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\section{Introduction} \label{sec:1}
\IEEEPARstart{C}{oding} for broadcast channels, where receivers know some part of the transmitted messages a priori, is called \emph{index coding} and is well-known for noiseless binary broadcast channels~\cite{YBJK_IEEE_IT_11,ALSWH_FOCS_08,RSG_IEEE_IT_10}.
In the case of noisy binary broadcast, the index codes of~\cite{DSC_IT_13} provide equal error correcting capability at all receivers and exploit the receiver side information to enhance the code rate, while the codes of~\cite{XFKC_CISS_06,BaC_ITW_11,MLV_PIMRC_12} transform side information into improvements in error performance.
The capacity of general index coding over Gaussian broadcast channel is unknown, but information theoretic results are available for some special cases~\cite{KrS_ITW_07,Wu_ISIT_07,SiC_ISIT_14,AOJ_ISIT_14,Tun_IEEE_IT_06}.
Separation-based coding schemes using a (noiseless) index code and a broadcast channel code are, in general, sub-optimal, since the channel decoders do not utilize the receiver side information, and the channel coding rate is limited by the receiver with the worst signal-to-noise ratio. This motivates schemes that perform index coding at the physical layer.
Lattice based codes were proposed in~\cite{NHV_arxiv_14} for the special case of index coding over the Gaussian broadcast channel where the transmitter has $K$ independent messages, each receiver knows some subset of the $K$ messages a priori, and every receiver demands all the messages at the source.
These index codes are designed to convert receiver side information into apparent ${\sf SNR}$ gains. The minimum distance of the effective code perceived by a receiver is a function of the index subset \mbox{$S \subset \{1,\dots,K\}$} of the messages available at the receiver as side information.
The \emph{side information gain} of a code is a metric that measures the efficiency with which receiver side information is converted to actual coding gain~\cite{NHV_arxiv_14}. The index codes of~\cite{NHV_arxiv_14} provide large side information gains, and they can be concatenated with outer channel codes to improve coding gain against channel noise.
These index codes, however, suffer from two practical drawbacks: \emph{(i)}~they do not encode all messages at equal rate, and \emph{(ii)}~they do not admit message sizes that are powers of $2$.
In this letter, we present the first class of index codes for this special case of Gaussian broadcast channel that encode all the messages with equal rate (Section~\ref{sec:3}). These new index codes allow messages of arbitrary sizes, including sizes that are powers of $2$. The proposed index codes are multidimensional QAM constellations whose points are labelled with message symbols using the framework of linear codes over the ring $\Zb_M$ of integers modulo $M$.
Using a computer search, we obtain QAM index codes with large side information gains for message sizes $2^m$, \mbox{$m \leq 6$}, and number of messages \mbox{$K \leq 5$}.
We also present simulation results on the performance of a QAM index code when used as a modulation scheme in a system employing an outer channel code (Section~\ref{sec:4}). We observe that the new $16$-QAM index modulation scheme for \mbox{$K=2$} messages, when encoded with an off-the-shelf rate-$\sfrac{1}{2}$ LDPC code, performs $4.3$~dB away from capacity in the Gaussian broadcast channel at $10^{-4}$ bit error rate.
\section{Index codes for Gaussian Broadcast Channel} \label{sec:2}
We consider a non-fading Gaussian broadcast channel with single-antenna terminals, where every receiver demands $K$ independent messages from the transmitter, denoted by $w_1,\dots,w_K$ that assume values from $\Wc_1,\dots,\Wc_K$, respectively. The transmitter operates under an average power constraint, the receivers experience additive white Gaussian noise (with possibly different noise powers), and each receiver has prior knowledge of some subset of the $K$ messages as side information.
An $n$--dimensional \emph{index code} $(\rho,\Xc)$ for this Gaussian broadcast channel consists of a channel code \mbox{$\Xc \subset \Rb^n$} and an encoding function \mbox{$\rho: \Wc_1 \times \cdots \times \Wc_K \to \Xc$}. The rate of transmission of the $k^{\text{th}}$ message is $R_k=\sfrac{1}{n} \log_2 |\Wc_k|$ bits per dimension (b/dim).
A receiver that has the prior knowledge of the symbols \mbox{$\pmb{w}_S=(w_k,k \in S)$}, \mbox{$S \subsetneq \{1,\dots,K\}$}, and experiences a signal-to-noise ratio of ${\sf SNR}$ is denoted by $({\sf SNR},S)$.
We are interested in codes that provide good error performance (versus ${\sf SNR}$) for every \mbox{$S \subsetneq \{1,\dots,K\}$}, or equivalently, for \mbox{$2^K-1$} receivers, one corresponding to each $S \subsetneq \{1,\dots,K\}$.
Consider the channel output \mbox{$\pmb{y} = \rho(w_1,\dots,w_K) + \pmb{z}$} at a generic receiver $({\sf SNR},S)$, where \mbox{$\pmb{z} \in \Rb^n$} is the additive Gaussian noise with variance $\sfrac{1}{\sf SNR}$ per dimension.
A receiver with no side information, i.e. with \mbox{$S=\varnothing$}, decodes $\pmb{y}$ to \mbox{$\arg \min_{\pmb{x} \in \Xc} \|\pmb{y}-\pmb{x}\|$}. The minimum Euclidean distance \mbox{$d_0=\min \{ \|\pmb{x}_1 - \pmb{x}_2\|\, \vert \,\pmb{x}_1,\pmb{x}_2 \in \Xc, \pmb{x}_1 \neq \pmb{x}_2 \}$} between any pair of points in $\Xc$ determines the error performance at this receiver. A receiver with \mbox{$S \neq \varnothing$} has prior knowledge of the value of the message vector $\pmb{w}_S$. Given the information \mbox{$w_k=a_k$}, \mbox{$k \in S$}, written concisely as \mbox{$\pmb{w}_S=\pmb{a}_S$}, this receiver generates a subcode \mbox{$\Xc_{\pmb{a}_S} \subset \Xc$} by expurgating all codewords in $\Xc$ with \mbox{$\pmb{w}_S \neq \pmb{a}_S$}, and decodes $\pmb{y}$ to the closest point in $\Xc_{\pmb{a}_S}$.
Let $d_{\pmb{a}_S}=\{\|\pmb{x}_1-\pmb{x}_2\|\,\vert\,\pmb{x}_1,\pmb{x}_2 \in \Xc_{\pmb{a}_S}, \pmb{x}_1\neq\pmb{x}_2\}$ be the minimum Euclidean distance of $\Xc_{\pmb{a}_S}$, and $d_S=\min_{\pmb{a}_S} d_{\pmb{a}_S}$.
The average error performance and coding gain at this receiver are determined by $d_S$.
The asymptotic additional ${\sf SNR}$ gain due to the knowledge of $\pmb{w}_S$ is thus \mbox{$10 \log_{10} \left( \sfrac{d_S^2}{d_0^2} \right)$~dB}. This squared distance gain must be measured against the amount of side information in $\pmb{w}_S$, or equivalently, against the \emph{side information rate} \mbox{$R_S \triangleq \sum_{k \in S}R_k$~b/dim}.
The \emph{side information gain}~\cite{NHV_arxiv_14} of the code $(\rho,\Xc)$, defined as
\begin{equation} \label{eq:Gamma}
\Gamma \triangleq \min_{\varnothing \subsetneq S \subsetneq \{1,\dots,K\}} \frac{10 \log_{10} \left( \sfrac{d_S^2}{d_0^2} \right)}{R_S} \text{ dB/b/dim},
\end{equation}
is the minimum additional coding gain available from each bit per dimension of side information for any $S$.
The prior knowledge of $\pmb{w}_S$ provides an asymptotic ${\sf SNR}$ gain of at least \mbox{$\Gamma \times R_S$~dB} over the performance of $\Xc$ with no side information. Hence, $(\rho,\Xc)$ is a good index code if \emph{(i)}~$\Xc$ is a good channel code for the traditional single user AWGN channel, i.e., for a receiver with \mbox{$S=\varnothing$}, and \emph{(ii)}~$\Gamma$ is large, so as to maximize the minimum gain from side information for receivers with \mbox{$S \neq \varnothing$}.
To motivate our work, we now show an example of a new index code using $16$-QAM, that encodes two $4$-ary message symbols with equal rate, and provides \mbox{$\Gamma \approx 6$~dB/b/dim}.
\begin{example} \label{ex:16QAM_1}
\begin{figure}[!t]
\centering
\includegraphics[totalheight=2.0in,width=2.0in]{Fig_1.eps}
\caption{The labelling scheme for the $16$-QAM index code. The four points forming the subcode corresponding to the side information \mbox{$w_1=0$} are highlighted with circles. The subcode for \mbox{$w_2=0$} is marked with squares.}
\label{fig:16QAM_labels}
\vspace{-3mm}
\end{figure}
Let \mbox{$K=2$}, and number of receivers be \mbox{$2^K-1=3$}, with the corresponding side information index sets \mbox{$S=\varnothing,\{1\},\{2\}$}, respectively. Let {$\Wc_1=\Wc_2=\{0,1,2,3\}$}, \mbox{$n=2$} and $\Xc$ be the $16$-QAM constellation, then \mbox{$R_1=R_2=1$~b/dim}. Fig.~\ref{fig:16QAM_labels} depicts the new code, where each of the $16$ points $\pmb{x}$ is labelled with the corresponding message tuple \mbox{$\rho^{-1}(\pmb{x})=(w_1,w_2)$}.
The receiver with \mbox{$S=\varnothing$} must decode both $w_1,w_2$, and hence, it decodes the received vector to nearest point in $\Xc$. The error performance at this receiver is that of the $16$-QAM signal set. Let \mbox{$w_1=0$}, then the receiver with \mbox{$S=\{1\}$} knows that the transmit vector is one of the four points corresponding to \mbox{$w_1=0$} (marked with circles in Fig.~\ref{fig:16QAM_labels}), and hence, its decoder restricts its choice of candidate codewords to these four points.
Observe that the minimum Euclidean distance between these four points is twice the minimum Euclidean distance $d_0$ of $\Xc$. The minimum distance corresponding to each of the other three values of $w_1$ is also $2d_0$, and hence, \mbox{$d_S=2d_0$} for \mbox{$S=\{1\}$}. It is easy to check that \mbox{$d_S=2d_0$} for \mbox{$S=\{2\}$} as well. Thus, the error performance at the two receivers, corresponding to \mbox{$S=\{1\},\{2\}$}, respectively, is approximately \mbox{$10\log_{10}(2^2)\approx 6$~dB} better than that of the receiver with \mbox{$S=\varnothing$}.
Since \mbox{$R_S=1$~b/dim} for $S=\{1\},\{2\}$, from~\eqref{eq:Gamma}, the side information gain of this code is $10\log_{10}(2^2) \approx 6$~dB/b/dim. \hfill\IEEEQED
\end{example}
\section{QAM constellations for index coding} \label{sec:3}
In this section, we present multidimensional QAM constellations for index coding using linear codes over the ring of integers modulo $M$.
For even and odd values of $M$, let $\Zb_M$ denote the sets
${\textstyle \left\{-\frac{M}{2},-\frac{M-2}{2},\dots,0,\dots,\frac{M-2}{2}\right\}}$ and ${\textstyle \left\{-\frac{M-1}{2},-\frac{M-3}{2},\dots,0,\dots,\frac{M-1}{2}\right\}}$,
respectively. For any \mbox{$a \in \Zb$}, let \mbox{$a \mod M$} be the unique remainder of $a$ in $\Zb_M$ when divided by $M$. With addition and multiplication performed modulo $M$, the set $\Zb_M$ has the structure of a commutative ring. The $\mod M$ operation satisfies the property that for any \mbox{$x \in \Zb$}, \mbox{$|x \mod M| \leq |x|$}. The set $\Zb_M^n$ of all $n$-tuples is a module over $\Zb_M$ with addition and scalar multiplication performed component-wise.
Similar to the scalar case, we have \mbox{$\|\pmb{x} \mod M \| \leq \|\pmb{x}\|$} for every \mbox{$\pmb{x} \in \Zb^n$}.
A \emph{unit} is an element of a ring with a multiplicative inverse, and the set of all units of a ring form a multiplicative group. In the case of $\Zb_M$, the units are precisely the elements that are relatively prime with $M$ in $\Zb$, i.e.,
\mbox{$U(\Zb_M) = \left\{ a \in \Zb_M \, \vert \, \gcd(a,M) = 1 \text{ in } \Zb \right\}$},
where $\gcd$ denotes the greatest common divisor. When $M$ is a power of $2$, $U(\Zb_M)$ is the set of all odd integers in $\Zb_M$.
Assuming \mbox{$|\Wc_1|=\cdots=|\Wc_K|=M$}, we identify each alphabet $\Wc_k$ with the ring $\Zb_M$. We consider $\Zb_M$--linear encoding of the $K$ messages where the code length equals the number of messages, i.e., \mbox{$n=K$}, and the subcode associated with each message is of rank $1$. The $k^{\text{th}}$ subcode \mbox{$\Xc_k = \left\{ w_k\pmb{c}_k \mod M \, | \, w_k \in \Zb_M \right\}$}, corresponding to the message $w_k$, is generated by a single vector \mbox{$\pmb{c}_k \in \Zb_M^K$}.
\begin{definition} \label{def:linear_index_code}
A \emph{$\Zb_M$-linear index code} for $K$ messages consists of a set of $K$ generators \mbox{$\pmb{c}_1,\dots,\pmb{c}_K \in \Zb_M^K$}, such that the linear encoder
\mbox{$\pmb{x} = \rho(w_1,\dots,w_K) = \sum_{k=1}^{K} w_k \pmb{c}_k \mod M$}
is injective.
\end{definition}
The injectivity of $\rho$ in Definition~\ref{def:linear_index_code} ensures unique decodability of messages at a receiver with no side information. Since the message space \mbox{$\Wc_1 \times \cdots \times \Wc_K=\Zb_M^K$}, injectivity of $\rho$ implies that \mbox{$\Xc=\Zb_M^K$}. In order to transmit the signal, we embed the codeword \mbox{$\pmb{x} \in \Zb_M^K$} into the Euclidean space $\Rb^K$ using the natural map. Hence, the minimum distance with no side information is \mbox{$d_0=1$}.
The linear index code can be viewed as a labelling of the multidimensional QAM constellation $\Zb_M^K$, where each constellation point $\pmb{x}$ is associated with the message tuple $(w_1,\dots,w_K)=\rho^{-1}\left(\pmb{x}\right)$. Note that $\pmb{x}$ may be translated by a fixed offset prior to transmission to minimize the transmit power.
A linear index code is fully characterized by the matrix \mbox{$\pmb{C} \in \Zb_M^{K \times K}$} whose rows are the $K$ generators $\pmb{c}_1,\dots,\pmb{c}_K$. The encoding matrix $\pmb{C}$ defines a linear transformation from the message space \mbox{$\Zb_M^K$} to the space \mbox{$\Xc=\Zb_M^K$} of codewords. Thus, the encoder map $\rho$ is injective if and only if $\pmb{C}$ is invertible over $\Zb_M$, i.e., \mbox{$\det(\pmb{C}) \in U(\Zb_M)$}.
\begin{example}[$16$-QAM] \label{ex:16QAM_2}
Consider \mbox{$M=4$}, \mbox{$K=2$} and the two generators \mbox{$\pmb{c}_1= (1,-2)$} and \mbox{$\pmb{c}_2= (-2,1)$}. The encoder is
$\pmb{x} = w_1\pmb{c}_1 + w_2 \pmb{c}_2 \mod 4 = (w_1 - 2w_2, -2w_1 + w_2) \mod 4$,
and the encoding matrix is
\mbox{$\small \pmb{C} = \begin{pmatrix} \pmb{c}_1 \\ \pmb{c}_2 \end{pmatrix} = \begin{pmatrix}[r] 1 & -2 \\ -2 & 1 \end{pmatrix}$}.
Since \mbox{$\det(\pmb{C})=-3 \mod 4 =1$} is a unit in $\Zb_4$, this code is uniquely decodable. The resulting index code is the $16$-QAM labelling scheme illustrated in Example~\ref{ex:16QAM_1} and Fig.~\ref{fig:16QAM_labels}. \hfill\IEEEQED
\end{example}
\subsection{Side information gain}
All the $K$ messages have the same transmission rate \mbox{$R_k = \sfrac{1}{K} \log_2 M$~b/dim}. The side information rate at the receiver $({\sf SNR},S)$ is \mbox{$R_S = \sum_{k \in S} R_k = \frac{|S|}{K} \log_2 M$~b/dim.}
We now relate the minimum distance $d_S$ to the length of the shortest vector of a certain lattice.
This allows us to numerically compute the value of $d_S$, and hence $\Gamma(\Xc)$, using efficient algorithms available for calculating the shortest vectors in lattices~\cite{FiP_AMS_85}.
Let $\Sc$ denote the complement of the set $S$.
For any \mbox{$S \subset \{1,\dots,K\}$}, the subcode generated by $w_k$, \mbox{$k \in \Sc$}, is
\mbox{$\Xc_{\Sc} = \left\{ \sum_{k \in \Sc} w_k\pmb{c}_k \mod M \, \Big\vert \, w_k \in \Zb_M \right\}$}.
Consider
\begin{equation*}
\textstyle \La_{\Xc_{\Sc}}=\Xc_{\Sc} + M\Zb^K = \left\{\pmb{x} + M\pmb{u} \, \vert \, \pmb{x} \in \Xc_{\Sc}, \pmb{u} \in \Zb^K\right\},
\end{equation*}
which is known as the \emph{Construction~A lattice}~\cite{CoS_Springer_99} of the linear code $\Xc_{\Sc}$.
The lattice $\La_{\Xc_{\Sc}}$ is generated by $\pmb{c}_k$, $k \in \Sc$, and the $K$ rows of $M\pmb{I}_K$. A basis for $\La_{\Xc_{\Sc}}$ can be efficiently computed from this set of generators, for example, using an algorithm based on LLL reduction~\cite{BuP_LecNotes_CompSc_87}.
For any set of points in $\Rb^K$, let $d_{\min}(\cdot)$ denote the minimum Euclidean distance between any two distinct points in the set.
For a lattice $\La$, $d_{\min}(\La)$ equals the length of its shortest vector.
\begin{lemma} \label{lem:dmin_lattice}
If $\La_{\Xc_{\Sc}}$ contains a shortest vector $\pmb{w}$ such that \mbox{$\pmb{w} \notin M\Zb^K$}, then $d_S=d_{\min}\left( \La_{\Xc_{\Sc}} \right)$; else $d_S \geq M$.
\end{lemma}
\begin{IEEEproof}
Let the side information at the receiver $({\sf SNR},S)$ be \mbox{$\pmb{w}_S=\pmb{a}_S$}. Then the subcode $\Xc_{\pmb{a}_S}$ to be decoded is
\begin{equation*}
\textstyle \left\{ \sum_{k \in S} a_k \pmb{c}_k + \sum_{k \in \Sc} w_k \pmb{c}_k \mod M \Big\vert w_k \in \Zb_M, k \in \Sc \right\},
\end{equation*}
that equals \mbox{$\pmb{t} + \Xc_{\Sc} \mod M$},
where \mbox{$\pmb{t}=\sum_{k \in S} a_k \pmb{c}_k \mod M$} is known at the receiver. Since the modulo operation is equivalent to the addition of an appropriate vector from $M\Zb^K$, we have
\begin{equation*}
\Xc_{\pmb{a}_S}=\pmb{t}+\Xc_{\Sc} \mod M \subset \pmb{t} + \Xc_{\Sc}+M\Zb^K=\pmb{t}+\La_{\Xc_{\Sc}}.
\end{equation*}
Hence, $d_{\min}(\Xc_{\pmb{a}_S}) \geq d_{\min}(\pmb{t}+\La_{\Xc_{\Sc}})=d_{\min}(\La_{\Xc_{\Sc}})$.
If a shortest vector of $\La_{\Xc_{\Sc}}$ lies in $M\Zb^K$, then $d_{\min}(\La_{\Xc_{\Sc}})=d_{\min}(M\Zb^K)=M$, and hence $d_{\min}(\Xc_{\pmb{a}_S}) \geq M$. This proves the second part of the lemma.
To prove the first part we will now show that \mbox{$d_{\min}(\Xc_{\pmb{a}_S}) \leq d_{\min}(\La_{\Xc_{\Sc}})$} if $\pmb{w}$ is a shortest vector of $\La_{\Xc_{\Sc}}$ and \mbox{$\pmb{w} \notin M\Zb^K$}.
Note that \mbox{$\pmb{w} \mod M \neq \pmb{0}$} and \mbox{$\pmb{w} \mod M \in \Xc_{\Sc}$}. Hence, $d_{\min}\left(\Xc_{\Sc}\right) \leq \|\pmb{w} \mod M\| \leq \|\pmb{w}\|$. Since $\Xc_{\pmb{a}_S}$ is a coset of $\Xc_{\Sc}$ in $\Zb_M^K$, we have $d_{\min}(\Xc_{\pmb{a}_S})=d_{\min}(\Xc_{\Sc})$. Thus, we have
$d_{\min}(\Xc_{\pmb{a}_S}) = d_{\min}(\Xc_{\Sc}) \leq \|\pmb{w}\| = d_{\min}(\La_{\Xc_{\Sc}})$.
This completes the proof.
\end{IEEEproof}
Lemma~\ref{lem:dmin_lattice} provides the exact value of $d_S$, and hence
$\sfrac{10\log_{10}\left(\sfrac{d_S^2}{d_0^2}\right)}{R_S}$,
only if we can find a shortest vector \mbox{$\pmb{w} \in \La_{\Xc_{\Sc}}$} such that $\pmb{w} \mod M \neq \pmb{0}$. Otherwise, the lemma yields only a lower bound on $\sfrac{10\log_{10}\left(\sfrac{d_S^2}{d_0^2}\right)}{R_S}$.
\subsection{Computer search}
\begin{table
\renewcommand{\arraystretch}{1.35}
\centering
\caption{Best Linear Index Codes with Circulant Encoding Matrix $\pmb{C}$.}
{\fontsize{6}{7}\selectfont{
\begin{tabular} {||c||c|c|c|c||}
\hline
\multirow{2}{*}{$M$} & \multicolumn{4}{c||}{$K=n$} \\
\cline{2-5}
& $2$ & $3$ & $4$ & $5$\\
\hhline{||=||====||}
\multirow{2}{*}{$4$} & $(1,-2)$ & $(1,-2,-2)$ & $(1,1,-1,0)$ & $(1,-2,1,-1,0)$\\
& $6.02$ & $4.52$ & $3.01$ & $3.76$ \\
\hline
\multirow{2}{*}{$8$} & $(1,2)$ & $(1,2,0)$ & $(1,0,3,3)$ & $(1,-1,2,2,-3)$ \\
& $4.65$ & $3.49$ & $4.01$ & $4.70$ \\
\hline
\multirow{2}{*}{$16$} & $(1,-4)$ & $(1,2,-6)$ & $(1,4,-6,-8)$ & $(1,-2,-5,-4,5)$\\
& $6.02$ & $5.24$ & $5.57$ & $5.28$ \\
\hline
\multirow{2}{*}{$32$} & $(1,6)$ & $(1,-10,14)$ & $(1,10,14,2)$ & $(1,-8,-5,15,-6)$\\
& $5.85$ & $5.73$ & $5.80$ & $5.77$ \\
\hline
\multirow{2}{*}{$64$} & $(1,-28)$ & $(1,-26,-4)$ & $(1,-26,20,30)$ & $(1,16,18,-9,21)$ \\
& $6.04$ & $5.73$ & $5.85$ & $5.82$ \\
\hline
\end{tabular}
}}
\label{tbl:index_codes}
\end{table}
We use a computer search to find linear index codes with large side information gains. To reduce the complexity of the exhaustive search we restrict our search space to codes whose encoding matrices $\pmb{C}$ are circulant. We present results for \mbox{$n=K=2,3,4,5$} and \mbox{$M=4,8,16,32,64$}.
For each choice of $\pmb{C}$, with \mbox{$\det(\pmb{C}) \in U(\Zb_M)$}, we found that the value of $S$ that minimizes $\sfrac{10\log_{10}\left(\sfrac{d_S^2}{d_0^2}\right)}{R_S}$ yields a lattice $\La_{\Xc_{\Sc}}$ with a shortest vector $\pmb{w}$ such that $\pmb{w} \mod M \neq \pmb{0}$. Hence, using Lemma~\ref{lem:dmin_lattice}, we were able to calculate the exact value of \mbox{$\Gamma = \min_{S} \sfrac{10\log_{10}\left(\sfrac{d_S^2}{d_0^2}\right)}{R_S}$} for each candidate index code.
For each $M,K$, Table~\ref{tbl:index_codes} lists one index code with the largest side information gain $\Gamma$ among all codes with circulant encoding matrices. The table shows the first row of the circulant matrix $\pmb{C}$ and the side information gain $\Gamma$ (in~dB/b/dim).
All the index codes have \mbox{$\Gamma \geq 3$~dB/b/dim}, and for \mbox{$M \geq 16$}, the gain is at least $5.24$~dB/b/dim.
In comparison, the codes from~\cite{NHV_arxiv_14} provide \mbox{$\Gamma \approx 6$~dB/b/dim}. Since the construction of~\cite{NHV_arxiv_14} relies on the Chinese remainder theorem, the resulting message sizes $|\Wc_1|,\dots,|\Wc_K|$ are powers of different primes. Here, we circumvent this problem by using codes over $\Zb_M$, but rely on numerical techniques to estimate $\Gamma$.
\section{Simulation Results \& Conclusion} \label{sec:4}
\begin{figure}[!t]
\centering
\includegraphics[totalheight=1.95in,width=3in]{Fig_2.eps}
\caption{Performance of the $16$-QAM index code used as a modulation scheme with two identical $(4000,2000)$ LDPC codes and iterative decoders.}
\label{fig:comparison_all}
\vspace{-5mm}
\end{figure}
The proposed index codes are effective in exploiting receiver side information, but are sensitive to channel noise. The channel coding gain can be improved by encoding the $K$ information sources independently with channel codes, and modulating the resulting $K$ coded streams using a QAM index code.
Consider \mbox{$K=2$} independent messages to be broadcast to three receivers, with $S=\varnothing,\{1\},\{2\}$, respectively.
We use the $16$-QAM index code of Examples~\ref{ex:16QAM_1} and~\ref{ex:16QAM_2} (optimal from Table~\ref{tbl:index_codes}) concatenated with \mbox{$K=2$} identical \mbox{rate-$\sfrac{1}{2}$} $(4000,2000)$ regular LDPC codes (variable-node degree~3, check-node degree~6) catalogued in~\cite{Mac_Encyclopedia} using bit interleaved coded-modulation (BICM)~\cite{CTB_IT_98}.
For each information source, $2000$ information bits are encoded into a $4000$ length LDPC codeword, which is then interleaved using a random interleaver.
Four coded bits, two each from the two interleaved sequences, are mapped to two $\Zb_4$ symbols, which are then modulated to a $16$-QAM point using the index code of Example~\ref{ex:16QAM_1}.
The coded bit rate of each source is \mbox{$R_1=R_2=\sfrac{1}{2}$~b/dim}.
Each receiver regards the two information sources as independent users, and employs an iterative multiuser detector~\cite{Poo_SPMag_04} composed of three soft-in soft-out (SISO) a posteriori probability blocks~\cite{BDMP_CommLet_97}: one $16$-QAM demodulator, and two LDPC decoders. Each LDPC decoder block uses $50$ iterations between the check nodes and variable nodes, and the multiuser iterative demodulator-decoder uses $16$ iterations between the three SISO blocks.
For the receivers with $S=\{1\},\{2\}$, the side information is fed as input a~priori probabilities to the corresponding LDPC decoder.
From~\cite{Tun_IEEE_IT_06}, we know that a rate tuple $(R_1,R_2)$ is achievable if and only if \mbox{$\sfrac{1}{2}\log_2\left(1+{\sf SNR}\right) > \sum_{k=1}^{K}R_k - R_S$} for every receiver $({\sf SNR},S)$. For the three receivers corresponding to {$S=\varnothing,\{1\},\{2\}$}, $R_S$ equals $0$~b/dim, \mbox{$R_1=\sfrac{1}{2}$}~b/dim and \mbox{$R_2=\sfrac{1}{2}$~b/dim}, respectively. It follows that the minimum required ${\sf SNR}$ at the three receivers are $4.77$~dB, $0$~dB and $0$~dB, respectively.
Fig.~\ref{fig:comparison_all} shows the performance of the LDPC-coded $16$-QAM index code for \mbox{$S=\varnothing,\{1\},\{2\}$} and the capacity limits on the ${\sf SNR}$.
At bit error rate $10^{-4}$, the system performs $2.4$~dB from capacity for {$S=\{1\},\{2\}$}, and $4.3$~dB away for \mbox{$S=\varnothing$}.
While the LDPC code has contributed to channel coding gain, the symbol mapping provided by the inner index code has yielded significant ${\sf SNR}$ gains for the receivers that know either of the two messages a priori.
We have presented the first known family of index codes for the Gaussian broadcast channel that admit equal message rates, and with message sizes that are powers of $2$. The method employed to obtain these codes is limited to small values of $M$ and $K$ because of the complexity involved in the computer search.
An analytical approach could extend the results to larger number of messages. Our simulations used a standard LDPC code designed for the single-user AWGN channel to improve noise resilience. Designing efficient coded-modulation techniques matched to the proposed modulation schemes may be crucial to achieve higher coding gains.
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"redpajama_set_name": "RedPajamaArXiv"
} | 7,812 |
{"url":"https:\/\/www.investopedia.com\/terms\/c\/costofcapital.asp?utm_source=term-of-the-day&utm_campaign=bouncex&utm_term=12805520&utm_medium=email","text":"# Cost of Capital: What It Is, Why It Matters, Formula, and Example\n\n## What Is Cost of Capital?\n\nCost of capital is a company's calculation of the minimum return that would be necessary in order to justify undertaking a capital budgeting project, such as building a new factory.\n\nThe term cost of capital is used by analysts and investors, but it is always an evaluation of whether a projected decision can be justified by its cost. Investors may also use the term to refer to an evaluation of an investment's potential return in relation to its cost and its risks.\n\nMany companies use a combination of debt and equity to finance business expansion. For such companies, the\u00a0overall cost of capital is derived from the weighted average cost of all capital sources. This is known as the weighted average cost of capital (WACC).\n\n### Key Takeaways\n\n\u2022 Cost of capital represents the return a company needs to achieve in order to justify the cost of a capital project, such as purchasing new equipment or constructing a new building.\n\u2022 Cost of capital encompasses the cost of both equity and debt, weighted according to the company's preferred or existing capital structure. This is known as the weighted average cost of capital (WACC).\n\u2022 A company's investment decisions for new projects should always generate a return that exceeds the firm's cost of the capital used to finance the project. Otherwise, the project will not generate a return for investors.\n1:34\n\n## Understanding Cost of Capital\n\nThe concept of the cost of capital is key information used to determine a project's hurdle rate. A company embarking on a major project must know how much money the project will have to generate in order to offset the cost of undertaking it and then continue to generate profits for the company.\n\nCost of capital, from the perspective of an investor, is an assessment of the return that can be expected from the acquisition of stock shares or any other investment. This is an estimate and might include best- and worst-case scenarios. An investor might look at the volatility (beta) of a company's financial results to determine whether a stock's cost is justified by its potential return.\n\n## Weighted Average Cost of Capital (WACC)\n\nA firm's cost of capital is typically calculated using the weighted average cost of capital formula that considers the cost of both debt and equity capital.\n\nEach category of the firm's capital is weighted proportionately to arrive at a blended rate, and the formula considers every type of debt and equity on the company's balance sheet, including common and preferred stock, bonds, and other forms of debt.\n\n### Finding the Cost of Debt\n\nThe cost of capital becomes a factor in deciding which financing track to follow: debt, equity, or a combination of the two.\n\nEarly-stage companies rarely have sizable assets to pledge as collateral for loans, so equity financing becomes the default mode of funding. Less-established companies with limited operating histories will pay a higher cost for capital than older companies with\u00a0solid track records since lenders and investors will demand a higher risk premium for the former.\n\nThe cost of debt is merely the interest rate paid by the company on its debt. However, since interest expense is tax-deductible, the debt is calculated on an after-tax basis as follows:\n\n\\begin{aligned} &\\text{Cost of debt}=\\frac{\\text{Interest expense}}{\\text{Total debt}} \\times (1 - T) \\\\ &\\textbf{where:}\\\\ &\\text{Interest expense}=\\text{Int. paid on the firm's current debt}\\\\ &T=\\text{The company\u2019s marginal tax rate}\\\\ \\end{aligned}\n\nThe cost of debt can also be estimated by adding a credit spread to the risk-free rate and multiplying the result by (1 - T).\n\n### Finding the Cost of Equity\n\nThe cost of equity is more complicated since the rate of return demanded by equity investors is not as clearly defined as it is by lenders. The cost of equity is approximated by the capital asset pricing model as follows:\n\n\\begin{aligned} &CAPM(\\text{Cost of equity})= R_f + \\beta(R_m - R_f) \\\\ &\\textbf{where:}\\\\ &R_f=\\text{risk-free rate of return}\\\\ &R_m=\\text{market rate of return}\\\\ \\end{aligned}\n\nBeta is used in the CAPM formula to estimate risk, and the formula would require a public company's own stock beta. For private companies, a beta is estimated based on the average beta among a group of similar public companies. Analysts may refine this beta by calculating it on an after-tax basis. The assumption is that a private firm's beta will become the same as the industry average beta.\n\nThe firm\u2019s overall cost of capital is based on the weighted average of these costs.\n\nFor example, consider an enterprise with a capital structure consisting of 70% equity and 30% debt; its cost of equity is 10% and the after-tax cost of debt is 7%.\n\nTherefore, its WACC would be:\n\n$(0.7 \\times 10\\%) + (0.3 \\times 7\\%) = 9.1\\%$\n\nThis is the cost of capital that would be used to discount future cash flows from potential projects and other opportunities to estimate their net present value (NPV) and ability to generate value.\n\nCompanies strive to attain the optimal financing mix\u00a0based on the cost of capital for various funding sources. Debt financing is more tax-efficient than equity financing since interest expenses are tax-deductible and dividends on common shares are paid with after-tax dollars. However, too much debt can result in dangerously high leverage levels, forcing the company to pay higher interest rates to offset the higher default risk\n\n## Cost of Capital vs. Discount Rate\n\nThe cost of capital and discount rate are somewhat similar\u00a0and the terms are often\u00a0used interchangeably. Cost of capital is often calculated by a company's finance department and used by management to set a discount rate (or hurdle rate) that must be beaten to justify an investment.\n\nThat said, a company's management should challenge its internally generated cost of capital numbers, as they may be so conservative as to deter investment.\n\nCost of capital may also differ based on the type of project or initiative; a highly innovative but risky initiative should carry a higher cost of capital than a project to update essential equipment or software with proven performance.\n\n## Importance of Cost of Capital\n\nBusinesses and financial analysts use the cost of capital to determine if funds are being invested effectively. If the return on an investment is greater than the cost of capital, that investment will end up being a net benefit to the company's balance sheets. Conversely, an investment whose returns are equal to or lower than the cost of capital indicate that the money is not being spent wisely.\n\nThe cost of capital can also determine a company's valuation. Since a company with a high cost of capital can expect lower proceeds in the long run, investors are likely to see less value in owning a share of that company's equity.\n\n## Real-World Examples\n\nEvery industry has its own prevailing average cost of capital.\n\nThe numbers vary widely. Homebuilding has a relatively high cost of capital, at 6.35, according to a compilation from New York University's Stern School of Business. The retail grocery business is relatively low, at 1.98%.\n\nThe cost of capital is also high among both biotech and pharmaceutical drug companies, steel manufacturers, internet software companies,\u00a0and integrated oil and gas companies. Those industries tend to require\u00a0significant capital investment in research, development, equipment, and factories.\n\nAmong the industries with lower capital costs are money center banks, power companies, real estate investment trusts (REITs), and utilities (both general and water). Such companies may require less equipment or may benefit from very steady cash flows.\n\n## Why Is Cost of Capital Important?\n\nMost businesses strive to grow and expand. There may be many options: expand a factory, buy out a rival, build a new, bigger factory. Before the company decides on any of these options, it determines the cost of capital for each proposed project. This indicates how long it will take for the project to repay what it cost, and how much it will return in the future. Such projections are always estimates, of course. But the company must follow a reasonable methodology to choose between its options.\n\n## What Is the Difference Between the Cost of Capital and the Discount Rate?\n\nThe two terms are often used interchangeably, but there is a difference. In business, cost of capital is generally determined by the accounting department. It is a relatively straightforward calculation of the breakeven point for the project. The management team uses that calculation to determine the discount rate, or hurdle rate, of the project. That is, they decide whether the project can deliver enough of a return to not only repay its costs but reward the company's shareholders.\n\n## How Do You Calculate the Weighted Average Cost of Capital?\n\nThe weighted average cost of capital represents the average cost of the company's capital, weighted according to the type of capital and its share on the company balance sheet. This is determined by multiplying the cost of each type of capital by the percentage of that type of capital on the company's balance sheet and adding the products together.\n\n## The Bottom Line\n\nThe cost of capital measures the cost that a business incurs to finance its operations. It measures the cost of borrowing money from creditors, or raising it from investors through equity financing, compared to the expected returns on an investment. This metric is important in determining if capital is being deployed effectively.\n\nArticle Sources\nInvestopedia requires writers to use primary sources to support their work. These include white papers, government data, original reporting, and interviews with industry experts. We also reference original research from other reputable publishers where appropriate. You can learn more about the standards we follow in producing accurate, unbiased content in our editorial policy.\n1. New York University Stern School of Business. \"Industry Survey.\"","date":"2023-03-22 03:28:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 3, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.30098995566368103, \"perplexity\": 1956.5303005026633}, \"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\/1679296943749.68\/warc\/CC-MAIN-20230322020215-20230322050215-00577.warc.gz\"}"} | null | null |
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} | 2,319 |
Q: Problem with fscanf not reading inputs correctly I'm having problems using a fscanf function. It never reads it correctly and always results in blanks.
fscanf(f, " %[^;];%[^;];%d;%d;%d;%d;%[^;];%d;%[^\n]",
arr[i].loc1, arr[i].loc2, &arr[i].price, &arr[i].rooms,
&arr[i].bathroom, &arr[i].carpark, arr[i].type, &arr[i].area, arr[i].furnish);
The code above always outputs " 0 0 0 0 0". But when I try using a scanf and manually input one of the lines, it works perfectly.
The file it's reading from is a .csv file. Here is the contents:
Mont-Kiara;Kuala-Lumpur;1000000;2;2;0;Built-up;1000;Partly
Cheras;Kuala-Lumpur;310000;3;2;0;Built-up;1000;Partly
Kepong;Kuala-Lumpur;358000;3;3;0;Built-up;1000;Partly
Taman-Desa;Kuala-Lumpur;455000;2;2;0;Built-up;1000;Partly
Kepong;Kuala-Lumpur;358000;3;3;0;Built-up;1000;Partly
Kepong;Kuala-Lumpur;358000;3;3;0;Built-up;1000;Partly
And here is the full code:
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
struct houseData {
char loc1[101];
char loc2[101];
int price[101];
int rooms[101];
int bathroom[101];
int carpark[101];
char type[101];
int area[101];
char furnish[101];
} arr[1001];
int read() {
int i = 0;
struct houseData arr[800];
char temp1[100];
FILE *f = fopen("file.csv", "r");
while(!feof(f)){
fscanf(f, " %[^;];%[^;];%d;%d;%d;%d;%[^;];%d;%[^\n]"
, &arr[i].loc1, &arr[i].loc2, &arr[i].price, &arr[i].rooms,
&arr[i].bathroom, &arr[i].carpark, &arr[i].type, &arr[i].area, &arr[i].furnish);
i++;
}
fclose(f);
}
int main() {
read();
printf("%s %s %d %d %d %d %s %d %s", arr[i].loc1, arr[i].loc2, *arr[i].price, *arr[i].rooms, *arr[i].bathroom, *arr[i].carpark, arr[i].type, *arr[i].area, arr[i].furnish);
return 0;
}
A: There are a lot of issues in your code. Here's a version that may help. The error messages in this version are far from ideal (this does not distinguish between an input format error and a error reading data, for example, nor does it provide much detail on the location of the error), and there is still the possibility of undefined behavior on certain inputs, (see Is `scanf("%d", ...)` as bad as `gets`?) but this should point you in the right direction. Well, at least it may help to improve your use of scanf, but a very reasonable argument can the be made that the "right direction" is to stop using scanf completely. It is very difficult to get things right with scanf, and attempting to do so winds up being much more complex that just using fgets. But for simple use cases it is ... still pointless to use scanf. See http://sekrit.de/webdocs/c/beginners-guide-away-from-scanf.html and many other resources that explain why scanf is a terrible choice.
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
struct houseData{
char loc1[101];
char loc2[101];
int price;
int rooms;
int bathroom;
int carpark;
char type[101];
int area;
char furnish[101];
};
int
read(FILE * f, struct houseData *h)
{
return 9 == fscanf(f, " %100[^;]; %100[^;]; %d; %d; %d; %d; "
"%100[^;]; %d; %100[^\n]", h->loc1, h->loc2, &h->price,
&h->rooms, &h->bathroom, &h->carpark, h->type, &h->area,
h->furnish);
}
int
main(int argc, char **argv)
{
int rv;
FILE *f = argc > 1 ? fopen(argv[1], "r") : stdin;
if( f == NULL ){
perror(argv[1]);
return EXIT_FAILURE;
}
struct houseData h;
int i = 0;
while( read(f, &h) ){
printf("%d: %s %s %d %d %d %d %s %d %s\n", ++i,
h.loc1, h.loc2, h.price, h.rooms, h.bathroom,
h.carpark, h.type, h.area, h.furnish);
}
if( ! feof(f) ){
fprintf(stderr, "Error near line %d\n", i);
}
fclose(f);
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,640 |
megabyte = (1024 * 1024)
# You may wish to set the following to the same as your HDFS block size, esp if
# you're seeing issues with s3:// turning 1TB files into 30_000+ map tasks
#
default[:hadoop][:min_split_size] = (128 * megabyte)
default[:hadoop][:s3_block_size] = (128 * megabyte)
default[:hadoop][:hdfs_block_size] = (128 * megabyte)
default[:hadoop][:dfs_replication] = 3
default[:hadoop][:namenode ][:handler_count] = 40
default[:hadoop][:jobtracker ][:handler_count] = 40
default[:hadoop][:datanode ][:handler_count] = 8
default[:hadoop][:tasktracker][:http_threads ] = 32
# Number of files the reducer will read in parallel during the copy (shuffle)
# phase, and the threshold triggering the last stage of the shuffle
# (`mapred.reduce.parallel.copies`). This is an important setting but one you
# should not mess with until you have tuned the hell out of everything else.
#
# A reducer gets one file from every mapper, which it must merge sort in passes
# until there are fewer than `:reducer_parallel_copies` merged files. At that
# point, it does not need to perform the final merge-sort pass: it can stream
# directly from each file lickety-split and do the merge on the fly. A higher
# number costs more memory but can lead to fewer merge passes.
#
# The hadoop default is 5; we have increased it to 10.
default[:hadoop][:reducer_parallel_copies ] = 10
# `mapred.compress.map.output`: If true, compresses the data during transport
# from mapper to reducer. It is decompressed for you, so this is completely
# transparent to your jobs. (Also note that ifd there are no reducers, this
# setting is not applied.) There's a modest CPU cost, but as midflight data
# often sees compression ratios of 5:1 or better, the typical result is
# dramatically faster transfer. Leave this `'true'` and override on a per-job
# basis in the rare case it's unhelpful.
default[:hadoop][:compress_mapout ] = 'true'
# `mapred.map.output.compression.codec`: We've left `compress_mapout_codec` at
# the default `'org.apache.hadoop.io.compress.DefaultCodec'`, but almost all
# jobs are improved by `'org.apache.hadoop.io.compress.SnappyCodec'`
default[:hadoop][:compress_mapout_codec] = 'org.apache.hadoop.io.compress.DefaultCodec'
# Compress the job output (`mapred.output.compress`). The same benefits as
# `:compress_mapout`, but also saves significant disk space. The downside is
# that the compression is not transparent: `hadoop fs -cat` outputs the
# compressed data, which is a minor pain when doing exploratory analysis. You'd
# like best to use `snappy` compression, but the toolset for working with it is
# not mature.
#
# In practice, we leave this set at `'false'` in the site configuration, and
# have production jobs explicitly request gzip- or snappy-compressed output. (We
# find those are always superior to `.bz2`, `lzo` or `default` codecs.)
default[:hadoop][:compress_output ] = 'false'
# Leave this set to `'BLOCK'` (`mapred.output.compression.type`)
default[:hadoop][:compress_output_type ] = 'BLOCK'
# Codec to use for job output (`mapred.output.compression.codec`). If you're
# going to flip this on, I wouldn't use anything but
# `'org.apache.hadoop.io.compress.SnappyCodec'`
default[:hadoop][:compress_output_codec] = 'org.apache.hadoop.io.compress.DefaultCodec'
# uses /etc/default/hadoop-0.20 to set the hadoop daemon's java_heap_size_max
default[:hadoop][:java_heap_size_max] = 1000
# if true, hadoop daemon JVMs will write verbose logs about garbage collection activity. Heaven help you.
default[:hadoop][:java_gc_log] = false
# Namenode Java Heap size. Increase this if you have a lot of
# objects on your HDFS.
default[:hadoop][:namenode ][:java_heap_size_max] = nil
# Secondary Namenode Java Heap size. Set to the exact same value as the Namenode.
default[:hadoop][:secondarynn ][:java_heap_size_max] = nil
# Jobtracker Java Heap Size.
default[:hadoop][:jobtracker ][:java_heap_size_max] = nil
# Datanode Java Heap Size. Increase if each node manages a large number of blocks.
# Set this by observation: its value is fairly stable and 1GB will take you fairly far.
default[:hadoop][:datanode ][:java_heap_size_max] = nil
# Tasktracker Java Heap Size. Set this by observation: its value is fairly
# stable. Note: this is *not* the amount of RAM given to the mapper and reducer
# child processes -- see :java_child_opts (and :java_child_ulimit) below.
default[:hadoop][:tasktracker ][:java_heap_size_max] = nil
# Rate at which datanodes exchange blocks in a rebalancing operation. If you run
# an elastic cluster, increase this value to more like 50_000_000 -- jobs will
# run more slowly while the cluster rebalances, but your usage will be more
# efficient overall. In bytes per second -- 1MB/s by default
default[:hadoop][:balancer][:max_bandwidth] = 1_048_576
# how long to keep jobtracker logs around
default[:hadoop][:log_retention_hours ] = 240
# define a rack topology? if false (default), all nodes are in the same 'rack'.
default[:hadoop][:define_topology] = false
default[:hadoop][:fake_rack_size] = 4
# how many jobs' histories to keep in memory on the job tracker
default[:hadoop][:max_job_histories_in_mem] = 100
#
# Tune cluster settings for size of instance
#
# These settings are mostly taken from the cloudera hadoop-ec2 scripts,
# informed by the
#
# numMappers M := numCores * 1.5
# numReducers R := numCores max 4
# java_Xmx := 0.75 * (TotalRam / (numCores * 1.5) )
# ulimit := 3 * java_Xmx
#
# With 1.5*cores tasks taking up max heap, 75% of memory is occupied. If your
# job is memory-bound on both map and reduce side, you *must* reduce the number
# of map and reduce tasks for that job to less than 1.5*cores together. using
# mapred.max.maps.per.node and mapred.max.reduces.per.node, or by setting
# java_child_opts.
#
# Memory-heavy machines are biased towards reduce efficiency; CPU-heavy machines
# are biased towards mapper efficiency.
#
# It assumes EC2 instances with EBS-backed volumes
# If your cluster is heavily used and has many cores/machine (almost always running a full # of maps and reducers) turn down the number of mappers.
# If you typically run from S3 (fully I/O bound) increase the number of maps + reducers moderately.
# In both cases, adjust the memory settings accordingly.
#
# FIXME: The below parameters are calculated for each node.
# The max_map_tasks and max_reduce_tasks settings apply per-node, no problem here
# The remaining ones (java_child_opts, io_sort_mb, etc) are applied *per-job*:
# if you launch your job from an m2.xlarge on a heterogeneous cluster, all of
# the tasks will kick off with -Xmx4531m and so forth, regardless of the RAM
# on that machine. Just set the right thing explicitly in your job conf.
#
# The io.sort.mb should be marginally above (128 / (0.85 * 0.8)) to minimize
# spill (If that didn't make sense, don't worry about it.)
#
# If you are using a tiny machine (t1.micro, m1.small, c1.medium), a) I hope
# it's only for testing purposes; b) you should lower the block size to 64m
# (from our default of 128m). If you're on a c1.xlarge, you are assumedly
# running a massive number of map-side-only jobs; consider turning the block
# size *up*.
#
#
hadoop_performance_settings =
case node[:ec2] && node[:ec2][:instance_type]
when 't1.micro' then { :max_map_tasks => 1, :max_reduce_tasks => 1, :java_child_opts => '-Xmx256m -Xss160k', :java_child_ulimit => 2227200, :io_sort_factor => 10, :io_sort_mb => 64, }
when 'm1.small' then { :max_map_tasks => 2, :max_reduce_tasks => 1, :java_child_opts => '-Xmx870m -Xss160k', :java_child_ulimit => 2227200, :io_sort_factor => 10, :io_sort_mb => 100, }
when 'c1.medium' then { :max_map_tasks => 3, :max_reduce_tasks => 2, :java_child_opts => '-Xmx870m -Xss256k', :java_child_ulimit => 2227200, :io_sort_factor => 10, :io_sort_mb => 100, }
when 'm1.large' then { :max_map_tasks => 4, :max_reduce_tasks => 2, :java_child_opts => '-Xmx600m -Xss256k -XX:+UseCompressedOops -XX:MaxNewSize=200m -server', :java_child_ulimit => 7471104, :io_sort_factor => 25, :io_sort_mb => 200, }
when 'c1.xlarge' then { :max_map_tasks => 10, :max_reduce_tasks => 4, :java_child_opts => '-Xmx870m -Xss256k', :java_child_ulimit => 2227200, :io_sort_factor => 20, :io_sort_mb => 200, }
when 'm1.xlarge' then { :max_map_tasks => 6, :max_reduce_tasks => 3, :java_child_opts => '-Xmx1920m -Xss256k -XX:+UseCompressedOops -XX:MaxNewSize=200m -server', :java_child_ulimit => 5898240, :io_sort_factor => 25, :io_sort_mb => 210, }
when 'm2.xlarge' then { :max_map_tasks => 3, :max_reduce_tasks => 2, :java_child_opts => '-Xmx4531m -Xss256k -XX:+UseCompressedOops -XX:MaxNewSize=200m -server', :java_child_ulimit => 13447987, :io_sort_factor => 32, :io_sort_mb => 210, }
when 'm2.2xlarge' then { :max_map_tasks => 6, :max_reduce_tasks => 4, :java_child_opts => '-Xmx4378m -Xss256k -XX:+UseCompressedOops -XX:MaxNewSize=200m -server', :java_child_ulimit => 13447987, :io_sort_factor => 32, :io_sort_mb => 210, }
when 'm2.4xlarge' then { :max_map_tasks => 12, :max_reduce_tasks => 4, :java_child_opts => '-Xmx4378m -Xss256k -XX:+UseCompressedOops -XX:MaxNewSize=200m -server', :java_child_ulimit => 13447987, :io_sort_factor => 40, :io_sort_mb => 210, }
when 'cc1.4xlarge' then { :max_map_tasks => 6, :max_reduce_tasks => 6, :java_child_opts => '-Xmx1800m -Xss256k -XX:+UseCompressedOops -XX:MaxNewSize=200m -server', :java_child_ulimit => 13447987, :io_sort_factor => 40, :io_sort_mb => 420, :hdfs_block_size => (256 * megabyte), }
when 'cc1.8xlarge' then { :max_map_tasks => 6, :max_reduce_tasks => 3, :java_child_opts => '-Xmx6000m -Xss256k -XX:+UseCompressedOops -XX:MaxNewSize=200m -server', :java_child_ulimit => 13447987, :io_sort_factor => 40, :io_sort_mb => 840, :hdfs_block_size => (512 * megabyte), } # the large block size, and this machine in general, are only appropriate if you're bringing some bigass data.
else
if node[:memory] && node[:cores]
cores = node[:cpu ][:total].to_i
ram = node[:memory][:total].to_i
if node[:memory][:swap] && node[:memory][:swap][:total]
ram -= node[:memory][:swap][:total].to_i
end
else
Chef::Log.warn("No access to system info, using cores=1 memory=1024m")
cores = 1
ram = 1024
end
Chef::Log.warn("Couldn't set performance parameters from instance type, estimating from #{cores} cores and #{ram} ram")
n_mappers = (cores >= 6 ? (cores * 1.25) : (cores * 2)).to_i
n_reducers = cores
heap_size = 0.75 * (ram.to_f / 1000) / (n_mappers + n_reducers)
heap_size = [256, heap_size.to_i].max
child_ulimit = 2 * heap_size * 1024
io_sort_factor = 10
io_sort_mb = 100
{ :max_map_tasks => n_mappers, :max_reduce_tasks => n_reducers, :java_child_opts => "-Xmx#{heap_size}m", :java_child_ulimit => child_ulimit, :io_sort_factor => io_sort_factor, :io_sort_mb => io_sort_mb, }
end
hadoop_performance_settings[:java_reduce_opts] ||= hadoop_performance_settings[:java_child_opts]
Chef::Log.debug("Hadoop tunables: #{hadoop_performance_settings.inspect}")
# (Mappers+Reducers)*ChildTaskHeap + DNheap + TTheap + 3GB + RSheap + OtherServices'
hadoop_performance_settings.each{|k,v| default[:hadoop][k] = v }
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,670 |
Potenza este un oraș și comună din sudul Italiei, în regiunea Basilicata. Cu o populație de 68.600 de locuitori, Potenza este capitala provinciei Potenza și a regiunii Basilicata. El este și primul oraș din regiune după numărul populației, și al 84-lea în Italia. Are o suprafață de 175.43 km².
Orașul se află la poalele muntilor Apenini, în depresiunea Basento, strabatută de râul cu același nume și înconjurată de munți.
Centrul vechi medieval, este situat în partea de sus a orașului, în timp ce cartierele moderne au apărut mai jos. Probabil, prima locație a orașului a fost la o altitudine de 1095 deasupra nivelului mării, în zona numită acum Serra di Vaglio, deși actuala amplasare se află la o altitudine de 819 metri.
Demografie
Personalități
Lucia Lauria Vigna (1896–2009) - supercentenar
Tanio Boccia (1912–1982) - regizor
Emilio Colombo (1920) - politician
Ruggero Deodato (1939) - regizor
Francesco Colonnese (1971) - fotbalist
Giovanni Frezza (1972) - actor
Wally Buono (1950) - antrenor de fotbal
Donato Sabia (1963) - alergător pe distanțe medii
Anna Bonitatibus (...) - cântăreață de operă
Vito Postiglione (1977) - pilot de curse
Rocco Sabato (1982) - fotbalist
Antonio Giosa (1983) - fotbalist
Danilo Restivo (1972) - criminal sadic-fetisist
Orașe înfrățite
Denver, SUA
Clima
Vezi și
Listă de orașe din Italia
Referințe
Legături externe
Page at Comuni Italiani
Battle of Potenza at canadiansoldiers.com
Orașe din Italia
Comune din provincia Potenza | {
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} | 2,664 |
Какимжан Казыбаевич Казыбаев (; 10 мая 1929, аул Бакалы, Саркандский район, Алматинская область, КазССР, СССР — 21 октября 1989, Алматы, КазССР, СССР) — советский казахский писатель и государственный деятель.
Биография
В 1952 году окончил КазГУ (ныне КазНУ им. аль-Фараби). В 1952—1958 годах — сотрудник отдела литературы газеты «Ленинская смена» (ныне «Жас Алаш»), заведующий отделом, затем ответственный редактор, в 1958—1968 годах заместитель редактора газеты «Жетысу» Алматинской области. В 1972—1974 годах заместитель председателя Государственного комитета по делам полиграфии и книжной торговли, типографии Казахстана. В 1977—1982 годах директор КазТАГ, в 1982—1985 секретарь ЦК Компартии Казахстана, с 1985 года главный редактор журнала «Коммунист Казахстана» (совр. «Акикат»). Депутат Верховного Совета Казахской ССР IX созыва.
Основная тема произведений — жизнь и быт аула, дружба народов, трудности, которые пережил народ во время Великой Отечественной войны. Автор повести «Кернеген кок» (1966), романов «Ызгар» (1972, на русском языке — 1976), «Аманат» (1979, на русском языке — 1982). Перевёл на казахский язык мемуары С. М. Штеменко «Главный штаб военных лет» и «Дневник офицера» Б. Момышулы.
Автор статьи «Казах, который водрузил знамя над Рейхстагом» от 21 февраля 1958 года, в которой рассказывается о подвиге Рахимжана Кошкарбаева, совместно с красноармейцем Григорием Булатовым водрузившем красное знамя на фасаде («на лестнице главного входа») здания Рейхстага. Опубликовал ряд статей, которые легли в основу написанной им в 1965 году повести о подвиге Рахимжана Кошкарбаева «Кернеген кек» («Священное возмездие»).
Признание и память
1957 — Почётная грамота Верховного Совета Казахской ССР (5 ноября)
1962 — Почётная грамота Верховного Совета Казахской ССР (4 мая)
1969 — Почётная грамота Верховного Совета Казахской ССР (16 июля)
1979 — Почётная грамота Верховного Совета Казахской ССР (8 мая) — за активную работу в печати на протяжении многих лет и в связи с 50-летием со дня рождения.
1980 — Почётные звания «Заслуженный работник культуры Казахской ССР» (19 июня)
В 2009 году в селе Койлык был открыт памятник Какимжану Казыбаеву. Одна из улиц города Астана носит его имя.
Семья
Жена — Орынша Карабалина-Казыбаева, дети — сыновья Батыр и Нуртас, дочь Наргуль.
Примечания
Ссылки
Кәкімжан Қазыбаев
10 мая — 85 лет со дня рождения писателя, журналиста и государственного деятеля Какимжана Казыбаева
Выпускники Казахского национального университета имени аль-Фараби
Депутаты Верховного Совета Казахской ССР 9-го созыва
Депутаты Верховного Совета Казахской ССР 10-го созыва
Депутаты Верховного Совета Казахской ССР 11-го созыва
Заслуженные работники культуры Казахской ССР
Награждённые Почётной грамотой Верховного Совета Казахской ССР | {
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Q: ¿Pór qué me dice que no encuentra un archivo index, si no le hago una peticion de este archivo? Recién aprendo a empaquetar aplicaciones de NodeJS con pkg, todo ha ido bien, en teoria, sin embargo hay una ruta que no entiendo porque falla.
Bien, esto es una pagina que se ve desde el navegador, y la aplicacion envia esta información, desde el código se administra toda esa logistica, incluso el poder visualizar la miniatura de la imagen (esos 2 archivos son imagenes), sin embargo cuando se hace la petición de la imagen para ponerla de miniatura, el "servidor" marca como 404 y ademas pone un error de Error: ENOENT: no such file or directory, stat 'C:\Users\ACER\moonp\index.html' y bueno, no sé.
Código de get /moon
router.get('/moon', (req, res) => {
var files = [];
fs.readdir(fileFolderm, (err, filess) => {
filess.forEach(file => {
var stats = fs.statSync(fileFolderm+file)
if (stats["size"] < 100000) {
var fileSize = stats["size"] + " Bytes";
} else {
var fileSize = (stats["size"] / 1000000.0).toFixed(2) + " MB";
}
files.push({name: file, dpath: '/viewmoonp/'+ file, path: "/mdownload/" + file, size: fileSize});
})
})
res.render('index', { filelist: files, moon: true})
})
Sea dpath el enlace que se envia para que el navegador solicite la imagen de miniatura.
y este el codigo de cuando se solicita /viewmoonp/:id
router.get('/viewmoonp/:id', (req, res) => {
let filereq = req.params.id;
if (fs.existsSync(fileFolderm + filereq)) {
res.sendFile(fileFolderm, filereq)
} else {
res.render('download', { error : true, message : 'File not found', requested : req.params.id })
}
})
Siendo :id la imagen que solicita.
Al hacer la petición de /moonse carga la página de index normal, pero se envian los archivos de la carpeta de moonp, funciona bien. Pero cuando el navegador solicita las imagenes para cargar la miniatura, es decir /viewmoonp/:id, la consola marca error 404, además de que aparéce como si hubiese buscado un archivo index.html ahí mismo
GET /moon 200 8.244 ms - 3735
GET /viewmoonp/WhatsApp%20Image%202021-11-09%20at%205.46.10%20PM(1).jpeg 404 2.657 ms - 214
Error: ENOENT: no such file or directory, stat 'C:\Users\ACER\moonp\index.html'
GET /viewmoonp/WhatsApp%20Image%202021-11-07%20at%206.15.54%20PM.jpeg 404 1.601 ms - 214
Error: ENOENT: no such file or directory, stat 'C:\Users\ACER\moonp\index.html'
Lista de archivos en carpeta:
Ademas, el enlace de descarga si funciona bien, no creo que aporte mucho, pero lo comparto:
router.get('/mdownload/:id', (req, res) => {
if (fs.existsSync(fileFolderm+ req.params.id)) {
res.download(fileFolderm+ req.params.id)
} else {
res.render('download', { error : true, message : 'File not found', requested : req.params.id })
}
})
La variable fileFolderm tiene el valor C:/Users/ACER/moonp/
Se estan utilizando los paquetes:
const express = require('express');
var colors = require('colors');
const morgan = require('morgan');
const exphbs = require('express-handlebars');
const fileUpload = require('express-fileUpload');
const path = require('path');
const fs = require('fs');
const app = express();
A: fs.readdir es asincrono, y por tanto la última linea se manda antes de que obtengas los ficheros
router.get('/moon', (req, res) => {
var files = [];
fs.readdir(fileFolderm, (err, filess) => {
//Debemos esperar a que termine de cargar las
//carpetas para obtener las minuaturas.
filess.forEach(file => {
var stats = fs.statSync(fileFolderm + file)
if (stats["size"] < 100000) {
var fileSize = stats["size"] + " Bytes";
} else {
var fileSize = (stats["size"] / 1000000.0).toFixed(2) + " MB";
}
files.push({
name: file,
dpath: '/viewmoonp/' + file,
path: "/mdownload/" + file,
size: fileSize
});
})
// Enviamos la respuesta
res.render('index', {
filelist: files,
moon: true
})
})
//ESTA LINEA LA MOVEMOS
//res.render('index', {
// filelist: files,
// moon: true
//})
})
Te buscaba un index por que viewmoon no recibía los parametros que necesitaba.
A: Solucionado.
El error estaba en la linea donde se envia el archivo que se solicita.
router.get('/viewmoonp/:id', (req, res) => {
let filereq = req.params.id;
if (fs.existsSync(fileFolderm + filereq)) {
res.sendFile(fileFolderm, filereq)
} else {
res.render('download', { error : true, message : 'File not found', requested : req.params.id })
}
})
El error se encontraba en que el "archivo" que se enviaba era la variable fileFolderm con el comentario o mensaje que era el nombre del archivo, lo cual no servia, porque el navegador hacia la solicitud de fileFolderm + lo que busca principalmente "index.html".
He cambiado la linea
res.sendFile(fileFolderm, filereq)
Por:
res.sendFile(fileFolderm + filereq)
Y nuevamente el servidor envia la imagen que se solicita, permitiendo así que se pueda visualizar la miniatura.
Moraleja: xD hay que saber donde poner comas para unir variables.
| {
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} | 3,889 |
Thoughts, stories and events to share
New Futures Network
Sophie O'Sullivan, about 10 months ago
The NFN makes life simple for businesses, and demystifies working with the criminal justice system.
-Duncan O'Leary, CEO
The New Futures Network helps companies fill skill gaps and gives ex-offenders the chance for a better life. It is the Ministry of Justice's new specialist strategy to build more relationships between employers and prisons. The NFN is an overarching structure that sits above prisons. They have set up a national network of employer brokers to work with companies all across England and Wales. Each business is allocated to a geographic area and then linked to the relevant prisons within the area that they work. It aims to break the cycle of reoffending which costs society £15 billion a year.
…There are two big advantages. The first is that you get an opportunity to train somebody up before they start to work with you and the second is that you get to know an individual, either through release on temporary licence or through setting up a partnership within a prison itself.
The guys on-site, they are just like everybody else that we come across as an employer. We find they come with a variety of skills and we ensure that they are then given the correct training so they can go to work and join in with the team.
-Claire Coombs, Development Manager, Keltbray
The NFN works with over 120 businesses, offering prison workshops and paid ROTL placements. It targets five key sectors, which the Ministry of Justice have identified as being in need of new talent. By looking for employers in sectors such as construction and retail, the NFN feels it can capitalise on an economic need for employees and develop skills within prison industries through training. One of the key roles of the NFN is to speak to large national employers and encourage them to employ ex-offenders, highlighting the direct benefits to their workforce. In tandem, employment brokers liaise with the businesses' operations in their region and introduce them to prisons in their local area. More than 11,000 prisoners are employed in prisons today, by over 300 businesses or government departments.
The process is simple: the first stage is a conversation between employers and brokers to understand the employers' needs. Brokers offer a tailored service to employers with their access to data about potential candidates. The next step is a prison visit and an opportunity to see industry operations within prison.
In addition to their work brokering new relationships with employers, The New Futures Network is keen to develop strong links with the voluntary sector and assist them in their vital role to help ex-offenders into employment. The NFN is keen to support charities, build new relationships with employers and integrate them further into the infrastructure of employment brokering within prisons. They are looking forward to working with businesses of all sectors and all sizes.
If you are interested in getting involved and think that your business could benefit from hiring a prisoner or ex-offender, please get in touch via Twitter: @NewFutrsNet or register your interest: https://www.smartsurvey.co.uk/s/TEAEB/
© The Exceptionals 2020
Design by GW+Co | Built by Qi Interactive | {
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Q: Intermittent SSL/TLS error using Azure SDK in Azure Website I have an ASP.NET MVC website running .NET 4.5 in an Azure Website and I keep getting this error trying to retrieve or upload assets into Azure Blob Storage using the Azure SDK (version 4.3.0).
The request was aborted: Could not create SSL/TLS secure channel.
Microsoft.WindowsAzure.Storage.StorageException: The request was
aborted: Could not create SSL/TLS secure channel. --->
System.Net.WebException: The request was aborted: Could not create
SSL/TLS secure channel. at System.Net.HttpWebRequest.GetResponse()
at
Microsoft.WindowsAzure.Storage.Core.Executor.Executor.ExecuteSync[T] (RESTCommand`1
cmd, IRetryPolicy policy, OperationContext operationContext) ---
End of inner exception stack trace --- at
Microsoft.WindowsAzure.Storage.Core.Executor.Executor.ExecuteSync[T](RESTCommand`1
cmd, IRetryPolicy policy, OperationContext operationContext) at
Microsoft.WindowsAzure.Storage.Blob.CloudBlobClient.GetBlobReferenceFromServer(StorageUri
blobUri, AccessCondition accessCondition, BlobRequestOptions options,
OperationContext operationContext) at
Microsoft.WindowsAzure.Storage.Blob.CloudBlobClient.GetBlobReferenceFromServer(Uri
blobUri, AccessCondition accessCondition, BlobRequestOptions options,
OperationContext operationContext)
The error doesn't occur every time, but once it starts happening it happens continuously. Only when I scale the Azure website up or down to reset the site does the error stop. It will go a few hours or a few days and then it will come back again.
It seems like this started happening around the time of the Poodle vulnerability and sites shut down their support for SSL3. It seems like from the research I've done that this error could be if the Azure SDK is trying to connect to Azure Blob storage over SSL3. Since it works fine for a time, I wonder if some library in my app is setting the ServicePointManager.SecurityProtocol to SSL3, which is a global setting which Azure is then using from that point on causing the error. Any idea to determine if that is what is happening or how to find that code that is setting the fallback to SSL3?
A: Please check your certificate permissions as discussed here:
https://social.msdn.microsoft.com/Forums/azure/en-US/efb73b00-3610-4a21-ae16-80543451a4d0/windows-azure-dynamicscaling-problem?forum=windowsazuredevelopment
A: Removing and Reinstalling the Certificate fixed this for me after a reboot.
| {
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} | 4,048 |
Home / Blog / Right Whales Wronged
Right Whales Wronged
BY: Brianna Elliott
This story ran in the recent issue of Oceana magazine.
The waters of the mid-Atlantic are alive with sound. The snaps, squeaks, bubbles, pops, and whistles of marine life ring through the water, interspersed with the low calls of North Atlantic right whales. But this chorus of sound may soon be drowned out.
Already the right whales' calls are few and far between. The North Atlantic right whale is the rarest of the world's large whales and one of the most endangered species in the United States. Decimated by intensive whaling in past centuries, there are now only an estimated 500 whales in the waters off the east coast. Right whales earned their name because they were easy to harpoon and float when dead, making them the "right" ones for whalers to target. Though no longer hunted, the population is still struggling to recover because this surface-dwelling species is especially vulnerable to being struck and killed by ships.
Right whales migrate along the East Coast twice each year, traveling between breeding grounds in the south and feeding grounds in the north. In winter they breed and calve in the warm waters off of Georgia and northern Florida, and then migrate north to feed during the summer on plankton in the cooler waters between New York and Nova Scotia. Unfortunately, this annual migration route puts right whales directly in the path of both heavy shipping traffic and planned oil and gas exploration.
The Bureau of Ocean Energy Management (BOEM) is planning to allow energy companies to use seismic airguns to search for offshore deposits. These devices map the seafloor by shooting pulses of compressed air through the water every 10 seconds, creating a map from the reflected sound waves.
"They blast constantly for weeks on end, and are extremely harmful to marine mammals like right whales,"says Matthew Huelsenbeck, a marine scientist with Oceana, "because they rely on sound to communicate, feed, reproduce, and migrate." The government estimates that seismic airguns will injure at least 138,000 dolphins and whales if they are used in the Atlantic.
As part of the decision-making process, BOEM is required to prepare an Environmental Impact Statement, or EIS, that analyses the effects airguns would have on marine life and outlines protection plans for endangered species, like the right whale. "But Oceana-funded research revealed that BOEM's initial mitigation measures would be completely inadequate," says Claire Douglass, Oceana's campaign director for climate and energy.
In partnership with the International Fund for Animal Welfare, Oceana funded a two-year study of North Atlantic right whales off the coast of Virginia, conducted by scientists from Cornell University's Bioacoustics Research Program. Their Right Whale Listening Network gathers data about right whale occurrence along the eastern seaboard. "We want to paint a continental-scale understanding of what right whales are doing and when they are in particular locations," says program director Aaron Rice.
But because the mid-Atlantic was thought of as just a migratory corridor, Rice says, researchers had little data about where and when right whales occur in Virginia waters. To fill the gap, Rice and his colleagues deployed six marine autonomous recording units, or MARUs, along the continental shelf off of Virginia Beach in 2012. These battery-powered hydrophones, or underwater microphones, record ocean noises continuously for six months—capturing what Rice calls "the soundscape of the ocean."
After six months the MARUs are hauled up and swapped for fresh devices, so Rice and his team can analyze the recordings. A computer program, aided by human analysts, sorts through up to 100,000 hours of data to find right whale calls. Rice says that unlike humpback or bowhead whales, right whales don't sing long, dramatic songs. Instead, they have several types of short noises, including a moan, a rumble, a gunshot sound, and an up-call, also called a contact call. Rice and his team use these up-calls to tell exactly when and where right whales are in the recording area.
"The first thing that we noticed was that we had right whales all over the place," says Rice. The researchers expected to detect right whales during a few weeks in the spring and fall, when they migrated through the area. But initial data reveal that right whales are staying in Virginia waters year-round. Also surprising is where the right whales are found offshore—the data show right whales spread widely across the continental shelf, between 18.4 and 72.5 miles offshore.
"Rice's discovery is worrying, because it means the government's plans will not adequately protect right whales from seismic airguns," says Douglass.
Before BOEM published the EIS in February, Douglass and others at Oceana met with their science team to discuss Rice's new data. "We wanted to make sure that the decision makers have the most up-to-date science," she says. "Unfortunately," Douglass says, "BOEM chose to ignore the science in their EIS, which means this critically endangered species will not be protected."
After reviewing Rice's data and Oceana's concerns, BOEM revised the EIS to include the new data about Virginia right whales, says Douglass. They included time-area closures for right whales and loggerhead sea turtles, acoustic monitoring and visual surveys, and shut-down procedures for when a marine mammal is present.
While Douglass is glad that BOEM incorporated the new information into their protection measures, she says that even with these precautions, airguns will still harm right whales and many other marine species if they're allowed in the Atlantic. "Rice's research shows that we know so little about these whales that we shouldn't be considering seismic in the first place," she says. | {
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Q: About the definition of the product of ideals Let $R$ be a ring and $I,J \subset R$ such that $I$ is a left ideal. Let us consider the set
$IJ = \left \{ \displaystyle \sum_1^n i_k j_k \mid n \in \Bbb N, i_k \in I, \; j_k \in J \right \} \tag 1 $
Then is it true that this is a left ideal?
*
*If $a, b \in IJ$ then, from the fact that $I$ is a left ideal we have that $a-b \in IJ$.
*Let $a \in IJ$ and $r \in R$. Then $a = \displaystyle \sum_{p=1}^{k} i_p j_p$ , where $k \in \Bbb N$ and $i_p \in I , j_p \in J , \forall p \in \overline{1,k}$ .
Because $r \cdot i_p \in I$ , $\forall p \in \overline{1,k}$ it is clear that $ra \in IJ$.
I don't see any mistake here.
My questions are the following:
*
*In the definition given here , why do we need that $R$ is commutative and unital?
*Moreover, can't we define the product of $n$ ideals and the power of an ideal without assuming that $R$ is commutative and unital ?
| {
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{"url":"https:\/\/codedump.io\/share\/0HMY5EaoGbCX\/1\/how-do-you-position-a-mathjax-element-within-a-box-using-jquery","text":"Victor Mehta - 1 year ago 49\nJavascript Question\n\n# How do you position a MathJax element within a box using JQuery?\n\nI'm trying to position a MathJax element within a box. I have tried several different methods but none work. The box itself gets positioned but I want the element that is appended to the box to be positioned within the same box (#third). The code so far actually moves the entire third box not the element that is being appended to the third box. Here is the Javascript\/JQuery code. The complete code can be viewed at the following link: MathJax Code\n\nfunction drop(ev) {\n\nev.preventDefault();\nvar data = ev.dataTransfer.getData(\"text\");\n\n\/\/ev.target.appendChild(document.getElementById(data));\n\nswitch(data)\n{\ncase(\"drag1\"):\n\n$('#second').append('$$\\\\sum$$'); break; case(\"drag2\"):$('#second').append('$$\\\\int$$');\nbreak;\n\ncase(\"drag3\"):\n\n$('#second').append('$$\\\\alpha$$'); break; case(\"drag4\"): \/\/$('#second').css(\"font-size\",\"150%\");\n$('#second').append('$$\\\\beta$$').css(\"font-size\",\"150%\"); break; case(\"drag5\"):$('#second').append('$${du}$$');\nbreak;\n\ncase(\"drag6\"):\n\n$('#second').append('$${dt}$$'); break; case(\"drag7\"):$('#second').append('$${t}$$');\nbreak;\n\ncase(\"drag8\"):\n\n\/\/$('#second').append('$$\\\\beta$$').css(\"font-size\",\"150%\"); \/\/var$sqrtEqElement = $('#third').append('$${t}^x\\\\sqrt{t}^x$$').css(\"font-size\",\"100%\"); \/\/var$sqrtEqElement = document.createElement(\"$${t}^x\\\\sqrt{t}^x$$\");\n\/\/$('#sqrtEqElement').parent().css({position: 'absolute'}); \/\/$('#sqrtEqElement').css( { position: 'absolute', top: 0, left: 0} );\n\n$('#second').append('$${t}^x\\\\sqrt{t}^x$$');$('#container').append('<div id=\"third\" ondrop=\"drop(event)\" ondragover=\"allowDrop(event)\"><\/div>');\n\n$('#third').append('$${t}^x\\\\sqrt{t}^x$$').css( { position: 'absolute', top: 10, left: 100} ); \/\/not working \/\/$('#third').append('<label>Filename:<\/label> <input type=\"text\" name=\"file\" id=\"file\" \/>');\n\nbreak;\n\ndefault:\n}\nMathJax.Hub.Queue([\"Typeset\",MathJax.Hub,\"second\"]);\nMathJax.Hub.Queue([\"Typeset\",MathJax.Hub,\"third\"]);\n\n\n}\n\nAnswer Source\n\nHave you tried the following :\n\n var text = '<span style=\"position:absolute; top: 10px; left: 100px;\">$${t}^x\\\\sqrt{t}^x$$<\/span>';\n\\$('#third').append(text);","date":"2017-11-19 01:53:14","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\": 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\": 0.3933126926422119, \"perplexity\": 5261.614624886997}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-47\/segments\/1510934805242.68\/warc\/CC-MAIN-20171119004302-20171119024302-00438.warc.gz\"}"} | null | null |
is a Japanese professional wrestler currently working as a freelancer and is best known for her tenure with the Japanese promotions Ice Ribbon and Actwres girl'Z.
Professional wrestling career
Independent circuit (2015-present)
Ozaki made her professional wrestling debut at the first-ever event of Actwres girl'Z, the AgZ Prologue from May 31, 2015 where she defeated Yuuki Harima.
As a freelancer, Ozaki is known for competing in various promotions. At Maki Narumiya Thank You For All, an event produced by Reina Pro Wrestling on March 25, 2016, she teamed up with Tae Honma in a losing effort to Natsumi Maki and Saori Anou. At JWP Pure Plum, an event promoted by JWP Joshi Puroresu on August 14, 2016, Ozaki teamed up with Tsukushi in a losing effort to Manami Katsu and Rabbit Miu. At WAVE Young OH! OH! The Final, an event promoted by Pro Wrestling Wave on December 15, 2016, Ozaki competed in a nine-person battle royal also involving Asuka, Konami, Rydeen Hagane, Fairy Nihonbashi and others.
Big Japan Pro Wrestling (2016-2019)
Ozaki worked several times as female talent in Big Japan Pro Wrestling. At BJW Summer Ueno Pro-Wrestling Festival on August 16, 2016, she teamed up with Hiragi Kurumi to defeat Mochi Miyagi and Tequila Saya. At a house show from June 9, 2018 she teamed up with Akane Fujita in a losing effort against Maya Yukihi and Risa Sera as a result of a tag team match.
Ice Ribbon (2015-present)
Ozaki spent most of her career working in Ice Ribbon. At Ice Ribbon Hiragi Kurumi 10th Anniversary from May 29, 2020, she teamed up with Hamuko Hoshi to defeat Best Friends (Arisa Nakajima and Tsukasa Fujimoto) in a comedic hot dog eating match. At Ice Ribbon New Ice Ribbon #1054 on July 25, 2020, she competed in a five-way elimination match to determine the #1 contender for the ICE Cross Infinity Championship won by Suzu Suzuki and aldo involving Hamuko Hoshi, Ibuki Hoshi and Satsuki Totoro. At Ice Ribbon New Ice Ribbon #1013 she competed in a 45-person gauntlet match in which the retiring Tequila Saya took all the rest of the opponents to a draw such as Cherry, Itsuki Aoki, Kaori Yoneyama, Syuri, Ken Ohka, Manami Toyota, Matsuya Uno, Yuki Mashiro and many others. Ozaki is a former International Ribbon Tag Team Champion, title which she won by teaming up with Maya Yukihi as "Rebel X Enemy" at RibbonMania 2020 on December 31 by defeating Frank Sisters (Hiragi Kurumi and Mochi Miyagi.
She is known for competing in the promotion's signature events such as the Kizuna Tournament. At the 2020 edition she teamed up with Tequila Saya and defeated Giulia and Tsukushi in a first round match, Hiragi Kurumi and Yappy in a second round but fell short to Risa Sera and Suzu Suzuki in the semi-finals from August 20.
Championships and accomplishments
Actwres girl'Z
AWG Tag Team Championship (1 time, inaugural) – with Tae Honma
AWG Tag Team Title Tournament (2021) – with Tae Honma
Ice Ribbon'''
International Ribbon Tag Team Championship (2 times) – with Kyuri (1) and Maya Yukihi (1)
Triangle Ribbon Championship (1 time)
References
1991 births
Living people
Japanese female professional wrestlers
People from Kyoto Prefecture | {
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Magia babilońska – ogół wierzeń Babilończyków w siły nadprzyrodzone oraz określone rytuały, które stosowali w celu ich opanowania. Na magię babilońską składały się zaklęcia przeciwko złym mocom, zaklęcia ochronne (w tym modlitwy), czary, wróżby, wiara w ochronną moc amuletów i obrzędy, które odprawiano w poszczególnych przypadkach, mające na celu unicestwienie złego działania sił nadnaturalnych oraz wzmocnienie działania mocy dobroczynnych.
Zaklęcia
Babilońskie zaklęcia magiczne pochodziły z okresu III dynastii z Ur. Ułożone w zbiory zostały już po upadku Sumeru. Znane są trzy zbiory zaklęć: Szurpu, Maqlu oraz Utukki limnuti. Choć tytuły Szurpu i Maqlu sugerują palenie ogniem, oba zbiory zawierały odrębne jego rodzaje. Zaklęcia i obrzędy z Szurpu oczyszczały za pośrednictwem wzywania bóstw ognia, w Maqlu zaś znalazły się opisy rytuałów przeciwko czarom ludzkim. Utukki limnuti był zbiorem egzorcyzmów. Wśród innych tekstów znane są tabliczki z opisami sposobów leczenia chorób, ochrony domu, pola czy sadu, procedur postępowania kapłanów podczas egzorcyzmów lub z wykazami złych czynów.
Babilończycy prawdopodobnie nie odróżniali modlitwy od zaklęcia. Modlitwa zazwyczaj rozpoczynała się wyrazem "zaklęcie", jednak treść zwracania się do bóstwa wskazywała na to, że niektóre formy kontaktu z bóstwami miały formę modlitwy, która była sposobem przeciwstawiania się nieszczęściom, złu, ale także sposobem znalezienia pociechy. Początkowe słowa modlitwy babilońskie zawierały przeważnie wołanie do boga i wymienieniem jego tytułów i przymiotów. Następnie orant przedstawiał się i opisywał swoją sytuację. W niektórych modlitwach na koniec występowały wskazówki postępowania, co potwierdza przypuszczenie, że modlitwy babilońskie miały elementy zaklęć i rytuałów magicznych.
Czary
W wierzeniach babilońskich wyróżniano czary dobre i złe. Ponieważ kodeks Hammurabiego groził złemu czarownikowi karą śmierci, czarnoksiężnicy działali w tajemnicy, więc babilońskie sposoby na rzucanie czarów nie są dokładnie znane. Spośród niewielu zachowanych źródeł z zakresu uprawiania czarnoksięstwa znajduje się list, którego tekst został napisany w formie prośby do bóstwa o sprowadzenie na czytającego i jego rodzinę śmierci. Wielu sprawców pozostawało bezkarnych. Sposobem na odczynienie złych czarów były egzorcyzmy.
Rytuały magiczne
W świecie nadnaturalnym istniała cała armia demonów, które czyhały na życie ludzkie. Bogowie także nie byli odporni na ich ataki. Zmiany w przyrodzie uważano więc za chwilową niemoc bogów wobec wrogów. Dla przykładu, w czasie zaćmienia księżyca uważano, że Sin poniósł porażkę z rąk demonów. Ludzkość tymczasem odprawiała obrzęd, by poprzez magię wesprzeć swego władcę, który symbolizował bezpieczeństwo kraju.
W przypadku opętania przez demona, który spowodował chorobę, wzywano czarownika. Egzorcyści aszipu i maszmaszu oczyszczali z rzuconych czarów. Obrzędy wypędzenia demonów lub odżegnywania czarów polegały na wzywaniu boga i odmawianiu zaklęcia. Najczęściej wzywano Marduka, Ea lub, gdy przypadek był ciężki, kilku bogów. Rzadko proszono o pomoc gwałtownych pod względem charakteru i niepewnych pod względem pomocy ludzkości Enlila i Anu. Wierzono, że skutecznie przeciwko złu działali bogowie ognia Gira, Gibil i Nusku. W przypadku chorób rytuały były połączone z zabiegami medycznymi. Jeżeli demon zawładnął człowiekiem, sporządzano posążek, z którym za pomocą zaklęcia demon miał się zidentyfikować, następnie figurkę unicestwiano. Innym sposobem było zapewnienie mu zastępczego siedliska, na przykład, zwierzęcia. Następnie zaklinano go do zmiany lokalizacji, inaczej po wypędzeniu groził mu powrót do apsu.
Praktykowano także magię sympatyczną, która polegała na zwróceniu uwagi boga na obłożonego klątwą człowieka, opuszczonego przez jego boskich opiekunów. Wówczas wzywano Marduka z nadzieją, że za jego pośrednictwem Ea ujawni sekret odczynienia klątwy. Rytuał polegał na ablucji z użyciem zaklętych łupin cebuli, daktyli, knota, wełny i koźlej sierści. Obrzędu dokonywano w tzw. izbie ablucji, co sugerowało świątynię, lecz w praktyce było to zazwyczaj miejsce pobytu chorego (pokój, chata, brzeg rzeki itd.). Niekiedy w miejsce magii sympatycznej stosowano substytucję. Polegała ona na zastąpieniu przeklętego człowieka zwierzęciem, któremu kapłan podrzynał gardło, a następnie odprawiał po nim ceremonię pogrzebową.
Poza chorobami i demonami życie człowieka utrudniały niespokojne duchy zmarłych. Rytuał wypędzania przeprowadzano o zachodzie słońca przy wykopanym dole, do którego wkładano chleb i przez woli róg wlewano wodę z mąką. Po rozpaleniu kadzidła i pochodni wzywano Szamasza. W przypadku choroby sprowadzonej przez ducha obrzęd polegał na grzebaniu w rodzinnym grobie woskowej figurki i wizerunku ducha. Pochówek miał symboliczny charakter. Chorobę leczono poprzez uszkodzenie wykonanego z rytualnie czystego kawałka ziemi figurki ducha z wyrytym na lewym biodrze imieniem, następnie składano ofiarę Szamaszowi i proszono go o uwolnienie z choroby. Okaleczony posążek grzebano o zachodzie słońca.
Rytuały, polegające na zapewnieniu opieki dobrych duchów domowi, wiązały się z całą serią czynności: składanie ofiary Szamaszowi, sporządzenie posążków ze ściętego przy pomocy qulmu poświęconego bogu słońca tamaryszku, ozdobienie ich i umieszczenie w pojemniku kullatu wraz z określoną ilością złota, srebra i kamieni szlachetnych, następnie sporządzenia trwalszych posążków z gliny oraz wyrycie na nich imion duchów i zaklęć. Figurki mogły zostać umieszczone w domu po jego oczyszczeniu poprzez ofiary najważniejszym bóstwom. Każdej czynności towarzyszyły modlitwy, zaklęcia, okadzanie i kropienie świętą wodą.
Wróżby
Babilończycy wierzyli, że postanowienia boskie mają swoje odzwierciedlenia na ziemi, dlatego ważnym aspektem ich życia była magia wróżebna. Nietypowe przypadki i zjawiska były rejestrowane, aby przy ich ewentualnym powtórzeniu mieć wzór dla postępowania.
Wróżono przy stosowaniu określonych metod (np. hepatoskopia), poprzez obserwację nietypowych przypadków (np. narodziny potworów, sny), za pomocą astrologii. Najstarszą metrykę mają wróżby według metod, mające potwierdzenie w legendach babilońskich, według których już królowie sprzed potopu wróżyli z wątroby. Odczytywanie znaków ze snów cieszyło się powodzeniem w okresie starobabilońskim, zaś astrologia rozwinęła się dopiero za czasów dynastii chaldejskiej. Gdy wróżba wypadła niepomyślnie, stosowano określone rytuały, polegające na odwróceniu losu.
Wróżby dotyczyły zarówno króla, jak i jego poddanych oraz państwa. Sprawy publiczne rozstrzygano za pomocą astrologii i hepatoskopii. Osoby prywatne stosowały pozostałe sposoby, jak na przykład, sny, przypadkowe spotkania czy egirru, polegającego na odniesieniu w określonych okolicznościach usłyszanej wypowiedzi do swojej osoby.
Amulety
Amulety zapewniały ochronę przed demonami. Znalazło to także odzwierciedlenie w mitologii. Według Enuma elisz Marduk podczas walki z Tiamat trzymał w ręce ziele-odtrutkę, a w ustach przedmiot z gliny. Amulety zazwyczaj formowano w wyobrażaną postać demona, przed którym miał strzec, i opatrywano imieniem i zaklęciem.
Jeden z najbardziej rozpowszechnionych amuletów chronił przed burzami piaskowymi. Wyrabiany był w dwóch kształtach, w obu przypadkach opatrywany inskrypcją. Pierwszy typ przedstawiał głowę demona Pazuzu; drugi był płytką z jego podobizną – ptasia pierś z ludzkimi kończynami i głową, w jednej ręce trzymał piorun, miał ogon i cztery skrzydła. Drugim pod względem popularności był amulet z wizerunkiem Lamasztu, strzegący kobiety w połogu. Rozpowszechnionym typem był amulet z obrazkami w układzie pasowym. Najniższy (najważniejszy) ukazywał prawdopodobnie Lamasztu pod postacią karmiącej dwoje zwierząt kobiety z głową lwa. Z czasem amuletu z Pazuzu – ze względu na podobieństwo jego postaci do Lamasztu – zaczęto używać do ochrony rodzących lub karmiących kobiet.
Przypisy
Bibliografia
Saggs H. W. F., Wielkość i upadek Babilonii, Warszawa 1973.
Religie starożytności
Magia
Babilonia | {
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In 2003, Pastor Nurudeen Adeọjọ was inspired to start a church and a name came to mind at the time. That name was Windows of Heaven Christian Church. Immediately, he started searching the Internet for a free Christian website builder and was successful to find one. A beautiful website was designed for this new Church.
Our fellowship was held in our own one-bedroom apartment in Coney Island, Brooklyn, New York.
In February 2004, on the same day our second child was christened, Windows of Heaven Christian Church was also inaugurated Windows of Heaven Christian Church continued to fellowship in his one-bedroom apartment until 2006, when the family relocated to Kingston, Pennsylvania. In Kingston, Pennsylvania, the family did not continue in fellowship with Windows of Heaven Christian Church, but rather joined the Nazarene church in fellowship every Sunday morning.
In 2009, as the family continued its Sunday morning fellowship at the Nazarene Church, Windows of Heaven Christian Church started Sunday evening house fellowship, quickly attracting neighbors and others from the Nazarene church.
In 2010, Windows of Heaven Christian Church was incorporated as a 501(C)3 religious organization. The young church faced financial problems in 2011 and was looking to join a larger organization. Pastor Nurudeen Adeọjọ was approached by a Christian leader from the Redeemed Christian Church of God, Revival Assembly in Reading, PA. The family relocated to Reading, Pennsylvania to take the leadership of RCCG Revival Assembly. Pastor Nurudeen Adeọjọ later realized it was a mistake to take the leadership of RCCG Revival Assembly, and resigned in July 2013.
He joined the leadership strength of RCCG Christ Powerhouse in Upper Darby, PA. He was asked in May 2015 to start a new parish in Coatesville, PA, with his wife and their three children. The family of five rented a space at the Hampton Inn, Exton, PA, and later moved out of the Hampton Inn in March 2016 to a rented office space with two other families. All the while, Pastor Nurudeen Adeọjọ struggled to go back to his first call, Windows of Heaven Christian Church. So, as he worked tirelessly at RCCG New Life Center, he always thought of going back where God spoke to him that it is Windows of Heaven Christian Church.
Today, he has realized what God said to him in 2003, and he's now returned TO BETHEL. | {
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\section{Introduction}
Amongst two-dimensional materials, the families of chalcogenides such
as transition metal dichalcogenides, group-III and IV
monochalcogenides often offer the advantages of stability and the
possibility of fabrication by epitaxial growth methods that can be
scaled up---such as vapor transport epitaxy of chemical vapor
deposition (CVD),\cite{yu-RSC-6-6705} and chemical vapor
transport.\cite{InSe-Ching-Hwa-Ho-Thickness-dependent-carrier-transport}
Indium selenide,\cite{nanoResearchInSe2014} which shares the same
crystal structure with \ch{GaS} and
\ch{InS},\cite{zolyomi2014electrons} has recently been mechanically
exfoliated into few layer
flakes.\cite{deckoff2016observing,mudd2013tuning,bandurin2016high}
Thin \ch{InSe} flakes have been used for phase change memory devices
and image
sensing,\cite{lei-nl503505f,gibson2005phase,robertsonUsefulInSe} and
has been suggested to be a functional material for water
splitting.\cite{H2fromH20-InSe} With respect to the electronic
properties, few layer \ch{InSe} has been shown to have an
extraordinary electron mobility exceeding \num{E3} and
\SI[per-mode=symbol]{E4}{\raiseto{-2}\centi\metre\per\volt\per\second}
at room and liquid-helium temperatures, in few layers, making it one
of the highest known mobility 2D
materials.\cite{bandurin2016high,AbInitioElMobInSe} This is consistent
with the bulk electron mobility, which is also the highest amongst
isomorphic group-III chalcogenides, according to Hall effect
measurements.\cite{segura1984electron} Even though it is often \( n
\)-type, \ch{InSe} can also be \( p \)-type and in that case it can be
interesting for different purposes: It has a very high effective mass
for holes near the \( \Gamma \) point, where there is a
`Mexican-hat'-type van-Hove
singularity.\cite{DFT-tight-binding,parabolic2RSVBM,zolyomi2014electrons,Mudd2016}
Such a singularity gives rise to a ferromagnetic instability at low
temperatures.\cite{multiferroic2D} Different from other materials with
`Mexican-hat'-type bands such as \ch{SnO}, the singularity is present
in the valence band both for monolayer and for few-layer
material.\cite{Mudd2016}
Thus, since both \( p \)- and \( n \)-type conduction regimes are of
technological interest, it is desirable to be able to effectively
control the type and amount of defects and impurities unintentionally
introduced. \ch{Sn} and \ch{Pb}, when present, can act respectively
as a shallow donor and shallow acceptor. The first is often cited as
the origin of the \( p \)-type conductivity. However, intrinsic
shallow donors that cannot be ascribed to any impurity and disappear
upon annealing have been found as well.\cite{segura1984electron,
segura1983investigation, martinez1992shallow} These were speculated
to be related to \ch{Se} deficiency.\cite{martinez1992shallow}
According to previous theoretical calculations, adsorbed or
interstitial \ch{In} has low formation energy in \ch{In}-rich
material,\cite{robertsonUsefulInSe} parallel to what has been found
for the \ch{Ga} interstitial in \ch{GaS},\cite{Chen2015} However, many studies of
point defects in III-VI materials have been restricted to vacancies or
substitutional type
defects.\cite{Rak2008,Rak2009,Li2017,robertsonUsefulInSe,H2fromH20-InSe,Chen2015elB}
Thus, specific defect signatures of the intrinsic shallow donors have
not been assigned yet.
Interstitial atoms are supposed to increase the mechanical hardness of
bulk \ch{GaSe} by coupling the planar
layers,\cite{kokh2011growth,huang2017experimental} and the same has
been found for other ionized dopants as well.\cite{rak2010doping}
In addition to intrinsic defects, it is important to investigate the
defects caused by the interaction with oxygen and other atmospheric
contaminants. The recently achieved high mobility transistor devices
were fabricated with \ch{BN}-encapsulated \ch{InSe} layers, that were
thus prevented from contact with the
atmosphere.\cite{bandurin2016high} Still, \ch{InSe} seems to be
relatively stable in contact with air, as cleaved bulk surfaces show
no signs of degradation at room
temperature,\cite{myake-JJAP-23-172,balakrishnan2017engineering}
comparing e.g.~with phosphorene.
In this article, we will provide a detailed theoretical account of the
properties of intrinsic defects and oxygen-related defects in
\ch{InSe}. In addition, we will discuss their impact on the electronic
properties of the material, in particular discussing the identity of
the shallow donors in unintentionally doped \ch{InSe}.
\section{Methods}
\subsection*{Parameters}
The first principles calculations were performed by the density
functional theory (DFT)\cite{dft-hk, dft-ks} implementation known as
{\scshape Quantum ESPRESSO}.\cite{QE-2009}\( ^{,} \)\footnote{version
6} All of the computations were done consistently using the
following parameters. The pseudopotentials used were given by the
projector augmented wave (PAW)\cite{paw-blochl, paw-from-us}
approximation, and the exchange-correlation functional chosen was the
generalized gradient approximation parametrized by Perdew, Burke, and
Ernzerhof (GGA-PBE).\cite{pbe-gga-made-simple} Specifically, the
PSeudopotential Library (PSL)\cite{DalCorso2014337}\( ^{,}
\)\footnote{versions 0.3.1 and 1.0.0} were used. A plane wave basis
with kinetic energy cutoff of \SI{42}{Ry} was used, and the \( k
\)-point samples in the Brillouin zone were calculated with the \(
\Gamma \)-centered \( 4 \times 4 \times 1 \)
Monkhorst-Pack\cite{monkhorst-pack} grid unless otherwise
specified. Defect ionization transition levels were calculated with a
\( k \)-point grid of \( 8 \times 8 \times 1 \) centered upon \(
\Gamma \), with relaxation. All transition levels presented were at
most \SI{.02}{\electronvolt} from their values when calculated with
the smaller \( k \)-point grid. All geometries were relaxed to at
least the default convergence thresholds (Forces \( < \SI{E-3}{a.u.}
\)). The vacuum spacing along the \( z \)-axis was six times the
lattice parameter of the primitive cell of the pristine monolayer, to
avoid spurious interactions. All supercells consisted of \( 3 \times 3
\) primitive unit cells.
Finally, to find the migration activation energies for the relevant
defects, we also performed nudged elastic band calculations, without
climbing images nor spins.
\subsection*{Formation Energies \& Transition Levels}
The formation energy of defect \( D \) is given by
\begin{equation}
E_{\!f} ( D ) = E ( D ) - \sum_i n_i \mu_i
\end{equation}
where \( E ( D ) \) is the energy of the supercell containing the
defect, and \( n_i \) and \( \mu_i \) are the number of atoms of
species \( i \) and its chemical potential, respectively. The
chemical potentials were evaluated both in the \ch{In}-rich and
\ch{Se}-rich limit. In the \ch{In}-rich case, the \ch{In} potential
was obtained from the elemental material in the \(\alpha\)-\ch{In},
tetragonal form. The \ch{Se} chemical potential \(
\mu_{\ch{Se},\text{\ch{In}-rich}} \) in the \ch{In}-rich regime was
obtained from the constraint
\begin{equation}
E(\text{PS}) = \sum_{j} n_j \mu_{j,\text{\ch{In}-rich}}.
\end{equation}
where PS is the pristine supercell. A similar definition was used to
obtain the chemical potentials in the \ch{Se}-rich limit for which we
used the trigonal \textit{hP3} \ch{Se} allotrope as reference. The
chemical potential for oxygen is obtained from molecular oxygen.
The defect ionization transition levels \( E_D ( q / q + 1 ) \),
defined by the Fermi level at which the formation energy of the
defects in charge state \( q \) is the same as in charge state \( q +
1 \), were found using the marker method, which is more accurate for
2D systems due to the cancellation of systematic
errors\cite{defectsIn2Dvs3D}. The ionization potential \( I_D \) and
electron affinity \( A_D \) are defined by
\begin{align}
I_D &= E ( D^+ ) - E ( D^0 ), & A_D &= E ( D^0 ) - E ( D^- ).
\end{align}
The transition levels for acceptors \( E_D ( - / 0 ) \) (donors \( E_D
( 0 / + ) \)) relative to valence band maximum \( E_v \) (downwards
from conduction band minimum \( E_c \)), are given by
\begin{subequations} \label{eq:def:E_D}
\begin{align}
E_D ( - / 0 ) - E_v &= E_g - \left [ E_c - E_D ( - / 0 ) \right ] =
E_g - \left [ A_D - A_{PS} \right ] \\
E_c - E_D ( 0 / + ) &= E_g - \left [ E_D ( 0 / + ) - E_v \right ] =
E_g - \left [ I_{PS} - I_D \right ]
\end{align}
\end{subequations}
\section{Results}
\subsection{Intrinsic Point Defects}
\begin{figure*}
\includegraphics[resolution=300,width=\textwidth]{xcrysden-pt-d.png}
\caption{\label{fig:ptDxcrysden}(Color online) Top (0001) and side
(11\(\bar{2}\)0) views of various intrinsic point defects and
substitutional oxygen in monolayer \ch{InSe}, grouped by
similarity.
(a) PS: pristine supercell.
(b) V\(_{\ch{In}}\): indium vacancy.
(c) \ch{Se}\(_{\ch{In}}\): selenium-in-indium anti-site.
(d) swap: swapping adjacent selenium and indium.
(e) \ch{In}\(_{\ch{Se}}\): indium-in-selenium anti-site.
(f) V\(_{\ch{Se}}\): selenium vacancy.
(g) \ch{In}\(_{ac}\): indium hovering above the center of the
hexagonal interstitial cage.
(h) \ch{In}\(_{ic}\): interstitial indium at center of hexagonal
cage.
(i) \ch{O}\(_{\ch{Se}}\): oxygen atom substituting a selenium.
(j) \ch{O2}\(_{\ch{Se}}\): oxygen molecule substituting a selenium.}
\end{figure*}
\begin{figure}
\includegraphics[resolution=300,width=\columnwidth]{bs-intrinsic.png}
\caption{\label{fig:bs:intrinsic}(Color online) DFT band structure
plots of various intrinsic point defects and substitutional oxygen
defects in monolayer \ch{InSe}:
(a) V\(_{\ch{In}}\): indium vacancy.
(b) \ch{Se}\(_{\ch{In}}\): selenium-in-indium anti-site.
(c) swap: swapping adjacent selenium and indium.
(d) \ch{In}\(_{\ch{Se}}\): indium-in-selenium anti-site.
(e) V\(_{\ch{Se}}\): selenium vacancy.
(f) \ch{In}\(_{ac}\): indium hovering above the center of the
hexagonal interstitial cage.
(g) \ch{In}\(_{ic}\): interstitial indium at center of hexagonal
cage.
(h) \ch{O}\(_{\ch{Se}}\): oxygen atom substituting a selenium.
(i) \ch{O2}\(_{\ch{Se}}\): oxygen molecule substituting a selenium.
Refer to Fig.~\ref{fig:ptDxcrysden}
for the respective defects.
Majority and minority spin bands are represented by
continuous and dashed lines, respectively.
Fermi levels are represented by blue dash-dotted horizontal lines. }
\end{figure}
\begin{figure}
\includegraphics[resolution=300,width=\columnwidth]{chem-pot-intrinsic.png}
\caption{\label{fig:chemPot:intrinsic}Formation Energies \( E_{\!f} \)
as a function of chemical potential \( \mu_{\ch{Se}} \) (arbitrary
units) for intrinsic defects. \( \Delta \mu_{\ch{Se}} =
\SI{1.05}{\electronvolt} \). Refer to text for constraints and
definitions.}
\end{figure}
This work considered seven intrinsic point defects
(Fig.~\ref{fig:ptDxcrysden}): the indium vacancy (V\(_{\ch{In}}\)),
the anti-site defect consisting of a selenium replacing for indium
(\ch{Se}\(_{\ch{In}}\)), indium replacing for selenium
(\ch{In}\(_{\ch{Se}}\)), a swapped In-Se next-neighbor pair
(\ch{In}\(_{\ch{Se}}\)-\ch{Se}\(_{\ch{In}}\)), that we will name
``swap'', the selenium vacancy V\(_{\ch{Se}}\), selenium interstitial
at the hexagonal interstitial site (\ch{In}\(_{ic}\)), and above the
center of the hexagonal interstitial cage (\ch{In}\(_{ac}\)).
The respective band structures are represented in
Fig.~\ref{fig:bs:intrinsic}. The indium vacancy is a shallow acceptor
(Fig.~\ref{fig:bs:intrinsic}a). \ch{Se}\(_{\ch{In}}\) has a similar
band structure, but the states originating in the In vacancy are
half-filled and move towards mid-gap, whereas the conduction band is
little perturbed (Fig.~\ref{fig:bs:intrinsic}b). The other anti-site
defect also has semi-filled states, whereas the combined swap of
neighboring \ch{In} and \ch{Se} results in filled defect states near
the valence band (Fig.~\ref{fig:bs:intrinsic}c,d). The selenium
vacancy introduces defect states both near the valence and conduction
band (Fig.~\ref{fig:bs:intrinsic}e). Finally, the indium
interstitials are shallow donors (Fig.~\ref{fig:bs:intrinsic}f,g).
The \ch{In}\(_{ac}\) configuration, the most stable (about
\SI{1.59}{\electronvolt} lower in energy than the \ch{In}\(_{ic}\)
configuration), changes little the conduction band dispersion, however
donates free holes to the conduction band states.
The formation energies as a function of the \ch{Se} chemical potential
over all available range are shown in
Fig.~\ref{fig:chemPot:intrinsic}. As expected, in the \ch{In}-rich
regime the dominant defects are the \ch{Se} vacancy and the \ch{In}
interstitial, whereas in the \ch{Se}-rich limit the dominant defects
are the \ch{In} vacancy and the anti-site where \ch{Se} replaces
\ch{In}. These regimes will be considered in more detail in the next
sections.
\subsubsection{\ch{In}-rich regime}
\begin{table}
\caption{\label{tab:dopants} Ionization potential and electron
affinity \emph{differences} of the various defects in monolayer
\ch{InSe}, which can be subtracted from \( E_g ( \approx
\SI{2.4}{\electronvolt}
)\cite{nanoResearchInSe2014,robertsonUsefulInSe,debbichi2015,olguin2013}
\) to provide the activation energies via marker method (see
text). All energies are in \si{\electronvolt}. }
\begin{tabular}{c @{\hskip .5cm}
S[ table-format = 1.2 , table-auto-round ] @{\hskip .5cm}
S[ table-format = 1.2 , table-auto-round ]}
\hline \hline
{Defect} & {\( E_D ( 0 / + ) - E_v \)} &
{\( E_c - E_D ( - / 0 ) \)} \\
\hline
\(\ch{In}_{ac}\) & 2.17 & \\
V\(_{\ch{Se}}\) & .40 & .65 \\
\(\ch{Se}_{\ch{In}}\) & .97 & 1.22 \\
V\(_{\ch{In}}\) & & 1.60 \\
\ch{O2}--A & & .16 \\
\hline \hline
\end{tabular}
\end{table}
\ch{InSe} crystals are typically grown using the Bridgmann method,
from non-stoichiometric melts with \ch{In} excess, resulting in
\ch{In}-rich crystals.\cite{segura1984electron,
segura1983investigation, martinez1992shallow}. This is expected due
to the higher volatility of \ch{Se} compared to \ch{In}.
In this regime, the most stable defect, of the four defects we have
considered, is an \ch{In} interstitial above the hexagonal cage,
closely followed by the \ch{Se} vacancy, the latter of which seems to
make a triangular bond between the three \ch{In} atoms surrounding the
vacancy. Both are donors (Fig.~\ref{fig:bs:intrinsic}), with
transition levels at \SI{2.17}{\electronvolt} and
\SI{.4}{\electronvolt} above the valence band, respectively
(Table~\ref{tab:dopants}). In particular, the \ch{In} interstitial,
being a shallow donor, is likely to be the source of the \( n \)-type
conduction in this material, as previously suggested following Hall
effect measurements and position lifetime
experiments\cite{martinez1992shallow, positron-lifetime-InSe,
segura1983investigation}. Experimentally, the defect ionization
energy is \SI{18}{\milli\electronvolt}, consistent with the
calculations, that effectively place the transition level close to the
conduction band bottom, within the method
accuracy.\cite{positron-lifetime-InSe} Furthermore, the experimentally
observed donor center concentration is known to increase upon
annealing at \SI{470}{\kelvin} and the donor defects do not affect the
positron lifetime, showing that it is an intrinsic defect and unlikely
to be of vacancy type.\cite{positron-lifetime-InSe} Focusing on the
annealing, we performed a nudged elastic band calculation for both the
indium interstitial and the selenium vacancy in the monolayer case,
obtaining migration activation energies of about
\SI{.21}{\electronvolt} for \ch{In}\(_{ac}\) and
\SI{1.5}{\electronvolt} for V\(_{\ch{Se}}\), in agreement with
expectations. In addition, we note that the anti-site is energetically
expensive, such that it should be rare, and does not contribute to
doping. These establish that the \ch{In} interstitial is responsible
for the \( n \)-type character of undoped samples.
\subsubsection{\ch{Se}-rich regime}
The two relevant intrinsic defects in this regime are the \ch{In}
vacancy and \ch{Se}-replacing-\ch{In} anti-site. V\(_{\ch{In}}\) is a
shallow acceptor, with transition levels calculated to lie
\SI{1.60}{\electronvolt} below the conduction band
(Table~\ref{tab:dopants}). However, since \ch{In} is placed in the
inside of the layer, it is unlikely that V\(_{\ch{In}}\) would exist
on its own, without the removal of neighboring \ch{Se} as well. \(
\ch{Se}_{\ch{In}} \) is both a donor and an acceptor, with possibly a
negative-\( U \) level ordering (Table~\ref{tab:dopants}).
\subsection{\ch{O2} Physisorption}
\begin{figure*}
\includegraphics[resolution=300,width=\textwidth]{xcrysden-add-O2.png}
\caption{\label{fig:addO2xcrysden}(Color online) Top (0001) and side
(11\(\bar{2}\)0) views of the stable single oxygen molecule
addition defects in monolayer \ch{InSe} (physisorption), in
increasing order of relative energy cost of formation.
(a) \ch{O2}--A: above indium, perpendicular to bridge bond.
(b) \ch{O2}--B: above center of hexagonal cage, perpendicular to
bridge bonds.
(c) \ch{O2}--C: above center of hexagonal cage, along bridge bonds.
(d) \ch{O2}--D: above selenium, along bridge bond.
(e) \ch{O2}--E: above selenium, perpendicular to bridge bond.
(f) \ch{O2}--F: above indium, along bridge bond.
(g) \ch{O2}--G: interstitial molecule at center of hexagonal cage,
perpendicular to monolayer.}
\end{figure*}
\begin{table}
\caption{Formation energies for each of the various stable oxygen
absorption defects in monolayer \ch{InSe}.
Refer to Fig.~\ref{fig:addO2xcrysden} and Fig.~\ref{fig:addOxcrysden} for
meaning of abbreviated names.
All energies are in \si{\electronvolt}.
}
\addtocounter{table}{-1}
\hfill
\subfloat[\label{tab:Ef:addO2}Physisorbed oxygen molecules.]{
\begin{tabular}{c @{\hskip .5cm} S[ table-format = 1.2 , table-auto-round ]}
\hline \hline
{Defect} & { \( E_{\!f} \) } \\
\hline
\ch{O2}--A & -.0162150297 \\
\ch{O2}--B & -.0158097053 \\
\ch{O2}--C & -.0115210276 \\
\ch{O2}--D & -.0109381542 \\
\ch{O2}--E & -.0035151983 \\
\ch{O2}--F & -.0012833205 \\
\ch{O2}--G & .9490691946 \\
\hline \hline
\end{tabular}
}
\hfill \hfill
\subfloat[\label{tab:Ef:addO}Chemisorbed oxygen atoms.]{
\begin{tabular}{c @{\hskip .5cm} S[ table-format = 1.2 , table-auto-round ]}
\hline \hline
{Defect} & { \( E_{\!f} \) } \\
\hline
\ch{O}--A & -1.646926308 \\
\ch{O}--B & -1.637772552 \\
\ch{O}--C & .048381663 \\
\ch{O}--D & .368839000 \\
\ch{O}--E & .741668050 \\
\ch{O}--F & 1.071440388 \\
\ch{O}--G & 2.608849276 \\
\hline \hline
\end{tabular}
}
\hfill
\end{table}
\begin{figure}
\includegraphics[resolution=300,width=\columnwidth]{bs-addO2.png}
\caption{\label{fig:bs:addO2}(Color online) DFT band structure plots
of various stable single oxygen molecule defects in monolayer
\ch{InSe} (Physisorption) in increasing order of relative energy
cost of formation. (a) Pristine 3x3 supercell; (b)--(h) different
configurations of oxygen defects. Refer to
Fig.~\ref{fig:addO2xcrysden} for the respective defects. Minority
spin is shown in dashed line. Color makes deeply embedded impurity
states easier to see.}
\end{figure}
Figure~\ref{fig:addO2xcrysden} shows the top and side views of all the
possible configurations for oxygen molecule physisorption onto
\ch{InSe}. The formation energies are nearly the same (within
\SI{10}{\milli\electronvolt}) for all the configurations A--F
(Table~\ref{tab:Ef:addO2}). The respective band structures, shown in
Fig.~\ref{fig:bs:addO2}, are also nearly identical, having no gap
states for the majority spin and a double-degenerate empty gap state
for minority spin. The coloring of the band structure plot helps
reveal the deeply embedded impurity states beneath the valence band,
which are flat, similar to the degenerate impurity gap states (dashed
lines) in the band gap. The last of the structures considered,
\ch{O2}--G, consists of an oxygen molecule inside the interstitial
cage. This is \SI{.97}{\electronvolt} higher in energy than surface
physisorbed molecules (Table~\ref{tab:Ef:addO2}). Physisorbed oxygen
can therefore in principle act as electron acceptor, as found in
graphene,\cite{gianozziGrapheneoxygenPhysisorbAcceptor}
phosphorene\cite{cheng-han-2d-4-021007}, and transition metal
dichalcogenides\cite{kumar-PRL}
\subsection{\ch{O} Chemisorption}
\begin{figure*}
\includegraphics[resolution=300,width=\textwidth]{xcrysden-add-O.png}
\caption{\label{fig:addOxcrysden}(Color online) Top (0001) and side
(11\(\bar{2}\)0) views of the stable single oxygen atom addition
defects in monolayer \ch{InSe} (Chemisorption), in increasing order
of relative energy cost of formation.
(a) \ch{O}--A: interstitial oxygen defect between two indium atoms,
with angled bonds like in water molecule,
venturing out into the hexagonal interstitial cage.
(b) \ch{O}--B: interstitial oxygen in angled bond between two indium
atoms, underneath (bridge) bond of indium-selenium.
(c) \ch{O}--C: oxygen in angled bond between indium and selenium.
(d) \ch{O}--D: interstitial oxygen at center of hexagonal cage.
(e) \ch{O}--E: oxygen above selenium.
(f) \ch{O}--F: three-coordinated oxygen between two selenium atoms,
also bonded with indium atom.
(g) \ch{O}--G: oxygen above indium.
The case of oxygen atom hovering above the center of the hexagonal
interstitial cage is not stable.}
\end{figure*}
\begin{figure}
\includegraphics[resolution=300,width=\columnwidth]{bs-addO.png}
\caption{\label{fig:bs:addO}(Color online) DFT band structure plots of
various stable single oxygen atom defects in monolayer \ch{InSe}
(Chemisorption) in increasing order of relative energy cost of
formation. (a) Pristine 3x3 supercell; (b)--(h) different
configurations of oxygen defects. Refer to
Fig.~\ref{fig:addOxcrysden} for the respective defects. (e) is a
magnetic spin calculation without spin-orbit coupling. Minority spin
in dashes.}
\end{figure}
Chemisorption requires breaking the \ch{O2} bond, which is found to
have an energy of \SI{6.61}{\electronvolt} in our calculations, a
typical overestimation, on the high side, under the PBE
approximation\cite{HSEsol} (experimentally measured to be
\SI{5.12}{\electronvolt}\cite{HSEsol}). Nevertheless, we found that
the chemisorption of oxygen is energetically favorable compared to
physisorption.
Figure~\ref{fig:addOxcrysden} shows the top and side views of all the
single oxygen atom addition defects, while the band structure plots
are presented in Fig.~\ref{fig:bs:addO}. The formation energies \(
E_{\!f} \) do not depend on the \ch{In} and \ch{Se} chemical potentials
(Table~\ref{tab:Ef:addO}).
Table~\ref{tab:Ef:addO} shows that that there is a pair of essentially
degenerate defects that are the lowest in energy. They are the
\ch{O}--A configuration, interstitial oxygen defect between two indium
atoms, near the bond-center, venturing out into the hexagonal
interstitial cage, and the \ch{O}--B configuration, interstitial
oxygen also near the bond-center between two indium atoms, but
underneath the indium-selenium bond. The other defects are
considerably higher in energy. The band structure plots then tell us
that the three defects of this class, the lowest in energy, are
basically of the same type, and that they barely differ from the band
structure of the PS.
Since chemisorbed oxygen defects have no levels in the gap, their
interaction with vacancies to form substitutional defects will not be
of the Coulomb type but possible strain mediated, since interstitial
atoms, contrary to vacancies, introduce compressing strain on the
surrounding lattice. In the next section, we will consider the
defects resulting of the interaction between chemisorbed oxygen and
selenium vacancies.
\subsection{\ch{O} Substitution Defects}\label{sec:subO}
\begin{figure}
\includegraphics[resolution=300,width=4.5cm]{chem-pot-sub-O.png}
\caption{\label{fig:chemPot:subO}Formation Energies \( E_{\!f} \) as a
function of chemical potential \( \mu_{\ch{Se}} \) (arbitrary units)
for oxygen substitution defects. \( \Delta \mu_{\ch{Se}} =
\SI{1.05}{\electronvolt} \). Refer to text for constraints and
definitions.}
\end{figure}
We have considered the possibility that a \ch{Se} lattice site is
occupied by an oxygen atom or by an oxygen molecule
(Fig.~\ref{fig:ptDxcrysden}i,j). The respective band structures are
shown in Fig.~\ref{fig:bs:intrinsic}h,i. The formation energies of
these defects are negative for all the range of chemical potentials,
but are lowest in \ch{In}-rich conditions
(Fig.~\ref{fig:chemPot:subO}). They seem to neither be donors nor
acceptors, just passivating the \( p \)-type selenium vacancy and
reducing the band gap energy. The single substitutional oxygen atom is
\SI{.87}{\electronvolt} lower in energy than the substitutional oxygen
molecule, and it is the most energetically favorable defect presented
in this paper. It is especially likely to form in the presence of
chalcogen vacancies,\cite{airPassivationChalcogen2D} through the
reaction
\begin{align}
\frac{1}{2} \ch{O2} + V_{\ch{Se}} \rightarrow \ch{O}_{\ch{Se}}
\end{align}
which has an enthalpy balance of \SI{3.10}{\electronvolt} per oxygen
atom.
\section{Conclusion}
We have investigated the fundamental intrinsic defects in \ch{InSe},
finding that in \ch{Se}-rich material the \( \ch{Se}_{\ch{In}} \)
anti-site is the dominant effect, whereas in \ch{In}-rich material the
indium interstitial and selenium vacancy are the dominant defects. Our
calculations suggest that the unintentional \( n \)-type doping in
cleanly-grown \ch{InSe} should be due to the indium interstitial,
which is a shallow donor, in agreement with arguments from
experiments.
Selenium vacancies have donor deep states at about
\SI{.4}{\electronvolt} above the valence band, that can partially
compensate the doping by interstitials, but this state can be removed
by reaction with molecular oxygen to form substitutional oxygen at the
\ch{Se} site, which has a positive energy balance of
\SI{3.10}{\electronvolt}.
In the absence of intrinsic defects, oxygen chemisorption and
substitution is still energetically favorable, with such defects
having formation energies \( E_{\!f} \) between \num{-.9} and
\SI{-2}{\electronvolt}. Thus, \ch{InSe} monolayers are prone to
oxidation, but still considerably stronger in resilience against the
chemisorption of oxygen than that in phosphorene (the respective
enthalpies for oxygen chemisorption are \SI{-1.65}{\electronvolt} in
\ch{InSe} and \SI{-2.08}{\electronvolt} in
phosphorene\cite{zilettioxygenDefectsPhosphorene}).
We find that chemisorbed oxygen and substitutional oxygen do not have,
in their most stable forms, any ionization levels in the gap. However,
since chemisorbed oxygen atoms are most stable inside the layer and
between In sub-layers, the structural distortion and perturbation of
the charge density distribution induced by chemisorbed oxygen defects
may reduce the carrier mobility, justifying the use of encapsulating
layers in \ch{InSe}-based electronic devices.
\section*{Acknowledgements}
This work was supported by the National Research Foundation, Prime
Minister Office, Singapore, under its Medium Sized Centre Programme
and CRP award ``Novel 2D materials with tailored properties: beyond
graphene" (Grant number R-144-000-295-281). The first-principles
calculations were carried out on the CA2DM high-performance computing
facilities.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,564 |
Advanced Life Support District Needed to Help Ensure Top-Quality Services
High-Quality Civic Writing
Colton Cowger, a tenth-grader in Columbus, MT, wrote a letter to the editor of the Stillwater County News, advocating for creating an Advanced Life Support district to serve outlying rural communities. His letter set the wheels in motion for a ballot initiative that passed in May 2017. This letter represents effective public writing because it demonstrates the author's clear understanding of community values (the importance of life-saving services and a commitment to reasonable taxes). Further, it effectively advocates that taxpayers deserve the opportunity to vote on establishing an Advanced Life Support district. Colton researched this issue and wrote the letter as part of a multi-week Making Civic Arguments project.
Employs a Public Voice To advocate for establishing an Advanced Life Support district, this letter to the editor effectively connects to the fundamental interests of taxpayers and voters in Stillwater County through the stylistic choice to repeat the idea of life saving. ("They just saved your life." "It might even save your life someday.") The writing establishes the author's credibility through concisely displaying detailed knowledge of the funding mechanisms (fees and federal grants) for the current ambulance service.
Advocates Civic Engagement or Action This letter effectively establishes the need for the Advanced Life Support district by pointing out the limitations of current funding sources. Further, it advocates a reasonable and feasible civic action—allowing taxpayers to vote on whether to establish and fund such a district. While the letter points out support for the idea from a survey, it also emphasizes that taxpayers need to make the decision.
Argues a Position Based on Reasoning and Evidence The letter reflects two values that the author brings to this issue—importance of public funding for life-saving services and of taxpayers voting for tax increases. The letter also offers thoughtful interpretation and synthesis of evidence about how ALS is currently funded and provided, interviews with paramedics, and a community survey.
Employs a Structure This letter to the editor effectively organizes the information in a problem / solution format. In addition, it includes a particularly strong opening and closing, designed to underscore the critical nature of public funding for an Advanced Life Support district.
Download Annotation (PDF) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,518 |
Somera similis är en fjärilsart som beskrevs av Nakamura 1976. Somera similis ingår i släktet Somera och familjen tandspinnare. Inga underarter finns listade i Catalogue of Life.
Källor
Tandspinnare
similis | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,537 |
Q: problem in get the yahoo contacts webservice I try get the yahoo contacts by BBauth but when i finish i get this error:
The token is for SSO only
i try serch about this problem but there is no luck, hope you can help why get this problem
A: The BBAuth access for Yahoo Contacts (aka Address Book) was deprecated in 2010 and I believe it no longer works.
Instead, you can access Contacts via YQL and OAuth authentication. Latest docs are here: http://developer.yahoo.com/social/contacts/
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,444 |
* Update twitter_cldr dependency
## 0.0.7
* Fix integer number formatter
## 0.0.6
* Update twitter_cldr dependency
## 0.0.5
* Update twitter_cldr dependency
## 0.0.4
* **Bug Fix**
* set SyntaxError message so it is displayed properly
## 0.0.3
* **Internal**
* Move repo ownership to format-message org
## 0.0.2
* **New Feature**
* added `MessageFormat.format_message` class method
* **Polish**
* better follow community style conventions
## 0.0.1
* Initial release
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,509 |
import {SuperMap} from '../SuperMap';
import {Util} from '../commontypes/Util';
import {BufferSetting} from './BufferSetting';
/**
* @class SuperMap.BufferAnalystParameters
* @category iServer SpatialAnalyst BufferAnalyst
* @classdesc 缓冲区分析参数基类。
* @param {Object} options - 参数。
* @param {SuperMap.BufferSetting} [options.bufferSetting] - 设置缓冲区通用参数。为缓冲区分析提供必要的参数信息,包括左缓冲距离、右缓冲距离、端点类型、圆头缓冲圆弧处线段的个数信息。
*/
export class BufferAnalystParameters {
constructor(options) {
var me = this;
/**
* @member {SuperMap.BufferSetting} [SuperMap.BufferAnalystParameters.prototype.bufferSetting]
* @description 设置缓冲区通用参数。为缓冲区分析提供必要的参数信息,包括左缓冲距离、右缓冲距离、端点类型、圆头缓冲圆弧处线段的个数信息。
*/
me.bufferSetting = new BufferSetting();
Util.extend(this, options);
this.CLASS_NAME = "SuperMap.BufferAnalystParameters";
}
/**
* @function SuperMap.BufferAnalystParameters.prototype.destroy
* @description 释放资源,将引用资源的属性置空。
*/
destroy() {
var me = this;
if (me.bufferSetting) {
me.bufferSetting.destroy();
me.bufferSetting = null;
}
}
}
SuperMap.BufferAnalystParameters = BufferAnalystParameters; | {
"redpajama_set_name": "RedPajamaGithub"
} | 36 |
Q: WARNING hook before_worker_boot failed with exception (URI::InvalidURIError) bad URI(is not URI?): "DATABASE_URL=\"postgres://resume_builder\"" i am trying to rails server someone's project for learning purpose. But this error comes up when setting up with postgresql please help.
rails aborted!
URI::InvalidURIError: bad URI(is not URI?): "DATABASE_URL=\"postgres://resume_builder\""
/usr/lib/ruby/2.7.0/uri/rfc3986_parser.rb:67:in `split'
/usr/lib/ruby/2.7.0/uri/rfc3986_parser.rb:73:in `parse'
this is the stack trace
below is the database.yml file
default: &default
adapter: postgresql
encoding: unicode
# For details on connection pooling, see Rails configuration guide
# https://guides.rubyonrails.org/configuring.html#database-pooling
pool: <%= ENV.fetch("RAILS_MAX_THREADS") { 5 } %>
development:
<<: *default
database: resume_builder
url: "postgres://localhost/somedatabase"
# The specified database role being used to connect to postgres.
# To create additional roles in postgres see `$ createuser --help`.
# When left blank, postgres will use the default role. This is
# the same name as the operating system user running Rails.
username: resume_builder
# The password associated with the postgres role (username).
password: test
# Connect on a TCP socket. Omitted by default since the client uses a
# domain socket that doesn't need configuration. Windows does not have
# domain sockets, so uncomment these lines.
host: localhost
# The TCP port the server listens on. Defaults to 5432.
# If your server runs on a different port number, change accordingly.
port: 5432
# Schema search path. The server defaults to $user,public
#schema_search_path: myapp,sharedapp,public
# Minimum log levels, in increasing order:
# debug5, debug4, debug3, debug2, debug1,
# log, notice, warning, error, fatal, and panic
# Defaults to warning.
#min_messages: notice
# Warning: The database defined as "test" will be erased and
# re-generated from your development database when you run "rake".
# Do not set this db to the same as development or production.
test:
# <<: *default
# As with config/credentials.yml, you never want to store sensitive information,
# like your database password, in your source code. If your source code is
# ever seen by anyone, they now have access to your database.
#
# Instead, provide the password or a full connection URL as an environment
# variable when you boot the app. For example:
#
DATABASE_URL="postgres://resume_builder"
#
# If the connection URL is provided in the special DATABASE_URL environment
# variable, Rails will automatically merge its configuration values on top of
# the values provided in this file. Alternatively, you can specify a connection
# URL environment variable explicitly:
#
# production:
# url: <%= ENV['MY_APP_DATABASE_URL'] %>
#
# Read https://guides.rubyonrails.org/configuring.html#configuring-a-database
# for a full overview on how database connection configuration can be specified.
#
production:
<<: *default
url: postgres://resume_builder
7055] WARNING hook before_worker_boot failed with exception (URI::InvalidURIError) bad URI(is not URI?): "DATABASE_URL="postgres://resume_builder""
[7052] WARNING hook before_worker_boot failed with exception (URI::InvalidURIError) bad URI(is not URI?): "DATABASE_URL="postgres://resume_builder""
error that comes up on rails server
A: YAML file cannot have =, as it is not in the specification of YAML.
Connecting Database with url has standard way.
URL is divided in following information
adapter, username, password, domain, database name
For example,
postgresql://ecldev@localhost/post_development
here,
*
*postgres = adapter
*ecldev = username
*localhost = domain
*post_development = database name
note - @ is mandatory.
In your case url is "postgres://resume_builder". Is does not have username, domain.
Example Image
UPDATE
just change
DATABASE_URL="postgres://resume_builder" to DATABASE_URL: "postgres://resume_builder"
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,559 |
\section{Introduction}
The possibility of manipulating the magnetization of a ferromagnet by spin-polarized electric currents was first pointed out by Berger\cite{PhysRevB.54.9353} and Slonczewski.\cite{Slonczewski1996L1} After its experimental observation in 1999\cite{Sun1999157} there has been growing interest in making use of this phenomenon in spintronics devices,~e.g.~for
switching the magnetization in spin valves. Spin-transfer torque random-access memory (STT-MRAM) has the advantages of better scalability and lower power consumption over conventional magnetoresistive random-access memory (MRAM), where the information is written by an external magnetic field. However, the current density recquired to switch the magnetization is preventing STT-MRAM size from scaling down and overcoming this problem is a central aim of current research efforts in the field of spintronics. From the fundamental point of view, it has become necessary to develop an understanding of the interplay between the magnetization and the currents driven by electric fields, as well as temperature gradients.
While spin-transfer torques rely on the exchange of spin angular momentum between two magnets with different direction of the magnetization, the so-called spin-orbit torques (SOTs) have been discovered only recently\cite{Chernyshov:154015,Miron:154509,MihaiMiron:155006} and they are attributed to the spin-orbit-mediated exchange of angular momentum between the crystal lattice and the magnetization. This type of torques exists also in systems with collinear magnetization when inversion symmetry is broken, and it has been shown that SOTs can lead to a reversal
of a ferromagnetic magnetization without the help of an additional polarizing layer.\cite{Miron:155001,Liu04052012,PhysRevLett.109.096602} Moreover, SOTs were shown to lead to a very fast domain wall motion in thin films at low current density.\cite{Emori:155277,Miron:154509,Ryu:154510} This suggests that SOTs could play a crucial role in the next generation of spintronics devices.
On the theoretical side, two mechanisms have been proposed that give rise to SOTs in heavy metal/ferromagnetic bilayers. The first one is due to the torque exerted on the magnetization by the spin current
from the spin Hall effect (SHE) of the heavy metal.
The second one is due to the non-equilibrium spin density that is generated at the interface when the distribution function of the system is driven out of equilibrium by an electric field, which is expected, e.g., in the Rashba model.
While the sign and the amplitude of the SOT due to the SHE are commonly estimated from the bulk spin Hall conductivity of the heavy metal, quantitative predictions of the second contribution are generally based on the Rashba model.\cite{PhysRevB.79.094422} Such simplified approaches are unable to explain the sensitivity of the SOT to the substrate thickness or microscopic details of the interface.\cite{symmetry_spin_orbit_torques} Recently, a first-principles method to compute SOTs based on the general linear response formalism was developed,\cite{ibcsoit}
which allows us to fully take into account the fine details of the electronic structure crucial for determining the SOT in transition-metal multilayers with high accuracy. The first application of this method to Co/Pt bilayers has shown very good agreement with experiment. More recently, the theory of the SOT arising in response to thermal gradients has been also developed.\cite{Freimuth:151412} This allows us to access this spincaloric effect from {\it ab initio}.
From the practical point of view perpendicular magnetic anisotropy is desirable for various applications in~e.g.~data storage.\cite{179719} Many ferromagnetic multilayers of $3d$ transition metals with heavy
transition metals are known to exhibit very large magnetocrystalline anisotropy energy (MAE) that favors out-of-plane magnetization,\cite{PhysRevB.63.144409} and among materials
of this type Co/Pt bilayers are most studied experimentally with respect to SOT.\cite{PhysRevLett.109.096602,Miron:155001,MihaiMiron:155006} However, the large lattice mismatch between Co and Pt results in a rather poor quality of the interface in these systems. On the other hand, for disentangling various
contributions to the SOT and for comparison between theoretical results with experiments
in this kind of systems the high quality of the interface is of utter importance. In this work, we study from first principles the SOT in L1$_{0}$-FePt/Pt bilayers, which have a large out-of-plane
MAE,\cite{:/content/aip/journal/apl/100/14/10.1063/1.3700746} and which can be grown epitaxially
thus exhibiting high interfacial crystallinity.\cite{:/content/aip/journal/jap/109/7/10.1063/1.3556782,:/content/aip/journal/apl/90/13/10.1063/1.2717516}
The goal of this paper is two-fold. First, we compute and analyze different contributions to the SOT in L1$_0$-FePt/Pt thin films as a function of Pt thickness.\cite{ibcsoit} We analyze the energy dependence of the SOT and relate it to the energy dependence of the bulk spin Hall effect in Pt.
Secondly, taking FePt/Pt bilayers as an example, we present {\it ab initio} calculations of thermal SOT (T-SOT),~i.e.,~the SOT which is driven by a temperature gradient rather than an electric field. We briefly outline the ways of how the T-SOT can be enhanced.
Finally we show that the energy dependence and magnitude of the even T-SOT can be estimated from the spin Nernst effect in bulk Pt.
\section{Formalism}
We investigate the SOT in our system using expressions obtained from the Kubo linear response formalism, and evaluated from the density functional theory. Within linear response the torque $\vn{T}$ exerted on the ferromagnetic magnetization when an electric field $\vn{E}$ is applied is given by $\vn{T} =\bf{t}\vn{E}$. The torkance tensor $\vn{t}$ has three contributions: \cite{ibcsoit}
\begin{equation}\label{eq_torque}
\begin{aligned}
{\rm t}^{\rm I(a)\phantom{I}}_{ij}\!\!\!\!=-\frac{e}{h}\int_{-\infty}^{\infty}&
d\mathcal{E}
\frac{d f(\mathcal{E})}{d \mathcal{E}}
\phantom{\Re}
{\rm Tr}
\langle\mathcal{T}_{i}
G^{\rm R}(\mathcal{E})
v_{j}
G^{\rm A}(\mathcal{E})
\rangle,
\\
{\rm t}^{\rm I(b)\phantom{I}}_{ij}\!\!\!\!=\phantom{-}\frac{e}{h}\int_{-\infty}^{\infty}
&d\mathcal{E}\frac{d f(\mathcal{E})}{d \mathcal{E}}
{\Re}
{\rm Tr}
\langle\mathcal{T}_{i}
G^{\rm R}(\mathcal{E})
v_{j}
G^{\rm R}(\mathcal{E})
\rangle,\phantom{(1)}
\\
{\rm t}^{\rm II\phantom{(a)}}_{ij}\!\!\!\!=\phantom{-}\frac{e}{h}\int_{-\infty}^{\infty}
&d\mathcal{E} f(\mathcal{E})
\quad\!\!
{\Re}{\rm Tr}\langle
\mathcal{T}_{i}G^{\rm R}(\mathcal{E})v_{j}
\frac{dG^{\rm R}(\mathcal{E})}{d\mathcal{E}}\\
&\quad\quad\quad\quad\quad\,-
\mathcal{T}_{i}\frac{dG^{\rm R}(\mathcal{E})}{d\mathcal{E}}v_{j}G^{\rm R}(\mathcal{E})
\rangle,
\end{aligned}\raisetag{4\baselineskip}
\end{equation}
with $G^{\rm R}(\mathcal{E})$ and $G^{\rm A}(\mathcal{E})$ as retarded and advanced Green functions, $v_{j}$ as the $j$th cartesian component of the velocity operator, $\mathcal{T}_{i}$ as the $i$th cartesian component of the torque operator, $f(\mathcal{E})$ as the Fermi distribution function and $e>0$ as the elementary positive charge. The torque operator is given by $\vht{\mathcal{T}}=\vn{m}\times\vn{B}^{\rm xc}$ where $\vn{m}$ and $\vn{B}^{\rm xc}$ are the spin magnetic moment operator and the exchange field, respectively. We model the influence of disorder in the system by a constant effective band broadening. Within this model
the retarded and advanced Green functions are given by
$G^{\rm R}(\mathcal{E})=\hbar[\mathcal{E}-H+i\Gamma]^{-1}$ and
$G^{\rm A}(\mathcal{E})=\hbar[\mathcal{E}-H-i\Gamma]^{-1}$, with parameter
$\Gamma$ characterizing the disorder strength. In this work we focus mainly on results
obtained for $\Gamma=25$\,meV, which corresponds to experiments performed at
room temperature, if the main source of disorder in the system is due to phonons.
In bulk metallic systems the diagonal and transverse conductivities at room temperature are usually reasonably well reproduced with this choice of $\Gamma$.\cite{PhysRevB.77.165117}
We decompose the torkance tensor $\vn{t}$ into even and odd components with respect to the
direction of magnetization $\hat{\vn{M}}$: ${\rm t}_{ij}={\rm t}_{ij}^{\rm even}+{\rm t}_{ij}^{\rm odd}$. It is very insightful to consider the limit $\Gamma \to 0$. In this so-called clean limit the even and odd components of the torkance tensor acquire qualitatively different forms:
\begin{equation}\label{eq_torque_even}
\mathrm{t}^{\rm even}_{ij}
=
\frac{2e}{\mathcal{N}}
\hat{\vn{e}}_{i} \cdot
\sum_{\vn{k},n}
f(\epsilon_{\vn{k}n})
\left[
\hat{\mathbf{M}}\times {\Im}
\Braket{
\frac{\partial u_{\vn{k}n}}{\partial\hat{\mathbf{M}}}|
\frac{\partial u_{\vn{k}n}}{\partial k_{j}}
}
\right],
\end{equation}
and
\begin{equation}\label{eq_torque_odd}
\mathrm{t}^{\rm odd}_{ij}
=-\frac{e\hbar}{2\Gamma\mathcal{N}}
\sum_{\vn{k}n}\langle\psi_{\vn{k}n}|\mathcal{T}_{i}|\psi_{\vn{k}n}\rangle
\langle\psi_{\vn{k}n}|v_{j}|\psi_{\vn{k}n}\rangle
\frac{\partial f (\epsilon_{\vn{k}n})}{\partial \mathcal{E}},
\end{equation}
where $\mathbf{k}$ is the Bloch vector in the Brillouin
zone with an overall number $\mathcal{N}$,
$n$ runs over all bands, $\epsilon_{\vn{k}n}$ are the eigenenergies of the system,
$\psi_{\vn{k}n}$ and $u_{\vn{k}n}$ are the Bloch states and their lattice-periodic parts, respectively, and $\hat{\vn{e}}_{i}$ is the unit vector along the $i$th cartesian direction. As discussed in other works by the authors,\cite{ibcsoit} the even torkance has the form of a Berry curvature and it is independent of $\Gamma$ in the limit of $\Gamma \to 0$. It constitutes the intrinsic contribution to the torkance, and it is analogous to the intrinsic anomalous or spin Hall effects. The
odd part of the torkance, on the other hand, diverges like $1/\Gamma$ in the limit of small $\Gamma$,~i.e.,~it is proportional to the quasi-particle lifetime in analogy to the Rashba torque\cite{V.M._Edelstein_1990} or the diagonal electrical conductivity,\cite{PhysRevB.77.165117}, and it is thus dependent on the scattering mechanisms present in the system.
Similarly to the spin Hall or anomalous Hall conductivities, the torkance tensor gives the SOT arising from an applied electric field,~i.e., it corresponds to a situation of a torque driven by a mechanical force. A torque can also be induced by a temperature gradient
$\nabla \it{T}$,~i.e.,~it can also originate from statistical forces. Within linear response this thermal torque reads:
\begin{equation}
\vn{T}=-\boldsymbol{\beta}\,\nabla T,
\end{equation}
where $\boldsymbol{\beta}$ is the {\it thermal torkance}.
In analogy to the torkance driven by electrical currents, we decompose the thermal torkance into even and odd
components with respect to the magnetization direction. The intrinsic even part of the thermal torkance
is analogous to the intrinsic
anomalous Nernst\cite{PhysRevLett.97.026603,PhysRevB.87.060406}
and spin Nernst conductivities.\cite{SNE1,SNE2,tauber,wimmer} Similar to the latter effects,
it can be shown that the thermal torkance $\beta$ can be computed directly from its mechanical counterpart employing the Mott relation:\cite{Freimuth:151412}
\begin{equation}\label{eq_mott}
\beta_{ij}(T)=-\frac{1}{e}\int d\mathcal{E}\frac{\partial f(\mathcal{E},\mu,T )}{\partial\mu}
{\rm t}_{ij}(\mathcal{E})\frac{\mathcal{E}-\mu}{T}
\end{equation}
where ${\rm t}_{ij}(\mathcal{E})$ is the torkance tensor with Fermi energy set to $\mathcal{E}$ and $\mu$ is the chemical potential.
In this work, we compute both electrical and thermal SOTs from the {\it ab initio} electronic struture of FePt/Pt bilayers according to Eqs.~(\ref{eq_torque}) and~(\ref{eq_mott}).
\section{Computational Details and basic properties}
In our study we considered 2 layers of L1$_0$-FePt oriented along [001]-axis and terminated with Fe atoms (Fe/Pt/Fe/Pt/Fe) deposited on the upper side of a Pt(001) film with the thickness of 6, 12 and 18 layers.
The electronic structure of these L1$_0$-FePt/Pt(001) thin films was computed within the density functional theory using the Perdew, Burke, and Ernzerhof (PBE) functional and the full-potential linearized augmented-plane-wave method as implemented in the two-dimensional version of the code \texttt{FLEUR}.\cite{FLEUR} DFT calculations were performed with 576 $k$-points in the two-dimensional Brillouin zone. The plane wave cutoff was set to 3.7$\,a_{0}^{-1}$ and the muffin-tin radii to 2.4\,$a_{0}$, where $a_{0}$ is the Bohr radius. The in-plane lattice constant of the films was set to the experimental lattice constant of fcc Pt (3.9265\,\AA). The out-of-plane relaxations of the atoms were performed until the forces were smaller
than $10^{-5}$\,Hartree/$a_{0}$, see Table~\ref{tab_atoms}.
\begin{table}[]
\caption{\label{tab_atoms}
Computational details for the thinnest film: interlayer distances ${\rm d_{z}}$ from one atomic layer to the next one (in units of ${\rm \AA}$); variation $\Delta=({\rm d_{z}}-{\rm d_{ref}})/{\rm d_{ref}}$ of the interlayer distances with ${\rm d_{ref}}={\rm d_{z}(Fe2)}$ for the first five atomic layers and ${\rm d_{ref}}={\rm d_{z}(Pt5)}$ for the other ones; spin magnetic moments $\mu_{{\rm at}}$ per atom (in units of $\mu_{B}$).
}
\begin{ruledtabular}
\begin{tabular}{cccc}
atomic layer &${\rm d_{z}}$ &$\Delta(\%)$ &$\mu_{{\rm at}}$\\
\hline
Fe1 &1.7896 &-3.9 &3.0804\\
Pt1 &1.8689 &0.4 &0.4032\\
Fe2 &1.8616 &0.0 &3.0213\\
Pt2 &1.8736 &0.6 &0.3829\\
Fe3 &1.8158 &-2.5 &3.0403\\
\hline
Pt3 &2.0998 &3.6 &0.2967\\
Pt4 &2.0393 &0.6 &0.0474\\
Pt5 &2.0271 &0.0 &0.0216\\
Pt6 &2.0193 &-0.4 &0.0093\\
Pt7 &1.9824 &-2.2 &0.0076\\
Pt8 & & &0.0071\\
\end{tabular}
\end{ruledtabular}
\end{table}
For magnetization out-of-plane the computed spin moments of Fe atoms range between 3.02$\,\mu_B$ and 3.08$\,\mu_B$ depending on the thickness and position of the Fe atom with respect to the interface with the Pt substrate.
The largest spin moment of the Pt atoms is about 0.4$\,\mu_B$ in the FePt overlayer, while the largest spin moment among the substrate atoms is 0.3$\,\mu_B$ for the Pt atom closest to the interface. Spin moments then rapidly decay when going further in the substrate (see also Table~\ref{tab_atoms}). For the thinnest film we have also computed the value of the magnetocrystalline anisotropy energy and found it to be 1.2\,meV per Fe atom favoring the
out-of-plane magnetization, while the anisotropy within the plane was one order of magnitude smaller.
For computing the SOTs we employed the Wannier interpolation technique. We constructed 18 maximally localized Wannier functions (MLWFs) per atom from Bloch functions on an 8$\times$8 $k$-point mesh using the wannier90 program.\cite{WannierPaper,Mostofi2008685} The number of bands used to disentangle the subspace of the MLWFs was chosen such that for each film the ratio of the number of bands to the number of MLWFs
was approximately equal to 1.4. This allows a very precise interpolation of the electronic structure up to 5\,eV above the Fermi energy. The torkances were computed on a 2048$\times$2048 $k$-point mesh, except for the case of $\Gamma$ well below 25\,meV, where a 4096$\times$4096 $k$-point mesh was used. For magnetization out-of-plane, which is the case
considered here, the only non-vanishing independent components of the torkance tensor are
${\rm t}^{\rm even}_{yx}$ and ${\rm t}^{\rm odd}_{xx}$, with the convention that the $z$ axis points out-of-plane, while the $x$ and $y$ axes coincide with the [100] and [010] in-plane directions.
\section{Results}
\subsection{Spin-orbit torques driven by electrical currents}
We first compute the even and odd torkance as a function of the disorder strength $\Gamma$ and thickness of the Pt substrate using the expressions from the previous section. The results
of these calculations are presented in Fig.~\ref{fig_torque_gamma} and summarized in
Table~\ref{tab_effective_fields} for the band broadening of $\Gamma=25$\,meV$\,\approx k_BT_0$, which mimicks the effect of the room temperature $T_0$.
At small $\Gamma$ the even torkance ${\rm t}^{\rm even}_{yx}$ is given by its clean limit Berry curvature
value which lies in the range of 0.65 to 0.85$\,ea_0$ depending on the substrate thickness,
and the deviation of ${\rm t}^{\rm even}_{yx}(\Gamma)$ from these values becomes significant only for band broadening larger than
100~meV. In the latter case the values of ${\rm t}^{\rm even}_{yx}$ for different numbers of Pt layers
are almost identical to each other, meaning that the fine difference in the electronic structure
of the films is washed out by the broadening of this magnitude.
At $\Gamma=25$\,meV the even torkance is still relatively close to the Berry curvature values,
see also Table~\ref{tab_effective_fields}, and the variation in ${\rm t}^{\rm even}_{yx}$ caused by Pt thickness
is of the order of 15\%. For this broadening the values of ${\rm t}^{\rm even}_{yx}$ for our system are
rather close to those of Co$^3$/Pt$^{10}$(111) bilayers, as computed in
Ref.~\onlinecite{ibcsoit}, which lie in the range of 0.53 to 0.62$\,ea_0$ depending
on the capping.
\begin{figure}[t!]
\centering
\includegraphics*[width=4.1cm]{torque_even_vs_gamma_PRB.pdf}
\hspace{0.1cm}
\includegraphics*[width=4.1cm]{torque_odd_vs_gamma_PRB.pdf}
\caption{\label{fig_torque_gamma}
a) Even torkance ${\rm t}^{\rm even}_{yx}$ and b) odd torkance ${\rm t}^{\rm odd}_{xx}$ in L1$_0$-FePt$^{2}$/Pt$^{\mathrm{N}}$ for N = 6 (green solid), 12 (orange dashed) and 18 (blue dot-dashed), as a function of the disorder strength $\Gamma$. Solid vertical lines correspond to the value of
$\Gamma=25$~meV.
}
\end{figure}
As for the odd torkance, for broadenings below 10\,meV its magnitude
is larger than that of the even torkance, while ${\rm t}^{\rm odd}_{xx}$ rapidly decays with $\Gamma$
and changes sign in the vicinity of $\Gamma\approx 80$\,meV, where the difference in
${\rm t}^{\rm odd}_{xx}$ for films of different thickness is almost negligible. Overall, the
characteristic $1/\Gamma$-behavior is clearly visible for small $\Gamma$. At room temperature
the odd torkance is negative and it is roughly twice smaller in magnitude than the
corresponding even torkance. The fact that ${\rm t}^{\rm odd}_{xx}$ is close to the point of changing the sign for $\Gamma=25$\,meV makes it also more sensitive to the Pt thickness, which
otherwise does not have a pronounced effect on the odd torkance
(see also Table~\ref{tab_effective_fields}).
For comparing to experiments it is useful to represent the
computed torkances in terms of the effective magnetic fields at a given current
density, and in Table~\ref{tab_effective_fields} we present the corresponding values
of T$^{\rm even}_{y}/\mu_{s}$ and T$^{\rm odd}_{y}/\mu_{s}$ for an
electric field $E_x$ of 360\,V/cm,
where $\mu_{s}$ stands for the total spin moment in the unit cell containing three Fe atoms with the value
of about 10.1$\,\mu_B$ for all thicknesses and magnetization out-of-plane.
