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=nn, labels=y))
optimizer = tf.train.AdamOptimizer().minimize(cost)
with tf.Session() as sess:
sess.run(tf.initialize_all_variables())
tf.set_random_seed(7)
# Train
for epoch in range(epochN):
epoch_loss = 0
ca, c = sess.run([optimizer, cost], feed_dict={'x:0': trainingX, 'y:0': trainingY,dropout:True})
epoch_loss += c
print('Epoch', epoch, 'loss', epoch_loss)
print('Training done')
scores = []
choices = []
for each_game in range(100):
score = 0
game_memory = []
prev_obs = []
env.reset()
for _ in range(goal_steps):
#env.render()
if len(prev_obs) == 0:
action = random.randrange(0, 2)
else:
action = np.argmax(sess.run([nn], feed_dict={'x:0':prev_obs.reshape(-1, len(prev_obs)),dropout:True}))
choices.append(action)
new_observation, reward, done, info = env.step(action)
prev_obs = new_observation
game_memory.append([new_observation, action])
score += reward
if done: break
scores.append(score)
print('Average Score:', sum(scores) / len(scores))
print('choice 1:{} choice 0:{}'.format(choices.count(1) / len(choices), choices.count(0) / len(choices)))
print(score_requirement)
training_data = initial_population()
playthegame(training_data) | {
"redpajama_set_name": "RedPajamaGithub"
} | 9,398 |
<html>
<head>
<title>Spring Transactions | Ebean</title>
<meta name="layout" content="_layout2/base-docs.html"/>
<meta name="bread1" content="Transactions" href="/docs/transactions"/>
<meta name="bread2" content="Spring" href="/docs/transactions/spring"/>
<#assign n0_docs="active">
<#assign n1_transactions="active">
<#assign n2_spring="active">
</head>
<body>
<h2>Spring transactions</h2>
<p>
The recommendation is to use Ebean's own transaction management but we can also use
spring transactions.
</p>
<p>
To do so include the dependency:
</p>
<pre content="xml">
<dependency>
<groupId>io.ebean</groupId>
<artifactId>ebean-spring-txn</artifactId>
<version>12.1.1</version>
</dependency>
</pre>
<p>
Set <code>SpringJdbcTransactionManager</code> as an external transaction by:
</p>
<pre content="java">
DatabaseConfig config = new DatabaseConfig();
config.setExternalTransactionManager(new SpringJdbcTransactionManager());
</pre>
<@next_edit "JTA Transactions" "/docs/transactions/jta" "/docs/transactions/spring.html"/>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,421 |
\section{Introduction}
Musical parody is a important part of Western society.
It is used as a form of free speech to advance political debate, give flattery, provide mockery, and to entertain.
In the United States of America parody is perceived as an important part of free speech to the point that it is protected by {\em fair use} laws, meaning that music can be copied if the lyrics are changed in a significant manner.
Musical parody takes advantage of familiar tunes to catch one's attention and bring awareness to the message captured in the new lyrics.
The author of this paper likes to motivate their seven-year old son to get ready to bed by making the announcement to the tune of songs he knows.
Is there really any more important use of speech than motivating one's child to go to bed so one can write a paper for arXiv?
Whereas making a song from scratch requires inventing new melodies and beats, song parody allows a person without any music writing experience to select new words that fit an existing syllable and rhyme scheme.
This is not to say that writing good music parody is easy.
The artist, ``Weird'' Al Yankovic is famous for publishing well-known parodies of famous musicians such as Michael Jackson, Madonna, and Nirvana.
Online social media platforms such as Twitter and Facebook have made parody more popular, allowing anyone to share parodies about topical themes, such as the showtune parodies by Randy Rainbow that mock politicians, computer science instructors teaching about machine learning topics,\footnote{\textit{ Overfitting Thriller} by Michael Littman and Charles Isbell \url{https://youtu.be/DQWI1kvmwRg}} or university professors singing about teaching remotely during a pandemic to the tune of {\em I will Survive} by Gloria Gaynor.\footnote{\textit{I Will Survive, Coronavirus version} by Michael Bruening \url{https://youtu.be/CCe5PaeAeew}}
In this paper, I introduce a system that can generate new parody lyrics for existing songs.
The system does not generate melodies---that is assumed to already exist because there is a song being parodied---nor does the system sing the lyrics it generates.
The new lyrics are textual and meant to be sung by the user to the original tune.
Figure~\ref{tab:weird} shows an example of original lyrics side by side with a famous human-written parody, and output from our system.
Our system can also produce a karaoke video so one can sing the new lyrics to original music.
I named the system {\em Weird AI Yankovic} to give homage to the greatest parody musician ever.
Also because if a san-serif font is used, it is hard to tell the difference between \textsf{Al} and \textsf{AI}.
It is best to read the rest of this paper in the voice of Al Yankovic.
\begin{table}[h!]
\centering
\footnotesize
\begin{tabular}{p{1.6in}|p{1.6in}|p{1.6in}}
{\bf {\em Beat It} by Michael Jackson} & {\bf {\em Eat It} by Weird Al Yankovic} & {\bf System output} \\
\hline
They told him don't you ever come around here &
How come you're always such a fussy young man? &
The best part is that each taco contains a small \\
Don't want to see your face, you better disappear &
Don't want no Captain Crunch, don't want no Raisin Bran &
To medium sized piece of sliced chicken nepal \\
The fire's in their eyes and their words are really clear &
Well, don't you know that other kids are starving in Japan? &
I don't think the food in question lasted awhile \\
So beat it, just beat it &
So eat it, just eat it &
I promise, just promise \\
\hline
\end{tabular}
\caption{Some examples of lyrics parodies. The original lyrics are on the left. Human-written parody lyrics are given in the center. Output from our system is give on the right.}
\label{tab:examples}
\end{table}
\section{Parody Lyrics as AI Challenge}
In it's most basic form, lyrics parody swaps one set of words that accompany a melody with a new set of words, preserving the number of syllables per line and the rhyme scheme indicating which lines rhyme with each other.
In doing so, the new lyrics are likely to also fit the melody, which remains unchanged, and will be recognizable to hearers.
The rhyme scheme and number of lyrics per line can be viewed as constraints on a language generation task.
Some examples of parody lyrics, human written and algorithmically generated, are shown in Table~\ref{tab:examples}.
AI lyric parody generation is a form of {\em controllable text generation}.
Many large-scale transformer-based neural language models, such as GPT-2~\cite{Radford2019LanguageMA}, XLNet~\cite{Yang2019XLNetGA}, T5~\cite{Raffel2019ExploringTL}, or even GPT-3~\cite{Brown2020LanguageMA}, are capable of producing fluent language.
However, neural language models predict a sequence of tokens based on a given sequence of tokens.
That's cool---one can provide a prompt and get a reasonable continuation.
However, the generation is not {\em controllable} because one cannot specify any constraints on what is produced,
such as:
the number of words in a sentence, the number of syllables per sentence, or whether certain words rhyme.
The reason that controllability is challenging is because generative language models do not perform look-ahead.
That is, local word choices do not take into account whether it makes it easier or harder to meet constraints that come in to play later, such as a rhyme.
Our instincts as deep learning researchers would be to train a neural language model from scratch, or fine-tune an existing transformer-based neural language model, to produce a given number of syllables and to end lines with a rhyme.
That would probably work; I dunno, I didn't try that.
Like, that just sounds hard.
Where do I get the corpus?
How do I label it?
Do I have to create a new model for each new song with different pattern of syllables or rhymes?
Can I train a general system and prompt it with the pattern?
An alternative approach is to provide a specialized sampling strategy that is sensitive to the constraints of syllable and rhyme.
The dirty secret of neural language modeling systems is that they can be thought of containing three components:
(1)~an encoder that compresses a prompt into a learned representation,
(2)~a decoder that decompresses a representation and produces a distribution over the vocabulary, and
(3)~a {\em sampler} that selects tokens from the distribution.
The simplest sampling strategy is greedy, taking the logit with the highest value.
Other sampling strategies include top-$k$, nucleus sampling (top-$p$)~\cite{Holtzman2020TheCC}, and beam search.
While I find that top-$k$ or top-$p$ work pretty well for most things I want to do, beam search
can increase the odds that later constraints are met as the width of the beam is increased, though it can become trapped in local maxima.
In this paper I show how a combination of forward generation with a neural language model, backward generation with a neural language model, and a specialized sampling strategy that is sensitive to syllables can produce parody lyrics that meet a set of given constraints regarding number of syllables per line and rhyme scheme between lines.
This paper presents the engineering considerations because I am not sure there are any scientific contributions.
I just wanted to make something that worked, and, frankly, it worked a lot better than I expected.
\section{The Weird AI Yankovic System}
This section walks through all the various parts of the system.
{\bf Constraints.}
Music parody starts with understanding the syllable and rhyme constraints from the original music.
Constraints should be easily provided by users.
The user provides a {\em scheme}, a list of line specifications where each line is a list of segments $(s_i, r_i, e_i)$ such that $s_i$ is the number of syllables, $r_i$ is an unique numerical identifier such that each segment with the identifier will rhyme, and $e_i$ is an optional parameter to end the segment with a period.
A line can consist of more than one of these segment tuples because of interior rhymes, used frequently in hip-hop, rap, and {\em Hamilton: the Musical}.
The rhyme identifier can be null, signifying that the line (or segment) does not need to rhyme with anything else.
The rhyme identifier can also be a word or phrase, indicating that the generator must use this exact word or phrase.
A {\em rhyme map} keeps track of words that must be rhymed with for each $r$; the rhyme map can be seeded if the user wants certain lines to rhyme with certain words.
\textbf{Context Prompt.}
The context prompt is another user input that provides a word, phrase, or sentence to cue a particular topic.
For example: ``my favorite food is tacos''.\footnote{The prompt doesn't always have to be true}
The prompt is provided as an initial input to see the language model generation for the first line of the lyrics (see below).
After that, each call to the generative language model is seeded with the prompt plus all subsequent lines that have been generated.
The original prompt does not appear in the final lyrics output.
\begin{wrapfigure}[23]{r}{2.25in}
\scriptsize
\vspace{-1.5\baselineskip}
\begin{tabular}{|p{2.1in}|}
\hline
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex,wide]
\item Let $s_1$ and $s_2$ be two potentially-rhyming phoneme sequences.
\item Replace ER with UH R in both sequences.
\item Let $v_1$ and $v_2$ be the last stressed vowels in $s_1$ and $s_2$.
\item Let $w_1$ and $w_2$ be last vowels in $s_1$ and $s_2$.
\item Let $s_1 = (a_1 v_1 x_1 w_1 c_1)$. Likewise, let $s_2 = (a_2 v_2 x_2 w_2 c_2)$.
\item Output NO under any of these circumstances:
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item $v_1 \neq v_2$
\item $w_1 \neq w_2$
\item \bm{$c_1 \neq c_2$}
\item $a_1 \neq \emptyset$ and $a_2\neq \emptyset$ and $a_1 = a_2$
\end{enumerate}
\item If $x_1$ and $x_2$ are single phonemes:
\begin{enumerate}[topsep=0pt,itemsep=-1ex,partopsep=1ex,parsep=1ex]
\item {\bf If \bm{$x1\sim x_2$}, then output YES.}
\item Otherwise, output NO.
\end{enumerate}
\item If $x_1$ and $x_2$ contain different numbers of vowels, output NO.
\item Let $p_1$ and $q_1$ be the first and last phonemes of $x_1$. Let $p_2$ and $q_2$ be the same for $x_2$.
\item {\bf If \bm{$(p_1 = p_2)$} and \bm{$(q_1 \sim q_2)$}, output YES.}
\item {\bf If \bm{$(p_1\sim p_2)$} and \bm{$(q_1 = q_2)$}, output YES.}
\item Otherwise, output NO.
\end{enumerate}\\
\hline
\end{tabular}
\caption{Near-rhyme detection algorithm~\cite{Ghazvininejad2016GeneratingTP} with modified lines highlighted.}
\label{tab:near-rhymes}
\end{wrapfigure}
\textbf{Near-Rhyme Dictionary.}
There are plenty of perfect-rhyme dictionaries.
A lot of music uses {\em near-rhymes}, which violate the rules of rhymes in subtle ways.
Whatever one thinks of the artist, Eminem, he once rhymed ``discuss me'', ``disgusting'', and ``just obscene''.
It is used quite frequently to amazing effect in rap and hip-hop.
But also Imagine Dragons, so take nothing for granted.
Anyway, figuring out whether two words are near-rhymes isn't as straight-forward as determining whether two words are perfect rhymes.
Ghazvininejad et al.~\cite{Ghazvininejad2016GeneratingTP} identified an algorithm for detecting near-rhymes, shown in
Figure~\ref{tab:near-rhymes}.
I made two changes.
The first was to delete line 6(c); I allow end consonants to be different because when lyrics are sung, the fina consonan of words are ofte softene, de-emphasize, or blende with the nex word-sound.
I also changed line 7(a), 10, and 11 where I should have used the sound of phonemes to determine similarity.
Instead, I created a set of rules to determine if two phonemes were similar.
I found this to work better for lyrics because I had greater control of what sounded good when sung out loud.
For example, I specify a rule that phonemes with `r' components should be marked as similar to a lot of vowels.
These design decisions gave my system a bit of a British sound, which is fine by me because I listen to a lot of pop music from the United Kingdom.\footnote{British people would probably disagree.}
I created a near-rhyme dictionary from the 20,000 most frequently used in the English language.
\textbf{Rhyme Selection.}
When the input scheme specifies, the system needs to pick a word that rhymes with one of the words in the rhyme map.
We can pick any word from the rhyme dictionary, but how do we know a rhyming word is going to be contextually relevant?
We need a way to rank these words on relevance.
I tried using the cosine similarity of language model contextual embeddings.
It turns out that there are some words that are just ``close'' to a lot of words and the system producing a lot of boring rhyme words.
Because the system would be using the rhyme word after some number of interstitial words that have not yet been generated,
I needed a way of guessing if any of the candidate words would be probable in future.
The system uses GPT-2 to generate $n$ sequences of length $m$.
As the system sample words to generate the sequences, it also measures each the rank of each rhyme candidate from the rhyme dictionary in the token distribution at each of the $m$ token positions in each of the $n$ sequences.
We finally pick the word from the rhyme dictionary with the highest average logit, which indicates that it is most likely to naturally occur during generation anyway.
I have not formally evaluated if this is significantly different from cosine similarity between candidate and a context prompt.
Anecdotally, I found that cosine similarity was resulting in more boring candidate choices and thus resulting in more boring lyrics;
the above look-ahead similarity technique seems to rank more interesting (more rare) words higher while not being too random.
\textbf{Backward Generation.}
Now that we have chosen a rhyme word to end a line,
how do we ensure that a line ends with a chosen rhyme word?
If we generate backward from that word, it will be guaranteed.
Fortunately, XLNet~\cite{Yang2019XLNetGA} can be induced to do just that.
Given the input ``\texttt{context\_1 ... context\_n MASK MASK ... MASK rhyme\_word}'', XLNet can fill each mask position one at a time.
XLNet attends to the context words at the beginning and to the rhyme word at the end when making it's decision about how to fill each mask position.
The system fills the masks starting from last and moving backward to the first, which seems to help with attending to the rhyme word.
But wait, how many mask positions should we have?
Because tokens don't correspond to syllables, one cannot say for sure.
The system constructs a prompt with one mask and incrementally produces prompts with more masks until it has tried $\lceil s \times 1.5\rceil$ masks where $s$ is the target number of syllables for the output line.
We have to go higher than $s$ because the language model can generate tokens corresponding to white space, punctuation, etc.
While we are generating, we disallow repeat tokens, punctuation, numbers, line breaks, or token corresponding to non-alphanumeric characters.
The system samples more tokens than necessary for each mask and iterates from the most likely to least likely, taking the first one that does not violate any of the restrictions listed above.
We repeat this entire iterative generation process $n \times 2$ times, ending the prompt with the rhyme word half the time and ending the prompt with the rhyme word followed by a period the other half.
This gives the system the option of a line that continues onto the next line or ending the line in a ``complete thought.''
Any lines that don't have the exact number of syllables are discarded, unless there are no lines with the requisite number of syllables, in which case, the system will allow lines with fewer syllables.
{\bf Forward Generation.}
When the rhyme scheme indicates that a line doesn't need to end in a rhyme, or when the line must end in a rhyme but there are no words that are yet associated with the line's rhyme index in the rhyme map (i.e., this is the first time a rhyme index is encountered) then the system uses GPT-2 to generate one token at a time.
Empirically we find that GPT-2 produces more fluent results than XLNet and is thus a better option when generating forward.
The system counts the number of syllables after each token is sampled and generation process stops after the specified number of syllables is generated.
If too many syllables are generated, the line is discarded.
This process is replicated $n$ times per line to give the system a number of options to pick from.
\textbf{Picking a Line.}
For each line in the song, either backward generation or forward generation is used and produces a number of candidate lines.
The system chooses randomly, proportional to the line's posterior probability when run through GPT-2.
One would think this would choose the most ``boring'' line, but believe me, the lines are already kind of weird.
Mathematically it is unclear how to distinguish between a weird but fluent sequence and non-fluent garbage, so my design decision was to go with more probable lines.
The rhyme selection process drives the interestingness by coming up with unusual words that have to be fit into the existing context of the song.
The system also has an interactive mode where the user can pick from the candidates.
\begin{wrapfigure}[20]{r}{3.0in}
\vspace{-1.0\baselineskip}
\includegraphics[width=3in]{searle}
\caption{Screenshot of the karaoke video generation, corresponding to Table~\ref{tab:weird}.
Lines are highlighted in sync with when the original lyrics would have been sung ({\em I Want to Hold Your Hand} by The Beatles).
}
\label{fig:karaoke}
\end{wrapfigure}
\textbf{Re-contextualization.}
As the number of lines grows, GPT-2 and XLNET are less likely to attend to the original prompt words and the topic can drift.
Re-contextualization is a process whereby the the system splices the prompt text into the song lyrics after a period so that the language model generators are more likely to attend to the prompt words and generation stays on topic.
The splicing of prompt words throughout the lyrics is just for the generators and do not appear in the final output.
\textbf{Post-processing.}
The user can pre-specify a number of post-processing macros.
One command is to repeat words in a line and append it to the end of a line.
You can see this in Table~\ref{tab:examples} in the final line of the generated output.
One can also repeat a line as a separate line.
One can also append or prepend ``oooh'', ``aaah'', ``yeah'', ``unh'', ``whoo'', ``shamon'', or whatever exclamation is necessary.
\textbf{Karaoke Video Generation.}
What good is this if you can't sing along with instrumental music in the background?
The system is also capable of generating a karaoke video.
Given a sound file of the melody and timing information about the original lyrics, the system creates a video that plays the music and shows each line of the newly generated lyrics at the appropriate time.
See Figure~\ref{fig:karaoke}.
\textbf{Fine-Tuning.}
One can fine-tune GPT-2 and XLNet if one wants to adopt the vocabular of a particular corpus.
The rhyme and syllable constraints will mostly dominate the form and style of the output, but the system will prefer different words.
I generally find that careful choice of a prompt is enough to bias the vocabulary choices.
Algorithm~\ref{algo:main} shows the Weird AI Yankovic main generation loop.
\begin{algorithm}[t]
\footnotesize
\SetKw{KwIn}{in}
\SetKw{KwOr}{or}
\SetKw{KwFalse}{false}
\SetKwFunction{genRhymeLines}{generate\_rhyme\_lines}
\SetKwFunction{genTerminalNonRhymeLines}{generate\_terminal\_non\_rhyme\_lines}
\SetKwFunction{genNonRhymeLines}{generate\_non\_rhyme\_lines}
\SetKwFunction{pickBestCandidate}{pick\_best\_candidate}
\KwData{$prompt$ is a string; $scheme$ is a list of lines where each line is a list of segments where $segment_i=\langle s_j, r_j, e_j\rangle$ for $j=1...n$ such that $s_j$ is the number of syllables in the segment, $r_j$ is rhyme index or string, and $e_j$ is an optional signifier that this segment should end a sentence; and $recontextualize?$ is a boolean.}
\KwResult{A list of strings constituting the lines of the lyrics}
$context\gets prompt$\\
\For{$line$ \KwIn $lines$}{
\If{$recontextualize?={\rm true}$}{
Insert $prompt$ in $context$ after last occurring period\\
}
\For{$segment$ \KwIn $line$}{
$target\_syllables\gets$ number of syllables specified in $segment$\\
$rhyme\_index\gets$ rhyme index specified in $segment$\\
$end?\gets$ true if $segment$ specifies the segment ends in a period\\
\If{$rhyme\_index$ is a string \KwOr $rhyme\_index$ \KwIn $rhyme\_map$}{
$end\_targets\gets$ pick rhyme words or use $rhyme\_index$\\
\For{$target$ \KwIn $end\_targets$}{
$candidates\gets candidates +$\genRhymeLines($target$, $context$, $target\_syllables$, $end?$)\\
}
}
\Else{
\If{$end?={\rm true}$}{
$candidates\gets candidates +$\genTerminalNonRhymeLines($context$, $target\_syllables$)\\
}
\Else{
$candidates\gets candidates +$\genNonRhymeLines($context$, $target\_syllables$)\\
}
}
$best\gets$\pickBestCandidate($candidates$, $context$)\\
$context=context+best$\\
$final\_segments\gets final\_segments + best$\\
}
$final\_lines = final\_lines + final\_segments$\\
}
\caption{The lyric generation loop.}
\label{algo:main}
\end{algorithm}
\section{Examples}
Table~\ref{tab:hamilton} shows a number of additional examples of system output alongside the original lyrics.