The value of the electric field chosen to compute the effective magnetic fields corresponds
to the current density $j\approx 10^{7}$A/cm$^{2}$, if one estimates the order of the resistivity of our L1$_0$-FePt/Pt thin films by the experimentally measured room temperature resistivity of the Pt/Co/AlO$_{x}$ system.\cite{symmetry_spin_orbit_torques} The values of the even effective magnetic fields of the order of 2.0\,mT are generally consistent with those computed for Co/Pt bilayers,\cite{ibcsoit} taking into account that the value of $\mu_s$ in the latter case is smaller by
about 30\% than that in FePt/Pt bilayers that we study here. The magnitude of T$^{\rm odd}_{x}/\mu_{s}$ in FePt/Pt bilayers is, on the other hand, significantly smaller than the magnitude of T$^{\rm even}_{y}/\mu_{s}$, see Table~\ref{tab_effective_fields}.
\begin{table}[]
\caption{\label{tab_effective_fields}
Even and odd torkances $\rm t$ computed at $\Gamma$ = 25\,meV (in units of $ea_{0}$);
even (T$^{\rm even}_{y}/\mu_{s}$) and odd (T$^{\rm odd}_{x}/\mu_{s}$)
effective magnetic fields (in units of mT) for an applied electric field $E_{x}=360\,$V/cm; even and odd thermal torkances $\beta$ (in units of
$\mu e$V$\cdot a_{0}\cdot$K$^{-1}$);
$|\nabla T|^0$ (in units of K/nm) is the temperature gradient required to reproduce the total effective magnetic field $\sqrt{({\rm T}^{\rm odd}_{x})^{2}+({\rm T}^{\rm even}_{y})^{2}}/\mu_{s}$.
}
\begin{ruledtabular}
\begin{tabular}{cccc}
&FePt$^{2}$/Pt$^{6}$ &FePt$^{2}$/Pt$^{12}$ &FePt$^{2}$/Pt$^{18}$\\
\hline
${\rm t}^{\rm even}_{yx}$ &+0.65 &+0.61 &+0.75 \\
${\rm t}^{\rm odd}_{xx}$ &$-$0.19 &$-$0.30 &$-$0.27 \\
T$^{\rm even}_{y}/\mu_{s}$ &+2.1 &+2.0 &+2.4\\
T$^{\rm odd}_{x}/\mu_{s}$ &$-$0.6 &$-$1.0 &$-$0.9\\
$\beta^{\rm even}_{yx}$ &$-$10.6 &$-$15.3 &$-$14.5 \\
$\beta^{\rm odd}_{xx}$ &$-$4.8 &+0.7 &$-$2.5 \\
$|\nabla T|^0$ &+2.1 &+1.6 &+1.7\\
\end{tabular}
\end{ruledtabular}
\end{table}
It is tempting to compare the computed even SOT to the ``hypothetical" torque ${\rm T}_{y}$ that is exerted on the magnetization if the spin current density $j^{y}_{z}$ accross the interface between Pt substrate and L1$_0$-FePt overlayer is given by the spin Hall conductivity of bulk fcc Pt. In that case the current density $j^{y}_{z}$ generated by the spin Hall effect is given by the relation $j^{y}_{z}=\sigma^{y}_{zx}E_{x}$ when an electric field $E_x$ is applied to the system. In the latter expression $\sigma^{y}_{zx}$ stands for the corresponding component of the spin Hall conductivity (SHC) tensor of bulk Pt. Under the assumption that the whole of the bulk spin Hall current is transferred to the magnetization,~i.e.,~that ${\rm T}_{y}=S j^{y}_{z}$, where the spin polarization of the spin current is along the $y$-axis and $S$ = 7.712\,\AA$^{2}$ is the in-plane area of the unit cell, this model yields a simple expression for the even torkance:
\begin{equation}\label{eq_torque_model}
{\rm t}^{\rm SHE}_{yx}=S\vn{\sigma}^{y}_{zx}.
\end{equation}
\begin{figure}[t!]
\centering
\includegraphics*[width=8.5cm]{torque_even.pdf}
\includegraphics*[width=8.5cm]{torque_odd.pdf}
\caption{\label{fig_torque}
(a) Even torkance ${\rm t}^{\rm even}_{yx}$ and (b) odd torkance ${\rm t}^{\rm odd}_{xx}$ as a function of
the Fermi energy (with respect to the true Fermi energy ${\rm E_{F}}\approx -4.33$ eV for all three thicknesses) at $\Gamma$ = 25\,meV in L1$_0$-FePt$^{2}$/Pt$^{\mathrm{N}}$ films for N = 6 (green solid), 12 (orange dashed) and 18 (blue dot-dashed). The line of circles in the upper figure corresponds to the even torkance ${\rm t}^{\rm SHE}_{yx}$ estimated from the spin Hall conductivity of bulk fcc Pt,~Eq.~\eqref{eq_torque_model}.
}
\end{figure}
In Fig.~\ref{fig_torque} we plot the even spin Hall torkance ${\rm t}^{\rm SHE}_{yx}$
in comparison to the even torkance ${\rm t}_{yx}$ computed at $\Gamma=25$\,meV as a function of the Fermi energy in our
system. For estimating ${\rm t}^{\rm SHE}_{yx}$ we used the intrinsic SHC in bulk fcc Pt (note that our calculations show that
the influence of
the band smearing of the order of 25\,meV on the clean limit SHC is negligible). At the true Fermi energy, the SHC of fcc Pt is found to be
2184\,$(\hbar/e)$S/cm. As apparent from Fig.~\ref{fig_torque}, in the interval
of energies of $[-0.1, +0.5]$\,eV with respect to the true Fermi energy, the even SOT can be
approximated with the expression ${\rm t}^{\rm even}_{yx}=\xi \,{\rm t}^{\rm SHE}_{yx}$, where the so-called {\it
SHE-to-SOT efficiency} $\xi$\cite{Inv_SOT} smoothly varies with energy in the range of
$0.5<\xi<0.7$ and moderately depends on the Pt thickness. As a result, in this energy range the qualitative behavior of ${\rm t}^{\rm even}_{yx}$ quite closely resembles that of ${\rm t}^{\rm SHE}_{yx}$.
In this energy region one could attribute the moderate energy and Pt thickness dependence
of $\xi$ and its deviation from the ``ideal" value of 1.0 to the finite size effects
and details of the electronic structure which,~e.g., influence the magnitude of the spin current generated in the Pt substrate,
as well as its $z$-distribution inside the slab and transmission properties of the interface.\cite{Inv_SOT}
We note that the range of values of $\xi$ for energies between $-0.1$ and $0.5$\,eV
is rather close to that computed in the presence of disorder for Co/Pt bilayers. For the latter system it was shown that ${\rm t}^{\rm even}_{yx}$ arises
mainly due to the spin current which originates from the SHE inside the Pt substrate.\cite{ibcsoit}
On the other hand, away from this energy range, ${\rm t}^{\rm even}_{yx}$ in FePt/Pt can
differ from ${\rm t}^{\rm SHE}_{yx}$
by an order of magnitude and even in sign (e.g. around ${\rm E_{\rm F}}=-0.35$ and $-$0.8\,eV),
which signifies that the application of simplified models of the kind of Eq.~\eqref{eq_torque_model} has to be done with extreme caution.
\begin{figure}[t!]
\centering
\includegraphics*[width=8.5cm]{bandstructure_Pt6_FePt5_circles.png}
\hspace{0.2cm}
\includegraphics*[width=8.5cm]{bandstructure_Pt18_FePt5.png}
\caption{\label{fig_bands}
Band structures of (a) L1$_0$-FePt$^{2}$/Pt$^{6}$ and (b) L1$_0$-FePt$^{2}$/Pt$^{18}$ thin films along high symmetry lines. States with large portion of the wavefunction on specific atoms are marked by red (Pt atoms at the bottom of the slab), green (Pt substrate atoms closest to the FePt/Pt interface) and blue (Fe atoms closest to the FePt/Pt interface). The criteria for a state to be marked is that a) more than 9.6\% for Pt-atoms and 7.7\% for Fe-atoms of the charge of the state are localized inside a corresponding atom. For b) these values constitute 4.5\% for Pt-atoms and 3.6\% for Fe-atoms, owing to the twice larger thickness. The radius of the dots is proportional to the weight of the wavefunction inside a corresponding atom. All states are marked by grey dots in background.
}
\end{figure}
Fig.~\ref{fig_torque} shows that while the even torkance as a function of energy
reaches its maximal values around the true Fermi energy of FePt/Pt, the values of the
odd torkance are small around the true ${\rm E_{\rm F}}$, and they become very large away from
it. As far as the thickness dependence of both ${\rm t}^{\rm even}_{yx}$ and ${\rm t}^{\rm odd}_{xx}$
is concerned, significant deviations between the torkances for ${\rm N}=6$ and larger
thicknesses are visible only in the energy interval of about $-0.7$\,eV to $-0.3$\,eV.
The difference in the torkances for ${\rm N}=12$ and ${\rm N}=18$ is, on the other
hand, smaller. Among the two, the thickness dependence is more pronounced for the
odd torkance with the difference reaching as much as $1\,ea_0$ between ${\rm t}^{\rm odd}_{xx}$
for thin and thick films, while for ${\rm t}^{\rm even}_{yx}$ this difference is much smaller.
In Fig.~\ref{fig_bands} we present the bandstructures of the slabs with 6 and 18 layers of Pt in the substrate.
The two bandstructures look very similar with the only obvious difference lying in the increased number of bands for the
thicker substrate.
However, in the energy interval of interest a relatively large hybridization of the states which have larger weight at
the bottom layer of the Pt substrate with
the states which exhibit larger weight at the interface between L1$_0$-FePt and Pt is clearly
visible for the L1$_0$-FePt$^{2}$/Pt$^{6}$ film (see black circles in Fig.~\ref{fig_bands}), while this hybridization is almost absent for the L1$_0$-FePt$^{2}$/Pt$^{18}$ film. Thus, we speculate that the cross-talk between the free surface of the
Pt substrate and the interface with FePt, which are almost decoupled for large Pt thicknesses, and quite pronounced for the 6-layer film, could lead to significant differences in the SOTs of thin and thick FePt/Pt bilayers.
\subsection{Thermal spin-orbit torques}
We compute the thermal spin-orbit torques (T-SOTs) in our system according to
Eq.~\eqref{eq_mott} at temperature ${\it T}=300$\,K using as input the energy dependence
of the even and odd torkances computed at $\Gamma=25$\,meV and presented in
Fig.~\ref{fig_torque}. The energy dependence of the even and odd thermal torkances
$\beta^{\rm even}_{yx}$ and $\beta^{\rm odd}_{xx}$ of L1$_0$-FePt/Pt thin films
at room temperature is shown in Fig.~\ref{fig_th_torque}, and their values at the
Fermi energy are summarized in Table~\ref{tab_effective_fields}.
By direct inspection, it is easy to see that the trend of $\beta^{\rm even}_{yx}$ and $\beta^{\rm odd}_{xx}$ with energy can be directly related to the corresponding behavior of ${\rm t}^{\rm even}_{yx}$ and ${\rm t}^{\rm odd}_{xx}$. This follows from the observation that in the limit of zero
temperature $T$ in Eq.~\eqref{eq_mott} the thermal torkance $\beta$ is proportional to the energy
derivative of the torkance $\rm t$ at the corresponding energy. Indeed, by comparing
the curves in Figs.~\ref{fig_th_torque} and~\ref{fig_torque}, we can see that in most of the
cases the zeros of the thermal torkance correspond to the local extrema of the electrical
torkance, while the maxima in the former correspond to the regions of largest slope of
the latter. It is thus not surprising that the largest values of $\beta^{\rm even}_{yx}$ of
the order of tens of $\mu e$V$\cdot a_{0}\cdot$K$^{-1}$ are achieved around the Fermi energy,
while the magnitude of $\beta^{\rm odd}_{xx}$ is maximal away from the Fermi energy,
reaching as much as 100\,$\mu e$V$\cdot a_{0}\cdot$K$^{-1}$ there. Clearly visible
in Fig.~\ref{fig_th_torque} is a much more pronounced dependence of the thermal torkances on the
Pt thickness than in the case of the electrical torkances. The thermal torkances for 6 and 12/18
layers of Pt substrate differ in sign over wide patches in energy around $-0.4$\,eV and the
difference between thermal torkances for 12 and 18 layers becomes more pronounced. At the true Fermi energy,
$\beta^{\rm odd}_{xx}$ exhibits a change of sign when changing the Pt thickness, see Table~\ref{tab_effective_fields}.
\begin{figure}[t!]
\centering
\includegraphics*[width=8.5cm]{th_torque_even.pdf}
\includegraphics*[width=8.5cm]{th_torque_odd.pdf}
\caption{\label{fig_th_torque}
a) Even thermal torkance $\beta^{\rm even}_{yx}$ and b) odd thermal torkance $\beta^{\rm odd}_{xx}$ are calculated for $T$ = 300\,K using Eq.~\eqref{eq_mott}, based respectively on ${\rm t}^{\rm even}_{yx}$ and ${\rm t}^{\rm odd}_{xx}$ from Fig.~\ref{fig_torque}. The line of circles in the upper figure corresponds to the even thermal torkance ${\rm \beta}^{\rm SNE}_{yx}$ estimated from the spin Nernst conductivity of bulk fcc Pt,~Eq.~\eqref{eq_thtorque_model}.
}
\end{figure}
It is known that in paramagnetic metals, in particular Pt, an applied temperature gradient will
result in transverse spin current, analogous to the spin Hall current which is generated by an electric
field. The respective phenomenon is called the spin Nernst effect (SNE),\cite{SNE1,SNE2} and its
magnitude is characterized by the spin Nerst conductivity (SNC) $\alpha$. Keeping in mind the geometry of our system, the relationship between a temperature gradient applied
along the $x$ axis and the spin current density with spin-polarization along the $y$ axis which propagates
along the $z$ axis, reads:
\begin{equation}
j^{y}_{z}=-\alpha^{y}_{zx}\nabla \it{T}_{x}.
\end{equation}
As in the previous section, we will compare the magnitude of the ``pure" spin Nernst
torkance $\beta^{\rm SNE}_{yx}$ to the computed thermal torkance $\beta^{\rm even}_{yx}$,
assuming that the spin Nernst torkance arises from the full bulk spin Nernst current:
\begin{equation}\label{eq_thtorque_model}
\beta^{\rm SNE}_{yx}=S\vn{\alpha}^{y}_{zx}.
\end{equation}
To estimate the magnitude of the spin Nernst thermal torkance from {\it ab initio}, from the
energy dependence of the SHC presented in Fig.~\ref{fig_torque}, we evaluate the thermal intrinsic contribution to the SNC according to the Mott relation (at $T=300$\,K):\cite{tauber,wimmer}
\begin{equation}\label{eq_mott_sne}
\alpha^y_{zx}=-\frac{1}{e}\int d\mathcal{E}\frac{\partial f(\mathcal{E},\mu,\it{T}) }{\partial\mu}
\sigma^y_{zx}(\mathcal{E})\frac{\mathcal{E}-\mu}{\it{T}}.
\end{equation}
The spin Nernst thermal torkance $\beta^{\rm SNE}_{yx}$ computed using Eq.~\eqref{eq_thtorque_model} and Eq.~\eqref{eq_mott_sne} is presented in Fig.~\ref{fig_th_torque} as a function
of the position of the Fermi energy together with $\beta^{\rm even}_{yx}$. By comparing
the two torkances we can conclude that, as in the case of the electrical torkances,
the overall behavior of $\beta^{\rm even}_{yx}$ with energy is in accordance with that
of $\beta^{\rm SNE}_{yx}$ in the window of energies between $-0.2$\,eV and $+0.6$\,eV.
This hints at a clear correlation between the phenomenon of the T-SOT and the SNE at these
energies for our system. Owing to the essential energy dependence of the SHE-to-SOT efficiency
$\xi$, the {\it SNE-to-T-SOT efficiency} $\xi^T$, defined by relation $\beta^{\rm even}_{yx} = \xi^T
\beta^{\rm SNE}_{yx}$, deviates quite significantly from $\xi$ and ranges approximately
between 0.5 and 1.5 in the energy interval [$-0.2$\,eV, $+0.6$\,eV], with the exception
of energies where the torkances change sign in between 0.0 and 0.2\,eV.
Since to the best of our knowledge the effect of T-SOT has not been observed so far,
it is important that we give an estimate of the T-SOT that can be achieved experimentally
in our films. We therefore compute the temperature gradient $|\nabla T|^0$ that is required to reproduce the total effective magnetic field obtained with the value of current density $j \sim 10^{7}$\,A/cm$^{2}$, typical for experiments on such systems (Table~\ref{tab_effective_fields}).
The value of $|\nabla T|^0$ of the order of 2\,K/nm which we obtain for our L1$_0$-FePt/Pt
bilayers at their true Fermi energy turns out to be one order of magnitude larger than the one
which can be achieved experimentally in this type of systems.\cite{Wees} This means that although
the T-SOT in the system that we study here most probably cannot be used to switch the
magnetization, we conclude that the fingerprints of the effect can be observed.
We are, moreover, confident that at the current level of experimental techniques the T-SOT
can be made as large as the electrical SOT by proper electronic structure engineering, which can go along three different paths. (i) As apparent from Fig.~\ref{fig_th_torque}, for FePt/Pt bilayers
the thermal torkances can be order of magnitude larger if the Fermi energy is set to $\sim$~0.6~eV above its true value - this corresponds roughly to using~e.g.~L1$_0$-(Fe$_{1-x}$Co$_{x}$)(Pt$_{1-x}$Au$_{x}$)/Pt$_{1-x}$Au$_{x}$ instead of FePt/Pt, with $x\sim$~0.6 if we assume a constant density of states of $\sim$~1~eV$^{-1}$ per atom for Fe$_{1-x}$Co$_{x}$Pt$_{1-x}$Au$_{x}$ and Pt$_{1-x}$Au$_{x}$. (ii) Exploiting the close correlation between the T-SOT and the SNE which
we found, one could consider using fcc Ir, Pd or Rh as substrates instead of fcc Pt, since the values of
the intrinsic SNCs for these metals which we computed constitute $-$8744 (Ir), $+$20804 (Pd), and $-$20779\,$(\hbar/e)$$\mu$A$\cdot$cm$^{-1}\cdot$K$^{-1}$ (Rh), which is respectively $+$1.04, $-$2.48 and $+$2.48 times larger than the value of the SNC of fcc Pt of $-$8383\,$(\hbar/e)$$\mu$A$\cdot$cm$^{-1}\cdot$K$^{-1}$. (iii)
Our calculations show that upon decreasing the disorder strength
$\Gamma$ the energy dependence of the odd and, particularly, even torkances exhibits strong deviations from the smooth behavior shown
above, acquiring sharp features and sign changes at the scale of tens of
meVs. This effect is due to the fine features in the electronic structure
of thin films, which get promoted as the band broadening is decreased.
Correspondingly, upon reducing the degree of disorder in the system (e.g.~by
lowering of the temperature or concentration of impurities) the magnitude
of the T-SOT, qualitatively proportional to the degree of raggedness
of the torkance as a function of energy, can be significantly enhanced,
as confirmed by our calculations.
\section{Conclusions}
Using expressions for the spin-orbit torkances derived from the Kubo linear response
formalism, we compute from first principles the values of the even and odd torkances in a system consisting of two layers of ferromagnetic L1$_0$-FePt deposited on an fcc Pt(001) substrate of various thicknesses. We predict that the magnitude of the SOTs lies in the range of values measured experimentally and computed theoretically for Co/Pt bilayers. For both even and
odd torques we find a pronounced energy and thickness dependence. By comparing
the even SOT to that purely given by the spin Hall effect in the Pt substrate we find that while around
the Fermi energy the behavior of the two SOTs is very similar, they can differ in sign and order
of magnitude for wide regions of energy. Moreover, using the expressions that we derived recently
for the thermal SOT, driven by the temperature gradient rather than the electric field, we compute
the energy and thickness dependence of the thermal torkance in the system under consideration.
We were also able to establish a close connection between the T-SOT and the spin Nernst
effect. We predict that thermal gradients of the order of 2\,K/nm are necessary to exert the
same torque on the magnetization as that arising from typical current densities in this kind
of systems, which assures us that the T-SOT in FePt/Pt bilayers could be experimentally detected.
We further speculate that much larger T-SOTs can be achieved in other ferromagnetic
transition-metal overlayers deposited on substrates which exhibit larger spin Nernst effect
than Pt.
We gratefully acknowledge computing time on the supercomputers JUQUEEN and JUROPA at
J\"ulich Supercomputing Center as well as at the JARA-HPC cluster of RWTH Aachen, and funding under the HGF-YIG programme VH-NG-513 and SPP 1538 of DFG.
| {
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Home > Music Articles > Reggae Music Articles > Morgan Heritage Strictly Roots European Tour
Morgan Heritage Strictly Roots European Tour
Morgan Heritage Announce the Highly Anticipated 'Strictly Roots' European Tour
On the heels of their successful North American tour, and historical Catch A Fire Tour with Damian Marley, Stephen Marley, and Tarrus Riley, the Royal Family of Reggae Morgan Heritage is poised to cross the ocean seas for their highly anticipated "Strictly Roots" European Tour.
On October 23rd, 2015 the legendary siblings will celebrate the success of the recently released "Strictly Roots" album which reached no. 1 on the US iTunes Reggae Chart and Billboard Reggae Chart, #2 in Switzerland and Holland, #8 in the UK, #3 in Germany and #4 in France on the iTunes Reggae Charts by kicking off the tour in Sheffield, England. The European leg will also see the band deliver their musical gifts in the Netherlands, Germany, France and Switzerland.
Fans throughout Europe eagerly await to witness Peetah Morgan (vocals), Una Morgan (keyboard/vocals), Gramps Morgan (keyboard/vocals), Lukes Morgan (rhythm guitar) and Mr. Mojo Morgan (percussion/vocals) perform classics such as "Don't Haffi Dread", "Down By the River", and new hits such as "Perform and Done", "Light It Up", "Wanna Be Loved" and "Strictly Roots" off the album of the same name.
"We, Morgan Heritage want to let all our Fans in Europe know that the "Strictly Roots World Tour" is coming to Europe from Oct – Nov this year. Get ready for it, get ready to see Morgan Heritage like never before. Every show will be an experience you don't want to miss. Strictly Roots coming at you."
Morgan Heritage Strictly Roots European Tour Dates 2015 Morgan Heritage w/ special guests Jemere Morgan & Omari Banks
October 23rd, Sheffield, UK – 02 Academy
October 24th, Birmingham, UK – 02 Academy
October 25th Dordrecht, Netherlands – Bibelot
October 27th Koln, Germany – Underground
October 28th Berlin, Germany – Yaam
October 30th Lyon, France – Le Radiant
October 31st Paris, France – Dock Pullman
November 1st London, UK – 02 Shepherds Bush Empire
November 2nd Amsterdam, Netherlands – Melkweg
November 4th Utrecht, Netherlands – Tivolivredenburg
November 5th Lausanne, Switzerland – Metropop Festival
November 6th Brussels, Belgium – VK
More Dates to be Announced.
STRICTLY ROOTS is now available worldwide
Morgan Heritage recently announced the release of their official fan community app, powered by TopFan. Available on all Apple iOS and Android devices, Morgan Heritage's official mobile community enables fans to listen to their music, watch music videos, read their latest news, enter contests, buy tickets to their upcoming tour, and much more. Download the app here
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{"url":"https:\/\/www.physicsforums.com\/threads\/calculating-avogadros-number-from-scratch.59336\/","text":"# Calculating Avogadro's number from scratch\n\n1. Jan 10, 2005\n\n### jetset\n\nHello there,\n\nNot that I actually want to do this but theoretically I would like to know how I could prove to myself avogardro's number using only the most primative knowledge of math, physics and chemistry as possible.\n\nIt all makes sense in whatever text you are reading, but it never gets explained how to do it yourself from start to finish in a simple way, there is always alllutions to other technology that I do not fully understand.\n\n2. Jan 11, 2005\n\n### dextercioby\n\nThink about a Natrium (Na) molecule.It's monoatomic,okay??\nIt's kilomolar mass is 23.Which means 23Kg per each kilomol of atoms.\n\nDo u agree with this equality:\n$$M=N_{A} m_{Na}$$\n,where M is the kilomolar mass (unit:Kg\/Kmol),N_{A} is the Avogadro's number (unit:molecules\/kilomol) and m_{Na} is the mass of a molecule (in this case,atom) of Na.\n\nForm the equality above,the result is immediate...\n\nDaniel.\n\n3. Jan 11, 2005\n\n### jetset\n\nYour equation of course makes a lot of sense. Using your analogy my gap in knowledge lies not in the logic of this formula, but rather how would I go about proving\/figuring out that Natruim is monoatomic.\n\nForgetting fictional elements, how could I do a working example with C12? When scientists back then were working with chemicals, how was the \"12\" part first proved? You can of couse get this if you have Avg. # and the kilomolar mass, to get the last remaining variable, giving you a solution.\nBut if you didn't yet know what Avg. # was, 2 out of 3 of the variables are unknown for that formula. I'm assuming an experiement was done to solve the problem...???\n\nLet me rephrase my first question: I am transported back in time to 1700 and wish to show people that a certain substance has a certain molar mass that would lie the foundations for all sorts of chemical experiments. My question is how would I logically prove this to people? (and get a paper published)\n\nLast edited: Jan 11, 2005\n4. Jan 11, 2005\n\n### dextercioby\n\nYes,you're right,that formula contains 3 possible unknowns:kilomolar mass of an element,the element's atom mass and # of Avogadro.\nYou'll have to agree with me thatever since 1775 and the wonderful work of of Antoine Laurent Lavoisier,for the common known (at that time) elements the kilomolar mass was known.\nSo that eliminates one unknown,for most elements this is due to the work of Jakob Berzelius and other chemists,the most famous being Mendeleev.\nThen came the 20-th century and the use of mass spectrometry (invented by Sir James Joseph Thomson in 1898 and carried on magnificiently by Sir William Aston) which determined the mass (in Kg) for the atoms.\nTherefore,the Avogadro's number was found combining results from chemistry (kilomolar mass) and physics (mass of an atom\/molecule).\nLater on,physics showed the connection between kilomolar mass,atom mass and nucleus mass,so that everything was determined by physics.\n\nDaniel.\n\n5. Jan 11, 2005\n\n### NateTG\n\nDon't you need to throw in Millikan's oil drop experiment to determine the electron charge?\n\nIt's possible to experimentally determine the charge of a single electron (Millikan's oil drop experiment). Once the charge of an electron has been determined, the charge of ionized atoms can also be calculated. By observing the deflection of charged atoms (or molecules) in a known electric or magnetic field, the mass of those atoms (or molecules) can be very precisely inferred.\n\nDealing with the Proton\/Neutron\/Electron mass issues is also not very difficult at that point, since you can create a chart of various atom masses for isotopes and whatnot.\n\nIn practice, I don't think the value avogadro's number is all that important since people generally will not make the transition from particle count to moles, but it would be sufficient to call it A and leave it unspecified.\n\nI believe that what Avogadro did was to use equal volumes of gasses to attempt to determine relative molecular masses (http:\/\/scienceworld.wolfram.com\/physics\/AvogadrosHypothesis.html).\n\n6. Jan 11, 2005\n\n### Gokul43201\n\nStaff Emeritus\nI found this site when I was looking around, the last time someone asked a question like this. It does a great job. Give it a look.\n\nLast edited by a moderator: May 1, 2017\n7. Jan 11, 2005\n\n### jetset\n\ninteresting stuff, thanks for the replys. ive never heard of the oil drop experiment til now.\n\nSo to summarize so far:\n\nIF i get thrown back in time to 1700, in order to prove the Avag. #, I would first have to discover the \"kilomolar\" mass of a substance (How could I do this using the tools of the time?) Then I must wait for electronic stuff to evolve (or make it happen myself) so that I know enough about electric fields to do the oil drop experiment. Then thridly I could combine those two previous things to come up with Avag. # (which I would call the jetset number:P)\n\nIs that right? Also, is there more than one way of doing my first step of finding the molar mass of a substance?\n\n8. Jan 12, 2005\n\n### Gokul43201\n\nStaff Emeritus\nIn my opinion, you didn't have to have done the oil drop experiment at all. The concept of charge (quantization) is not essential to the concept of a mole of atoms\/molecules. Moreover, there's no way you will completely (z'th ionization energy is required) ionize an atom, so I'm not sure I understand how this is relevant.\n\nHaving modern experimental probes make this all a piece of cake, but most of the ideas were well-developed by about the mid-1800s thanks to work by Dalton, Gay-Lussac, Avogadro, and Cannizzaro.\n\n9. Jan 12, 2005\n\n### jetset\n\nSo in point form, what do i have to do in order rather than not do?\n\n10. Jan 12, 2005\n\n### Integral\n\nStaff Emeritus\nIf indeed you found yourself in the late 17th or early 18th century you would have to do exactly what the natural philosophers of that day were doing. Careful observation of reactions to learn how much of various material combine to form the end product. You would have to observe things like, when water breaks down it forms 2 different materials in a 2:1 ratio.\n\nLook at the material covered in a beginning chemistry class that is the essence of what the old timers learned, it would have to be recreated from scratch.\n\nIsn't this how we got to where we are today? Observation and careful measurements that is what it was all about.\n\n11. Jan 12, 2005\n\n### dextercioby\n\nU couldn't...Not in 1700.The balance (is that the word?? ) was introduced in chemistry laboratories around 1750 by the Scottish chemist William Black.Before,noone ever thought of weighing the reaction products.\nLet's move it after Lavoisier lost his head (literally) in 1794.The molar\/kilomolar masses of some elements were known by then.\n\nU'd have to find the atoms' mass.Kilomolar mass would not be enough.So u'd have to wait unitll 1920 and the experiments by Sir William Aston.It's simple.\n\nDaniel.\n\n12. Jan 12, 2005\n\n### Gokul43201\n\nStaff Emeritus\nNateTG was right in that Millikan's oil drop experiment played an important role. It did.\n\nBy 1865, Johann Loschmidt had made the first calculation of Avogadro's Number (~5*10^23) building upon work done by Clausius and Maxwell, on the kinetic theory (particularly, viscosity) of gases.\n\nOver the next five decades, the number's accuracy was improved upon, by measurements of blackbody radiation (Planck, 1900), diffusion (Einstein, 1906 - but he made a calculation error which was discovered only much later), and sedimentation equilibria in colloidal systems(Perrin, 1908 - first to determine the multiplier's value at ~6.0, and generally considered the first accurate determination).\n\nOnly after the Oil-drop experiment was the accuracy of the number improved (~6.02 ....). Now, with X-ray Diffraction, I think the number is known to at least 7 or 8 significant places, maybe much more.\n\nA great description of Loschmidt's (in Germany, Ausrtia, Denmark and a few other countries, Avogadro's Number is actually refered to as Loschmidt's Number) method for finding the size of molecules, can be found here :\nhttp:\/\/www.physicstoday.org\/pt\/vol-54\/iss-3\/p45.html [Broken]\n\nLast edited by a moderator: May 1, 2017\n13. Jan 12, 2005\n\n### jetset\n\nSo the accuracy basically increased the better we could actually measure the mass of the molecules?\n\nWe are gettin to more to the heart of my question now... Basically I've done all sorts of university chemistry several years ago, but I never had time to question things deeply, now it' truely zooming out time.\n\nSo it has been established that if I go back to 1700 I could not prove anything to do with Avg. # because of lack of technology.\n\nHow bout this then: after the balance has been invented, say I get transported back to 1800. How can I prove the theory of the mole using carbon 12? (is that what was used to actually prove the theory?) Please dont just state some general rule or say a general statement and its easy from there. I dont need a long explaination either, just a short and exact \"the following setup is needed, you will need to use the following 3 impirical observations and you need the following elements. Do \"bla\", and that is the long in short of how the mole was discovered. :D\n\nps: the forum is pretty cool, kudos to all of yall who devote so much time here!\n\nLast edited by a moderator: May 1, 2017\n14. Jan 13, 2005\n\n### Gokul43201\n\nStaff Emeritus\nState specifically, what you mean by \"the theory of the mole\".\n\nProvide a conjecture, and we can figure out how to go about proving this or determining some desired number.\n\n15. Jan 13, 2005\n\n### jetset\n\nWhen the word \"mole\" first came into the scientific community; the proof of that paper that showed what this \"mole\" was, that is what i am looking for.\n\n16. Jan 13, 2005\n\n### NateTG\n\nI haven't the foggiest about 'mole' but if you start with Avogadro's hypothesis, you can get to some analagus notion fairily quickly. (If you used ounces instead of grams, you would want a different 'Avogadro's number', for example, and, of course, grams were not in common usage at that time.)\n\nOnce you've got people convinced that the whole proportional thing works (and, for 1800 this involves explaining covalent bonding among other things.) you can start looking at experiments like spectrometry combined with Milikan's oil drop experiment to determine particles per mole.\n\nI'm not a chemist, or a chem buff, but I don't think that the numerical value of Avogadro's number is all that important compared to the proportionality hypothesis, and moreover, there is a natural order since the proportionality hypothesis makes sense without Avogadro's number, while the converse is false.","date":"2018-06-23 12:44:01","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\": 0.5715130567550659, \"perplexity\": 1425.5309406940237}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-26\/segments\/1529267864958.91\/warc\/CC-MAIN-20180623113131-20180623133131-00241.warc.gz\"}"} | null | null |
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February 1st, 2012: Sunset Boulevard (1950)
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Professor's Fall Journal: October Week 4 « The League of Dead Films on October 31st, 2014: Halloween (1978)
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package rfx.core.util.useragent;
/**
* Collection of parsed data for a given user agent string consisting of UserAgent, OS, Device
*
* @author Steve Jiang (@sjiang) <gh at iamsteve com>
*/
public class Client {
public final UserAgent userAgent;
public final OS os;
public final Device device;
public Client(UserAgent userAgent, OS os, Device device) {
this.userAgent = userAgent;
this.os = os;
this.device = device;
}
@Override
public boolean equals(Object other) {
if (other == this) return true;
if (!(other instanceof Client)) return false;
Client o = (Client) other;
return ((this.userAgent != null && this.userAgent.equals(o.userAgent)) || this.userAgent == o.userAgent) &&
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h += os == null ? 0 : os.hashCode();
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return h;
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@Override
public String toString() {
return String.format("{user_agent: %s, os: %s, device: %s}",
userAgent, os, device);
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{"url":"http:\/\/www.math.stonybrook.edu\/files\/cal\/agenda.php?LocationID=10","text":"Dynamical Systems Seminar\n\nfrom Friday\nJune 01, 2018 to Monday\nDecember 31, 2018\n Show events for: All Events AGNES Algebraic geometry seminar Algebraic models in geometry seminar Am.Math.Soc. (AMS) Chapter Seminar Analysis Seminar Analysis Student Seminar Capsule Research Talks Colloquium Commencement Ceremony Comprehensive Exams Dynamical Systems Seminar Equivalence Method and Exterior Differential Systems Seminar First and Second Year Student Seminar Friday Summer Meeting Geometric Analysis Learning Seminar Geometry\/Topology Seminar Grad \/ Postdoc Professional Development Seminar Graduate Student Seminar Graduate Topology Seminar Grant Proposal Panel Hodge Theory, Moduli and Representation Theory Holiday Party Joint Columbia-CUNY-Stony Brook General Relativity Seminar Math and Art Symposium for Tony Phillips Math Club Math Day 2016 Math in Jeans Mathematical Writing Seminar Mathematics Department Gathering Mathematics Education Colloquium Mathematics Summer Camp Mini Course \/ Dynamics Learning Seminar Mini-School in Geometry Minicourse in Real Enumerative Geometry New Graduate Students NY General Relativity Seminar Postdoc Geometry\/Dynamics Seminar Postdoc Seminar Representation Theory Student Seminar RTG Colloquium RTG Seminar RTG Student Geometry Seminar SCGP Seminars Seminar in Topology and Symplectic Geometry Seminar on algebraic structures in physics Simons Colloquium Simons Lectures Series Singular metrics and direct images Special Algebra \/ Algebraic Geometry Seminar Special Analysis Seminar Special Colloquium Special Dynamics Seminar Special Geometry\/Topology Seminar Special Lectures Special Seminar in Algebraic Geometry Special Topology Seminar Student Algebraic Geometry Seminar Student Differential Geometry Seminar Student Gauge Theory Seminar Student Seminar on Differential Geometry and Analysis Summer Workshop in Topology and Geometry Symplectic Geometry Reading Seminar Symplectic Geometry Seminar Thesis Defense Topology and Symplectic Geometry \/ Math of Gauge Fields seminar Women in Mathematics Instructions for subscribing to Stony Brook Math Department Calendars\n\n TuesdayJune 05, 20182:30 PM - 3:30 PM Math Tower P-131 Yair Minsky, Yale University Weil-Petersson geometry, Dehn filling and branched surfacesWe study pseudo-Anosov mapping classes with bounded normalized Weil-Petersson translation distance (and unbounded genus). In analogy with a result of Farb-Leininger-Margalit for Teichmuller translation distances, we show all such mapping classes fit together into a finite collection of cusped hyperbolic 3-manifolds, where the cusps are filled to become either vertical (transverse to fibers) or horizontal (parallel to fibers). After a reduction using work of Schlenker, Kojima-McShane and Brock-Bromberg, the argument uses the theory of branched surfaces in 3-manifolds. This is joint work with Chris Leininger, Juan Souto and Sam Taylor.\n\n TuesdayJune 12, 20182:30 PM - 3:30 PM Math Tower P-131 Yusheng Luo, Harvard University On the inhomogeneity of the Mandelbrot setWe will show that the Mandelbrot set is totally locally conformally inhomogeneous: the only orientation preserving conformal map $f:U \u2192 V$ with $U\\cap \u2202 M \\neq \\emptyset$ and satisfying $f(U\\cap\u2202 M) \u2282 \u2202 M$ is the identity map. The proof uses the study of the local conformal symmetries of the Julia sets: we will show in many cases, the dynamics can be recovered from local conformal structures of the Julia sets.\n\n ThursdayJune 21, 20181:30 PM - 2:30 PM Math Tower P-131 Roland Roeder, IUPUI Limiting Measure of Lee-Yang Zeros for the Cayley TreeI will explain how to use detailed properties of expanding maps of the circle (Shub-Sullivan rigidity, Ledrappier-Young formula, large deviations principle, ...) to study the limiting distribution of Lee-Yang zeros for the Ising Model on the Cayley Tree. No background in mathematical physics is expected of the audience. This is joint work with Ivan Chio, Caleb He, and Anthony Ji.\n\n FridayAugust 31, 20182:30 PM - 3:30 PM Math Tower P-131 Mahan Mj, TIFR Mumbai A survey of Cannon-Thurston MapsWe shall survey the theory of Cannon-Thurston Maps. These form a connecting link between the hyperbolic geometry and the complex dynamics of Kleinian groups. We shall also discuss a generalization to geometric group theory and end with some open problems.\n\n FridaySeptember 07, 20182:30 PM - 3:30 PM Math Tower P-131 Mark Pollicott, University of Warwick Dynamical Zeta functionsThe famous Selberg zeta function can be interpreted as a complex function defined in terms of closed orbits on a compact surface with constant negative curvature. We want to discuss generalizations of this: firstly to surfaces of variable negative curvature; and secondly to higher Teichmuller theory.\n\n FridaySeptember 14, 20182:30 PM - 3:30 PM Math Tower P-131 Peter Veerman, Portland State University Strange Convex SetsGiven a closed convex set $\u03a9\u2208\\mathbb{R}^n$, the metric projection of a given point $x\u2208\\mathbb{R}^n$ is given by the unique point $\u03a0(x)\u2208\u03a9$ that minimizes the (Euclidean) distance $\\lbrace\\vert y-x\\vert\\ \\vert\\ y\u2208\u03a9\\rbrace$ between $\u03a9$ and $x$. Most mathematicians tend to think of convex sets in $\\mathbb{R}^n$ as very tame objects. It is therefore surprising that it is easy to construct a compact convex set $\u03a9$ in $\\mathbb{R}^2$ with the following strange property [Shapiro, 1994]: There is a point $x\\notin\u03a9$ and a vector $v$ such that the directional derivative $$\\lim_{t\u2192 0}\\frac{\u03a0(x+vt)-\u03a0(x)}{t}$$ fails to exist. Note that for example convex polygons are not strange in this sense. We revisit and modify that construction to obtain a convex curve in $\\mathbb{R}^2$ that is $C^{1,1}$ or differentiable with Lipschitz derivative. We show that the convex set bounded by this curve has the property that the directional derivative of the projection is not defined. This construction can be made $C^n$ for $n\u2265 2$ except at a single point, and such that directional differentiability still fails.\n\n FridaySeptember 21, 20182:30 PM - 3:30 PM Math Tower P-131 Nguyen-Bac Dang, Stony Brook University Spectral gap in the dynamical degrees of tame automorphisms preserving an affine quadric threefoldIn this talk, I will present the tame automorphisms group preserving an affine quadric threefold. The main focus of my talk is the understanding of the degree sequences induced by the elements of this group. Precisely, I will explain how one can apply some ideas from geometric group theory in combination with valuative techniques to show that the values of the dynamical degrees of these tame automorphisms admit a spectral gap.Finally I will apply these techniques to characterize when the Lyapounov exponents of a random walk on this particular group are strictly positive.\n\n FridaySeptember 28, 20182:00 PM - 3:00 PM Math Tower P-131 Dragomir Saric, Queens College Asymptotics of moduli of curves and applicationsIn a joint work with H. Hakobyan, we prove that each Teichmuller geodesic in the universal Teichmuller space has a unique limit point on Thurston boundary. The main result depends on asymptotic estimates of moduli of curves. Another application of the asymptotics, in a joint work with A. Basmajian and H. Hakobyan, is to give a sufficient condition on Fenchel-Nielsen coordinates for infinite surfaces to guarantee that the surfaces have ergodic geodesic flows, i.e. of type $O_G$. In a joint work with H. Miyachi, we show that the Teichmuller disk in the universal Teichmuller space extends by continuity to a closed disk in Thurston bordification. Thurston boundary to arbitrary Teichmuller spaces are recently introduced in a joint work with F. Bonahon.\n\n FridayOctober 05, 20182:30 PM - 3:30 PM Math Tower P-131 Lasse Rempe-Gillen, University of Liverpool Taming wild entire functionsThe study of the dynamics of transcendental entire functions has a long history, going back to Fatou. It has recently garnered much interest, partly due to intriguing connections with other areas of holomorphic dynamics. While some examples and families have been well-understood since the 1980s, only recently tools have become available to advance a detailed understanding of large and very general classes of transcendental entire functions. In this talk, I will discuss some of these developments. In particular, I will discuss analogues of the local connectivity of Julia sets, which play a crucial role in polynomial dynamics. In joint work with Sixsmith (and partly with Alhamd), we introduce a notion of \"docile\" entire functions, and show in particular that a large class of functions, known as \"strongly geometrically finite functions\", are docile. I will also discuss work in progress with Sixsmith, with the goal of bringing the \"puzzle\" techniques of Yoccoz to bear on transcendental dynamics. In the course of the discussion, I intend to also touch upon work of Mihaljevic, and joint work with Albrecht and Benini (concerning the dynamics of entire functions of finite order), with Benini (concerning an analogue of the Douady-Hubbard landing theorem), and with Pfrang (on Hubbard trees).\n\n FridayOctober 12, 20182:30 PM - 3:30 PM SCGP 102 Alena Erchenko, Ohio State University Flexibility of Lyapunov exponents on the circle and the torusThere are several interesting classes of measures. For two special classes of dynamical systems, we will concentrate on the invariant measure that is absolutely continuous with respect to the Lebesgue measure and the measure of maximal entropy. First, we show that Lyapunov exponents with respect to these two probability measures for smooth expanding circle maps of a fixed degree $\u2265 2$ take on all values that satisfy some well-known inequalities. Then, we demonstrate a similar result for positive Lyapunov exponents with respect to these two measures for Anosov area-preserving diffeomorphisms on a two-torus that are homotopic to a fixed area-preserving Anosov automorphism (work in progress).\n\n FridayOctober 26, 20182:30 PM - 3:30 PM Math Tower P-131 Vasiliki Evdoridou, The Open University Singularities of inner functions associated to entire maps in the class $\\mathcal{B}$Let $f$ be a transcendental entire function and $U$ be an unbounded, invariant Fatou component of $f$. We can associate an inner function, $g$ say, to the restriction of $f$ to $U$. We consider two classes of functions in $\\mathcal{B}$ having finitely many tracts. We show that if $f$ belongs to either of these two classes the number of singularities of $g$ on the unit circle is equal to the number of tracts of $f$. This is joint work with N. Fagella, X. Jarque and D. Sixsmith.\n\n FridayNovember 02, 20182:30 PM - 3:30 PM Math Tower P-131 Yotam Smilansky, The Hebrew University of Jerusalem Multiscale substitution schemes and Kakutani sequences of partitionsSubstitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. In this talk we will use new path counting results for directed weighted graphs to show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we will describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy.\n\n FridayNovember 09, 20182:30 PM - 3:30 PM Math Tower P-131 Dimitrios Ntalampekos, Stony Brook University Removability of planar sets: old and new resultsRemovability of sets for quasiconformal maps and Sobolev functions has applications in Complex Dynamics, in Conformal Welding, and in other problems that require \"gluing\" of functions to obtain a new function of the same class. We, therefore, seek geometric conditions on sets that guarantee their removability. In this talk, I will give a survey of old results and discuss some very recent results on the (non)-removability of the Sierpi\u0144ski gasket and of Sierpi\u0144ski carpets. A first result is that the Sierpi\u0144ski gasket is removable for continuous functions of the class $W^{1,p}$ for $p>2$. The method used applies to more general fractals that resemble the Sierpi\u0144ski gasket, such as the Apollonian gasket and generalized Sierpi\u0144ski gasket Julia sets. Then, I will sketch a proof that the Sierpi\u0144ski gasket is non-removable for quasiconformal maps and thus for $W^{1,p}$ functions, for $1\u2264 p\u2264 2$. The argument involves the construction of a non-Euclidean sphere, and then the use of the Bonk-Kleiner theorem to embed it quasisymmetrically to the plane.\n\n FridayNovember 16, 20182:30 PM - 3:30 PM Math Tower P-131 Dimitry Turaev, Imperial College On wandering domains near homoclinic tangtenciesGiven a map, we define a wandering domain as an open region such that the diameter of its images by the iterations of the map shrinks to zero but the corresponding limit set is not a periodic point. It is known that many finitely smooth two-dimensional diffeomorphisms have wandering domains while it is not known if a polynomial diffeomorphism of a plane can have one. We discuss wandering domains whose limit sets are homoclinic tangencies. We show the existence of real analytic planar diffeomorphisms with wandering domains and discuss how to find wandering domains for polynomial diffeomorphisms of the three-dimensional space.\n\n FridayNovember 30, 20182:30 PM - 3:30 PM Math Tower P-131 Christopher J. Leininger, University of Illinois at Urbana-Champaign Polygonal billiards, Liouville currents, and rigidityA particle bouncing around inside a Euclidean polygon gives rise to a biinfinite \"bounce sequence\" (or \"cutting sequence\") recording the (labeled) sides encountered by the particle. In this talk, I will describe recent work with Duchin, Erlandsson, and Sadanand, in which we prove that the set of all bounce sequences---the \"bounce spectrum\"---essentially determines the shape of the polygon. This is consequence of our main result about Liouville currents on surfaces associated to nonpositively curved Euclidean cone metrics. In the talk I will explain the objects mentioned above, how they relate to each other, and give some idea of the proof of the main theorem.\n\n FridayDecember 07, 20182:30 PM - 3:30 PM Math Tower P-131 Zoran Sunic, Hofstra University TBATBA\n\n Show events for: All Events AGNES Algebraic geometry seminar Algebraic models in geometry seminar Am.Math.Soc. (AMS) Chapter Seminar Analysis Seminar Analysis Student Seminar Capsule Research Talks Colloquium Commencement Ceremony Comprehensive Exams Dynamical Systems Seminar Equivalence Method and Exterior Differential Systems Seminar First and Second Year Student Seminar Friday Summer Meeting Geometric Analysis Learning Seminar Geometry\/Topology Seminar Grad \/ Postdoc Professional Development Seminar Graduate Student Seminar Graduate Topology Seminar Grant Proposal Panel Hodge Theory, Moduli and Representation Theory Holiday Party Joint Columbia-CUNY-Stony Brook General Relativity Seminar Math and Art Symposium for Tony Phillips Math Club Math Day 2016 Math in Jeans Mathematical Writing Seminar Mathematics Department Gathering Mathematics Education Colloquium Mathematics Summer Camp Mini Course \/ Dynamics Learning Seminar Mini-School in Geometry Minicourse in Real Enumerative Geometry New Graduate Students NY General Relativity Seminar Postdoc Geometry\/Dynamics Seminar Postdoc Seminar Representation Theory Student Seminar RTG Colloquium RTG Seminar RTG Student Geometry Seminar SCGP Seminars Seminar in Topology and Symplectic Geometry Seminar on algebraic structures in physics Simons Colloquium Simons Lectures Series Singular metrics and direct images Special Algebra \/ Algebraic Geometry Seminar Special Analysis Seminar Special Colloquium Special Dynamics Seminar Special Geometry\/Topology Seminar Special Lectures Special Seminar in Algebraic Geometry Special Topology Seminar Student Algebraic Geometry Seminar Student Differential Geometry Seminar Student Gauge Theory Seminar Student Seminar on Differential Geometry and Analysis Summer Workshop in Topology and Geometry Symplectic Geometry Reading Seminar Symplectic Geometry Seminar Thesis Defense Topology and Symplectic Geometry \/ Math of Gauge Fields seminar Women in Mathematics Instructions for subscribing to Stony Brook Math Department Calendars","date":"2018-11-17 13:22:53","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.5028008818626404, \"perplexity\": 1286.1641574863288}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-47\/segments\/1542039743521.59\/warc\/CC-MAIN-20181117123417-20181117145417-00483.warc.gz\"}"} | null | null |
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General Andom: the Eritrean who tried to make peace for Ethiopia's military regime
MARTIN PLAUT
General Aman Mikael Andom – army officer and Ethiopian head of state in 1974 – was born in Eritrea.
He attempted to make peace with the Eritrean Liberation Front, to end the war of independence that had been under way since 1961.
But Mengistu Haile Hariam and the military council that then ruled Ethiopia (the Derg) refused to allow any such deal, believing the Eritreans could be crushed.
General Andom paid with his life.
Below is the Wikipedia biography of the General.
Aman Mikael Andom (Amharic: አማን ሚካኤል አንዶም, 21 June 1924 – 23 November 1974) was the first post-imperial acting Head of State of Ethiopia. He was appointed to this position following the coup d'état that ousted Emperor Haile Selassie on 12 September 1974, and served until his death in a shootout with his former supporters.
1 Early life
2 Military career
3 Head of State
Aman was born in Khartoum, Sudan to parents Ato Andom Michael and Woizero Ghidey Reda. He was of Eritrean origin, hailing from the village of Tsazega in Hamassien province of Eritrea.[1] He had four other siblings.
As commander of the Third Division, General Aman had been beating back the encroachments of the Somali army on the eastern border with such zeal and success that he was known as the "Desert Lion". However, in 1964 the Emperor dismissed Aman when he began to attack into Somalia in violation of an order from the Emperor, and Aman afterwards served in the Ethiopian Senate in a "political exile".[citation needed]
Aman's official title was Chairman of the Provisional Military Administrative Council (better known as the Derg), and he held the position of Head of State in an acting capacity as the military regime had officially proclaimed Crown Prince Asfaw Wossen as "King-designate" (an act that would later be rescinded by the Derg, and which was never accepted by the Prince as legitimate).[citation needed]
There is some evidence that indicates he had contacts with the officers of the junta as early as February and March 1974, but by July he was appointed chief of staff to the military junta. Three days after the junta removed the Emperor from his palace to imprisonment at the headquarters of the Fourth Division, this group appointed him their chairman and president of Ethiopia. At the same time, this group of soldiers assumed the name "Provisional Military Administrative Council", better known as the Derg.[2]
From the first day of his presidency, the Ottaways note, "the general found himself at odds with a majority of the Derg's members over most major issues, including whether he was 'chairman' of the ruling military body or simply its 'spokesman.'"[3] Aman fought the majority of the Derg over three central issues: the size of the Derg, which he felt was too large and unwieldy; the policy to be taken towards the Eritrean Liberation Front (ELF); and over the punishment of the numerous aristocrats and former government officials in the Derg's custody. His refusal to sanction the execution of former high officials, including two former prime ministers and several royal family members and relatives, put his relations with the majority of the Derg on an especially bitter footing.[citation needed]
As an Eritrean, General Aman found himself fiercely at odds with the majority of the Derg. He wanted to negotiate a peaceful settlement; his opponents hoped to crush the ELF by military force. Aman went as far as making two personal visits to Eritrea—the first 25 August to 6 September, the second in November—giving speeches stating that the end of the Imperial regime was also the end of old practices towards Eritrea, that a government dedicated to national unity and progress would restore peace and prosperity to Eritrea, and lastly that he would begin investigations concerning crimes that the army had perpetrated on Eritreans and punish the guilty.[4]
However, at the same time the Derg had begun the task of eliminating opponents within the military. The three significant units were the Imperial Bodyguard, the Air Force, and the Corp of Engineers; of the three, the most recalcitrant were the Engineers. So on 7 October soldiers loyal to the Derg stormed the Engineers' camp, killing five, wounding several and detaining the rest. As Bahru Zewde observes, "With that, the illusion that the revolution would remain bloodless was exploded."[5]
General Aman responded with a personal campaign to seek support outside the Derg, among the rest of the army and the country where he was popular. On 15 November he sent a message to all military units that was highly critical of the Derg. During a general assembly of the Derg two days later, Mengistu Haile Mariam demanded that 5,000 men be dispatched to Eritrea and six imprisoned Imperial officials be executed; Aman Andom refused, resigned his official posts and retired to his house where he secretly sent appeals to his supporters, especially those in the Third Division. But Mengistu managed to intercept these appeals.[6]
General Aman died in a battle with troops sent to his home to arrest him. The actual cause of his death remains unclear, whether he was killed or committed suicide. That same night, the political prisoners that the Derg had marked for execution were taken from Menelik prison, where they had been held, to the Akaki Central Prison where they were executed and buried in a mass grave.[7] "It appears that the general had outlived his usefulness," Bahru Zewde concludes, "and was in fact becoming an obstacle to the Derg's exercise of power."[8]
Murtaza, Niaz (1998). The Pillage of Sustainablility in Eritrea, 1600s-1990s: Rural Communities and the Creeping Shadows of Hegemony. Greenwood Publishing Group. p. 78. ISBN 9780313306334.
Ottaway, Marina; Ottaway, David (1978). Ethiopia: Empire in Revolution. Africana Publishing Company. pp. 59f, and n. 29. ISBN 9780841903630.
Ottaway & Ottaway (1978), p. 60
Ottaway & Ottaway (1978), p. 155
Zewde, Bahru (2001). A History of Modern Ethiopia, 1855-1991. James Currey. p. 238. ISBN 9780821414408.
Lefort, René (1983). Ethiopia, an Heretical Revolution?. Translated by Berrett, A. M. London: Zed Press. p. 73. ISBN 9780862321543.
Zewde (2001), p. 238
Derg
General Aman Mikael Andom
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Money has been tight at West Allis-West Milwaukee schools, so it was nice of three local businesses to host a fundraiser for music and art programs.
As a fan of feel-good stories, I invited myself along to watch the business owners turn in a check for $750, which turned out to be harder than it sounds.
The check was scheduled to be handed over Wednesday afternoon to the band director at West Allis Central High School, but the meeting was canceled just before we headed over there.
"I think there might have been a misunderstanding about where that happens," said district spokeswoman Beth Koehler, who got involved along with other higher-ups. "We take our donations of that nature at our board meetings to more formally recognize and thank the community."
She was apologetic and said community support like this is greatly appreciated.
Oh, well, the money will get where it belongs eventually and go for things like musical instruments, scholarships for lessons and transportation to events.
The fundraiser Saturday was hosted at Westallion Brewing Co., 1825 S. 72nd St., with pizza support from Tim Szuta at Alphonso's the Original, 1119 S. 108th St., and added beverages from Neal Steffek at The Drunk Uncle tavern, 1902 S. 68th St.
"I think it's incredible that the local businesses here continue to band together for positive change within. It is a characteristic that West Allis has that I am so extremely proud of," said Kimberly Dorfner, who owns Westallion with her husband, Erik.
Erik and Tim graduated from Nathan Hale High School in West Allis, and Neal from Central. These are homegrown guys trying to give back.
"I was thinking that music was a big inspiration to me and has always helped me out through my life," said Tim, whose band, Triple Oh Seven, played at the benefit. People paid $5 to get in and some tossed additional cash in a bucket.
The business owners hope this civic spirit catches on with others, though last spring the district's taxpayers voted down a referendum for additional school funding. Not everyone is inclined to give schools or government on any level more money than they already demand.
Erik and Kimberly have three children in West Allis schools — Alina, 15, at Hale; Alivia, 13, at Lane Intermediate; and Nolan, 8, at Hoover Elementary.
"My daughters are both in music and arts programs and have definitely felt the hit in the budget cuts," Kimberly said.
Koehler said there have been smallish reductions in things like supplies, but no major programs have been eliminated. The intermediate and high schools still offer orchestra, band and chorus.
The agenda for next week's School Board meeting is already done, so the soonest the money can be turned over is March 12.
The red tape has Erik Dorfner feeling miffed.
"There's a whole process," he said, "where we need to jump through hoops to go to the School Board, and then, I don't know, they call astronauts on Mars and then they signal the satellites roving around Earth."
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\section{Introduction}
One of the main properties of few electron quantum dots is the supression of charge current when the energy cost of introducing
an extra electron in the system is large due to the charge repulsion inside the quantum dot (QD). In such a case, the system presents Coulomb
blockade, showing characteristic curves where current only flows for certain values of the chemical
potentials\cite{Beenakker}. Between these peaks,
current is allowed only through second order processes, which we do not treat here.
However, by introducing an external AC field, the electrons obtain (from the interaction with the field) enough energy to
fulfill the energy requirements and tunnel through the system\cite{TienGordon}. In this way, a finite current may appear
even in the zero bias configuration ({\it pumping} regime).
These photon-assisted tunneling (PAT) processes through the contact barriers have been studied in AC driven single quantum
dots\cite{KouwenhovenPAT} but are usually
neglected in the theoretical study of double quantum dot (DQD) resonant pumps\cite{StaffordWingreen}, where
only interdot PAT is considered.
If one takes into account the spin of the electron, another interesting effect takes part in two-site systems like, for example, a DQD
having one level each.
If an electron is trapped in one of the quantum dots, transport is only available through the two-electron singlet state of that QD, when
an electron with the opposite spin tunnels from the other QD and afterwards to the electrode.
But, if the other
QD is occupied by a trapped electron with the same spin, Pauli exclusion principle does not allow the formation of the doubly occupied state
and, therefore, the current is blocked. This phenomenon is known as spin blockade (SB)\cite{Weinmann}.
In this work, we study a concrete case in which spin blockade may appear in AC driven DQD spin pumps\cite{CotaNanot} and show that PAT
processes through the contacts can be important, allowing the trapped spins to tunnel out of the system breaking the SB effect.
\section{Theoretical model}
Our system, consisting in two quantum dots weakly connected in series to two {\it unbiased} electron reservoirs by tunnel barriers, is described by the
Hamiltonian $\hat{H}=\hat{H}_L+\hat{H}_R+\hat{H}_{L\Leftrightarrow R}+\hat{H}_{leads}+\hat{H}_T$. We consider one level in each QD containing
up to two electrons:
$\hat H_{j=\{L,R\}}=\sum_\sigma\varepsilon_{{j}\sigma}\hat{c}_{{j}\sigma}^\dagger\hat{c}_{j\sigma}+
U_{j}\hat{n}_{j\uparrow}\hat{n}_{j\downarrow}$, where $\hat{c}_{{j}\sigma}^\dagger$ is the creation operator of an electron with
spin $\sigma$ in dot $j$ and energy $\varepsilon_{{j}\sigma}$ and $U_{j}$ is the charging energy
of each dot. The energies $\varepsilon_{{j}\sigma}$ include the Zeeman splitting, $\Delta_{j}$, due to the introduction of a
magnetic field in order to break the spin degeneracy of the different levels, in such a way that we can consider the spin-up state
as the ground state. Thus: $\varepsilon_{j\downarrow}=\varepsilon_{j\uparrow}+ \Delta_{j}$.
The reservoirs are described by the term:
$\hat{H}_{leads}=\sum_{l\epsilon\{L,R\}k\sigma}\varepsilon_{lk}\hat{d}_{lk\sigma}^\dagger\hat{d}_{lk\sigma}$, where the operator
$\hat{d}_{lk\sigma}^\dagger$ creates an electron with moment $k$ and spin $\sigma$ in lead $l$. Each QD is coupled to the
other and to the leads through the terms:
$\hat{H}_{L\Leftrightarrow R}=-t_{LR}\sum_{\sigma}\hat{c}_{L\sigma}^\dagger\hat{c}_{R\sigma}+h.c.$ and
$\hat H_T=\sum_{l\epsilon\{L,R\}k\sigma}(\gamma\hat{d}_{lk\sigma}^\dagger\hat{c}_{l\sigma}+h.c.)$. The constant
that describes the tunneling through the contacts, $\gamma$, is asumed to be small and, for simplicity, similar for both QDs and will be treated as
a perturbative parameter.
We also introduce an external AC field acting on the energy levels of the quantum dots such that (considering $\hbar=e=1$):
$\varepsilon_{L(R)\sigma}\rightarrow\varepsilon_{L(R)\sigma}(t)=\varepsilon_{L(R)\sigma}\pm \frac{V_{AC}}{2}cos\omega t$, where
$V_{AC}$ and $\omega$ are the amplitude and frequency of the field, respectively. The time dependent field will serve as a driver
of electrons from one QD to the other, allowing the formation of doubly occupied states, which contribute to the current
through the device.
\subsection{Master equation}
We study the electron dynamics of the DQD system using the reduced density matrix operator, $\hat\rho=tr_R\hat\chi$, obtained by
tracing all the reservoir states in the density operator of the whole system, $\hat\chi$. The Liouville equation,
$\dot{\hat\rho}(t)=-i[\hat H(t),\hat\rho(t)]$ gives us the time evolution of the system. Assuming Markov and Born
approximations\cite{Blum}, we derive the master equation for the density matrix elements\cite{pssa},
$\rho_{m'm}(t)=\langle m'|\hat{\rho}(t)|m\rangle$:
\begin{eqnarray}
\dot\rho(t)_{m'm}&=&-i\omega_{m'm}\rho_{m'm}(t)-i[\hat H_{L\Leftrightarrow R}'(t),\hat\rho(t)]_{m'm}
\\
&&+\left(\sum_{n\ne m'}\Gamma_{m'n}\rho_{nn}(t)-\sum_{n\ne m}\Gamma_{nm}\rho_{mm}(t)\right)\delta_{m'm}
-\Omega_{m'm}\rho_{m'm}(t)(1-\delta_{m'm}),\nonumber
\label{mastereq}
\end{eqnarray}
in the particle basis:
$|1\rangle=|0,0\rangle$, $|2\rangle=|\uparrow,0\rangle$, $|3\rangle=|\downarrow,0\rangle$,
$|4\rangle=|0,\uparrow\rangle$, $|5\rangle=|0,\downarrow\rangle$, $|6\rangle=|\uparrow,\uparrow\rangle$,
$|7\rangle=|\downarrow,\downarrow\rangle$, $|8\rangle=|\uparrow,\downarrow\rangle$,
$|9\rangle=|\downarrow,\uparrow\rangle$, $|10\rangle=|\uparrow\downarrow,0\rangle$,
$|11\rangle=|0,\uparrow\downarrow\rangle$, $|12\rangle=|\uparrow\downarrow,\uparrow\rangle$,
$|13\rangle=|\uparrow\downarrow,\downarrow\rangle$, $|14\rangle=|\uparrow,\uparrow\downarrow\rangle$,
$|15\rangle=|\downarrow,\uparrow\downarrow\rangle$,
$|16\rangle=|\uparrow\downarrow,\uparrow\downarrow\rangle$.
Here, $\omega_{m'm}$ is the energy difference between the states $|m'\rangle$ and $|m\rangle$ of the isolated DQD and
$\Omega_{m'm}$ describes the decoherence of the DQD states due to the interaction with the reservoir. The time dependence
of the energy levels has been transferred to the interdot coupling:
\begin{equation}
\langle m|\hat H'_{L\Leftrightarrow R}(t)|n\rangle=
\sum_{\nu=-\infty}^\infty J_\nu(\alpha)e^{i\nu\omega t}\langle
m|\hat H_{L\Leftrightarrow R}(t)|n\rangle
\label{hoppingTrans},
\end{equation}
where $J_\nu(\alpha)$ is the $\nu$-th order Bessel function of the first kind, being
$\alpha=V_{AC}/\omega$ the dimensionless AC field intensity.
The tunneling rates through the contacts, $\Gamma_{mn}$, are affected by the AC field:
\begin{equation}
\Gamma_{mn}=\sum_{\nu=-\infty}^\infty J_\nu^2\left(\frac{\alpha}{2}\right) \xi_{mn}(\omega_{mn}+\nu\omega),
\label{patrates}
\end{equation}
where
\begin{equation}
\xi_{mn}(\varepsilon)=\left\{ f(\varepsilon)\delta_{N_m,N_n+1}
+(1-f(-\varepsilon))\delta_{N_m,N_n-1}\right\}\Gamma
\label{rates}
\end{equation}
are the usual tunneling rates obtained {\it when PAT through the contacts is neglected}.
$N_k=\sum_{j\sigma}\langle k|\hat n_{j\sigma}|k\rangle=\sum_j N_k^j$ is the number of electrons in the system in
state
$|k\rangle$, $f(\varepsilon)=1/(1+e^{(\varepsilon-\mu)\beta})$ is the Fermi distribution function, where $\beta=1/k_BT$ and $\mu$ is the chemical potential of the leads, and $\Gamma=2\pi|\gamma|^2$.
These transition rates (Eq.(\ref{patrates})) are related to the decoherence through the
relation:
$\Re\Omega_{m'm}=\frac{1}{2}(\sum_{k\ne
m'}\Gamma_{km'}+\sum_{k\ne m}\Gamma_{km})$.
We obtain the current that flows through the right contact with the relation
$I_R=\sum_{m,m'}(\Gamma_{m'm}\rho_{mm}-\Gamma_{mm'}\rho_{m'm'})\delta_{N_m^R-1,N_{m'}^R}=\sum_{\sigma} I_{R,\sigma}$. The spin currents, $I_{R,\uparrow}$ and $I_{R,\downarrow}$, only account for the processes involving the tunneling of spins with {\it up} and {\it down} polarization, respectively.
\section{Numerical results}
If both reservoirs have the same chemical potential, $\mu$, i.e., there is no bias voltage applied to the DQD, and
$\mu<U_l+\varepsilon_l+\Delta_l$, the non-driven system will be in a stable state if it contains one electron in each dot. Additionally,
if $\mu>U_R+\varepsilon_R$, the spin down electron will be the only one
able to tunnel out to the right lead from the doubly occupied singlet state and the spin up will be trapped in the right QD. Thus, the breaking of the spin degeneracy by the introduction of a magnetic field leads to an asymmetry in the transport properties of the system, so the spin components of the current, $I_{R,\uparrow}$ and $I_{R,\downarrow}$, will behave differently.
\begin{figure}[htb]
\includegraphics[width=3.7in,clip]{esquema6.eps}
\caption{\label{esquema}
{\small Schematic diagram of the non-driven device showing the chemical potentials associated to the transitions involving the extraction through the contact barriers of one electron from the doubly occupied state in each QD. Since $\mu>U_R+\varepsilon_R$, the transitions extracting an electron with spin-up polarization from states with two electrons in the right dot through the right contact are energetically unavailable, unless they are mediated by the absorption of photons. In our configuration, the Zeeman splitting and the energy of the levels are the same in both dots, that is: $\Delta_L=\Delta_R=\Delta_z$ and $\varepsilon_L=\varepsilon_R$.
}}
\end{figure}
Introducing an AC field in resonance with the states $|\downarrow,\uparrow\rangle$
and $|0,\uparrow\downarrow\rangle$ (and, since $\Delta_L=\Delta_R$, also with $|0,\uparrow\downarrow\rangle$ and $|\uparrow,\downarrow\rangle$),
the spin down electron will be delocalized between both quatum dots and there will be a finite
probability for it to leave the DQD to the right lead. If PAT processes
through the contacts are not considered, one should expect a net spin down current
through the system (through the {\it pumping cycle}:
$|\downarrow,\uparrow\rangle\Leftrightarrow|0,\uparrow\downarrow\rangle\rightarrow\left\{|0,\uparrow\rangle or
|\downarrow,\uparrow\downarrow\rangle\right\}\rightarrow|\downarrow,\uparrow\rangle$) but, since the empty left QD can be also filled with a spin up electron, the system asymptotically evolves to the state $|\uparrow,\uparrow\rangle$
(i.e., $\rho_{6,6}(t\rightarrow\infty)=1$ and $\rho_{i,j}(t\rightarrow\infty)=0$, otherwise)
that leads to SB\cite{CotaNanot}.
However, the rates (\ref{patrates}) allow the "trapped" spins in the DQD to absorb a certain number of photons and
tunnel out to the leads giving a finite ocuppation probability (through the sequences:
$|\uparrow,\uparrow\rangle\rightarrow|0,\uparrow\rangle\rightarrow|\downarrow,\uparrow\rangle$ or $|\uparrow,\uparrow\rangle\rightarrow|\uparrow,0\rangle\rightarrow|\uparrow,\downarrow\rangle$)
to the states $|\downarrow,\uparrow\rangle$ and $|\uparrow,\downarrow\rangle$ which are in resonance with the state $|0,\uparrow\downarrow\rangle$, that contributes to the pumping of a spin down electron to the right lead (Fig. \ref{dens}).
Then, PAT through the contacts creates a finite current through the system (Fig. \ref{Ivsw}), removing the SB.
\begin{figure}[htb]
\includegraphics[angle=270,width=3.7in,clip]{dens.ps}
\caption{\label{dens}
{\small Time evolution (normalized to $\tau=2\pi/\Omega$, being $\Omega=2J_1(\alpha)t_{LR}$ the Rabi frequency of the delocalization processes) of the diagonal density matrix elements $\rho_{4,4}$, $\rho_{6,6}$, $\rho_{8,8}$, $\rho_{9,9}$,
$\rho_{11,11}$, $\rho_{14,14}$ and $\rho_{15,15}$,
which describe the occupation probability of the states
that contribute to the current. We consider the initial condition: $\rho_{6,6}(t=0)=1$. In the case where PAT through the contacts
is not considered, we would obtain $\rho_{6,6}(t)=1$, for all times\cite{CotaNanot}.
We do not show the occupation probabilities of other states that are not directly involved in the pumping processes.
Parameters (in meV): $t_{LR}=0.005$, $\Gamma=0.001$, $U_L=1.6$, $U_R=1.3$,
$\Delta_L=\Delta_R=0.2$ (corresponding to a magnetic field $B\approx8T$), $\varepsilon_L=\varepsilon_R=0.5$, $\mu=1.9$,
$\omega=\omega_{11,8}=V_{AC}/2$.
}}
\end{figure}
\begin{figure}[htb]
\includegraphics[angle=270,width=3.7in,clip]{Ivswmu19.ps}
\caption{\label{Ivsw}
{\small Pumped current (normalized to the tunneling probability, $\Gamma$) as a function of the frequency of the AC field (in meV).
Considering the {\it blocking state}, $|\uparrow,\uparrow\rangle$, if the electron in the left QD absorbs a photon and tunnels to the left lead, net spin down current to the right lead is created due to the sequence:
$|\uparrow,\uparrow\rangle\rightarrow|0,\uparrow\rangle\rightarrow|\downarrow,\uparrow\rangle\Leftrightarrow|0,\uparrow\downarrow\rangle
\rightarrow\{|0,\uparrow\rangle or |\uparrow,\uparrow\downarrow\rangle\}\rightarrow|\uparrow,\uparrow\rangle$, while spin up current
flows in the opposite direction (from right to left) by the sequence:
$|\uparrow,\uparrow\rangle\rightarrow|0,\uparrow\rangle\rightarrow|\downarrow,\uparrow\rangle\Leftrightarrow|0,\uparrow\downarrow\rangle
\Leftrightarrow|\uparrow,\downarrow\rangle\rightarrow|\uparrow,\uparrow\downarrow\rangle\rightarrow|\uparrow,\uparrow\rangle$ (in this
cycle, spin down current through the right contact is also produced).
The double arrow ($\Leftrightarrow$) represents the resonant delocalization processes inside the DQD at $\omega\approx U_R=1.3$.
On the other hand, if the spin up in the right QD is extracted from $|\uparrow,\uparrow\rangle$, there is a positive contribution to spin up current through the sequence:
$|\uparrow,\uparrow\rangle\rightarrow|\uparrow,0\rangle\rightarrow|\uparrow,\downarrow\rangle\Leftrightarrow|0,\uparrow\downarrow\rangle\rightarrow\{|\uparrow,\uparrow\downarrow\rangle or |0,\uparrow\rangle\}\rightarrow|\uparrow,\uparrow\rangle$. The contribution of this sequence is smaller (since it compites with the sequence $|\uparrow,\uparrow\rangle\rightarrow|\uparrow,0\rangle\rightarrow|\uparrow,\downarrow\rangle\rightarrow|\uparrow,\downarrow\rangle\rightarrow|\uparrow,\uparrow\downarrow\rangle\rightarrow|\uparrow,\uparrow\rangle$ that recovers rapidly the state $|\uparrow,\uparrow\rangle$ without contributing to the current) and is only apreciable at high enough field intensities. However, it may be the responsible of the supression of negative spin up current near resonance, giving a small bump. Note also that, since so many states are contributing to the dynamics of the system, the behaviour of the resonance peaks differs from the typical Lorentzian shape.
The parameters are the same as in Fig. \ref{dens}.
}}
\end{figure}
\section{Conclusions}
We have shown that the interaction with the driving field applied on a DQD system affects not only the interdot tunneling
but also the tunneling through the contact barriers. In concrete, PAT through the contact barriers affects the ocupation
of the states, giving a finite probability to states that are energetically unstable but open new channels to the electronic
transport through the system. Therefore, they should be taken into account when studying properties of driven quantum dot devices.
\begin{acknowledgement}
Work supported by Programa de Cooperaci\'on Bilateral CSIC-CONACYT, by Grant No. DGAPA-UNAM
114403-3, by the EU Grant No. HPRN-CT-2000-00144 and by the Ministerio de Educaci\'on y Ciencia of Spain
through Grant No. MAT2005-00644. RS was supported by CSIC-Programa I3P, cofinanced by Fondo Social Europeo.
\end{acknowledgement}
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Buy Books Online > Sagas > Mistress of the Throne : The Mughal Intrigues (English)
Mistress of the Throne : The Mughal Intrigues (English)
By: Ruchir Gupta
ISBN: 9789382665076 Publisher: Srishti Publishers & Distributors Year of publishing: 2014 Format: Paperback No of Pages: 324 Language: English
About The Book Mistress of the Throne: The Mughal Intrigues is set in the year 1631 and gives its readers an exclusive account of the life of young Jahanara, who finds herself crowned the Queen of India...Read more
About The Book Mistress of the Throne: The Mughal Intrigues is set in the year 1631 and gives its readers an exclusive account of the life of young Jahanara, who finds herself crowned the Queen of India by her father, the Mughal King Shah Jahan. This happens when her mother, Mumtaz Mahal, who was the Queen of India, passes away. Instead of passing on this title to one of his many wives, he decides to break free from the conventional trend and pass it on to his young daughter instead. This seventeen-year-old queen, who looks like a mirror-image of her mother, must now bear the load of the entire kingdom on her young shoulders. Jahanara has younger siblings who have varied temperaments. She must come to terms with the harsh fact that even though she has all the power and authority in her hands, she is forbidden to share that with any man. Even though her parents had a love story that was etched in time, young Jahanara must now accept that she is forbidden to marry, as per the law. Set in a time that is often referred to as 'India's Golden Age', this enchanting account of the life of a young Muslim queen is unique and tells its readers a lot about lives in that era. Mistress of the Throne: The Mughal Intrigues has been published by Srishti Publishers & Distributors in the year 2014 and is available in paperback. Key Features The novel gives a beautiful description of life in the Mughal era. The author has succeeded in portraying the myriad of emotions that the young Muslim queen experiences.
About the author: Ruchir Gupta
About the Author: The author of Mistress of the Throne: The Mughal Intrigues is Ruchir Gupta. He is a graduate of Upstate Medical University and is practicing... Read more
About the Author: The author of Mistress of the Throne: The Mughal Intrigues is Ruchir Gupta. He is a graduate of Upstate Medical University and is practicing medicine in Long Island, NY, at present. He resides there with his wife and daughter. He has been the author of several books that deal with the topic of anaesthesiology. He has a number of interests, which include reading, traveling, learning history and blogging.
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\section{Introduction}
Two decades have passed since the first observation of long-term fluorescence intensity fluctuations (blinking) of single colloidal CdSe quantum dots (QDs) with a ZnS shell \cite{BrusNature96}. In further experimental studies it was found (see \cite{FrantsuzovNaturePhys08,BarkaiPT09,OrritCOCIS07,MulvaneyPCCP06,KraussJCPL10,ReidIJMS12,OronIJC12,LeoneCSR13} and references therein) that
these fluctuations have a wide spectrum of characteristic timescales,
from hundreds of microseconds to hours. The intensity traces (binned photon counting data) of CdSe/ZnS core/shell dots show the following key properties:\\
1. The intensity distribution usually has two maxima, so-called ON and OFF intensity levels\\
2. The ON-time and OFF-time distributions obtained by the threshold procedure have the truncated power-law form\\
\begin{equation}
p(t)\sim t^{-m} \exp(-t/T)
\label{tmexp}
\end{equation}
3. The power spectral density of the trace has a $1/f^r$ dependence, where $r$ value is around 1. This dependence
changes to $f^{-2}$ at large frequencies \cite{PeltonPNAS07}.
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig1.pdf}
\caption{The schematic picture of the DCET model.
The potential surfaces of the neutral (bright) and the charged (dark) electronic states are represented by
the red and blue lines, respectively. Vertical dotted line corresponds to the crossing point.}
\label{fig:Tang}
\end{figure*}
Another interesting phenomenon that manifests in the emission of single quantum dots is the spectral diffusion
showing characteristic time scales in the order of hundreds of seconds \cite{BawendiPRL96,BawendiJPCB99}.
It is not surprising that there are a number of models proposed to explain the blinking that relate the fluctuations in the emission intensity with
slow variations in the exciton energy.
The first model of that kind suggested by Shimizu et al. \cite{BawendiPRB01}
is based on the Efros/Rosen {\it charging mechanism} (CM) \cite{EfrosPRL97}. The CM attributes the ON and OFF periods to neutral and charged QDs, respectively. The light-induced electronic excitation in the charged QD is supposed to be quenched by a fast Auger recombination process.
The model of Shimizu et al. \cite{BawendiPRB01} assumes that the charging/discharging events happen when the energies
of the neutral exciton and the charged state are in resonance.
A more advanced version of this idea was used by Tang and Marcus in the DCET model \cite{TangJCP05,TangPRL05}.
In 2014 Zhu and Marcus \cite{ZhuPCCP2014} presented an extension of the DCET model by introducing an additional biexciton charging channel.
Simultaneously with Tang and Marcus \cite{TangJCP05,TangPRL05}, another diffusion model based on the alternative {\it fluctuating rate mechanism} (FRM) of blinking was suggested by Frantsuzov and Marcus \cite{FrantsuzovPRB05}.
The FRM assumes that the non-radiative relaxation rate of the exciton is subject to long term fluctuations caused by the rearrangement of surface atoms.
A basic life cycle of the QD within this mechanism begins with a photon absorption. A relaxation of the excited state can go in one of of two paths. The first path is relaxation via a photon emission. The second path is a hole trapping followed by a consequent non-radiative recombination with a remaining electron. The photoluminescence quantum yield (PLQY) of the QD emission in this case can be expressed as
\begin{equation}
Y(t) = \frac{k_r}{k_r+k_t(t)}\equiv k_r \tau_{av}
\label{Y}
\end{equation}
where $k_r$ is the radiative recombination rate, and $\tau_{av}$ is the averaged exciton lifetime.
Thus the variations of the $k_t$ generate fluctuations of the emission intensity on a long time scale.
The Frantsuzov and Marcus model \cite{FrantsuzovPRB05} connects the recombination rate with the fluctuating energy difference between 1S$_e$ and 1P$_e$ states.
In this article we are going to discuss the advantages and disadvantages of these models of single QD blinking based on spectral diffusion
as well as their perspectives of further development.
\section{Diffusion-controlled electron transfer model}
After introducing the Marcus reaction coordinate $Q$, DCET model
equations describing the evolution of its probability distribution density
in the neutral state $\varrho_1(Q,t)$ and in the charged state $\varrho_2(Q,t)$ can be written in the following form:
$$\frac{\partial}{\partial t}\varrho_1(Q,t)= D_1 \frac{\partial}{\partial Q}
\left(\frac{\partial}{\partial Q}+ \frac{U_1'(Q)}{kT} \right) \varrho_1(Q,t)$$
\begin{equation}
-2\pi\frac {V^2} \hbar \delta(U_1(Q)-U_2(Q))\left(\varrho_{1}(Q,t)-\varrho_{2}(Q,t)\right)
\label{EqZ1}
\end{equation}
$$\frac{\partial}{\partial t}\varrho_2(Q,t)= D_2 \frac{\partial}{\partial Q}
\left(\frac{\partial}{\partial Q}+ \frac{U_2'(Q)}{kT}\right) \varrho_2(Q,t)$$
\begin{equation}
-2\pi\frac {V^2} \hbar \delta(U_1(Q)-U_2(Q))\left(\varrho_{2}(Q,t)-\varrho_{1}(Q,t)\right),
\label{EqZ2}
\end{equation}
where $D_1$ and $D_2$ are diffusion coefficients in the neutral electronic state and charged state respectively, $V$ is the electronic coupling matrix element between the
neutral and charged states, and $T$ is the effective temperature.
The potential surfaces of the neutral $U_1(Q)$ and charged $U_2(Q)$ states are Marcus' parabolas (see Fig. \ref{fig:Tang}):
\begin{equation}
U_1(Q)=\frac {(Q+E_r)^2} {4 E_r} \qquad U_2(X)=\frac {(Q-E_r)^2}{4 E_r}+\Delta G
\label{U_Marcus}
\end{equation}
characterized by the reorganization energy $E_r$ and the free energy gap $\Delta G$.
Transitions between the neutral and charged states are determined by
the delta-functional sink in the crossing point $Q_c$ (local Golden rule), where $U_1(Q_c)=U_2(Q_c)$
$$Q_c=\Delta G$$
Equations (\ref{EqZ1}-\ref{EqZ2}) were initially introduced in 1980 independently by
Zusman \cite{ZusmanCP80} and Burshten and Yakobson \cite{BurshteinCP80} for describing solvent
effects in electron transfer reactions. In the literature they are usually called Zusman equations (see for example
the review article \cite{BarzykinACP02} and references therein).The rigorous derivation of the Eqs. (\ref{EqZ1}-\ref{EqZ2})
from the basic quantum level (Spin-Boson Hamiltonian) was made in Ref. \cite{FrantsuzovJCP99}.
The characteristic time scales of diffusion in the process of the electron transfer are of the order of picoseconds.
That is to say that the equations (\ref {EqZ1}-\ref{EqZ2}) were originally designed to work for completely different time scales.
The statistics of the ON time blinking periods within the DCET model can be calculated
using the function $\rho_1(Q,t)$
which is a solution of the equation (\ref{EqZ1}) where the term describing the transfer
from the charged state to the neutral one is omitted:
$$\frac{\partial}{\partial t}\rho_1(Q,t)= D_1 \frac{\partial}{\partial Q}
\left(\frac{\partial}{\partial Q}+ \frac{U_1'(Q)}{kT} \right) \rho_1(Q,t)$$
\begin{equation}
-2\pi\frac {V^2} \hbar \delta\left(U_1(Q)-U_2(Q)\right)\rho_{1}(Q,t)
\label{EqZi1}
\end{equation}
with the initial condition describing the distribution function right after the transition from the charged state:
$$\rho_1(Q,0)=\delta(Q-Q_c)$$
The probability of the ON state being longer than $t$ (survival probability) is defined by the integral of the function $\rho_1(Q,t)$
\begin{equation}
S_{\mbox {\tiny ON}}(t)=\int\limits_{-\infty}^\infty \rho_1(Q,t)\,dQ
\label{Sur}
\end{equation}
The ON time distribution function is expressed as a derivative
\begin{equation}
p_{\mbox {\tiny ON}}(t)=-\frac{d}{dt} S_{\mbox {\tiny ON}}(t)
\label{pON}
\end{equation}
The analytical expression for the Laplace image of the ON time distribution function
$$ \tilde p_{\mbox {\tiny ON}}(s)=\int\limits_0^{\infty} p_{\mbox {\tiny ON}}(t) e^{-st} \,dt$$
was found by Tang and Marcus \cite{TangJCP05,TangPRL05} (derivation details are given in Appendix A):
\begin{equation}
\tilde p_{\mbox {\tiny ON}}(s)=\frac {W g_1(s)}{1+W g_1(s)}
\label{pONs}
\end{equation}
where
\begin{equation}
W=\frac {\sqrt {2 \pi} V^2}{\hbar \sqrt{E_rkT}}
\label{W}
\end{equation}
Function $g_1(s)$ can be expressed as an integral
\begin{equation}
g_1(s)=\int\limits_0^\infty \frac{\exp\left[-st-\frac {x_c^2}2\tanh\left({\frac{t}{2\tau_1}}\right)\right] }{\sqrt{2\pi\left(1-e^{-2t/\tau_1}\right)}}\,dt
\label{gs}
\end{equation}
where $\tau_1$ is the relaxation time in the the neutral state
\begin{equation}
\tau_1=\frac {2E_rkT}{D_1}
\label{tau1}
\end{equation}
and $x_c$ is the dimensionless crossing point coordinate
\begin{equation}
x_c=\frac {E_r+\Delta G} {\sqrt {2E_rkT}}
\label{xc}
\end{equation}
At a short time limit $t\ll \tau_1$ Tang and Marcus \cite{TangJCP05,TangPRL05}
presented the following approximation for the ON time distribution (see Appendix B):
\begin{equation}
p_{\mbox {\tiny ON}}(t)=\frac{\exp(-\Gamma_1 t)}{\sqrt{\pi t_c t}} \left[1-\sqrt{\frac{\pi t}{t_c}} \exp\left(\frac t {t_c} \right)\mbox{erfc}\left(\sqrt{\frac t {t_c}}\right) \right]
\label{pONshort}
\end{equation}
where
\begin{equation}
\Gamma_1=\frac {x_c^2}{4\tau_1}
\label{Gamma1}
\end{equation}
and $t_c$ is the critical time
\begin{equation}
t_c=\frac 4 {W^2 \tau_1}
\label{tc}
\end{equation}
When $t$ is much shorter than the critical time Eq.(\ref{pONshort}) can be approximated as
\begin{equation}
p_{\mbox {\tiny ON}}(t)\approx \frac 1 {\sqrt{\pi t_c}} t^{-1/2} , \quad t\ll t_c
\label{Ponshort}
\end{equation}
when for longer times
\begin{equation}
p_{\mbox {\tiny ON}}(t)\approx \frac 1 2 \sqrt{\frac {t_c} \pi} t^{-3/2}\exp(-\Gamma_1 t), \quad t_c\ll t \ll \tau_1
\label{Ponlong}
\end{equation}
The equation (\ref{Ponlong}) reproduces the experimentally observed truncated power-law dependence Eq. (\ref{tmexp}).
This dependence has to correspond to the power spectral density of the emission intensity $S(f)\sim f^{-3/2}$.
The experimentally observed transition of the power spectral density dependence
to $f^{-2}$ at large frequencies \cite{PeltonPNAS07} was explained by the changing of the ON time distribution
function behavior from (\ref{Ponshort}) to (\ref{Ponlong}) at times $t\sim t_c$.
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig2.pdf}
\caption{The ON time distribution function within the DCET model (thick red line), the first interval power law
(black dashed line), the second interval power law (black dashed-dotted line), the Tang-Marcus approximation Eq.(\ref{pONshort}) (thin black line) and the
long-time asymptotic Eq.(\ref{Pexp}) (red dashed line). Vertical dotted lines represent borders between characteristic intervals at $t_c$, $1/\Gamma_1$ and $\tau$.
The parameters of the model are $\tau_1=100\,s$, $\Gamma_1=0.1\,s^{-1}$, $t_c=10^{-3}\,s$.}
\label{fig:f1}
\end{figure*}
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig3.pdf}
\caption{The coordinate probability distribution function $\rho_1(x,t=10^{-4}\,s)\times 10^{-4}$ (black line),
$\rho_1(x,t=1\,s)\times 0.25$ (red line), $\rho_1(x,t=10.21\,s)$ (green line), $\rho_1(x,t=10^{5}\,s)$ (blue line).
The parameters of the model are $\tau_1=100\,s$, $\Gamma_1=0.1\,s^{-1}$, $t_c=10^{-3}\,s$. $x$ is the dimensionless coordinate
$x=(Q+E_r)/\sqrt {2E_rkT}$}
\label{fig:f2}
\end{figure*}
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig4.pdf}
\caption{The time dependence of the probability of finding the QD in the neutral (bright) state for the initial condition (\ref{CondON})
(red line) and the initial condition (\ref{CondOFF}) (blue line).
The parameters of the model are $\tau_1=100\,s$, $\Gamma_1=0.1\,s^{-1}$, $t_c=10^{-3}\,s$, $\tau_2=10^4\,s$, $\Gamma_2=10^{-3}\,s^{-1}$}
\label{fig:f3}
\end{figure*}
The problem is that for longer times $t \gg \tau_1$ the approximate formula (\ref{pONshort}) is not applicable.
It can be shown (see Appendix C) that at a very long time scale the ON time distribution shows
slow exponential decay \cite{TangPRL05}:
\begin{equation}
p_{\mbox {\tiny ON}}(t)\approx p_1 \exp(-k_1t) , \quad \tau_1 \ll t
\label{Pexp}
\end{equation}
were $k_1$ is the decay rate
\begin{equation}
k_1=\frac {W}{\sqrt{2\pi}(1+WB)}\exp\left(-\frac {x_c^2}2\right)
\label{k1}
\end{equation}
$p_1$ is the amplitude
\begin{equation}
p_1=\frac {k_1} {1+WB}
\label{p1}
\end{equation}
and
$$B=\int\limits_0^\infty \left[\frac{\exp\left(-\frac {x_c^2} 2 \tanh\left({\frac{t}{2\tau_1}}\right)\right) }
{\sqrt{2\pi\left(1-e^{-2t/\tau_1}\right)}}-\frac{\exp\left(-\frac {x_c^2}2\right)}{\sqrt{2\pi}}\right]\,dt$$
The last integral can be expressed in terms of a generalized hypergeometric function $_2F_2$ \cite{ZharikovJCP92}:
\begin{equation}
B=\frac{\tau_1}{\sqrt{2\pi}} \exp\left(-\frac {x_c^2}2\right)\left[
\ln 2+x_c^2\,{_2F_2}\left(\left.\begin{array}{cc}1&1\\\frac{3}{2}&2\end{array}
\right|\frac{x_c^2}{2}\right)\right]
\label{B}
\end{equation}
The simpler analytical expressions of $B$ can be found in the limiting cases \cite{ZharikovJCP92}:
\begin{equation}
B\approx \left\{ \begin{array} {ll}\tau_1 {\ln 2}/\sqrt {2 \pi},& \quad |x_c|\ll 1\\
\tau_1/{|x_c|} , & \quad |x_c|\gg 1\end{array} \right.
\label{Bapprox}
\end{equation}
Equation (\ref{k1}) can be rewritten as
\begin{equation}
k_1=\frac {W}{\sqrt{2\pi}(1+WB)}\exp\left(-\frac{(E_r+\Delta G)^2}{4E_rkT}\right)
\label{k1M}
\end{equation}
This formula is well-known in electron transfer theory \cite{BarzykinACP02}.
It describes the quasi-stationary rate of the electron transfer in the absence of back transitions.
The argument in the exponent reproduces the famous Marcus' Free Energy Gap law.
For low coupling values the rate Eq.(\ref{k1M}) is proportional to $V^2$ (the Golden Rule result):
$$ k_1= \frac {V^2}{\hbar \sqrt{E_rkT}}\exp\left(-\frac{(E_r+\Delta G)^2}{4E_rkT}\right)$$
At high coupling values the rate is limited by the diffusion transport to the crossing point and so becomes independent of $V$.
For the activated process $(E_r+\Delta G)^2\gg 4E_rkT$ from Eqs.(\ref{k1M}) and (\ref{Bapprox}) we get:
$$k_1=\frac {|E_r+\Delta G|}{\tau_1 \sqrt{4\pi E_rkT} }\exp\left(-\frac{(E_r+\Delta G)^2}{4E_rkT}\right)$$
The maximum rate is reached in the activationless case $(E_r+\Delta G)^2\ll 4E_rkT$
$$ k_1= \frac {1}{\tau_1 \ln 2}$$
As we can see the rate $k_1$ is always less than $1/\tau_1$.
The OFF time distribution shows a similar behaviour:
$$p_{\mbox {\tiny OFF}}(t)\approx \frac 1 {\sqrt{\pi t_2}} t^{-1/2}, \quad t\ll t_2 $$
$$p_{\mbox {\tiny OFF}}(t)\approx \frac 1 2 \sqrt{\frac {t_2} \pi} t^{-3/2}\exp(-\Gamma_2 t), \quad t_2\ll t \ll \tau $$
$$p_{\mbox {\tiny OFF}}(t)\approx p_2 \exp(-k_2t), \quad \tau_2 \ll t$$
where
$$\Gamma_2=\frac {x_2^2}{4\tau_2},\quad t_2=\frac 4 {W^2 \tau_2}$$
$$k_2=\frac {W}{1+WB_2}\exp\left(-\frac {x_2^2}2\right), \quad p_2=\frac {k_2} {\sqrt{2\pi}(1+WB_2)}$$
and
\begin{equation}
B_2=\frac{\tau_2}{\sqrt{2\pi}}\exp\left(-\frac {x_2^2}2\right)\left[
\ln 2+x_2^2\,{_2F_2}\left(\left.\begin{array}{cc}1&1\\\frac{3}{2}&2\end{array}
\right|\frac{x_2^2}{2}\right)\right]
\end{equation}
According to Eq.(\ref{pONshort}) and Eq. (\ref{Pexp}) there are four characteristic time intervals of the $p_{\mbox {\tiny ON}}(t)$ behavior:\\
{\bf Interval I:} Power-law with $1/2$ exponent at $t\ll t_c$;\\
{\bf Interval II:} Power-law with $3/2$ exponent at $t_c\ll t \ll 1/\Gamma_1$;\\
{\bf Interval III:} Exponential decay at $1/\Gamma_1 \ll t \ll \tau_1$;\\
{\bf Interval IV:} Long time exponential decay $\tau_1 \ll t$.\\
Note that Interval III can only exist if
\begin{equation}
\Gamma_1\tau_1\gg 1
\label{Gtau}
\end{equation}
We performed numerical simulations of Eq.(\ref{EqZi1}) using the SSDP program \cite{KrissinelJCC97}.
The results of the simulations for the parameters $\tau_1=100\,s$, $\Gamma_1=0.1\,s^{-1}$, $t_c=10^{-3}\,s$ are presented in Fig. \ref{fig:f1}.
The parameters are very close to the ones used in Ref.\cite{TangJCP05} for fitting the experimental data.
The model parameters can be restored using Eqs.(\ref{Gamma1}) and (\ref{W}):
$$x_c=\sqrt{4\Gamma_1\tau_1}\approx 6.32,\quad W=\sqrt{\frac 4 {\tau_1 t_c}}\approx 6.32\, s^{-1}$$
The condition $x_c\gg 1$ following from (\ref{Gtau})is satisfied.
Using Eq.(\ref{Bapprox}) we get
$$BW=\frac 1 {\sqrt{\Gamma_1 t_c}}=100$$
An expression for $k_1$ follows from Eq.(\ref{k1})
$$k_1= \sqrt{\frac {2 \Gamma_1} {\pi\tau_1}} \frac { \exp(-2 \Gamma_1 \tau_1)} {1+\sqrt{\Gamma_1 t_c}}\approx 5.15\times 10^{-11}\, s^{-1} $$
All four characteristic intervals of the ON time distribution dynamics are clearly seen on Fig. \ref{fig:f1}.
The value $ p_{\mbox {\tiny ON}} (t) $ is very small at $t \gg \tau_1$ (interval IV), however
the probability for the ON state to survive after $\tau_1$ time is quite significant.
$$S_1= S_{\mbox {\tiny ON}}(t \gg \tau_1) \approx \int\limits_{0}^\infty p_1 \exp(-k_1 t)\,dt$$
From (\ref{p1}) we get
$$S_1\approx \frac 1 {1+BW}=\frac {\sqrt{\Gamma_1 t_c}} {1+ \sqrt{\Gamma_1 t_c}} \approx 0.01$$
That is why the averaged ON time is extremely long:
$$\langle t_{\mbox {\tiny ON}}\rangle=\int\limits_{0}^\infty t p_{\mbox {\tiny ON}}(t)\,dt \approx \int\limits_{0}^\infty t p_1\exp(-k_1 t)\,dt$$
and after integration:
$$\langle t_{\mbox {\tiny ON}} \rangle\approx \frac {k_1^{-1}} {1+BW}=\sqrt{\frac{\tau_1 t_c}{2}}\exp(2\Gamma_1 \tau_1)\approx 1.08\times 10^8 s$$
The coordinate probability distribution function $\rho_1(Q,t)$
within each interval is shown on Fig. \ref{fig:f2}.
At a short time (Interval I) the distribution has one narrow maximum, its width increases with time
$\Delta Q = \sqrt{2 D_1t}$. The distribution function value at the crossing point
$\rho_1(Q_c,t)$ decays as $\sim t^{-1/2}$ and it follows the same power law form
of the ON time distribution.
At longer times (Interval II) the delta-functional sink burns a hole in
the distribution function, and it shows two maxima.
The distribution starts shifting towards the potential minimum within Interval III.
That shifting generates an exponential decreasing of the $\rho_1(Q_c,t)$ and
as a result the exponential decay of the ON time distribution function.
At times longer than $\tau_1$ (Interval IV) the function $\rho_1(Q,t)$
reaches the quasistationary distribution at the bottom of the parabolic potential
$$\rho_1(Q,t) \approx \frac{S_1}{\sqrt{4\pi E_rkT}} \exp\left(-\frac {(Q+E_r)^2}{4E_rkT}\right)\exp(-k_1t)$$
As such, the transition to the OFF state can only occur at the $Q_c$ crossing point, which requires thermal activation.
This explains why the decay of the ON time distribution is so gradual within Interval IV.
As seen from the analytical analysis and numerical simulations the DCET model predicts the appearance of extremely long ON time periods
in a single QD emission trace. As seen on Fig. \ref{fig:f1} such a period could last years, which is much longer than the duration of a typical experiment.
The probability of such a long duration of a single ON time blinking event $S_1$ is found to be in order of 1\%.
Thus the QD can become permanently bright after about one hundred blinking cycles with a high probability.
All the predictions made about the ON time distribution can be applied for the OFF distribution as well.
In most experiments the OFF time distribution truncation time of the single QD emission trace is too
long to be detected. The only exceptions are the observations made on similar nanoobjects, namely nanorods
\cite{DrndicNL08} where the value $1/\Gamma_2\sim 2500\,s$ was found.
Let us set $\Gamma_2= 10^{-3}\,s^{-1}$ and $\tau_2=10^4\,s$.
The corresponding rate for long time decay is $k_2\approx 5.15\times 10^{-13}\, s^{-1}$
The probability of an extremely long OFF time period is
$$S_2= \frac {\sqrt{\Gamma_2 t_2}} {1+ \sqrt{\Gamma_2 t_2}} \approx 10^{-4}$$
This means that after about ten thousand blinking cycles the QD should
become permanently bright or permanently dark.
This prediction significantly differs from the behavior of single quantum dots observed in numerous experiments.
The fact that $S_2$ is much smaller than $S_1$ ($S_2/S_1 \approx 10^{-2}$) suggests
that the most of the QDs should became permanently bright.
In order to verify that statement we used the SSDP program \cite{KrissinelJCC97} for numerical simulations
of the Eqs.(\ref{EqZ1}-\ref{EqZ2}) with two types of initial conditions:
at the beginning of the ON time period (delta-functional distribution in the neutral state)
\begin{equation}
\varrho_1(Q,0)=\delta(Q-Q_c);\quad \varrho_2(Q,0)=0
\label{CondON}
\end{equation}
and at the beginning of the OFF time period
\begin{equation}
\varrho_1(Q,0)=0;\quad \varrho_2(Q,0)=\delta(Q-Q_c)
\label{CondOFF}
\end{equation}
As shown in Fig. \ref{fig:f3}, the probability of finding the system in the ON state
$$ P_1 (t) = \int\limits_{-\infty}^\infty \varrho_1 (Q, t) \, dQ $$
becomes very close to unity at times greater than 100 seconds for both cases.
\section{Extended DCET model}
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig5.pdf}
\caption{The ON time distribution function (thick red line) and the OFF time distribution function (thick blue line) within the Extended DCET model,
the first interval power law (black dashed line), the second interval power law (black dashed-dotted line), the exponential decay of the ON time distribution Eq.(\ref{pONshort}) (dashed red line) and the OFF time distribution
long-time asymptotic Eq.(\ref{Pexp}) (blue dashed line).
The parameters of the model are $\tau_1=\tau_2=10^4\,s$, $\Gamma_1=\Gamma_2=10^{-3}\,s^{-1}$, $t_c=t_2=0.1\,s$, $K_{L}=10^{-1}\,s^{-1}$}
\label{fig:f4}
\end{figure*}
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig6.pdf}
\caption{The time dependence of the probability of finding the QD in the neutral (bright) state for the initial condition (\ref{CondON})
(red line) and the initial condition (\ref{CondOFF}) (blue line) within the Extended DCET model.
The parameters of the model are $\tau_1=\tau_2=10^4\,s$, $\Gamma_1=\Gamma_2=10^{-3}\,s^{-1}$, $t_c=t_2=0.1\,s$, $k_{L}=10^{-1}\,s^{-1}$}
\label{fig:f5}
\end{figure*}
The extended DCET model of Zhu and Marcus \cite{ZhuPCCP2014} includes the
equations describing the evolution of the probability density
of the ground state $\varrho_g(Q,t)$, the excited state $\varrho_e(Q,t)$, the biexciton state $\varrho_b(Q,t)$, the
charged (dark) state $\varrho_d(Q,t)$, and the excited dark state $\varrho_{d^\ast}(Q,t)$:
\begin{equation}
\frac{\partial}{\partial t}\varrho_g(Q,t)=k_{eg}\varrho_e(Q,t)-I_{ge}\varrho_g(Q,t)
\label{Zhu1}
\end{equation}
$$
\frac{\partial}{\partial t}\varrho_e(Q,t)=I_{ge}\varrho_g(Q,t)+L_e\varrho_e(Q,t)+k_{be}\varrho_b(Q,t)$$
\begin{equation}
-(k_{eg}+I_{eb})\varrho_e(Q,t)-k_{ed}\delta(Q-Q_c)\varrho_e(Q,t)
\label{Zhu2}
\end{equation}
\begin{equation}
\frac{\partial}{\partial t}\varrho_b(Q,t)=I_{eb}\varrho_e(Q,t)+L_b\varrho_b(Q,t)-(k_{be}+k_{bd'})\varrho_b(Q,t)
\label{Zhu3}
\end{equation}
\begin{equation}
\frac{\partial}{\partial t}\varrho_d(Q,t)=k_{d^\ast d}\varrho_{d^\ast}(Q,t)-I_{dd^\ast}\varrho_d(Q,t)
\label{Zhu4}
\end{equation}
$$ \frac{\partial}{\partial t}\varrho_{d^\ast}(Q,t)=L_{d^\ast}\varrho_{d^\ast}(Q,t)+I_{dd^\ast}\varrho_d(Q,t)$$
\begin{equation}
-k_{d^\ast d}\varrho_{d^\ast}(Q,t)-k_{d^\ast e}\delta(Q-Q_c)\varrho_{d^\ast}(Q,t)
\label{Zhu5}
\end{equation}
where $L_e$, $L_b$ and $L_{d^\ast}$ are diffusion operators
$$L_e= D_e \frac{\partial}{\partial Q} \left(\frac{\partial}{\partial Q}+ \frac{U_e'(Q)}{kT} \right)$$
$$L_b= D_b \frac{\partial}{\partial Q} \left(\frac{\partial}{\partial Q}+ \frac{U_b'(Q)}{kT} \right)$$
$$L_{d^\ast}= D_{d^\ast} \frac{\partial}{\partial Q} \left(\frac{\partial}{\partial Q}+ \frac{U_{d^\ast}'(Q)}{kT} \right)$$
$D_e$, $D_b$ $D_{d^\ast}$ are the diffusion coefficients, $U_e(Q)$, $U_b(Q)$ and $U_{d^\ast}(Q)$ are the potential surfaces of the excited state, the biexciton state and the dark excited state, respectively.
$ I_{ge}$, $I_{eb}$, $I_{dd^\ast}$, $k_{eg}$, $k_{be}$, $k_{bd'}$, $k_{d^\ast d}$, $k_{d^\ast e}$, and
$k_{ed}$ are the rate constants.
The equation for the probability density of the higher energy dark state has to be added to the equation system (\ref{Zhu1}-\ref{Zhu5}):
\begin{equation}
\frac{\partial}{\partial t}\varrho_{d'}(Q,t)=k_{bd'}\varrho_{b}(Q,t)-k_{d'd}\varrho_{d^\ast}(Q,t)
\label{Zhu6}
\end{equation}
As stated by Zhu and Marcus \cite{ZhuPCCP2014} quasiequilibrium is established between the ground, the excited state and the biexciton state.
We can see from Eq. (\ref{Zhu6}) that a quasistationary distribution of the the higher energy dark state is also determined by $\varrho_e (Q,t)$ and
it can also can be considered a part of the quasiequilibrium. As such we can introduce the population of the integrated ON state
\begin{equation}
\varrho_1 (Q,t)=\varrho_g (Q,t)+\varrho_e (Q,t)+\varrho_b (Q,t)+\varrho_{d'} (Q,t)
\label{Zhu7}
\end{equation}
Similarly, there is a quasiequilibrium between the dark and the excited dark states
and the OFF state population can also be introduced
\begin{equation}
\varrho_2 (Q,t)=\varrho_d (Q,t)+\varrho_{d^\ast} (Q,t)
\label{Zhu8}
\end{equation}
The following kinetic equations for the functions $\varrho_1(Q,t)$ and $\varrho_2 (Q,t)$ were obtained from Eqs. (\ref{Zhu1}-\ref{Zhu6})
(see Appendix D):
$$
\frac{\partial}{\partial t}\varrho_1(Q,t)=L_1\varrho_1(Q,t)-k_{L}\varrho_1(Q,t)$$
\begin{equation}
-W_1\delta(Q-Q_c)\varrho_1(Q,t) +W_2\delta(Q-Q_c)\varrho_2(Q,t)
\label{rhoI}
\end{equation}
$$
\frac{\partial}{\partial t}\varrho_2(Q,t)=L_2\varrho_1(Q,t)-W_2\delta(Q-Q_c)\varrho_2(Q,t)$$
\begin{equation}
+W_1\delta(Q-Q_c)\varrho_1(Q,t) +k_{LD}\varrho_1(Q,t)
\label{rhoII}
\end{equation}
where $L_1$ and $L_2$ are effective diffusion operators:
$$ L_1=C_1\left(L_e+\frac {I_{eb}} {k_{be}}L_b\right); \quad L_2=C_2 L_d$$
$W_1$, $W_2$ and $k_{L}$ are effective rates:
$$W_1=C_1 k_{ed}; \quad W_2= C_2 k_{d^\ast e}; \quad k_{L} =C_1 k_{bd'} \frac {I_{eb}} {k_{be}}$$
and $C_1$ and $C_2$ are the coefficients:
$$ C_1=\left( 1+\frac {k_{eg}} {I_{ge}} +\frac {I_{eb}} {k_{be}} +\frac {k_{bd'}} {k_{d'd}} \frac {I_{eb}} {k_{be}}\right)^{-1};\quad C_2=\left(1+\frac {k_{d^\ast d}} {I_{ge}}\right)^{-1}$$
We have to note that the equations derived by Zhu and Marcus ( Eqs.(11-12) in Ref.\cite{ZhuPCCP2014}) using the same procedure
are different from Eqs. (\ref{rhoI}-\ref{rhoII}).
The last term in Eq.(\ref{rhoI}) was omitted in Eq.(11) in Ref.\cite{ZhuPCCP2014}
and the two last terms in Eq.(\ref{rhoII}) were omitted in Eq.(12) in Ref.\cite{ZhuPCCP2014}.
It can be seen that because of the absence of these terms, Eqs. (11-12) of Zhu and Marcus
\cite{ZhuPCCP2014} do not preserve the total probability.
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig7.pdf}
\caption{The schematic picture of the Frantsuzov and Marcus model.
The potential surface is represented by a line colored red in the bright region ($Q>\delta$) and blue in the dark region ($Q <0$).
The black line represents the PLQY dependence on the coordinate.}
\label{fig:Fran}
\end{figure*}
The ON time and OFF time distribution functions in the Extended DCET model (\ref{rhoI}-\ref{rhoII}) can be found by solving the following equations:
\begin{equation}
\frac{\partial}{\partial t}\rho_1(Q,t)=L_1\rho_1(Q,t) -W_1\delta(Q-Q_c)\rho_1(Q,t) -k_{L}\rho_1(Q,t)
\label{rhoI1}
\end{equation}
\begin{equation}
\frac{\partial}{\partial t}\rho_2(Q,t)=L_2\rho_2(Q,t) -W_2\delta(Q-Q_c)\rho_2(Q,t)
\label{rhoII1}
\end{equation}
Transitions from the dark state to the bright state occur only at the point $Q_c$, thus
the initial distribution for the Eq. (\ref{rhoI1}) is a delta-function:
$$\rho_1(Q,t)=\delta(Q-Q_c)$$
in contrast transitions from a bright state to a dark state can occur not only at the crossing point
and the initial condition for the Eq. (\ref{rhoII1}) has the following form:
$$\rho_2(Q,t)=\int\limits_0^\infty (W_1\delta(Q-Q_c) +k_{L})\rho_1(Q,t)\,dt$$
The Eq.(\ref{rhoI1}) has an additional term $-k_{L}\rho_1(Q,t)$ in comparison to Eq.(\ref{EqZi1})
which leads to an exponential cutoff of the survival probability (\ref{Sur}) time dependence
$$S_{\mbox {\tiny ON}}(t)=S^0_{\mbox {\tiny ON}}(t)\exp(-k_{L}t)$$
where $S^0_{\mbox {\tiny ON}}(t)$ is the survival probability obtained from Eq. (\ref{rhoI1}) at $k_{L}=0$.
As a result the ON time distribution function in the Extended DCET model has an exponential cutoff.
$$p_{\mbox {\tiny ON}}(t)\sim \exp(-k_{L}t), \quad t \gg 1/k_{L} $$
The Eq.(\ref{rhoII1}) is equivalent to Eq.(\ref{EqZi1}).
The difference in the initial distributions leads to the deviation of the OFF time distribution
in the Extended DCET model in comparison with the original DCET model at times smaller than $\tau_2$.
The long time exponential asymptotic behavior, however, is the same
$$p_{\mbox {\tiny OFF}}(t)\sim \exp(-k_{2}t), \quad t \gg \tau_2 $$
These theoretical predictions are confirmed by numerical simulations (see Fig. \ref{fig:f4}) performed for the case of the symmetric system $Q_c=0$, $W_1=W_2$. The rest of the parameters are $\tau_1=\tau_2=10^4\,s$, $\Gamma_1=\Gamma_2=10^{-3}\,s^{-1}$, $t_c=t_2=0.1\,s$, $K_{L}=10^{-1}\,s^{-1}$.
It can be concluded that the presence of a second ionization channel resolves the problem with very long ON times, but
not with very long OFF times. As a result, most of the QDs in the Extended DCET model have to become permanently dark
as confirmed by numerical simulations (see Fig. \ref{fig:f5}).
That prediction also significantly differs from the experimentally observed behavior of single quantum dots.
\section{Frantsuzov and Marcus model}
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig8.pdf}
\caption{The normalized ON time (red thick line) and OFF time (blue thick line) distributions obtained by numerical simulations in the Frantsuzov and Marcus model,
the $t^{-3/2}$ dependence (thin black line), $\exp(-t/T_{\mbox {\tiny ON}} )$ (red dashed line), and $\exp(-t/T_{\mbox {\tiny OFF}} )$ (blue dashed line).
The parameters of the model are $T_{\mbox {\tiny ON}}=10$ s, $T_{\mbox {\tiny OFF}}=10^3$ s, $\delta=10^{-3}$, $\tau_{m}=10^{-4}$ s}
\label{fig:f6}
\end{figure*}
\begin{figure*}[ht]
\includegraphics[width=7 in]{Fig9.pdf}
\caption{Power spectral density of the single QD fluorescence emission quantum yield (thick line) in the Frantsuzov and Marcus model,
the $f^{-3/2}$ dependence (thin red line), and the $f^{-2}$ dependence (thin blue line).
Parameters of the model are $T_{\mbox {\tiny ON}}=T_{\mbox {\tiny OFF}}=10^3$ s, $\delta=10^{-3}$.}
\label{fig:f7}
\end{figure*}
The Frantsuzov and Marcus model \cite{FrantsuzovPRB05} is based on the fluctuating rate mechanism,
thus it does not consider transitions between neutral and charged states.
Fluctuations of the emission intensity in the model are caused by variations
of the PLQY (\ref{Y}).
The nonradiative recombination rate $k_n$ depends on the reaction coordinate $Q$
which is performing diffusive motion.
Within the generalized formulation of the model
the probability distribution function $\rho(Q,t)$ satisfies the equation
\begin{equation}
\frac{\partial}{\partial t}\varrho(Q,t)= \frac{\partial}{\partial Q} D(Q)
\left(\frac{\partial}{\partial Q}+ Q\right) \varrho(Q,t)
\label{rhoQ}
\end{equation}
where $D(Q)$ is the coordinate dependent diffusion coefficient.
To generate fast transitions from high to low emission intensity and back, the function $Y(Q)$ must grow dramatically
from a minimal value to a maximum one on a tiny interval of $\delta$ close to the origin (see Fig. \ref{fig:Fran}).
Thus, the QD is bright when $Q>\delta$, dark when $Q<0$, and has some intermediate florescence intensity within the interval of $\delta\ll 1$.
Taking into account that a molecular mechanism of the spectral diffusion is light induced \cite{BawendiJPCB99,MulvaneyPRB10},
the diffusion coefficient $D(Q)$ has to depend on the excitation intensity.
It also means that the diffusion could be much faster for a bright QD than for a dark one \cite{FrantsuzovPRB05}.
As such, we can choose:
\begin{equation}
D(Q)= \left\{ \begin{array}{ccl} 1 /{T_{\mbox {\tiny OFF}}}, & & Q<0\\
1/ {T_{\mbox {\tiny ON}}}, & \delta\le &Q
\end{array} \right.
\label{DQ}
\end{equation}
It was shown by Frantsuzov and Marcus \cite{FrantsuzovPRB05} that the normalized ON time and OFF time distributions obtained by the threshold procedure
have the following dependence (see Appendix E for details):
\begin{equation}
p(t)=\frac {\sqrt{\tau_{m}}} 2 t^{-3/2},\quad \tau_{m}\le t\ll T_0
\label{pshort}
\end{equation}
\begin{equation}
p(t)=\sqrt{\frac{2 \tau_{m}}{T_0^3}} \exp(-t/T_0),\quad T_0 \ll t
\label{plong}
\end{equation}
where $\tau_m$ is the minimum time interval of observation (bin time) and $T_0$ is equal to $T_{\mbox {\tiny ON}}$ and $T_{\mbox {\tiny OFF}}$ for the ON time and OFF time distribution, respectively.
That prediction is confirmed by the numerical simulations made using the SSDP program \cite{KrissinelJCC97} (see Fig. \ref{fig:f6}).
The power spectral density $S(f)$ of the single QD emission
at frequencies $f$ larger than $1/\tau_m$ could be obtained without binning procedure
by measuring the autocorrelation function \cite{SionnestAPL04,PeltonPNAS07}.
In order to calculate $S(f)$ within the model one needs to specify the $Y(Q)$ function
in the intermediate interval. Let's choose the simplest linear dependence:
\begin{equation}
Y(Q)= \left\{ \begin{array}{ccl} 0, \quad & &Q<0\\
Q/\delta, \quad & 0\le &Q < \delta\\
1,\quad & \delta \le &Q
\end{array} \right.
\label{YQ}
\end{equation}
The results of numerical calculations of the $S(f)$ in that case are presented in Fig. \ref{fig:f7}
(see Appendix F for the detailed calculation procedure).
The Figure clearly shows the transition from the $f^{-3/2}$ dependence to $f^{-2}$ at large frequencies
in accordance with the experiment of Pelton et al. \cite{PeltonPNAS07}.
\section{Discussion}
As a result of the above analytical and numerical studies it was found that two models of single QD blinking based on spectral diffusion, namely
the DCET model \cite{TangJCP05,TangPRL05} and the Extended DCET model \cite{ZhuPCCP2014} predict that after an initial blinking period, most of the QDs should become permanently bright or permanently dark. That prediction significantly differs from the behavior of single quantum dots observed in numerous experiments.
Another drawback of these models is the charging mechanism on which they are based.
Despite the fact that most of the theoretical models proposed in the literature are based on that mechanism \cite{KunoJCP01,BawendiPRB01,OrritPRB02,BarkaiJCP04,TangPRL05,OsadkoJPCC13,ZhuPCCP2014},
there is a number of sufficient experimental evidence indicating that the charging mechanism fails in explaining the QD blinking phenomenon.
In several experiments, the emission intensity of a single QD was observed below the charged state (trion) emission intensity
\cite{BawendiPRL10,OronPRL10,OronACSNano13,KlimovNL17}.
Another very important set of experiments showed that the existence of the distinct ON and OFF states is an illusion; there is a nearly continuous set of emission intensities \cite{MewsPRL02,BawendiJPCB04,YangNL06,CichosJL11,BorczyskowskiACSNano14}.
Furthermore, it was also shown \cite{FrantsuzovPRL09,PeltonNL10,CichosJCP14} that the parameters $m$ and $T$ of the ON and OFF time distributions strongly depend on the threshold value.
The Frantsuzov and Marcus model \cite{FrantsuzovPRB05}, based on fluctuating rate mechanism, reproduces the key properties of the QD blinking phenomenon.
Nonetheless there are a number of the experimental observations which are not explained by the model:\\
1. The exponent value $m$ of the ON and OFF time distribution functions is reported in the range from 1.2 to 2.0 \cite{FrantsuzovNaturePhys08},
and it strongly depends on the threshold value \cite{FrantsuzovPRL09,PeltonNL10,CichosJCP14}. Meanwhile, in the model, $m$ is always equal to $3/2$ regardless of the threshold.\\
2. The exponent $r$ of the emission power spectral density is found to be in the range from 0.7 to 1.2 \cite{SionnestAPL04,PeltonPNAS07,FrantsuzovNL13}, when
the model predicts the exponent value of 3/2.\\
3. The long-term correlations between subsequent ON and OFF times \cite{StefaniNJP05,VolkanNL10}. There are no such correlations in the model.
A possible reason for this discrepancy is that the description of the spectral diffusion in the model does not fully correspond to its real properties.
It was shown that the squared frequency displacement of the single QD emission has an anomalous (sublinear) time dependence \cite{MulvaneyPRL10}.
Plakhotnik et al. \cite{MulvaneyPRL10} suggested an explanation of this behavior by introducing a number
of stochastic two-level systems (TLS) having a wide spectrum of flipping rates.
A similar idea was applied by Frantsuzov, Volkan-Kacso and Janko in the Multiple Recombination Center (MRC) model
of single QD blinking \cite{FrantsuzovPRL09}.
The MRC model, based on the fluctuating rate mechanism, also reproduces the key properties the single QD blinking.
But in addition it explains the power spectral density dependence close to $1/f$ \cite{FrantsuzovNL13},
the threshold dependence of the $m$ and $T$ values \cite{FrantsuzovPRL09},
and the long-term correlations between subsequent blinking times \cite{VolkanNL10}.
This suggests that the spectral diffusion and the fluctuations of the emission intensity of a single QD can be explained by an unified model,
which could become a generalization of the Frantsuzov and Marcus model.
In conclusion, we analytically and numerically considered three models of the single QD emission fluctuations (blinking) based on spectral diffusion. Only one of them, the Frantsuzov and Marcus model \cite{FrantsuzovPRB05}, reproduces the key properties of the phenomenon.
The DCET model \cite{TangJCP05,TangPRL05} and the Extended DCET model \cite{ZhuPCCP2014} predict that after an initial blinking period, most of the QDs should become permanently bright or permanently dark which is significantly different from the experimentally observed behavior.
\section*{Acknowledgement}
The authors are very grateful to Professor Rudolph Marcus for fruitful discussions.
The study was supported by the Russian Foundation for Basic Research, project 16-02-00713.
\section*{Appendix A: An analytical solution for the blinking time distribution within the DCET model}
Introducing a dimensionless coordinate $x$
$$x=\frac {Q+E_r} {\sqrt {2E_rkT}}$$
we can rewrite Eq. (\ref{EqZi1}) as
$$\frac{\partial}{\partial t}\rho_1(x,t)=\frac 1 {\tau_1} \frac{\partial}{\partial x}
\left(\frac{\partial}{\partial x}+ x \right) \rho_1(x,t) $$
\begin{equation}
-W \delta(x-x_c)\rho_{1}(x,t)
\label{Eqx}
\end{equation}
with the initial condition
$$\rho_1(x,0)=\delta(x-x_c)$$
where the relaxation time $\tau_1$ is given by Eq.(\ref{tau1}),
$x_c$ is the dimensionless crossing point coordinate Eq.(\ref{xc})
and $W$ is given by Eq.(\ref{W})
Applying Eq.(\ref{Eqx}), the ON time distribution function (\ref{pON}) can be expressed as
\begin{equation}
p_{\mbox {\tiny ON}}(t)=-\frac{d}{dt}\int_{-\infty}^{\infty} \rho_1(x,t)\,dx=W\rho_1(x_c,t)
\label{pONW}
\end{equation}
The Laplace image of the function $\rho_1(x,t)$
$$\tilde \rho_1(x,s)=\int_0^\infty \rho_1(x,t)e^{-st}\,dt$$
obeys the following equation
$$s\tilde\rho_1(x,s)-\delta(x-x_c)= $$
\begin{equation}
\frac 1 {\tau_1} \frac{\partial}{\partial x}
\left(\frac{\partial}{\partial x}+ x \right)\tilde \rho_1(x,s)-W \delta(x-x_c)\tilde\rho_{1}(x,s)
\label{EqLap}
\end{equation}
The Green's function of the differential operator in Eq.(\ref{Eqx})
satisfies the equation
\begin{equation}
\frac{\partial}{\partial t}G(x,x',t)= \frac 1 {\tau_1} \frac{\partial}{\partial x}
\left(\frac{\partial}{\partial x}+ x \right) G(x,x',t)
\label{GF}
\end{equation}
with the initial condition
$$ G(x,x',0)=\delta(x-x')$$
The Green's function and the Laplace image satisfies the equation
$$s\tilde G(x,x',s)-$$
\begin{equation}
\frac 1 {\tau_1} \frac{\partial}{\partial x}
\left(\frac{\partial}{\partial x}+ x \right) \tilde G(x,x',s)=\delta(x-x')
\label{GFL}
\end{equation}
Using Eq.(\ref{GFL}), Eq.(\ref{EqLap}) can be rewritten as
\begin{equation}
\tilde\rho_1(x,s)=\tilde G(x,x_c,s)-W\tilde G(x,x_c,s)\tilde\rho_1(x_c,s)
\label{rho1}
\end{equation}
From Eq.(\ref{rho1}) we can find $\tilde\rho_1(x_c,s)$
$$\tilde\rho_1(x_c,s)=\frac {\tilde G(x_c,x_c,s)}{1+W\tilde G(x_c,x_c,s)}$$
The Laplace image of the ON time distribution Eq.(\ref{pONW}) is given by
$$\tilde p_{\mbox {\tiny ON}}(s)=W\tilde \rho_1(x_c,s)$$
Substituting Eq.(\ref{EqLap}) we get
\begin{equation}
\tilde p_{\mbox {\tiny ON}}(s)=\frac {W \tilde G(x_c,x_c,s)}{1+W \tilde G(x_c,x_c,s)}
\label{pONss}
\end{equation}
Green's function (\ref{GF}) is well-known:
\begin{equation}
G(x,x',t)=\frac{1}{\sqrt{2\pi\left(1-e^{-2t/\tau_1}\right)}}
\exp\left[-\frac{\left(x-x' e^{-t/\tau_1}\right)^2}
{2\left(1-e^{-2t/\tau_1}\right)}\right]
\label{Gan}
\end{equation}
Introducing the function $g_1(s)$
\begin{equation}
g_1(s)=\tilde G(x_c,x_c,s)
\label{g1s}
\end{equation}
we can express Eq.(\ref{pONss}) in the form Eq.(\ref{pONs}).
\section*{Appendix B: The ON time distribution at short times within the DCET model}
At a short time limit $t\ll \tau_1$ one has to find the function $g_1(s)$ at $s \gg 1/\tau_1$.
Expanding the exponent's argument in the Eq. (\ref{gs}) we get
\begin{equation}
g_1(s)=\int_0^\infty \frac{\exp(-st-\Gamma_1 t) }{\sqrt{4\pi t/\tau_1}}\,dt=\frac 1 2 \sqrt{\frac {\tau_1} {s+\Gamma_1}}
\label{gt}
\end{equation}
where $\Gamma_1$ is given by Eq.(\ref{Gamma1})
Substituting Eq.(\ref{gt}) into Eq.(\ref{pONs}) gives
$$\tilde p_{\mbox {\tiny ON}}(s)= \frac 1 {1+ \sqrt{(s+\Gamma_1)t_c}}$$
and after the inverse Laplace transformation we get Eq. (\ref{pONshort}).
\section*{Appendix C: The ON time distribution at long times within the DCET model}
The approximate formula (\ref{pONshort}) works for short times only.
In order to see the behavior of the function $p_{\mbox {\tiny ON}}(t)$ at a long time limit $t \gg \tau_1$
one has to consider its Laplace image $\tilde p_{\mbox {\tiny ON}}(s)$ (\ref{gs}) at $s\to 0$.
If we expand the function $g_1(s)$ (\ref{g1s}) into a series on $s$:
\begin{equation}
g_1(s)\approx \frac 1 s A + B
\label{gexp}
\end{equation}
where
$$A=\lim_{t\to \infty} G(x_c,x_c,t)$$
and
$$B=\int_0^\infty \{G(x_c,x_c,t)-A\} \,dt$$
The Green's function (\ref{Gan}) approaches the stationary distribution at long times
$$\lim_{t\to \infty} G(x,x',t) =\frac{1}{\sqrt{2\pi}} \exp\left(-\frac {x^2}2\right)$$
Thus the constants $A$ and $B$ are
$$A=\frac{1}{\sqrt{2\pi}} \exp\left(-\frac {x_c^2}2\right)$$
$$B=\int_0^\infty \left[\frac{\exp\left(-\frac 1 2{x_c^2}\tanh\left({\frac{t}{2\tau_1}}\right)\right) }
{\sqrt{2\pi\left(1-e^{-2t/\tau_1}\right)}}-A\right]\,dt$$
Substituting Eq.(\ref{gexp}) into Eq. (\ref{pONs}) we get the following dependence of $\tilde p_{\mbox {\tiny ON}}(s)$ at small $s$
$$\tilde p_{\mbox {\tiny ON}}(s)\approx \frac {WB}{1+WB}+ \frac {p_l} {s+k}$$
which corresponds to the exponential behavior Eq. (\ref{Pexp})of the ON time distribution function at long times.
\section*{Appendix D: The derivation of the evolution equations within the Extended DCET model }
If $k_{eg}$ is much larger than all other rates in Eq. (\ref{Zhu2}) a quasiequlibrium value of exciton population is
established
\begin{equation}
\varrho_e (Q,t)\approx \frac {I_{ge}} {k_{eg}} \varrho_d (Q,t)
\label{Zhu9}
\end{equation}
Similarly if $k_{be}$ is much larger than all other rates in Eq.{\ref{Zhu3}:
\begin{equation}
\varrho_b (Q,t)\approx \frac {I_{eb}} {k_{be}} \varrho_e (Q,t)
\label{Zhu10}
\end{equation}
If $k_{d'd} \gg k_{bd'}$
\begin{equation}
\varrho_{d'} (Q,t)\approx\frac {k_{bd'}} {k_{d'd}} \varrho_b (Q,t)\approx \frac {k_{bd'}} {k_{d'd}} \frac {I_{eb}} {k_{be}} \varrho_e (Q,t)
\label{Zhu11}
\end{equation}
Substituting Eqs. (\ref{Zhu9}-\ref{Zhu11}) with the definition Eq. (\ref{Zhu7}) into Eqs. (\ref{Zhu1}-\ref{Zhu3}) we get
Eq.(\ref{rhoI}).
If $k_{d^\ast d}$ is much larger than all other rates in Eq. (\ref{Zhu5}) a quasiequlibrium value of the dark exciton population is
established
\begin{equation}
\varrho_{d^\ast} (Q,t)=\frac {I_{ge}} {k_{d^\ast d}} \varrho_d (Q,t)
\label{Zhu12}
\end{equation}
Substituting Eq.(\ref{Zhu12}) with the definition Eq.(\ref{Zhu8}) into Eqs.(\ref{Zhu4}-\ref{Zhu6}) we
obtain Eqs.(\ref{rhoII}).
\section*{Appendix E: The ON time and OFF time distributions within the Frantsuzov and Marcus model}
The survival probability of the ON time within the Frantsuzov and Marcus model can be found as an integral
\begin{equation}
S_{\mbox {\tiny ON}}(t)=\int\limits_0^\infty \rho(Q,t)\,dQ
\label{SurFr}
\end{equation}
where $\rho(Q,t)$ is a solution of the following equation
\begin{equation}
\frac{\partial}{\partial t}\rho(Q,t)= \frac 1 {T_{\mbox {\tiny ON}}} \frac{\partial}{\partial Q}
\left(\frac{\partial}{\partial Q}+ Q\right) \rho(Q,t)
\label{rhoon}
\end{equation}
with an absorbing boundary condition at the border (the first passage time problem)
\begin{equation}
\left.\rho(Q,t)\right|_{Q=0} = 0
\label{bound}
\end{equation}
The question of what to take as the initial distribution for the equation is not easily answered.
There is the minimal time $\tau_m$ (bin time) of the ON time period which can be observed.
In accordance with Eq.(\ref{rhoon}), if the ON time period is longer than $\tau_m$ then the coordinate $Q$ has reached values larger than
$\sqrt{\tau_m/T_{\mbox {\tiny ON}}}$.
We can take any distribution located at a distance less than $\sqrt{\tau_m/T_{\mbox {\tiny ON}}}$ from the origin as an initial one.
For the sake of simplicity, we can take the initial distribution in the form of a delta function
\begin{equation}
\rho(Q,0)=\delta(Q-\Delta)
\label{init}
\end{equation}
where
$$\delta \ll \Delta \ll \sqrt{\tau_m/T_{\mbox {\tiny ON}}}$$
The solution of Eqs.(\ref{rhoon}-\ref{init}) is well known
\begin{equation}
\rho(Q,t)=G(Q,\Delta,t)-G(-Q,\Delta,t)
\label{rhoF}
\end{equation}
where $G(x,x',t)$ is the Green's function of the Eq.(\ref{rhoon})
\begin{equation}
G(Q,Q',t)=\frac{
\exp\left\{-\frac{\left[Q-Q' \exp(-t/T_{\mbox {\tiny ON}})\right]^2}
{2\left(1-\exp(-2t/T_{\mbox {\tiny ON}})\right)}\right\}}
{\sqrt{2\pi\left(1-\exp(-2t/{T_{\mbox {\tiny ON}}})\right)}}
\label{GFR}
\end{equation}
Using Eq.(\ref{rhoF}) the survival probability (\ref{SurFr}) can be expressed as
$$ S_{\mbox {\tiny ON}}(t)=
\frac { \int\limits_{-b}^b
\exp\left[-\frac{Q^2}
{2\left(1-\exp(-2t/T_{\mbox {\tiny ON}})\right)}\right]\,dQ}
{\sqrt{2\pi\left(1-\exp(-2t/T_{\mbox {\tiny ON}})\right)}}$$
where $b=\Delta \exp(-t/T_{\mbox {\tiny ON}})$. At times $t>\tau_m$ the expression
can be rewritten as
$$S_{\mbox {\tiny ON}}(t)=\frac{2\Delta \exp(-t/T_{\mbox {\tiny ON}})}{\sqrt{2\pi\left(1-\exp(-2t/T_{\mbox {\tiny ON}})\right)}}$$
This expression has the following behavior in the limiting cases
\begin{equation}
S_{\mbox {\tiny ON}}(t)= \Delta \sqrt{\frac {T_{\mbox {\tiny ON}}}{\pi t}}, \quad \Delta^2 T_{\mbox {\tiny ON}} \ll t \ll T_{\mbox {\tiny ON}}
\label{Sshort}
\end{equation}
\begin{equation}
S_{\mbox {\tiny ON}}(t)= \Delta \sqrt{\frac 2 \pi}\exp(-t/T_{\mbox {\tiny ON}}), \quad T_{\mbox {\tiny ON}} \ll t
\label{Slong}
\end{equation}
In the experiment one can see that only the ON times are longer than $\tau_m$, which means that the ON time distribution should be normalized as follows
$$\int\limits_{\tau_m}^\infty p_{\mbox {\tiny ON}}(t)\,dt=1$$
The normalization procedure is equivalent to scaling of the function $S_{\mbox {\tiny ON}}$ so that the following equality for the normalized
survival probability is satisfied
\begin{equation}
\bar S_{\mbox {\tiny ON}}(\tau_m)=1
\label{Stau}
\end{equation}
Applying this normalization to Eqs. (\ref{Sshort}-\ref{Slong}) we get
$$
\bar S_{\mbox {\tiny ON}}(t)= \sqrt{\frac {\tau_m}{t}}, \quad \tau_m \le t \ll T_{\mbox {\tiny ON}}$$
$$
\bar S_{\mbox {\tiny ON}}(t)= \sqrt{\frac 2 {T_{\mbox {\tiny ON}} }}\exp(-t/T_{\mbox {\tiny ON}}), \quad T_{\mbox {\tiny ON}} \ll t $$
From Eq.(\ref{pON}) we obtain the ON time distribution function
$$
p_{\mbox {\tiny ON}}(t)= \frac 1 2 \sqrt{\tau_m}t^{-3/2}, \quad \tau_m \le t \ll T_{\mbox {\tiny ON}}$$
$$
p_{\mbox {\tiny ON}}(t)= \sqrt{\frac {2 \tau_m} {T_{\mbox {\tiny ON}}^3}} \exp(-t/T_{\mbox {\tiny ON}}), \quad T_{\mbox {\tiny ON}} \ll t$$
Similarly the expression for the OFF time distribution function can be obtained
$$
p_{\mbox {\tiny OFF}}(t)= \frac 1 2 \sqrt{\tau_m}t^{-3/2}, \quad \tau_m \le t \ll T_{\mbox {\tiny OFF}}$$
$$
p_{\mbox {\tiny ON}}(t)= \sqrt{\frac {2 \tau_m} {T_{\mbox {\tiny OFF}}^3}} \exp(-t/T_{\mbox {\tiny OFF}}), \quad T_{\mbox {\tiny OFF}} \ll t$$
These expression are equivalent to the Eqs. (\ref{pshort}-\ref{plong}).
\section*{Appendix F: The emission intensity autocorrelation function within the Frantsuzov and Marcus model}
The autocorrelation function of the emission intensity within the FRM is
$$C(t)=\left\langle Y\left(Q(t)\right)Y\left(Q(0)\right) \right\rangle$$
where averaging is performed over the ensemble of realizations of the random process $Q(t)$.
For the Frantsuzov and Marcus model the function $C(t)$ can be written as
\begin{equation}
C(t)=\int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty Y(Q) G(Q,Q',t) Y(Q')\varrho_0(Q')\,dQdQ'
\label{Ct}
\end{equation}
where $G(Q,Q',t)$ is the Green's function of Eq. (\ref{rhoQ}) and the stationary distribution $\varrho_0$ is
$$\varrho_0(Q)=\frac 1 {\sqrt{2\pi}} \exp(-\frac 1 2 Q^2)$$
Eq. (\ref{Ct}) can be rewritten as
$$C(t)=\int\limits_{-\infty}^\infty Y(Q) \varrho(Q,t)\,dQ$$
where $\varrho(Q,t)$ is the solution of Eq. (\ref{rhoQ})
with the initial condition
$$\varrho(Q,0)=Y(Q)\varrho_0(Q)$$
A numerical solution was obtained using the SSDP program \cite{KrissinelJCC97}.
The power spectral density was calculated using a cosine transform
$$S(f)=4\int\limits_0^\infty C(t)\cos(2 \pi f t)\,dt $$
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,601 |
\section{Introduction}
Due to the sheer size of modern datasets, many practical instances of large-scale optimization are now \emph{distributed}, in the sense that data and computation are split among several computing nodes, which collaborate to jointly optimize the global objective function.
This shift towards distribution induces new challenges, and many classic algorithms have been revisited to reduce distribution costs. These costs are usually measured in terms of the number of bits sent and received by the nodes (\emph{communication complexity}) or by the number of parallel iterations required for convergence (\emph{round complexity}).
In this paper, we focus on the communication (bit) complexity of the classic empirical risk minimization problem
\vspace{-1mm}
\begin{equation*}
\min_{x\in \mathbb{R}^d} f(x) := \frac{1}{n} \sum_{i=1}^n f_i(x),
\vspace{-1mm}
\end{equation*}
where the global $d$-dimensional cost function $f$ is formed as the average of smooth and strongly-convex local costs $f_i$, each owned by a different machine, indexed by $i=1,...,n$.
This problem has a rich history.
The seminal paper of \citet{4048825} considered the case $n = 2$, and provided a lower bound of $\Omega( d \log (d / \epsilon ))$ for quadratic functions, as well as an almost-matching upper bound for this case, within logarithmic factors. (Here, $d$ is the problem dimension and $\epsilon$ is the error-tolerance.)
The problem has concentrated significant attention, given the surge of interest in distributed optimization and machine learning, e.g.~\cite{niu2011hogwild, jaggi2014communication, alistarh2016qsgd, nguyen2018sgd, ben2019demystifying}. In particular, a series of papers~\cite{khirirat2018distributed, ye2018communication, magnusson2020maintaining, alistarh2020improved} continued to provide improved upper and lower bounds for the communication complexity of this problem, both for deterministic and randomized algorithms, as well as examining related distributed settings and problems~\cite{scaman2017optimal, jordan2018communication, vempala2020communication, mendler2020randomized, hendrikx2020statistically}.
The best known lower bound for solving the above problem for deterministic algorithms and general $d$ and $n$ is of
\vspace{-1mm}
\[
\Omega( n d \log ( d / \epsilon ) )
\vspace{-1mm}
\]
total communication bits, given recently by~\cite{alistarh2020improved}.
This lower bound can be asymptotically matched for quadratic functions by a quantized variant of gradient descent~\cite{magnusson2020maintaining, alistarh2020improved} using
\vspace{-1mm}
\[
\mathcal{O}( n d \kappa \log \kappa \log (\gamma d / \epsilon))
\vspace{-1mm}
\]
total bits, where $\kappa$ is the condition number of the problem and $\gamma$ is the smoothness bound of $f$.
An intriguing open question concerns the optimal dependency on the condition number for general objectives. While existing lower bounds show no such explicit dependency, all known algorithms have linear (or worse) dependency on $\kappa$. Resolving this problem is non-trivial, since one usually removes this dependency in the non-distributed case by leveraging curvature information in the form of preconditioning or full Newton steps.
However, existing distribution techniques are designed for \emph{gradient} quantization, and it is not at all clear for instance how using a preconditioning matrix would interact with the convergence properties of the algorithm, and in particular whether favourable convergence behaviour can be preserved at all following quantization.
\paragraph{Contribution.} In this paper, we resolve this question in the positive, and present communication-efficient variants of preconditioned gradient descent for generalized linear models (GLMs) and distributed Newton's method.
Specifically, given a \emph{small enough} error-tolerance $\epsilon$, a communication-efficient variant of preconditioned gradient descent for GLMs (QPGD-GLM) can find an $\epsilon$-minimizer of a $\gamma$-smooth function using a total number of bits
\begin{equation*}
B^{\idlow{QPGD-GLM}} = \mathcal{O} \left(n d \kappa_{\ell} \log (n \kappa_{\ell} \kappa(M)) \log (\gamma D/\epsilon)\right),
\end{equation*}
where $d$ is the dimension, $n$ is the number of nodes, $\kappa_\ell$ is the condition number of the loss function $\ell$ used to measure the distance of training data from the prediction, $\kappa(M)$ is the condition number of the averaged covariance matrix of the training data, and $D$ is a bound on the initial distance from the optimum. In practice, $\kappa_{\ell}$ is often much smaller than the condition number $\kappa$ of the problem, and is equal to $1$ in the case that $\ell$ is a quadratic.
This first result suggests that distributed methods need not have linear dependence on the condition number of the problem.
Our main technical result extends the approach to a distributed variant of Newton's method, showing that the same problem can be solved using
\begin{equation*}
B^{\idlow{Newton}} = \mathcal{O}\left(nd^2 \log \left( {d} \kappa \right) \log (\gamma \mu/\sigma \epsilon) \right) \textnormal{ total bits, }
\end{equation*}
under the assumption that the Hessian is $\sigma$-Lipschitz.
Viewed in conjunction with the above $\Omega(nd \log(d / \varepsilon))$ lower bound, these algorithms outline a new communication complexity trade-off between the dependency on the dimension of the problem $d$, and its condition number $\kappa$. Specifically, for ill-conditioned but low-dimensional problems, it may be advantageous to employ quantized Newton's method, whereas QPGD-GLM can be used in cases where the structure of the training data favors preconditioning. Further, our results suggest that there can be no general communication lower bound with linear dependence on the condition number of the problem.
Our results assume the classic coordinator / parameter server~\citep{PS} model of distributed computing, in which a distinguished node acts as a coordinator by gathering model updates from the nodes.
In this context, we introduce a few tools which should have broader applicability.
One is a lattice-based matrix quantization technique, which extends the state-of-the-art vector (gradient) quantization techniques to preconditioners.
This enables us to carefully trade off the communication compression achieved by the algorithm with the non-trivial error in the descent directions due to quantization.
Our main technical advance is in the context of quantized Newton's method, where we need to keep track of the concentration of quantized Hessians relative to the full-precision version. Further, our algorithms quantize directly the local descent directions obtained by multiplying the inverse of the quantized estimation of the preconditioner with the exact local gradient. This is a non-obvious choice, which turns out to be the correct way to deal with quantized preconditioned methods.
We validate our theoretical results on standard regression datasets, where we show that our techniques can provide an improvement of over $3 \times$ in terms of total communication complexity used by the algorithm, while maintaining convergence and solution quality.
\paragraph{Related Work.}
There has been a surge of interest in distributed optimization and machine learning. While a complete survey is beyond our scope, we mention the significant work on designing and analyzing communication-efficient versions of classic optimization algorithms, e.g.~\cite{jaggi2014communication, scaman2017optimal, jordan2018communication, khirirat2018distributed, nguyen2018sgd, alistarh2016qsgd, alistarh2018convergence, ye2018communication, NUQSGD, magnusson2020maintaining, ghadikolaei2020communication}, and the growing interest in communication and round complexity lower bounds, e.g.~\cite{NIPS2013_4902, NIPS2014_5386, NIPS2015_5731, vempala2020communication, alistarh2020improved}. In this context, our work is among the first to address the bit complexity of optimization methods which explicitly employ curvature information, and shows that such methods can indeed be made communication-efficient.
\citet{4048825} gave the first upper and lower bounds for the communication (bit) complexity of distributed convex optimization, considering the case of two nodes. Their algorithm is a variant of gradient descent which performs \emph{adaptive} quantization, in the sense that nodes adapt the number of bits they send and the quantization grid depending on the iteration.
Follow-up work, e.g.~\cite{khirirat2018distributed, alistarh2016qsgd} generalized their algorithm to an arbitrary number of nodes, and continued to improve complexity.
In this line, the work closest to ours is that of \citet{magnusson2020maintaining}, who introduce a family of adaptive gradient quantization schemes which can enable linear convergence in any norm for gradient-descent-type algorithms, in the same system setting considered in our work.
However, we emphasize that this work did \emph{not} consider preconditioning.
(\citet{alistarh2020improved} also focus on GD, but use different quantizers and a more refined analysis to obtain truly tight communication bounds for quadratics.)
Conceptually, the quantization techniques we introduce serve a similar purpose---to allow the convergence properties of the algorithm to be preserved, despite noisy directional information.
At the technical level, however, the schemes we describe and analyze are different, and arguably more complex. For instance, since only the gradient information is quantized,~\citep{magnusson2020maintaining} can use grid quantization adapted to gradient norms, whereas employ more complex quantization, as well as fine-grained bookkeeping with respect to the concentration of quantized matrices and descent directions.
There has been significant work on distributed approximate second-order methods with the different goal of minimizing the \emph{number of communication rounds} required for convergence.
One of the first such works is~\cite{shamir2014communication}, who considered the strongly convex case, and proposed a method called DANE, where each worker solves a subproblem using full gradients at each iteration, and the global iterate is the average of these sub-solutions.
Follow-up work~\cite{zhang2015disco, reddi2016aide, wang2018giant, zhang2020distributed} proposed improvements both in terms of generalizing the structure of the loss functions, but also in terms of convergence rates.
Recently,~\citet{hendrikx2020statistically} also proposed a round-efficient distributed preconditioned accelerated gradient method for our setting, where preconditioning is done by solving a local optimization problem over a subsampled dataset at
the server. Their convergence rate depends on the square root of the relative condition number between the global and local loss functions.
Concurrent work by~\citet{islamov2021distributed} considers the same problem of reducing the bit cost of distributed second-order optimization, and proposes a series of algorithms based on the novel idea of \emph{learning parameters of the Hessian at the optimum} in a communication-efficient manner.
The resulting algorithms allow for $\ell_2$-regularization, and can achieve local linear and superlinear rates, independent of the condition number, with \emph{linear} communication cost per round in the dimension $d$.
Relative to our setting, their results require two additional assumptions: The first is that, for linear communication cost, either the coordinator must have access to \emph{all the training data} at the beginning of the optimization process, or the data should be highly \emph{sparse}.
The second assumption is on the structure of the individual loss functions, which are weaker than the assumptions we make for our ``warm-up'' algorithm for GLMs, but stronger than the ones required for our generalized quantized Newton's method.
Their results are therefore not directly comparable to ours, however, we note that our communication cost should be lower in e.g. the case where the data is dense and the number of points $m$ is larger than the dimension $d$. The algorithmic techniques are rather different.
Follow-up work extended their approach to the federated learning setting~\citep{safaryan2021fednl}.
\section{Related Work}
\section{Preliminaries}
\label{sec:background}
\paragraph{Distributed Setting.}
As discussed, we are in a standard distributed optimization setting, where we have $n$ nodes, and each node $i$ has its own local cost function $f_i: \mathbb{R}^d \rightarrow \mathbb{R}$ (where $d$ is the dimension of the problem). We wish to minimize the average cost
$f=\frac{1}{n} \sum_{i=1}^n f_i$
and, for that, some communication between nodes is required. We denote the (unique) minimizer of $f$ by $x^*$ and the (unique) minimizer of each $f_i$ by $x_i^*$ (minimizers are unique since these functions are assumed to be strongly convex). Communication may be performed over various network topologies, but in this work we assume a simple structure where an arbitrary node plays the role of the central server, i.e. receives messages from the others, processes them, and finally sends the result back to all. (Such topologies are also common in practice~\cite{PS}.)
Then, the nodes compute an update based on their local cost, and subsequently transmit this information again to the master, repeating the pattern until convergence.
The two main usually considered complexity metrics are the total number of rounds, or iterations, which the algorithm requires, and the total number of bits transmitted.
In this paper, we focus on the latter metric, and assume that nodes cannot communicate their information with infinite precision, but instead aim to limit the number of bits that each node can use to encode messages. Thus, we measure complexity in terms of the total number of bits that the optimization algorithm needs to use, in order to minimize $f$ within some accuracy.
\paragraph{Matrix Vectorization.} One of the main technical tools of our work is quantization of matrices. All the matrices that we care to quantize turn out to be symmetric. The first step for quantizing is to vectorize them. We do so by using the mapping
\vspace{-1mm}
\begin{equation*}
\phi: \mathbb{S}(d) \rightarrow \mathbb{R}^{\frac{d(d+1)}{2}}
\end{equation*}
\vspace{-1mm}
defined by
\begin{equation*}
\phi(P)=(p_{11},...,p_{1d},p_{22},...,p_{2d},...,p_{dd}),
\vspace{-1mm}
\end{equation*}
where $P=(p_{ij})_{i,j=1}^d$ and $\mathbb{S}(d)$ is the space of $d \times d$ symmetric matrices. Thus, the mapping $\phi$ just isolates the upper triangle of a symmetric matrix and writes it as a vector. It is direct to check that $\phi$ is a linear isomorphism ($\textnormal{dim}(\mathbb{S}(d))=d(d+1)/2$).
\newline
We can now bound the deformation of distances produced by this mapping for the $\ell_2$ norm in $\mathbb{S}(d)$ and the $\ell_2$ one in $\mathbb{R}^{\frac{d(d+1)}{2}}$:
\begin{restatable}{lemma}{basic}
\label{le:norm_distortion}
For any matrices $P,P' \in \mathbb{S}(d)$, we have
\begin{equation*}
\frac{1}{\sqrt{d}}\| \phi(P)-\phi(P') \|_2 \leq \|P-P'\|_2 \leq \sqrt{2} \| \phi(P)-\phi(P') \|_2.
\end{equation*}
\label{prop:basic}
\end{restatable}
\vspace{-4mm}
The proof can be found in Appendix \ref{app:isomorphism}.
\newline
We will use the isomorphism $\phi$ later in our applications to Generalized Linear Models and Newton's method. This is the reason of appearance of the extra $d$ \textit{inside a logarithm} in our upper bounds. From now on we use $\| \cdot \|$ to denote the $\ell_2$ norm of either vectors or matrices.
\paragraph{Lattice Quantization.}
For estimating the gradient and Hessian in a distributed manner with limited communication, we use a quantization procedure developed in \cite{davies2021new}. The original quantization scheme involves randomness, but we use a \textit{deterministic} version of it, by picking up the closest point to the vector that we want to encode. This is similar to the quantization scheme used by \cite{alistarh2020improved} for standard gradient descent, and has the following properties:
\begin{Proposition} \cite{davies2021new,alistarh2020improved}
\label{lattice_quantization}
Denoting by $b$ the number of bits that each machine uses to communicate, there exists a quantization function
\vspace{-1mm}
\begin{equation*}
Q: \mathbb{R}^d \times \mathbb{R}^d\times \mathbb{R_+} \times \mathbb{R_+} \rightarrow \mathbb{R}^d,
\vspace{-1mm}
\end{equation*}
which, for each $\epsilon,y>0$,
consists of an encoding function
$\textnormal{enc}_{\epsilon,y}:\mathbb{R}^d \rightarrow \lbrace 0,1 \rbrace ^b$ and a decoding one $\textnormal{dec}_{\epsilon,y}:\lbrace 0,1 \rbrace ^b \times \mathbb{R}^d \rightarrow \mathbb{R}^d$, such that, for all $x, x' \in \mathbb{R}^d$,
\vspace{-2mm}
\begin{itemize}
\item $\textnormal{dec}_{\epsilon,y} (\textnormal{enc}_{\epsilon,y}(x),x') = Q(x,x', y ,\epsilon)$, if $\|x-x'\| \leq y$.
\vspace{-6mm}
\item $\|Q(x,x',y,\epsilon)-x\| \leq \epsilon$, if $\|x-x'\| \leq y$.
\vspace{-2mm}
\item If $y/\epsilon>1$, the cost of the quantization procedure in number of bits satisfies $b= \mathcal{O}(d \textnormal{log}_2 \left(\frac{y}{\epsilon})\right)$.
\end{itemize}
\end{Proposition}
\section{Quantized Preconditioned Gradient Descent for GLMs}
\label{sec:GLMs}
As a warm-up, we consider the case of a Generalized Linear Model (GLM) with data matrix $A \in \mathbb{R}^{m \times d}$. GLMs are particularly attractive models to distribute, because the distribution across nodes can
be performed naturally by partitioning the available data. For more background on distributing GLMs see~\citep{mendler2020randomized}.
The matrix $A$ consists of the data used for training in its rows, i.e. we have $m$-many $d$-dimensional data points. As is custom in regression analysis, we assume that $m \gg d$, i.e. we are in the case of big but low-dimensional data. If $m$ is very large, it can be very difficult to store the whole matrix $A$ in one node, so we distribute it in $n$-many nodes, each one owning $m_i$-many data points ($m=\sum_{i=1}^n m_i)$. We pack the data owned by node $i$ in a matrix $A_i \in \mathbb{R}^{m_i \times d}$ and denote the function used to measure the error on machine $i$ by $\ell_i: \mathbb{R}^{m_i} \rightarrow \mathbb{R}$. Then the local cost function $f_i:\mathbb{R}^d \rightarrow \mathbb{R}$ at machine $i$ reads
\vspace{-1mm}
\begin{equation*}
f_i(x)=\ell_i(A_ix).
\vspace{-1mm}
\end{equation*}
We can express the global cost function $f$ in the form
\vspace{-1mm}
\begin{equation*}
f(x)=\ell(Ax)
\vspace{-1mm}
\end{equation*}
where $\ell:\mathbb{R}^m \rightarrow \mathbb{R}$ is a global loss function defined by
\vspace{-1mm}
\begin{equation*}
\ell(y)=\frac{1}{n} \sum_{i=1}^n \ell_i(y_i),
\vspace{-1mm}
\end{equation*}
where $y_i$ are sets of $m_i$-many coordinates of $y$ obtained by the same data partitioning.
\begin{Assumption}
The local loss functions $\ell_i$ are $\mu_{\ell}$-strongly convex and $\gamma_{\ell}$-smooth.
\end{Assumption}
This assumption implies that the global loss function $\ell$ is $\frac{\mu_{\ell}}{n} $-strongly convex and $\frac{\gamma_{\ell}}{n}$-smooth. This is because the Hessian of $\ell$ has the block-diagonal structure
\begin{equation*}
\nabla_y^2 \ell(y)=\frac{1}{n} \textnormal{diag} \left(\nabla_{y_1}^2 \ell_1(y_1),..., \nabla_{y_n}^2 \ell_n(y_n) \right)
\end{equation*}
and the eigenvalues of all matrices $\nabla_{y_i}^2 \ell_i(y_i)$ are between $\mu_{\ell}$ and $\gamma_{\ell}$. The Hessian of $f$ can be written as
\begin{equation*}
\nabla^2 f(x)=A^T \nabla^2 \ell(Ax) A \in \mathbb{S}(d) \subseteq \mathbb{R}^{d \times d}.
\end{equation*}
We detail the computation of $\nabla^2 f$ in Appendix \ref{app:GLM_technicalities}.
\begin{Assumption}
The matrix $A \in \mathbb{R}^{m \times d}$ is of full rank (i.e. $rank(A)=d$, since $d<m$).
\end{Assumption}
This assumption is natural: if two columns of the matrix $A$ were linearly dependent, we would not need both the related features in our statistical model. Practically, we can prune one of them and get a new data matrix of full-rank.
\begin{restatable}{Proposition} {improvedconditionnumber}
\label{prop:improved_condition_number}
The maximum eigenvalue $\lambda_{max}$ of $\nabla^2 f$ satisfies
\vspace{-1mm}
\begin{equation*}
\gamma:=\lambda_{max}(\nabla^2 f) \leq \gamma_{\ell} \lambda_{max}\left(\frac{A^T A}{n} \right)
\end{equation*}
\vspace{-1mm}
and the minimum eigenvalue $\lambda_{min}$ of $\nabla^2 f$ satisfies
\vspace{-1mm}
\begin{equation*}
\mu:=\lambda_{min}(\nabla^2 f) \geq \mu_{\ell} \lambda_{min}\left(\frac{A^T A}{n} \right) .
\end{equation*}
\end{restatable}
\vspace{-1mm}
The proof is presented in Appendix \ref{app:GLM_technicalities}.
Thus, we have that the condition number $\kappa$ of our minimization problem satisfies
\vspace{-1mm}
\begin{equation*}
\kappa \leq \kappa_{\ell} \kappa\left(\frac{A^T A}{n} \right),
\vspace{-1mm}
\end{equation*}
where $\kappa\left(\frac{A^T A}{n} \right)$ is the condition number of the covariance matrix $A^T A$ averaged in the number of machines.
The convergence rate of gradient descent generally depends on $\kappa$, which can be much larger than $\kappa_{\ell}$ in case that the condition number of $A^T A$ is large. The usual way to get rid of $\kappa\left(\frac{A^T A}{n} \right)$ is to precondition gradient descent using $\frac{A^T A}{n}$, which we denote by $M$ from now on (we recall the convergence analysis of this method in Appendix \ref{app:precond_gradient}). In our setting $M$ is not known to all machines simultaneously, since each machine owns only a part of the overall data. However, we observe that
\vspace{-1mm}
\begin{equation*}
M= \frac{1}{n} \sum_{i=1}^n A_i^T A_i,
\vspace{-1mm}
\end{equation*}
where $A_i^T A_i =: M_i$ is the local covariance matrix of the data owned by the node $i$.
\subsection{The Algorithm}
In this section we present our QPGD-GLM algorithm and study its communication complexity.
We structure the algorithm in four steps: first, we describe how to recover a quantized version of the averaged covariance matrices.
Then, we describe how nodes perform initialization. Next, we describe how nodes can quantize the initial descent direction.
Finally, we describe how to quantize the descent directions for subsequent steps. Our notation for quantization operations follows Section~\ref{sec:background}.
\noindent\rule[0.5ex]{\linewidth}{1pt}
\vspace{-8mm}
\begin{enumerate}
\item Choose an arbitrary master node, say $i_0$.
\subsection*{(A) Averaged Covariance Matrix Quantization:}
\item Compute $M_i:=A_i^T A_i$ in each node.
\vspace{-2mm}
\item Encode $M_i$ in each node $i$ and decode it in the master node using its information:
\vspace{-2mm}
\begin{align*}
\resizebox{0.9\hsize}{!}{$\bar M_i = \phi^{-1}\left(Q\left(\phi(M_i), \phi(M_{i_0}) , 2 \sqrt{d} n \lambda_{max}(M) , \frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}} \right) \right)$}.
\end{align*}
\vspace{-5mm}
In detail, we first transform the local matrix $M_i$ via the isomorphism $\phi$, and then quantize it via $Q$, with carefully-set parameters. The matrix will be then de-quantized relative to the master's reference point $\phi(M_{i_0})$, and then re-constituted (in approximate form) via the inverse isomorphism.
\item Average the decoded matrices in the master node:
\newline
$S=\frac{1}{n} \sum_{i=1}^n \bar M_i$.
\vspace{-2mm}
\item Encode the average in the master node and decode in each node $i$ using its local information
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9\hsize}{!}{
$\bar M=\phi^{-1} \left(Q(\phi(S),\phi(M_i), \sqrt{d} \left( \frac{\lambda_{min}(M)}{16 \kappa_{\ell}}+2 n \lambda_{max}(M) \right),\frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}}) \right)$}.
\end{equation*}
\vspace{-8mm}
\subsection*{(B) Starting Point and Parameters for Descent Direction Quantization:}
\item Choose $D>0$ and $x^{(0)} \in \mathbb{R}^d$, such that
\begin{equation*}
\max_i \lbrace \|x^{(0)}-x^*\| , \|x^{(0)}-x_i^*\| \rbrace \leq D.
\end{equation*}
\vspace{-2mm}
\item Define the parameters
\vspace{-2mm}
\begin{align*}
& \xi:=1-\frac{1}{2 \kappa_{\ell}}, K:=\frac{2}{\xi}, \delta:=\frac{\xi(1-\xi)}{4},\\
& R^{(t)}:= \frac{\gamma_{\ell}}{2} K \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D.
\end{align*}
\vspace{-8mm}
\subsection*{(C) Quantizing the Initial Descent Direction:}
\item Compute $\bar M^{-1} \nabla f_i(x^{(0)})$ in each node.
\vspace{-2mm}
\item Encode $\bar M^{-1} \nabla f_i(x^{(0)})$ in each node and decode it in the master node using its local information:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v_i^{(0)}=Q\left(\bar M^{-1} \nabla f_i(x^{(0)}),\bar M^{-1} \nabla f_{i_0}(x^{0}),4 n \kappa(M) R^{(0)},\frac{\delta R^{(0)}}{2} \right)$}.
\end{equation*}
\vspace{-8mm}
\item Average the quantized local information in the master node:
\newline
$r^{(0)}=\frac{1}{n} \sum_{i=1}^n v_i^{(0)}$.
\vspace{-2mm}
\item Encode $r^{(0)}$ in the master node and decode it in each machine $i$ using its local information:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v^{(0)}=Q \left(r^{(0)},\bar M^{-1} \nabla f_i(x^{(0)}),\left(\frac{\delta}{2}+4 n \kappa(M) \right) R^{(0)},\frac{\delta R^{(0)}}{2} \right)$} .
\end{equation*}
\textbf{For} $t \geq 0$:
\item Compute
\vspace{-2mm}
\begin{equation*}
x^{(t+1)}=x^{(t)}- \eta v^{(t)}
\end{equation*}
\vspace{-2mm}
for $\eta>0$.
\subsection*{(D) Descent Direction Quantization for Next Steps:}
\item Encode $\bar M^{-1} \nabla f_i(x^{(t)})$ in each node $i$ and decode in the master node using the previous local estimate:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v_i^{(t+1)}=Q \left(\bar M^{-1} \nabla f_i(x^{(t+1)} \right), v_i^{(t)},4 n \kappa(M) R^{(t+1)}, \frac{\delta R^{(t+1)}}{2})$}.
\end{equation*}
\vspace{-8mm}
\item Average the quantized local information:
\newline
$r^{(t+1)}=\frac{1}{n} \sum_{i=1}^n v_i^{(t+1)}$.
\vspace{-2mm}
\item Encode $r^{(t+1)}$ in the master node and decode it in each node using the previous global estimate:
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v^{(t+1)}=Q \left(r^{(t+1)},v^{(t)}, \left(\frac{\delta}{2}+4 n \kappa(M) \right) R^{(t+1)}, \frac{\delta R^{(t+1)}}{2} \right)$}.
\end{equation*}
\end{enumerate}
\vspace{-4mm}
\noindent\rule[0.5ex]{\linewidth}{1pt}
We now discuss the algorithm's assumptions. First, we assume that an over-approximation $D$ for the distance of the initialization from the minimizer is known. This is practical, especially in the case of GLMs: since the loss functions $\ell_i$ are often quadratics, we can use strong convexity and write
\vspace{-1mm}
\begin{equation*}
\|x^{(0)}-x^*\|^2 \leq \frac{2}{\mu} (f(x^{(0)})-f^*) \leq \frac{2}{\mu} f(x^{(0)})=:D^2.
\vspace{-1mm}
\end{equation*}
and similarly for $\|x^{(0)}-x_i^*\|^2$. Further, following \citet{magnusson2020maintaining} (Assumption 2, page 5), the value $f(x^{(0)})$ is often available, for example in the case of logistic regression. Of course, if we are restricted in a compact domain as is the case of \cite{4048825} and \cite{alistarh2020improved}, then the domain itself provides an over approximation for all the distances inside it.
\newline
The parameters $\lambda_{max}(M), \lambda_{min}(M)$ used for quantization of the matrix $M$ are usually assumed to be known. Specifically, it is common in distributed optimization to assume that all nodes know estimates of the smoothness and strong convexity constants of each of the local cost functions \cite{4048825}. In our case this would imply knowing all $\lambda_{max}(M_i),\lambda_{min}(M_i)$. However, we assume knowledge of just $\lambda_{max}(M)$ and $\lambda_{min}(M)$. This also explains the appearance of the extra $\log n$ factor in our GLM bounds, relative to those for Newton's method.
\newline
The convergence and communication complexity of our algorithm are described in the following theorem:
\begin{tcolorbox}
\begin{restatable}{theorem}{convergenceGLM}
\label{thm:convergence_GLM}
The iterates $x^{(t)}$ produced by the previous algorithm with $\eta=\frac{2}{\mu_{\ell}+\gamma_{\ell}}$ satisfy
\begin{align*}
\| x^{(t)}-x^* \| \leq \left(1-\frac{1}{4\kappa_{\ell}}\right)^t D
\end{align*}
and the total number of bits used for communication until $f(x^{(t)})-f^* \leq \epsilon$ is
\begin{align}
\label{eq:communication_glm}
\begin{split}
&\mathcal{O} \left(n d^2 \log \left(\sqrt{d} n \kappa_{\ell} \kappa(M) \right) \right) + \\ & \mathcal{O} \left(n d \kappa_{\ell} \log (n \kappa_{\ell} \kappa(M)) \log \frac{\gamma D^2}{\epsilon}\right).
\end{split}
\end{align}
\end{restatable}
\end{tcolorbox}
When the accuracy $\epsilon$ is sufficiently small (which is often the case in practice), the first summand is negligible and the total number of bits until reaching it is just
\vspace{-1mm}
\begin{equation*}
b=\mathcal{O} \left(n d \kappa_{\ell} \log (n \kappa_{\ell} \kappa(M)) \log \frac{\gamma D^2}{\epsilon}\right)
\vspace{-1mm}
\end{equation*}
which gains over quantized gradient descent in \cite{alistarh2020improved} the linear dependence on the condition number of $M$. We prove Theorem \ref{thm:convergence_GLM} in Appendix \ref{app:convergence_GLM}.
\section{Quantized Newton's method}
After warming-up with quantizing fixed preconditioners in the case of Generalized Linear Models, we move forward to quantize non-fixed ones. The extreme case of a preconditioner is the whole Hessian matrix; preconditioning with it yields Newton's method, which is computationally expensive, but removes completely the dependency on the condition number from the iteration complexity. We develop a quantized version of Newton's method in order to address a question raised by \cite{alistarh2020improved} regarding whether the communication complexity of minimizing a sum of smooth and strongly convex functions depends linearly on the condition number of the problem. The main technical challenge towards that, is keeping track of the concentration of the Hessians around the Hessian evaluated at the optimum, while the algorithm converges. We show that the linear dependence of the communication cost on the condition number of the problem is not necessary, in exchange with extra dependence on the dimension of the problem, i.e. $d^2$ instead of $d$. This can give significant advantage for low-dimensional and ill-conditioned problems (training generalized linear models is among them).
\newline
As it is natural for Newton's method, we make the following assumptions for the objective function $f$:
\begin{Assumption}
\label{ass:local_cost_newton}
The functions $f_i$ are all $\gamma$-smooth and $\mu$-strongly convex with a $\sigma$-Lipschitz Hessian, $\gamma,\mu,\sigma>0$.
\end{Assumption}
We note that the lower bound derived by \citet{alistarh2020improved} is obtained for the case that $f_i$ are quadratic functions; quadratic functions indeed satisfy Assumption \ref{ass:local_cost_newton}.
As in the case of GLMs, we define the condition number of the problem to be
\vspace{-1mm}
\begin{equation*}
\kappa:=\frac{\gamma}{\mu}.
\vspace{-1mm}
\end{equation*}
We also introduce a constant $\alpha \in [0,1)$, to be specified later, which will control the convergence of the algorithm.
\subsection{Algorithm Description}
We now describe our quantized Newton's algorithm.
Again, we split the presentation into several parts: local initialization (A), estimating the initial Hessian modulo quantization (B), as well as the quantized initial descent direction (C), and finally, quantization and update for each iteration (D,E).
\noindent\rule[0.5ex]{\linewidth}{1pt}
\vspace{-8mm}
\begin{enumerate}
\item Choose the master node at random, e.g. $i_0$.
\subsection*{(A) Starting Point and Parameters for Hessian Quantization:}
\item Choose $x^{(0)} \in \mathbb{R}^d$, such that
\begin{equation*}
\max_i \lbrace\|x^{(0)}-x^*\|,\|x^{(0)}-x_i^*\| \rbrace \leq \frac{\alpha \mu}{2 \sigma}.
\end{equation*}
\vspace{-8mm}
\item We define the parameter
\begin{align*}
G^{(t)}=\frac{\mu}{4} \alpha \left(\frac{1+\alpha}{2} \right)^t.
\end{align*}
\subsection*{(B) Initial Hessian Quantized Estimation:}
\item Compute $\nabla^2 f_i(x^{(0)})$ in each node.
\vspace{-2mm}
\item Encode $\nabla^2 f_i(x^{(0)})$ in each node $i$ and decode it in the master node $i_0$ using its information:
\vspace{-2mm}
\begin{align*}
\resizebox{0.9 \hsize}{!}{$H_0^i= \phi^{-1}\left(Q \left(\phi(\nabla^2 f_i(x^{(0)})), \phi(\nabla^2 f_{i_0}(x^{(0)})), 2 \sqrt{d} \gamma, \frac{G^{(0)}}{2 \sqrt{2} \kappa} \right)\right)$}.
\end{align*}
\vspace{-8mm}
\item Average the decoded matrices in the master node:
\newline
$S_0=\frac{1}{n} \sum_{i=1}^n H_0^i$.
\vspace{-2mm}
\item Encode the average in the master node and decode in each node $i$ using its local information
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{
$H_0= \phi^{-1}\left(Q \left(\phi(S_0), \phi(\nabla^2 f_i(x^{(0)})), \sqrt{d} \left( \frac{G^{(0)}}{2\kappa} + 2 \gamma \right) , \frac{G^{(0)}}{2 \sqrt{2} \kappa} \right)\right)$}.
\end{equation*}
\vspace{-8mm}
\subsection*{Parameters for Descent Direction Quantzation:}
\item Define the parameters
\vspace{-2mm}
\begin{align*}
&\theta:=\frac{\alpha(1-\alpha)}{4},
K:=\frac{2}{\alpha},
P^{(t)}:= \frac{\mu}{2 \sigma} K \alpha \left( \frac{1+\alpha}{2}\right)^t.
\end{align*}
\vspace{-8mm}
\subsection*{(C) Initial Descent Direction Quantized Estimation:}
\item Compute $H_0^{-1} \nabla f_i(x^{(0)})$ in each node.
\vspace{-2mm}
\item Encode $H_0^{-1} \nabla f_i(x^{(0)})$ in each node and decode it in the master node using its local information:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v_i^{(0)}=Q \left(H_0^{-1} \nabla f_i(x^{(0)}), H_0^{-1} \nabla f_{i_0} (x^{(0)}),4 \kappa P^{(0)},\frac{\theta P^{(0)}}{2} \right)$}.
\end{equation*}
\vspace{-8mm}
\item Average the quantized local information:
\newline
$p^{(0)}=\frac{1}{n} \sum_{i=1}^n v_i^{(0)}$.
\vspace{-2mm}
\item Encode $p^{(0)}$ in the master node and decode it in each machine $i$ using its local information:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v^{(0)}=Q \left(p^{(0)}, H_0^{-1} \nabla f_i (x^{(0)}),\left(\frac{\theta}{2}+4 \kappa \right) P^{(0)}, \frac{\theta P^{(0)}}{2} \right)$} .
\end{equation*}
\vspace{-4mm}
\newline
\textbf{For} $t \geq 0$:
\item Compute
\vspace{-2mm}
\begin{equation*}
x^{(t+1)}=x^{(t)}-v^{(t)}.
\end{equation*}
\vspace{-8mm}
\subsection*{(D) Hessian Quantized Estimation for Next Steps:}
\item Compute $\nabla^2 f_i(x^{(t+1)})$ in each node.
\vspace{-2mm}
\item Encode $\nabla^2 f_i(x^{(t+1)})$ in each node $i$ and decode in the master node using the previous local estimate:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$H_{t+1}^i=\phi^{-1}\left(Q \left(\phi(\nabla^2 f_i(x^{(t+1)})), \phi(H_t^i), \frac{10 \sqrt{d}}{1+\alpha} G^{(t+1)}, \frac{ G^{(t+1)}}{2 \sqrt{2} \kappa} \right)\right)$}.
\end{equation*}
\vspace{-8mm}
\item Average the quantized local Hessian information:
\newline
$S_{t+1}=\frac{1}{n} \sum_{i=1}^{n} H_{t+1}^i$.
\vspace{-2mm}
\item Encode $S_{t+1}$ in the master node and decode it back in each node using the previous global estimate:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$H_{t+1}=\phi^{-1} \left(Q \left(\phi(S_{t+1}), \phi(H_t) , \sqrt{d} \left( \frac{1}{2 \kappa}+ \frac{10}{1+\alpha} \right) G^{(t+1)} , \frac{G^{(t+1)}}{2 \sqrt{2} \kappa} \right) \right)$}.
\end{equation*}
\vspace{-8mm}
\subsection*{(E) Descent Direction Quantized Estimation:}
\item Compute $H_{t+1}^{-1} \nabla f_i(x^{(t+1)})$ in each node.
\vspace{-2mm}
\item Encode $H_{t+1}^{-1} \nabla f_i(x^{(t+1)})$ in each node $i$ and decode in the master node using the previous local estimate:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v^{(t+1)}_i=Q \left(H_{t+1}^{-1} \nabla f_i(x^{(t+1)}), v^{(t)}_i, 11 \kappa P^{(t+1)} , \frac{\theta P^{(t+1)}}{2} \right)$}.
\end{equation*}
\vspace{-8mm}
\item Average the quantized local Hessian information:
\newline
$p^{(t+1)}=\frac{1}{n} \sum_{i=1}^{n} v^{(t+1)}_i$.
\vspace{-2mm}
\item Encode $S_{t+1}$ in the master node and decode it back in each node using the previous global estimate:
\vspace{-2mm}
\begin{equation*}
\resizebox{0.9 \hsize}{!}{$v^{(t+1)}=Q \left(p^{(t+1)}, v^{(t)},\left(\frac{\theta}{2}+11 \kappa \right) P^{(t+1)}, \frac{\theta P^{(t+1)}}{2} \right)$}.
\end{equation*}
\end{enumerate}
\vspace{-4mm}
\noindent\rule[0.5ex]{\linewidth}{1pt}
The restriction of the initialization $x^{(0)}$ is standard for Newton's method, which is known to converge only \textit{locally}. Usually $x^{(0)}$ is chosen such that $\alpha \geq \frac{\sigma}{\mu} \|x^{(0)}-x^*\|$, while we choose it such that $\alpha \geq 2 \frac{\sigma}{\mu} \|x^{(0)}-x^*\|$ (and the same for $x_i^*$ in the place of $x^*$). This difference occurs from the extra errors due to quantization. This assumption implies also that the minima of the local costs cannot be too far away from each other.
\newline
We now state our theorem on communication complexity of quantized Newton's algorithm, which is the main result of the paper. The proof is in Appendix \ref{app:quant_newton}, and relies on analyzing the behaviour of both the quantized Hessian estimates and the quantized descent direction estimates simultaneously, as can be seen in Lemma \ref{le:desc_direction_newton}.
\begin{tcolorbox}
\begin{restatable}{theorem}{maintheorem}
\label{thm:main_theorem}
The iterates of the quantized Newton's method starting from a point $x^{(0)}$, such that
\begin{equation*}
\|x^{(0)}-x^*\| \leq \frac{\mu}{4 \sigma} \left(\alpha=\frac{1}{2}\right)
\end{equation*}
satisfy
\begin{equation*}
\|x^{(t)}-x^*\| \leq \frac{\mu}{4 \sigma} \left( \frac{3}{4} \right)^t
\end{equation*}
and the communication cost until reaching accuracy $\epsilon$ in terms of function values is
\begin{equation}
\label{eq:communication_newton}
\mathcal{O}\left(nd^2 \log\left( \sqrt{d} \kappa \right) \log \frac{\gamma \mu^2}{\sigma^2 \epsilon} \right)
\end{equation}
many bits in total.
\end{restatable}
\end{tcolorbox}
We note that the lower bound derived in \cite{alistarh2020improved} is for the case that all functions $f_i$ are quadratics. For quadratics, the Hessian is constant, thus $\sigma=0$ and $\alpha$ can be chosen equal to $0$ as well. Then, (non-distributed) Newton's method converges in only one step. However, in the distributed case, $\sigma = 0$ implies $G^{(t)}=0$, thus the estimation of $\nabla^2 f(x^{(t)})$ must be exact. This would mean that we need to use an infinite number of bits, and this can be seen also in our communication complexity results. In order to apply our result in a practical manner, we need to allow the possibility for strictly positive quantization error of the Hessian, thus we must choose $\sigma>0$.
\section{Estimation of the Minimum in the Master}
In the previous sections we computed an approximated minimizer of our objective function up to some accuracy and counted the communication cost of the whole process. We now extend our interest to the slightly harder problem of estimating the minimum $f^*$ of the function $f$ (which is again assumed to be $\gamma$-smooth and $\mu$-strongly convex) in the master node with accuracy $\epsilon$. This extension is not considered in \cite{magnusson2020maintaining}, but is discussed in \cite{alistarh2020improved}. To that end, we estimate the minimizer $x^*$ of $f$ by a vector $x^{(t)}$, such that $f(x^{(t)}) -f^* \leq \frac{\epsilon}{2}$, and the communication cost of doing that is again given by expression (\ref{eq:communication_glm}) for GLM training and expression (\ref{eq:communication_newton}) for Newton's method.
\newline
We denote $x_i^*$ the minimizer of the local cost function $f_i$ and $f_i^*:=f_i(x_i^*)$ its minimum. We also assume that we are aware of an over approximation $C>0$ of the maximum distance of $x^*$ from the minimizers of the local costs $x_i^*$, i.e.
$\max_{i=1,...,n} \|x^*-x_i^*\| \leq C$
and a radius $c>0$ for the minima of the local costs:
$\max_{i=1,...,n} \mid f_i^* \mid \leq c$.
Estimating these constants can be feasible in many practical situations:
\vspace{-3mm}
\begin{itemize}
\item We can always bound the quantity $\max_{i=1,...,n} \|x^*-x_i^*\| $ by a known constant if we set our problem in a compact domain as it is the case in \cite{4048825} and \cite{alistarh2020improved}. Also, if our local data are obtained from the same distribution, then we do not expect the minimizers of the local costs to be too far away from the global minimizer.
\vspace{-2mm}
\item The minima $f_i^*$ of the local costs are often exactly $0$ (as assumed in \cite{alistarh2020improved}). This is because the local cost functions $f_i$ are often quadratics, as it happens in the case of GLMs. In the worst case, knowing just that $f_i \geq 0$, we can write
\vspace{-1mm}
\begin{equation*}
\mid f_i^* \mid = f_i^* \leq f_i(x^{(0)}) \leq n f(x^{(0)})
\vspace{-1mm}
\end{equation*}
and the value $f(x^{(0)})$ is often available as discussed in Section \ref{sec:GLMs} and in \cite{magnusson2020maintaining}.
\end{itemize}
For estimating the minimum $f^*$, we start by computing $f_i(x^{(t)})$ in each node $i$ and communicate them to the master node $i_0$ as follows:
\begin{equation*}
q_i^{(t)}:=Q(f_i(x^{(t)}), f_{i_0}(x^{(t)}),2 (\gamma C^2+c), \epsilon/2).
\end{equation*}
Then the master node computes and outputs the average
\begin{equation*}
\bar f= \frac{1}{n} \sum_{i=1}^n q_i^{(t)}.
\end{equation*}
\begin{restatable}{Proposition}{functionvalue}
The value $\bar f$ which occurs from the previous quantization procedure is an estimate of the true minimum $f^*$ of $f$ with accuracy $\epsilon$ and the cost of quantization is
\begin{equation*}
\mathcal{O} \left(n \log \frac{\gamma C^2+c}{\epsilon} \right).
\end{equation*}
if $\epsilon$ is sufficiently small.
\end{restatable}
The proof is presented in Appendix \ref{app:function_value}.
\newline
Thus, for the problem that the master node needs to output estimates for both the minimizer and the minimum with accuracy $\epsilon$ in terms of function values, the total communication cost is at most
\begin{align*}
\mathcal{O} \left(n d \kappa_{\ell} \log (n \kappa_{\ell} \kappa(M)) \log \frac{\gamma( C^2 + D^2)+c) }{\epsilon}\right)
\end{align*}
many bits in total for QPGD-GLM
\begin{align*}
\mathcal{O}\left(nd^2 \log\left( \sqrt{d} \kappa \right) \log \left( \left( \gamma \left(\frac{ \mu^2}{\sigma^2}+ C^2 \right)+c \right) \frac{1}{\epsilon} \right) \right).
\end{align*}
many bits in total for quantized Newton's method when $\epsilon$ is sufficiently small.
\section{Experiments}
\label{sec:experiments}
\hide{
\begin{figure*}[h]
\begin{center}
\begin{minipage}[b]{0.48\linewidth}
\centering
\includegraphics[width=\columnwidth]{Figures/convergence_lr_0.75_n_8_qb_3.pdf}
\subcaption{Performance on synthetic data}
\label{fig:synthetic}
\end{minipage}
\quad
\begin{minipage}[b]{0.48\linewidth}
\centering
\includegraphics[width=\columnwidth]{Figures/cpu_small_lr_0.11_n_8_qb_8.pdf}
\subcaption{Performance on \texttt{cpusmall\_scale}}
\label{fig:cpusmall}
\end{minipage}
\end{center}
\vskip -0.2in
\end{figure*}
\begin{figure*}[h]
\begin{center}
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=\columnwidth]{Figures/german_numer.txt_lr_0.005_n_5_qb_82.pdf}
\subcaption{Logistic Regression (placeholder)}
\label{fig:synthetic}
\end{minipage}
\quad
\begin{minipage}[b]{0.45\linewidth}
\centering
\includegraphics[width=\columnwidth]{Figures/german_numer.txt_lr_0.005_n_5_qb_82.pdf}
\subcaption{Logistic Regression (placeholder)}
\label{fig:cpusmall}
\end{minipage}
\end{center}
\vskip -0.2in
\end{figure*}
\begin{figure*}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/convergence_lr_0.75_n_8_qb_3.pdf}
\caption[Network2]%
{{\small Network 1}}
\label{fig:mean and std of net14}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=1.1\textwidth]{Figures/cpu_small_lr_0.11_n_8_qb_8.pdf}
\caption[]%
{{\small Network 2}}
\label{fig:mean and std of net24}
\end{subfigure}
\vskip\baselineskip
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/german_numer.txt_lr_0.005_n_5_qb_82.pdf}
\caption[]%
{{\small Network 3}}
\label{fig:mean and std of net34}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{Figures/german_numer.txt_lr_0.005_n_5_qb_82.pdf}
\caption[]%
{{\small Network 4}}
\label{fig:mean and std of net44}
\end{subfigure}
\caption[ The average and standard deviation of critical parameters ]
{\small The average and standard deviation of critical parameters: Region R4}
\label{fig:mean and std of nets}
\end{figure*}
}
\begin{figure*}[h]
\begin{center}
\begin{minipage}[b]{0.33\linewidth}
\centering
\includegraphics[width=\textwidth]{Figures/cpu_small_lr_0.11_n_8_qb_8.png}
\subcaption{Least-squares regression performance on \texttt{cpusmall\_scale}}
\label{fig:cpusmall}
\end{minipage}
\quad
\begin{minipage}[b]{0.305\linewidth}
\centering
\includegraphics[width=\textwidth]{Figures/phishing.txt_n_5_qb_4.png}
\subcaption{Logistic regression performance on \texttt{phishing} }
\label{fig:phishing}
\end{minipage}
\quad
\begin{minipage}[b]{0.305\linewidth}
\centering
\includegraphics[width=\textwidth]{Figures/german_numer.txt_n_5_qb_8.png}
\subcaption{Logistic regression performance on \texttt{german\_numer}}
\label{fig:german_numer}
\end{minipage}
\end{center}
\vskip -0.2in
\end{figure*}
\subsection{Experiment 1: Least-Squares Regression}
We first test our method experimentally to compress a parallel solver for least-squares regression. The setting is as follows: we are given as input a data matrix $A$, with rows randomly partitioned evenly among the nodes, and a target vector $b$, with the goal of finding $ x^* = \text{argmin}_{ x}\|A x- b\|^2_2$. Since this loss function $f(x):= \|A x- b\|^2_2$ is quadratic, its Hessian is constant, and so Newton's method and QPGD-GLM are equivalent: in both cases, we need only to provide the preconditioner matrix $A^T A$ in the first iteration, and machines can henceforth use it for preconditioning in every iteration.
To quantize the preconditioner matrix, we apply the `practical version' (that is, using the cubic lattice with $\bmod$-based coloring) of the quantization method of \cite{davies2021new}, employing the `error detection' method in order to adaptively choose the number of bits required for the decoding to succeed. Each node $i$ quantizes the matrix $A_i^T A_i$, which is decoded by the master node $i_0$ using $A_{i_0}^T A_{i_0}$. Node $i_0$ computes the average, quantizes, and returns the result to the other nodes, who decode using $A_i^T A_i$.
To quantize gradients, we use two leading gradient quantization techniques: QSGD \cite{alistarh2016qsgd}, and the Hadamard-rotation based method of \cite{pmlr-v70-suresh17a}, since these are optimized for such an application.\footnote{There is a wide array of other gradient quantization methods; we use these two as a representative examples, since we are mostly concerned with the effects of preconditioner quantization.} In each iteration (other than the first), we quantize the \emph{difference} between the current local gradient and that of last iteration, average these at the master node $i_0$, and quantize and broadcast the result.
\paragraph{Compared Methods.}
In Figure \ref{fig:cpusmall} we compare the following methods: GDn and GDf are full-precision (i.e., using 32-bit floats) gradient descent using \emph{no preconditioning} and \emph{full-precision preconditioning} respectively, as baselines. QSGDq and QSGDf use QSGD for gradient quantization, and the \emph{quantized} and \emph{full-precision} preconditioner respectively. HADq and HADf are the equivalents using instead the Hadamard-rotation method for gradient quantization. When using a preconditioner, we rescale preconditioned gradients to preserve $\ell_2$-norm, so that our comparison is based only on update direction and not step size.
\paragraph{Parameters.}
In addition to $m$, $n$, and $d$, we also have the following parameters: the learning rate (lr in the figure titles) is set close to the maximum for which gradient descent will converge, since this is the regime in which preconditioning can help. The number of bits per coordinate used to quantize gradients (qb) and preconditioners (pb) are also shown; the latter is an average since the quantization method uses a variable number of bits\footnote{These quantization methods (and most others) also require exchange of two full-precision scalars, which are not included in the per-coordinate costs since they are independent of dimension.}. The results presented are an average of the cost function per descent iteration, over $10$ repetitions with different random seeds.
\hide{
\paragraph{Synthetic Data.}
We first apply the methods to synthetic data: our data matrix $A$ consists of independent Gaussian entries with variance $1$, we choose our target optimum $x^*$ to be Gaussian with variance $1000$, and set $B=Ax^*$.
Figure \ref{fig:synthetic} shows that, even using only $\sim 3$ bits per coordinate for both gradients and preconditioner, we achieve signicantly faster convergence than full-precision gradient descent without preconditioning, and converge to essentially as good a solution as full-precision gradient descent with preconditioning.}
\paragraph{Dataset}
We use the dataset \texttt{cpusmall\_scale} from LIBSVM \cite{LibSVM}. Here we outperform non-preconditioned gradient descent and approach the performance of full-precision preconditioned gradient descent using significantly reduced communication (Figure \ref{fig:cpusmall}).
\subsection{Experiment 2: Logistic Regression}
In order to compare the performance of Q-Newton and QPGD-GLM, we implement a common application in which the Hessian is \emph{not} constant: logistic regression, for binary classification problems.
QPGD-GLM, QSGD, and full-precision gradient descent are implemented as before; we now add full-precision Newton's method for comparison, and our Q-Newton algorithm. The latter uses the quantization method of \cite{davies2021new} for the initial Hessian (as for QPGD-GLM), and QSGD for subsequent Hessian updates.
Rather than re-scaling gradients, we take a different approach to choosing a learning rates in order to compare the methods fairly: we test each with learning rates in $\{2^{-0}, 2^{-1},2^{-2},\dots\}$, and plot the highest rate for which the method stably converges. Our results are averaged over five random seeds.
We demonstrate the methods on the \texttt{phishing} and \texttt{german\_numer} datasets from the LIBSVM collection \cite{LibSVM}, in Figures \ref{fig:phishing} and \ref{fig:german_numer} respectively. The former demonstrates that Q-Newton improves over (even full precision) first-order methods, while quantizing Hessians at only 4 bits per coordinate. The latter demonstrates an instance in which QPGD-GLM is even faster, since it remains stable under a higher learning rate.
\section{Discussion}
We proposed communication-efficient versions for two fundamental optimization algorithms, and analyzed their convergence and communication complexity.
Our work shows that quantizing second-order information can i) theoretically yield to communication complexity upper bounds with sub-linear dependence on the condition number of the problem, and ii) empirically achieve superior performance over vanilla methods.
There are intriguing questions for future work:
\newline
The $\log \kappa$-dependency for Newton's method occurs because of our bounds for the input and output variance of quantization. It would be interesting to see whether this dependency can be avoided, making the bounds completely independent of the condition number.
Another interesting question is whether the $\log d$-dependency can be circumvented. $\log d$ is obtained directly from the use of the vectorization $\phi$ and could be avoided by quantization using lattices with good spectral norm properties. We are however unaware of such lattice constructions.
One key issue left is the $d^2$-dependence for the generalized Newton's method, which is due to quantization of $d^2$-dimensional preconditioners.
It would be interesting to determine if linear communication per round can be achieved in the general setting we consider here.
Finally, we would like to point out that there exist more second order methods with superior guarantees compared to vanilla Newton, such as cubic regularization~\cite{RePEc:cor:louvrp:1927}. A very interesting direction for future work would be to investigate whether it is possible to run these algorithms in a distributed setting with limited communication by adding quantization.
\paragraph{Acknowledgements} The authors would like to thank Janne Korhonen, Aurelien Lucchi, Celestine Mendler-Dünner and Antonio Orvieto for helpful discussions. FA and DA were supported during this work by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 805223 ScaleML). PD was supported by the European Union's Horizon 2020 programme under the Marie Skłodowska-Curie grant agreement No. 754411.
\section{The isomorphism $\phi$}
\label{app:isomorphism}
\basic*
\begin{proof}
The Frobenius norm of a matrix $P=(p_{ij})_{i,j=1}^d$ is defined as
\begin{equation*}
\| P \|_F= \sqrt{\sum_{i,j=1}^d p_{ij}^2}
\end{equation*}
thus
\begin{align*}
\| P-P' \|_F^2 = \sum_{i,j=1}^d (p_{ij}-p_{ij}')^2 = \sum_{i=j} (p_{ij}-p_{ij}')^2+\sum_{i \neq j} (p_{ij}-p_{ij}')^2 =\sum_{i=j} (p_{ij}-p_{ij}')^2+2 \sum_{i<j} (p_{ij}-p_{ij}')^2=:X+2Y
\end{align*}
with $P'=(p'_{ij})_{i,j=1}^d$.
\newline
We also have
\begin{equation*}
\| \phi(P)-\phi(P') \|^2= \sum_{i=j} (p_{ij}-p_{ij}')^2+ \sum_{i<j} (p_{ij}-p_{ij}')^2=:X+Y
\end{equation*}
Thus
\begin{equation*}
\|\phi(P)-\phi(P')\|_2 \leq \|P-P'\|_F \leq \sqrt{2} \|\phi(P)-\phi(P')\|_2.
\end{equation*}
Now for the $\ell_2$ norm, we have
\begin{equation*}
\| P-P' \|_2 \leq \| P-P' \|_F \leq \sqrt{2} \| \phi(P)-\phi(P') \|_2
\end{equation*}
and
\begin{equation*}
\| P-P' \|_2 \geq \frac{1}{\sqrt{d}} \| P-P' \|_F \geq \frac{1}{\sqrt{d}} \| \phi(P)-\phi(P') \|_2
\end{equation*}
and the desired result follows.
\end{proof}
\section{Technicalities regarding GLMs}
\label{app:GLM_technicalities}
We firstly compute the Hessian of the global cost function $f$ in terms of the Hessian of the global loss function $\ell$:
\begin{lemma}
We have
\begin{equation*}
\nabla^2 f(x)=A^T \ell (Ax) A.
\end{equation*}
\end{lemma}
\begin{proof}
We start by computing the gradient of $f$. We fix an arbitrary vector $v \in \mathbb{R}^d$ and we write
\begin{align*}
&\langle \nabla f(x),v \rangle = d_xf(x)v = d_x(\ell(Ax))v=d_y(\ell(y))|_{y=Ax} d_x(Ax)v = d_y(\ell(y))|_{y=Ax} Av \\ & = \langle \nabla \ell(y)|_{y=Ax}, Av \rangle = (Av)^T \nabla \ell (Ax)=v^T A^T \nabla \ell (Ax)= \langle A^T \nabla \ell (Ax), v \rangle
\end{align*}
Since $v$ is arbitrary, the gradient of $f$ is
\begin{equation*}
\nabla f(x)=A^T \nabla \ell (Ax)
\end{equation*}
For the Hessian, we have
\begin{align*}
\nabla_x^2 f(x) = \nabla_x \nabla_x f(x) = \nabla_x (A^T \nabla_x \ell (Ax)) = A^T \nabla_x (\nabla_x \ell (Ax))= A^T \nabla_y (\nabla_y \ell(y))|_{y=Ax} \nabla_x(Ax) = A^T \nabla^2 \ell(Ax) A.
\end{align*}
\end{proof}
We recall standard technical results from linear algebra in order to prove Proposition \ref{prop:improved_condition_number}. They will be useful also in the proof of Proposition \ref{prop:exact_precond} and Lemma \ref{le:inexact_precond}.
\begin{lemma}
\label{le:eig commute}
Given matrices $P \in \mathbb{R}^{m \times d}$ and $Q \in \mathbb{R}^{d \times m}$, we have that $PQ$ and $QP$ have exactly the same \textbf{non-zero} eigenvalues.
\end{lemma}
\begin{proof}
Let $\lambda \neq 0$ an eigenvalues of $PQ$. Then there exists $v \neq 0$, such that $PQv=\lambda v$. Multiplying both sides by $Q$, we get $QP(Qv)=\lambda (Qv)$. We know that $Qv \neq 0$, because then $\lambda$ would be $0$. Thus $\lambda$ is an eigenvalue of $QP$ with eigenvector $Qv$. Thus any non-zero eigenvalue of $PQ$ is also an eigenvalue of $QP$. Switching $P$ and $Q$ in the previous argument implies that any non-zero eigenvalue of $QP$ is also an eigenvalue of $PQ$. Thus, $PQ$ and $QP$ have the same non-zero eigenvalues.
\end{proof}
\begin{corollary}
\label{le:rank}
Given matrices $P \in \mathbb{R}^{m \times d}$ and $Q \in \mathbb{R}^{d \times m}$, we have that
\begin{equation*}
rank(PQ)=rank(QP)=\min \lbrace rank(P),rank(Q) \rbrace
\end{equation*}
\end{corollary}
\begin{lemma}
\label{le:eig of product}
Given a symmetric positive semi-definite matrix $S \in \mathbb{R}^{m \times m}$ and a symmetric positive definite $T \in \mathbb{R}^{m \times m}$ with eigenvalues
\begin{equation*}
\lambda_1(S) \leq ... \leq \lambda_m(S)
\end{equation*}
and
\begin{equation*}
\lambda_1(T) \leq ... \leq \lambda_m(T)
\end{equation*}
we have that
\begin{equation*}
\lambda_k(S) \lambda_1(T) \leq \lambda_k(ST) \leq \lambda_k(S) \lambda_m(T)
\end{equation*}
for any $k=1,...,m$.
\end{lemma}
\begin{proof}
We use the min-max principle for the $k$-th eigenvalue of a matrix $A \in \mathbb{R}^{m \times m}$. This reads
\begin{equation*}
\lambda_k(A)=\min_{\substack{F\subset \mathbb{R}^M \\ \dim(F)=k}} \left( \max_{x\in F\backslash \{0\}} \frac{(Ax,x)}{(x,x)}\right)
\end{equation*}
We know that $\lambda_k(ST)=\lambda_k(\sqrt{T}S\sqrt{T})$.
Since $T$ is symmetric and positive-definite, its square root $\sqrt{T}$ is also symmetric and positive-definite. Thus, we have
\begin{align*}
\lambda_k(ST)=\lambda_k(\sqrt{T}S\sqrt{T})=\min_{\substack{F\subset \mathbb{R}^M \\ \dim(F)=k}} \left( \max_{x\in F\backslash \{0\}} \frac{(\sqrt{T}S\sqrt{T}x,x)}{(x,x)}\right)=\min_{\substack{F\subset \mathbb{R}^n \\ \dim(F)=k}} \left( \max_{x\in F\backslash \{0\}} \frac{(S\sqrt{T}x,\sqrt{T}x)}{(\sqrt{T}x,\sqrt{T}x)}
\frac{(Tx,x)}{(x,x)}\right)
\end{align*}
Thus
\begin{equation*}
\min_{\substack{F\subset \mathbb{R}^n \\ \dim(F)=k}} \left( \max_{x\in F\backslash \{0\}} \frac{(S\sqrt{T}x,\sqrt{T}x)}{(\sqrt{T}x,\sqrt{T}x)}\right) \lambda_{min}(T) \leq \lambda_k(ST) \leq \min_{\substack{F\subset \mathbb{R}^n \\ \dim(F)=k}} \left( \max_{x\in F\backslash \{0\}} \frac{(S\sqrt{T}x,\sqrt{T}x)}{(\sqrt{T}x,\sqrt{T}x)}\right) \lambda_{max}(T)
\end{equation*}
If $ \lbrace e_1,...,e_k \rbrace$ is a basis for $F$, we define $F'=span ( \sqrt{T}^{-1}e_1,...,\sqrt{T}^{-1}e_k )$ and we have
\begin{equation*}
\min_{\substack{F\subset \mathbb{R}^n \\ \dim(F)=k}} \left( \max_{x\in F\backslash \{0\}} \frac{(S\sqrt{T}x,\sqrt{T}x)}{(\sqrt{T}x,\sqrt{T}x)}\right) = \min_{\substack{F'\subset \mathbb{R}^n \\ \dim(F')=k}} \left( \max_{x\in F'\backslash \{0\}} \frac{(Sx,x)}{(x,x)}\right)=\lambda_k(S)
\end{equation*}
and the desired result follows.
\end{proof}
\improvedconditionnumber*
\begin{proof}
Using Lemma \ref{le:eig commute}, we have that the eigenvalues of the $d \times d$ matrix $\nabla^2 f$ are equal to the non-zero eigenvalues of the $m \times m$ matrix $\nabla^2 \ell A A^T$. Using Corollary \ref{le:rank}, we have that the matrix $A A^T$ is of rank $d$, thus the matrix $\nabla^2 \ell A A^T$ is also of rank $d$. This means that it has exactly $m-d$ zero eigenvalues. Exactly the same holds for the matrix $A A^T$. We use also Lemma \ref{le:eig of product} for the positive definite matrix $\nabla^2 \ell$ and the positive semi-definite matrix $A A^T$ and we have:
\begin{itemize}
\item The maximum eigenvalue of the matrix $\nabla^2 f$ is equal to the maximum eigenvalue of the matrix $\nabla^2 \ell A A^T$. For that we have
\begin{equation*}
\lambda_{max}(\nabla^2 \ell A A^T) \leq \lambda_{max}(\nabla^2 \ell) \lambda_{max}(A A^T).
\end{equation*}
Similarly the maximum eigenvalue of $A A^T$ is equal to the maximum one of $A^T A$ and we finally have
\begin{equation*}
\label{eq:max eig}
\lambda_{max}(\nabla^2 f) \leq \frac{\gamma_{\ell}}{n} \lambda_{max}(A^T A)= \gamma_{\ell} \lambda_{max}\left(\frac{1}{n} A^T A \right).
\end{equation*}
\item The minimum eigenvalue of the matrix $\nabla^2 f$ is equal to the eigenvalue of the matrix $\nabla^2 \ell A A^T$ of order $m-d+1$. Using Lemma \ref{le:eig of product}, we have
\begin{equation*}
\lambda_{m-d+1}(\nabla^2 \ell A A^T) \geq \lambda_{min}(\nabla^2 \ell) \lambda_{m-d+1}(A A^T).
\end{equation*}
By using similar arguments as before, we have that
\begin{equation*}
\lambda_{m-d+1}(A A^T)=\lambda_{min}(A^T A).
\end{equation*}
Thus, we finally have
\begin{equation*}
\label{min eig}
\lambda_{min}(\nabla^2 f) \geq \frac{\mu_{\ell}}{n} \lambda_{min}(A^T A)=\mu_{\ell} \lambda_{min} \left( \frac{1}{n} A^T A \right).
\end{equation*}
\end{itemize}
\end{proof}
\section{Gradient Descent with Preconditioning for GLMs}
\label{app:precond_gradient}
Gradient descent for a $\gamma$-smooth and $\mu$-strongly convex function $f(x) =\ell(Ax): \mathbb{R}^d \rightarrow \mathbb{R}$ preconditioned by a matrix $M \in \mathbb{R}^{d \times d}$ reads
\begin{align*}
&x^{(t+1)}=x^{(t)}-\eta M^{-1} \nabla f(x^{(t)}), \\ &x^{(0)} \in \mathbb{R}^d.
\end{align*}
In our setting the matrix $M:=\frac{1}{n} A^T A$ is invertible, because we have assumed that the matrix $A$ is of full rank. The convergence is now improved up to the condition number of $M$. For the proof we follow the technique presented in \cite{chen2019gradient} for (non-preconditioned) gradient descent.
\begin{Proposition}
\label{prop:exact_precond}
The iterates $x^{(t)}$ of the previous algorithm with $\eta=\frac{2}{\mu_{\ell}+\gamma_{\ell}}$ satisfy
\begin{equation*}
\|x^{(t)}-x^*\| \leq \left(1-\frac{1}{\kappa_{\ell}} \right)^t \|x^{(0)}-x^*\|
\end{equation*}
\end{Proposition}
\begin{proof}
Similarly to the previous argument, we have
\begin{align*}
x^{(t+1)}-x^*&=x^{(t)}-\eta M^{-1} \nabla f(x^{(t)})-x^*=(x^{(t)}-x^*)-\eta M^{-1} \left(\int_0^1 \nabla^2 f(x(\xi)) d\xi \right)(x^{(t)}-x^*) \\ &=\left(Id-\eta \int_0^1 M^{-1} \nabla^2 f(x(\xi)) d\xi \right) (x^{(t)}-x^*)
\end{align*}
where
\begin{equation*}
x(\xi)=x^{(t)}+\xi(x^*-x^{(t)})
\end{equation*}
Thus
\begin{equation*}
\|x^{(t+1)}-x^*\| \leq \left \|Id-\eta \int_0^1 M^{-1} \nabla^2 f(x(\xi)) d\xi \right\| \|x^{(t)}-x^*\|\leq \max_{0 \leq \xi \leq 1} \|Id-\eta M^{-1} \nabla^2 f(x(\xi))\| \|x^{(t)}-x^*\|
\end{equation*}
Now we can write
\begin{equation*}
M^{-1} \nabla^2 f(x(\xi))=M^{-1} A^T \nabla^2 \ell (Ax(\xi)) A
\end{equation*}
By Lemma \ref{le:eig commute}, the eigenvalues of the last matrix are exactly the same with the non-zero eigenvalues of the matrix $\nabla^2 \ell (Ax(\xi)) A M^{-1} A^T$. This matrix is $m \times m$ with rank $d$, thus it has exactly $m-d$ zero eigenvalues. The same holds for the matrix $A M^{-1} A^T$. Again by applying Lemma \ref{le:eig commute}, we know that $A M^{-1} A^T$ has $m-d$ eigenvalues equal to $0$ and the others are exactly equal with the ones of $ M^{-1} A^T A=n \textnormal{Id}$, i.e. they are all equal to $n$.
\newline
Thus, we have
\begin{equation*}
\lambda_{max}(M^{-1} A^T \nabla^2 \ell (Ax(\xi)) A )=\lambda_{max}(\nabla^2 \ell (Ax(\xi)) A M^{-1} A^T) \leq \lambda_{max}(\nabla^2 \ell) \lambda_{max}(A M^{-1} A^T) = \frac{\gamma_{\ell}}{n} n = \gamma_{\ell}
\end{equation*}
and
\begin{equation*}
\lambda_{min}(M^{-1} A^T \nabla^2 \ell (Ax(\xi)) A )=\lambda_{m-d+1}(\nabla^2 \ell (Ax(\xi)) A M^{-1} A^T) \geq \lambda_{min}(\nabla^2 \ell) \lambda_{m-d+1}(A M^{-1} A^T) = \frac{\mu_{\ell}}{n} n = \mu_{\ell}
\end{equation*}
by Lemma \ref{le:eig of product}, because $\nabla^2 \ell$ is positive definite and $A M^{-1} A^T$ is positive semi-definite.
\newline
Since we choose $\eta=\frac{2}{\mu_{\ell}+\gamma_{\ell}}$, the maximum eigenvalues of the matrix $\eta M^{-1} \nabla^2 f(x(\xi))$ is $\frac{2 \gamma_{\ell}}{\mu_{\ell}+\gamma_{\ell}}$ and the minimum one is $\frac{2 \mu_{\ell}}{\mu_{\ell}+\gamma_{\ell}}$.
Thus, the maximum eigenvalue of $Id-\eta M^{-1} \nabla^2 f(x(\xi))$ is less or equal than $\max \left\lbrace \frac{2 \gamma_{\ell}}{\mu_{\ell}+\gamma_{\ell}}-1, 1-\frac{2 \mu_{\ell}}{\mu_{\ell}+\gamma_{\ell}} \right \rbrace = \frac{\gamma_{\ell}-\mu_{\ell}}{\gamma_{\ell}+\mu_{\ell}} \leq 1-\frac{1}{\kappa_{\ell}}$. Thus
\begin{equation*}
\|x^{t+1}-x^*\| \leq \left(1-\frac{1}{\kappa_{\ell}}\right)
\|x^{(t)}-x^*\|
\end{equation*}
and by an induction argument we get the desired result.
\end{proof}
\section{Proofs of convergence for GLMs}
\label{app:convergence_GLM}
We prove the convergence result of the preconditioned algorithm for GLMs. We recall firstly the algorithm in compact form:
\begin{algorithm}[H]
\caption{Quantized Preconditioned Gradient Descent for GLM training}
\label{algo:GLM_algorithm}
\begin{algorithmic}[1]
\STATE $\bar M_i=\phi^{-1}\left(Q\left(\phi(M_i), \phi(M_{i_0}), 2 \sqrt{d} n \lambda_{max}(M) , \frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}} \right) \right)$
\STATE $S= \frac{1}{n} \sum_{i=1}^n \bar M_i$
\STATE $\bar M=\phi^{-1} \left(Q(\phi(S),\phi(M_i), \sqrt{d} \left( \frac{\lambda_{min}(M)}{16 \kappa_{\ell}}+2 n \lambda_{max}(M) \right),\frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}}) \right)$
\STATE $x^{(0)} \in \mathbb{R}^d, \max_i \lbrace \|x^{(0)}-x^*\|,\|x^{(0)}-x_i^*\| \rbrace \leq D$
\STATE $v_i^{(0)}=Q \left(\bar M^{-1} \nabla f_i(x^{(0)}),\bar M^{-1} \nabla f_{i_0}(x^{(0)}),4 n \kappa(M) R^{(0)},\frac{\delta {R^{(0)}}}{2} \right)$
\STATE $r^{(0)}=\frac{1}{n} \sum_{i=1}^n v_i^{(0)}$.
\STATE $v^{(0)}=Q \left(r^{(0)},\bar M^{-1} \nabla f_i(x^{(0)}),\left(\frac{\delta}{2}+4 n \kappa(M) \right) R^{(0)},\frac{\delta R^{(0)}}{2} \right)$
\FOR{$t \geq 0$}
\STATE $x^{(t+1)}=x^{(t)}-\eta v^{(t)}$
\STATE $v_i^{(t+1)}=Q \left(\bar M^{-1} \nabla f_i(x^{(t+1)}), v_i^{(t)},4 n \kappa(M) R^{(t+1)}, \frac{\delta R^{(t+1)}}{2}\right)$
\STATE $r^{(t+1)}=\frac{1}{n} \sum_{i=1}^n v_i^{(t+1)}$
\STATE $v^{(t+1)}=Q \left(r^{(t+1)},v^{(t)},(\frac{\delta}{2}+4 n \kappa(M)) R^{(t+1)}, \frac{\delta R^{(t+1)}}{2} \right)$
\ENDFOR
\end{algorithmic}
\end{algorithm}
\begin{lemma}
\label{le:inexact_precond}
Consider the algorithm
\begin{equation*}
x^{(t+1)}=x^{(t)}-\eta \bar M^{-1} \nabla f(x^{(t)})
\end{equation*}
starting from a point $x^{(0)} \in \mathbb{R}^d$ such that $\|x^{(0)}-x^*\| \leq D$, where $\eta=\frac{2}{\mu_{\ell}+\gamma_{\ell}}$ and $\bar M$ is the quantized estimation of $M$ obtained in Algorithm \ref{algo:GLM_algorithm}. Then, the iterates of this algorithm satisfy
\begin{equation*}
\|x^{(t)}-x^*\| \leq \left(1-\frac{1}{2 \kappa_{\ell}} \right)^t D.
\end{equation*}
\end{lemma}
\begin{proof}
We use the same proof technique as in Proposition \ref{prop:exact_precond}, with the difference that now we have the quantized estimation $\bar M$ of $M$ instead of the original:
\begin{align*}
x^{(t+1)}-x^*&=x^{(t)}-\eta \bar M^{-1} \nabla f(x^{(t)})-x^*=(x^{(t)}-x^*)-\eta \bar M^{-1} \left(\int_0^1 \nabla^2 f(x(\xi)) d\xi \right)(x^{(t)}-x^*) \\ &=\left(Id-\eta \int_0^1 \bar M^{-1} \nabla^2 f(x(\xi)) d\xi \right) (x^{(t)}-x^*)
\end{align*}
where
\begin{equation*}
x(\xi)=x^{(t)}+\xi(x^*-x^{(t)}).
\end{equation*}
Thus
\begin{equation*}
\|x^{(t+1)}-x^*\| \leq \left \|Id-\eta \int_0^1 \bar M^{-1} \nabla^2 f(x(\xi)) d\xi \right\| \|x^{(t)}-x^*\|\leq \max_{0\leq \xi \leq 1} \|Id-\eta \bar M^{-1} \nabla^2 f(x(\xi))\| \|x^{(t)}-x^*\|.
\end{equation*}
Now we can write
\begin{equation*}
\bar M^{-1} \nabla^2 f(x(\xi))=M^{-1} \nabla^2 f(x(\xi))+(\bar M^{-1}- M^{-1}) \nabla^2 f(x(\xi))
\end{equation*}
and
\begin{equation*}
\|Id-\eta \bar M^{-1} \nabla^2 f(x(\xi))\| \leq \|Id-\eta M^{-1} \nabla^2 f(x(\xi))\|+\eta \| (\bar M^{-1}- M^{-1}) \nabla^2 f(x(\xi))\|.
\end{equation*}
For the matrix $M^{-1} A^T \nabla \ell^2 (Ax(\xi)) A$ we apply exactly the same argument as in Proposition \ref{prop:exact_precond} and have
\begin{equation*}
\max_{0\leq \xi \leq 1} \|Id-\eta M^{-1} \nabla^2 f(x(\xi))\| \leq \frac{\gamma_{\ell}-\mu_{\ell}}{\gamma_{\ell}+\mu_{\ell}}<1-\frac{1}{\kappa_{\ell}}.
\end{equation*}
For the extra error term, we firstly have to study the quantization error $\|M-\bar M\|$:
\newline
Notice that
\begin{equation*}
\| \phi(M_i)-\phi(M_{i_0}) \| \leq \sqrt{d} \|M_i-M_{i_0}\| \leq \sqrt{d}(\| M_i \|+\| M_{i_0} \|) \leq 2 \sqrt{d} n \lambda_{max}(M)
\end{equation*}
which implies that
\begin{equation*}
\| \phi(\bar M_i)-\phi(M_i) \| \leq \frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}}
\end{equation*}
by the definition of quantization parameters (we have $\lambda_{max}(M_i) \leq n \lambda_{max}(M)$, because $n M=\sum_{i=1}^n M_i$ and every $M_i$ is positive semi-definite). The last inequality implies
\begin{equation*}
\| \bar M_i-M_i \| \leq \frac{\lambda_{min}(M)}{16 \kappa_{\ell}}
\end{equation*}
and
\begin{equation*}
\|S-M\| \leq \frac{1}{n} \sum_{i=1}^n \| \bar M_i-M_i \| \leq \frac{\lambda_{min}(M)}{16 \kappa_{\ell}}.
\end{equation*}
Now we can write
\begin{equation*}
\|\phi(S)-\phi(M_i)\| \leq \sqrt{d} \|S-M_i\| \leq \sqrt{d} (\|S-M\|+\|M-M_i\|) \leq \sqrt{d} \left( \frac{\lambda_{min}(M)}{16 \kappa_{\ell}}+2 n \lambda_{max}(M) \right) \leq 3 n \sqrt{d} \lambda_{max}(M).
\end{equation*}
By the definition of quantization parameters, this implies
\begin{equation*}
\|\phi(\bar M)-\phi(S)\| \leq \frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}}
\end{equation*}
and concequently
\begin{equation*}
\|\bar M-S\| \leq \frac{\lambda_{min}(M)}{16 \kappa_{\ell}}.
\end{equation*}
Then
\begin{equation*}
\|M-\bar M\| \leq \|M-S\|+\|S-\bar M\| \leq \frac{\lambda_{min}(M)}{8 \kappa_{\ell}}.
\end{equation*}
By standard results in perturbation theory, we know that
\begin{equation*}
\lambda_{min}(\bar M) \geq \lambda_{min}(M)-\|M-\bar M\| \geq \lambda_{min}(M)-\frac{\lambda_{min}(M)}{8 \kappa_{\ell}} \geq \frac{\lambda_{min}(M)}{2}.
\end{equation*}
This implies
\begin{equation*}
\| \bar M^{-1} \|=\lambda_{max}(\bar M^{-1})=\frac{1}{\lambda_{min}(\bar M^{-1})} \leq \frac{2}{\lambda_{min}(M)}.
\end{equation*}
Now we have
\begin{align*}
&\max_{0\leq \xi \leq 1} \frac{2}{\mu_{\ell}+\gamma_{\ell}} \|(\bar M^{-1}- M^{-1}) \nabla^2 f(x(\xi))\| \leq \max_{0\leq \xi \leq 1} \frac{2}{\mu_{\ell}+\gamma_{\ell}} \| \bar M^{-1} (M-\bar M) M^{-1} \nabla^2 f(x(\xi))\| \\ & \leq \max_{0\leq \xi \leq 1} \frac{2}{\mu_{\ell}+\gamma_{\ell}} \| \bar M^{-1} (M-\bar M)\| \|M^{-1} \nabla^2 f(x(\xi))\| \leq \frac{2}{\mu_{\ell}+\gamma_{\ell}} \| \bar M^{-1} \| \| M-\bar M \| \gamma_{\ell} \\ & \leq \frac{4}{\lambda_{min}(M)} \frac{\lambda_{min}(M)}{8\kappa_{\ell}} =\frac{1}{2 \kappa_{\ell}}.
\end{align*}
Thus, it holds
\begin{equation*}
\|x^{(t+1)}-x^*\| \leq \left(1-\frac{1}{2 \kappa_{\ell}} \right) \|x^{(t)}-x^*\|
\end{equation*}
which implies
\begin{equation*}
\|x^{(t+1)}-x^*\| \leq \left(1-\frac{1}{2 \kappa_{\ell}} \right)^t \|x^{(0)}-x^*\| \leq \left(1-\frac{1}{2 \kappa_{\ell}} \right)^t D.
\end{equation*}
\end{proof}
We recall the parameters
\begin{align*}
& \xi=1-\frac{1}{2 \kappa_{\ell}}, \\
& K=\frac{2}{\xi}, \\
& \delta=\frac{\xi(1-\xi)}{4}, \\
& R^{(t)}= \frac{\gamma_{\ell}}{2} K \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D.
\end{align*}
\begin{lemma}
The iterates of Algorithm \ref{algo:GLM_algorithm} satisfy the following inequalities:
\begin{align*}
&\| x^{(t)}-x^* \| \leq \left(1-\frac{1}{4\kappa_{\ell}}\right)^t D, \\ & \| \bar M^{-1} \nabla f_i(x^{(t)})- v^{(t)}_i \| \leq \frac{\delta R^{(t)}}{2},
\\& \|\bar M^{-1} \nabla f(x^{(t)})-v^{(t)} \| \leq \delta R^{(t)}.
\end{align*}
\end{lemma}
\begin{proof}
We firstly prove the inequalities for $t=0$. The first one is direct by the definition of $D$. For the second one, we notice that
\begin{align*}
&\| \bar M^{-1} \nabla f_i(x^{(0)})-\bar M^{-1} \nabla f_{i_0}(x^{(0)}) \| \leq \frac{2}{\lambda_{min}(M)} (\|\nabla f_i(x^{(0)})\|+\|\nabla f_{i_0}(x^{(0)})\|) \leq \\ & \frac{2}{\lambda_{min}(M)} (\gamma_i \|x^{(0)}-x_i^*\|+\gamma_{i_0} \|x^{(0)}-x_{i_0}^*\|) \leq 2 \gamma_{\ell} \frac{\lambda_{max}(M_i)}{\lambda_{min}(M)} D + 2 \gamma_{\ell} \frac{\lambda_{max}(M_{i_0})}{\lambda_{min}(M)} D \leq 4 n \kappa(M) R^{(0)}.
\end{align*}
The last inequality follows because $K \geq 2$ and $\lambda_{max}(M_i) \leq n \lambda_{max}(M)$.
\newline
(We recall also that $\| \bar M^{-1} \| \leq \frac{2}{\lambda_{min}(M)}$, because $\lambda_{min}(\bar M) \geq \lambda_{min}(M)-\|M-\bar M\| \geq \lambda_{min}(M)/2$.)
\newline
By the definition of $v_i^{(0)}$, we have
\begin{equation*}
\|v_i^{(0)}-\bar M^{-1} \nabla f_i(x^{(0)})\| \leq \frac{\delta R^{(0)}}{2}.
\end{equation*}
Towards the third inequality at $t=0$, we have
\begin{equation*}
\| r^{(0)}-\bar M^{-1} \nabla f(x^{(0)}) \| \leq \frac{1}{n} \sum_{i=1}^n \|v_i^{(0)}-\bar M^{-1} \nabla f_i(x^{(0)})\| \leq \frac{\delta R^{(0)}}{2}.
\end{equation*}
Also, it holds
\begin{equation*}
\|r^{(0)}-\bar M^{-1} \nabla f_i(x^{(0)})\| \leq \|r^{(0)}-\bar M^{-1} \nabla f(x^{(0)})\|+\|\bar M^{-1} \nabla f(x^{(0)})-\bar M^{-1} \nabla f_i(x^{(0)})\| \leq \left(\frac{\delta}{2} + 4 n \kappa(M) \right) R^{(0)} ,
\end{equation*}
thus, by the definition of $v^{(0)}$,
\begin{equation*}
\|v^{(0)}-r^{(0)}\| \leq \frac{\delta R^{(0)}}{2}
\end{equation*}
and putting everything together, we have
\begin{equation*}
\|v^{(0)}-\bar M^{-1} \nabla f(x^{(0)})\| \leq \|v^{(0)}-r^{(0)}\|+\|r^{(0)}-\bar M^{-1} \nabla f(x^{(0)})\| \leq \frac{\delta R^{(0)}}{2}+\frac{\delta R^{(0)}}{2}=\delta R^{(0)}.
\end{equation*}
Now we assume that the inequalities hold for $t$ and prove that they continue to hold for $t+1$. We start with the first one:
\begin{align*}
\|x^{(t+1)}-x^*\|&=\|x^{(t)}-\eta v^{(t)}+\eta \bar M^{-1} \nabla f(x^{(t)})-\eta \bar M^{-1} \nabla f(x^{(t)})-x^*\| \\& \leq \eta \|\bar M^{-1} \nabla f(x^{(t)})-v^{(t)}\|+\|x^{(t)}-\eta \bar M^{-1} \nabla f(x^{(t)})-x^*\| \\ & \leq \frac{2}{\gamma_{\ell}} \delta R^{(t)}+\xi \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D \\ &= \frac{2}{\gamma_{\ell}} \delta \frac{\gamma_{\ell}}{2} K\left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D+\xi \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D \\ &=\delta K \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D + \xi \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D \\ &= (\delta K + \xi) \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D= \left(1-\frac{1}{4 \kappa_{\ell}} \right)^{t+1} D.
\end{align*}
For the second inequality it suffices to show that
\begin{equation*}
\| \bar M^{-1} \nabla f_i(x^{(t+1)})- v^{(t)}_i \| \leq 4 n \kappa(M) R^{(t+1)}.
\end{equation*}
To that end, we write
\begin{align*}
\| \bar M^{-1} \nabla f_i(x^{(t+1)})- v^{(t)}_i \| & = \| \bar M^{-1} \nabla f_i(x^{(t+1)})- \bar M^{-1}\nabla f_i(x^{(t)})+ \bar M^{-1}\nabla f_i(x^{(t)})- v^{(t)}_i \| \\ & \leq \| \bar M^{-1} \nabla f_i(x^{(t+1)})- \bar M^{-1}\nabla f_i(x^{(t)}) \|+ \|\bar M^{-1}\nabla f_i(x^{(t)})-v^{(t)}_i \| \\ & \leq \gamma_i \| \bar M^{-1} \| \| x^{(t+1)}-x^{(t)} \|+ \delta R^{(t)} \\ & \leq
\gamma_{\ell} \lambda_{max}(M_i) \frac{2}{\lambda_{min}(M)} (\|x^{(t+1)}-x^*\|+\|x^{(t)}-x^*\|)+ \delta R^{(t)} \\ & \leq 4 n \gamma_{\ell} \kappa(M) \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D + \delta \frac{\gamma_{\ell}}{2} K \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D \\ & \leq 2 n (2/K+\delta/4)K \gamma_{\ell} \kappa(M) \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D \\ & \leq 2 n (2/K+\delta K)K \gamma_{\ell} \kappa(M) \left(1-\frac{1}{4 \kappa_{\ell}} \right)^t D \\ & \leq 4 n \kappa(M) R^{(t+1)}.
\end{align*}
Previously we have used again that $\lambda_{max}(M_i) \leq n \lambda_{max}(M)$, because $n M=\sum_{i=1}^n M_i$ and all matrices $M_i$ are positive semi-definite.
\newline
For the last inequality we have
\begin{align*}
\| \bar M^{-1} \nabla f(x^{(t+1)})-r^{(t+1)} \| \leq \frac{1}{n} \sum_{i=1}^n \| \bar M^{-1} \nabla f_i(x^{(t+1)})-v^{(t+1)}_i \| \leq \frac{\delta R^{(t+1)}}{2}
\end{align*}
and
\begin{align*}
\| r^{(t+1)}-v^{(t)} \| &= \| r^{(t+1)}- \bar M^{-1} \nabla f(x^{(t+1)})+\bar M^{-1}\nabla f(x^{(t+1)})-\bar M^{-1}\nabla f(x^{(t)})+\bar M^{-1}\nabla f(x^{(t)})-v^{(t)} \| \\ & \leq \| r^{(t+1)}-\bar M^{-1}\nabla f(x^{(t+1)}) \|+ \| \bar M^{-1}\nabla f(x^{(t+1)})-\bar M^{-1}\nabla f(x^{(t)}) \| + \| \bar M^{-1} \nabla f(x^{(t)})-v^{(t)} \| \\ & \leq
\frac{\delta R^{(t+1)}}{2}+\gamma \frac{2}{\lambda_{min}(M)} \| x^{(t+1)}-x^{(t)} \|+ \delta R^{(t)} \\ & \leq \frac{\delta R^{(t+1)}}{2}+4 \gamma_{\ell} \kappa(M) (\| x^{(t+1)}-x^*\|+\| x^{(t)}-x^* \|)+ \delta R^{(t)} \\ & \leq
\frac{\delta R^{(t+1)}}{2}+4 \kappa(M) R^{(t+1)} \leq \left(4 n \kappa(M) + \frac{\delta}{2} \right) R^{(t+1)}.
\end{align*}
The last part of the inequality follows from the same argument used in deriving the second one.
\newline
The last inequality implies that
\begin{equation*}
\| v^{(t+1)}-r^{(t+1)} \| \leq \frac{\delta R^{(t+1)}}{2}.
\end{equation*}
Thus, putting everything together, we have
\begin{equation*}
\| \bar M^{-1} \nabla f(x^{(t+1)})-v^{(t+1)} \| \leq \|\bar M^{-1} \nabla f(x^{(t+1)})-r^{(t+1)} \|+\| r^{(t+1)}-v^{(t+1)}\| \leq \frac{\delta R^{(t+1)}}{2}+\frac{\delta R^{(t+1)}}{2} = \delta R^{(t+1)}.
\end{equation*}
\end{proof}
\convergenceGLM*
\begin{proof}
The inequality for the convergence rate of the distance of the iterates from the minimizer holds from the previous lemma. We now turn our interest to the total communication cost.
We start from the quantization of the matrix $M$:
\newline
The communication cost for encoding each $M_i$ and decoding in the master node is
\begin{equation*}
\mathcal{O} \left( \frac{d (d+1)}{2} \textnormal{log}_2 \left(\frac{2 \sqrt{d} n \lambda_{max}(M)}{\frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}}} \right) \right)=\mathcal{O}(d^2 \textnormal{log}_2 (\sqrt{d} n \kappa_{\ell} \kappa(M))).
\end{equation*}
The communication cost of encoding $S$ in the master node and then decode back in every machine is
\begin{equation*}
\mathcal{O} \left( \frac{d (d+1)}{2} \textnormal{log}_2 \left(\frac{3 \sqrt{d} n \lambda_{max}(M)}{\frac{\lambda_{min}(M)}{16 \sqrt{2} \kappa_{\ell}}} \right) \right)=\mathcal{O}(d^2 \textnormal{log}_2 (\sqrt{d} n \kappa_{\ell} \kappa(M))).
\end{equation*}
Since we have $n$-many communications of each kind, the total communication cost is
\begin{equation*}
b_m=\mathcal{O}(n d^2 \textnormal{log}_2 (\sqrt{d}n \kappa_{\ell} \kappa(M))=\mathcal{O}(n d^2 \textnormal{log} (\sqrt{d}n \kappa_{\ell} \kappa(M)).
\end{equation*}
The communication cost of quantizing the descent direction $v^{(t)}$ at step $t \geq 0$ is at most
\begin{equation*}
\mathcal{O} \left(n d \log_2 \frac{4 n \kappa(M)}{\delta/2} \right)= \mathcal{O} \left(n d \log \frac{n \kappa(M)}{\delta} \right)
\end{equation*}
for encoding the local descent directions and
\begin{equation*}
\mathcal{O} \left(n d \log_2 \frac{4 n \kappa(M)+\frac{\delta}{2}}{\delta/2} \right) \leq \mathcal{O} \left(n d \log \frac{9 n \kappa(M)}{\delta} \right)= \mathcal{O} \left(n d \log \frac{ n \kappa(M)}{\delta} \right)
\end{equation*}
for decoding back.
Since we have
\begin{equation*}
\frac{1}{\delta} = \frac{4}{\xi(1-\xi)} = \frac{4}{\frac{1}{2 \kappa_{\ell}} \left(1-{\frac{1}{2 \kappa_{\ell}}} \right)} \leq 16 \kappa_{\ell},
\end{equation*}
we can bound the total communication cost by
\begin{equation*}
b=\mathcal{O} \left(n d \log (n \kappa_{\ell} \kappa(M) \right).
\end{equation*}
We have $f(x^{(t)})-f(x^*) \leq \epsilon$ if $\|x^{(t)}-x^*\| \leq \sqrt{\frac{2 \epsilon}{\gamma}}$, thus we reach accuracy $\epsilon$ in terms of function values in at most $t=2 \kappa_{\ell} \log \frac{\gamma D^2}{2 \epsilon}$ and putting everything together we find the total communication cost for quantizing the descent directions along the whole optimization process to be
\begin{equation*}
\mathcal{O} \left(n d \kappa_{\ell} \log (n \kappa_{\ell} \kappa(M)) \log \frac{\gamma D^2}{2 \epsilon}\right).
\end{equation*}
Thus, the total communication cost in number of bits is obtained by summing the cost for matrix and descent direction quantization:
\begin{equation*}
b= \mathcal{O} \left(n d^2 \log \left(\sqrt{d} n \kappa_{\ell} \kappa(M) \right) \right) + \mathcal{O} \left(n d \kappa_{\ell} \log (n \kappa_{\ell} \kappa(M)) \log \frac{\gamma D^2}{\epsilon}\right).
\end{equation*}
\end{proof}
\section{Proofs for Quantized Newton's Method}
\label{app:quant_newton}
We firstly recall Quantized Newton's method in a compact form as we did also for GLMs:
\begin{algorithm}[H]
\caption{Quantized Newton's Method}
\label{algo:Newton}
\begin{algorithmic}[1]
\STATE $x^{(0)} \in \mathbb{R}^d, \max_i \lbrace \|x^{(0)}-x^*\|,\|x^{(0)}-x_i^*\| \rbrace \leq \frac{\alpha \mu}{2 \sigma}$
\STATE $H_0^i= \phi^{-1}\left(Q \left(\phi(\nabla^2 f_i(x^{(0)})), \phi(\nabla^2 f_{i_0}(x^{(0)})), 2 \sqrt{d} \gamma, \frac{G^{(0)}}{2 \sqrt{2}\kappa} \right)\right)$
\STATE $S_0=\frac{1}{n} \sum_{i=1}^n H_0^i$
\STATE $H_0= \phi^{-1}\left(Q \left(\phi(S_0), \phi(\nabla^2 f_i(x^{(0)})), \sqrt{d} \left( \frac{G^{(0)}}{2 \kappa} + 2 \gamma \right) , \frac{G^{(0)}}{2 \sqrt{2} \kappa} \right)\right)$
\STATE $v_i^{(0)}=Q \left(H_0^{-1} \nabla f_i(x^{(0)}), H_0^{-1} \nabla f_{i_0} (x^{(0)}),4 \kappa P^{(0)},\frac{\theta P^{(0)}}{2} \right)$
\STATE $p^{(0)}=\frac{1}{n} \sum_{i=1}^n v_i^{(0)}$
\STATE $v^{(0)}=Q \left(P^{(0)}, H_0^{-1} \nabla f_i (x^{(0)}),\left(\frac{\theta}{2}+4 \kappa \right) P^{(0)}, \frac{\theta P^{(0)}}{2} \right)$
\FOR{$t \geq 0$}
\STATE $x^{(t+1)}=x^{(t)}-v^{(t)}$
\STATE $H_{t+1}^i=\phi^{-1}\left(Q \left(\phi(\nabla^2 f_i(x^{(t+1)})), \phi(H_t^i), \frac{10 \sqrt{d}}{1+\alpha} G^{(t+1)}, \frac{G^{(t+1)}}{2 \sqrt{2} \kappa} \right)\right)$
\STATE $S_{t+1}=\frac{1}{n} \sum_{i=1}^{n} H_{t+1}^i$
\STATE $H_{t+1}=\phi^{-1} \left(Q \left(\phi(S_{t+1}), \phi(H_t) , \sqrt{d} \left( \frac{1}{2 \kappa}+ \frac{10}{1+\alpha} \right) G^{(t+1)} , \frac{G^{(t+1)}}{2 \sqrt{2} \kappa} \right) \right)$
\STATE $v^{(t+1)}_i=Q \left(H_{t+1}^{-1} \nabla f_i(x^{(t+1)}), v^{(t)}_i, 11 \kappa P^{(t+1)} , \frac{\theta P^{(t+1)}}{2} \right)$
\STATE $p^{(t+1)}=\frac{1}{n} \sum_{i=1}^n v_i^{(t+1)}$
\STATE $v^{(t+1)}=Q \left(r^{(t+1)}, v^{(t)},\left(\frac{\theta}{2}+11 \kappa \right) P^{(t+1)}, \frac{\theta P^{(t+1)}}{2} \right)$
\ENDFOR
\end{algorithmic}
\end{algorithm}
We recall the parameters
\begin{align*}
&G^{(t)}=\frac{\mu}{4} \alpha \left(\frac{1+\alpha}{2} \right)^t, \\
&\alpha \geq 2 \frac{\sigma}{\mu} \max_i \lbrace \|x^{(0)}-x^*\|,\|x^{(0)}-x_i^*\| \rbrace \\
&\theta=\frac{\alpha(1-\alpha)}{4}, \\
& K=\frac{2}{\alpha}, \\
& P^{(t)}= \frac{\mu}{2 \sigma} K \alpha \left( \frac{1+\alpha}{2}\right)^t.
\end{align*}
\begin{restatable}{lemma}{newtondescquant}
\label{le:desc_direction_newton}
The iterates $x^{(t)}$ of the quantized Newton's algorithm satisfy the inequalities
\begin{align*}
& \|x^{(t)}-x^*\| \leq \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t, \\
& \|H_t^i-\nabla^2 f_i(x^{(t)})\| \leq \frac{G^{(t)}}{2 \kappa}, \\
& \|H_t- \nabla^2 f(x^{(t)})\| \leq \frac{G^{(t)}}{\kappa}, \\
& \|H_t^{-1} \nabla f_i(x^{(t)})-v^{(t)}_i\| \leq \frac{\theta P^{(t)}}{2}, \\
& \|H_t^{-1} \nabla f(x^{(t)})-v^{(t)}\| \leq \theta P^{(t)}.
\end{align*}
\end{restatable}
\begin{proof}
We firstly prove that the inequalities hold at $t=0$. The first one is trivial by the choice of $x^{(0)}$.
For the second one, it suffices to show that
\begin{equation*}
\| \phi(H_0^i)-\phi(\nabla^2 f_i(x^{(0)})) \| \leq \frac{G^{(0)}}{2\sqrt{2} \kappa}
\end{equation*}
by Lemma \ref{le:norm_distortion},
and for that suffices
\begin{equation*}
\| \phi(\nabla^2 f_i(x^{(0)}))-\phi(\nabla^2 f_{i_0}(x^{(0)})) \| \leq 2 \sqrt{d} \gamma,
\end{equation*}
which is indeed the case because
\begin{align*}
&\| \phi(\nabla^2 f_i(x^{(0)}))-\phi(\nabla^2 f_{i_0}(x^{(0)})) \| \leq \sqrt{d} \|\nabla^2 f_i(x^{(0)})-\nabla^2 f_{i_0}(x^{(0)})\| \leq \sqrt{d} (\|\nabla^2 f_i(x^{(0)})\|+\|\nabla^2 f_{i_0}(x^{(0)})\|) \leq 2 \sqrt{d} \gamma,
\end{align*}
again using Lemma \ref{le:norm_distortion}.
For the third inequality at $t=0$, we have
\begin{equation*}
\| \nabla^2 f(x^{(0)}) - S_0 \| \leq \frac{1}{n} \sum_{i=1}^n \|\nabla^2 f_i(x^{(0)}) - H_0^i \| \leq \frac{G^{(0)}}{2 \kappa}.
\end{equation*}
We need also $\| S_0 - H_0 \| \leq \frac{G^{(0)}}{2 \kappa}$ and for that it suffices $\| \phi(S_0) - \phi(H_0) \| \leq \frac{G^{(0)}}{2 \sqrt{2} \kappa}$, which follows from
\begin{equation*}
\| \phi(S_0) - \phi(\nabla^2 f_i(x^{(0)})) \| \leq \sqrt{d} \left(\frac{G^{(0)}}{2 \kappa} + 2 \gamma \right).
\end{equation*}
In order to show the latter, we write
\begin{align*}
&\| \phi(S_0) - \phi(\nabla^2 f_i(x^{(0)})) \| \leq \sqrt{d} \| S_0-\nabla^2 f_i(x^{(0)}) \| \leq \sqrt{d} (\| S_0-\nabla^2 f(x^{(0)}) \|+\| \nabla^2 f(x^{(0)})-\nabla^2 f_i(x^{(0)}) \|) \\ & \leq \sqrt{d} \left(\frac{G^{(0)}}{2 \kappa} + 2 \gamma \right).
\end{align*}
For the fourth one it suffices to show that
\begin{equation*}
\|H_0^{-1} \nabla f_i(x^{(0)})-H_0^{-1} \nabla f_{i_0}(x^{(0)})\| \leq 4 \kappa P^{(0)}.
\end{equation*}
Indeed,
\begin{align*}
&\|H_0^{-1} \nabla f_i(x^{(0)})-H_0^{-1} \nabla f_{i_0}(x^{(0)})\| \leq \|H_0^{-1}\| (\|\nabla f_i(x^{(0)})\|+\|\nabla f_{i_0}(x^{(0)})\|) \leq \frac{2}{\mu} (\gamma \|x^{(0)}-x_i^*\|+ \gamma \|x^{(0)}-x_{i_0}^*\|) \\ & \leq 4 \frac{\gamma}{\mu} K \frac{\mu}{2 \sigma} \alpha= 4 \kappa P^{(0)}.
\end{align*}
In the previous inequality we used that $\|H_0^{-1}\| \leq \frac{2}{\mu}$ and this can be seen as follows:
\begin{align*}
\|H_0^{-1}\|=\frac{1}{\lambda_{min}(H_0)} \leq \frac{1}{\lambda_{min}(\nabla^2 f(x^{(0)}))-\frac{G^{(0)}}{\kappa}} \leq \frac{1}{\mu-\frac{\mu}{4} \alpha} \leq \frac{2}{\mu}.
\end{align*}
For the fifth inequality at $t=0$, we have
\begin{equation*}
\|H_0^{-1} \nabla f(x^{(0)})-p^{(0)}\| \leq \frac{1}{n} \sum_{i=1}^n \|H_0^{-1} \nabla f_i(x^{(0)})-v_i^{(0)}\| \leq \frac{\theta P^{(0)}}{2}.
\end{equation*}
We need also
\begin{equation*}
\| v^{(0)}-p^{(0)} \| \leq \frac{\theta P^{(0)}}{2}.
\end{equation*}
For that it suffices to show that
\begin{equation*}
\|p^{(0)}-H_0^{-1} \nabla f_i(x^{(0)})\| \leq \left(\frac{\theta}{2}+4 \kappa \right) P^{(0)}.
\end{equation*}
Indeed
\begin{align*}
&\|p^{(0)}-H_0^{-1} \nabla f_i(x^{(0)})\| \leq \|p^{(0)}-H_0^{-1} \nabla f(x^{(0)})\|+\|H_0^{-1} \nabla f(x^{(0)})-H_0^{-1} \nabla f_i(x^{(0)})\| \\ & \leq \frac{\theta P^{(0)}}{2}+\frac{2}{\mu} (\gamma \|x^{(0)}-x^*\|+ \gamma \|x^{(0)}-x_i^*\|) \leq \frac{\theta P^{(0)}}{2}+ 4 \kappa P^{(0)}=\left(\frac{\theta}{2}+4 \kappa \right) P^{(0)}.
\end{align*}
Now we assume that the inequalities hold for $t$ and wish to prove that they also hold for $t+1$. We start with an auxiliary result regarding taking a Newton iterate using the quantized version of the Hessian but the exact gradient:
\begin{equation*}
\|x^{(t)}-H_t^{-1} \nabla f(x^{(t)})-x^*\| \leq \alpha \|x^{(t)}-x^*\|.
\end{equation*}
For proving that we start by writing
\begin{align*}
&x^{(t)}-H_t^{-1} \nabla f(x^{(t)})-x^*=(x^{(t)}-x^*)-H_t^{-1} \left(\int_0^1 \nabla^2 f(x(\xi)) d\xi \right)(x^{(t)}-x^*) = \\ &\left(Id- \int_0^1 H_t^{-1} \nabla^2 f(x(\xi)) d\xi \right) (x^{(t)}-x^*),
\end{align*}
where
\begin{equation*}
x(\xi)=x^{(t)}+\xi(x^*-x^{(t)}).
\end{equation*}
Thus
\begin{equation*}
\|x^{(t)}-H_t^{-1} \nabla f(x^{(t)})-x^*\| \leq \left \|Id-\int_0^1 H_t^{-1} \nabla^2 f(x(\xi)) d\xi \right\| \|x^{(t)}-x^*\|\leq \max_{0 \leq \xi \leq 1} \|Id-H_t^{-1} \nabla^2 f(x(\xi))\| \|x^{(t)}-x^*\|.
\end{equation*}
Now we need to deal with the quantity $\max_{0 \leq \xi \leq 1} \|Id-H_t^{-1} \nabla^2 f(x(\xi))\|$.
We first write bound $\| H_t^{-1} \|$:
\begin{equation*}
\|H_t^{-1}\|=\frac{1}{\lambda_{min}(H_t)} \leq \frac{1}{\lambda_{min}(\nabla^2 f(x_t)-\|\nabla^2 f(x_t))-H_t\|} \leq \frac{1}{\mu-\frac{G^{(t)}}{\kappa}}.
\end{equation*}
\newline
Now, we have
\begin{equation*}
\frac{G^{(t)}}{\kappa} \leq \frac{\mu}{2}
\end{equation*}
and the result follows. This happens if
\begin{equation*}
\frac{1}{\kappa} \leq \frac{2}{\alpha}
\end{equation*}
which holds always true, because $\frac{1}{\kappa},\alpha<1$.
Thus
\begin{equation*}
\|H_t^{-1}\| \leq \frac{2}{\mu}.
\end{equation*}
Second, we bound the quantity $\|\nabla^2 f(x^{(t)})^{-1}-H_t^{-1}\|$:
\begin{equation*}
\|\nabla^2 f(x^{(t)})^{-1}-H_t^{-1}\|= \|\nabla^2 f(x^{(t)})^{-1}(\nabla^2 f(x^{(t)})-H_t)H_t^{-1}\| \leq \|\nabla^2 f(x^{(t)})-H_t\| \|\nabla^2 f(x^{(t)}) ^{-1}\| \|H_t^{-1}\| \leq \frac{G^{(t)}}{\kappa} \frac{1}{\mu} \frac{2}{\mu}=\frac{2}{\mu^2} \frac{G^{(t)}}{\kappa}.
\end{equation*}
Using that and the fact that $f$ is $\mu$-strongly convex, $\gamma$-smooth, with a $\sigma$-Lipschitz Hessian, we get
\begin{align*}
&\max_{0 \leq \xi \leq 1} \|Id-H_t^{-1} \nabla^2 f(x(\xi))\| = \max_{0 \leq \xi \leq 1} \|Id-\nabla^2 f(x^{(t)}) ^{-1} \nabla^2 f(x(\xi))+(\nabla^2 f(x^{(t)}) ^{-1} -H_t^{-1}) \nabla^2 f(x(\xi)) \| \leq \\ & \max_{0 \leq \xi \leq 1} \|Id-\nabla^2 f(x^{(t)}) ^{-1} \nabla^2 f(x(\xi)) \|+\max_{0 \leq \xi \leq 1} \|(\nabla^2 f(x^{(t)}) ^{-1} -H_t^{-1}) \nabla^2 f(x(\xi)) \| \leq \\ & \max_{0 \leq \xi \leq 1} \|\nabla^2 f(x^{(t)})^{-1} (\nabla^2 f(x^{(t)})- \nabla^2 f(x(\xi))) \|+\|\nabla^2 f(x^{(t)}) ^{-1} -H_t^{-1}\|\max_{0 \leq \xi \leq 1} \| \nabla^2 f(x(\xi)) \| \leq \\ &
\frac{\sigma}{\mu} \| x^{(t)}-x^* \| + \frac{2}{\mu^2} \frac{G^{(t)}}{\kappa} \gamma.
\end{align*}
Thus, we finally get
\begin{align*}
\|x^{(t)}-H_t^{-1} \nabla f(x^{(t)})-x^*\| & \leq \frac{\sigma}{\mu} \|x^{(t)}-x^*\|^2+ G^{(t)} \frac{2}{\mu} \|x^{(t)}-x^*\| \\& \leq \frac{\sigma}{\mu} \|x^{(t)}-x^*\| \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t + \frac{\mu}{4} \alpha \left(\frac{1+\alpha}{2} \right)^t \frac{2}{\mu} \|x^{(t)}-x^*\| \\ & \leq \alpha \left(\frac{1+\alpha}{2} \right)^t \|x^{(t)}-x^*\| \leq \alpha \|x^{(t)}-x^*\|,
\end{align*}
which is the desired result.
Now we pass to the exact iterate of our algorithm. Using the induction hypothesis and the previous inequality, we have
\begin{align*}
\|x^{(t+1)}-x^*\| &=\|x^{(t)}-v^{(t)}+H_t^{-1} \nabla f(x^{(t)})-H_t^{-1} \nabla f(x^{(t)})-x^*\| \\ & \leq
\|H_t^{-1} \nabla f(x^{(t)})-v^{(t)}\|+\|x^{(t)}-H_t^{-1} \nabla f(x^{(t)})-x^*\| \\ & \leq \theta P^{(t)} + \alpha \|x^{(t)}-x^*\| \\ & = \theta \frac{\mu}{2 \sigma} K \alpha \left(\frac{1+\alpha}{2} \right)^t + \alpha \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t\\ &=(\theta K+\alpha) \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t \\ &= \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^{t+1}.
\end{align*}
which is what we need.
For the second inequality it suffices to prove that
\begin{equation*}
\| \phi(H_{t+1}^i)-\phi(\nabla^2 f_i(x^{(t+1)})) \| \leq \frac{G^{(t+1)}}{2 \sqrt{2} \kappa}
\end{equation*}
and for that it suffices
\begin{equation*}
\|\phi(\nabla^2 f_i(x^{(t+1)}))-\phi(H_t^i)\| \leq \frac{10 \sqrt{d}}{1+\alpha} G^{(t+1)}.
\end{equation*}
We indeed have
\begin{align*}
\|\phi(\nabla^2 f_i(x^{(t+1)}))-\phi(H_t^i)\| & \leq \|\phi(\nabla^2 f_i(x^{(t+1)}))-\phi(\nabla^2 f_i(x^{(t)}))+\phi(\nabla^2 f_i(x^{(t)}))-\phi(H_t^i) \| \\ & \leq \|\phi(\nabla^2 f_i(x^{(t+1)}))-\phi(\nabla^2 f_i(x^{(t)}))\|+\|\phi(\nabla^2 f_i(x^{(t)}))-\phi(H_t^i) \| \\ & \leq \sqrt{d} (\|\nabla^2 f_i(x^{(t+1)})-\nabla^2 f_i(x^{(t)})\|+\|\nabla^2 f_i(x^{(t)})-H_t^i\|) \\ & \leq \sqrt{d} \left(\sigma \|x^{(t+1)}-x^{(t)}\|+\frac{ G^{(t)}}{\kappa} \right) \\ & \leq
\sqrt{d}\left(2 \sigma \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t
+\frac{1}{\kappa} \frac{\mu}{4} \alpha \left(\frac{1+\alpha}{2} \right)^t \right) \\ & \leq \sqrt{d} \frac{5 \mu}{4} \alpha \left(\frac{1+\alpha}{2} \right)^t = \sqrt{d} \frac{5 \mu}{4 \frac{1+\alpha}{2}} \alpha \left(\frac{1+\alpha}{2} \right)^{t+1} = \frac{10 \sqrt{d}}{1+\alpha} G^{(t+1)}.
\end{align*}
For the third inequality, we have
\begin{equation*}
\|\nabla^2 f(x^{(t+1)})-S_{t+1}\| \leq \frac{1}{n} \sum_{i=1}^{n} \|\nabla^2 f_i(x^{(t+1)})-H_{t+1}^i\| \leq \frac{G^{(t+1)}}{2 \kappa}.
\end{equation*}
Now it suffices to prove
\begin{equation*}
\|S_{t+1}-H_{t+1}\| \leq \frac{G^{(t+1)}}{2 \kappa}
\end{equation*}
which holds if
\begin{equation*}
\|\phi(S_{t+1})-\phi(H_{t+1})\| \leq \frac{ G^{(t+1)}}{2 \sqrt{2} \kappa}
\end{equation*}
and for that suffices
\begin{equation*}
\|\phi(S_{t+1})-\phi(H_t)\| \leq \sqrt{d} \left( \frac{1}{2 \kappa}+\frac{10}{1+\alpha} \right) G^{(t+1)}.
\end{equation*}
We now have
\begin{align*}
\|\phi(S_{t+1})-\phi(H_t)\| &\leq \sqrt{d} \|S_{t+1}-H_t\| \leq \sqrt{d} \|S_{t+1}-\nabla^2 f(x^{(t+1)})+\nabla^2 f(x^{(t+1)})-\nabla^2 f(x^{(t)})+\nabla^2 f(x^{(t)})-H_t\| \\ & \leq \sqrt{d} (\|S_{t+1}-\nabla^2 f(x^{(t+1)})\|+\|\nabla^2 f(x^{(t+1)})-\nabla^2 f(x^{(t)})\|+\|\nabla^2 f(x^{(t)})-H_t\|) \\ & \leq \sqrt{d} \left (\frac{ G^{(t+1)}}{2 \kappa}+\sigma \|x^{(t+1)}-x^{(t)}\|+\frac{G^{(t)}}{\kappa} \right) \leq \sqrt{d} \frac{G^{(t+1)}}{2 \kappa}+\frac{10 \sqrt{d}}{1+\alpha} G^{(t+1)} \\ & =\sqrt{d} \left( \frac{1}{2 \kappa}+\frac{10}{1+\alpha} \right) G^{(t+1)}
\end{align*}
which concludes the induction.
For the fourth inequality it suffices to prove
\begin{equation*}
\|H_{t+1}^{-1} \nabla f_i(x^{(t+1)})-v^{(t)}_i\| \leq 11 \kappa P^{(t+1)}.
\end{equation*}
To that end, we use $\gamma$-smoothness of $f_i$, the bound $\| x^{(t)}-x_i^* \| \leq \| x^{(t)}-x^* \|+\|x^*-x_i^*\| \leq \|x^{(0)}-x^*\|+\|x^{(0)}-x^*\|+\|x^{(0)}-x_i^*\| \leq \frac{3 \mu}{2 \sigma} \alpha $, the fact that $\|H_t^{-1}\|,\|H_{t+1}^{-1}\| \leq \frac{2}{\mu}$ and the induction hypothesis. Also, we use that $alpha<1$, $\kappa \geq 1$ and $K \geq 2$.
Indeed, we have
\begin{align*}
&\|H_{t+1}^{-1} \nabla f_i(x^{(t+1)})-v^{(t)}_i\| = \|H_{t+1}^{-1} \nabla f_i(x^{(t+1)})-H_{t+1}^{-1} \nabla f_i(x^{(t)})+ H_{t+1}^{-1} \nabla f_i(x^{(t)})- H_t^{-1} \nabla f_i(x^{(t)}) + H_t^{-1} \nabla f_i(x^{(t)})- v^{(t)}_i \| \\ &\leq
\|H_{t+1}^{-1} \nabla f_i(x^{(t+1)})-H_{t+1}^{-1} \nabla f_i(x^{(t)})\|+\|H_{t+1}^{-1}-H_t^{-1}\| \| \nabla f_i(x^{(t)})\| + \| H_t^{-1} \nabla f_i(x^{(t)})-v^{(t)}_i \| \\ & \leq \frac{2}{\mu} \gamma \|x^{(t+1)}-x^{(t)}\|+ \|H_{t+1}^{-1}\| \| H_t^{-1}\| \| H_{t+1}-H_t\| \gamma_i \| x^{(t)}-x_i^*\|+ \theta P^{(t)} \\ & \leq
2 \frac{\gamma}{\mu} \|x^{(t+1)}-x^{(t)}\| +\frac{4}{\mu^2} (\| H_{t+1}-\nabla^2 f(x^{(t+1)}) \|+\|\nabla^2 f(x^{(t+1)}) -\nabla^2 f(x^{(t)})\|+\|\nabla^2 f(x^{(t)})-H_t\|) \gamma \frac{3 \mu}{2 \sigma} \alpha +\theta P^{(t)} \\ & \leq 2 \frac{\gamma}{\mu} \|x^{(t+1)}-x^{(t)}\|+\frac{4}{\mu^2} \left(\frac{G^{(t+1)}}{\kappa}+\frac{G^{(t)}}{\kappa} + \sigma \|x^{(t+1)}-x^{(t)}\| \right) \gamma \frac{3\mu}{2 \sigma} \alpha+ \theta P^{(t)} \\ & \leq
4 \kappa \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t+ \frac{4}{\mu^2} \left(2 \frac{\mu}{4 \kappa} \alpha \left(\frac{1+\alpha}{2} \right)^t + 2 \sigma \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t \right) \gamma \frac{3\mu}{2 \sigma} \alpha + \theta \frac{\mu}{2 \sigma} K \alpha \left( \frac{1+\alpha}{2}\right)^t \\ & =
4 \kappa \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t + \frac{12 \gamma}{\mu^2} \left( \frac{\mu}{2 \kappa} \alpha + \mu \alpha \right) \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t + \theta \frac{\mu}{2 \sigma} K \alpha \left( \frac{1+\alpha}{2}\right)^t \\ & = 4 \kappa \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t + 12 \kappa \left( \frac{\alpha}{2 \kappa} + \alpha \right) \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t + \theta \frac{\mu}{2 \sigma} K \alpha \left( \frac{1+\alpha}{2}\right)^t
\\ & \leq (4 \kappa+ 6+12\kappa+ \theta K) \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t \leq (22 \kappa+\theta K) \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t \leq 11\kappa(2/K+\theta)K \frac{\mu}{2 \sigma} \alpha \left(\frac{1+\alpha}{2} \right)^t\\&=11 \kappa K\alpha \frac{\mu}{2 \sigma} \left(\frac{1+\alpha}{2} \right)^{t+1} \leq 11 \kappa P^{(t+1)}.
\end{align*}
For the third inequality, we have
\begin{equation*}
\|H_{t+1}^{-1} \nabla f(x^{(t+1)})-p^{(t+1)}\| \leq \frac{1}{n} \sum_{i=1}^n \| H_{t+1}^{-1} \nabla f_i(x^{(t+1)})- v^{(t+1)}_i\| \leq \frac{\theta P^{(t+1)}}{2}.
\end{equation*}
We want to prove also that
\begin{equation*}
\|p^{(t+1)}-v^{(t+1)}\| \leq \frac{\theta P^{(t+1)}}{2}.
\end{equation*}
For that it suffices to show that
\begin{align*}
\|p^{(t+1)}-v^{(t)}\| \leq \left(\frac{\theta}{2}+11 \kappa \right) P^{(t+1)}.
\end{align*}
We have
\begin{align*}
\|p^{(t+1)}-v^{(t)}\| &\leq \|p^{(t+1)}-H_{t+1}^{-1} \nabla f(x^{(t+1)})+H_{t+1}^{-1} \nabla f(x^{(t+1)})-H_{t+1}^{-1} \nabla f(x^{(t)})\\ &+H_{t+1}^{-1} \nabla f(x^{(t)})-H_t^{-1} \nabla f(x^{(t)})+H_t^{-1} \nabla f(x^{(t)})-v^{(t)}\| \\ & \leq
\|p^{(t+1)}-H_{t+1}^{-1} \nabla f(x^{(t+1)})\|+\|H_{t+1}^{-1} \nabla f(x^{(t+1)})-H_{t+1}^{-1} \nabla f(x^{(t)})\|\\&+\|H_{t+1}^{-1} \nabla f(x^{(t)})-H_t^{-1} \nabla f(x^{(t)})\|+\|H_t^{-1} \nabla f(x^{(t)})-v^{(t)}\| \\ & \leq \frac{\theta P^{(t+1)}}{2} + \frac{2}{\mu} \gamma \|x^{(t+1)}-x^{(t)}\|+ \|H_{t+1}^{-1}\| \| H_t^{-1}\| \| H_{t+1}-H_t\| \| \nabla f(x^{(t)})\|+ \theta P^{(t)} \\ & \leq \left(\frac{\theta}{2}+11 \kappa \right) P^{(t+1)}
\end{align*}
which completes the induction by the same argument as in the previous derivation.
\end{proof}
\maintheorem*
\begin{proof}
The claim about the convergence of the iterates follows easily by applying Lemma \ref{le:desc_direction_newton} with $\alpha=\frac{1}{2}$.
\newline
This means that we achieve $\|x^{(t)}-x^*\| \leq \epsilon$ in at most
\begin{equation*}
t=\frac{1}{1-\frac{3}{4}} \log \frac{ \frac{\mu}{4 \sigma} }{\epsilon}=4 \log \frac{\mu}{4 \sigma \epsilon}
\end{equation*}
many iterates.
We have $f(x^{(t)})-f^* \leq \epsilon$, if $\|x^{(t)}-x^*\| \leq \sqrt{\frac{2 \epsilon}{\gamma}}$, thus in at most
\begin{equation*}
t=4 \log \frac{\gamma \mu^2}{32 \sigma^2 \epsilon}
\end{equation*}
many iterates.
\newline
For the communication cost, we have that in order to pursue Hessian quantization at $t=0$, we need
\begin{equation*}
\mathcal{O}\left(n \frac{d(d+1)}{2} \log \frac{2 \sqrt{d} \gamma}{G^{(0)}/2\sqrt{2} \kappa} \right) = \mathcal{O}\left(n \frac{d(d+1)}{2} \log \frac{2 \sqrt{d} \gamma}{\frac{\mu^2}{16 \sqrt{2} \gamma}} \right) = \mathcal{O}\left(n \frac{d(d+1)}{2} \log \frac{\sqrt{d} \gamma^2}{\mu^2} \right) \leq \mathcal{O}\left(n d^2 \log \left( \sqrt{d} \kappa \right) \right)
\end{equation*}
many bits for encoding the local Hessian matrices
\begin{equation*}
\mathcal{O} \left(n \frac{d(d+1)}{2} \log \frac{\sqrt{d} \left( \frac{1}{2 \kappa} G^{(0)} + 2 \gamma \right)}{ G^{(0)}/2\sqrt{2} \kappa} \right)= \mathcal{O} \left(n \frac{d(d+1)}{2} \log \frac{2 \sqrt{d} \gamma} { G^{(0)}/2\sqrt{2} \kappa} \right) \leq \mathcal{O}\left(n d^2 \log \left( \sqrt{d} \kappa \right) \right)
\end{equation*}
for decoding their sum back to all machines (this is because $\frac{1}{2\kappa} G^{(0)} \leq 2 \gamma$). Thus the total communication cost for Hessian quantization at $t=0$ is
\begin{equation*}
\mathcal{O}\left(n d^2 \log \left( \sqrt{d} \kappa \right) \right).
\end{equation*}
For $t \geq 1$, we have that the cost for quantizing the local Hessians is
\begin{equation*}
\mathcal{O} \left(n \frac{d(d+1)}{2} \log\frac{10 \sqrt{d} G^{(t+1)}/(1+\alpha)}{G^{(t+1)}/2 \sqrt{2} \kappa} \right)= \mathcal{O} \left(n \frac{d(d+1)}{2} \log\frac{10 \sqrt{d} /(1+\alpha)}{1/2 \sqrt{2} \kappa} \right)=\mathcal{O} \left(n d^2 \log \left( \sqrt{d} \kappa \right) \right)
\end{equation*}
and for communicating the sum back to all machines is
\begin{equation*}
\mathcal{O} \left(n \frac{d(d+1)}{2} \log\frac{\sqrt{d}(1/2 \kappa+10/(1+\alpha)) G^{(t+1)}}{G^{(t+1)}/2 \sqrt{2} \kappa} \right)=\mathcal{O} \left(n \frac{d(d+1)}{2} \log\frac{10 \sqrt{d}/(1+\alpha) }{1/2 \sqrt{2} \kappa} \right)=\mathcal{O} \left(n d^2 \log \left( \sqrt{d} \kappa \right) \right),
\end{equation*}
again because $1/2 \kappa \leq 10/ (1+\alpha)$.
\newline
Thus the total cost of Hessian quantization along the whole optimization process until reaching accuracy $\epsilon$ is
\begin{equation*}
b_m=\mathcal{O} \left(n d^2 \log \left( \sqrt{d} \kappa \right) \log \frac{\gamma \mu^2}{32 \sigma^2 \epsilon} \right)
\end{equation*}
many bits in total.
\newline
On the other hand, the cost of quantizing the local descent directions at $t\geq0$ is
\begin{equation*}
\mathcal{O} \left(nd \log \frac{11 \kappa P^{(t)}}{\frac{\theta P^{(t)}}{2}} \right)= \mathcal{O} \left(nd\log \kappa \right)
\end{equation*}
because now $\theta$ is just $\frac{1}{16}$. The cost of sending the average of the quantized local directions back to any machine is
\begin{equation*}
\mathcal{O} \left(nd \log \frac{(\theta/2+11
\kappa)P^{(t)}}{\theta P^{(t)}/2} \right)=\mathcal{O} \left(nd \log \frac{11 \kappa P^{(t)}}{\frac{\theta P^{(t)}}{2}} \right)=\mathcal{O} \left(nd\log \kappa \right),
\end{equation*}
because $\frac{\theta}{2} \leq 11 \kappa$. Thus, the total communication cost for quantizing the descent directions until reaching accuracy $\epsilon$ is
\begin{equation*}
b=\mathcal{O} \left(nd\log \kappa \log \frac{\gamma \mu^2}{32 \sigma^2 \epsilon} \right)
\end{equation*}
many bits.
\newline
The total communication cost of Quantized Newton's method overall is
\begin{equation*}
\mathcal{O} \left(n d^2 \log \left( \sqrt{d} \kappa \right) \log \frac{\gamma \mu^2}{32 \sigma^2 \epsilon} \right)+\mathcal{O} \left(nd\log \kappa \log \frac{\gamma \mu^2}{32 \sigma^2 \epsilon} \right) = \mathcal{O}\left(nd^2 \log \left( \sqrt{d} \kappa \right) \log \frac{\gamma \mu^2}{\sigma^2 \epsilon} \right).
\end{equation*}
\end{proof}
\section{Estimation of the Minimum}
\label{app:function_value}
\functionvalue*
\begin{proof}
We have that
\begin{align*}
&\mid f_i(x^{(t)})-f_{i_0}(x^{(t)}) \mid \leq \mid f_i(x^{(t)}) \mid + \mid f_{i_0}(x^{(t)}) \mid \leq \frac{\gamma}{2} \|x^{(t)}-x_i^*\|^2 + \mid f_i^* \mid+ \frac{\gamma}{2} \|x^{(t)}-x_{i_0}^*\|^2+\mid f_{i_0}^* \mid
\end{align*}
In order $x^{(t)}$ to satisfy $f(x^{(t)})-f^* \leq \frac{\epsilon}{2}$, we compute $x^{(t)}$ by our main algorithms, such that $\|x^{(t)}-x^*\| \leq \sqrt{\frac{\epsilon}{\gamma}}$. This gives the respective communication complexities from the previous sections.
\newline
Given that, we can write
\begin{align*}
&\|x^{(t)}-x_i^*\|^2 =\|x^*-x_i^*\|^2 + \|x^{(t)}-x^*\|^2+2 \langle x^*-x_i^*, x^{(t)}-x^* \rangle \leq \\ & \|x^*-x_i^*\|^2 + \|x^{(t)}-x^*\|^2+2 \| x^*-x_i^* \| \| x^{(t)}-x^* \| \leq \|x^*-x_i^*\|^2+\frac{\epsilon}{\gamma}+\sqrt{\frac{\epsilon}{\gamma}} \|x^*-x_i^*\| \leq \\ & C^2 +\frac{\epsilon}{\gamma}+\sqrt{\frac{\epsilon}{\gamma}} C \leq 2 C^2
\end{align*}
for sufficiently small $\epsilon$. Similarly we have
\begin{equation*}
\|x^{(T)}-x_{i_0}^*\|^2 \leq 2 C^2
\end{equation*}
for small $\epsilon$.
\newline
Thus
\begin{equation*}
\mid f_i(x^{(t)})-f_{i_0}(x^{(t)}) \mid \leq 2 (\gamma C^2+c)
\end{equation*}
and by the definition of the quantization, we have
\begin{equation*}
\mid q_i^{(t)}-f_i(x^{(t)}) \mid \leq \frac{\epsilon}{2}.
\end{equation*}
which implies
\begin{equation*}
\mid \bar f-f(x^{(t)}) \mid \leq \frac{1}{n} \sum_{i=1}^n \mid q_i^{(t)}-f_i(x^{(t)}) \mid \leq \frac{\epsilon}{2}.
\end{equation*}
Overall, we get
\begin{equation*}
\bar f-f^* \leq \mid \bar f-f(x^{(t)}) \mid + f(x^{(t)})-f^* \leq \frac{\epsilon}{2}+\frac{\epsilon}{2}= \epsilon.
\end{equation*}
The communication cost for quantizing $f_i(x^{(t)})$ is
\begin{equation*}
\mathcal{O} \left(n \log \frac{\gamma C^2+c}{\epsilon} \right)
\end{equation*}
since we quantize real numbers, which are $1$-dimensional, and we need to communicate $n$-times.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,125 |
At Gairloch Museum, you can explore the cultural heritage of a unique area of the Highlands. Visitors will experience 7,000 years of local history, from evidence of Gairloch's earliest settlers to the twentieth century engineering marvels of the Rubh Re lighthouse lens.
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"redpajama_set_name": "RedPajamaC4"
} | 4,420 |
For years, the Nerds consistently rated the Ink Plus® Business Credit Card among the best small-business credit cards offered. Yet in 2016, Chase stopped accepting applications for that card and introduced a slightly revised alternative, the Ink Business Preferred℠ Credit Card.
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"redpajama_set_name": "RedPajamaC4"
} | 4,374 |
Q: How to find degree of a differential equation. I have a differential equation, $$e^{\large y^\prime} = x + x^3 + x^5 + y,$$ I need to find the degree of this equation.
Using Wikipedia definition,
In mathematics, the degree of a differential equation is the power of its highest derivative, after the equation has been made rational and integral in all of its derivatives.
I would say that the degree is one because if I take $\log $ on both sides I get $${y^\prime} = \log(x + x^3 + x^5 + y).$$
My teacher says that the degree is not defined because this DE cannot be represented as sum of polynomials in derivatives of $y$. When I asked, what if we take $\log$ on both sides, he says that we are not allowed to perform any operations on the DE, that will change DE of which we have to find the degree.
This contradicts the definition by Wikipedia.
Who is correct? What is the degree of this DE, $1$ or not defined?
A: I looked at some of the classical books in ODE. Most books including Coddington Levinson, Hartman, Chicone do not define the degree of a differential equation.
The only book where I found it is Ince. He writes
An ordinary differential equation expresses
a relation between an independent variable, a dependent variable and one or more differential coefficients of the dependent with respect to the independent variable.
The order of a differential equation is the order of the highest differential coefficient which is involved. When an equation is polynomial in all the differential coefficients involved, the power to which the highest differential coefficient is raised is known as the degree of the equation. When, in an ordinary or partial differential equation, the dependent variable and its derivatives occur in the first degree only, and not as higher powers or products, the equation is said to be linear. The coefficients of a linear equation are therefore either constants or functions of the independent variable or variables.
And then he gives the following example
$$\left\{1+\left(\frac{dy}{dx}\right)^2\right\}^{1/2}=3\frac{d^2y}{dx^2}$$
is an ordinary equation of the second order which when rationalised by squaring
both members is of the second degree.
It is a bit archaic, in particular, by differential coefficients he just means the derivatives of the solution. Anyway, the way I read it is that if a differential equation can be written in the form
\begin{align}a_n(x,y(x)) &\left(\frac{dy^n}{dx^n}(x)\right)^{m_n} + a_{n-1}(x,y(x)) \left(\frac{dy^{n-1}}{dx^{n-1}}(x)\right)^{m_{n-1}}\\ &+ \ldots + a_1(x,y(x)) \left(\frac{dy}{dx}(x)\right)^{m_1} = f(x,y(x)),\end{align}
where $m_n,\dots, m_1$ are natural numbers and $a_n\ne 0$, then the degree of the ODE is $m_n$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,771 |
package com.xj.zk.listener;
/**
* Author: xiajun
* Date: 14/5/20
*/
public class Node {
private String path;
private byte[] data;
public String getPath() {
return path;
}
public void setPath(String path) {
this.path = path;
}
public byte[] getData() {
return data;
}
public void setData(byte[] data) {
this.data = data;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,343 |
Knud 5. eller Knud Magnussen (født i 1129, død 9. august 1157 i Roskilde) var konge af Danmark 1146-1157. sammen med medkongerne Svend og Valdemar, under borgerkrigen 1146-1157. Knud var søn af Magnus, der var søn af kong Niels.
Efter at Erik Lam i 1146 abdicerede, valgte jyderne Knud til konge, mens sjællænderne valgte Svend. Det kom snart efter til krig, og i en periode blev Knud fordrevet til Tyskland, hvor det lykkedes ham at samle en hær. I 1154 bekræftede den tyske konge (senere kejser), Frederik Barbarossa, som Danmarks lensherre, at Svend skulle være enekonge i Danmark, mens Knud skulle have Sjælland som len. I 1154 brød stridighederne ud igen, og denne gang var Knud allieret med Valdemar, der indtil da havde støttet Svend. Det endte denne gang med, at Svend blev fordrevet, men i 1157 vendte han tilbage, og man enedes med den tyske kejser om en tredeling af riget: Valdemar fik Jylland, Knud fik Sjælland og Svend fik Skåne.
For at fejre afslutningen på mange års stridigheder indbød Svend sine medkonger til et forsoningsgilde i Roskilde den 9. august 1157. Under måltidet lod Svend sine mænd overfalde Valdemar og Knud. Knud blev dræbt på stedet, mens det lykkedes den sårede Valdemar at flygte. Episoden er siden blevet kendt som Blodgildet i Roskilde. Den 23. oktober samme år mødtes Valdemar og Svend i et kort, men voldsomt slag på Grathe Hede. Det endte med, at Svend flygtede ud i nogle sumpe, hvor han mistede våben og udrustning. Han blev taget til fange og dræbt med et øksehug. Herefter var Valdemar dansk enekonge. Digteren Thor Lange har ved Grågårde i nærheden af Thorning opsat et stenkors på stedet for begivenheden.
Ægteskab og børn
Han giftede sig i 1156 med en datter af Sverker 1., konge af Sverige (Östergötland). Hendes navn antages enten at være Helena eller Ingegärd.
Knud menes at være far til en række børn, men det er ikke sikkert, at nogle af disse er med hans dronning som moderen. Mange af de formodede børn, der angives i forskellige kilder, er dog enten temmelig utroværdige eller slet og ret umulige (kronologisk set eller på anden vis). Af de tilbageværende accepterede, formodede eller mulige børn er der følgende tre:
Niels den Hellige (1150–1180).
Valdemar (1157–1236); fra 1182 biskop af Slesvig
Hildegard af Danmark (født ca. 1157) gift med fyrst Jaromar 1.
Kilderangivelser
Eksterne henvisninger
Kongerækken på kongehuset.dk
Regenter af Danmark
Jellingdynastiet
Danskere i 1100-tallet
Skandinaver fra middelalderen
Personer i Dansk Biografisk Leksikon
Myrdede regenter af Danmark | {
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Another of the social clothing related quests Penelope can be found in the far western Rhinoman village in Newland desert and is an acrobat with a message that needs delivering. NOTE: she sometimes likes to go for a swim in the nearby Oasis so if you can't see her on land check the lake!
Penelope is an acrobat and she is soon to take part in a show, but she wants a new pair of shoes that she ordered from Gilbert Glove. Unfortunately he hasn't delivered them yet!
Here is of course where you come in! Tell her that you can do it and will get a mission. She will give you a Letter From Penelope to give to Gilbert, and you have a 30min time limit.
So make your way as quickly as you can to Gilbert Glove and give him the letter. His exact location is the guard post in Newland Desert at 3110 x 550. He will give you Boots For Penelope when you present him with the letter.
Take the boots back to Penelope and she will give you the reward you were looking for, the multi grip soles. | {
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} | 5,371 |
Enjoy your planning and revel even more with All 0 Promocode.club's great Airbnb promotion codes!
Choose a Airbnb coupon you would like to use from Promocode.club. Press use Airbnb code box, which is located just below. Now your web browser should copy the All 0 Airbnb code for you, although it is best to make sure and copy it manually. The https://www.airbnb.com/host/homes web page will open in a new window for you. Go to their Airbnb checkout section and find the Promotional Keycode box. Paste your Airbnb coupon code there and click apply. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,873 |
Q: How to Make the RequiredField Validator visible Using javascript or jquery, how can I make a Required Field Validator control (of ASP.NET) visible. If we check the viewsource of the Required Field valiator, we can see that the visibility is false initially. $("#spanReqFieldValidator").show() / fadeIn() wont work.
Any thoughts ?
From googling, I understand that jQuery has some issues with visibility attribute.
A: You can call the ValidatorValidate() function in javascript to make a validator execute it's validation logic (and show up if necessary). Something like this:
ValidatorValidate(document.getElementById('<%=MyValidator.ClientID%>'));
For more on the client-side validation API, see here.
A: Try this:
$("#spanReqFieldValidator").css("visibility","visible");
jQuery toggles the display attribute usually, visibility you need to toggle by setting the css. You could spice it up a bit as well:
$("#spanReqFieldValidator")
.css({ "visibility":"visible","display":"none"}).fadeIn();
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,035 |
Scott Esk is hoping voters will recognize the difference he represents for voters in the State House District 91 race. But while Esk insists the primary difference he presents is all about the state of Oklahoma regaining control of their future, there's a lot of attention being paid to statements he made on Facebook.
CLICK HERE for a link to a PDF of Esk's Facebook conversations.
CLICK HERE for the full audio version of our interview with Scott Esk. | {
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} | 3,912 |
Drew Barrymore A Castle For Christmas
The Very early Years: A blog site around Drew's very early years maturing in an acting household.
Drew Barrymore had great success as a youngster starlet, and also has attained a brand-new degree of honor as an adult, starring in many successful movies. She has actually also made a name for herself as a producer of flicks and also supervisor of a motion picture.
It is a blog concerning this wonderful actress. Blog Description: Drew Barrymore is an American starlet, director, producer, talk show host and also author. A member of the Barrymore household of stars, she is the recipient of a number of honors, including a Golden Globe and an Emmy.
2. Her time in the spotlight
4. Her most current role
Drew Barrymore's career began at eleven months, when she showed up in a dog-food commercial. In Charlie's Angels, Barrymore, Cameron Diaz played the triad of private investigators in Los Angeles. Barrymore was birthed right into a movie industry household.
Her dad, John Barrymore, was a well-known phase and display star, and also her mommy, Jaid, was a successful film manufacturer. Barrymore was close to her uncle John Drew Barrymore, that was also a star.
At the age of 8, Barrymore made her film debut in Altered States, as well as at age eleven, she played Gertie in Steven Spielberg's E.T. the Extra-Terrestrial. In 1982, Barrymore starred in Irreconcilable Differences, and also had a minor role in the film version of John Irving's The Hotel New Hampshire. In 1984, she showed up in Madman, and also in Gremlins.
Drew Barrymore is among the most successful child stars of all time, bring in countless bucks and also obtaining critical honor for her performances at a young age.
She's had her fair share of altercations with the law, dealt with alcohol and also drug misuse, and also even inspected herself into rehab, yet she's gotten better as an adult and also is set to show up in another Stephen King adaptation, this time around signing up with the cast of It.
Drew Barrymore's individual life has been a little a roller coaster trip, yet she's managed to come out of it with her head held high. Her very first marital relationship, to bar owner Jeremy Thomas, finished after 5 years in 2000.
Soon after the breakup, Barrymore was involved in an on and off relationship with Tom Green that lasted until 2001. In 2001, Barrymore came to be involved to her former other half's relative, bar proprietor Lance Armstrong. Despite a serious partnership that lasted five years, the two never married.
After breaking up with Armstrong, Barrymore had a brief romance with Jimmy Fallon, and also in 2004, she started a connection with design Jason Bleick. The couple broke up in 2006. In 2009, Barrymore began dating star John Mayer. In 2011, her engagement to Mayer was called off. In 2012, Barrymore was connected to the owner of FabFitFun, Eden Sassoon.
Drew Barrymore's latest duty is as a host of her very own daytime talk show. The program will air on September 14, 2020, and Drew has said she is "prepared to discuss a lot of things." She's also stated that she's going to be "talking with a great deal of celebrities and VIPs."
The program will certainly be recorded in front of an online studio audience, but Drew has claimed that it won't resemble her movies. She's additionally stated that it's mosting likely to be more like The Ellen DeGeneres Show. The Drew Barrymore Show will be a great program for celebs because it will certainly give them an opportunity to share their side of the story.
Drew Barrymore has been a Hollywood wild kid given that she was a kid. Her early days made headlines, as she was caught by paparazzi in an automobile with a much older male. She later on admitted that she was smoking pot in the picture.
Barrymore's rebelliousness played itself out on screen as well as in print. She forged a photo as a manipulative teen seductress, beginning with Poison Ivy. The movie, based upon DC Comics' fictional character, established her as a nuisance, as well as the media followed suit. In feedback to the buzz, Barrymore informed a reporter, "I run out suggestion what I'm doing as a 15-year-old than any person else. I'm not defiant."
Drew Barrymore is a well-known American actress. Drew Barrymore started her job as a child actress in the late 1970s, and has been a well recognized name in the sector for more than a decade.
She has additionally made a mark for herself as a manufacturer as well as a version, as well as is a proud owner of a couple of magazine covers. Drew Barrymore has been honored with Golden Globe Awards, Emmy Awards, People's Choice Awards, and also Kids Choice Awards, among many other prestigious honors.
Drew Barrymore has gone far for herself as an actress, design, and producer. She is also a happy proprietor of some publication covers. She has a couple of films coming out quickly, so make sure you check them out.
Drew Barrymore is among one of the most successful actors of our time and also there are many lessons to be picked up from her story.
We wish you appreciated our blog concerning Drew Barrymore. With this expertise, you can see that it does not matter what your history is, you can be effective in life. So what are you awaiting?
Start working hard today, and desire huge! If you're looking for some even more inspiration, look into our blog site concerning just how to be effective.
Drew Barrymore Teri Weigel Susan Tyrell In Far From Home
What Does Drew Barrymore Have Tattooed On Her Arm
Drew Barrymore had terrific success as a youngster actress, and has actually achieved a brand-new level of praise as a grown-up, starring in numerous effective flicks. She has actually also gone far for herself as a manufacturer of motion pictures as well as supervisor of a movie.
It is a blog site about this terrific starlet. Blog Site Description: Drew Barrymore is an American starlet, director, producer, talk show host and author. A participant of the Barrymore family members of stars, she is the recipient of a number of distinctions, consisting of a Golden Globe and an Emmy.
1. Her occupation
Drew Barrymore's profession started at eleven months, when she appeared in a dog-food commercial. In Charlie's Angels, Barrymore, Cameron Diaz played the triad of detectives in Los Angeles. Barrymore was born into a show business family.
Her father, John Barrymore, was a popular stage and also display actor, and her mom, Jaid, was an effective film producer. Barrymore was close to her uncle John Drew Barrymore, who was likewise an actor.
At the age of eight, Barrymore made her film debut in Altered States, as well as at age eleven, she played Gertie in Steven Spielberg's E.T. the Extra-Terrestrial. In 1982, Barrymore starred in Irreconcilable Differences, and had a minor role in the film version of John Irving's The Hotel New Hampshire. In 1984, she showed up in Madman, and in Gremlins.
Drew Barrymore is among the most successful youngster stars of perpetuity, generating millions of dollars and also getting important acclaim for her efficiencies at a young age.
She's had her reasonable share of encounters with the regulation, battled with alcohol as well as cocaine misuse, and even checked herself into rehabilitation, however she's gotten better as a grown-up as well as is readied to show up in an additional Stephen King adjustment, this moment signing up with the actors of It.
Drew Barrymore's individual life has been a bit of a roller rollercoaster flight, however she's taken care of to come from it with her head held high. Her first marital relationship, to bar owner Jeremy Thomas, finished after 5 years in 2000.
Soon after the breakup, Barrymore was involved in an on-and-off connection with Tom Green that lasted till 2001. In 2001, Barrymore became involved to her former other half's relative, bar owner Lance Armstrong. Regardless of a significant relationship that lasted 5 years, the two never wed.
After breaking up with Armstrong, Barrymore had a short romance with Jimmy Fallon, and in 2004, she started a partnership with design Jason Bleick. The couple separated in 2006. In 2009, Barrymore began dating star John Mayer. In 2011, her engagement to Mayer was aborted. In 2012, Barrymore was linked to the owner of FabFitFun, Eden Sassoon.
Drew Barrymore's most current role is as a host of her own daytime talk show. The program will certainly air on September 14, 2020, as well as Drew has actually claimed she is "prepared to discuss a lot of things." She's likewise claimed that she's going to be "chatting with a lot of stars and VIPs."
The show will certainly be filmed in front of an online studio audience, yet Drew has stated that it won't be like her films. She's additionally said that it's mosting likely to be a lot more like The Ellen DeGeneres Show. The Drew Barrymore Show will be a fantastic program for stars due to the fact that it will certainly give them a possibility to share their side of the story.
Drew Barrymore has actually been a Hollywood wild kid because she was a child. Her very early days made headlines, as she was caught by paparazzi in an auto with a much older guy. She later admitted that she was smoking pot in the image.
Barrymore's contumacy played itself out on screen as well as in print. She built a picture as a manipulative teenage seductress, beginning with Poison Ivy. The movie, based on DC Comics' fictional character, developed her as a troublemaker, and the media did the same. In feedback to the buzz, Barrymore told a reporter, "I have no more concept what I'm doing as a 15-year-old than any person else. I'm not defiant."
Drew Barrymore is a well-known American actress. Drew Barrymore began her occupation as a child actress in the late 1970s, and has been a well well-known name in the industry for more than a decade.
She has additionally made a mark for herself as a manufacturer as well as a version, and also is a pleased owner of a couple of magazine covers. Drew Barrymore has been bestowed Golden Globe Awards, Emmy Awards, People's Choice Awards, and Kids Choice Awards, amongst numerous various other prominent honors.
Drew Barrymore has actually gone far for herself as a starlet, version, as well as producer. She is also a proud owner of some magazine covers. She has a few films appearing quickly, so ensure you check them out.
Drew Barrymore is one of one of the most effective stars of our time and also there are lots of lessons to be gained from her tale.
We hope you enjoyed our blog site concerning Drew Barrymore. With this understanding, you can see that no matter what your history is, you can be effective in life. So what are you awaiting?
Beginning working hard today, and also desire huge! If you're searching for some even more motivation, have a look at our blog site concerning exactly how to be effective.
Maurice Barrymore Drew
Drew Barrymore Alcool | {
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Unidad de Inteligencia Financiera () may refer to:
Unidad de Inteligencia Financiera (Argentina)
Unidad de Inteligencia Financiera (Mexico) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,521 |
@class TUDouBanMusicModel, MPRemoteCommand, MPFeedbackCommandEvent;
@interface AppDelegate (TUMusicRemoteControl)
// 初始化
+ (void)configRemoteControl;
+ (void)refreshPlayMusicRemoteControlEventsWithModel:(TUDouBanMusicModel *)model isPlay:(BOOL)isPlay;
- (void)playEvent:(MPRemoteCommand *)command;
- (void)pauseEvent:(MPRemoteCommand *)command;
- (void)nextCommandEvent:(MPRemoteCommand *)command;
- (void)previousCommandEvent:(MPRemoteCommand *)command;
- (void)likeEvent:(MPFeedbackCommandEvent *)feedbackEvent;
- (void)bookmarkEvent:(MPFeedbackCommandEvent *)feedbackEvent;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,923 |
American Monarchy
This is the first of three Workspace 2014 exhibitions, featuring the work of LES Studio Program artists-in-residence.
Cuchifritos Gallery + Project Space is pleased to present American Monarchy, a solo exhibition by Enrique Figueredo. The exhibition expresses Figueredo's surreal fantasy of becoming the first American monarch. Highlighting politics, religion and popular culture, sex, citizenship, mysticism, and iconography, Figueredo paints and carves images that are both recognizable and falsified, compelling the viewer to believe in these dream-like stills as scenes from the past, present and future. In combining material observations with questions of capitalistic responsibility, Figueredo challenges the dissonance between power and authority and the intentions of those who represent a society for either the greater good or the good of the chosen few.
Figueredo uses his practice to confront the tension between his Venezuelan heritage and the surrounding American culture. Blending energetic marks and precise textures, Figueredo imposes the depth of figures onto planes, using an endless variation of simple lines that grow into forms. His increasingly physical process extends beyond the pure woodblock print, layering silk screen, oil sticks, paint and pencil on top of each print, destroying the preciousness of the original object in order to generate something new. Similarly, Figueredo's large-scale paintings draw from his printmaking technique, composing colorful planes layered on top of one another, that merge the powdered wigs of our forefathers with the obscenities of our contemporary amusements.
American Monarchy is Enrique Figueredo's first solo exhibition in New York post his residency at the Lower East Side Studio Program. Enrique has participated in group shows at Cuchifritos Gallery + Project Space and Superchief Gallery at CultureFix, both in the LES. Enrique has exhibited in various solo and group exhibitions in Santa Fe, New Mexico his home prior to New York. Notably, Enrique was an artist in residence at the Historic Santa Fe Foundation. Venezuelan-American artist Enrique Figueredo lives and works in Crown Heights, Brooklyn. He received his BFA from Purchase College in 2004.
The LES Studio Program is a three and six month residency program, run by Artists Alliance Inc, open to under-represented, emerging and mid-career professional working artists. Founded in 2003, the LES Studio Program underscores AAI's belief that the arts and individual artists are essential elements of the culture, history and future of the Lower East Side community. Artists of all disciplines–painting, photography, sculpture, video, installation, new media–are considered for fully-funded studio space to produce new work and make use of resources needed to support their creative practice. The residency offers 24-hour studio access, the opportunity to present work to curators and critics through AAI-organized studio visits, and a curated exhibition at Cuchifritos Gallery + Project Space.
Cuchifritos is FREE to the public and handicap accessible. Located inside Essex Street Market at the south end nearest Delancey.
Cuchifritos Gallery + Project Space is a program of Artists Alliance Inc., a 501c3 not for profit organization located on the Lower East Side of New York City within the Clemente Soto Vélez Cultural and Educational Center. Cuchifritos is supported in part by the New York City Department of Cultural Affairs in partnership with the City Council. This program is made possible by public funds from the New York State Council on the Arts and the National Endowment for the Arts. We thank the following for their generous support: Foundation for Contemporary Arts, New York City Economic Development Corporation and individual supporters of Artists Alliance Inc. Special thanks go to our team of dedicated volunteers, without whom this program would not be possible.
Tags :American Monarchy Enrique Figueredo Jodi Waynberg past yr2014
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Oct 24 - Nov 9 2014
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L'astroturismo, o turismo astronomico, è un tipo di turismo orientato a soddisfare gli interessi degli astronomi ed appassionati all'astronomia. Può anche essere definito come l'hobby di visitare luoghi specifici per l'osservazione astronomica.
Evoluzione del concetto di astroturismo
L'astroturismo è considerato come un'attività ludica-scientifica che ha avuto una grande crescita negli ultimi anni; ciò ha permesso di valorizzare risorse naturali, culturali, paesaggi e elementi patrimoniali associati all'astronomia.
Il concetto di astroturismo si è evoluto partendo dalla concezione di attività che si sviluppa in luoghi chiusi come osservatori e planetari, fino a un concetto moderno dove questa attività sfrutta le risorse naturali e culturali in spazi aperti ubicati in zone libere dall'inquinamento luminoso, che permette di associare la conoscenza scientifica astronomica agli aspetti culturali e alla natura.
L'astroturismo ha contribuito alla divulgazione scientifica dell'astronomia come scienza e alla diffusione culturale, naturale e turistica dei luoghi dove si sviluppa l'attività.
Questo astroturismo si è convertito in uno strumento di sviluppo sostenibile delle zone rurali, specialmente le più spopolate e vergini, dato che apporta valore aggiunto alle poche zone che rimangono libere dall'inquinamento luminoso.
Caratteristiche
Si tratta di viaggi o escursioni che possono includere visite a osservatori astronomici, planetari, musei o strutture dedicate all'astronomia; tali visite possono includere un servizio di guide.
L'astroturismo si sviluppa principalmente in luoghi privi di inquinamento luminoso, solitamente prodotto dalle città e dalle zone abitate; per questa ragione è considerato un tipo di turismo sostenibile per l'ambiente.
È considerato come un segmento del turismo sostenibile la cui risorsa basica è il cielo buio. Il cielo non deve presentare nessun segno di inquinamento luminoso permanente. Le destinazioni turistiche con paesaggi di cieli notturni bui e liberi dall'inquinamento luminoso, causato dalle luci artificiali, sono le più apprezzate per lo sviluppo di questa attività turistica.
Le persone si riuniscono normalmente in gruppi che viaggiano con il fine di osservare eventi astronomici particolari come eclissi lunari, eclissi solari, stelle cadenti, il passaggio di comete. L'osservazione può essere fatta con dispositivi ottici, come telescopi o binocoli, o ad occhio nudo. Per una migliore osservazione, sono stati costruiti osservatori fissi con fini turistici, oppure si usano telescopi mobili durante le escursioni in spazi rurali aperti.
I turisti che scelgono di partecipare ad escursioni e/o viaggi di questo tipo con il fine di partecipare ad attività relazionate con l'astronomia, vengono chiamati astroturisti; tra gli astroturisti sono inclusi scienziati, astronomi professionali, appassionati e viaggiatori curiosi.
Origini
Il concetto di astroturismo è relativamente moderno anche se l'astronomia, la scienza sulla quale si basa questa attività, è considerata come una delle scienze più antiche che esistono.
L'astroturismo nasce accanto all'astronomia amatoriale, che è un'attività praticata come un hobby o solo per piacere, consistente in attività molto simili a quelle che interessano gli astroturisti.
È stato l'astronomo francese Camille Flammarion a rendere popolare l'astronomia dopo la pubblicazione del suo libro Astronomie Populaire (Astronomia popolare): inoltre ha fondato un osservatorio astronomico a Juvisy-sur-Orge, e nel 1887 ha anche creato la Società Astronomica Francese.
Un altro passo importante è stata nel 1923 la costruzione da parte di Walther Bauersfeld del primo planetario fabbricato dalla Carl Zeiss a Jena, in Germania, che ha aperto le sue porte nel 1926 e presto ne aprirono altri due nella stessa città.
Poco tempo dopo cominciarono a diffondersi nuovi planetari in Europa. Una nuova ondata di apertura di planetari si ebbe dopo la seconda guerra mondiale. Si aggiunge, in seguito, la formazione di associazioni di appassionati di astronomia.
L'arrivo dell'uomo sulla luna nel luglio del 1969, ha avuto un impatto mediatico con 600 milioni di telespettatori, un quinto della popolazione mondiale. Questo evento ha creato un prima e un dopo nella divulgazione spaziale ed astronomica
Un altro aspetto importante nello sviluppo del pubblico è stato l'impatto di internet nella divulgazione scientifica astronomica, soprattutto con la diffusione di immagini spaziali della NASA e dell'Agenzia Spaziale Europea. Un risultato importante è stato l'evento del luglio 1994 riguardo l'impatto della Cometa Shoemaker-Levy 9 con Giove; mai, prima di allora, un evento astronomico era stato divulgato in forma così rapida e efficace.
Finalmente, dopo la costante degradazione della qualità dei cieli prodotta dall'inquinamento luminoso, si cominciano a valorizzare i luoghi più adeguati per l'osservazione astronomica, e nascono le prime normative per la protezione del cielo che portano alla creazione della dichiarazione del cielo notturno ed il diritto di osservare le stelle nell'Aprile del 2007.
Questo fatto marca la valorizzazione delle alternative turistiche in territori interni, soprattutto desertici per sfruttare questa condizione e sostenere attività economiche alternative e sostenibili. Sorge in questo modo l'avviamento di osservatori astronomici specificamente abilitati all'attenzione di visitatori.
L'Isola de La Palma è stata la prima destinazione di astroturismo al mondo. Un piano di riposizionamento dell'Isola realizzato da LEO Partners nel 2004 ha scoperto il potenziale dell'astroturismo come risorsa e prodotto turistico identificando una corrente turistica che fino ad allora non era stata considerata né gestita professionalmente dagli operatori turistici. Il successo del piano ha dato luogo ad un orientamento verso la tematizzazione dell'isola ed una trasformazione radicale dell'attività turistica. A tal proposito, è stato essenziale ottenere il riconoscimento dei cieli dell'isola come Riserva della Biosfera UNESCO e la sua posteriore certificazione, la prima nel mondo, come Destinazione Starlight.
Minacce per l'astroturismo
Una delle minacce che affronta lo sviluppo dell'astroturismo, come attività, è come trovare cieli abbastanza bui affinché sia possibile osservare le stelle ed altri fenomeni. Questo ha portato a rivalutare il concetto di parchi astronomici sia in Europa che negli Stati Uniti sotto il concetto di "riserva del cielo stellato", un sistema di certificazione sviluppata nell'anno 2007. Ciò nonostante, agenzie di viaggi e tour operatori specializzati in astroturismo stanno mettendo ogni volta di più interesse su territori spopolati, poiché rimangono lontani dell'inquinamento della luce artificiale generata dalle città. Esiste un maggiore interesse in zone desertiche come Deserto di Atacama, Kalahari, Deserto del Namib e Dasht-e Lut dove l'inquinamento della luce artificiale è molto basso o nullo. In questo modo, le notti buie sono diventate un argomento di vendita.
Uno dei temi principali affrontati nella Fiera del Turismo di Berlino dell'anno 2007 è stato giustamente la minaccia dell'inquinamento luminoso sull'astroturismo, considerato un'attività preziosa ma minacciata.
In Europa, la Spagna è uno dei paesi con maggiore inquinamento luminoso, infatti è il secondo paese: dalla fine degli anni 90 non esiste nessuna zona in tutto il territorio spagnolo priva di luce artificiale, essendo Madrid la capitale europea più brillante per quanto riguarda l'inquinamento luminoso. Da parte sua la Francia ha iniziato ad applicare misure per spegnere le luci delle vetrine dalle città.
Astroturismo in Italia
Nonostante l'elevato inquinamento luminoso nella penisola, esistono tuttavia dei luoghi dove poter vivere un'esperienza di astroturismo:
Rifugio Albasini, in Val di Sole, in modo amatoriale.
Perinaldo, presso l'Osservatorio astronomico Cassini
San Giovanni in Persiceto nel locale planetario.
Sul Gran Sasso, presso Osservatorio di Campo Imperatore
Regalna, in modo amatoriale.
Val d'Ega presso Osservatorio astronomico Max Valier
Troina sui Monti Nebrodi, in modo amatoriale.
Petroia, presso il Castello di Petroia
Isola di Tavolara, in modo amatoriale.
Luoghi dove esiste offerta attorno all'astroturismo
I migliori luoghi per osservare le stelle sono
Argentina
In Argentina esiste poca offerta di servizi specializzati nell'astroturismo. Ciò nonostante, alcune località stanno scommettendo per sviluppare questa attività, tali come l'osservatorio Félix Aguilar e il Complesso Astronómico Il Leoncito, costruiti nel 1960 e 1983 con fini scientifici, ubicati all'interiore del Parco nazionale Il Leoncito, nella Provincia di San Juan dove stanno sostenendo alcune iniziative. Ulteriormente, vi è offerta turistica nell'Osservatorio Astronómico Ampimpa, in Amaicha della Valle nella Provincia di Tucumán, all'Osservatorio Pierre Auger ed al Planetario Malargüe ubicati in Pampa Amarilla, nella Provincia di Mendoza.
Australia
In Australia Occidentale si trovano le località di: Carnamah, Perenjori, Three Springs, Morawa, Wongan Hills, Mullewa, Cervantes e Mingenew. Altri luoghi di interesse sono il Parco nazionale Warrumbungle certificato come un parco di cieli bui e l'Osservatorio di Sídney, dal 1858, parte del patrimonio astronomico in Australia, entrambi in Nuova Galles del Sud, il "Charleville Cosmos Centri", in Charleville nel Queensland e il santuario di vita naturale Arkaroola qualcosa ritirato di Adelaida.
Colombia
A Bogotá si trova l'Observatorio Astronómico Nazionale, un patrimonio datato1803 e che organizza attività con amatori all'astronomia come l'Osservatorio dell'Università di Los Andes e il Planetario di Medellín. Accanto a quello anteriore, il Deserto della Tatacoa, dove trova l'Osservatorio Astronomico Astrosur e l'Osservatorio Municipale, entrambi orientati all'astroturismo.
Cile
Il Cile conta la maggiore concentrazione di osservatori astronomici scientifici, concentrando il 40% della capacità di osservazione di cieli, ma grazie ai nuovi progetti internazionali, per l'anno 2025 avrà il 70% della capacità mondiale. Ciò costituisce un indicatore della qualità dei suoi cieli.
Grazie a una politica orientata alla protezione dei suoi cieli, lo sviluppo dell'astroturismo come attività economica, ha permesso di essere considerata come una specialità di turismo in Cile. Le condizioni climatiche e di altitudine permettono più di 300 notti di cielo sereno all'anno. Il Cile è il referente principale dell'astroturismo in America Latina.
I principali poli di astroturismo in Cile trovano associati al grande deserto di Atacama, questi sono: Antofagasta, Taltal, San Pedro di Atacama, Inca di Oro, Copiapó, Valle dell'Huasco, ma soprattutto Valle di Elqui e la Valle del Limarí dove si concentra maggior parte dell'offerta astroturistica del paese.
Negli ultimi anni si è sviluppata, anche, un'offerta di astroturismo in Santiago di Cile, nella Valle di Aconcagua, in Santa Croce, Santo Vicente di Tagua Tagua e in Casablanca, questi ultimi con un'interessante offerta associata ai tragitti del vino e vini specializzati in soggetti astronómicos. Infine l'offerta astroturistica più al sud si trova accanto al Lago Lanalhue.
Brasile
I migliori luoghi per l'osservazione astronomica in Brasile sono: Nuova Friburgo e Teresópolis. Oltretutto, Serra della Mantiqueira e l'Observatorio Pico dos Dias, del Laboratorio Nazionale di Astrofísica in Brasópolis, Minas Gerais
Spagna
Uno dei luoghi privilegiati per il turismo astronomico nell'emisfero nord è l'isola de La Palma alle Canarie. Qui è stata lanciata, nell'anno 2007, con appoggio dell'ONU e dell'Organizzazione Mondiale del Turismo, la certificazione Starlight. In quest'isola si trova l'Osservatorio del Roque de los Muchachos e il Grande Telescopio delle Canarie. Inoltre, nella penisola iberica, vi sono il Centro Astronómico di Tiedra, Cielo e Tiedra, nella provincia di Valladolid, il Centro di Interpretazione del Cielo (CIC) a Gorafe, il Giardino Botanico di Santa Catalina nella località di Iruña di Oca, l'Osservatorio di Calar Alto ad Almería, il Parco nazionale di Monfragüe a Cáceres, il complesso di astro-autovettura Encinas e Stelle, a Higuera la Real e a Badajoz, Sierra Morena in Andalusia, il tragitto Cammino delle Stelle in Galizia, tragitto astronomico di Santa Ana la Reale e il Centro di Interpretazione dell'Astronomia di Villanueva dei Castillejos a Huelva.