If there are exact word matches between the new lyrics and the original lyrics, as in end of {\em Mad World} (third example), it is because I used the input constraints to force word choices.
All examples were first runs with a given set of inputs.
One way to measure the success of a creative AI system is the {\em curation coefficient}, the number of runs necessary before a human feels comfortable sharing a generated output~\cite{Colton2012ComputationalCT}.
That is, how many runs produce content that are not worth sharing with a public audience?
In general I find that I will get a set of lyrics that is amusing and coherent enough to share on Twitter in the first 3-5 runs.
To get a more accurate estimate of the curation coefficient, I ran the system on the same set of inputs as many times as necessary to produce 15 sets of lyrics I would be willing to share on Twitter.com.\footnote{Lyrics were shared publicly at on my Twitter feed: \url{https://twitter.com/mark_riedl/status/1304242039337504768}} \footnote{If you are a researcher from a distant future where Twitter.com doesn't exist, please accept my apologies.}
I configured the system to rewrite the last lines of the song ``Weird Science'' by Oingo Boingo using the prompt
``I've created a monster.''
I required 31 runs, resulting a curation coefficient of $\sim$2.06.
Table~\ref{tab:weird} shows the complete set of runs, with the 15 shared runs in black and those not shared in red.
The input scheme is also given in the bottom right corner.
Since these are three-line runs, I would expect to see a higher curation coefficient for longer lyrics.
The most common reasons for rejection are (a)~linguistic disfluencies (e.g., ``An order to help you out in''), (b)~sheer jibberish (e.g., ``And my character is not
particularly throughout
of your game world''), (c)~nonsense words (e.g., ``and find out fout''), and (d)~anything even remotely racist or sexist.
I acknowledge that this experiment is not very scientific and experimenter bias may be present, which is why I give the full set of runs so one can judge for oneself whether the coefficient should be higher or lower.
We do not provide a more formal evaluation of our system.
However, the system was released publicly and within a day I received an email requesting a ``fart'' mode.
Further research is required to determine if this is an effective metric for evaluating creative language generation systems.
\section{Conclusions}
Weird AI Yankovic is a demonstration that stylistic control in neural text generation can be achieved through sampling and a combination of forward and backward generation.
Whereas a lot of AI research focuses on data and novel neural model encoder and decoder architectures, we show that careful design of the sampling algorithm is an equally important part of practical and effective neural text generation.
Being able to generate while attending to a historical context and also a word at the end of the sequence is useful for controllable text generation because it allows a system to make token-by-token decisions based on constraints regarding the end of the sequence while attending to text that comes before the masks.
One of the reasons why the Weird AI Yankovic system might be considered successful is because creative systems are designed to embrace failure.
Indeed, parody is about expressing an idea with constraint on the style and shape of the language that would not be present in ordinary everyday communication.
The result is often awkward language usage when human do it.
Seeing the awkward solution that succeeds in meeting all the constraints provides a cathartic pleasure release in the audience.
This is accentuated in generative neural language model, which can fail in unexpected ways and non-human ways.
Indeed, one of the appeals of the Weird AI Yankovic, if there are in fact any, is seeing the train wreck of text that is output and deriving delight in the close-but-not-quite-sensical results.
Weird AI Yankovic is a case study in design trade-offs---when to handle failure modes and when to permit failures---where improvements that reduce the likelihood of failed outputs also make the outputs ``safer'' but also less interesting.
\begin{table}[t]
\centering
\scriptsize
\begin{tabular}{p{1.75in}|p{1.75in}|l}
{\bf Original Lyrics} & {\bf Generated Lyrics} & {\bf Scheme}\\
\hline
{\bf {\em My Shot} by Lin Manuel Miranda} & \multicolumn{2}{l}{\bf Prompt: ``I want to start a revolution''} \\
\hline
I am not throwing away my shot &
But no mean I want to start from taught & $(9, 1)$\\
I am not throwing away my shot &
Religious belief espoused throughout & $(9, 1)$\\
Hey yo, I'm just like my country &
Hey yo a society to & $(6, 2)$\\
I'm young, scrappy and hungry &
Blindly pursue. But no sir & $[(4, 2), (3, 2)]$\\
And I'm not throwing away my shot &
Give me a benefit of the doubt. & $(9, 1, {\rm :end})$ \\
I'ma get a scholarship to King's College &
On February & $(11, 3)$\\
I prob'ly shouldn't brag, but dag, I amaze and astonish &
And March we will have to Maria to our honorary & $[(6, 4), (2, 4), (7, 3)]$\\
The problem is I got a lot of brains but no polish &
Canons in every reverie and subsidiary & $(14, 3)$\\
I gotta holler just to be heard &
It is very difficult indeed & $(9, 5)$\\
With every word, I drop knowledge &
Not to concede. Arbitrary & $[(4, 5), (4, 3)]$\\
I'm a diamond in the rough, a shiny piece of coal &
And unreasonable of the just honorable & $[(7, 21), (6, 6)]$\\
Tryin' to reach my goal my power of speech, unimpeachable &
Dishonorable imposition on the honourable. & $[(5, 6), (10, 6, {\rm :end})]$\\
Only nineteen but my mind is older &
It is difficult indeed to do this. & $(10, 7)$\\
These New York City streets get colder, I shoulder &
But yet we have all agreed to chris' will and chris' & $[(9, 7), (3, 7)]$\\
Every burden, every disadvantage &
In our hearts it is only a little more & $(12, 8)$\\
I have learned to manage, I don't have a gun to brandish &
Certain that the coup d'or will not come or that the coup d'or & $[(6, 8), (8, 8)]$\\
I walk these streets famished &
Will come with a guitar & $(6, 8)$\\
The plan is to fan this spark into a flame &
In hand and maintained that this is more certain. & $[(2, 9), (3, 9), (2, 12), (4, 10)]$\\
But damn, it's getting dark, so let me spell out the name &
Sustained that this is chris' way of eliminating & $[(2, 9), (4, 12), (7, 10)]$\\
I am the A-L-E-X-A-N-D-E-R we are meant to be. &
Any chance of us all coming to terms with this. & $(12, 11, {\rm :end})$\\
& & \texttt{rhyme: index $1\rightarrow {\rm shot}$}\\
& & \texttt{post: prepend ``hey yo''}\\
& & \texttt{~~~~~~~on line 3}\\
\hline
{\bf {\em Sound of Silence} by Simon \& Garfunkel} & \multicolumn{2}{l}{\bf Prompt: ``Hello darkness, my old friend.''} \\
\hline
Hello darkness, my old friend &
But Darkness never explained. & $(7, 1)$ \\
I've come to talk with you again &
But once gained, Darkness never gained. & $(8, 1, {\rm :end})$\\
Because a vision softly creeping &
My shadow has vanished from & $(8, 2)$\\
Left its seeds while I was sleeping &
This plane of existence. See em. & $(8, 2)$\\
And the vision that was planted in my brain &
On the horizon and look at Darkness in & $(11, 3)$\\
Still remains &
The between. & $(3, 3, {\rm :end})$\\
Within the sound of silence &
See em. In the silence. & $(6, {\rm silence}, {\rm :end})$\\
\hline
{\bf {\em Mad World} by Tears for Fears} & \multicolumn{2}{l}{\bf Prompt: ``All around me are familiar faces. Mad world.''} \\
\hline
All around me are familiar faces &
Of course I do remember them very & $(10, 1)$\\
Worn out places, worn out faces &
The contrary. Old Jack Kerry. & $[(4, 1), (4, 1, {\rm :end})]$\\
Bright and early for their daily races &
The Henry factory. Wilmer's diary. & $(10, 1)$\\
Going nowhere, going nowhere &
In my mind the image of the. & $[(4, 2), (4, 2, {\rm :end})]$\\
Their tears are filling up their glasses &
Is one of them which my mother & $(9, 3)$\\
No expression, no expression & has
of me and my father will die. & $[(4, 4), (4, 4, {\rm :end})]$\\
Hide my head I want to drown my sorrow &
Are not so different from them but one & $(10, 5)$\\
No tomorrow, no tomorrow &
Hamburger bun. They are the un. & $[(4, 5), (4, 5, {\rm :end})]$\\
& \\
And I find it kind of funny &
Or something like that. Its funny. & $(8, {\rm funny})$ \\
I find it kind of sad &
But very strange. Its sad. & $(6, {\rm sad}, {\rm :end})$\\
The dreams in which I'm dying are the best I've ever had &
Just never know what other people think of me. And Brad. & $(14, 6)$\\
I find it hard to tell you 'cause I find it hard to take &
On another hand its not just what they say but how & $(13, 7)$\\
When people run in circles it's a very, very &
With other people their opinions are those things & $(13, 8)$\\
Mad world, mad world &
Mad world, mad world. & $(4, {\rm Mad~world, mad~world})$\\
& & \texttt{rhyme: index $6\rightarrow {\rm sad}$}\\
\hline
{\bf {\em Can't Touch This} by M.C. Hammer} & \multicolumn{2}{l}{\bf Prompt: ``You cannot touch this.''} \\
\hline
My, my, my my music hits me so hard & And and and and should not touch this again & $[(1, {\rm None}), (6, 1)]$\\
Makes me say, ``Oh my Lord'' & Do not touch to the tongue. & $(3, 1, {\rm :end})$\\
Thank you for blessin' me & The person in question & $(6, 2)$\\
With a mind to rhyme and two hype feet & And no one trying indigestion & $[(3, 3), (2, 3), (4, 2, {\rm :end})]$\\
It feels good, when you know you're down & Do it in anger because they & $(8, 4)$\\
A super dope homeboy from the Oaktown & Should not touch the tongue say it ye and yea. & $(10, 4, {\rm :end})$\\
And I'm known as such & I said unto you & $(5, 5)$\\
And this is a beat, uh, you can't touch & And if anything, uh, you pursue. & $[(5, {\rm None}), (3, 5, {\rm :end})]$\\
& & \texttt{post: repeat 1st word $3\times$},\\
& & \texttt{~~~~~~~insert ``uh'' line 8}\\
\hline
\end{tabular}
\caption{Example first runs with different input constraints and prompts.}
\label{tab:hamilton}
\end{table}
\section{Societal Implications}
As a neural language generation system, our system faces the same potential pitfalls as other contemporary language learning systems.
Namely, it is prone to echoing the biases present in the dataset, including
{\em prejudicial} biases or descriptions of non-normative behavior.
Prejudicial biases are biases that are unwanted because they result in language that demeans or infers hatred or violence toward certain people.
Non-normative behavior~\cite{peng} is that which is not considered outside the norms of a particular group of people.
In the United States this might include descriptions of public sex or murder.
The system as described in it's current manifestation has little practical value and thus presents little chance of harm at scale.
As with any broad capability technology, it can be put to purposes that are benign, malicious, or negligent that have not been envisioned by the author.
One general concern in artificial intelligence is the prospect of automating jobs.
Unlike mundane tasks where there is little opportunity for human improvisational problem solving, ``creative'' AI systems introduce the prospect of more specialized forms of work currently believed to be uniquely considered only possible by humans.
Weird AI Yankovic is not good enough to completely replace human musical artists.
Part of the reason is that GPT-2 and XLNet, while reasonably good at producing fluent language, do not have any particular understanding of what they are generating or how the resultant language will impact a human audience.
Systems such as GPT-3 may make significant gains in fluency but are theoretically limited with respect to its ability to generate text with an intended impact on the reader/audience.
However, the above discussion assumes that human consumers of music---or any type of creative expression---will accept AI-generated creative content, even if it of equal objective quality to human-generated content.
One of the reasons we value creative expression is because of the tacit acknowledgement of the effort that went into the creative expression.
I hypothesize that we will not value computational effort as equivalent to human effort---humans must make tradeoffs on how they spend their finite time and resources in a way whereas computational effort is cheap.
One use that this work may be put to is human-AI co-creation tool, wherein a human works with a computational system in a mixed-initiative interaction.
Although Weird AI Yankovic provides an interactive mode where a user can manually select candidate lines.
However, this does not align with actual needs or creative processes of human artists.
There is a common misconception that any fully autonomous creative system can be simply folded into a mixed-initiative framework to make an AI creativity support.
Kristen Stewart, of the Twighlight movie fame, wrote a paper documenting how difficult it is to use AI creative systems that were not designed for co-creation when trying to achieve a desired aesthetic solution~\cite{Joshi2017BringingIT}.
Weird AI Yankovic in interactive mode has more in common with a party game than a creativity support tool.
Finally, one societal impact of Weird AI Yankovic is that one may accused of copyright infringement.
Despite that fact that parody is protected in the United State of America as fair use, this does not stop those with vested interested in the copyright of an original work of music using tactics to limit parody.
The International Federation of the Phonographic Industry (IFPI), which manages copyright protection for a number of music artists, including Michael Jackon's estate, sent Twitter.com a take-down request targeting one om tweets because it contained a link to a karaoke video generated by the Weird AI Yankovic system.
There is a legal gray zone wherein the lyrics, which are textually overlaid on a black screen (see Figure~\ref{fig:karaoke} for an example) while an original instrumental track plays.
Can one derive value from listening to the instrumental track without reading the parody lyrics and thus deprive a copyright holder from monetary value?
Further complicating the issue, fair use laws are not uniform internationally, meaning that generated lyrics may be protected in some countries but not others.
\begin{table}[t]
\centering
\scriptsize
\begin{tabular}{p{1.6in}|p{1.6in}|p{1.6in}}
\multicolumn{3}{l}{\bf Curated runs}\\
\hline
For my own personal use & And don't forget that i'm just & I'll have it in the hands of\\
Only to reintroduce & Constantly chasing the guste. & My eternally belove\\
That beast to the ooh, weird science & To prove my point ooh, weird science & Waswolf boyfriend ooh, weird science\\
\hline
Have to make him do something & For those of you who don't know & When he went missing last week\\
Bad like this twentysomething & Exactly what molding dough & He was surrounded by chic\\
A couple of ooh, weird science & Of that molder ooh, weird science & Was going to ooh, weird science\\
\hline
Are you really that big with & I will kill it and live on & And i know how to control.\\
Other people messing with & A hundred million yuan. & And he started with rock'n'roll.\\
Just a little ooh, weird science & In the meantime ooh, weird science & I want you to ooh, weird science\\
\hline
And this monster will only & The first one is called the black & And now he'll come after us\\
Ever be sad and lonely. & Cat but he is a chirac. & All and then we never guess\\
You must defeat ooh, weird science & It is called the ooh, weird science & And the next time ooh, weird science\\
\hline
To make her feel good about & And now we are ready to & Think about all those guys who\\
Having the thing she had sought. & Step onboard the ship that threw & Never get seasonal flu\\
Would love to see ooh, weird science & Canisters out ooh, weird science & Had so many ooh, weird science\\
\hline
\multicolumn{3}{l}{\bf Rejected runs}\\
\hline
\textcolor{red}{And we are all living in} & \textcolor{red}{And i have created two.} & \textcolor{red}{And here we go again with}\\
\textcolor{red}{The world we are living in.} & \textcolor{red}{And those two have not been threw} & \textcolor{red}{Without the monster and with}\\
\textcolor{red}{So if you're like ooh, weird science} & \textcolor{red}{And will not throw ooh, weird science} & \textcolor{red}{Would be nothing ooh, weird science}\\
\hline
\textcolor{red}{You i will be a giant.} & \textcolor{red}{Of my own creation by} & \textcolor{red}{And i don't really care if}\\
\textcolor{red}{You that is what i wiant.} & \textcolor{red}{Simply using the cute shy} & \textcolor{red}{This might be written by ziff.}\\
\textcolor{red}{And it will be ooh, weird science} & \textcolor{red}{And the little ooh, weird science} & \textcolor{red}{Has also done ooh, weird science}\\
\hline
\textcolor{red}{Is a simple program which} & \textcolor{red}{Have created some monster} & \textcolor{red}{That you can use on your own}\\
\textcolor{red}{Can helps you create a fritch} & \textcolor{red}{More like thesanta santa} & \textcolor{red}{Side and also in a sloan.}\\
\textcolor{red}{Of different ooh, weird science} & \textcolor{red}{Are more like that ooh, weird science} & \textcolor{red}{Would also have ooh, weird science}\\
\hline
\textcolor{red}{And my character is not} & \textcolor{red}{And we are going to do} & \textcolor{red}{Have you ever seen that one}\\
\textcolor{red}{Particularly throughout} & \textcolor{red}{About with what they pursue.} & \textcolor{red}{Little black terrebonne}\\
\textcolor{red}{Of your game world ooh, weird science} & \textcolor{red}{Also it will ooh, weird science} & \textcolor{red}{Would grow into ooh, weird science}\\
\hline
\textcolor{red}{I'm here to help you out in} & \textcolor{red}{And i'm gonna try it out} & \textcolor{red}{If people are afraid that}\\
\textcolor{red}{An order to help out in} & \textcolor{red}{There to see and find out fout} & \textcolor{red}{The inmarsat inmarsat}\\
\textcolor{red}{And out of this ooh, weird science} & \textcolor{red}{For myself if ooh, weird science} & \textcolor{red}{Have something to ooh, weird science}\\
\hline
\textcolor{red}{But let me tell you something} & \multicolumn{2}{c}{$(7, 1)$}\\
\textcolor{red}{About this twentysomething} & \multicolumn{2}{c}{$(7, 1)$}\\
\textcolor{red}{That came out of ooh, weird science} & \multicolumn{2}{c}{$[(4, {\rm None}), (4,$ ``ooh, weird~science''$)]$}\\
\hline
\end{tabular}
\caption{Fifteen curated runs of Weird AI Yankovic generating the final lines of ``Weird Science'' by Oingo Boingo using the scheme in the lower right corner.
}
\label{tab:weird}
\end{table}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 31 |
Q: VirtualBox connect USB device to guest I have Virtualbox 4.1.18 installed in a laptop with windows seven pro 64 bits.
As many others, I have been strugglying for hours..days sharing an usb device (alfa card) with the guest machine (BT5). The virtualbox guest additions are updated to 4.1.18 as well in the guest machine. I followed these steps but still the usb is not recognized in the guest (lsusb, lspci). Other usb devices (like a mouse or a webcam) are recognized despite they are not filtered. So I also tried not to filter the usb NIC, but still the same. So I'm back using the VB filter (with USB2.0 extension).
Using the virtualbox GUI to connect the usb device, I get the famous error message :
USB device 'Manufacturer_Realtek_RTL8187_ RTL8187_Wireless' with UUID {xxxx} is busy with a previous request. Please try again later."
Could this issue be related to the host driver for this usb device ? Or what else would I miss ?
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Q: AngularJS: Removing the parent element I have li item that repeat, according to $scope.items list. In each li I have a checkbox. What I want to do is to catch the change event of this checkbox and do my work there. Executed in $scope.change function.
When my work done, I want to remove the row of the checked checkbox.
My code so far:
<!doctype html>
<html>
<head>
<script src="http://code.angularjs.org/1.2.3/angular.min.js"></script>
<script>
var app = angular.module('myapp', []);
app.controller('mainController', function($scope) {
$scope.items = ["item1", "item2", "item3"];
$scope.change = function() {
// My work here.
// ...
// Work is done. remove the caller checkbox.
this.parent.remove(); // <--- BOOM.
}
});
</script>
</head>
<body ng-app="myapp">
<div ng-controller="mainController">
<ul>
<li ng-repeat="item in items">
<input type="checkbox" ng-model="checked" ng-change="change()">
</li>
</ul>
</div>
</body>
</html>
Live version is here: http://plnkr.co/edit/05IBKp6aVoj1SqFNi5JJ
My problem is in this code-line:
this.parent.remove(); // <--- BOOM.