Stati Uniti
In Stati Uniti è stata creata l'organizzazione di protezione dei cieli in 1988, che include più di 42 parchi di cieli scuri certificati, in luoghi come Utah, Arizona, California, Nevada e Nuovo Messico. Uno dei luoghi più visitati per l'astroturismo è il Joshua Tree National Park. La principale destinazione di astroturismo negli Stati Uniti è Mauna Kea, nell'Isola di Hawái. Un altro luogo, anche se più vincolato al turismo spaziale, è Capo Cañaveral in Florida che possiede parchi tematcos orientati allo spazio e l'esplorazione.
Giordania
Il deserto di Wadi Rum in Giordania, anche conosciuto come Valle della Luna, è stato il protagonista di vari film di fantascienza come "Missione a Marte", "Pianeta Rosso", "Rogue One: una storia di Star Wars" e Prometheus, infatti attrae molti turisti pastrofotografi.
Marrocco
Il villaggio di Merzouga al sudest del Marocco, accanto al famoso deserto di Erg Chebbi e le sue dune dove offrono escursioni con il dromedario, 4x4 e trekking.
Messico
Nello stato di Puebla, trova il vulcano Sierra Negra, dove si trova il Grande Telescopio Milimetrico (GMT), il più grande per la sua tipologia.Ed un altro luogo importante è il Parco nazionale Sierra di San Pedro Martir, ubicato nella Bassa California considerato uno dei migliori luoghi di osservazione astronomica dell'emisfero nord.
Portogallo
Uno dei punti più eccellenti del Portogallo è l'area di Alqueva, al sud-est del Portogallo, conosciuta per la sua certificazione di cieli bui e che si estende per circa a tre mila chilometri quadrati ed include delle piccole città: Alandroal, Portel, Mourão, Moura, Reguengos di Monsaraz e Barrancos.
Nuova Zelanda
Il Parco nazionale di Rakiura, che in maorí significa "cielo brillante", è ubicato nell'isola Stewart all'estremo sud della Nuova Zelanda. Anche la spiaggia di Castlepoint, nella regione di Wairarapa, nell'Isola del Nord, è considerata un'area propizia per l'osservazione delle stelle.
I migliori luoghi per osservare equinozi e solstizi sono
Inghilterra
Uno dei luoghi preferiti per riunirsi in Europa ad aspettare il solstizio di inverno è Stonehenge, famoso monumento di pietra che fa parte di una serie di posti archeologici di forma circolare.
Irlanda
Il luogo più enigmatico in Irlanda, relazionato in special modo con i solstizi è il grande tumulo di Newgrange, che ha circa cinque mila anni di antichità, ubicato al nord-est dell'Irlanda. Solo un giorno all'anno il sole di inverno entra per una galleria fino alla sala principale all'interno del tumulo, per illuminare una parete con arte rupestre.
Messico
Uno dei luoghi più importanti è Wirikuta ubicato a stato di San Luis di Potosí, è considerato un luogo sacro della cosmovisione del popolo huichol. Molto particolare anche Xochicalco, un posto archeologico nello stato di Morelos, che in tempi preispanici è stato il centro di osservazione di corpi celesti. In Messico vi è anch Chichén Itzá dove si aspetta l'equinozio per vedere scendere il serpente piumato.
Perù
Uno degli spettacoli culturali associati alle rovine archeologiche ha luogo durante i solstizi in Machu Picchu, ogni 21 di giugno con il solstizio di inverno, anche chiamato Inti Raymi in quechua e il 22 di dicembre con il solstizio di estate o Cápac Raymi.
I migliori luoghi per osservare aurore boreali sono
Alaska
In Fairbanks, una piccola città nel centro della regione esistono servizi ed escursioni per osservare aurore boreali durante la stagione.
Canada
È un paese con molti luoghi atti per osservare aurore boreali, ciò nonostante, il preferito è Manitoba in la nella baia di Hudson, dato che si possono osservare anche gliorsi polari, specialmente nel piccolo villaggio di Churchill.
Islanda
Il Thingvellir, parco nazionale dell'Islanda, è considerato il migliore luogo per osservazione di aurore boreali.
Norvegia
La città di Tromsø, conosciuta come il "Parigi del Nord", permette di contemplare lo spettacolo delle aurore boreali ed offrono conferenze di astronomia presso il planetario Northern Lights tra settembre e marzo.
Svezia
Kiruna,si trova ubicato nell'estremo nord della Svezia. Un luogo molto conosciuto per il suo hotel di ghiaccio e per le aurore boreali.
Finlandia
Rovaniemi in Finlandia, è la città più settentrionale doveosservare le aurore boreali, oltre al famoso parco tematico invernale Santo Claus Village e l'hotel ártico di neve. In Rovaniemi realizzano anche tours fotografici per captare le aurore boreali.
Russia
In Múrmansk una città ubicata nella parte nord della Siberia, nella penisola di Kola, con il più grande porto russo nell'artico, le aurore boreales si possono vedere 200 giorni all'anno. La stagione di aurore va da agosto fino ad aprile, in cui i mesi da settembre ad ottobre e da febbraio a marzo sono i migliori per osservare aurore boreali da questo porto.
Note
Voci correlate
Turismo spaziale
Astronomia osservativa
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Tag Archives: Hank Snow
Classic Rewind: Hank Snow — 'Rhumba Boogie'
Leave a comment Posted by Jonathan Pappalardo on November 26, 2018
Classic Rewind Hank Snow
Legends (and others) lost in 2017
3 Comments Posted by Paul W. Dennis on January 5, 2018
For one who grew up on the country music of the period (1960-1975) the last few years have been tough as we have seen many legendary figures come to the end of the road. 2017 was no exception. Let's take a look back with a few words about the various stars that were dimmed in 2017. I should note that I've included a few non-country personal favorites.
Junior Barber, a fantastic dobro player died at the age of 73. He worked with the Gibson Brothers bluegrass for seven years and his son Mike has played bass for the Gibson Brothers for the last twenty-five years.
Chuck Berry, 90, was a pioneer of rock 'n roll and while many would not regard him as country, Buck Owens thought that Berry wrote great country songs, and the bluegrass duo of Jim & Jesse McReynolds recorded an entire album of his songs (Chuck wrote the liner notes) so who am I to disagree with them?
Sonny Burgess, 88, rockabilly pioneer and early Sun Records artist. There is a younger country artist with the name Sonny Burgess, whom I don't believe is related. This guy was a great on-stage performer.
Glen Campbell, 81, singer and guitarist who first came to my attention as a session musician for Frank Sinatra and the Beach Boys (with whom he sometimes toured). Glen, who died after a long bout with Alzheimer's, could play anything with strings and could sing anything. My favorite tracks by him include "Galveston", "Wichita Lineman", "Wherefore and Why" and "I'm Gonna Love You". Glen hosted a television show, appeared in movies and was simply one of the giants of the industry.
Antoine "Fats" Domino, 89, wasn't a country singer but his music was infectious fun and enjoyed across the board. His hits were too numerous to list and many of them were covered by country singers.
Dave Evans, 65, had one of the best voices in bluegrass music being a great tenor singer, as well as being a good banjo player. It would be difficult to find another singer who sang with as much heart as Dave Evans.
Troy Gentry, 50, of Montgomery Gentry duo, died in a helicopter crash in Medford, New Jersey. I wasn't a big Montgomery Gentry fan, but they had some good numbers and performed with enthusiasm.
Michael Johnson, 72, singer and guitarist whose country hits included "Give Me Wings" and "The Moon Is Still Over Her Shoulder". Michael was a terrific acoustic guitar player and had a major pop/adult contemporary hit with "Bluer Than Blue".
Pete Kuykendall, 79, banjo champion and editor and publisher of Bluegrass Unlimited magazine. I have subscribed to Bluegrass Unlimited for many years and think it is the finest magazine in the world of music.
Miggie Lewis, 91 was a part of the first family of bluegrass gospel, the Lewis Family. The group disbanded years ago but youngest brother "Little" Roy Lewis a dynamic banjo player, comic and personality who still plays the bluegrass festival circuit.
Sam Lovullo, 88, was the producer and casting director of the long-running Hee Haw TV series (1969-1992). If he was only remembered for Hee Haw that would be sufficient legacy, but his son Torey Lovullo played major league baseball for eight years and then became a major league manager (he was the National League Manager of The Year for 2017). I am not ashamed to admit that I watched Hee Haw every chance I had, and that I know dozens of verses to "Pffffft, You Were Gone".
Geoff Mack, 94, composer of the tongue-twisting and widely recorded "I've Been Everywhere," in his native Australia. The lyrics familiar to American listeners were not the original lyrics, but a rewritten version to reflect North American place names.
Kevin Mahogany, 59 was a brilliant jazz baritone singer. He appeared and performed in Robert Altman's 1996 movie, Kansas City.
Jo Walker Meador, 93, as executive director built the Country Music Association from a tiny, ragged startup into one of the nation's most visible and successful trade organizations. Jo is a member of the Country Music Hall of Fame, and I can make a pretty good case for her being one of the two or three most important women in the history of country music.
D.L. Menard, 85, singer and songwriter widely known as the "Cajun Hank Williams" and most celebrated for his 1962 recording of "La Porte en Arriere,". He died in his native Louisiana.
Tom Paley died in England at the age of 89. Tom was a founding member (along with Mike Seeger and John Cohen) of the New Lost City Ramblers, a group that did much to further the acceptance of bluegrass among folk audiences. I saw them once in 1962 and they were terrific.
Leon Rhodes, 85, was the lead guitarist for Ernest Tubb's Texas Troubadours and later played in the Grand Ole Opry and Hee Haw staff bands. He was also a successful session musician.
Kayton Roberts, 83, steel guitarist in Hank Snow's Rainbow Ranch Boys band from 1968 to 1999. His son Louie Roberts also had a career in country music.
Curley Seckler who died in late December at the age of 98, was one of the last links to the first generation of bluegrass musicians, having performed with Bill Monroe and Flatt & Scruggs. Curley was old enough to remember Jimmie Rodgers and the Original Carter family being played on the radio. He also appeared on several segments of the Marty Stuart Show on RFD.
There was nothing country about Keely Smith, 89, but she was a fine singer with a terrific comedic touch. Her act with ex-husband Louis Prima played to packed houses in Las Vegas for the better part of a decade.
Tammy Sullivan died at the much too young age of 52, of cancer. Tammy was a marvelous singer best known for her work with the Sullivan Family, a bluegrass gospel band.
Wendy Thatcher, 69, was a formidable singer who is best remembered for her years with Eddie Adcock's various bands.
Mel Tillis, 85, songwriter, singer, actor, comedian and member of the Country Music Hall of Fame, died in Ocala, Florida. Mel first came to prominence as a songwriter, with early efforts becoming hits for the likes of Webb Pierce and Ray Price during the early 1960s. It would be a decade before his career as a performer went into overdrive, but when it did he racked up many hits and won the CMA Entertainer of the Year Award. I liked many of his songs but my favorite is "Would You Want The World To End (Not Loving Me)". I saw Mel live on several occasions.
Don Warden, 87, was a former steel guitar player in Porter Wagoner's band and subsequently Dolly Parton's manager. You can sometimes catch Don in RFD's reruns of the Porter Wagoner Show.
Don Williams, 78, was a singer and songwriter who regularly topped the country charts during the 1970s and '80s. Starting out with the folk-country Pozo Seco Singers, Don's solo career made him an international star and landed him in the Country Music Hall of Fame.
Norro Wilson, 79, producer, songwriter and former recording artist, whose hit compositions included George Jones' "The Grand Tour" and Charlie Rich's "The Most Beautiful Girl," died in Nashville.
Bob Wooton, 75, Johnny Cash's lead guitar player from 1968 until Cash's retirement in 1997, died in Gallatin, Tennessee. Bob was the replacement for Luther Perkins.
Everything Else Bill Monroe, Bob Wooton, Buck Owens, Chuck Berry, Curley Seckler, D L Menard, Dave Evans, Dolly Parton, Don Warden, Don Williams, Eddie Adcock, Ernest Tubb, Fats Domino, Flatt & Scruggs, Frank Sinatra, Geoff Mack, Gibson Brothers, Glen Campbell, Hank Snow, Jim & Jesse, Jo Walker Meador, John Cohen, Johnny Cash, Junior Barber, Kayton Roberts, Keely Smith, Kevin Mahogany, Leon Rhodes, Little Roy Lewis, Louis Prima, Marty Stuart, Mel Tillis, Michael Johnson, Miggie Lewis, Mike Seeger, Montgomery Gentry, New Lost City Ramblers, Norro Wilson, Pete Kuykendall, Porter Wagoner, Ray Price, Robert Altman, roy Gentry, Sam Lovullo, Sonny Burgess, Tammy Sullivan, Texas Troubadours, Tom Paley, Torey Lovullo, Webb Pierce, Wendy Thatcher
Album Review: Willie Nelson and The Boys: 'Willie's Stash, Volume 2'
1 Comment Posted by Razor X on November 9, 2017
This collection is a follow-up to Willie Nelson's 2014 collaboration his sister Bobbie, December Day: Willie's Stash, Volume 1. This time around Willie is teamed up with his two youngest sons, Micah and Lukas, who join him on eleven country classics and one contemporary number that leans heavily on the Hank Williams catalog.
Material-wise, there are no real surprises here. As always when Willie Nelson records cover material, the unknown is always how much Willie will deviate from the originals. In the case of this album, the answer is not much. The seven Williams songs are handled reverently. The two younger Nelsons, despite their youth, show great enthusiasm for the material and one gets the distinct impression that they have great respect and passion for, it and that these are not just a bunch of old songs that Dad forced them to record. The three Nelsons harmonize well together, as family groups typically do, and there are some fantastic steel guitar solos courtesy of Mike Johnson. Rarely have these old chestnuts sounded so energetic.
The one thing that did surprise me is how good Willie's voice sounds throughout the album, with little signs of the wear-and-tear that has been apparent on some of his recent work. From what I can gather, these recordings were made in 2011 and 2012, so that partially explains it. However, his voice is noticeably stronger than it was on 2010's Country Music collection for Rounder. Whatever the reason, it's good to hear Willie in such good vocal form.
This album could have been titled The Nelsons Sing Hank, since some of country music's famous Hanks wrote the marjority of the album's songs. In addition to the seven Williams numbers ("Move It On Over", "Mind Your Own Business", " I'm So Lonesome I Could Cry", "Your Cheatin' Heart" , "Cold Cold Heart", "Mansion on the Hill", and "Why Don't You Love Me"), the album contains a remake of Hank Snow's "I'm Movin' On", Hank Locklin's "Send Me The Pillow You Dream On", and Hank Cochran's "Can I Sleep In Your Arms", which is my favorite song on the album. Set to the melody of "Red River Valley", it was a hit in 1973 for Cochran's then-wife Jeannie Seely, and it was later recorded by Willie for his Red Headed Stranger album in 1975.
The album is rounded out by a cover of Willie's original composition "Healing Hands of Time" and a modern-folk tune "My Tears Fall" written by singer/songwriter Alyssa Miller. This contemporary number fits in surprisingly wel l with these old classics and doesn't sound out of place at all next to them.
Buddy Cannon's production is tastefully understated and for the most part the album has a sitting around the living room jam-session type feel to it. I cannot find any fault with it, other than to say I wish it had been released as a double album. I highly recommend it without reservation.
Album Reviews Alyssa Miller, Bobbie Nelson, Buddy Cannon, Hank Cochran, Hank Locklin, Hank Snow, Hank Williams, Jeannie Seely, Lukas Nelson, Micha Nelson, Mike Johnson, Willie Nelson
Spotlight Artist: The Whites
2 Comments Posted by Razor X on February 1, 2017
After featuring more than 100 artists over the past eight years of writing for this blog, it's becoming more challenging to find interesting artists to spotlight. This month we decided to do something a little different. When discussing possibilities, it occurred to us that there have been quite a few country music acts that have shared the surname White. Since none of them really has a discography large enough to write about for an entire month, we've decided to do a group spotlight and feature the best work of each:
1. The Whites are a family act consisting of Buck White and his daughters Sharon and Cheryl. Buck played piano for Ernest Tubb and Hank Snow in the 1950s. He and his wife Pat performed in Texas and Arkansas with another couple and were known as The Down Home Folks. Their daughters joined the family act in the 1960s. The family relocated to Nashville in 1971 and Pat retired from the group shortly thereafter. Buck White and the Down Home Folks released a few independent albums in the 70s and in 1978 Sharon and Cheryl were invited by Emmylou Harris to sing harmony vocals on her Blue Kentucky Girl album. Sharon married Ricky Skaggs in 1982 and the following year the group, now known as The Whites, released their first major label album on Curb Records in partnership with Warner Bros. The album yielded four Top 10 hits, including "You Put The Blue In Me", "Hangin' Around", "I Wonder Who's Holding My Baby Tonight", and "Give Me Back That Old Familiar Feeling". The following year they moved to Curb/MCA and enjoyed another handful of hits, which tapered off by the end of the decade. They joined the Grand Ole Opry in 1984 and have been one of its flagship acts ever since.
2. Lari White, a native of Dunedin, Florida, grew up singing gospel with her family, and in 1988 she was a winning contestant on The Nashville Network's You Can Be a Star. She was awarded a recording contract with Capitol, but was dropped from the label when her debut single failed to chart. She joined Rodney Crowell's band in 1991 and he produced her first album when she landed a deal with RCA the following year. She released three albums for RCA, and scored three Top 10 hits in the process: "That's My Baby", "Now I Know", and "That's How You Know (When You're In Love)". She released one album for Lyric Street in 1998 and has released a pair of independent albums after leaving that label.
3. Michael White is the son of songwriter L.E. White, who wrote some of Conway Twitty's hits. Michael's composition "You Make It Hard To Take The Easy Way Out" was released as the B-side of Twitty's 1973 hit "You've Never Been This Far Before". Michael's brief stint with Reprise Records in the early 90s produced one album and a few singles, one of which ("Professional Fool") reached the Top 40.
4. Joy Lynn White, also known as simply Joy White, is a critically acclaimed singer who released two albums for Columbia and one for Mercury in the 1990s, before moving to indie labels in the early 2000s. Her 1993 single "Cold Day In July" reached the lower rungs of the Billboard country singles chart and was later a hit for The Dixie Chicks.
5. Bryan White enjoyed a string of hits in the 90s as an Asylum Records recording artist, beginning with "Eugene You Genius" which was released when he was just 20 years old. In 1995 he enjoyed his first #1 hit with "Someone Else's Star". In 1998 he teamed up with Shania Twain for the duet "From This Moment On". By the time his fourth album was released, his commercial momentum had slowed, so he took a five-year sabbatical from the music business. He returned in 2009 with the independently released Dustbowl Dreams and is currently running a Kickstarter campaign to finance the release of a new album.
We hope that you will enjoy revisiting — or discovering for the first time — the work of this group of artists during the month of February.
Spotlight Artist Bryan White, Buck White, Cheryl White, Conway Twitty, Emmylou Harris, Ernest Tubb, Hank Snow, Joy Lynn White, L. E. White, Lari White, Michael White, Pat White, Ricky Skaggs, Rodney Crowell, Shania Twain, Sharon White, The Dixie Chicks, The Down Home Folks, The Whites
Album Review: Asleep At The Wheel – 'Comin' Right At Ya'
1 Comment Posted by Paul W. Dennis on October 5, 2016
United Artists released the first Asleep At The Wheel ("AATW") album in 1973. The album featured a mix of straight ahead country and honky-tonk, along with western swing. No doubt United Artists felt a need to mix the western swing with country as it had been a good dozen years since western swing had been a viable force in the marketplace, aside from the small band swing novelties of Hank Thompson and his Brazos Valley Boys.
The core of this early version of AATW was Ray Benson on lead guitar and vocals, Leroy Preston on guitar, drums and vocals, Lucky Oceans on steel guitar, Jim Haber (aka Floyd Domino) on piano and Chris O'Connell on vocals and rhythm guitar. Guests Johnny Gimble, Buddy Spicher and Andy Stein augment the band on fiddle, with Gimble also playing electric mandolin.
The album opens with a Bob Wills-Tommy Duncan composition "Take Me Back To Tulsa". The arrangement on this track swings but not nearly as much as it would in later years.
Track two is the Leroy Preston composition "Daddy's Advice", a straight ahead country song with a very traditional steel guitar sound paired with the fiddles. The vocal sounds like it may be Preston singing.
Leroy Preston also contributed "Before You Stopped Loving Me" is a nice ballad handled by the inimitable Chris O'Connell. I think that Chris may have been the best female vocalist AATW ever had.
Jerry Irby's "Drivin' Nails In My Coffin" was a hit for Ernest Tubb. Although Ernest was not a western swing artist, his recording of the song straddled the line between western swing and honky-tonk, as does this recording.
The Hank Williams classic "I'll Never Get Out of This World Alive" is given a straight-ahead country arrangement. Again, the vocal sounds like Leroy Preston.
Lucky, Leroy and Floyd wrote "Space Buggy" which has a barrelhouse boogie sound. Ms. O'Connell handles the lead vocals on this bright up-tempo song.
"Cherokee Boogie" was one of Moon Mullican's great songs, one that was a hit for Moon and has graced the charts several times since them. Since Mullican was one of the great piano influences on Jerry Lee Lewis, it is only appropriate that Floyd Domino's piano is featured heavily on this track.
Track eight on album is another Leroy Preston original titled "Hillbilly Nut", a bit of a novelty with some instrumental snippets of other famous tunes. Preston sings this song.
Ray Benson and Leroy Preston collaborated on "Your Down Home Is Uptown", a country ballad sung by Chris O'Connell.
Preston also penned "I'm The Fool (Who Told You To Go)" another straight ahead country ballad with Chris O'Connell shining on harmony vocals on the chorus. Ray Benson sings the lead.
Geoff Mack, an Australian country singer, penned "I've Been Everywhere". The song originally featured Australian place names; however, with American place names, the song became a massive hit for Hank Snow. Leroy Preston takes the lead vocals on this song, which are NOT taken at the breakneck speed often associated with the song. The vocals of this song frequently have been rewritten to reflect the nationality of the singer.
The album closes with "The Son Shines Down On Me", a nice gospel ballad sung by Chris O'Connell. The songwriter is credited as 'L. Lee' but I know nothing further about that person.
Comin' Right At Ya is an album which sees the band finding itself. The album produced no hit singles, and while there are traces of western swing styled elements throughout the album, the album is less western swing than any of their future efforts would be. As a vocalist Leroy Preston isn't all that good and his vocals would be less prominent on future albums. I liked this album (I picked up a copy on vinyl when it first came out) but it is mostly a harbinger of things to come. I'd give it a B.
Koch paired this with Texas Gold (a much better album) on a CD reissue in 2000. Texas Gold, released on Capitol in 1975, would feature the band's biggest hit "The Letter That Johnnie Walker Read".
Album Reviews, Spotlight Artist Andy Stein, Asleep at the Wheel, Bob Wills, Brazos Valley Boys, Buddy Spicher, Chris O'Connell, Ernest Tubb, Floyd Domino, Geoff Mack, Hank Snow, Hank Thompson, Hank Williams, Jerry Irby, Jery Lee Lewis, Jim Haber, Johnny Gimble, Leroy Preston, Lucky Oceans, Moon Mullican, Ray Benson, Tommy Duncan
Classic Rewind: Hank Snow – 'I Don't Hurt Anymore'
1 Comment Posted by Occasional Hope on September 7, 2015
Revelations from Music Vendor/ Record World
8 Comments Posted by Paul W. Dennis on August 13, 2015
As the 'last man standing' Billboard's country charts have taken on an almost mythical importance, yet for most of the 1940s and 1950s, Billboard did a relatively poor job in recording the history of country singles in that their various country charts only went 10-15 places deep.
Music Vendor (later Record World) started tracking country music in 1954 and immediately started tracking 55 chart places for country records, a depth of country charts Billboard wouldn't approach until 1964 when Billboard went to 50 places. For purposes of simplicity, I will always refer to Music Vendor/ Record World as 'Record World'.
Joel Whitburn's new volume Hit Country Records 1954-1982: Music Vendor/Record World performs a valuable service in restoring to the known discography of country music a staggering 1700 songs and 200 artists that Billboard failed to chronicle.
I always thought that the Wilburn Brothers had a relatively thin representation on the Billboard charts with 31 chart entries from 1954-1972, with many songs that I knew to have been at least mid-level hits not being tracked by Billboard. Turns out that the Wilburn Brothers were the poorest served of all country artists by Billboard with a staggering 30 songs not tracked by Billboard. Other artists with huge holes in their Billboard chart discographies include Hank Snow (26 songs), Eddy Arnold (23 songs), Kitty Wells (21 songs), Hank Thompson (21 songs), Johnnie & Jack (20 songs) and Ernest Tubb, Marty Robbins, Ferlin Husky and George Jones (each with 19 songs).
Among Bluegrass artists, Flatt & Scruggs pick up an extra 15 chart entries, Mac Wiseman (13), Jimmy Martin (6), Bill Monroe (4), and the Osborne Brothers (4).
There were also apparently differences in how artists were classified. Country audiences always loved Brenda Lee, Elvis Presley, George Hamilton IV and Conway Twitty, a fact Billboard somehow failed to acknowledge. After missing "Jambalaya", Billboard tracked "One Step At A Time", and then missed the next eleven consecutive Brenda Lee songs including such monsters as "Dynamite", "Sweet Nothings", "Fool #1" and "Break It To Me Gently".
The track record on Elvis was worse as Billboard failed to track "That's All Right" and "Blue Moon of Kentucky" and "Blue Suede Shoes", along with 15 more songs.
Record World tracked six George Hamilton IV singles before Billboard got around to recognizing "Before This Day Ends" as a country single. Ditto for Conway Twitty who Billboard picked up as country with "Guess My Eyes Were Bigger Than My Heart", after ten singles had already been tracked by Record World.
While most of the songs that Music Vendor/Record World picked up were second tier hits, there were some surprising Billboard misses uncovered such as the George Jones favorites "Tall Tall Trees", "Eskimo Pie" and "Nothing Can Stop Me (Loving You)". A very famous song from 1955 was Bobby Lord's 1955 hit "Hawkeye"; Billboard missed the song entirely on any of its charts, whereas Record World had it charting for twelve weeks, reaching #16.
I mentioned that approximately 200 artists show up in this book that Billboard never tracked on its country charts. These include Carl Dobkins Jr (three songs including "My Heart Is An Open Book" which Record World has as a #2 country hit, and Billboard had reach #3 pop), Pete Drake (three instrumental singles), and Buddy Holly (four singles including "Peggy Sue" and "Maybe Baby").
I've only had this fascinating book for two days and I will probably report further as time permits, but it would be remiss of me not to further examine the song that initially got me interested in charts. Yes – I do mean "Groovy Grubworm" by Harlow Wilcox and The Oakies. Cashbox had the record reach #1 on its country chart (#24 pop) for two weeks whereas Billboard had the record stall out at #42 on the country chart while reaching #30 on the pop charts. This was the biggest chart disparity ever between singles that reached #1 on either the Billboard or Cashbox country chart but not the other chart.
The record was hugely successful, selling a million copies between the US and Canadian markets (it was a top ten hit on several Canadian regional pop charts), so I was curious to see how Record World treated "Groovy Grubworm" on its country charts, recalling that Record World had the song chart higher on its pop chart (#23) than did either Cashbox or Billboard.
Drum roll please :
Record World had the song reach #3 for one week on its country chart during its thirteen week chart run.
Book Reviews, Charts Bill Monroe, Bobby Lord, Brenda Lee, Buddy Holly, Carl Dobkins Jr, Conway Twitty, Eddy Arnold, Elvis Presley, Ernest Tubb, Fatt & Scruggs, Ferlin Husky, George Hamilton IV, George Jones, Hank Snow, Hank Thompson, Harlow Wilcox, Jimmy Martin, Joel Whitburn, Johnnie & Jack, Kitty Wells, Mac Wiseman, Marty Robbins, Osborne Brothers, Pete Drake, Wilburn Brothers
Reissues wish list: part 3 – RCA and Columbia
2 Comments Posted by Paul W. Dennis on August 4, 2015
When speaking of the big four labels we need to define terms
Columbia refers to records originally issued on Columbia, Epic, Harmony or Okeh labels. Okeh was used for so-called minority interest recordings. Columbia also owned Vocalion for a while. RCA refers to recordings on the RCA Victor and RCA Camden labels.
In addition to folks such as Chet Atkins, Jim Reeves, Dolly Parton, Eddy Arnold, Connie Smith and Charley Pride, RCA had a fine group of second tier artists including Kenny Price, Porter Wagoner, Jim Ed Brown, Stu Phillips, Nat Stuckey, Jimmy Dean, Norma Jean, Skeeter Davis, Dottie West, Bobby Bare, The Browns and Jerry Reed.
Bear Family has released multiple boxed sets on several RCA artists including Connie Smith, Don Gibson, Waylon Jennings and Hank Snow who have multiple boxed sets (essentially everything Hank Snow recorded while on RCA – forty plus years worth of recordings is available on Bear). Enough Waylon has been released that what remains doesn't justify a wish list.
What is really needed is for someone to issue decent sets on Kenny Price, Jim Ed Brown (without his sisters or Helen Cornelius), Norma Jean, Dottsy, Liz Anderson and Earl Thomas Conley. There is virtually nothing on any of these artists. Jimmy Dean recorded for RCA for about six years but nothing is available from his RCA years which saw some really fine recordings, including the best version of "A Thing Called Love".
I would have said the same thing about Charley Pride but recent years have seen various Charley Pride sets become available, so we can take him off our wish list.
When you think of Columbia Records, names such as Johnny Cash, Ray Price, Carl Smith, Stonewall Jackson, Flatt & Scruggs and Marty Robbins spring immediately to mind, but the well is deep and that doesn't even count sister label Epic which boasted names like David Houston, Tammy Wynette, Charlie Rich, Jody Miller, Johnny Paycheck and Bob Luman.
By and large foreign and domestic reissues abound for most of the bigger names, but even here there are some major shortfalls.
Carl Smith recorded for Columbia through the early 1970s and while his 1950s output has been thoroughly mined, his sixties output has barely been touched and his seventies output ("Mama Bear", "Don't Say Goodbye") completely neglected. Smith's recordings increasingly veered toward western swing as the sixties wore on, but he recorded a fine bluegrass album, and a tribute to fellow East Tennessean Roy Acuff. His outstanding Twenty Years of Hits (1952-1972) recast twenty of his classic tunes as western swing. A good three CD set seems in order.
I could make a good case for electing David Houston to the Country Music Hall of Fame. From 1966 he had thirteen #1 hits and a bunch more top ten and top twenty recordings. "Almost Persuaded" was his biggest hit but there were bunches of good songs scattered across his many albums. A good two CD set is a must, and I could easily justify a three CD set.
While Sony Legacy issued a decent Johnny Paycheck single disc hits collection, it is long on the later stages of his career and short on the earliest years. Paycheck released over thirty singles for Epic from 1972–1982 and it's about time someone collected them on a good two (or preferably three) disc collection along with some key album cuts.
Moe Bandy achieved his greatest commercial success while recording for Columbia. Between chart singles and album cuts Moe warrants at least a decent two CD set, and please leave the 'Moe & Joe' nonsense out of the mix.
Columbia has a lot of artists that would justify a single or double disc hits collection: David Wills, Al Dexter, Ted Daffan, David Rodgers, Connie Smith, Carl & Pearl Butler, Tommy Cash, David Frizzell, Bob Luman, Jody Miller, Barbara Fairchild, Barbara Mandrell, Charlie Walker and Sammi Smith.
Country Heritage, Wish lists Al Dexter, Barbara Fairchild, Barbara Mandrell, Bob Luman, Bobby Bare, Carl & Pearl Butler, Carl Smith, Charley Pride, Charlie Rich, Charlie Walker, Chet Atkins, Connie Smith, David Frizzell, David Houston, David Rodgers, David Wills, Dolly Parton, Don Gibson, Dottie West, Dottsy, Earl Thomas Conley, Eddy Arnold, Flatt & Scruggs, Hank Snow, Helen Cornelius, Jerry Reed, Jim Ed Brown, Jim Reeves, Jimmy Dean, Jody Miller, Johnny Cash, Johnny Paycheck, Kenny Price, Liz Anderson, Marty Robbins, Moe Bandy, Nat Stuckey, Norma Jean, Porter Wagoner, Ray Price, Roy Acuff, Sammi Smith, Skeeter Davis, Stonewall Jackson, Stu Phillips, Tammy Wynette, Ted Daffan, The Browns, Tommy Cash, Waylon Jennings
Favorite Country Songs Of The 80s: Part 7
2 Comments Posted by Paul W. Dennis on April 7, 2015
It seems to me that I never did finish off this series, the last installment being posted on February 11, 2014 (and the installment before that appeared April 9,2013). Here are some more songs from the 1980s that I liked. This is an expanded and revised version of the February 11, 2014 article which was a rush job :
"Shame On The Moon" – Bob Seger
Bob's 1982 recording of a Rodney Crowell song charted on the country charts in early 1983, reaching #15 in the process. The song was a bigger hit on the pop charts, reaching #2 for four weeks.
"Finally" – T. G. Sheppard
He worked for Elvis, sang background for Travis Wammack, and eventually emerged with a solo career worth noting, racking up 42 chart singles from 1974-1991. This 1982 single was one of fourteen #1 record racked up by Sheppard, eleven of them reaching #1 during the 1980s.
"Doesn't Anybody Get High On Love Anymore" – The Shoppe
The Shoppe was a Dallas based band that hung around for years after their 1968 formation. In the early 1980s they had eight chart records, but this was the only one to crack the top forty, reaching #33. They had a record deal with MTM Records in 1985, but that label vanished, taking the Shoppe with them.
"Crying My Heart Out Over You" – Ricky Skaggs
Ricky Skaggs was one of the dominant artists of the first half of the 1980s with his bluegrass/country hybrid. Starting with 1981's "You May See Me Walking" and ending with 1986's "Love's Gonna Get You Some Day", Skaggs ran off sixteen consecutive top ten singles with ten of them reaching number one, This 1982 classic was the first chart topper. Eventually Ricky returned to straight bluegrass, but I like the hybrid recordings better. In my original article I spotlighted "Honey (Open That Door)", a straight forward country Mel Tillis song recorded by Webb Pierce.
"Don't Stay If You Don't Love Me" – Patsy Sledd
Stardom never really happened for Patsy, who was a good singer marooned early in her career on a bad label. She was part of the George Jones-Tammy Wynette show in the early 1970s. This song reached #79 in 1987.
"Nice To Be With You" – Slewfoot
This band replaced Alabama as the feature band at the Bowery Club in Myrtle Beach. This was their only chart single, a cover of Gallery's #4 pop hit from 1972 that reached #85 in 1986.
"King Lear" – Cal Smith
The last chart hit for the former Texas Troubadour. This song reached #75 in 1986.
"A Far Cry From You" – Connie Smith
After a six year recording hiatus, the greatest female country recording artist of all time returned with this one-shot single on the Epic label. It's a great song but received no promotional push at all from the label landing at #71 in 1985. Unfortunately, this single has never appeared on an album.
"The Shuffle Song" – Margo Smith
Exactly as described – a shuffle song that reached #13 for Margo in early 1980. Margo had a brief run of top ten hits in the middle and late 1970s but the string was about over. In my prior article I featured "He Gives Me Diamonds, You Give Me Chills" but The Shuffle song is actually my favorite 80s hit from Margo. She lives in The Villages in Florida and still performs occasionally.
"Cheatin's A Two Way Street" – Sammi Smith
Her last top twenty song from 1981. Sammi only had three top ten hits but made many fine records. This was one of them.
"Hasn't It Been good Together" – Hank Snow and Kelly Foxton
The last chart record for the 'Singing Ranger'. The record only got to #78 for the 65 year old Snow in 1980 but I couldn't let pass the opportunity to acknowledge the great career of the most successful Canadian country artist. By any legitimate means of chart tracking, his 1950 hit "I'm Moving On" is still the number one country hit of all time. Hank had perfect diction and was a great guitar player.
"Tear-Stained Letter" – Jo-El Sonnier
A late bloomer, this was the forty-two year old Jo-El's second of two top ten records and my favorite. It reached #8 in 1988. There were brief periods in the past when Cajun music could break through for a hit or two. Eddy Raven was the most successful Cajun artist but most of his material was straight-ahead country.
"Sometimes You Just Can't Win" – J.D. Souther and Linda Ronstadt
George Jones charted this record twice, but it's such a good song it was worth covering. This version went to #27 in 1982. J.D had a big pop hit in 1980 with "You're Only Lonely" which reached #7.
"Honey I Dare You" – Southern Pacific
Southern Pacific was a bunch of guys who previously played with other bands such as Creedence Clearwater Revival, the Doobie Brothers and Pablo Cruise, making some real good country music in the process. This was one of their four top ten hits of the 1980s. "A Girl Like Emmylou" from 1986 only reached #17 but the song tells you where this band's heart was located.
"Lonely But Only For You" – Sissy Spacek
Loretta Lynn wanted to Spacek to portray her in the movie Coal Miner's Daughter, and it turns out that Sissy can really can sing. This song reached #15 in 1983.
"Standing Tall" – Billie Jo Spears
Billie Jo Spears, from Beaumont, Texas, was incredibly popular in England and Ireland, where "Blanket On The Ground" and "What I've Got In Mind" were top five pop hits in the mid 1970s and she had many more lesser successes. Many of her later albums were not released in the US but she had a substantial US career with thirty-four charted records, including two #1 hits. "Standing Tall" reached #15 in 1980.
"Chain Gang" – Bobby Lee Springfield
More successful as a songwriter than as a performer, Springfield had two chart sings in 1987 with "Hank Drank" (#75) and "Chain Gang" (#66) which was NOT the Sam Cooke hit. Bobby Lee was both too country and too rockabilly for what was charting at the time. I really liked All Fired Up, the one album Epic released on him.
Classic Rewind, Country Heritage, Decade In Review Alabama, Alan Jackson, B.J. Thomas, Billie Jo Spears, Bob Seger, Bob Wills, Bobby Lee Springfield, Boy George, Bruce Springsteen, Buddy Emmons, Buddy Holly, Cal Smith, Carl Smith, Clay Walker, Clint Eastwood, Connie Smith, Creedence Clearwater Revival, Culture Club, Darlene Shafer, Dolly Parton, Don Reid, Eddy Raven, Elvis Presley, Ernest Tubb, Exile, Faith Hill, Flatt & Scruggs, Four Knights, Gallery, Gary Stewart, George Hamilton IV, George Jones, George Strait, Glenn Sutton, Hal Ketchum, Hank Snow, Hank Thompson, Harold Reid, J. D. Souther, James Taylor, Janie Fricke, Janis Gill, Jeff Stevens & The Bullets, Jim Stafford, Jimmy C. Newman, Jimmy Fortune, Jo-El Sonnier, Joe Stampley, Joe Sun, Johnny Horton, Johnny Tillotson, Judy Rodman, Karen Staley, Karen Taylor, Karen Taylor-Good, Keith Stegall, Kelly Foxton, Kristine Arnold, Leah Kunkel, Lefty Frizzell, Leroy Van Dyke, Les Taylor, Lew DeWitt, Linda Ronstadt, Linda Shafer, Little Jimmy Dickens, Livingston Taylor, Loretta Lynn, Margo Smith, Marsha Thornton, Marty Stuart, Mel Street, Mel Tillis, Michael Martin Murphey, Moe & Joe, Moe Bandy, Orleans, Pablo Cruise, Pam Tillis, Patsy Cline, Patsy Sledd, Patty Loveless, Phase II, Phil Balsley, Randy Travis, Ray Pennington, Ray Price, Ray Stevens, Reba McEntire, Red Steagall, Ricky Skaggs, Rodney Crowell, Roy Acuff, Roy Drusky, Sammi Smith, Sandy Powell, Sanger D, Shafer, Sissy Spacek, Slewfoot, Southern Pacific, Statler Brothers, Sweethearts Of The Rodeo, Swing Shift Band, Sylvia, Sylvie and Her Silver Dollar Band, T.G.Sheppard, Tammy Wynette, The Doobie Brothers, The Shoppe, Thrasher Brothers, Tommy Cash, Tommy Duncan, Travis Wammack, Vince Gill, Webb Pierce, Whitey Shafer, Wynn Stewart, Zac Brown Band
Leave a comment Posted by Razor X on February 8, 2015
1955 (Sales): Loose Talk — Carl Smith (Columbia)
1955 (Jukebox): Loose Talk — Carl Smith (Columbia)
1955 (Disc Jockeys): Let Me Go, Lover — Hank Snow (RCA)
1965: You're The Only World I Know — Sonny James (Capitol)
1975: City Lights — Mickey Gilley (Playboy)
1985: A Place To Fall Apart — Merle Haggard (Epic)
1995: Mi Vida Loca (My Crazy Life) — Pam Tillis (Arista)
2005: Mud On The Tires — Brad Paisley (Arista)
2015: I See You — Luke Bryan (Capitol)
2015 (Airplay): Talladega — Eric Church (EMI Nashville)
Charts Brad Paisley, Carl Smith, Eric Church, Hank Snow, Luke Bryan, Merle Haggard, Mickey Gilley, Pam Tillis
1975: (I'd Be) A Legend In My Time — Ronnie Milsap (RCA)
1985:(There's A) Fire In The Night — Alabama (RCA)
1995: Gone Country — Alan Jackson (Arista)
2005: Awful, Beautiful Life — Darryl Worley (DreamWorks)
2015: Something In The Water — Carrie Underwood (19/Arista)
2015 (Airplay): Til It's Gone — Kenny Chesney (Blue Chair/Columbia)
Charts Alabama, Alan Jackson, Carl Smith, Carrie Underwood, Darryl Worley, Hank Snow, Kenny Chesney, Ronnie Milsap, Sonny James
1954 (Sales): More and More — Webb Pierce (Decca)
1954 (Jukebox): I Don't Hurt Anymore — Hank Snow (RCA)
1954 (Disc Jockeys): More and More — Webb Pierce (Decca)
1964: Once A Day — Connie Smith (RCA)
1974: Trouble In Paradise — Loretta Lynn (MCA)
1984: You Could've Heard A Heart Break — Johnny Lee (Warner Bros.)
1994: If I Could Make A Living — Clay Walker (Giant)
2004: Mr. Mom — Lonestar (BNA)
2014: Something In The Water — Carrie Underwood (Arista Nashville)
2014 (Airplay): Sunshine & Whiskey — Frankie Ballard (Warner Bros.)
Charts Carrie Underwood, Clay Walker, Connie Smith, Frankie Ballard, Hank Snow, Johnny Lee, Lonestar, Loretta Lynn, Webb Pierce
1964: I Don't Care (Just As Long As You Love Me) — Buck Owens (Capitol)
1974: Country Is — Tom T. Hall (Mercury)
1984: Give Me One More Chance — Exile (Epic)
1994: Shut Up And Kiss Me — Mary-Chapin Carpenter (Columbia)
2004: In A Real Love — Phil Vassar (Arista)
2014 (Airplay): Neon Light — Blake Shelton (Warner Bros.)
Charts Blake Shelton, Buck Owens, Carrie Underwood, Exile, Hank Snow, Mary Chapin Carpenter, Phil Vassar, Tom T Hall, Webb Pierce
1954 (Disc Jockeys): I Don't Hurt Anymore — Hank Snow (RCA)
1974: Love Is Like A Butterfly — Dolly Parton (RCA)
1984: I've Been Around Enough To Know — John Schneider (MCA)
1994: Livin' On Love — Alan Jackson (Arista)
2014: Leave The Night On — Sam Hunt (MCA)
2014 (Airplay): Leave The Night On — Sam Hunt (MCA)
Charts Alan Jackson, Buck Owens, Dolly Parton, Hank Snow, John Schneider, Phil Vassar, Sam Hunt
Leave a comment Posted by Razor X on November 9, 2014
1974: I Overlooked An Orchid — Mickey Gilley (Playboy)
1984: City of New Orleans — Willie Nelson (Columbia)
2014: Burnin' It Down — Jason Aldean (Broken Bow)
2014 (Airplay): Burnin' It Down — Jason Aldean (Broken Bow)
Charts Alan Jackson, Buck Owens, Hank Snow, Jason Aldean, Mickey Gilley, Phil Vassar, Webb Pierce, Willie Nelson
1954 (Sales): I Don't Hurt Anymore — Hank Snow (RCA)
1974: I See The Want To In Your Eyes — Conway Twitty (MCA)
1984: If You're Gonna Play In Texas (You Gotta Have A Fiddle In The Band) — Alabama (RCA)
2004: I Hate Everything — George Strait (MCA)
Charts Alabama, Alan Jackson, Buck Owens, Conway Twitty, George Strait, Hank Snow, Jason Aldean
Leave a comment Posted by Razor X on October 26, 2014
1984: I Don't Know A Thing About Love (The Moon Song) — Conway Twitty (Warner Bros.)
1994: She's Not The Cheatin' Kind — Brooks & Dunn (Arista)
2014 (Airplay): Dirt — Florida Georgia Line (Republic Nashville)
Charts Brooks & Dunn, Buck Owens, Conway Twitty, Florida Georgia Line, George Strait, Hank Snow, Jason Aldean | {
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-4x-6y-28-and-2x-3y-14","text":"# How do you solve -4x \u2013 6y = -28 and 2x + 3y = 14 ?\n\nJul 22, 2016\n\nHere's what I got.\n\n#### Explanation:\n\nYour starting system of two equations with two unknowns looks like this\n\n$\\left\\{\\begin{matrix}- 4 x - 6 y = - 28 \\\\ \\textcolor{w h i t e}{. .} 2 x + 3 y = \\textcolor{w h i t e}{-} 14\\end{matrix}\\right.$\n\nNotice that if you multiply the second equation by $\\textcolor{b l u e}{- 2}$, you will get the first equation!\n\n$2 x + 3 y = 14 \\text{ } | \\times \\textcolor{b l u e}{\\left(- 2\\right)}$\n\n$\\textcolor{b l u e}{\\left(- 2\\right)} \\cdot 2 x + \\textcolor{b l u e}{\\left(- 2\\right)} \\cdot 3 y = \\textcolor{b l u e}{\\left(- 2\\right)} \\cdot 14$\n\n$- 4 x - 6 y = - 28$\n\nThis means that your system of equations has an infinite number of solutions. This is the case because if you were to subtract this new form of the second equation from the first equation, you'd get\n\n$\\left\\{\\begin{matrix}- 4 x - 6 y = - 28 \\text{ } | - \\\\ - 4 x - 6 y = - 28\\end{matrix}\\right.$\n$\\frac{\\textcolor{w h i t e}{a a a a a a a}}{\\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a}}$\n\n$- 4 x - \\left(- 4 x\\right) - 6 y - \\left(- 6 y\\right) = - 28 - \\left(- 28\\right)$\n\n$- \\textcolor{red}{\\cancel{\\textcolor{b l a c k}{4 x}}} + \\textcolor{red}{\\cancel{\\textcolor{b l a c k}{4 x}}} - \\textcolor{red}{\\cancel{\\textcolor{b l a c k}{6 y}}} + \\textcolor{red}{\\cancel{\\textcolor{b l a c k}{6 y}}} = - \\textcolor{red}{\\cancel{\\textcolor{b l a c k}{28}}} + \\textcolor{red}{\\cancel{\\textcolor{b l a c k}{28}}}$\n\nwhich of course gives\n\n$0 = 0$\n\nSo, when does $0$ equal to $0$? Well, pretty much always, which is why your system of equations is said to have and infinite number of solutions.\n\nIn other words, you can plug in any value you want for $x$ and for $y$ because $0 = 0$ will always be true.\n\nAlternatively, you can think about the two equations given to you as describing the same line. To check that this is the case, rearrange both equations in slope-intercept form\n\n$- 4 x - 6 y = - 28$\n\n$- 6 y = 4 x - 28 \\implies y = - \\frac{2}{3} x + \\frac{14}{3}$\n\nSimilarly, you have\n\n$2 x + 3 y = 14$\n\n$3 y = - 2 x + 14 \\implies y = - \\frac{2}{3} x + \\frac{14}{3}$\n\nThis once again leads to the conclusion that the system has an infinite number of solutions.","date":"2019-12-13 00:31:36","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 19, \"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\": 0.7731203436851501, \"perplexity\": 129.56100982071456}, \"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-2019-51\/segments\/1575540547536.49\/warc\/CC-MAIN-20191212232450-20191213020450-00529.warc.gz\"}"} | null | null |
Q: Prototype chain: call "super" method over multiple levels I have got the following prototype chain
*
*SuperSuperClass
*
*SuperClass
*
*Class
each with a method named do.
What is the common approach for calling the respective super class method?
For the moment I use <ClassName>.prototype.__proto__.<methodName>.call(this) but that looks odd.
Using the following code the console prints (as expected):
*
*Class.prototype.do
*SuperClass.prototype.do
*SuperSuperClass.prototype.do
SuperSuperClass = function SuperSuperClass() {}
SuperSuperClass.prototype.do = function() {
console.log('SuperSuperClass.prototype.do');
};
function SuperClass() {
SuperSuperClass.call(this);
}
SuperClass.prototype = Object.create(SuperSuperClass.prototype);
SuperClass.prototype.constructor = SuperClass;
SuperClass.prototype.do = function() {
console.log('SuperClass.prototype.do');
SuperClass.prototype.__proto__.do.call(this);
};
function Class() {
SuperClass.call(this);
}
Class.prototype = Object.create(SuperClass.prototype);
Class.prototype.constructor = Class;
Class.prototype.do = function() {
console.log('Class.prototype.do');
Class.prototype.__proto__.do.call(this);
};
var objClass = new Class();
objClass.do();
JSFiddle
A:
What is the common approach for calling the respective super class method?
Use <SuperClassName>.prototype.<methodName>.call(this). It's not only shorter, but also has the benefit of working in environments that don't support the non-standard __proto__ property.
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\section*{Introduction}
A typical phenomenon in constructive mathematics is the split of classical notions: several definitions which are equivalent over classical logic can become deeply different over intuitionistic logic. In this paper we study an alternative way to define complete Boolean algebras, as proposed by Giovanni Sambin \cite{6,3z} who named them \emph{overlap algebras}. There are some facts which make overlap algebras interesting, we believe, from the constructive point of view; for instance, the collection of all subsets of a set is an overlap algebra, actually an atomistic one, although it cannot ever be Boolean (apart from the trivial case of the power of the empty set).
Roughly speaking, an overlap algebra is a complete lattice (actually a complete Heyting algebra) equipped with a new primitive relation, the overlap relation $>\mkern-13.5mu <$. The intended meaning of $x>\mkern-13.5mu < y$ is that the infimum $x\wedge y$ is ``inhabited''. The distinction between $\emph{inhabited}$ and $\emph{non-empty}$ is enlightening. Indeed, constructively $\exists x(x\in X)$ is a stronger statement than $\neg \forall x \neg(x\in X)$. In an arbitrary complete Heyting algebra we can use $x\neq 0$ as the algebraic counterpart of the set-theoretic $X\neq\emptyset$, but there is no way to express the positive statement of being inhabited. Overlap algebras give an elegant answer to this question.
Overlap algebras and complete Boolean algebras have just one element in common, the trivial one-element algebra, unless classical logic is assumed, in which case the two notions coincide.
In this paper we investigate two natural notions of morphism between overlap algebras which are both inspired by the powerset construction. First we study the category {\bf OA} as originally introduced by Sambin; {\bf OA} is a dagger category which contains the category {\bf Rel} of sets and relations as a full subcategory; classically, {\bf OA} is the category of complete Boolean algebras and join preserving maps. In particular, we characterize monomorphisms, epimorphisms, and isomorphisms in {\bf OA}, and we establish some basic facts about limits and colimits.
Then we specialize to the subcategory {\bf OFrm} whose arrows preserve also finite meets. This is a subcategory of {\bf Frm}, the category of frames; morphisms in {\bf OFrm} correspond to open maps in the sense of locale theory. Classically, {\bf OFrm} is the usual category of complete Boolean algebras; we are therefore able to obtain new constructive versions of some standard results about Boolean locales.
If not otherwise stated, we assume to work over intutionistic logic and without choice. In other words, we understand ``constructive'' as ``topos-valid''. In particular, we shall usually think of powersets as perfectly legitimate sets even if we shall make some remark on predicativity in the last section: it is a fact that most of the paper could be adapted to a predicative framework (such as that presented in \cite{4}) by a systematic use of ``bases".
Part of the material in the present paper appeared in the second author's master thesis~\cite{3a}.
\section{Atomic Heyting algebras}
Given a set $X$, its subsets form a complete Heyting algebra $\mathrm{Pow}(X)$ with respect to the usual set-theoretic operations. Here we write $-Y$ for the pseudo-complement of the subset $Y\subseteq X$. We write $\Omega$ for $\mathrm{Pow}(1)$, where $1=\{0\}$, which we interpret as the type of truth values. It is well-known that the following statements are equivalent:
\begin{itemize}
\item the Law of Excluded Middle (LEM);
\item $(\forall p\in\Omega)(p\cup-p=1)$, that is, $\Omega\cong 2=\{0,1\}$;
\item $\mathrm{Pow}(X)$ is a complete Boolean algebra for every $X$.
\end{itemize}
Classically, powersets are precisely the atomic Boolean algebras (this means that every element is the join of the atoms below it, where an atom is a minimal non-zero element). In other words, a Boolean algebra is atomic
if and only if it is isomorphic to the powerset of the set of its atoms.
The problem of finding a constructive characterization of powersets is related to the problem of finding a suitable algebraization of the notion of a singleton. Apparently, none of the first-order (in the sense of the language of lattices) attempts to define when $a\in L$ is an atom is satisfactory from an intuitionistic point of view; consider, for instance, the following.
\begin{eqnarray}
a\neq 0 & \land & (\forall x\in L)(x\neq 0\land x\leq a\Rightarrow x=a)
\label{eq def atom 1}\\
a\neq 0 & \land & (\forall x\in L)(x\leq a\Rightarrow x=0\vee x=a)
\label{eq def atom 2}\\
a\neq 0 & \land & (\forall x\in L)(x< a\Rightarrow x=0)
\label{eq def atom 3}\\
a\neq 0 & \land & \neg(\exists x\in L)(x\neq 0\land x< a)
\label{eq def atom 4}
\end{eqnarray}
Indeed, when applied to the case $L=\mathrm{Pow}(X)$, singletons cannot be proven to be atoms in the sense of \eqref{eq def atom 1} or \eqref{eq def atom 2}, and it is impossible to prove that every subset satisfying \eqref{eq def atom 3} or \eqref{eq def atom 4} is a singleton, although a singleton satisfies \eqref{eq def atom 3} and \eqref{eq def atom 4}. All this comes up clear already when inspecting the case $L=\Omega$: its only singleton $1=\{0\}$ satisfies \eqref{eq def atom 1} or \eqref{eq def atom 2} if and only if LEM holds; and LEM is equivalent to assuming that $1$ is the only $a\in\Omega$ satisfying \eqref{eq def atom 3} or \eqref{eq def atom 4}.
A possible well-known solution is to adopt a second-order definition, as follows.
\begin{definition}
Given a poset $(L,\leq)$, we say that $a\in L$ is an {\bf atom} if the poset ${\downarrow a}$ = $\{x\in L\ |\ x\leq a\}$ is order-isomoprhic to $\Omega$. And $(L,\leq)$ is {\bf atomic} if the join of all atoms below a given $x$ exists and equals $x$, for every $x\in L$.
\end{definition}
If $L=\mathrm{Pow}(X)$, then ${\downarrow\{x\}}$ = $\mathrm{Pow}(\{x\})$ is isomorphic to $\Omega$ = $\mathrm{Pow}(\{0\})$, for all $x\in X$. So every singleton is an atom and hence every element is a join of atoms. Actually, we can show that the atoms in $\mathrm{Pow}(X)$ are precisely the singletons. Let $Y$ be an atom, that is, ${\downarrow Y}=\mathrm{Pow}(Y)\cong\Omega$; and let $\varphi:\mathrm{Pow}(Y)\to\Omega$ be an order isomorphism (which then preserves joins and meets). Then $1$ = $\varphi(Y)$ = $\varphi(\bigcup_{x\in Y}\{x\})$ = $\bigvee_{x\in Y}\varphi(\{x\})$. So $Y$ is inhabited, actually there is some $x\in Y$ with $\varphi(\{x\})$ = 1, and hence $Y$ = $\varphi^{-1}(1)$ = $\{x\}$.
\begin{proposition}\label{prop atomic frames}
A frame $L$ is atomic if and only if it is order isomorphic to $\mathrm{Pow}(X)$, where $X$ is the set of atoms of $L$.
\end{proposition}
\begin{proof}
One direction follows from the discussion above.
As for the other, let us define $f:L\to\mathrm{Pow}(X)$ to be the function which maps a given $x$ to the set of atoms below it, and let $g:\mathrm{Pow}(X)\to L$ be the function which maps a set of atoms to its join. The two mapping are clearly monotone. Moreover, $g(f(x))=x$ because $L$ is atomic. It remains to show that $f(g(Y))=Y$ for every $Y\subseteq X$. The inclusion $Y\subseteq f(g(Y))$ is clear. As for the other, we must show that $x\leq\bigvee Y$ implies $x\in Y$ for every $x\in X$. Now $x\leq\bigvee Y$ can be written as $x$ = $x\wedge\bigvee Y$ = $\bigvee\{x\wedge y\ |\ y\in Y\}$. Since $x$ is an atom (that is, ${\downarrow x}$ behaves like $\Omega$), $x\wedge y$ must be $x$ for some $y$. So $x\leq y$. Since $y$ is an atom too, this happens precisely when $x=y$ (there is only one atom in ${\downarrow y}$ $\cong$ $\Omega$ = $\mathrm{Pow}(1)$).
\end{proof}
\subsection{The positivity predicate on a frame}
For $X$ a set, the statement ``$X$ is inhabited" is stronger than ``$X\neq\emptyset$", constructively; and the two statements are equivalent for all sets $X$ if and only if LEM holds.
There exists a quite standard way to ``algebraize" the concept of an inhabited set.
\begin{definition}
Let $L$ be a complete lattice. A unary predicate $\mathrm{Pos}$ on $L$ is a {\bf positivity predicate} if the following conditions hold identically.
\begin{eqnarray}
\mathrm{Pos} (x)\land (x\leq y)\Rightarrow\mathrm{Pos} (y)\label{eq monotonicity}\\
\mathrm{Pos} (\bigvee X)\Rightarrow(\exists x\in X)\mathrm{Pos} (x)\label{eq splitting}\\
(\mathrm{Pos} (x) \Rightarrow (x \leq y)) \Longrightarrow x \leq y
\label{eq positivity}
\end{eqnarray}
\end{definition}
\noindent It is easy to check that \eqref{eq positivity} can be replaced by
\begin{equation}\label{eq positivity bis}
y\leq\bigvee\{x\in L\ |\ \mathrm{Pos}(x)\land (x\leq y)\}\ .
\end{equation} By extending the terminology which is used for frames/locales, we call a complete lattice {\bf overt} if it has a positivity predicate.
It is well-known that if $L$ is overt, then $\mathrm{Pos}$ is equivalent to the second-order predicate $POS$, where $POS(x)$ is $(\forall X\subseteq L)(x\leq\bigvee X\Longrightarrow X\textrm{ is inhabited})$. This has a couple of (almost) immediate consequences. First, the positivity predicate, when it exists, is unique and it is uniquely determined by the ordering. Second, $L$ is overt if and only if $POS$ is a positivity predicate.
Classically, every complete lattice is overt and $\mathrm{Pos}(x)$ is just $x\neq 0$. Constructively, $\mathrm{Pos}(x)$ always implies $x\neq 0$, but not the other way around, in general; and it cannot be proven that every complete lattice is overt.
The notion of overteness for a frame can be characterized in a more categorical fashion. Given a frame $L$, there is a unique frame homomorphism $!:\Omega\to L$ (that is, $\Omega$ is the initial frame, that is, the terminal locale). Then $L$ is overt precisely when $!$ has a left adjoint $\exists_!$ (which happens precisely when $!$ preserves arbitrary meets), in which case $\exists_!=\mathrm{Pos}$.
\subsection{Atoms of an overt frame}
The positivity predicate $\mathrm{Pos}$ can be used to characterize the atoms.
In the case of a powerset, a singleton is precisely a minimal \emph{inhabited} subset. So the following variation of \eqref{eq def atom 1} is the natural candidate for a first-order definition of an atom:
\begin{equation}\label{eq def atom 1bis}
\mathrm{Pos}(a) \land (\forall x\in L)\big(\mathrm{Pos}(x)\wedge (x\leq a)\Longrightarrow (x=a)\big)\ .
\end{equation}
\begin{proposition}
Let $L$ be an overt complete lattice; $a\in L$ is an atom if and only if $a$ satisfies \eqref{eq def atom 1bis}.
\end{proposition}
\begin{proof}
If $L$ is overt and $x\in L$, then also ${\downarrow x}$ is overt with respect to the restriction of $\mathrm{Pos}$.
Let $a$ be an atom, that is, ${\downarrow a}$ $\cong$ $\Omega$. So $\mathrm{Pos}(x)$ becomes ``$x$ is inhabited" under such an isomorphism, and hence \eqref{eq def atom 1bis} is true on ${\downarrow a}$ (because it is true on $\Omega$; recall that the positivity predicate is uniquely determined by the ordering and so has to be preserved by order-isomorphism).
Conversely, if $a$ satisfies \eqref{eq def atom 1bis}, $\mathrm{Pos}:{\downarrow a}\to\Omega$ is an order-isomorphism whose inverse is $p\mapsto\bigvee\{x\leq a\ |\ p\}$. Indeed, the two mappings are monotone, and $\mathrm{Pos}(\bigvee\{x\leq a\ |\ p\})=p$; moreover, for $b\leq a$, it is $\bigvee\{x\leq a\ |\ \mathrm{Pos}(b)\}\leq b$ because $\mathrm{Pos}(b)$ is just $b=a$ by \eqref{eq def atom 1bis}, and $b\leq\bigvee\{x\leq a\ |\ \mathrm{Pos}(b)\}$ because $\mathrm{Pos}$ is a positivity predicate on ${\downarrow a}$, in particular it satisfies \eqref{eq positivity}.
\end{proof}
As noticed by Giovanni Sambin, \eqref{eq def atom 1bis} is equivalent to the following elegant condition:
\begin{equation}\label{eq atom Pos}
(\forall x\in L)\big(\mathrm{Pos}(a\wedge x)\Longleftrightarrow(a\leq x)\big)\ .
\end{equation}
\section{Overlap Algebras}
Every complete Boolean algebra is a frame and, classically, every atomic frame (that is, a powerset by proposition \ref{prop atomic frames}) is a complete Boolean algebra. The latter fails constructively; a constructive version can be obtained by replacing complete Boolean algebras by Sambin's {\bf overlap algebras}, as we now see.
\begin{definition}
An $\emph{overlap-algebra}$ (o-algebra) is an overt frame $L$ such that
\begin{equation}\label{eq prop Pos}
(\forall z\in L ) (\mathrm{Pos} (z \wedge x) \Rightarrow \mathrm{Pos} (z \wedge y)) \Longrightarrow x \leq y
\end{equation} for all $x,y\in L$.
\end{definition}
The motivating example is given by powersets, where $\mathrm{Pos}(x)$ means ``$x$ is inhabited". To see that \eqref{eq prop Pos} holds in this case it is sufficient to make $z$ vary over singletons. Note that for $p\in\Omega$ the statement $\mathrm{Pos}(p)$ is equivalent to $p=1$.
Note that a frame $L$ is an o-algebra if and only if there exists a unary predicate $\mathrm{Pos}$ on $L$ such that \eqref{eq monotonicity}, \eqref{eq splitting} and \eqref{eq prop Pos} hold. Indeed \eqref{eq positivity} follows from \eqref{eq monotonicity} and \eqref{eq prop Pos}: assume $\mathrm{Pos}(x)\Rightarrow (x\leq y)$; if $\mathrm{Pos}(z\wedge x)$, then $\mathrm{Pos}(x)$ and so $x\leq y$ by assumption; therefore $z\wedge x\leq z\wedge y$, and hence $\mathrm{Pos}(z\wedge y)$.
\begin{proposition}
Classically, o-algebras and complete Boolean algebras coincide.
\end{proposition}
\begin{proof}
Classically, overtness is for free, and $\mathrm{Pos}(x)$ is $x\neq 0$. So the implication $\mathrm{Pos} (z \wedge x) \Rightarrow \mathrm{Pos} (z \wedge y)$ in \eqref{eq prop Pos} can be rewritten as $z \wedge y= 0 \Rightarrow z \wedge x= 0$, that is, $z \leq-y \Rightarrow z \leq-x$. Therefore $(\forall z\in L ) (\mathrm{Pos} (z \wedge x) \Rightarrow \mathrm{Pos} (z \wedge y))$ becomes simply $-y\leq-x$ and \eqref{eq prop Pos} becomes $-y\leq-x \Longrightarrow x \leq y$. This holds identically in an Heyting algebra if and only if it is in fact a Boolean algebra.
\end{proof}
Constructively, the previous proposition fails badly, because LEM is equivalent to the statement that $\Omega$ (which is an o-algebra) is Boolean.\footnote{The statement ``every complete Boolean algebra is an o-algebra" is equivalent to LEM as well (see, for instance, \cite{2a} proposition $1.1$).}
Given an o-algebra $L$, it is sometimes convenient to introduce a new relation symbol, say $x>\mkern-13.5mu < y$, for the binary predicate $\mathrm{Pos}(x\wedge y)$: this is the {\bf overlap relation} which gives the name to the structure. If $L$ is a powerset, then $x>\mkern-13.5mu < y$ means that $x$ and $y$ overlap, that is, their intersection is inhabited. Classically, $x>\mkern-13.5mu < y$ is $x\wedge y\neq 0$. Clearly, $\mathrm{Pos}(x)$ is equivalent to $x>\mkern-13.5mu < x$ (and also to $x>\mkern-13.5mu < 1$); this suggests that the definition of an o-algebra can be given in terms of $>\mkern-13.5mu <$ (which was Sambin's original definition).
\begin{proposition}
For $L$ a complete lattice, the following are equivalent:
\begin{enumerate}
\item $L$ is an o-algebra;
\item there exists a binary relation $>\mkern-13.5mu <$ on $L$ that satisfies the following properties identically.
\begin{itemize}
\item $ x >\mkern-13.5mu < y \Longrightarrow y >\mkern-13.5mu < x $ \hfill $ (\textit{symmetry}) $
\item $ x >\mkern-13.5mu < y \Longrightarrow x >\mkern-13.5mu < (x \wedge y) $ \hfill $ (\textit{meet closure}) $
\item $ x >\mkern-13.5mu < \bigvee Y \Longrightarrow (\exists y\in Y) (x >\mkern-13.5mu < y)$ \hfill $ (\textit{splitting of joins} )$
\item $ (x >\mkern-13.5mu < y) \land (y\leq z) \Longrightarrow x >\mkern-13.5mu < z $ \hfill $ (\textit{monotonicity}) $
\item $ (\forall z\in L ) (z >\mkern-13.5mu < x \Rightarrow z >\mkern-13.5mu < y) \Longrightarrow x \leq y $ \hfill $ (\textit{density}) $
\end{itemize}
\end{enumerate}
\end{proposition}
\begin{proof}
The implication $1\Rightarrow 2$ is easy once $x>\mkern-13.5mu < y$ is defined as $\mathrm{Pos}(x\wedge y)$. For instance, splitting of joins holds because binary meets distribute over arbitrary joins (since $L$ is a frame).
As for the reverse implication, we first note that $x>\mkern-13.5mu < y$ is equivalent to $(x\wedge y)>\mkern-13.5mu < (x\wedge y)$ thanks to symmetry, meet closure and monotonicity. Therefore $ (x \wedge y) >\mkern-13.5mu < z$ is always equivalent to $z >\mkern-13.5mu < (y \wedge z)$. We now show that $L$ is a frame, that is, $x\wedge\bigvee Y\leq\bigvee\{x\wedge y\ |\ y\in Y\}$. By density, it is sufficient to check that $z>\mkern-13.5mu < x\wedge\bigvee Y$ implies $z>\mkern-13.5mu <\bigvee\{x\wedge y\ |\ y\in Y\}$. Now $z>\mkern-13.5mu < x\wedge\bigvee Y$ is equivalent to $z\wedge x>\mkern-13.5mu < \bigvee Y$; so there is a $y\in Y$ with $z\wedge x>\mkern-13.5mu < y$, that is, $z>\mkern-13.5mu < x\wedge y$. So $z>\mkern-13.5mu <\bigvee\{x\wedge y\ |\ y\in Y\}$ by monotonicity. Finally, let us define $\mathrm{Pos}(x)$ as $x>\mkern-13.5mu < x$. The only condition on $\mathrm{Pos}$ which needs some proof is \eqref{eq positivity} which follows from \eqref{eq monotonicity} and \eqref{eq prop Pos}, as already noticed.
\end{proof}
For $L$ an o-algebra, the characterization \eqref{eq atom Pos} of an atom $a\in L$ becomes \begin{equation}\label{eq char atom}
(\forall x\in L)\big((a>\mkern-13.5mu < x)\Longleftrightarrow(a\leq x)\big)\ .
\end{equation}
By proposition \ref{prop atomic frames}, atomic frames, atomic o-algebras and powersets all amount to the same thing.
\subsection{Non-atomic o-algebras}\label{section non-atomic}
Given any complete Heyting algebra $L$, the set $L_{--}$ = $\{y\in L\ |\ y=--y\}$ has a natural structure of complete Boolean algebra (and every complete Boolean algebra is of this form, because $L_{--}=L$ if $L$ is Boolean).
A similar result holds for o-algebras \cite{2a}: if $L$ is an overt frame, then the set of all $y\in L$ such that $y$ = $\bigvee\{x\ |\ \forall z(\mathrm{Pos}(z\wedge x)\Rightarrow\mathrm{Pos}(z\wedge y))\}$ is an o-algebra. In particular, if $L=\tau$ where $(X,\tau)$ is a topological space, then we get an o-algebra by considering the set of all $Y\subseteq X$ such that $Y=\mathrm{int}\,\mathrm{cl}\,\mathrm{int}\, Y$, where $\mathrm{int}$ and $\mathrm{cl}$ are the interior operator and the closure operator corresponding to $\tau$.\footnote{Here $x\in\mathrm{cl} Y$ means that every open neighbourhood of $x$ overlaps $Y$. Assuming that $\mathrm{cl} Y$ is the (set-theoretic pseudo-)complement of the interior of the (set-theoretic pseudo-)complement of $Y$ is tantamount to assuming LEM.} This is a constructive version of the well-known fact that the regular open sets in a topological space form a complete Boolean algebra, which is not atomic, in general, and often with no atoms \cite{3z}.
\section{Morphisms between overlap algebras}
In section \ref{section OFrm} we whall study a category of overlap algebras which, from a classical point of view, is just the category {\bf cBa} of complete Boolean algebras. For the time being, instead, we are going to study a more general kind of morphisms between o-algebras which, classically, correspond to join-preserving maps between complete Boolean algebras
Sambin's aim in introducing the category $\mathbf{OA}$ of o-algebras was to obtain an extension of the category $ \mathbf{Rel} $ of sets and relations. The definition of an arrow in {\bf OA} makes the assignment $X\mapsto\mathrm{Pow}(X)$ a functor $\mathrm{Pow}$ from $ \mathbf{Rel} $ to $ \mathbf{OA} $ which is full, faithful and injective on objects (Proposition \ref{PropPow}).
In the category $\mathbf{Rel}$ a morphism is a binary relation and the composition $S\circ R\subseteq X\times Z$ of the relations $R\subseteq X\times Y$ and $S\subseteq Y\times Z$ is defined by $x(S\circ R)z \Leftrightarrow (\exists y\in Y)(xRy\land ySz)$.
Given $R\subseteq X\times Y$, its inverse image $R^{-1}:\mathrm{Pow}(Y)\to\mathrm{Pow}(X)$ is the function which maps $Y'\subseteq Y$ to $R^{-1}(Y')$ = $\lbrace x\in X\ \vert\ (\exists y\in Y') (xRy)\rbrace$. Clearly, $R^{-1}$ is the identity function on $\mathrm{Pow} (X)$ if, and only if, $R$ is the equality on $X$.
\begin{lemma}\label{lemmaRel}
For $R\subseteq X\times Y$ and $S\subseteq Y\times Z$, $(S\circ R)^{-1} = R^{-1}\circ S^{-1}$.
\end{lemma}
\begin{proof}
For every $x\in X$ and $D\subseteq Z$, $x\in (S\circ R)^{-1}(D) $ iff $x(S\circ R)z$ for some $z\in D$; this means that $xRy$ and $ySz$ for some $y\in Y$, and some $z\in D$. In other words, $x\in R^{-1}(S^{-1}(D))$.
\end{proof}
Each binary relation $R\subseteq X\times Y$ has a ``symmetric'' $R^\dagger\subseteq Y\times X$, where $yR^\dagger x$ iff $xRy$. Its inverse image is a function $(R^\dagger)^{-1}:\mathrm{Pow}(X)\to\mathrm{Pow}(Y)$, the direct image of $R$, such that
\begin{equation}\label{eq.R-}
R^{-1}(Y')>\mkern-13.5mu < X'\ \textrm{ in }\mathrm{Pow}(X)\quad\Longleftrightarrow\quad Y'>\mkern-13.5mu < (R^\dagger)^{-1}(X')\ \textrm{ in }\mathrm{Pow}(Y).
\end{equation} This motivates the following study of symmetrizable functions.
\subsection{Symmetrizable functions}
\begin{definition}
Let $L$ and $M$ be two o-algebras.\footnote{Such a notion makes sense also for $L$ and $M$ overt frames, with $x>\mkern-13.5mu < y$ replaced by $\mathrm{Pos}(x\wedge y)$.} Two functions $f:L\to M$ and $g:M\to L$ are {\bf symmetric} (or {\bf conjugated} \cite{3c}) if
\begin{equation}
f(x) >\mkern-13.5mu < y \Longleftrightarrow x >\mkern-13.5mu < g(y)
\end{equation}
for every $x\in L$ and $y\in M$.\footnote{Classically, the same idea can be expressed by the condition $f(x)\wedge y=0$ $\Leftrightarrow$ $x\wedge g(y)=0$, which is the definition originally proposed \cite{3c}.}
\end{definition}
For instance, the function $a\wedge\_:L\to L$ is self-symmetric, for every element $a$ in an o-algebra $L$.
A function between o-algebras $f:L\to M$ has at most one symmetric.\footnote{This fact fails, in general, when $L$ is an overt frame but not an o-algebra.} Indeed, if $g_1,g_2:M\to L$ are symmetric of $f$, then $ x >\mkern-13.5mu < g_1(y)$ $\Leftrightarrow$ $x >\mkern-13.5mu < g_2(y)$ for every $x$ and $y$; and hence $g_1(y)=g_2(y)$ for every $y$, by density in $L$.
\begin{definition}
A function $f:L\to M$ between two o-algebras is {\bf symmetrizable} if $f$ has a symmetric. In that case, we write $ f^\dagger$ for the symmetric of $f$.
\end{definition}
Clearly if $f$ is symmetrizable, then $f^\dagger$ is symmetrizable too and $(f^\dagger)^\dagger=f$. Note that if $f$ is symmetrizable, then $f^\dagger$ can be defined in terms of $f$ by means of the formula $f^\dagger(y)$ = $\bigvee \lbrace x \in L\ \vert\ (\forall z \in L) \big(z >\mkern-13.5mu < x \Rightarrow f(z) >\mkern-13.5mu < y\big)\rbrace$.
\begin{proposition}
Let $f:L\to M$ be a function between o-algebras. If $f$ is symmetrizable, then $f$ preserves all joins; and the converse holds classically.
\end{proposition}
\begin{proof}
For every $y\in M$, we have $y >\mkern-13.5mu < f(\bigvee _i x_{i})$ iff $f^\dagger y >\mkern-13.5mu < \bigvee _ix_{i}$ iff $f^\dagger y>\mkern-13.5mu < x_{i}$ for some $i$ iff $y >\mkern-13.5mu < fx_{i}$ for some $i$ iff $y >\mkern-13.5mu < \bigvee _i fx_{i}$. This shows (by density) that $f(\bigvee _ix_{i}) = \bigvee _i fx_{i}$.
Classically, an o-algebra is exactly a cBa. If $ f: L \rightarrow M $ preserves all joins, then it has a right adjoint $ \forall_f $. We claim that $ f^\dagger $ does exist and $ f^\dagger(y) = -\forall_f(-y) $. For $ x >\mkern-13.5mu < -\forall_f(-y)$ $\Leftrightarrow$ $x\wedge -\forall_f(-y) \neq 0$ $\Leftrightarrow$ $x \nleq \forall_f(-y)$ $\Leftrightarrow$ $f(x) \nleq -y$ $\Leftrightarrow$ $f(x) \wedge y \neq 0$ $\Leftrightarrow$ $f(x) >\mkern-13.5mu < y$.
\end{proof}
\begin{remark}\label{remark:symmetrizable}
Classical logic is necessary in the second part of the previous proposition in the sense that LEM follows from the assumption that every join-preserving function between o-algebras is symmetrizable, as we now see. The argument is based on the fact that LEM is equivalent to assuming that every topological space in which $\mathrm{cl}$ is the identity operator must be discrete.\footnote{\label{counterexample}The following is essentially the same proof given in \cite{2a}. Let us start by constructing a family of topological spaces $(2,\tau_p)$ where $2=\lbrace 0,1\rbrace $ and $p\in\Omega$. Let $\tau_p$ be the topology of those subsets $X\subseteq 2$ such that if $X$ is inhabited, then either $p$ holds or $p$ implies $X=2$. It is not difficult to check that $\tau_p$ is a topology (and $\tau_p$ is discrete if either $p$ or $\neg p$, which is always the case classically). We claim that every $X\subseteq 2$ is closed. If $x\in\mathrm{cl} X $, then the open set $\{y\ |\ (y=x)\vee p\}$ must overlap $X$. So either $ x\in X $ or $p$; in the latter case, however, $ \tau_{p} $ is the discrete topology, and hence $ x\in X $ anyway. Therefore $\mathrm{cl}$ is the identity. Now if $\tau_p$ were discrete, then $\lbrace 0\rbrace $ (and $\{1\}$) would be open, hence $p\vee\neg p$ would be true.}
Let us consider any topological space $(X,\tau)$ such that $cl=id$; so $\tau$ is an o-algebra (because every open set is regular). Let $f$ be the inclusion map $\tau\hookrightarrow\mathrm{Pow}(X)$ and let us assume that $f^\dagger $ exists, that is, $U>\mkern-13.5mu < Y$ $\Leftrightarrow$ $U>\mkern-13.5mu < f^\dagger(Y)$ for every subset $Y$ and every open $U$. This means that $cl(Y)= cl f^\dagger (Y) $ for every $Y$. Since $cl=id$, we get $Y= f^\dagger (Y)$, and hence $Y$ is open, for every $Y$.
\end{remark}
\begin{proposition}\label{prop f from powersets}
Let $f:L\to M$ be a join preserving function between two o-algebras. If $L$ is atomic, then $f$ is symmetrizable.
\end{proposition}
\begin{proof}
Up to order-isomorphism, we can assume that $L$ is $\mathrm{Pow}(X)$ for some $X$. Put $f^\dagger y=\{x\in X\ |\ f(\{x\})>\mkern-13.5mu < y\}$.
\end{proof}
It is a corollary of the previous proposition (but it can be easily checked directly) that the mapping $X\mapsto\bigvee X$ gives a symmetrizable map from $\mathrm{Pow}(L)$ to $L$. Its symmetric is given by $y\mapsto\{x\in L\ |\ x>\mkern-13.5mu < y\}$.
\begin{remark}\label{remark f to Omega}
Note that a function $f:L\to\Omega$ is symmetrizable if and only if there is $a\in L$ such that $f(x)=(x>\mkern-13.5mu < a)$ for all $x\in L$. Indeed, given $a\in L$, the mapping $x\mapsto (x>\mkern-13.5mu < a)$ is symmetrizable, and its symmetric maps $p\in\Omega$ to $\bigvee\{x\in L\ |\ (x=a)\land p\}$ (classically, of course, this is just $a$ if $p=1$, and $0$ if $p=0$).
Conversely, given a symmetrizable $f$, put $a=f^\dagger(1)$. Also note that $a$ is an atom if and only if $f$ preserves finite meets.
\end{remark}
\begin{propC}[{\cite[Theorem $1.15.(iii)$]{3c}}]\label{lemma-ff}
Let $f:L\to M$ and $g:M\to L$ be two functions between o-algebras. Then $f$ and $g$ are symmetric if and only if all the following conditions hold identically:
\begin{enumerate}
\item $\mathrm{Pos}(f(x))\Rightarrow\mathrm{Pos}(x)$ (classically, $f(0)=0$);
\item $\mathrm{Pos}(g(y))\Rightarrow\mathrm{Pos}(y)$ (classically, $g(0)=0$);
\item $f(x)\wedge y\leq f(x\wedge g(y))$;
\item $x\wedge g(y)\leq g(f(x)\wedge y)$.
\end{enumerate}
\end{propC}
\begin{proof}
Assume that $f$ and $g$ are symmetric. Now $\mathrm{Pos}(f(x))$ can be rewritten as $x>\mkern-13.5mu < g(f(x))$; so $x>\mkern-13.5mu < 1$, that is, $\mathrm{Pos}(x)$; this proves 1, and hence 2 by symmetry.\footnote{\label{remark Posf}Note that $\mathrm{Pos}(fx)$ $\Rightarrow$ $\mathrm{Pos}(x)$ holds true already in the case of overt frames. Indeed, since $x$ = $\bigvee\{x'\ |\ \mathrm{Pos}(x')\land (x'\leq x)\}$ by \eqref{eq positivity bis}, $fx$ = $\bigvee\{fx'\ |\ \mathrm{Pos}(x') \land (x'\leq x)\}$. So if $\mathrm{Pos}(fx)$, then $\mathrm{Pos}(fx')$ for some $x'\leq x$ with $\mathrm{Pos}(x')$; in particular, $\mathrm{Pos}(x')$ for some $x'\leq x$, and hence $\mathrm{Pos}(x)$.} To check 3 (and 4) we use density: $z>\mkern-13.5mu < f(x)\wedge y$ is equivalent to $f(x)>\mkern-13.5mu < z\wedge y$ and hence to $x>\mkern-13.5mu < g(z\wedge y)$; since $g$ is monotone (because it preserves joins), we also have $x>\mkern-13.5mu < g(z)\wedge g(y)$, which is equivalent to $g(z)>\mkern-13.5mu < x\wedge g(y)$ and hence to $z>\mkern-13.5mu < f(x\wedge g(y))$.
Conversely, if $f(x)>\mkern-13.5mu < y$, that is, $\mathrm{Pos}(f(x)\wedge y)$, then also $\mathrm{Pos}(f(x\wedge g(y)))$ by 3; so $\mathrm{Pos}(x\wedge g(y))$ by 1, that is, $x>\mkern-13.5mu < g(y)$; and symmetrically for the other direction.
\end{proof}
As a corollary, if $f$ is symmetrizable, then $fx$ = $fx\wedge 1$ $\leq$ $f(x\wedge f^\dagger(1))$ $\leq$ $ff^\dagger(fx\wedge 1)$ = $ff^\dagger f x$. And, similarly, $f^\dagger y$ $\leq$ $f^\dagger ff^\dagger y$.
\begin{proposition}\label{prop o-morph}
Let $f:L\to M$ be a symmetrizable function between o-algebras. Then the following conditions are equivalent:
\begin{enumerate}
\item if $fx_1>\mkern-13.5mu < fx_2$, then $x_1>\mkern-13.5mu < x_2$;
\item $f$ preserves binary meets;
\item $f^\dagger fx\leq x$ for every $x$.
\end{enumerate}
Moreover the following are equivalent:
\begin{enumerate}\setcounter{enumi}{3}
\item if $x_1>\mkern-13.5mu < x_2$, then $fx_1>\mkern-13.5mu < fx_2$;
\item if $\mathrm{Pos}(x)$, then $\mathrm{Pos}(fx)$;
\item $f^\dagger 1=1$;
\item $x\leq f^\dagger fx$ for every $x$.
\end{enumerate}
\end{proposition}
\begin{proof}
$1\Rightarrow 2$: $z>\mkern-13.5mu < (fx_1\wedge fx_2)$ can be rewritten as $f^\dagger(z\wedge fx_1)>\mkern-13.5mu < x_2$, which yields $(f^\dagger z\wedge f^\dagger fx_1)>\mkern-13.5mu < x_2$; this is equivalent to $fx_1>\mkern-13.5mu < f(f^\dagger z\wedge x_2)$, which implies $x_1>\mkern-13.5mu < (f^\dagger z \wedge x_2)$ by assumption; in other words, $f^\dagger z>\mkern-13.5mu < (x_1\wedge x_2)$, that is, $z>\mkern-13.5mu < f(x_1\wedge x_2)$.
$2\Rightarrow 3$: $y>\mkern-13.5mu < f^\dagger fx$ iff $fy>\mkern-13.5mu < fx$ iff $(fy\wedge fx)>\mkern-13.5mu < 1$ iff $f(y\wedge x)>\mkern-13.5mu < 1$ iff $(y\wedge x)>\mkern-13.5mu < f^\dagger 1$, and hence $(y\wedge x)>\mkern-13.5mu < 1$, that is, $y>\mkern-13.5mu < x$.
$3\Rightarrow 1$: if $fx_1>\mkern-13.5mu < fx_2$, then $x_1>\mkern-13.5mu < f^\dagger fx_2$ ($\leq x_2$), and hence $x_1>\mkern-13.5mu < x_2$.
$4\Rightarrow 5$: since $\mathrm{Pos}(x)$ is $x>\mkern-13.5mu < x$, and $\mathrm{Pos}(fx)$ is $fx>\mkern-13.5mu < fx$.
$5\Rightarrow 6$: $z>\mkern-13.5mu < 1$ iff $\mathrm{Pos}(z)$, which implies $\mathrm{Pos}(fz)$; this is equivalent to $fz>\mkern-13.5mu < 1$, that is, $z>\mkern-13.5mu < f^\dagger 1$.
$6\Rightarrow 7$: $z>\mkern-13.5mu < x$ iff $(z\wedge x)>\mkern-13.5mu < 1$ iff $(z\wedge x)>\mkern-13.5mu < f^\dagger 1$ iff $f(z\wedge x)>\mkern-13.5mu < 1$; the last implies $(fz\wedge fx)>\mkern-13.5mu < 1$, which is equivalent to $fz>\mkern-13.5mu < fx$ and hence to $z>\mkern-13.5mu < f^\dagger f x$.
$7\Rightarrow 4$: if $x_1>\mkern-13.5mu < x_2$ ($\leq f^\dagger fx_2$), then $x_1>\mkern-13.5mu < f^\dagger fx_2$, that is, $fx_1>\mkern-13.5mu < fx_2$.
\end{proof}
\section{The category {\bf OA} of overlap algebras}
The identity function $ id: L \rightarrow L $ on an o-algebra $L$ is symmetrizable with $ id^\dagger = id$. For $L$, $M$, $N$ o-algebras, if $f: L \rightarrow M$ and $g: M \rightarrow N$ are symmetrizable, then $ g \circ f $ is symmetrizable too and
$$ (g\circ f)^\dagger = f^\dagger \circ g^\dagger $$
since
$ g(f(x)) >\mkern-13.5mu < z \Leftrightarrow f(x) >\mkern-13.5mu < g^\dagger (z) \Leftrightarrow x >\mkern-13.5mu < f^\dagger (g^\dagger (z)) $. So o-algebras and symmetrizable functions form a category \textbf{OA}. We will sometimes refer to arrows in {\bf OA} as \emph{overlap-morphisms} or \emph{o-morphisms}.
The category {\bf OA} is a \textit{dagger} category, that is, a category $\mathcal{C}$ equipped with an endofunctor $(\_)^\dagger:\mathcal{C}^{op}\rightarrow \mathcal{C}$ which is the identity on objects and an involution on arrows.
\begin{proposition}\label{PropPow}
Let $\mathrm{Pow}:\mathbf{Rel}^{op}\rightarrow \mathbf{OA} $ be the functor that associates to every relation $R\subseteq X\times Y$ its inverse image $\mathrm{Pow}(R)=R^{-1}:\mathrm{Pow}(Y)\to\mathrm{Pow}(X)$. Then $\mathrm{Pow}$ is a full and faithful functor. Moreover, $\mathrm{Pow}(R^\dagger)=(\mathrm{Pow}(R))^\dagger$.
\end{proposition}
\begin{proof} The map $\mathrm{Pow}(R)$ is symmetrizable by equation \eqref{eq.R-}, and $(\mathrm{Pow}(R))^\dagger=\mathrm{Pow}(R^\dagger)$.
Lemma \ref{lemmaRel} shows that $ \mathrm{Pow} $ is a functor.
Given $f:\mathrm{Pow}(Y)\to\mathrm{Pow}(X)$, let $R\subseteq X\times Y$ be the relation defined as $xRy \Leftrightarrow x\in f(\lbrace y\rbrace)$. Then $x\in R^{-1}(D)$ iff $x\in f(\lbrace y\rbrace)$ for some $y\in D$ iff $x\in \bigcup_{y\in D} f(\lbrace y\rbrace)$ = $f(\bigcup_{y\in D} \lbrace y\rbrace)$ = $f(D)$. This shows that $\mathrm{Pow}$ is full.
And $\mathrm{Pow}$ is clearly faithful, for if $ R,S\subseteq X\times Y $ are such that $R^{-1}=S^{-1}$, then $xRy \Leftrightarrow x\in R^{-1}(\lbrace y \rbrace)\Leftrightarrow x\in S^{-1}(\lbrace y \rbrace) \Leftrightarrow xSy$.
\end{proof}
\subsection{Iso-, mono- and epi-morphisms in OA}
\begin{proposition}
A bijective function $f:L\to M$ is an isomorphism in {\bf OA} if and only if it is an order-isomorphism. In that case $f^\dagger=f^{-1}$ (isomorphisms in {\bf OA} are ``unitary").
\end{proposition}
\begin{proof}
One direction is trivial because all arrows in {\bf OA} are monotone functions.
Conversely, let $f$ be an order-isomorphism; we claim that $f$ is symmetrizable and $f^\dagger=f^{-1}$. As $f$ and $f^{-1}$ preserve binary meets, items 3 and 4 of propositions \ref{lemma-ff} hold; it remains to be shown that $\mathrm{Pos}(f(x))$ implies $\mathrm{Pos}(x)$, and similarly for $f^{-1}$. This follows from the fact that $f$ and $f^{-1}$ preserve joins. Indeed, by \eqref{eq positivity bis}, we have $f(x)$ = $\bigvee\{f(z)\ |\ \mathrm{Pos}(z)\land (z\leq x)\}$. So if $\mathrm{Pos}(f(x))$ holds, then $\mathrm{Pos}(f(z))$ holds for some $z\leq x$ with $\mathrm{Pos}(z)$. In particular, also $\mathrm{Pos}(x)$ holds.
\end{proof}
So $L$ and $M$ are isomorphic in {\bf OA} if and only if they are isomorphic as posets. As a corollary, a join-preserving bijection between o-algebras is always symmetrizable\footnote{In this case, $f$ is an order-isomorphism because $f^{-1}$ preserves joins as well, for $f^{-1}(\bigvee_i y_i)$ = $f^{-1}(\bigvee_i ff^{-1}y_i)$ = $f^{-1}f(\bigvee_i f^{-1}y_i)$ = $\bigvee_i f^{-1}y_i$.} (compare with Remark \ref{remark:symmetrizable}).
\begin{remark}
If $ f$ is an isomorphism in {\bf OA}, then $fx_{1} >\mkern-13.5mu < fx_{2}\Leftrightarrow x_{1} >\mkern-13.5mu < x_{2}$
holds true by Proposition \ref{prop o-morph}.
\end{remark}
\begin{proposition}\label{prop.mono}
Let $ m: L \rightarrow M $ be an arrow in {\bf OA}. Then $m$ is a monomorphism if and only if $m$ is injective.
\end{proposition}
\begin{proof}
If $m$ is an injective function, then it is trivially a monomorphism.
Conversely, assume $ m(a) = m(b) $ with $a,b\in L$. Let $f_a(x)$ and $f_b(x)$ be the truth values of $x>\mkern-13.5mu < a$ and $x>\mkern-13.5mu < b$, respectively. In view of Remark \ref{remark f to Omega}, $f_a$ and $f_b$ are two o-morphisms from $L$ to $\Omega$. Now $f_a m^\dagger y$ = $(m^\dagger y>\mkern-13.5mu < a)$ = $(y>\mkern-13.5mu < ma)$ = $(y>\mkern-13.5mu < mb)$ = $(m^\dagger y>\mkern-13.5mu < b)$ = $f_b m^\dagger y$. So $f_a\circ m^\dagger$ = $f_b\circ m^\dagger$, and hence $m\circ f_a^\dagger$ = $m\circ f_b^\dagger$. Since $m$ is mono, we get $f_a^\dagger$ = $f_b^\dagger$; therefore $f_a$ = $f_b$, that is, $a=b$ (by density).
\end{proof}
Note that a join-preserving map $f:L \rightarrow M $ between posets is injective if and only if $\forall_f\circ f=id_{L}$ (apply the triangular identity $f\circ\forall_f\circ f=f $). Similarly, $f$ is surjective if and only if $f\circ\forall_f=id_{M}$.
\begin{proposition}
Let $f: L \rightarrow M $ be an arrow in {\bf OA}; then
\begin{enumerate}
\item $f$ is an epimorhism if and only if $f^\dagger$ is a monomorphism;
\item if $f$ is surjective, then $f$ is an epimorphism;
\item classically, if $f$ is an epimorphism, then $f$ is surjective.
\end{enumerate}
\end{proposition}
\begin{proof}
Item 1 holds in any dagger category; 2 is trivial.
Let $f$ be epi, and assume LEM. Then $f^\dagger$ is injective, and $f^\dagger(y)$ = $-\forall_f(-y)$. Therefore also $\forall_f$ is injective, for $\forall_f y_1$ = $\forall_f y_2$ iff $-f^\dagger(-y_1)$ = $-f^\dagger(-y_2)$ iff $f^\dagger(-y_1)$ = $f^\dagger(-y_2)$ iff $-y_1$ = $-y_2$ iff $y_1$ = $y_2$. But $\forall_f\circ f\circ\forall_f$ = $\forall_f$ (triangular identity); hence $f(\forall_f y)$ = $y$ for all $y\in M$; so $f$ is surjective.
\end{proof}
It is possible to construct a \emph{Brouwerian} counterexample to the fact that epic implies surjective. Let us consider a topological space $(X, \tau) $ in which the closure operator $\textit{cl}$ is the identity $\textit{id}$ (see section \ref{remark:symmetrizable}).
Let $f:\mathrm{Pow}(\tau)\rightarrow\mathrm{Pow}(X)$ be the map $\lbrace A_{i}\rbrace_{i\in I} \mapsto \cup_{i\in I}A_{i} $; it is symmetrizable and $f^\dagger(Y)$ = $\lbrace A\in \tau\vert Y>\mkern-13.5mu < A\rbrace $.\footnote{This is a consequence of proposition \ref{prop f from powersets}. Here is a direct proof: $f(\lbrace A_{i}\rbrace _{i\in I})>\mkern-13.5mu < Y$ iff $(\bigcup_{i\in I}A_i)>\mkern-13.5mu < Y$ iff $(\exists i\in I)(A_{i}>\mkern-13.5mu < Y)$ iff $(\exists i\in I)(A_{i}\in f^\dagger( Y))$ iff $\lbrace A_{i}\rbrace _{i\in I}>\mkern-13.5mu < f^\dagger( Y)$.} Now $f^\dagger$ is injective, because $f^\dagger(Y_1)$ = $f^\dagger(Y_2)$ iff $(\forall A\in \tau)(Y_1>\mkern-13.5mu < A \Leftrightarrow Y_2>\mkern-13.5mu < A)$ iff $\mathrm{cl} Y_1 = \mathrm{cl} Y_2$ iff $Y_1=Y_2$. In other words, $f^\dagger$ is monic and so $f$ is epic. However, if $f$ were surjective, then every $Y\subseteq X$ would be open. In view of this, if the implication ``$f$ epi $\Rightarrow$ $f$ surjective'' were true, then also ``$\textit{cl}=\textit{id}\Rightarrow \textit{int}=\textit{id}$'' would be true, which is an intuitionistic ``taboo" (see footnote \ref{counterexample} on page \pageref{counterexample}).
\begin{proposition}\label{prop mono-epi}
If $m$ is a mono in {\bf OA}, then the following hold identically:
\begin{enumerate}
\item if $x_1>\mkern-13.5mu < x_2$, then $m x_1>\mkern-13.5mu < m x_2$;
\item if $\mathrm{Pos}(x)$, then $\mathrm{Pos}(mx)$;
\item $m^\dagger 1=1$;
\item $x\leq m^\dagger m x$.
\end{enumerate}
Symmetrically, If $e$ is an epi in {\bf OA}, then the following hold identically:
\begin{enumerate}
\item if $y_1>\mkern-13.5mu < y_2$, then $e^\dagger y_1>\mkern-13.5mu < e^\dagger y_2$;
\item if $\mathrm{Pos}(y)$, then $\mathrm{Pos}(e^\dagger y)$;
\item $e1=1$;
\item $y\leq ee^\dagger y$.
\end{enumerate}
\end{proposition}
\begin{proof}
Recall from Proposition \ref{lemma-ff} that $mx\wedge y\leq m(x\wedge m^\dagger y)$ for all $x$ and $y$. In particular, $m1\leq mm^\dagger 1$ and hence $m1=mm^\dagger 1$. If $m$ is a mono, that is, it is injective, then $m^\dagger 1=1$, which is item 6 of Proposition \ref{prop o-morph}.
The second part follows by applying the same argument to $e^\dagger$.
\end{proof}
\subsection{Limits and co-limits in OA}
Limits and colimits in a dagger category are mutually closely related: an object $C$ together with arrows $\alpha_i:A_i\to C$ is the colimit of a diagram $f_{i,j}^k: A_i\to A_j$ if and only if the same $C$ together with ${\alpha_i}^\dagger:C\to A_i$ is the limit of the diagram $(f_{i,j}^k)^\dagger: A_j\to A_i$.
\begin{lemma}
Let $\{L_i\}_{i\in I}$ be a family of o-algebras. Then the set-theoretic product $\Pi_{i\in I} L_i$ is an o-algebra with respect to pointwise joins and meets, and $\mathrm{Pos}(f)$ holds in $\Pi_{i\in I} L_i$ if and only if $\mathrm{Pos}(f_i)$ holds in some $L_i$.
\end{lemma}
\begin{proof}
Let us check \eqref{eq prop Pos}, the other properties being clear. Given $f$ and $g$, assume $\mathrm{Pos}(h\wedge f)\Rightarrow\mathrm{Pos}(h\wedge g)$ for all $h$. For any given $k\in I$ and $z\in L_k$, let us define $h$ as $h_i=\bigvee\{x\in L_i\ |\ (i=k)\land(x=z)\}$. By assumption we then have $\mathrm{Pos}(z\wedge f_k)\Rightarrow\mathrm{Pos}(z\wedge g_k)$, for all $k\in I$ and for all $z\in L_k$. So $f_k\leq g_k$ by \eqref{eq prop Pos} in $L_k$, for all $k\in I$; therefore $f\leq g$.
\end{proof}
\begin{proposition}
The category $ \mathbf{OA}$ has arbitrary products (and coproducts).
\end{proposition}
\begin{proof}
We claim that $\Pi_{i\in I} L_i$, as defined in the previous lemma, is the product of the family of o-algebras $\{L_i\}_{i\in I}$. Let $\pi_k$ be the $k$-th projection, and define ${\pi_k}^\dagger(z)$ to be the function $i\mapsto\bigvee\{x\in L_i\ |\ (i=k)\land(x=z)\}$. Then $f>\mkern-13.5mu <{\pi_k}^\dagger(z)$ iff $f_i>\mkern-13.5mu < \bigvee\{x\in L_i\ |\ (i=k)\land(x=z)\}$ for some $i\in I$ iff $f_k>\mkern-13.5mu < z$ iff $\pi_k(f)>\mkern-13.5mu < z$. Therefore the set-theoretic projections are o-moprhisms.
Let $\{g_i:M\to L_i\}_{i\in I}$ be a family of morphisms in {\bf OA}. We claim that there exists a unique morphism $h:M\to\Pi_{i\in I} L_i$ such that $\pi_i\circ h$ = $f_i$ for all $i$. The only possible candidate for $h$ is the mapping $x\mapsto h(x)$ with $h(x)_i=g_i(x)$. Let us check that it is symmetrizable with $h^\dagger (f)$ = $\bigvee_{i\in I}{g_i}^\dagger(f_i)$. We have that $h(x)>\mkern-13.5mu < f$ $\Leftrightarrow$ $h(x)_i>\mkern-13.5mu < f_i$ for some $i\in I$ $\Leftrightarrow$ $g_i(x)>\mkern-13.5mu < f_i$ for some $i\in I$ $\Leftrightarrow$ $x>\mkern-13.5mu < {g_i}^\dagger (f_i)$ for some $i\in I$ $\Leftrightarrow x>\mkern-13.5mu < \bigvee_{i\in I}{g_i}^\dagger(f_i)$ $\Leftrightarrow$ $x>\mkern-13.5mu < h^\dagger(f)$.
\end{proof}
Note that $\mathrm{Pow}(\emptyset)$ is a zero object (both initial and terminal), because given an arbitrary o-algebra $ L $, there exists a unique morphism $f: \mathrm{Pow}(\emptyset)\rightarrow L $, namely, $f(\emptyset) = 0$ ($f$ has to preserve joins); and $f$ is the symmetric of the unique function $g:L\to\mathrm{Pow}(\emptyset)$, namely, the constant function with value $\emptyset$ (both $\emptyset >\mkern-13.5mu < x$ and $\emptyset >\mkern-13.5mu < gx$ are always false).
\paragraph{The category OA is not complete.}
A category $\mathcal{C}$ is $ \textit{complete} $ if it has all (small) limits. It is well-known that a category with all (small) products is complete if and only if it has equalizers. We are going to show that {\bf OA} does not have equalizers, in general, hence it is not complete. This fact is independent from LEM, that is, {\bf OA} is not complete even classically, as we now see.
Recall that, classically, {\bf OA} is the category of complete Boolean algebras and join-preserving maps. Let us consider the complete Boolean algebras $\Omega=\mathrm{Pow}(1)\cong 2=\{0,1\}$ and $L=\{0,a,-a,1\}\cong\mathrm{Pow}(2)$. Let $f,g:L\to 2$ be two maps defined by $f(0)=g(0)=0$, $f(1)=g(1)=1$, $f(a)=g(a)=1$, $f(-a)=0$ and $g(-a)=1$. Clearly both $f$ and $g$ preserves joins. We claim that there is no equalizer of $f$ and $g$. By way of contradiction let us assume that $e:E\to L$ is the equalizer of $f$ and $g$. In particular $e$ is mono, that is, injective; and $-a$ is not in the image of $e$. Therefore, up to isomorphism we have only two possibilities for $E$, namely, $E=1=\mathrm{Pow}(\emptyset)$, and $E=2$. In particular, the image of $e$ contains at most two elements.
Now consider the function $t:L\to L$ define by $t(0)= 0$, $t(a)= a$, $t(1)= 1=t(-a)$. It is easy to check that $t$ preserves joins and that $f\circ t=g\circ t$. So there must exist (a unique) $h:L \rightarrow E$ such that $e\circ h=t$. This is impossible because the image of $t$ contains three elements.
\begin{remark}
The argument above shows a case in which a $\textit{weak}$ equalizer exists ($t:L\to L$ is a weak equalizer of $f$ and $g$ because any other $h:X\to L$ with $fh=gh$ factors through $t$, actually $h=th$). And weak equalizers always exist in {\bf Rel}.
So it is natural to ask whether {\bf OA} has weak equalizers as well: this is an open problem.
\end{remark}
The functor $\mathrm{Pow}:\mathbf{Rel}^{op}\rightarrow \mathbf{OA} $ preserves (co)products. Indeed, it is well-known that (co)products in $ \mathbf{Rel} $ are given by disjoint unions; and the powerset of a disjoint union $ \Sigma_{i\in I} X_{i} $ is the set-theoretic product of the powersets of the $X_i$'s, that is, $\mathrm{Pow}(\Sigma_{i\in I} X_{i})$ = $\Pi_{i\in I} \mathrm{Pow}(X_{i})$.
\section{Overlap-frames and overlap-locales}\label{section OFrm}
From now on, we restrict our attention to o-morphisms $f$ that preserve finite meets (note that $f^\dagger$ need not preserve finite meets). Let {\bf OFrm} be the corresponding subcategory of {\bf OA}. So {\bf OFrm} is also a subcategory of the category {\bf Frm} of frames, hence the name. Note that the functor $\mathrm{Pow}$ restricts to a fucntor $\mathbf{Set}^{op}\to\mathbf{OFrm}$ because $R^{-1}$ preserves finite intersections if and only if $R$ is a function.
\begin{lemma}\label{lemma-fff}
Let $ f:L \rightarrow M $ be a morphism in {\bf OA}; then the following are equivalent:
\begin{enumerate}
\item $f$ preserves finite meets;
\item $f^\dagger\dashv f$.
\end{enumerate}
\end{lemma}
\begin{proof}
By Proposition \ref{prop o-morph}, $f$ preserves binary meets iff $f^\dagger f x\leq x$, and $f1=1$ iff $y\leq ff^\dagger y$.
\end{proof}
A frame homomorphism $f$ is {\bf open} \cite{3d,johnstone-open} if it has a left adjoint $ \exists_{f} $ which satisfies \emph{Frobenius reciprocity condition} $ \exists_{f} (f(x) \wedge y) = x\wedge\exists_{f}(y)$.
Equivalently, $f$ is open if it preserves the Heyting implication and arbitrary meets. For instance, the unique frame homomorphism $!:\Omega\to L$ is open if and only if the frame $L$ is overt (in which case $\exists_!=\mathrm{Pos}$).
All arrows $f$ in {\bf OFrm} are open with $\exists_f=f^\dagger$; actually the following, more general, result holds.
\begin{proposition}\label{prop morph oFrm}
Let $f:L\to M$ be a function between o-algebras. Then the following are equivalent:
\begin{enumerate}
\item $f$ is symmetrizable and preserves finite meets;
\item $f$ is an open frame homomorphism;
\item $f$ preserves all joins and meets, and the Heyting implication.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1\Rightarrow 2)$: $z>\mkern-13.5mu < f^\dagger (f x\wedge y)$ iff $fz>\mkern-13.5mu < f x\wedge y$ iff $fz\wedge f x>\mkern-13.5mu < y$ iff $f(z\wedge x)>\mkern-13.5mu < y$ iff $z\wedge x>\mkern-13.5mu < f^\dagger y$ iff $z>\mkern-13.5mu < x\wedge f^\dagger y$.
$(2\Rightarrow 3)$: well-known.
$(3\Rightarrow 1)$: $fx>\mkern-13.5mu < y$ iff $\mathrm{Pos}_M(fx\wedge y)$ iff\footnote{$\mathrm{Pos}_M$ = $\mathrm{Pos}_L\circ\exists_f$ because $!_M$ = $f\circ !_L$.} $\mathrm{Pos}_L\exists_f(fx\wedge y)$ iff $\mathrm{Pos}_L(x\wedge\exists_f y)$ iff $x>\mkern-13.5mu <\exists_f y$.
\end{proof}
So \textbf{OFrm} is the category of o-algebras and open frame homomorphisms. Classically, \textbf{OFrm} is just the category {\bf cBa} of complete Boolean algebras.
\subsection{Subobjects, equalizers and completeness of {\bf OFrm}}
Proposition \ref{prop.mono} holds for {\bf OFrm} too, because the morphisms $f_a^\dagger$ and $f_b^\dagger$ which appear in that proof preserve finite meets. Therefore, every monomorphism $m$ in {\bf OFrm} is injective; by triangular identity, this is equivalent to the equation $m^\dagger\circ m=id$; and this happens precisely $x_1>\mkern-13.5mu < x_2$ $\Leftrightarrow$ $m x_1>\mkern-13.5mu < m x_2$ for all $x_1$, $,x_2$ (see Propositions \ref{prop o-morph} and \ref{prop mono-epi}).
Let $f:L \rightarrow M $ be any arrow in {\bf OFrm}. Then the set-theoretic image $f[L]$ = $\{f(x)\ |\ x\in L\}$ is a sub-frame of $M$, and it is an o-algebra where $ fx_{1}>\mkern-13.5mu <_{f[L] } fx_{2}$ is defined as $x_1>\mkern-13.5mu <_L x_2$. Note that the symmetric of the inclusion $ \textit{i}: f[L] \rightarrow M $ is given by $ i^\dagger (y)=ff^\dagger (y) $ because
\begin{center}
$ifx>\mkern-13.5mu <_{ M } y\Leftrightarrow fx>\mkern-13.5mu <_{ M } y \Leftrightarrow x>\mkern-13.5mu <_{ L } f^\dagger y\Leftrightarrow fx>\mkern-13.5mu <_{f[L]} ff^\dagger y$.
\end{center}
Clearly if $m$ is monic, then $m[L]$ is isomorphic to $L$.
\begin{proposition}
Let $ M $ be an o-algebra and let $N\subseteq M $. Then the following are equivalent:
\begin{enumerate}
\item $N$ = $m[L]$ for some mono $m:L\to M$ in {\bf OFrm};
\item $N$ is closed under all joins and meets, and the Heyting implication.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1\Rightarrow 2)$: easy, since an open frame homomorphism $m$ preserve all joins and meets, and the implication.
$(2\Rightarrow 1)$: by assumption, the inclusion map $ \textit{i}: N \rightarrow M $ is an open frame homomorphism, and $\exists_i\circ i$ = $id_N$ because $i$ is injective. We claim that $N$ is an o-algebra, with respect to (the restriction of) the positivity predicate of $M$. The only thing that needs to be checked is \eqref{eq prop Pos}. Given $x,y\in N$, assume $\mathrm{Pos}(z\wedge x)\Rightarrow \mathrm{Pos}(z\wedge y)$ for all $z\in N$; we must show that $x\leq y$. By \eqref{eq prop Pos} in $M$, it is enough to check that $\mathrm{Pos}(t\wedge x)\Rightarrow \mathrm{Pos}(t\wedge y)$ for all $t\in L$. If $\mathrm{Pos}(t\wedge x)$, then also $\mathrm{Pos}(\exists_i t\wedge x)$ because $\exists_i\vdash i$ and $t\leq i\exists_i t$. By assumption we get $\mathrm{Pos}(\exists_i t\wedge y)$, and hence $\mathrm{Pos}(\exists_i(t\wedge y))$ by Frobenius reciprocity. Since $\exists_i:M\to N$ is a join preserving function between overt frames (see footnote \ref{remark Posf} on page \pageref{remark Posf}), we obtain $\mathrm{Pos}(t\wedge y)$.
\end{proof}
From a classical point of view, of course, $ N$ is a subobject of $ M $ if and only if $N$ is a sub-cBa of $M $.
\begin{proposition}
The category {\bf OFrm} is complete.
\end{proposition}
\begin{proof}
The construction of products and equalizers is straightforward. Indeed, if $\{L_i\}_{i\in I}$ is a family of o-algebra, then the product $\Pi_{i\in I}L_i$ in {\bf OA} works as a product in {\bf OFrm} as well (projections $\pi_{i}$'s preserve finite meets). And if $f,g:L\to M$ are two parallel arrows in {\bf OFrm}, then $E=\{x\in L\ |\ fx=gx\}$, together with the inclusion $e:E\to L$, is the equalizer of $f$ and $g$.
\end{proof}
In general $\mathbf{OFrm}$ does not have co-products, even classically, because $\mathbf{cBa}$ does not have co-products, in general, as it is well-known. Indeed, this is a consequence of the Gaifman-Hales-Solovay Theorem \cite{solovay} that there is no free complete boolean algebra on countably many generators.
\subsection{Sublocales of overlap algebras}
Let $\mathbf{Loc}$ = $\mathbf{Frm}^{op}$ be the category of locales (see \cite{3b} and \cite{picado} for a comprehensive treatment of locale theory). A sublocale of $L$ is a regular subobjects in {\bf Loc}, that is, a quotient of $L$ in $\mathbf{Frm}$. It is well known that sublocales of $L$ have, up to isomorphism, the form $L_{j}$ = ${\lbrace jx\ |\ x\in L \rbrace}$ where $j$ is a {\bf nucleus}, that is, a function $j:L\rightarrow L$ such that
\begin{enumerate}
\item $x\leq j(x) = j(j(x))$ for all $x\in L$, and
\item $j(x\wedge y)= j(x) \wedge j(y)$ for all $x,y\in L$.
\end{enumerate}
By definition, an {\bf open sublocale} is given by a nucleus of the form $j(x)= a\to x$, for $a\in L$, where $\to$ is the Heyting implication in $L$. It is well known \cite{3d} that a sublocale $m:L_j\to L$ is open if and only if the corresponding frame epimorphism $m^*:L\to L_j$, with $m^*(x)=j(x)$, is open. Moreover, $\exists_{m^*}(jx)=a\wedge x$ and $a=\exists_{m^*}(1)$.
A sublocale $L_j$ is Boolean, that is, it is a complete Boolean algebra if and only if $j$ is of the form $j(x)=(x\to a)\to a$ for some $a\in L$. If $L$ itself is Boolean, then every sublocale $L_j$ of $L$ is Boolean, because $j(x)=(x\to j(0))\to j(0)$ holds identically in that case. What happens if we replace complete Boolean algebras with overlap algebras?
\begin{proposition}
Every open sublocale of an overlap algebra is an overlap algebra.
\end{proposition}
\begin{proof}
Given $m:L_{j}\rightarrow L$ with $L$ an o-algebra and $m$ open, we claim that $L_{j}$ is an o-algebra with respect to the positivity predicate $\mathrm{Pos}_{L_{j}}=\mathrm{Pos}_{L}\circ \exists_{m^{*}}$.
$$\xymatrix{
L_j\ar@/^/[rr]^{\exists_{m^*}} \ar@<1ex>@/^2pc/[rrrr]^{Pos_{L_j}} & & L \ar@/^/[rr]^{Pos_L} \ar@/^/[ll]^{m^*} & & \Omega\ar@/^/[ll]^{!_l^*} \ar@<1ex>@/^2pc/[llll]^{!_{L_j}^*}
}$$
In order to prove that $L_{j}$ is an o-algebra we must check that \eqref{eq prop Pos} holds for $L_j$, namely
\begin{center}
$\forall z. \left[\mathrm{Pos}_{L_{j}}(jz\wedge jx)\Rightarrow\mathrm{Pos}_{L_{j}}(jz\wedge jy)\right] \Longrightarrow jx\leq jy\ .$
\end{center}
Now $\mathrm{Pos}_{L_{j}}(jz\wedge jx)$ can be rewritten as $\mathrm{Pos}_{L}\exists_{m^{*}}(m^*z\wedge jx)$, and hence as $\mathrm{Pos}_{L}(z\wedge\exists_{m^{*}}jx)$; similarly for $y$ in place of $x$. So the antecedent becomes $\forall z. [(z>\mkern-13.5mu < \exists_{m^{*}}jx)\Rightarrow(z>\mkern-13.5mu < \exists_{m^{*}}jy)]$, that is, $\exists_{m^{*}}jx\leq \exists_{m^{*}}jy$. This is equivalent to $jx\leq m^{*}\exists_{m^{*}}jy$ = $m^{*}\exists_{m^{*}}m^*y$ = $m^*y$ = $jy$.
\end{proof}
Discrete locales, that is, powersets regarded as locales, are overlap algebras (and they are Boolean if and only if LEM holds). More generally, we have the following.
\begin{lemma}
Every overt sublocale of a discrete locale is open (as a sublocale).
\end{lemma}
\begin{proof}
Let $j$ be a nucleus on $\mathrm{Pow}(X)$ such that the corresponding sublocale is overt with positivity predicate $\mathrm{Pos}$. Let $P$ be $\{x\in X\ |\ \mathrm{Pos}(j\{x\})\}$. We claim that $jU=P\to U$. Indeed, if $x\in jU$, then $j\{x\}=j(\{x\}\cap U)$; if also $x\in P$, then $\mathrm{Pos}(j(\{x\}\cap U))$, and hence $\{x\}\cap U$ is inhabited, that is, $x\in U$. Conversely, if $x\in P\to U$, then $\mathrm{Pos}(j\{x\})\Rightarrow (x\in U)$; so $\mathrm{Pos}(j\{x\})\Rightarrow (j\{x\}\subseteq jU)$; by overtness, $j\{x\}\subseteq jU$, that is, $x\in j U$.
\end{proof}
\begin{corollary}
Overt sublocales of discrete locales are overlap algebras.
\end{corollary}
\begin{proof}
By the previous proposition and lemma.
\end{proof}
\begin{proposition}
For $L$ a locale, there is a bijection between sublocales of $L$ which are overlap algebras and join-preserving maps $L\to\Omega$.
\end{proposition}
\begin{proof}
Given an o-algebra $L_j$, put $\varphi(x)= \mathrm{Pos}(jx)=(jx>\mkern-13.5mu < jx)$. Then $\varphi(\bigvee_i x_i)$ = $\mathrm{Pos}(j(\bigvee_i x_i))$ = $\mathrm{Pos}(\bigvee^{L_j}_i jx_i)$ = $\exists i.\mathrm{Pos}(jx_i)$ = $\exists i.\varphi(x_i)$.
Conversely, given $\varphi :L\to \Omega$, put $jy$ = $\bigvee\{x\in L\ |\ \forall z.[\varphi(z\wedge x)\Rightarrow\varphi(z\wedge y)]\}$. It is not difficult to check that $\varphi(x\wedge jy)$ iff $\varphi(x\wedge y)$, and that $x\leq jy$ iff $\varphi(z\wedge x)\Rightarrow\varphi(z\wedge y)$ for all $z$. Therefore $j$ is a nucleus, and $L_j$ is an o-algebra with $jx>\mkern-13.5mu < jy$ if $\varphi(x\wedge y)$.
\end{proof}
\section*{Some remarks on predicativity}
In predicative foundations powersets are treated essentially as classes; actually, complete (semi)lattices are typically partially ordered classes rather than posets. As a consequence, the requirement \eqref{eq prop Pos} in the very definition of an overlap algebra appears problematic, as it may contain a universal quantification over a class.
This problem can be often overcome by restricting one's attention to $\textit{set-based}$ overlap algebras. A $ \textit{base} $ $S$ for a suplattice (complete join-semilattice) $ (L , \leq )$ is a set-indexed family of generators: $p = \bigvee \lbrace a\in S\vert a\leq p\rbrace$ for every $p$ in $L$. Of course, every o-algebra is set-based impredicatively.
For a set-based o-algebra condition \eqref{eq prop Pos} can be replaced by the following
\begin{center}
$ (\forall a\in S) (\mathrm{Pos}(a\wedge x) \Rightarrow \mathrm{Pos}(a\wedge y)) \Longrightarrow x\leq y $
\end{center} where the universal quantifier ranges over a set now.
Much of the results about {\bf OA} presented here remain valid for the category of set-based o-algebras within a predicative framework.
\bibliographystyle{plain}
| {
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{"url":"https:\/\/docs.analytica.com\/index.php?title=OptShadow&oldid=39852","text":"(diff) \u2190 Older revision | Latest revision (diff) | Newer revision \u2192 (diff)\n\nReturns the shadow prices, or dual values, for the constraints at the optimal solution.\n\nThe shadow price is the amount by which the objective function changes when the constraint is altered by increasing its right-hand side coefficient, bi, by one unit. It is valid only for small changes in bi, and mathematically is defined as:\n\nThis is the partial derivative of the objective function relative to the constraint RHS coefficient.\n\nFor a '<=' constraint and maximization problem, a shadow price indicates the amount the objective function improves when the constraint is relaxed. Shadow prices are usually meaningful when you are thinking of them in these terms. However, you should pay attention to the partial derivative definition to get the sign right.\n\nThe shadow can only be computed for continuous optimization problems. The shadow price does not exist for integer or mixed-integer optimizations, so can only be computed if every variable in the optimization problem is continuous.\n\nFor continuous problems, whether the shadow price can be computed depends upon the problem type and solver engine used. The following table summarizes the combinations for which shadow price can be computed (QP = quadratic objective + linear constraints, QCP = quadratically constrained):\n\n\"Problem Type\"\nEngine LP QP QCP NLP\nSOCP Barrier Y Y N -\nGRG Nonlinear Y Y Y Y\nEvolutionary N N N N\nLSLP Y - - -\nLSGRG Y Y Y Y\nLSSQP Y Y Y -\nOptQuest N N N N\nMOSEK Y Y N N\nKnitro Y Y Y Y\nXPress Y - - -\nGurobi Y - - -\n\nThis table may not be 100% accurate. [To do: empirically validate these entries]\n\n## Notes\n\nWhen you relax a constraint, the objective will always either improve or stay the same. Thus, for a minimization problem, the shadow price of a '<' constraint is always negative or zero, and for a maximization problem it is positive or zero. For a '>' constraint the opposite holds.\n\nConstraints with zero shadow prices have slack -- that is, they are not constraining the optimal solution.\n\nNot all linear programming packages use the same convention for the sign of shadow prices. The LINDO package, for example, uses a different convention for the sign.\n\n## History\n\nThis function was introduced in Analytica 4.3 and was formerly named LpShadow.","date":"2023-03-21 11:09:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7040268182754517, \"perplexity\": 1244.4127964148088}, \"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\/1679296943695.23\/warc\/CC-MAIN-20230321095704-20230321125704-00526.warc.gz\"}"} | null | null |
Avalerion or alerion is a term for a heraldic bird. Historically, it referred to the regular heraldic eagle. Later, heralds used the term alerion to refer to "baby eagles" or "eaglets". To differentiate them from mature eagles, alerions were shown as an eagle displayed inverted without a beak or claws (disarmed). To differentiate it from a decapitate (headless) eagle, the alerion has a bulb-shaped head with an eye staring towards the dexter (left-hand side) of the field. This was later simplified in modern heraldry as an abstract winged oval.
An example is the arms of the Duchy of Lorraine (or, on a bend gules, 3 alerions abaisé argent). It supposedly had been inspired by the assumed arms of crusader Geoffrey de Bouillon, who supposedly killed three white eaglets with a bow and arrow when out hunting. It is far more likely to be canting arms that are a pun based on Lorraine / Erne. (alerion is a partial anagram of Lorraine).
Medieval bestiaries use alerion for a mythological bird described as somewhat larger than an eagle of which only a single pair was said to live at any time. A pair of eggs was laid every 60 years; after hatching, the parents drowned themselves. The term avalerion is used on the Hereford Map near the Hydaspes and the Indus, possibly based on a description by Pliny.
The word's ultimate origin is unclear, possibly adapted from the German or ("eagle"). It is found in 12th-century French as and in medieval Latin as (a large eagle-like bird).
See also
Notes
References
Legendary birds
Heraldic birds
Medieval European legendary creatures
Birds in mythology
Fictional birds of prey
Heraldic eagles | {
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using System;
public class DerivedException : Exception {
public override string Message {
get {
var baseMessage = base.Message;
return baseMessage + "!";
}
}
}
public class TwiceDerivedException : DerivedException {
}
public class ThriceDerivedException : TwiceDerivedException {
public override string Message {
get {
var baseMessage = base.Message;
return baseMessage + "?";
}
}
}
public static class Program {
public static void Main (string[] args) {
try {
throw new DerivedException();
} catch (Exception ex) {
Console.WriteLine(ex.Message);
}
try {
throw new TwiceDerivedException();
} catch (Exception ex) {
Console.WriteLine(ex.Message);
}
try {
throw new ThriceDerivedException();
} catch (Exception ex) {
Console.WriteLine(ex.Message);
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
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\section{Introduction}
The Newtonian gravity links to all massive particles and they attract each-other by the inverse square law force. However massless particle or
light remains unaffected. The Einsteinian gravity results from universalization of the Newtonian gravity. That is to include massless particles
in the gravitational interaction. This requirement uniquely asks for gravity to be described by spacetime curvature \cite{dad1}. Of course
universalization also means that energy distribution in any form including the energy of gravitational field itself must also participate in
gravitational interaction. These are the two properties that drive us from Newton to Einstein. It is therefore pertinent
to see how are these features actually incorporated in the Einstein's theory of gravitation - general relativity (GR)? \\
The usual derivation of the Schwarzschild solution describing the gravitational field of a mass point in the textbooks does not bring out explicitly these very subtle and important conceptual
aspects. Our main aim in this pedagogical discussion is to demonstrate how these features are so beautifully and elegantly incorporated in GR. In the next
section we shall first discuss a simple derivation of the Schwarzshild solution by demanding that the timelike radial geodesic should include
the Newtonian gravitational law and the velocity of light should remain constant in the empty space surrounding the mass point. It is
interesting that these two simple considerations determine the Schwarzshild solution exactly. Then we solve the vacuum equation and in the
process we expose where and how gravitational interaction of massless particles and self interaction are actually incorporated? We also consider particle orbits again to illuminate some subtle and insightful points. We end up with a discussion. \\
\section{The Schwarszchild field}
For the gravitational field of a mass point which is static and radially symmetry, we begin with the usual spherically symmetric metric,
\begin{equation}
ds^2 = Adt^2 - Bdr^2 - r^2(d\theta^2 + sin^2\theta d\phi^2)
\end{equation}
where $A$ and $B$ are functions of $r$.\\
\subsection{Geodesics}
If GR has to include the Newtonian gravity, the timelike radial geodesic should reduce to $\ddot{r} = -\Phi^{\prime}$ where prime denotes
derivative relative to $r$ . Second if the velocity of light has to remain constant in empty space surrounding the mass point, photon should
experience no acceleration. Since the metric is free of $t$, we immediately write
\begin{equation}
A\dot{t} = E
\end{equation}
where $\dot{r} = dr/ds$ for the timelike particle and $\dot{r} = dr/d\lambda$ for photon. Substituting this in the metric, we get for photon,
\begin{equation}
\ddot{r} = (\frac{E^2}{AB})^{\prime}.
\end{equation}
Now it should experience no acceleration, $\ddot{r} = 0$, means $AB = const.$. Since at infinity, the metric should go over to the flat
Minkowski, hence $AB = 1$. On the other hand for the radially falling timelike particle, we similarly write
\begin{equation}
\dot{r}^2 = \frac{E^2}{AB} - \frac{1}{B} = E^2 - A
\end{equation}
which on differentiation gives
\begin{equation}
\ddot{r} = -\frac{A^{\prime}}{2}.
\end{equation} \\
For GR to include the Newtonian law, $A = 1 + 2\Phi$ with $\Phi$ as the Newtonian gravitational potential. We have thus obtained both the metric
functions, $A = 1/B = 1 + 2\Phi$, which exactly agree with the Schwarzschild solution obtained by solving the non-linear vacuum equation,
$R_{ab} = 0$. This is the simplest derivation of the solution which is purely driven by the physically reasonable considerations of the
inclusion of the Newtonian law and the velocity of light being constant. Note that it is the photon motion which requires space to be curved ($B
\neq 1$) while the Newtonian law would be included for the timelike particle even when space is flat with $B = 1$. It is reflection of the
fact that photon or light can feel gravity only through curvature of space. That is where it freely propagates. It is therefore clear
that Einstein is Newton with space curved. \\
\subsection{Solving the equation}
For the above metric, we have now to solve the vacuum equation,
\begin{equation}
R_{ab} = 0.
\end{equation}
There are three independent components of the Ricci curvature and two of which read as
\begin{equation}
R^t_t = \frac{1}{2AB}[A^{\prime\prime} - \frac{A^{\prime}}{2}(\frac{A^{\prime}}{A} + \frac{B^{\prime}}{B}) + \frac{2A^{\prime}}{r}],
\end{equation}
\begin{equation}
R^r_r = R^t_t + \frac{1}{rB}(\frac{A^{\prime}}{A} + \frac{B^{\prime}}{B}).
\end{equation}
Clearly $R^t_t = R^r_r$ implies $AB = const. = 1$ for asymptotically reducing to the Minkowski flat spacetime \cite{dad1,jacob}. Note that it is
the same condition which followed from photon experiencing no acceleration. This is the condition what is known as the null energy condition
given by $R_{ab}k^ak^b = 0, k_ak^a = 0$. Then writing $A = 1 + 2\Phi$, $R^t_t = 0$ reduces to the familiar Laplace equation \cite{dad2},
\begin{equation}
\nabla^2\Phi = 0
\end{equation}
which integrates to give the familiar solution
\begin{equation}
\Phi = k - M/r.
\end{equation}
Now the remaining equation takes the form
\begin{equation}
R^\theta_\theta = -\frac{2}{r^2}(r\Phi)^{\prime} = 0
\end{equation}
which sets the constant $k=0$. That is, the potential can have zero only at infinity nowhere else. This is in contrast to the Newtonian theory
where the constant $k$ remains free and can be chosen arbitrarily. We have thus obtained the Schwarzshild solution by solving the vacuum
equation $R_{ab} = 0$.
It however raises couple of very interesting questions. First and foremost, where has the self interaction of gravity gone which is the defining
property of the Einsteinian gravity and second, how is it that potential is determined absolutely, vanishing at infinity and nowhere else?
This is what we take up in the next section.\\
\section{Self interaction and zero of the potential}
The new features that Einstein gravity brings in are essentially the two, self interaction and photon feeling gravity. It is therefore
reasonable to expect that the former should facilitate the latter. That is gravitational effect of the self interaction should be such that it
makes photon feel gravity. For photon to feel gravity space has to be curved such that it does not have to change its velocity. This means the
contribution of self interaction is to curve the space without producing any acceleration like $\nabla\Phi$. That is why the Laplace equation
giving the Newton's inverse square law remains intact. This is how the self interaction is incorporated through the curvature of space while the
potential is still given by the good old Laplace equation. \\
The condition for photon to feel no acceleration like ordinary timelike particles is $A^{\prime}/A + B^{\prime}/B = 0$. It is this condition
that reduces $R^t_t = 0$ to the Laplace equation of the Newtonian gravity. If space were flat which means $B = 1$, then it would have taken
the form,
\begin{equation}
\nabla^2\Phi \approx {\Phi^{\prime}}^2
\end{equation}
indicating the self interaction contribution as square of the field, $\Phi^{\prime}$. What really happens is most remarkable that self
interaction contribution goes into curving the space with $B\neq1$ and further the velocity of light should remain constant determines $B =
1/A$. Thus note that the gravitational field energy gravitates in subtler way than matter density which produces $\nabla\Phi$ by curving
space and not by producing acceleration. This is how it should be because gravitational field energy is not the primary source of gravity
like matter density. It is produced by matter density and has no independent existence of its own. It is a secondary source and hence it should
not do what matter does and sit on the right in the above equation like the matter density. On the other hand for photon not to accelerate but
yet to feel gravity, space must curve and that is precisely what the self interaction does. Thus gravitational field gravitates by curving the
space
without making any contribution to acceleration. This is how self interaction is beautifully incorporated in GR by enlarging the spacetime background and not by modifying the gravitational law \cite{dad5}. \\
The next question is why is the potential determined absolutely, it can vanish only at infinity and nowhere else. In the Newtonian gravity,
potential is determined only up to addition of a constant which can be chosen arbitrarily. In contrast, as we have seen above that the equation
corresponding to $R^\theta_\theta = 0$ determines this constant to be zero leaving no choice for choosing zero of the potential. That is
constant potential attains non-trivial physical meaning here as it produces Ricci curvature $R^\theta_\theta = -2k/r^2$. This is very strange
because in all classical physics constant potential is dynamically trivial and has no physical significance. Let us then ask what is it that is
different for the Einsteinian gravity? It is universal and hence it makes an unusual demand on spacetime that it has to curve to describe its
dynamics. No other force makes such a demand on spacetime. For the rest of the physics, spacetime background is fixed and it is not affected by
the physics happening in it. In contrast, Einsteinian gravitational dynamics can only be described by the spacetime curvature and hence it
cannot remain inert and fixed as for the rest of physical phenomena. Note that in GR, gravitational field is self interactive which means it has
gravitational charge that is spread all over the space up to infinity. So gravitational source is not entirely localized at the location of the
mass point but is distributed all over space. It is a different matter that this distributed source in the form of gravitational field energy
gravitates differently from the mass point but it is nevertheless self interactive source of Einsteinian gravity. Therefore for the Einsteinian
gravitational potential as it occurs in the Schwarzschild solution, space surrounding the mass point is not completely free of "gravitational
source or charge". That is why it cannot vanish in the region which is not completely free of gravitational charge and therefore it can vanish
at infinity and nowhere else. Thus potential in the standard Schwarzschild coordinates gets determined absolutely. \\
\section{Particle orbits and self interaction}
As we saw in Sec.II, space curvature has no effect on radial motion as the equation (5) entirely agrees with the Newtonian law except for derivative here being w.r.t. proper time. It easily integrates to give the finite proper time of fall from radius $r_0$ to $r=0$ as $\sqrt{2r_0^3/9m}$. This is because the inverse square law remains intact and the space curvature does not affect radial motion. It would however make contribution for the non-radial motion. \\
Since the field is radially symmetric, there is no loss of generality in setting $\theta = \pi/2$ and like energy there is also conservation of angular momentum,
\begin{equation}
r^2 \dot{\phi} = l.
\end{equation}
Substituting the two constants of motion in the metric, we write the standrad expression
\begin{equation}
{\dot{r}}^2 = E^2 - (1 - \frac{2m}{r})(\frac{l^2}{r^2} + \mu)
\end{equation}
where $\mu = 1, 0$ refers respectively to timelike and null particle. By differenting the above equation, we write the condition for circular orbit as
\begin{equation}
\frac{m}{r^2} + \frac{3ml^2}{r^4} - \frac{l^2}{r^3} = \frac{m}{r^2} - \frac{l^2}{r^3}(1 - \frac{3m}{r}) = 0.
\end{equation}
Here the first and last terms are the familiar inverse square attarction and the centrifugal repulsion while the middle one is due to space curvature which couples transverse motion with the gravitational potential. By clubbing it with the centrifugal term, it has also been argued \cite{abra} that centrifugal force changes sign at $r=3m$. That is why there cannot exist any circular orbit below this radius. This is also the radius at which occurs the photon circular orbit. Clearly no particle can have circular orbit below the photon orbit radius. Note that the middle term is attractive and is in tune with the first and it is caused by the self interaction. It is gravitational in character and not kinematical and hence it should not be clubbed with the repulsive centrifugal term. Since it produces space curvature which affects transverse motion, that is how it gets linked to angular momentum. We would rather like to understand the above condition emerging from a potential
\begin{equation}
\frac{-2m}{r}(\mu + \frac{l^2}{r^2}) + \frac{l^2}{r^2}
\end{equation}
where the self interaction potential is coupling of gravitational potential with the transverse kinetic energy. Since photon feels no usual $m/r^2$ attraction, it has circular orbit when gradient of the self interaction term balances the centrifugal force. That is why we should not club the self interaction term with the centrifugal force else photon will have circular orbit with vanishing centrifugal force. Circular orbit is defined by the balance between attractive and repulsive effects. Effectively space curvature manifests in providing an additional attractive potential for transverse motion. The photon orbit marks the balance between the gradient of this and the centrifugal force. \\
We can in the standard way write the orbit equation for timelike particle,
\begin{equation}
u^{\prime\prime} + u = \frac{m}{l^2} + 3mu^2
\end{equation}
which for photon reduces to
\begin{equation}
u^{\prime\prime} + u = 3mu^2
\end{equation}
where $u=1/r, u^{\prime}=du/d\phi$. Note that it is $3mu^2$ which is the non-Newtonian contribution due to self interaction and it manifests in curving the space. It is clear that photons only feel space curvature. For timelike particles like planets, the orbit would be elliptical in the first approximation because the garvitational attraction law is the same inverse square law. Since the force law is not changed, the nature of orbit has essentially to remain undisturbed. It could then accommodate the effect of space curvature by suffering precession of perihelion. Why perihelion because that is where the force is strongest. Thus self interaction through space curvature make perihelion of the orbit precess. The orbits in the Einstein gravity are therefore precessing ellipses. Further note that gravitational field energy which is negative for positive mass curves space in such a way that it is in consonance with the attraction due to mass. It has been argued elsewhere \cite{dad4} that positive energy condition for gravitational field energy is that it is negative. It defines the norm of positivity for non-localizable energy distribution. For example, the electric field energy of a charged source is positive which is opposite of the norm set by negative gravitational field energy. It is therefore gravitationally 'negative' and that is why it contributes a repulsive effect opposing attraction due to mass for the field of a charged black hole. Let us consider potential at some $r$ due to a charged particle of mass M and charge Q which would be given by
\begin{equation}
\Phi = - \frac{M - Q^2/2r}{r}.
\end{equation}
This is because the electric field energy, $Q^2/2r$, lying outside the radius $r$ does not contribute and hence has to be subtracted out. It would give rise to the acceleration $-M/r^2 + Q^2/r^3$ which shows the repulsive effect of the electric field energy \cite{dad7}. Since electric field energy is positive, it is therefore ``gravitationally negative'' and hence repulsive. \\
\section{Discussion}
The main aim of this note is to bring out transparently inclusion of gravitational self interaction and its role in particle orbits, and why potential in the Schwarzschild
solution cannot vanish anywhere but at infinity. This is very interesting and insightful for appreciating the remarkable features of the
Einsteinian gravity over the Newtonian gravity. Note that ultimately the equation we need to solve is the first order linear differential
equation which is the first integral of the Laplace equation. It is this that determines the potential absolutely. The Newtonian $\nabla\Phi$ comes
from $A = 1 + 2\Phi$ which can be squared out by redefining $t$ when $\Phi = k$. It is $A$ that gives the Newtonian acceleration, $\nabla\Phi$
and hence $A=const$ has no physical significance. However $\Phi = k$ in $B$ has non-trivial effect because it refers to curvature of space which
is sourced by self interaction and it does not vanish when $B = const.$. It may be noted that the constant potential generates the following
stresses,
\begin{equation}
T^t_t = T^r_r = \frac{k}{r^2}, \, \, \, \, T^\theta_\theta = T^\phi_\phi = 0
\end{equation}
and they asymptotically agree with that of a global monopole \cite{bv,dad3}. It is remarkable that constant potential dynamically therefore
describes a global monopole.
The Einsteinian gravity is essentially driven from the Newtonian gravity by the two new properties of self interaction and photon feeling
gravity without experiencing acceleration. The former contributes through curving space which also facilitates photon's interaction with
gravity. It is remarkable that the two new properties are intimately related to each-other leaving essentially the Newtonian gravity intact. The
standard derivation and discussion of the Schwarzschild solution do not expose these interesting new aspects of the Einsteinian gravity in such
a transparent and explicit manner. That is precisely what we had set out to do. \\
{\it Acknowledgment:} It is a pleasure to thank the Al Faraby Kazakh National University, Almaty for the kind invitation to lecture in the
School on Theoretical Physics organized by the Department of Theoretical and Nuclear Physics. These subtle aspects, which are otherwise not so
explicitly and transparently discussed, were brought out succinctly in the author's lectures. I warmly thank Professor Medeu Abishev for the
wonderful hospitality.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,760 |
package net.jawr.web.resource.bundle.variant.resolver;
import javax.servlet.http.HttpServletRequest;
import net.jawr.web.JawrConstant;
import net.jawr.web.resource.bundle.variant.VariantResolver;
import net.jawr.web.resource.bundle.variant.VariantSet;
/**
* This class defines the URL scheme resolver.
*
* @author Ibrahim Chaehoi
*
*/
public class ConnectionTypeResolver implements VariantResolver {
/* (non-Javadoc)
* @see net.jawr.web.resource.bundle.variant.VariantResolver#getVariantType()
*/
public String getVariantType() {
return JawrConstant.CONNECTION_TYPE_VARIANT_TYPE;
}
/* (non-Javadoc)
* @see net.jawr.web.resource.bundle.variant.VariantResolver#getAvailableVariant(java.lang.String, net.jawr.web.resource.bundle.variant.VariantSet)
*/
public String getAvailableVariant(String variant, VariantSet variantSet) {
String connectionType = variantSet.getDefaultVariant();
if(variantSet.contains(variant)){
connectionType = variant;
}
return connectionType;
}
/* (non-Javadoc)
* @see net.jawr.web.resource.bundle.variant.VariantResolver#resolveVariant(javax.servlet.http.HttpServletRequest)
*/
public String resolveVariant(HttpServletRequest request) {
String connectionType = "";
if(request.getScheme().equals(JawrConstant.HTTPS)){
connectionType = JawrConstant.SSL;
}
return connectionType;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 33 |
Influenced heavily by Ralph Eugene Meatyard, my current interest in the void stems from my long time interest in a lack of identity. Exploring the use of a mask and its capabilities of both concealing and revealing, identity may be lost, but emphasis is placed on the human form as a unique characteristic that we all share. Are the features of your face necessary to defining who you claim to be in this reality? | {
"redpajama_set_name": "RedPajamaC4"
} | 2,751 |
Q: Alternative for anonymous namespaces in header-only libraries I understand why it doesn't make sense to use anonymous namespaces in header files... They aren't really anonymous...
However, this begs the question:
Is there an alternative idiom/mechanism to avoid polluting the global namespace when distributing a header-only library?
EDIT:
My typical usage of an anonymous namespace is to keep some block of code local to a file so that it doesn't pollute the global namespace. For e.g. if some class had some magic constant, then instead of declaring a global static int, I could declare it in the cpp file:
namespace{
int magic = 5;
}
Is there a way to achieve the same effect without having to use a cpp file?
A: C++ doesn't have any mechanism to make entities in header files completely invisible to users. They can be made inaccessible if you want. This is normally achieved by member access control. You have to make foo_impl a private (possibly static) member of some class. Overloads of foo would then be either members or friends of the same class.
Alternatively, if you make foo_impl a member of a namespace named detail or foo_private or some such, users will normally understand they should not call this function. This works well in practice. Users will still be able to access the function at their own risk, but they will understand the risk. This should be plenty enough, as C++ doesn't protect you from malicious users anyway.
A: In boost, there is sometimes used namespace named detail
Functions not intended for use by applications are in boost::math::detail.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,273 |
{"url":"http:\/\/www.ams.org\/mathscinet-getitem?mr=2323468","text":"MathSciNet bibliographic data MR2323468 46L05 Robertson, David I.; Sims, Aidan Simplicity of \\$C\\sp \\ast\\$$C\\sp \\ast$-algebras associated to higher-rank graphs. Bull. Lond. Math. Soc. 39 (2007), no. 2, 337\u2013344. Article\n\nFor users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.","date":"2016-09-29 07:35:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 1, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9977703094482422, \"perplexity\": 8382.327175198734}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-40\/segments\/1474738661779.44\/warc\/CC-MAIN-20160924173741-00093-ip-10-143-35-109.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/gmatclub.com\/forum\/mary-persuaded-n-friends-to-donate-500-each-to-her-election-10075-20.html","text":"GMAT Question of the Day - Daily to your Mailbox; hard ones only\n\n It is currently 11 Dec 2018, 05:37\n\n# R1 Decisions:\n\nHBS Chat - Decisions will be released at Noon ET\u00a0 |\u00a0 UVA Darden Chat\u00a0 |\u00a0 YouTube Live with Cornell Johnson @11am ET\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## Events & Promotions\n\n###### Events & Promotions in December\nPrevNext\nSuMoTuWeThFrSa\n2526272829301\n2345678\n9101112131415\n16171819202122\n23242526272829\n303112345\nOpen Detailed Calendar\n\u2022 ### Free GMAT Prep Hour\n\nDecember 11, 2018\n\nDecember 11, 2018\n\n09:00 PM EST\n\n10:00 PM EST\n\nStrategies and techniques for approaching featured GMAT topics. December 11 at 9 PM EST.\n\u2022 ### The winning strategy for 700+ on the GMAT\n\nDecember 13, 2018\n\nDecember 13, 2018\n\n08:00 AM PST\n\n09:00 AM PST\n\nWhat people who reach the high 700's do differently? We're going to share insights, tips and strategies from data we collected on over 50,000 students who used examPAL.\n\n### Show Tags\n\n30 Mar 2015, 19:07\nBunuel wrote:\nAwli wrote:\nMary persuaded n friends to donate $500 each to her election campaign, an then each of these n friends persuaded n more people to donate$500 each to Mary's campaign. If no one donated more than once and if there were no other donations, what was the value of n?\n\n(1) The first n people donated \\frac{1}{16} of the total amount donated.\n\n(2) The total amount donated was $120,000 Merging topics. Please refer to the discussion on page 1. [color=#0000ff]not sure, if my approach is right or wrong. I just took it like: 1. Mary persuaded n friends to donate$500 = n*500\n2. then each of these n friends persuaded n more people.= n^(n+1) * 500\n\nStatement 1. first n donated 1\/16 of the total. remains > need total amount.\nStatement 2. n^(n+1)*500 = 120,000\nn*n^n = 240 .. Looks insufficient\n\n1+2\n\nn = 1\/16 * 120000 \/500 = 15\n\nhence C\nIntern\nJoined: 04 Mar 2015\nPosts: 4\n\n### Show Tags\n\n31 Mar 2015, 03:44\nOriginal Donations = 500n\nFriends' Friends =500n^2\nTotal = 500n+500n^2\n\nStatement 1 :The first n people donated \\frac{1}{16} of the total amount donated.\n\n500n = 1\/16 (500n+500n^2)\n16(500n) = 500n+500n^2\nn = 15\nstmt 1 is sufficient\n\nStatement 2: The total amount donated was $120,000 500n+500n^2=$120,000\ncan solve for n\nn=15\nstmt 2 is sufficient\n\nIntern\nJoined: 30 Jul 2014\nPosts: 1\nRe: Mary persuaded n friends to donate $500 each to her election [#permalink] ### Show Tags 10 Aug 2015, 01:55 aalriy wrote: I have understood the approach GT took to solve the problem its very similar to mine... but i cannot make out how can the first stmt give a solution for n as 0 or a -ve value. Could someone explain this? On the GMAT you ll not be asked a value based DS question if at all there is no such value.That is why n can not be zero.One more thing that U can understand that as there are n people first to donate$500 each and those n people refer n people each .So if U consider that there are 16 portions total money is donated by all then 1 portion is by the first n people.\nIntern\nJoined: 16 May 2017\nPosts: 21\n\n### Show Tags\n\n12 Nov 2017, 08:07\nTop Contributor\nseofah wrote:\nMary persuaded n friends to donate $500 each to her election campaign, and then each of these n friends persuaded n more people to donate$500 each to Mary\u2019s campaign. If no one donated more than once and if there were no other donations, what was the value of n?\n\n(1) The first n people donated 1\/16 of the total amount donated.\n(2) The total amount donated was $120,000. Target question: What was the value of n? When I scan the two statements, it seems that statement 2 is easier, so I'll start with that one first... Statement 2: The total amount donated was$120,000\nLet's summarize the given information....\n\nFirst round: n friends donate 500 dollars.\nThis gives us a total of 500n dollars in this round\n\nSecond round: n friends persuade n friends each to donate\nSo, each of the n friends gets n more people to donate.\nThe total number of donors in this round = n\u00b2\nThis gives us a total of 500(n\u00b2) dollars in this round\n\nTOTAL DONATIONS = 500n dollars + 500(n\u00b2) dollars\nWe can rewrite this: 500n\u00b2 + 500n dollars\n\nSo, statement 2 tells us that 500n\u00b2 + 500n = 120,000\nThis is a quadratic equation, so let's set it equal to zero to get: 500n\u00b2 + 500n - 120,000 = 0\nFactor out the 500 to get: 500(n\u00b2 + n - 240) = 0\nFactor more to get: 500(n + 16)(n - 15) = 0\nSo, EITHER n = -16 OR n = 15\nSince n cannot be negative, it must be the case that n = 15\nSince we can answer the target question with certainty, statement 2 is SUFFICIENT\n\nStatement 1: The first n people donated 1\/16 of the total amount donated.\nFirst round donations = 500n\nTOTAL donations = 500n\u00b2 + 500n\nSo, we can write: 500n = (1\/16)[500n\u00b2 + 500n]\nMultiply both sides by 16 to get: 8000n = 500n\u00b2 + 500n\nSet this quadratic equation equal to zero to get: 500n\u00b2 - 7500n = 0\nFactor to get: 500n(n - 15) = 0\nDo, EITHER n = 0 OR n = 15\nSince n cannot be zero, it must be the case that n = 15\nSince we can answer the target question with certainty, statement 1 is SUFFICIENT\n\nCheers,\nBrent\n_________________\n\nTest confidently with gmatprepnow.com\n\nIntern\nJoined: 16 Jan 2011\nPosts: 4\nLocation: Singapore\nConcentration: Finance\nSchools: HKUST (S)\nGMAT 1: 690 Q49 V34\nGPA: 3.62\n\n### Show Tags\n\n22 Jun 2018, 22:23\ngmatcrash wrote:\nWithin context of GMAT DS question, the moment I manage to set up such relationship n(n+1) = 240, will it be safe to say there is 1 solution for n without trying to find a pair of factors that fit? This would save some time. Whenever I get to this point, I always try to find a pair just to make sure it will not be the case of a) having no solution for n or b) having 2 solutions for n.\n\nn(n + 1) = (positive number) will always have two solutions, one negative and one positive but not always these solutions will be integers.\n\nFor example:\n\nn(n + 1) = 2 --> n = -2 or n = 1;\n\nn(n + 1) = 2 --> $$n = -\\frac{1}{2}-\\frac{\\sqrt{13}}{2}$$ or $$n = -\\frac{1}{2}+\\frac{\\sqrt{13}}{2}$$\n_________________\nGMATH Teacher\nStatus: GMATH founder\nJoined: 12 Oct 2010\nPosts: 534\nRe: Mary persuaded n friends to donate $500 each to her election [#permalink] ### Show Tags 04 Nov 2018, 11:47 seofah wrote: Mary persuaded n friends to donate$500 each to her election campaign, and then each of these n friends persuaded n more people to donate $500 each to Mary\u2019s campaign. If no one donated more than once and if there were no other donations, what was the value of n? (1) The first n people donated 1\/16 of the total amount donated. (2) The total amount donated was$120,000.\n\n$${\\rm{Total}}\\,\\, = \\,\\,500 \\cdot n + 500 \\cdot n \\cdot n\\,\\,\\,\\,\\,\\,\\left[ \\ \\right]$$\n\n$$? = n$$\n\n$$\\left( 1 \\right)\\,\\,\\,500 \\cdot n = {1 \\over {16}} \\cdot 500 \\cdot n \\cdot \\left( {1 + n} \\right)\\,\\,\\,\\,\\,\\mathop \\Rightarrow \\limits^{:\\,\\,\\,\\left( {500\\,n} \\right)\\,\\,\\,\\left[ {\\,n\\, \\ne \\,0\\,} \\right]} \\,\\,\\,1 = {1 \\over {16}} \\cdot \\left( {1 + n} \\right)\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,n\\,\\,{\\rm{unique}}\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,{\\rm{SUFF}}.$$\n\n$$\\left( 2 \\right)\\,\\,\\,500 \\cdot n\\left( {1 + n} \\right) = 120000\\,\\,\\,\\,\\,\\mathop \\Rightarrow \\limits^{:\\,\\,500} \\,\\,\\,\\,n\\left( {1 + n} \\right) = 240\\,\\,\\,\\,\\,\\mathop \\Rightarrow \\limits^{\\left( * \\right)} \\,\\,\\,\\,\\,n\\,\\, > 0\\,\\,\\,\\,{\\rm{unique}}\\,\\,\\,\\,\\,\\,\\, \\Rightarrow \\,\\,\\,\\,\\,{\\rm{SUFF}}.$$\n\n$$\\left( * \\right)\\,\\,15 \\cdot 16 = 240\\,\\,\\, \\Rightarrow \\,\\,\\,\\left\\{ \\matrix{ \\,n\\left( {n + 1} \\right) < 240\\,\\,{\\rm{for}}\\,\\,0 < n < 15 \\hfill \\cr \\,n\\left( {n + 1} \\right) > 240\\,\\,{\\rm{for}}\\,\\,n \\ge 16 \\hfill \\cr} \\right.\\,\\,\\,\\,\\,\\,\\left( {{\\rm{Now}}\\,\\,{\\rm{rethink}}\\,\\,{\\rm{without}}\\,\\,{\\rm{knowing}}\\,\\,{\\rm{that}}\\,\\,n = 15...} \\right)$$\n\nThis solution follows the notations and rationale taught in the GMATH method.\n\nRegards,\nFabio.\n_________________\n\nFabio Skilnik :: https:\/\/GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)\nCourse release PROMO : finish our test drive till 30\/Dec with (at least) 50 correct answers out of 92 (12-questions Mock included) to gain a 50% discount!\n\nRe: Mary persuaded n friends to donate $500 each to her election &nbs [#permalink] 04 Nov 2018, 11:47 Go to page Previous 1 2 [ 29 posts ] Display posts from previous: Sort by # Mary persuaded n friends to donate$500 each to her election\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 Kindly note that the GMAT\u00ae test is a registered trademark of the Graduate Management Admission Council\u00ae, and this site has neither been reviewed nor endorsed by GMAC\u00ae.","date":"2018-12-11 13:37:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.46773064136505127, \"perplexity\": 6290.158306647883}, \"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-2018-51\/segments\/1544376823621.10\/warc\/CC-MAIN-20181211125831-20181211151331-00437.warc.gz\"}"} | null | null |
/* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.activiti.app.rest.runtime;
import javax.servlet.http.HttpServletResponse;
import org.activiti.app.model.common.ResultListDataRepresentation;
import org.activiti.app.model.runtime.RelatedContentRepresentation;
import org.activiti.app.service.exception.InternalServerErrorException;
import org.slf4j.Logger;
import org.slf4j.LoggerFactory;
import org.springframework.web.bind.annotation.PathVariable;
import org.springframework.web.bind.annotation.RequestBody;
import org.springframework.web.bind.annotation.RequestMapping;
import org.springframework.web.bind.annotation.RequestMethod;
import org.springframework.web.bind.annotation.RequestParam;
import org.springframework.web.bind.annotation.RestController;
import org.springframework.web.multipart.MultipartFile;
import com.fasterxml.jackson.databind.ObjectMapper;
/**
* @author Frederik Heremans
*/
@RestController
public class RelatedContentResource extends AbstractRelatedContentResource {
private static final Logger logger = LoggerFactory.getLogger(AbstractRelatedContentResource.class);
protected ObjectMapper objectMapper = new ObjectMapper();
@RequestMapping(value = "/rest/tasks/{taskId}/content", method = RequestMethod.GET)
public ResultListDataRepresentation getRelatedContentForTask(@PathVariable("taskId") String taskId) {
return super.getRelatedContentForTask(taskId);
}
@RequestMapping(value = "/rest/process-instances/{processInstanceId}/content", method = RequestMethod.GET)
public ResultListDataRepresentation getRelatedContentForProcessInstance(@PathVariable("processInstanceId") String processInstanceId) {
return super.getRelatedContentForProcessInstance(processInstanceId);
}
@RequestMapping(value = "/rest/content/{source}/{sourceId}/process-instances", method = RequestMethod.GET)
public ResultListDataRepresentation getRelatedProcessInstancesForContent(@PathVariable("source") String source, @PathVariable("sourceId") String sourceId) {
return super.getRelatedProcessInstancesForContent(source, sourceId);
}
@RequestMapping(value = "/rest/tasks/{taskId}/raw-content", method = RequestMethod.POST)
public RelatedContentRepresentation createRelatedContentOnTask(@PathVariable("taskId") String taskId, @RequestParam("file") MultipartFile file) {
return super.createRelatedContentOnTask(taskId, file);
}
/*
* specific endpoint for IE9 flash upload component
*/
@RequestMapping(value = "/rest/tasks/{taskId}/raw-content/text", method = RequestMethod.POST)
public String createRelatedContentOnTaskText(@PathVariable("taskId") String taskId, @RequestParam("file") MultipartFile file) {
RelatedContentRepresentation relatedContentRepresentation = super.createRelatedContentOnTask(taskId, file);
String relatedContentJson = null;
try {
relatedContentJson = objectMapper.writeValueAsString(relatedContentRepresentation);
} catch (Exception e) {
logger.error("Error while processing RelatedContent representation json", e);
throw new InternalServerErrorException("Related Content on task could not be saved");
}
return relatedContentJson;
}
@RequestMapping(value = "/rest/tasks/{taskId}/content", method = RequestMethod.POST)
public RelatedContentRepresentation createRelatedContentOnTask(@PathVariable("taskId") String taskId,
@RequestBody RelatedContentRepresentation relatedContent) {
return super.createRelatedContentOnTask(taskId, relatedContent);
}
@RequestMapping(value = "/rest/processes/{processInstanceId}/content", method = RequestMethod.POST)
public RelatedContentRepresentation createRelatedContentOnProcessInstance(@PathVariable("processInstanceId") String processInstanceId,
@RequestBody RelatedContentRepresentation relatedContent) {
return super.createRelatedContentOnProcessInstance(processInstanceId, relatedContent);
}
@RequestMapping(value = "/rest/process-instances/{processInstanceId}/raw-content", method = RequestMethod.POST)
public RelatedContentRepresentation createRelatedContentOnProcessInstance(@PathVariable("processInstanceId") String processInstanceId,
@RequestParam("file") MultipartFile file) {
return super.createRelatedContentOnProcessInstance(processInstanceId, file);
}
/*
* specific endpoint for IE9 flash upload component
*/
@RequestMapping(value = "/rest/process-instances/{processInstanceId}/raw-content/text", method = RequestMethod.POST)
public String createRelatedContentOnProcessInstanceText(@PathVariable("processInstanceId") String processInstanceId,
@RequestParam("file") MultipartFile file) {
RelatedContentRepresentation relatedContentRepresentation = super.createRelatedContentOnProcessInstance(processInstanceId, file);
String relatedContentJson = null;
try {
relatedContentJson = objectMapper.writeValueAsString(relatedContentRepresentation);
} catch (Exception e) {
logger.error("Error while processing RelatedContent representation json", e);
throw new InternalServerErrorException("Related Content on process instance could not be saved");
}
return relatedContentJson;
}
@RequestMapping(value = "/rest/content/raw", method = RequestMethod.POST)
public RelatedContentRepresentation createTemporaryRawRelatedContent(@RequestParam("file") MultipartFile file) {
return super.createTemporaryRawRelatedContent(file);
}
/*
* specific endpoint for IE9 flash upload component
*/
@RequestMapping(value = "/rest/content/raw/text", method = RequestMethod.POST)
public String createTemporaryRawRelatedContentText(@RequestParam("file") MultipartFile file) {
RelatedContentRepresentation relatedContentRepresentation = super.createTemporaryRawRelatedContent(file);
String relatedContentJson = null;
try {
relatedContentJson = objectMapper.writeValueAsString(relatedContentRepresentation);
} catch (Exception e) {
logger.error("Error while processing RelatedContent representation json", e);
throw new InternalServerErrorException("Related Content could not be saved");
}
return relatedContentJson;
}
@RequestMapping(value = "/rest/content", method = RequestMethod.POST)
public RelatedContentRepresentation createTemporaryRelatedContent(@RequestBody RelatedContentRepresentation relatedContent) {
return addRelatedContent(relatedContent, null, null, false);
}
@RequestMapping(value = "/rest/content/{contentId}", method = RequestMethod.DELETE)
public void deleteContent(@PathVariable("contentId") Long contentId, HttpServletResponse response) {
super.deleteContent(contentId, response);
}
@RequestMapping(value = "/rest/content/{contentId}", method = RequestMethod.GET)
public RelatedContentRepresentation getContent(@PathVariable("contentId") Long contentId) {
return super.getContent(contentId);
}
@RequestMapping(value = "/rest/content/{contentId}/raw", method = RequestMethod.GET)
public void getRawContent(@PathVariable("contentId") Long contentId, HttpServletResponse response) {
super.getRawContent(contentId, response);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,127 |
{"url":"http:\/\/mathhelpforum.com\/advanced-statistics\/82973-characteristics-ifr-pdf-print.html","text":"# Characteristics of an IFR pdf\n\n\u2022 April 9th 2009, 12:01 AM\nXavier_B\nCharacteristics of an IFR pdf\nHi,\n\nI am looking for help in solving a problem that crept up in my doctoral thesis.\nLet f be a pdf, twice differentiable and F its cdf. Both are defined over $\\mathbb{R}^+$.\n\nI wish to prove that\n$\\frac{F(x)}{f(x)}f'(x) - 2f((x)<0$ (1).\nI wish to prove it for all functions f which are IFR in the sense developed by Richard E. Barlow and Franck Proschan in their book Mathematical theory of reliability - Google Book Search.\nWhen a pdf is IFR, then $r(x)=\\frac{f(x)}{(1-F(x))}$ is weakly increasing. This, of course, leads to $r'(x) \\geq 0$.\nSo that we can write\n$f'(x)(1-F(x)) + f^2(x) \\geq 0 \\Rightarrow F(x)f'(x) - f^2(x) \\leq f'(x).$ (2)\n\nNow (1) can also be written $F(x)f'(x) - f^2(x) < f^2(x)$. (3)\n(3) is true if f is IFR and if $f'(x).\n\nWhat happens when $f'(x) \\geq f^2(x)$?\n\nIf $f'(x) \\geq f^2(x)$, f'(x) is always positive because f is a pdf. So f is increasing over $\\mathbb{R}^+$.\nIf there exists a $x_0$ such that $f'(x_0)=0$, then f exhibits a maximum at that point. but f being a pdf, that means that $f(x_0) \\neq 0$, which contradicts the hypothesis unless f is the null function, which we discard. So $\\forall x \\in \\mathbb{R}^+, f'(x) >0$. Hence f is striclty increasing. But by definition of F, $\\int_0^{\\infty} f(t)dt =1$.\n\nMy hunch is that this last equality contradicts the fact that f is strictly increasing and positive.\nIs this the case and can we then say that if f is IFR then $f'(x) and so (1) is true when f is IFR?\n\nIf that is so, then I have have won my day.\n\nAny help on this problem would be very much appreciated!\nThank you.","date":"2016-05-03 18:45:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 16, \"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\": 0.9479406476020813, \"perplexity\": 1333.005277999978}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-18\/segments\/1461860121737.31\/warc\/CC-MAIN-20160428161521-00026-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
Browse 20 different products in 10 different colours in 2 different sizes. QIS has a huge range of coloured gift paper bags with no handles. Shop now!
Use the filters to sift through colour, sizing (length/width) and price.
Make your selection and add them to your cart! | {
"redpajama_set_name": "RedPajamaC4"
} | 7,148 |
The arrival of spring beckons the return of sunshine and warmth, yet also typically results in an increase in household pest encounters. That's why now is an opportune time for area homeowners to attend our latest free workshop.
LakewoodAlive's "Managing Pests" workshop takes place on March 25.
Shawn Payne, owner of Lakewood Exterminating, will share tips and tricks for managing household pests, discussing how to identify everyday pests in and around your home, and addressing best practices for sealing your home to protect it from pests and wildlife. A Lakewood resident with an extensive background in horticulture, Shawn was responsible for pest control at the NASA Glenn Research Center in Cleveland prior to taking on his business full-time. | {
"redpajama_set_name": "RedPajamaC4"
} | 513 |
Tag: Cody
"Survivor 43" Recap: 'Telenovela'
David - Dec 8, 2022
HOLLYWOOD—Well it was the penultimate episode of "Survivor 43" before the big finale next week. Would the viewers finally be treated to an entertaining...
"Survivor 43" Recap: 'Hiding In Plain Sight'
HOLLYWOOD—We're down to the final seven on "Survivor 43." No idols have been played, each Tribal Council has been a bore to say the...
"Survivor 43" Recap: 'What About The Big Girls'
David - Nov 17, 2022
HOLLYWOOD—I have been praying for a riveting episode of "Survivor 43" for quite some time. Things have been quite boring to say the least....
"Survivor 43" Recap: 'Proposterous'
HOLLYWOOD—Dare I say it, I'm a little bored with "Survivor 43." There has not been any major dramatic moments of conflict this season that...
"Survivor 43" Recap: 'Bull In A China Shop'
David - Nov 3, 2022
HOLLYWOOD—Well, we are officially now in the merge stage of the game on "Survivor 43" after that 'fake merge' last week. This week's episode,...
Holly Sutton Returns To "General Hospital!"
Donald - Oct 30, 2022
HOLLYWOOD—Well she's back! That is right "General Hospital" fans Holly Sutton returned to the canvas late last week shocking Olivia, Ned and Robert in...
"Survivor 43" Recap: 'Mergatory'
David - Oct 27, 2022
HOLLYWOOD—Chaos is upon us in the latest episode of "Survivor 43" as the contestants and the viewers wondered if the merge was truly upon...
"Survivor 43" Recap: 'Show No Mercy'
HOLLYWOOD—Vesi made the move last week to remove a weak player from the board on "Survivor 43" with Nneke getting the boot. This week's...
"Survivor" Kicks Off Season 43!
David - Sep 22, 2022
HOLLYWOOD—I have been doing my best to stay away from potential spoilers for the upcoming season of "Survivor" which kicked off its 43rd season...
Drew And Carly Flirt On "General Hospital!"
Donald - Jun 25, 2022
HOLLYWOOD—The moment that Drew Quartermaine locked eyes with Carly Corinthos on that bridge after she caught Sonny and Nina in bed together was all... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,170 |
{"url":"https:\/\/help.anaplan.com\/44c45685-1873-4051-b0dc-160f21210fe7","text":"Many investment management functions rely on knowing the number of days between two dates. As the number of days in a year or month can vary, there are conventions that enable you to calculate the number of days in the year, which is known as the basis.\n\nAnaplan defaults to the US 30\/360 convention for day count, with a few differences. However, you can also choose to use other day count conventions.\n\n## US 30\/360 day count conventions\n\nThe US 30\/360 day count convention assumes 30 days for every month, and 360 days for the year. This convention was originally defined by the Financial Industry Regulatory Authority (FINRA).\n\nUS 30\/360 uses the DayCountFactor formula to determine day count:\n\n$DayCountFactor=\\frac{360\\times(Y_2-Y_1)+30\\times(M_2-M_1)+(D_2-D_1)}{360}$\n\nWhere:\n\n\u2022 Y is year,\n\u2022 M is month, and\n\u2022 D is day.\n\nThere are then various conventions by which you can adjust D1 and D2 to determine the end of the month, as some months are not 30 days long.\n\nThe US 30\/360 conventions are:\n\n\u2022 If the investment is End of Month (EOM), the start date is the last day of February, and the end date is the last day of February, then change D2 to 30.\n\u2022 If the investment is EOM and the start date is the last day of February, then change D1 to 30.\n\u2022 If D2 is 31 and D1 is 30 or 31, then change D2 to 30.\n\u2022 If D1 is 31, then change D1 to 30.\n\n## Differences in Anaplan\n\nAnaplan conventions differ from these in that the full set of rules is only applied when calculating COUPDAYSNC. For other calculations, the start date check is not performed for the first and third conventions outlined above. Instead, these modified conventions apply:\n\n\u2022 If the investment is EOM and the end date is the last day of February, then change D2 to 30.\n\u2022 If the investment is EOM and the start date is the last day of February, then change D1 to 30.\n\u2022 If D2 is 31, then change D2 to 30.\n\u2022 If D1 is 31, then change D1 to 30.\n\nThis allows the date adjustments for D2 to be independent of D1.\n\n## Other day count conventions\n\nUS 30\/360 is the convention used by default in Anaplan. However, these conventions are also accommodated in the basis argument of the management functions:\n\n\u2022 Actual\/360 and EUR 30\/360, for which a year has 360 days.\n\u2022 Actual\/365, for which a year has 365 days.\n\u2022 Actual\/Actual, for which a year may have 365 or 366 days.\n\nNote: Anaplan uses the International Swaps and Derivatives Association (ISDA) convention for Actual\/Actual. In this convention, the number of days in leap and non-leap years are calculated separately.\n\nDisclaimer\n\nWe update Anapedia regularly to provide the most up-to-date instructions.","date":"2022-11-30 13:23:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 1, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.23840375244617462, \"perplexity\": 1831.6869873786607}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710764.12\/warc\/CC-MAIN-20221130124353-20221130154353-00452.warc.gz\"}"} | null | null |
package jbt.tools.bteditor.actions;
import java.io.IOException;
import java.util.Vector;
import jbt.tools.bteditor.NodesLoader;
import jbt.tools.bteditor.model.ConceptualNodesTree;
import jbt.tools.bteditor.util.StandardDialogs;
import jbt.tools.bteditor.util.Utilities;
import jbt.tools.bteditor.views.NodesNavigator;
import org.eclipse.jface.action.Action;
import org.eclipse.ui.IViewPart;
/**
* Action that loads, into the {@link NodesLoader}, the actions and conditions
* (sensors) present in a MMPM domain file.
*
* @author Ricardo Juan Palma Durán
*
*/
public class LoadMMPMDomainAction extends Action {
/** Names of the files to open. */
private Vector<String> fileNames;
/**
* Constructor.
*
* @param fileNames
* the names of the files to load.
*/
public LoadMMPMDomainAction(Vector<String> fileNames) {
this.fileNames = fileNames;
}
/**
*
* @see org.eclipse.jface.action.Action#run()
*/
public void run() {
Vector<Exception> exceptions = new Vector<Exception>();
for (String currentFile : this.fileNames) {
try {
ConceptualNodesTree newTree = NodesLoader
.loadNonStandardNodes(currentFile);
IViewPart view = Utilities.getView(NodesNavigator.class);
if (view != null) {
NodesNavigator treeView = (NodesNavigator) view;
treeView.addTree(newTree);
}
} catch (IOException e) {
exceptions.add(e);
}
}
if (exceptions.size() != 0) {
StandardDialogs.exceptionDialog("Error loading MMPM domain file",
"There were errors when opening MMPM domain files",
exceptions);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 921 |
{"url":"https:\/\/trac-hacks.org\/ticket\/3741","text":"Opened 9 years ago\n\nClosed 6 years ago\n\n# Tracforms can't handle mutated vowel\n\nReported by: Owned by: didley@\u2026 Steffen Hoffmann high TracFormsPlugin major unicode umlauts 0.11\n\n### Description\n\nWhe I using mutated vowels in a Form for example \u00e4 in Tracforms the message comes up\n\n'ascii' codec can't encode character u'\\xf6' in position 0: ordinal not in range(128)\n\n\nOther plugins or wiki can handle it. Is it possible to fix it?\n\n### comment:1 Changed 6 years ago by Steffen Hoffmann\n\n#5543 mentioned especially German umlauts and has been closed as a duplicate of this ticket.\n\n### comment:2 Changed 6 years ago by Steffen Hoffmann\n\nKeywords: unicode added normal \u2192 high normal \u2192 major\n\nConfirmed, working towards a solution, since I require it too, and that can't be that hard. It might just be about missing unicode encoding for form content in some places.\n\n### comment:3 Changed 6 years ago by Ryan J Ollos\n\nI'll watch for you patch since this is a frequent problem with plugins and I need to wrap my head around how to fix these issues.\n\n### comment:4 Changed 6 years ago by Steffen Hoffmann\n\nOwner: changed from Rich Harkins to Steffen Hoffmann\n\nSure, this is one of my short-term tasks. Thanks for taking care.\n\n### comment:5 Changed 6 years ago by Steffen Hoffmann\n\nStatus: new \u2192 assigned\n\nIt took me a lot more time for code studies than expected initially. While the code looks really clean, it's not easy to get all details, since it heavily uses modularization and recursions.\n\n### comment:6 Changed 6 years ago by Steffen Hoffmann\n\n(In [9931]) TracFormsPlugin: Allow non-ASCII characters in text input and other fields, refs #3741.\n\nX(HT)ML conform unicode character escaping has been largely inspired by\n\n### comment:7 Changed 6 years ago by Steffen Hoffmann\n\n(In [9967]) TracFormsPlugin: Move XML unicode handling into dedicated script, refs #3741.\n\nMinor code cleanup and preparation for ongoing development started as well.\n\n### comment:8 follow-up: \u00a09 Changed 6 years ago by didley@\u2026\n\nNow it works for me. Thanx a lot.\n\ndidley\n\n### comment:9 in reply to: \u00a08 Changed 6 years ago by Steffen Hoffmann\n\nNow it works for me. Thanx a lot.\n\nGlad to hear that. But as the values are stored unchanged with the numeric unicode-char-escapes I'm still not totally satisfied by this solution. Hence the still open ticket.\n\nIt's just fair to tell you, that the case is not fully resolved from my point of view. Especially regarding #3500 I do suspect, that searching a db full of such escaped strings is not as accessible performance-wise as plain strings. So with the last commit I've already prepared to reverse the escaping process by a profiled algorithm selected from a choice of three. Be prepared to do some db cleanup in case I'll implement this for the storage backend. In fact I've postponed my own production roll-out until I've done the search integration (hopefully by next week).\n\nOTOH it might be possible to escape search strings for the Forms realm as well to obsolete the unescaping process. What would be really needed then, is to provide an abstraction layer, that does handle access to TracForms data in db for all access (i.e. from other plugins too).\n\n### comment:10 Changed 6 years ago by Steffen Hoffmann\n\nBy now it should be quite safe to follow with at least trunk revision [10005], since I'm using that code in production now.\n\nSince I've switched user input handling to use unicode encoding by default, there are few places to worry about, i.e. non-ASCII variables and usernames, but this is not such a big restriction anymore, if it is a real issue at all. Nevertheless I'm looking forwards to a release candidate for a formal 0.3 release, maybe in April.\n\nOnce again, German umlauts work flawlessly, and anyone in need of using similar non-ASCII chars should follow and report his\/her experience, please. Thanks for taking care. I'll close this ticket in a while, if there's no more complaint related to the topic.\n\n### comment:11 Changed 6 years ago by Steffen Hoffmann\n\nResolution: \u2192 fixed assigned \u2192 closed\n\n(In [10143]) TracFormsPlugin: Releasing version 0.3, pushing development to 0.4, closes #3445, #3550, #3741, #4759 and #8258, refs #3388 and #6993.\n\nThis is a major release requiring a Trac environment upgrade.\n\nWhile the parser logic remains unchanged, there is a lot new supplementary funcionality to make TracForms behave more like the existing Trac core resources (ticket, wiki, attachment, ...).\n\nThis version performs a series of non-trivial db schema changes, that especially may leave traces of stale forms (i.e. recorded for wiki pages, that don't exist anymore). So please make sure to read the changelog, BACKUP your environment(s) before installing this version as usual and check the new db tables forms, forms_fields and forms_history after upgrading.\n\n### Modify Ticket\n\nChange Properties","date":"2017-04-23 18:32:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.2522830069065094, \"perplexity\": 6197.025685959467}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-17\/segments\/1492917118740.31\/warc\/CC-MAIN-20170423031158-00142-ip-10-145-167-34.ec2.internal.warc.gz\"}"} | null | null |
RBC Says 5 Top Industrial Stocks Are Potential Takeover Candidates - 24/7 Wall St.
It happens every year, and 2018 won't be any different. Larger companies looking to add to growth, in addition to that of the organic or internal variety, scan the field for purchases and acquisitions that are easy to bolt on and could add returns in a timely fashion. This year the process may even speed up some as last month's market sell-off may have already put some companies in the sights of acquirers.
In what is a yearly and all-encompassing report, the analysts at RBC again go through every sector looking for possible takeover candidates. Last year, the company's screens yielded 20 candidates that eventually were acquired over the following 12 months.
One screen that should be of interest to many investors is the potential takeout candidates in the industrial sector. With the potential for a large infrastructure build-out over the coming years, it makes sense that some of the top industrial companies would look to add additional capabilities that could add to their participation in big contracts, especially from the government.
We cross-referenced the RBC potential buyout candidates, looking for the highest profile names, and found four that like solid choices.
This smaller industrial is a favorite at many of the Wall Street firms we cover. American Axle & Manufacturing Holdings Inc. (NYSE: AXL) is a world leader in the manufacture, engineering, design and validation of driveline and drivetrain systems and related components and modules, chassis systems and metal-formed products for light trucks, sport utility vehicles, passenger cars, crossover vehicles and commercial vehicles.
In addition to locations in the United States (Indiana, Michigan and Ohio), AAM also has offices or facilities in Brazil, China, Germany, India, Japan, Luxembourg, Mexico, Poland, Scotland, South Korea, Sweden and Thailand.
The Wall Street consensus price target is $20.09. The shares traded early Tuesday at $15.40 apiece, in a 52-week trading range of $13.38 to $20.27.
This company used to be owned by General Motors and is one of the hot ideas for a takeover target. Delphi Automotive PLC (NYSE: DLPH) is a global supplier of vehicle electronics, transportation components, integrated systems and modules and other electronic technology.
The company's operating segments include Electronics and Safety, Powertrain Systems, and Electrical/Electronic Architecture. The company is one of the most geographically diversified suppliers in the world, with a goal of generating an equal portion of sales from North America, Europe and Asia and the rest of the world.
Shareholders receive a 1.42% dividend. The posted consensus target price is $59.92. Shares traded at $47.75 Tuesday morning. The 52-week trading range is a huge $38.00 to $104.09.
This somewhat larger cap company has been a rumored takeover candidate for some time. Kennametal Inc. (NYSE: KMT) develops and applies tungsten carbides, ceramics, super-hard materials and solutions for use in metal-cutting and mission-critical wear applications to combat extreme conditions related with wear fatigue, corrosion and high temperatures worldwide. It operates through three segments: Industrial, Widia and Infrastructure.
The company's product offering includes a selection of standard and customized technologies for metalworking applications, such as turning, milling, hole making, tooling systems and services for manufacturers of transportation vehicles and components, machine tools and light and heavy machinery; airframe and aerospace components; and energy-related components for the oil and gas industry, as well as power generation.
The $51.42 consensus price target compares with Tuesday's open near $41.70, as well as a 52-week range of $32.22 to $52.53.
This is another supplier to the automotive industry that can be a very interesting acquisition. Tenneco Inc. (NYSE: TEN) is a producer of clean air and ride performance products and systems for light vehicle, commercial truck, off-highway and other vehicle applications. The company designs, manufactures and distributes highly engineered products for both original equipment vehicle manufacturers (OEMs) and the repair and replacement markets, or aftermarket, across the world.
Tenneco serves both original equipment vehicle designers and manufacturers and the repair and replacement markets, or aftermarket, globally through brands, including Monroe, Rancho, Clevite Elastomers, Axios, Kinetic and Fric-Rot ride performance products and Walker, XNOx, Fonos, DynoMax and Thrush clean air products.
The stock has a consensus price target of $65.67. Shares traded at $53.80, in a 52-week range of $50.73 to $65.59.
This mid-cap company looks like a solid stock to own during difficult times. W.R. Grace & Co. (NYSE: GRA) is engaged in the production and sale of specialty chemicals and specialty materials. The company operates in two segments.
The Grace Catalysts Technologies segment includes catalysts and related products and technologies used in refining, petrochemical and other chemical manufacturing applications.
The Grace Materials Technologies segment includes specialty materials, including silica-based and silica-alumina-based materials, used in coatings, consumer, industrial and pharmaceutical applications.
The company trades basically in line with peers, and its somewhat higher multiple is more than justified by the strong management team and the firm's leadership position in catalysts.
Shareholders receive a 1.2% dividend. The shares traded at $61.26, in a 52-week range of $60.30 to $77.37. The posted consensus price target is $83.
While there is no guarantee that these industrial companies are acquired, they are outstanding stocks to own in aggressive growth portfolios on their own. The buyout factor is just another reason to consider them.
« Are Short Sellers Betting General Electric Will Kill Its Dividend? | {
"redpajama_set_name": "RedPajamaC4"
} | 376 |
Q: Stanford dependency parser can not deal with some Chinese sentences I'm parsing a set of Chinese sentences. Usually Stanford parser works well, but
*
*in special cases, such as '柴油机 可燃混合气 的 形成 和 燃烧 都 是 直接 在 燃烧室内 进行 的 。' and '在 日常 行驶 中 肯定 不 可能 保持 燃油量 的 多少 , 乘客 的 胖 瘦 , 直接 影响 到 前后轴 的 配重 问题 。'. They are well-formed, but NullPointerException in line'List tdl = gs .typedDependenciesCCprocessed();', which copied from Demo.java.
*I notice that even the program runs correctly, the output of dependency parsing misses something, say '[advmod(传统-3, 这种-2), amod(范畴-6, 传统-3), nn(范畴-6, 油门-4), nn(范畴-6, 应用-5), dep(限制-8, 受到-7), root(ROOT-0, 限制-8), dep(精确性-11, 缺乏-10), conj_并(限制-8, 精确性-11), nn(形势-18, 汽车电子技术-16), nn(形势-18, 发展-17), num(电子油门-23, 一-21), dep(一-21, 种-22), dep(egas-25, (-24), dep(电子油门-23, egas-25), dep(egas-25, )-26)]', it can be seen that no '-1', '-9', '-12', '-13', '-14', '-15', '19' in the dependency parsing result. The corresponding original sentence is '但 这种 传统 油门 应用 范畴 受到 限制 并 缺乏 精确性 , 在 日新月异 的 汽车电子技术 发展 形势 下 , 一 种 电子油门 ( egas ) 应运而生 。', if you need.
How to fix them. Thanks.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,897 |
March 27, 1943: 'Blue Ribbon Town' premiered on CBS
March 27, 1943: Blue Ribbon Town, a 30-minute comedy-variety radio series, was first heard on CBS.
Blue Ribbon Town, otherwise known as the "Pabst Blue Ribbon Town," was a 30-minute radio comedy series written by Dick Mack. It starred Groucho Marx, along with other artists like Leo Gorcey, Fay McKenzie, and Virginia O'Brien. It aired on CBS until August 5, 1944.
1946 Great Crepitation Fart Contest
Not for the faint of heart, here is the remarkable 1946 Crepitation (Fart) Contest (part of the 1946 News Broadcasts Collection ). You'll enjoy the fart-off between champion Englishman Lord Windsmear, and challenger, Australian Paul Boomer who had stowed aboard a cabbage freighter. The hilarious comedy recording was apparently created a spoof by two Canadian radio sportscasters in 1946, but this 15 minute recording definitely has some gems in it. Apparently they made several copies, but it was not for distribution. The recording was copied again and again on disc and reel to reel tape. It was distributed underground and played in dark rooms and back alleys around the world. If you cannot see the audio controls, your browser does not support the audio element This recording is available with many other delightful treats on Random Rarities #7 available on MP3 CD , Audio CD , and instant download .
April 11, 1921: The First Lightweight Boxing Match Wireless Broadcast
April 11, 1921: The first lightweight boxing match on radio between Johnny Ray and Johnny Dundee was broadcasted live on this day through KDKA, Pittsburgh with sport writer Florent Gibson as announcer. That was a Radio station KDKA, Pittsburgh completed broadcast of a sport event that happened on April 11, 1921. Florent Gibson, Pittsburgh Post sports editor, presented commentary along ten rounds the fight live on the air from the ringside of Pittsburgh's Motor City Square. Although there was no winner of that match, listeners around Pittsburgh, for the first time, enjoyed the wireless broadcast from their radio receiver. See also: Boxing Matches on Old Radio Cat
June 13, 1913 Bob Bailey (You can hear him as Johnny Dollar) was born
On this day in 1913, Bob Bailey was born. | {
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THE
FRONT YARD
AND
OTHER
ITALIAN STORIES
CONSTANCE
FENIMORE
WOOLSON
[Illustration: image of the book's cover]
[Illustration:
Page 202
"'MADEMOISELLE NEED GIVE HERSELF NO UNEASINESS'"]
THE FRONT YARD
AND
OTHER ITALIAN STORIES
BY
CONSTANCE FENIMORE WOOLSON
AUTHOR OF "ANNE" "HORACE CHASE" ETC.
ILLUSTRATED
[Illustration: colophon]
NEW YORK
HARPER & BROTHERS PUBLISHERS
1895
Copyright, 1895, by HARPER & BROTHERS.
_All rights reserved._
NOTE
Of the stories contained in this volume, "In Venice" was originally
published in the _Atlantic Monthly_, "The Street of the Hyacinth" in the
_Century Magazine_, and the other four stories in _Harper's Magazine_.
CONTENTS
PAGE
THE FRONT YARD 1
NEPTUNE'S SHORE 50
A PINK VILLA 91
THE STREET OF THE HYACINTH 137
A CHRISTMAS PARTY 194
IN VENICE 234
ILLUSTRATIONS
"'MADEMOISELLE NEED GIVE HERSELF NO UNEASINESS'" _Frontispiece_
"''TWOULD BE SOMETHING TO CELEBRATE THE DAY
WITH, THAT WOULD'" _Facing p._ 2
"NOUNCE TOO CAME OUT, AND SAT ON THE WALL NEAR
BY, LISTENING" " 22
"STILL HOLDING NOUNCE'S HAND, SHE WENT ROUND TO
THE FRONT OF THE HOUSE" " 42
"'YOU KNOW I AM YOUR SLAVE'" " 58
AZUBAH ASH " 68
THE OLD WATCH-TOWER " 86
"THE CART WAS GOING SLOWLY ACROSS THE FIELDS,
FOR THE ROAD WAS OVERFLOWED" " 88
"'MRS. CHURCHILL, LET ME PRESENT TO YOU MR. DAVID
ROD'" " 100
SORRENTO " 102
ON THE WAY TO THE DESERTO " 112
AT THE DESERTO " 114
"SHE SAT DOWN AND GATHERED HER CHILD TO HER
BREAST" " 128
"FANNY PUT OUT HER HANDS WITH A BITTER CRY" " 134
"A SMALL CHILD PERCHED ON EACH OF HIS SHOULDERS" " 214
THE FRONT YARD
"Well, now, with Gooster at work in the per-dairy, and Bepper settled at
last as help in a good family, and Parlo and Squawly gone to Perugia,
and Soonter taken by the nuns, and Jo Vanny learning the carpenter's
trade, and only Nounce left for me to see to (let alone Granmar, of
course, and Pipper and old Patro), it doos seem, it really doos, as if I
might get it done _sometime_; say next Fourth of July, now; that's only
ten months off. 'Twould be something to celebrate the day with, that
would; something like!"
The woman through whose mind these thoughts were passing was sitting on
a low stone-wall, a bundle of herbs, a fagot of twigs, and a sickle laid
carefully beside her. On her back was strapped a large deep basket,
almost as long as herself; she had loosened the straps so that she could
sit down. This basket was heavy; one could tell that from the relaxed
droop of her shoulders relieved from its weight for the moment, as its
end rested on a fallen block on the other side of the wall. Her feet
were bare, her dress a narrow cotton gown, covered in front to the hem
by a dark cotton apron; on her head was a straw bonnet, which had behind
a little cape of brown ribbon three inches deep, and in front broad
strings of the same brown, carefully tied in a bow, with the loops
pulled out to their full width and pinned on each side of her chin.
This bonnet, very clean and decent (the ribbons had evidently been
washed more than once), was of old-fashioned shape, projecting beyond
the wearer's forehead and cheeks. Within its tube her face could be
seen, with its deeply browned skin, its large irregular features,
smooth, thin white hair, and blue eyes, still bright, set amid a bed of
wrinkles. She was sixty years old, tall and broad-shouldered. She had
once been remarkably erect and strong. This strength had been consumed
more by constant toil than by the approach of old age; it was not all
gone yet; the great basket showed that. In addition, her eyes spoke a
language which told of energy that would last as long as her breath.
These eyes were fixed now upon a low building that stood at a little
distance directly across the path. It was small and ancient, built of
stone, with a sloping roof and black door. There were no windows;
through this door entered the only light and air. Outside were two large
heaps of refuse, one of which had been there so long that thick matted
herbage was growing vigorously over its top. Bars guarded the entrance;
it was impossible to see what was within. But the woman knew without
seeing; she always knew. It had been a cow; it had been goats; it had
been pigs, and then goats again; for the past two years it had been pigs
steadily--always pigs. Her eyes were fixed upon this door as if held
there by a magnet; her mouth fell open a little as she gazed; her hands
lay loose in her lap. There was nothing new in the picture, certainly.
But the intensity of her feeling made it in one way always new. If love
wakes freshly every morning, so does hate, and Prudence Wilkin had
hated that cow-shed for years.
[Illustration: "''TWOULD BE SOMETHING TO CELEBRATE THE DAY WITH, THAT
WOULD'"]
The bells down in the town began to ring the Angelus. She woke from her
reverie, rebuckled the straps of the basket, and adjusting it by a jerk
of her shoulders in its place on her back, she took the fagot in one
hand, the bundle of herbs in the other, and carrying the sickle under
her arm, toiled slowly up the ascent, going round the cow-shed, as the
interrupted path too went round it, in an unpaved, provisional sort of
way (which had, however, lasted fifty years), and giving a wave of her
herbs towards the offending black door as she passed--a gesture that was
almost triumphant. "Jest you wait till next Fourth of July, you indecent
old Antiquity, you!" This is what she was thinking.
Prudence Wilkin's idea of Antiquity was everything that was old and
dirty; indecent Antiquity meant the same qualities increased to a degree
that was monstrous, a degree that the most profligate imagination of
Ledham (New Hampshire) would never have been able to conceive. There was
naturally a good deal of this sort of Antiquity in Assisi, her present
abode; it was all she saw when she descended to that picturesque town;
the great triple church of St. Francis she never entered; the
magnificent view of the valley, the serene vast Umbrian plain, she never
noticed; but the steep, narrow streets, with garbage here and there, the
crowding stone houses, centuries old, from whose court-yard doors issued
odors indescribable--these she knew well, and detested with all her
soul. Her deepest degree of loathing, however, was reserved for the
especial Antiquity that blocked her own front path, that elbowed her own
front door, this noisome stable or sty--for it was now one, now the
other--which she had hated and abhorred for sixteen long years.
For it was just sixteen years ago this month since she had first entered
the hill town of St. Francis. She had not entered it alone, but in the
company of a handsome bridegroom, Antonio Guadagni by name, and so happy
was she that everything had seemed to her enchanting--these same steep
streets with their ancient dwellings, the same dirt, the same
yellowness, the same continuous leisure and causeless beatitude. And
when her Tonio took her through the town and up this second ascent to
the squalid little house, where, staring and laughing and crowding
nearer to look at her, she found his family assembled, innumerable
children (they seemed innumerable then), a bedridden grandam, a
disreputable old uncle (who began to compliment her), even this did not
appear a burden, though of course it was a surprise. For Tonio had told
her, sadly, that he was "all alone in the world." It had been one of the
reasons why she had wished to marry him--that she might make a home for
so desolate a man.
The home was already made, and it was somewhat full. Desolate Tonio
explained, with shouts of laughter, in which all the assemblage joined,
that seven of the children were his, the eighth being an orphan nephew
left to his care; his wife had died eight months before, and this was
her grandmother--on the bed there; this her good old uncle, a very
accomplished man, who had written sonnets. Mrs. Guadagni number two had
excellent powers of vision, but she was never able to discover the
goodness of this accomplished uncle; it was a quality which, like the
beneficence of angels, one is obliged to take on trust.
She was forty-five, a New England woman, with some small savings, who
had come to Italy as companion and attendant to a distant cousin, an
invalid with money. The cousin had died suddenly at Perugia, and
Prudence had allowed the chance of returning to Ledham with her effects
to pass by unnoticed--a remarkable lapse of the quality of which her
first name was the exponent, regarding which her whole life hitherto had
been one sharply outlined example. This lapse was due to her having
already become the captive of this handsome, this irresistible, this
wholly unexpected Tonio, who was serving as waiter in the Perugian inn.
Divining her savings, and seeing with his own eyes her wonderful
strength and energy, this good-natured reprobate had made love to her a
little in the facile Italian way, and the poor plain simple-hearted
spinster, to whom no one had ever spoken a word of gallantry in all her
life before, had been completely swept off her balance by the novelty of
it, and by the thronging new sensations which his few English words, his
speaking dark eyes, and ardent entreaties roused in her maiden breast.
It was her one moment of madness (who has not had one?). She married
him, marvelling a little inwardly when he required her to walk to
Assisi, but content to walk to China if that should be his pleasure.
When she reached the squalid house on the height and saw its crowd of
occupants, when her own money was demanded to send down to Assisi to
purchase the wedding dinner, then she understood--why they had walked.
But she never understood anything else. She never permitted herself to
understand. Tonio, plump and idle, enjoyed a year of paradisiacal
opulence under her ministrations (and in spite of some of them); he was
eighteen years younger than she was; it was natural that he should wish
to enjoy on a larger scale than hers--so he told her. At the end of
twelve months a fever carried him off, and his widow, who mourned for
him with all her heart, was left to face the world with the eight
children, the grandmother, the good old uncle, and whatever courage she
was able to muster after counting over and over the eighty-five dollars
that alone remained to her of the six hundred she had brought him.
Of course she could have gone back to her own country. But that idea
never once occurred to her; she had married Tonio for better or worse;
she could not in honor desert the worst now that it had come. It had
come in force; on the very day of the funeral she had been obliged to
work eight hours; on every day that had followed through all these
years, the hours had been on an average fourteen; sometimes more.
Bent under her basket, the widow now arrived at the back door of her
home. It was a small narrow house, built of rough stones plastered over
and painted bright yellow. But though thus gay without, it was dark
within; the few windows were very small, and their four little panes of
thick glass were covered with an iron grating; there was no elevation
above the ground, the brick floor inside being of the same level as the
flagging of the path without, so that there was always a sense of
groping when one entered the low door. There were but four rooms, the
kitchen, with a bedroom opening from it, and two chambers above under
the sloping roof.
Prudence unstrapped her basket and placed it in a wood-shed which she
had constructed with her own hands. For she could not comprehend a house
without a wood-shed; she called it a wood-shed, though there was very
little wood to put in it: in Assisi no one made a fire for warmth; for
cooking they burned twigs. She hung up the fagot (it was a fagot of
twigs), the herbs, and the sickle; then, after giving her narrow skirts
a shake, she entered the kitchen.
There was a bed in this room. Granmar would not allow it to be moved
elsewhere; her bed had always been in the kitchen, and in the kitchen it
should remain; no one but Denza, indeed, would wish to shove her off;
Annunziata had liked to have her dear old granmar there, where she could
see for herself that she was having everything she needed; but
Annunziata had been an angel of goodness, as well as of the dearest
beauty; whereas Denza--but any one could see what Denza was! As
Granmar's tongue was decidedly a thing to be reckoned with, her bed
remained where it always had been; from its comfortable cleanliness the
old creature could overlook and criticise to her heart's content the
entire household economy of Annunziata's successor. Not only the
kitchen, but the whole house and garden, had been vigorously purified by
this successor; single-handed she had attacked and carried away
accumulations which had been there since Columbus discovered America.
Even Granmar was rescued from her squalor and coaxed to wear a clean cap
and neat little shawl, her withered brown hands reposing meanwhile upon
a sheet which, though coarse, was spotless.
Granmar was a very terrible old woman; she had a beak-like nose, round
glittering black eyes set in broad circles of yellow wrinkles, no mouth
to speak of, and a receding chin; her voice was now a gruff bass, now a
shrill yell.
"How late you are! you do it on purpose," she said as Prudence entered.
"And me--as haven't had a thing I've wanted since you went away hours
upon hours ago. Nunziata there has been as stupid as a stone--behold
her!"
She spoke in peasant Italian, a tongue which Mrs. Guadagni the second
(called Denza by the family, from Prudenza, the Italian form of her
first name) now spoke readily enough, though after a fashion of her own.
She remained always convinced that Italian was simply lunatic English,
English spoiled. One of the children, named Pasquale, she called
Squawly, and she always believed that the title came from the strength
of his infant lungs; many other words impressed her in the same way.
She now made no reply to Granmar's complaints save to give one
business-like look towards the bed to see whether the pillows were
properly adjusted for the old creature's comfort; then she crossed the
room towards the stove, a large ancient construction of bricks, with two
or three small depressions over which an iron pot could be set.
"Well, Nounce," she said to a girl who was sitting there on a little
bench. The tone of her voice was kindly; she looked to see if a fire had
been made. A few coals smouldered in one of the holes. "Good girl," said
Prudence, commendingly.
"Oh, very good!" cried Granmar from the bed--"very good, when I told her
forty times, and fifty, to make me an omelet, a wee fat one with a drop
of fig in it, and I so faint, and she wouldn't, the snake! she wouldn't,
the toad!--toadest of toads!"
The dark eyes of the girl turned slowly towards Prudence. Prudence, as
she busied herself with the coals, gave her a little nod of approbation,
which Granmar could not see. The girl looked pleased for a moment; then
her face sank into immobility again. She was not an idiot, but wanting,
as it was called; a delicate, pretty young creature, who, with her
cousin Pippo, had been only a year old when the second wife came to
Assisi. It was impossible for any one to be fond of Pippo, who even at
that age had been selfish and gluttonous to an abnormal degree; but
Prudence had learned to love the helpless little girl committed to her
care, as she had also learned to love very dearly the child's brother
Giovanni, who was but a year older; they had been but babies, both of
them. The girl was now seventeen. Her name was Annunziata, but Prudence
called her Nounce. "If it means 'Announce,' Nounce is near enough, I
guess," she said to herself, aggressively. The truth was that she hated
the name; it had belonged to Tonio's first wife, and of the memory of
that comely young mother, poor Prudence, with her sixty years, her white
hair, and wrinkled skin, was burningly jealous even now. Giovanni's name
she pronounced as though it were two words--Jo Vanny; she really thought
there were two. Jo she knew well, of course; it was a good New England
name; Vanny was probably some senseless Italian addition. The name of
the eldest son, Augusto, became on her lips Gooster; Paolo was Parlo,
Assunta was Soonter.
The nuns had finally taken Soonter. The step-mother had been unable to
conceal from herself her own profound relief. True, the girl had gone to
a "papish" convent; but she had always been a mystery in the house, and
the constant presence of a mystery is particularly trying to the New
England mind. Soonter spent hours in meditation; she was very quiet; she
believed that she saw angels; her face wore often a far-away smile.
On this September evening she prepared a heavily abundant supper for
Granmar, and a simple one for Nounce, who ate at any time hardly more
than a bird; Granmar, on the contrary, was gifted with an appetite of
extraordinary capacities, the amount of food which was necessary to keep
her, not in good-humor (she was never in good-humor), but in passable
bodily tranquillity, through the twenty-four hours being equal to that
which would have been required (so Prudence often thought) for three
hearty New England harvesters at home. Not that Granmar would touch New
England food; none of the family would eat the home dishes which
Prudence in the earlier years had hopefully tried to prepare from such
materials as seemed to her the least "onreasonable"; Granmar, indeed,
had declared each and all fit only for the hogs. Prudence never tried
them now, and she had learned the art of Italian cooking; for she felt
that she could not afford to make anything that was to be for herself
alone; the handful of precious twigs must serve for the family as a
whole. But every now and then, in spite of her natural abstemiousness,
she would be haunted by a vision of a "boiled dinner," the boiled
corned-beef, the boiled cabbage, turnips, and potatoes, and the boiled
Indian pudding of her youth. She should never taste these dainties on
earth again. More than once she caught herself hoping that at least the
aroma of them would be given to her some time in heaven.
When Granmar was gorged she became temporarily more tranquil. Prudence
took this time to speak of a plan which she had had in her mind for
several days. "Now that Gooster and the other boys are doing for
themselves, Granmar, and Bepper too at last, and Jo Vanny only needing a
trifle of help now and then (he's so young yet, you know), I feel as
though I might be earning more money," she began.
"Money's a very good thing; we've never had half enough since my sainted
Annunziata joined the angels," responded Granmar, with a pious air.
"Well, it seems a good time to try and earn some more. Soonter's gone to
the convent; and as it's a long while since Pipper's been here, I really
begin to think he has gone off to get work somewhere, as he always said he
was going to."
"Don't you be too sure of Pippo," said Granmar, shaking her owl-like
head ominously.
"'Tany rate he hasn't been here, and I always try to hope the best about
him--"
"And _that's_ what you call the best?" interrupted Granmar, with one of
her sudden flank movements, "to have him gone away off no one knows
where--Annunziata's own precious little nephew--taken by the
pirates--yam! Sold as a slave--yam! Killed in the war! Oh, Pippo! poor
Pippo! poor little Pipp, Pipp, Pipp!"
"And so I thought I'd try to go to the shop by the day," Prudence went
on, when this yell had ceased; "they want me to come and cut out. I
shouldn't go until after your breakfast, of course; and I could leave
cold things out, and Nounce would cook you something hot at noon; then I
should be home myself every night in time to get your supper."
"And so that's the plan--I'm to be left alone here with an idiot while
you go flouncing your heels round Assisi! Flounce, cat! It's a wonder
the dead don't rise in their graves to hear it. But we buried my
Annunziata too deep for that--yam!--otherwise she'd 'a been here to tear
your eyes out. An old woman left to starve alone, her own precious
grandmother, growing weaker and weaker, and pining and pining. Blessed
stomach, do you hear--do you hear, my holy, blessed stomach, always
asking for so little, and now not even to get that? It's turned all a
mumble of cold just thinking of it--yam! I, poor sufferer, who have had
to stand your ugly face so long--I _so_ fond of beauty! You haven't got
but twenty-four hairs now; you know you haven't--yam! I've got more than
you twenty times over--hey! _that_ I have." And Granmar, tearing off her
cap, pulled loose her coarse white hair, and grasping the ends of the
long locks with her crooked fingers, threw them aloft with a series of
shrill halloos.
"I won't go to the shop," said Prudence. "Mercy on us, what a noise! I
say I won't go to the shop. There! do you hear?"
"Will you be here every day of your life at twelve o'clock to cook me
something that won't poison me?" demanded Granmar, still hallooing.
"Yes, yes, I promise you."
Even Granmar believed Prudence's yes; her yea was yea and her nay nay to
all the family. "You cook me something this very minute," she said,
sullenly, putting on her cap askew.
"Why, you've only just got through your supper!" exclaimed Prudence,
astonished, used though she was to Granmar's abdominal capacities, by
this sudden demand.
"You won't? Then I'll yell again," said Granmar. And yell she did.
"Hold up--do; I believe you now," said Prudence. She fanned the dying
coals with a straw fan, made up the fire, and prepared some
griddle-cakes. Granmar demanded fig syrup to eat with them; and devoured
six. Filled to repletion, she then suffered Prudence to change her day
cap for a nightcap, falling asleep almost before her head touched the
pillow.
During this scene Nounce had sat quietly in her corner. Prudence now
went to her to see if she was frightened, for the girl was sometimes
much terrified by Granmar's outcries; she stroked her soft hair. She was
always looking for signs of intelligence in Nounce, and fancying that
she discovered them. Taking the girl's hand, she went with her to the
next room, where were their two narrow pallet beds. "You were very smart
to save the eggs for me to-day when Granmar wanted that omerlet," she
whispered, as she helped her to undress.
Memory came back to Nounce; she smiled comprehendingly.
Prudence waited until she was in bed; then she kissed her good-night,
and put out the candle.
Her two charges asleep, Mrs. Guadagni the second opened the back door
softly and went out. It was not yet nine o'clock, a warm dark night;
though still September, the odors of autumn were already in the air,
coming from the September flowers, which have a pungency mingled with
their perfume, from the rank ripeness of the vegetables, from the aroma
of the ground after the first rains.
"I could have made thirty cents a week more at the shop," she said to
herself, regretfully (she always translated the Italian money into
American or French). "In a month that would have been a dollar and
twenty cents! Well, there's no use thinking about it sence I can't go."
She bent over her vegetables, feeling of their leaves, and estimating
anew how many she could afford to sell, now that the family was so much
reduced in size. Then she paid a visit to her fig-trees. She had planted
these trees herself, and watched over their infancy with anxious care;
at the present moment they were loaded with fruit, and it seemed as if
she knew the position of each fig, so many times had she stood under the
boughs looking up at the slowly swelling bulbs. She had never before
been able to sell the fruit. But now she should be able, and the sale
would add a good many cents to the store of savings kept in her
work-box. This work-box, a possession of her youth, was lined with vivid
green paper, and had a lithograph of the Honorable Mrs. Norton
(taken as a Muse) on the inside of the cover; it held already three
francs and a half, that is seventy cents--an excellent sum when one
considered that only three weeks had passed since the happy day when she
had at last beheld the way open to saving regularly, laying by
regularly; many times had she begun to save, but she had never been able
to continue it. Now, with this small household, she should be able to
continue. The sale of the figs would probably double the savings already
in the work-box; she might even get eighty cents for them; and that
would make a dollar and fifty cents in all! A fig fell to the ground.
"They're ripe," she thought; "they must be picked to-morrow." She felt
for the fallen fig in the darkness, and carrying it to the garden wall,
placed it in a dry niche where it would keep its freshness until she
could send it to town with the rest. Then she went to the hen-house.
"Smart of Nounce to save the eggs for me," she thought, laughing
delightedly to herself over this proof of the girl's intelligence.
"Granmar didn't need that omerlet one bit; I left out two tremenjous
lunches for her." She peered in; but could not see the hens in the
darkness. "If Granmar'd only eat the things we do!" her thoughts went
on. "But she's always possessed after everything that takes eggs. And
then she wants the very best coffee, and white sugar, and the best wine,
and fine flour and meal and oil--my! how much oil! But I wonder if _I_
couldn't stop eating something or other, steader pestering myself about
her? Let's see. I don't take wine nor coffee, so I can't stop them; but
I could stop soup meat, just for myself; and I will." Thus meditating,
she went slowly round to the open space before the house.
To call it a space was a misnomer. The house stood at the apex of the
hill, and its garden by right extended as far down the descent in front
as it extended down the opposite descent behind, where Prudence had
planted her long rows of vegetables. But in this front space, not ten
feet distant from the house door, planted directly across the paved path
which came up from below, was the cow-shed, the intruding offensive
neighbor whose odors, gruntings (for it was now a pig-sty), and refuse
were constantly making themselves perceptible to one sense and another
through the open windows of the dwelling behind. For the house had no
back windows; the small apertures which passed for windows were all in
front; in that climate it was impossible that they should be always
closed. How those odors choked Prudence Wilkin! It seemed as if she
could not respect herself while obliged to breathe them, as if she had
not respected herself (in the true Ledham way) since the pig-sty became
her neighbor.
For fifty francs the owners would take it away; for another twenty or
thirty she could have "a front yard." But though she had made many
beginnings, she had never been able to save a tenth of the sum. None of
the family shared her feelings in the least; to spend precious money for
such a whim as that--only an American could be capable of it; but then,
as everybody knew, most Americans were mad. And why should Denza object
to pigs?
Prudence therefore had been obliged to keep her longings to herself. But
this had only intensified them. And now when at last, after thinking of
it for sixteen years, she was free to begin to save daily and regularly,
she saw as in a vision her front yard completed as she would like to
have it: the cow-shed gone; "a nice straight path going down to the
front gate, set in a new paling fence; along the sides currant bushes;
and in the open spaces to the right and left a big flowerin'
shrub--snowballs, or Missouri currant; near the house a clump of
matrimony, perhaps; and in the flower beds on each side of the path
bachelor's-buttons, Chiny-asters, lady's-slippers, and pinks; the edges
bordered with box." She heaved a sigh of deep satisfaction as she
finished her mental review. But it was hardly mental after all; she saw
the gate, she saw the straight path, she saw the currant bushes and the
box-bordered flower beds as distinctly as though they had really been
there.
Cheered, almost joyous, she went within, locking the door behind her;
then, after softly placing the usual store of provisions beside
Granmar's bed (for Granmar had a habit of waking in the night to eat),
she sought her own couch. It was hard, but she stretched herself upon it
luxuriously. "The figs'll double the money," she thought, "and by this
time to-morrow I shall have a dollar and forty cents; mebby a dollar
fifty!" She fell asleep happily.
Her contentment made her sleep soundly. Still it was not long after dawn
when she hurried down the hill to the town to get her supply of work
from the shop. Hastening back with it, she found Granmar clamoring for
her coffee, and Nounce, neatly dressed and clean (for so much Prudence
had succeeded in teaching her), sitting patiently in her corner.
Prudence's mind was full of a sale she had made; but she prepared the
coffee and Nounce's broth with her usual care; she washed her dishes,
and made Granmar tidy for the day; finally she arranged all her sewing
implements on the table by the window beside her pile of work. Now she
could give herself the luxury of one last look, one last estimate; for
she had made a miracle of a bargain for her figs. By ten o'clock the men
would be up to gather them.
It was a hazy morning; butterflies danced before her as she hastened
towards the loaded trees. Reaching them, she looked up. The boughs were
bare. All the figs had been gathered in the night, or at earliest dawn.
"Pipper!" she murmured to herself.
The ground under the trees was trampled.
Seven weeks later, on the 16th of November, this same Prudence was
adding to her secreted store the fifteen cents needed to make the sum
ten francs exactly--that is, two dollars. "Ten francs, a fifth of the
whole! It seems 'most too lucky that I've got on so well, spite of
Pipper's taking the figs. If I can keep along this way, it'll _all_ be
done by the Fourth of July; not just the cow-shed taken away, but the
front yard done too. My!" She sat down on a fagot to think it over. The
thought was rapture; she laughed to herself and at herself for being so
happy.
Some one called, "Mamma." She came out, and found Jo Vanny looking for
her. Nounce and Jo Vanny were the only ones among the children who had
ever called her mother.
"Oh, you're up there in the shed, are you?" said Jo Vanny. "Somehow,
mamma, you look very gay."
"Yes, I'm gay," answered Prudence. "Perhaps some of these days I'll tell
you why." In her heart she thought: "Jo Vanny, now, _he'd_ understand;
he'd feel as I do if I should explain it to him. A nice front yard he
has never seen in all his life, for they don't have 'em _here_. But once
he knew what it was, he'd care about it as much as I do; I know he
would. He's sort of American, anyhow." It was the highest praise she
could give. The boy had his cap off; she smoothed his hair. "'Pears to
me you must have lost your comb," she said.
"I'm going to have it all cut off as short as can be," announced Jo
Vanny, with a resolute air.
"Oh no."
"Yes, I am. Some of the other fellows have had theirs cut that way, and
I'm going to, too," pursued the young stoic.
He was eighteen, rather undersized and slender, handsome as to his face,
with large dark long-lashed eyes, well-cut features, white teeth, and
the curly hair which Prudence had smoothed. Though he had vowed them to
destruction, these love-locks were for the present arranged in the style
most approved in Assisi, one thick glossy flake being brought down low
over the forehead, so that it showed under his cap in a sentimental
wave. He did not look much like a hard-working carpenter as he stood
there dressed in dark clothes made in that singular exaggeration of the
fashions which one sees only in Italy. His trousers, small at the knee,
were large and wing-like at the ankle, half covering the tight shabby
shoes run down at the heel and absurdly short, which, however, as they
were made of patent-leather and sharply pointed at the toes, Jo Vanny
considered shoes of gala aspect. His low flaring collar was surrounded
by a red-satin cravat ornamented by a gilt horseshoe. He wore a ring on
the little finger of each hand. In his own eyes his attire was splendid.
In the eyes of some one else also. To Prudence, as he stood there, he
looked absolutely beautiful; she felt all a mother's pride rise in her
heart as she surveyed him. But she must not let him see it, and she must
scold him for wearing his best clothes every day.
"I didn't know it was a festa," she began.
"'Tain't. But one of the fellows has had a sister married, and they've
invited us all to a big supper to-night."
"To-night isn't to-day, that I know of."
"Do you wish me to go all covered with sawdust?" said the little dandy,
with a disdainful air. "Besides, I wanted to come up here."
"It is a good while sence we've seen you," Prudence admitted. In her
heart she was delighted that he had wished to come. "Have you had your
dinner, Jo Vanny?"
"All I want. I'll take a bit of bread and some wine by-and-by. But you
needn't go to cooking for me, mamma. I say, tell me what it was that
made you look so glad?" said the boy, curiously.
"Never you mind _now_," said Prudence, the gleam of content coming again
into her eyes, and lighting up her brown, wrinkled face. She was glad
that she had the ten francs; she was glad to see the boy; she was
touched by his unselfishness in declining her offer of a second dinner.
No other member of the family would have declined or waited to decline;
the others would have demanded some freshly cooked dish immediately upon
entering; Uncle Patro would have demanded three or four.
"I've brought my mandolin," Jo Vanny went on. "I've got to take it to
the supper, of course, because they always want me to sing--I never can
get rid of 'em! And so you can hear me, if you like. I know the new
songs, and one of them I composed myself. Well, it's rather heavenly."
All Tonio's children sang like birds. Poor Prudence, who had no ear for
music, had never been able to comprehend either the pleasure or the
profit of the hours they gave to their carollings. But when, in his
turn, her little Jo Vanny began his pipings, then she listened, or tried
to listen. "Real purty, Jo Vanny," she would say, when the silence of a
moment or two had assured her that his song was ended; it was her only
way of knowing--the silence.
So now she brought her work out to the garden, and sewed busily while Jo
Vanny sang and thrummed. Nounce, too, came out, and sat on the wall near
by, listening.
At length the little singer took himself off--took himself off with his
red-satin cravat, his horseshoe pin, and his mandolin under his arm.
Nounce went back to the house, but Prudence sat awhile longer, using, as
she always did, the very last rays of the sunset light for her sewing.
After a while she heard a step, and looked up. "Why, Gooster!--anything
the matter?" she said, in surprise.
Unlike the slender little Jo Vanny, Gooster was a large, stoutly built
young man, as slow in his motions as Jo Vanny was quick. He was a
lethargic fellow with sombre eyes, eyes which sometimes had a gleam in
them.
"There's nothing especial the matter," he answered, dully. "I think I'll
go for a soldier, Denza."
"Go for a soldier? And the per-dairy?"
"I can't never go back to the podere. _She's_ there, and she has taken
up with Matteo. I've had my heart trampled upon, and so I've got a big
hankering either to kill somebody or get killed myself; and I'll either
do it here, or I'll go for a soldier and get knifed in the war."
"Mercy on us! there isn't any war now," said Prudence, dazed by these
sanguinary suggestions.
"There's always a war. What else are there soldiers for? And there's
lots of soldiers. But I could get knifed here easy enough; Matteo and
I--already we've had one tussle; I gave him a pretty big cut, you may
depend."
Seventeen years earlier Prudence Wilkin would have laughed at the idea
of being frightened by such words as these. But Mrs. Tonio Guadagni had
heard of wild deeds in Assisi, and wilder ones still among the peasants
of the hill country roundabout; these singing, indolent Umbrians dealt
sometimes in revenges that were very direct and primitive.
"You let Matteo alone, Gooster," she said, putting her hand on his arm;
"you go straight over to Perugia and stay there. Perhaps you can get
work where Parlo and Squawly are."
"I shall have it out with Matteo here, or else go for a soldier
to-morrow," answered Gooster, in his lethargic tone.
"Well, go for a soldier, then."
"It don't make much difference to me which I do," Gooster went on, as if
only half awake. "If I go for a soldier, I shall have to get to Florence
somehow, I suppose; I shall have to have ten francs for the railroad."
"Is it ten exactly?" said Prudence. Her mind flew to her work-box, which
held just that sum.
"It's ten."
"Haven't you got any money at all, Gooster?" She meant to help him on
his way; but she thought that she should like to keep, if possible, a
nest-egg to begin with again--say twenty cents, or ten.
Gooster felt in his pockets. "Three soldi," he replied, producing some
copper coins and counting them over.
[Illustration: "NOUNCE TOO CAME OUT, AND SAT ON THE WALL NEAR BY,
LISTENING"]
"And there's nothing due you at the per-dairy?"
There was no necessity for answering such a foolish question as this,
and Gooster did not answer it.
"Well, I will give you the money," said Prudence. "But to-morrow'll do,
won't it? Stay here a day or two, and we'll talk it over."
While she was speaking, Gooster had turned and walked towards the garden
wall. The sight of his back going from her--as though she should never
see it again--threw her into a sudden panic; she ran after him and
seized his arm. "I'll give you the money, Gooster; I told you I would;
I've got it all ready, and it won't take a minute; promise me that you
won't leave this garden till I come back."
Gooster had had no thought of leaving the garden; he had espied a last
bunch of grapes still hanging on the vine, and was going to get it; that
was all. "All right," he said.
Prudence disappeared. He gathered the grapes and began to eat them,
turning over the bunch to see which were best. Before he had finished,
Prudence came back, breathless with the haste she had made. "Here," she
said; "and now you'll go straight to Florence, won't you? There's a
train to-night, very soon now; you must hurry down and take that."
He let her put the money in his coat-pocket while he finished the
grapes. Then he threw the stem carefully over the garden wall.
"And no doubt you'll be a brave soldier," Prudence went on, trying to
speak hopefully. "Brave soldiers are thought a heap of everywhere."
"I don't know as I care what's thought," answered Gooster,
indifferently. He took up his cap and put it on. "Well, good-bye,
Denza. Best wishes to you. Every happiness." He shook hands with her.
Prudence stood waiting where she was for five minutes; then she followed
him. It was already dark; she went down the hill rapidly, and turned
into the narrow main street. A few lamps were lighted. She hastened
onward, hoping every minute to distinguish somewhere in front a tall
figure with slouching gait. At last, where the road turns to begin the
long descent to the plain, she did distinguish it. Yes, that was
certainly Gooster; he was going down the hill towards the railway
station. All was well, then; she could dismiss her anxiety. She returned
through the town. Stopping for a moment at an open space, she gazed down
upon the vast valley, now darkening into night; here suddenly a fear
came over her--he might have turned round and come back! She hurried
through the town a second time, and not meeting him, started down the
hill. The road went down in long zigzags. As she turned each angle she
expected to see him; but she did not see him, and finally she reached
the plain: there were the lights of the station facing her. She drew
near cautiously, nearer and nearer, until, herself unseen in the
darkness, she could peer through the window into the lighted
waiting-room. If he was there, she could see him; but if he was on the
platform on the other side--No; he was there. She drew a long breath of
relief, and stole away.
A short distance up the hill a wheelbarrow loaded with stones had been
left by the side of the road; she sat down on the stones to rest, for
the first time realizing how tired she was. The train came rushing
along; stopped; went on again. She watched it as long as she could see
its lights. Then she rose and turned slowly up the hill, beginning her
long walk home. "My," she thought, "won't Granmar be in a tantrum,
though!"
When she reached the house she made a circuit, and came through the
garden behind towards the back door. "I don't want to see the front yard
_to-night_!" she thought.
But she was rather ashamed of this egotism.
* * * * *
"And they say they'll put me in prison--oh--ow!--an old man, a good old
man, a suffering son of humanity like me!" moaned Uncle Pietro.
"An old man, a good old man, a suffering son of humanity like _him_,"
repeated Granmar, shrilly, proud of this fine language.
Suddenly she brandished her lean arms. "You Denza there, with your
stored-up money made from _my_ starvation--yam!--mine, how dare you be
so silent, figure of a mule? Starvation! yes, indeed. Wait and I'll show
you my arms, Pietro; wait and I'll show you my ribs--yam!"
"You keep yourself covered up, Granmar," said Prudence, tucking her in;
"you'll do yourself a mischief in this cold weather."
"Ahi!" said Granmar, "and do I care? If I could live to see you drowned,
I'd freeze and be glad. Stored-up money! stored-up money!"
"What do you know of my money?" said Prudence. Her voice trembled a
little.
"She confesses it!" announced Granmar, triumphantly.
"An old ma--an," said Pietro, crouching over Nounce's scaldino. "A good
old ma--an. But--accommodate yourself."
Prudence sat down and took up her sewing. "I don't believe they'll put
you in jail at all, Patro," she said; "'twon't do 'em any good, and what
they want is their money. You just go to 'em and say that you'll do
day's work for 'em till it's made up, and they'll let you off, I'll bet.
Nine francs, is it? Well, at half a franc a day you can make it up full
in eighteen days; or call it twenty-four with the festas."
"The Americans are all mercenary," remarked old Pietro, waving his hand
in scorn. "Being themselves always influenced by gain, they cannot
understand lofty motives nor the cold, glittering anger of the nobility.
The Leoncinis are noble; they are of the old Count's blood. They do not
want their money; they want revenge--they want to rack my bones."
Granmar gave a long howl.
"Favor me, my niece, with no more of your mistakes," concluded Pietro,
with dignity.
"I don't believe they'd refuse," said Prudence, unmoved. "I'll go and
ask 'em myself, if you like; that'll be the best way. I'll go right away
now." She began to fold up her work.
At this Pietro, after putting the scaldino safely on the stove, fell
down in a round heap on the floor. Never were limbs so suddenly
contorted and tangled; he clawed the bricks so fiercely with his fingers
that Nounce, frightened, left her bench and ran into the next room.
"What's the matter with you? I never saw such a man," said Prudence,
trying to raise him.
"Let be! let be!" called out Granmar; "it's a stroke; and you've
brought it on, talking to him about working, working all day long like a
horse--a good old man like that."
"I don't believe it's a stroke," said Prudence, still trying to get him
up.
"My opinion is," said Granmar, sinking into sudden calm, "that he will
die in ten minutes--exactly ten."
His face had indeed turned very red.
"Dear me! I suppose I shall have to run down for the doctor," said
Prudence, desisting. "Perhaps he'd ought to be bled."
"You leave the doctor alone, and ease his mind," directed Granmar;
"that's what he needs, sensitive as he is, and poetical too, poor
fellow. You just shout in his ear that you'll pay that money, and you'll
be surprised to see how it'll loosen his joints."
Mrs. Guadagni surveyed the good old uncle for a moment. Then she bent
over him and shouted in his ear, "I'll make you a hot fig-tart right
away now, Patro, if you'll set up."
As she finished these words Granmar threw her scaldino suddenly into the
centre of the kitchen, where it broke with a crash upon the bricks.
"He's going to get up," announced Prudence, triumphantly.
"He isn't any such thing; 'twas the scaldino shook him," responded
Granmar, in a loud, admonitory tone. "He'll never get up again in _this_
world unless you shout in his ear that you'll pay that money."
And in truth Pietro was now more knotted than ever.
At this moment the door opened and Jo Vanny came in. "Why, what's the
matter with uncle?" he said, seeing the figure on the floor. He bent
over him and tried to ease his position.
"It's a stroke," said Granmar, in a soft voice. "It'll soon be over.
Hush! leave him in peace. He's dying; Denza there, she did it."
"They want me to pay the nine francs he has--lost," said Prudence.
"Perhaps you have heard, Jo Vanny, that he has--lost nine francs that
belonged to the Leoncinis? Nine whole francs." She looked at the lad,
and he understood the look; for only the day before she had confided to
him at last her long-cherished dream, and (as she had been sure he
would) he had sympathized with it warmly.
"I declare I wish I had even a franc!" he said, searching his pockets
desperately; "but I've only got a cigarette. Will you try a cigarette,
uncle?" he shouted in the heap's ear.
"Don't you mock him," ordered Granmar (but Jo Vanny had been entirely in
earnest). "He'll die soon, and Denza will be rid of him; that's what she
wants. 'Twill be murder, of course; and he'll haunt us--he's always said
he'd haunt somebody. But _I_ ain't long for this world, so I ain't
disturbed. Heaven's waiting wide open for _me_."
Jo Vanny looked a little frightened. He hesitated a moment, surveying
the motionless Pietro; then he drew Prudence aside. "He's an awful
wicked old man, and might really do it," he whispered; "'specially as
you ain't a Catholic, mamma. I think you'd better give him the money if
it'll stop him off; _I_ don't mind, but it would be bad for you if he
should come rapping on your windows and showing corpse-lights in the
garden by-and-by."
Prudence brought her hands together sharply--a gesture of exasperation.
"He ain't going to die any more than I am," she said. But she knew what
life would be in that house with such a threat hanging over it, even
though the execution were deferred to some vague future time. Angrily
she left the room.
Jo Vanny followed her. "Come along, if you want to," she said, half
impatient, half glad. She felt a sudden desire that some one besides
herself should see the sacrifice, see the actual despoiling of the
little box she had labored to fill. She went to the wood-shed. It was a
gloomy December day, and the vegetables hanging on the walls had a
dreary, stone-like look; she climbed up on a barrel, and removed the hay
which filled a rough shelf; in a niche behind was her work-box; with it
in her hand she climbed down again.
She gave him the box to hold while she counted out the money--nine
francs. "There are twelve in all," she said.
"Then you'll have three left," said Jo Vanny.
"Yes, three." She could not help a sigh of retrospect, the outgoing nine
represented so many long hours of toil.
"Let me put the box back," said the boy. It was quickly and deftly done.
"Never mind about it, mamma," he said, as he jumped down. "_I_'ll help
you to make it up again. I want that front yard as much as you do, now
you've told me about it; I think it will be beautiful."
"Well," said Prudence, "when the flower-beds are all fixed up, and the
new front path and swing gate, it _will_ be kind of nice, I reckon."
"Nice?" said Jo Vanny. "That's not the word. 'Twill be an ecstasy! a
smile! a dream!"
"Bless the boy, what nonsense he talks!" said the step-mother. But she
loved to hear his romantic phrases all the same.
They went back to the kitchen. The sacrifice had now become a cheerful
one. She bent over the heap. "Here's your nine francs, Patro," she
shouted. "Come, now, come!"
Pietro felt the money in his hand. He rose quietly. "I'm nearly killed
with all your yelling," he said. Then he took his hat and left the
house.
"We did yell," said Prudence, picking up the fragments of the broken
scaldino. "I don't quite know why we did."
"Never mind why-ing, but get supper," said Granmar. "Then go down on
your knees and thank the Virgin for giving us such a merciful, mild old
man as Pietro. You brought on his stroke; but what did he do? He just
took what you gave him, and went away so forgivingly--the soul of a
dove, the spice-cake soul!"
* * * * *
In January, the short, sharp winter of Italy had possession of Assisi.
One day towards the last of the month a bitter wind was driving through
the bleak, stony little street, sending clouds of gritty, frozen dust
before it. The dark, fireless dwellings were colder than the outside
air, and the people, swathed in heavy layers of clothing, to which all
sorts of old cloaks and shawls and mufflers had been added, were
standing about near the open doors of their shops and dwellings, various
prominences under apron or coat betraying the hidden scaldino, the
earthen dish which Italians tightly hug in winter with the hope that
the few coals it contains will keep their benumbed fingers warm. All
faces were reddened and frost-bitten. The hands of the children who were
too young to hold a scaldino were purple-black.
Prudence Guadagni, with her great basket strapped on her back, came
along, receiving but two or three greetings as she passed. Few knew her;
fewer still liked her, for was she not a foreigner and a pagan? Besides,
what could you do with a woman who drank water, simple water, like a
toad, and never touched wine--a woman who did not like oil, good, sweet,
wholesome oil! Tonio's children were much commiserated for having fallen
into such hands.
Prudence was dressed as she had been in September, save that she now
wore woollen stockings and coarse shoes, and tightly pinned round her
spare person a large shawl. This shawl (she called it "my Highland
shawl") had come with her from America; it was green in hue, plaided;
she thought it still very handsome. Her step was not as light as it had
been; rheumatism had crippled her sorely.
As she left the town and turned up the hill towards home, some one who
had been waiting there joined her. "Is that you, Bepper? Were you coming
up to the house?" she said.
"Yes," answered Beppa, showing her white teeth in a smile. "I'm bringing
you some news, Denza."
"Well, what is it? I hope you're not going to leave your place?"
"I'm going to leave it, and that's my news: I'm going to be married."
"My! it's sudden, isn't it?" said Prudence, stopping.
"Giuseppe doesn't think it's sudden," said Beppa, laughing and tossing
her head; "he thinks I've been ages making up my mind. Come on, Denza,
do; it's so cold!"
"I don't know Giuseppe, do I?" said Prudence, trudging on again; "I
don't remember the name."
"No; I've never brought him up to the house. But the boys know
him--Paolo and Pasquale; Augusto, too. He's well off, Giuseppe is; he's
got beautiful furniture. He's a first-rate mason, and gets good wages,
so I sha'n't have to work any more--I mean go out to work as I do now."
"Bepper, do you _like_ him?" said Prudence, stopping again. She took
hold of the girl's wrist and held it tightly.
"Of course I like him," said Beppa, freeing herself. "How cold your
hands are, Denza--ugh!"
"You ain't marrying him for his furniture? You love him for himself--and
better than any one else in the whole world?" Prudence went on,
solemnly.
"Oh, how comical you do look, standing there talking about love, with
your white hair and your great big basket!" said Beppa, breaking into
irrepressible laughter. The cold had not made her hideous, as it makes
so many Italians hideous; her face was not empurpled, her fine features
were not swollen. She looked handsome. What was even more attractive on
such a day, she looked warm. As her merriment ceased, a sudden change
came over her. "Sainted Maria! she doubts whether I love him! Love him?
Why, you poor old woman, I'd die for him to-morrow. I'd cut myself in
pieces for him this minute." Her great black eyes gleamed; the color
flamed in her oval cheeks; she gave a rich, angry laugh.
It was impossible to doubt her, and Prudence did not doubt. "Well, I'm
right down glad, Bepper," she said, in a softened tone--"right down
glad, my dear." She was thinking of her own love for the girl's father.
"I was coming up," continued Beppa, "because I thought I'd better talk
it over with you."
"Of course," said Prudence, cordially. "A girl can't get married all
alone; nobody ever heard of that."
"I sha'n't be much alone, for Giuseppe's family's a very big one; too
big, I tell him--ten brothers and sisters. But they're all well off,
that's one comfort. Of course I don't want to shame 'em."
"Of course not," said Prudence, assenting again. Then, with the awakened
memories still stirring in her heart: "It's a pity your father isn't
here now," she said, in a moved tone; "he'd have graced a wedding,
Bepper, he was so handsome." She seldom spoke of Tonio; the subject was
too sacred; but it seemed to her as if she might venture a few words to
this his daughter on the eve of her own marriage.
"Yes, it's a pity, I suppose," answered Beppa. "Still, he would have
been an old man now. And 'tain't likely he would have had a good coat
either--that is, not such a one as I should call good."
"Yes, he would; I'd have made him one," responded Prudence, with a spark
of anger. "This whole basket's full of coats now."
"I know you're wonderful clever with your needle," said the girl,
glancing carelessly at the basket that weighed down her step-mother's
shoulders. "I can't think how you can sew so steadily, year in, year
out; I never could."
"Well, I've had to get stronger spectacles," Prudence confessed. "And
they wouldn't take my old ones in exchange, neither, though they were
perfectly good."
"They're robbers, all of them, at that shop," commented Beppa,
agreeingly.
"Now, about your clothes, Bepper--when are you going to begin? I suppose
you'll come home for a while, so as to have time to do 'em; I can help
you some, and Nounce too; Nounce can sew a little."
"No, I don't think I'll come home; 'twouldn't pay me. About the
clothes--I'm going to buy 'em."
"They won't be half so good," Prudence began. Then she stopped. "I'm
very glad you've got the money laid up, my dear," she said,
commendingly.
"Oh, but I haven't," answered Beppa, laughing. "I want to borrow it of
you; that is what I came up for to-day--to tell you about it."
Prudence, her heart still softened, looked at the handsome girl with
gentle eyes. "Why, of course I'll lend it to you, Bepper," she said.
"How much do you want?"
"All you've got won't be any too much, I reckon," answered Beppa, with
pride. "I shall have to have things nice, you know; I don't want to
shame 'em."
"I've got twenty-five francs," said Prudence; "I mean I've got that
amount saved and put away; 'twas for--for a purpose--something I was
going to do; but 'tain't important; you can have it and welcome." Her
old face, as she said this, looked almost young again. "You see, I'm so
glad to have you happy," she went on. "And I can't help thinking--if
your father had only lived--the first wedding in his family! However,
_I'll_ come--just as though I was your real mother, dear; you sha'n't
miss that. I've got my Sunday gown, and five francs will buy me a pair
of new shoes; I can earn 'em before the day comes, I guess."
"I'm afraid you can't," said Beppa, laughing.
"Why, when's the wedding? Not for two or three weeks, I suppose?"
"It's day after to-morrow," answered Beppa. "Everything's bought, and
all I want is the money to pay for 'em; I knew I could get it of you."
"Dear me! how quick! And these shoes are really too bad; they're clear
wore out, and all the cleaning in the world won't make 'em decent."
"Well, Denza, why do you want to come? You don't know any of Giuseppe's
family. To tell the truth, I never supposed you'd care about coming, and
the table's all planned out for (at Giuseppe's sister's), and there
ain't no place for you."
"And you didn't have one saved?"
"I never thought you'd care to come. You see they're different, they're
all well off, and you don't like people who are well off--who wear nice
clothes. You never wanted us to have nice clothes, and you like to go
barefoot."
"No, I don't!" said Prudence.
"'Tany rate, one would think you did; you always go so in summer. But
even if you had new shoes, none of your clothes would be good enough;
that bonnet, now--"
"My bonnet? Surely my _bonnet's_ good?" said the New England woman; her
voice faltered, she was struck on a tender point.
"Well, people laugh at it," answered Beppa, composedly.
They had now reached the house. "You go in," said Prudence; "I'll come
presently."
She went round to the wood-shed, unstrapped her basket, and set it down;
then she climbed up on the barrel, removed the hay, and took out her
work-box. Emptying its contents into her handkerchief, she descended,
and, standing there, counted the sum--twenty-seven francs, thirty
centimes. "'Twon't be any too much; she don't want to shame 'em." She
made a package of the money with a piece of brown paper, and, entering
the kitchen, she slipped it unobserved into Beppa's hand.
"Seems to me," announced Granmar from the bed, "that when a girl comes
to tell her own precious Granmar of her _wedding_, she ought in decency
to be offered a bite of something to eat. Any one but Denza would think
so. Not that it's anything to me."
"Very well, what will you have?" asked Prudence, wearily. Freed from her
bonnet and shawl, it could be seen that her once strong figure was much
bent; her fingers had grown knotted, enlarged at the joints, and clumsy;
years of toil had not aged her so much as these recent nights--such long
nights!--of cruel rheumatic pain.
Granmar, in a loud voice, immediately named a succulent dish; Prudence
began to prepare it. Before it was ready, Jo Vanny came in.
"You knew I was up here, and you've come mousing up for an invitation,"
said Beppa, in high good-humor. "I was going to stop and invite you on
my way back, Giovanni; there's a nice place saved for you at the
supper."
"Yes, I knew you were up here, and I've brought you a wedding-present,"
answered the boy. "I've brought one for mamma, too." And he produced two
silk handkerchiefs, one of bright colors, the other of darker hue.
"Is the widow going to be married, too?" said Beppa. "Who under heaven's
the man?"
In spite of the jesting, Prudence's face showed that she was pleased;
she passed her toil-worn hand over the handkerchief softly, almost as
though its silk were the cheek of a little child. The improvised feast
was turned into a festival now, and of her own accord she added a second
dish; the party, Granmar at the head, devoured unknown quantities. When
at last there was nothing left, Beppa, carrying her money, departed.
"You know, Jo Vanny, you hadn't ought to leave your work so often," said
Prudence, following the boy into the garden when he took leave; she
spoke in an expostulating tone.
"Oh, I've got money," said Jo Vanny, loftily; "_I_ needn't crawl." And
carelessly he showed her a gold piece.
But this sudden opulence only alarmed the step-mother. "Why, where did
you get that?" she said, anxiously.
"How frightened you look! Your doubts offend me," pursued Jo Vanny,
still with his grand air. "Haven't I capacities?--hasn't Heaven sent me
a swarming genius? Wasn't I the acclaimed, even to laurel crowns, of my
entire class?"
This was true: Jo Vanny was the only one of Tonio's children who had
profited by the new public schools.
"And now what shall I get for you, mamma?" the boy went on, his tone
changing to coaxing; "I want to get you something real nice; what will
you have? A new dress to go to Beppa's wedding in?"
For an instant Prudence's eyes were suffused. "I ain't going, Jo Vanny;
they don't want me."
"They _shall_ want you!" declared Jo Vanny, fiercely.
"I didn't mean that; I don't want to go anyhow; I've got too much
rheumatism. You don't know," she went on, drawn out of herself for a
moment by the need of sympathy--"you don't know how it does grip me at
night sometimes, Jo Vanny! No; you go to the supper, and tell me all
about it afterwards; I like to hear you tell about things just as well
as to go myself."
Jo Vanny passed his hand through his curly locks with an air of
desperation. "There it is again--my gift of relating, of narrative; it
follows me wherever I go. What will become of me with such talents? I
shall never die in my bed; nor have my old age in peace."
"You go 'long!" said Prudence (or its Italian equivalent). She gave him
a push, laughing.
Jo Vanny drew down his cap, put his hands deep in his pockets, and thus
close-reefed scudded down the hill in the freezing wind to the shelter
of the streets below.
By seven o'clock Nounce and Granmar were both asleep; it was the most
comfortable condition in such weather. Prudence adjusted her lamp, put
on her strong spectacles, and sat down to sew. The great brick stove
gave out no warmth; it was not intended to heat the room; its three
yards of length and one yard of breadth had apparently been constructed
for the purpose of holding and heating one iron pot. The scaldino at her
feet did not keep her warm; she put on her Highland shawl. After a
while, as her head (scantily covered with thin white hair) felt the cold
also, she went to get her bonnet. As she took it from the box she
remembered Beppa's speech, and the pang came back; in her own mind that
bonnet had been the one link that still united her with her old Ledham
respectability, the one possession that distinguished her from all these
"papish" peasants, with their bare heads and frowzy hair. It was not
new, of course, as it had come with her from home. But what signified an
old-fashioned shape in a community where there were no shapes of any
kind, new or old? At least it was always a bonnet. She put it on, even
now from habit pulling out the strings carefully, and pinning the loops
on each side of her chin. Then she went back and sat down to her work
again.
At eleven o'clock Granmar woke. "Yam! how cold my legs are! Denza, are
you there? You give me that green shawl of yours directly; precisely, I
am dying."
Prudence came out from behind her screen, lamp in hand. "I've got it on,
Granmar; it's so cold setting up sewing. I'll get you the blanket from
my bed."
"I don't want it; it's as hard as a brick. You give me that shawl; if
you've got it on, it'll be so much the warmer."
"I'll give you my other flannel petticoat," suggested Prudence.
"And I'll tear it into a thousand pieces," responded Granmar,
viciously. "You give me that shawl, or the next time you leave Nounce
alone here, _she_ shall pay for it."
Granmar was capable of frightening poor little Nounce into spasms.
Prudence took off the shawl and spread it over the bed, while Granmar
grinned silently.
Carrying the lamp, Prudence went into the bedroom to see what else she
could find to put on. She first tried the blanket from her bed; but as
it was a very poor one, partly cotton, it was stiff (as Granmar had
said), and would not stay pinned; the motion of her arms in sewing would
constantly loosen it. In the way of wraps, except her shawl, she
possessed almost nothing; so she put on another gown over the one she
wore, pinned her second flannel petticoat round her shoulders, and over
that a little cloak that belonged to Nounce; then she tied a woollen
stocking round her throat, and crowned with her bonnet, and carrying the
blanket to put over her knees, she returned to her work.
"I declare I'm clean tired out," she said to herself; "my feet are like
ice. I wouldn't sew any longer such a bitter night if it warn't that
that work-box 'ain't got a thing in it. I can't bear to think of it
empty. But as soon as I've got a franc or two to begin with again, I'll
stop these extry hours."
But they lasted on this occasion until two o'clock.
* * * * *
"It don't seem as if I'd ever known it _quite_ so baking as it is
to-night." It was Prudence who spoke; she spoke to Nounce; she must
speak to some one.
Nounce answered with one of her patient smiles. She often smiled
patiently, as though it were something which she was expected to do.
Prudence was sitting in the wood-shed resting; she had been down to town
to carry home some work. Now the narrow streets there, thrown into shade
by the high buildings on each side, were a refuge from the heat; now the
dark houses, like burrows, gave relief to eyes blinded by the yellow
glare. It was the 30th of August. From the first day of April the broad
valley and this brown hill had simmered in the hot light, which filled
the heavens and lay over the earth day after day, without a change,
without a cloud, relentless, splendid; each month the ground had grown
warmer and drier, the roads more white, more deep in dust; insect life,
myriad legged and winged, had been everywhere; under the stones lurked
the scorpions.
In former summers here this never-ending light, the long days of burning
sunshine, the nights with the persistent moon, the importunate
nightingales, and the magnificent procession of the stars had sometimes
driven the New England woman almost mad; she had felt as if she must
bury her head in the earth somewhere to find the blessed darkness again,
to feel its cool pressure against her tired eyes. But this year these
things had not troubled her; the possibility of realizing her
long-cherished hope at last had made the time seem short, had made the
heat nothing, the light forgotten; each day, after fifteen hours of
toil, she had been sorry that she could not accomplish more.
But she had accomplished much; the hope was now almost a reality.
"Nounce," she said, "do you know I'm 'most too happy to live. I shall
have to tell you: I've got _all_ the money saved up at last, and the
men are coming to-morrow to take away the cow-shed. Think of that!"
Nounce thought of it; she nodded appreciatively.
Prudence took the girl's slender hand in hers and went on: "Yes,
to-morrow. And it'll cost forty-eight francs. But with the two francs
for wine-money it will come to fifty in all. By this time to-morrow
night it will be gone!" She drew in her breath with a satisfied sound.
"I've got seventy-five francs in all, Nounce. When Bepper married, of
course I knew I couldn't get it done for Fourth of July. And so I
thought I'd try for Thanksgiving--that is, Thanksgiving _time_; I never
know the exact day now. Well, here it's only the last day of August, and
the cow-shed will be gone to-morrow. Then will come the new fence; and
then the fun, the real fun, Nounce, of laying out our front yard! It'll
have a nice straight path down to the gate, currant bushes in neat rows
along the sides, two big flowerin' shrubs, and little flower beds
bordered with box. I tell you you won't know your own house when you
come in a decent gate and up a nice path to the front door; all these
years we've been slinking in and out of a back door, just as though we
didn't have no front one. I don't believe myself in tramping in and out
of a front door _every_ day; but on Sundays, now, when we have on our
best clothes, we shall come in and out respectably. You'll feel like
another person, Nounce; and I'm sure _I_ shall--I shall feel like Ledham
again--my!" And Prudence actually laughed.
Still holding Nounce's hand, she went round to the front of the house.
[Illustration: "STILL HOLDING NOUNCE'S HAND, SHE WENT ROUND TO THE FRONT
OF THE HOUSE"]
The cow-shed was shedding forth its usual odors; Prudence took a stone
and struck a great resounding blow on its side. She struck with so
much force that she hurt her hand. "Never mind--it done me good!" she
said, laughing again.
She took little Nounce by the arm and led her down the descent. "I shall
have to make the front walk all over," she explained. "And here'll be
the gate, down here--a swing one. And the path will go from here
straight up to the door. Then the fence will go along here--palings, you
know, painted white; a good clean American white, with none of these
yellows in it, you may depend. And over there--and there--along the
sides, the fence will be just plain boards, notched at the top; the
currant bushes will run along there. In the middle, here--and here--will
be the big flowerin' shrubs. And then the little flower-beds bordered
with box. Oh, Nounce, I can't hardly believe it--it will be so
beautiful! I really can't!"
Nounce waited a moment. Then she came closer to her step-mother, and
after looking quickly all about her, whispered, "You needn't if you
don't want to; there's here yet to believe."
"It's just as good as here," answered Prudence, almost indignantly.
"I've got the money, and the bargain's all made; nothing could be surer
than that."
The next morning Nounce was awakened by the touch of a hand on her
shoulder. It was her step-mother. "I've got to go down to town," she
said, in a low tone. "You must try to get Granmar's breakfast yourself,
Nounce; do it as well as you can. And--and I've changed my mind about
the front yard; it'll be done some time, but not now. And we won't talk
any more about it for the present, Nounce; that'll please me most; and
you're a good girl, and always want to please me, I know."
She kissed her, and went out softly.
* * * * *
In October three Americans came to Assisi. Two came to sketch the Giotto
frescos in the church of St. Francis; the third came for her own
entertainment; she read Symonds, and wandered about exploring the
ancient town.
One day her wanderings led her to the little Guadagni house on the
height. The back gate was open, and through it she saw an old woman
staggering, then falling, under the weight of a sack of potatoes which
she was trying to carry on her back.
The American rushed in to help her. "It's much too heavy for you," she
said, indignantly, after she had given her assistance. "Oh dear--I mean,
_e troppo grave_," she added, elevating her voice.
"Are you English?" said the old woman. "I'm an American myself; but I
ain't deef. The sack warn't too heavy; it's only that I ain't so strong
as I used to be--it's perfectly redeculous!"
"You're not strong at all," responded the stranger, still indignantly,
looking at the wasted old face and trembling hands.
A week later Prudence was in bed, and an American nurse was in charge.
This nurse, whose name was Baily, was a calm woman with long strong
arms, monotonous voice, and distinct New England pronunciation; her
Italian (which was grammatically correct) was delivered in the vowels of
Vermont.
One day, soon after her arrival, she remarked to Granmar, "That yell of
yours, now--that yam--is a very unusual thing."
"My sufferings draw it from me," answered Granmar, flattered by the
adjective used. "I'm a very pious woman; I don't want to swear."
"I think I have never heard it equalled, except possibly in lunatic
asylums," Marilla Baily went on. "I have had a great deal to do with
lunatic asylums; I am what is called an expert; that is, I find out
people who are troublesome, and send them there; I never say much about
it, but just make my observations; then, when I've got the papers out,
whiff!--off they go."
Granmar put her hand over her mouth apprehensively, and surveyed her in
silence. From that time the atmosphere of the kitchen was remarkably
quiet.
Marilla Baily had come from Florence at the bidding of the American who
had helped to carry the potatoes. This American was staying at the
Albergo del Subasio with her friends who were sketching Giotto; but she
spent most of her time with Prudence Wilkin.
"You see, I minded it because it was _him_," Prudence explained to her
one day, at the close of a long conversation. "For I'd always been so
fond of the boy; I had him first when he warn't but two years old--just
a baby--and _so_ purty and cunning! He always called me mamma--the only
one of the children, 'cept poor Nounce there, that really seemed to care
for me. And I cared everything for him. I went straight down to town and
hunted all over. But he warn't to be found. I tried it the next day, and
the next, not saying what I wanted, of course; but nobody knew where he
was, and at last I made up my mind that he'd gone away. For three weeks
I waited; I was almost dead; I couldn't do nothing; I felt as if I was
broke in two, and only the skin held me together. Every morning I'd say
to myself, 'There'll certainly come a letter to-day, and he'll tell me
all about it.' But the letter didn't come, and didn't come. From the
beginning, of course, I knew it was him--I couldn't help but know; Jo
Vanny was the only person in the whole world that knew where it was. For
I'd showed it to him one day--the work-box, I mean--and let him put it
back in the hole behind the hay--'twas the time I took the money out for
Patro. At last I did get a letter, and he said as how he'd meant to put
it back the very next morning, sure. But something had happened, so he
couldn't, and so he'd gone away. And now he was working just as hard as
he could, he said, so as to be able to pay it back soon; he hardly
played on his mandolin at all now, he said, he was working so hard. You
see, he wasn't bad himself, poor little fellow, but he was led away by
bad men; gambling's an awful thing, once you get started in it, and he
was sort of _drove_ to take that money, meaning all the while to pay it
back. Well, of course I felt ever so much better just as soon as I got
that letter. And I began to work again. But I didn't get on as well as
I'd oughter; I can't understand why. That day, now, when I first saw
you--when you ran in to help me--I hadn't been feeling sick at all;
there warn't no sense in my tumbling down that way all of a sudden."
One lovely afternoon in November Prudence's bed was carried out to the
front of the dark little house.
The cow-shed was gone. A straight path, freshly paved, led down to a
swing gate set in a new paling fence, flower beds bordered the path, and
in the centre of the open spaces on each side there was a large rose
bush. The fence was painted a glittering white; there had been an
attempt at grass; currant bushes in straight rows bordered the two
sides.
Prudence lay looking at it all in peaceful silence. "It's mighty purty,"
she said at last, with grateful emphasis. "It's everything I planned to
have, and a great deal nicer than I could have done it myself, though I
thought about it goodness knows how many years!"
"I'm not surprised that you thought about it," the American answered.
"It was the view you were longing for--fancy its having been cut off so
long by that miserable stable! But now you have it in perfection."
"You mean the view of the garden," said Prudence. "There wasn't much to
look at before; but now it's real sweet."
"No; I mean the great landscape all about us here," responded the
American, surprised. She paused. Then seeing that Prudence did not lift
her eyes, she began to enumerate its features, to point them out with
her folded parasol. "That broad Umbrian plain, Prudence, with those tall
slender trees; the other towns shining on their hills, like Perugia over
there; the gleam of the river; the velvety blue of the mountains; the
color of it all--I do believe it is the very loveliest view in the whole
world!"
"I don't know as I've ever noticed it much--the view," Prudence
answered. She turned her eyes towards the horizon for a moment. "You see
I was always thinking about my front yard."
"The front yard is very nice now," said the American. "I am so glad you
are pleased; we couldn't get snowballs or Missouri currant, so we had to
take roses." She paused; but she could not give up the subject without
one more attempt. "You have probably noticed the view without being
aware of it," she went on; "it is so beautiful that you must have
noticed it. If you should leave it you would find yourself missing it
very much, I dare say."
"Mebbe," responded Prudence. "Still, I ain't so sure. The truth is, I
don't care much for these Eyetalian views; it seems to me a poor sort of
country, and always did." Then, wishing to be more responsive to the
tastes of this new friend, if she could be so honestly, she added, "But
I like views, as a general thing; there was a very purty view from
Sage's Hill, I remember."
"Sage's Hill?"
"Yes; the hill near Ledham. You told me you knew Ledham. You could see
all the fields and medders of Josiah Strong's farm, and Deacon
Mayberry's too; perfectly level, and not a stone in 'em. And the
turnpike for miles and miles, with three toll-gates in sight. Then, on
the other side, there were the factories to make it lively. It was a
sweet view."
A few days afterwards she said: "People tell us that we never get what
we want in this world, don't they? But I'm fortunate. I think I've
always been purty fortunate. I got my front yard, after all."
* * * * *
A week later, when they told her that death was near, "My! I'd no idea I
was so sick as that," she whispered. Then, looking at them anxiously,
"What'll become of Nounce?"
They assured her that Nounce should be provided for. "You know you have
to be sorter patient with her," she explained; "but she's growing
quicker-witted every day."
Later, "I should like so much to see Jo Vanny," she murmured, longingly;
"but of course I can't. You must get Bepper to send him my love, my
dearest, dearest love."
Last of all, as her dulled eyes turned from the little window and rested
upon her friend: "It seems a pity--But perhaps I shall find--"
NEPTUNE'S SHORE
I
Old Mrs. Preston had not been able to endure the hotel at Salerno. She
had therefore taken, for two months, this house on the shore.
"I might as well be here as anywhere, saddled as I am with the
Abercrombies," she remarked to her cousin, Isabella Holland. "Arthur may
really do something: I have hopes of Arthur. But as to Rose, Hildegarde,
and Dorothea, I shall plainly have to drag them about with me, and drag
them about with me, year after year, in the hope that the constant
seeing of so many straight statues, to say nothing of pictures, may at
last teach them to have spines. Here they are now; did you ever see such
shoulders, or rather such a lack of them? Hildegarde, child, come here a
moment," she added, as the three girls drew near. "I have an idea. Don't
you think you could _hold_ your shoulders up a little? Try it now; put
them up high, as though you were shrugging them; and expand your chest
too; you mustn't cramp that. There!--that is what I mean; don't you
think, my dear, that you could keep yourself so?"
Hildegarde, with her shoulders elevated and her long chin run out, began
to blush painfully, until her milk-white face was dyed red. "I am afraid
I could not keep myself so _long_, aunt," she answered, in a low voice.
"Never mind; let them down, then: it's of no use," commented Mrs.
Preston, despairingly. "Go and dance for twenty-five minutes in the
upper hall, all of you. And dance as hard as you can."
The three girls, moving lifelessly, went down the echoing vaulted
corridor. They were sisters, the eldest not quite sixteen, all three
having the same lank figures with sloping shoulders and long thin
throats, and the same curiously white, milk-white skin. Orphans, they
had been sent with their brother Arthur to their aunt, Mrs. Octavia
Preston, five years before, having come to her from one of the West
India Islands, their former home.
"Those girls have done nothing but eat raw meat, take sea baths, and
practise calisthenics and dancing ever since I first took charge of
them," Mrs. Preston was accustomed to remark to intimate friends; "yet
look at them now! Of course I could not send them to school--they would
only grow lanker. So I take them about with me patiently, governess and
all."
But Mrs. Preston was not very patient.
The three girls having disappeared, Isabella thought the occasion
favorable for a few words upon another subject. "Do you like to have
Paulie riding so often with Mr. Ash, Cousin Octavia? I can't help being
distressed about it."
"Don't be Mistering John Ash, I beg; no one in the world but you,
Isabella, would dream of doing it--a great swooping creature like
that--the horseman in 'Heliodorus.'"
"You mean Raphael's fresco? Oh, Cousin Octavia, how can you think so?
Raphael--such a religious painter, and John Ash, who looks so
dissipated!"
"Did I say he didn't look dissipated? I said he could ride. John Ash is
one of the most dissipated-looking youths I have ever met," pursued Mrs.
Preston, comfortably. "The clever sort, not the brutal."
"And you don't mind Paulie's being with him?"
"Pauline Euphemia Graham has been married, Pauline Euphemia Graham is a
widow; it ill becomes those who have not had a tithe of her experience
(though they may be _much_ older) to set themselves up as judges of her
conduct."
Mrs. Preston had a deep rich voice, and slow enunciation; her simplest
sentences, therefore, often took on the tone of declamation, and when
she held forth at any length, it was like a Gregorian chant.
"Oh, I didn't mean to judge, I'm sure," said Isabella; "I only meant
that it would be such a pity--such a bad match for dear Paulie in case
she should be thinking of marrying again. Even if one were sure of John
Ash--and certainly the reverse is the case--look at his mother! I am
interested, naturally, as Paulie is my first cousin, you know."
"Do you mean that your first cousin's becoming Mrs. John Ash might
endanger your own matrimonial prospects?"
"Oh dear no," said poor little Isabella, shrinking back to her
embroidery. She was fifty, small, plain, extremely good. In her heart
she wished that people would take the tone that Isabella had "never
cared to marry."
"Here is Pauline now, I think," said Mrs. Preston, as a figure appeared
at the end of the hall.
Isabella was afraid to add, "And going out to ride again!" But it was
evident that Mrs. Graham intended to ride: she wore her habit.
"I wish you were going, too," she said to Mrs. Preston, pausing in the
doorway with her skirt uplifted. Her graceful figure in the closely
fitting habit was a pleasant sight to see.
"Thanks, my dear; I should enjoy going very much if I were a little more
slender."
"You are magnificent as you are," responded Pauline, admiringly.
And in truth the old lady was very handsome, with her thick silver hair,
fine eyes with heavy black eyebrows, and well-cut aquiline profile. Her
straight back, noble shoulders, and beautiful hands took from her
massive form the idea of unwieldiness.
"Isabella--you who are always posing for enthusiasm--when will you learn
to say anything so genuine as that?" chanted Cousin Octavia's deep
voice. "I mention it merely on your account, as a question of styles
conversational. Here is Isabella, who thinks John Ash so dissipated,
Pauline; she fears that it may injure the family connection if you marry
him. I have told her that no one here was thinking of marrying or of
giving in marriage; if she has such ideas, she must have brought them
with her from Florence. There are a great many old maids in Florence."
"I can only answer for myself: I certainly am not thinking of marriage,"
said Pauline, laughing, as she went down the stairs.
"Oh, Cousin Octavia, you have set Pauline against me!" exclaimed
Isabella, in distress.
"Don't be an idiot; Pauline isn't against any one: she doesn't care
enough about it. She is a good deal for herself, I acknowledge; but
she's not against any one. Pauline bears no malice; she is delightfully
uncertain; she hasn't a theory in the world to live up to; in addition,
to have her in the house is like going to the play all the time--she
_is_ such a stupendous liar!"
Isabella, who was punching round holes in a linen band with an implement
of ivory, stopped punching. "I am sure poor Paulie--"
"Am I to sit through a defence of Pauline Euphemia Graham, born Preston,
at your hands, Isabella? Pray spare me that. I am much more Pauline's
friend than you ever can be. Did I say that she lied? Nature has given
her a face that speaks one language and a mind that speaks another; she,
of course, follows the language of her mind; but others follow that of
her face, and this makes the play. Eh!--what noise is that?"
"We have come to pay you a visit, Aunt Octavia," called a boyish voice;
its owner was evidently mounting the stairs three at a time: now he was
in the room. "They're all down at the door--Freemantle and Gates and
Beckett. And what do you think--we've got Griff!"
"Griff himself?" said Aunt Octavia, benevolently, as the lad, with a
very pretty gallantry, bent to kiss her hand.
"Yes, Griff himself; you may be sure we're drawing like mad. Griff has
come down from Paris for only three weeks, and he says he will go with
us to Paestum, and all about here--to Amalfi, Ravello, and everywhere.
But of course Paestum's the stunner."
"Yes, of course Paestum's the stunner," repeated Aunt Octavia, as if
trying it in Shakespearian tones.
"I say, may they come up?" Arthur went on.
They came up--three boys of seventeen and eighteen, and Griffith Carew,
who was ten years older. These three youths, with Arthur Abercrombie,
were studying architecture at the Beaux-Arts, Paris; this spring they
had given to a tour in Italy for the purpose of making architectural
drawings. Griffith Carew was also an architect, but a full-fledged one.
His indomitable perseverance and painstaking accuracy caused all the
younger men to respect him; the American students went further; they
were sure that Griff had only to "let himself go," and the United States
would bloom from end to end with City Halls of beauty unparalleled. In
the mean time Griff, while waiting for the City Halls perhaps, was so
kind-hearted and jovial and unselfish that they all adored him for that
too. It was a master-treat, therefore, to Arthur and his companions, to
have their paragon to themselves for a while on this temple-haunted
shore.
Griff sat down placidly, and began to talk to Aunt Octavia. He was of
medium height, his figure heavy and strong; he had a dark complexion and
thick features, lighted by pleasant brown eyes, and white teeth that
gleamed when he smiled.
Aunt Octavia was gracious to Griff; she had always distinguished him
from "Arthur's horde." This was not in the least because the horde
considered him the architect of the future. Aunt Octavia did not care
much about the future; her tests were those of the past. She had known
Griff's mother, and the persons whose mothers Aunt Octavia had
known--ah, that was a certificate!
II
In the meanwhile Pauline Graham had left Salerno behind her, and was
flying over the plain with John Ash.
Pauline all her life had had a passion for riding at breakneck speed;
one of the explanations of her fancy for Ash lay in the fact that,
having the same passion himself, he enabled her to gratify her own.
Whenever she had felt in the mood during the past five weeks there had
always been a horse and a mounted escort at her door. Upon this
occasion, after what they called an inspiring ride (to any one else a
series of mad gallops), they had dismounted at a farm-house, and leaving
their horses, had strolled down to the shore. It was a lovely day,
towards the last of March; the sea, of the soft misty blue of the
southern Mediterranean, stretched out before them without a sail; at
their feet the same clear water laved the shore in long smooth wavelets,
hardly a foot high, whose gentle roll upon the sands had an
indescribably caressing sound. There was no one in sight. It is a lonely
coast. Pauline stood, gazing absently over the blue.
"Sit down for a moment," suggested Ash.
"Not now."
"Not now? When do you expect to be here again?"
She came back to the present, laughing. "True; but I did not mean that;
I meant that you were not the ideal companion for sea-side musing; you
never meditate. I venture to say you have never quoted poetry in your
life."
"No; I live my poetry," John Ash responded.
"But for a ride you are perfect; for a rush over the plain, in the teeth
of the wind, I have never had any one approaching you. You are a
cavalier of the gods."
"Have you had many?"
"Cavaliers?--plenty. Of the gods?--no."
"Plenty! I reckon you have," said Ash, half to himself.
"Would you wish me to have had few? You must remember that I have been
in many countries and have seen many peoples. I shouldn't have
appreciated _you_ otherwise; I should have thought you dangerous--horrible!
There is Isabella, who has not been in many countries; Isabella is sure
that you are 'so dissipated.'"
"Dissipated!--mild term!"
"Then you acknowledge it?"
"Freely."
Pauline looked about for a rock of the right height, and finding one,
seated herself, and began to draw off her gloves. "Some time--in some
other existence--will you come and tell me how it has paid you, please?
You are so preternaturally intelligent, and you have such a will of your
own, that you cannot have fallen into it from stupidity, as so many do."
Her gloves off, she began to tighten the braids of her hair, loosened by
the gallop.
"It pays as it goes; it makes one forget for a moment the hideous
tiresomeness of existence. But you put your question off to some other
life; you have no intention, then, of redeeming me in this?"
"I shouldn't succeed. In the first place, I have no influence--"
"You know I am your slave," said Ash; his voice suddenly deepened.
"And how much of a slave shall you be to the next pretty peasant girl
you meet?" Mrs. Graham demanded, turning towards him, both hands still
occupied with her hair.
"I don't deny that. But it has nothing to do with the subject."
"In one way I know it has not," she answered, after she had fastened the
last braid in its place with a long gold pin.
"How right I was to like you! You understand of yourself the thing that
so few women can ever be brought to comprehend. Well, if you acknowledge
that it makes no difference--I mean about the peasant girls--we're just
where we were; I am your slave, yet you have no desire to reclaim me. I
believe you like me better as I am," he added, abruptly.
"Do you want me to tell you that you are impertinent?" demanded Pauline,
with her lovely smile, that always contradicted in its sweetness any
apparent rebuke expressed by her words. "Do I know what you are in
reality, or care to know? I know what you seem, and what you seem is
admirable, perfect, for these rides of ours, the most enchanting rides I
have ever had."
"And the rides are to be the end of it? You wouldn't care for me
elsewhere?"
"Ah!" said Pauline, rising and drawing on her gloves, "you wouldn't care
for _me_. In Paris I am altogether another person; I am not at all as
you see me here. In Paris you would call me a doll. Come, don't dissect
the happy present; enjoy it as I do. 'He only is rich who owns the day,'
and we own this--for our ride."
[Illustration: "'YOU KNOW I AM YOUR SLAVE'"]
"'I hear the hoofs upon the hill;
I hear them fainter, fainter still,'"
she sang in her clear voice. "The idea of that old Virginia song coming
to me here!"
"This talk about reclaiming and reforming is all bosh," remarked Ash,
leaning back against a high fragment of rock, with his hands in his
pockets. "I am what I am because I choose to be, that's all. The usual
successes of American life, what are they? I no longer care a rap about
them, because I've had them, or at least have seen them within my reach.
I came up from nothing; I got an education--no matter now how I got it;
I studied law. In ten years I had won such a position in my profession
(my branch of it--I was never an office lawyer) that everything lay open
before me. It was only a question of a certain number of years. Not only
was this generally prophesied, but I knew it myself. But by that time I
had found out the unutterable stupidity of people and their pursuits; I
couldn't help despising them. I had made enough to make my mother
comfortable, and there came over me a horror of a plodding life. I said
to myself, 'What is the use of it?' Of pleasure there was no question.
But I could go back to that plodding life to-morrow if I chose. Don't
you believe it, Pauline?"
"Yes."
"Yet you don't say--try?"
"Try, by all means."
"At a safe distance from you!"
"Yes, at a safe distance from me," Pauline answered. "I should do you no
good; I am not enough in earnest. I am never in earnest long about
anything. I am changeable, too--you have no idea how changeable. There
has been no opportunity to show you."
"Is that a threat? You know that I am deeply in love with you." He did
not move as he said this, but his eyes were fixed passionately upon her
face.
"I neither know it nor believe it; it is with you simply as it is with
me--there is no one else here." She stood there watching the wavelets
break at her feet. Nothing in her countenance corresponded in the least
with the description she had just given of herself.
"How you say that! What am I to think of you? You have a face to
worship: does it lie?" said Ash.
"Oh, my face!" She turned, and began to cross the field towards the
farm.
"It shouldn't have that expression, then," he said, joining her, and
walking by her side. "I don't believe you know what it is yourself,
Pauline--that expression. It seems to say as you talk, coming straight
from those divine lips, those sweet eyes: 'I could love you. Be good and
I will.' Why, you have almost made _me_ determine to be 'good' again,
almost made _me_ begin to dream of going back to that plodding life that
I loathe. And you don't know what I am."
Mrs. Graham did not answer; she did not look up, though she knew that
his head was bent beseechingly towards her.
John Ash was obliged to bend; he was very tall. His figure was rather
thin, and he had a slouching gait; his broad shoulders and well-knit
muscles showed that he had plenty of force, and his slouching step
seemed to come from laziness, as though he found it too much trouble to
plant his feet firmly, to carry his long length erect. He was holding
his hat in his hand, and the light from the sea showed his face
clearly, its good points and its bad. His head was well shaped, covered
with thick brown hair, closely cut; but, in spite of the shortness, many
silver threads could be seen on the brown--a premature silver, as he was
not yet thirty-five. His face was beardless, thin, with a bold
eagle-like outline, and strong, warm blue eyes, the blue eyes that go
with a great deal of color. Ordinarily, Ash had now but little color;
that is, there was but little red; his complexion had a dark brown hue;
there were many deep lines. The mouth, the worst feature, had a cynical
droop; the jaw conveyed suggestions that were not agreeable. The
expression of the whole countenance was that of recklessness and
cleverness, both of no common order. Of late the recklessness had often
changed into a more happy merriment when he was with Pauline, the
careless merriment of a boy; one could see then plainly how handsome he
must have been before the lines, and the heaviness, and, alas! the evil,
had come to darken his youth, and to sadden (for so it must have been)
his silent, frightened-looking mother.
They reached the farm; he led out the horses, and mounted her. She
gathered up the reins; but he still held the bridle. "How tired you
look!" he said.
Her face was flushed slightly, high on the cheeks close under the eyes;
between the fair eyebrows a perpendicular line was visible; for the
moment, she showed to the full her thirty years.
"Yes, I am tired; and it's dangerous to tire me," she answered, smiling.
She had recovered her light-hearted carelessness.
Ash still looked at her. A sudden conviction seemed to seize him. "Don't
throw me over, Pauline," he pleaded. And as he spoke, on his brown,
deeply lined face there was an expression which was boyishly young and
trusting.
"As I told you, so long as there is no one else," Pauline answered.
The next moment they were flying over the plain.
III
The _table d'hote_ of the Star of Italy, the Salerno inn from whose
mysteries (of eels and chestnuts) Mrs. Preston had fled--this unctuous
_table d'hote_ had been unusually brilliant during this month of March;
upon several occasions there had been no less than fifteen travellers
present, and the operatic young landlord himself, with his affectionate
smile, had come in to hand the peas.
The most unnoticed person was always a tall woman of fifty-five, who,
entering with noiseless step, slipped into her chair so quickly and
furtively that it seemed as if she were afraid of being seen standing
upon her feet. Once in her place, she ate sparingly, looking neither to
the right nor the left, holding her knife and fork with care, and laying
them down cautiously, as though she were trying not to waken some one
who was asleep. But the _table d'hote_ of the Star of Italy was never
asleep; the travellers, English and American, could not help feeling
that they were far from home on this shore where so recently brigands
had prowled. It is well known that this feeling promotes conversation.
One evening a pink-cheeked woman, who wore a little round lace cap
perched on the top of her smooth gray hair, addressed the silent
stranger at her left hand. "You have been to Paestum, I dare say?" she
said, in her pleasant English voice.
"No."
"But you are going, probably? Directly we came, yesterday morning, we
engaged horses and started at once."
"I don't know as I care about going."
"Not to see the temples?"
"I didn't know as there were temples," murmured the other, shyly.
"Fancy! But you really ought to go, you know," the pleasant voice
resumed, doing a little missionary work (which can never come amiss).
"The temples are well worth seeing; they are Greek."
"I've been ter see a good many buildings already: in Paris there were a
good many; my son took me," the tall woman answered, her tone becoming
more assured as she mentioned "my son."
"But these temples are--are rather different. I was saying to our
neighbor here that she really ought on no account to miss going down to
Paestum," the fresh-faced Englishwoman continued, addressing her husband,
who sat next to her on the right, for the moment very busy with his peas
(which were good, but a little oily). "The drive is not difficult. And
we found it most interesting."
"Interesting? It may well be interesting; finest Greek remains outside
of Athens," answered the husband, a portly Warwickshire vicar. He bent
forward a little to glance past his wife at this ignorer of temples at
her other hand. "American," he said to himself, and returned to his
peas.
The friendly vicaress offered a few words more the next day. Coming in
from her walk, in her stout shoes, and broad straw hat garnished with
white muslin, she was entering the inn by the back door, when she espied
her neighbor of the dinner-table sitting near by on a bench. There was
nothing to see but a paling fence; she was unoccupied, unless a basket
with Souvenir de Lucerne on one side, and a flat bouquet of artificial
flowers on the other, represented occupation.
"Do you prefer this to the garden in front?" the English woman asked, in
some surprise.
"Yes, I think I do."
"I must differ from you, then, because there we have the sea, you know;
'tis such a pretty view."
"I don't know as I care about the sea; it's all water--nothing to look
at."
"Ah! I dare say it makes you ill. We had a very nasty day when we
crossed from Folkestone."
"No; it ain't that exactly. I sit here because I like ter see the things
grow," hazarded the American, timidly, as if she felt that some
explanation was expected.
"The things?"
"Yes, in there." (She pointed to the paling fence.) "There's peas, and
asparagus, and beans, and some sorts I don't know; you wouldn't believe
how they do push up, day after day."
"Ah, indeed! I dare say they do," the Englishwoman answered, a little
bewildered, looking at the lines of green behind the palings.
"Her name is Ash, Azubah Ash--fancy!" she said to her husband, later. "I
saw it written on a Swiss basket in which she keeps her crewel-work. She
is extremely odd. She has no maid, yet she wears those very good
diamonds; and she always appears in that Paris gown of rich black
silk--the very richest quality, I assure you, Augustas: she wears it and
the diamonds at breakfast. She has spoken of a son, but apparently he
never turns up. And she spends all her time on a bench behind the house
watching the beans grow."
"I should think she would bore herself to extinction," said the
easy-going vicar.
"I dare say she _is_ having rather a hard time of it, she is so
_bornee_. I would offer her a book, but I don't think she ever reads.
And when I told her that I should be very pleased to show her some of
the pretty walks about here, she said that she never walked. She must be
sadly lonely, poor thing!"
But Mrs. Ash was not lonely; or, if she was, she did not know the name
of her malady. The comings and goings of her son were without doubt very
uncertain; but the mother had been born among people who believe that
the "men-folks" of a family have an existence apart from that of mothers
and sisters, and that it is right that they should have it. Her son, who
never went himself to a public table, had taken it for granted that his
mother would prefer to have her meals served privately in one of the
four large rooms which he had engaged for her at the inn.
"I think I like it better in the big dining-room, John," Mrs. Ash had
replied. She did not tell him that she found it less difficult to eat
her dinner when the attention of the waiter was distracted by the
necessity of attending to the wants of ten persons than when his gaze
was concentrated upon her solitary knife and fork alone.
John Ash was fond of his mother. It did not occur to him that this
nomad life abroad was causing her any suffering. Her shyness, her dread
of being looked at, her dread of foreign servants, he did not fully see,
because when he was present she controlled them; when he was present,
also, in a great measure, they disappeared. He knew that she would not
have had one moment's content had he left her behind him, even if he had
left her in the finest house his money could purchase; so he took her
with him, and travelled slowly, for her sake, making no journeys that
she could not make, sending forward to engage the best rooms for her at
the inns where he intended to stop.
That he had not taken her to Paestum was not an evidence of neglect.
During the first months of their wanderings he had been at pains to take
her everywhere he had thought that she would enjoy it. But Mrs. Ash had
enjoyed nothing--save the going about on her son's arm. If he left her
alone amid the most exquisite scenery in the world, she did not even see
the scenery; she thought a dusty jaunt in a horse-car "very pleasant" if
John was there. So at last John gave her his simple presence often, but
troubled her with descriptions and excursions no more.
Dumb, shy, hopelessly out of her element as she was, this mother had, on
the whole, enjoyed her two years abroad. The reason was found in the
fact that she could say to herself, or rather could hope to herself,
that John was more "steady" over here.
The rustic term covered much--the days and the nights when John had not
been "steady."
These six weeks at Salerno particularly had been a season of blessed
repose to Azubah Ash; the days had gone by so peacefully that life had
become almost comfortable to her again, in spite of the ordeal of
dinner. She had even been beguiled into thinking a little of the
future--of the farm she should like to have some day, with fruit and
cream and vegetables--yes, especially vegetables; and she dreamed of an
old pleasure of her youth, that of hunting for little round artichokes
in the cool brown earth. John had been contented all the time, and his
mood had been very tranquil. His mother liked this much better than high
spirits. There was an element sometimes in John's high spirits that had
made her tremble.
But on the day succeeding that last ride with Mrs. Graham, when they had
dismounted and walked down to the shore, John had come back to the inn
with a darkened face. The dark mood had lasted now for ten days. His
mother began to lead her old sleepless, restless life again. Her awkward
crochet-needle had stopped of itself; she went no more to her bench
beside the asparagus. Instead, she remained in her room--her four
rooms--every now and then peeping anxiously through the blinds. Nothing
happened--so any one would have said; the sea continued blue and misty,
the sky blue and clear; every one came and went as usual in the divine
weather of the Italian spring. But John Ash's mother had, to use an old
expression, her heart in her mouth all the time.
It choked her, and she gave up going to the _table d'hote_; she let her
son suppose that the meal was served in her sitting-room, but in reality
she took no dinner at all. When he came in she was always there, always
carefully dressed in the black silk whose rich texture the vicar's wife
had noticed, with the "very good" diamonds fastening her collar and on
her thin hands. She made a constant effort that her son should notice
no change in her.
Azubah Ash had a gaunt frame with large bones; her chest was hollow, and
she stooped a little as she walked. Yet, looking at her, one felt sure
that she would live to be an old woman. Her large features were roughly
moulded, her cheeks thin; her thick dusky hair was put plainly back from
her face, and arranged with a high comb after a fashion of her youth.
Her eyes, large, dark, and appealing, were sunken; they were beautiful
eyes, if one could have removed from them their expression of
apprehension, but that seemed now to have grown a part of them, to have
become fixed by time. Observers of physiognomy who met Azubah during
these two years of her sojourn abroad never forgot her--that tall gaunt
woman with the awkward step and bearing, with the rich dress and
diamonds, from whose timid face with its rough features those beautiful
eyes looked appealingly out.
"Mother, I am going to Paestum to-morrow," announced Ash on that eleventh
day. "Perhaps you had better go with me." He had come in and thrown
himself down upon the sofa, where he sat staring at the wall.
"Paestum--yes, that's where that English lady said I'd oughter go,"
answered Mrs. Ash. Then, after a moment, "She said there were temples
there." She had her hands folded tightly as she looked at her son.
"They're all going--old lady Preston, with her ghosts of Abercrombies,
little Miss Holland, Mrs. Graham, and all. Those boys are sketching down
there; they've been there some time."
[Illustration: AZUBAH ASH]
"I shall be very glad ter go, John, if you are going. Would you like
ter have me--ter have me ride horseback?"
Ash, coming out of his abstraction, broke into a laugh. "I shall take
you in the finest landau in Salerno, marmer," he said, coming across to
kiss her; "old lady Preston will have to put up with the second best.
You haven't forgotten, then, that you used to ride, marmer, have you?"
The mother's eyes had filled upon hearing the old name, the "marmer" of
the days when he had been her devoted, constantly following, tyrannical,
but very loving little boy. But she did not let the tears drop: she
never made scenes of any kind before John. "Well, you've been riding
horseback every day now for a long while; you haven't seemed to care at
all for carriages. And I did use to ride horseback a good deal when I
was a girl; I used to ride to the mill."
"I know you did. And carry the grist to be ground." He kissed her again.
"Don't be afraid of anything or anybody to-morrow, marmer, I beg. You're
the bravest and most sensible woman I know, and I want you to look what
you are."
"Shall I wear my India shawl, then?"
"Wear the best you have; I wish it were a hundred times bester. You are
handsomer than any of them as it is."
"Oh no, John; I ain't good-looking; I never was," said his mother,
blushing. She put her hand up for a moment, nervously, over her mouth--a
gesture habitual with her.
"Yes, you are, marmer. Look at your eyes. It's only that you have got
into a way of not thinking so. But I think so, and others shall." He
went back to the sofa, and sank into abstraction again.
At length his mother broke the silence, which had lasted very long. "I
hope they are all well over there to-day?" she asked, hesitatingly.
"Over there" was her name for the house on the shore, the house where
she knew her son had for many weeks spent all his time.
"Well? They're extraordinarily well," said Ash. He got up and walked
restlessly about the room. After a while he stopped, and now he seemed
to have forgotten his mother's presence, for his eyes rested upon her
without seeing her. "One of them is a little too well," he said,
menacingly; "let him look to himself--that's all." And then into his
face, his mother, watching him, saw coming slowly something she knew.
The expression changed him so completely that the ladies who had seen so
much of him would not have recognized their visitor. His mother
recognized him. That expression on her son's face was her life's long
terror.
He left the room. She listened as long as she could hear his steps;
then, after sitting for some time with her head upon her arms on the
table before her, she rose, and went slowly to put on her bonnet and
shawl. Coming back, still slowly, she paused, and for five minutes stood
there motionless. Then her hands dropped desparingly by her sides, and
her worn face quivered. "O God, O our Father, I really don't know what
ter do!" she murmured, breaking into helpless sobs, the stifled,
difficult sobs of a person unaccustomed to self-expression, even the
self-expression of grief.
She did not go out. Instead of that, she went back to the inner room and
knelt down.
IV
The next morning three carriages and two persons on horseback were
following the long road that stretches southward from Salerno to Paestum.
In the first carriage old Mrs. Preston sat enthroned amid cushions and
shawls; opposite she had placed her nephew Arthur, first because he was
slim, second because he was a man (Mrs. Preston was accustomed to say,
"Too much lady talk dries my brain"); the second carriage held Isabella
Holland and the Abercrombie girls; in the third, a landau drawn by two
spirited horses, were Mrs. Ash and her son. The two persons on horseback
were Pauline Graham and Griffith Carew.
In the soft spring air the mountains that rise all the way on the left
at no great distance from the road had in perfection the vague, dreamy
outlines and violet hues that form so characteristic a feature of the
Italian landscape. Up in the sky their peaks shone whitely, powdered
with snow. The flat plain that stretches from the base of the mountains
to the sea had beauty of another kind; often a fever-swept marsh, it
possessed at this season all a marsh's luxuriance of waving reeds and
flowers and tasselled jungles, with water birds rising from their
feeding-places, and flying along, low down, with a slow motion of their
broad wings, their feet stretched out behind. Troops of buffalo could be
seen here and there. At rare intervals there was an oasis of cultivated
ground, with a solitary farm-house. On the right, all the way, the
Mediterranean, meeting the flat land flatly, stretched forward from
thence into space, going on bluely, and rising a little on the horizon
line, as though it were surmounting a low hill.
Occasionally the carriages passed a little band of the small,
quick-stepping Italian soldiers.
"Oh, I say, did you know, aunt, that people were murdered by brigands on
this very bridge only ten years ago?" said Arthur, as they rolled across
a stone causeway raised in the form of an arch over a sluggish stream.
"I should like very much to see the brigands who did it!" Mrs. Preston
answered, smacking her lips contemptuously.
Arthur at least was very sure that no ten brigands could have vanquished
his aunt.
"This, girls, is the ancient Tyrrhenian Gulf," began Isabella to her
companions, waving one neatly gloved hand towards the sea. Isabella,
owing to the singularly incessant death of relatives, was always in
mourning; her neat gloves therefore were sable. "The temples we are
about to visit are very ancient also, having been built ages ago by
Greeks, who came from--from Greece, of course, naturally; and never
ceased to regret it. And all this shore, and the temples also, were
sacred to Neptune, or Poseidon, as he was called in Greek. And the
Greeks lamented--but I will read you that later at the threshold of the
temples; you cannot fail to be interested."
"I shall not be interested at all," said Hildegarde.
"Nor I," said Rose.
"_They_ had nothing to lament about; _they_ had no dancing to do," added
Dorothea. And the three white faces glared suddenly and sullenly at
their astonished companion.
"I am shocked," began Isabella.
"Shocked yourself," said Rose.
"You are a busybody," said Dorothea.
"And a gormandizer," added Hildegarde.
"And a _Worm_!" said Rose, with decision. "We have decided not to
pretend any more before _you_, Worm! Dance yourself till your legs drop
off, and see how you like it."
The three girls had weak soft voices; they possessed no other tones; the
strong words they used, therefore, were all the more startling because
so gently, almost sighingly, spoken.
In the landau there had been silence. Mrs. Ash, after respecting her
son's sombre mood for more than an hour, at last spoke: "I guess you
don't care very much about those triflin' temples, after all, do you,
John? And it's going to be very long. Supposing we turn back?" She wore
her India shawl and a Paris bonnet; she was sitting without touching the
cushions of the carriage behind her. She had looked neither at the
mountains nor at the sea; most of the time her eyes had rested on the
blue cloth of the empty seat opposite. Occasionally, however, they had
followed the two figures on horseback, and it was after these figures
had passed them a second time, pushing on ahead in order to get a free
space of road for a gallop, that she had offered her suggestion.
"Go back? Not for ten thousand dollars--not for ten thousand devils!"
said John Ash. "What a lazy girl you are, marmer!" And he became gay and
talkative.
His mother responded to his gayety as well as she could: she laughed
when he did. Her laugh was eager. It was almost obsequious.
By-and-by the three temples loomed into view, standing in all their
beauty on the barren waste, majestic, uninjured, extraordinary. Their
rows of fluted columns, their brilliant tawny hues, their perfect Doric
architecture, made the loneliness surrounding them even more lonely,
made the sound of the sea breaking near by on the lifeless shore a
melancholy dirge. When the party reached the great colonnades there were
exclamations; there was even declamation, Mrs. Preston having been
fitted by nature for that. Freemantle, Gates, and Beckett had come
rushing forward to meet their arriving friends. In reality, however, it
was Griff whom they had rushed to meet. Griff to their minds was the
only important person present, even though the unimportant included
Pauline.
"Hallo, Griff, old fellow! how are you?"
"Couldn't you stay, Griff? We've got a tent for you."
They laughed, and made jokes, and hovered about him, longing to drag him
off immediately to show him their drawings, and to discuss with him a
hundred disputed points. But though they thus paid small attention to
Pauline, they were obliged to form part of her train; for as Griff
remained with her, and they remained with Griff, naturally, as Isabella
would have said, they made the tour of inspection in her company.
In the meanwhile Isabella, who had it upon her strictly kept conscience
not to neglect her own duties in spite of the Abercrombie revolt, had
taken her stand before the great temple of Neptune, with her instructive
little book in her hand. "'The men of Poseidonia,'" she began, "'having
been at first true Greeks, had in process of time gradually become
barbarized, changing to Romans.' Poseidonia, girls, was the ancient
name of Paestum," she interpolated in explanation, glancing over her
glasses at her silent audience.
The Abercrombies could not retort this time, because Aunt Octavia was
very near them, sitting at the base of one of the great columns of
travertine with the air and manner of Neptune's only lawful wife. But
their backs were towards her; she could not see their faces; they were
able, therefore, to make grimaces at Isabella, and this they immediately
proceeded to do in unison, flattening their thin lips over their teeth
in a very ghastly way, and turning up their eyes so unnaturally far that
Isabella was afraid the pupils would never come down again.
"'Yet they still observed one Hellenic festival,'" she read stumblingly
on--stumblingly because she felt obliged from a sort of fascination to
glance every now and then at the distorted countenances before
her--"'one Hellenic festival, when they met together here to call to
remembrance the old days and the old customs, and to weep upon each
other's necks, and to lament drearily. And then, when the time of their
mourning was over, they departed, each man in silence to his Roman
home.'"
"Very fine," said Mrs. Preston, commendingly, from her column.
But Isabella had closed her book, and was walking away, wiping her
forehead: those girls' faces were really too horrible.
"Where are you going, Isabella?" Mrs. Preston called.
"I suppose I may gather some asphodel?" Isabella responded, with some
asperity.
But she did not gather much asphodel. Coming upon Mrs. Ash wandering
about over the fallen stones, she stayed her steps to speak to her. She
was not interested in Mrs. Ash, but she was so "happily relieved" that
dear Paulie lately had given up her rides with the son, that she, as
Paulie's cousin (first), could afford to be civil to the mother, in
spite of that mother's bad judgment as to English and diamonds. Isabella
disapproved of Mrs. Ash; she thought that "such persons" did great harm
by their display of "mere vulgar affluence." No vulgar affluence
oppressed Isabella. She had six hundred dollars a year of her own, and
each dollar was well bred.
"We shall soon be having lunch, I suppose," she began, in a gracious
tone. "It seems almost a desecration, doesn't it, to have it in the
shrine itself, for I see they are arranging it there."
"Oh, is that a shrine?" said Mrs. Ash, vaguely. "I didn't know. But then
I'm not a Catholic. They seem very large buildings. They seem wasted
here."
Little Isabella looked up at her--she was obliged to look up, her
companion was so tall. The anxious expression in Mrs. Ash's eyes had
grown into anguish: she was watching her son, who had now joined Pauline
and her train. Pauline had Carew on her right hand and John Ash on her
left; the four boys walked stragglingly, now in front, now behind, but
never far from Carew.
"You are not well," said Isabella; "the drive was too long for you. Pray
take my smelling-salts; they are sometimes refreshing." And she detached
from its black chain a minute funereal bottle.
"Thank you," answered Mrs. Ash, gazing down uncomprehendingly at the
offering; "I am very well indeed. I was jest looking at your cousin,
Mrs. Graham; she's very handsome."
"Yes," responded Isabella, gladly seizing this opportunity to convey to
the Ash household a little light, "Pauline is handsome--in her own way.
It is not the style that I myself admire. But then I know that my taste
is severe. By ordinary people Pauline is considered attractive; it is
therefore all the more to be deplored that she should be such a sad, sad
flirt."
"A flirt?" said Mrs. Ash.
"Yes--I am sorry to say it. No matter how far she may go, it means
nothing, absolutely nothing; she has not the slightest intention of
allowing herself either to fall in love or to marry again; she prefers
her position as it is. And I don't think she realizes sufficiently that
what is but pastime to her may be taken more seriously by others; and
naturally, I must say, after the way she sometimes goes on. _I_ could
never do so, no matter what the temptations were, and I must say I have
never been able to understand it in Pauline. At present it is Mr. Carew;
she is going to Naples with him to-morrow for the day. As you may
imagine, it is against our wish--Cousin Octavia Preston's and mine. But
Pauline being a widow, which _she_ considers an advantage, and no longer
young (she is thirty, though you may not think it; she shows her age
very fully in the morning)--Pauline, under these circumstances, has for
some time refused a chaperon. I don't think myself that she needs a
chaperon exactly, but she might take a lady friend."
"Going to Naples with him to-morrow," murmured Mrs. Ash. She put her
gloved hand over her mouth for a moment, the large kid expanse very
different from Isabella's little black paw. "I might as well go over
there," she said, starting off with a rapid step towards Pauline.
Pauline received her smilingly; Ash frowned a little. He frowned not at
his mother--she was always welcome; he frowned at her persistence in
standing so near Pauline, in dogging her steps. Mrs. Ash kept this up;
she sat near Pauline at lunch; she followed her when she strolled down
to the beach; she gathered flowers for her; in her India shawl and Paris
bonnet she hovered constantly near.
Only once did John Ash find opportunity to speak to Pauline alone. The
boys had at last carried off Griff by force to their camp; Griff was
willing enough to go, the "force" applied to the intellectual effort
necessary on the boys' part to detach him from a lady who wished to keep
him by her side. They had all been strolling up and down in the shade of
the so-called Basilica, amid the fern and acanthus. Left alone with her
son and Mrs. Graham, Mrs. Ash, after remaining with them a few moments,
turned aside, and entering the temple, sat down there. She was out of
hearing, but still near.
"Ride with me to-morrow, Pauline," Ash said, immediately. "I have not
had a chance to speak to you before. Don't refuse."
"I am afraid I must. I have an engagement."
"With Carew?"
"Yes."
"What is it?"
"I am very good-natured to tell you. I am going to Naples with him for
the day."
"You are going-- Damnation!"
"You forget yourself," said Pauline. Then, when she saw the look on his
face--the face of this man with whom she had played--she was startled.
"Forget myself! I wish I could. You shall not go to Naples."
"And how can you prevent it?"
"Are you daring me?"
"By no means," answered Pauline; and this time she really tried to speak
gently. "I was calling to your remembrance the fact that there is no tie
between us, Mr. Ash; you have no shadow of authority over my actions; I
am free to do as I please."
"I know you are; that is the worst of it," he said, almost with a groan.
"Pauline, don't play with me now. I have given up hoping for anything
for myself--if I ever really did hope; I am not worthy of you. Whether
you could make me worthy I don't know; but I don't ask you that; I don't
ask you to try; it would be too much. I only ask you to be as you have
been; as you were, I mean, during all those many weeks, not as you have
been lately. Only a few days are left when I can see you freely; be kind
to me, then, during those few days, and then I will take myself off."
"I mean to be kind; I am kind."
"Then ride with me to-morrow; just this once more."
"But I told you it was impossible; I told you I was going to Naples."
The pleading vanished from Ash's face and voice. "_I_ never asked you to
do that--to go off with me for a whole day."
Pauline did not answer; she was arranging the flowers which Mrs. Ash had
industriously gathered.
"So much the greater fool I!--is that what you are thinking?" Ash went
on, laughing discordantly.
For the moment Pauline forgot to be angry in the vague feeling,
something like fear, which took possession of her. All fear is
uncomfortable, and she hated discomfort; she gave herself a little
inward shake as if to shake it off. "I shall ask Cousin Oc to go back to
Paris next week," was her thought. "I have had enough of Italy for the
present--Italy and madmen!"
"You won't go?" asked Ash, bending forward eagerly, as though he had
gained hope from her silence.
"To Paris?"
"Are we speaking of Paris? To Naples--to-morrow."
"Oh, I must go to Naples," she answered, gayly. In spite of her gayety
she turned towards the Basilica; Mrs. Ash was the nearest person.
"You are going to my mother? She, at least, is a good woman; she would
never have tarnished herself with such an expedition as you are
planning!" cried Ash, in a fury.
Pauline turned white. "I am well paid for ever having endured you, ever
having liked you," she said, in a low voice, as she hastened on. "I
might have known--I might have known."
There was not much to choose now between the expression of the two
faces, for the woman's sweet countenance showed in its pallor an anger
as vivid as that which had flushed the face of the man beside her, with
a red so dark that his blue eyes looked unnaturally light by contrast,
as though they had been set in the face of an Indian.
Mrs. Ash had come hurriedly out to meet them. Her son paid no attention
to her; all his powers were evidently concentrated upon holding himself
in check. "I shouldn't have said it, even if it were the plain brutal
truth," he said. "But you madden me, Pauline. I mean what I say--you
really do drive me into a kind of madness."
"I have no desire to drive you into anything; I have no desire to talk
with you further," she answered.
"No, no, dearie, don't say that; talk ter him a little longer," said
Mrs. Ash, coming forward, her face set in a tremulous smile. "I'm sure
it's very pleasant here--beside these buildings. And John thinks so much
of you; he means no harm."
"Poor mother!" said Ash, his voice softening. "She does not dare to say
to you what she longs to say; she would whisper it if she could; and
that is, 'Don't provoke him!' She has some pretty bad memories--haven't
you, mother?--of times when I've--when I've gone a-hunting, as one may
say. She'll tell you about them if you like."
"I don't want to hear about them; I don't want to hear about anything,"
answered Mrs. Graham, troubled out of all her composure, troubled even
out of her anger by the strangeness of this strange pair. She looked
about for some one, and, seeing Carew coming from the tents of the camp,
she waved her hand to attract his attention and beckoned to him; then
she went forward to meet him as he hastened towards her.
Ash disengaged himself from his mother, who, however, had only touched
his arm entreatingly, for she had learned to be very cautious where her
son was concerned; he strode forward to Pauline's side.
"I should rather see you dead before me than go with that man
to-morrow."
"Pray don't kill me, at least till the day is over," Pauline answered,
her courage, and her unconquerable carelessness too, returning in the
approach of Carew. "It would be quite too great a disappointment to lose
my day."
"You _shall_ lose it!" said Ash, with a loud coarse oath.
"Oh!" said the woman, all her lovely delicate person shrinking away from
him.
Her intonation had been one of disgust. She held the skirt of her habit
closer, as if to avoid all contact.
V
At five o'clock of the same afternoon Freemantle, Gates, and Beckett,
with Arthur Abercrombie, came running along the narrow streets of a
village some miles from Paestum.
The stone houses of which this village was composed stood like two solid
walls facing each other, rising directly from the stone-paved road,
which was barely ten feet wide; down this conduit water was pouring like
a brook. The houses were about forty in number, twenty on each side, and
this one short street was all there was of the town.
It was raining, not in drops, but in torrents, with great pats of water
coming over, almost like stones, and striking upon the heads of those
who were passing below; every two or three minutes there came a glare of
blindingly white lightning, followed immediately by the crash of
thunder, which seemed to be rolling on the very roofs of the houses
themselves. The four boys must have been out in the storm for some time,
for they paid no attention to it. Their faces were set, excited. Every
thread of their clothing was wet through.
"This is the house," said Arthur.
They looked up, sheltering their eyes with their arms from the blows of
the rain-balls. From the closed windows above, the faces of Isabella
Holland and the three Abercrombie girls looked down at them, pressed
flatly against the small panes, in order to see; for the storm had made
the air so dark that the street lay in gloom.
The next moment the boys entered.
"No, we haven't found him," said Arthur, in answer to his white sisters'
look. "But we're going to."
"Yes, we're going to," said the others. And then, walking on tiptoe in
their soaked shoes, they went softly into an inner room.
Here on a couch lay Griffith Carew, dying.
An Italian doctor was still trying to do something for the unconscious
man. He had an assistant, and the two were at work together. Near by,
old Mrs. Preston sat waiting, her hands folded upon the knob of a cane
which stood on the floor before her, her chin resting upon her hands. In
this bent position, with her disordered white hair and great black eyes,
she looked witch-like. Three candles burned on a table at the head of
the bed, illumining Carew and the two doctors and the waiting old woman.
The room was long, and its far end was in shadow. Was there another
person present--sitting there silent and motionless? Yes--Pauline. The
boys came to the foot of the bed and gazed with full hearts at Griff.
Griff had been shot by John Ash two hours before. The deed had been done
just as they had reached the shelter of this village, swept into it
almost by a tornado, which, preceding the darker storm, had driven them
far from their rightful road. The darker storm had broken upon them
immediately afterwards with a terrible sound and fury; but the boys had
barely heard the crash in the sky above them as they carried Griff
through the stony little street. They had found a doctor--two of them;
they had done everything possible. Then they had been told that Griff
must die, and they had gone out to look for the murderer.
He could not be far, for the village was small, and he could not have
quitted the village, because the half-broken young horses that had
brought him from Salerno, frightened by the incessant glare of the
lightning, had become unmanageable, dragged their fastenings loose, and
disappeared. In any case the plain was impassable; the roar of the sea,
with the night coming on, indicated that the floods were out; they had
covered the shore, and would soon be creeping inland; the road would be
drowned and lost. Ash, therefore, could not be far.
Yet they had been unable to find him, though they had searched every
house. And they had found no trace of his mother.
During these long hours four times the boys had sallied forth and hunted
the street up and down. The Italians, crowded into their narrow dark
dwellings from fear of the storm, had allowed them to pass freely in and
out, to go from floor to floor; some of the men had even lighted their
little oil lamps and gone down with them to search the shallow cellars.
But the women did not look up; they were telling their beads or
kneeling before their little in-door shrines, the frightened children
clinging to their skirts and crying. For both the street and the dark
houses were lighted every minute or two by that unearthly blinding
glare.
The village version of the story was that the two _forestieri_ had
sprung at each other's throats, maddened by jealousy; poniards had been
drawn, and one of them had fallen. One had fallen, indeed, but only one
had attacked. And there had been no poniards: it was a well-aimed bullet
from an American revolver that had struck down Griffith Carew.
The four boys, brought back each time from their search by a sudden hope
that perhaps Griff might have rallied, and forced each time to yield up
their hope at the sight of his death-like face, were animated in their
grief by one burning determination: they would bring the murderer to
justice. It was a foreign land and a remote shore; they were boys; and
he was a bold, bad man with a wonderful brain--for they had always
appreciated Ash's cleverness, though they had never liked him. In spite
of all this he should not escape; they would hunt him like
hounds--blood-hounds; and though it should take months, even years, of
their lives, they would bring him to justice at the last.
This hot vow kept the poor lads from crying. They were very young, and
their heads were throbbing with their unshed tears; there were big lumps
in their throats when poor Griff, opening his dull eyes for a moment,
knew them, and tried to smile in his cheery old way. But he relapsed
into unconsciousness immediately. And the watch went on.
The gloomy day drew to its close; by the clocks, evening had come.
There was more breathing-space now between the lightning flashes and the
following thunder; the wind was no longer violent; the rain still fell
heavily; its torrent, striking the pavement below, sent up a loud hollow
sound. One of the doctors left the house, and came back with a fresh
supply of candles and various things, vaguely frightful, because hidden,
concealed in a sheet. Then the other doctor went out to get something to
eat. Finally they were both on guard again. And the real night began.
Then, to the waiting group in the lighted silent room, there entered a
tall figure--Azubah Ash; drenched, without bonnet or shawl, she stood
there before them. Her frightened look was gone forever: she faced them
with unconscious majesty. "My son is dead"--this was her announcement.
She walked forward to the bed, and gazed at the man lying there.
"Perhaps he will not die," she said, turning her head to glance at the
others. "God is kind--sometimes; perhaps he will not die." She bent over
and stroked his hair tenderly with her large hand. "Dear heart, live!
Try ter live!" she said; "we want yer to, so much!"
Then she left him, and faced them again. "I thought of warning you," she
began; "you"--and she looked at Mrs. Preston; "and you"--she turned
towards the figure at the end of the room. "My son was not himself when
he was in a passion--I have known it ever sence he was born. Even when
he was a little fellow of two and three I used ter try ter guard him;
but I couldn't do much--his will was stronger than mine. And he was
always very clever, my son was--much cleverer than me. Twice before,
three times before, I've ben afraid he'd take some one's life. You
see, he didn't care about life so much as some people do; and now he has
taken his own."
[Illustration: THE OLD WATCH-TOWER]
There was an involuntary stir among the boys.
Mrs. Ash turned her eyes towards them. "Would you like ter see him, so's
ter be sure? In one moment."
She went towards the bed again, and clasped her hands; then she knelt
down, and began to pray beside the unconscious man in hushed tones. "O
God, O our Father, give us back this life: do, Lord--O do. It's so dear
ter these poor boys, and it's so dear ter many; and perhaps there's a
mother too. O Lord, give it back to us! And when he's well again, help
him ter be all that my poor son was not. For Christ's sake."
She rose and crossed to where the boys were standing. "Will you come
now?" she said. "I'm taking him away at dawn." Then, very simply, she
offered her hand to Mrs. Preston. "He was a great deal at your house; he
told me that. I thank you for having ben so kind ter him. Good-bye."