My target is to remove the parent li.
Questions:
*
*How this can be done right?
*When I using the this keyword (in controller.change function), is this something that I can use with JQuery syntax? Something like $(this).parent().remove();?
A: You can delete the item from $scope.items and it will be automatically removed and you don't need to use jQuery.
I updated the plunker http://plnkr.co/edit/3aYHQspeLAj3VryQAhBT?p=preview
<ul>
<li ng-repeat="item in items">
<input type="checkbox" ng-model="checked" ng-change="change(item)">
</li>
and in JS
$scope.change = function(item) {
// Work is done. remove the caller checkbox.
var index = $scope.items.indexOf(item);
$scope.items.splice(index, 1);
}
A: Please have a look at this plunker, I have used ng-click to detect the change and I have passed the $event as the parameter.
<!doctype html>
<html>
<head>
<script src="http://code.angularjs.org/1.2.3/angular.min.js"></script>
<script src="http://code.jquery.com/jquery-2.2.0.min.js"></script>
<script>
var app = angular.module('myapp', []);
app.controller('mainController', function($scope) {
$scope.items = ["item1", "item2", "item3"];
$scope.change = function(e) {
// My work here.
// ...
console.log($(this));
console.log(e.target);
// Work is done. remove the caller checkbox.
$(e.target).parent().remove(); // Not working
}
});
</script>
</head>
<body ng-app="myapp">
<div ng-controller="mainController">
<ul>
<li ng-repeat="item in items">
<input type="checkbox" ng-model="checked" ng-click="change($event)">
</li>
</ul>
</div>
</body>
</html>
Remove li by passing the $event.
A: Please do following change.
HTML:
<ul> <li ng-repeat="item in items"> <input type="checkbox" ng-model="checked" ng-change="change($index)"> </li>
JS.
$scope.change = function(index) {
$scope.items.splice(index, 1);
}
Que.2
You can do this using jqLite and directive.
A: Changed:
html:
<ul>
<li ng-repeat="item in items">
<input type="checkbox" ng-model="checked" ng-click="change($event)">
</li>
</ul>
js:
$scope.change = function (e) {
angular.element(e.target).parent().remove();
};
A: I am assuming you are trying to implement a simple checklist functionality.
You could pass the $index for onChange function and then use array.splice() method to perform the following action.
<!doctype html>
<html>
<head>
<script src="http://code.angularjs.org/1.2.3/angular.min.js"></script>
<script>
var app = angular.module('myapp', []);
app.controller('mainController', function($scope,$timeout) {
$scope.items = ["item1", "item2", "item3"];
$scope.prevItem ={
value: null,
index: null
}
$scope.change = function(index) {
$timeout(function(){
// Your work here.
$scope.prevItem.value = $scope.items[index];
$scope.prevItem.index = index;
$scope.items.splice(index, 1);
}, 300);
}
$scope.undoCheck = function(){
$scope.items.splice($scope.prevItem.index, 0, $scope.prevItem.value);
}
});
</script>
</head>
<body ng-app="myapp">
<div ng-controller="mainController">
<ul>
<li ng-repeat="item in items">
<input type="checkbox" ng-model="checked" ng-change="change($index)">
{{item}}
</li>
</ul>
<button ng-click="undoCheck()">Undo Previous Check</button>
</div>
</body>
</html>
I have also added undo functionality to just help you get more clarity on splice function.
The timeout function is added to just show the check on the checkbox before removing it.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 107 |
Tensiometer bezeichnet Messgeräte für:
die Feuchtigkeit im Boden, siehe Tensiometer (Bodenfeuchte)
die Oberflächenspannung von Flüssigkeiten, siehe Tensiometer (Oberflächenspannung)
die Spannung einzelner Speichen an Speichenlaufrädern, siehe Tensiometer (Speichenspannung) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,053 |
\section{Introduction}
Statistical evidence against a hypothesis often relies on the asymptotic normality of a test statistic, as in the case of the commonly used Wald or score tests.
Many authors ignore the asymptotic nature of the argument and assume that in finite samples the distribution of the test statistic is indeed normal.
This perfunctory approach generates misleading beliefs about the $p$-value distribution, such as i) the distribution of the $p$-values under the null is exactly uniform or that ii) the cumulative distribution function (henceforth, cdf) of the $p$-value under the alternative is concave.
However, there are important exceptions from these rules, e.g. discrete tests are not normally distributed in any finite sample settings, so that the distribution of the $p$-values under the null is certainly not uniform.
Similarly, it is not obvious that the cdf is concave under the alternative as we will illustrate with some examples.
Testing procedures aimed at controlling the family-wise error rate (FWER) or the false discovery rate (FDR, see \cite{BH}) typically assume that i) or ii) holds.
In \cite{Cao2013}, the authors examine the optimality of FDR control procedures when i) or ii) are violated and provide alternative conditions to maintain said optimality.
Clearly, a more precise characterization of the $p$-value distribution that accounts for the approximation error is pivotal in controlling the occurrence of false discoveries.
Complicating matters even further, the issue of calibrating the location and variance of the test statistic is often overlooked, particularly under the alternative.
Under the alternative the test statistic can be improperly re-scaled since often the variance of the test statistic is obtained under the null.
While under the null, the test statistic may not have zero mean and may also not be correctly standardized, thus making the standard Gaussian approximation suspect.
The problem of biases in the variance and expectation is aggravated in the presence of a large number of nuisance parameters.
For instance, while it has been demonstrated in \cite{DiCicio} that the profile score statistic has a location and variance bias under the null, in Section \ref{subsec:score} we show that the variance bias can persist under the alternative.
These concerns motivate us to perform a systematic study of the $p$-value distribution in the presence of information or location biases under the null and alternative, while accounting for the approximation error resulting from the use of asymptotic arguments.
We explore how certain asymptotically non-vanishing and vanishing biases in the variance and location of the test statistic can occur in finite samples, violating the assumptions generally placed on the null and alternative distributions of the $p$-values.
We study both continuous and discrete distributions supported on lattices.
In doing so we include all approximation errors, including those induced by discreteness, to fully characterize the behaviour of the distributions of $p$-values under the null and alternative.
This work extends the results of \cite{hung}, who studied the distribution of $p$-values under alternative assuming the test statistic is normally distributed, to a broader framework.
We focus on univariate test statistics for a one dimensional parameter of interest based on sums of independent random variables, possibly in the presence of a large number of nuisance parameters.
These types of test statistics are commonly used to infer the significance of individual coefficients in most regression models.
The results of the paper are in the same vein as those found in \cite{hall2013bootstrap} and \cite[\S~3]{kolassa1994series}, whose objective was the coverage properties of confidence intervals.
We expand their results to the $p$-values, motivated by the multitude of scientific investigations that rely on the $p$-value distribution rather than confidence intervals.
We begin with a simple example illustrating how the standard assumptions on the null and alternative distributions of the $p$-values can be violated in practice.
\begin{example}
We wish to test the null hypothesis $H_0: \beta = 0.01 $ against the alternative $H_1: \beta = 0.01/1.05$, where $\beta$ is the rate parameter of a gamma distribution, based on 750 observations $x_1, \cdots, x_{750}$, assuming that the shape parameter is known to be $\alpha = 0.01$. From the central limit theorem, we know that the test statistic
\begin{align*}
S_n = \sqrt{n} \left(\frac{\bar{X} - 1}{\sigma} \right) \rightarrow N(0, 1),
\end{align*}
so we are able to obtain a two-sided $p$-value based on the standard normal distribution.
We plot the histograms of the $p$-values obtained under the null and alternative in Figure \ref{fig:test}.
The plots are obtained by simulation using 100,000 replications.
We see on Figure \ref{fig:test} that the distribution of the $p$-values obtained from the simulations does not adhere to its expected behaviour under the null or the alternative.
The upper left plot in Figure \ref{fig:test} shows a marked departure from the $U(0,1)$ distribution expected from the null. Thus, a typical rejection rule which assumes uniformity of the $p$-value distribution under the null will not provide type I error control for certain choices of $\alpha$.
For example, if we desire a $10^{-4}$ significance level, we obtain a type I error approximately equal to $1.579*10^{-3}$, which is fifteen times higher than the nominal level.
Under a local alternative, the upper right plot in Figure 1 shows that the $p$-value distribution may not be stochastically smaller than a $U(0,1)$.
The resulting lack of concavity of the distribution $p$-value under the alternative can violate the typical assumption that the false negative rate is strictly decrease and the FDR is increasing in the nominal control level $\alpha$ in the multiple testing setting; see \cite{Cao2013}.
Note that the cause for this poor calibration is not the low sample size.
\end{example}
\begin{figure}[ht]
\centering
\subfloat{\includegraphics[width = 2in]{null_1_1.png}}
\subfloat{\includegraphics[width = 2in]{alt_1_105.png}}\\
\subfloat{\includegraphics[width = 2in]{gamma_saddle}}
\subfloat{\includegraphics[width = 2in]{gamma_saddle_alt}}
\caption{Distribution of $p$-values under $H_0$ and $H_1$ in Example 1. {\it Upper left}: $p$-values obtained under $H_0$ from the normal approximation. {\it Upper right}: $p$-values obtained under $H_1$ from the normal approximation. {\it Lower left and right}: corrected $p$-values for the null and alternative, respectively, using the saddlepoint approximation that is introduced in Section 3. The number of samples is 750. The upper left panel clearly does not exhibit uniformity and upper right panel's distribution does not appear to have a concave cdf. We plot the theoretical prediction from Theorem 1 in blue for the upper left and upper right panels.}
\label{fig:test}
\end{figure}
To assure the reader that the above example is not a singular aberration, we present in Figure \ref{fig:lung} the histogram of over 13 million $p$-values from the genome-wide association study of lung cancer generated from the UK Biobank data \citep{biobank}. These $p$-values are produced by the Neale Lab \citep{neale}, based off 45,637 participants and 13,791,467 SNPs; SNPs with minor allele frequency less than 0.1\% and INFO scores less than 0.8 were excluded from the analysis.
We note that the histogram exhibits a similar behaviour to the one seen in Example 1, i.e., the distribution of $p$-values exhibits a secondary mode that is far from zero.
\begin{figure}[ht]
\centering
\includegraphics[width=8cm, height = 5cm]{lung_cancer}
\caption{Empirical p-value distribution based on a genome wide association study ($n=45,637$) of lung cancer.
}
\label{fig:lung}
\end{figure}
Figure \ref{shapes} briefly summarizes the shapes that the density of the $p$-value distribution might take for two-tailed tests, based on the results in Theorems 1 and 2 that are introduced in Section 2. The descriptions in Table 1
verbalise the various mathematical conditions that can lead to the four shapes in Figure \ref{shapes}.
In practice it is possible to have combinations of the shapes listed in Figure \ref{shapes}, as the observed test statistics may not be identically distributed and can be drawn from a mixture of the null and alternatives hypotheses.
\begin{figure}[H]
\centering
\subfloat[Shape 1]{\includegraphics[width = 2in]{shape_1.png}}
\subfloat[Shape 2]{\includegraphics[width = 2in]{shape_2.png}}\\
\subfloat[Shape 3]{\includegraphics[width = 2in]{shape_3.png}}
\subfloat[Shape 4]{\includegraphics[width = 2in]{shape_4.png}} \caption{General chart of the behaviour of $p$-values under the null and alternative for a two tailed test. Shapes 1 to 3 were obtained from simulating from the null and alternative from Example 1 with different parameters $\alpha$ and $\beta$. Shape 4 was obtained when using a misspecified variance, as detailed in Section 2.2. }\label{shapes}
\end{figure}
\begin{table}[h]
\caption{Description of the test statistic's characteristics and the resulting shapes (as shown in Figure \ref{shapes}) of the $p$-value distribution under the null or alternative hypothesis.}
\fbox
\begin{tabular}{ | m{3em} | m{6cm}| m{6cm} |}
\hline
Shape & Null & Alternative \\
\hline
1 & The typical uniform shape. & Possible if effect size is small. \\
\hline
2 & Possible if variance is misspecified, underestimated. & Typical behaviour. \\
\hline
3 & Possible if test statistic has large higher order cumulants, see Example 1. & Possible if effect size is small and the higher order cumulants are large, see Example 1. \\
\hline
4 & Possible if variance is misspecified, overestimated, see Example 3. & Possible if variance is misspecified, overestimated and the effect size is small, see Example 3. \\
\hline
\end{tabular}}
\end{table}
Section 2 contains the main theoretical results of this paper, Theorems 1 and 2, which characterize the distribution of $p$-values under the null and alternative.
Section 2.1 examines the $p$-value distribution resulting from the score test, while Section 2.2 studies specific examples.
Section 3 provides numerical results and considers some remedies aimed at calibrating the $p$-value distribution.
Section 4 closes the paper with a discussion of the implications of our results and some recommendations to practitioners.
\section{Distribution of $p$-values under Non-Normality}
All theoretical details and proofs, as well as a brief introduction of the concepts needed for the proof of Theorems 1 and 2, are deferred to the Supplementary Materials.
We consider the case where the test statistic, $S_n$, can be discrete and may also have a non-zero mean under the null and a non-unit variance under the null or alternative.
We assume that $\psi$ is a one dimensional parameter of interest and $\lambda$ is a vector of nuisance parameters.
Without loss of generality, let the statistic $S_n$ either be used to test the null hypothesis $H_0: \psi = \psi_0$ for a two-sided test or $H_0: \psi \geq \psi_0 $ for a one-sided test.
All results are given in terms of the cdf.
Theorems 1 and 2 deal with the case where $S_n$'s distribution is continuous and discrete respectively.
We first consider the case where the statistic $S_n$ admits a density.
We typically assume that $S_n$ has been appropriately calibrated such that $\mathrm{E}[S_n]= 0$ under the null, and $\mathrm{E}[S_n] \neq 0$ under the alternative hypothesis.
We let $p(S_n)$ denote the $p$-value obtained from the test statistic $S_n$.
However, as discussed in the introduction, the mean of $S_n$ may not be exactly $0$ under the null due to a location bias.
We also would expect that the variance of the test statistic should be 1 under the null and alternative, which may not be the case for all test statistics however; see Example 3.
The location bias complicates the precise determination of whether $S_n$'s distribution should be considered under the null or the alternative.
However, note that Theorem 1 statement is applicable under both the null and alternative, since its conclusion depends only on the expectation, variance, and the other cumulants of the statistic $S_n$, regardless of the true hypothesis.
We first introduce some notations:
\begin{itemize}
\item[(i)] $Z_p$ is the $p$-th quantile of a standard normal distribution.
\item[(ii)] $\rho_{n,i}$ is the $i$-th order standardized cumulant of $S_n$, and $\rho_n$ is a vector containing all cumulants.
\item[(iii)] $\phi(x)$ is the standard normal density.
\end{itemize}
\begin{theorem}\label{th:cont_approx}
Let $X_1,\ldots, X_n$ be a sequence of continuous, independent random variables. Set $S_n= \sqrt{n} (\bar{X}_n - a_n)/b_n$ where $\bar{X}_n = n^{-1}\sum_{i = 1}^n X_i$, and let $\{a_n\}_{n\ge 0}$, $\{b_n\}_{n\ge 0}$ be two sequences of real numbers. Let $\mathrm{E}[S_n]= \mu_n$, ${\mathrm{Var}}(S_n) = v_n^2$, and $\rho_n$ denote the cumulants of $(S_n - \mu_n)/v_n$. Then the CDF of the one-sided $p$-value is
\begin{align}
\mathbb{P} (p(S_n) < t) = \Phi\left(\frac{Z_t - \mu_n}{v_n}\right) -E_2\left(\frac{Z_t - \mu_n}{v_n}, \rho_{n} \right) + O\left(n^{-3/2}\right),
\end{align}
and the CDF of the two-sided $p$-value is:
\begin{align}
\mathbb{P} (p(S_n) < t) &= 1 + \Phi\left(\frac{Z_{t/2} - \mu_n}{v_n} \right) - \Phi\left(\frac{-Z_{t/2} - \mu_n}{v_n}\right)\nonumber \\
&+E_2\left(\frac{Z_{t/2} - \mu_n}{v_n} , \rho_n\right) - E_2\left( \frac{-Z_{t/2} - \mu_n}{v_n}, \rho_n\right) + O\left(n^{-3/2} \right), \label{eq:cont_one}
\end{align}
where,
\begin{align}
E_2(t, \rho_n) = -\phi(t)\Big\lbrace \frac{\rho_{n,3} H_2(t)}{6} + \frac{\rho_{n,4} H_3(t)}{24} + \frac{(\rho_{n,3} )^2 H_5(t)}{72} \Big\rbrace, \label{eq:cont_two}
\end{align}
and $H_j(t)$ denotes the j-th Hermite polynomial.
\end{theorem}
\begin{remark}
The $j$-th order Hermite polynomial is a polynomial of $j$-th degree defined through the differentiation of a standard normal density. A table of the Hermite polynomials is given in the Supplementary Materials.
\end{remark}
\begin{remark}
Should an approximation to the probability density of the $p$-value distribution be desired, it can be obtained from differentiating Equations (\ref{eq:cont_one}) and (\ref{eq:cont_two}).
\end{remark}
In general $E_2 = O(1/n^{1/2})$, however in the
the case that $\mu_n = 0$, $E_2(Z_{t/2}/ v_n , \rho_n) - E_2( -Z_{t/2}/v_n, \rho_n) = O(1/n)$ for two-sided tests due to cancellations which occur in the difference of the odd Hermite polynomials.
We refer to terms in $E_2(t, \rho_n)$ as the higher order terms.
Therefore, supposing $\mu_n = 0$ under the null, meaning the sequence $a_n = \mathbb{E}[\bar{X}_n]$, we obtain the following corollary:
\begin{corollary}
Assume the setting and notation from Theorem \ref{th:cont_approx} and suppose that under the null we have $\mathrm{E}[S_n]= 0$, and $\mathrm{Var}(S_n) = 1$. The CDF of the distribution of the $p$-values for a one-sided test under the null is
\begin{align}
\mathbb{P} \left(p(S_n) < t \right) = t + O\left( n^{-1/2}\right),
\end{align}
and the CDF of the distribution of the $p$-values for two-sided test under the null is
\begin{align}
\mathbb{P} \left(p(S_n) < t \right) &= t + O\left(n^{-1} \right).
\end{align}
\end{corollary}
Corollary 1 shows that the two-sided test is preferable unless there is a scientific motivation for using the one-sided test.
The case when $S_n$ has a discrete distribution supported on a lattice is covered in Theorem \ref{th-discrete}.
\begin{theorem}
\label{th-discrete}
Let $X_1, \cdots, X_n $ be a sequence of independent discrete random variables where $X_i$ has mean $m_i$. Suppose that $X_i - m_i$ is supported on a lattice of the form $c + j\cdot d$, for $j \in \mathbb{Z}$ and for all $1\le i \le n$. Assume $d$ is the largest number for which this property holds.
Set $S_n= \sqrt{n} (\bar{X}_n - a_n)/b_n$, where $\bar{X}_n = n^{-1}\sum_{i = 1}^n X_i$, $\mathrm{E}[S_n]= \mu_n$, $\mathrm{Var}(S_n) = v_n^2$, $\rho_n$ as the cumulants of $(S_n - \mu_n)/v_n$ and $d_n = d \: v_n /(\sqrt{n}b_n) $.
Then the CDF of the one-sided $p$-value is
\begin{align*}
\mathbb{P} (p(S_n) < t) &= \Phi\left( \frac{Z_t - \mu_n}{v_n} \right) +E_2 \left( \frac{ Z_t - \mu_n}{v_n}, \rho_n \right)
+ C_2\left(\frac{ Z_t - \mu_n}{v_n}, \rho_n \right) + O\left(n^{-3/2}\right),
\end{align*}
and the CDF of the two-sided $p$-value is
\begin{align*}
\mathbb{P} (p(S_n) < t) &=1 + \Phi\left( \frac{Z_{t/2} - \mu_n}{v_n} \right) - \Phi\left(\frac{-Z_{t/2} - \mu_n}{v_n} \right)
+E_2\left(\frac{Z_{t/2} - \mu_n}{v_n}, \rho_n \right) \\
&- E_2 \left(\frac{-Z_{t/2} - \mu_n}{v_n}, \rho_n \right) +C_2 \left(\frac{Z_{t/2} - \mu_n}{v_n}, \rho_n \right)
- C_2\left(\frac{-Z_{t/2} - \mu_n}{v_n}, \rho_n \right) +O \left(n^{-3/2} \right),
\end{align*}
where,
\begin{align*}
C_2(t, \rho_n) = - d_n Q_1\Big( \frac{ t - \sqrt{n}c}{d_n}\Big) \Big( 1 + \frac{\rho_{n,3} H_3(t)}{6} \Big) + \frac{ d_n^2}{2} Q_2\Big( \frac{ t - \sqrt{n}c}{d_n}\Big),
\end{align*}
and $Q_j(t)$ are periodic polynomials with a period of $1$. On $[0,1)$, they are defined by
\begin{align*}
Q_1(t) = t - \frac{1}{2}, \quad Q_2(t) = t^2 - 2t +\frac{1}{6},
\end{align*}
and $E_2(t, \rho_n)$ and $H_j(t)$ are defined as in Theorem 1.
\end{theorem}
\begin{corollary}
Assume the setting and notation from Theorem \ref{th-discrete} and suppose that under the null $\mathrm{E}[S_n]= 0$, and $Var(S_n) = 1$. Then the $p$-values obtained from one or two-sided tests satisfy
\begin{align}
\mathbb{P} (p(S_n) < t) = t + O\left(n^{-1/2}\right).
\end{align}
\end{corollary}
Note that the convergence is slower by a factor of $n^{-1/2}$ compared to the continuous case for a two-sided test. This is due to the jumps in the CDF which are of order $O(n^{-1/2})$.
\begin{remark}
Under the alternative, the $p$-value distribution depends on the effect size, $\mu_n$, as well as the magnitude of the higher order cumulants.
However, for large values of $\mu_n$ the impact of the higher order terms will be negligible, as $E_2$ is a product of an exponential function and a polynomial function which decays to 0 asymptotically in $\mu_n$. We explore this further in Example 4.
\end{remark}
\begin{remark}
When performing multiple hypothesis testing corrections, the $p$-values of interest are often extremely small.
Therefore from Corollary 1 and 2, we see that a large amount of samples is needed to guarantee the level of accuracy required since the approximation error is additive.
\end{remark}
\subsection{An Application of the Main Theorems: The Score Test} \label{subsec:score}
We examine the broadly used score test statistic, also known as the Rao statistic.
The popularity of the score statistic is due to its computational efficiency and ease of implementation.
In the presence of nuisance parameters, the score statistic is defined through the profile likelihood.
Suppose that the observations $y_i$'s are independent then
\[ l_\text{pro}(\psi) = \sup_{\lambda} l(\psi, \lambda; Y) = l(\psi, \hat\lambda_\psi; Y), \]
where $\hat\lambda_\psi$ denotes the constrained maximum likelihood estimator. The score statistic is defined as:
\[ S_n(\psi_0) = \frac{l_{\text{pro}}^\prime(\psi_0)}{ \lbrace l^{\prime\prime}_{\text{pro} } (\psi_0) \rbrace^{1/2}}= \sum_{i = 1}^n \frac{ \frac{d}{d\psi} l(\psi, \hat\lambda_\psi; y_i)}{ \left\lbrace\sum_{i = 1} \frac{d^2}{d\psi^2} l(\psi, \hat\lambda_\psi; y_i)\right\rbrace^{1/2}} \xrightarrow{D} N(0,1), \]
under the usual regularity assumptions.
Due to the form of $S_n$, we may apply Theorem 1 or 2.
The presence of nuisance parameters induces a bias in the mean and variance of the score statistic; see \cite{profile_bias} and \cite{DiCicio}.
Thus, it is not the case that the mean of the score statistic is 0 and the variance is 1 under the null, as the profile likelihood does not behave like a genuine likelihood and does not satisfy the Bartlett identities.
In general this problem is compounded if the number of nuisance parameters is increased, as we illustrate below.
We only discuss the location bias, since the formulas for the information or variance bias are much more involved and compromise the simplicity of the arguments.
From \cite{profile_bias}, the bias of the profile score under the null is:
\begin{align}
\mathbb{E}\lbrace l^\prime_{\text{pro}}(\psi_0) \rbrace &= \alpha_n + O\left( n^{-1} \right),
\end{align}
where the term $\alpha_n = O(1)$. The form of $\alpha_n$ is given in the Supplementary Materials.
To estimate the effect of the dimension of the nuisance parameter on the size of the bias, we use a similar argument as \cite{laplace}, in which they count the number of nested summations that depends on $k$, the number of parameters in the model, to estimate the rate of growth of a function in $k$.
From the expression of $\alpha_n$ given in the Supplementary Materials, we obtain at most 4 nested summations which depends on $k$ therefore the bias of the profile score is of order $O(k^4)$ in the worst case scenario.
The rather large location bias can be impactful as it may induce a perceived significance when $k$ is large, an example using Weibull regression is given in Section 3.3.
A similar argument can be applied to the information bias; see \cite{DiCicio} for a comprehensive discussion on the form of these biases.
The information bias for the score statistic can also be highly influential under the alternative.
In that case, the expected value of the score statistic is non-zero, which is desirable, but the variance of the statistic $S_n$ can be either over- or underestimated.
Since the true parameter value is not $\psi_0$, there is no guarantee that ${l_{\text{pro}}^{\prime\prime}(\psi_0)}$ gives the correct standardization.
If the estimated variance is larger than the true variance of the score, then it is possible to obtain Shape 4 in Figure \ref{shapes}, which violates the concavity assumption for $p$-value distribution's CDF under the alternative.
Further, if we assume that under the null the $p$-value distribution is uniform, then this also violates the monotonicity assumption required by \cite{Cao2013} for the optimality of FDR control.
Example \ref{ex:score_glm} illustrates this phenomenon using the score test in a generalised linear model.
\begin{example} \label{ex:score_glm}
Assume the following regression model based on the linear exponential family where the density of the observations $y_1, \cdots, y_n$ are independent and follows
\[ h(y_i| \beta, X_i) = \exp\lbrace a(X_i\beta) y_i+ b(X_i\beta) + D(y_i) \rbrace, \]
where $X_i$ is a vector of covariates associated with each $y_i$, and $\beta = (\beta_0, \beta_1, \cdots, \beta_k)$ is a vector of regression coefficients.
Let $f(X\beta) = E[y|X]$ denote the mean function.
\cite{score_reg} studied the score statistic for testing the global null $\beta_1 = \beta_2= \cdots = \beta_k = 0$ and linked the resulting statistic to linear regression.
A similar analysis can be performed for different hypothesis, such as inference for a parameter of interest in the presence of nuisance parameters to produce a more general result whose derivation is consigned to the Supplementary Materials. The resulting score statistic takes the form
\[S_n = \lbrace y - f(X\hat\beta_{\text{null}}) \rbrace^\top W X \left\lbrace X^\top D X \right\rbrace^{-1} X^\top W \lbrace y - f(X\hat\beta_{\text{null}} ) \rbrace \xrightarrow[]{D} \chi^2_{q}, \]
where $f(X\hat\beta_{\text{null}})$ is a vector whose $i$-th entry is $f(X_i\hat\beta_{\text{null}})$, $q$ is the number of constraints in the null hypothesis, and $\hat\beta_{\text{null}}$ denotes the constrained maximum likelihood estimate under the null. W and D are square diagonal matrices of dimension $n$ whose entries are $[W]_{ii} = a^\prime(X_i\hat\beta_{\text{null}})$ and $[D]_{ii} = a^\prime(X_i\hat\beta_{\text{null}}) f^\prime(X_i\hat\beta_{\text{null}}) $ for $i = 1, \dots, n$. Using a suitable change of variable, the statistic $S_n$ can be related to weighted linear regression.
In the common case where we wish to test for $\beta_j = 0$, the score statistic can be re-written in the form:
\[S_n = [ (X^\top D X)^{-1} ]_{jj}^{1/2} \sum_{i = 1}^n a^\prime(X_i \hat\beta_{\text{null}}) x_{ij} \lbrace y_i - f(X_i \hat\beta_{\text{null}}) \rbrace \xrightarrow{D} N(0,1), \]
under the null. Under the alternative we may write
\[S_n = (1 + c_n )\tilde{S}_n + d_n, \]
where $\tilde{S}_n$ converges in distribution to a standard normal. The scaling factor $c_n = O(1)$ is an information bias and $d_n$ plays the role of the effect size and will increase to infinity as the number of samples increases.
However, if $\beta_j \approx 0 $ then $d_n$ can be quite small, meaning that the effect of the scaling factor $c_n$ can be consequential.
For an example of this see Example 3, where the effect size is not large enough to offset the scaling factor.
\end{example}
\begin{remark}
Although the likelihood ratio statistic can be written as a summation of independent random variables, the limiting distribution of the likelihood ratio test is a gamma random variable, therefore Theorem 1 or 2 are not directly applicable.
It may be possible to modify the baseline density used in the Edgeworth expansions to obtain a result based on Laguerre polynomials.
This can also be useful when examining the asymptotic behaviour of test statistics for testing vector parameters of interest, as these test statistics often have a gamma distributed limiting distribution.
\end{remark}
\begin{remark}
The bias issue discussed within this section is also present for the Wald test statistics, even if it can not be represented by a summation of independent random variables.
It is rarely the case that the maximum likelihood estimate is unbiased, and the same applies for the estimate of the variance of the maximum likelihood estimate.
Generally the problem worsens as the number of nuisance parameters increases.
\end{remark}
\subsection{Numerical Examples of Application of the Main Theorems}
We illustrate the results of the main results with some numerical examples to demonstrate how various problems in the distributions of the $p$-value can occur.
We first examine a discrete case where the statistic $S_n$ does not admit a density.
We note that when the $E_2$ term is negligible, our results on the distribution of the $p$-values coincide with those obtained by \cite{hung} when the exact normality of the test statistic holds.
On the contrary, when the additional terms are not negligible or the variance is incorrectly specified,
the behaviour of the distribution of $p$-values can be quite different.
The exact size of the difference depends on the behaviour of the Hermite polynomials, the higher order cumulants and the variance.
We consider the following examples in order to illustrate some of the ramifications.
\begin{example}\label{ex:linkage}
Consider a simple linkage analysis of sibling pairs who share the same trait of interest, a common problem in statistical genetics.
The underlying principle is that genes that
are responsible for the trait are expected to be over-shared between relatives, while the null hypothesis states that the trait similarity does not impact allele sharing, i.e. independence between the trait and gene.
The problematic distribution of $p$-values in this example is caused by the discrete nature of the problem along with a misspecified variance under the alternative.
Since the offsprings are from the same parents, under the null we would expect the number of shared alleles to be either 0, 1 or 2 with probability $\theta_{null} = (p_0, p_1, p_2) = (0.25,0.5,0.25)$ based on Mendel's first law of segregation.
However, under the alternative we can expect the sharing level to be higher than expected.
Assume that we have $n$ affected sibling pairs.
Let $x_i$ be the number alleles shared amongst the $i$-th affected sibling pair.
Then under the null $E[x_i] = 1$, $Var[x_i] = 0.5$ and we let $y_i = (x_i - 1)/\sqrt{0.5}$.
We consider the following well known non-parametric linkage test
\begin{align*}
S_n = \sum_{i = 1}^n \frac{y_i}{\sqrt{n}} \sim N(0,1),
\end{align*}
see \cite{laird2010fundamentals}. The above can be compared to a score test as only the information under the null was used.
Under the alternative, the distribution of the test can be misspecified, since a different distribution of allele sharing will yield a different variance.
Consider the simple example when the distribution of the numbers of shared alleles follows a multinomial distribution with $\theta_{alt1} = (0.09,0.8,0.11)$.
The variance of this distribution is $0.4 < 0.5$. Yet another alternative in which $\theta_{alt2} = (0.29,0.4,0.31)$ yields the variance $0.6 > 0.5$; in both case there is oversharing.
We include visualizations of the $p$-value distribution under the two alternatives in Figure \ref{fig:linkage}. Theorem 2 is used to produce the approximation given by the blue curve, due to the discrete nature of the test statistic.
\begin{figure}
\centering
\subfloat[Score test, $\theta = \theta_{alt1}$]{\includegraphics[width = 2in]{linkage_1.png}}
\subfloat[Score test, $\theta = \theta_{alt2}$ ]{\includegraphics[width = 2in]{linkage_2.png}}
\\
\subfloat[Wald test, $\theta = \theta_{alt1}$]{\includegraphics[width = 2in]{linkage_wald_2.png}}
\subfloat[Wald test, $\theta = \theta_{alt2}$ ]{\includegraphics[width = 2in]{linkage_wald_1.png}}
\caption{Plots for Example 3, examining the behaviour of the $p$-value distribution for non-parametric linkage analysis for the score test (upper panel) and the Wald test (lower panel). The simulation is performed with $n =400$, and $100,000$ replications. Samples of sibling pairs are generated from a multinomial distribution with $\theta_{alt1} = (0.09,0.8, 0.11) $ for the two plots on the left panel, and $\theta_{alt2} = (0.29,0.4,0.31)$ for the two plots on the right panel. For the score test, both histograms have spikes due to the discrete nature of the problem. The discrete version of the Edgeworth approximation, plotted in blue, is used as the test statistic is supported on a lattice. The histograms of the $p$-values obtained form the Wald test look much better than their score test counterparts. }
\label{fig:linkage}
\end{figure}
\end{example}
In this case the problem can be resolved by considering a Wald type test where the variance is calculated from the maximum likelihood estimate $\hat{\theta} = ( \#(x_i = 0)/n, \#(x_i = 1)/n, \#(x_i = 2)/n )$, and use:
\[S_n^\prime = \sum_{i = 1}^n \frac{x_i - 1}{\sqrt{n\widehat{\text{var}}(x_i)}}.
\]
We plot the results of applying the Wald test in Figure \ref{fig:linkage}.
The solution is quite simple in this case, but in more complex models it is more computationally expensive to calculate the variance estimate under the alternative. \\
\noindent\textbf{Example 1 revisited.} The abnormal distribution of $p$-values in this scenario is caused by a large numerical value of $\rho_{n,3}$ and $\rho_{n,4}$.
Going back to Example 1, we look at the theoretically predicted behaviour of the $p$-values under the null and alternative.
Figure \ref{fig:test} shows the histograms of the empirical $p$-values obtained by simulation versus the theoretical prediction given in Theorem 1, shown as the blue curve.
Without accounting for the higher order terms in the expansion we would have expected the null distribution to be uniform, however, using Theorem 1, we obtain a much more accurate description of the $p$-value distribution.
In the bottom panel of Figure \ref{fig:test} we also show a corrected version of the $p$-values approximation using the the saddlepoint approximation which will be introduced in Section 3.
The estimation of small p-values based on the standard normal approximation can be drastically optimistic. We report in Table \ref{tb:deltas}
the differences between the exact and the approximate $p$-value obtained from Example 1 for the 5 smallest $p$-values.
The smallest $p$-values from the normal approximation are not on the same scale as the exact $p$-values, the smallest approximate $p$-value being five-fold times smaller than its exact counterpart.
In contrast, the $p$-values produced by the saddlepoint approximation are very close to the exact ones.
\begin{table}[b]
\caption{ \label{tb:deltas} Table of $p$-values obtained from Example 1 under the null. The exact $p$-values are obtained from the density of the gamma distribution, the approximate $p$-values are obtained from the normal approximation.}
\centering
\begin{tabular}{rrrrr}
\hline
ID & rank & $p$-value exact & $p$-value approx. & $p$-val saddlepoint \\
\hline
60326 & 1 & 1.04E-05 & 1.04E-10 & 1.04E-05 \\
91132 & 2 & 1.46E-05 & 3.06E-10 & 1.47E-05 \\
83407 & 3 & 2.12E-05 & 9.66E-10 & 2.12E-05 \\
97470 & 4 & 3.31E-05 & 3.75E-09 & 3.32E-05 \\
2573 & 5 & 3.80E-05 & 5.66E-09 & 3.81E-05 \\
\hline
\end{tabular}
\end{table}
\begin{example}
We examine the influence of the effect size $\mu_n$ on the distribution of the $p$-values under the alternative using the same set-up as in Example 1. In our simulations we increase the effect size $\mu_n$ by changing the value of $\beta$, while keeping $\alpha$ fixed. The results are displayed in Figure \ref{fig:altnernaives}.
\begin{figure}[H]
\centering
\subfloat[Subfigure 1 list of figures text][Small effect size ]{\includegraphics[width = 1.75in]{alt_1025.png}}
\subfloat[Subfigure 2 list of figures text][Medium effect size ]{\includegraphics[width = 1.75in]{alt_105.png}}
\subfloat[Subfigure 1 list of figures text][Large effect size ]{\includegraphics[width = 1.75in]{alt_1075.png}} \\
\subfloat[Subfigure 1 list of figures text][Small effect size ]{\includegraphics[width = 1.75in]{alt_1025_s.png}}
\subfloat[Subfigure 2 list of figures text][Medium effect size ]{\includegraphics[width = 1.75in]{alt_105_s.png}}
\subfloat[Subfigure 1 list of figures text][Large effect size ]{\includegraphics[width = 1.75in]{alt_1075_s.png}}
\caption{Distribution of the approximated $p$-values (top panel) and the corrected $p$-values (bottom panel), under three different alternatives with $\alpha_1 =\alpha_2 = \alpha_3 =0.01$ and $\beta_2 = 0.01/1.025$, $\beta_1 = 0.01/1.05$ and $\beta_3 = 0.01/1.1$, from left to right.
By increasing the effect size, the approximate $p$-values starts to behave in an expected manner.
While the corrected $p$-values obtained by using the saddlepoint approximation is well behaved for all effect sizes.}
\label{fig:altnernaives}
\end{figure}
As discussed in Remark 3, for large effect sizes $\mu_n$, the distribution of $p$-values generated from the test statistic follows the expected trend, where there is a concentration of $p$-values around $0$ and the density decreases in a monotone fashion to 1.
Conversely, should $\mu_n$ be small then the behaviour under the alternative can be quite different from what we would expect, as illustrated by the top-right plot in Figure \ref{fig:altnernaives}.
\end{example}
\section{Additional Examples and Possible Remedies}
We provide additional examples of problematic $p$-value distributions, and we explore some possible remedies based on high order asymptotics.
We also provide additional examples of problematic $p$-value distributions.
A commonly used tool for higher order asymptotics is the saddlepoint approximation, which is a density approximation that can be integrated to obtain tail probabilities, e.g. $p$-values.
For a good survey of the saddlepoint approximation and its applications in statistics, we refer the reader to \cite{reid1988} or for a more technical reference, we suggest \cite{jensen1995saddlepoint} or \cite{kolassa1994series}.
The saddlepoint approximation can be most easily obtained for a sum or average of independent random variables, $X_1, \dots, X_n$. The density approximation then results in an approximation of the cumulative distribution through a tail integration argument,
\begin{align}
P(\bar{X} < s) = \Phi(r_s)\lbrace 1 + O(n^{-1}) \rbrace, \label{accuracy_saddle}
\end{align}
where $r_s$ is a quantity constructed from the saddlepoint and the cumulants of the distribution of the $X_i$'s. This can be used for conditional inference in generalized linear models by approximating the distribution of the sufficient statistics in a exponential family model; see \cite{davison}.
Another more broadly applicable tail approximation is the normal approximation to the $r^\star$ statistic \citep{barndorff1989asymptotic}, which is obtained by adding a correction factor to $r$, the likelihood root. It can be used in regression settings for inference on a scalar parameter of interest.
Let $r =\text{sign}(\hat\psi) [2 \lbrace l_{\text{pro}}(\hat\psi) - l_{\text{pro}}(\psi_0) \rbrace]^{1/2}$ denote the likelihood root, and in what follows the quantity $Q$ varies depending on the model.
\begin{align*}
P(r < s) = \Phi\left\lbrace r + \frac{1}{r} \log\left(\frac{Q}{r}\right)\right\rbrace \left\lbrace 1 +O\left( n^{-3/2} \right) \right\rbrace.
\end{align*}
Using the above, we also obtain an improved approximation to the true distribution of the likelihood root.
For a discussion of $r^\star$ see \cite{reid_wald}.
The proposed methods require two model fits, one under the alternative and one under the null in order to obtain $r$, contrary to the score test.
The methods listed here are by no means comprehensive since there are a variety of other candidates which may be of use, such as the often applied Firth correction \citep{firth} or other forms of bias correction obtainable by adjusting the score equation \citep{kosmidis2020mean}.
\subsection{The Gamma example}
We apply the saddlepoint approximation to Example 1 and display the results in Figure \ref{fig:test}.
Considering the null $H_0: \alpha = \beta = 0.01$ (the two plots on the left panel), there is a spike around 0 for $p$-values obtained using the CLT (top left plot).