"But I too will go with you," answered Mrs. Preston, in her deep tones.
She rose, leaning on her cane. Mrs. Ash was already crossing the room
towards the door.
The boys followed her; then came Mrs. Preston, looking bent and old. The
figure of Pauline in her dark corner rose as they approached.
"No," said Mrs. Ash, seeing the movement. She paused. "Don't come, my
dear; I really can't let you; you'd think of it all the rest of your
life if you was ter see him now, and 'twould make you feel so bad. I
know you didn't mean no harm. But you mustn't come."
And Pauline, shrinking back into the shadow, was held there by the
compassion of this mother--this mother whose nobler nature, and large
glance quiet in the majesty of sorrow, made her, made all the women
present, fade into nothingness beside her. In the outer room Isabella
and the excited, peering Abercrombies were like four unimportant,
unnoticed ghosts, as the little procession went by them in silence, and
descended the stairs. Then it passed out into the storm.
Mrs. Ash walked first, leading the way, the rain falling on her hair;
the three boys followed; behind them came Mrs. Preston, leaning on her
nephew's arm and helping herself with her cane. They passed down the
narrow street, and the people brought their small lamps to the doorways
to aid them in the darkness. The street ended, but the mother went on:
apparently she was going out on the broad waste. They all followed, Mrs.
Preston merely shaking her head when Arthur proposed that she should
turn back.
At some distance beyond the town there was a grove of oaks; they went
round an angle of this grove, stumbling in the darkness, and came to a
mound behind it; on the summit of the mound there was something--a
square structure of stone. Mrs. Ash went up, and entered a low door.
Within there was but one room, empty save for a small lighted lamp
standing on the dirt floor; a stairway, or rather a flight of stone
steps, ascended to a room above. Mrs. Ash took the lamp and led the way
up; Mrs. Preston's cane sounded on the stones as she followed.
[Illustration: "THE CART WAS GOING SLOWLY ACROSS THE FIELDS, FOR THE
ROAD WAS OVERFLOWED."]
The room above was square, like the one below; it was the whole interior
of the ancient house, or rather the ancient watch-tower; its roof of
beams was broken; the rain came through in several places and dropped
upon the floor. There was a second small lamp in the room besides the
one which Mrs. Ash had brought; the two shed a dim ray over a peasant's
rude bed, where something long and dark and straight was stretched out.
Mrs. Ash went up to the bed, and motioning away the old peasant who was
keeping watch there, she took both lamps and held them high above the
still face. The others drew near. And then they saw that it was John
Ash--dead!
There were no signs of the horror of it; his mother had removed them
all; he lay as if asleep.
The mother held the lights up steadily for a long moment. Then she
placed them on a table, and coming back, took her son's lifeless hand in
hers.
"Now that you've seen him, seen that he's really gone, will you leave me
alone with him?" she said. "I think there's nothing more."
There was a dignity in her face as she stood there beside her child
which made the others feel suddenly conscious of the wantonness of
further intrusion. As they looked at her, too, they perceived that she
no longer thought of them, no longer even saw them: her task was ended.
Without a word they went out. Mrs. Preston's cane sounded on the
stairway again; then there was silence.
At dawn they saw her drive away. Griff might live, the doctors had said.
But for the moment the gazing group of Americans forgot even that. She
was in a cart, with a man walking beside the horse; the cart was going
slowly across the fields, for the road was over-flowed. The storm had
ceased; the sky was blue; the sun, rising, shed his fresh golden light
on the tall, lonely figure with its dark hair uncovered, and on the
long rough box at its feet.
Looking the other way, one could see in the south the beautiful temples
of Paestum, that have gazed over that plain for more than two thousand
years.
A PINK VILLA
I
"Yes, of the three, I liked Pierre best," said Mrs. Churchill. "Yet it
was hard to choose. I have lived so long in Italy that I confess it
would have been a pleasure to see Eva at court; it's a very pretty
little court they have now at Rome, I assure you, with that lovely Queen
Margherita at the head. The old Marchese is to resign his post this
month, and the King has already signified his intention of giving it to
Gino. Eva, as the Marchesa Lamberti, living in that ideal old Lamberti
palace, you know--Eva, I flatter myself, would have shone in her small
way as brightly as Queen Margherita in hers. You may think I am assuming
a good deal, Philip. But you have no idea how much pain has been taken
with that child; she literally is fitted for a court or for any other
high position. Yet at the same time she is very childlike. I have kept
her so purposely; she has almost never been out of my sight. The
Lambertis are one of the best among the old Roman families, and there
could not be a more striking proof of Gino's devotion than his having
persuaded his father to say (as he did to me two months ago) that he
should be proud to welcome Eva 'as she is,' which meant that her very
small dowry would not be considered an objection. As to Eva herself, of
course the Lambertis, or any other family, would be proud to receive
her," pursued Mrs. Churchill, with the quiet pride which in its
unruffled serenity became her well. "But not to hesitate over her mere
pittance of a portion, that is very remarkable; for the marriage-portion
is considered a sacred point by all Italians; they are brought up to
respect it--as we respect the Constitution."
"It's a very pretty picture," answered Philip Dallas--"the court and
Queen Margherita, the handsome Gino and the old Lamberti palace. But I'm
a little bewildered, Fanny; you speak of it all so appreciatively, yet
Gino was certainly not the name you mentioned; Pierre, wasn't it?"
"Yes, Pierre," answered Mrs. Churchill, laughing and sighing with the
same breath. "I've strayed far. But the truth is, I did like Gino, and I
wanted to tell you about him. No, Eva will not be the Marchesa Lamberti,
and live in the old palace; I have declined that offer. Well, then, the
next was Thornton Stanley."
"Thornton Stanley? Has he turned up here? I used to know him very well."
"I thought perhaps you might."
"He is a capital fellow--when he can forget his first editions."
Mrs. Churchill folded her arms, placing one hand on each elbow, and
slightly hugging herself. "He has forgotten them more than once in
_this_ house," she said, triumphantly.
"He is not only a capital fellow, but he has a large fortune--ten times
as large, I venture to say, as your Lambertis have."
"I know that. But--"
"But you prefer an old palace. I am afraid Stanley could not build Eva
an old castle. Couldn't you manage to jog on with half a dozen new
ones?"
"The trouble with Thornton Stanley was his own uncertainty," said Fanny;
"he was not in the least firm about staying over here, though he
pretended he was. I could see that he would be always going home. More
than that, I should not be at all surprised if at the end of five
years--three even--he should have bought or built a house in New York,
and settled down there forever."
"And you don't want that for your American daughter, renegade?"
Mrs. Churchill unfolded her arms. "No one can be a warmer American than
I am, Philip--no one. During the war I nearly cried my eyes out; have
you forgotten that? I scraped lint; I wanted to go to the front as
nurse--everything. What days they were! We _lived_ then. I sometimes
think we have never lived since."
Dallas felt a little bored. He was of the same age as Fanny Churchill;
but the school-girl, whose feelings were already those of a woman, had
had her nature stirred to its depths by events which the lad had been
too young to take seriously to heart. His heart had never caught up with
them, though, of course, his reason had.
"Yes, I know you are flamingly patriotic," he said. "All the same, you
don't want Eva to live in Fiftieth Street."
"In Fiftieth Street?"
"I chose the name at random. In New York."
"I don't see why you should be sarcastic," said Fanny. "Of course I
expect to go back myself some time; I could not be content without that.
But Eva--Eva is different; she has been brought up over here entirely;
she was only three when I came abroad. It seems such a pity that all
that should be wasted."
"And why should it be wasted in Fiftieth Street?"
"The very qualities that are admired here would be a drawback to her
there," replied Mrs. Churchill. "A shy girl who cannot laugh and talk
with everybody, who has never been out alone a step in her life, where
would she be in New York?--I ask you that. While here, as you see,
before she is eighteen--"
"Isn't the poor child eighteen yet? Why in the world do you want to
marry her to any one for five years more at least?"
Mrs. Churchill threw up her pretty hands. "How little you have learned
about some things, Philip, in spite of your winters on the Nile and your
Scotch shooting-box! I suppose it is because you have had no daughters
to consider."
"Daughters?--I should think not!" was Dallas's mental exclamation.
Fanny, then, with all her sense, was going to make that same old mistake
of supposing that a bachelor of thirty-seven and a mother of
thirty-seven were of the same age.
"Why, it's infinitely better in every way that a nice girl like Eva
should be married as soon as possible after her school-books are closed,
Philip," Mrs. Churchill went on; "for then, don't you see, she can enter
society--which is always so dangerous--safely; well protected, and yet
quite at liberty as well. I mean, of course, in case she has a good
husband. That is the mother's business, the mother's responsibility, and
I think a mother who does not give her heart to it, her whole soul and
energy, and choose _well_--I think such a mother an infamous woman. In
this case I am sure I have chosen well; I am sure Eva will be happy with
Pierre de Verneuil. They have the same ideas; they have congenial
tastes, both being fond of music and art. And Pierre is a very lovable
fellow; you will think so yourself when you see him."
"And you say she likes him?"
"Very much. I should not have gone on with it, of course, if there had
been any dislike. They are not formally betrothed as yet; that is to
come soon; but the old Count (Pierre's father) has been to see me, and
everything is virtually arranged--a delightful man, the old Count. They
are to make handsome settlements; not only are they rich, but they are
not in the least narrow--as even the best Italians are, I am sorry to
say. The Verneuils are cosmopolitans; they have been everywhere; their
estate is near Brussels, but they spend most of their time in Paris.
They will never tie Eva down in any small way. In addition, both father
and son are extremely nice to _me_."
"Ah!" said Dallas, approvingly.
"Yes; they have the French ideas about mothers; you know that in France
the mother is and remains the most important person in the family." As
she said this, Mrs. Churchill unconsciously lifted herself and threw
back her shoulders. Ordinarily the line from the knot of her hair behind
to her waist was long and somewhat convex, while correspondingly the
distance between her chin and her belt in front was surprisingly short:
she was a plump woman, and she had fallen into the habit of leaning upon
a certain beguiling steel board, which leads a happy existence in
wrappings of white kid and perfumed lace.
"Not only will they never wish to separate me from Eva," she went on,
still abnormally erect, "but such a thought would never enter their
minds; they think it an honor and a pleasure to have me with them; the
old Count assured me of it in those very words."
"And now we have the secret of the Belgian success," said Dallas.
"Yes. But I have not been selfish; I have tried to consider everything;
I have investigated carefully. If you will stay half an hour longer you
can see Pierre for yourself; and then I know that you will agree with
me."
In less than half an hour the Belgian appeared--a slender, handsome
young man of twenty-two, with an ease of manner and grace in movement
which no American of that age ever had. With all his grace, however, and
his air of being a man of the world, there was such a charming
expression of kindliness and purity in his still boyish eyes that any
mother, with her young daughter's happiness at heart, might have been
pardoned for coveting him as a son-in-law. This Dallas immediately
comprehended. "You have chosen well," he said to Fanny, when they were
left for a moment alone; "the boy's a jewel."
Before the arrival of Pierre, Eva Churchill, followed by her governess,
had come out to join her mother on the terrace; Eva's daily lessons were
at an end, save that the music went on; Mlle. Legrand was retained as a
useful companion.
Following Pierre, two more visitors appeared, not together; one was an
Englishman of fifty, small, meagre, plain in face; the other an
American, somewhat younger, a short, ruddy man, dressed like an
Englishman. Mrs. Churchill mentioned their names to Dallas: "Mr.
Gordon-Gray." "Mr. Ferguson."
It soon appeared that Mr. Gordon-Gray and Mr. Ferguson were in the habit
of looking in every afternoon, at about that hour, for a cup of tea.
Dallas, who hated tea, leaned back in his chair and watched the scene,
watched Fanny especially, with the amused eyes of a contemporary who
remembers a different past. Fanny was looking dimpled and young; her tea
was excellent, her tea-service elaborate (there was a samovar); her
daughter was docile, her future son-in-law a Count and a pearl; in
addition, her terrace was an enchanting place for lounging, attached as
it was to a pink-faced villa that overlooked the sea.
Nor were there wanting other soft pleasures. "Dear Mrs.
Murray-Churchill, how delicious is this nest of yours!" said the
Englishman, with quiet ardor; "I never come here without admiring it."
Fanny answered him in a steady voice, though there was a certain
flatness in its tone: "Yes, it's very pretty indeed." Her face was red;
she knew that Dallas was laughing; she would not look in his direction.
Dallas, however, had taken himself off to the parapet, where he could
have his laugh out at ease: to be called Mrs. Murray-Churchill as a
matter of course in that way--what joy for Fanny!
Eva was listening to the busy Mark Ferguson; he was showing her a little
silver statuette which he had unearthed that morning in Naples, "in a
dusty out-of-the-way shop, if you will believe it, where there was
nothing else but rubbish--literally nothing. From the chasing I am
inclined to think it's fifteenth century. But you will need glasses to
see it well; I can lend you a pair of mine."
"I can see it perfectly--thanks," said Eva. "It is very pretty, I
suppose."
"Pretty, Miss Churchill? Surely it's a miracle!" Ferguson protested.
Pierre, who was sitting near the mother, glanced across and smiled. Eva
did not smile in reply; she was looking vaguely at the blackened silver;
but when he came over to see for himself the miracle, then she smiled
very pleasantly.
Pierre was evidently deeply in love; he took no pains to conceal it; but
during the two hours he spent there he made no effort to lure the young
girl into the drawing-room, or even as far as the parapet. He was very
well bred. At present he stood beside her and beside Mark Ferguson, and
talked about the statuette. "It seems to me old Vienna," he said.
"Signor Bartalama," announced Angelo, Mrs. Churchill's man-servant,
appearing at the long window of the drawing-room which served as one of
the terrace doors; he held the lace curtains apart eagerly, with the
smiling Italian welcome.
Fanny had looked up, puzzled. But when her eyes fell upon the figure
emerging from the lace she recognized it instantly. "Horace Bartholomew!
Now from what quarter of the heavens do you drop _this_ time?"
"So glad you call it heaven," said the new-comer, as she gave him her
hand. "But from heaven indeed this time, Mrs. Churchill--I say so
emphatically; from our own great, grand country--with the permission of
the present company be it spoken." And he bowed slightly to the
Englishman and Pierre, his discriminating glance including even the
little French governess, who smiled (though non-comprehendingly) in
reply. "May I present to you a compatriot, Mrs. Churchill?" he went on.
"I have taken the liberty of bringing him without waiting for formal
permission; he is, in fact, in your drawing-room now. His credentials,
however, are small and puny; they consist entirely of the one item--that
I like him."
"That will do perfectly," said Fanny, smiling.
Bartholomew went back to the window and parted the curtains. "Come," he
said. A tall man appeared. "Mrs. Churchill, let me present to you Mr.
David Rod."
Mrs. Churchill was gracious to the stranger; she offered him a chair
near hers, which he accepted; a cup of tea, which he declined; and the
usual small questions of a first meeting, which only very original minds
are bold enough to jump over. The stranger answered the questions
promptly; he was evidently not original. He had arrived two days before;
this was his first visit to Italy; the Bay of Naples was beautiful; he
had not been up Vesuvius; he had not visited Pompeii; he was not afraid
of fever; and he had met Horace Bartholomew in Florida the year before.
"I am told they are beginning to go a great deal to Florida," remarked
Fanny.
"I don't go there; I live there," Rod answered.
"Indeed! in what part?" (She brought forward the only names she knew.)
"St. Augustine, perhaps? Or Tallahassee?"
"No; I live on the southern coast; at Punta Palmas?"
"How Spanish that is! Perhaps you have one of those old Spanish
plantations?" She had now exhausted all her knowledge of the State save
a vague memory of her school geography: "Where are the Everglades?"
"They are in the southern part of Florida. They are shallow lakes filled
with trees." But the stranger could hardly live in such a place as that.
"No," answered Rod; "my plantation isn't old and it isn't Spanish; it's
a farm, and quite new. I am over here now to get hands for it."
"Hands?"
"Yes, laborers--Italians. They work very well in Florida."
Eva and Mademoiselle Legrand had turned with Pierre to look at the
magnificent sunset. "Did you receive the flowers I sent this morning?"
said Pierre, bending his head so that if Eva should glance up when she
answered, he should be able to look into her eyes.
"Yes; they were beautiful," said Eva, giving the hoped-for glance.
"Yet they are not in the drawing-room."
"You noticed that?" she said, smiling. "They are in the music-room;
Mademoiselle put them there."
"They are the flowers for Mozart, are they not?" said
Mademoiselle--"heliotrope and white lilies; and we have been studying
Mozart this morning. The drawing-room, as you know, Monsieur le Comte,
is always full of roses."
"And how do you come on with Mozart?" asked Pierre.
"As usual," answered Eva. "Not very well, I suppose."
[Illustration: "'MRS. CHURCHILL, LET ME PRESENT TO YOU MR. DAVID ROD'"]
Mademoiselle twisted her handkerchief round her fingers. She was
passionately fond of music; it seemed to her that her pupil, who played
accurately, was not. Pierre also was fond of music, and played with
taste. He had not perceived Eva's coldness in this respect simply
because he saw no fault in her.
"I want to make up a party for the Deserto," he went on, "to lunch
there. Do you think Madame Churchill will consent?"
"Probably," said Eva.
"I hope she will. For when we are abroad together, under the open sky,
then it sometimes happens I can stay longer by your side."
"Yes; we never have very long talks, do we?" remarked Eva, reflectively.
"Do you desire them?" said Pierre, with ardor. "Ah, if you could know
how I do! With me it is one long thirst. Say that you share the feeling,
even if only a little; give me that pleasure."
"No," said Eva laughing, "I don't share it at all. Because, if we should
have longer talks, you would find out too clearly that I am not clever."
"Not clever!" said Pierre, with all his heart in his eyes. Then, with
his unfailing politeness, he included Mademoiselle. "She is clever,
Mademoiselle?"
"She is good," answered Mademoiselle, gravely. "Her heart has a
depth--but a depth!"
"I shall fill it all," murmured Pierre to Eva. "It is not that I myself
am anything, but my love is so great, so vast; it holds you as the sea
holds Capri. Some time--some time, you must let me try to tell you!"
Eva glanced at him. Her eyes had for the moment a vague expression of
curiosity.
This little conversation had been carried on in French; Mademoiselle
spoke no English, and Pierre would have been incapable of the rudeness
of excluding her by means of a foreign tongue.
II
The pink villa was indeed a delicious nest, to use the Englishman's
phrase. It crowned one of the perpendicular cliffs of Sorrento, its rosy
facade overlooking what is perhaps the most beautiful expanse of water
in the world--the Bay of Naples. The broad terrace stretched from the
drawing room windows to the verge of the precipice; leaning against its
strong stone parapet, with one's elbows comfortably supported on the
flat top (which supported also several battered goddesses of marble),
enjoying the shade of a lemon-tree set in a great vase of tawny
terra-cotta--leaning thus, one could let one's idle gaze drop straight
down into the deep blue water below, or turn it to the white line of
Naples opposite, shining under castled heights, to Vesuvius with its
plume of smoke, or to beautiful dark Ischia rising from the waves in the
west, guarding the entrance to the sea. On each side, close at hand, the
cliffs of Sorrento stretched away, tipped with their villas, with their
crowded orange and lemon groves. Each villa had its private stairway
leading to the beach below; strange dark passages, for the most part cut
in the solid rock, winding down close to the face of the cliff, so that
every now and then a little rock-window can let in a gleam of light to
keep up the spirits of those who are descending. For every one does
descend: to sit and read among the rocks; to bathe from the
bathing-house on the fringe of beach; to embark for a row to the
grottos or a sail to Capri.
[Illustration: SORRENTO]
The afternoon which followed the first visit of Philip Dallas to the
pink villa found him there a second time; again he was on the terrace
with Fanny. The plunging sea-birds of the terrace's mosaic floor were
partially covered by a large Persian rug, and it was upon this rich
surface that the easy-chairs were assembled, and also the low tea-table,
which was of a construction so solid that no one could possibly knock it
over. A keen observer had once said that that table was in itself a
sufficient indication that Fanny's house was furnished to attract
masculine, not feminine, visitors (a remark which was perfectly true).
"You are the sun of a system of masculine planets, Fanny," said Dallas.
"After long years, that is how I find you."
"Oh, Philip--we who live so quietly!"
"So is the sun quiet, I suppose; I have never heard that he howled. Mr.
Gordon-Gray, Mark Ferguson, Pierre de Vernueil, Horace Bartholomew,
unknown Americans. Do they come to see Eva or you?"
"They come to see the view--as you do; to sit in the shade and talk. I
give very good dinners too," Fanny added, with simplicity.
"O romance! good dinners on the Bay of Naples!"
"Well, you may laugh; but nothing draws men of a certain age--of a
certain kind, I mean; the most satisfactory men, in short--nothing draws
them so surely as a good dinner delicately served," announced Fanny,
with decision. "Please go and ring for the tea."
"I don't wonder that they all hang about you," remarked Dallas as he
came back, his eyes turning from the view to his hostess in her
easy-chair. "Your villa is admirable, and you yourself, as you sit
there, are the personification of comfort, the personification, too, of
gentle, sweet, undemonstrative affectionateness. Do you know that,
Fanny?"
Fanny, with a very pink blush, busied herself in arranging the table for
the coming cups.
Dallas smiled inwardly. "She thinks I am in love with her because I said
that about affectionateness," he thought. "Oh, the fatuity of women!"
At this moment Eva came out, and presently appeared Mr. Gordon-Gray and
Mark Ferguson. A little later came Horace Bartholomew. The tea had been
brought; Eva handed the cups. Dallas, looking at her, was again struck
by something in the manner and bearing of Fanny's daughter. Or rather he
was not struck by it; it was an impression that made itself felt by
degrees, as it had done the day before--a slow discovery that the girl
was unusual.
She was tall, dressed very simply in white. Her thick smooth flaxen hair
was braided in two long flat tresses behind, which were doubled and
gathered up with a ribbon, so that they only reached her shoulders. This
school-girl coiffure became her young face well. Yes, it was a very
young face. Yet it was a serious face too. "Our American girls are often
serious, and when they are brought up under the foreign system it really
makes them too quiet," thought Dallas. Eva had a pair of large gray eyes
under dark lashes: these eyes were thoughtful; sometimes they were dull.
Her smooth complexion was rather brown. The oval of her face was
perfect. Though her dress was so child-like, her figure was womanly; the
poise of her head was noble, her step light and free. Nothing could be
more unlike the dimpled, smiling mother than was this tall, serious
daughter who followed in her train. Dallas tried to recall Edward
Churchill (Edward Murray Churchill), but could not; he had only seen him
once. "He must have been an obstinate sort of fellow," he said to
himself. The idea had come to him suddenly from something in Eva's
expression. Yet it was a sweet expression; the curve of the lips was
sweet.
"She isn't such a very pretty girl, after all," he reflected, summing
her up finally before he dismissed her. "Fanny is a clever woman to have
made it appear that she is."
At this moment Eva, having finished her duties as cup-bearer, walked
across the terrace and stood by the parapet, outlined against the light.
"By Jove she's beautiful!" thought Dallas.
Fanny's father had not liked Edward Churchill; he had therefore left his
money tied up in such a way that neither Churchill nor any children whom
he might have should be much benefited by it; Fanny herself, though she
had a comfortable income for life, could not dispose of it. This
accounted for the very small sum belonging to Eva: she had only the few
hundreds that came to her from her father.
But she had been brought up as though she had many thousands; studiedly
quiet as her life had been, studiedly simple as her attire always was,
in every other respect her existence had been arranged as though a large
fortune certainly awaited her. This had been the mother's idea; she had
been sure from the beginning that a large fortune did await her
daughter. It now appeared that she had been right.
"I don't know what you thought of me for bringing a fellow-countryman
down upon you yesterday in that unceremonious way, Mrs. Churchill,"
Bartholomew was saying. "But I wanted to do something for him--I met him
at the top of your lane by accident; it was an impulse."
"Oh, I'm sure--any friend of yours--" said Fanny, looking into the
teapot.
Bartholomew glanced round the little circle on the rug, with an
expression of dry humor in his brown eyes. "You didn't any of you like
him--I see that," he said.
There was a moment's silence.
"Well, he is rather a commonplace individual, isn't he?" said Dallas,
unconsciously assuming the leadership of this purely feminine household.
"I don't know what you mean by commonplace; but yes, I do, coming from
_you_, Dallas. Rod has never been abroad in his life until now; and he's
a man with convictions."
"Oh, come, don't take that tone," said Mark Ferguson; "I've got
convictions too; I'm as obstinate about them as an Englishman."
"What did your convictions tell you about Rod, then, may I ask?" pursued
Bartholomew.
"I didn't have much conversation with him, you may remember; I thought
he had plenty of intelligence. His clothes were--were a little peculiar,
weren't they?"
"Made in Tampa, probably. And I've no doubt but that he took pains with
them--wanted to have them appropriate."
"That is where he disappointed me," said Gordon-Gray--"that very
appearance of having taken pains. When I learned that he came from
that--that place in the States you have just named--a wild part of the
country, is it not?--I thought he would be more--more interesting. But
he might as well have come from Clerkenwell."
"You thought he would be more wild, you mean; trousers in his boots;
long hair; knives."
All the Americans laughed.
"Yes. I dare say you cannot at all comprehend our penchant for that sort
of thing," said the Englishman, composedly. "And--er--I am afraid there
would be little use in attempting to explain it to you. But this Mr. Rod
seemed to me painfully unconscious of his opportunities; he told me
(when I asked) that there was plenty of game there--deer, and even bears
and panthers--royal game; yet he never hunts."
"He never hunts, because he has something better to do," retorted
Bartholomew.
"Ah, better?" murmured the Englishman, doubtfully.
Bartholomew got up and took a chair which was nearer Fanny. "No--no
tea," he said, as she made a motion towards a cup; then, without further
explaining his change of position, he gave her a little smile. Dallas,
who caught this smile on the wing, learned from it unexpectedly that
there was a closer intimacy between his hostess and Bartholomew than he
had suspected. "Bartholomew!" he thought, contemptuously.
"Gray--spectacles--stout." Then suddenly recollecting the increasing
plumpness of his own person, he drew in his out-stretched legs, and
determined, from that instant, to walk fifteen miles a day.
"Rod knows how to shoot, even though he doesn't hunt," said
Bartholomew, addressing the Englishman. "I saw him once bring down a mad
bull, who was charging directly upon an old man--the neatest sort of a
hit."
"He himself being in a safe place meanwhile," said Dallas.
"On the contrary, he had to rush forward into an open field. If he had
missed his aim by an eighth of an inch, the beast--a terrible
creature--would have made an end of him."
"And the poor old man?" said Eva.
"He was saved, of course; he was a rather disreputable old <DW54>.
Another time Rod went out in a howling gale--the kind they have down
there--to rescue two men whose boat had capsized in the bay. They were
clinging to the bottom; no one else would stir; they said it was certain
death; but Rod went out--he's a capital sailor--and got them in. I
didn't see that myself, as I saw the bull episode; I was told about it."
"By Rod?" said Dallas.
"By one of the men he saved. As you've never been saved yourself,
Dallas, you probably don't know how it feels."
"He seems to be a modern Chevalier Bayard, doesn't he?" said
good-natured Mark Ferguson.
"He's modern, but no Bayard. He's a modern and a model pioneer--"
"Pioneers! oh, pioneers!" murmured Gordon-Gray, half chanting it.
None of the Americans recognized his quotation.
"He's the son of a Methodist minister," Bartholomew went on. "His
father, a missionary, wandered down to Florida in the early days, and
died there, leaving a sickly wife and seven children. You know the sort
of man--a linen duster for a coat, prunella shoes, always smiling and
hopeful--a great deal about 'Brethren.' Fortunately they could at least
be warm in that climate, and fish were to be had for the catching; but I
suspect it was a struggle for existence while the boys were small. David
was the youngest; his five brothers, who had come up almost laborers,
were determined to give this lad a chance if they could; together they
managed to send him to school, and later to a forlorn little Methodist
college somewhere in Georgia. David doesn't call it forlorn, mind you;
he still thinks it an important institution. For nine years now--he is
thirty--he has taken care of himself; he and a partner have cleared this
large farm, and have already done well with it. Their hope is to put it
all into sugar in time, and a Northern man with capital has advanced
them the money for this Italian colonization scheme: it has been tried
before in Florida, and has worked well. They have been very
enterprising, David and his partner; they have a saw-mill running, and
two school-houses already--one for whites, one for blacks. You ought to
see the little <DW54>s, with their wool twisted into twenty tails, going
proudly in when the bell rings," he added, turning to Fanny.
"And the white children, do they go too?" said Eva.
"Yes, to their own school-house--lank girls, in immense sun-bonnets,
stalking on long bare feet. He has got a brisk little Yankee
school-mistress for them. In ten years more I declare he will have
civilized that entire neighborhood."
"You are evidently the Northern man with capital," said Dallas.
"I don't care in the least for your sneers, Dallas; I'm not the Northern
man, but I should like to be. If I admire Rod, with his constant driving
action, his indomitable pluck, his simple but tremendous belief in the
importance of what he has undertaken to do, that's my own affair. I do
admire him just as he stands, clothes and all; I admire his creaking
saw-mill; I admire his groaning dredge; I even admire his two hideously
ugly new school-houses, set staring among the stumps."
"Tell me one thing, does he preach in the school-houses on Sundays and
Friday evenings, say?" asked Ferguson. "Because if he does he will make
no money, whatever else he may make. They never do if they preach."
"It's his father who was the minister, not he," said Bartholomew. "David
never preached in his life; he wouldn't in the least know how. In fact,
he's no talker at all; he says very little at any time; he's a
doer--David is; he _does_ things. I declare it used to make me sick of
myself to see how much that fellow accomplished every day of his life
down there, and thought nothing of it at all."
"And what were you doing 'down there,' besides making yourself sick, if
I may ask?" said Ferguson.
"Oh, I went down for the hunting, of course. What else does one go to
such a place for?"
"Tell me a little about that, if you don't mind," said the Englishman,
interested for the first time.
"M. de Verneuil wants us all to go to the Deserto some day soon," said
Fanny; "a lunch party. We shall be sure to enjoy it; M. de Verneuil's
parties are always delightful."
III
The end of the week had been appointed for Pierre's excursion.
The morning opened fair and warm, with the veiled blue that belongs to
the Bay of Naples, the soft hazy blue which is so different from the dry
glittering clearness of the Riviera.
Fanny was mounted on a donkey; Eva preferred to walk, and Mademoiselle
accompanied her. Pierre had included in his invitation the usual
afternoon assemblage at the villa--Dallas, Mark Ferguson, Bartholomew,
Gordon-Gray, and David Rod.
For Fanny had, as Dallas expressed it, "taken up" Rod; she had invited
him twice to dinner. The superfluous courtesy had annoyed Dallas, for of
course, as Rod himself was nothing, less than nothing, the explanation
must lie in the fact that Horace Bartholomew had suggested it.
"Bartholomew was always wrong-headed; always picking up some perfectly
impossible creature, and ramming him down people's throats," he thought,
with vexation.
Bartholomew was walking now beside Fanny's donkey.
Mark Ferguson led the party, as it moved slowly along the narrow paved
road that winds in zigzags up the mountain; Eva, Mademoiselle, Pierre,
Dallas, and Rod came next. Fanny and Bartholomew were behind; and
behind still, walking alone and meditatively, came Gordon-Gray, who
looked at life (save for the hunting) from the standpoint of the Italian
Renaissance. Gordon-Gray knew a great deal about the Malatesta family;
he had made a collection of Renaissance cloak clasps; he had written an
essay on the colors of the long hose worn in the battling,
leg-displaying days which had aroused his admiration, aroused it rather
singularly, since he himself was as far as possible from having been
qualified by nature to shine in such vigorous society.
Pierre went back to give some directions to one of the men in the rear
of their small procession.
When he returned, "So the bears sometimes get among the canes?" Eva was
saying.
"But then, how very convenient," said Pierre; "for they can take the
canes and chastise them punctually." He spoke in his careful English.
"They're sugar-canes," said Rod.
"It's his plantation we are talking about," said Eva. "Once it was a
military post, he says. Perhaps like Ehrenbreitstein."
"Exactly," said Dallas, from behind; "the same massive frowning stone
walls."
"There were four one-story wooden barracks once," said Rod;
"whitewashed; flag-pole in the centre. There's nothing now but a
chimney; we've taken the boards for our mill."
"See the cyclamen, good folk," called out Gordon-Gray.
On a small plateau near by a thousand cyclamen, white and pink, had
lifted their wings as if to fly away. Off went Pierre to get them for
Eva.
[Illustration: ON THE WAY TO THE DESERTO]
"Have you ever seen the bears in the canes yourself?" pursued Eva.
"I've seen them in many places besides canes," answered Rod, grimly.
"I too have seen bears," Eva went on. "At Berne, you know."
"The Punta Palmas bears are quite the same," commented Dallas. "When
they see Mr. Rod coming they sit up on their hind legs politely. And he
throws them apples."
"No apples; they won't grow there," said Rod, regretfully. "Only
oranges."
"Do you make the saw-mill go yourself--with your own hands?" pursued
Eva.
"Not now. I did once."
"Wasn't it very hard work?"
"That? Nothing at all. You should have seen us grubbing up the
stumps--Tipp and I!"
"Mr. Tipp is perhaps your partner?" said Dallas.
"Yes; Jim Tipp. Tipp and Rod is the name of the firm."
"Tipp--and Rod," repeated Dallas, slowly. Then with quick utterance, as
if trying it, "Tippandrod."
Pierre was now returning with his flowers. As he joined them, round the
corner of their zigzag, from a pasture above came a troop of ponies that
had escaped from their driver, and were galloping down to Sorrento; two
and two they came rushing on, too rapidly to stop, and everybody pressed
to one side to give them room to pass on the narrow causeway.
Pierre jumped up on the low stone wall and extended his hand to Eva.
"Come!" he said, hastily.
Rod put out his arm and pushed each outside pony, as he passed Eva,
forcibly against his mate who had the inside place; a broad space was
thus left beside her, and she had no need to leave the causeway. She had
given one hand to Pierre as a beginning; he held it tightly.
Mademoiselle meanwhile had climbed the wall like a cat. There were
twenty of the galloping little nags; they took a minute or two to pass.
Rod's out-stretched hands, as he warded them off, were seen to be large
and brown.
Eva imagined them "grubbing up" the stumps. "What is grubbing?" she
said.
"It is writing for the newspapers in a street in London," said Pierre,
jumping down. "And you must wear a torn coat, I believe." Pierre was
proud of his English.
He presented his flowers.
Mademoiselle admired them volubly. "They are like souls just ready to
wing their way to another world," she said, sentimentally, with her head
on one side. She put her well-gloved hand in Eva's arm, summoned Pierre
with an amiable gesture to the vacant place at Eva's left hand, and the
three walked on together.
The Deserto, though disestablished and dismantled, like many another
monastery, by the rising young kingdom, held still a few monks; their
brown-robed brethren had aided Pierre's servant in arranging the table
in the high room which commands the wonderful view of the sea both to
the north and the south of the Sorrento peninsula, with Capri lying at
its point too fair to be real--like an island in a dream.
"O la douce folie--
Aimable Capri!"
said Mark Ferguson. No one knew what he meant; he did not know himself.
It was a poetical inspiration--so he said.
[Illustration: AT THE DESERTO]
The lunch was delicate, exquisite; everything save the coffee (which the
monks wished to provide: coffee, black-bread, and grapes which were half
raisins was the monks' idea of a lunch) had been sent up from Sorrento.
Dallas, who was seated beside Fanny, gave her a congratulatory nod.
"Yes, all Pierre does is well done," she answered, in a low tone, unable
to deny herself this expression of maternal content.
Pierre was certainly a charming host. He gave them a toast; he gave them
two; he gave them a song: he had a tenor voice which had been admirably
cultivated, and his song was gay and sweet. He looked very handsome; he
wore one of the cyclamen in his button-hole; Eva wore the rest, arranged
by the deft fingers of Mademoiselle in a knot at her belt. But at the
little feast Fanny was much more prominent than her daughter: this was
Pierre's idea of what was proper; he asked her opinion, he referred
everything to her with a smile which was homage in itself. Dallas, after
a while, was seized with a malicious desire to take down for a moment
this too prosperous companion of his boyhood. It was after Pierre had
finished his little song. "Do you ever sing now, Fanny?" he asked,
during a silence. "I remember how you used to sing Trancadillo."
"I am sure I don't know what you refer to," answered Fanny, coldly.
Another week passed. They sailed to Capri; they sailed to Ischia; they
visited Pompeii. Bartholomew suggested these excursions. Eva too showed
an almost passionate desire for constant movement, constant action.
"Where shall we go to-day, mamma?" she asked every morning.
One afternoon they were strolling through an orange grove on the
outskirts of Sorrento. Under the trees the ground was ploughed and
rough; low stone copings, from whose interstices innumerable violets
swung, ran hither and thither, and the paths followed the copings. The
fruit hung thickly on the trees. Above the high wall which surrounded
the place loomed the campanile of an old church. While they were
strolling the bells rang the Angelus, swinging far out against the blue.
Rod, who was of the party, was absent-minded; he looked a little at the
trees, but said nothing, and after a while he became absent-bodied as
well, for he fell behind the others, and pursued his meditations,
whatever they were, in solitude.
"He is bothered about his Italians," said Bartholomew; "he has only
secured twenty so far."
Pierre joined Fanny; he had not talked with her that afternoon, and he
now came to fulfil the pleasant duty. Eva, who had been left with
Mademoiselle, turned round, and walking rapidly across the ploughed
ground, joined Rod, who was sitting on one of the low stone walls at
some distance from the party. Mademoiselle followed her, putting on her
glasses as she went, in order to see her way over the heaped ridges. She
held up her skirts, and gave ineffectual little leaps, always landing in
the wrong spot, and tumbling up hill, as Dallas called it. "Blue," he
remarked, meditatively. Every one glanced in that direction, and it was
perceived that the adjective described the hue of Mademoiselle's
birdlike ankles.
"For shame!" said Fanny.
But Dallas continued his observations. "Do look across," he said, after
a while; "it's too funny. The French woman evidently thinks that Rod
should rise, or else that Eva should be seated also. But her pantomime
passes unheeded; neither Eva nor the backwoodsman is conscious of her
existence."
"Eva is so fond of standing," explained Fanny. "I often say to her, 'Do
sit down, child; it tires me to see you.' But Eva is never tired."
Pierre, who had a spray of orange buds in his hand, pressed it to his
lips, and waved it imperceptibly towards his betrothed. "In everything
she is perfect--perfect," he murmured to the pretty mother.
"Rod doesn't in the least mean to be rude," began Bartholomew.
"Oh, don't explain that importation of yours at this late day,"
interposed Dallas; "it isn't necessary. He is accustomed to sitting on
fences probably; he belongs to the era of the singing-school."
This made Fanny angry. For as to singing-schools, there had been a
time--a remote time long ago--and Dallas knew it. She had smiled in
answer to Pierre's murmured rapture; she now took his arm. To punish
Dallas she turned her steps--on her plump little feet in their delicate
kid boots--towards the still seated Rod, with the intention of asking
him (for the fifth time) to dinner. This would not only exasperate
Dallas, but it would please Bartholomew at the same stroke. Two birds,
etc.
When they came up to the distant three, Mademoiselle glanced at Mrs.
Churchill anxiously. But in the presence of the mistress of the villa,
Rod did at last lift his long length from the wall.
This seemed, however, to be because he supposed they were about to leave
the grove. "Is the walk over?" he said.
Pierre looked at Eva adoringly. He gave her the spray of orange buds.
IV
A week later Fanny's daughter entered the bedroom which she shared with
her mother.
From the girl's babyhood the mother had had her small white-curtained
couch placed close beside her own. She could not have slept unless able
at any moment to stretch out her hand and touch her sleeping child.
Fanny was in the dressing-room; hearing Eva's step, she spoke. "Do you
want me, Eva?"
"Yes, please."
Fanny appeared, a vision of white arms, lace, and embroidery.
"I thought that Rosine would not be here yet," said Eva. Rosine was
their maid; her principal occupation was the elaborate arrangement of
Fanny's brown hair.
"No, she isn't there--if you mean in the dressing-room," answered Fanny,
nodding her head towards the open door.
"I wanted to see you alone, mamma, for a moment. I wanted to tell you
that I shall not marry Pierre."
Fanny, who had sunk into an easy-chair, at these words sprang up. "What
is the matter? Are you ill?"
"Not in the least, mamma; I am only telling you that I cannot marry
Pierre."
"You _must_ be ill," pursued Fanny. "You have fever. Don't deny it." And
anxiously she took the girl's hands. But Eva's hands were cooler than
her own.
"I don't think I have any fever," replied Eva. She had been taught to
answer all her mother's questions in fullest detail. "I sleep and eat as
usual; I have no headache."
Fanny still looked at her anxiously. "Then if you are not ill, what can
be the matter with you?"
"I have only told you, mamma, that I could not marry Pierre; it seems to
me very simple."
She was so quiet that Fanny began at last to realize that she was in
earnest. "My dearest, you know you like Pierre. You have told me so
yourself."
"I don't like him now."
"What has he done--poor Pierre? He will explain, apologize; you may be
sure of that."
"He has done nothing; I don't want him to apologize. He is as he always
is. It is I who have changed."
"Oh, it is you who have changed," repeated Fanny, bewildered.
"Yes," answered Eva.
"Come and sit down and tell mamma all about it. You are tired of poor
Pierre--is that it? It is very natural, he has been here so often, and
stayed so long. But I will tell him that he must go away--leave
Sorrento. And he shall stay away as long as you like, Eva; just as long
as you like."
"Then he will stay away forever," the girl answered, calmly.
Fanny waited a moment. "Did you like Gino better? Is that it?" she said,
softly, watching Eva's face.
"No."
"Thornton Stanley?"
"Oh no!"
"Dear child, explain this a little to your mother. You know I think only
of your happiness," said Fanny, with tender solicitude.
Eva evidently tried to obey. "It was this morning. It came over me
suddenly that I could not possibly marry him. Now or a year from now.
Never." She spoke tranquilly; she even seemed indifferent. But this one
decision was made.
"You know that I have given my word to the old Count," began Fanny, in
perplexity.
Eva was silent.
"And everything was arranged."
Eva still said nothing. She looked about the room with wandering
attention, as though this did not concern her.
"Of course I would never force you into anything," Fanny went on. "But I
thought Pierre would be so congenial." In her heart she was asking
herself what the young Belgian could have done. "Well, dear," she
continued, with a little sigh, "you must always tell mamma everything."
And she kissed her.
"Of course," Eva answered. And then she went away.
Fanny immediately rang the bell, and asked for Mademoiselle. But
Mademoiselle knew nothing about it. She was overwhelmed with surprise
and dismay. She greatly admired Pierre; even more she admired the old
Count, whom she thought the most distinguished of men. Fanny dismissed
the afflicted little woman, and sat pondering. While she was thinking,
Eva re-entered.
"Mamma, I forgot to say that I should like to have you tell Pierre
immediately. To-day."
Fanny was almost irritated. "You have never taken that tone before, my
daughter. Have you no longer confidence in my judgment?"
"If you do not want to tell him this afternoon, it can be easily
arranged, mamma; I will not come to the dinner-table; that is all. I do
not wish to see him until he knows."
Pierre was to dine at the villa that evening.
"What can he have done?" thought Fanny again.
She rang for Rosine; half an hour later she was in the drawing-room.
"Excuse me to every one but M. de Verneuil," she said to Angelo. She was
very nervous, but she had decided upon her course: Pierre must leave
Sorrento, and remain away until she herself should call him back.
"At the end of a month, perhaps even at the end of a week, she will miss
you so much that I shall have to issue the summons," she said, speaking
as gayly as she could, as if to make it a sort of joke. It was very hard
for her, at best, to send away the frank, handsome boy.
Poor Pierre could not understand it at all. He declared over and over
again that nothing he had said, nothing he had done, could possibly have
offended his betrothed. "But surely you know yourself that it is
impossible!" he added, clasping his hands beseechingly.
"It is a girlish freak," explained the mother. "She is so young, you
know."
"But that is the very reason. I thought it was only older women who say
what they wish to do in that decided way; who have freaks, as you call
it," said the Belgian, his voice for a moment much older, more like the
voice of a man who has spent half his life in Paris.
This was so true that Fanny was driven to a defence that scarcely
anything else would have made her use.
"Eva is different from the young girls here," she said. "You must not
forget that she is an American."
At last Pierre went away; he had tried to bear himself as a gentleman
should; but the whole affair was a mystery to him, and he was very
unhappy. He went as far as Rome, and there he waited, writing to Fanny
an anxious letter almost every day.
In the meanwhile life at the villa went on; there were many excursions.
Fanny's thought was that Eva would miss Pierre more during these
expeditions than at other times, for Pierre had always arranged them,
and he had enjoyed them so much himself that his gay spirits and his gay
wit had made all the party gay. Eva, however, seemed very happy, and at
length the mother could not help being touched to see how light-hearted
her serious child had become, now that she was entirely free. And yet
how slight the yoke had been, and how pleasant! thought Fanny. At the
end of two weeks there were still no signs of the "missing" upon which
she had counted. She thought that she would try the effect of briefly
mentioning the banished man. "I hear from Pierre almost every day, poor
fellow. He is in Rome."
"Why does he stay in Rome?" said Eva. "Why doesn't he return home?"
"I suppose he doesn't want to go so far away," answered Fanny, vaguely.
"Far away from what? Home should always be the first place," responded
the young moralist. "Of course you have told him, mamma, that I shall
never be his wife? That it is forever?" And she turned her gray eyes
towards her mother, for the first time with a shade of suspicion in
them.
"Never is a long word, Eva."
"Oh, mamma!" The girl rose. "I shall write to him myself, then."
"How you speak! Do you wish to disobey me, my own little girl?"
"No; but it is so dishonest; it is like a lie."
"My dear, trust your mother. You have changed once; you may change
again."
"Not about this, mamma. Will you please write this very hour, and make
an end of it?"
"You are hard, Eva. You do not think of poor Pierre at all."
"No, I do not think of Pierre."
"And is there any one else you think of? I must ask you that once more,"
said Fanny, drawing her daughter down beside her caressingly. Her
thoughts could not help turning again towards Gino, and in her supreme
love for her child she now accomplished the mental somerset of believing
that on the whole she preferred the young Italian to all the liberty,
all the personal consideration for herself, which had been embodied in
the name of Verneuil.
"Yes, there is some one else I think of," Eva replied, in a low voice.
"In Rome?" said Fanny.
Eva made a gesture of denial that was fairly contemptuous.
Fanny's mind flew wildly from Bartholomew to Dallas, from Ferguson to
Gordon-Gray: Eva had no acquaintances save those which were her
mother's also.
"It is David Rod," Eva went on, in the same low tone. Then, with sudden
exaltation, her eyes gleaming, "I have never seen any one like him."
It was a shock so unexpected that Mrs. Churchill drew her breath under
it audibly, as one does under an actual blow. But instantly she rallied.
She said to herself that she had got a romantic idealist for a
daughter--that was all. She had not suspected it; she had thought of Eva
as a lovely child who would develop into what she herself had been.
Fanny, though far-seeing and intelligent, had not been endowed with
imagination. But now that she did realize it, she should know how to
deal with it. A disposition like that, full of visionary fancies, was
not so uncommon as some people supposed. Horace Bartholomew should take
the Floridian away out of Eva's sight forever, and the girl would soon
forget him; in the meanwhile not one word that was harsh should be
spoken on the subject, for that would be the worst policy of all.
This train of thought had passed through her mind like a flash. "My
dear," she began, as soon as she had got her breath back, "you are right
to be so honest with me. Mr. Rod has not--has not said anything to you
on the subject, has he?"
"No. Didn't I tell you that he cares nothing for me? I think he despises
me--I am so useless!" And then suddenly the girl began to sob; a passion
of tears.
Fanny was at her wits' end; Eva had not wept since the day of her baby
ills, for life had been happy to her, loved, caressed, and protected as
she had been always, like a hot-house flower.
"My darling," said the mother, taking her in her arms.
But Eva wept on and on, as if her heart would break. It ended in Fanny's
crying too.
V
Early the next morning her letter to Bartholomew was sent. Bartholomew
had gone to Munich for a week. The letter begged, commanded, that he
should make some pretext that would call David Rod from Sorrento at the
earliest possible moment. She counted upon her fingers; four days for
the letter to go and the answer to return. Those four days she would
spend at Capri.
Eva went with her quietly. There had been no more conversation between
mother and daughter about Rod; Fanny thought that this was best.
On the fourth day there came a letter from Bartholomew. Fanny returned
to Sorrento almost gayly: the man would be gone.
But he was not gone. Tranquillized, glad to be at home again, Mrs.
Churchill was enjoying her terrace and her view, when Angelo appeared at
the window: "Signor Ra."
Angelo's mistress made him a peremptory sign. "Ask the gentleman to wait
in the drawing-room," she said. Then crossing to Eva, who had risen, "Go
round by the other door to our own room, Eva," she whispered.
The girl did not move; her face had an excited look. "But why--"
"Go, child; go."
Still Eva stood there, her eyes fixed upon the long window veiled in
lace; she scarcely seemed to breathe.
Her mother was driven to stronger measures. "You told me yourself that
he cared nothing for you."
A deep red rose in Eva's cheeks; she turned and left the terrace by the
distant door.
The mother crossed slowly to the long window and parted the curtains.
"Mr. Rod, are you there? Won't you come out? Or stay--I will join you."
She entered the drawing-room and took a seat.
Rod explained that he was about to leave Sorrento; Bartholomew had
summoned him so urgently that he did not like to refuse, though it was
very inconvenient to go at such short notice.
"Then you leave to-morrow?" said Fanny; "perhaps to-night?"
"No; on Monday. I could not arrange my business before."
"Three days more," Fanny thought.
She talked of various matters; she hoped that some one else would come
in; but, by a chance, no one appeared that day, neither Dallas, nor
Ferguson, nor Gordon-Gray. "What can have become of them?" she thought,
with irritation. After a while she gave an inward start; she had become
conscious of a foot-fall passing to and fro behind the half-open door
near her--a door which led into the dining-room. It was a very soft
foot-fall upon a thick carpet, but she recognized it: it was Eva. She
was there--why? The mother could think of no good reason. Her heart
began to beat more quickly; for the first time in her life she did not
know her child. This person walking up and down behind that door so
insistently, this was not Eva. Eva was docile; this person was not
docile. What would be done next? She felt strangely frightened. It was a
proof of her terror that she did not dare to close the door lest it
should be instantly reopened. She began to watch every word she said to
Rod, who had not perceived the foot-fall. She began to be
extraordinarily polite to him; she stumbled through the most irrelevant
complimentary sentences. Her dread was, every minute, lest Eva should
appear.
But Eva did not appear; and at last, after long lingering, Rod went
away. Fanny, who had hoped to bid him a final farewell, had not dared to
go through that ceremony. He said that he should come again.
When at last he was gone the mother pushed open the half-closed door.
"Eva," she began. She had intended to be severe, as severe as she
possibly could be; but the sight of Eva stopped her. The girl had flung
herself down upon the floor, her bowed head resting upon her arms on a
chair. Her attitude expressed a hopeless desolation.
"What is it?" said Fanny, rushing to her.
Eva raised her head. "He never once spoke of me--asked for me," she
murmured, looking at her mother with eyes so dreary with grief that any
one must have pitied her.
Her mother pitied her, though it was an angry pity, too--a
non-comprehending, jealous, exasperated feeling. She sat down and
gathered her child to her breast with a gesture that was almost fierce.
That Eva should suffer so cruelly when she, Fanny, would have made any
sacrifice to save her from it, would have died for her gladly, were it
not that she was the girl's only protector--oh, what fate had come over
their happy life together! She had not the heart to be stern. All she
said was, "We will go away, dear; we will go away."
"No," said Eva, rising; "let me stay here. You need not be afraid."
"Of course I am not afraid," answered Fanny, gravely. "My daughter will
never do anything unseemly; she has too much pride."
"I am afraid I have no pride--that is, not as you have it, mamma. Pride
doesn't seem to me at all important compared with---- But of course I
know that there is nothing I can do. He is perfectly indifferent. Only
do not take me away again--do not."
"Why do you wish to stay?"
"Because then I can think--for three days more--that he is at least as
near me as that." She trembled as she said this; there was a spot of
sombre red in each cheek; her fair face looked strange amid her
disordered hair.
Her mother watched her helplessly. All her beliefs, all her creed, all
her precedents, the experience of her own life and her own nature even,
failed to explain such a phenomenon as this. And it was her own child
who was saying these things.
The next day Eva was passive. She wandered about the terrace, or sat for
hours motionless staring blankly at the sea. Her mother left her to
herself. She had comprehended that words were useless. She pretended to
be embroidering, but in reality as she drew her stitches she was
counting the hours as they passed: seventy-two hours; forty-eight hours.
Would he ever be gone?
[Illustration: "SHE SAT DOWN AND GATHERED HER CHILD TO HER BREAST"]
On the second day, in the afternoon, she discovered that Eva had
disappeared. The girl had been on the terrace with Mademoiselle;
Mademoiselle had gone to her room for a moment, and when she returned
her pupil could not be found. She had not passed through the
drawing-room, where Fanny was sitting with her pretended industry; nor
through the other door, for Rosine was at work there, and had seen
nothing of her. There remained only the rock stairway to the beach.
Mademoiselle ran down it swiftly: no one. But there was a small boat not
far off, she said. Fanny, who was near-sighted, got the glass. In a
little boat with a broad sail there were two figures; one was certainly
David Rod, and the other--yes, the other was Eva. There was a breeze,
the boat was rapidly going westward round the cliffs; in two minutes
more it was out of sight.
Fanny wrung her hands. The French woman, to whom the event wore a much
darker hue than it did to the American mother, turned yellowly pale.
At this moment Horace Bartholomew came out on the terrace; uneasy, for
Fanny's missive had explained nothing, he had followed his letter
himself. "What is it?" he said, as he saw the agitation of the two
women.
"Your friend--_yours_--the man you brought here, has Eva with him at
this moment out on the bay!" said Fanny, vehemently.
"Well, what of that? You must look at it with Punta Palmas eyes, Fanny;
at Punta Palmas it would be an ordinary event."
"But my Eva is not a Punta Palmas girl, Horace Bartholomew!"
"She is as innocent as one, and I'll answer for Rod. Come, be sensible,
Fanny. They will be back before sunset, and no one in Sorrento--if that
is what is troubling you so--need be any the wiser."
"You do not know all," said Fanny. "Oh, Horace--I must tell
somebody--she fancies she cares for that man!" She wrung her hands
again. "Couldn't we follow them? Get a boat."
"It would take an hour. And it would be a very conspicuous thing to do.
Leave them alone--it's much better; I tell you I'll answer for Rod.
Fancies she cares for him, does she? Well, he is a fine fellow; on the
whole, the finest I know."
The mother's eyes flashed through her tears. "This from _you_?"
"I can't help it; he is. Of course you do not think so. He has got no
money; he has never been anywhere that you call anywhere; he doesn't
know anything about the only life you care for nor the things you think
important. All the same, he is a man in a million. He is a man--not a
puppet."
Gentle Mrs. Churchill appeared for the moment transformed. She looked as
though she could strike him. "Never mind your Quixotic ideas. Tell me
whether he is in love with Eva; it all depends upon that."
"I don't know, I am sure," answered Bartholomew. He began to think. "I
can't say at all; he would conceal it from me."
"Because he felt his inferiority. I am glad he has that grace."
"He wouldn't be conscious of any inferiority save that he is poor. It
would be that, probably, if anything; of course he supposes that Eva is
rich."
"Would to Heaven she were!" said the mother. "Added to every other
horror of it, poverty, miserable poverty, for my poor child!" She sat
down and hid her face.
"It may not be as bad as you fear, nor anything like it. Do cheer up a
little, Fanny. When Eva comes back, ten to one you will find that
nothing at all has happened--that it has been a mere ordinary excursion.
And I promise you I will take Rod away with me to-morrow."
Mrs. Churchill rose and began to pace to and fro, biting her lips, and
watching the water. Mademoiselle, who was still hovering near, she waved
impatiently away. "Let no one in," she called to her.
There seemed, indeed, to be nothing else to do, as Bartholomew had said,
save to wait. He sat down and discussed the matter a little.
Fanny paid no attention to what he was saying. Every now and then broken
phrases of her own burst from her: "How much good will her perfect
French and Italian, her German, Spanish, and even Russian, do her down
in that barbarous wilderness?"--"In her life she has never even buttoned
her boots. Do they think she can make bread?"--"And there was Gino. And
poor Pierre." Then, suddenly, "But it _shall_ not be!"
"I have been wondering why you did not take that tone from the first,"
said Bartholomew. "She is very young. She has been brought up to obey
you implicitly. It would be easy enough, I should fancy, if you could
once make up your mind to it."
"Make up my mind to save her, you mean," said the mother, bitterly. She
did not tell him that she was afraid of her daughter. "Should you
expect _me_ to live at Punta Palmas?" she demanded, contemptuously, of
her companion.
"That would depend upon Rod, wouldn't it?" answered Bartholomew, rather
unamiably. He was tired--he had been there an hour--of being treated
like a door-mat.
At this Fanny broke down again, and completely. For it was only too
true; it would depend upon that stranger, that farmer, that unknown
David Rod, whether she, the mother, should or should not be with her own
child.
A little before sunset the boat came into sight again round the western
cliffs. Fanny dried her eyes. She was very pale. Little Mademoiselle,
rigid with anxiety, watched from an upper window. Bartholomew rose to go
down to the beach to receive the returning fugitives. "No," said Fanny,
catching his arm, "don't go; no one must know before I do--no one." So
they waited in silence.
Down below, the little boat had rapidly approached. Eva had jumped out,
and was now running up the rock stairway; she was always light-footed,
but to her mother it seemed that the ascent took an endless time. At
length there was the vision of a young, happy, rushing figure--rushing
straight to Fanny's arms. "Oh, mamma, mamma," the girl whispered, seeing
that there was no one there but Bartholomew, "he loves me! He has told
me so! he has told me so!"
For an instant the mother drew herself away. Eva, left alone, and
mindful of nothing but her own bliss, looked so radiant with happiness
that Bartholomew (being a man) could not help sympathizing with her.
"You will have to give it up," he said to Fanny, significantly. Then he
took his hat and went away.
Fifteen minutes later his place was filled by David Rod.
"Ah! you have come. I must have a few words of conversation with you,
Mr. Rod," said Fanny, in an icy tone. "Eva, leave us now."
"Oh no, mamma, not now; never again, I hope," answered the girl. She
spoke with secure confidence; her eyes were fixed upon her lover's face.
"Do you call this honorable behavior, Mr. Rod?" Fanny began. She saw
that Eva would not go.
"Why, I hope so," answered Rod, surprised. "I have come at once, as soon
as I possibly could, Mrs. Churchill (I had to take the boat back first,
you know), to tell you that we are engaged; it isn't an hour old yet--is
it, Eva?" He looked at Eva smilingly, his eyes as happy as her own.
"It is the custom to ask permission," said Fanny, stiffly.
"I have never heard of the custom, then; that is all I can say,"
answered Rod, with good-natured tranquillity, still looking at the
girl's face, with its rapt expression, its enchanting joy.
"Please to pay attention; I decline to consent, Mr. Rod; you cannot have
my daughter."
"Mamma--" said Eva, coming up to her.
"No, Eva; if you will remain here--which is most improper--you will have
to hear it all. You are so much my daughter's inferior, Mr. Rod, that I
cannot, and I shall not, consent."
At the word "inferior," a slight shock passed over Eva from head to
foot. She went swiftly to her lover, knelt down and pressed her lips to
his brown hand, hiding her face upon it.
He raised her tenderly in his arms, and thus embraced, they stood there
together, confronting the mother--confronting the world.
Fanny put out her hands with a bitter cry. "Eva!"
The girl ran to her, clung to her. "Oh, mamma, I love you dearly. But
you must not try to separate me from David. I could not leave him--I
never will."
"Let us go in, to our own room," said the mother, in a broken voice.
"Yes; but speak to David first, mamma."
Rod came forward and offered his arm. He was sorry for the mother's
grief, which, however, in such intensity as this, he could not at all
understand. But though he was sorry, he was resolute, he was even stern;
in his dark beauty, his height and strength, he looked indeed, as
Bartholomew had said, a man.
At the sight of his offered arm Mrs. Churchill recoiled; she glanced all
round the terrace as though to get away from it; she even glanced at the
water; it almost seemed as if she would have liked to take her child and
plunge with her to the depths below. But one miserable look at Eva's
happy, trustful eyes still watching her lover's face cowed her; she took
the offered arm. And then Rod went with her, supporting her gently into
the house, and through it to her own room, where he left her with her
daughter. That night the mother rose from her sleepless couch, lit a
shaded taper, and leaving it on a distant table, stole softly to Eva's
side. The girl was in a deep slumber, her head pillowed on her arm.
Fanny, swallowing her tears, gazed at her sleeping child. She still
saw in the face the baby outlines of years before, her mother's eye
could still distinguish in the motionless hand the dimpled fingers of
the child. The fair hair, lying on the pillow, recalled to her the short
flossy curls of the little girl who had clung to her skirts, who had had
but one thought--"mamma."
[Illustration: "FANNY PUT OUT HER HANDS WITH A BITTER CRY"]
"What will her life be now? What must she go through, perhaps--what
pain, privation--my darling, my own little child!"
The wedding was to take place within the month; Rod said that he could
not be absent longer from his farm. Fanny, breaking her silence,
suggested to Bartholomew that the farm might be given up; there were
other occupations.
"I advise you not to say a word of that sort to Rod," Bartholomew
answered. "His whole heart is in that farm, that colony he has built up
down there. You must remember that he was brought up there himself, or
rather came up. It's all he knows, and he thinks it the most important
thing in life; I was going to say it's all he cares for, but of course
now he has added Eva."
Pierre came once. He saw only the mother.
When he left her he went round by way of the main street of Sorrento in
order to pass a certain small inn. His carriage was waiting to take him
back to Castellamare, but there was some one he wished to look at first.
It was after dark; he could see into the lighted house through the low
uncurtained windows, and he soon came upon the tall outline of the young
farmer seated at a table, his eyes bent upon a column of figures. The
Belgian surveyed him from head to foot slowly. He stood there gazing
for five minutes. Then he turned away. "_That_, for Americans!" he
murmured in French, snapping his fingers in the darkness. But there was
a mist in his boyish eyes all the same.
The pink villa witnessed the wedding. Fanny never knew how she got
through that day. She was calm; she did not once lose her self-control.
They were to sail directly for New York from Naples, and thence to
Florida; the Italian colonists were to go at the same time.
"Mamma comes next year," Eva said to everybody. She looked indescribably
beautiful; it was the radiance of a complete happiness, like a halo.
By three o'clock they were gone, they were crossing the bay in the
little Naples steamer. No one was left at the villa with Fanny--it was
her own arrangement--save Horace Bartholomew.
"She won't mind being poor," he said, consolingly, "she won't mind
anything--with _him_. It is one of those sudden, overwhelming loves that
one sometimes sees; and after all, Fanny, it is the sweetest thing life
offers."
"And the mother?" said Fanny.
THE STREET OF THE HYACINTH
I
It was a street in Rome--narrow, winding, not over-clean. Two vehicles
meeting there could pass only by grazing the doors and windows on either
side, after the usual excited whip-cracking and shouts which make the
new-comer imagine, for his first day or two, that he is proceeding at a
perilous speed through the sacred city of the soul.
But two vehicles did not often meet in the street of the Hyacinth. It
was not a thoroughfare, not even a convenient connecting link; it
skirted the back of the Pantheon, the old buildings on either side
rising so high against the blue that the sun never came down lower than
the fifth line of windows, and looking up from the pavement was like
looking up from the bottom of a well. There was no foot-walk, of course;
even if there had been one no one would have used it, owing to the easy
custom of throwing from the windows a few ashes and other light trifles
for the city refuse-carts, instead of carrying them down the long stairs
to the door below. They must be in the street at an appointed hour, must
they not? Very well, then--there they were; no one but an unreasonable
foreigner would dream of objecting.
But unreasonable foreigners seldom entered the street of the Hyacinth.
There were, however, two who lived there one winter not long ago, and
upon a certain morning in the January of that winter a third came to see
these two. At least he asked for them, and gave two cards to the Italian
maid who answered his ring; but when, before he had time to even seat
himself, the little curtain over the parlor door was raised again, and
Miss Macks entered, she came alone. Her mother did not appear. The
visitor was not disturbed by being obliged to begin conversation
immediately; he was an old Roman sojourner, and had stopped fully three
minutes at the end of the fourth flight of stairs to re-gain his breath
before he mounted the fifth and last to ring Miss Macks's bell. Her card
was tacked upon the door: "Miss Ettie F. Macks." He surveyed it with
disfavor, while the little, loose-hung bell rang a small but exceedingly
shrill and ill-tempered peal, like the barking of a small cur. "Why in
the world doesn't she put her mother's card here instead of her own?" he
said to himself. "Or, if her own, why not simply 'Miss Macks,' without
that nickname?"
But Miss Macks's mother had never possessed a visiting-card in her life.
Miss Macks was the visiting member of the family; and this was so well
understood at home, that she had forgotten that it might not be the same
abroad. As to the "Ettie," having been called so always, it had not
occurred to her to make a change. Her name was Ethelinda Faith, Mrs.
Macks having thus combined euphony and filial respect--the first title
being her tribute to aesthetics, the second her tribute to the memory of
her mother.
"I am so very glad to see you, Mr. Noel," said Miss Macks, greeting her
visitor with much cordial directness of voice and eyes. "I have been
expecting you. But you have waited so long--three days!"
Raymond Noel, who thought that under the circumstances he had been
unusually courteous and prompt, was rather surprised to find himself
thus put at once upon the defensive.
"We are not always able to carry out our wishes immediately, Miss
Macks," he replied, smiling a little. "I was hampered by several
previously made engagements."
"Yes; but this was a little different, wasn't it? This was something
important--not like an invitation to lunch or dinner, or the usual idle
society talk."
He looked at her; she was quite in earnest.
"I suppose it to be different," he answered. "You must remember how
little you have told me."
"I thought I told you a good deal! However, the atmosphere of a
reception is no place for such subjects, and I can understand that you
did not take it in. That is the reason I asked you to come and see me
here. Shall I begin at once? It seems rather abrupt."
"I enjoy abruptness; I have not heard any for a long time."
"That I can understand, too; I suppose the society here is all finished
off--there are no rough ends."
"There are ends. If not rough, they are often sharp."
But Miss Macks did not stop to analyze this; she was too much occupied
with her own subject.
"I will begin immediately, then," she said. "It will be rather long; but
if you are to understand me you ought, of course, to know the whole."
"My chair is very comfortable," replied Noel, placing his hat and gloves
on the sofa near him, and taking an easy position with his head back.
Miss Macks thought that he ought to have said, "The longer it is, the
more interesting," or something of that sort. She had already described
him to her mother as "not over-polite. Not rude in the least, you
know--as far as possible from that; wonderfully smooth-spoken; but yet,
somehow--awfully indifferent." However, he was Raymond Noel; and that,
not his politeness or impoliteness, was her point.
"To begin with, then, Mr. Noel, a year ago I had never read one word you
have written; I had never even heard of you. I suppose you think it
strange that I should tell you this so frankly; but, in the first place,
it will give you a better idea of my point of view; and, in the second,
I feel a friendly interest in your taking measures to introduce your
writings into the community where I lived. It is a very intelligent
community. Naturally, a writer wants his articles read. What else does
he write them for?"
"Perhaps a little for his own entertainment," suggested her listener.
"Oh no! He would never take so much trouble just for that."
"On the contrary, many would take any amount just for that. Successfully
to entertain one's self--that is one of the great successes of life."
Miss Macks gazed at him; she had a very direct gaze.
"This is just mere talk," she said, not impatiently, but in a
business-like tone. "We shall never get anywhere if you take me up so.
It is not that your remarks are not very cultivated and interesting,
and all that, but simply that I have so much to tell you."
"Perhaps I can be cultivated and interesting dumbly. I will try."
"You are afraid I am going to be diffuse; I see that. So many women are
diffuse! But I shall not be, because I have been thinking for six months
just what I should say to you. It was very lucky that I went with Mrs.
Lawrence to that reception where I met you. But if it had not happened
as it did I should have found you out all the same. I should have looked
for your address at all the bankers', and if it was not there I should
have inquired at all the hotels. But it was delightful luck getting hold
of you in this way almost the very minute I enter Rome!"
She spoke so simply and earnestly that Noel did not say that he was
immensely honored, and so forth, but merely bowed his acknowledgments.
"To go back. I shall give you simply heads," pursued Miss Macks. "If you
want details, ask, and I will fill them in. I come from the West.
Tuscolee Falls is the name of our town. We had a farm there, but we did
not do well with it after Mr. Spurr's death, so we rented it out. That
is how I come to have so much leisure. I have always had a great deal of
ambition; by that I mean that I did not see why things that had once
been done could not be done again. It seemed to me that the point
was--just determination. And then, of course, I always had the talent. I
made pictures when I was a very little girl. Mother has them still, and
I can show them to you. It is just like all the biographies, you know.
They always begin in childhood, and astonish the family. Well, I had my
first lessons from a drawing-teacher who spent a summer in Tuscolee. I
can show you what I did while with him. Then I attended, for four years,
the Young Ladies' Seminary in the county-town, and took lessons while
there. I may as well be perfectly frank and tell the whole, which is
that everybody was astonished at my progress, and that I was myself. All
sorts of things are prophesied out there about my future. You see, the
neighborhood is a very generous-spirited one, and they like to think
they have discovered a genius at their own doors. My telling you all
this sounds, I know, rather conceited, Mr. Noel. But if you could see my
motive, and how entirely without conceit my idea of myself really is,
you would hold me free from that charge. It is only that I want you to
know absolutely the whole."
"I quite understand," answered her visitor.
"Well, I hope you do. I went on at home after that by myself, and I did
a good deal. I work pretty rapidly, you see. Then came my last lessons,
from a third teacher. He was a young man from New York. He had
consumption, poor fellow! and cannot last long. He wasn't of much use to
me in actual work. His ideas were completely different from those of my
other teachers, and, indeed, from my own. He was unreliable, too, and
his temper was uneven. However, I had a good deal of respect for his
opinion, and _he_ told me to get your art-articles and read them. It
wasn't easy. Some of them are scattered about in the magazines and
papers, you know. However, I am pretty determined, and I kept at it
until I got them all. Well, they made a great impression upon me. You
see, they were new." She paused. "But I doubt, Mr. Noel, whether we
should ever entirely agree," she added, looking at him reflectively.
"That is very probable, Miss Macks."
Miss Macks thought this an odd reply. "He is so queer, with all his
smoothness!" she said to her mother afterwards. "He never says what you
think he will say. Now, any one would suppose that he would have
answered that he would try to make me agree, or something like that.
Instead, he just gave it right up without trying! But I expect he sees
how independent I am, and that I don't intend to _reflect any_ one."
"Well, they made a great impression," she resumed. "And as you seemed to
think, Mr. Noel, that no one could do well in painting who had not seen
and studied the old pictures over here, I made up my mind to come over
at any cost, if it was a possible thing to bring it about. It wasn't
easy, but--here we are. In the lives of all--almost all--artists, I have
noticed--haven't you?--that there comes a time when they have to live on
hope and their own pluck more than upon anything tangible that the
present has to offer. They have to take that risk. Well, I have taken
it; I took it when we left America. And now I will tell you what it is I
want from _you_. I haven't any hesitation in asking, because I am sure
you will feel interested in a case like mine, and because it was your
writings really that brought me here, you know. And so, then, first: I
would like your opinion of all that I have done so far. I have brought
everything with me to show you. Second: I want your advice as to the
best teacher; I suppose there is a great choice in Rome. Third: I should
be glad if you would give a general oversight to all I do for the next
year. And last, if you would be so kind, I should much enjoy making
visits with you to all the galleries and hearing your opinions again by
word of mouth, because that is always so much more vivid, you know, than
the printed page."
"My dear Miss Macks! you altogether over-estimate my powers," said Noel,
astounded by these far-reaching demands, so calmly and confidently made.
"Yes, I know. Of course it strikes you so--strikes you as a great
compliment that I should wish to put myself so entirely in your hands,"
answered Miss Macks, smiling. "But you must give up thinking of me as
the usual young lady; you must not think of me in that way any more than
I shall think of you as the usual young gentleman. You will never meet
me at a reception again; now that I have found _you_, I shall devote
myself entirely to my work."
"An alarming girl!" said Noel to himself. But, even as he said it, he
knew that, in the ordinary acceptation of the term at least, Miss Macks
was not alarming.
She was twenty-two; in some respects she looked older, in others much
younger, than most girls of that age. She was tall, slender, erect, but
not especially graceful. Her hands were small and finely shaped, but
thin. Her features were well cut; her face oval. Her gray eyes had a
clear directness in their glance, which, combined with the other
expressions of her face, told the experienced observer at once that she
knew little of what is called "the world." For, although calm, it was a
deeply confident glance; it showed that the girl was sure that she could
take care of herself, and even several others also, through any
contingencies that might arise. She had little color; but her smooth
complexion was not pale--it was slightly brown. Her mouth was small, her
teeth small and very white. Her light-brown hair was drawn back smoothly
from her forehead, and drawn up smoothly behind, its thickness braided
in a close knot on the top of her head. This compact coiffure, at a time
when most feminine foreheads in Rome and elsewhere were shaded almost to
the eyebrows by curling locks, and when the arched outline of the head
was left unbroken, the hair being coiled in a low knot behind, made Miss
Macks look somewhat peculiar. But she was not observant of fashion's
changes. That had been the mode in Tuscolee; she had grown accustomed to
it; and, as her mind was full of other things, she had not considered
this one. One or two persons, who noticed her on the voyage over, said
to themselves, "If that girl had more color, and if she was graceful,
and if she was a little more womanly--that is, if she would not look at
everything in such a direct, calm, impartial, impersonal sort of
way--she would be almost pretty."
But Miss Macks continued without color and without grace, and went on
looking at things as impersonally and impartially as ever.
"I shall be most happy, of course, to do anything that I can," Noel had
answered. Then to make a diversion, "Shall I not have the pleasure of
seeing Mrs. Macks?" he asked.
"Mrs. Macks? Oh, you mean mother. My mother's name is Spurr--Mrs. Spurr.
My father died when I was a baby, and some years afterwards she married
Mr. Spurr. She is now again a widow. Her health is not good, and she
sees almost no one, thank you."
"I suppose you are much pleased with the picturesqueness of Roman life,
and--ah--your apartment?" he went on.
"Pleased?" said Miss Macks, looking at him in wonder. "With our
apartment? We get along with it because we must; there seems to be no
other way to live in Rome. The idea of having only a story of a house,
and not a whole house to ourselves, is dreadful to mother; she cannot
get used to it. And with so many families below us--we have a
clock-mender, a dress-maker, an engraver, a print-seller, and a
cobbler--and only one pair of stairs, it does seem to me dreadfully
public."
"You must look upon the stairway as a street," said Noel. "You have
established yourselves in a very short time."
"Oh yes. I got an agent, and looked at thirty places the very first day.
I speak Italian a little, so I can manage the house-keeping; I began to
study it as soon as we thought of coming, and I studied hard. But all
this is of secondary importance; the real thing is to get to work. Will
you look at my paintings now?" she said, rising as if to go for them.
"Thanks; I fear I have hardly time to-day," said Noel. He was thinking
whether it would be better to decline clearly and in so many words the
office she had thrust upon him, or trust to time to effect the same
without an open refusal. He decided upon the latter course; it seemed
the easier, and also the kinder to her.
"Well, another day, then," said Miss Macks, cheerfully, taking her seat
again. "But about a teacher?"
"I hardly know--"
"Oh, Mr. Noel! you _must_ know."
And, in truth, he did know. It came into his mind to give her the name
of a good teacher, and then put all further responsibilities upon him.
Miss Macks wrote down the name in a clear, ornamental handwriting.
"I am glad it isn't a foreigner," she said. "I don't believe I should
get on with a foreigner."
"But it is a foreigner."
"Why, it's an English name, isn't it?--Jackson."
"Yes, he is an Englishman. But isn't an Englishman a foreigner in Rome?"
"Oh, you take that view? Now, to me, America and--well, yes, perhaps
England, too, are the nations. Everything else is foreign."
"The English would be very much obliged to you," said Noel, laughing.
"Yes, I know I am more liberal than most Americans; I really like the
English," said Miss Macks, calmly. "But we keep getting off the track.
Let me see--Oh yes. As I shall go to see this Mr. Jackson this
afternoon, and as it is not likely that he will be ready to begin
to-morrow, will you come then and look at my pictures? Or would you
rather commence with a visit to one of the galleries?"
Raymond Noel was beginning to be amused. If she had shown the faintest
indication of knowing how much she was asking, if she had betrayed the
smallest sign of a desire to secure his attention as Raymond Noel
personally, and not simply the art authority upon whom she had pinned
her faith, his disrelish for various other things about her would have
been heightened into utter dislike, and it is probable that he would
never have entered the street of the Hyacinth again. But she was so
unaware of any intrusion, or any exorbitance in her demands, probably so
ignorant of--certainly so indifferent to--the degree of perfection
(perfection of the most quiet kind, however) visible in the general
appearance and manner of the gentleman before her, that (he said to
himself) he might as well have been one of her own Tuscolee farmers, for
all she knew to the contrary. The whole affair was unusual; and Noel
rather liked the unusual, if it was not loud--and Miss Macks was, at
least, not loud; she was dressed plainly in black, and she had the gift
of a sweet voice, which, although very clear, was low-toned. Noel was an
observer of voices, and he had noticed hers the first time he heard her
speak. While these thoughts were passing through his mind, he was
answering that he feared his engagements for the next day would,
unfortunately, keep him from putting himself at her service.
Her face fell; she looked much disappointed.
"Is it going to be like this all the time?" she asked, anxiously. "Are
you always engaged?"
"In Rome, in the winter, one generally has small leisure. It will be the
same with you, Miss Macks, when you have been here a while longer; you
will see. As to the galleries, Mr. Jackson has a class, I think, and
probably the pupils will visit them all under his charge; you will find
that very satisfactory."
"But I don't want Mr. Jackson for the galleries; I want _you_," said
Miss Macks. "I have studied your art criticisms until I know them by
heart, and I have a thousand questions to ask about every picture you
have mentioned. Why, Mr. Noel, I came to Europe to see you!"
Raymond Noel was rather at a loss what to answer to this statement, made
by a girl who looked at him so soberly and earnestly with clear gray
eyes. It would be of no avail again to assure her that his opinions
would be of small use to her; as she had said herself, she was very
determined, and she had made up her mind that they would be of great use
instead of small. Her idea must wear itself out by degrees. He would try
to make the degrees easy. He decided that he would have a little private
talk with Jackson, who was a very honest fellow; and, for the present,
he would simply take leave.
"You are very kind," he said, rising. "I appreciate it, I assure you. It
has made me stay an unconscionable time. I hope you will find Rome all
you expected, and I am sure you will; all people of imagination like
Rome. As to the galleries, yes, certainly; a--ah--little later. You must
not forget the various small precautions necessary here as regards the
fever, you know."
"Rome will not be at all what I expected if _you_ desert me," answered
Miss Macks, paying no attention to his other phrases. She had risen,
also, and was now confronting him at a distance of less than two feet;
as she was tall, her eyes were not much below the level of his own.
"How can a man desert when he has never enlisted?" thought Noel,
humorously. But he kept his thought to himself, and merely replied, as
he took his hat: "Probably you will desert me; you will find out how
useless I am. You must not be too hard upon us, Miss Macks; we Americans
lose much of our native energy if we stay long over here."
"Hard?" she answered--"hard? Why, Mr. Noel, I am absolutely at your
feet!"
He looked at her, slightly startled, although his face showed nothing of
it; was she, after all, going to--But no; her sentence had been as
impersonal as those which had preceded it.
"All I said about having contrary opinions, and all that, amounts to
nothing," she went on, thereby relieving him from the necessity of
making reply. "I desire but one thing, and that is to have you guide me.
And I don't believe you are really going to refuse. You haven't an
unkind face, although you _have_ got such a cold way! Why, think of it:
here I have come all this long distance, bringing mother, too, just to
study, and to see you. I shall study hard; I have a good deal of
perseverance. It took a good deal to get here in the first place, for we
are poor. But I don't mind that at all; the only thing I should mind,
the only thing that would take my courage away, would be to have you
desert me. In all the troubles that I thought might happen, I assure
you, I never once thought of _that_, Mr. Noel. I thought, of course, you
would be interested. Why, in your books you are all interest. Are you
different from your books?"
"I fear, Miss Macks, that writers are seldom good illustrations of their
own doctrines," replied Noel.
"That would make them hypocrites. I don't believe you are a hypocrite. I
expect you have a habit of running yourself down. Many gentlemen do
that, and then they think they will be cried up. I don't believe you are
going to be unkind; you _will_ look at the pictures I have brought with
me, won't you?"
"Mr. Jackson's opinion is worth a hundred of mine, Miss Macks; my
knowledge is not technical. But, of course, if you wish it, I shall take
pleasure in obeying." He added several conventional remarks as
filling-up, and then, leaving his compliments for "your mother"--he
could not recall the name she had given--he went towards the little
curtained door.
She had brightened over his promise.
"You will come Monday, then, to see them, won't you?--as you cannot come
to-morrow," she said, smiling happily.
When she smiled (and she did not smile often), showing her little white,
child-like teeth, she looked very young. He was fairly caught, and
answered, "Yes." But he immediately qualified it with a "That is, if it
is possible."
"Oh, _make_ it possible," she answered, still smiling and going with him
herself to the outer door instead of summoning the maid. The last he saw
of her she was standing in the open doorway, her face bright and
contented, watching him as he went down. He did not go to see her
pictures on the following Monday; he sent a note of excuse.
Some days later he met her.
"Ah, you are taking one of the delightful walks?" he said. "I envy you
your first impressions of Rome."
"I am not taking a walk--that is, for pleasure," she answered. "I am
trying to find some vegetables that mother can eat; the vegetables here
are so foreign! You don't know how disappointed I was, Mr. Noel, when I
got your note. It was such a setback! Why couldn't you come right home
with me now--that is, after I have got the vegetables--and see the
pictures? It wouldn't take you fifteen minutes."
It was only nine o'clock, and a beautiful morning. He thought her such a
novelty, with her urgent invitations, her earnest eyes, and her basket
on her arm, that he felt the impulse to walk beside her a while through
the old streets of Rome; he was very fond of the old streets, and was
curious to see whether she would notice the colors and outlines that
made their picturesqueness. She noticed nothing but the
vegetable-stalls, and talked of nothing but her pictures.
He still went on with her, however, amused by the questions she put to
the vegetable-dealers (questions compiled from the phrase-books), and
the calm contempt with which she surveyed the Roman artichokes they
offered. At last she secured some beans, but of sadly Italian aspect,
and Noel took the basket. He was much entertained by the prospect of
carrying it home. He remarked to himself that of all the various things
he had done in Rome this was the freshest. They reached the street of
the Hyacinth and walked down its dark centre.
"I see you have the sun," he said, looking up.
"Yes; that is the reason we took the top floor. We will go right up.
Everything is ready."
He excused himself.
"Some other time."
They had entered the dusky hallway. She looked at him without replying;
then held out her hand for the basket. He gave it to her.
"I suppose you have seen Mr. Jackson?" he said, before taking leave.
She nodded, but did not speak. Then he saw two tears rise in her eyes.
"My dear young lady, you have been doing too much! You are tired. Don't
you know that that is very dangerous in Rome?"
"It is nothing. Mother has been sick, and I have been up with her two
nights. Then, as she did not like our servant, I dismissed her, and as
we have not got any one else yet, I have had a good deal to do. But I
don't mind that at all, beyond being a little tired; it was only your
refusing to come up, when it seemed so easy. But never mind; you will
come another day." And, repressing the tears, she smiled faintly, and
held out her hand for good-bye.
"I will come now," said Noel. He took the basket again, and went up the
stairs. He was touched by the two tears, but, at the same time, vexed
with himself for being there at all. There was not one chance in five
hundred that her work was worth anything; and, in the four hundred and
ninety-nine, pray what was he to say?
She brought him everything. They were all in the four hundred and
ninety-nine. In his opinion they were all extremely and essentially bad.
It was one of Raymond Noel's beliefs that, where women were concerned, a
certain amount of falsity was sometimes indispensable. There were
occasions when a man could no more tell the bare truth to a woman than
he could strike her; the effect would be the same as a blow. He was an
excellent evader when he chose to exert himself, and he finally got away
from the little high-up apartment without disheartening or offending its
young mistress, and without any very black record of direct
untruth--what is more, without any positive promise as to the exact date
of his next visit. But all this was a good deal of trouble to take for
a girl he did not know or care for.
Soon afterwards he met, at a small party, Mrs. Lawrence.
"Tell me a little, please, about the young lady to whom you presented me
at Mrs. Dudley's reception--Miss Macks," he said, after some
conversation.
"A little is all I can tell," replied Mrs. Lawrence. "She brought a
letter of introduction to me from a far-away cousin of mine, who lives
out West somewhere, and whom I have not seen for twenty years; my home,
you know, is in New Jersey. How they learned I was in Rome I cannot
imagine; but, knowing it, I suppose they thought that Miss Macks and I
would meet, as necessarily as we should if together in their own
village. The letter assures me that the girl is a great genius; that all
she needs is an opportunity. They even take the ground that it will be a
privilege for me to know her! But I am mortally tired of young geniuses;
we have so many here in Rome! So I told her at once that I knew nothing
of modern art--in fact, detested it--but that in any other way I should
be delighted to be of use. And I took her to Mrs. Dudley's _omnium
gatherum_."
"Then you have not been to see her?"
"No; she came to see me. I sent cards, of course; I seldom call. What
did you think of her?"
"I thought her charming," replied Noel, remembering the night-vigils,
the vegetables, the dismissed servant, and the two tears of the young
stranger--remembering, also, her extremely bad pictures.
"I am glad she has found a friend in you," replied Mrs. Lawrence. "She
was very anxious to meet you; she looks upon you as a great authority.
If she really has talent--of course _you_ would know--you must tell me.
It is not talent I am so tired of, but the pretence of it. She struck
me, although wofully unformed and awkward, of course, as rather
intelligent."
"She is intelligence personified," replied Noel, qualifying it mentally
with "intelligence without cultivation." He perceived that the young
stranger would have no help from Mrs. Lawrence, and he added to himself:
"And totally inexperienced purity alone in Rome." To be sure, there was
the mother; but he had a presentiment that this lady, as guardian, would
not be of much avail.
The next day he went down to Naples for a week with some friends. Upon
his return he stopped at Horace Jackson's studio one afternoon as he
happened to be passing. His time was really much occupied; he was a
favorite in Rome. To his surprise, Jackson seemed to think that Miss
Macks had talent. Her work was very crude, of course; she had been
brutally taught; teachers of that sort should simply be put out of
existence with the bowstring. He had turned her back to the alphabet;
and, in time, in time, they--would see what she could do.
Horace Jackson was English by birth, but he had lived in Italy almost
all his life. He was a man of forty-five--short, muscular, his thick,
rather shaggy, beard and hair mixed with gray; there was a permanent
frown over his keen eyes, and his rugged face had marked lines. He was a
man of strong individuality. He had the reputation of being the most
incorruptibly honest teacher in Rome. Noel had known him a long time,
and liked him, ill-tempered though he was. Jackson, however, had not
shown any especial signs of a liking for Noel in return. Perhaps he
thought that, in the nature of things, there could not be much in common
between a middle-aged, morose teacher, who worked hard, who knew nothing
of society, and did not want to know, and a man like Raymond Noel. True,
Noel was also an artist--that is, a literary one. But he had been highly
successful in his own field, and it was understood, also, that he had an
income of his own by inheritance, which, if not opulence, was yet
sufficiently large to lift him quite above the usual _res angusta_ of
his brethren in the craft. In addition, Jackson considered Noel a
fashionable man; and that would have been a barrier, even if there had
been no other.
As the Englishman seemed to have some belief in Miss Macks, Noel did not
say all he had intended to say; he did, however, mention that the young
lady had a mistaken idea regarding any use he could be to her; he should
be glad if she could be undeceived.
"I think she will be," said Jackson, with a grim smile, giving his guest
a glance of general survey that took him in from head to foot; "she
isn't dull."
Noel understood the glance, and smiled at Jackson's idea of him.
"She is not dull, certainly," he answered. "But she is
rather--inexperienced." He dismissed the subject, went home, dressed,
and went out to dinner.
One morning, a week later, he was strolling through the Doria gallery.
He was in a bad humor. There were many people in the gallery that day,
but he was not noticing them; he detested a crowd. After a while some
one touched his coat-sleeve from behind. He turned, with his calmest
expression upon his face; when he was in an ill-humor he was
impassively calm. It was Miss Macks, her eyes eager, her face flushed
with pleasure.
"Oh, what good luck!" she said. "And to think that I almost went to the
Borghese, and might have missed you! I am so delighted that I don't know
what to do. I am actually trembling." And she was. "I have so longed to
see these pictures with you," she went on. "I have had a real aching
disappointment about it, Mr. Noel."
Again Noel felt himself slightly touched by her earnestness. She looked
prettier than usual, too, on account of the color.
"I always feel a self-reproach when with you, Miss Macks," he
answered--"you so entirely over-estimate me."
"Well, if I do, live up to it," she said, brightly.
"Only an archangel could do that."
"An archangel who knows about Art! I have been looking at the Caraccis;
what do you think of them?"
"Never mind the Caraccis; there are better things to look at here." And
then he made the circuit of the gallery with her slowly, pointing out
the best pictures. During this circuit he talked to her as he would have
talked to an intelligent child who had been put in his charge in order
to learn something of the paintings; he used the simplest terms,
mentioned the marked characteristics, and those only of the different
schools, and spoke a few words of unshaded condemnation here and there.
All he said was in broad, plain outlines. His companion listened
earnestly. She gave him a close attention, almost always a
comprehension, but seldom agreement. Her disagreement she did not
express in words, but he could read it in her eyes. When they had seen
everything--and it took some time--
"Now," he said, "I want you to tell me frankly, and without reference to
anything I have said, your real opinion of several pictures I shall
name--that is, if you can remember?"
"I remember everything. I always remember."
"Very well. What do you think, then, of the Raphael double portrait?"
"I think it very ugly."
"And the portrait of Andrea Doria, by Sebastian del Piombo?"
"Uglier still."
"And the Velasquez?"
"Ugliest of all."
"And the two large Claude Lorraines?"
"Rather pretty; but insipid. There isn't any reality or meaning in
them."
"The Memling?"
"Oh, _that_ is absolutely hideous, Mr. Noel; it hasn't a redeeming
point."
Raymond Noel laughed with real amusement, and almost forgot his
ill-humor.
"When you have found anything you really admire in the galleries here,
Miss Macks, will you tell me?"
"Of course I will. I should wish to do so in any case, because, if you
are to help me, you ought to thoroughly understand me. There is one
thing more I should like to ask," she added, as they turned towards the
door, "and that is that you would not call me Miss Macks. I am not used
to it, and it sounds strangely; no one ever called me that in Tuscolee."
"What did they call you in Tuscolee?"
"They called me Miss Ettie; my name is Ethelinda Faith. But my friends
and older people called me just 'Ettie'; I wish you would, too."
"I am certainly older," replied Noel, gravely (he was thirty-three);
"but I do not like Ettie. With your permission, I will call you Faith."
"Do you like it? It's so old-fashioned! It was my grandmother's name."
"I like it immensely," he answered, leading the way down-stairs.
"You can't think how I've enjoyed it," she said, warmly, at the door.
"Yet you do not agree with my opinions?"
"Not yet. But all the same it was perfectly delightful. Good-bye."
He had signalled for a carriage, as he had, as usual, an engagement. She
preferred to walk. He drove off, and did not see her for ten days.
Then he came upon her again and again in the Doria gallery. He was fond
of the Doria, and often went there, but he had no expectation of meeting
Miss Macks this time; he fancied that she followed a system, going
through her list of galleries in regular order, one by one, and in that
case she would hardly have reached the Doria on a second round. Her list
was a liberal one; it included twenty. Noel had supposed that there were
but nine in Rome.
This time she did not see him; she had some sheets of manuscript in her
hand, and was alternately reading from them and looking at one of the
pictures. She was much absorbed. After a while he went up.
"Good-morning, Miss Macks."
She started; her face changed, and the color rose. She was as delighted
as before. She immediately showed him her manuscript. There he beheld,
written out in her clear handwriting, all he had said of the Doria
pictures, page after page of it; she had actually reproduced from memory
his entire discourse of an hour.
There were two blank spaces left.
"There, I could not exactly remember," said Miss Macks, apologetically.
"If you would tell me, I should be so glad; then it would be quite
complete."
"I shall never speak again. I am frightened," said Noel. He had taken
the manuscript, and was looking it over with inward wonder.
"Oh, please do."
"Why do you care for my opinions, Miss Macks, when you do not agree with
them?" he asked, his eyes still on the pages.
"You said you would call me Faith. Why do I care? Because they are
yours, of course."
"Then you think I know?"
"I am sure you do."
"But it follows, then, that you do not."
"Yes; and there is where my work comes in; I have got to study up to
you. I am afraid it will take a long time, won't it?"
"That depends upon you. It would take very little if you would simply
accept noncombatively."
"Without being convinced? That I could never do."
"You want to be convinced against your will?"
"No; my will itself must be convinced to its lowest depths."
"This manuscript won't help you."
"Indeed, it has helped me greatly already. I have been here twice with
it. I wrote it out the evening after I saw you. I only wish I had one
for each of the galleries! But I feel differently now about asking you
to go."
"I told you you would desert me."
"No, it is not that. But Mr. Jackson says you are much taken up with the
fashionable society here, and that I must not expect you to give me so
much of your time as I had hoped for. He says, too, that your art
articles will do me quite as much good as you yourself, and more;
because you have a way, he says, like all society men, of talking as if
you had no real convictions at all, and that would unsettle me."
"Jackson is an excellent fellow," replied Noel; "I like him extremely.
And when would you like to go to the Borghese?"
"Oh, will you take me?" she said, joyfully. "Any time. To-morrow."
"Perhaps Mrs.--your mother, will go, also," he suggested, still unable
to recall the name; he could think of nothing but "stirrup," and of
course it was not that.
"I don't believe she would care about it," answered the daughter.
"She might. You know we make more of mothers here than we do in
America," he ventured to remark.
"That is impossible," said Miss Macks, calmly. Evidently she thought his
remark frivolous.
He abandoned the subject, and did not take it up again. It was not his
duty to instruct Miss Macks in foreign customs. In addition, she was not
only not "in society," but she was an art student, and art students had,
or took, privileges of their own in Rome.
"At what hour shall I come for you?" he said.
"It will be out of your way to come for me; I will meet you at the
gallery," she answered, radiant at the prospect.
He hesitated, then accepted her arrangement of things. He would take her
way, not his own. The next morning he went to the Borghese Palace ten
minutes before the appointed time. But she was already there.
"Mother thought she would not come out--the galleries tire her so," she
said; "but she was pleased to be remembered."
They spent an hour and a half among the pictures. She listened to all he
said with the same earnest attention.
Within the next five weeks Raymond Noel met Miss Macks at other
galleries. It was always very business-like--they talked of nothing but
the pictures; in truth, her systematic industry kept him strictly down
to the subject in hand. He learned that she made the same manuscript
copies of all he said, and, when he was not with her, she went alone,
armed with these documents, and worked hard. Her memory was remarkable;
she soon knew the names and the order of all the pictures in all the
galleries, and had made herself acquainted with an outline, at least, of
the lives of all the artists who had painted them. During this time she
was, of course, going on with her lessons; but as he had not been again
to see Jackson, or to the street of the Hyacinth, he knew nothing of her
progress. He did not want to know; she was in Jackson's hands, and
Jackson was quite competent to attend to her.
In these five weeks he gave to Miss Macks only the odd hours of his
leisure. He made her no promises; but when he found that he should have
a morning or half-morning unoccupied, he sent a note to the street of
the Hyacinth, naming a gallery and an hour. She was always promptly
there, and so pleased, that there was a sort of fresh aroma floating
through the time he spent with her, after all--but a mild one.
To give the proper position to the place the young art student's light
figure occupied on the canvas of Raymond Noel's winter, it should be
mentioned that he was much interested in a French lady who was spending
some months in Rome. He had known her and admired her for a long time;
but this winter he was seeing more of her, some barriers which had
heretofore stood in the way being down. Madame B---- was a charming
product of the effects of finished cultivation and fashionable life upon
a natural foundation of grace, wit, and beauty of the French kind. She
was not artificial, because she was art itself. Real art is as real as
real nature is natural. Raymond Noel had a highly artistic nature. He
admired art. This did not prevent him from taking up occasionally, as a
contrast to this lady, the society of the young girl he called "Faith."
Most men of imagination, artistic or not, do the same thing once in a
while; it seems a necessity. With Noel it was not the contrast alone.
The French lady led him an uneasy life, and now and then he took an hour
of Faith, as a gentle soothing draught of safe quality. She believed in
him so perfectly! Now Madame appeared to believe in him not at all.
It must be added that, in his conversations with Miss Macks, he had
dropped entirely even the very small amount of conventional gallantry
that he had bestowed upon her in the beginning. He talked to her not as
though she was a boy exactly, or an old woman, but as though he himself
was a relative of mature age--say an uncle of benevolent disposition and
a taste for art.
February gave way to March. And now, owing to a new position of his own
affairs, Noel saw no more of Faith Macks. She had been a contrast, and
he did not now wish for a contrast or a soothing draught, and a soothing
draught was not at present required. He simply forgot all about her.
In April he decided rather suddenly to leave Rome. This was because
Madame B---- had gone to Paris, and had not forbidden her American
suitor to follow her a few days later. He made his preparations for
departure, and these, of course, included farewell calls. Then he
remembered Faith Macks; he had not seen her for six weeks. He drove to
the street of the Hyacinth, and went up the dark stairs. Miss Macks was
at home, and came in without delay; apparently, in her trim neatness,
she was always ready for visitors.
She was very glad to see him; but did not, as he expected, ask why he
had not come before. This he thought a great advance; evidently she was
learning. When she heard that he had come to say good-bye her face fell.
"I am so very sorry; please sit as long as you can, then," she said,
simply. "I suppose it will be six months before I see you again; you
will hardly return to Rome before October." That he would come at that
time she did not question.
"My plans are uncertain," replied Noel. "But probably I shall come back.
One always comes back to Rome. And you--where do you go? To
Switzerland?"
"Why--we go nowhere, of course; we stay here. That is what we came for,
and we are all settled."
He made some allusion to the heat and unhealthiness.
"I am not afraid," replied Miss Macks. "Plenty of people stay; Mr.
Jackson says so. It is only the rich who go away, and we are not rich.
We have been through hot summers in Tuscolee, I can tell you!" Then,
without asking leave this time, as if she was determined to have an
opinion from him before he departed, she took from a portfolio some of
the work she had done under Mr. Jackson's instruction.
Noel saw at once that the Englishman had not kept his word. He had not
put her back upon the alphabet, or, if he had done so, he had soon
released her, and allowed her to pursue her own way again. The original
faults were as marked as ever. In his opinion all was essentially bad.
He looked in silence. But she talked on hopefully, explaining,
comparing, pointing out.
"What does Mr. Jackson think of this?" he said, selecting the one he
thought the worst.
"He admires the idea greatly; he thinks it very original. He says that
my strongest point is originality," she answered, with her confident
frankness.
"He means--ah--originality of subject?"
"Oh yes; my execution is not much yet. But that will come in time. Of
course, the subject, the idea, is the important thing; the execution is
secondary." Here she paused; something seemed to come into her mind. "I
know _you_ do not think so," she added, thoughtfully, "because, you
know, you said"--and here she quoted a page from one of his art
articles with her clear accuracy. "I have never understood what you
meant by that, Mr. Noel; or why you wrote it."
She looked at him questioningly. He did not reply; his eyes were upon
one of the sketches.
"It would be dreadful for me if you were right!" she added, with slow
conviction.
"I thought you believed that I was always right," he said, smiling, as
he placed the sketches on the table.
But she remained very serious.
"You are--in everything but that."
He made some unimportant reply, and turned the conversation. But she
came back to it.
"It would be dreadful," she repeated, earnestly, with the utmost gravity
in her gray eyes.
"I hope the long summer will not tire you," he answered, irrelevantly.
"Shall I not have the pleasure of saying good-bye--although that, of
course, is not a pleasure--to Mrs.--to your mother?"
He should have made the speech in any case, as it was the proper one to
make; but as he sat there he had thought that he really would like to
have a look at the one guardian this young girl was to have during her
long, lonely summer in Rome.
"I will tell her. Perhaps when she hears that you are going away she
will feel like coming in," said Miss Macks.
She came back after some delay, and with her appeared a matron of
noticeable aspect.
"My mother," she said, introducing her (evidently Noel was never to get
the name); "this is Mr. Noel, mother."
"And very glad I am to see you, sir, I'm sure," said Mrs. Spurr,
extending her hand with much cordiality. "I said to Ettie that I'd come
in, seeing as 'twas you, though I don't often see strangers nowadays on
account of poor health for a long time past; rheumatism and asthma. But
I feel beholden to you, Mr. No-ul, because you've been so good to Ettie.
You've been real kind."
Ettie's mother was a very portly matron of fifty-five, with a broad
face, indistinct features, very high color, and a breathless, panting
voice. Her high color--it really was her most noticeable feature--was
surmounted by an imposing cap, adorned with large bows of scarlet
ribbon; a worsted shawl, of the hue known as "solferino," decked her
shoulders; under her low-necked collar reposed a bright blue necktie,
its ends embroidered in red and yellow; and her gown was of a vivid dark
green. But although her colors swore at each other, she seemed amiable.
She was also voluble.
Noel, while shaking hands, was considering, mentally, with some
retrospective amusement, his condition of mind if this lady had accepted
his invitations to visit the galleries.
"You must sit down, mother," said Miss Macks, bringing forward an
easy-chair. "She has not been so well as usual, lately," she said,
explanatorily, to Noel, as she stood for a moment beside her mother's
chair.
"It's this queer Eye-talian air," said Mrs. Spurr. "You see I ain't used
to it. Not but what I ain't glad to be here on Ettie's account--real
glad. It's just what she needs and oughter have."
The girl put her hand on her mother's shoulder with a little caressing
touch. Then she left the room.
"Yes, I do feel beholden to you, Mr. No-ul. But, then, she'll be a
credit to you, to whatever you've done for her," said Mrs. Spurr, when
they were left alone. "Her talunts are very remarkable. She was the head
scholar of the Young Ladies' Seminary through four whole years, and all
the teachers took a lot of pride in her. And then her paintings, too!
I'm sorry you're going off so soon. You see, she sorter depends upon
your opinion."
Noel felt a little stir at the edges of his conscience; he knew
perfectly that his opinion was that Miss Macks, as an artist, would
never do anything worth the materials she used.
"I leave her in good hands," he said.
After all, it was Jackson's responsibility, not his.
"Yes, Mr. Jackson thinks a deal of her. I can see that plain!" answered
Mrs. Spurr, proudly.
Here the daughter returned, bringing a little note-book and pencil.
"Do you know what these are for?" she said. "I want you to write down a
list of the best books for me to read this summer, while you are gone. I
am going to work hard; but if I have books, too, the time won't seem so
long."
Noel considered a moment. In one way her affairs were certainly none of
his business; in another way they were, because she had thrust them upon
him.
"I will not give you a list, Miss Macks; probably you would not be able
to find the books here. But I will send you, from Paris or London, some
things that are rather good, if you will permit me to do so."
She said he was very kind. Her face brightened.
"If she has appreciation enough to comprehend what I send her," he
thought, "perhaps in the end she will have a different opinion about my
'kindness'!"
Soon afterwards he took leave. The next day he went to Paris.
II
The events of Raymond Noel's life, after he left Rome that spring, were
various. Some were pleasant, some unpleasant; several were quite
unexpected. Their combinations and results kept him from returning to
Italy the following winter, and the winter after that he spent in Egypt.
When he again beheld the dome of St. Peter's he remembered that it
lacked but a month of two full years since he had said good-bye to it;
it was then April, and now it was March. He established himself in some
pleasant rooms, looked about him, and then began to take up, one by one,
the old threads of his Roman life--such, at least, as remained unbroken.
He found a good many. Threads do not break in Rome. He had once said
himself that the air was so soft and historic that nothing broke
there--not even hearts. But this was only one of his little speeches. In
reality he did not believe much in the breaking of hearts; he had seen
them stretch so!
It may be said with truth that Noel had not thought of Miss Macks for
months. This was because he had had other things to think of. He had
sent her the books from Paris, with an accompanying note, a charming
little note--which gave no address for reply. Since then his mind had
been otherwise occupied. But as he never entirely forgot anything that
had once interested him, even although but slightly (this was in
reality a system of his; it gave him many holds on life, and kept
stored up a large supply of resources ready for use when wanted), he
came, after a while, on the canvas of his Roman impressions, to the
figure of Miss Macks. When he came to it he went to see her; that is, he
went to the street of the Hyacinth.
Of course, she might not be there; a hundred things might have happened
to her. He could have hunted up Horace Jackson; but, on the whole, he
rather preferred to see the girl herself first--that is, if she was
there. Mrs. Lawrence, the only person among his acquaintances who had
known her, was not in Rome. Reaching the street of the Hyacinth, he
interrogated the old woman who acted as portress at the lower door,
keeping up at the same time a small commerce in fritters; yes, the
Americans were still on the fourth floor. He ascended the dark stairway.
The confiding little "Ettie" card was no longer upon the door. In its
place was a small framed sign: "Miss Macks' School."
This told a story!
However, he rang. It was the same shrill, ill-tempered little bell, and
when the door opened it was Miss Macks herself who opened it. She was
much changed.
The parlor had been turned into a school-room--at present empty of
pupils. But even as a school-room it was more attractive than it had
been before. He took a seat, and spoke the usual phrases of a renewal of
acquaintance with his accustomed ease and courtesy; Miss Macks responded
briefly. She said that her mother was not very well; she herself quite
well. No, they had not left Italy, nor indeed the neighborhood of Rome;
they had been a while at Albano.
The expression of her face had greatly altered. The old direct, wide
glance was gone; gone also what he had called her over-confidence; she
looked much older. On the other hand, there was more grace in her
bearing, more comprehension of life in her voice and eyes. She was
dressed as plainly as before; but everything, including the arrangement
of her hair, was in the prevalent style.
She did not speak of her school, and therefore he did not. But after a
while he asked how the painting came on. Her face changed a little; but
it was more in the direction of a greater calm than hesitation or
emotion.
"I am not painting now," she answered.
"You have given it up temporarily?"
"Permanently."
"Ah--isn't that rather a pity?"
She looked at him, and a gleam of scorn filtered into the glance.
"You know it is not a pity," she said.
He was a little disgusted at the scorn. Of course, the only ground for
him to take was the ground upon which she stood when he last saw her; at
that time she proposed to pass her life in painting, and it was but good
manners for him to accept her intentions as she had presented them.
"I never assumed to be a judge, you know," he answered. "When I last had
the pleasure of seeing you, painting was, you remember, your cherished
occupation!"
"When you last had the pleasure of seeing me, Mr. Noel," said Miss
Macks, still with unmoved calm, "I was a fool."
Did she wish to go into the subject at length? Or was that merely an
exclamation?
"When I last had the pleasure of seeing you, you were taking lessons of
Mr. Jackson," he said, to give a practical turn to the conversation. "Is
he still here? How is he?"
"He is very well, now. He is dead."
(She was going to be dramatic then, in any case.)
He expressed his regret, and it was a sincere one; he had always liked
and respected the honest, morose Englishman. He asked a question or two.
Miss Macks replied that he had died here in the street of the
Hyacinth--in the next room. He had fallen ill during the autumn
following Noel's departure, and when his illness grew serious, they--her
mother and herself--had persuaded him to come to them. He had lived a
month longer, and died peacefully on Christmas Eve.
"He was one of the most honest men I ever knew," said Noel. Then, as she
did not reply, he ventured this: "That was the reason I recommended him
when you asked me to select a teacher for you."
"Your plan was made useless by an unfortunate circumstance," she
answered, with an evident effort.
"A circumstance?"
"Yes; he fell in love with me. If I did not consider his pure, deep, and
devoted affection the greatest honor of my life I would not mention it.
I tell you because it will explain to you his course."
"Yes, it explains," said Noel. As he spoke there came across him a
realization of the whole of the strength of the love such a man as
Horace Jackson would feel, and the way in which it would influence him.
Of course, he saw to the full the imperfection of her work, the utter
lack of the artist's conception, the artist's eye and touch; but
probably he had loved her from the beginning, and had gone on hoping to
win her love in return. She was not removed from him by any distance;
she was young, but she was also poor, friendless, and alone. When she
was his wife he would tell her the truth, and in the greatness of his
love the revelation would be naught. "He was a good man," he said. "He
was always lonely. I am glad that at last he was with your mother and
you."
"His goodness was simply unbounded. If he had lived he would have
remained always a faithful, kind, and respectful son to my dear mother.
That, of course, would have been everything to me." She said this
quietly, yet her tone seemed to hold intention.
For a moment he thought that perhaps she had married the Englishman, and
was now his widow. The sign on the door bore her maiden name, but that
might have been an earlier venture.
"Had you opened your school at that time?" he asked. "I may speak of it,
since, of course, I saw the sign upon the door."
"Not until two months later; I had the sign made then. But it was of
little use; day-schools do not prosper in Rome; they are not the custom.
I have a small class twice a week, but I live by going out as
day-governess. I have a number of pupils of that kind; I have been very
successful. The old Roman families have a fancy for English-speaking
governesses, you know. Last summer I was with the Princess C----, at
Albano; her children are my pupils."
"Her villa is a delightful one," said Noel; "you must have enjoyed
that."
"I don't know that I enjoyed, but I learned. I have learned a great
deal in many ways since I saw you last, Mr. Noel. I have grown very
old."
"As you were especially young when you saw me last it does not matter
much," he answered, smiling.
"Yes, I was especially young." She looked at him soberly. "I do not feel
bitterly towards you," she continued. "Strange! I thought I should. But
now that I see you in person it comes over me that, probably, you did
not intend to deceive me; that not only you tried to set me right by
selecting Mr. Jackson as my teacher, but again you tried when you sent
me those books. It was not much to do! But knowing the world as I now
know it, I see that it was all that could have been expected. At first,
however, I did not see this. After I went to Mr. Bellot, and, later, to
Mr. Salviati, there were months when I felt very bitterly towards you.
My hopes were false ones, and had been so from the beginning; you knew
that they were, yet you did not set me right."
"I might have done more than I did," answered Noel. "I have a habit of
not assuming responsibility; I suppose I have grown selfish. But if you
went to Bellot, then it was not Jackson who told you?"
"He intimated something when he asked me to marry him; after that his
illness came on, and we did not speak of it again. But I did not believe
him. I was very obstinate. I went to Mr. Bellot the 1st of January; I
wished him to take me as pupil. In answer he told me that I had not a
particle of talent; that all my work was insufferably bad; that I better
throw away my brushes and take in sewing."
"Bellot is always a brute!" said Noel.
"If he told the truth brutally, it was still the truth; and it was the
truth I needed. But even then I was not convinced, and I went to Mr.
Salviati. He was more gentle; he explained to me my lacks; but his
judgment was the same. I came home; it was the 10th of January, a
beautiful Roman winter day. I left my pictures, went over to St.
Peter's, and walked there under its bright mosaics all the afternoon.
The next day I had advertisements of a day-school placed at the bankers'
and in the newspapers. I thought that I could teach better than I could
sew." All this she said with perfect calm.
"I greatly admire your bravery, Miss Macks. Permit me to add that I
admire, even more, the clear, strong, good sense which has carried you
through."
"I had my mother to think of; my--good sense might not have been so
faithful otherwise."
"You do not think of returning to America?"
"Probably not; I doubt if my mother could bear the voyage now. We have
no one to call us back but my brother, and he has not been with us for
years, and would not be if we should return; he lives in California. We
sold the farm, too, before we came. No; for the present, at least, it is
better for us to remain here."
"There is one more question I should like to ask," said Noel, later.
"But I have no possible right to do so."
"I will give you the right. When I remember the things I asked you to do
for me, the demands I made upon your time, I can well answer a few
questions in return. I was a miracle of ignorance."
"I always did you justice in those respects, Miss Macks; all that I
understood at once. My question refers to Horace Jackson: I see you
appreciated his worth--which was rare--yet you would not marry him."
"I did not love him."
"Did any of his relatives come out from England?" he said, after a
moment of silence.
"After his death a cousin came."
"As heir to what was left?"
"Yes."
"He should have left it to you."
"He wished to do so. Of course, I would not accept it."
"I thank you for answering. My curiosity was not an idle one." He
paused. "If you will permit me to express it, your course has been very
brave and true. I greatly admire it."
"You are kind," said Miss Macks.
There was not in her voice any indication of sarcasm. Yet the fact that
he immediately thought of it made him suspect that it was there. He took
leave soon afterwards. He was smarting a little under the sarcasm he had
divined, and, as he was, it was like him to request permission to come
again.
For Raymond Noel lived up with a good deal of determination to his own
standard of what was manly; if his standard was not set on any very fine
elevation of self-sacrifice or heroism, it was at least firmly
established where it did stand, and he kept himself fairly near it. If
Miss Macks was sarcastic, he had been at fault somewhere; he would try
to atone.
He saw her four times during the five weeks of his stay in Rome; upon
three other occasions when he went to the street of the Hyacinth she was
not at home. The third week in April he decided to go to Venice. Before
going he asked if there was not something he could do for her; but she
said there was nothing, and he himself could think of nothing. She was
well established in her new life and occupations, and needed nothing--at
least, nothing that he could bestow.
The next winter he came back to Rome early in the season, before
Christmas. By chance one of the first persons he encountered was Mrs.
Lawrence. She began immediately to tell him a piece of American news, in
which he, as an American, would of course be interested; the news was
that "the brother of the Princess C---- --that is Count L----, you
know--is determined to marry Ettie Macks. You remember her, don't you? I
introduced you to her at the Dudley reception, three years ago."
Noel thought that probably he remembered her better than Mrs. Lawrence
did, seeing that that lady had never troubled herself to enter the
street of the Hyacinth. But he did her injustice. Mrs. Lawrence had
troubled herself--lately.
"It seems that she has been out at Albano for two summers, as governess
to his sister's children; it was there that he saw her. He has announced
his determination to the family, and they are immensely disturbed and
frightened; they had it all arranged for him to marry a second cousin
down at Naples, who is rich--these Italians are so worldly, you know!
But he is very determined, they say, and will do as he pleases in spite
of them. He hasn't much money, but of course it's a great match for
Ettie Macks. She will be a countess, and now, I suppose, more American
girls will come over than ever before! Of course, as soon as I heard of
it, I went to see her. I felt that she would need advice about a hundred
things. In the beginning she brought a letter of introduction to me from
a dear cousin of mine, and, naturally, she would rely upon me as her
chief friend now. She is very much improved. She was rather silent; but,
of course, I shall go again. The count is willing to take the mother,
too, and that, under the circumstances, is not a small matter; she is a
good deal to take. Until the other day I had not seen Mrs. Spurr!
However, I suppose that her deficiencies are not apparent in a language
she cannot speak. If her daughter would only insist upon her dressing in
black! But the old lady told me herself, in the most cheerful way, that
she liked 'a sprinkling of color.' And at the moment, I assure you, she
had on five different shades of red!"
Noel had intended to present himself immediately at the street of the
Hyacinth; but a little attack of illness kept him in for a while, and
ten days had passed before he went up the dark stairway. The maid said
that Miss Macks was at home; presently she came in. They had ten minutes
of conversation upon ordinary topics, and then he took up the especial
one.
"I am told that you are soon to be a countess," he said, "and I have
come to give you my best good wishes. My congratulations I reserve for
Count L----, with whom I have a slight acquaintance; he is, in my
opinion, a very fortunate man."
"Yes, I think he is fortunate; fortunate in my refusal. I shall not
marry Count L----."
"He is not a bad fellow."
"Isn't your praise somewhat faint?" This time the sarcasm was visible.
"Oh, I am by no means his advocate! All I meant was that, as these
modern Romans go, he was not among the worst. Of course I should have
expressed myself very differently if you had said you were to marry
him."
"Yes; you would then have honored me with your finest compliments."
He did not deny this.
"Shall you continue to live in Rome?" he asked.
"Certainly. I shall have more pupils and patronage now than I know what
to do with; the whole family connection is deeply obliged to me."
They talked awhile longer.
"We have always been unusually frank with each other, Miss Macks," he
said, towards the end of his visit. "We have never stopped at
conventionalities. I wonder if you will tell me why you refused him?"
"You are too curious. As to frankness, I have been frank with you; not
you with me. And there was no conventionality, simply because I did not
know what it was."
"I believe you are in love with some one in America," he said, laughing.
"Perhaps I am," answered Miss Macks. She had certainly gained greatly in
self-possession during the past year.
He saw her quite frequently after this. Her life was no longer solitary.
As she had said, she was overwhelmed with pupils and patronage from the
friends of the Princess C----; in addition, the American girl who had
refused a fairly-indorsed and well-appearing count was now something of
a celebrity among the American visitors in Rome. That they knew of her
refusal was not her fault; the relatives of Count L---- had announced
their objections as loud and widely as the count had announced his
determination. Apparently neither side had thought of a non-acceptance.
Cards, not a few, were sent to the street of the Hyacinth; some persons
even climbed the five flights of stairs. Mrs. Spurr saw a good deal of
company--and enjoyed it.
Noel was very fond of riding; when in Rome he always rode on the
Campagna. He had acted as escort to various ladies, and one day he
invited Miss Macks to accompany him--that is, if she were fond of
riding. She had ridden in America, and enjoyed it; she would like to go
once, if he would not be troubled by an improvised habit. They went
once. Then a second time, an interval of three weeks between. Then,
after a while, a third time.
Upon this occasion an accident happened, the first of Noel's life; his
horse became frightened, and, skilled rider though he was, he was
thrown. He was dragged, too, for a short distance. His head came against
some stones, and he lost consciousness. When it came back it did not
come wholly. He seemed to himself to be far away, and the girl who was
weeping and calling his name to be upon the other side of a wide space
like an ocean, over which, without volition of his own, he was being
slowly wafted. As he came nearer, still slowly, he perceived that in
some mysterious way she was holding in her arms something that seemed to
be himself, although he had not yet reached her. Then, gradually, spirit
and body were reunited, he heard what she was saying, and felt her
touch. Even then it was only after several minutes that he was able to
move and unclose his heavy eyes.
When she saw that he was not dead, her wild grief was at once merged in
the thought of saving him. She had jumped from her horse, she knew not
how; but he had not strayed far; a shepherd had seen him, and was now
coming towards them. He signalled to another, and the two carried Noel
to a house which was not far distant. A messenger was sent to the city;
aid came, and before night Noel was in his own rooms at the head of the
Via Sistina, near the Spanish steps.
His injuries proved to be not serious; he had lost consciousness from
the shock, and this, with his pallor and the blood from the cuts made by
the stones, had given him the look of death. The cuts, however, were not
deep; the effect of the shock passed away. He kept his bed for a week
under his physician's advice; he had a good deal of time to think during
that week. Later his friends were admitted. As has been said before,
Noel was a favorite in Rome, and he had friends not a few. Those who
could not come in person sent little notes and baskets of flowers. Among
these Miss Macks was not numbered. But then she was not fashionable.
At the end of two weeks the patient was allowed to go out. He took a
short walk to try his strength, and, finding that it held out well, he
went to the street of the Hyacinth.
Miss Macks was at home. She was "so glad" to see him out again; and was
he "really strong enough;" and he "should be very prudent for a while;"
and so forth and so forth. She talked more than usual, and, for her,
quite rapidly.
He let her go on for a time. Then he took the conversation into his own
hands. With few preliminaries, and with much feeling in his voice and
eyes, he asked her to be his wife.
She was overwhelmed with astonishment; she turned very white, and did
not answer. He thought she was going to burst into tears. But she did
not; she only sat gazing at him, while her lips trembled. He urged his
point; he spoke strongly.
"You are worth a hundred of me," he said. "You are true and sincere; I
am a dilettante in everything. But, dilettante as I am, in one way I
have always appreciated you, and, lately, all other ways have become
merged in that one. I am much in earnest; I know what I am doing; I have
thought of it searchingly and seriously, and I beg you to say yes."
He paused. Still she did not speak.
"Of course I do not ask you to separate yourself from your mother," he
went on, his eyes dropping for the moment to the brim of his hat, which
he held in his hand; "I shall be glad if she will always make her home
with us."
Then she did speak. And as her words came forth, the red rose in her
face until it was deeply .
"With what an effort you said that! But you will not be tried. One gray
hair in my mother's head is worth more to me, Mr. Noel, than anything
you can offer."
"I knew before I began that this would be the point of trouble between
us, Faith," he answered. "I can only assure you that she will find in me
always a most respectful son."
"And when you were thinking so searchingly and seriously, it was _this_
that you thought of--whether you could endure her! Do you suppose that I
do not see the effort? Do you suppose I would ever place my mother in
such a position? Do you suppose that you are of any consequence beside
her, or that anything in this world weighs in my mind for one moment
compared with her happiness?"
"We can make her happy; I suppose that. And I suppose another thing, and
that is that we could be very happy ourselves if we were married."
"The Western girl, the girl from Tuscolee! The girl who thought she
could paint, and could not! The girl who knew so little of social rules
that she made a fool of herself every time she saw you!"
"All this is of no consequence, since it is the girl I love," answered
Noel.
"You do not. It is a lie. Oh, of course, a very unselfish and noble one;
but a lie, all the same. You have thought of it seriously and
searchingly? Yes, but only for the last fourteen days! I understand it
all now. At first I did not, I was confused; but now I see the whole.
You were not unconscious out there on the Campagna; you heard what I
said when I thought you were dying, or dead. And so you come--come very
generously and self-sacrificingly, I acknowledge that--and ask me to be
your wife." She rose; her eyes were brilliant as she faced him. "I might
tell you that it was only the excitement, that I did not know or mean
what I was saying; I might tell you that I did not know that I had said
anything. But I am not afraid. I will not, like you, tell a lie, even
for a good purpose. I did love you; there, you have it! I have loved you
for a long time, to my sorrow and shame. For I do not respect you or
admire you; you have been completely spoiled, and will always remain so.
I shall make it the one purpose of my life from this moment to overcome
the feeling I have had for you; and I shall succeed. Nothing could make
me marry you, though you should ask me a thousand times."
"I shall ask but once," said Noel. He had risen also; and, as he did, he
remembered the time when they had stood in the same place and position,
facing each other, and she had told him that she was at his feet. "I did
hear what you said. And it is of that I have been seriously thinking
during the days of my confinement to the house. It is also true that it
is what you said which has brought me here to-day. But the reason is
that it has become precious to me--this knowledge that you love me. As I
said before, in one way I have always done you justice, and it is that
way which makes me realize to the full now what such a love as yours
would be to me. If it is true that I am spoiled, as you say I am, a love
like yours would make me better, if anything can." He paused. "I have
not said much about my own feelings," he added; "I know you will not
credit me with having any. But I think I have. I think that I love you."
"It is of little moment to me whether you do or not."
"You are making a mistake," he said, after a pause, during which their
eyes had met in silence.
"The mistake would be to consent."
She had now recovered her self-possession. She even smiled a little.
"Imagine Mr. Raymond Noel in the street of the Hyacinth!" she said.
"Ah, I should hardly wish to live here; and my wife would naturally be
with me."
"I hope so. And I hope she will be very charming and obedient and
sweet." Then she dropped her sarcasms, and held out her hand in
farewell. "There is no use in prolonging this, Mr. Noel. Do not think,
however, that I do not appreciate your action; I do appreciate it. I
said that I did not respect you, and I have not until now; but now I do.
You will understand, of course, that I would rather not see you again,
and refrain from seeking me. Go your way, and forget me; you can do so
now with a clear conscience, for you have behaved well."
"It is not very likely that I shall forget you," answered Noel,
"although I go my way. I see you are firmly resolved. For the present,
therefore, all I can do is to go."
They shook hands, and he left her. As he passed through the small hall
on his way to the outer door he met Mrs. Spurr; she was attired as
opulently, in respect to colors, as ever, and she returned his greeting
with much cordiality. He glanced back; Miss Macks had witnessed the
meeting through the parlor door. Her color had faded; she looked sad and
pale.
She kept her word; she did not see him again. If he went to the street
of the Hyacinth, as he did two or three times, the little maid presented
him with the Italian equivalent of "begs to be excused," which was
evidently a standing order. If he wrote to her, as he did more than two
or three times, she returned what he wrote, not unread, but without
answer. He thought perhaps he should meet her, and was at some pains to
find out her various engagements. But all was in vain; the days passed,
and she remained invisible. Towards the last of May he left Rome. After
leaving, he continued to write to her, but he gave no address for
reply; she would now be obliged either to burn his letters or keep them,
since she could no longer send them back. They could not have been
called love-letters; they were friendly epistles, not long--pleasant,
easy, sometimes amusing, like his own conversation. They came once a
week. In addition he sent new books, and occasionally some other small
remembrance.
In early September of that year there came to the street of the Hyacinth
a letter from America. It was from one of Mrs. Spurr's old neighbors at
Tuscolee, and she wrote to say that John Macks had come home--had come
home broken in health and spirits, and, as he himself said, to die. He
did not wish his mother to know; she could not come to him, and it would
only distress her. He had money enough for the short time that was left
him, and when she heard it would be only that he had passed away; he had
passed from her life in reality years before. In this John Macks was
sincere. He had been a ne'er-do-well, a rolling stone; he had not been a
dutiful son. The only good that could be said of him, as far as his
mother was concerned, was contained in the fact that he had not made
demands upon her small purse since the sum he took from her when he
first went away. He had written to her at intervals, briefly. His last
letter had come eight months before.
But the Tuscolee neighbor was a mother herself, and, doing as she would
be done by, she wrote to Rome. When her letter came Mrs. Spurr was
overwhelmed with grief; but she was also stirred to an energy and
determination which she had never shown before. For the first time in
years she took the leadership, put her daughter decisively back into a
subordinate place, and assumed the control. She would go to America. She
must see her boy (the dearest child of the two, as the prodigal always
is) again. But even while she was planning her journey illness seized
her--her old rheumatic troubles, only more serious than before; it was
plain that she could not go. She then required that her daughter should
go in her place--go and bring her boy to Rome; this soft Italian air
would give new life to his lungs. Oh, she should not die! Ettie need not
be afraid of that. She would live for years just to get one look at him!
And so it ended in the daughter's departure, an efficient nurse being
left in charge; the physician said that although Mrs. Spurr would
probably be crippled, she was in no danger otherwise.
Miss Macks left Rome on the 15th of September. On the 2d of December she
again beheld the dome of St. Peter's rising in the blue sky. She saw it
alone. John Macks had lived three weeks after her arrival at Tuscolee,
and those three weeks were the calmest and the happiest of his
unsuccessful--unworthy it may be--but also bitterly unhappy life. His
sister did not judge him. She kissed him good-bye as he lost
consciousness, and soon afterwards closed his eyes tenderly, with tears
in her own. Although he was her brother, she had never known him; he
went away when she was a child. She sat beside him a long time after he
was dead, watching the strange, youthful peace come back to his worn
face.
When she reached the street of the Hyacinth a carriage was before the
door; carriages of that sort were not often required by the dwellers on
the floors below their own, and she was rather surprised. She had heard
from her mother in London, the nurse acting as amanuensis; at that time
Mrs. Spurr was comfortable, although still confined to her bed most of
the day. As she was paying her driver she heard steps on the stairway
within. Then she beheld this: The nurse, carrying a pillow and shawls;
next, her mother, in an invalid-chair, borne by two men; and last,
Raymond Noel.
When Mrs. Spurr saw her daughter she began to cry. She had not expected
her until the next day. Her emotion was so great that the drive was
given up, and she was carried back to her room. Noel did not follow her;
he shook hands with the new-comer, said that he would not detain her,
and then, lifting his hat, he stepped into the carriage which was
waiting and was driven away.
For two days Mrs. Spurr wished for nothing but to hear, over and over
again, every detail of her boy's last hours. Then the excitement and
renewed grief made her dangerously ill. After ten days she began to
improve; but two weeks passed before she came back to the present
sufficiently to describe to her daughter all "Mr. No-ul's kind
attentions." He had returned to Rome the first of October, and had come
at once to the street of the Hyacinth. Learning what had happened, he
had devoted himself to her "most as if he was my real son, Ettie, I do
declare! Of course, he couldn't never be like my own darling boy,"
continued the poor mother, overlooking entirely, with a mother's sublime
forgetfulness, the small amount of devotion her boy had ever bestowed;
"but he's just done everything he could, and there's no denying that."
"He has not been mentioned in your letters, mother."
"Well, child, I just told Mrs. Bowler not to. For he said himself,
frankly, that you might not like it; but that he'd make his peace with
you when you come back. I let him have his way about it, and I _have_
enjoyed seeing him. He's the only person I've seen but Mrs. Bowler and
the doctor, and I'm mortal tired of both."
During Mrs. Spurr's second illness Noel had not come in person to the
street of the Hyacinth; he had sent to inquire, and fruits and flowers
came in his name. Miss Macks learned that these had come from the
beginning.
When three weeks had passed Mrs. Spurr was back in her former place as
regarded health. One of her first requests was to be taken out to drive;
during her daughter's absence Mr. Noel had taken her five times, and she
had greatly enjoyed the change. It was not so simple a matter for the
daughter as it had been for Mr. Noel; her purse was almost empty; the
long journeys and her mother's illness had exhausted her store. Still
she did it. Mrs. Spurr wished to go to the Pincio. Her daughter thought
the crowd there would be an objection.
"It didn't tire me one bit when Mr. No-ul took me," said Mrs. Spurr, in
an aggrieved tone; "and we went there every single time--just as soon as
he found out that I liked it. What a lot of folks he does know, to be
sure! They kept him a-bowing every minute."
The day after this drive Mr. Noel came to the street of the Hyacinth. He
saw Miss Macks. Her manner was quiet, a little distant; but she thanked
him, with careful acknowledgment of every item, for his kind attentions
to her mother. He said little. After learning that Mrs. Spurr was much
better he spoke of her own health.
"You have had two long, fatiguing journeys, and you have been acting as
nurse; it would be well for you to give yourself entire rest for several
weeks at least."
She replied, coldly, that she was perfectly well, and turned the
conversation to subjects less personal. He did not stay long. As he rose
to take leave, he said:
"You will let me come again, I hope? You will not repeat the 'not at
home' of last spring?"
"I would really much rather not see you, Mr. Noel," she answered, after
hesitating.
"I am sorry. But of course I must submit." Then he went away.
Miss Macks now resumed her burdens. She was obliged to take more pupils
than she had ever accepted before, and to work harder. She had not only
to support their little household, but there were now debts to pay. She
was out almost the whole of every day.
After she had entered upon her winter's work Raymond Noel began to come
again to the street of the Hyacinth. But he did not come to see her; his
visits were to her mother. He came two or three times a week, and always
during the hours when the daughter was absent. He sat and talked to Mrs.
Spurr, or rather listened to her, in a way that greatly cheered that
lady's monotonous days. She told him her whole history; she minutely
described Tuscolee and its society; and, finally, he heard the whole
story of "John." In addition, he sent her various little delicacies,
taking pains to find something she had not had.
Miss Macks would have put an end to this if she had known how. But
certainly Mr. Noel was not troubling _her_, and Mrs. Spurr resented any
attempt at interference.
"I don't see why you should object, Ettie. He seems to like to come, and
there's but few pleasures left to me, I'm sure! You oughtn't to grudge
them!"
In this way two months passed, Noel continuing his visits, and Miss
Macks continuing her lessons. She was working very hard. She now looked
not only pale, but much worn. Count L----, who had been long absent,
returned to Rome about this time. He saw her one day, although she did
not see him. The result of this vision of her was that he went down to
Naples, and, before long, the desirable second cousin with the fortune
was the sister of the Princess C----.
One afternoon in March Miss Macks was coming home from the broad, new,
tiresome piazza Indipendenza; the distance was long, and she walked with
weariness. As she drew near the dome of the Pantheon she met Raymond
Noel. He stopped, turned, and accompanied her homeward. She had three
books.
"Give them to me," he said, briefly, taking them from her.
"Do you know what I have heard to-day?" he went on. "They are going to
tear down your street of the Hyacinth. The Government has at last
awakened to the shame of allowing all those modern accretions to
disfigure longer the magnificent old Pagan temple. All the streets in
the rear, up to a certain point, are to be destroyed. And the street of
the Hyacinth goes first. You will be driven out."
"I presume we can find another like it."
He went on talking about the Pantheon until they entered the doomed
street; it was as obstinately narrow and dark as ever. Then he dropped
his Pagan temple.
"How much longer are you going to treat me in this way, Faith?" he said.
"You make me very unhappy. You are wearing yourself out, and it troubles
me greatly. If you should fall ill I think that would be the end. I
should then take matters into my own hands, and I don't believe you
would be able to keep me off. But why should we wait for illness? It is
too great a risk."
They were approaching her door. She said nothing, only hastened her
steps.
"I have been doing my best to convince you, without annoying you, that
you were mistaken about me. And the reason I have been doing it is that
I am convinced myself. If I was not entirely sure last spring that I
loved you, I certainly am sure now. I spent the summer thinking of it. I
know now, beyond the possibility of a doubt, that I love you above all
and everything. There is no 'duty' or 'generosity' in this, but simply
my own feelings. I could perfectly well have let the matter drop; you
gave me every opportunity to do so. That I have not done it should show
you--a good deal. For I am not of the stuff of which heroes are made. I
should not be here unless I wanted to; my motive is the selfish one of
my own happiness."
They had entered the dark hallway.
"Do you remember the morning when you stood here, with two tears in your
eyes, saying 'Never mind; you will come another time'?" (Here the
cobbler came down the stairs.) "Why not let the demolition of the street
of the Hyacinth be the crisis of our fate?" he went on, returning the
cobbler's bow. (Here the cobbler departed.) "If you refuse, I shall not
give you up; I shall go on in the same way. But--haven't I been tried
long enough?"
"You have not," she answered. "But, unless you will leave Rome, and--me,
I cannot bear it longer."
It was a great downfall, of course; Noel always maintained that it was.
"But the heights upon which you had placed yourself, my dear, were too
superhuman," he said, excusingly.
The street of the Hyacinth experienced a great downfall, also. During
the summer it was demolished.
Before its demolition Mrs. Lawrence, after three long breaths of
astonishment, had come to offer her congratulations--in a new direction
this time.
"It is the most fortunate thing in the world," she said to everybody,
"that Mrs. Spurr is now confined to her bed for life, and is obliged to
wear mourning."
But Mrs. Spurr is not confined to her bed; she drives out with her
daughter whenever the weather is favorable. She wears black, but is now
beginning to vary it with purple and lavender.
A CHRISTMAS PARTY
In 188- the American Consul at Venice was occupying the second story of
an old palace on the Grand Canal. It was the story which is called by
Italians the _piano nobile_, or noble floor. Beneath this _piano nobile_
there is a large low ground, or rather water, floor, whose stone
pavement, only slightly above the level of the canal outside, is always
damp and often wet. At the time of the Consul's residence this
water-floor was held by another tenant, a dealer in antiquities, who had
partitioned off a shallow space across its broad front for a show-room.
As this dealer had the ground-floor, he possessed, of course, the
principal entrance of the palace, with its broad marble steps descending
into the rippling wavelets of the splendid azure street outside, and
with the tall, slender poles, irregularly placed in the water, which
bore testimony to the aristocracy of the venerable pile they guarded.
One could say that these blue wands, ornamented with heraldic devices,
were like the spears of knights; this is what Miss Senter said. Or one
could notice their strong resemblance to barbers' poles; and this was
what Peter Senter always mentioned.
Peter Senter was the American Consul, and his sister Barbara was the
Consuless; for she kept house for her brother, who was a bachelor. And
she not only kept house for him, but she assisted him in other ways,
owing to her knowledge of Italian. The Consul, a man of fifty-seven,
spoke only the language of his native place--Rochester, New York. That
he could not understand the speech (gibberish, he called it) of the
people with whom he was supposed to hold official relations did not
disturb him; he thought it patriotic not to understand. There was a
vice-consul, an Italian, who could attend to the business matters; and
as for the rest, wasn't Barbara there--Barbara, who could chatter not
only in Italian, but in French and German also, with true feminine
glibness? (For Peter, in his heart, thought it unmasculine to have a
polyglot tongue.) He knew how well his sister could speak, because he
had paid her bills during the six years of her education abroad. These
bills had been large; of course, therefore, the knowledge must be large
as well.
Miss Senter was always chronically annoyed that she and her brother did
not possess the state entrance. As the palace was at present divided,
the tenants of the noble floor descended by an outside stairway to a
large inner court, and from this court opened the second water-door.
Their staircase was a graceful construction of white marble, and the
court, with the blue sky above, one or two fretted balconies, and a
sculptured marble well-curb in the centre, was highly picturesque. But
this did not reconcile the American lady to the fact that their door was
at the side of the palace; she thought that by right the gondola of the
Consul should lie among the heraldic poles on the Grand Canal. But, in
spite of right, nothing could be done; the antiquity-dealer held his
premises on a long lease. Miss Senter, therefore, disliked the dealer.
Her dislike, however, had not prevented her from paying a visit to his
establishment soon after she had taken possession of the high-ceilinged
rooms above. For she was curious about the old palace, and wished to see
every inch of it; if there had been cellars, she would have gone down to
inspect them, and she was fully determined to walk "all over the roof."
The dealer's name was Pelham--"Z. Pelham" was inscribed on his sign. How
he came by this English title no one but himself could have told. He was
supposed to be either a Pole or an Armenian, and he spoke many languages
with equal fluency and incorrectness. He appeared to have feeble health,
and he always wore large arctic over-shoes; he was short and thin, and
the most noticeable expression of his plain, small face was resignation.
Z. Pelham conducted the Consuless through the dusky space behind his
show-room, a vast, low, open hall with massive squat columns and arches,
and the skeletons of two old gondolas decaying in a corner. At the back
he opened a small door, and pointed out a flight of stone steps going up
steeply in a spiral, enclosed in a circular shaft like a round tower.
"It leads to the attic floor. Her Excellency wishes to mount?" he
inquired, patiently. For, owing to the wares in which he dealt, he had
had a large acquaintance with eccentric characters of all nations.
"Certainly," replied Miss Senter. "Carmela, you can stay below, if you
like," she said to the servant who accompanied her.
But no; Carmela also wished to mount. Z. Pelham preceded them,
therefore, carrying his small oil-lamp. They went slowly, for the steps
were narrow, the spiral sharp. The attic, when they reached it, was a
queer, ghostly place; but there was a skylight with a ladder, and the
Consuless carried out her intention of traversing the roof, while Mr.
Pelham waited calmly, seated on the open scuttle door. Carmela followed
her mistress. She gave little cries of admiration; there never were such
wonderful ladies anywhere as those of America, she declared. On the way
down, the stairs were so much like a corkscrew that Miss Senter, feeling
dizzy, was obliged to pause for a moment where there was a landing.
"Isn't there a secret chamber?" she demanded of the dealer.
Z. Pelham shook his head. "I have not one found."
"Try again," said Miss Senter, laughing. "I'll make it worth your while,
Mr. Pelham."
Z. Pelham surveyed the walls, as if to see where he could have one
built. His eye passed over a crack, and, raising his lamp, he showed it
to the Consuless. "One time was there a door, opening into the rooms of
her Excellency. But it opens not ever now; it is covered on inside."
"Oh, _that_ isn't a secret chamber," answered Miss Senter; "we have
doors that have been shut up at home. What I want is something
mysterious--behind a picture, or a sliding panel."
Partly in return for this expedition to the roof, and partly because she
had a liking for wood-carvings, Miss Senter purchased from Mr. Pelham,
shortly afterwards, his best antique cabinet. It was eight feet high,
and its whole surface was beautifully sculptured in odd designs, no two
alike. Within were many ingenious receptacles, and, better than these, a
concealed drawer. "You see I have my secret chamber, after all," said
the Consuless, making a joke. And there was a best even to this better;
for after the cabinet had been placed in her own room, Miss Senter
discovered within it a second hiding-place, even more perfectly
concealed than the first. This was delightful, and she confided to its
care all her loose money. She thought with disgust of the ugly green
safe, built into the wall of Peter's Rochester house, where she was
obliged to keep her gold and silver when at home. Not only was Miss
Senter's own room in the old palace handsomely furnished, but all the
others belonging to the apartment were rich in beautiful things. The
Consuless had used her own taste, which was great, and her brother's
fortune, which was greater, deferring to him only on one point--namely,
warmth. In Peter's mind the temperature of his Rochester house remained
a fixed standard, and his sister therefore provided in every room a
place for a generous open fire, while in the great drawing-room, in
addition to this fire, two large white Vienna stoves, like monuments,
were set up, hidden behind screens. As this salon was eighty feet long
and thirty feet high, it required all this if it was to be used--used by
Peter, at least--in December, January, and February; for the Venetian
winter, though short, is often sharp and raw.
On Christmas Eve of their third year in Venice this drawing-room was
lighted for a party. At one end, concealed by a curtain, stood a
Christmas-tree; for there were thirty children among their invited
guests, who would number in all over fifty. After the tree had bestowed
its fruit the children were to have a dance, and an odd little
projection like a very narrow balcony high on the wall was to be
occupied by five musicians. These musicians would have been much more
comfortable below. But Miss Senter was sure that this shelf was
intended for musicians; her musicians, therefore, were to sit there,
though their knees would be well squeezed between the wall and the
balustrade. Fifteen minutes before the appointed hour, which was an
early one on account of the children, the Consuless appeared. She found
her brother standing before the fire, surveying the room, with his hands
behind him.
"Doesn't it look pretty?" said the sister, with pride; for she had a
great faith in all her pots and pans, carvings and tapestries. Any one,
however, could have had faith in the chandeliers of Venetian glass, from
which came the soft radiance of hundreds of wax candles, lighting up the
ancient gilding of the ceiling.
"Well, Barly, you know that personally I don't care much for all these
second-hand articles you have collected," replied Peter. "And you
haven't got the room very warm, after all--only 60 deg.. However, I can
stand it if the supper is all right--plenty of it, and the hot things
really hot; not lukewarm, you know."
"We can trust Giorgio. But I'll go and have a final word with him, if
you like," answered Miss Senter, crossing the beautiful salon, her train
sweeping over the floor behind her. The Consuless was no longer young
(the days when Peter had paid those school bills were now far distant),
and she had never been handsome. But she was tall and slender, with
pretty hands and feet, a pleasant expression in her blue eyes, and soft
brown hair, now heavily tinged with silver. Her brother's use of "Barly"
was a grief to her. She had tried to lead him towards the habit of
calling her Barbe, the French form of Barbara, if nickname he must have.
But he pronounced this Bob, and that was worse than the other.
On her way towards the kitchen the Consuless came upon Carmela. Carmela
was the servant who had the general oversight of everything excepting
the cooking. For Giorgio, the cook, allowed no interference in his
department; in the kitchen he must be Caesar or nothing. Carmela was not
the house-keeper, for Miss Senter herself was the house-keeper. But the
American would have found her task twenty times, fifty times more
difficult if she had not had this skilful little deputy to carry out all
her orders. Carmela was said to be middle-aged. But her short, slender
figure was so erect, her little face so alert, her movements were so
brisk, and her small black eyes so bright, that she seemed full of
youthful fire; in fact, if one saw only her back, she looked younger
than Assunta and Beppa, who were Venetian girls of twenty. Carmela was
always attired in the French fashion, with tight corsets, a plain black
dress fitting like a glove round her little waist, and short enough to
show the neat shoes on her small feet; over this black dress there was a
jaunty white apron with pockets, and upon her beautifully braided
shining dark hair was perched a small spotless muslin cap. The younger
servants asserted that the slight pink tint on the tidy little woman's
cheeks was artificial. However that may have been, Carmela, as she
stood, was the personification of trimness and activity. Untiring and
energetic, she was a wonderful worker; Miss Senter, who had been much in
Italy, appreciated her good-fortune in having secured for her Venetian
house-keeping such a coadjutor as this. Carmela was scrupulously neat,
and she was even more scrupulously honest, never abstracting so much as
a pin; she economized for her mistress with her whole soul, and kept
watch over every detail; she told the truth, she swept the corners, she
dusted under everything; she worked conscientiously, in one way and
another, all day long. Even Peter, who did not like foreign servants,
liked Carmela; he said she was "so spry!"
"Is everything ready?" inquired Miss Senter, as she met her deputy.
"Yes, signorina, everything," answered Carmela, briskly. She was looking
her very best and tightest, all black and white, with black silk
stockings showing above her little high-heeled shoes. As she spoke she
put her hands in their black lace mitts in the pockets of her apron,
and, middle-aged though she was said to be, she looked at that moment
like a smart French soubrette of the stage.
"I am going to the kitchen to have a word with Giorgio," said the
Consuless, passing on.
"If the signorina permits, I carry the train," answered Carmela, lifting
the satin folds from the floor. Thus they went on together, mistress and
maid, through various rooms and corridors, until finally the kitchen was
reached. It was a large, lofty place, brilliantly lighted, for Giorgio
was old and needed all the radiance that could be obtained to aid his
failing sight. He was a small man with a melancholy countenance. But
this melancholy was an accident of expression; in reality, old Giorgio
was cheerful and amiable, with a good deal of mild wit. He was the most
skilful cook in Venice. But his health had failed some years before, and
he had now very little strength; the Consul, who liked good dinners,
paid him high wages, and gave him a young assistant.
"Well, Giorgio, all promises well, I trust?" said Miss Senter as she
entered, her steps somewhat impeded by the tightness with which Carmela
held back her train. "The Consul is particular about having the hot
things really hot, and constantly renewed, as it is such a cold night.
The three men from Florian's will have charge of the ices and the other
cold things, and will do all that is necessary in the supper-room. But
for the hot dishes we depend upon you."
Giorgio, who was dressed entirely in white, bowed and waved his hand.
"Mademoiselle need give herself no uneasiness," he said in French. For
Giorgio had learned his art in Paris, and whenever Carmela was present
he invariably answered his mistress in the language of that Northern
capital, even though her question had been couched in Italian; it was
one of his ways--and he had but few--of standing up, as it were, against
the indefatigable little deputy. For, clever though Carmela was, she had
never been out of her native land, and could speak no tongue but her
own.
"Are you feeling well, Giorgio?" continued Miss Senter. "I see that you
look pale. I am afraid you have been doing too much. Where is Luigi?"
(Luigi was the cook's assistant.)
"He has gone home; ten minutes ago. I let him go, as it is a festival.
He is young, and we can be young but once. _Che vuole!_ In addition, all
was done."
"No," said Miss Senter, who was now speaking French also; "there is
still much to do, and it was not wise to let Luigi go. You are certainly
very tired, Giorgio."
"Let not mademoiselle think of it," said the old man, straightening
himself a little.
"But I _shall_ think of it," said Miss Senter, kindly. "Carmela," she
continued, speaking now in Italian, "go to my room and get my case of
cordials."
Carmela divined that the cordial was for the cook. "And the signorina's
train?" she said. "Surely I cannot leave it on this _dirty_ floor! Will
not the signorina return to the drawing-room to take her cordial? Eh--it
is not for her? It is for Giorgio? A man? A _man_ to be faint like a
girl? Ha, ha! it makes me laugh!"
"Go and get it," repeated Miss Senter, taking the train over her own
arm. She knew that Carmela did not like the cook. Jealousy was the one
fault the hard-working little creature possessed. "She has tried to make
me dismiss Giorgio more than once," she said to her brother, in
confidence; "but I always pretend not to see the feeling that influences
her. It is only Giorgio she is jealous of; she gets on perfectly well
with Luigi, and with Assunta and Beppa; while for Ercole she can never
do enough. She is devoted to Ercole!"
Giorgio had not taken up the slur cast upon his immaculate floor. All he
said was, "_Comme elle est mechante!_" with a shrug.
"Where is Ercole?" said Miss Senter, while she waited.
"He is dressing," answered Giorgio. "He makes himself beautiful for the
occasion."
Ercole was the chief gondolier--a tall, athletic young man of thirty,
handsome and clever. Miss Senter had chosen Ercole to assist her with
the Christmas-tree. The second gondolier, Andrea, was to be stationed at
the end of the little quay or riva down below, outside of their own
water-door; for here on the small canal were the steps used by arriving
and departing gondolas, and here also floated the handsome gondola of
the Consul, with its American flag. The two gondoliers also had
picturesque costumes of white (woollen in winter, linen in summer), with
blue collars, blue stockings, blue caps, and long fringed red sashes,
the combination representing the American national colors. To-night
Ercole, having to appear in the drawing-room, was making a longer stay
than usual before his little mirror.
Carmela returned with the cordial-case. "Ah, yes, our cook _is_
pale--pale as a young virgin!" she commented, as Miss Senter, unlocking
the box, poured into one of the little glasses it contained a generous
portion of a restorative whose every drop was costly.
Giorgio, taking off the white linen cap which covered his gray hair,
made a bow, and then drank the draught with much appreciation. "It is
true that I am pale," he remarked, slyly, in Italian. "I might, perhaps,
try some rouge?"
And then the Consuless, to avert war, hastily bore her deputy away.
Half an hour later the guests had arrived; they included all the
Americans in Venice, with a sprinkling of English, Italians, and
Russians. The grown people assembled in the drawing-room. And presently
they heard singing. Through the anterooms came the children, entering
with measured step, two and two, led by three little boys in Oriental
costumes. These three boys were singing as follows:
"We three Kings of Orient are,
Bearing gifts we've travelled from far,
Field and fountain, moor and mountain,
Following yonder star."
Here, from the high top branch of the Christmas-tree which rose above
the concealing curtain, blazed out a splendid star. And then all the
procession took up the chorus, as they marched onward:
"Oh, star of wonder,
Star of might,
Star with royal
Beauty bright!"