In contrast, we see a marked improvement of the overall
behaviour of the $p$-value distribution after the proposed correction (bottom left plot).
\subsection{Logistic Regression in Genetic Association Studies}
We apply the normal approximation to $r^\star$ to a simulated genome-wide association study to further illustrate the practical use of the proposed correction.
We consider a logistic regression model that links the probability of an individual suffering from a disease to that individual's single nucleotide polymorphism (SNP), a genetic ordinal variable coded as 0, 1 or 2, and other covariates such as age and sex.
Formally, let the disease status of the individual be $Y_i$, which is either $0$ (individual is healthy) or $1$ (individual is sick) and $\pi_i = E[Y_i]$ denote the probability of individual $i$ having the disease and let $X_{i, s}$ denote the genetic covariate of interest of the $i$-th individual, while $X_{i,j}, j = 1, 2$, $j\ne s$ are the other covariates.
The regression model is:
\begin{align*}
\text{logit}(\pi_i) = X^i_s \beta_s + \sum_{j = 1}^2 X^i_j \beta_{j} + \beta_0.
\end{align*}
We consider the difficult case where the disease is uncommon in the population and the SNPs of interest are rare, i.e. most observed values of $X_{i,s}$ are 0.
It is known that in this situation the single-SNP test performs poorly, and pooled analyses of multiple SNPs have been proposed \citep{pooled}.
However for the purpose of this study, we assume that the individual SNPs are of interest.
We consider a simulated example to demonstrate the effectiveness of the correction.
We generate a sample of 3,000 individuals, their genetic variable $X_s$ are simulated from a $Binomial(2, 0.025)$, a binary variable $X_1$ from a $Binomial(1,0.5)$ and finally $X_2$ from a $N(20, 1)$.
We let $\beta_0 = -3.5$, $\beta_s = 0$, $\beta_1 = 0.02$ and $\beta_2 = 0.02$.
With this set of parameters we would expect on average $ \approx 4.6\%$ of the cohort to be in the diseased group, based on the expected value of the covariates, i.e. approximately 137 participants with $Y_i = 1$.
For each replication of the simulation, we re-generate the labels from the logistic model.
Figure \ref{fig:6} shows that the correction works well under the null.
\begin{figure}[ht]
\centering
\subfloat{\includegraphics[width=2.75in]{Wald_null}}
\subfloat{\includegraphics[width=2.75in]{cond_inf_null}}
\caption{Empirical distribution of the null $p$-values from a logistic regression association study of SNPs with with low minor allele frequency and a low number of diseased individuals.
Left histogram displays the $p$-values from the Wald test under the null. The right histogram displays the $p$-values histogram obtained from $r^\star$ under the null. }
\label{fig:6}
\end{figure}
This example suggests that the usefulness of the proposed higher order corrections is not limited to small sample scenarios, as note by \cite{zhou2018efficiently} who used the saddlepoint approximation in case control studies with extreme sample imbalance.
Naively we would expect that with 3,000 participants, of which 137 are in the diseased group, the Wald test should behave correctly. However, the skewed distribution of the SNP values severely reduces the accuracy of the test.
The use of $r^\star$ corrects the distribution of the $p$-values as shown in Figure \ref{fig:6} (right plot) where the distribution of the $p$-values under the null ($\beta_s=0$) is approximately Unif(0, 1) as expected.
In the example above it is clear that even though we have 3,000 individuals, of which 137 are affected by the disease, the standard approximation performs very poorly.
This seems to suggest that in our particular example, the effective sample size is lower than 137 for the diseased group.
Next we consider a simple regression with a single genetic covariate in order to illustrate the loss in information resulting from the sparsity of the minor allele.
We use the available Fisher information about the parameter of interest as a measure of effective sample size.
The standard deviation of the parameter of interest obtained from the inverse information matrix is
\begin{align*}
var(\hat\beta_s) &= \frac{\sum_{i = 1}^{n} \hat{P_i} (1 - \hat{P_i})}{\sum_{i = 1}^{n} \hat{P_i} (1 - \hat{P_i})\sum_{i = 1}^{n} x_i^2 \hat{P_i} (1 - \hat{P_i}) - (\sum_{i = 1}^{n} x_i \hat{P_i} (1 - \hat{P_i}) )^2},\\
&\approx \frac{1}{\sum_{i = 1}^{n} x_{i,s} \hat{P_i} (1 - \hat{P_i})},
\end{align*}
where $\hat{P}_i$ is the predicted probability of an individual being diseased and the approximation is valid under the assumption that the allele frequency is low enough such that we observe very few 1's and almost no 2's.
The information about the parameter $\beta_s$ is increasing in terms of $x_i \hat{P_i}(1 - \hat{P_i})$.
It is apparent that the rate of increase in information is limited by the sparsity of the rare allele.
In order to have more information about the parameter, we would need to observe more individuals who have the rare allele, i.e. $X_i \ne 0$.
\subsection{Logistic Regression - Data from the 1000 Genome Project}
We consider an additional logistic regression example as this type of model is broadly used in statistical genetics.
Using phase 3 data from the 1000 Genome project \citep{1000genome}, we construct an artificial observational study in order to study how these approximations behave on real genome-wide genetic data.
In our simulations, we take the 2504 individuals within the database and assign the $i$-th individual a label of $0$ or $1$ based on the following logistic model, where $\pi_i = P(Y_i = 1)$:
\[ \text{logit}(\pi_i) = \sum_{j = 1}^4 X^i_j \beta_{j} + \beta_\text{Sex}*I( \text{Sex}_i = \text{male}) + \beta_0, \]
where $\text{Sex}_i$ is the biological sex of the $i$-th individual. Four other covariates are included, where $X^i_j$ are independent for all $i, j$ and follow a standard normal distribution.
The model coefficients are set to
\[(\beta_0, \beta_1, \beta_2, \beta_3, \beta_4, \beta_{\text{Sex}}) = (-3.25, 0.025, -0.025, 0.025, -0.03, 0.1).\]
Once we assign a label to the $i$-th individual we keep it fixed throughout the simulation.
We then fit a logistic model using the SNPs for which the minor allele frequency is at least $1\%$ on chromosome $10$, and ethnicity as additional covariates.
We use the Wald test, and $r^\star$, but do not consider the cases where perfect separation occurs, as both methods considered here cannot deal with this issue.
We plot some of the results for the Wald test and $r^\star$.
We focus on rare variants with MAF $\leq 2.5\%$ and semi-common variants with $2.5\% <$ MAF $\leq 10\%$, as the remaining common variants are expected to behave well.
In total $160,580$ SNPs fall into the rare variant category while $176,350$ SNPs fall into the semi-common variant category.
\begin{figure}[h]
\centering
\subfloat[Subfigure 1 list of figures text][Rare variants, Wald]{\includegraphics[width =2in]{wald_low_MAF_10.png}}
\subfloat[Subfigure 1 list of figures text][Rare variants, $r^\star$]{\includegraphics[width = 2in]{rstar_low_MAF_10.png}}\\
\subfloat[Subfigure 1 list of figures text][Semi-common variants, Wald ]{\includegraphics[width = 2in]{wald_mid_MAF_10.png}}
\subfloat[Subfigure 1 list of figures text][Semi-common variants, $r^\star$ ]{\includegraphics[width = 2in]{rstar_mid_MAF_10.png}}
\caption{Distribution of $p$-values for Wald test and $r^\star$. The null distribution was simulated by using sex and four other randomly generated covariates. We fit a logistic regression model using SNPs from chromosome 10, with $160,580$ being rare ($\text{MAF}\leq 2.5\%$ ) and $176,350$ semi-common variants ($2.5\% < \text{MAF}\leq 10\%$ ).}
\label{altnernaives}
\end{figure}
As expected, the two tests behave better for semi-common SNPs than rare SNPs (bottom vs. top panel of Figure 7), producing $p$-values that more closely follow the Unif(0,1) distribution. Among the two tests, the proposed $r^\star$ method clearly out-performs the traditional Wald test.
However, this application also points out the limitation of $r^\star$ as the correction for rare variants is not sufficient (top right plot), and further improvement of the method in this case is of future interest.
\subsection{Weibull survival regression}
Consider an example where there is a large number of nuisance parameters, leading to an inconsistent estimate of the variance.
We examine a Weibull survival regression model in which all of the regression coefficients, except the intercept, are set to $0$ by simulating $y_i \sim \text{Weibull}(1,2)$, independently of any covariate.
We set the number of observations, $n$ to 200 and the number of covariates to $50$, and generated the covariates as IID standard Gaussian, and test for whether the first (non-intercept) regression coefficient is 0.
We perform 10,000 replications and plot the histogram of the $p$-values, and compare the Wald test to the $r^\star$ correction.
\begin{figure}[H]
\centering
\subfloat{\includegraphics[width=0.35\linewidth]{weibull.png}}
\subfloat{ \includegraphics[width=0.35\linewidth]{weibull_r.png}}
\caption{On the left, a histogram of the $p$-values produced by the Wald test for $\beta_1 = 0$ under the null with $n = 200$ and $p = 50$ with no censoring. On the right, a histogram of the $p$-value obtained from the $r^\star$ correction. 10,000 replications were performed.}
\label{regression}
\end{figure}
In Figure \ref{regression} we see a high concentration of $p$-values around $0$ for the Wald test, leading to increased I error.
The corrective procedure brings the distribution under the null much closer to uniformity.
We see that naively adding more and more information into the model while trying to perform inference on a one dimensional parameter of interest is problematic as it creates a perceived significance of the parameter of interest under the null.
\section{Discussion and Conclusion}
We characterize the distribution of $p$-values when the test statistic is not well approximated by a normal distribution by using additional information contained in the higher order cumulants of the distribution of the test statistic.
We also demonstrate that there are issues beyond failure to converge to normality in the that the expectation and variance of the test statistics can be misspecified, and these issues can persist even in large sample settings.
In doing so we have extended the previous work done by \cite{hung} to greater generality, examining the score test in exponential models in the presence of nuisance parameters.
We also examine some possible remedies for making the $p$-value distribution adhere more closely to their usual required behaviour such as uniformity under the null or concavity of the CDF under the alternative.
These assumptions are very important to justify the usage of current FWER and FDR procedures.
The proposed remedies may not solve all problems
relating to the $p$-value distribution in the finite sample settings, but they do at least partially correct some of the flaws.
We suggest the use of the proposed saddlepoint approximation or the normal approximation to $r^\star$ in practice, because a) the exact distribution of a test statistic is often unknown, b) the usual CLT approximation may not be adequate, and c) the high order methods are easy to implement.
This will ensure a closer adherence to the assumptions usually needed to conduct corrective procedures used in FWER control or FDR control.
\section*{Acknowledgement}
The first author would like to thank Nancy Reid, Michele Lambardi di San Miniato and Arvind Shrivats for the help and support they provided. We also thank the Natural Sciences and Engineering Research Council, the Vector Institute and the Ontario government for their funding and support.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,145 |
{"url":"https:\/\/mathematica.stackexchange.com\/questions\/75873\/differentiation-of-an-unknown-function","text":"# Differentiation of an unknown function\n\nI have to take the partial differentiation of an unknown function. For example, take the unknown function to be $g(x)$. Then it's derivative w.r.t $x$ is $g'(x)$.\n\nBy default, Mathematica differentiates the function. I want to keep the result of differentiation as $d(g(x))$ and not $g'(x)$. Is there any way to achieve this?\n\nMore precisely, I am using Conjugate[g[x]] as the unknown function and I want the output should be displayed only as d[Conjugate[g[x]] and not as Conjugate'[x]g'[x].\n\nAlso, can I handle the conjugate more efficiently than just carrying it all along in the code?\n\nEdited because the goal was changed in the comment:\n\nThis can be done by directly defining the outcome of Derivative when applied to g in the two combinations that you seem to be interested in:\n\nDerivative[1][g][x_] := d[g[x]]\n\nDerivative[1][Conjugate][g[x_]] := Conjugate[d[g[x]]]\/d[g[x]];\nDerivative[1][Conjugate][d[x_]] := Conjugate[d[d[x]]]\/d[d[x]]\n\nDerivative[1][d][x_] := d[d[x]]\/d[x];\nDerivative[1][d][x_Symbol] := d[d[x]]\n\n\nOn the second line, I used the fact that g is a generic function whose derivative under a Conjugate by default invokes the chain rule. All I do then is to reverse the chain rule by dividing by the factor d[g[x]] that the chain rule will produce. This leaves only the factor I want, and I then replace that by the desired outcome d[Conjugate[g[x]]].\n\nThe analogous thing is done for d to allow higher derivatives. The exception is when d[x] is encountered where x is the differentiation variable (which isn't in the question, but I expect may happen). Then there is no chain rule needed, and I therefore specify a separate rule for it with the pattern x_Symbol.\n\nHere is the test:\n\nD[g[x], x]\n\n(* ==> d[g[x]] *)\n\nD[Conjugate[g[x]], x]\n\n(* ==> d[Conjugate[g[x]]] *)\n\nD[g[x], x, x]\n\n(* ==> d[d[g[x]]] *)\n\nD[d[g[x]], x]\n\n(* ==> d[d[g[x]]] *)\n\nD[d[x], x]\n\n(* ==> d[d[x]] *)\n\nD[Conjugate[g[x]], x]\n\n(* ==> Conjugate[d[g[x]]] *)\n\nD[Conjugate[g[x]], x, x]\n\n(* ==> Conjugate[d[d[g[x]]]] *)\n\n\nNow the remaining issue is to replace the repeated application of d by formatting of the type d^2 g[x] for d[d[g[x]]]. I'll wait to see if this is really desired before doing it.\n\n\u2022 Thanks for your input. This is exactly what I need. Can you please modify the output of D[Conjugate[g[x]], x] as Conjugate[d[g[x]] and not as d[Conjugate[g[x]]] . You are free to change the above mentioned approach of yours if needed. Anyways, Many Many thanks for your input. \u2013\u00a0Shivam Sahu Mar 5 '15 at 12:41\n\u2022 You just hve to do this: Derivative[1][Conjugate][g[x_]] := Conjugate[d[g[x]]]\/d[g[x]] (but that's not what you asked in the question. \u2013\u00a0Jens Mar 5 '15 at 14:51\n\u2022 Thanks. This serves my purpose well. Can you please also help me with obtaining the double derivative of g[x] and Conjugate[g[x]] as d^2 g[x] and Conjugate[d^2[g[x]]]. I tried it all this time but I am unable to obtain the above desired representation. \u2013\u00a0Shivam Sahu Mar 7 '15 at 17:48\n\u2022 What you're asking now is very different from the original question because it aims for a new formatting that is inconsistent with Mathematica syntax (because of the squares). It would require box-level manipulations as I did here. \u2013\u00a0Jens Mar 7 '15 at 18:36\n\nIf it is just the displayed form you are after, you can also go with HoldForm like so:\n\nHoldForm@D[Conjugate[g[x]],x]\n\n\nThis will carry over throughout the notebook without further ado, until you call ReleaseHold on it.\n\nI hope this might be of some help to you.","date":"2019-10-20 12:31:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.6754785180091858, \"perplexity\": 943.0273056126181}, \"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-2019-43\/segments\/1570986707990.49\/warc\/CC-MAIN-20191020105426-20191020132926-00387.warc.gz\"}"} | null | null |
{"url":"http:\/\/mathematica.stackexchange.com\/questions\/30617\/generate-non-singular-matrix-of-nn-dimension","text":"Generate Non-Singular Matrix of $n*n$ dimension\n\nI want to create a function that will make an $n*n$ matrix which is non-singular that will always retrieve the same matrix (meaning that the matrix will not be random each time).\n\nI tried using the table function with different expressions using $i$ and $j$ but they always ended up singular because they had similar rules.\n\nSo please don't give me a random matrix, the matrix needs to have a pattern that if two people used the function on different computers with the same $n$ value, they would get the same matrix\n\nP.S. Keep the matrix in the Real set of numbers (also Integers) and don't make them too high so that its not too big to work with (I want the values to be ideally under 20)\n\n-\nI couldn't resist: IdentityMatrix[n] \u2013\u00a0 Kuba Aug 17 '13 at 9:25\nhmm yeah thats not gonna work, I'm doing Matrix Cryptography and I am going to pretend that the person I send the encrypted matrix knows the encoding matrix and using the Identity Matrix to encode it, won't be that encrypted. \u2013\u00a0 user9053 Aug 17 '13 at 9:27\nI know it is not what are you looking for but it fits the question :) that's I couldn't resist to post it :) \u2013\u00a0 Kuba Aug 17 '13 at 9:28\ntrue true, well thanks, I'll update it in the OP, but the matrix should be only Integers, so Nassers answer won't work \u2013\u00a0 user9053 Aug 17 '13 at 9:30\nI am confused. You say but the matrix should be only Integers, so Nassers answer won't work but HankelMatrix is all integers. May be you meant should NOT be integers? \u2013\u00a0 Nasser Aug 17 '13 at 9:36\n\nA non-singular matrix is a matrix with full rank. You can use any orthogonal basis, for example:\n\nTable[HermiteH[i, j], {i, 5}, {j, 5}]\n\n\nFor a 5x5 non-singular matrix. There are several more basis generating function in Mathematica.\n\nSince these numbers can be a bit big, here's a way of generating non-singular, non-random matrices with entries between -1 and 1:\n\northMatrix[n_] := Orthogonalize[RandomReal[1, {n, n}]]\n\n\nTo make it non-random we use RandomSeed:\n\nSeedRandom[1337]; orthMatrix[10]\n\n-\nwow, that has HUGH condition number! MatrixConditionNumber[ Table[HermiteH[i, j], {i, 5}, {j, 5}]] \/\/ N gives 431620. en.wikipedia.org\/wiki\/Condition_number that will cause about 5 decimal point accuracy loss? \u2013\u00a0 Nasser Aug 17 '13 at 9:17\n@Nasser We don't really know what the requirements for the matrix is. I'm just pointing out that HermiteH, LaguerreL, BesselJ etc. all solves the problem in a straightforward manner. \u2013\u00a0 Pickett Aug 17 '13 at 9:23\nSure. But high condition number will cause numerical problems when used for Ax=b problems. But there are many ways to generate random matrices that are not singular. just googled it. So, fixing the seed to fixed value before starting the random number generator will cause the same matrix to be generated each time. \u2013\u00a0 Nasser Aug 17 '13 at 9:26\n@Nasser Added such a method. \u2013\u00a0 Pickett Aug 17 '13 at 9:41\nUsing your last method with the orthMatrix, my values are all decimals, is there a way to make them integers under a maximum value? \u2013\u00a0 user9053 Aug 17 '13 at 9:56\n\nmaybe Hankel?\n\nN[LinearAlgebraMatrixConditionNumber[HankelMatrix[#]]] & \/@ Range[4, 32]\n\n{4.09288, 5.13808, 6.11956, 7.14252, 8.1322, 9.14612, 10.1395, \\\n11.1489, 12.1443, 13.151, 14.1477, 15.1527, 16.1502, 17.1541, \\\n18.1521, 19.1552, 20.1536, 21.1562, 22.1548, 23.157, 24.1559, \\\n25.1577, 26.1567, 27.1583, 28.1574, 29.1588, 30.1581, 31.1593, \\\n32.1586}\n\n-\nI updated it in the OP, but I forgot to mention that all values are integers :\/ thanks anyways \u2013\u00a0 user9053 Aug 17 '13 at 9:33\nbut it IS all integers. That what Hankel matrix is. The above is NOT the matrix, these are the conditions numbers. Showing the matrix is stable numerically. \u2013\u00a0 Nasser Aug 17 '13 at 9:37\noh, ok, I tested it out in Mathematica and it seems like the best for my scenario so far, I will keep this option open \u2013\u00a0 user9053 Aug 17 '13 at 9:44\n\nJust for fun:\n\nn = 5;\nPartition[Prime \/@ Range[n^2], n]\n\n{{2, 3, 5, 7, 11},\n{13, 17, 19, 23, 29},\n{31, 37, 41, 43, 47},\n{53, 59, 61, 67, 71},\n{73, 79, 83, 89, 97}}\n\n\na little improvement:\n\nPrime@Array[Plus, {n, n}]\n\n{{3, 5, 7, 11, 13},\n{5, 7, 11, 13, 17},\n{7, 11, 13, 17, 19},\n{11, 13, 17, 19, 23},\n{13, 17, 19, 23, 29}}\n\n\nit will generate lower values because rows will overlap in n-1 posistions. Also I used the fact that Prime is listable.\n\n-\nThanks, this is my favourite, and this will work for all values of n? \u2013\u00a0 user9053 Aug 17 '13 at 9:18\n@user9053 it suppose to but for huge n it may be slow and generate huge values :) I don't have big experience with primes so you have to check if it is good for your purposes :) Please, do not accept so fast, the question is quite new, let's do not discourage others :) you can still upvote if you like it \u2013\u00a0 Kuba Aug 17 '13 at 9:21\nHigh condition number also ! \u2013\u00a0 Nasser Aug 17 '13 at 9:22\nI'll unaccept the answer for the moment, and if it's the best answer later I'll accept it \u2013\u00a0 user9053 Aug 17 '13 at 9:32\n@user9053 Ok ;). Also, take a look at my edit, it will generate lower values. \u2013\u00a0 Kuba Aug 17 '13 at 9:34\n\nOK, so I decided on a method similar to Anon's solution\n\nI am just going to use\n\nSeedRandom[value];EncodingMatrix=RandomInteger[1,{dimension,dimension}]\n\n\nHowever I wanted to find an optimal seed for my scenario. I generally wanted to have all values of $n$ from 1 - 20 so I made the following program to find it out\n\nGetEncodingMatrix[Dimensions_,Seed_]:=(SeedRandom[Seed];RandomInteger[1,{Dimension,Dimension}])\nTestValid[n_]:=(state=\"True\";For[i=1,i<21,i++,If[Det[GetEncodingMatrix[i,n]]==0,state=\"False\"]];state)\nn=1;While[n<5000,If[TestValid[n]==\"true\",Print[n]];n++]\n`\n\n1918, 3381, 3385\n\n$\\therefore$ My seed value should work for those values.\n\nIf anyone refers back to this and needs to make it work for every value of $n$, I recommend any of the other three options as they are all excellent\n\n-","date":"2014-08-21 22:01:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.3270317018032074, \"perplexity\": 1293.1623085777594}, \"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-2014-35\/segments\/1408500821666.77\/warc\/CC-MAIN-20140820021341-00327-ip-10-180-136-8.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
Silhouetted against the Galactic background, Infrared Dark Clouds
(IRDCs) are opaque at wavelengths $\sim$ 10 $\mu$m (P\'erault et
al. 1996; Egan et al. 1998), cold ($T < 20$~K; Carey et al. 1998;
Pillai et al. 2006), and dense ($\rm n_H \geq 10^{3} - 10^{5} cm^{3}$;
Teyssier et al. 2002; Rathborne et al. 2006; Butler \& Tan 2009,
hereafter BT09; Peretto \& Fuller 2010). They are likely to be the
precursors of massive stars and star clusters as they have similar
physical conditions, such as mass surface densities, as regions with
such star formation activity (Rathborne et al. 2006; Tan 2007; Zhang
et al. 2009; Ragan et al. 2009). CO emission
from these clouds may be useful for understanding their dynamics
(e.g. Hernandez \& Tan 2011, hereafter HT11), but could be affected by
depleted gas phase abundances due to freeze-out onto dust grains,
especially in the coldest, highest density regions.
Gas phase depletion of CO, averaged along the line of sight, has been
observed in the cold ($T\lesssim 10$~K) centers of relatively low-mass
and nearby starless cores, (e.g. Willacy et al. 1998; Caselli et
al. 1999; Kramer et al. 1999; Bergin et al. 2002; Whittet et al. 2010;
Ford \& Shirley 2011). Typically, depletion is characterized by
measuring the depletion factor, $f_D$, defined as the ratio of CO
column density {\it expected} assuming standard gas phase abundances
given the column of material observed from either the mm dust
continuum emission or near infrared (NIR) dust extinction to the {\it
observed} CO column density (typically from $\rm C^{17}O$ or $\rm
C^{18}O$). Caselli et al. (1999) estimated the expected CO column
based on mm dust continuum emission, which has the advantage of being
able to probe to high column densities, but is sensitive to the
adopted dust temperature and emissivity. They concluded depletion
affected a region at the core center containing about $2\:M_\odot$ of
gas, where $n_{\rm H}\gtrsim 10^5\:{\rm cm^{-3}}$, with depletion
factors of up to $\sim$10 where the mass surface density is
$\Sigma\simeq 0.6\:{\rm g\:cm^{-2}}$. Kramer et al. (1999) estimated
the expected CO column based on NIR extinction, which does not require
knowing the dust temperature, but does require there to be a
sufficient areal density of background stars detectable in the NIR.
They found depletion factors of up to $\sim 2.5$ for regions with
$A_V\sim 20-30$~mag, corresponding to $\Sigma\sim 0.1-0.15\:{\rm
g\:cm^{-2}}$.
Massive protostellar cores and clumps are typically more distant and
difficult to study, but CO depletion has been reported by Fontani et
al. (2006) from a study of 10 sources with median $f_D\simeq 3.2$ (but
a dispersion of about a factor of 10), Thomas \& Fuller (2008) from a
study of 10 sources with a mean $f_D\simeq 1.3$ and Lo et al. (2011)
from a study of 1 source with $f_D\sim 10$. These results rely on
estimates of the expected CO column density based on mm dust continuum
emission, are derived only for single pointings to the sources, and
can depend on radiative transfer modeling of the unresolved source
density and temperature structure (Thomas \& Fuller 2008; Lo et
al. 2011). Source to source comparisons are hampered by possible
isotopic abundance variations affecting these rare CO isotopologues.
The above sources already contain massive protostars, but it is not
clear if the depletion signal arises from the immediate surrounding
envelope or from nearby unresolved starless cores. Some of the massive
protostars studied produce ultra-compact \ion{H}{2} regions and
photodissociation of molecules could be occurring in localized
regions, which would mimic depletion.
We expect CO depletion to be widespread in the dense regions of IRDCs,
potentially affecting: the physical properties one derives from CO
emission; the mid and far infrared opacities of dust grains as CO ice
mantles build up; and thus the initial conditions of star and planet
formation in these regions. Individual resolved IRDCs, assumed to have
uniform isotopic abundances, may also be useful laboratories in which to
study the depletion process as a function of local gas conditions.
In this paper, we present IRAM 30m observations of $\ceto$ $J=1
\rightarrow 0$ and $J=2 \rightarrow 1$ emission from the filamentary
IRDC G035.30-00.33 (Cloud H in BT09; near kinematic distance of
$d=2.9$~kpc). To look for evidence of depletion, the $\ceto$-derived
mass surface density, $\Sigma_{\rm C18O}$, is compared with the small
median filter (SMF) mid-infrared (MIR) extinction mapping derived mass
surface density, $\Sigsmf$ (BT09; Butler \& Tan 2011, hereafter
BT11). This work is motivated by the study of HT11, who used $\thco$
molecular line emission from the Galactic Ring Survey (GRS) to
estimate the mass surface densities of two highly filamentary IRDCs,
including Filament H. Assuming a constant value of $\tex=15$~K, HT11
found tentative evidence for CO depletion, but could not exclude the
possibility that other effects, such as systematic changes in the
excitation temperature or the contribution of high opacity cores, were
the cause of the observed decrease of $\Sigma_{\rm 13CO}/\Sigsmf$ with
increasing $\Sigma$. With our new higher-resolution, multi-transition
$\ceto$ data, we are able to exclude or mitigate these effects, as
well as resolving higher mass surface density structures to probe a
larger range of conditions where depletion may be occurring.
\section{Mass Surface Density from MIR Extinction Mapping}\label{S:SMF}
The 8~$\rm \mu m$ SMF mass surface density, $\Sigsmf$, map was derived
at 2$\arcsec$ resolution from the {\it Spitzer} IRAC band 4 (Galactic
Legacy Mid-Plane Survey Extraordinaire [GLIMPSE]; Benjamin et
al. 2003) image by comparing the observed intensity at each position
with the expected background intensity, estimated by interpolating the
intensities of surrounding nearby regions where median filter
smoothing is used to define the background model (see Figure 1a and
1b). Following BT09, a dust opacity of $\kappa_{\rm 8 \mu m}=7.5\:{\rm
cm^{2}\:g^{-1}}$ was adopted, similar to the filter response and
background spectrum weighted mean IRAC band 4 opacity expected from
the Ossenkopf \& Henning (1994) thin ice mantle moderately coagulated
grain model with a gas-to-dust mass ratio of 156. This value is
somewhat higher than values adopted by other dust models (e.g. 125 is
used for the Weingartner \& Draine 2001), although a recent estimate
from depletion studies finds a gas-to-dust ratio of 141 (Draine 2011,
p265). In any case, as described below, our study of CO depletion
compares relative abundances as a function of $\Sigma$ in the IRDC and
so is independent of this choice of overall normalization.
A correction for foreground emission also needs to be estimated. BT09
made this correction by estimating the amount of foreground emission
from a physical model of the Milky Way and given a measured kinematic
distance (assumed to be near) of the cloud. Battersby et al. (2010)
have pointed out an additional source of foreground from scattering in
the IRAC array. BT11 have developed a more accurate empirical method
for estimating the foreground emission, based on the presence of
independent saturated (high optical depth) cores, and here we use this
new method. For the region we analyze in this particular IRDC, the
values of $\Sigsmf$ are increased by about 10\% from those presented
by BT09.
$\Sigsmf$ in the filament is derived from comparison with adjacent
regions, which are assumed to have negligible MIR extinction. In
reality, we know from molecular line observations (e.g. $\thco$ from
the GRS analyzed by HT11), that these regions do have some material
present associated with the IRDC. We refer to this as the IRDC
``envelope''. The presence of the envelope and other systematic
uncertainties associated with estimation of the MIR background
intensity mean that $\Sigsmf$ becomes unreliable when $\lesssim
0.01\:{\rm g\:cm^{-2}}$. For our comparison with the mass surface
density derived from $\ceto$ emission, the $\Sigma_{\rm SMF}$ map is
regridded to the much lower resolution of the CO data (see below) and
all pixels with $\Sigsmf<0.01\:{\rm g\:cm^{-2}}$ are excluded from the
analysis. Methods of accounting for the envelope material are
discussed further in \S\ref{S:comparison}.
As noted by BT09, we must also account for locations of bright MIR
emission. Wherever the observed MIR intensity is greater than the
adopted background model an unphysical negative value of $\Sigma$ will
be estimated. Negative values of $\Sigma$ are allowed up to levels
comparable with the observed noise, but more extreme values, which are
mostly due to discrete MIR bright sources, have $\Sigsmf$ set to zero.
This causes an underestimation of the mass surface density in these
regions. We identify and exclude from further analysis remaining
(i.e. $\geq 0.01\:{\rm g\:cm^{-2}}$) pixels in $\Sigsmf$ map (smoothed
to the CO resolution) that have more than 20\% of their area occupied
by zero or negative values. In Figure~1b, these excluded pixels are
indicated with ``X'' and ``O'' symbols for the CO(1-0) and CO(2-1)
resolutions, respectively. Their exclusion is due either to the
presence of a MIR bright source or in regions where the background
modeling is inaccurate, which can sometimes occur near the edge of the
filament. Only a relatively small number of pixels are affected by
this exclusion.
In fact, our final results would not have varied significantly if this
exclusion had not been implemented.
\begin{sidewaysfigure}
\begin{center}$
\begin{array}{lcrr}
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1a.eps}
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1c.eps}
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1e.eps}
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1g.eps}
\\
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1b.eps}
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1d.eps}
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1f.eps}
\includegraphics[height=2.in,trim=0 0 0 0, angle=0]{f1h.eps}
\end{array}$
\end{center}
\caption{
\small
Morphology and depletion maps of the IRDC. {\it ($\rm a$) Top left:}
{\it Spitzer} GLIMPSE IRAC 8~$\mu$m image, with linear intensity scale
in MJy~$\rm Sr^{-1}$. The image has 1.2\arcsec\ pixels and the PSF has
a FWHM of 2\arcsec. {\it (b) Bottom left:} Mass surface density,
$\Sigma_{\rm SMF}$, with linear intensity scale in $\rm g\:cm^{-2}$,
derived from the image in panel (a) using the small median filter
(SMF) MIR extinction mapping method of Butler \& Tan (2009;
2011). Regions with $\Sigma_{\rm SMF}>0.01\:{\rm g\:cm^{-2}}$ but
which are $>20\%$ affected by artifacts in the extinction map
(e.g. due to MIR bright sources) are excluded from analysis and shown
by ``X'''s and ``O'''s for CO(1-0) and (2-1) resolution grids,
respectively.
{\it (c) Top middle left:} Integrated intensity map of $\ceto$
($J=2\rightarrow 1$) emission over the velocity range of $40-50 \kms$,
i.e. the gas associated with the IRDC (HT11), in linear units of $\rm
K\:km\:s^{-1}$ and a pixel scale of 5\arcsec. {\it (d) Bottom middle
left:} The mean excitation temperature map weighted by the column
density in K, with pixel size of 11\arcsec. {\it (e) Top middle
right:} Relative depletion factor ($f_D^\prime$) map for Case 1 (no
CO envelope subtraction). {\it (f) Bottom middle
right:} Relative depletion factor ($f_D^\prime$) map for Case 1 HiRes (no
CO envelope subtraction, $\Sigma$ derived at the CO(2-1) resolution).
{\it (g) Top right:} Relative depletion factor
map for Case 2 (CO envelope contribution estimated via interpolation
across strips 2, 3 and 4 then subtracted; note we consider this process
unreliable for strip 1). {\it (h) Bottom right:} Relative depletion factor
map for Case 2 HiRes.
}
\label{6paneldata}
\end{sidewaysfigure}
\section{Mass Surface Density from $\ceto$ Emission}
\subsection{Observations}
The $\ceto$ $J=1 \rightarrow 0$ and $J=2 \rightarrow 1$ lines were
mapped using the IRAM (Instituto de Radioastronomia Milimetrica) 30m
antenna in Pico Veleta, Spain in August and December 2008. An area
of $2'\times4'$ was mapped using the On-The-Fly (OTF) method towards
G035.39-00.33 with a central position of $\alpha(J2000)=18^h57^m08^s$,
$\delta(J2000)=02^{\circ}10'30"$ ($l=35.517^{\circ}$,
$b=-0.274^{\circ}$). While the $\ceto$ $J=1 \rightarrow 0$ transition
was observed with the ABCD receivers with typical single side band
(SSB) rejections $>10$~dB, the $\ceto$ $J=2 \rightarrow 1$ lines
emission was mapped by using the HERA multi-beam receiver.
Off-positions for both transition lines were set to ($1830\arcsec$,$658\arcsec$).
The beam size at $\sim 110$ GHz for the $J=1 \rightarrow 0$
transitions is 22$\arcsec$, while at $\sim 220$ GHz the $J=2
\rightarrow 1$ beam size is 11$\arcsec$ . The VESPA spectrometer
provided spectral resolutions of 20kHz and 80kHz for the $J=1
\rightarrow 0$ and $J=2 \rightarrow 1$ lines respectively, which
correspond to velocity resolutions of $\sim0.05 \kms$ and $\sim0.1
\kms$. For this study, all spectra were resampled to the same velocity
resolution of $0.2 \kms$. The typical system temperatures were 150-220 K.
Intensities were calibrated in units of antenna temperature (T$_{\rm A}^*$),
and converted into a main beam brightness temperature, $\rm T_{B,\nu}$,
via $T_A\equiv \eta f_{\rm clump}T_{B,\nu}$, where $\eta$ is a main beam
efficiency and $f_{\rm clump}$ is the beam dilution factor. We use $\eta=0.64$
for the $J=1 \rightarrow 0$ transition, and $\eta=0.52$ for the $J=2 \rightarrow 1$
transition. The typical 1$\sigma$ RMS noise of the data is $0.2 K \kms$ over the
velocity range of $40-50 \kms$. Since the $\ceto$ emission is extended over the filament,
we assume $f_{\rm clump}=1$. Figure \ref{6paneldata}c presents the
morphology of Filament H as seen in $\ceto$ $J=2 \rightarrow 1$
emission.
\subsection{Mass Surface Density and $\tex$ Estimates}
We estimate the column density of $\ceto$ molecules, ${\rm d} N_{\rm
C18O}$, in the velocity interval ${\rm d}v$, from their emission
through the general equation:
\beq
\frac{{\rm d}N_{\rm C18O}(v)}{{\rm d}v} = \frac{8\pi}{A \lambda_0^3} \frac{g_l}{g_u} \frac{\tau_\nu}{1-{\rm exp}\left(-h\nu/kT_{\rm ex}\right)} \frac{Q_{\rm rot}}{g_l {\rm exp}(-E_l/kT_{\rm ex})}.
\label{eq:dN}
\eeq
Here $Q_{\rm rot}$ is the partition function for linear molecules
given by $Q_{\rm rot}= \sum_{J=0}^{\infty} (2J+1) {\rm
exp}(-E_J/kT_{\rm ex})$ with $E_J = J(J+1)h B$, where $\rm J$ is the
rotational quantum number and B is the $\ceto$ rotational constant
equal to $5.4891\times 10^{10}\:{\rm s}^{-1}$. $h\nu /k=5.269, 10.54$~K for
$J=1 \rightarrow 0$ and $J=2 \rightarrow 1$ transitions, respectively. At $\rm 7.5$~K,
$Q_{\rm rot}=3.205$.
$A$ is the Einstein coefficient, $6.266, 60.11 \times10^{-8}{\rm s^{-1}}$ for
$J=1 \rightarrow 0$ and $J=2 \rightarrow 1$, respectively. $\lambda_0$ is the
wavelength of the transition,
$0.273, 0.137$ cm for $J=1 \rightarrow 0$ and $J=2 \rightarrow 1$, respectively. $g_l$
and $g_u$ are the statistical weights of the lower and upper levels,
and $\tau_\nu$ is the optical depth of the line at frequency $\nu$,
i.e. at velocity $v$. The excitation temperature, $T_{\rm ex}$, is
assumed to be the same for all rotational levels. Details on the
estimation of $\tex$ are given below.
The optical depth, $\tau_\nu$, is derived through the detection equation:
\beq
T_{B,\nu} = \frac{h\nu}{k} [f(T_{\rm ex}) - f(T_{\rm bg})] \left[1 - e^{-\tau_\nu}\right]
\label{eq:detection}
\eeq
where $T_{B,\nu}$ is the main beam brightness temperature at frequency $\nu$,
$f(T)\equiv [{\rm exp}(h\nu/[kT])-1]^{-1}$, and $T_{\rm bg}$ is the
background temperature of $2.725$~K. For the observable, $T_{B,\nu}$, and for an
assumed $T_{\rm ex}$, $\tau_\nu$ can be solved for directly through
equation (\ref{eq:detection}). Therefore, we can solve for the column
density per unit velocity, ${\rm d}N_{\rm 18CO}/{\rm d}v$, at each
$l,b,v$ position.
While care is taken to account for the optical depth in our column
density estimates, for reference we also state the case of the
optically thin limit of the $\ceto$ ($J=1\rightarrow 0$) column
density. If $\tau_\nu$ is small, then equation (\ref{eq:detection})
reduces to $T_{B,\nu}=(h\nu/k)[f(T_{\rm ex}) - f(T_{\rm
bg})]\tau_\nu$. Inserting into equation (\ref{eq:dN}) gives:
\begin{eqnarray}
\frac{{\rm d}N_{\rm C18O}(v)}{{\rm d}v} & = & 6.571\times 10^{14} \frac{Q_{\rm rot}}{f(T_{\rm ex}) - f(T_{\rm bg})} [1-{\rm exp}(-h\nu/ k T_{\rm ex})]^{-1}\frac{T_A/K}{\eta f_{\rm clump}} \:{\rm cm^{-2} km^{-1}s}\\
& \rightarrow & 9.758\times 10^{14}\frac{T_A^*/K}{\eta f_{\rm clump}} \:{\rm cm^{-2} km^{-1}s}\:\:(T_{\rm ex}=7.5~{\rm K}).
\label{eq:thin}
\end{eqnarray}
As in HT11, an inspection of the $\ceto$ emission in $l,b,v$ space
indicates that the gas associated with the filament is in the range of
$40-50 \kms$. The total column density per pixel is then calculated
over the entire velocity range of the filament, $N_{\rm C18O} = \int
dN_{\rm C18O}$.