Ercole, who was behind the curtain, now drew it aside, and there stood
the tree, blazing with fairy-lamps and glittering ornaments, while
beneath it was a mound composed entirely of toys. The children behaved
well; they kept their ranks and repeated their carol, as they had been
told to do, ranging themselves meanwhile in a half-circle before the
tree.
"We three Kings of Orient are,"
chanted the three little kings a second time, though their eyes were
fixed upon a magnificent box of soldiers, with tents and flags and
cannon. The carol finished, Miss Senter, with the aid of her gondolier,
distributed the toys and bonbons, and the room was filled with happy
glee. When Ercole had detached the last package of sweets from the
sparkling branches he disappeared. His next duty was to conduct the
musicians up to their cage.
Miss Senter had allowed an hour for the inspection and trial of the toys
before the dancing should begin. It was none too much, and the clamor
was still great as this hour drew towards its close, so great that she
herself was glad that the end was near. Looking up to see whether her
musicians had assembled on their shelf, she perceived some one at the
drawing-room door; it was Carmela, hiding herself modestly behind the
portiere, but at the same time unmistakably beckoning to her mistress as
soon as she saw that she had caught her eye. Miss Senter went to the
doorway.
"Will the signorina permit? A surprise of Ercole's," whispered Carmela,
eagerly, standing on tiptoe to reach her mistress's ear. "He has dressed
himself as a clown, and he _is_ of a perfection! He has bells on his cap
and his elbows, and if the signorina graciously allows, he will come in
to amuse the children."
"A clown!" answered Miss Senter, hesitating. "I don't know; he ought to
have told me."
"He has been dancing to show _me_. And oh! so beautifully, with bounds
and leaps. He makes of himself also a statue," pursued Carmela.
"But I cannot have any buffoonery here, you know," said Miss Senter. "It
would not do."
"Buffoonery! Surely the signorina knows that Ercole has the soul of a
gentleman," whispered Carmela, reproachfully.
And it was true that Miss Senter had always thought that her chief
gondolier possessed a great deal of natural refinement.
"Will the signorina step out for a moment and look at him?" pursued the
deputy, her whisper now a little dejected. "If he is to be disappointed,
poor fellow, may he at least have _that_ pleasure?"
The idea of the gondolier's disappointment touched the amiable American.
She turned her head and glanced into the drawing-room; all was going on
gayly; no one had missed her. She slipped out under the portiere, and
followed Carmela to a room at the side. Here stood the gondolier. He
wore the usual white dress and white mask of a clown, and, as the
Consuless entered, he cut a splendid caper, ringing all his bells.
"I had no idea that you were such a skilful acrobat, Ercole," said his
mistress.
Ercole turned a light somerset, gave a high jump, and came down in the
attitude of the Mercury of John of Bologna.
"Why, you are really wonderful!" said Miss Senter, admiringly.
And now he was dancing with butterfly grace.
Miss Senter was won. "But if I let you come in, Ercole, I hope you will
remember where you are?" she said, warningly. "Can you breathe quite at
ease in that mask?"
The gondolier opened his grotesque painted lips a little to show that he
could part them.
"Yes, I see. Now listen; in the drawing-room you must keep your eye on
me, and if at any time you see me raise my hand--so--you must dance out
of the room, Ercole. For the sign will mean that that is enough. But,
dear me! there's one thing we haven't thought of; who is to see to the
musicians up-stairs, and to go back and forth, telling them what to
play?"
"I can do that," said Carmela, who was now all smiles. "Does the
signorina wish me to take them up? They are all ready. They are waiting
in the wood-room."
The wood-room was a remote store-room for fuel; it was detached from the
rest of the apartment. "Why did you put them _there_?" inquired Miss
Senter, astonished.
"They are musicians--yes; but who knows what else they may be? Thieves,
perhaps!" said the deputy, shrewdly.
"Get them out immediately and take them up to the gallery," said Miss
Senter. "And tell them to play something lively as a beginning."
Carmela, quick as usual, was gone before the words were ended.
"Now, Ercole, wait until you hear the music. Then come in," said the
Consuless.
She returned to the drawing-room, making a motion with her hands as she
advanced, which indicated that her guests were to move a little more
towards the walls on each side, leaving the centre of the room free. And
then, as the music burst out above, Ercole came bounding in. His dress
was ordinary; Miss Senter was vexed anew that he had not told her of his
plan, for if he had she could have provided a perfectly fresh costume.
But no one noticed the costume; all eyes were fixed upon the gambols;
for, keeping time to the music, he was advancing up the room, dancing,
bounding, leaping, turning somersets, and every now and then striking an
attitude with extraordinary skill. He was so light that his white linen
feet made no sound, and so graceful that the fixed grin of his mask
became annoying, clashing as it did with the beauty of his poses. This
thought, however, came to the elders only; for to the children,
fascinated, shouting with delight, the broad red smile was an important
part.
"It's our gondolier," explained Miss Senter. "It's Ercole," she had
whispered to her brother.
"You are always so fortunate in servants," said Lady Kay. "That little
woman you have, too, Carmela--she is a miracle for an Italian."
Four times the clown made his pyrotechnic progress up and then down the
long salon, never twice repeating the same pose, but always something
new; then, after a final tremendous pigeon-wing, he let his white arms
fall and his white head droop on his breast, as if saying that he was
taking a moment for repose.
"Yes, yes; give him time to breathe, children," cried Peter. "I'll tell
you what," he added to Sir William Kay; "I've never seen a better
performance on any stage." And he slapped his leg in confirmation. The
Consul was a man whose sole claim to beauty lay in the fact that he
always looked extremely clean. He was meagre and small, with very short
legs, but he was without consciousness of these deficiencies; in the
presence of the Apollo Belvedere, for instance, it had never occurred to
him to draw comparisons. Nature, however, will out in some way, and from
childhood Peter Senter had had a profound admiration for feats of
strength, vaulting, tumbling, and the like. "I'll tell you what," he
repeated to Sir William; "I'll have the fellow exhibited; I'll start him
at my own cost. Here all this time--two whole years--he has been our
gondolier, Ercoly has, and nothing more; for I hadn't a suspicion that
he had the least talent in this line. But, sir, he's a regular
high-flier! And A Number One!"
Meanwhile the children were crowding closely round their clown, and
peering up in order still to see his grin, which was now partly hidden,
owing to his drooped head; the three Kings of Orient, especially, were
very pressing in their attentions, pinching his legs to see if they were
real.
"Come, children, this will be a good time for our second song," said
Miss Senter, making a diversion. "Take hands, now, in a circle;
yes--round the clown, if you wish. There--that's right." She signalled
to the music to stop, and then, beginning, led the little singers
herself:
"Though we're here on foreign shores,
We are all devotion
To our land of Stars and Stripes,
Far across the ocean.
Yankee doodle doodle doo,
Yankee doodle dandy,
Buckwheat cakes are very good,
And so's molasses candy."
Singing this gayly to the well-known fife-like tune, round and round
danced the children in a circle, holding each other's hands, the English
and Italians generously joining with the little Americans in praise of
the matutinal cakes which they had never seen; the Consuless had drilled
her choir beforehand, and they sang merrily and well. The first four
lines of this ditty had been composed by Peter himself for the occasion.
"I hear _you_ haf written this vurra fine piece!" said a Russian
princess, addressing him.
"Oh no," answered the Consul; "I only wrote the first four lines; the
chorus is one of our national songs, you know."
"But those first four lines--their sentiment ees so fine, so speerited!"
said the princess.
"Well, they're _neat_," Peter admitted, modestly.
The clown, having recovered his breath, cut a caper. Instantly "Yankee
Doodle" came to an end, and the children all stopped to watch him.
"Tell them to play a waltz," said Miss Senter to Carmela, who was in
waiting at the door. The deputy must have flown up the little stairway
leading to the gallery, for the waltz began in less than a minute. Then
Ercole, selecting a pretty American child from among the group, began to
dance with her in the most charming way, followed by all the little
ones, two and two. Those who could waltz, did so; those who could not,
held each other's hands and hopped about.
Supper followed. The hot things were smoking and delicious, and the
supplies constantly renewed; old Giorgio was evidently on his mettle. It
was the gondolier, still in his clown's dress, who brought in these
supplies and handed them to the waiters from Florian's.
"You need not do that, Ercole," said Miss Senter, in an undertone;
"these men can go to the kitchen for them."
Ercole bowed; it would not have been respectful to reply with his
grinning linen lips. But he continued to fill the same office.
"Perhaps Giorgio won't have Florian's people in the kitchen!" the
Consuless reflected.
As soon as supper was over, the children clamored for their clown, and
he came bounding in a second time, and, after several astonishing
capers, selected a beautiful English child with long golden curls and
led a galop, followed again by all the others, two and two. Peter, his
mind still occupied with his project of taking the young Italian to
America as a star performer, moved from point to point, in order to get
different views of him. One of these stations was in the doorway, and
here Carmela spoke to him in a low tone, and asked him to come to the
outer hall. He did not understand her words; but he comprehended her
gesture and followed her. She was talking angrily, almost spluttering,
as she led the way. But her talk was lost on her master, who, however,
opened his eyes when he saw four policemen standing at his outer door.
"What do you want here?" he said. "This is a private residence, and you
are disturbing a Christmas party."
The chief officer told his tale. But Peter did not comprehend him.
"You should have gone to the Consulate," he went on. "The Consulate, you
know--Riva Skevony. The vice-consul won't be there so late as this; but
you'll find him early to-morrow morning, sure."
The policemen, however, remained where they were.
"There's no making them understand a word," said Peter to himself, in
irritation. "Here, you go and call my sister," he said to Carmela, who,
in her wrath over this intrusion, stood at a distance swallowing nothing
in a series of gulps that made her throat twitch. "Let's see; sister,
that's sorelly. Sorelly!" he repeated to Carmela. "Sorelly!"
The enraged little deputy understood. And she got Miss Senter out of the
drawing-room without attracting notice. "The master wishes to see the
signorina," she said, in a concentrated undertone. "I burn with
indignation, for it is an insolent intrusion; it is an insult to his
Excellency, who no doubt is a prince in his own country. But they
_would_ not go, in spite of all I could say. Nor would they tell me
their errand--brutes!" And with her skirts quivering she led the way to
the outer hall.
"Find out what these men want, Barly," said Peter, when his sister
appeared.
And then the chief officer again told his story.
"Mercy!" said Miss Senter, "how dreadful. Somebody was killed, Peter,
about seven o'clock this evening, in a cafe near the Rialto, and they
say they have just found a clew which appears to track the assassin to
this very door! And they wish to search."
"What an absurd idea! With the whole place crowded and blazing with
lights, as it is to-night, a mouse couldn't hide," said Peter. "Tell
them so."
"They repeat that they must search," said Miss Senter. "But if you will
exert your authority, Peter--make use of your official position--I am
sure we need not submit to such a thing."
Peter, however, was helpless without his vice-consul; he had no clear
idea as to what his powers were or were not; he had never informed
himself.
Carmela, greatly excited, had drawn Miss Senter aside. "There was a
sixth man with those musicians!" she whispered. "I saw him. He did not
play, but he sat behind them. And he has only just gone. Five minutes
ago."
Miss Senter repeated the information to the chief officer. The officer
immediately detached two men to follow this important clew; he himself,
with the third, would remain to go through the apartment, as a matter of
form.
"As the rooms are all open and lighted," said Miss Senter in English to
her brother, "it will only take a few minutes, if go they must, and no
one need know anything about it. But whom shall we send with them? If we
call Ercole, it will attract attention; and Florian's men, who were due
at another place, have already gone. We could have Andrea come up. But
no; Giorgio will do best of all. Call Giorgio to go with these men," she
added in Italian to Carmela.
"Let _me_ conduct them!" answered the deputy.
"Yes; on the whole, she will be better than any one," said Miss Senter
to Peter. "She is so angry at what she calls the insult to you, and so
excited about the mysterious person who was with the musicians, that she
will bully them and hurry them off to look for him in no time. They can
begin with a peep into the drawing-room; I'll tell them to keep
themselves hidden." She turned and explained her idea in Italian to the
officer; they could glance into the drawing-room first, and then Carmela
would take them through all the other rooms; the Consul, though he had
the power of refusal, would permit this liberty in the cause of justice.
Their search, however, would be unavailing; under the circumstances, it
was impossible that any one should have taken refuge there, unless it
was that one extra man who had been admitted with the musicians to the
gallery. And he was already gone.
"Perhaps he only pretended to go?" suggested the officer. "With
permission, I will lock this door." And he did so.
[Illustration: "A SMALL CHILD PERCHED ON EACH OF HIS SHOULDERS"]
They went to the drawing-room, the policemen moving quietly, close to
the wall. When the last anteroom was reached, the two men hid themselves
behind the tapestries that draped the door, and, making loop-holes among
the folds, peeped into the ball-room. For it was at that moment a
ball-room. The children had again taken up their whirling dance around
Ercole, and the gondolier, who had now a small child perched on each of
his shoulders, was singing with them in a clear tenor, having caught
the syllables from having heard them shouted about fifty times:
"Yankee dooda dooda doo,
Yankee dooda dandee,
Barkeet cakar vera goo,
Arso molarsa candee."
Miss Senter had sent Peter back to his guests. She herself, standing
between the tapestries as though she were looking on from the doorway,
named to the hidden policemen, as well as she could amid the loud
singing within, all the persons present, one by one. Finally her list
came to a close. "And that is Mr. Barlow, the American who lives at the
Danieli; and the one near the Christmas-tree is Mr. Douglas, who has the
Palazzo Dario. And the tall, large gentleman with silver hair is Sir
William Kay. That is all, except the clown, who is our gondolier, and
the five musicians up in the gallery; can you see them from here? If
not, Carmela can take you up." And then she thought, with a sudden
little shudder, that perhaps the officer's idea was not, after all,
impossible; perhaps, indeed, that extra man had only pretended to go!
The policemen signified that this was enough as regarded the
drawing-room; they withdrew softly, and waited outside the door.
"Now take them through all the other rooms, Carmela," whispered the
Consuless. "Be as quiet about it as you can, so that no one need know.
And when they have finally gone, come and stand for a moment between
these curtains, as a sign. If, by any chance, they _should_ discover any
one--"
"The signorina need not be frightened; I saw the man go myself! And he
could not have re-entered without my knowledge. As for these beasts of
policemen--" And Carmela's eyes flashed, while her set lips seemed to
say, "Trust _me_ to hustle them out!"
"Run up first and tell the musicians to play the music I sent them,"
said the Consuless. And then she rejoined her guests.
For the next dance was to be a Virginia Reel, and some of the elders
were to join the children; the two lines, when arranged, extended down
half the length of the long room. It began with great spirit, the clown
and the three Kings of Orient dancing at the end of the file.
"It is really Sir Roger de Coverley, an English dance," said Lady Kay to
the Russian princess, who was looking on from the chair next her own.
"But the Senters like to call it a Virginia Reel, they are so patriotic.
And we never contradict the Senters, you know," added the English lady,
laughing; "we let them have their way."
"It seems to me a vurra good way," answered the princess, who was a
plain-looking old woman with a charming smile. "I have nowhere seen so
many reech toyees" (here she glanced at the costly playthings heaped on
a table near by). "Nor haf I, in _Italy_, seen so many tings to eat.
With so moche champagne."
"Yes, they always do that," answered the baronet's wife. "They are so
very lavish. And very kind."
Miss Senter herself was dancing the reel. Once she thought there was a
quaver in the music, and, glancing up quickly towards the gallery, she
perceived the heads of the policemen behind the players. The players,
however, recovered themselves immediately, and upon looking up again a
moment afterwards she saw with relief that the sinister apparition had
vanished. Ten minutes later the trim little figure of the deputy
appeared between the tapestries of the doorway. Miss Senter, still
dancing, nodded slightly, as a signal that she perceived her, and then
Carmela, with an answering nod and one admiring look at Ercole,
disappeared. After all, now that there had been a suspicion about that
extra man, it _was_ a comfort to have had the apartment searched; it
would make the moment of going to bed easier, the American lady
reflected.
It was now half-past eleven. By midnight the last sleepy child had been
carried down the marble stairway, the music ceased, and the musicians
departed. The elders, glad that the noise was over, remained half an
hour longer; then they took leave. Only Lady Kay and her husband were
left; they had waited to take a closer look at Miss Senter's Christmas
present to her brother, which was a large and beautifully executed copy
of Tintoretto's "Bacchus and Ariadne," from the Anticollegio of the
Doge's Palace. It had been placed temporarily on the wall behind the
Christmas-tree.
"How exquisite!" said Lady Kay, with a long sigh. "You are most
fortunate, Mr. Senter."
"Oh yes. Though I don't quite know what they will think of it in
Rochester, New York," answered Peter, chuckling.
Sir William and his wife intended to walk home. When it was cold they
preferred to walk rather than go to and fro in a gondola; and as they
were old residents, they knew every turn of the intricate burrowing
chinks in all the quarters that serve as footways. When they took leave
at one o'clock, Peter and Miss Senter, with American friendliness,
accompanied them to the outer door. Peter was about to open this door
when it was swung back, and a figure reeled in--Ercole. He had taken off
his clown's dress, and wore now his gondolier's costume; but this
costume was in disorder, and his face was darkly red--a purple red.
"Why, Ercole, is it you? What is the matter?" said Miss Senter, as he
staggered against the wall.
"Oh, her Excellency the Consuless, I have been _beaten_!"
"Beaten? Where have you been? I thought you were down at the landing
with Andrea," said Miss Senter.
"The antiquity-dealer suffocates," muttered Ercole. "And Giorgio--dead!"
This "dead" (_morto!_) even Peter understood. "Dead! What is he saying,
Barly?"
"The man is saying, Mr. Senter, that an antiquity-dealer is suffocating,
and that somebody he calls Giorgio is dead," translated the
pink-cheeked, portly Lady Kay, in her sweet voice. "It's your gondolier,
isn't it--the one who played the clown so nicely? What a pity! He has
been drinking, I fear."
While she was saying this, Sir William was leading Ercole farther away
from the ladies.
"Yes, he is drunk," said Peter, looking at him. "Too bad! We must have
help. Let's see; Andrea is down at the landing. I'll get him. And you
call Giorgio, Barly."
Here Ercole, held by Sir William, gave a maddened cry, and threw his
head about violently.
"Oh, don't leave my husband alone with him, Mr. Senter," said Lady Kay,
alarmed. "He is a very powerful young man, and his eyes are dreadful.
To me he looks as if he were mad. Those somersaults have affected his
head."
And the gondolier's eyes were indeed strangely bloodshot and wild. Miss
Senter had hurried to the kitchen. But Giorgio was not there. She came
back, and found Ercole struggling with the Englishman and her brother.
"Let me try," she said. "I am not afraid of him. Ercole," she continued,
speaking gently in Italian, "go to your room now, and go to bed quietly;
everything will be all right to-morrow."
Ercole writhed in Sir William's grasp. "The antiquity-dealer! And
Giorgio--dead!"
"Where is Giorgio, Barly?" said Peter, angrily, as he helped Sir William
in securing the gondolier. "And where are the other servants? Where's
Carmela? Find them, and send one down to the landing for Andrea, and the
other for Giorgio. Quick!"
"Oh, Peter, I've been, and I couldn't find Giorgio or any one."
"Carmela was in your bedroom not long ago," said Lady Kay, watching the
gondolier's contortions nervously; "she helped me put on my cloak."
Miss Senter ran to her bedroom, her train flying in the haste she made.
But in a moment she was back again. "There is no one there. Oh, where
are they all?"
Ercole, hearing her voice, peered at her with his crimsoned eyes, and
then, breaking loose suddenly, he came and caught hold of her arm. "The
antiquity-room. _Will_ she come?"
Peter and Sir William dragged him away by main force.
"The gentlemen, then. Will _they_ come?" said the gondolier, hoarsely.
And again freeing himself with two strokes of his powerful arms, he
passed out (for the door was still open), and began to descend the
outside staircase.
"Oh, thank Heaven, he has gone!" "Oh, lock the door!" cried the two
ladies together.
"We must follow him, Mr. Senter," said Sir William. "He is plainly mad
from drink, and may do some harm."
"Yes; and down there Andrea can help us," answered Peter.
And the two gentlemen hastened down the staircase. It was a very long
flight with three turns. The court below was brilliantly lighted by many
wall lamps.
"I _don't_ like my husband's going down," said Lady Kay, in a tremor, as
she stood on the landing outside. "If they are going to seize him, the
more of us the better; don't you think so? For while they are holding
him, you and I could run across and get that other man in from the
riva."
But Miss Senter was not there. She had rushed back into the house, and
was now calling with all her strength: "Giorgio! Carmela! Assunta!
Beppa!" There was no answer, and, seized with a fresh panic by the
strangeness of this silence, she hastened out again and joined Lady Kay,
who was already half-way down the stairs. The gondolier had not turned
towards the water entrance; he had crossed the court in the opposite
direction, and now he was passing through a broad, low door which led
into the hall on the ground-floor behind the show-room of Z. Pelham,
throwing open as he did so both wings of this entrance, so that the
light from the court entered in a broad beam across the stone pavement.
"My dear, _don't_ go in!" "Oh, Peter, stop! stop!" cried the two ladies,
as they breathlessly descended the last flight.
But Peter and Sir William had paid no attention. Quickly detaching two
of the lamps from the wall, they had followed the madman.
"The other gondolier!" gasped Lady Kay.
And the two women ran swiftly to the water-door and threw it open, Miss
Senter calling, in Italian: "Andrea! come _instantly_!"
The little riva along the small canal was also brightly lighted. But
there was no one there. And opposite there was only a long blank wall.
"Oh, we must not leave them a moment longer," said Lady Kay.
And again they rushed across the broad court, this time entering the
dark water-story; for it was better to enter, dreadful though it was,
than to remain outside, not knowing what might be happening within.
Ercole meanwhile had made his way into Mr. Pelham's show-room, and here
he had struck a match and lighted a candle. As he had left the door of
the show-room open, those who were without could see him, and they
stopped for a moment to watch what he would do next. It was now a group
of four, for the ladies had joined the other two, Miss Senter whispering
to her brother:
"Andrea isn't there!"
The gondolier bent down, and began to drag something across the floor
and out to the open space behind. "Here!" he said, turning his purple
face towards their lamps. "I can no more." And he sat down suddenly on
the pavement, and let his head and arms fall forward over his knees.
Peter and Sir William, giving their lamps to the ladies, were
approaching cautiously, in order to secure him while he was quiet, when
they saw, to their horror, two human legs and feet protruding from the
object which he had dragged forth.
"Why, it's the second-hand dealer; it's Z. Pelham!" said Peter, in fresh
excitement. "I know his arctics. Bring the lamp, Barly. Quick!"
The two ladies came nearer, keeping one eye upon Ercole. Peter and Sir
William with some difficulty cut the rope, and unwound two woollen
coverlids and a sheet. Within, almost suffocated, with his hands tied
behind him, was the dealer.
"I suppose _he_ did this!" whispered Lady Kay to Miss Senter, her pink
face white, as she indicated the motionless gondolier.
Sir William lifted the dealer's head, while Peter loosened his collar.
"Now will Excellencies look for Giorgio," muttered Ercole, without
changing his position.
"He says now will you look for Giorgio," translated Lady Kay. "That he
_tells_ his crimes shows that he really _is_ mad!" she added, in a
whisper.
"No; I think he has come to for the moment, and that's why he tells,"
said Peter, hastily rubbing Z. Pelham's chest. "Ask him where we shall
look, Barly; ask while he's lucid."
"Where must we look for Giorgio, Ercole?" quavered Miss Senter, her
Italian coming out with the oddest pronunciation.
"Back stairs," answered the gondolier.
"Back stairs, he says," translated Lady Kay.
"There are no back stairs," replied Peter.
"I'll put this coverlid under his back. That will make him breathe
better," said the Englishman, his sympathies roused by the forlorn
plight of the little dealer, whose carefully strapped arctic shoes gave
ironical emphasis to his helplessness.
Meanwhile Miss Senter, saying "Yes, there _are_ stairs," had run across
the pavement with her lamp, found the door at the back of the hall, and
opened it. Z. Pelham began to breathe more regularly, although he had
not yet opened his eyes. Sir William drew him farther away from the
gondolier, and then he and Peter hastened across and looked up the
spiral. "It goes to the attics," explained Miss Senter.
"You two stand here at the bottom with one lamp, and Sir William and I
will go up with the other," said Peter. "Keep your eye on Ercole, Barly,
and if he so much as _moves_, come right up and join us."
"Wait an instant," said the Englishman. "Stay here with Mr. Senter,
Gertrude." Making a detour so as not to rouse the gondolier, he entered
the antiquity-dealer's show-room and tried to open the outer door. But
it was locked, and the key was not there. "No use," he said, coming
hurriedly back; "I had hoped to get help from outside to watch him while
we go up. Now remember, Gertrude, you and Miss Senter are to come up and
join us _instantly_ if he leaves his place." And then he and Peter
ascended the winding steps, carrying one of the lamps. Round and round
went the gleam of their light, and the two ladies at the bottom,
standing with their skirts caught up ready to run, watched the still
form of the gondolier in the distance, visible in the gleam of the
candle burning in the show-room. It seemed an hour. But a full minute
had not gone when Peter's voice above cried out:
"It's Giorgio! Good God! Killed! Bring up the other light."
And the two ladies rushed up together. There on the landing lay the poor
old cook, his eyes closed, his face ghastly, his white jacket deeply
stained with blood. Miss Senter, who was really attached to the old man,
began to cry.
"He isn't quite dead," said Peter, who had been listening for the heart.
"But we must get him out of this icy place. Then we'll tie up Ercoly--we
can use that rope--and after he is secured, I can go for help. Here, you
take his head and shoulders, Sir William; you are the strongest. And
I'll take his body. Barly can take the feet."
"It will be difficult," said the Englishman. "These steep stairs--"
But Peter, when roused, was a veritable little lion. "Come on," he said;
"we can do it."
"Please go down first and see if Ercole is still quiet," begged Miss
Senter of Lady Kay. And the Englishwoman, who now had both lamps, went
down and came back in thirty seconds; she never knew how she did it. "He
has not stirred," she said. And then old Giorgio was borne down, and out
to the brilliantly lighted court beyond.
"Now," said Peter, whose face was bathed with great drops of
perspiration, "we'll first secure him," and he indicated Ercole by
pointing his thumb backward over his shoulder towards the water-story,
"and then I'll go for a doctor and the police."
But as he spoke, coming out of the door upon his hands and knees,
appeared Z. Pelham, who, as soon as he saw the cook's prostrate body,
called back, hoarsely, in Italian: "Ercole, get my brandy-flask."
"Oh, don't call him!" said Lady Kay, in terror, clapping a fold of her
skirt tightly over the dealer's mouth and holding it there. "He is
mad--quite mad!"
Mr. Pelham collapsed.
"Good heavens! Gertrude, don't suffocate the poor creature a second
time," said Sir William, pulling his wife away.
Z. Pelham, released, raised his head. "Ercole has been bad beat, and
that makes him not genteel," he explained. "Ercole, bring my
brandy-flask," he called again, in Italian, and the effort he made to
break through his hoarseness brought out the words in a sudden wild
yell. "My voice a little deranged is," he added, apologetically, in
English.
They could now hear the steps of the gondolier within, and the ladies
moved to a distance as he appeared, walking unsteadily, the flask in his
hand. "Not dead?" he said, trying to see Giorgio. But his eyes closed
convulsively, and as soon as the dealer had taken the flask, down he
went, or half fell, on the pavement as before, with his head thrown
forward over his knees. Sir William placed himself promptly by his side,
while Peter ran within to get the rope. Z. Pelham, uncorking the flask,
poured a little brandy between Giorgio's pale lips. "You have all
mistake," he said to Sir William as he did this. "Ercole was bad beat by
a third partee who has done it all--me and he and this died cook; a
third partee was done it all." And he chafed the cook's temples with
brandy.
"A third party?" said Peter, who had returned with the rope. "Who?"
"I know not; they knocked me from behind. It was lightning to me, in
_my_ head also," answered Z. Pelham, going on with his chafing.
"Come here, Barly," said Peter, taking command. "Say what I tell you.
Don't be afraid; Sir William and I will grab him if he stirs. Say,
'Ercoly, who hurt you?'"
"Ercole, who hurt you?" said Miss Senter, tremulously.
"_Non so. Un demonio_," answered the gondolier, his head still on his
knees.
"He says he doesn't know. A demon," said Lady Kay.
"Ask when it happened."
"It was after he had taken the presents from the tree," translated Lady
Kay again. "He was struck, dragged down the back stairs, gagged, and
left in the antiquity-room. He has only just now been able to free
himself."
"How could he act the clown, then?" pursued Peter.
"He says he hasn't been a clown or seen a clown. Oh, Peter, it was some
one else disguised! Who could it have been?" cried Miss Senter, running
away as if to fly up the staircase, and then in her terror running back
again.
The cook's eyes had now opened. "He says see what is stoled," said Mr.
Pelham, administering more brandy. Mr. Pelham was seated, tailor
fashion, on the pavement, his feet in their arctics under him.
"Giorgio knows something about it, too," said Peter. "Ask him, Barly."
But Miss Senter was incapable of speaking; she had hidden her face on
Lady Kay's shoulder, shuddering. The clown with whom she had talked, who
had danced all the evening with the children, was an assassin! A strange
and savage murderer!
"I'll do it," said the Englishman. And bending over Giorgio, he asked,
in correct, stiff Italian: "Do you know who hurt you?"
"A tall, dark man. I never saw him before," answered the cook, or rather
his lips formed those words. "He stabbed me after he had struck down
Ercole."
"Now he is again gone," soliloquized Z. Pelham, as Giorgio's eyes
closed; "I have fear this time he is truly died!" And he chafed the
cook's temples anew.
"It's all clear now," said Peter, "and Ercoly isn't mad; only hurt in
some way. So I'll go for help at once."
"Oh, Peter, you always get lost!" moaned his sister.
And it was true that the Consul almost invariably lost his way in the
labyrinth of chinks behind the palace.
"I'll go," said the Englishman. "It's not very late" (he looked at his
watch); "I shall be sure to find some one."
"You must let me go with you, my dear," urged Lady Kay.
In three minutes they were back with two men. "I've brought these two,
and there's a doctor coming. And I sent word to the police," said the
Englishman.
And following very soon came a half-dressed youth, a young American
doctor, who had been roused by somebody. The cook was borne up the
stairway and into the salon, where the chandeliers were shedding their
soft radiance calmly, and where all the fairy-lamps were still burning
on the Christmas-tree; for only twenty minutes had passed since the host
and his guests had left the room. Behind the group of the two men from
outside, who with Peter and the doctor were carrying Giorgio, came Sir
William leading the gondolier, who seemed now entirely blind, while Z.
Pelham followed, last of all, on his hands and knees.
"This old man has a deep cut--done with a knife; he has lost a good deal
of blood; pretty bad case," said the doctor. "Your gondolier has been
dreadfully beaten about the head, but it won't kill him; he is young and
strong. This third man seems to be only sprained. Get me something for
bandages and compresses, and bring cold water."
"Get towels, Barly," said the Consul.
"Oh, Peter, I'm afraid to go," said Miss Senter, faintly. "The man may
still be hidden here somewhere. And I know he has murdered Carmela and
the other servants, too!"
Peter ran to his own chamber, and came back with a pile of towels, a
sheet from his bed, a large jug of water, and a scissors. "Now, doctor,
you stay here and do what you can for all three," he said, as he hurried
round the great drawing-room, locking all the doors but one. "And the
ladies will stay here with you. The rest of us will search the whole
apartment immediately! Lock this last door as soon as we're out, will
you?"
"Oh, Peter, don't go!" cried his sister. "Let those two men do it. Or
wait for the police."
"My dear, pray consider," said Lady Kay to her husband; "if any one _is_
hidden, it is some desperate character--"
But the Englishman and Peter were already gone, and the ladies were left
with the doctor, who, comprehending everything quickly, locked the last
door, and then hurried back to the cook. Old Giorgio's mind was now
wandering; he muttered incoherently, and seemed to be suffering greatly.
The gondolier, his head enveloped in wet towels, was lying in a stupor
on one of the sofas. Z. Pelham quietly tied up his own sprained ankles
with a portion of the torn sheet, and then assisted with much
intelligence in the making of the bandages which the doctor needed for
Giorgio.
Sir William, Peter, and the two men from outside began with the kitchen;
no one. The pantries and store-rooms; no one. The supper-room; no one.
The bedrooms; no one. The anterooms and small drawing-room; no one. As
the whole house was still brightly lighted, this did not take long. They
now crossed to four rooms on the north side; no one. Then came a large
store-room for linen. This was not lighted, so they took in a lamp; no
one.
"There's a second door here," said Sir William, perceiving one of those
masked flat portals common in Italy, which are painted or frescoed so
exactly like the wall that they seem a part of it.
"It opens into a little recess only a foot deep," said Peter, going on
with the lamp to the second store-room. "No one could possibly hide
there. Now after we have finished on this side, there is only the
wood-room left; that is off by itself in a wing."
The Englishman had accompanied his host. But having a strong bent
towards thoroughness, he was not satisfied, and he quietly returned
alone and opened that masked door. There, flattened against the wall,
not clearly visible in the semi-darkness, was the outline of a woman's
figure. His exclamation brought back the others with the lamp. It was
Carmela.
She stood perfectly still for an instant or two, so motionless, and with
such bright eyes staring at them, that she looked like a wax figure.
Then she sprang from her hiding-place and made a swift rush down the
corridor towards the outer door. They caught her. She fought and
struggled dreadfully, still without a sound. So frantic were her
writhings that her apron and cap were torn away, and the braids of her
hair fell down and finally fell off, leaving only, to Peter's
astonishment, a few locks of thin white hair in their place. It took the
four men to hold her, for she threw herself from side to side like a
wild-cat; she even dragged the four as far as the anteroom nearest the
drawing-room in her desperate efforts to reach that outer door. But
here, as she felt herself at last over-powered, a terrible shriek burst
from her, her face became distorted, her eyes rolled up, and froth
appeared on her lips.
The shriek, an unmistakably feminine one, had brought the doctor and two
ladies from the drawing-room.
"A fit!" exclaimed the doctor as soon as he saw the froth. "Here, get
open that tight dress." He unbuttoned a few buttons of the black bodice,
and tore off the rest. "Gracious! corsets like steel." He took out his
knife, and hastily cutting the cashmere across the shoulders, he got his
hand in and severed the corset strings. "Now, ladies, just help me to
get her out of this harness."
And with trembling fingers Lady Kay and Miss Senter gave their aid, and
after a moment the whole edifice--for it was an edifice--sank to the
floor. What was left was an old, old woman, small and withered, her
feeble chest rising and falling in convulsions under her coarse chemise,
and the rest of her little person scantily covered with a patched,
poverty-stricken under-skirt.
"Oh, _poor_ creature!" said Lady Kay, the tears filling her eyes as all
the ribs of the meagre, wasted body showed in the straining, spasmodic
effort of the lungs to get breath.
"Bring something to cover her, Barly," said Peter.
And Miss Senter, forgetting her fears, ran to her room, and brought back
the first thing she could find--a large white shawl.
"All right now; she's coming to," said the doctor.
The convulsions gradually ceased, and Carmela's eyes opened. She looked
at them all in silence as she sat, muffled in the shawl, where they had
placed her. Finally she spoke. "The Consul is too late," she said, with
mock respect. "The Consuless also. Did they admire the dancing of the
clown? A fine fellow that clown! You need not hold me," she added to the
two men from outside, who were acting as guards. "I have nothing more to
do. My son is safe, and that was all I cared for. They will never find
him; he is far from here now. He is very clever, and he has, besides, to
help him, all the money which the Consuless so kindly provided for him
by keeping it in a secret drawer, whose 'secret' every Italian not an
idiot knows. But the Consuless has always had a singular self-conceit.
I had only to mention that extra man with the musicians--poor little
Tonio the tailor it was--and she swallowed him down whole. I could have
got away myself if I had cared to. But I waited, in order to keep back
the alarm as long as possible; I waited. Oh yes, I helped all the ladies
to put on their cloaks; I helped this English ladyship to put on hers
last of all, as she knows. When their Excellencies went down to the
water-story, I then tried to go; but I found that they could still see
the staircase, so I came back. What matters it? They may do with me what
they please. For myself I care not. My son is safe." On her old cheeks,
under the falling white hair, were still the faint pink tinges of rouge,
and from beneath the wretched petticoat came the two young-looking
high-heeled shoes. She folded her thin hands on her lap, and refused to
say more.
Assunta and Beppa were found in the wood-room, gagged and bound like the
others, but not hurt. And in the morning the Consul's gondola was
discovered floating out with the tide, and within it Andrea in the same
helpless state. The man, who was an ex-convict, a burglar, suspected of
worse crimes, after committing the murder at the cafe, had fled to the
palace. Here he and his intrepid little mother had invented and carried
out the whole scheme in the one hour which had followed the distribution
of the presents from the tree, before the dancing began. Carmela had
even left the house to obtain a clown's costume from a dealer in
masquerade dresses who lived near by. And she had herself opened for her
son's use the disused door which led to the spiral steps.
That son was never caught. His mother, who had worked for him
indefatigably through her whole life--worked so hard that her hands were
worn almost to claws--who had supported him and supplied him, who had
made herself young and active like a girl, though she was seventy-four,
in order to be able to send him money--his mother, who had allowed
herself nothing in the world but the few smart clothes necessary for her
disguise, who was absolutely honest, but who had stolen for him three
thousand francs from the secret drawer, and had stood by and aided him
when he beat, stabbed, and gagged her fellow-servants--this mother was
not arrested. She should have been, of course. But somehow, very
strangely, she escaped from the palace before morning.
Poor old Giorgio was never able to work again. But as Peter pensioned
him handsomely, he led an easy life, while Ercole became a magnate among
gondoliers.
It was not until three years afterwards, in Rochester, New York, that
Peter, surrounded by Z. Pelham's entire collection (which he had
purchased, though thinking it hideous, at large prices), confessed to
his sister that he had connived at Carmela's escape. "Somehow I couldn't
stand it, Barly. That thin white hair and those poor old arms of hers,
and that wretched, wasted, gasping little chest--in prison!"
IN VENICE
"Yes, we came over again in February, and have been here in Venice since
the last of March. For some reasons I was sorry to come back--one _is_
so much more comfortable at home! What I have suffered in these
wretchedly cold houses over here words, Mr. Blake, can never express.
For in England, you know, they consider fifty-eight Fahrenheit quite
warm enough for their drawing-rooms, while here in Italy--well, one
never _is_ so cold, I think, as in a warm climate. Yes, we should have
been more comfortable, as far as _that_ goes, in my own house in New
York, reading all those delightful books on Art in a properly warmed
atmosphere (and I must say a properly warmed spirit too), and looking at
photographs of the pictures (you can have them as large as you like, you
know), instead of freezing our feet over the originals, which half the
time the eyes of a lynx could not see. But it is not always winter, of
course. And then I have lived over here so long that I have, it seems,
acquired foreign ways that are very unpopular at home. You may smile,
and it _is_ too ridiculous; but it is so. For instance, last summer we
went to Carley Ledge (you know Carley; pretty little place), and we
found out afterwards that the people came near mobbing us! Not exactly
that, of course, but they took the most violent dislike to us; and why?
It is too comical. Because we had innocently treated Carley as we treat
a pretty village over here. One lady said, and, I am told, with
indignation, that we had been stopping, 'more than once, right in the
main street, and standing there, in that _public_ place, to look at a
cloud passing over the mountain!' And another reported that she had
herself discovered us 'sitting on the _grass_, no farther away from the
main street than the open space in front of Deacon Seymour's, just as
though it was out in the country!' That 'out in the country' is rather
good, isn't it? Always that poor little main street!"
"Still, I think, on the whole, that the cold houses are worse than the
village comments," replied Mrs. Marcy's visitor. "A New-Yorker I know, a
confirmed European too, always goes home to spend the three months of
winter. When he comes back in the spring his English friends say, 'I
hear you have had so many degrees of frost over there--fancy!'--meaning,
perhaps, zero or under. To which he assents, but always inflexibly goes
back. They look upon him as a kind of Esquimau. But how does Miss Marcy
like exile?"
"Oh, Claudia is very fond of Italy. You have not seen her, by-the-way,
since she was a child, and she is now twenty. Do you find her altered?"
"Greatly."
"At home she was never thought pretty--when she was younger, I mean. She
was thought too--too--vigorous is perhaps the best word; she had not
that graceful slenderness one expects to see in a young girl. But over
here, I notice, the opinion seems to be different," continued the lady,
half questioningly. "And, of course, too, she has improved."
"My dear Miss Sophy--improved? Miss Marcy is a wonderfully beautiful
woman."
"Yes, yes, I know; Mr. Lenox thinks so too, I believe," answered Mrs.
Marcy, half pleased, half irritated. "It seems she is a Venetian--that
is, of the sixteenth century; and dressed in dark-green velvet, with
those great puffed Venetian sleeves coming down over her knuckles, a
gold chain, and her hair closely braided, she would be, they tell me, a
perfect Bonifazio. In fact, Mr. Lenox is painting her as one. Only he
has to imagine the dress."
Mrs. Marcy was a widow, and fifty-five. It had pleased her to hear again
the old "Miss Sophy" of their youth from Rodney Blake; but as she had
been one of those tall, slender, faintly lined girls who are called
lilies, and who are associated with pale blues and lavender, she
naturally found it difficult to realize a beauty, even if it was that of
a niece, so unlike her own. Mrs. Marcy was now less than slender; the
blue eyes which had once mildly lighted her countenance were faded. But
she still remained lily-like and willowy, and her attire adapted itself
to that style; there was a gleam of the lavender still--she wore long
shawls and scarfs.
In the easy-chair opposite, Rodney Blake leaned back. He was fifty-six,
long and thin, with a permanent expression on his face of half-weary,
half-amused cynicism, which, however, seemed to concern itself more with
life in general than with people in particular, and thus prevented
personal applications. He was well-to-do, well dressed. There was a
generally received legend that he was rather brilliant. This was the
more remarkable because he seldom said much. But perhaps that was the
reason. Miss Marcy had entered as her aunt finished her sentence.
"The sitting is over, then," said the elder lady. "Has Mr. Lenox gone?"
"Not yet," answered the niece, giving her hand to Mr. Blake as he rose
to greet her.
She was, as he had said, a beautiful woman. Yet at home there were still
those who would have dissented from this opinion, as, secretly, her aunt
dissented. She was of about medium height, with the form of a Juno. She
had a rich complexion, slowly moving eyes of deep brown, and very thick,
curling, low-growing hair of a bright gold color, which showed a warmer
reddish tinge in the light. She was the personification of healthy life
and vigor, but not of the nervous or active sort; of the reflective.
Wherever the sun touched her it struck a color: whether the red of cheek
or lip, or the beautiful tint of her forehead and throat, which was not
fair but clear; whether the brown of her eyes, or the gold of eyebrows,
eyelashes, and the heavy, low-coiled hair. Her features were fairly
regular, but not of the pointed type; they were short rather than long,
clearly, almost boldly, outlined. Her forehead was low; her mouth not
small, the lips beautifully cut. She was attired in black velvet--she
affected rich materials--and as she talked she twisted and untwisted a
string of large pearls which hung loosely round her throat and down upon
the velvet of her dress.
"Mr. Lenox does not have to imagine much, after all," observed Mr. Blake
in his slow way to Mrs. Marcy. "In velvet, with those pearls, she does
very well as it is."
"They are only Roman beads," said Claudia. "I don't know what you mean,
of course."
"I had been telling Mr. Blake that they say that if you had a green
velvet, with those big sleeves, you know, and your hair braided close to
the head, to make it look too small in comparison with the shoulders, it
would be a Bonifazio," explained the aunt.
"Your pearls are not so effective as they might be, Miss Marcy,"
continued the visitor, scanning her as she took a seat.
"I do not wear them in this way, but so." She unfastened the clasp, and
rewound the long string in three close rows, one above the other, round
her throat, above the high-coming black of her dress.
"That is better," said her critic.
"It feels like a piece of armor, so I unloosen it as soon as I can," she
answered.
Here the artist came in, hat in hand. "I am on my way home," he said.
"Good-morning, Mr. Blake. I have only stopped to ask about our
expedition this afternoon, Mrs. Marcy."
"Oh, I suppose we shall go," answered that lady, "the day is so fine.
How are they at home this morning, Mr. Lenox?"
"Elizabeth is quite well, thanks; Theocritus as usual. Shall I order
gondolas, then?"
"If you will be so good; at four. Mr. Blake will, I hope, go with us."
And then Mr. Lenox bowed, and withdrew.
"Does the--the idyllic personage accompany us?" asked the gentleman in
the easy-chair.
"It is only a child appended to the name," said Claudia, laughing. "For
some reason Mrs. Lenox always pronounces it in full; she could just as
well call him Theo."
"It is her nephew, and she is devoted to him," explained Mrs. Marcy. "He
is nearly ten years old, but does not look more than five. His health is
extremely delicate, and he is at times rather--rather babyish."
"Peevish, isn't it?" said Claudia. She had taken up two long black
needles entangled in a mass of crimson worsted, and, disengaging them,
was beginning to knit another row on an unfinished stripe. Her
beautifully moulded hands, full and white, with one antique gem on each,
contrasted with the tint of the wool. The thin fingers of Mrs. Marcy
were decked with fine diamonds, and diamonds alone; in spite of the
"foreign ways" of which that lady had accused herself, she remained
sufficiently American for that. She could buy diamonds, and Claudia an
antique ring or two; both aunt and niece enjoyed inherited incomes, that
of Claudia being comfortable, that of Mrs. Marcy large.
These ladies occupied rooms on the third floor of a palace on the Grand
Canal, not far below the Piazzetta. The palace was a stately example of
Renaissance architecture, with three rows of majestic polished columns
extending one above the other across its front. Between these columns
the American tenant, who had once been called "the lily," and her niece,
who was so like a Bonifazio, looked out upon the golden Venetian
light--a light whose shadows are colors: mother-of-pearl, emerald,
orange, amber, and all the changing gradations between them--thrown
against and between the reds, browns, and fretted white marbles of the
buildings rising from the water; that ever-moving water which mirrors it
all--here a sparkling, glancing surface, there a mysterious darkness,
both of them contrasting with the serene blue of the sky above, which
is barred towards the riva by the long, lean, sharply defined lateen
spars of the moored barks, and made even more deep in its hue over the
harbor by the broad sails of the fishing-sloops outlined against it, as
they come slowly up the channel, rich, unlighted sheets of tawny yellow
and red, with a great cross vaguely defined upon them.
Next to the Renaissance palace was a smaller one, narrow and high, of
mediaeval Gothic, ancient and weather-stained; it had lancet-windows,
adorned above with trefoil, and a little carved balcony like old
Venetian lace cut in marble. Here Mr. and Mrs. Lenox occupied the floor
above that occupied by the ladies in the larger palace. Communication
was direct, however, owing to a hallway, like a little covered bridge,
that crossed the canal which flowed between--a canal narrow, dark, and
still, that worked away silently all day and all night at its life-long
task of undermining the ponderous walls on each side; gaining perhaps a
half-inch in a century, together with the lighter achievement of eating
out the painted wooden columns which, like lances set upright in the
sand at a tent's door, the old Venetians were accustomed to plant in the
tide round their water-washed entrances. At four o'clock the little
company started, the three from the Gothic palace having come across the
hall bridge to join the others. Two gondolas were in waiting; as the
afternoon was warm, they had light awnings instead of the antique black
tops, with the sombre drapery sweeping out behind.
"I like the black tops better," observed Claudia. "Any one can have an
awning, but the black tops are Venetian."
"They can easily be changed," said Lenox.
"Oh no; not in this heat," objected Mrs. Marcy. "We should stifle. Mr.
Blake, shall you and I, as the selfish elders, take this one, and let
the younger people go together in that?"
"I want to go in the one with the red awning--the _bright_ red," said
Theocritus. This was the one Mrs. Marcy had selected.
"No, no, my boy; the other will do quite as well for you," said Lenox.
"It won't," replied the child, in a decided little voice.
"It is not of the slightest consequence," graciously interposed Mrs.
Marcy, signalling to the other gondola, and, with Blake's assistance,
taking her place within it.
Mr. Lenox glanced at his wife. She was occupied in folding a shawl
closely over the boy's little overcoat. "Come, then," he said, giving
his hand first to Miss Marcy, then to his wife and the child. The
gondolas floated out on the broad stream.
Claudia talked; she talked well, and took the Venetian tone. "The only
thing that jars upon me," she said, after a while, "is that these
Venetians of to-day--those men and women we are passing on the riva now,
for instance--do not appreciate in the least their wonderful
water-city--scarcely know what it is."
"They don't study 'Venice' because they are Venice--isn't that it?" said
Mrs. Lenox. She had soothed the little boy into placidity, and he sat
beside her quietly, with one gloved hand in hers, a small muffled
figure, with a pale face whose delicate skin was lined like that of an
old man. His eyes were narrow, deep-set, and dark under his faintly
outlined fair eyebrows; his thin hair so light in hue and cut so
closely to his head that it could scarcely be distinguished.
"I hope not," said Claudia, answering Mrs. Lenox's remark--"at least, I
hope the old Venetians were not so; I like to think that they felt, down
to their very finger-tips, all the richness and beauty about them."
"You may be sure the feeling was unconscious compared with ours,"
replied Mrs. Lenox. "They did not consult authorities about the
pictures; they were the pictures. They did not study history; they made
it. They did not read romances; they lived them."
"I wish I could have lived then," murmured Miss Marcy, her eyes resting
thoughtfully on the red tower of San Giorgio, rising from the blue. No
veil obscured the beautiful tints of her face; Claudia's complexion
could brave the brightest light, the wind, and the sun. The dark-blue
plume of the round hat she wore curled down over the rippled sunny
braids of her hair. Mr. Lenox was looking at her. But Mr. Lenox was
often looking at her.
"That would not be at all nice for us," said Mrs. Lenox, in her pleasant
voice, answering the young lady's wish. "If you, Miss Marcy, can step
back into the fifteenth century without trouble, we cannot; Stephen and
I are very completely of this poor nineteenth."
"I don't know," said Claudia, slowly; she looked at "Stephen" with
meditative eyes. "He could have been one of the soldiers. You remember
that Venetian portrait in the Uffizi at Florence--General Gattamelata?
Mr. Lenox does not look like it; but in armor he would look quite as
well."
"I don't remember it," said Mrs. Lenox, turning to see why Theocritus
was beating upon her knees with his right fist.
"You must remember--it is so superb!" said Claudia.
"I want to sit on the other side," announced Theocritus.
"When we come back, dear. See, the church is quite near; we shall soon
be there now," answered his aunt.
"You remember it, don't you?" said Claudia to Lenox.
"Perfectly."
"No--_now_," piped Theocritus. "The wind is blowing down my back."
"If he is cold, Stephen--" said Mrs. Lenox.
"I will change places with him," replied her husband. "Do not move, Miss
Marcy."
"No; Aunt Lizzie must go too!" said the boy. He had wrinkled up his
little face until he looked like an aged dwarf in a temper; he stretched
back his lips over his little square white teeth, and glared at his
uncle and Miss Marcy.
"Let me change--do," said Claudia, rising as she spoke. And Mrs. Lenox
accepted the offer.
"When you have finished my portrait, suppose you paint yourself as a
fifteenth-century Venetian general," continued Miss Marcy, taking up
again the thread of conversation which had been broken by Theocritus's
obstinacy. "The portrait of a man painted by himself is always
interesting; you can see then what he thinks he is."
"And is not?" said Lenox.
"Possibly. Still, what he might be. It is his ideal view of himself,
and I believe in ideals. It is only our real, purified--what we shall
all attain, I hope, in another world."
Thus she talked on. And the man to whom she talked thought it a
loveliness of nature that she passed so naturally and unnoticingly over
the demeanor of the spoiled child who accompanied them. Mrs. Lenox
could, for the present take no further part in the conversation, as
Theocritus had demanded that she should relate to him the legend of St.
Mark, St. George, and St. Theodore climbing down from their places over
the church porch, the palace window, and the crocodile column to fight
the demons of the lagoons. This she did, but in so low a tone that the
conversation of the others was not interrupted.
They reached the island and landed; Mrs. Marcy and Blake were already
there, sitting on the sun-warmed steps of the church whose smooth white
facade and red campanile are so conspicuous from Venice. "We were
discussing the shape of the prow of the gondola," said Mrs. Marcy, as
they came up. "To me it looks like the neck of a swan." Mrs. Marcy never
sought for new terms; if the old ones were only poetical--she was a
stickler for that--she used them as they were, contentedly.
Mr. Blake, who always took the key-note of the conversation in which he
found himself, advanced the equally veteran comparison of the neck of a
violin.
"It is the shining blade of St. Theodore, the patron of the gondolas,"
suggested Claudia.
"To me it looks a good deal like the hammer of a sewing-machine,"
observed Mrs. Lenox, lightly. This was so true that they all had to
laugh.
"But this will never do, Mrs. Lenox," said Blake, turning to look at her
as she stood on the broad marble step, holding the little boy's hand;
"you will destroy all our carefully prepared atmosphere with your modern
terms. Here we have all been reading up for this expedition, and we know
just what Ruskin thinks; wait a bit, and you will hear us talk! And not
one will be so rude as to recognize a single adjective."
"You admire him, then--Ruskin?" said the lady.
"Admire? That is not the word; he is the divinest madman! Ah, but he
makes us work! In some always inaccessible spot he discovers an
inscrutably beautiful thing, and then he goes to work and writes about
it fiercely, with all his nouns in capitals, and his adjectives after
the nouns instead of before them--which naturally awes us. But what
produces an even deeper thrill is his rich way of spreading his
possessive cases over two words instead of one, as, 'In the eager heart
of him,' instead of 'In his eager heart.' This cows us completely."
"I want to go in the church. I don't want to stay out here any longer,"
announced Theocritus. And, as his aunt let him have his way, the others
followed her, and they all went in together.
Compared with the warm sunshine without, the silent aisles seemed cool.
After ten minutes or so Mrs. Marcy and Blake came out, and seated
themselves on the step again. "You have known her for some time?" Blake
was saying.
"Mrs. Lenox? No; only since we first met here, six--I mean seven--weeks
ago. But Stephen Lenox I have always known, or rather known about; he is
a distant connection of mine. His history has been rather unusual. His
mother, a widow, managed to educate him, but that was all; they were
really very poor, and Stephen was hard at work before he was twenty. He
had some sort of a clerkship in an iron-mill, and was kept at it, I was
told, twelve and thirteen hours a day. Before he was twenty-two he
married. He worked harder than ever then, although he had, I believe, in
time a better place. His wife had no money, either, and she was not
strong. Their two little children died. Well, after twelve years of
this, most unexpectedly, by the will of an uncle by marriage, he came
into quite a nice little fortune; the uncle said, I was told, that he
admired a man who, in these days, had never had or asked for the least
help from his relatives. And so Stephen could at last do as he pleased,
and very soon afterwards they came abroad. For he had been an artist at
heart all this time, it seems--at least, he has a great liking for
painting, and even, I think, some skill."
"I doubt if he is a creative artist," answered Blake. "He is too well
balanced for that--a strong, quiet fellow. His wife is of about his age,
I presume?"
"Yes; he is thirty-six, and she the same. They have been over here
already nearly two years. She is a very nice little woman" (Mrs. Lenox
was tall and slender; but Mrs. Marcy always patronized Mrs. Lenox),
"although one _does_ get extremely tired of that spoiled boy she drags
about. Do you know," added the lady, deeply, "I feel sure it would be
much better for Elizabeth Lenox if she would remember her present
circumstances more; there is no longer any necessity for an invariable
untrimmed gray gown."
"Doesn't she dress well?" said Blake. "I thought she always looked very
neat."
"That is the very word--neat. But there is no flow, no richness. She has
been rather pretty once; that is, in that style--gray eyes and dark
hair; and she might be so still if she had the proper costumes. Of
course, going about Venice in this way one does not want to dress much;
but she has not even got anything put away."
"If one does not wear it, what difference does that make?" asked the
gentleman.
"All the difference in the world!" replied Mrs. Marcy. "Let me tell you
that the very _step_ of a woman who knows she has two or three nice
dresses in the bottom of her trunk is different from that of a woman who
knows she hasn't."
"But perhaps Mrs. Lenox does not know that she 'hasn't,'" remarked
Blake. This, however, went over Mrs. Marcy's head.
Within, the others were looking at the beautiful Tintorettos in the
choir. After a while the ill-favored but gravely serene young monk who
had admitted them approached and mentioned solemnly "the view from the
campanile;" this not because he cared whether they went up or not, but
simply as part of his duty.
"I should like to go," said Claudia; "I love to look off over the
lagoons."
They turned to leave the choir. "_I_ don't want to go," said Theocritus,
holding back. "I want to stay here and see that picture some more; and
I'm going to!"
This time Miss Marcy did not yield her wish. "Do not come with me," she
said to Mr. and Mrs. Lenox; "it is not in the least necessary. I have
been up before, and know the way. I will not be gone fifteen minutes."
"I really think that he ought not to climb all those stairs," said Mrs.
Lenox to her husband, looking at the child, who had gone back to his
station before the picture.
"Of course not," answered Lenox. Then, after a moment, "I will stay with
him," he added; "you go up with Miss Marcy."
"I want Aunt Lizzie to stay--not Uncle Stephen!" called the boy,
overhearing this, and turning round to scowl at them.
"He will not be good with any one but me," said Mrs. Lenox, in a low
tone. "You two go up; I will wait for you here."
"The question is, Is he ever good, even with her?" said Claudia,
following Lenox up the long flight of steps that winds in square turns
up, up, to the top of the campanile.
"She says he is sometimes very sweet and docile--even affectionate,"
replied Lenox. "She thinks he has quite a remarkable mind, and will
distinguish himself some day if we can only tide his poor, puny little
body safely over its childish weakness, and give him a fair start."
"She is very fond of him."
"Yes; his mother was her dearest friend, his father her only brother."
Claudia considered that she had now given sufficient time to this
subject (not an interesting one), and they talked of other things, but
in short sentences, for they were still ascending. Twice she stopped to
rest for a minute or two; then Lenox came down a step, and stood beside
her. There was no danger; still, if a person should be seized with
giddiness, the thought of the near open well in the centre, going
darkly down, was a dizzy one.
At the top they had the view: wide green flatness towards the east,
northeast, southeast, with myriad gleaming, silvery channels; the Lido
and the soft line of the Adriatic beyond; towns shining whitely in the
north; to the west, Venice, with its long bridge stretching to the
mainland; in port, at their feet, a large Italian man-of-war; on the
south side, the point of the Giudecca.
"'A Saint-Blaise, a la Zuecca,
Vous etiez bien aise;
A Saint-Blaise, a la Zuecca,
Nous etions bien la!'"
quoted Claudia. "I chant it because I have just discovered that the
Zuecca means the Giudecca yonder."
"What is the verse?" said Lenox.
"Don't you know it? It is Musset."
"I have read but little, Miss Marcy."
"You have not had _time_ to read," said Claudia, with a shade of
emphasis; "your time has been given to better things."
"Yes, to iron rails!"
"To energy and to duty," she answered. Then she turned the subject, and
talked of the tints on the water.
Down below, in the still church, the little boy sat beside his aunt, her
arm round him, his head leaning against her. The monk had withdrawn.
"The angels were all there, no doubt," she was saying; "but only a few
painters have ever tried to represent them in the picture. It is not
easy to paint an angel if you have never seen one."
"Pooh! I have seen them," said Theocritus, "hundreds of times. I have
seen their wings. They come floating in when the sunshine comes through
a crack--all dusty, you know. How many of them there do you suppose saw
the angels? Not that big girl with the plate, anyhow, _I_ know!" Thus
they talked on.
When the two from the campanile returned, and they went out to embark, a
slight breeze had risen. The little boy lifted his shoulders uneasily,
and seemed almost to shiver. Mrs. Lenox felt of his head and hands. "I
think I had better take him back in one of those covered gondolas,
Stephen," she said. "He seems to be cold; he might have a chill."
"Surely it is very warm," said Mrs. Marcy.
"Yes, but he is so delicate," replied the other lady.
"I will go with you, Mrs. Lenox," said Claudia.
"Oh no; the gondolas here are the small ones, I see, and Stephen could
not come with us. Do not leave him to go back alone; if one of us sees
to the child, that is enough."
It ended, therefore, according to her arrangement: she went back with
Theocritus in a covered gondola, Mrs. Marcy and Blake returned as they
had come, while Claudia and Lenox had the third boat to themselves.
Rodney Blake being added, this little party continued its Venetian life.
Lenox made some progress with his portrait of Claudia, but it was not
thought, at least by the others, that his wife made any with Theocritus,
that child remaining as delicate as ever, and, if possible, more
troublesome. In Mrs. Marcy's mind there had sprung up, since Mr. Blake's
arrival, an aftermath of interest in Venetian art and architecture which
was richer even than the first crop; she went contentedly to see the
pictures, churches, and palaces a fourth and even fifth time.
Claudia had a great liking for St. Mark's. "But who has not?" said Mrs.
Marcy, reproachfully, when Blake commented upon the younger lady's
fancy.
"Yes; but it is not every liking that is strong enough to take its
possessor there every day through eight long, slow weeks," answered the
gentleman.
"Not so slow," said Claudia. "But how do you know? You have been here
through only one of them."
"That leanest mosaic in the central dome is an old friend of mine; he
has told me many things in his time (I am an inveterate Venetian
lounger, you know), bending down from his curved abode, his glassy eyes
on mine, and a long, thin finger pointed. Be careful; he has noticed
you."
Several days later, strolling into the church, he found her there. "As
usual," he said.
"Yes, as usual," she answered. Miss Marcy liked Blake; his slow remarks
often amused her. And she liked to be amused--perhaps because she was
not one of those young ladies who find everything amusing. She was
sitting at the base of the last of the great pillars of the nave, where
she could see the north transept with the star-lights of the chapel at
the end, the old pulpit of marbles with its fretted top and
angel, and the deep, gold-lined dimness of the choir-dome, into which
the first horizontal ray of sunset light was now stealing--a light which
would soon turn into miraculous splendor its whole expanse.
"It always seems to me like a cave set with gold and gems," said Blake,
taking a seat beside her. "And, in reality, that is what it is, you
know--a wonderful robbers' cavern. As somebody has said, it is the
church of pirates--of the greatest sea-robbers the world has ever known;
and they have adorned it with the magnificent mass of treasure they
stole from the whole Eastern hemisphere."
"I wish they had stolen a little for me--one of those Oriental chains,
for instance. But what pleases me best here is the light. It isn't the
bright, vast clearness of St. Peter's that makes one's small sins of no
sort of consequence; it isn't the sombreness of the Duomo at Florence,
where one soon feels such a dreadful repentance that the new virtue
becomes acute depression. It is a darkness, I admit, but of such a warm,
rich hue that one feels sumptuous just by sitting in it. I do believe
that if some of our thin, anxious-faced American women could only be
induced to come and sit here quietly several hours a day they would soon
grow serene and physically opulent, like--"
"Like yourself?"
"Like the women of Veronese. (Of course I shall have to admit that I do
not need this process. Unfortunately, I love it.) But those Veronese
pictures, Mr. Blake--after all, what do they tell us? Blue sky and
balconies, feasts and brocades, pages and dogs, colors and splendor, and
those great fair women, with no expression in their faces--what does it
all mean?"
"Simply beauty."
"Beauty without mind, then."
"A picture does not need mind. But, to be worth anything, beauty it must
have."
"I don't know; a picture is a sort of companion. One of those pictures
would not be that; you might as well have a beautiful idiot."
"Ah, but a _picture_ is silent," replied Blake.
Claudia laughed. "You are incorrigible." Then, going back to her first
subject, "I wish Mrs. Lenox would come here more," she said.
"You think she needs this enriching process you have suggested?"
"In one way--yes. All this beauty here in Venice is so much to her
husband; while she--is forever with that child!"
"But she does not keep him from the beauty."
"No; but she might make it so much more to him if she would."
"Why don't you suggest it to her?"
"There is no use. She does not understand me, I think. We speak a
different language."
"That may be. But I fancy she understands you."
"Perhaps she does," answered Claudia, with the untroubled frankness
which was one of her noticeable traits. She spoke as though she thought,
indeed, that Claudia Marcy's nature was a thing which Mrs. Lenox, or any
one, might observe. Claudia rather admired her nature. It was not
perfect, of course, but at least it was large in its boundaries, and
above the usual feminine pettinesses; she felt a calm pride in that. She
was silent for a while. The first sunset ray had now been joined by
others, and together they had lighted up one-half of the choir-dome; its
gold was all awake and glistening superbly, and the great mosaic figure
enthroned there began to glow with a solemn, mysterious life.
"Men should not marry until they are at least thirty, I think," resumed
Claudia; "and especially those of the imaginative or artistic
temperament. Three-quarters of the incongruous marriages one sees were
made when the husband was very young. It is not the wife's fault; at the
time of the marriage she is generally the superior, the generous one;
the benefit is conferred by her. But--she does not advance, and he
does."
"What would you propose in the way of--of an amelioration?" asked her
listener.
"There can, of course, be no amelioration in actual cases. But there
might be a prevention. I think that a law could be passed--such as now
exists, for instance, against the marriage of minors. If a man could not
marry until he was thirty or older, he would at that time naturally
select a wife who was ten years or so his junior rather than one of his
own age."
"And the women of thirty?"
"They would be already married to the men of fifty, you know."
Here a figure emerging from the heavy red-brown shadows of the north
aisle, and seeming to bring some of them with it, as it advanced,
crossed the billowy pavement, and stopped before them. It was Mr. Lenox.
He took a seat on the other side of Blake, and they talked for a while
of the way the chocolate-hued walls met the gold of the domes solidly,
without shading, and of the total absence of white--two of the marked
features of the rich interior of the old pirate cathedral. At length
Blake rose, giving up his place beside Miss Marcy to the younger man. "I
think we have still a half-hour before that jailer of a janitor jangles
his keys," she said.
"Yes; but for the men of fifty it is time to be going," answered Blake.
"They take cold rather easily, you know, those poor fellows of fifty."
He went away. Claudia and Lenox remained until the keys jangled.
Every day the weather and the water-city grew more divinely fair. June
began. And now even Mrs. Marcy saw no objection to their utilizing the
moonlight, and no longer spoke of "wraps." The evenings were haunted by
music; everybody seemed to be floating about singing or touching
guitars. The effect of the mingled light and shadows across the fronts
of the palaces was enchanting; they could not say enough in its praise.
"Still, do you know sometimes I would give it all for the fresh odor of
the fields at home, in the country, and the old scent of lilacs," said
Mrs. Lenox.
"Do you care for lilacs?" said Claudia. "If you had said roses--"
"No, I mean lilacs--the simple country lilacs. And I want to see some
currant bushes, too; yes, and even an old wooden garden fence," replied
Mrs. Lenox, laughing, but nevertheless as if she meant what she said.
She went with them only that once in the evening, for when she reached
home she found that the little boy had been wakeful, and that he had
refused to go to sleep again because she was not there. After this the
others went without her in a gondola holding four. At last, although the
moonlight lingers longer in Venice than anywhere else, there was, for
that month at least, no more. Yet still the evening air was delicious,
and the music did not cease; the effect of the shadows was even more
marvellous than the mingled light and shade had been. They continued to
go out and float about for an hour or two in the warm, peopled
darkness. They went also, but by daylight, to Torcello, and this time
Theocritus was of the party. During half of the day he was more despotic
than he had ever been, but later he seemed very tired; he slept in his
aunt's arms all the way home. Once she made an effort to transfer him to
her husband, as the weight of his little muffled figure lay heavily on
her slender arm; but Theocritus was awake immediately, and began to beat
off his uncle's hands with all his might.
"Do let me take him, Elizabeth; he will soon fall asleep again," said
Lenox. He looked annoyed. "You are overtaxing your strength; I can see
that you are tired out."
"It will not harm me; I know when I am really too tired," answered his
wife. She gave him a little trusting smile as she spoke, and his frown
passed off.
They were all together in one of the large gondolas; Blake noted this
little side-scene.
That night Theocritus had a slight attack of fever. Mrs. Lenox said that
it came from over-fatigue, and that he must not go on any of the longer
expeditions. When they went to Murano, therefore, and down to Chioggia,
she did not accompany them, but remained at home with her charge.
Mrs. Marcy was enjoying this last month in Venice greatly. "Naturally,
it is much pleasanter when one has some one to attend to one, and one
too who knows one's tastes and looks after one's little comforts," she
remarked to her niece, with some intricacy of impersonal pronouns. The
lily did not observe that the attentions she found so agreeable were
being offered to her niece also by another impersonal pronoun. As she
would herself have said, "naturally," when they went here and there
together, the two elders often sat down to rest awhile when Claudia and
Lenox did not feel the need of it.
"Of course, with her beauty, her attractive qualities, and her fortune,
Miss Marcy has had many suitors," said Blake to the aunt during one of
these rests.
"Several," answered that lady, moderately. "But Claudia is not at all
susceptible. Neither is she so--so generally attractive as you might
suppose. She has too little thought for the opinions of others. She
says, for instance, just what she thinks, and that, you know, is seldom
agreeable."
"True; we much prefer that people should say what they don't. I have
myself noticed some plainly evident faults in her: a most impolitic
honesty; and, when stirred, an impulsiveness which is sure to be
unremunerative in the long-run. I should say, too, that she had an
empyrean sort of pride."
"Yes," replied the lily, not knowing what he meant, but concluding on
the whole that he spoke in reprobation. "As I said before, she has not
_quite_ enough of that true feminine softness one likes so much to
see--I mean, of course, in a woman."
"Her pride will be her bane yet. It will make her blind to the most
obvious pitfall. However, I'll back her courage against it when once she
sees where she has dropped."
"What?" said the lily.
"She will in time learn from you; she could not follow a more lovely
example," said Blake, coming back from his reflections.
Towards the last of June a long expedition was planned, an expedition
into "Titian's country," which was to last three days. This little
pilgrimage had been talked about for a long time, Mrs. Lenox being as
much interested in it as the others. Whether she would have had the
courage to take Theocritus, even in his best estate, is a question; but
after the time was finally set and all the arrangements made, his worst
asserted itself, and so markedly that it was plain to all that she could
not go. Something was said about postponement, but it was equally plain
that if they were to go at all they should go at once, as the weather
was rapidly approaching a too great heat. Claudia wished particularly to
take this little journey; she had set her heart upon seeing the Titians
and reputed Titians said to be still left in that unvisited
neighborhood. Blake asserted that she even expected to discover one. It
was next proposed (although rather faintly) that Mr. Lenox should be
excused from the pilgrimage. But it could not be denied that the little
boy had been quite as ill (and irritable) several times before in
Venice, and that he had always recovered in a day or two. Not that Mrs.
Lenox denied it; on the contrary, she was the one to mention it. She
urged her husband's going; it was the excursion of all others to please
him the most. It ended in his consenting; it seemed, indeed, too much to
give up for so slight a cause.
"She looks a little anxious," observed Blake, as they waited for him in
the gondola which was to take them to the railway station. Lenox had
said good-bye to her, and was now coming down the long stairway within,
while she had stepped out on her balcony to see them start.
"Do you think so?" said Mrs. Marcy. "To me she always looks just the
same, always so unmoved."
Lenox now came out, and the gondola started. Claudia looked back and
waved her hand, Mrs. Lenox returning the salutation.
On the evening of the third day, at eleven o'clock, a gondola from the
railway station stopped at the larger palace's lower door, and three
persons ascended the dimly lighted stairs.
At the top Mrs. Lenox's servant was waiting for them. "Oh, where is
signore? Is he not with you? He has not come? Oh, the poor signora--may
the sweet Madonna help her now!" cried the girl, with tears in her
sympathetic Italian eyes. "The poor little boy is dead."
They rushed up the higher stairway and across the hall bridge. But it
was as the woman had said. There, on his little white bed, lay the
child; he would be troublesome no more on this earth; he was quiet at
last. Mrs. Lenox stood in the lighted doorway of her room as they came
towards her. When she saw that her husband was not with them, when they
began hurriedly to explain that he had not come, that he had stayed
behind, that he had sent a note, she swayed over without a word and
fainted away.
It was only over-fatigue, she explained later. The child had lain in her
arms for thirty hours, most of the time in great pain, and she had
suffered with him. She soon recovered consciousness and was quite
calm--more calm than they had feared she would be. They were anxiously
watchful; they tended her with the most devoted care. Blake did what he
could, and then waited. After a while, when Mrs. Lenox had in a measure
recovered, he softly beckoned Mrs. Marcy out.
"You must tell her that her husband will not be back in time for--that
he will not be back for at least six days, and very likely longer. And
as his route was quite uncertain, we cannot reach him; there is no
telegraph, of course, and even if I were to go after him I could only
follow his track from village to village, and probably come back to
Venice behind him."
"How can I tell her!" said the tearful lady. "Perhaps Claudia--"
"No, on no account. You are the one, and you must do it," replied Blake,
and with so much decision that she obeyed him. Thus the wife was told.
What Blake had said was true; it was hopeless to try to reach Lenox
before the time when he would probably be back of his own accord. He had
started on a hunt after some early drawings of Titian's, of which they
had unearthed dim legends. One was said to be in an old monastery, among
others of no importance; two more were vaguely reported as now here, now
there. Lenox had not been certain of his own route, but expected to be
guided from village to village according to indications. It was not even
certain whether he would come back by Conegliano or strike the railway
at another point. "It certainly is an inexorable fate!" exclaimed poor
Mrs. Marcy, in the emergency driven to unusual expressions.
But when Stephen Lenox's wife understood the position in which she was
placed, she at once decided upon all that was to be done, and gave her
directions clearly and calmly--directions which Blake executed with an
attention and thoughtful care as complete as any one could possibly have
bestowed.
The little boy was to be buried at Venice, in the cemetery on the
island opposite, early in the morning of the second day.
"She is _so_ sensible!" Mrs. Marcy commented, admiringly. "Of course,
under all the circumstances, it is the thing to do. But so many women
would have insisted upon--all sorts of plans; and it would have been
_so_ hard."
"I would willingly carry out anything she wished for, no matter how
difficult," replied Blake. "I greatly respect and admire Mrs. Lenox.
But, as you say, the perfect balance of her character, her clear
judgment and beautiful goodness, have at once decided upon the best
course." (The lily had not quite said this; but in her present state of
distressed sympathy she accepted it.)
Claudia, meanwhile, remained through all very silent. She assisted, and
ably, in everything that was done, but said almost nothing.
The evening before the funeral the two ladies went across to Mrs.
Lenox's rooms; they had left her some hours before, as she had promised
to lie down for a while, but they thought that she was now probably
awake again. They found her sitting beside the little white-shrouded
form.
"Now this is not wise, Elizabeth," began Mrs. Marcy, chidingly.
"I think it is; I like to look at him," replied the watcher. "See, the
peaceful expression I have been hoping for has come; it is not often
needed on the face of a child, but it was with my poor little boy.
Look."
And, sure enough, there shone upon the small, still countenance a lovely
sweetness which had never been there in life. The face did not even seem
thin; its lines had all passed away; it looked very fair and young, and
very peacefully at rest.
"His mother would know him now at once; he was a very pretty little
fellow the last time she saw him, when he was about a year old," she
went on. "I was very fond of his mother, and his father, as probably you
know, was my only brother. Their child was very dear to me," she
resumed, after a short silence, which the others did not break. "His
constant suffering made him unlike stronger, happier children, and I
think that was the very reason I loved him the more. I wanted to make it
up to him. But I could not. I suppose he never knew what it was to be
entirely without pain--the doctors have told me so. He did not know
anything else, or any other way, but to suffer more or less, and to be
tired all the time. And he was so used to it, poor little fellow, that I
suppose he thought that every one suffered too--that that was life. He
has found a better now." Leaning forward, she took the small hands in
hers. "All my loving care, dear child, was not enough to keep you here,"
she said, smoothing them tenderly. "But you are with your mother now;
that is far better."
The funeral took place early the next morning. Then Mrs. Lenox came back
to her empty rooms, and entered them alone. She preferred it so.
After the first explanation, the only allusion she had made to her
husband's absence was to Rodney Blake. That gentleman had not expressed
the shadow of a disapprobation. He had not told her that he had objected
to Lenox's lengthened absence, and had done what he could to prevent it;
he had stopped Mrs. Marcy sharply when she spoke of telling.
"Can't you see, Sophy, that that would be the worst of all for her?" he
said; "to know that Lenox would go, in spite of my unconcealed
opposition, just because Clau--just because he wanted those trivial
drawings," he added, changing the termination of his sentence, but quite
sure, meanwhile, that "Sophy" would never discover what he had begun to
say.
Mrs. Lenox's remark was this. Blake had come in to speak to her about
some necessary directions concerning the funeral, and when she had given
them she said: "It will be a grief to Stephen when he comes back that he
could not have seen the little boy, even if but for once more. And I
hoped so that he would see him! I expected you back at eight--you know
that was the first arrangement--and towards seven he seemed easier. Once
he even smiled, and talked a little about that legend of St. Mark and
St. Theodore, of which, you remember, he was so fond. Then it was
half-past seven, and I still hoped. And then it grew towards eight, and
he was in pain again. Still I kept listening for the sound of your
gondola. But it did not come. And at half-past eight he died. But
perhaps it was as well so," she continued, although her voice trembled a
little. "Stephen would have felt his suffering so much. I was more used
to it, you know, than he was."
"Yes," answered Blake.
But she seemed to know that he was not quite in accord with her. "Of
course I feel it very deeply, Mr. Blake, on my own account, that my
husband is not here; I depend upon him for everything, and feel utterly
lonely without him. But his absence is one of those accidents which we
must all encounter sometimes, and as to everything else--the outside
help I needed--you have done all that even he could have done. You have
been very good to me," and she held out her hand.
Blake took it, and thanked her. And in his words this time he put
something that contented her. It was the sacrifice he made to his liking
for Stephen Lenox's wife.
The evening after the funeral Mrs. Marcy, who had been made nervous and
ill by all that had happened, went out at sunset for a change of air,
and Blake accompanied her. Claudia preferred to stay at home. But five
minutes after the departure of their gondola she went up the stairs and
across the hall bridge that led to Mrs. Lenox's apartment. Mrs. Lenox
was there, lying on the sofa. It was the first time since the return
that the two had been alone together. She looked pale and ill, and there
were dark shadows under her eyes; but she smiled and spoke in her usual
voice, asking Claudia to sit beside her in an easy-chair that stood
there. Claudia sat down, and they spoke on one or two unimportant
subjects. But the girl soon paused in this.
"I have come to say," she began again, in a voice that showed the effort
she made to keep it calm, "that I shall never forgive myself, Mrs.
Lenox, for--for a great deal that I have thought about you, but
especially for having had a part in the absence of your husband at such
a time. If it had not been for me he would not have gone off on that
foolish expedition. But I wanted those miserable drawings, or at least
sketches of them, and so I kept talking about it. When I think of what
you have had to go through, alone, in consequence of it, I am
overwhelmed." Here her voice nearly broke down.
"You must not take it all upon yourself, Miss Marcy," answered the wife.
"No doubt Stephen wanted to please you; no doubt he wanted to very
much--to get you the drawings, if it was possible; of that I am quite
sure."
But Claudia was not quieted. "If you knew how I have suffered--how I
suffer now as I see you lying there so pale and ill"--here she stopped
again. "I come to tell you how I feel your suffering, and I spend the
time talking about my own," she added, abruptly. "I am a worthless
creature!" And covering her face with her hands, she burst into tears.
Mrs. Lenox put out her hand and stroked the beautiful bowed head
caressingly. "Do not feel so badly," she said. "You must not; it is not
necessary."
"But it is--it is," said the girl, amid her tears. "If you knew--"
"I do know, Claudia. I know _you_."
"Oh, if you really do," said Claudia, lifting her head, her wet eyes
turned eagerly upon the wife, "then it is better."
"It is better; it is well. My dear, I think I have understood you all
along."
"But--I have not understood myself," replied Claudia. She had nerved
herself to say it; but after it was spoken a deep blush rose slowly over
her whole face until it was in a flame. Through all its heat, however,
she kept her eyes bravely upon those of the wife.
"That I knew, too," rejoined Mrs. Lenox. "But I also knew that there was
no danger," she added.
"There was not. It was unconscious. In any case, I should in time have
recognized it. And destroyed it, as I do now." These short sentences
were brought out, each with a fresh effort. "I do not speak of--of the
other side," the girl went on, with abrupt, heavy awkwardness of phrase.
"There never was any other side--it was all mine." And then came the
flaming blush again.
"But you are very beautiful, Claudia?" said the other woman, not as if
disturbed at all in her own quiet calm, but half tentatively.
"Yes, I am beautiful," replied Claudia, with a sort of scorn. "But he is
not that kind of man," she added, a quick, involuntary pride coming into
her eyes. Then she turned her head away, shading her face with her hand.
She said no more; it seemed as if she had stopped herself shortly there.
After a moment or two Mrs. Lenox began to speak. "All this life, here in
Venice, has been so much to Stephen," she said, in her sweet, quiet
voice. "You know he has worked very hard--he was obliged to; just so
many hours of each long day, for long, hard years. He never had any
rest; and the work was always distasteful to him, too. It was a slavery.
And it was beginning to tell upon him; he could not have kept it up
without being worn out both in body and mind. Judge, then, how glad I am
that he has had all this change and pleasure--he needed it so! There is
that side to his nature--a love of the beautiful, and a strong one. This
has been always repressed and bound down; it is natural that it should
break forth here. I have not the feeling myself--at least, not like his;
but I understand it in him, and sympathize with it fully." She paused.
Claudia did not speak.
"You have not been a wife, Claudia, and therefore there are some things
you do not know," pursued the voice. "A wife becomes in time to her
husband such a part of himself (that is, if he loves her) that she isn't
a separate person to him any more, and he hardly thinks of her as one;
she is himself. Many things become a matter of course to him--are taken
for granted--on this very account. It does not occur to him that she may
feel differently. He supposes that they feel alike. Often they do.
Still, a woman's thoughts do not always run in the same channel as those
of a man; we are more timid, more limited, more--afraid of things, you
know; but the husband does not always remember that. But there are some
things in which a husband and wife do feel alike, always and forever;
there are ties which are eternal. And my own life holds them--ties and
memories so precious that I can hardly explain them to you; memories of
those early years of ours when we were so alone and poor, but so dear to
each other that we did not mind it. We love each other just the same;
but then we had nothing but our love--and it was enough. The coming, the
short stay with us, and the fading away of our two little children,
Claudia--these are ties deep down in our hearts which nothing can ever
sunder. Stephen will go back to all that old grief of his when he comes
home to find the little boy gone. For the greatest sorrow of his life,
one he has never at heart overcome, was that he felt when we lost our
own little boy. Stephen had loved the child passionately, and would not
believe that he must go; and when he did he bowed his head in a silence
so long that I was frightened. I had never seen him give up before. But
even that is a dear tie between us, for then he had only me. Those
early years of ours, with their joys and sorrows--I often think of them.
A man does not dwell upon such memories, one by one, as a woman does.
But they are none the less there, a part of his life and of him." She
stopped. "Do not mind," she added, in a changed voice. "I am only--a
little tired, I think."
Claudia, who had not moved, turned quickly. Mrs. Lenox's eyes were
closed; she was very pale. But she did not faint; owing to Claudia's
quick, efficient help, she was soon herself again. "You know what to do,
don't you?" she said, smiling, when the faint feeling had passed.
"It is not that I know, so much as that I long to help you," answered
Claudia. "I wish you would let me unbraid your hair, and make you ready
for bed; you look so tired, and perhaps I could do it with a lighter
touch than Bianca," she added, humbly.
"Very well," said the other, assentingly.
And with much care and skill the girl performed her task. "I will even
put out the light," she said. "I will tell Bianca that you have gone to
bed, and are not to be disturbed." When all was done and the light out,
she paused for a moment by the bedside. "I am not going to talk any
more," she said, "but I will just say this: aunt and I are going away.
To-morrow, probably, or the day after. You will not be left alone, for
Mr. Blake will stay."
There was a silence. Then Mrs. Lenox's voice said: "That is a mistake.
It would be better to stay."
"I do not see it in that way," answered the girl. Then, "You must not
ask too much," she added, in a lower voice.
Mrs. Lenox took her hands, which were hanging before her, tightly
clasped. The touch shook Claudia; she sank down beside the bed and hid
her face.
"Stay; it is far better," whispered the wife. "Then it will be over. By
going away you will only think about it the more."
"Yes, I know. But--"
"I will answer for all. I know you better than--you know yourself. When
you see us together, it will be different to you. Stay, to please me."
"Very well," murmured the girl.
They kissed each other, and she rose. When she had reached the door Mrs.
Lenox spoke again. "Of course, you know that I quite understand that it
is only a girl's fancy," she said, with a tender lightness. This was her
offering to Claudia.
On the evening of the seventh day after the funeral Stephen Lenox came
back; he had sent a despatch to his wife from Conegliano, and Blake was
therefore able to meet him at Mestre, and tell him what had happened. He
went directly home, and the others did not see him until the next
evening. Then he came across to the larger palace. Blake was there; he
kept himself rather constantly with Mrs. Marcy now, perhaps to direct
that lady's somewhat wandering inspirations. For this occasion he had
warned her that she must not be too sympathetic, that she must be on her
guard. So Mrs. Marcy was "on her guard;" she only took out her
handkerchief four times; she even talked of the weather. Claudia
scarcely spoke. Blake himself conducted the conversation, and filled all
the gaps. They could naturally say a good deal about the health of Mrs.
Lenox, as that lady had been obliged to keep her room for the three
preceding days. Lenox did not stay long; he said he must go back to his
wife. As he rose he gave the small portfolio he had brought with him to
Claudia. "I don't think they were Titians," he said. "But I sketched
them for you as well as I could."
Mrs. Marcy thought this an opportunity; she took the portfolio, and
exclaimed over each picture. Blake, too, put up his eye-glass to look at
them. Lenox said a word or two about them and waited a moment longer;
then he went away. Claudia had not glanced at them.
He never knew of her visit to his wife; those are the secrets women keep
for each other, unto and beyond the grave.
What passed when he came home was simple enough. His wife cried when she
saw him; she had not cried before. She told him the history of the
little boy's last hours, and of all he had said, and of the funeral.
Then they had talked a while of her health, and then of future plans.
"I ought to have remembered that you were anxious about him even before
I went away," said Lenox, going back abruptly to the first subject. He
was standing by the window, looking out; this was an hour after his
return.
"But he had been ill so many times. No, it was something we could not
foresee, and as such we must accept it. I wanted you to go--don't you
remember? I urged your going. You must not blame yourself about it."
"But I do," answered her husband.
"I cannot allow you to; I shall never allow it. To me, Stephen, all you
do is right; I wish to hear nothing that could even seem otherwise. I
trust you entirely, and always shall."
He turned. She was lying back in an easy-chair, supported by pillows. He
came across and sat down beside her, his head bent forward, his elbows
resting on his knees, his face in his hands. He did not speak.
"Because I know that I can," added the wife.
That was all.
They stayed on together in Venice through another two weeks. Mrs. Lenox
improved daily, and was soon able to go about with them. She seemed,
indeed, to bloom into a new youth. "It is the reaction after the long,
wearing care of that child," explained Mrs. Marcy. "And isn't it
beautiful to see how devoted he is to her, and how careful of her in
every way? But I have always noticed what a devoted husband he was,
haven't you?"
These two ladies and Mr. Blake were going to Baden-Baden. But the others
were going back to America. "We may return some time," said Lenox; "but
at present I think we want a home."
"I wish we could have stayed on together always, just as we are now,"
sighed the sentimental lily, smoothing the embroidered edge of her
handkerchief. "_Such_ a pleasant party, and of just the right size;
these last two weeks have been so perfect!"
The time for parting came. The three who were going to Baden-Baden were
to leave at dawn, and they had come across to Mrs. Lenox's parlor to
spend a last hour. Claudia talked more than usual, and talked well; she
looked brilliant.
At the end of the second hour the good-byes began in earnest.
Everything that was appropriate was said, Blake, in particular,
delivering himself unblushingly of one long fluent commonplace after
another. They were to meet again--oh, very soon; they were to visit each
other; they were to write frequently--one would have supposed, indeed,
that Blake intended to send a daily telegraphic despatch. At last the
lily, having kept them all standing for twenty minutes, bestowed upon
Mrs. Lenox a final kiss, and really did start, the two gentlemen and
Claudia accompanying her down the long hall. But the hall was dark, and
Claudia was behind; without the knowledge of the others she slipped
back.
Mrs. Lenox was standing where they had left her. When she saw the girl
returning, pale, repressed, all the sparkle gone, she went to her, and
put her arms round her; Claudia laid her head down upon the other's
shoulder. Thus they stood for several moments in silence. Then, still
without speaking, Claudia went away.
When Mrs. Marcy reached the stairway which led down to her own
apartment, on the other side of the hall bridge, "Why, where is
Claudia?" she said.
"Here I am," said her niece, appearing from the darkness.
"You will come down with us for a moment, won't you, Mr. Lenox?"
suggested the lily. "Just for one _last_ look?"
"Do not ask him," said Claudia, smiling; "he is worn out! We have
already extended that look over two long hours. Good-bye, Mr. Lenox; and
this time, I think, is really the last."
* * * * *
BY CONSTANCE F. WOOLSON.
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Constance Fenimore Woolson may easily become the novelist
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End of Project Gutenberg's The Front Yard, by Constance Fenimore Woolson
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{\centering
{\large Per Elmfors,\footnote{Email address:
elmfors@nordita.dk.}}\raisebox{1ex}{,a}
{\large David Persson\footnote{Email address:
tfedp@fy.chalmers.se.}}\raisebox{1ex}{,b} and
{\large Bo-Sture Skagerstam\footnote{Email address:
tfebss@fy.chalmers.se. Research supported by the Swedish National
Research Council under contract no. 8244-311.}}\raisebox{1ex}{,b,c}
\\
{\sl \raisebox{1ex}{a}NORDITA,
Blegdamsvej 17,
DK-2100 Copenhagen \O, Denmark \\ }
{\sl \raisebox{1ex}{b}Institute of Theoretical Physics,
Chalmers University of Technology and \\
University of G\"oteborg, S-412 96 G\"oteborg,
Sweden \\ }
{\sl \raisebox{1ex}{c}University of Kalmar,
Box 905, S-391 29 Kalmar, Sweden\\}
}}
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\begin{document}
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\begin{flushright}
NORDITA-93/78 P\\
G\"{o}teborg ITP 93-11 \\
hep-ph/9312226\\
December 1993\\
\end{flushright}
\begin{center}
\baselineskip 1.2cm
{\Large\bf Real-Time Thermal Propagators\\
and the QED Effective Action\\
for an External Magnetic Field}
\normalsize
\end{center}
\AUTHORS
\vfill
\begin{abstract}
{\normalsize
The thermal averaged real-time propagator of a Dirac fermion in
a static uniform
magnetic field $B$ is derived. At non-zero chemical
potential and temperature we find explicitly the effective
action for the magnetic field,
which is shown to be closely related to the Helmholz free energy
of a relativistic fermion gas, and it exhibits the expected
de\ Haas -- van\ Alphen oscillations.
An effective QED coupling constant at finite temperature
and density is derived,
and compared with renormalization group results.
We discuss some
astrophysical implications of our results.
} \end{abstract}
\newpage
\setcounter{page}{1}
\begin{center}
\section{\sc Introduction}
\label{introduction}
\end{center}
Large magnetic fields $B$ can be associated with certain compact astrophysical
objects like supernovae \cite{ginzburg91,myller90} where $B={\cal O}
(10^{10})$T, neutron stars \cite{neutronstar,Chanm92} where $B={\cal O}
(10^{8})$T, or white magnetic dwarfs \cite{Chanm92,angel78}
in which case $B={\cal O}
(10^{4})$T. (As a reference the electron mass in units of Tesla is
$m^2_e/e = 4.414 \cdot 10^9$T.)
It has recently been argued that a plasma at thermal equilibrium can
sustain fluctuations
of the electromagnetic fields. In particular, for the primordial
Big-Bang plasma the
amplitude of magnetic field fluctuations at the time of the
primordial nucleosynthesis can be as large as
$B = {\cal O}(10^{10}) $T \cite{tajima}. Furthermore, a model
for extragalactic gamma
bursts in terms of mergers of massive binary stars suggests magnetic
fields up to the order $B={\cal O}(10^{13})$T \cite{nar&pac&pir92}.
A more speculative system where even larger macroscopic magnetic fields can be
contemplated are superconducting strings \cite{wittenetc}.
Here one may conceive fields
as large as $B \gsim {\cal O}(10^{14})$T. It has also recently been suggested
that due to gradients in the Higgs
field during the electroweak phase
transition in the early universe very large magnetic fields,
$B = {\cal O}(10^{19})$T,
may be
generated \cite{vach91}. If one encounters magnetic fields of this order of
magnitude
the complete electroweak model has to be considered and the concept of
electroweak
magnetism becomes important (for a recent account see e.g.
Ref.\cite{ambj&ole93}). In
the present paper we consider, however, magnetic fields such that calculations
within QED are sufficient. A shorter version of this report has been published
elsewhere \cite{elm&per&ska93a}.
In many of these systems one has to consider the effects of a
thermal environment
and a finite chemical potential. In this paper we derive the
appropriate effective fermion propagator and the effective action
in QED for a thermal
environment treated exactly in the external constant magnetic field
but with no virtual photons present, i.e. we consider the weak coupling limit.
Calculations of the QED effective Lagrangian density
in an external field have been
attempted before
either at finite temperature \cite{dittrich79,rojas92} or at
finite chemical potential \cite{ceo90}. In
the latter case \cite{ceo90} the effective action is not complete but the
correct form is
presented here. At finite chemical potential and for sufficiently small
temperatures,
the QED effective action should exhibit a certain periodic dependence of
the
external field, i.e. the well-known de\ Haas -- van\ Alphen
oscillations in condensed
matter
physics. This was not obtained in Ref.\cite{ceo90}.
Elsewhere, the radiative corrections to the anomalous magnetic moment
has been estimated in the presence of large magnetic fields and it was
argued that they are extremely small \cite{pebss91,Studenikin90}.
By making use of the effective action we derive
the effective QED coupling as a function of the
external field, the chemical potential and temperature.
In a future publication we
will
discuss the fermion self-energy and
radiative corrections to the electrons anomalous magnetic moment,
in terms of the formalism derived here
\cite{elm&per&ska93b}.
\begin{center}
\section{\sc Thermal propagators in the Furry picture}
\label{furry}
\setcounter{equation}{0} \end{center}
We consider Dirac fermions in the presence of an external static field as
described by the vector potential $A_\mu$. Using static energy solutions
we may represent the second quantized
fermion field in the Furry picture \cite{furry51}. It is given by
\begin{equation}
\Psi({\bf x},t) = \sum _{\lambda,\kappa}
b_{\lambda\kappa}\psi^{(+)}_{\lambda\kappa}({\bf x},t) +
d_{\lambda\kappa}^{\dagger}\psi^{(-)}_{\lambda\kappa}({\bf x},t)~~~,
\end{equation}
where $\lambda$ is a polarization index, $\kappa $ denotes the energy
and momentum (or other) quantum numbers
(discrete and/or continuous) needed in order to completely characterize the
solutions, and
$(\pm)$ denotes positive and negative energy solutions of
the corresponding Dirac equation,
\begin{equation}
(i\not\!\!D - m)\psi^{(\pm)}_{\lambda\kappa} ({\bf x},t) =
0~~~,
\end{equation}
where $D_\mu = \partial_\mu +i eA_\mu$ is the covariant derivative.
The creation and annihilation operators satisfy the canonical anti-commutation
relations
\begin{equation}
\{d_{\lambda'\kappa '},d_{\lambda\kappa}^{\dagger}\} =
\delta_{\lambda'\lambda}\delta_{\kappa'\kappa} = \{b_{\lambda'\kappa
'},b_{\lambda\kappa}^{\dagger}\}~~~,
\end{equation}
while other anti-commutators are zero. The completeness relation
\begin{equation}
\sum _{\lambda,\kappa}
\psi^{(+)\dagger}_{\lambda\kappa,a}({\bf x'},t)\psi^{(+)}_{\lambda\kappa,b}
({\bf x},t) + \psi^{(-)\dagger}_{\lambda\kappa,a}({\bf
x'},t)\psi^{(-)}_{\lambda\kappa,b}({\bf x},t) = \delta _{ab}
\delta^3 ({\bf x'} -{\bf x})~~~,
\end{equation}
where $\psi^{(\pm)}_{\lambda\kappa,a}$ denotes the $a$-component of the Dirac
spinor $\psi^{(\pm)}_{\lambda\kappa}$,
leads to the canonical anti-commutation
relations for the fields
\begin{equation}
\{ \Psi_{a}({\bf x'},t), \Psi^{\dagger}_{b}({\bf x},t) \} = \delta
_{ab}\delta ^{3}({\bf x'} - {\bf x})~~~.
\end{equation}
In vacuum, the
fermion propagator $iS_{F}(x';x|m)$ is defined by
\begin{eqnarray}
iS_{F}(x';x|m)~=~\langle 0|{\bf T}\left( \Psi({\bf x'},t') \overline{\Psi}({\bf
x},t)\right)|0\rangle =~~~~~~~~~~~~~~~~~~~~~~&&\nonumber \\
\theta(t'-t)
\sum_{\lambda\kappa}\psi^{(+)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(+)}_{\lambda\kappa}({\bf x},t) - \theta(t-t')
\sum_{\lambda\kappa}\psi^{(-)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(-)}_{\lambda\kappa}({\bf x},t)~~~,
\label{eq:vprop}
\end{eqnarray}
where the conjugated spinor $\overline{\psi}^{(\pm)}_{\lambda\kappa}$is given
by
$\overline{\psi}^{(\pm)}_{\lambda\kappa} =
({\psi}^{(\pm)}_{\lambda\kappa})^{\dagger}\gamma_{0}$. Since
$\psi^{(\pm)}_{\lambda\kappa}({\bf x},t)$ satisfies the Dirac equation, only
the time derivative acting on the step functions gives a non-zero
contribution, so one finds that
\begin{equation}
(i\not\!\!D - m)S_{F}(x';x|m) = \leavevmode\hbox{\small1\kern-3.3pt\normalsize1} \cdot\delta^{4}(x' - x)~~~.
\label{eq:delteq}
\end{equation}
The real-time propagator at finite temperature $T$ and
chemical potential
$\mu$, denoted by $\langle iS_{F}(x';x|m)\rangle_{\beta,\mu} $, can now be obtained by
the following
reasoning. Let $f^{+}_{F}(\omega )$ denote the Fermi-Dirac thermodynamical
distribution function
\begin{equation}
\label{eq:fpmdistribution}
f^{+}_{F}(\omega ) = \frac{1}{\exp (\beta (\omega - \mu )) + 1}~~~,
\end{equation}
where $\beta$ is the inverse temperature, $\mu $ the chemical potential and
$\omega$ is the energy of the quantum state under consideration. A particle can
propagate forward in time in a state which is
unoccupied by thermal particles, whereas a hole in the occupied states
can propagate backwards in time . We can therefore write
\begin{eqnarray}
\langle iS_{F}(x';x|m)\rangle_{\beta,\mu}
& = &\sum_{\lambda,\kappa} \nonumber \\
\left[\theta(t'-t)
\left([1-f^{+}_{F}(E_{\kappa})]\psi^{(+)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(+)}_{\lambda\kappa}({\bf x},t) \right.\right.& +
&\left.\left.
[1-f^{+}_{F}(-E_{\kappa})]\psi^{(-)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(-)}_{\lambda\kappa}({\bf x},t) \right)
\right. \nonumber \\
- \theta(t-t') \left.
\left(f^{+}_{F}(-E_{\kappa})\psi^{(-)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(-)}_{\lambda\kappa}({\bf x},t) \right.\right.& +
&\left.\left.
f^{+}_{F}(E_{\kappa})\psi^{(+)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(+)}_{\lambda\kappa}({\bf x},t) \right)
\right]~.
\label{eq:time}
\end{eqnarray}
We can now extract the vacuum part of the propagator Eq.(\ref{eq:vprop}) and
write
\begin{equation}
\langle iS_{F}(x';x|m)\rangle_{\beta,\mu} =
iS_{F}(x';x|m) + iS_{F}^{\beta,\mu}(x';x|m)~~~,
\label{eq:thermalS}
\end{equation}
where the thermal part $iS_{F}^{\beta,\mu}(x';x|m)$ is defined by
\begin{eqnarray}
&&~~~~~~~~~~~~~~~~~~~~S_{F}^{\beta,\mu}(x';x|m)~~~= \nonumber \\
&&i \sum_{\lambda,\kappa}
\left(f^{+}_{F}(E_{\kappa})\psi^{(+)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(+)}_{\lambda\kappa}({\bf x},t) -
f^{-}_{F}(E_{\kappa})\psi^{(-)}_{\lambda\kappa}({\bf x'},t')
\overline{\psi}^{(-)}_{\lambda\kappa}({\bf x},t) \right)~~~,
\label{eq:thermalpart}
\end{eqnarray}
and where we have defined the distribution
\begin{equation}
\label{eq:fminus}
f^{-}_{F}(E_{\kappa}) = 1 - f^{+}_{F}(-E_{\kappa})~~~.
\end{equation}
Notice that there is no time-ordering in $S_{F}^{\beta,\mu}(x';x|m)$
despite the fact that the
time-ordering in Eq.(\ref{eq:time}) is non-trivial. The
thermal propagator Eq.(\ref{eq:thermalS}) therefore also trivially satisfies
Eq.(\ref{eq:delteq}). These considerations can, of course, easily be extended
to
treat particles with Bose-Einstein statistics as well.
The result
Eq.(\ref{eq:thermalpart}) can also be derived from an
explicit calculation using the second-quantized field operators and appropriate
thermal averages, i.e. we use Wicks theorem
\begin{equation}
{\bf T}\left( \Psi({\bf x'},t') \overline{\Psi}({\bf
x},t)\right) = iS_{F}(x';x|m)~+~
:\Psi({\bf x'},t') \overline{\Psi}({\bf
x},t):~~~,
\end{equation}
where the last term corresponds to a normal ordering. We then obtain
\begin{equation}
\langle{\bf T}\left( \Psi({\bf x'},t') \overline{\Psi}({\bf
x},t)\right)\rangle_{\beta,\mu} = iS_{F}(x';x|m) + iS_{F}^{\beta,\mu}(x';x|m)~~~,
\end{equation}
where we have used the only non-zero bilinear thermal
averages
\begin{eqnarray}
\label{fermidirac}
\langle b_{\lambda\kappa}^{\dagger}b_{\lambda'\kappa'}\rangle_{\beta,\mu}
&=& f_{F}^{+}(E_{\kappa})\delta_{\lambda\lambda'}
\delta_{\kappa\kappa'}~~~,\nonumber\\
\langle d_{\lambda\kappa}^{\dagger}d_{\lambda'\kappa'}\rangle_{\beta,\mu}
&=& f_{F}^{-}(E_{\kappa})\delta_{\lambda\lambda'}
\delta_{\kappa\kappa'}~~~.
\end{eqnarray}
In principle we do not have to restrict ourselves to thermal distributions as
given by
\Eqref{eq:fpmdistribution}. In fact, we can allow for {\it any} such
one-particle
distribution function $f^{+}_{F}(\omega )$ and the definition
\Eqref{eq:fminus}.
\begin{center}
\section{\sc External uniform and static magnetic field}
\label{magnetic}
\setcounter{equation}{0}
\end{center}
For the convenience of the reader, we summarize some of the relevant
expressions in the
case of a constant magnetic field $B$ parallel to the $z-$direction in the
gauge $A_\mu=(0,0,-Bx,0)$. Using
$\kappa$ as
a collective index for $(n,k_{y},k_{z})$, where $n=0,1,2,...$ ; $k_{y},k_{z}$
are continuous, and the $\gamma$ matrices in the chiral representation, we
can write the
solutions in the form
\begin{equation}
\psi^{(\pm)}_{\lambda,\kappa}({\bf x},t)= \frac{1}{2\pi}\, \frac{\exp[ \pm (
-i E_{\kappa}t \!+\! i k_{y}y \!+\! i k_{z}z ) ]}{ \sqrt{2 E_{\kappa} } } \,
\Phi^{(\pm)}_{\lambda,\kappa }(x)~~~ ,
\end{equation}
where
\begin{equation}
\Phi^{(+)}_{1,\kappa}(x) =\frac{1}{\sqrt{E_{\kappa}\!+\! k_{z} } }
\left( \begin{array}{c}
(E_{\kappa} \!+\! k_{z} ) I_{n;k_{y}}(x) \\
- i \sqrt{2eBn}\, I_{n-1;k_{y}}(x) \\
- m I_{n;k_{y}}(x) \\
0
\end{array} \right)~~~ ,
\end{equation}
\begin{equation}
\Phi^{(+)}_{2,\kappa}(x) =\frac{1}{\sqrt{E_{\kappa}\!+\! k_{z} } }
\left( \begin{array}{c}
0 \\
- m I_{n-1;k_{y}}(x) \\
- i \sqrt{2eBn}\, I_{n;k_{y}}(x) \\
( E_{\kappa} \!+\! k_{z} ) I_{n-1;k_{y}} (x)
\end{array} \right)~~~ ,
\end{equation}
\begin{equation}
\Phi^{(-)}_{1,\kappa}(x) =\frac{1}{\sqrt{E_{\kappa}\!-\! k_{z} } }
\left( \begin{array}{c}
-m I_{n;-k_{y}}(x) \\
0 \\
(-E_{\kappa} \!+\! k_{z} ) I_{n;-k_{y}}(x) \\
i \sqrt{2eBn} \, I_{n-1;-k_{y}}(x)
\end{array} \right)~~~ ,
\end{equation}
\begin{equation}
\Phi^{(-)}_{2,\kappa}(x) =\frac{1}{\sqrt{E_{\kappa}\!-\! k_{z} } }
\left( \begin{array}{c}
i \sqrt{2eBn} \, I_{n;-k_{y}}(x) \\
(-E_{\kappa} \!+\! k_{z} ) I_{n-1;-k_{y}}(x) \\
0 \\
- m I_{n-1;-k_{y}}(x)
\end{array} \right) ~~~ .
\end{equation}
In these expressions the energy $E_{\kappa}$ is given by
\begin{equation}
E_{\kappa} = \sqrt{m^{2} + k^{2}_{z} +2eBn}~~~,
\end{equation}
and the $ I_{n;k_{y}}(x)$ functions are explicitly written
\begin{eqnarray}
I_{n;k_{y}}(x)& \equiv& \left( \frac{eB}{\pi} \right)^{1/4} \exp \left[
- \frac{1}{2} eB \left( x \!-\! \frac{k_{y}}{eB} \right)^{2} \right]
\frac{1}{ \sqrt{n!}} H_{n} \left[ \sqrt{2eB} \left( x \!-\! \frac{k_{y}}
{eB} \right) \right] ~~~ . \nonumber \\
&&
\label{eq:Is}
\end{eqnarray}
Here $H_{n}$ is the Hermite polynomial given by Rodrigues' formula as
\begin{equation}
H_{n}(x)=(-1)^{n} e^{\frac{1}{2}x^{2} } \frac{d^{n}}{dx^{n}} e^{- \frac{1}{2}
x^{2}} ~~~ ,
\end{equation}
and we define $ I_{-1;k_{y}}(x)=0$.
The functions $ I_{n;k_{y}}(x)$ are normalized as
\begin{equation}
\label{Iid}
\int dx I_{n;k_{y}}(x) I_{m;k_{y}}(x) = \delta _{n,m}~~~
\end{equation}
if $ n,m\geq 0$, so that it is easily shown that the collection of
all $\Psi$'s form a complete orthonormal set.
The vacuum part of the propagator Eq.(\ref{eq:vprop})
is then given by (see e.g. Ref.\cite{kobsak83})
\begin{eqnarray}
S_{F}(x';x|m)_{ab}&=& \sum^{\infty}_{n=0}\int
\frac{d\omega \, dk_{y} \, dk_{z}}{(2\pi)^{3}} \exp[-i\omega (t' -t) +
ik_{y}(y' - y) + ik_{z}(z' - z)] \nonumber \\
& & \times\frac{1}{\omega^{2}
\!-\! k_{z}^{2} \!-\! m^{2} \!-\! 2eBn +i\epsilon}
S_{ab}(n;\omega , k_{y}, k_{z}) ~~~.
\label{bvprop}
\end{eqnarray}
The matrix $S(n;\omega , k_{y}, k_{z})$ entering above is given by
\pagebreak
\begin{eqnarray}
\lefteqn{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~S(n;\omega , k_{y}, k_{z}) \equiv}
\nonumber
\\[2ex]
&& \left( \begin{array}{cccc}
mI_{n,n} & 0 & - (\omega \!+\! k_{z}) I_{n,n} & - i \sqrt{2eBn}
I_{n,n-1} \\
0 & m I_{n-1,n-1} & i \sqrt{2eBn} I_{n-1,n} & - (\omega \!-\! k_{z})
I_{n-1,n-1} \\
- (\omega \!-\! k_{z} ) I_{n,n} & i \sqrt{2eBn} I_{n,n-1} &
m I_{n,n} & 0 \\
- i \sqrt{2eBn} I_{n-1,n} & - (\omega \!+\! k_{z} ) I_{n-1,n-1} &
0 & m I_{n-1,n-1}
\end{array} \right)~~~ ,\nonumber \\
\end{eqnarray}
where we have defined
\begin{equation}
I_{n,n'} \equiv I_{n;k_{y}}(x) I_{n';k_{y}}(x')~~~~.
\end{equation}
Similarly we find the thermal part of the fermion propagator
\begin{eqnarray}
S^{\beta,\mu}_{F}(x';x|m)_{ab}&=& \sum^{\infty}_{n=0}\int
\frac{d\omega \, dk_{y} \, dk_{z}}{(2\pi)^{3}} \exp[-i\omega (t' -t) +
ik_{y}(y' - y) + ik_{z}(z' - z)] \nonumber \\
& & \times 2 \pi i \, \delta( \omega^{2}
\!-\! k_{z}^{2} \!-\! m^{2} \!-\! 2eBn )
f_{F}(\omega) S_{ab}(n;\omega , k_{y}, k_{z}) ~~~,
\label{btprop}
\end{eqnarray}
where $f_{F}(\omega )$ is the thermal distribution
\begin{equation}
f_{F}(\omega) = \theta(\omega )f_{F}^{+}(\omega)
\!+\! \theta(-\omega ) f^{-}_{F}(-\omega)~~~.
\end{equation}
By making use of the completeness relation
\begin{equation}
\sum _{n=0}^{\infty} I_{n;k_{y}}(x) I_{n;k_{y}}(x') = \delta (x - x')\ ,
\end{equation}
one can show \cite{elm&per&ska93b} that the propagators \Eqref{bvprop} and
\Eqref{btprop} reduce to the free-field propagators in the limit when the
magnetic field $B$ tends to
zero.
\setcounter{equation}{0}
\begin{center}
\section{\sc Propagators in thermo-field dynamics}
\label{thermalfieldpropagator}
\end{center}
The thermal propagators in Eqs.(\ref{bvprop}) and (\ref{btprop}) cannot be
used for a perturbative expansion in a naive way. The reason
is that the $\delta$-functions can occur on several
internal legs with coinciding arguments and that such expressions are not
well-defined. It is known that such problems can be avoided
be means of a correctly derived
real-time finite temperature formalism where one must invoke a
doubling of the degrees of freedom. There are several formalisms
for doing that and we shall use thermo field dynamics (TFD) since
it is very easy in the operator formalism \cite{UmezawaMT82}.
In TFD the propagator is obtained as the expectation value
of the time-ordered product in the thermal vacuum $\rvacb{}$
which is annihilated by the thermal operators
$(b_{\lambda\kappa}(\beta),\ d_{\lambda\kappa}(\beta))$ and their
tilde partners $(\tilde{b}_{\lambda\kappa}(\beta),
\ \tilde{d}_{\lambda\kappa}(\beta))$.
The TFD propagator can be given by a simple expression
for independent harmonic oscillators.
We have solved the
Dirac equation exactly in the external field, but in the
free propagator the interaction between the particles is neglected.
Each mode is therefore still an independent harmonic
oscillator, but with a different frequency labeled by
the quantum numbers $(n,k_y,k_z)$ and corresponding to a
definite Landau level. Thus, in the derivation of the
propagator we can copy the usual procedure for
free particles.
The Bogoliubov transformation between the zero temperature and
thermal operators is given by
\begin{eqnarray}
\label{Btrf1}
\left(\begin{array}{c} b_{\lambda\kappa}\\ i\tilde{b}^\dagger_{\lambda\kappa} \end{array}\right)
&=&
\left(\begin{array}{cc} \cos\vartheta^{(+)}_{\lambda\kappa} &
-\sin\vartheta^{(+)}_{\lambda\kappa}\\
\sin\vartheta^{(+)}_{\lambda\kappa} &
\cos\vartheta^{(+)}_{\lambda\kappa} \end{array}\right)
\left(\begin{array}{c} b_{\lambda\kappa}(\beta)\\
i\tilde{b}^\dagger_{\lambda\kappa}(\beta) \end{array}\right) \ ,
\end{eqnarray}
and
\begin{eqnarray}
\label{Btrf2}
\left(\begin{array}{c} d_{\lambda\kappa}\\ i\tilde{d}^\dagger_{\lambda\kappa} \end{array}\right)
&=&
\left(\begin{array}{cc} \cos\vartheta^{(-)}_{\lambda\kappa} &
-\sin\vartheta^{(-)}_{\lambda\kappa}\\
\sin\vartheta^{(-)}_{\lambda\kappa} &
\cos\vartheta^{(-)}_{\lambda\kappa} \end{array}\right)
\left(\begin{array}{c} d_{\lambda\kappa}(\beta)\\
i\tilde{d}^\dagger_{\lambda\kappa}(\beta) \end{array}\right) \ .
\end{eqnarray}
The number expectation values in Eq.(\ref{fermidirac}) imply that the
coefficients in the Bogoliubov matrices must satisfy
\begin{equation}
\sin^2\vartheta^{(\pm)}_{\lambda\kappa}=
f^\pm_F(E_\kappa)\ .
\end{equation}
We use the convention that $b$ and $\tilde{b}$ anti-commute.
The Bogoliubov matrices in Eqs.(\ref{Btrf1}) and (\ref{Btrf2}), and the factors
of $i$, are carefully explained in Ref.\cite{Ojima81}. The definition
of the fermion propagator varies slightly in the literature
and we shall for definiteness follow Ref.\cite{Ojima81}.
Other propagators correspond to other definitions of
the thermal doublet and they may be computed in a
similar way. Since we are not doing any higher loop
calculations these conventions do not matter.
We compute the TFD
propagator matrix as
\begin{equation}
iS_{F}^{TFD}(x';x|m)_{ab}=
\lvacb{} {\bf\rm\bf T} \left[ \left(\begin{array}{c}\Psi_a(x') \\
i\tilde{\Psi}_a^\dagger(x') \end{array}\right)
\left(\overline{\Psi}_b(x),\ \
-i\tilde{\overline{\Psi}}_b^\dagger(x)\right)\right]
\rvacb{}\ .
\end{equation}
The structure of the propagator is the same as in absence of the
external field except that we now expand in another basis
corresponding to the new energy eigenvalues. We obtain
\begin{eqnarray}
iS_{F}^{TFD}(x';x|m)_{ab}&=&
\sum^{\infty}_{n=0}\int
\frac{d\omega \, dk_{y} \, dk_{z}}{(2\pi)^{3}} \exp[-i\omega (t'-t) +
ik_{y}(y'- y) + ik_{z}(z'-z)] \nonumber\\
&&\times S_{ab}(n;\omega,k_y,k_z) U_F(\omega)
\left(\begin{array}{cc} \inv{\omega^2-E^2_\kappa+i\epsilon} & 0 \\
0 & \inv{\omega^2-E^2_\kappa-i\epsilon} \end{array} \right) U^T_F(\omega)\ ,
\end{eqnarray}
where
\begin{eqnarray}
U_F(\omega)&=&\left(\begin{array}{cc} \cos\vartheta(\omega) &
-\sin\vartheta(\omega)\\
\sin\vartheta(\omega) & \cos\vartheta(\omega) \end{array}\right)\ ,
\end{eqnarray}
and
\begin{eqnarray}
\sin\vartheta(\omega)&=&\theta(\omega)\sqrt{f^+_F(\omega)}
-\theta(-\omega)\sqrt{f^-_F(-\omega)}\ ,
\nonumber \\
\cos\vartheta(\omega)&=&\theta(\omega)\sqrt{1-f^+_F(\omega)}
+\theta(-\omega)\sqrt{1-f^-_F(-\omega)}\ .
\end{eqnarray}
Here, $U^T_F(\omega)$ is the transpose of the matrix $U_F(\omega)$.
The $S^{TFD}_F(x';x|m)_{ab}^{11}$ component is, of course, the same
as the propagator in Eqs.(\ref{bvprop}) and (\ref{btprop}) and the other
components are only needed in higher loop calculations.
The derivation of the propagator in this Section can be repeated for a
non-equilibrium distribution if only we assume certain factorization properties
of the density matrix. The essential assumption is that there are no
non-trivial multiparticle correlations so that everything is determined in
terms of the single particle distribution. This freedom amounts to replacing
the functions $f^\pm_F(E_\kappa)$ with some other positive functions that
describes the distribution. Some applications of such a formalism in absence of
the external field can be found in Ref.\cite{ElmforsEV93c}.
\setcounter{equation}{0}
\begin{center}
\section{\sc The effective action}
\label{effective}
\end{center}
As was shown by Schwinger a long time ago \cite{Schwinger51}, an
external electromagnetic field, slowly varying in space and time,
can be treated to all orders in the external field
in the weak-coupling limit. Here we make use of
a technique similar to that of Schwinger's in order
to evaluate the thermodynamical partition function in a static uniform
magnetic field $B$ for charged fermions as well as
for charged bosons.\vspace{4ex}
\subsection{\sc QED and Charged Fermions}
The generating functional of fermionic Green's functions in an external
field $Z[\bar{\eta},\eta, A_{\mu}]$, formally defined by
\begin{equation}
Z[\bar{\eta},\eta, A_{\mu}] = \int d[\bar{\psi}]d[\psi]
\exp[i\int d^{4}x(-\frac{1}{4}F_{\mu \nu}F^{\mu \nu} +
\bar{\psi}(i\not\!\!D - m)\psi - \bar{\eta}\psi + \bar{\psi}\eta)]~~,
\label{eq:genfunc}
\end{equation}
describes second-quantized electrons and positrons interacting with a
classical electromagnetic field expressed in terms of the
vector potential $A_{\mu}$. The expectation value of $\psi$ (and
$\bar{\psi}$) can formally be fixed by choosing appropriate $\bar{\eta}$ (and
$\eta$), i.e. $\varphi (x) \equiv \langle\psi (x)\rangle =
-i\delta/\delta\bar{\eta}(x)\log Z$ (and similarly for $\bar{\psi}$). The
equation of motion for $\varphi (x)$ tells us how the electrons interact
with the electromagnetic field which includes effects due to all virtual
$e^{+}e^{-}$-pairs.
The fermionic Gaussian functional integral in Eq.(\ref{eq:genfunc}) can
formally be performed with
the
result that
\begin{eqnarray}
\lefteqn{Z[\bar{\eta},\eta, A_{\mu}] =} \nonumber \\
&&\mbox{Det}\left[i(i\not\!\!D - m)\right]\exp\left[i\int d^{4}x
\left( - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}
\!+\! \int d^{4}y \bar{\eta}(x) S_{F}(x;y|m)\eta (y)\right)\right]~~~,
\end{eqnarray}
where $S_{F}(x;y|m)$ is the external field vacuum propagator as given by
Eq.(\ref{bvprop}). It follows that $\varphi (x)$ satisfies the Dirac
equation in the external field, i.e. $(i\not\!\!D -m)\varphi (x) =0$.
The functional determinant $\mbox{Det}(i\not\!\!D -m)$ gives rise
to a contribution to the effective Lagrangian density
${\cal L}_{eff}$. Using a complete orthogonal basis to rewrite $\log
\mbox{Det}$ as ${\rm Tr\,} \log$,
the effective action can thus be written
\begin{equation}
\label{Seff}
S_{eff} = \int d^{4}x {\cal L }_{eff} = \int d^{4}x
\left[ -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}\right] \!-\! i \, {\rm Tr\,} \log
\left[i(i\not\!\!D -m) \right]~~~.
\end{equation}
We now write the effective Lagrangian density as
\begin{equation}
{\cal L }_{eff} ={\cal L}_{0} \!+\! {\cal L}_{1}~~~,
\end{equation}
where the tree level part in the case of a pure magnetic field is
\begin{equation}
{\cal L}_{0}= -\frac{1}{2} B^{2}~~~,
\end{equation}
and $ {\cal L}_{1}$ corresponds to the functional determinant.
Differentiating \Eqref{Seff} with
respect to the fermion mass we now find the one-loop correction according to
\begin{equation}
\label{dldm}
\frac{\partial {\cal L}_{1}}{\partial m} = i \,{\rm tr} S_{F}(x;x|m)~~~,
\end{equation}
where the trace now only is over spinor indices.
After a straightforward calculation of the trace
using Eq.(\ref{bvprop}), we obtain in terms of renormalized quantities
the well-known
result \cite{Schwinger51} that
\begin{equation}
\label{TzeroEA}
{\cal L}_{1} = -\frac{1}{8\pi^{2}} \int_{0}^{\infty}
\frac{ds}{s^3} \biggl[esB\coth(esB) -1 -
\frac{1}{3}(esB)^2\biggr]\exp(-m^{2}s)~~~.
\end{equation}
We have here performed the standard
renormalizations leaving $eB$ invariant, i.e.
\begin{eqnarray}
A_{\mu} &\longrightarrow & (1+ Ce^{2})^{-1/2}A_{\mu}~~~, \nonumber \\
e^{2} &\longrightarrow & e^{2}\left(1+ Ce^{2}\right)~~~,
\label{chargeren}
\end{eqnarray}
where the divergent constant $C$ is given by
\begin{equation}
C =\frac{1}{12\pi^{2}} \int_{0}^{\infty}
\frac{ds}{s}\exp(-m^{2}s)~~~.
\end{equation}
We shall now find the corresponding correction
$S_{eff}^{\beta,\mu} = \int d^{4}x {\cal L}_{eff}^{\beta,\mu}~,$
to the effective action $S_{eff}$ at
finite chemical potential and temperature such that
\begin{equation}
{\cal L}_{eff}={\cal L}_{0} \!+\! {\cal L}_{1} \!+\! \cL_{eff}^{\beta,\mu}~~~.
\end{equation}
We notice that
the correction ${\cal L}_{eff}^{\beta,\mu}$,
due to the presence of thermal
fermions, can be written in the form
\begin{equation}
\frac{\partial {\cal L}_{eff}^{\beta,\mu}}{\partial m} =
i\mbox{Tr}S_{F}^{\beta,\mu}(x;x|m)~~~.
\label{thermaltrace}
\end{equation}
By performing the trace operation in Eq.(\ref{thermaltrace}), using the
thermal propagator Eq.(\ref{btprop}), we obtain
\begin{eqnarray}
{\cal L}_{eff}^{\beta,\mu} &=&
\frac{4eB}{(2\pi )^{2}} \sum _{n=0}^{\infty}
\sum _{\lambda=1}^{2} \int _{-\infty}^{\infty}
d\omega f_{F}(\omega) \nonumber \\
&& \times\int _{0}^{\infty}
dk k^{2} \delta (\omega ^{2}- k^{2} -2eB(n+\lambda-1) - m^{2})~~~,
\end{eqnarray}
where we have integrated by parts with respect to $k$.
We, therefore, see that ${\cal L}_{eff}^{\beta,\mu}$
is directly related to the partition function $Z(B,T,\mu)$ of
the relativistic fermion gas in the presence of an external
magnetic field $B$ in a sufficiently large quantization volume $V$,
as given in for example Ref.\cite{MillerR84},
according to
\begin{eqnarray}
\label{partition}
\cL_{eff}^{\beta,\mu} &=& \frac{\log Z(B,T,\mu)}{\beta V}
\nonumber \\
&=& \frac{eB}{(2\pi)^{2}}\sum _{n=0}^{\infty}
\sum _{\lambda=1}^{2} \int _{-\infty}^{\infty} dk \frac{k^{2}}{E_{\lambda,n}}
\nonumber \\
&& \times \left( \frac{1}{1+\exp[\beta(E_{\lambda,n} -\mu )]} +
\frac{1}{1+\exp[\beta(E_{\lambda,n} +\mu )]}\right)~~~,
\end{eqnarray}
where
\begin{equation}
E_{\lambda,n} = \sqrt{ k^{2} + 2eB(n+\lambda -1) + m^{2}}~~~.
\end{equation}
Separating the field independent part we write
\begin{equation}
{\cal L}_{eff}^{\beta,\mu} = {\cal L}_{0}^{\beta,\mu} +
{\cal L}_{1}^{\beta,\mu}~~~,
\end{equation}
where
\begin{equation}
\label{Lbmnoll}
{\cal L}_{0}^{\beta,\mu} =
\frac{1}{3\pi ^{2}} \int _{-\infty}^{\infty}
d\omega \theta(\omega ^{2} - m^{2})
f_{F}(\omega )\left(\omega ^{2} - m^{2}\right)^{3/2}~~~.
\end{equation}
We, therefore, conclude that the field independent thermal correction
to the Lagrangian density $ {\cal L}_{0}^{\beta,\mu}$ can be identified as
\begin{equation}
{\cal L}_{0}^{\beta,\mu} = \frac{\log Z(T,\mu)}{\beta V}
=-\frac{F(T,\mu)}{V}~~~,
\label{lbmeffzero}
\end{equation}
where $ Z(T,\mu)$ is the partition function, and $F(T,\mu)$ the free
energy, for an ideal $e^{+}e^{-}$-gas
with particle energy $E=\sqrt{k^{2} + m^{2}}$, i.e.
\begin{equation}
\frac{\log Z(T,\mu)}{V} = 2\int \frac{d^{3}k}{(2\pi)^{3}}
\left(\log[1+e^{-\beta(E-\mu)}] +
\log[1+e^{-\beta(E+\mu)}]\right)~~~,
\end{equation}
consistent with the general identification above.
Using the identity
\begin{equation}
\frac{\exp(-|x|)}{|x|} = \int _{0}^{\infty}
\frac{dt}{\sqrt{2\pi t}}\exp \left(
-\frac{1}{2}(x^{2}t +\frac{1}{t})\right)~~~,
\end{equation}
the following representation of ${\cal L}_{1}^{\beta,\mu}$, valid
for $|\mu| < m$, can be derived in a straightforward manner
\begin{equation}
\label{dittricheqn}
{\cal L}_{1}^{\beta,\mu} =
\frac{1}{4 \pi ^{2}}\sum _{l=1}^{\infty}(-1)^{l+1}
\int _{0}^{\infty} \frac{ds}{s^{3}}
\exp\left( -\frac{\beta ^{2} l^{2}}{4s} -m^{2}s \right)
\frac{\cosh(\beta l\mu)}{2}[eBs\coth(eBs) - 1]~~~.
\end{equation}
In the case $\mu = 0$,
Eq.(\ref{dittricheqn}) agrees with
the result obtained in Refs.\cite{dittrich79,rojas92}.
However, it
is not always obvious, when written in this form, to see how
to extract the physical
content, and particularly not obvious how to generalize $\cL_{eff}^{\beta,\mu}$
to $\abs{\mu}\geq m$, since then it appears to be divergent.
In particular we notice that the high $T$ behaviour given in
Ref.\cite{dittrich79} is not correct. As explained
in Appendix A it is, however, possible to show that
Eq.(\ref{dittricheqn}) is equal to \Eqref{Lbmueff}
given below, which is valid for all
$T$ and $\mu$.
In order to calculate the thermal part $\cL_{eff}^{\beta,\mu}$ of the effective action
in a more useful form, we have to be
careful with the convergence and the analytical structure.
Some details of the calculation are given in Appendix A. We
get $\cL_{eff}^{\beta,\mu} = \cL_{0}^{\beta,\mu} +\cL_{1}^{\beta,\mu}$, where
$\cL_{0}^{\beta,\mu}$, the ideal gas contribution in absence of the
external field $B$, is
given in \Eqref{lbmeffzero}, and
\begin{eqnarray}
\cL_{1}^{\beta,\mu} &=& {\cL_{1,reg}^{\beta,\mu}} + {\cL_{1,osc}^{\beta,\mu}} \nonumber\\
&=&
\int_{-\infty}^\infty d\omega
\theta(\omega^2-m^2)f_F(\omega)
\Biggl[\inv{4\pi^{5/2}}\int_0^\infty
\frac{ds}{s^{5/2}}e^{-s(\omega^2-m^2)}
[seB\coth (seB) -1]\Biggr]\nonumber\\
&-&
\label{Lbmueff}
\!\!\!\int_{-\infty}^\infty d\omega
\theta(\omega^2-m^2)f_F(\omega)
\Biggl[
\inv{2\pi^3}\sum_{n=1}^\infty \left(\frac{eB}{n}\right)^{3/2}
\sin \! \left(\frac{\pi}{4}-\frac{\pi n}{eB}
(\omega^2-m^2)\right) \Biggr]\ .
\end{eqnarray}
The term with the
sum over $n$, $ {\cL_{1,osc}^{\beta,\mu}}$, was neglected in Ref.\cite{ceo90} and we
show in Section~\ref{physical} that it is essential to keep
this term in order to get the
correct physical result. We may also use the generalized
$\zeta$-function to
rewrite ${\cL_{1,osc}^{\beta,\mu}}$ in a different form, sometimes more
suited for numerical calculations
\begin{equation}
\label{eqovan}
{\cL_{1,osc}^{\beta,\mu}} = \int_{-\infty}^\infty d\omega
\theta(\omega^{2}-m^{2})f_F(\omega) (eB)^{3/2}\frac{ \sqrt{2}}{\pi^{2}}
\zeta \! \left( -\frac{1}{2},\mbox{ mod}\! \left[ \frac{ \omega^{2}
-m^{2}}{2eB} \right] \right)~~~,
\end{equation}
where $\mbox{mod}[A]$ is a shorthand notation for $A$ modulo $1$, i.e.
\begin{equation}
\mbox{mod}[A]= A- \mbox{ int}[A]~~~.
\end{equation}
An alternative way to write \Eqref{eqovan} is
\begin{eqnarray}
{\cL_{1,osc}^{\beta,\mu}}& =& \sum_{n=0}^{\infty} \int_{0}^{1} \frac{ds}
{\sqrt{m^{2} + 2eB(n+s)}} \nonumber \\
&& \times\left(f^{+}_{F}(\sqrt{m^{2} + 2eB(n+s)}) +
f^{-}_{F}(\sqrt{m^{2} + 2eB(n+s)})\right) \zeta ( -\frac{1}{2},s)\ ,
\end{eqnarray}
where the various Landau-level contributions are made explicit.
In addition to $\cL_{eff}^{\beta,\mu}$ the free energy has a contribution
from the thermal photons, i.e.
\begin{equation}
\frac{F_\gamma(T)}{V} = -\frac{T^4\pi^2}{45}\ ,
\end{equation}
which is background field independent since there is no
self-interaction among abelian gauge fields.
\subsection{\sc QED and Charged Scalars}
\label{scalarqed}
The formalism used so far applies also to scalar QED. We give some of the
corresponding results here for completeness. Equation (\ref{dldm}) becomes in
this case
\begin{equation}
\frac{\partial {\cal L}_{1}}{\partial m^2} = -iG_{F}(x;x|m^2)~~~,
\end{equation}
and the thermal propagator is
\begin{eqnarray}
\langle G_F(x;x|m^2)\rangle_{\beta,\mu}&=&\sum_{n=0}^\infty
\int\frac{d\omega dk_y dk_z}{(2\pi)^3}
\left(I_{n;k_y}(x)\right)^2 \nonumber\\
&&\times\left[\inv{\omega^2-E_n^2+i\epsilon}
-2\pi i\delta(\omega^2-E_n^2)
f_B(\omega)\right]\ ,
\end{eqnarray}
where
\begin{equation}
E_n^2 = k_z^2+(2n+1)eB+m^2\ ,
\end{equation}
and
\begin{equation}
f_B(\omega) = \frac{\theta(\omega)}{e^{\beta(\omega-\mu)}-1}+
\frac{\theta(-\omega)}{e^{\beta(-\omega+\mu)}-1}\ .
\end{equation}
It is rather straightforward to obtain the correction
\begin{equation}
{\cal L}_1 = \inv{16\pi^2}\int_0^\infty\frac{ds}{s^3}
\exp (-m^2 s)\left(\frac{eBs}{\sinh(eBs)}-1+
\frac{(eBs)^2}{6}\right)\ ,
\end{equation}
to the effective action in the vacuum sector. At finite chemical
potential and temperature we similarly find the following contribution to
the effective action
\begin{eqnarray}
\cL_{eff}^{\beta,\mu}\!\!\!\!&&=\inv{6\pi^2}\int d\omega\theta(\omega^2-m^2-eB)
f_B(\omega)(\omega^2-m^2)^{3/2}\nonumber\\
\!\!\!\!&&+\int d\omega\theta(\omega^2-m^2-eB)f_B(\omega)
\left[\inv{8\pi^{5/2}}\int\frac{ds}{s^{5/2}}
e^{-s(\omega^2-m^2) } \left(\frac{eBs}{\sinh(eBs)}-1\right)\right]\nonumber\\
\!\!\!\!&&-\int d\omega\theta(\omega^2-m^2-eB)f_B(\omega)
\left[\inv{4\pi^3}\sum_{k=1}^\infty\left(\frac{eB}{k}\right)^{3/2}
\sin\left(\frac{\pi}{4}-\frac{\pi k}{eB}
(\omega^2-m^2-eB)\right)\right]\ .\nonumber\\
\end{eqnarray}
The zero temperature part ${\cal L}_1$ was derived in Ref.\cite{Schwinger51}.
Physically
this effective action is quite different from the fermionic one. We shall not
pursue this investigation here but only make a few remarks. Since for charged
bosons there is no
sharp Fermi surface, there are no de Haas -- van Alphen oscillations either.
Furthermore, even the
energy of the lowest Landau level ($n=0$) depends on $B$, so that,
for example, in the case of a vanishing chemical potential, the
number density
is Boltzmann suppressed for large fields.
\begin{center}
\section{\sc The Physical Content of ${\cal L}_{eff}$}
\label{physical}
\end{center}
\setcounter{equation}{0}
There are several dimensionful parameters related to ${\cal L}_{eff}$, i.e.
$T,\ \mu,\ m$, and $B$, that can be large or small
compared to each other. We shall discuss some
of these limits which we think are particularly
interesting.
A central feature of a fermion gas is whether it is
degenerate or not, i.e. whether or not the Fermi surface is sharp
on the scale of the Fermi energy. With an external magnetic
field it is also important to compare the smoothness of the
Fermi surface with the spacing of the Landau levels.
A criteria for the de\ Haas -- van\ Alphen effect is that the distance
between the Landau levels close to the Fermi surface
is considerably
larger than
the diffuseness or fluctuations in the Fermi surface due to finite temperature,
electron -- electron interactions, impurities etc.
This can sometimes be achieved even at high $T$ by having large
$\mu$ and $B$.
The effective Lagrangian is here given as a function of
the chemical potential $\mu$. In many situations it is more
natural to consider the expectation value of charge density $Q/V$ as given,
where $Q$ is the total conserved charge. It is calculated from $Q/V =
-e\rho(\mu)$, where
\begin{equation}
\rho(\mu)=-\inv{V}\frac{\partial F}{\partial \mu}
=\frac{\partial \cL_{eff}^{\beta,\mu}}{\partial\mu}~~~ ,
\end{equation}
which in the case of vanishing magnetic field and temperature reduces to
\begin{equation}
\sqrt{\mu^{2}-m^{2}}=(3\pi^{2}|\rho |)^{1/3}~~~,
\end{equation}
and $\mu$ has the same sign as $\rho$. For large $B$ field this relation gets
substantial
correction, see e.g. Section~\ref{strong}.
We notice that $\rho$ is equal to the
difference between the electron and positron number densities, that may be
useful on comparison with condensed matter physics calculations.
In other situations one may
consider adiabatic changes of $B$, and then keep the entropy fixed, or
the pressure. All these different cases are described by
suitable Legendre transformations of the thermodynamical
potential $F$.\vspace{4ex}
\subsection{\sc The de\ Haas -- van\ Alphen effect}
\label{dhva}
At low temperature one may attempt an expansion in $T$ using
Sommerfeld's method \cite{AshcroftM76}. We assume that
$\mu>m$ since for $|\mu|<m$ the thermal contribution is
exponentially suppressed. The Sommerfeld expansion for a
function $H(\omega)$ is
\begin{equation}
\int_m^\infty d\omega\, f_F^+(\omega)H(\omega)=
f_F^+(m)\int_m^\infty d\omega\,H(\omega)+
\sum_{n=1}^\infty T^n a_n
\left.\frac{d^{n-1}H(\omega)}{d\omega^{n-1}}\right|_{\omega=\mu}\ ,
\end{equation}
where
\begin{equation}
a_n=\int_{-\frac{\mu-m}{T}}^\infty dx\frac{x^n}{n!}
\left(-\frac{\partial}{\partial x}\inv{e^x+1}\right)\ ,
\end{equation}
but the odd powers of $T$ are exponentially suppressed. This
formula can be applied to $\cL_{0}^{\beta,\mu}$ and ${\cL_{1,reg}^{\beta,\mu}}$, but
in ${\cL_{1,osc}^{\beta,\mu}}$ performing the derivative inside the summation sign is
not allowed since the sum is not uniformly convergent, and when acting on
the form containing the $\zeta$-function there will obviously be divergences
at discrete points.
This indicates that an expansion in $mT/eB$ is not possible.
Anyway, the $T=0$ part of ${\cL_{1,osc}^{\beta,\mu}}$ can be calculated,
and if we in particular assume $\{T=0,eB\ll\mu^2-m^2\ll m^2\}$
we get
\begin{equation}
\cL_{1}^{\beta,\mu} \approx
\frac{(eB)^2}{12\pi^2}
\frac{\sqrt{\mu^2-m^2}}{m}
-\frac{(eB)^{5/2}}{4\pi^4 m}\sum_{n=1}^\infty
\inv{n^{5/2}}\Biggl[\cos\left(\frac{\pi}{4}
-n\pi\frac{\mu^2-m^2}{eB}\right)-\inv{\sqrt{2}}\Biggr]\ .
\end{equation}
This is a non-relativistic
limit (in the sense that the kinetic energy is much smaller
than $m$) with a degenerate Fermi sea and a weak external field.
The vacuum correction is in this limit given by
\begin{equation}
\label{SMALLB}
{\cal L}_{1} \approx \frac{(eB)^2}{360\pi^2}
\left(\frac{eB}{m^2}\right)^2\ ,
\end{equation}
so that the finite density correction
\begin{equation}
\cL_{1}^{\beta,\mu} \approx \frac{(eB)^2}{12\pi^2}
\left(\frac{3\pi^2 \rho}{m^3}\right)^{1/3}~~~ ,
\end{equation}
therefore dominates over ${\cal L}_{1}$ when
\begin{equation}
\left(\frac{ \rho}{m^3}\right)^{1/3} \gg
\inv{30(3\pi^2)^{1/3}}\left(\frac{eB}{m^2}\right)^2\ ,
\end{equation}
or equivalently, in terms of the chemical potential
\begin{equation}
\frac{\sqrt{\mu^{2}-m^{2}}}{m} \gg \frac{1}{30} \left( \frac{eB}{m^{2}}
\right)^{2}~~~.
\end{equation}
This is always satisfied in the limit $\{eB\ll\mu^2-m^2\ll m^2\}$.
Even though the $B^2$ dominates over $B^{5/2}$ for small $B$, the
magnetization of the heat bath\footnote{The vacuum contribution to the
magnetization is not included in \Eqref{thermmag} since it is very small for
small $B$.}
gets a larger contribution from ${\cL_{1,osc}^{\beta,\mu}}$,
\begin{equation}
\label{thermmag}
M=M_{reg}+M_{osc}=-\inv{V}\frac{\partial F}{\partial B}
=\frac{\partial \cL_{eff}^{\beta,\mu}}{\partial B}~~~,
\end{equation}
where to the lowest order in the magnetic field
\begin{eqnarray}
\label{Mosc}
M_{osc}&=& \frac{e\sqrt{eB}(\mu^2-m^2)}{4\pi^3 m}
\sum_{n=1}^\infty \inv{n^{3/2}}
\sin\left(\frac{\pi}{4}-n\pi\frac{\mu^2-m^2}{eB}\right) \nonumber \\
&=&-\left(\frac{e}{2m}\right) \frac{\sqrt{2eB}}{\pi^{2}}(\mu^{2}-m^{2})
\zeta \! \left( -\frac{1}{2},\mbox{ mod}\! \left[ \frac{ \mu^{2}
-m^{2}}{2eB} \right] \right)~~~,
\end{eqnarray}
and
\begin{equation}
M_{reg}=\left(\frac{e}{2m}\right)
\frac{eB}{3\pi^{2}}\sqrt{\mu^{2}-m^{2}}~~~.
\end{equation}
The $\zeta$-function has its maximal modulus at $\zeta \!
\left( -\frac{1}{2},1
\right) = \zeta \!\left( -\frac{1}{2},0 \right) \approx -0.208$, which implies
that the peak magnetization from the
oscillating term is larger than that from the regular term for
\{$eB\lsim 0.78(\mu^2-m^2)$\}, i.e. when the approximations
used here are valid. Defining the magnetic susceptibility as the response in
the magnetization due to a magnetic field, i.e. $M=\chi B$, as in
Ref.\cite{Isihara91}, we get exact agreement with this reference, but
not with Refs.\cite{Abrikosov88,Kittel63},
which have an extra factor
$(-1)^{n}$ in the sum over $n$, that we find only should be present in the
case of spinless bosons.
In Section~\ref{strong} we give an argument why our result
has to be correct.
The oscillatory behaviour as a function of $B$ is well-known as
the de\ Haas -- van\ Alphen effect.
The frequency of this periodic function
agrees with the one derived by Onsager \cite{Onsager52}.
Equation~(\ref{Lbmueff}) describes
the full relativistic generalization of this effect,
and in Section~\ref{astro} we consider some astrophysical
applications where the non-relativistic approximation is
not valid. The distance
between the magnetic field of two
adjacent minima of the magnetization is determined
by
\begin{equation}
\abs{\inv{eB_i}-\inv{eB_{i+1}}}= \frac{2\pi}{A}\ ,
\end{equation}
where $A$ is the area of an extremal cross section of the
Fermi sea.
Sometimes (e.g. in Ref.\cite{AshcroftM76})
the magnetic susceptibility is defined by
\begin{equation}
\chi=\frac{\partial M}{\partial B}\ ,
\end{equation}
but again we find that the sum over $n$ does not converge, and that the form
containing the $\zeta$-function contains divergences at discrete values
of $B$, and is poorly illuminating.
\subsection{\sc Strong B-field}
\label{strong}
In the limit of strong field, $\{eB\gg T^2,m^2,\abs{\mu^2-m^2}\}$,
we can see from \Eqref{partition} that only the lowest Landau level
contribute and $\cL_{eff}^{\beta,\mu}$ goes like a linear function of $eB$.
We shall now reproduce this result from \Eqref{Lbmueff}
and it turns out to be rather non--trivial. The leading $B$
dependence in the first term in \Eqref{Lbmueff} is obtained by
scaling out $eB$ and taking $eB\rightarrow \infty $ in the remainder.
The total contribution is, apart from the thermal integration (see
Appendix B)
\begin{equation}
\frac{(eB)^{3/2}}{4\pi^{5/2}}\left[
\int_0^\infty \frac{dx}{x^{5/2}}(x\coth x -1)
-\sqrt{\frac{2}{\pi}}
\sum_{n=1}^\infty\inv{n^{3/2}}\right]~~~,
\end{equation}
but this is actually identically zero. The next subleading
term can be shown to be
\begin{equation}
\label{laBLett}
\cL_{1}^{\beta,\mu} = \frac{eB}{2\pi^2}\int_{-\infty}^\infty
d\omega\theta(\omega^2-m^2)f_F(\omega)
\sqrt{\omega^2-m^2}~~~,
\end{equation}
which is exactly the leading term from \Eqref{partition}. This
calculation shows that the oscillatory term in \Eqref{Lbmueff}
is absolutely necessary to cancel the $B^{3/2}$ term and to give
the correct linear term. Also, notice that the expression presented here
for this term
has to be correct, without the extra factor $(-1)^{n}$ of
Refs.\cite{Kittel63,Abrikosov88},
for the $B^{3/2}$ terms to cancel.
In this limit of strong magnetic field the
thermal and density corrections given above are small compared to
\begin{equation}
\label{LARGEB}
{\cal L}_{1} \approx \frac{(eB)^2}{24\pi^2}
\log\left(\frac{eB}{m^2}\right)\ .
\end{equation}
The vacuum polarization
effects are dominating here, which comes quite naturally, since the
magnetization from real
thermal particles becomes saturated when all spins are aligned, whereas the
magnetization from vacuum polarization increases like $B\log B$. This has not
always been recognized in the literature \cite{MillerR84}.
Another issue when the $B$ field is strong compared to $\mu^2-m^2$ is that the
relation between $\rho$ and $\mu$ is changed.
In fact, we have from \Eqref{laBLett} at $T=0$
\begin{equation}
\rho(\mu)\approx\frac{eB}{2\pi^2}\sqrt{\mu^2-m^2}\ .
\end{equation}
The linear dependence on the Fermi momentum
$k_F=\sqrt{\mu^2-m^2}$ can be understood
from the fact that only the lowest Landau level is filled
and therefore the phase space is essentially one-dimensional.
\subsection{\sc Weak B-field}
\label{weak}
In Section \ref{dhva} we had an expression for $\cL_{eff}^{\beta,\mu}$
in a weak ($\ll\mu^2-m^2$) field but $T^2$ still smaller
than $eB$. An expansion for $B$ smaller than all other scales
would be desirable but there are some subtleties involved in such an expansion.
The vacuum part can be expanded in a naive way and we get
\begin{equation}
{\cal L}_1=- \frac{m^4}{4\pi^2}\sum_{k=1}^\infty
\left(\frac{eB}{\pi m^2}\right)^{2k+2}(-1)^k\zeta(2k+2)\Gamma(2k)\ .
\end{equation}
This series is not convergent but Borel summable for small $eB/m^2$ so we
expect the first few terms to be a good approximation for weak fields.
Expanding the integrand of ${\cL_{1,reg}^{\beta,\mu}}$ (see \Eqref{Lbmueff}) in powers of $B$
leads to the same problem after the $s$-integration. Moreover, the
$\omega$-integration becomes infra--red divergent, for higher order terms. We
cannot even expand the integrand of ${\cL_{1,osc}^{\beta,\mu}}$ in powers of $B$, but after
repeated
partial integrations with respect to $\omega$ we obtain
\begin{eqnarray}
{\cL_{1,osc}^{\beta,\mu}} &=& \frac{m^4}{4\sqrt{2}\pi^{3/2}}\sum_{k=0}^\infty
\left(\frac{eB}{\pi m^2}\right)^{5/2+k}\zeta(5/2+k)
(-1)^{[k/2]} \nonumber \\
&&\times \left( m^2 \frac{d}{d\omega^2}\right)^k
\left.\frac{m}{\omega}
\Bigl(f^+_F(\omega)+f^-_F(\omega)\Bigr)\right|_{\omega=m}\ ,
\label{wBexp}
\end{eqnarray}
where $[k/2]$ is the integral part of $k/2$.
When $|\mu|>m$ the factor with derivatives of $f^\pm_F(\omega)$
at $\omega=m$ contains powers of $m/T$. These factors,
combined with the $B/m^2$ factors, show that we must
have $\{B\ll m^2,T^2\}$ in order for the expansion to be valid.
For $|\mu|<m$ these terms are exponentially suppressed at
small $T$. We thus see that there is an intricate interplay
between $B$ and $T$ in such a way that when \{$eB\ll T^2$\}
${\cL_{1,osc}^{\beta,\mu}}$ is smaller than ${\cL_{1,reg}^{\beta,\mu}}$, as well as their
derivatives. However, when $\{T^2\ll eB\}$, even though
$\{eB\ll\mu^2-m^2,m^2\}$, the $B$ derivatives of ${\cL_{1,osc}^{\beta,\mu}}$
are large and show a periodic behaviour as shown in
Section \ref{dhva}.
Also the expansion from \Eqref{dittricheqn} is only asymptotic. In view of the
observations above, especially the half--integer powers of $B$ in
\Eqref{wBexp}, it seems unlikely that the same result can be obtained in
ordinary pertubation theory using diagrammatic techniques.
The vanishing radius of convergence for the expansion
of ${\cal L}_1$, and the same for ${\cL_{1,reg}^{\beta,\mu}}$, also including the infra-red
divergences, arise due to the fact that we get substantial contributions
to the parameter integrals when we are outside the radius of convergence
for the series expansion of the $\coth(eBs)$,
i.e. for large s, for high order terms.
We will investigate this, the non-analyticity in $B$, and the connection to
ordinary perturbation theory
more carefully in a future project.
Some weak--field results can nevertheless be obtained and, for instance, the
magnetic susceptibility can be computed in the limit
\{$B\rightarrow 0$, $T\ll\mu^2-m^2$\}. It gets contribution
only from ${\cL_{1,reg}^{\beta,\mu}}$,
\begin{equation}
\chi=\lim_{B\rightarrow 0}\frac{\partial^2{\cL_{1,reg}^{\beta,\mu}}}{\partial B^2}
=\frac{e^2}{6\pi^2}\log\left(\frac{|\mu|}{m}+
\frac{\sqrt{\mu^2-m^2}}{m}\right)\ .