The column densities for both transitions, $N_{\rm C18O}$, are
converted to a total mass surface density $\Sigma_{\rm C18O}$, by
assuming the abundance ratios of $\rm n_{16O}/n_{18O}=327$ from Wilson
\& Rood (1994) and $\rm n_{12CO}/n_{H2}=2 \times 10^{-4}$ from Lacy et
al. (1994). Thus, our assumed abundance ratio of $\ceto$ to $\rm H_2$
is $\rm 6.12 \times 10^{-7}$ and $\Sigma$ for each pixel is then
given by:
\begin{equation} \Sigma_{\rm C18O}=7.652 \times 10^{-2} \frac{N_{\rm C18O}}{10^{16}{\rm cm^{-2}}}\: {\rm g\: cm^{-2}},
\label{eq:Sigtot}
\end{equation}
assuming a mass per H nucleus of $\mu_{\rm H}=2.34 \times 10^{-24} {\rm g}$, i.e. $\Sigma= 1\:{\rm g\:cm^{-2}}$ is equivalent to $N_{\rm H} = 4.27\times 10^{23}\:{\rm cm^{-2}}$.
In order to accurately derive the mass surface density of the
filament, an estimate of the excitation temperature, $\tex$, is
needed. To perform this estimate throughout the filament, we varied
the assumed temperature at each $l,b,v$ position until the ratio between
the column densities derived from both transitions were in
agreement. To do this, we first defined $R_{2,1}$ as the ratio
between the $J=2 \rightarrow 1$ and $J=1 \rightarrow 0$ column
densities:
\begin{equation} R_{2,1}\equiv \frac{{\rm d}N_{\rm C18O,21}}{{\rm d}N_{\rm C18O,10}}.
\label{eq:Tratio}
\end{equation}
This method is similar to the one used in Kramer et al. (1999), except
they averaged over the velocity profile of their cloud. The higher
resolution $J=2 \rightarrow 1$ data was convolved with a beam of
22$\arcsec$ and regridded to match the resolution and pixel scale of
the $J=1 \rightarrow 0$ data. For all $l,b,v$ positions above a noise
limit of $3\sigma$ in both transitions, $R_{2,1}$ was calculated first
assuming a $\tex=30$~K. Then, $\tex$ was iteratively decreased until
$R_{2,1}$ converged to unity. This step provided a three dimensional
grid containing estimates of $\tex$ for all positions above the noise
limit. Next, for all positions below the noise threshold, their
$\tex$ was estimated by taking the mean excitation temperature at the
corresponding $l,b$ position. Finally, for any remaining $l,b,v$
positions without an estimated excitation temperature, the mean $\tex$
of 7.2 K resulting from the previous steps was used. Positions left
for this final step are mainly in the outer regions of the filament
where the emission is weak and/or the noise is high. The column
density weighted $\tex$ map is shown in Figure \ref{6paneldata}d.
\section{Comparison of $\Sigceto$ and $\Sigsmf$:\\ Evidence for CO Depletion}\label{S:comparison}
In Figure \ref{6paneldata}, we present the morphology of the
filamentary IRDC H. The goal of this section is to compare $\Sigceto$
and $\Sigsmf$. The simplest way of doing this, which we refer to as
Case 1, involves a straightforward pixel by pixel comparison of these
values, smoothing the $\Sigsmf$ data to the resolution of the
$\ceto$(1-0) observations, for which we have derived accurate
excitation temperature information. Note, that only pixels with
$\Sigsmf$ and $\Sigceto\geq 0.01\:{\rm g\:cm^{-2}}$ are
considered. Also, pixels for which $\Sigsmf$ is affected by bright MIR
emission are excluded (see \S\ref{S:SMF}). We also perform a
comparison at the higher angular resolution of the $\ceto$(2-1)
observations, which we refer to as Case 1 HiRes, assuming $T_{\rm ex}$
at this higher angular resolution can be estimated from the values
derived at the (1-0) resolution. For both these versions of Case 1, we
refer to $\Sigceto$ as $\Sigma_{\rm C18O,TOT}$, since it is derived
from all the $\ceto$ emission associated with the IRDC and its
surrounding GMC.
However, as is apparent from Figure \ref{6paneldata}, the $\ceto$
emission is more extended than the $\rm 8\mu m$ extinction map from
Butler \& Tan (2009; 2011). This is because, as discussed above, the
extinction map is derived from an ``on-off'' comparison with adjacent
regions, which help define the background MIR intensity that is
expected to be behind the filament. Thus the MIR extinction mapping
method becomes insensitive to material present in these adjacent,
lower column density (``envelope'') regions. A fair comparison between
$\Sigsmf$ and $\Sigceto$ would allow for this envelope material. We
thus define ``filament'' and ``envelope'' regions based on the 8~$\rm
\mu m$ image of the IRDC. Following HT11, the filament is defined to
be a rectangular strip centered at $\alpha(J2000)=18^h57^m08.02^s$,
$\delta(J2000)=02^{\circ}10'35.7\arcsec$, 2.05\arcmin \ wide in R.A. and
4.47\arcmin \ long in Dec. The outline of this filament region is shown
by a red box in the panels of Figure 1. The envelope region is defined
to be made up of two adjacent rectangular regions on either side of
the filament. These are shown as blue rectangles in Figure 1 and are
each 0.56\arcmin \ wide in R.A. and 4.47\arcmin \ long in Dec. Note, that
because of the limited area mapped by our observations, these envelope
regions are narrower than those considered by HT11.
For our Case 2, we assume that the $\ceto$ material present in the
envelope regions is also present at the similar levels towards the
filament region, and so attempt to subtract this emission from the
$\ceto$ spectrum of the filament, before then comparing to
$\Sigsmf$. To carry out this subtraction we divide the filament and
envelope into four E-W strips (1 to 4 from N to S) (see Figure 1). In
each strip, the mean column density per unit velocity is evaluated for
the filament (based on 66 $\ceto$(1-0) pixels) and the two adjacent
envelope regions (based on 18 $\ceto$(1-0) pixels each) (see Figure
\ref{envspectra18}), using the $\tex$ estimates described
previously. The envelope spectra are averaged and then subtracted from
the filament. The total column of this envelope-subtracted spectrum is
evaluated and used to derive $\Sigma_{\rm C18O,FIL}$. This is of
course an approximate method for accounting for the envelope material:
one can see from Figure \ref{envspectra18} that the envelope spectra
on either side of the filament can be quite different, especially for
strips 1 and 2. The uncertainty in the envelope-subtracted spectrum
becomes large when the envelope spectra are of similar strength as
that of the filament, as is the case for strip 1. Thus we do not
regard the results of envelope subtraction for strip 1 as being
reliable, and we exclude these pixels from the Case 2 analysis. As
with Case 1, we also perform a Case 2 HiRes analysis, using
$\Sigma_{\rm C18O,FIL}$ estimated at the higher resolution of the
CO(2-1) data, adopting values of $\tex$ evaluated at the CO(1-0)
resolution.
With these Case 1 and 2 methods, we now compare the pixel by pixel
values of $\Sigceto$ with $\Sigsmf$ derived from MIR extinction
mapping. As noted in HT11, these measurements of $\Sigma$ are
essentially independent of cloud distance uncertainties. Figure
\ref{Sigcomp18}a presents $\Sigma_{\rm C18O,TOT}$ versus $\Sigsmf$,
i.e. Case 1 of no envelope subtraction. The best fit power law
relation to the CO(1-0) resolution data of $\Sigma_{\rm C18O,TOT}/{\rm
g\:cm^{-2}} = A (\Sigma_{\rm SMF}/{\rm g\:cm^{-2}})^\alpha$ has
$\alpha=0.452\pm0.054$ and $A=0.146\pm0.023$. For Case 1 HiRes (i.e. at
the CO(2-1) resolution, adopting CO(1-0) resolution $\tex$ estimates) we find
$\alpha = 0.463\pm 0.025$ and $A = 0.151\pm0.010$. These results are
summarized in Table~\ref{tab:depletion}.
These uncertainties are derived assuming that the errors of each
individual measurement are as follows:
for $\Sigceto$, a fixed value of $0.0024\:{\rm g\:cm^{-2}}$ (derived from
the 1$\sigma$ RMS noise of $\rm 0.2 K \kms$ over the velocity range of
$40-50 \kms$) and a 20\% error to account for uncertainties in $\tex$
assumed to be 1~K at the typical temperature of 7~K; for $\Sigsmf$,
a 15\% error plus a systematic
error of $\rm 0.01 g\:cm^{-2}$ (BT09). At the resolution of the CO
pixels (11\arcsec for CO(1-0) and 5\arcsec for CO(2-1)), the $\Sigsmf$
measurements are independent, but the $\Sigceto$ results are not since
the telescope beam is about twice the pixel scale. Thus the above
quoted uncertainties of the power law fits assume, conservatively,
only 25\% of the pixels are used (although the derived values of the
parameters are based on fits to all of the pixels).
We argue below that $\Sigsmf$ is a more accurate measure of the true
mass surface density in IRDCs than $\Sigceto$, since one does not
expect large changes in MIR dust opacities in these environments,
based on the Ossenkopf \& Henning (1994) dust models. If this is true,
then if $\ceto$ were also an accurate tracer of mass surface density,
then we should see a one-to-one relation between $\Sigceto$ and
$\Sigsmf$, i.e. $\alpha\simeq 1$, even if $A$ (the value of
$\Sigceto/\Sigsmf$ when $\Sigsmf=1\:{\rm g\:cm^{-2}}$) is not exactly
unity because of systematic uncertainties in the absolute values of
$\ceto$ abundance or MIR dust opacities. We measure
$\alpha=0.452\pm0.054$ for Case 1 and $\alpha=0.463\pm0.025$ for Case 1
HiRes, which are significantly ($10\sigma$ and $21\sigma$) different
from one, and we interpret these results as being evidence for CO
depletion from the gas phase.
To illustrate that these results do not depend on the choice of dust
opacity per unit gas mass, we have repeated the analysis but with a
gas-to-dust mass ratio of 100 (rather than our fiducial value of
156). We find $\alpha=0.509\pm0.073$ (about $7\sigma$ different from
$\alpha=1$) for Case 1 and $\alpha=0.552\pm0.035$ (about $13\sigma$ different
from $\alpha=1$) for Case 1 HiRes. Note that we do not expect to derive
exactly the same values of $\alpha$ as before since we have a fixed
threshold of $\Sigma\geq 0.01\:{\rm g\:cm^{-2}}$ to include points in
the analysis and so reducing the gas-to-dust mass ratio causes us to
lose some data points near this limit.
Figure \ref{Sigcomp18}b shows the ratio $\Sigma_{\rm
C18O,TOT}/\Sigsmf$ versus $\Sigsmf$ for our fiducial Case 1 and Case
1 HiRes analyses, with the derived power law relations overlaid. For
$0.01<\Sigsmf/{\rm g\:cm^{-2}} <0.03$ the mean values of $\Sigma_{\rm
C18O,TOT}/\Sigsmf$ are 1.316 and 1.471 for Case 1 and Case 1 HiRes,
respectively. By the time $\Sigsmf\gtrsim 0.1\:{\rm g\:cm^{-2}}$,
$\Sigma_{\rm C18O,TOT}/\Sigsmf$ has declined to values of $\lesssim
0.4$.
In Case 2 we attempt to account for the IRDC envelope: we consider
that we can do this reliably only for strips 2, 3 and 4, where the
envelope is relatively weak compared to the filament. Figure
\ref{Sigcomp18}c presents $\Sigma_{\rm C18O,FIL}$ versus $\Sigsmf$ for
Case 2. The best fit power law relation to the CO(1-0) resolution data
of $\Sigma_{\rm C18O,FIL}/{\rm g\:cm^{-2}} = A (\Sigma_{\rm SMF}/{\rm
g\:cm^{-2}})^\alpha$ has $\alpha=0.239\pm0.080$ and
$A=0.074\pm0.017$. For Case 2 HiRes (i.e. at the CO(2-1) resolution,
adopting CO(1-0) $\tex$ estimates) we find $\alpha = 0.317 \pm 0.038$
and $A = 0.090\pm 0.010$. These uncertainties assume the same
measurement uncertainties as Case 1, except an additional systematic
error of $\rm 0.01 g\:cm^{-2}$ has been applied to $\Sigma_{\rm
C18O,FIL}$ due to uncertainties associated with envelope
subtraction. Again these results indicate a significant ($10\sigma$
and $18\sigma$ for Case 2 and Case 2 HiRes, respectively) departure
from a one-to-one ($\alpha=1$) relation, which we again interpret as
evidence for CO depletion. The results with a gas-to-dust mass ratio
of 100 are $\alpha=0.303\pm0.11$ (about $6\sigma$ different from
$\alpha=1$) for Case 2 and $\alpha=0.372\pm0.048$ (about $13\sigma$
different from $\alpha=1$) for Case 2 HiRes.
Figure \ref{Sigcomp18}d shows the ratio $\Sigma_{\rm
C18O,FIL}/\Sigsmf$ versus $\Sigsmf$ for Case 2 and Case 2 HiRes,
with the above power law relations overlaid. For $0.01<\Sigsmf/{\rm
g\:cm^{-2}} <0.03$ the mean values of $\Sigma_{\rm
C18O,FIL}/\Sigsmf$ are 1.099 and 1.238 for Case 2 and Case 2 HiRes,
respectively. These values are smaller than their equivalents for Case
1, as is to be expected now that we are allowing for the molecular
envelope. The values are also very close to unity, suggesting that our
adopted $\ceto$ abundances and dust opacity per unit gas mass are
reasonable. Again, by the time $\Sigsmf\gtrsim 0.1\:{\rm g\:cm^{-2}}$,
$\Sigma_{\rm C18O,TOT}/\Sigsmf$ has declined to values of $\lesssim
0.4$.
\subsection{Alternatives to CO Depletion}
There are several physical processes that could be responsible for the
observed trend of decreasing $\Sigceto/\Sigsmf$ with increasing
$\Sigsmf$. One possibility could be that our corrections for the
optical depth of the $\ceto$ emission are systematically
underestimated near the center of the filament where the column
density is large. However, the largest optical depth corrections in
the highest column density locations increase the column by only 30\%
(the highest optical depths are $\sim 1$, but lower when averaged over
the whole column), so this effect is unlikely to be driving the
observed trend.
HT11 suggested their observed trend of decreasing $\Sigma_{\rm 13CO}/\Sigsmf$
with increasing $\Sigsmf$ could potentially result if at the same time
there is a systematic decrease in the excitation temperature of about
5~K. However, from our $\tex$ estimates, we find no strong negative
temperature gradient within the IRDC towards the mass surface density
peaks. In fact, $\tex$ increases slightly towards to the center of the
filament, probably as the densities become greater than the effective
critical densities and the lower CO levels can thermalize. Thus, we
exclude trends in $\tex$ as causing the observed variation of
$\Sigceto/\Sigsmf$.
Fractionation of $\ceto$ could in principle change the local abundance
of this molecule, but the most important way in which this can be
achieved is via isotope selective photodissociation at cloud edges,
which would not be able to explain the trends of decreasing $\ceto$
abundance that we see running from $\Sigma\simeq 0.02\:{\rm
g\:cm^{-2}}$ ($A_V\simeq 4$~mag) to $\simeq 0.2\:{\rm g\:cm^{-2}}$
($A_V\simeq 40$~mag).
Another possibility to be considered is systematic changes in $\rm
8\:\mu m$ dust opacities for gas at higher densities. If the opacity
was to increase (e.g. due to grain coagulation and/or ice mantle
formation and growth), then this could explain our observed trend of
decreasing $\Sigceto/\Sigsmf$ with increasing $\Sigma$. The Ossenkopf
\& Henning (1994) dust models do show an increase of $\kappa_{\rm 8\mu
m}$ of 19\% going from the uncoagulated thin ice mantle model to the
uncoagulated thick ice mantle (all volatiles depleted) model. Maximal
coagulation (corresponding to that expected after $10^5$~yr at
densities of $10^8\:{\rm cm^{-3}}$or after $\sim 10^8$~yr at densities
of $\sim 10^5\:{\rm cm^{-3}}$ , which is probably more that can be
expected to have occurred since the observed densities of IRDC cores
are $\lesssim 10^{5}\:{\rm cm^{-3}}$; BT09) raises $\kappa_{\rm 8\mu
m}$ by an additional 17\%. Thus, ice mantle growth and grain
coagulation appears to be able to account for only a small fraction of
the observed variation of $\Sigceto/\Sigsmf$.
We conclude the most likely cause of the trend of decreasing
$\Sigceto/\Sigsmf$ with increasing $\Sigsmf$ is CO depletion due to
freeze out onto dust grains. This would cause a systematic reduction
in the amount of CO gas observed in higher mass surface density
regions, which are likely to also be of higher volume
density.
\subsection{CO Depletion and Implications}
Following the definitions of \S1 and the notation of Fontani
et al. (2006), the depletion factor is
\begin{equation}
f_D\equiv\frac{X^E_{\rm CO}}{X^O_{\rm CO}}=\frac{\Sigma_{\rm SMF}}{\Sigma_{\rm C18O}},
\label{eq:depfact}
\end{equation}
where $X^E_{\rm CO}$ is the expected abundance of CO relative to $\rm
H_2$ given standard gas phase abundances, $X^O_{\rm CO}$ is the
observed abundance and the last equality assumes that $\Sigma_{\rm
SMF}$ estimated from MIR extinction mapping is an accurate measure
of the true mass surface density (this assumption is discussed further
below). Given the uncertainties in the absolute values of the $\ceto$
abundance and the MIR dust opacity per unit gas mass, we renormalize
$f_D$ to be unity for the regions of the IRDC with $0.01<\Sigsmf/{\rm
g\:cm^{-2}} <0.03$ and refer to this renormalized value as the
relative depletion factor $f_D^\prime = B f_D$, where the scaling
factor, $B=1.316, 1.471, 1.099, 1.238$ for Case 1, Case 1 HiRes, Case
2, Case 2 HiRes, respectively. We show maps of $f_D^\prime$ for these
four cases in Figure \ref{6paneldata}e-h. We note that the values of
$f_D^\prime$ presented here, peaking at values $\simeq 5$, are mass
surface density weighted averages and thus lower limits to the maximum
values of the depletion factor that apply in the densest regions of
the cloud.
We conclude that with high ($\sim 10 \sigma$) significance, widespread
CO depletion is occurring in this IRDC, with depletion factors of up
to $\sim 5$ (see Table~\ref{tab:depletion}). These values are larger
than those seen towards more evolved cores and clumps already
containing massive protostars (Fontani et al. 2006; Thomas \& Fuller
2008). Our measurement of CO depletion suffers from fewer systematic
uncertainties, especially since we do not require knowledge of the
dust temperature.
Each pixel in the lower resolution depletion maps (11\arcsec, half the
$\ceto$(1-0) angular resolution) corresponds to a length of 0.155~pc
at the cloud distance of 2.9~kpc, and so contains a mass of
$11.4(\Sigma/0.1{\rm g\:cm^{-2}})\:M_\odot$. Thus, hundreds of solar
masses appear to be affected by depletion along the filament (the
total SMF-derived mass in the 4 strips is $580\pm 230\:M_\odot$,
HT11), including a particularly prominent massive core or clump in
strip 2 and a larger clump partially in strip 4 and extending to the
south.
Thus, IRDC G035.30-00.33 is one of the most massive clouds in which CO
depletion has been detected by direct CO-based and non-CO-based
measurements of mass surface density. Our results also suggest that CO
depletion will be a common occurrence in IRDCs, since the values of
$\Sigma\sim 0.1\:{\rm g\:cm^{-2}}$ in this cloud are quite typical
(e.g. BT09). CO is therefore an imperfect tracer of a significant
fraction of the mass of IRDCs (not just the coldest, densest
cores). Accurate accounting for depletion and/or use of species
suffering minimal depletion, such as $\rm NH_3$ and $\rm N_2H^+$, are
required for more accurate dynamical studies of these clouds.
An estimate of the CO depletion timescale due to freeze-out onto dust
grains is $t_D\simeq 8000/(n_{\rm H_2,5}S)\:{\rm yr}$, where $n_{\rm
H_2,5}$ is the number density of $\rm H_2$ molecules in units of
$10^5\:{\rm cm^{-3}}$ and $S$ is the sticking probability (of order
unity; e.g. Tielens \& Allamandola 1987) for CO on grains. We can
apply this to the thinnest region of the IRDC: the $\sim 5\arcsec$
(0.070~pc) wide filament near the center of strip 3, which appears to
have significant CO depletion with $f_D^\prime\sim 3-4$. Assuming the
depth of the filament, which has $\Sigsmf\simeq 0.2\:{\rm
g\:cm^{-2}}$, is similar to its width, then $n_{\rm H_2,5}=2.0$ and
$t_D\simeq 4000$~yr. This provides a lower limit to the age of this
part of the IRDC. The free-fall time, $t_{\rm
ff}=(3\pi/[32G\rho])^{1/2}$, for this density is $6.9\times
10^4\:{\rm yr}$, i.e. much longer. However, if the filament has been
created by larger scale supersonic flows, then one might expect the
high density gas to have been present for about the flow crossing time
across the width of the filament. Velocities of $\sim 10\:{\rm
km\:s^{-1}}$ may be relevant in models of GMC-GMC collisions (Tan
2000) or if the large-scale SiO emission seen towards this filament
(Jim\'enez-Serra et al. 2010) has been created by such flows. The flow
crossing time at this speed for this part of the IRDC is only
$6800$~yr. Thus the fact that we see CO depletion in these very thin
filaments of the IRDC can help to constrain models for the cloud's
formation. For models in which the cloud lifetime is less than the
flow crossing time across the filament, a constraint is placed on the
flow speed. For the thinnest region of this IRDC, this corresponds to
flow speeds $\lesssim 17\:{\rm km\:s^{-1}}$.
\begin{figure*}[!tb]
\begin{center}$
\begin{array}{cc}
\includegraphics[width=3in,angle=0]{f2a.eps} &
\includegraphics[width=3in,angle=0]{f2b.eps}
\end{array}$
\end{center}
\caption{
\small
Velocity structure of the $\ceto$ molecules associated with the IRDC
and its envelope. The column density distribution, ${\rm d}N_{\rm
C^{18}O}/{\rm d}v$, has been derived from the $\ceto$(1-0) and (2-1)
spectra, local estimates of $\tex$ and including optical depth
corrections. {\it ($\rm a$) Left:} the 4 sets of profiles (offset to
display from top to bottom and labeled 1 to 4) correspond to the 4
strips shown in Figure~\ref{6paneldata}. The dotted, red line is the
summed contribution from gas from the central region of each strip,
corresponding to the IRDC ``filament'' (see Figure~\ref{6paneldata}
and text). The dot-dashed and long-dashed blue lines show summed
contribution from the gas from the eastern and western envelope
regions, respectively. {\it (b) Right:} Illustration of envelope
subtraction (Case 2, see text). For the same strips as in (a), we
subtract the average of the eastern and western envelopes
(short-dashed blue lines) from the filament (dotted red lines), to
leave an estimate of the material in the filament (solid black
lines). We consider this process unreliable for strip 1, where the
envelope contains a similar amount of material as the filament. }
\label{envspectra18}
\end{figure*}
\begin{figure*}[!tb]
\begin{center}$
\begin{array}{cc}
\includegraphics[width=3in]{f3a.eps} &
\includegraphics[width=3in]{f3c.eps}\\
\includegraphics[width=3in]{f3b.eps} &
\includegraphics[width=3in]{f3d.eps}
\end{array}$
\end{center}
\caption{
\small
Evidence for CO depletion. {\it (a) Top Left:} Comparison of
$\Sigma_{\rm C^{18}O,TOT}$ (i.e. Case 1) and $\Sigsmf$ for all
$\ceto$(1-0) (crosses) and $\ceto$(2-1) (dots) pixels for which both
$\Sigceto$ and $\Sigsmf>0.01\:{\rm g\:cm^{-2}}$ and the pixel is
$<20\%$ affected by $\Sigsmf$ artifacts, e.g. due to MIR bright
sources. The dotted line shows the condition $\Sigma_{\rm
C^{18}O,TOT}=\Sigsmf$. The solid, dashed lines show the best-fit
power law relations to the $\ceto$(1-0), $\ceto$(2-1) resolution data,
respectively. {\it (b) Bottom Left:} Ratio $\Sigma_{\rm
C^{18}O,TOT}/\Sigsmf$ (i.e. Case 1) versus $\Sigsmf$, with the same
symbol and line notation as in (a). The horizontal solid, dashed lines
from $0.01<\Sigsmf/{\rm g\:cm^{-2}} <0.03$ indicate the mean values of
the data in this range for the $\ceto$(1-0), $\ceto$(2-1) resolution
data, respectively. The cross in the upper-right corner indicates
typical estimated uncertainties. {\it (c) Top Right:} Same as (a), but
now estimating $\Sigma_{\rm C^{18}O,FIL}$ from molecular gas
associated with the filament after envelope subtraction (Case 2) in
strips 2, 3 and 4. {\it (d) Bottom Right:} Same as (b), but for Case
2. Both (b) and (d) show that $\Sigceto/\Sigsmf$ decreases by up to a
factor of $\sim 5$ as $\Sigsmf$ increases from $\sim 0.02\:{\rm g\:cm^{-2}}$
up to $\sim 0.2\:{\rm g\:cm^{-2}}$.}
\label{Sigcomp18}
\end{figure*}
\begin{table}
\centering
\caption{Parameters of Depletion Factor Analysis}
\label{tab:depletion}
\begin{tabular}{lcccc}
\hline
\hline
Case & $\alpha$ & $A$ & $B$ & $f_D^\prime$(max) \\
\hline
Case 1 & $0.452\pm0.054$ & $0.146\pm0.023$ & 1.316 & 3.5 (at $\Sigsmf=0.16\:{\rm g\:cm^{-2}}$) \\
Case 1 HiRes & $0.463\pm 0.025$ & $0.151\pm0.010$ & 1.471 & 4.6 (at $\Sigsmf=0.20\:{\rm g\:cm^{-2}}$) \\
Case 2 & $0.239\pm0.080$ & $0.074\pm0.017$ & 1.099 & 3.8 (at $\Sigsmf=0.16\:{\rm g\:cm^{-2}}$)\\
Case 2 HiRes & $0.317 \pm 0.038$ & $0.090\pm 0.010$ & 1.238 & 4.9 (at $\Sigsmf=0.20\:{\rm g\:cm^{-2}}$)\\
\hline
\end{tabular}
\end{table}
\acknowledgments We thank E. van Dishoeck and R. Visser for helpful
discussions and the comments of an anonymous referee, which helped
improve the paper. AKH acknowledges support from a SEAGEP Dissertation
Fellowship. JCT acknowledges support from NSF CAREER grant
AST-0645412; NASA Astrophysics Theory and Fundamental Physics grant
ATP09-0094; NASA Astrophysics Data Analysis Program ADAP10-0110 and a
Faculty Enhancement Opportunity grant from the University of Florida.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,076 |
Sámuel Gyarmathi () (July 15, 1751, Kolozsvár — March 4, 1830, Kolozsvár) was a Hungarian linguist, born in Cluj (then Kolozsvár, Transylvania). He is best known for his systematic demonstration of the comparative history of the Finno-Ugric languages in the book Affinitas linguae hungaricae cum linguis fennicae originis grammatice demonstrata (1799), which rested on the earlier work of János Sajnovics.
Life and works
Gyarmathi studied to be a teacher in Nagyenyed (Aiud) before training to be a doctor in Vienna, after which he practised medicine in Transylvania. In 1789 he read of a competition offering a prize for linguistic research in a Hungarian newspaper and spent the next two years working on his Okoskodva tanító magyar nyelvmester (Hungarian Grammar Taught Rationally). The Transylvanian Diet made funds available for its publication and it appeared in two volumes in 1794.
Through the success of this book, Gyarmathi joined the household of Count Gergely Bethlen as a family physician and tutor to the Bethlen children. His position gave him plenty of leisure for his research into languages and allowed him to accompany Bethlen's son on a trip to the University of Göttingen in Germany, then a leading centre for comparative linguistics. Here he made the acquaintance of the historian August Ludwig von Schlözer, who was a specialist in Northern and Eastern Europe.
In Göttingen, Gyarmathi developed the theories of János Sajnovics, which had shown a relationship between Hungarian and Sami (Lapp). The result of Gyarmathi's studies was Affinitas, published in Göttingen in 1799. In the first part of the work, Gyarmathi compares Hungarian, Finnish and Sami. In the second, he treats of the similarities between Hungarian and Estonian. In the third, he covers several other Uralic languages. Affinitas sought to show that these languages were part of the same family, by demonstrating similarities in grammatical structure between them. The book was immediately recognised as a major contribution to linguistics.
After leaving Göttingen, Gyarmathi became a teacher/administrator at the Calvinist College in Zilah (Zalău), before returning to work as the family physician to the Bethlens in 1810. His last major work was Vocabularium, published in Vienna in 1816. This is a word list that compares Hungarian vocabulary with 57 other languages. It also contains valuable information on the Szekler dialect of Transylvania. Gyarmathi died in Cluj at the age of 79.
See also
Comparative method
Notes
References
Editorial material in Sámuel Gyarmathi Grammatical Proof of the Affinity of the Hungarian Language With Languages of Fennic Origin (a translation of the Affinitas by Victor Egon Hanzeli, Amsterdam Classics in Linguistics Vol.19, 1983)
External links
Affinitas linguae hungaricae cum linguis fennicae originis grammatice demonstrata (1799)
1751 births
1830 deaths
18th-century Hungarian people
19th-century Hungarian people
18th-century linguists
19th-century linguists
Linguists from Hungary
Historical linguists
Hungarian Finno-Ugrists
Writers from Cluj-Napoca | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,875 |
__author__ = 'jfernandez'
"""
Imports all steps already defined and implemented in 'install_product' feature
"""
from component.install_product.features.install_product import *
from commons.utils import wait_for_software_installed, generate_content_installed_by_product
from commons.fabric_utils import execute_content_in_file
def _check_product_attributes_installed_helper(attributes):
"""
For each attribute, this method will validate if that attribute is in the file created by the recipe/manifest in
the remote VM.
:param attributes: Product instance attributes
"""
for attribute in attributes:
assert_true(execute_content_in_file(world.file_name,
generate_content_installed_by_product(world.product_name,
world.product_version,
attributes,
installator=world.cm_tool)),
"Attribute value not found in product installed [{}]".format(attribute[VALUE]))
@step(u'a installed product with name "([^"]*)" and release "([^"]*)"')
def installed_product(step, product_name, product_version):
a_created_product_with_name_group1(step, product_name, product_version)
i_install_the_product_in_the_vm(step)
task_is_created(step)
the_task_is_performed(step)
@step(u'the task is performed')
def the_task_is_performed(step):
the_task_has_finished_with_status_group1(step, TASK_STATUS_VALUE_SUCCESS)
@step(u'the product is installed')
def the_product_is_installed(step):
world.file_name = PRODUCT_FILE_NAME_FORMAT.format(product_name=world.product_name,
product_version=world.product_version,
installator=world.cm_tool)
assert_true(wait_for_software_installed(status_to_be_finished=True, file_name=world.file_name),
"ERROR: SOFTWARE IS NOT INSTALLED")
@step(u'the product with attributes is installed')
def the_product_with_attributes_is_installed(step):
""" Step: Checks if the product has been installed using default attribute values defined in the scenario """
the_product_is_installed(step)
_check_product_attributes_installed_helper(world.instance_attributes)
@step(u'the product instance has been installed without attributes')
def the_product_instance_has_been_installed_without_attributes(step):
""" Step: Checks if the product has been installed using default attribute values defined in recipe/manifest """
the_product_is_installed(step)
_check_product_attributes_installed_helper(DEFAULT_ATTRIBUTE[ATTRIBUTE])
@step(u'the task has finished with status "(.*)" after "(.*)" checks')
def the_task_has_finished_with_status_group1_after_group2_checks(step, status, checks):
"""
Step: Waits for task execution. It will check after 5 seconds the task status. Number of checks by param 'checks'
"""
finished = wait_for_task_finished(vdc_id=world.tenant_id, task_id=world.task_id,
status_to_be_finished=status, headers=world.headers, seconds=int(checks))
assert_true(finished, 'Task is not in the correct status. Expected: {}'.format(status))
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,993 |
Q: Running a Powershell Script with no errors and no resuults I found this script here on Stack Overflow by Pmental
I am new too PowerShell, but when I run the script, (even as admin) the script seems to run but I get no output either to the screen or to any file.
I used: Get-ProcessPlus and Get-ProcessPlus using Firefox with no errors and no results. What am I doing wrong?
Any help would be great.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,715 |
Q: Fragment doesn't show properly in ViewPager I have an Activity with a traditionnal viewpager component. This viewpager contains MainFragment class (it's a standard fragment). This is the layout of this fragment:
<FrameLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:id="@+id/fragment_container"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_weight="1">
</FrameLayout>
Ps: I need to use this framelayout because I can switch with 2 fragments inside according an user data.
In this MainFragment I want to show a specific fragment according an user data:
@Override
public void onViewCreated(View view, Bundle savedInstanceState) {
super.onViewCreated(view, savedInstanceState);
final Bundle bundle = getArguments();
int mydata = bundle.getInt(BUNDLE_KEY_MY_DATA);
if (mydata == 1)
addFragment(Fragment1.newInstance(), Fragment1.TAG);
else if (mydata == 2)
addFragment(Fragment2.newInstance(), Fragment2.TAG);
}
My addFragment method:
protected void addFragment(@NonNull Fragment fragment,
@NonNull String fragmentTag) {
getFragmentManager()
.beginTransaction()
.add(R.id.fragment_container, fragment, fragmentTag)
.disallowAddToBackStack()
.commit();
}
But there is a strange behavior during execution (no crash!). For example in my tab (associated with the viewpager) there are these elements: "1" "2" "3" "4" "5" "6" "7"
So the default position is on element "4"; it works perfectly. Now if I swipe for example to "3", the Fragment1 (or Fragment2 doesn't matter) doesn't appear. To show it, I must to swipe again (so to "2"), then come back to "3" and the fragment appears correctly.
Thanks for your help guys!
A: Use Fragment.getChildFragmentManager() in your addFragment().
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 474 |
I write to be read. Although of course it's scary sending work out: people who don't know me will be thinking about and making judgements on my writing – good and bad. But that's why I do it.
I know lots of writers who have trouble sharing their work, even amongst the safe environment of the writing group I'm in, let alone sending their manuscripts out to agents. Of course when Our Endless Numbered Days gets a poor review (and all books do), it hurts and for five minutes I wonder why I'm putting myself through this. And then I come across someone who has loved my novel, wants to talk about the twists, what they have spotted, and the ending. And I realise it has touched them in some small way and that makes it worthwhile.
I think my relatively thick skin comes from my first degree in sculpture. At least twice a week for three years, us sculpture students participated in 'crits'. All ten of us and our lecturer or head of department stood around a student's piece and said what we thought was working and what could be improved. There were always many things in everyone's work which could be made better, but most of the time (as long as the criticism was considered and a justification was made) I liked hearing what my fellow students had to say. Best of all was when I had a piece in an exhibition and I got to hear what the public thought.
Just like putting my sculpture on display, knowing my book will be read by strangers; that the words from my head will go into someone else's and change in the process, is why I write.
Claire Fuller's debut novel, Our Endless Numbered Days, is published by Fig Tree in the UK and last week won the Desmond Elliott prize (huge congratulations, Claire!). She has also written many short stories and pieces of flash fiction; several of which have been published and won competitions including the BBC's Opening Lines. She has an MA in Creative and Critical Writing and lives in Winchester, England with her husband and children.
Find out more or say hi to her on Twitter.
This entry was posted in Why I Write on July 6, 2015 by Nicci Cloke. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,215 |
Archive for 'Mumbai'
Mumbai aftermath
Allahpundit has an extensive roundup of the news on the terrorist attacks in Mumbai, India. Still waiting on the all-clear at the Taj hotel.
This was a sophisticated attack. The terrorists had thoroughly cased the locations, and had checked into the hotel days in advance. The terrorists targeted Jews, killing five, including a rabbi and his wife. Americans were also targeted, including a Virginia man and his daughter who were killed.
There is no fool-proof means of preventing such attacks, in India or anywhere else. The only effective long-term strategy is to identify terrorist groups and their members, place them under surveillance, and try to disrupt their plans by arresting them for weapons charges, immigration violations and the like.
Posted in Mumbai, terrorism | Leave a Comment »
Two Americans killed in Mumbai
A father and his daughter:
The U.S. State Department said on Friday two Americans were among those killed in the attacks by militants in the Indian financial capital Mumbai.
Spokesman Gordon Duguid said the department had notified the families of the victims. He did not give details or identify the dead Americans.
Earlier, a group called the Synchronicity Foundation said two Americans who were in India as part of a meditation program had died in the attacks.
The Synchronicity Foundation said in a statement on its website that Alan Scherr and his 13-year-old daughter Naomi were killed.
It gave no details about how they were killed but said Scherr and his family had been involved in the Synchronicity community in Faber, Virginia for more than a decade.
Allahpundit has a fresh thread.
UPDATE: Five Jewish hostages killed at Chabad House.
Death toll estimated at 145.
UPDATED & BUMPED: Hostage standoffs continue:
Fresh gunfire and explosions were heard late Thursday in Mumbai as police battled terrorists at three sites almost 24 hours after the first wave of violence hit the city.
Fresh explosions have been heard at the Taj Mahal hotel, where police are trying to free hostages.
Indian Prime Minister Manmohan Singh suggested the group behind the terrorist attacks, which killed 125 people, was based outside the country.
CNN reporters said regular gun fire and blasts could be heard Thursday at the Oberoi and Taj Mahal hotels and a Jewish center in the city. . . .
"It is evident that the group which carried out these attacks, based outside the country, had come with single-minded determination to create havoc in the financial capital of the country," [Singh] said. . . .
Authorities found 8 kilograms (17 pounds) of RDX, one of the most powerful kinds of military explosives, at a restaurant near the Taj, indicating that the attackers may have been planning more violence.
Gunmen also remained holed up in a building called Chabad House, where several Jewish families live. Rabbi Gabriel Holtzberg, the city's envoy for the community, was being held inside with his wife, a member of the Hasidic Jewish movement said. The couple's 18-month-old baby was released unharmed.
Possibility of an al-Qaeda link is unclear:
Christine Fair, senior political scientist and a South Asia expert at the RAND Corporation, was careful to say that the identity of the terrorists could not yet be known. But she insisted the style of the attacks and the targets in Mumbai suggested the militants were likely to be Indian Muslims and not linked to Al Qaeda or Lashkar-e-Taiba, another violent South Asian terrorist group.
"There's absolutely nothing Al Qaeda-like about it," she said of the attack. "Did you see any suicide bombers? And there are no fingerprints of Lashkar. They don't do hostage-taking and they don't do grenades." By contrast, [security expert Sajjan] Gohel in London said "the fingerprints point to an Islamic Al Qaeda-affiliated terrorist group."
Allahpundit has a fresh thread today.
PREVIOUSLY: India is hit with the kind of large-scale coordinated attacks that are an al-Qaeda trademark:
At least 101 people have been killed in attacks by gunmen in Mumbai , police said on Thursday. At least six foreigners have been killed and the death figure has gone up to 101 now," Ramesh Tayde, a senior police officer told from Mumbai's control room.
In one of the most violent terror attacks on Indian soil, Mumbai came under an unprecedented night attack as terrorists used heavy machine guns, including AK-47s, and grenades to strike at the city's most high-profile targets — the hyper-busy CST (formerly VT) rail terminus; the landmark Taj Hotel at the Gateway and the luxury Oberoi Trident at Nariman Point; the domestic airport at Santa Cruz; the Cama and GT hospitals near CST; the Metro Adlabs multiplex and Mazgaon Dockyard — killing at least 80 and sending more than 900 to hospital, according to latest reports.
The attacks have taken a tragic toll on the city's top police brass: The high-profile chief of the anti-terror squad Hemant Karkare was killed; Mumbai's additional commissioner of police (east) Ashok Kamte was gunned down outside the Metro; and celebrated encounter specialist Vijay Salaskar was also killed.
The Indian Mujahideen group that has taken credit for the attack has clear ties to Pakistan, the Weekly Standard reports:
Indian intelligence believes the Indian Mujahideen is a front group created by Lashkar-e-Taiba and the Harkat ul Jihad al Islami to confuse investigators and cover the tracks of the Students' Islamic Movement of India, or SIMI, a radical Islamist movement. The groups receive support from Pakistan's Inter-Service Intelligence and are al Qaeda affiliates.
Hot Air has an extensive roundup, including this CNN video:
The Counter-Terrorism Blog names the group Lashkar-e-Toiba as a likely sponsor of this attack. The terrorists were seeking American and British hostages at the luxury hotels in Mumbai:
Two terrorists carrying guns tonight took 15 people, half of them foreigners, hostage on the roof of the luxury Taj Hotel, one of the hostages who managed to escape said.
…Replying to a question, Patel said the terrorists wanted to know if any one of the hostages was carrying American and British passports.
They clearly wanted foreigners, he added.
Via Ace of Spades, where the flaming skull alert is in effect.
Posted in India, Mumbai, terrorism | Leave a Comment » | {
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"redpajama_set_name": "RedPajamaCommonCrawl"
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