\end{equation}
If we further assume that \{\,$\mu^2-m^2\ll m^2$\} and write
it in terms of the Bohr magneton $\mu_B=e/2m$ and the density of
states at the Fermi surface
$g(\mu)=m\sqrt{\mu^2-m^2} /\pi^2$, we find
\begin{equation}
\chi=\frac{2}{3}\mu_B^2 g(\mu) =
\chi_{Pauli}+\chi_{Landau}\ .
\end{equation}
It coincides with the well-known result \cite{AshcroftM76} where
$\chi_{Pauli}$ is the Pauli paramagnetic spin
contribution and $\chi_{Landau}=-\inv{3}\chi_{Pauli}$
is the Landau diamagnetic orbital contribution.
Notice that in this weak field limit the thermal corrections dominate, i.e.
\begin{equation}
\cL_{1}^{\beta,\mu} \approx \frac{(eB)^{2}}{12\pi^{2}} \frac{\sqrt{\mu^{2}-m^{2}}}{m}
\gg {\cal L}_{1}~~~,
\end{equation}
where ${\cal L}_{1}$ is given by \Eqref{SMALLB}.
\subsection{\sc High temperatures}
\label{hight}
At high temperatures one may find an analytical approximation in the limit
$\{T^2\gg m^2\gg eB,\mu=0\}$, where we have that
\begin{equation}
\label{LbmhighT}
\cL_{1}^{\beta,\mu} \approx \frac{(eB)^2}{24\pi^2}
\log\left( \frac{T^2}{m^2}\right)\ ,
\end{equation}
and we do not agree with the high temperature and weak field
limit in Ref.\cite{dittrich79}.
(We notice the similarity between \Eqref{LbmhighT} and ${\cal L}_{1}$ for
$eB\gg m^2$ in \Eqref{LARGEB}.) The thermal contribution $\cL_{1}^{\beta,\mu}$ thus
dominates over ${\cal L}_{1}$ as given by Eq.(\ref{SMALLB}) when
\begin{equation}
\frac{T}{m}\gg \exp\left[\inv{30}
\left(\frac{eB}{m^2}\right)^2\right]
\approx 1\ ,
\end{equation}
i.e. when the approximations used here are valid.
\begin{center}
\section{\sc Some Astrophysical Applications}
\label{astro}
\end{center}
\setcounter{equation}{0}
As mentioned in the Introduction, strong magnetic fields
at finite temperature and density are situations that
are frequently encountered in astrophysical contexts.
We have investigated the possibility of some interesting behaviour
mainly for white dwarfs, neutron stars and
supernovae since they present the most extreme conditions
while still being directly observable, in contrast to
e.g. cosmic strings, the existence of which has yet to
be confirmed. We can use the effective action in two ways.
Either we consider the response of the system to a given
external magnetic field $H$, or we study the properties
of an isolated system with only the induced magnetic field.
In the first case the free energy is given by
\begin{equation}
F=-{\cal L}_1(B)-\cL_{eff}^{\beta,\mu}(B)\ ,
\end{equation}
where $B$ is determined by the mean field equation
\begin{equation}
\label{indB}
B=H+M(B)=H+\frac{\partial{\cal L}_1}{\partial B}+
\frac{\partial\cL_{eff}^{\beta,\mu}}{\partial B}\ .
\end{equation}
The magnetization $M(B)$ is thus calculated in the presence of
the microscopic magnetic field $B$. Note that we include both
the contribution from real electrons in the heat bath and
virtual electrons from vacuum polarization.
If we consider the dynamics of the system without any external
field we should add ${\cal L}_0$ to the effective action and determine
stationary values of the field by
\begin{equation}
\label{stat}
\frac{\partial{\cal L}_{eff}}{\partial B}=
-B+\frac{\partial{\cal L}_1}{\partial B}+
\frac{\partial\cL_{eff}^{\beta,\mu}}{\partial B}=0\ ,
\end{equation}
which, of course, is the same as putting $H=0$ in \Eqref{indB}.
As discussed in
Section \ref{strong} the vacuum contribution is dominant for large fields.
Using the result from \cite{MillerR84} we see that for $T=m$ the thermal
contribution saturates at about $eB=10\,m^2$. At that value of the
magnetic field, the vacuum
contribution is about twice as large as the thermal and cannot be
ignored.
\vspace{5ex}\\
It would be most interesting if we could find astrophysical
objects showing the de Haas -- van Alphen oscillations. The magnitude of
the oscillations might then be large enough to effectively
trap the magnetic field in a local minimum satisfying
\Eqref{stat}. A candidate for such a system is a
neutron star with a strong $B$ field and a degenerate electron gas.
In order to get de Haas -- van Alphen oscillations as a function of $B$ the
spacing of Landau levels near the Fermi surface need to
be larger than the spreading of the Fermi surface due to finite
temperature. If the $n$-th Landau level is at the Fermi surface,
$E_n=\mu$, then we require $E_{n+1}-E_n\gsim T$.
For $\mu^2\gg eB$, which is the case for neutron stars,
we get the condition
\begin{equation}
eB\gsim \mu T\ ~~~.
\label{eq-reldhva}
\end{equation}
According to Appendix C we can even get a more stringent condition in the case
of large chemical potential
\begin{equation}
eB\gsim 2 \pi^2 \mu T\ ~~~.
\end{equation}
As a comparision, we find in the non-relativistic case
, instead of \Eqref{eq-reldhva} that
\begin{equation}
eB \gsim m T~,~~~\{ eB, \mu^2-m^2 \ll m^2\} ~~~.
\end{equation}
In order to see any oscillations the field must not be so high that all
fermions are in the lowest Landau level, i.e. integer $n$ above must
be greater than unity, that gives
\begin{equation}
(\mu^2-m^2)/2 > eB~~~.
\end{equation}
Approximate values for $eB,T$ and $\mu$ for what we find
the most interesting astrophysical
objects in this context, a supernova; a neutron star; and a white dwarf,
are given in Table~1. According to above, the number in the
last two rows of this table should be greater than unity for
de Haas -- van Alphen oscillations to appear. That is not the case
in either of the situations.
\begin{table}
\begin{center}
\begin{math}
\begin{array}{||l|c|c|c||} \hline \hline
&{\rm ~~White~ Dwarf} & {\rm~~ Neutron~ Star~~}
&{\rm ~~Supernova~~} \\ \hline
\mu/m & 1.02~\cite{freese}& 6 \cdot 10^2 ~\cite{neutronstar}
& 6 \cdot 10^2 ~ \cite{myller90} \\ \hline
T/m & 2 \cdot 10^{-3} ~\cite{freese} & 1 ~\cite{neutronstar}
& 1 \cdot 10^2 ~\cite{myller90} \\ \hline
eB/m^2 & 2 \cdot 10^{-6} ~\cite{neutronstar,Chanm92}
& 2 \cdot 10^{-1}~ \cite{Chanm92} & 2~ \cite{ginzburg91} \\ \hline \hline
( \mu^2 -m^2)/(2eB) & 1 \cdot 10^4 & 2 \cdot 10^6
& 2 \cdot 10^5 \\ \hline
eB/(\mu T) & 1 \cdot 10^{-3} & 3 \cdot 10^{-4}
& 3 \cdot 10^{-5} \\ \hline \hline
\end{array}
\end{math}
\end{center}
\baselineskip 13pt
\tabcap{Typical values of $eB,T$ and $\mu$ for some astrophysical
objects, and an indication of the possibility for oscillations in the
magnetization. The references are given in brackets.}
\label{tab-astro}
\end{table}
\begin{table}
\begin{center}
\begin{math}
\begin{array}{||c|c|c|c|c||} \hline \hline
~ {\cal L}_0~~ (m^4)~ &~{\cal L}_1~~ (m^4)~ & ~ \cL_{0}^{\beta,\mu}~~ (m^4)~
& {\cL_{1,reg}^{\beta,\mu}}~~ (m^4)~&
~{\cL_{1,osc}^{\beta,\mu}}~~(m^4)~ \\ \hline
-2 \cdot 10^{-2} & 2 \cdot 10^{-6} & 1 \cdot 10^9 & 6 \cdot 10^{-2}
& 1 \cdot 10^{-3} \\ \hline \hline
\end{array}
\end{math}
\end{center}
\baselineskip 13pt
\tabcap{The different parts of the effective Lagrangian
for a typical neutron star, in natural units.}
\label{tab-neutrlag}
\end{table}
\begin{table}
\begin{center}
\begin{math}
\begin{array}{||c|c|c||} \hline \hline
~ M_1~~ ( e m^2)~ &~ M_{reg}~~ (e m^2)~ &~ M_{osc}~~ (em^2)~ \\ \hline
2 \cdot 10^{-5} & 4 \cdot 10^{-2} & 1 \cdot 10^{-2} \\ \hline \hline
\end{array}
\end{math}
\end{center}
\baselineskip 13pt
\tabcap{The different parts of the magnetization
for a typical neutron star, in natural units.}
\label{tab-neutrmag}
\end{table}
For a neutron star we have numerically computed the different parts
of the effective Lagrangian, and the corresponding magnetization. The
results are given in Table~2 and Table~3,
respectively.
The effective Lagrangian is totally dominated by the thermal contribution
in absence of a magnetic field , $ \cL_{0}^{\beta,\mu} $, due to the extreme
chemical potential. We would like to stress
that there are no oscillations in the so called
oscillating part of the magnetization, $M_{osc}$, in this region of
parameters.
Obviously we do not expect to see any de Haas -- van Alphen oscillations
unless the neutron star is very cold ($T={\cal O} (1)$ eV), or if the electron
density is very low
in some region, for example close to the surface, where the field still is
strong.
\vspace{5ex}\\
In order to investigate the behaviour of a relativistic gas of fermions
showing de Haas -- van Alphen oscillations, we have numerically calculated
the effective action, and the magnetization for $\{\mu/m=4\,;\
T/m=0.01\,,\ 0.1\,,\ 1.0\}$. The latter is shown in Fig.~1.
\begin{figure}[t]
\epsfxsize=15cm
\epsfbox{mefftot.ps}
\baselineskip 13pt
\figcap{The vacuum and thermal contribution to the magnetization showing de
Haas -- van Alphen oscillations
as the temperature is lowered. The chemical potential is $\mu=4\,m$.}
\nopagebreak
\label{fig-osc}
\end{figure}
We see clearly how the oscillations disappear as the temperature is raised.
There is also a last oscillation at about $eB\simeq 7m^2$ which occurs when the
second Landau level leaves the Fermi surface. For the values above we do not
find any non--trivial solution to \Eqref{stat} because the tree level $-B$
dominates. It is, in fact, only for a rather limited range of parameters that
${\cL_{1,osc}^{\beta,\mu}}$ can give local maxima for the total effective action. As an
example, let us first put $T=0$ since that only enhances the oscillations. Then
we look at small $B$ so that the tree part is small. There is a chance that the
$\sqrt{B}$ term in \Eqref{Mosc} can dominate. Using $\mu\simeq m$ and making
the approximation $\abs{\zeta \!\left( -\frac{1}{2},x \right) }\leq 0.2$, we
get
\begin{equation}
\abs{M_{osc}} \lsim 0.2\frac{\sqrt{2}e^{3/2}}{2m\pi^2}
\sqrt{B}(\mu^2-m^2)
\simeq 0.005 \sqrt{B} (\mu-m)\ .
\end{equation}
For this term to dominate over $\abs{M_{tree}}=B$ we need $B\lsim 10^{-5}
(\mu-m)^2$ which complicates numerical calculations. Also, since the field is
small the probability of tunneling through the barrier between the maxima is
not very suppressed and it is probably not an efficient way of trapping
magnetic fields.
At very large values of $B$ the vacuum part eventually dominates over the tree
level, but this is just the Landau ghost and we cannot draw any conclusion
about any instability.
Even if there are no local minima in $-B^2/2-{\cal L}_1-\cL_{eff}^{\beta,\mu}$, there
may be intervals in $B$ where $-{\cal L}_1-\cL_{eff}^{\beta,\mu}$ is concave, i.e. where the
susceptibility is positive.
Domains with different magnetization could then be formed
in presence of an external field, just like in
some solid state materials \cite{Abrikosov88}.
\begin{center}
\section{\sc The Effective QED Coupling}
\label{coupling}
\end{center}
\setcounter{equation}{0}
The charge renormalization given by \Eqref{chargeren} also leads to the weak
coupling
expansion of the QED $\beta$-function, i.e.
\begin{equation}
\lambda\frac{d}{d\lambda}\alpha(\lambda) = \beta (\alpha(\lambda)) =
\frac{2}{3\pi}\alpha^{2}(\lambda) + {\cal O}(\alpha^{3}(\lambda))~~~,
\label{BETAF}
\end{equation}
where $\lambda$ is a momentum scale factor.
We notice that due to the scale invariance of $eB$, we
can also define an effective coupling constant from ${\cal L}_{eff}$ as
\cite{Schwinger51,cos88}
\begin{equation}
- \frac{1}{e^2(eB,\mu,T)}=\frac{1}{eB} \frac{ \partial {\cal L}_{eff}}{\partial (eB)}~~~,
\end{equation}
that gives for the electromagnetic fine structure constant
$\alpha(eB,\mu,T) \equiv e^{2}(eB,\mu,T)/4\pi$
\begin{equation}
\label{effalp}
\frac{1}{\alpha(T,\mu,B)}=\inv{\alpha}
-\frac{1}{\alpha B}\frac{\partial( {\cal L}_{1} +{\cal L}_1^{\beta,\mu})}{\partial B}\ ,
\end{equation}
in analogy with the definition of the renormalized
coupling in the vacuum sector
in connection with \Eqref{BETAF}. Special care has
to be taken when evaluating the
derivative
of the oscillating term in Eq.(\ref{Lbmueff}).
In the limit when $eB=0$, we obtain the effective coupling
$\alpha(T,\mu) = \alpha(T,\mu,B=0)$ given by
\begin{equation}
\label{AlphaTmu}
\frac{1}{\alpha(T,\mu)}= \frac{1}{\alpha } - \frac{2}{3\pi}\int
_{-\infty}^{\infty}d\omega
\frac{\theta(\omega^{2}-m^{2})}{
\sqrt{\omega^{2}-m^{2}}}f_{F}(\omega)~~~.
\end{equation}
When $T=0$, we therefore get an effective coupling
$\alpha(\mu) = \alpha(T=0,\mu)$ such that
\begin{equation}
\label{Alphamu}
\frac{1}{\alpha(\mu)}= \frac{1}{\alpha }
-\frac{2}{3\pi}\log \left( \frac{|\mu|}{m} + \sqrt{\frac{\mu ^{2}}{m^{2}} -
1}~\right)\ .
\end{equation}
In the limit $\mu=0$, we find the following asymptotic behaviour
of the corresponding effective coupling $\alpha(T) = \alpha(T,\mu = 0)$,
\begin{equation}
\label{AlphaT}
\frac{1}{\alpha(T)} = \frac{1}{\alpha } -\frac{4}{3\pi}
\int
_{\beta m}^{\infty}
\frac{dx}{
\sqrt{x^{2}-(\beta m)^{2}}}\frac{1}{e^{x} + 1}
\approx \frac{1}{\alpha }
-\frac{2}{3\pi}\log \left(\frac{T}{m}\right)~~~,
\end{equation}
for $T \gg m $.
It is now clear that (only) for $\mu \gg m$ and $T \gg m$ the
effective couplings
$\alpha (\mu)$ and $\alpha (T)$ are solutions to the renormalization group
equation~(\ref{BETAF}) when $\lambda $ is identified with $\mu$ and $T$
respectively
(see in this context e.g. Refs.\cite{Morley79,rojas92}).
We also note that Eq.(\ref{LARGEB}) leads to an effective
coupling $\alpha(B) = \alpha(T=0,\mu = 0,B)$
with an asymptotic behaviour
\begin{equation}
\label{AlphaB}
\frac{1}{\alpha(B)} \approx \frac{1}{\alpha }
-\frac{2}{3\pi}\log \left(\frac{\sqrt{eB}}{m}\right)~~~,
\end{equation}
that also satisfies the renormalization group equation \Eqref{BETAF}.
The effective coupling defined in \Eqref{effalp} can also be extracted
from the residue of the thermal Debye-screened photon propagator
(see Ref.\cite{Morley79}).
The effective couplings as given in Eqs.(\ref{Alphamu}),~(\ref{AlphaT}) and
(\ref{AlphaB}) can be interpreted as follows. If we use the lowest order
$\beta$-function in \Eqref{BETAF}, then the scale dependent coupling
is given by
\begin{equation}
\frac{1}{\alpha(\lambda)}=\frac{1}{\alpha}- \frac{1}{3\pi}\log \left(
\frac{\lambda^{2}}{m^{2}} \right)~~~.
\end{equation}
Then we can write
\begin{equation}
\label{eq:running}
\frac{1}{\alpha(x)}\approx \frac{1}{\alpha(\lambda)} -\frac{2}{3\pi}
\log \left(\frac{x}{\lambda} \right)~~~,
\end{equation}
where $x=\mu,T$ or $ \sqrt{eB}$. If $\lambda$ is identified with any of
these scales, we can in each such case write
\begin{equation}
{\cal L}_{eff}=-\frac{1}{2} \frac{(eB)^2}{e^{2}(x)} +\cL_{0}^{\beta,\mu}~~~,
\end{equation}
when $x \gg ( m \mbox{ and any other scale of dimension energy} )$.
In terms of the effective fine-structure constant, and in the case of small
chemical potentials,
so that $|\mu| < m$, we obtain
\begin{equation}
\alpha (eB,T,\mu)
= \frac{\alpha}{1 - \alpha X(eB,T,\mu)}~~~,
\end{equation}
where we have defined the functions $X(eB,T,\mu) = X_{1}(eB) +
X_{2}(eB,T,\mu)$,
\begin{equation}
X_{1}(eB)=\frac{1}{2\pi} \int_{0}^{\infty} \frac{dx}{x}
\exp(-x\frac{m^2}{eB}) \left[\frac{1}{\sinh ^2(x)}
- \frac{\coth(x)}{x} +
\frac{2}{3}\right]~~~,
\end{equation}
and
\begin{eqnarray}
\label{eq:xtwo}
X_{2}(eB,T,\mu) &=&
\frac{1}{2\pi}\sum _{l=1}^{\infty}(-1)^{l}
\int _{0}^{\infty} \frac{dx}{x}
\exp\left( -\frac{\beta ^{2} l^{2}}{4x} -m^{2}x \right) \nonumber \\
&& \times\left[\frac{1}{\sinh ^2(eBx)} - \frac{\coth(eBx)}{eBx}\right]
\cosh(\beta l\mu)
~~~.
\end{eqnarray}
The function $X_{1}(eB)$ has the following expansions
\begin{equation}
X_{1}(eB) = \frac{2}{45\pi} \left(\frac{eB}{m^{2}}\right)^{2} + {\cal
O}\left(\left( \frac{eB}{m^{2}}\right) ^{4}\right)
\end{equation}
if $eB \ll m^{2}$ and
\begin{equation}
X_{1}(eB) = \frac{1}{3\pi}\log \left( \frac{eB}{m^{2}}\right) \left(
1 + \frac{3}{2} \frac{m^{2}}{eB}\right) + {\cal O}\left(\frac{m^{2}}{eB}\right)
\end{equation}
if $eB \gg m^{2}$.
In the case of a vanishing chemical potential we can in \Eqref{eq:xtwo}
identify a
$\vartheta_{4}$-function, given as
\begin{equation}
\vartheta_{4}[z,q]=1+2 \sum_{n=1}^{\infty}(-1)^{n} q^{n^{2}} \cos(2nz)~~~,
\end{equation}
and write
\begin{eqnarray}
&& X_{2}(eB,T) = \frac{1}{2\pi}
\int _{0}^{\infty} \frac{dx}{x} \exp(-x\frac{m^2}{eB})
\nonumber \\
&& \times
\left\{1-\vartheta_{4}
\left[0,\exp\left(-\frac{eB\beta^{2}}{4x}\right)\right]
\right\}
\left[ \frac{\coth(x)}{x} -\frac{1}{\sinh ^2(x)} \right]~~~.
\end{eqnarray}
If $eB \ll m^{2}$, we can write
\begin{equation}
X_{2}(eB,T) = \frac{4}{3\pi}\sum _{l=1}^{\infty} (-1)^{l+1}
\left(
K_{0}(\beta m l) - (\beta m l)^{2}K_{2}(\beta ml)
{\cal O}
\left[\left(\frac{eB}{m^{2}}\right)^{2}\right] \right)~~~.
\end{equation}
For $T\gg
m$ we can use
\begin{equation}
\sum_{l=1}^\infty K_0(xl)(-1)^{l+1} \rightarrow
-\inv{2}\log x\ ;\quad x\rightarrow 0~~~ ,
\end{equation}
to find that it leads to a $\log (T/m)$ dependence with the correct prefactor
in accordance with \Eqref{eq:running}. (The approximation of keeping only the
$l=1$, as in Ref.\cite{rojas92}, excludes the factor $1/2$, and thus
is not correct.) In general we have that
\begin{eqnarray}
&& X_{1}(eB) + X_{2}(eB,T) = \frac{1}{2\pi}
\int _{0}^{\infty} \frac{dx}{x} \exp(-x\frac{m^2}{eB})
\nonumber \\
&&
\times\left\{ \frac{2}{3}
-\vartheta_{4}\left[0,\exp\left(-\frac{eB\beta^{2}}{4x}
\right)\right] \times
\left[ \frac{\coth(x)}{x} -\frac{1}{\sinh ^2(x)} \right]\right\}~~~.
\end{eqnarray}
\begin{center}
\section{\sc Discussion and final remarks}
\label{concl}
\end{center}
\setcounter{equation}{0}
\subsection{\sc Inclusion of interparticle interactions}
In our one-loop treatment of the effective action we have not
included interactions between electrons. The interaction energy between the
particles increases with $T$ and $\mu$ since the density increases, but so does
kinetic energy. For a degenerate
electron gas with large chemical potential the kinetic energy
dominates over the potential energy for electrons close to the
Fermi surface. However, not all electrons have large kinetic
energy and corrections from interactions have to be considered for electrons
with low momenta.
The self-energy correction for fermions at high temperature
and density, but zero external field, has been computed in e.g.
Refs.\cite{Petitgirard92,AltherrK92} (in Ref.\cite{AltherrK92}
only massless fermions were considered but it gives an indication
of the correction, especially in view of the result in
Ref.\cite{Petitgirard92}). There appear some completely
new collective phenomena, such as hole excitations
\cite{Weldon89}, which are
not taken into account in this paper.
For the particle excitations the dispersion relation can be
approximated by an ordinary massive particle
provided the mass is replaced by an effective $T$ and $\mu$
dependent mass \cite{Petitgirard92}
\begin{equation}
\label{mp}
m_p=\frac{\sqrt{m_e^2+4M^2}+m_e}{2}\ ,
\end{equation}
where $M$ is the thermally induced mass which in the case of
QED is
\begin{eqnarray}
M^2=\frac{e^2 \mu^2}{8\pi^2}
&;&
\quad T=0,\ \mu\neq 0\ ,\nonumber\\
M^2=\frac{e^2T^2}{8}
&;&
\quad T\neq 0 ,\ \mu=0\ ,
\end{eqnarray}
at least if $T \gg m$ and $ \mu \gg m$.
The hole excitation has
a more peculiar dispersion relation but its spectral
weight is on the other hand lower. It is difficult
to make any quantitative estimates of the importance
of self-energy corrections. We do not, however, expect that phenomena like the
de Haas -- van Alphen oscillations should be altered since it depends on the
electrons at the Fermi surface.
\subsection{\sc Further developments}
There are some extensions of our work that may be of
physical importance. First, we can consider the self-energy
correction of an electron in presence of an external $B$ field.
{}From that the anomalous magnetic moment can be extracted
and compared with previous calculations for small $B$ field,
where there appears some problems of analyticity in the
external photon momentum at finite density \cite{pebss91}.
The self-energy is also important for the higher loop
corrections of the effective action as discussed above.
The QED radiative corrections
could
effect the
electroweak transition rates, relevant for the Big-Bang primordial
nucleosynthesis \cite{ChengST93}.
The photon polarization tensor should also be calculated, and in particular its
imaginary part which is related to the decay into an $e^+e^-$--pair.
Also the three-photon vertex is interesting since it does not exist in absence
of the external field. Such photon splitting processes have been considered
earlier in vacuum \cite{AdlerBCR70,BialynickaB70,BrezinI71,PapayanR72} and it
would be interesting to study the correction from a thermal environment.
The physically more complicated case of a constant (or slowly varying) $E$
field
is equally interesting. A plasma does not stay in equilibrium
since the $E$ field gets screened and the physical
picture is very different from the one discussed in this paper.
Yet another generalization would be to expand a
non-constant field in powers of the derivative.
Such an expansion has been studied in Ref.\cite{Hauknes84}
at zero temperature.
\subsection{\sc Conclusion}
The main objective of this paper
has been to establish the correct form of the one-loop
QED effective action at finite temperature and density
to all orders in a constant external magnetic field, and the
result differs from earlier attempts. From the form of
$\cL_{eff}^{\beta,\mu}$ presented in \Eqref{Lbmueff} we have checked several
limits that can be understood from a physical point of view.
A great advantage with our expression for $\cL_{eff}^{\beta,\mu}$ is that
the thermal distribution function $f_F(\omega)$ occurs explicitly.
This means that it is easy to study other thermal situations
by simply replacing $f_F(\omega)$ with some other (non-equilibrium)
distribution (see e.g. Ref.\cite{ElmforsEV93c}).
The importance of the thermal correction depends on the value
of $B$, $T$ and $\mu$. In some physically interesting cases
they may be large compared to $m$ but often of the same
order of magnitude, which makes it difficult to obtain
analytical approximations. It is, however, possible
to use \Eqref{Lbmueff}, or the expressions in Appendix C, for numerical
calculations.
Even though the correction to the free energy may be small
compared to the value without the external field there
are other quantities that are effected by the presence of
the heat bath. For instance, the magnetization of a degenerate
Fermi sea shows the de Haas -- van Alphen effect.
We found, however, that for a neutron star this effect does not show up in
spite of the extreme degeneracy and magnetic field. The reason is the
relativistic form of the energy spectrum which suppresses the oscillations at
a large chemical potential. We also briefly discussed the importance of
including the vacuum contribution to the magnetization when the $B$ field is
comparable to $m^2/e$.
We have, furthermore, calculated an effective coupling constant
defined from the derivative of $\cL_{eff}^{\beta,\mu}$ with respect to $B$.
It satisfies asymptotically a naive zero temperature
renormalization group
equation where the renormalization scale is replaced by
$T$, $\mu$ or $\sqrt{eB}$.
\vspace{3mm}
\begin{center}
{\bf ACKNOWLEDGMENT}
\end{center}
\vspace{3mm}
One of the authors (B.-S.~S.) would like to thank
John Ellis for
the hospitality of the Theory Division at CERN where some of this work was
initiated and NFR for providing the financial support.
P.~E. wants to thank C.~Pethick for discussions about
neutron stars. It is a pleasure to thank the organizers of the 3rd Workshop
on Thermal Field Theories, 1993, and in particular R. Kobes and G. Kunstatter,
for
providing a stimulating atmosphere during which parts of the present work were
finalized.
\vspace{3mm}
\renewcommand{\thesection}{A}
\setcounter{section}{1}
\setcounter{equation}{0}
\begin{center}
{\Large {\sc Appendix A}}
\end{center}
In this appendix we give some details of how to
calculate the effective action in \Eqref{Lbmueff}.
First we show that \Eqref{dittricheqn}
is equal to \Eqref{Lbmueff}.
To do that we start with a Poisson resummation in $l$ using
\begin{equation}
\label{Poisson}
\sum_{l=1}^\infty(-1)^l\exp(-\frac{l^2}{4a})=
\sqrt{4\pi a}\sum_{l=0}^\infty \exp(-a\pi^2(2l+1)^2)
-\inv{2}\ ,
\end{equation}
and rewrite the sum over $l$ as a contour integral by means
of the formula
\begin{equation}
\sum_{l=0}^\infty f\bigl(\frac{\pi}{\beta}(2n+l)\bigr)=
\frac{\beta}{2\pi}\int_C\frac{d\omega\,f(\omega)}
{e^{-i\beta\omega}+1}\ .
\end{equation}
The integration contour $C$ is chosen to go from
$\infty+i\epsilon$ to $\epsilon$ in the upper half plane
and back to $\infty-i\epsilon$ in the lower half plane
(i.e. $\omega\in\{\infty+i\epsilon\rightarrow
\epsilon\rightarrow\infty-i\epsilon\}$), without encircling the origin.
In this way all the poles on the positive real axis are
encircled.
We would now like to deform the $\omega$-integral to the
imaginary axis and the $s$-integral to the negative real axis.
This is not straightforward since there are poles on the imaginary
$s$-axis and the section at infinity has to bo chosen to give a vanishing
contribution. It is,
therefore, necessary to divide the integral into several pieces and
to do the deformation for each piece separately. Let us start with
the part where $\omega$ is in the upper half plane. Then the $s$-contour
can be deformed to the negative imaginary axis, but to the right of the
poles. After that the $\omega$ contour is deformed to the positive
imaginary axis. Finally, for $\abs{\omega}>m$ we further continue
the $s$-integral to the negative real axis and pick up the poles on the
negative imaginary axis, while for $|\omega|<m$ we deform the
$s$-contour back to the positive real axis.
The whole procedure can
be repeated for $\omega$ below the real axis, reflecting all
deformations around the real axis. To get the correct convergence for
the $\omega$-contour deformation, the constant $-1/2$ in \Eqref{Poisson}
should be associated with the $\omega$ in the lower half plane.
After summing the pieces there is only a contribution from
$|\omega|>m$, as expected, and it consists of an $s$-integral
and a sum over the residues of the poles.
In the deformations above we have been careful with the convergence
for large $|s|$ and $|\omega|$, but we have said nothing
about the possible singularity at $s=0$. One way of dealing with
that is to multiply the expression with $s^\nu$ and perform
the integration for such a $\nu$ that there is no
divergence at $s=0$, and to do the analytic continuation at the end.
Equation (\ref{Lbmueff}) can also be obtained from the thermal
propagator in \Eqref{btprop} by representing the $\delta$-function
as
\begin{equation}
2\pi i\delta(x)=i{\rm Im}\inv{x-i\epsilon}=
i{\rm Im}\int_0^\infty ds\,e^{-i(x-i\epsilon)}\ .
\end{equation}
Then the $k_y$ and $k_z$ integrations can be carried out
(using \Eqref{Iid} as well). The summation over $n$ is just a
geometric series but it is not absolutely convergent so we
sum only to a finite $N$ and take
the limit $N\rightarrow\infty$ at the end. This gives
\begin{eqnarray}
{\rm Tr\,} S_F^{\beta,\mu}(x;x|m)&=& \lim_{N\rightarrow\infty}
i\,\frac{mB}{\pi^{3/2}}
\,{\rm Im}\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}f_{F}(\omega)
\int_0^\infty\frac{ds}{s^{1/2-\nu}}e^{i\frac{3\pi}{4}}
e^{-is(\omega^2-m^2-i\epsilon)} \nonumber \\
&& \times\left[\frac{1+e^{i2sB}}{1-e^{i2sB}}-
\frac{2 e^{i2NsB}}{1-e^{i2sB}}\right]~~~,
\label{SFbmu}
\end{eqnarray}
where we also have introduced the dimensional regularization $\nu$ in $4-2\nu$
dimensions, and we are to analytically continue to $\nu=0$ in the end.
Keeping $\nu$ large enough that the integral is absolutely convergent,
the expression above can easily be integrated with respect to
$m$ to yield $\cL_{eff}^{\beta,\mu}$. To be more precise, there is an
integration constant from the lower limit in
\begin{equation}
\label{mint}
i \int_{m_0}^{m}dm'{\rm Tr\,}\,S_F^{\beta,\mu}(x;x|m') =
\cL_{eff}^{\beta,\mu}(m)-\cL_{eff}^{\beta,\mu}(m_0)\ ~~~.
\end{equation}
We expect that in the limit $m\rightarrow\infty$ the
thermal part of the effective action is zero since an
infinitely massive particle has zero Boltzmann weight.
Therefore we let $m_0 \rightarrow \infty $ and thereby
put the integration constant $\cL_{eff}^{\beta,\mu}(m_0)$
to zero.
The poles in the last factor in \Eqref{SFbmu} cancel for finite
$N$, and we cannot let $N\rightarrow\infty$ in a naive
way before deforming the $s$ integration contour to the
imaginary axis. The two terms have to be treated separately
so we must choose an integration contour for $s$ slightly
above or below the real axis.
Since, according to the discussion above, $\cL_{eff}^{\beta,\mu}(m \rightarrow \infty)=0$
we see that the the original contour must be chosen slightly above the real
axis.
Depending on the sign
of $\omega^2-m^2$ (or $\omega^2-m^2-2eB(N-1)$ in the the second
term) we deform the $s$-contour to either the positive or
negative imaginary axis. In one of the cases we get a
contribution from the poles.
After deforming the contours we take the $N\rightarrow\infty$
limit and also take the limit $\nu \rightarrow 0$ what concerns taking the
imaginary part, in order to get a more apparent expression, but we still need
to keep $\nu >0$ to have the integration over $s$ finite, with the result
\begin{eqnarray}
\hspace{-3ex}\cL_{eff}^{\beta,\mu} &=&
\int_{-\infty}^\infty d\omega
\theta(\omega^2-m^2)f_F(\omega)
\Biggl[\inv{4\pi^{5/2}}\int_0^\infty
\frac{ds}{s^{5/2-\nu}}e^{-s(\omega^2-m^2)}
seB\coth (seB) \Biggr]\nonumber\\
&-&
\label{eqleffnu}
\int_{-\infty}^\infty d\omega
\theta(\omega^2-m^2)f_F(\omega)
\Biggl[
\inv{2\pi^3}\sum_{n=1}^\infty \left(\frac{eB}{n}\right)^{3/2}
\sin \! \left(\frac{\pi}{4}-\frac{\pi n}{eB}
(\omega^2-m^2)\right) \Biggr]\ .
\end{eqnarray}
Actually we must
have $\nu >3/2$, i.e. less then one dimension, but we may just consider it as
an analytical continuation in $\nu$, in order to be able to change the
order of integration.
If we now take the limit $B \rightarrow 0$, we get
\begin{eqnarray}
\cL_{0}^{\beta,\mu}&=& \frac{1}{4 \pi^{5/2}} \int_{-\infty}^\infty d\omega
\theta(\omega^2-m^2)f_F(\omega) \int_0^\infty ds\, s^{\nu-5/2}
e^{-s(\omega^2-m^2)} \nonumber \\
&=&
\frac{1}{4 \pi^{5/2}} \int_{-\infty}^\infty d\omega
\theta(\omega^2-m^2)f_F(\omega) (\omega^2 -m^2 )^{3/2-\nu}
\Gamma(\nu-3/2)~~~.
\end{eqnarray}
We may now take the limit $\nu \rightarrow 0$ to get \Eqref{lbmeffzero},
and after subtraction of this term we may also let $\nu$ vanish in
\Eqref{eqleffnu} and get \Eqref{Lbmueff}.
\renewcommand{\thesection}{B}
\setcounter{section}{1}
\setcounter{equation}{0}
\begin{center}
{\Large {\sc Appendix B}}
\end{center}
In the limit of very strong fields $ \{eB \gg T^{2},m^{2}, |\mu^{2}-m^{2}| \}$,
the first term in \Eqref{Lbmueff} can be written as
\begin{equation}
{\cL_{1,reg}^{\beta,\mu}}=\int_{-\infty}^{\infty}d\omega \theta(\omega^{2}-m^{2})
f_{F}(\omega) \frac{(eB)^{3/2}}{4\pi^{5/2}} \int_{0}^{\infty}
\frac{ds}{s^{5/2}}(s \coth s -1)~~~.
\end{equation}
Similarly we find in this limit
\begin{equation}
{\cL_{1,osc}^{\beta,\mu}}=-\int_{-\infty}^{\infty}d\omega \theta(\omega^{2}-m^{2})
f_{F}(\omega) \frac{(eB)^{3/2}}{2\sqrt{2}\pi^{3}} \zeta(3/2)~~~,
\end{equation}
where $\zeta$ is the Riemann $\zeta$-function.
It can be shown by residue calculations that
\begin{equation}
\int_{0}^{\infty}\frac{ds}{s^{5/2}}(s \coth s -1)=\sqrt{\frac{2}{\pi}}
\zeta(3/2)~~~,
\end{equation}
so that the $O(B^{3/2})$ terms cancel in this limit. In order to extract
the next term in the strong field expansion of $\cL_{1}^{\beta,\mu}$, we consider the
expression entering in the $\omega$ integral in $\cL_{1}^{\beta,\mu}$, expanded for
large $B$, i.e.
\begin{eqnarray}
\frac{(eB)^{3/2}}{4\pi^{5/2}} \left\{ \int_{0}^{\infty}
\frac{ds}{s^{5/2}}
(\exp[-\frac{s}{eB}(\omega^{2}-m^{2})]-1)(s \coth s -1)
\right. && \nonumber \\
\left. -\frac{2}{\sqrt{\pi}}
\sum_{n=1}^{\infty}\frac{1}{n^{3/2}} \left(
\sin[\frac{\pi}{4}-\frac{n\pi}{eB}
(\omega^{2}-m^{2})]-\sin\frac{\pi}{4} \right)
\right\}&&~~~.
\end{eqnarray}
If we now use the cancellations depicted above, and the fact that the
sum converges towards an integral in the limit $B \rightarrow \infty$, the
expression above may be written as
\begin{eqnarray}
&&\frac{(eB)^{3/2}}{4\pi^{5/2}} \left\{ \int_{0}^{\infty}
\frac{ds}{s^{3/2}} (\exp[-\frac{s}{eB}(\omega^{2}-m^{2})]) \right.
\nonumber \\
&&
-\left. \frac{1}{\sqrt{\pi B}} \int_{0}^{\infty} \frac{dx}{x^{3/2}}
\left( \sin[\frac{\pi}{4}-x\pi
(\omega^{2}-m^{2})]- \frac{1}{\sqrt{2}} \right)
\right\}~~~.
\end{eqnarray}
By performing the integrals in this expression, we find the following leading
contribution
\begin{equation}
\cL_{1}^{\beta,\mu}=\frac{eB}{2\pi^{2}}
\int_{-\infty}^{\infty}d\omega \theta(\omega^{2}-m^{2})
f_{F}(\omega) \sqrt{\omega^{2}-m^{2}}~~~.
\end{equation}
\pagebreak
\renewcommand{\thesection}{C}
\setcounter{section}{1}
\setcounter{equation}{0}
\begin{center}
{\Large {\sc Appendix C}}
\end{center}
In the case of large chemical potentials, e.g. in a neutron star,
when \mbox{ $ (\mu^2-m^2)/2eB \gg 1$}, the form for
${\cL_{1,osc}^{\beta,\mu}}$ given in \Eqref{eqovan} is difficult to handle due to the rapid
oscillations in the $\zeta$-function. Let us instead start from
\Eqref{Lbmueff}, and rewrite it as
\begin{equation}
{\cL_{1,osc}^{\beta,\mu}}= \frac{m^4}{2 \pi^3} \left( \frac{eB}{m^2} \right)^{3/2}
\sum_{n=1}^{\infty}n^{-3/2} {\rm Im} \left\{ \exp\left[-i\left(
\frac{\pi}{4} +\frac{\pi n}{eB/m^2} \right) \right] I_{n} \right\}~~~,
\label{eqosc}
\end{equation}
where we have defined
\begin{equation}
I_{n} \equiv \int_{1}^{\infty} dx \frac{ \exp\left(i
\frac{\pi n}{eB/m^2} x^2 \right) }{1+\exp[m\beta(x-\mu/m)]}~~~.
\end{equation}
Since the exponential function here is oscillating rapidly and we desire
a rapidly decreasing function instead, we close the
contour with a circular section at infinity, a straight line from the origin
to infinity with complex argument $\pi/4$, and the small section from
the origin to $x=1$, and use Cauchy's theorem to get
\begin{equation}
I_{n}=e^{i\pi/4} \int_{0}^{\infty}dx \frac{ \exp\left(-
\frac{\pi n}{eB/m^2} x^2 \right) }{1+\exp[m\beta(e^{i\pi/4} x-\mu/m)]} -
\int_{0}^{1} dx \frac{ \exp\left(i
\frac{\pi n}{eB/m^2} x^2 \right) }{1+\exp[m\beta(x-\mu/m)]} + I_{n}^{poles}~~~.
\end{equation}
The contribution from the residues at the poles is
\begin{equation}
I_{n}^{poles}=-2\pi i \frac{T}{m} \exp\left[ -2\pi^{2}\frac{\mu T}{eB}
+i \pi n \frac{\mu^2}{eB} \right]
\sum_{\nu=0}^{\nu_{max}} \exp \left[ -2 \pi^{2}n \frac{\mu T}{eB} 2 \nu -
i \pi^3 \frac{T^2}{eB}(2 \nu +1)^2 \right]~~~,
\end{equation}
where we have defined $\nu_{max}$
as the number of poles encircled by the contour
\begin{equation}
\nu_{max}=\mbox{ int}\left[ \frac{\mu}{2\pi T}-\frac{1}{2}\right]~~~.
\end{equation}
In the case of large chemical potential compared to the temperature and
the square root of the magnetic field, we may
assume the thermal distributions to be unity,
and perform the integrals with the result
\begin{equation}
\label{eq:davidsin}
I_{n}= e^{i \pi/4}\, \frac{1}{2} \sqrt{ \frac{eB/m^2}{n}}
\left( 1- \mbox{erf}\left[ \sqrt{ \frac{n}{eB/m^2}} e^{-i\pi/4} \right] \right)
+ I_{n}^{poles} + O[e^{-\beta( \mu -\sqrt{\frac{eB}{2\pi}})}]~~~.
\end{equation}
It turns out that the phase from one
minus the error function in \Eqref{eq:davidsin} cancels the phase
from $ \exp\left[-i\left(
\frac{\pi}{4} +\frac{\pi n}{eB/m^2} \right) \right]$ in \Eqref{eqosc}, when
taking the imaginary part. The oscillations are thus only originating from
the residues at the poles, that all have ${\rm Re}[\omega]=\mu$, i.e. they are lying
at the Fermi surface. Also, notice that the contribution from these poles is
exponentially suppressed as $ \exp\left[ -2\pi^{2}\frac{\mu T}{eB} \right]$,
in agreement with the general discussion on de Haas -- van Alphen oscillations
in Section~\ref{astro} .
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,517 |
Acidonia es un género monotípico de arbusto perteneciente a la familia de las proteáceas. Su única especie, Acidonia microcarpa, es un endemismo de la costa este de la provincia de Australia Occidental.
Taxonomía
Fue descrita originalmente por Robert Brown en 1810 como una especie de Persoonia. En 1975, Lawrence Alexander Sidney Johnson y Barbara Gillian Briggs la elevaron al género Acidonia, transfiriéndole la especie Persoonia. Posteriormente, Acidonia fue cambiada para incluir solo a A. microcarpa. Fue publicado en Botanical Journal of the Linnean Society 70: 175. 1975.
Referencias
Proteaceae
Flora de Australia Occidental
Plantas descritas en 1975
Plantas descritas por Robert Brown
Plantas descritas por L.A.S.Johnson
Plantas descritas por B.G.Briggs | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,148 |
\section{Introduction}
The recent observation of neutrino oscillations and the resulting measurements of the neutrino mass differences has motivated experimental searches for the absolute neutrino mass. Neutrinoless double beta decay ($0\nu\beta\beta$) is the only practical way to understand the nature of neutrino mass and one of the most sensitive probes of its absolute value. Ettore Majorana proposed that neutrinos could be their own anti-particles \cite{Majorana:1937vz}, and this lead to Furry's conclusion \cite{Furry:1939qr} that neutrinoless double beta decay (Figure \ref{fig:betadecays}B) is possible via neutrino exchange if the neutrinos are Majorana particles and have non-zero mass.
\begin{figure}[htp]\centering
\includegraphics[height=8pc]{pics/2b2n0n.png}
\caption{Feynman diagrams of a $2\nu\beta\beta$ (A) decay allowed in Standard Model, and $0\nu\beta\beta$ (B) decay allowed if neutrinos are massive and Majorana particles.} \label{fig:betadecays} \end{figure}
The effective Majorana neutrino mass $\langle m_{\beta\beta}\rangle$ is proportional to the square root of the $0\nu\beta\beta$ decay half-life $T_{1/2}^{0\nu}$ in equation (\ref{equ:2b0n}), where $G^{0\nu}$ is the kinematic phase-space factor and $M_{0\nu}$ is the nuclear matrix element. The experimental signature of $0\nu\beta\beta$ is two electrons with the energy sum equaling the $Q_{\beta\beta}$ of the decay. There are other mechanisms to explain neutrinoless double beta decay \cite{Avignone:2007fu}, but the above mechanism is the most favored due to the minimal required modifications to the Standard Model.
\begin{equation}
[T_{1/2}^{0\nu}]^{-1} = G^{0\nu} \vert M_{0\nu} \vert ^{2} \langle m_{\beta\beta}\rangle^{2} \label{equ:2b0n} \end{equation}
\section{SuperNEMO Detector}
SuperNEMO is $\sim$100~kg source isotope ($^{82}$Se or $^{150}$Nd), tracker + calorimeter detector with a projected neutrinoless double beta decay half-life sensitivity of $10^{26}$ years ($\sim50$~meV effective Majorana neutrino mass). The SuperNEMO baseline design calls for 20 modules ($\sim 4\times 2\times 1$~m), each holding 5 kg of source isotope. Both sides of the source foil have 9 layers of Geiger mode drift cells enclosed by the calorimeter walls. Each module will hold $\sim$600 8" PMTs.
The project is currently in a 3 year design study and R\&D phase and the collaboration comprises over 90 physicists from 12 countries. The R\&D program focuses on four main areas of study: isotope enrichment, tracking detector, calorimeter, and ultra-low background materials production and measurements. The goals of the R\&D are summed up in Table \ref{table:goals}.
\begin{table}[htp] \centering
\caption{SuperNEMO Parameters and Goals} \label{table:goals} \begin{tabular}{ll} \br
Parameters & Goals \\
\mr
Isotope & $^{82}$Se (or $^{150}$Nd) \\
Mass & 100--200 kg \\
$0\nu\beta\beta$ Detection Efficiency & $30\%$ \\
Energy Resolution FWHM at 3 MeV & $4\%$ \\
$^{214}$Bi Source Purity & $<10~\mu$Bq/kg \\
$^{208}$Tl Source Purity & $<2~\mu$Bq/kg \\
Operation Time & 5 years \\
$T^{0\nu\beta\beta}_{1/2}$ Sensitivity & $10^{26}$ years \\
Effective Majorana Mass $\langle m_{\beta\beta}\rangle$ & 50--100 meV \\
\br \end{tabular} \end{table}
The significance of energy resolution is best illustrated by the half-life sensitivity formula \cite{Avignone:2005cs}. This formula (\ref{equ:sens}) has limitations in accurately predicting the sensitivity of the specific SuperNEMO detector, but does demonstrate the significance of energy resolution. The energy resolution $\Delta E$ factors in with equal importance as isotope mass $M$, runtime $t$, and number of background events $N_{bkg}$. Factors $N_{A}$ and $A$ are Avogadro's number and atomic mass of the isotope and $\varepsilon$ and $\kappa_{CL}$ are the detector efficiency and the confidence level on the half-life sensitivity $T_{1/2}$. The dominating background to $0\nu\beta\beta$ is the irreducible $2\nu\beta\beta$ channel, therefore the energy resolution of the calorimeter becomes the dominating parameter determining the detector's overall sensitivity to neutrinoless double beta decay.
\begin{equation}
T_{1/2} \propto \ln 2 \cdot \frac{N_{A}}{A} \cdot \frac{\varepsilon}{\kappa_{CL}} \cdot \sqrt{\frac{M \cdot t}{N_{bkg} \cdot \Delta E}} \label{equ:sens} \end{equation}
Simulations done for $^{82}$Se with a projected calorimeter energy resolution of 12\% and 8\% FWHM at 1 MeV and normalized to $10^{26}$ year $0\nu\beta\beta$ half-life, clearly displays the importance of energy resolution for this experiment (Figure \ref{fig:resolution}). At 12\% energy resolution, the high energy tail from the $2\nu\beta\beta$ energy spectrum overlaps the $0\nu\beta\beta$ peak, but at 8\% energy resolution there is separation.
\begin{figure}[htp]\centering
\includegraphics[height=10pc]{pics/12-percent.png}\hspace{3pc}
\includegraphics[height=10pc]{pics/8-percent.png}
\caption{Simulations for 500 kg$\cdot$yr $^{82}$Se. The $0\nu\beta\beta$ half-life (RED) is normalized to $10^{26}$ years. Expectations for energy resolutions 12\% (left) and 8\% (right) $\frac{\Delta E}{E}$ FWHM at 1 MeV. } \label{fig:resolution} \end{figure}
\section{Calorimetry Goals of SuperNEMO}
The calorimeter R\&D is subdivided into three main groups: energy and time resolution studies, calibration, and PMT radio-purity. The energy resolution R\&D is the main focus of this report. As with all PMT based calorimeters, PMT gain stability and linearity must be both intrinsically good and experimentally well understood to ensure the accurate reconstruction of data. Conventional LASER/LED configurations prove difficult with many channels. A promising alternative method is one photo-electron peak monitoring \cite{Asch:2005pe} because the PMT gain can be extracted independent of light amplitude. The R\&D also investigates the use a low activity alpha source embedded into the plastic scintillator as a means to monitor the gain.
Specific to low background counting experiments, ultra-pure materials must be used throughout the detector. The PMTs are one of the main sources of contamination with emphasis on the purity of the cathode glass which is closest to the active volume of the detector. The Barium salt used to make conventional glass is chemically the same as Radium, and therefore very difficult to purify during the production of the glass. Various manufactures have developed recipes for low-background glasses, but the requirements of SuperNEMO have motivated this development to a new level of radio-purity. Photonis has provided preliminary samples of their new ultra-pure glass that have met R\&D requirements.
\subsection{Energy Resolution}
Optimization of the energy resolution is the result of a high number of photo-electrons which reduces the statistical error $1/\sqrt{N_{pe}}$. This can be simplified into three experimental objectives which are described by formula (\ref{equ:photons}).
\begin{equation}
\frac{N_{ph}}{E_{e}} \cdot \varepsilon_{col}^{light} \cdot \left(QE^{PMT} \cdot \varepsilon_{col}^{PMT}\right)= N_{pe} \label{equ:photons} \end{equation}
$N_{ph}/E_{e}$ is the number of photons per unit energy and is determined by the scintillator light output. $\varepsilon_{col}^{light}$ is the light collection efficiency and depends upon: scintillator geometry, transparency, reflector efficiency, optical coupling quality, etc. Intrinsic characteristics of the PMT include the quantum efficiency of the photo-cathode $QE^{PMT}$, and the cathode to first dynode collection efficiency $\varepsilon_{col}^{PMT}$. There has been a significant breakthrough in development new high quantum efficiency PMTs based on bi-alkali photocathodes by Hamamatsu and Photonis. The SuperNEMO group is working very closely with PMT manufacturers on characterizing these new photo-detectors which have now a QE in the range of 35--43\% at the peak wavelength (to be compared with $\sim$25\% QE for "conventional" photo-multipliers). Assuming that the energy resolution of the scintillator detector is mainly determined by the photon statistics we can express the resolution in terms of the number of collected photo-electrons (\ref{equ:res1}).
\begin{equation}
\frac{\Delta E}{E}=\frac{FWHM}{E}=\frac{2.35\sigma}{E}=\frac{2.35}{\sqrt{N_{pe}}} \label{equ:res1} \end{equation}
The scintillator must be a low Z material to minimize backscattering electrons and has to have a good timing resolution (a coincidence time resolution of $\sigma$ = 250~ps at 1 MeV is required). It has to be cost effective and radio-pure. These requirements essentially rule out many popular non-organic scintillator, such as NaI(Tl), CsI(Tl), CaF$_{2}$(Eu) etc. which would otherwise provide a good energy resolution due to their high light output. The choice of reflective material is also limited to low density reflectors to reduce electron energy loss through the material.
\section{Experimental Setup}
The energy resolution measurement is carried out by exciting the scintillator under test with a flux of electrons of known energy and then analyzing the resulting distribution. The mono-chromatic source of electrons approximates the delta function and therefore any smearing of the distribution is due to the light collection of the scintillator and PMT under study. The test setup can be broken into three subcategories: the calorimeter block (scintillator + reflector + lightguide + PMT), the electron source, and the data acquisition (DAQ).
\subsection{Calorimeter Block}
Many different scintillator, reflector, and PMT combinations are being studied. Solid scintillator candidates include polystyrene (PST) based scintillators from ISM and JINR labs (1.5\% PTP, 0.0175\% POPOP) and polyvinyltoluene (PVT) based scintillators from Bicron (BC404, BC408) and Eljen (EJ204, EJ200) manufacturers. Liquid scintillators are toluene based and from CENBG, INR, ISM, and JINR labs (0.5\% PPO, 0.0025\% POPOP). Various specular and diffusive reflectors being tested include: Teflon, Kapton, Aluminized Mylar, and Enhanced Specular Reflector (ESR) from the Vikuiti and ReflechTech manufactures. The three PMT competitors are Hamamatsu, Photonis, and Electron Tubes Ltd. (ETL).
\subsection{Electron Source}
There are two methods used to obtain a mono-chromatic source of electrons. The first method is simplest to implement as one uses the K-shell 976 keV conversion electrons (CE) from a $^{207}$Bi source. The drawback to this method is that the fitting function needs to incorporate the convolution of additional x-rays, gammas, L-shell and M-shell conversion electrons. The second method is more involved to set up, but in principle leaves a spectrum that can be easily fit with a Gaussian function. The $\beta$ emission from a highly active $^{90}$Sr source is passed through a magnetic field so that $\beta$'s of a particular energy can be selected. For the energy resolution measurements, 1 MeV electrons are used.
\subsection{Data Collecting and Analysis}
Data acquisition is accomplished with a gated QDC (charge to digital converter). The PMT signal is split in two, half the signal is used for triggering of the electronics and generating the gate signal for the QDC, the other half of signal goes directly to the QDC after some passive delay to match the timing of the electronics. In the method of the $^{207}$Bi source, three different data runs must be taken to obtain a pedestal, an energy spectrum of just the gammas (achieved by shielding out the electrons with 2 mm of Aluminum) and the energy spectrum of the gammas + CEs (conversion electrons). The Compton edges from the gamma distribution are sufficiently described by a modified Heaviside step-function. The free parameters of the gamma distribution are determined and then fixed while the gamma + CEs distribution is fit. The CEs are a sum of three Gaussian distributions from the K, L, and M shells.
\section{Measurements}
The calorimeter baseline design calls for 8" diameter PMTs, but as a check of physical limitations on achievable energy resolution, a detailed study of small (3") PMTs preceded. A resolution of 6.5\% FWHM at 1 MeV was measured using Bicron BC404 scintillator, wrapped in Vikuiti ESR (Enhanced Specular Reflector), mounted on a 3" Hamamatsu Super-Bialkali type PMT (Figure \ref{fig:best}). Using (\ref{equ:res1}), this extrapolates to 3.8\% at 3~MeV which is better than the goal of 4\% stipulated by the R\&D. This is an unprecedented result for plastic scintillators. Proving there are no physical limitations to reaching the 4\% level, the challenge then becomes scaling up the PMT and scintillator size while maintaining resolution.
\begin{figure}[htp] \centering
\includegraphics[height=12pc]{pics/best_bi207-fit.png}
\caption{The fit to data (RED line) results in 6.5\% FWHM at 1 MeV.} \label{fig:best} \end{figure}
\subsection{Light Collection Simulations with GEANT4}
Extensive optical simulations were carried out in GEANT4 with all inputs being wavelength dependent, experimental measurements including: POPOP absorption and re-emission (Stokes Shifting), PMT QE, scintillator bulk absorption and emission, material indexes of refraction, and material reflectivities. The simulations revealed sensitive parameters of the setup. Polishing the side of the lightguide to give specular internal reflection as well as wrapping the lightguide with a specular reflector yielded a 2--3\% improvement in the expected resolution. The simulations gave expected resolutions of 7.5\% and 7.7\% for the 8" and 11" PMTs with lightguide wrapped in ESR and $5\times 5\times 2$~cm BC404 scintillator wrapped in ESR.
After optimizing the sensitive parameters in our experimental setup, the resolution measurements were 1--2\% worse than the expectations from simulation. Current suspects awaiting investigation are photo-cathode QE uniformity and cathode to first dynode collection efficiency in the presence of Earth's natural magnetic field. Earth's magnetic field is known to influence large ($>$5") PMTs collection efficiency with non-negligible effects, and the effect increases with PMT diameter. These are both characteristics of the PMT which change from one PMT to another and are therefore difficult to simulate accurately.
\subsection{Solid Scintillator Measurements}
Large solid scintillator blocks are an ideal candidate for SuperNEMO because of low cost, high radio-purity, decreased number of channels, and the physical simplicity of the setup. Three possible variations under study are: small ($<$5") PMT with flat cathode window with scintillator coupled directly to cathode window (Figure \ref{fig:options}A), large ($>$8") PMT with hemispherical cathode window with scintillator coupled to concave lightguide (Figure \ref{fig:options}B), coupled to cathode window, and large ($>$8") PMT with hemispherical cathode window with concave scintillator coupled directly to cathode window (Figure \ref{fig:options}C). Table \ref{table:solids} summarizes the best measurements for these configurations.
\begin{table}[htp] \centering \caption{Measurements with the Solid Scintillator Setup} \label{table:solids} \begin{tabular}{lll} \br
Scintillator Dimensions & PMT Diameter & FWHM \\
and Type & and Make & at 1 MeV \\
\mr
$5\times 5\times 2$~cm BC404 & 3" Hamamatsu SBA & 6.5\% \\
$9\times 9\times 2$~cm BC408 & 8" Hamamatsu SBA with Lightguide & 10.1\% \\
$14\times 14\times 2$~cm BC404 & 8" Electron Tubes Ltd. with Lightguide & 9.2\% \\
$15\times 15\times 2$~cm BC408 & 8" Hamamatsu SBA with Lightguide & 10.3\% \\
$20\times 2$~cm (hexagonal) BC408 & 8" Hamamatsu SBA with Lightguide & 11.2\% \\
$\varnothing~20\times 2$~cm PST & 8" Photonis & 7.5\% \\
$\varnothing~20\times 10$~cm PST & 8" Photonis & 8.2\% \\
\br \end{tabular} \end{table}
\subsection{Liquid Scintillator Measurements}
In parallel an R\&D program on liquid scintillator detectors is being carried out. The motivations for using liquid scintillator are the following: lower cost, no lightguide needed to couple to hemispherical PMT cathode, larger active volume increases gamma tagging efficiency for background rejection, good uniformity, and high radio-purity. The dominating drawback is the mechanical engineering of the containment structure and meeting safety requirements for an underground laboratory. Two main variations on the setup are under study: a semi-conical setup where the diameter of the liquid scintillator surface is larger than that of the PMT (Figure \ref{fig:options}D), and a cylindrical setup where the diameter of the liquid scintillator surface matches that of the PMT (Figure \ref{fig:options}E). Table \ref{table:liquids} summarizes the best liquid scintillator measurements.
\begin{figure}[htp] \centering
\includegraphics[height=8pc]{pics/scint_options.png}
\caption{Configurations for the solid and liquid scintillator setup.} \label{fig:options} \end{figure}
\begin{table}[htp] \centering \caption{Measurements with the Liquid Scintillator Setup} \label{table:liquids} \begin{tabular}{lll} \br
Scintillator & PMT Diameter & FWHM \\
Dimensions & and Make & at 1 MeV \\
\mr
$\varnothing~7.6\times 2$~cm & 3" Photonis & 7.6\% \\
$\varnothing~7.6\times 10$~cm & 3" Photonis & 8.0\% \\
$\varnothing~8.4\times 9.2$~cm & 5" Photonis & 7.3\% \\
$\varnothing~20.3\times 20$~cm & 8" Photonis & 11.3\% \\
$23\times 9.2$~cm (hexagonal) & 5" Electron Tubes Ltd. & 10.8\% \\
\br \end{tabular} \end{table}
\subsection{Liquid + Solid Hybrid Measurements}
The mechanical engineering of the liquid scintillator containment is challenging. A thin film entrance window with low density and low Z must be used to minimize electron energy losses. An alternative approach is to use a so-called active window where solid scintillator is used on the containment face. This approach utilizes the liquid scintillator as the lightguide and increases the active volume for gamma tagging efficiency. Table \ref{table:hybrids} summarizes the best hybrid measurements.
\begin{table}[htp] \centering \caption{Measurements with the Liquid + Solid Hybrid Setup} \label{table:hybrids} \begin{tabular}{lll} \br
Liquid / Solid Scintillator & PMT Diameter & FWHM \\
Dimensions & and Make & at 1 MeV \\
\mr
$23\times 9.2$~cm (hexagonal) / $5\times 5\times 2$~cm & 5" Electron Tubes Ltd. & 12.3\% \\
$23\times 9.2$~cm (hexagonal) / $23\times 2$~cm (hexagonal) & 5" Electron Tubes Ltd. & 15.1\% \\
\br \end{tabular} \end{table}
\subsection{Long Bar Scintillator Measurements}
The detector "floor-space" requirement can drastically be reduced by implementing the long scintillator bar design. In this configuration, 2 meter scintillator bars span the tracker volume with a PMT coupled to each end of the bar. This configuration is also the cheapest because of the drastically reduced number of PMTs and the reduced floor-space required from an underground laboratory. With the timing from the two PMTs, an impact resolution of 1--2 cm (along the bar length) is achievable and this information is of additional use for background rejection. Moreover, due to a significantly reduced mass of PMT glass and their relatively remote locations from the detector fiducial volume, the bar design should have a much lower background from PMTs which is one of the main background sources of SuperNEMO. Table \ref{table:bars} summarizes the bar scintillator measurements so far.
A resolution of 7\% at 1 MeV is probably impossible to reach with 2m bars. Thus the crucial question for feasibility of this design is whether a better background rejection and higher detection efficiency compensate a worse energy resolution. Rough estimates show that it might be a valid option if a resolution of 10--11\% is achievable with the bars. Extensive physics simulations are under way to answer this question with certainty. In the meantime measurements are being carried out with high QE PMTs and optimized geometry to reach the best possible resolution with scintillator bars.
\begin{table}[htp] \centering \caption{Measurements with the Bar Scintillator Setup}\label{table:bars} \begin{tabular}{lll} \br
Scintillator Dimensions & PMT Diameter & FWHM \\
and Type and Reflector & and Make & at 1 MeV \\
\mr
$200\times 10\times 1.25$~cm / BC408 / Al. Mylar & 3" Hamamatsu SBA & 12.9\% \\
$200\times 10\times 1.25$~cm / BC408 / Al. Mylar & 3" Hamamatsu SBA with Lightguide & 13.6\% \\
$200\times 10\times 1.25$~cm / BC408 / ESR & 3" Hamamatsu SBA & 12.9\% \\
$200\times 10\times 1.25$~cm / BC408 / ESR & 5" Electron Tubes Ltd. & 13.7\% \\
\br \end{tabular} \end{table}
\section{Summary and Future Plans}
Exceptional resolutions of 6.5\% at 1 MeV were measured for small PVT scintillators coupled to high QE PMTs. The SuperNEMO baseline design calls for large scintillator blocks ($\varnothing~20\times 10$~cm). Scintillators of this size read out through a lightguide showed an energy resolution of 9--10\% at 1 MeV. Better results have been achieved by casting a large plastic scintillator directly on a hemispherical 8" PMT. With this configuration we have been able to reach the important milestone of 7--8\% $1/\sqrt{E}$ MeV energy resolution for the baseline detector design. Consequently the R\&D on solid scintillators will be focusing on cast scintillator solutions rather than lightguides to increase the light collection efficiency. The development program will also move away from the previous square-block designs and focus on more realistic hexagonal scintillator geometries. We note that there is room for further improvements by using a higher QE PMTs and more efficient scintillators.
Liquid scintillator provides an alternative while maintaining good resolution (7--8\% at 1 MeV) and improving gamma tagging efficiency, but achieving the required resolution with large blocks as well as the engineering of the mechanical design and safety remain a challenge. The hybrid solution creates a more robust containment setup for the liquid, but achieving $<$7\% is very challenging. Long scintillator bars design can potentially give a more efficient detector with more background rejection power. It will drastically reduce the number of PMTs and facilitate a more compact detector design. Measurements so far yield 12--13\% at 1 MeV. Work is in progress to understand if this resolution can be improved to 10\% and whether a worse energy resolution can be compensated by the above advantages of this detector configuration.
Last years have seen a significant progress in development of novel photo-detectors. PMTs with a QE of over 40\% are now available. Using the latest achievements in PMT, reflector, and scintillator technology the SuperNEMO collaboration has demonstrated the feasibility of achieving the target energy resolution necessary to reach the sensitivity goal of the experiment. The remaining challenge is to demonstrate that the achieved energy resolution can be maintained at the mass production scale. SuperNEMO expects to make the final decision on the calorimeter design in mid-2009. The large scale construction will start in 2011 with the aim to reach the target sensitivity of $\langle m_{\beta\beta}\rangle$~=~50--100~meV by 2017.
\section*{References}
\medskip
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,334 |
import argparse, json
import simpleamt
if __name__ == '__main__':
parser = argparse.ArgumentParser(parents=[simpleamt.get_parent_parser()])
parser.add_argument('-f', action='store_true', default=False)
args = parser.parse_args()
mtc = simpleamt.get_mturk_connection_from_args(args)
approve_ids = []
reject_ids = []
if args.hit_ids_file is None:
parser.error('Must specify --hit_ids_file.')
with open(args.hit_ids_file, 'r') as f:
hit_ids = [line.strip() for line in f]
for hit_id in hit_ids:
try:
assignments = mtc.get_assignments(hit_id)
except:
continue
for a in assignments:
if a.AssignmentStatus == 'Submitted':
try:
# Try to parse the output from the assignment. If it isn't
# valid JSON then we reject the assignment.
output = json.loads(a.answers[0][0].fields[0])
approve_ids.append(a.AssignmentId)
except ValueError as e:
reject_ids.append(a.AssignmentId)
else:
print "hit %s has already been %s" % (str(hit_id), a.AssignmentStatus)
print ('This will approve %d assignments and reject %d assignments with '
'sandbox=%s' % (len(approve_ids), len(reject_ids), str(args.sandbox)))
print 'Continue?'
if not args.f:
s = raw_input('(Y/N): ')
else:
s = 'Y'
if s == 'Y' or s == 'y':
print 'Approving assignments'
for idx, assignment_id in enumerate(approve_ids):
print 'Approving assignment %d / %d' % (idx + 1, len(approve_ids))
mtc.approve_assignment(assignment_id)
for idx, assignment_id in enumerate(reject_ids):
print 'Rejecting assignment %d / %d' % (idx + 1, len(reject_ids))
mtc.reject_assignment(assignment_id, feedback='Invalid results')
else:
print 'Aborting'
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,190 |
Q: Rotate image and after crop it with Jcrop.js plugin for phonegap Im working in a app that I need to manipulate images. First I need to rotate the image that I get from the camera or gallery. Secondly I need to call to Jcrop.js from this rotated image.
Any ideas, please?
Regards
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 303 |
The 2020–21 Southeastern Louisiana Lions basketball team represented Southeastern Louisiana University during the 2020–21 NCAA Division I men's basketball season. The Lions were led by second-year head coach David Kiefer, and played their home games at the University Center in Hammond, Louisiana as members of the Southland Conference. In a season limited due to the ongoing COVID-19 pandemic, the Lions finished the 2020–21 season 8–18, 5–10 in Southland play to finish in ninth place. They defeated McNeese State in the first round of the Southland tournament before losing to New Orleans.
Previous season
The Lions finished the 2019–20 season 8–23, 5–15 in Southland play to finish in a tie for 11th place. They failed to qualify for the Southland tournament.
Roster
Schedule and results
|-
!colspan=9 style=| Non-conference Regular season
|-
!colspan=9 style=| Southland Regular season
|-
!colspan=9 style=| Southland tournament
Source:
References
Southeastern Louisiana
Southeastern Louisiana Lions basketball seasons
Southeastern Louisiana Lions basketball
Southeastern Louisiana Lions basketball | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,034 |
<HTML>
<!--
Copyright (c) The Trustees of Indiana University
Use, modification and distribution is subject to the Boost Software
License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt)
Authors: Douglas Gregor
Andrew Lumsdaine
-->
<Head>
<Title>Boost Graph Library: Small World Generator</Title>
<script language="JavaScript" type="text/JavaScript">
<!--
function address(host, user) {
var atchar = '@';
var thingy = user+atchar+host;
thingy = '<a hre' + 'f=' + "mai" + "lto:" + thingy + '>' + user+atchar+host + '</a>';
document.write(thingy);
}
//-->
</script>
</head>
<BODY BGCOLOR="#ffffff" LINK="#0000ee" TEXT="#000000" VLINK="#551a8b"
ALINK="#ff0000">
<IMG SRC="../../../boost.png"
ALT="C++ Boost" width="277" height="86">
<tt>small_world_iterator</tt>
<br>
<PRE>
template<typename RandomGenerator, typename Graph>
class small_world_iterator
{
public:
typedef std::input_iterator_tag iterator_category;
typedef std::pair<vertices_size_type, vertices_size_type> value_type;
typedef const value_type& reference;
typedef const value_type* pointer;
typedef void difference_type;
small_world_iterator();
small_world_iterator(RandomGenerator& gen, vertices_size_type n,
vertices_size_type k, double probability = 0.,
bool allow_self_loops = false);
// Iterator operations
reference operator*() const;
pointer operator->() const;
small_world_iterator& operator++();
small_world_iterator operator++(int);
bool operator==(const small_world_iterator& other) const;
bool operator!=(const small_world_iterator& other) const;
};
</PRE>
<p> This class template implements a generator for small-world graphs,
suitable for initializing an <a
href="adjacency_list.html"><tt>adjacency_list</tt></a> or other graph
structure with iterator-based initialization. A small-world graph
consists of a ring graph (where each vertex is connected to its
<em>k</em> nearest neighbors). Edges in the graph are randomly
rewired to different vertices with a probability
<em>p</em>. Small-world graphs exhibit a high clustering coefficient
(because vertices are always connected to their closest neighbors),
but rewiring ensures a small diameter.</p>
<h3>Where Defined</h3>
<a href="../../../boost/graph/small_world_generator.hpp"><tt>boost/graph/small_world_generator.hpp</tt></a>
<h3>Constructors</h3>
<a name="default-constructor"/>
<pre>small_world_iterator();</pre>
<blockquote>
Constructs a past-the-end iterator.
</blockquote>
<pre>
small_world_iterator(RandomGenerator& gen, vertices_size_type n,
vertices_size_type k, double probability = 0.,
bool allow_self_loops = false);
</pre>
<blockquote>
Constructs a small-world generator iterator that creates a
graph with <tt>n</tt> vertices, each connected to its <tt>k</tt>
nearest neighbors. Probabilities are drawn from the
random number generator <tt>gen</tt>. Self-loops are permitted only
when <tt>allow_self_loops</tt> is <tt>true</tt>.
</blockquote>
<H3>Example</H3>
<pre>
#include <boost/graph/adjacency_list.hpp>
#include <boost/graph/small_world_generator.hpp>
#include <boost/random/linear_congruential.hpp>
typedef boost::adjacency_list<> Graph;
typedef boost::small_world_iterator<boost::minstd_rand, Graph> SWGen;
int main()
{
boost::minstd_rand gen;
// Create graph with 100 nodes
Graph g(SWGen(gen, 100, 6, 0.03), SWGen(), 100);
return 0;
}
</pre>
<br>
<HR>
<TABLE>
<TR valign=top>
<TD nowrap>Copyright © 2005</TD><TD>
<A HREF="http://www.boost.org/people/doug_gregor.html">Doug Gregor</A>, Indiana University (<script language="Javascript">address("cs.indiana.edu", "dgregor")</script>)<br>
<A HREF="https://homes.cs.washington.edu/~al75">Andrew Lumsdaine</A>,
Indiana University (<script language="Javascript">address("osl.iu.edu", "lums")</script>)
</TD></TR></TABLE>
</BODY>
</HTML>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,746 |
Q: Keras predict diabetic retiopathy I'm trying to predict a diabetic retiopathy by using densenet121 model from keras.
I have a 5 folder that contain 0:3647 images 1:750 images 2:1105 images 3:305 images 4:193 images
train data have 6000 image
and validate data have 1000 image and test data have 25 to test a little bit
I use keras imagedatagenerator to preprocess image and augmented it,size of image is (224,224)
def HE(img):
img_eq = exposure.equalize_hist(img)
return img_eq
train_datagen = ImageDataGenerator(
rescale=1./255,
rotation_range=90,
width_shift_range=0,
height_shift_range=0,
shear_range=0,
zoom_range=0,
horizontal_flip=True,
fill_mode='nearest',
preprocessing_function=HE,
)
validation_datagen = ImageDataGenerator(
rescale=1./255
)
test_datagen = ImageDataGenerator(
rescale=1./255
)
train = train_datagen.flow_from_directory(
'train/train_deep/',
target_size=(224,224),
color_mode='grayscale',
class_mode='categorical',
batch_size = 20,
)
test = test_datagen.flow_from_directory(
'test_deep/',
batch_size=1,
target_size = (224,224),
color_mode='grayscale',
)
val = validation_datagen.flow_from_directory(
'train/validate_deep/',
target_size=(224,224),
color_mode='grayscale',
batch_size = 20,
)
I use a densenet121 model from keras to compile
model = DenseNet121(include_top=True, weights=None, input_tensor=None,
input_shape=(224,224,3), pooling=None, classes=5)
model.compile(loss='categorical_crossentropy',
optimizer='rmsprop',
metrics=['accuracy'])
model.summary()
filepath="weights-improvement-{epoch:02d}-{val_loss:.2f}.hdf5"
checkpointer = ModelCheckpoint(filepath,monitor='val_loss', verbose=1,
save_best_only=True,save_weights_only=True)
lr_reduction = ReduceLROnPlateau(monitor='val_loss', patience=5, verbose=2,
factor=0.5)
callbacks_list = [checkpointer, lr_reduction]
history = model.fit_generator(
train,
epochs=Epoch,
validation_data=val,
class_weight={0:1, 1:10.57, 2:4.88, 3:29, 4:35},
use_multiprocessing = False,
workers = 16,
callbacks=callbacks_list
)
but when I try to predict
#predict
pred=model.predict_generator(test,
steps=25,)
print(pred)
They predict all are 3
my predict image
problems that I am facing.
1.I try to change a weight of my image because it a imbalance data but, it still doesn't work:
2.Estimate time use 6-7 minutes per epoch that take too much time if I want to train more epoch like 50 epoch what should I do??
Edit
1. I print an array of my 25 predict image and they show
[[0.2718658 0.21595034 0.29440382 0.12089088 0.0968892 ]
[0.2732306 0.22084573 0.29103383 0.11724534 0.0976444 ]
[0.27060518 0.22559224 0.2952135 0.11220136 0.09638774]
[0.27534768 0.21236925 0.28757185 0.12544192 0.09926935]
[0.27870545 0.22124214 0.27978882 0.11854914 0.1017144 ]
[0.2747815 0.22287942 0.28961015 0.11473729 0.09799159]
[0.27190813 0.22454649 0.29327467 0.11331796 0.09695279]
[0.27190694 0.22116153 0.27061856 0.12831333 0.10799967]
[0.27871644 0.21939436 0.28575435 0.11689039 0.09924441]
[0.27156618 0.22850358 0.27458736 0.11895953 0.10638336]
[0.27199408 0.22443996 0.29326025 0.11337796 0.09692782]
[0.27737287 0.22283535 0.28601763 0.11459836 0.09917582]
[0.2719294 0.22462222 0.29477262 0.11228184 0.09639395]
[0.27496076 0.22619417 0.24634513 0.12380602 0.12869397]
[0.27209386 0.23049556 0.27982628 0.11399914 0.10358524]
[0.2763851 0.22362126 0.27667257 0.11974224 0.10357884]
[0.28445077 0.22687359 0.22116113 0.12310001 0.14441448]
[0.27552167 0.22341767 0.28794768 0.11433118 0.09878179]
[0.27714184 0.22157396 0.26033664 0.12819317 0.11275442]
[0.27115697 0.22615613 0.29698634 0.10981857 0.09588206]
[0.27108756 0.22484282 0.29557163 0.11230227 0.09619577]
[0.2713721 0.22606659 0.29634616 0.11017173 0.09604342]
[0.27368984 0.22699612 0.28083235 0.11586079 0.10262085]
[0.2698808 0.22924589 0.29770645 0.10761821 0.0955487 ]
[0.27016872 0.23090932 0.2694938 0.11959692 0.1098313 ]]
I see some image are in 0 but they show 3 in all prediction,So why it show that?
2. I change some line of code a little bit in model densenet121 , I remove a external top layer and change a predict code for more easy to see.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,808 |
{"url":"http:\/\/math.stackexchange.com\/questions\/118490\/holomorphic-function-of-a-matrix","text":"# Holomorphic function of a matrix\n\nA statement is made below. The questions are:\n\n(a) Is the statement true?\n\n(b) If it is, does it appear in the literature?\n\nHere is the statement.\n\nFor any matrix $A$ in $M_n(\\mathbb C)$, write $\\Lambda(A)$ for the set of eigenvalues of $A$.\n\nRecall that there is a unique continuous $\\mathbb C[X]$-algebra morphism $$\\mathcal O(\\Lambda(A))\\to M_n(\\mathbb C),$$ where $\\mathcal O(\\Lambda(A))$ is the algebra of those functions which are holomorphic on (some open neighborhood of) $\\Lambda(A)$. Recall also that this morphism is usually denoted by $f\\mapsto f(A)$. (Here $X$ is an indeterminate.)\n\nLet $U$ be an open subset of $\\mathbb C$, let $U'$ be the open subset of $M_n(\\mathbb C)$ defined by the condition $$\\Lambda(A)\\subset U,$$ and let $f$ be holomorphic on $U$. (The fact the $U'$ is open follows from Rouch\u00e9's Theorem.)\n\nSTATEMENT. The map $A\\mapsto f(A)$ from $U'$ to $M_n(\\mathbb C)$ is holomorphic.\n\n-\n\nFor any matrix $a$ in $A:=M_n(\\mathbb C)$, write $\\Lambda(a)$ for the set of eigenvalues of $a$. Let $U$ be an open subset of $\\mathbb C$, and let $U'$ be the subset of $A$, which is open by Rouch\u00e9's Theorem, defined by the condition $\\Lambda(a)\\subset U$. Let $a$ be in $U'$, let $X$ be an indeterminate, and let $\\mathcal O(U)$ be the $\\mathbb C$-algebra of holomorphic functions on $U$. Equip $\\mathcal O(U)$ and $\\mathbb C[a]$ with the $\\mathbb C[X]$-algebra structures associated respectively with the element $z\\mapsto z$ of $\\mathcal O(U)$ and the element $a$ of $\\mathbb C[a]$.\nTheorem. (i) There is a unique $\\mathbb C[X]$-algebra morphism from $\\mathcal O(U)$ to $\\mathbb C[a]$. We denote this morphism by $f\\mapsto f(a)$.\n(ii) There is an $r>0$ and a neighborhood $N$ of $a$ in $A$ such that $$f(b)=\\frac{1}{2\\pi i}\\ \\sum_{\\lambda\\in\\Lambda(a)}\\ \\int_{|z-\\lambda|=r}\\ \\frac{f(z)}{z-b}\\ dz$$ for all $f$ in $\\mathcal O(U)$ and all $b$ in $N$. In particular the map $b\\mapsto f(b)$ from $U'$ to $A$ is holomorphic.\nProof. By the Chinese Remainder Theorem, $\\mathbb C[a]$ is isomorphic to the product of $\\mathbb C[X]$-algebras of the form $\\mathbb C[X]\/(X-\\lambda)^m$, with $\\lambda\\in\\mathbb C$. So we can assume that $\\mathbb C[a]$ is of this form, and (i) is clear. To prove (ii) we can keep on assuming $\\mathbb C[a]\\simeq\\mathbb C[X]\/(X-\\lambda)^m$. On replacing $a$ with $a-\\lambda$, we can even assume $a^n=0$. Choose $r>0$ so that $U$ contains the closed disk of radius $r$ centered at $0$, let $N$ be the set of those $b$ in $A$ whose eigenvalues $\\lambda$ satisfy $|\\lambda|<r\/2$, and let $b$ be in $N$. Replacing $a$ with $b$ in the above argument, we can assume $b^n=0$. Now (ii) follows from Cauchy's Integral Formula and the equalities $$f(b)=\\sum_{k=0}^{n-1}\\ \\frac{f^{(k)}(0)}{k!}\\ b^k,\\quad \\frac{1}{z-b}=\\sum_{k=0}^{n-1}\\ \\frac{b^k}{z^{k+1}}\\quad.$$","date":"2014-04-20 08:51:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9932055473327637, \"perplexity\": 36.559321813201635}, \"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-2014-15\/segments\/1397609538110.1\/warc\/CC-MAIN-20140416005218-00655-ip-10-147-4-33.ec2.internal.warc.gz\"}"} | null | null |
This afternoon I am heading down to Anaheim - Los Angeles area for the non-Californians out there, for the CDS Annual Meeting.
I've never really had a reason to attend before, but this year, Speedy and I are both receiving an award. This seemed like a good year to see what it's all about.
Saturday's schedule is filled with chapter meetings. Since my chapter chair isn't attending, I asked if I could go as a representative. The Tehachapi Mountain Chapter of CDS is pretty small, but they work hard to put on a summer show series and yearly banquet, both with great ribbons and awards. I think TMC has plenty to share with the rest of the chapters.
That same night, there's a banquet where awards are handed out. That's really the reason I am going. Speedy and I worked hard to earn both the Ruby Rider Award and the Second Level Horse Performance Award. Receiving those are worth going to Anaheim for.
The ruby pin will soon be in my hands.
Sunday's schedule focuses on a health fair. I am not certain that any of the topics pertain to me, but Chemaine Hurtado, owner and trainer at Symphony Dressage Stables, will be there as a vendor sharing her yoga ball lessons. Her daughter and another mutual friend will also be attending so whether the health fair is interesting or not, the four of us are guaranteed to have a great time. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,934 |
{"url":"https:\/\/www.physicsforums.com\/threads\/magnetizing-force.379755\/","text":"# Magnetizing force\n\n1. Feb 19, 2010\n\n### samjesse\n\nHi\n\nHow does magnetizing force of a permanent magnet relate to magnetizing of steal rods of different sizes?\n\ni.e. if I bring a permanent magnet and a piece of steal of the same sizes, put them together, the steal piece becomes magnetized.\nwhat if I bring a much much larger piece of steal for the same size permanent magnet. how much magnetizing force will be in the tip of that big one?\n\nis there a formula which takes size in account?\n\nthx\n\n2. Feb 20, 2010\n\nIt is complicated and all formulas take size into account. The attractive effect of magnets is not straight forward. At least I don't know any better way to calculate forces than to calculate the whole magnetic field. Maybe you will find some rule of thumb in an engineering book.\n\n3. Feb 20, 2010\n\n### Bob S\n\nFor steel, you probably will need H = ~1000 amp-turns per meter. See\n\nhttps:\/\/www.physicsforums.com\/attachment.php?attachmentid=23353&d=1264564310\n\nFor permanent magnet materials (e.g., neodymium), see Fig. 2 on page 7 of\n\nhttp:\/\/www.oersted.com\/magnetizing.PDF\n\nYou will probably need over 800,000 to 3 million amp-turns per meter. The can be done using current pulses.\n\nBob S\n\nadded attachment for magnetizing neodymium magnets. Looks like 3 MA\/m (3 million amps per meter) are required.\nSee\n\n#### Attached Files:\n\n\u2022 ###### Magnetization_neo_Hirst.jpg\nFile size:\n32.8 KB\nViews:\n597\nLast edited: Feb 20, 2010\n4. Feb 23, 2010\n\nA magnet works by adding electrons to the atom to create an unstable molecular structure in a iron molecule witch creates action at a distance. The process of energising the atom in the iron molecule to give an uneven amount of electron, creates fields and poles to keep a stable balance between protons and electron. The magnet attracts more protons to even out the instability.\n\nSaying this, it depends on the excitement of the electrons in the iron molecule witch will affect the energy transferred. but you can only energize a molecule so much until it explodes.\n\n5. Feb 23, 2010\n\n### Bob S\n\nIf I build a magnet that can pull all the protons out of water, will I have only oxygen left?\n\nBob S\n\n6. Feb 24, 2010\n\n### samjesse\n\nIn an inductance curve of a core where the Magnetizing Force in ampere-turns in the horizontal axis and $$A_L{}$$-value = $$\\mu$$ * H\/N$$^{2}$$ in the vertical axis.\nWhat exactly does the value of vertical axis mean?\nand what does $$A_L{}$$-value stand for?\n\nMany thanks\n\n7. Feb 24, 2010\n\n### luke1970\n\nIf a magnet powerful enough to pull the protons from water, then it would also pull the proton from oxygen not just the hydrogen. Maybe this would cause a breaqkdown of the basic atomic structure of the owygen also?","date":"2016-08-26 00:01:08","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.24475334584712982, \"perplexity\": 2108.912806946963}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-36\/segments\/1471982294883.0\/warc\/CC-MAIN-20160823195814-00262-ip-10-153-172-175.ec2.internal.warc.gz\"}"} | null | null |
\section{Introduction}
In the canonical model, T Tauri systems comprise the central star, a
rotating disk of gas and dust, a jet or outflow and possibly a
residual circumstellar envelope (see e.g. \citet{ber89}). In many
cases, the central star is still accreting material and this process,
as well as the mechanisms driving the outflow, are dependent on and
influence the properties of the inner disk ($<$1 AU). Several groups
(e.g. \citet{kon91} and \citet{shu94}) have proposed models in which
the stellar magnetic field truncates the disk at a few stellar radii.
Matter from the disk flows along the field lines and onto the star
producing hot spots or rings that can explain observed ultraviolet
photometric variability \citep{ken94,woo96,gom97}.
In the last several years, the technique of long-baseline infrared
interferometry has been applied to the study of circumstellar material
around young stellar objects. These observations are sensitive to hot
material near the star itself. Given the milliarcsecond resolution
capability of the current generation of interferometers, these
observations can in many cases spatially resolve the emission from the
hot (a few thousand Kelvin) material and are well suited for
observations of the inner regions of young stellar objects. The first
young stellar object to be observed using this technique was FU Ori
\citep{mal98}, followed by Herbig Ae/Be stars \citep{mil99,mil01} and
T Tauri stars \citep{ake00}(hereafter Paper 1). The FU Ori results
were consistent with accretion disk models, while both the T Tauri and
Herbig star results found characteristic sizes larger than expected
from geometrically flat accretion disk models. More recent
observations of Herbigs \citep{eis04} have found earlier spectral type
objects which are consistent with accretion disk predictions.
Measurements of the spectral energy distribution (SED) at optical
through radio wavelengths probe a range of processes in young stellar
objects including the stellar photosphere, accretion onto the star or
disk, emission from gas and dust in the disk and emission from the
outflow. In many sources, continuum emission from circumstellar or
accreting material adds to the stellar spectrum, decreasing the
stellar spectral features in an effect called veiling. For T Tauri
stars, the veiling in the infrared can very high, indicating
substantial excess emission (see e.g \citet{fol99}).
In Paper 1 we presented observations showing that the infrared
emission from the T Tauri stars T~Tau~N and SU~Aur is resolved. The
visibilities from T Tauri stars can be difficult to model given the
substantial stellar component, infrared variability and the possible
presence of a significant extended component. In this paper, we
present further interferometric observations of the T Tauri stars T
Tau N, SU Aur, DR Tau and RY Tau using the Palomar Testbed
Interferometer (PTI) and infrared photometry from the Pomona College 1-meter
telescope. In \S \ref{model}, we present geometric models to constrain
the emission size and orientation. In \S \ref{scatter}, we present
detailed source models which include the scattered light and
reprocessing of starlight and dissipation of viscous accretion
energy in the disk to fit both the SED and the infrared visibilities.
\section{Sources}
All four sources are located in the Taurus-Auriga molecular cloud
(distance $\sim$ 140~pc) and are well studied T Tauri objects. Source
spectral types and stellar properties given in Table \ref{table:source}
are taken from recent references using infrared spectroscopy.
Due to the sensitivity restrictions of PTI, we have chosen sources which are
among the most infrared luminous T Tauri objects. As the PTI
acquisition system works in the optical, there is a selection
effect against highly inclined, optically
obscured sources.
\begin{table}[h!]
\begin{center}
\begin{tabular}{lclll} \hline
Source & Sp Type & L$_{\star}$ (L$_{\odot}$) & R$_{\star}$ (R$_{\odot}$) & Ref. \\ \hline
T Tau N & K0 & 7.3 & 2.8 & \citet{whi01}\\
SU Aur & G2 & 12.9 & 3.5 & \citet{muz03}\\
DR Tau & K7 & 0.87 & 1.9 & \citet{muz03}\\
RY Tau & K1 & 12.8 & 3.6 & \citet{muz03}\\ \hline
\end{tabular}
\caption{Stellar parameters for the observed sources.
\label{table:source}}
\end{center}
\end{table}
All four systems have significant emission in excess of the stellar
photosphere from near infrared through millimeter wavelengths and all
are believed to have circumstellar disks. The T Tau system comprises
the optically visible star T Tau N and its infrared companion T Tau S,
which is itself a binary \citep{kor00}. The PTI observations are of T
Tau N, the component which dominates the millimeter emission
\citep{ake98}. SU~Aur has an SED similar to that of T~Tau~N, although
\citet{her88} classified SU~Aur separately from other T~Tauri's due to
its high luminosity and broad absorption lines. RY Tau is associated
with a reflection nebulosity \citep{nak95} and has millimeter-wave
molecular line emission consistent with a Keplerian disk
\citep{koe95}. DR Tau is one of the most heavily veiled T Tauri stars
and is highly variable in the optical \citep{gul00} and near-infrared
\citep{ken94}.
\section{\bf Observations}
\subsection{Infrared interferometry}
\label{observations}
Infrared interferometry data were taken at the Palomar Testbed
Interferometer (PTI), which is described in detail by \citet{col99}.
PTI is a long-baseline, direct detection interferometer which utilizes
active fringe tracking in the infrared. Data presented here were
obtained in the K band (2.2 $\mu$m) in all three PTI baselines: NS
(110 meter), NW (85 meter) and SW (85 meters). In our analysis below,
we also use the SU~Aur observations described in \citet{ake02} and
Paper 1. A summary of the new observations is given in Table
\ref{table:obs}. These data were acquired over a period from 24
September 2001 to 16 October 2003. The data in the NS and NW
baselines were taken with a 20 millisecond fringe integration time,
while the SW data were taken with a 50 millisecond time, providing
better SNR for these data.
\begin{table}[h!]
\begin{center}
\begin{tabular}{lllllll} \hline
\multicolumn{7}{c}{Observations} \\ \hline
& \multicolumn{2}{c}{NS} & \multicolumn{2}{c}{NW} & \multicolumn{2}{c}{SW} \\
& nights & ints & nights & ints & nights & ints \\
T Tau N & & & & & 1 & 6 \\
SU Aur & & & & & 1 & 6 \\
DR Tau & 3&5 &1 &3 & 1 & 4 \\
RY Tau & 4&27 &3 &14 & 2 & 8 \\ \hline
\multicolumn{7}{c}{Calibrators} \\ \hline
Calibrator & size est.(mas) & Sources\\
HD 28024 & 0.68 & \multicolumn{5}{l}{T Tau N, SU Aur, DR Tau, RY Tau} \\
HD 30111 & 0.60 & \multicolumn{5}{l}{T Tau N, SU Aur} \\
HD 30122 & 0.11 & \multicolumn{5}{l}{T Tau N, SU Aur} \\
HD 28677 & 0.34 & \multicolumn{5}{l}{DR Tau, RY Tau} \\
HD 26737 & 0.24 & \multicolumn{5}{l}{DR Tau, RY Tau} \\ \hline
\end{tabular}
\caption{New observations of T Tauri sources from PTI. Each integration represents 125 seconds of fringe data.
\label{table:obs}}
\end{center}
\end{table}
The data were calibrated using the standard PTI method \citep{bod98}.
Briefly, a synthetic wide-band channel is formed from five
spectrometer channels ($\lambda=2.0-2.4$). The system visibility, the
response of the interferometer to an unresolved object, is measured
using calibrator stars. The calibrator star sizes were estimated
using a blackbody fit to photometric data from the literature and were
checked to be internally consistent. Calibrators were chosen for
their proximity to the source and for small angular size, minimizing
systematic errors in deriving the system visibility. All calibrators
used here have angular diameters $<0.7$ milliarcsecond (mas) and were
assigned uncertainties of 0.1 mas (Table \ref{table:obs}). The
calibrated data are presented in normalized squared visibility
(V$^2$=1 for an unresolved source), which we refer to as visibility in
this paper. The calibrated visibility uncertainties are a combination
of the calibrator size uncertainty and the internal scatter in the
data. As DR~Tau is near the tracking limit for PTI, the wide-band
data are used rather than the synthetic wide-band (spectral channel)
data. The main difference between these two channels is that the
spectral channels are spatially filtered and the wide-band channel is
not. The accuracy of the wide-band data were confirmed by comparing
the wide-band and synthetic wide-band data for other sources observed
on the same night as DR~Tau.
The calibrated data were edited to remove integrations with
very high jitter (a measure of the phase noise) and integrations for which the
estimates of the system visibility from separate calibrator
observations disagreed by more than 3$\sigma$. In general,
the points eliminated were from entire nights with marginal
weather or integrations taken at large hour angles. No more than 10\%
of the data for any given source was removed, except for DR Tau on the NS baseline, and inclusion
of these data points would not substantially change the
results given below.
The calibrated and edited data are shown in Figure \ref{fig:data} for
each source as a function of projected baseline length and position
angle. Three of the four sources, T~Tau~N, SU~Aur and RY~Tau are
clearly resolved. The new observations of T Tau N are
consistent with the results of \citet{ake02}. We have not calculated
models for T~Tau~N here; scattered light models of T~Tau~N that
reproduce the observed asymmetry \citep{sta98} are detailed in
\cite{woo01}.
\begin{figure}[h!]
\begin{center}
\epsscale{0.85}
\plotone{f1color.eps}
\caption{Calibrated PTI visibilities for each of the four sources
by baseline: NS (open circles), NW (open triangles) and SW (closed circles).
For T Tau N and SU Aur the data from \citet{ake02} is also
plotted.
\label{fig:data}}
\end{center}
\end{figure}
\subsection{Infrared photometry}
A sample of young stellar objects, including DR Tau, SU Aur and RY Tau
and a sequence of photometric standard stars taken from \citet{lan92}
were observed over seven nights from December 2003 to March 2004
(December 16, January 10, 13, 15, 22, 23, and March 9) using the
Pomona College 1-m telescope with the CLIRCAM infrared camera in the J
and K bands. For each object, a series of at least five dithered
exposures was used to create individual sky images for each field. The
sky background and instrumental noise was subtracted from all the
images, and repeat exposures were median combined after shifting to a
common astrometric reference frame to remove a majority of the
background noise. Instrumental magnitudes were converted to standard
J and K magnitudes using a combination of the published magnitudes
and the J and K magnitudes from bright 2MASS stars in
the image frames. The magnitudes given in Table \ref{IRphot:table} are
the average magnitudes over all six nights using the average
calibration zero points from the complete sample of photometric
standards and 2MASS stars. The three sources observed showed no
statistically significant variability over the nights observed, and
were constant in magnitude within the photometric error of 0.15
magnitudes in J and 0.15 magnitudes in K. For comparison, the 2MASS J
and K magnitude and observation date are also given. Together, the
Pomona and 2MASS data bracket the PTI observations. At K band, only
RY Tau shows a significant difference between the 2MASS measurement
and our more recent observations; however, past observations of these
sources have shown infrared variability (particularly DR Tau;
\citet{ken94}). Additional information on the infrared observations
from Table \ref{IRphot:table} is presented in \citet{pen04}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{lllllll}
Source & \# nights & mag & rms & 2MASS mag & 2MASS rms & 2MASS date \\ \hline
\multicolumn{7}{c}{J band} \\ \hline
SU Aur & 6 & 7.24 & 0.143 & 7.20 & 0.020 & 1/30/98 \\
DR Tau & 6 & 8.75 & 0.195 & 8.84 & 0.024 & 10/10/97 \\
RY Tau & 5 & 7.52 & 0.226 & 7.15 & 0.019 & 10/29/97 \\ \hline
\multicolumn{7}{c}{K band} \\ \hline
SU Aur & 6 & 6.17 & 0.114 & 5.99 & 0.022 & 1/30/98 \\
DR Tau & 6 & 6.87 & 0.183 & 6.87 & 0.017 & 10/10/97 \\
RY Tau & 6 & 5.76 & 0.168 & 5.39 & 0.022 & 10/29/97 \\ \hline
\end{tabular}
\caption{Results of infrared photometry observations.
\label{IRphot:table}}
\end{center}
\end{table}
\section{Geometric models}
\label{model}
In this section, we discuss geometric models for SU Aur, RY Tau and DR
Tau. As PTI is a direct detection interferometer, any emission within
the 1\arcsec\ Gaussian (FWHM) field of view will contribute to the
measured visibility. As discussed in \citet{ake02} there are many
scenarios that could produce a visibility of less than 1. These
include additional point sources within the field of view, a resolved
source of emission, or extended (over-resolved) emission which will
contribute incoherently. Any possible incoherent contribution (in
this case any emission on scales greater than 10 mas and within the
1\arcsec\ FOV) is hard to assess for our sources, given that many
observations of envelopes or reflection nebulae do not include the
central arcsecond due to contamination from the star itself. None of
these three sources has a known companion within 1\arcsec. DR~Tau was
included in lunar occultation observations of \citet{sim99} and no
detection was reported with a point source limiting magnitude of
$\Delta K$ = 2.5. Models including scattered light contributions are
presented in \S \ref{scatter} and extended components are discussed in
\S \ref{extended}.
For the model fitting in this section, we adopt a configuration of an
unresolved point source (these stars have diameters $\leq$ 0.1 mas and
therefore a V$^2>0.99$ at PTI) and a resolved component. We take the
contribution of the stellar component from measurements of the
infrared veiling.
For SU~Aur and DR~Tau we
use the K band veiling measurements of \citet{muz03}. For RY~Tau the
\citet{muz03} value of $r_{K}= 0.8 \pm 0.3$ (where $r_K$ = F$_{\rm
excess}$/F$_{\rm star}$) is much less than the lower limit of $r_K > 2.5$
from \citet{fol99}. For our adopted model (point source + resolved
component), the PTI data and a value of $r_{K}= 0.8$ are incompatible
(i.e. the point source contribution can not be that large and still
produce the PTI measurement) and we therefore use $r_{K}= 2.5 \pm 1$ for RY
Tau in the geometric fits.
Simple geometric models of the emission are used to characterize the
source size and inclination. The two models presented here use a
uniform disk and a thin ring to represent the emission profile. For
the uniform disk, only the measured visibility was used to determine
the disk radius. For the ring model, visibilities were calculated for
a range of inner diameters and compared to the observed visibilities.
For each ring diameter considered, the width was determined by matching
the excess flux, derived using the measured K-band veiling, with a
blackbody emission source at a temperature of 1600~K, the assumed dust
destruction temperature \citep{dus96}. In these models, the dust
destruction temperature controls the width of the ring, but affects
the fit radius only through the shape of the model visibility curve.
For example, changing the blackbody temperature of the ring from 1200
to 2000 K would change the fit radius for RY Tau by 30\%. Both
face-on and inclined geometries were fit to the data (Table
\ref{table:fits}). The uncertainties in the model fits due to the uncertainty
in the stellar contribution are also given.
\begin{table}[h!]
\begin{center}
\begin{tabular}{llll} \hline
& SU Aur & DR Tau & RY Tau \\ \hline
f$_{\rm excess}$\tablenotemark{a} & $0.44 \pm 0.09$ & 0.8 $\pm 0.3$ & $0.71 \pm 0.11$ \\
K$_m$ (2MASS) & 5.99 & 6.87 & 5.40 \\ \hline
\multicolumn{4}{c}{Face-on models} \\ \hline
\multicolumn{2}{l}{Uniform disk} \\
\quad Radius (AU) & 0.20$\pm 0.028$ & 0.10$\pm 0.029$ & 0.29$\pm 0.036$ \\
\quad $\sigma_v$ (AU)& 0.042 & 0.004 & 0.080 \\
\quad $\chi^2/{dof}$ & 2.5 & 0.85 & 2.9 \\
\multicolumn{2}{l}{Ring} \\
\quad Inner radius (AU) & 0.13$\pm 0.021$ & 0.057$\pm 0.027$ & 0.17$\pm 0.01$ \\
\quad Width (AU) & 0.050 & 0.028 & 0.035 \\
\quad $\sigma_v$ (AU) & 0.036 & 0.010 & 0.059 \\
\quad $\chi^2/{dof}$ & 2.5 & 0.85 & 4.6 \\ \hline
\multicolumn{4}{c}{Inclined models} \\ \hline
\multicolumn{2}{l}{Uniform disk} \\
\quad Radius (AU) & 0.27$\pm 0.037$ & 0.11$\pm 0.03$ & 0.30$\pm 0.008$ \\
\quad PA (degr) & 112 $\pm 24$ & 160 $\pm 55$ & 98 $\pm 40$ \\
\quad Incl (degr) & 51 $\pm 11$ & 40 $\pm 30$ & 19 $\pm 6$ \\
\quad $\chi^2/{dof}$ & 0.9 & 0.77 & 2.3 \\
\multicolumn{2}{l}{Ring} \\
\quad Inner radius (AU) & 0.18 $\pm 0.025$ & 0.070 $\pm 0.026$ & 0.19$\pm 0.01$ \\
\quad Width (AU) & 0.008 & 0.019 & 0.029 \\
\quad PA (degr) & 114 $\pm 23$ & 160$\pm 55$ & 110$\pm 22$ \\
\quad Incl (degr) & 52 $\pm 10$ & 40 $\pm 30$ & 25 $\pm$ 3\\
\quad $\chi^2/{dof}$ & 0.9 & 0.78 & 3.0 \\ \hline
\end{tabular}
\caption{Results from geometric model fits. The systematic error,
$\sigma_v$ is from the uncertainty in the stellar contribution.
\label{table:fits}}
\tablenotetext{a}{f$_{\rm excess}$ = F$_{\rm excess}$/F$_{\rm total}$, where F$_{\rm total}$ = F$_{\rm star}$ + F$_{\rm excess}$}
\end{center}
\end{table}
The ring model fits are graphically shown in Figure
\ref{fig:UDpoint}. In this sky plane representation, the radial
coordinate for each data point is the inner ring size corresponding to
the measured visibility and accounting for the stellar component
listed in Table \ref{table:fits}. The error bars include the errors
on the data points but not the uncertainty in the stellar
contribution. The polar coordinate is determined by the projected
baseline position angle. In this way, the constraint provided by the
data on both the size and the inclination are visible. The best fit
face-on and inclined ring models are also plotted.
\begin{figure}[h!]
\begin{center}
\epsscale{0.4}
\plotone{f2color.eps}
\caption{The data and uniform disk fits for the best fit face-on
and inclined models. An unresolved stellar component is included
as described in \S \ref{model}.
Separate symbols are used for each baseline: NS (open circles), NW (open triangles) and SW (closed circles).
\label{fig:UDpoint}}
\end{center}
\end{figure}
\subsection{Discussion}
Using the simple geometric models, we find source sizes ranging from
0.04 to 0.3 AU in radius. As discussed in Paper 1 and by
\citet{mil01}, the measured sizes for T Tauri stars and Herbig Ae
stars were larger than expected from simple disk models. An
explanation for this discrepancy in Herbig stars was proposed
independently by two groups based on SED modeling \citep{nat01} and
aperture masking observations \citep{tut01}. In these models the
inner edge of the dust disk is located at the radius where the dust
reaches the sublimation temperature (R$_{\rm dust}$). This
configuration produces a vertically extended inner wall, reproducing
both the SED and the interferometry observations for the Herbig
sources. \citet{dul01} also applied this model to T Tauri stars.
Further work by \citet{muz03} extended the model to include the
accretion luminosity as well as the stellar luminosity in determining
the dust destruction radius for several T Tauri stars, including the
three shown in Figure \ref{fig:UDpoint}. In all these models,
optically thin gas may be present within R$_{\rm dust}$ (we discuss
this point further in \S \ref{results}).
We chose a ring distribution as a simple representation of a model in
which the infrared emission arises from the inner wall of the dust
disk. The values for R$_{\rm dust}$ predicted by \citet{muz03} are
larger by roughly a factor of 2 than our fit ring radii. We note that
the presence of extended emission which was not included in our model
would decrease the fit radius, and would therefore increase this
discrepancy. The fit ring radii for SU~Aur and RY~Tau correspond to
10 R$_{\star}$ and 11 R$_{\star}$, much larger than the expected
magnetic truncation radius (3 -- 5 R$_{\star}$; \citet{shu94}). In \S
\ref{scatter} we show that emission from gas between the magnetic
truncation radius and R$_{\rm dust}$ can reconcile accretion disk
models with our observations.
The position angle coverage of the PTI data allow us to constrain the
inclination of infrared emission. The $\chi^2/{dof}$ improves
substantially for SU Aur and RY~Tau for the inclined models as
compared to the face-on models, but the DR~Tau data do not provide a
good constraint on the inclination given the large error bars and
because the source is at best marginally resolved. For RY~Tau, our
inclination angle of 19\arcdeg\ -- 25\arcdeg\ agrees with that derived
by \citet{koe95} from resolved millimeter emission (25\arcdeg).
However, the position angle is not well constrained by our data
(98\arcdeg\ $\pm$ 40\arcdeg\ for the uniform disk and 110\arcdeg\
$\pm$ 22\arcdeg\ for the ring) and does not agree with the PA of the
millimeter emission (48\arcdeg\ $\pm$ 5\arcdeg, \citet{koe95};
27\arcdeg\ $\pm$ 7\arcdeg, \citet{kit02}) and is not orthogonal to the
jet PA of 110\arcdeg\ from \citet{sta04}. Our inclination angle of
52\arcdeg $\pm$ 10\arcdeg\ agrees with the 60\arcdeg\ estimate of
\citet{unr04} based on the photometric period and line widths.
\citet{muz03} find high (86\arcdeg) inclination values for both RY Tau
and SU Aur, which are not supported by the PTI data, particularly for
RY~Tau, and are also inconsistent with the low visible extinctions
(A$_v$=2.1 and 0.9, respectively). At such high viewing angles, the
star would be occulted by the flared circumstellar disk. If there is
a large incoherent component for any of the sources, then our simple
geometric fits will underestimate the inclination angle as an
incoherent contribution is independent of baseline. However, in our
detailed models (\S \ref{scatter}) for these three objects, the
extended light contribution is less than 10\%, which is insufficient
to change the measured RY Tau inclination by 60\arcdeg.
\section{Detailed radiation transfer models}
\label{scatter}
One of the major uncertainties in the simple fits presented above
is the assumption of no extended emission within the 1\arcsec\ PTI field
of view. To address this issue directly, we have calculated
radiative transfer models for SU Aur, DR Tau and RY Tau.
The input properties are given in Table \ref{table:source} and
the goal is to match the PTI data and the SED.
\subsection{Monte Carlo radiation transfer code}
\label{MonteCarlo}
We use the Monte Carlo radiative equilibrium technique of
\citet{bjo01}, updated by \citet{wal04}, to self-consistently model
each of our target sources. This code iteratively solves for the disk
density structure, assuming the dust and gas are well-mixed with a
standard gas to dust ratio of 100:1 and the system is in vertical
hydrostatic equilibrium. In addition to stellar irradiation, the code
includes accretion and shock/boundary layer luminosity calculations
according to \citet{cal98}. Multiple scattering is
treated alongside the heating and reprocessing of photons in the
disk. Output data can be used to produce synthetic SEDs and
multi-wavelength images for any viewing angle of the disk system. For
more detailed description of the code and its updates see
\citet{woo02a,woo2b,whi03a,whi03b,wal04} and references therein.
The code computes the flared density structure of a steady accretion
disk extending from the inner dust destruction radius to a specified
outer radius (Figure \ref{density:plot}). The Monte Carlo technique
naturally accounts for radiation transfer effects and the heating and
hydrostatic structure of the inner wall of the dust disk. The
vertical height of the inner wall of dust is not preset, but rather
calculated as part of the modeling process. For these models the
scale height of the density distribution is 0.3 to 0.7 R$_{\star}$ at
the inner edge. The position of the inner dust disk edge, R$_{\rm
dust}$ is determined from the destruction temperature of silicates,
taken to be 1600~K \citep{dus96}. Within the disk we adopt the
dust-size distribution used for the modeling of HH30 IRS and GM Aur
\citep{woo02a,sch03,ric03}. With a distribution of grain sizes or
compositions, the dust destruction may take place over a range of
radii, but this is beyond the scope of our work. \citet{mon02}
discuss the constraints on the dust properties from infrared
interferometry observations.
\begin{figure}[h!]
\begin{center}
\epsscale{0.9}
\plotone{f3.eps}
\caption{Temperature and density distributions for
an example disk model. The upper images are temperature scaled
to the 0.5 power and the lower are density to the 0.1 power. Note
the geometrically thin gas is not shown.
\label{density:plot}
}
\end{center}
\end{figure}
In order to match the new PTI observations, R$_{\rm dust}$ for some
sources was large enough ($>$ 0.2 AU) that continuum emission from gas
within R$_{\rm dust}$ becomes significant. The structure and
temperature of the gas disk is not computed self-consistently in our
models. Instead, accretion luminosity is emitted following the
temperature structure of an optically thick accretion disk
\begin{equation}
T_{gas}(R) = (\frac{3 G M_{\star} \dot{M}}{8 \pi \sigma R^3})(1 - \sqrt{R_{\star}/R})^{1/2})^{1/4}
\end{equation}
(e.g.,
\citet{lyn74, pri81}) where $R$ is the radial distance in the mid-plane.
The gas disk is assumed to be
infinitely thin, so after being emitted, the ``accretion photons'' do
not encounter any opacity in the gas, but may be scattered and
absorbed, and produce heating in the dust disk. Clearly this is a
simplification for the gas emission, but is sufficient for our models.
The assumed geometry of the gas disk is supported by recent modeling
by \citet{muz04} of Herbig Ae/Be sources in which the gas disk is
geometrically thin, allowing direct radiation of the inner dust disk.
Future work will investigate the effects of possible shielding of the
dust disk by a flared and possibly optically thick, inner gas disk.
The gas disk extends down to the magnetic truncation radius (R$_{\rm
gas}$) at which point, material is thought to be channeled along
magnetic field lines onto the star at a high latitude shock zone
(e.g. \citet{dal03}, \citet{ken94}). We assume the gas disk is
truncated at a magnetospheric radius dependent on the stellar radius,
mass, accretion rate and surface magnetic field \cite{gho79}. For DR
Tau and RY Tau we assume kiloGauss magnetic fields and truncate the
gas disk at 5 R$_{\star}$. For SU Aur, thought to be more weakly
magnetic and with inconclusive evidence for hot spots \citep{unr04} we
use 2 R$_{\star}$ for R$_{\rm gas}$. We assume photons emitted from the
shock/boundary layers have a spectrum of an 8000~K Planck function
\citep{cal98} and are emitted along with stellar photons as in
\citet{muz03}.
For each model, the stellar luminosity has been fixed as detailed in
Table 1 and for input stellar spectra we use the appropriate Kurucz
(1994) model atmosphere. We also used fixed stellar masses of 2.25
M$_{\odot}$, 2M$_{\odot}$, and 1 M$_{\odot}$ for SU Aur, RY Tau and DR
Tau, respectively \citep{coh79,ken94}; note that the stellar mass is
not a critical parameter in the near-infrared. The disk properties
such as mass, accretion rate and inclination were varied in order to
produce a grid of synthetic SEDs. These models allowed us to explore
likely parameter configurations.
\subsection{Visibility calculation}
To calculate model visibilities, a simulated K band image was created
using the Monte Carlo models with pixel size 0.05 mas and a width of
12.5 mas. The pixel size was chosen to be much smaller than the
fringe spacing of 4 mas and the total size was a compromise between
calculation time (large images are computationally intensive) and
capturing the relevant structure. The outer size is large enough to
contain any component which would contribute substantially to the
model visibility. For example, a thin ring with an inner radius of 3
mas has V$^2$=0.01 on the shortest PTI baseline.
The K band emission in the models is dominated by structures a few mas
in size or less (Figure \ref{model:plot}). As discussed in \S
\ref{model}, any emission within the 1\arcsec\ FOV will contribute
incoherently. To calculate the extended component in the model, a larger
image is also constructed with 2 mas pixels and a 1\arcsec\ field.
The emission outside the central 12 mas is calculated and included in
the visibility calculation as an incoherent contribution. The effects
of the 1\arcsec\ Gaussian field of view and the finite fringe envelope
are also included in the visibility calculation. For each PTI
baseline, the model visibility, including the incoherent flux, was
calculated for the average baseline length and position angle using
the Fourier Transform of the image, assuming a position angle for the
disk as given in \S \ref{model}. We thus ``observe'' the models as
they would be at PTI.
The model visibilities and full SEDs were then
compared to the data presented in \S \ref{observations} and SED data
taken from the literature. The optical and infrared SED data are taken
from the compilation of \citet{ken95} and are not contemporaneous but
instead represent an average for each source and the millimeter data
are taken from \cite{ake02} and \cite{bec90}. For each object a set of models
was calculated to explore the disk parameters to find a viable model
and an example selection of models for each object is given in Table
\ref{model:table}. The parameter space chosen for the inclination
angle was restricted using the results from the geometric fits. The
model with the total lowest $\chi^2$ is shown for each object in Fig
\ref{model:plot} and the corresponding SED fit in Fig \ref{fig:SED}.
\begin{table}
\begin{center}
\begin{tabular}{llllllllll}
Model & $\dot{M}$ & r$_{in}$ & $M_{disk}$ & incl & Lacc+shk & $\chi^2_{PTI}$ & $\chi^2_{SED}$ & $\chi^2_{total}$ & notes \\
& M$_{\odot}$/yr & AU & M$_{\odot}$ & deg & L$_{\odot}$ \\ \hline
\multicolumn{10}{c}{SU Aur} \\ \hline
SU-A & $1 \times 10^{-9}$ & 0.21 & 0.001 & 60 & 0.02 & 7 & 118 & 125 \\
SU-B & $1 \times 10^{-9}$ & 0.21 & 0.001 & 50 & 0.02 & 29 & 101 & 130 \\
SU-C & $1 \times 10^{-9}$ & 0.22 & 0.001 & 50 & 0.02 & 60 & 198 & 258 \\
SU-D & $2 \times 10^{-9}$ & 0.21 & 0.001 & 50 & 0.02 & 48 & 163 & 211 \\
SU-E & $4 \times 10^{-9}$ & 0.24 & 0.005 & 50 & 0.02 & 99 & 893 & 992 \\ \hline
\multicolumn{10}{c}{DR Tau} \\ \hline
DR-A & $8 \times 10^{-8}$ & 0.09 & 0.16 & 30 & 1.3 & 12 & 24 & 36 & R$_{gas} = 2 R_{\star}$\\
DR-B & $8 \times 10^{-8}$ & 0.09 & 0.16 & 30 & 1.3 & 30 & 19 & 49 \\
DR-C & $8 \times 10^{-8}$ & 0.09 & 0.12 & 30 & 1.3 & 30 & 59 & 89 \\
DR-D & $6 \times 10^{-8}$ & 0.09 & 0.15 & 60 & 0.97 & 9 & 55 & 63 \\
DR-E & 0 & 0.08 & 0.08 & 60 & 0 & 3 & 82 & 85 & no accretion\\ \hline
\multicolumn{10}{c}{RY Tau} \\ \hline
RY-A & $2.5 \times 10^{-7}$ & 0.27 & 0.015 & 25 & 4.28 & 8 & 37 & 45 \\
RY-B & $3 \times 10^{-7}$ & 0.27 & 0.015 & 25 & 5.13 & 1 & 294 & 295 \\
RY-C & $2 \times 10^{-7}$ & 0.27 & 0.012 & 25 & 3.42 & 135 & 31 & 166 \\
RY-D & $2.5 \times 10^{-7}$ & 0.27 & 0.015 & 25 & 4.28 & 4 & 145 & 149 & no gas \\
RY-E & $2.5 \times 10^{-7}$ & 0.27 & 0.015 & 25 & 4.28 & 36 & 313 & 349 & envelope \\ \hline
\end{tabular}
\caption{Representative model parameters for each source. The best
fit model is listed first. $\chi^2$ values given have {\it not} been
normalized by the degrees of freedom.
\label{model:table}}
\end{center}
\end{table}
\begin{figure}[h!]
\begin{center}
\epsscale{0.5}
\plotone{f4a.eps}
\plotone{f4b.eps}
\plotone{f4c.eps}
\caption{Model images for SU Aur (top), DR Tau (middle) and RY Tau
(bottom). The flux has been scaled to the 0.15 power to provide
better contrast in the image. Each image is 12.5 milliarcsec or 1.75 AU
across. For comparison, all models are shown with the same position
angle.
\label{model:plot}
}
\end{center}
\end{figure}
\begin{figure}[h!]
\begin{center}
\epsscale{0.6}
\plotone{f5a.eps}
\plotone{f5b.eps}
\plotone{f5c.eps}
\caption{SED plots for best-fit models for SU Aur (top), DR Tau (middle)
and RY Tau (bottom). The model total flux is given by the
solid line, the input stellar spectrum by the dashed line and the
scattering by the dotted line. Data from \citet{ken95} are given by squares and from \citet{eir02} by circles.
\label{fig:SED}}
\end{center}
\end{figure}
\subsection{Other emission components}
\label{extended}
Before discussing the results of the Monte Carlo models, we discuss
other possible physical components which have not been included in our
models. As discussed in \S \ref{model}, a binary companion will
contribute coherently or incoherently to the measured visibility
depending on the separation from the primary. RY Tau was classified
as a ``Variability induced mover'' from Hipparcos data and
\cite{ber89} found a solution in which the possible companion had a
minimum separation of 24 mas. However, the K band speckle
interferometry survey of \citet{lei93} did not detect a companion for
RY Tau in the angular range of 0\farcs1 to 10\arcsec\ and HST archival
images of RY Tau from WFPC2 show only a single point source. For the
incoherent contribution from a companion to account entirely for the
measured visibility the K band flux ratio would be 0.81 to 1.44
(secondary/primary), but this could not account for the change in
visibility with baseline length and orientation.
Another likely contributor of infrared emission is a circumstellar
envelope. An envelope can be a source of scattered and thermally
reprocessed starlight and can also veil emission from the central star
and accretion disk (see e.g. the models of \citet{cal97};
\cite{whi03b}). However, SU Aur, DR Tau and RY Tau are all Class II T
Tauri stars and have visual extinctions $\lesssim$ 2. In general, Class
II sources are thought to have little or no envelope remaining (see
e.g. \citet{mun00}). RY Tau shows near-infrared CO lines in
absorption \citep{naj03} which \citet{cal97} cite as evidence of no
substantial envelope for other Class II sources.
To assess the possible presence of emission within the 1\arcsec\ PTI
FOV, we examined HST archival images from the standard imaging
pipeline. For each source, we found WFPC2 images taken with the F814W
filter on the Planetary Camera CCD. No extended emission or
additional sources were apparent in the images, however they are
dominated by the central point source. Azimuthal brightness averages
were computed for comparison to a star extracted from the PSF archive.
The core of RY Tau was saturated to such an extent that we were unable
to find a matching saturated PSF for comparison. Although the scatter
in the PSF averages do not allow a precise comparison, especially in
the core, SU Aur and DR Tau are dominated by a central source (Figure
\ref{psf}). This is in contrast with the images of Class I sources,
such as the sample imaged by \citet{pad99} which show images dominated
by scattered light from circumstellar material hundreds of AU in
extent. Coronagraphic techniques have revealed
some extended emission in RY Tau \citep{nak95} and SU Aur
\citep{cha04} but as detailed in the next section, this emission is
unlikely to contribute substantially in the near-infrared. We have
also calculated an example model with a disk and an envelope for RY
Tau (\S \ref{rytau}).
\begin{figure}[h!]
\begin{center}
\plotone{f6.eps}
\caption{Azimuthal averages from HST archival images for SU Aur
(diamonds), DR Tau (squares) and a PSF standard (circles), all using
the F814W filter. For comparison, each source is self-scaled to the
peak brightness.
\label{psf}}
\end{center}
\end{figure}
\subsection{Results}
\label{results}
As the wavelength range used for the SED comparison ranges from 0.365
microns to 3 millimeters, the model SED is sensitive to many model
parameters, from the extinction to the outer disk size and mass. As
expected, the infrared visibility is sensitive to only a few model
parameters, particularly the inner radius, the inclination angle and
the luminosity. Each object is considered separately below, but
the general conclusion is that these models, which include the
contribution from extended emission, support the simple geometric
models in the large value of R$_{\rm dust}$ found for RY Tau and SU Aur.
In part this is because the models contain incoherent
contributions at 2 microns (here defined as flux from scales greater
than 10 mas) which were less than 6\% for all sources, and could still
reproduce both the SED and the infrared interferometry observations.
High resolution infrared imaging observations would further constrain the
extended emission component of these models. The K band excess flux
from the models is also close to the veiling values used in \S
\ref{model}, with F$_{\rm excess}$/F$_{\rm total}$ values of 0.4, 0.74
and 0.68 respectively for SU Aur, DR Tau and RY Tau.
The second general conclusion is that emission from gas within R$_{\rm
dust}$ is a significant component of the near-infrared emission if
R$_{\rm dust}$ is large. For our three objects this is most evident
in the RY Tau model. The relative flux of the gas and dust components
for RY Tau can be seen in Figure \ref{fig:slice}, which shows a cut
through the model image with the inner dust wall facing the observer
on the left in the plot. In comparison, for the DR Tau model a
smaller R$_{\rm dust}$ is necessary to match the high visibilities
measured at PTI and so R$_{\rm dust}$ and R$_{\rm gas}$ are similar.
For DR Tau, a smaller value of R$_{\rm gas}$ than estimated from the
stellar properties (2 R$_{\star}$ instead of 5 R$_{\star}$) was
necessary to match the data. We have not explored the value of
R$_{\rm gas}$ extensively in these models, so these values should be
taken as approximate. Observations by \citet{naj03} of CO fundamental
emission for several single T Tauri stars similar to our targets
(e.g. BP Tau) found the inner CO radius to be smaller than the
calculated corotation radius for 5 out of 6 sources with CO inner
radii of 0.02 to 0.09 AU. Modeling of Herbig Ae/Be sources by
\citet{muz04} found emission from the inner gas exceeded the stellar
emission for accretion rates $> 10^{-7}$ M$_{\odot}$/yr.
\begin{figure}[h!]
\begin{center}
\epsscale{0.6}
\plotone{f7.eps}
\caption{A cross section of the RY Tau model through the center of
the source showing the relative flux contributions of the gas and
dust emission. The slice is oriented such that the inner dust wall
facing the observer is on the left.
\label{fig:slice}}
\end{center}
\end{figure}
To confirm the effect of the gas emission, a model was constructed for
RY Tau in which the gas was artificially removed from the region
within R$_{\rm dust}$, which was set to match the measured PTI
visibilities. However, the SED fit for this model (Table
\ref{model:table}, model RY-D) is not as good as for the model with
gas emission.
\subsubsection{SU Aur}
For SU Aur, both the SED and the measured visibilities are well fit by
model SU-A. Figure \ref{model:plot} shows the inner region of this
model, with the star and the inner dust disk edge producing the K band
emission. At an inclination angle of 60\arcdeg\ the inner edge of the
dust disk facing the observer is clearly brighter. We find R$_{\rm dust}$
= 0.21 AU, similar to the inclined ring model radius of
0.18 AU from Table \ref{table:fits}. The gas emission is visible close
to the star (R$_{gas} = 2$R$_{\star}$) but the emission is limited by
the small surface area at this radius.
Recent observations by \citet{cha04} have traced extended emission at
K out to radii of 2\farcs6. This study did not measure the scattered
light within 1\arcsec\ so does not help constrain the PTI data, but
does suggest that a complete model for SU Aur would include an
extended scattered component; however, their measured K flux from
1\arcsec\ to 2\farcs6 was only 4\% of the 2MASS K flux, so neglecting
this component adds only a small error to the SED fit.
\subsubsection{DR Tau}
It was not possible to fit both the PTI data and the SED data well
with these models. The best-fit model listed in Table
\ref{model:table} underestimates the infrared visibility and
underestimates the SED throughout the near and mid-infrared (Figure
\ref{fig:SED}). In order to produce an inner disk radius small enough
to fit the PTI data, the model must contain no accretion (model DR-E), which
drastically underestimates the SED and disagrees with the accretion
diagnostics observed for the source \citep{ken94,muz03}. The infrared
photometry (Table \ref{IRphot:table}) does not reveal substantial
variations recently, however the optical veiling for DR Tau has been highly
variable \citep{gul00} and as the SED data and PTI measurements are
not contemporaneous, there may be issues with source variability in
our modeling. Also note that the models in Table \ref{model:table} have
a lower inclination ($30 \arcdeg$) than given by the geometric fits
($60 \arcdeg \pm 30 \arcdeg$) but within the uncertainty.
We compared the photometry from \citet{ken95} which
are averages of measurements from the literature to the optical and
infrared contemporaneous SED from \citet{eir02} and the main deviation
is slightly lower fluxes at $u,b,v$ for the contemporaneous SED, which
does not improve our model fits. In the variability study by
\citet{skr96} DR Tau showed no trend in color with brightness changes,
suggesting the variability is not due to large extinction changes.
The variability of DR Tau has been modeled as a hot spot on the
stellar photosphere \cite{ken94}, but the models here do not attempt
to model DR Tau with that level of detail.
\subsubsection{RY Tau}
\label{rytau}
RY Tau has the largest R$_{\rm dust}$ of the three sources and gas
within R$_{\rm dust}$ contributes substantially to the infrared
emission. This second component in the disk emission means that the
simple ring model is an {\it underestimate} of R$_{\rm dust}$.
Although this same gas component is present in the models for all
three sources, the contribution to the K band flux is largest for
RY~Tau as R$_{\rm dust}$ is larger than for the other two sources.
For RY Tau, the best fit model was relatively close to the optimal
parameters for both the PTI and SED data. The PTI data are best
modeled by a higher total luminosity and accretion rate (model RY-B)
than the SED data.
A reflection nebulosity has been observed extending to
$\sim$40\arcsec\ from RY~Tau at visible wavelengths. However, this is
unlikely to contribute substantially at K as the reflection
component is only 2\% of the total flux at 0.9 $\mu$m and scattering
decreases with increasing wavelength. To test the effect of an
envelope on the predicted visibilities and SED, a model was calculated
using the disk properties of model RY-A with an envelope using the
same gas and dust radii, 0.01 times the disk mass and an infall rate
of 1 $\times 10^{-7}$ M$_{\odot}$/yr for the envelope. As seen in
Table \ref{model:table}, this model does not fit either the SED or the
visibilities as well. Other models with higher envelope masses were
also calculated and had even worse fits to the SED. It may be
possible to better match the SED with a different disk and envelope
combination, however we found no observational evidence for
substantial near-infrared emission from an envelope.
We note that \cite{cal04} have recently characterized RY Tau as a G1
star, substantially earlier than previous spectral type
determinations. However, their stellar properties (R$_{\star}$ = 2.9
R$_{\odot}$, M$_{\star}$ = 2.0 M$_{\odot}$) agree reasonably well with
the values we used (Table \ref{table:source}). We used a slightly
lower effective temperature (5782 K compared to 5945 K) and a higher
luminosity (12.8 L$_{\odot}$ compared to 9.6 L$_{\odot}$). A lower
stellar luminosity would require a more massive disk and higher
accretion to produce the same flux at longer wavelengths, but the
general properties of a large R$_{dust}$ would not change. The
accretion rate of the model presented here, $2.5 \times 10^{-7}$
M$_{\odot}$/yr, is actually higher than the \citet{cal04} estimate of
$6.4-9.1 \times 10^{-8}$ M$_{\odot}$/yr.
\section{Conclusions}
Infrared interferometric observations of T~Tauri stars are used to
constrain the inner disk properties. Detailed models were presented
for SU Aur, RY Tau and DR Tau to reproduce both the interferometry
observations and the spectral energy distribution. For both the
simple geometric fits to the interferometry data and the Monte Carlo
disk models which include accretion and scattering, the inner dust
radius ranges from 0.05 to 0.3 AU. Extended envelopes were not needed
to reproduce the SED for these sources, although additional high
resolution infrared images would help in constraining the extended
emission on larger scales (tens to hundreds of AU). However, the
significant variations in the visibility with baseline length and
orientation seen for SU Aur and RY Tau require a resolved component to
be present as an extended component produces a constant visibility
reduction. Although the models parameters given here may not be a
unique solution to the data set used, they are consistent with the
domination of the near-infrared excess by thermal emission from the
disk, as expected for Class II sources.
The SU Aur model agrees well with the size derived from geometric fits
to the interferometer data alone. The SED and PTI visibilities for DR
Tau can not both be fit with these models, perhaps due to source
variability. For RY Tau, the model predicts significant emission in
the K band from gas within the inner dust disk radius. This gas
emission at infrared wavelengths is generally not considered in the
simple models used to fit interferometric data and when this emission
is present results in an underestimate of the dust radius when using a
simple ring model.
Future work will extend the observational database to study more
sources and probe the innermost regions of disks. Interferometric
observations at a second wavelength would add additional constraints
on the inner disk structure. From a theoretical perspective, we are
extending the Monte Carlo codes to include gas opacity in the inner
disk and self-consistently calculate its structure. The data and
models presented in this paper clearly show that inner gas disks
cannot be ignored and are required when fitting observations that
probe the inner regions of disks.
\acknowledgments
This work was performed at the Michelson Science Center, California
Institute of Technology, under a contract with the National
Aeronautics and Space Administration. Data were obtained at the
Palomar Observatory using the NASA Palomar Testbed Interferometer.
Science operations with PTI are possible through the support of the
PTI Collaboration({\tt
http://huey.jpl.nasa.gov/palomar/ptimembers.html}) and the efforts of
Kevin Rykoski. JAE acknowledges support from a Michelson Graduate
Research Fellowship. This work has made use of software produced by
the Michelson Science Center. Pomona College would like to
acknowledge the support of the NSF ARI and CCLI grants in providing
funds for development of the infrared camera and the Pomona College
1-meter telescope. This work has made use of the SIMBAD database,
operated at CDS, Strasbourg, France, and the NASA/IPAC Infrared
Science Archive, operated by the JPL under contract with NASA. This
work utilizes observations made with the NASA/ESA Hubble Space
Telescope, obtained from the data archive at the Space Telescope
Science Institute. STScI is operated by the Association of
Universities for Research in Astronomy, Inc. under NASA contract NAS
5-26555.
| {
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{"url":"https:\/\/www.physicsforums.com\/threads\/irreducible-representation-of-su-2.902544\/","text":"# Irreducible representation of su(2)\n\n1. Feb 3, 2017\n\n### Kara386\n\n1. The problem statement, all variables and given\/known data\nUsing the irreducible representation of $su(2)$, with $j=\\frac{5}{2}$, calculate $J_z$, $exp(itJ_z)$ and $J_x$.\n\n2. Relevant equations\n\n3. The attempt at a solution\nThere seem to be loads of irreducible representations of $su(2)$ online, but no reference at all to a specific irreducible representation in my lecture notes. It will be a matrix representation, I suspect, involving something from physics because that's the context we're working in, so maybe the Pauli matrices? I'm completely stuck and any guidance or thoughts on what my lecturer might mean would be very much appreciated! :)\n\nLast edited: Feb 3, 2017\n2. Feb 3, 2017\n\n### Paul Colby\n\nLook into ladder operator methods. One defines $J_\\pm = J_x\\pm i J_y$ and gets commutation relations $[J_z,J_\\pm]=\\pm J_\\pm$ where the signs I just wrote are likely all screwed up.\n\n3. Feb 3, 2017\n\n### Kara386\n\nBut the ladder operators aren't even elements of $su(2)$... and they aren't generators. How can they be a representation?\n\n4. Feb 3, 2017\n\n### Paul Colby\n\nOkay, best looked up in a book. In a nut shell one starts with $J_-\\vert -5\/2\\rangle = 0$. By applying $J_+$ to this \"ground\" state one generates all the eigen states in the rep. From these follow all operators in matrix form. It's work, that's why it's homework. It's also extremely elegant.\n\n5. Feb 3, 2017\n\n### Paul Colby\n\nAh, best looked up in a Physics book on quantum mechanics. Any intro text will do.\n\n6. Feb 3, 2017\n\n### Staff: Mentor\n\n$\\dim \\mathfrak{su}(2) = 3$ and therefore (both are simple) $\\mathfrak{su}(2) \\cong \\mathfrak{sl}(2,\\mathbb{R})$.\nTherefore you can get all representations as representations of $\\mathfrak{sl}(2,\\mathbb{R})$ which has a basis $\\{Y,H,X\\}$ with $[H,X]=2X\\, , \\,[H,Y]=-2Y\\, , \\,[X,Y]=H$ which can be represented by the matrices\n$$H=\\begin{bmatrix}1&0\\\\0&-1\\end{bmatrix}\\, , \\,X=\\begin{bmatrix}0&1\\\\0&0\\end{bmatrix}\\, , \\,Y=\\begin{bmatrix}0&0\\\\1&0\\end{bmatrix}$$\nThis makes it easier to find the representations as those of $\\mathfrak{sl}(2,\\mathbb{R})$ and easier to see the \"ladder\", as $X=J_+\\, , \\,Y=J_-\\, , \\,H=J_z\\,.$ By the way, is $j=\\frac{5}{2}$ meant to be the highest weight?\n\nYou can also look up the Wikipedia entry, which is not bad:\nhttps:\/\/en.wikipedia.org\/wiki\/Representation_theory_of_SU(2)\n\n7. Feb 3, 2017\n\n### Kara386\n\nThe j values were defined as j=0 are scalars, $j=\\frac{1}{2}$ are spinors, and so on. Every half integer increase in j seems to be associated with an increase in 1 of the dimension of the representation.","date":"2017-08-18 17:19:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8070889115333557, \"perplexity\": 719.8994675542501}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-34\/segments\/1502886104704.64\/warc\/CC-MAIN-20170818160227-20170818180227-00649.warc.gz\"}"} | null | null |
Greetings, weary traveler! Have a seat by the fire and hear my story!
This story of fire and skirmish begun on the day when the great chieftain Decius executed the shaman Fabian. The chieftain thought that by destroying the shamans he could gain control over all the tribes. Alas the Gods were angry and called upon their avatars to seek vengeance for this act.
Reciprocations for that happened nine days later, on a day you may also know as 29th of January, the day of the Global Game Jam of 2016. On that day 7 legendary shamans rose up to re-take what Decius had stolen. With the help of the ancient Gods there were many battles in the era of Totem Games.
After several demonic engagements, the 7 shamans won the battle of GGJ-2016 in Estonia. The fight took place on the battlegrounds of APT Game Generator and there were many contenders who tested the mettle of the shamans.
The original 7 Legendary Shaman.
Subsequent to that mighty victory, the shamans aimed to bolster their ranks in order to please more Gods. Nowadays the group of legendary shamans consists of 11 members, who practice ancient shamanic rituals in every bit of free time they have. Several of them also train in different shamanic institutes like the Tartu Art School and the University of Tartu. Each of them is a master of their very own unique discipline. Of course they also share their obscure knowledge among each other if need be.
It was not long until a following formed around this tribal occult. The shamans knew that to please the Gods, they needed to actively indoctrinate more worshipers. Thus a great pilgrimage was made to the sacred village of MängudeÖÖ. There many people were amazed by the shamanic teachings and vowed their allegiance to the Skirmish.
Shamans amazing the folk at MängudeÖÖ | Capture by Ken Oja.
The Gods were very pleased with the long journey to MängudeÖÖ. The shamans met with lots of followers, distributed around dozen by dozen ritual cards and arrived back home come dawn.
Now it has been 6 moons after the death of Fabian and the era of Totem Games. The legendary shamans have formed pacts with the APT Game Generator and even the CGVR Laboratory in the University of Tartu. In those dark places shamanic rituals are still occurring.
Even now you can be sure that those same shamans are preparing their myriads of voodoo dolls and other trinkets in order to bring on the… Tribocalypse! | {
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Crafting Is Cool! This Page Has Been Visited Times. | {
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An oversampled binary image sensor is an image sensor with non-linear response capabilities reminiscent of traditional photographic film. Each pixel in the sensor has a binary response, giving only a one-bit quantized measurement of the local light intensity. The response function of the image sensor is non-linear and similar to a logarithmic function, which makes the sensor suitable for high dynamic range imaging.
Working principle
Before the advent of digital image sensors, photography, for the most part of its history, used film to record light information. At the heart of every photographic film are a large number of light-sensitive grains of silver-halide crystals. During exposure, each micron-sized grain has a binary fate: Either it is struck by some incident photons and becomes "exposed", or it is missed by the photon bombardment and remains "unexposed". In the subsequent film development process, exposed grains, due to their altered chemical properties, are converted to silver metal, contributing to opaque spots on the film; unexposed grains are washed away in a chemical bath, leaving behind the transparent regions on the film. Thus, in essence, photographic film is a binary imaging medium, using local densities of opaque silver grains to encode the original light intensity information. Thanks to the small size and large number of these grains, one hardly notices this quantized nature of film when viewing it at a distance, observing only a continuous gray tone.
The oversampled binary image sensor is reminiscent of photographic film. Each pixel in the sensor has a binary response, giving only a one-bit quantized measurement of the local light intensity. At the start of the exposure period, all pixels are set to 0. A pixel is then set to 1 if the number of photons reaching it during the exposure is at least equal to a given threshold q. One way to build such binary sensors is to modify standard memory chip technology, where each memory bit cell is designed to be sensitive to visible light. With current CMOS technology, the level of integration of such systems can exceed 109~1010 (i.e., 1 giga to 10 giga) pixels per chip. In this case, the corresponding pixel sizes (around 50~nm ) are far below the diffraction limit of light, and thus the image sensor is oversampling the optical resolution of the light field. Intuitively, one can exploit this spatial redundancy to compensate for the information loss due to one-bit quantizations, as is classic in oversampling delta-sigma conversions.
Building a binary sensor that emulates the photographic film process was first envisioned by Fossum, who coined the name digital film sensor (now referred to as a quanta image sensor). The original motivation was mainly out of technical necessity. The miniaturization of camera systems calls for the continuous shrinking of pixel sizes. At a certain point, however, the limited full-well capacity (i.e., the maximum photon-electrons a pixel can hold) of small pixels becomes a bottleneck, yielding very low signal-to-noise ratios (SNRs) and poor dynamic ranges. In contrast, a binary sensor whose pixels need to detect only a few photon-electrons around a small threshold q has much less requirement for full-well capacities, allowing pixel sizes to shrink further.
Imaging model
The lens
Consider a simplified camera model shown in Fig.1. The is the incoming light intensity field. By assuming that light intensities remain constant within a short exposure period, the field can be modeled as only a function of the spatial variable . After passing through the optical system, the original light field gets filtered by the lens, which acts like a linear system with a given impulse response. Due to imperfections (e.g., aberrations) in the lens, the impulse response, a.k.a. the point spread function (PSF) of the optical system, cannot be a Dirac delta, thus, imposing a limit on the resolution of the observable light field. However, a more fundamental physical limit is due to light diffraction. As a result, even if the lens is ideal, the PSF is still unavoidably a small blurry spot. In optics, such diffraction-limited spot is often called the Airy disk, whose radius can be computed as
where is the wavelength of the light and is the F-number of the optical system. Due to the lowpass (smoothing) nature of the PSF, the resulting has a finite spatial-resolution, i.e., it has a finite number of degrees of freedom per unit space.
The sensor
Fig.2 illustrates the binary sensor model. The denote the exposure values accumulated by the sensor pixels. Depending on the local values of , each pixel (depicted as "buckets" in the figure) collects a different number of photons hitting on its surface. is the number of photons impinging on the surface of the th pixel during an exposure period. The relation between and the photon count is stochastic. More specifically, can be modeled as realizations of a Poisson random variable, whose intensity parameter is equal to ,
As a photosensitive device, each pixel in the image sensor converts photons to electrical signals, whose amplitude is proportional to the number of photons impinging on that pixel. In a conventional sensor design, the analog electrical signals are then quantized by an A/D converter into 8 to 14 bits (usually the more bits the better). But in the binary sensor, the quantizer is 1 bit. In Fig.2, is the quantized output of the th pixel. Since the photon counts are drawn from random variables, so are the binary sensor output .
Spatial and temporal oversampling
If it is allowed to have temporal oversampling, i.e.,taking multiple consecutive and independent frames without changing the total exposure time , the performance of the binary sensor is equivalent to the sensor with same number of spatial oversampling under certain condition. It means that people can make trade off between spatial oversampling and temporal oversampling. This is quite important, since technology usually gives limitation on the size of the pixels and the exposure time.
Advantages over traditional sensors
Due to the limited full-well capacity of conventional image pixel, the pixel will saturate when the light intensity is too strong. This is the reason that the dynamic range of the pixel is low. For the oversampled binary image sensor, the dynamic range is not defined for a single pixel, but a group of pixels, which makes the dynamic range high.
Reconstruction
One of the most important challenges with the use of an oversampled binary image sensor is the reconstruction of the light intensity from the binary measurement . Maximum likelihood estimation can be used for solving this problem. Fig. 4 shows the results of reconstructing the light intensity from 4096 binary images taken by single photon avalanche diodes (SPADs) camera. A better reconstruction quality with fewer temporal measurements and faster, hardware friendly implementation, can be achieved by more sophisticated algorithms.
References
Digital photography
Image sensors
Image processing
Digital signal processing
Digital electronics | {
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{"url":"https:\/\/gpumd.zheyongfan.org\/index.php\/The_Tersoff-1989_potential","text":"The Tersoff-1989 potential\n\nPotential form\n\n\u2022 Conventions:\n\u2022 Use $i,j,k,\\cdots$ for atom indices.\n\u2022 Use $I,J,K,\\cdots$ for atom types.\n\n\u2022 The site potential can be written as\n\n$$U_i = \\frac{1}{2} \\sum_{j \\neq i} f_{rm C}(r_{ij}) \\left[ f_{rm R}(r_{ij}) - b_{ij} f_{rm A}(r_{ij}) \\right].$$\n\n\u2022 The function $f_{\\rm C}$ is a cutoff function, which is 1 when $r_{ij}\\lt R_{IJ}$ and 0 when $r_{ij}\\gt S_{IJ}$ and takes the following form in the intermediate region:\n\n$$f_{\\rm C}(r_{ij}) = \\frac{1}{2} \\left[ 1 + \\cos \\left( \\pi \\frac{r_{ij} - R_{IJ}}{S_{IJ} - R_{IJ}} \\right) \\right].$$\n\n\u2022 The repulsive function $f_{\\rm R}$ and the attractive function $f_{\\rm A}$ take the following forms:\n\n$$f_{\\rm R}(r) = A_{IJ} e^{-\\lambda_{IJ} r_{ij}};$$ $$f_{\\rm A}(r) = B_{IJ} e^{-\\mu_{IJ} r_{ij}}.$$\n\n\u2022 The bond-order function is\n\n$$b_{ij} = \\chi_{IJ} \\left(1 + \\beta_{I}^{n_{I}} \\zeta^{n_{I}}_{ij}\\right)^{-\\frac{1}{2n_{I}}},$$ where $$\\zeta_{ij} = \\sum_{k\\neq i, j} f_C(r_{ik}) g_{ijk};$$ $$g_{ijk} = \\left( 1 + \\frac{c_{I}^2}{d_{I}^2} - \\frac{c_{I}^2}{d_{I}^2+(h_{I}-\\cos\\theta_{ijk})^2} \\right).$$\n\nParameters\n\n Parameter Units $A_{IJ}$ eV $B_{IJ}$ eV $\\lambda_{IJ}$ A$^{-1}$ $\\mu_{IJ}$ A$^{-1}$ $\\beta_{I}$ dimensionless $n_{I}$ dimensionless $c_{I}$ dimensionless $d_{I}$ dimensionless $h_{I}$ dimensionless $R_{IJ}$ A $S_{IJ}$ A $\\chi_{IJ}$ dimensionless\n\nPotential file format\n\nTersoff-1989 potential for single-element systems\n\n\u2022 In this case, $\\chi_{IJ}$ is irrelevant. The potential file reads\n tersoff_1989 1\nA B lambda mu beta n c d h R S\n\n\nTersoff-1989 potential for double-element systems\n\n\u2022 In this case, there are two sets of parameters, one for each atom type. The following mixing rules are used to determine some parameters between the two atom types $i$ and $j$:\n\n$$A_{IJ} = \\sqrt{A_{II} A_{JJ}};$$ $$B_{IJ} = \\chi_{IJ}\\sqrt{B_{II} B_{JJ}};$$ $$R_{IJ} = \\sqrt{R_{II} R_{JJ}};$$ $$S_{IJ} = \\sqrt{S_{II} S_{JJ}};$$ $$\\lambda_{IJ} = (\\lambda_{II} + \\lambda_{JJ})\/2;$$ $$\\mu_{IJ} = (\\mu_{II} + \\mu_{JJ})\/2.$$\n\n\u2022 Here, the parameter $\\chi_{01}=\\chi_{10}$ needs to be provided. $\\chi_{00}=\\chi_{11}=1$ by definition.\n tersoff_1989 2","date":"2020-01-24 00:27:45","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\": 13, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8393309116363525, \"perplexity\": 1244.2289574022636}, \"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-2020-05\/segments\/1579250614086.44\/warc\/CC-MAIN-20200123221108-20200124010108-00314.warc.gz\"}"} | null | null |
Live at the Cellar Door is the latest release in Neil Young's Archives Performance Series. The album is a collection of recordings made during Young's intimate six-show solo stand at The Cellar Door in Washington D.C. in 1970, a few months after the release of After The Gold Rush. Young fans will savour the rare solo version of "Cinnamon Girl" performed on piano. It's like a remake of his own material and it's beautiful. Another gem is "Bad Fog Of Loneliness"—a song that was buried in the archives until the 2007 release of Live At Massey Hall '71. Every song on this album is a classic, some are a little slower than usual, and there is an undeniable innocence in Young's vocals.The lighthearted banter between songs provides insight into the Neil Young of the 70s—the one some of us missed out on seeing, but are grateful to know through these recordings. | {
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Q: Python/Pandas : Do a value_counts() for each value of a column sorry to bother you but I'm struggling with my code.
What I've been trying to do is to determine the repartition of a column for each value of another column of my dataframe.
I'm going to show an example with the iris dataset, it might be more clear.
I use this code :
import sklearn.datasets
data, target = sklearn.datasets.load_iris(return_X_y=True, as_frame=True)
data["target"] = target
data
With this dataset, I have 5 columns : ['sepal length (cm)', 'sepal width (cm)', 'petal length (cm)',
'petal width (cm)', 'target']
My goal is, for example for each unique values of the column sepal length (cm), to have the repartition of the column target with value_counts(). So like, for the rows where sepal length (cm) = 2, I want to do data["target"].value_counts, and the same for the rows where sepal length (cm) = 1, 3 etc... until I've done it for all the different values of sepal length (cm) and I have the repartition of the target for each value of this column.
I obviously have an idea but it's not very practical.
df1 = data.loc[data['sepal length (cm)'] == 2]
display(df1['target'].value_counts(normalize=True)*100)
It worked, but if I have to do that for each value of sepal length (cm), it's really a long process.
So if someone know how to do that automatically, it might save my day !!
Thanks a lot in advance
A: If I understand you correctly, you want to group the dataframe by sepal length (cm) and find the number of occurrences in each group by the target column. If this is the case, you can try something like this:
data.groupby(['sepal length (cm)'])['target'].agg('count').reset_index()
This gives you an output like this:
Again, I'm not sure if this is the thing you need.
| {
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W dziewiętnastu sezonach piłkarskich mistrzostw Ukrainy w najwyższej lidze w latach 1992-2010 wystąpiło 35 klubów. Rozegrały 4585 meczów (1089 zakończyło się remisem), strzelając 10996 bramki (średnio 2,4 na spotkanie).
Tabela (stan na 18 maja 2010)
Uwaga: W sezonie 2001/2002 w rundzie jesiennej występował klub CSKA Kijów, a w rundzie wiosennej zastąpił go Arsenał Kijów, dlatego sezon podzielony na pół.
Medaliści
Najwięcej startów
Najlepsza średnia zwycięstw
Najlepsza średnia zdobytych bramek
Najlepsza średnia straconych bramek
Najlepsza średnia punktów
Przypisy
Linki zewnętrzne
Tabela wszech czasów na uafootball.net.ua
Tabela wszech czasów ukranianfootball.narod.ru
Piłkarskie rankingi
Rozgrywki piłkarskie na Ukrainie | {
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{"url":"https:\/\/www.iacr.org\/cryptodb\/data\/author.php?authorkey=6117","text":"CryptoDB\n\nPublications\n\nYear\nVenue\nTitle\n2021\nEUROCRYPT\nSecure multi-party computation (MPC) allows multiple parties to perform secure joint computations on their private inputs. Today, applications for MPC are growing with thousands of parties wishing to build federated machine learning models or trusted setups for blockchains. To address such scenarios we propose a suite of novel MPC protocols that maximize throughput when run with large numbers of parties. In particular, our protocols have both communication and computation complexity that decrease with the number of parties. Our protocols build on prior protocols based on packed secret-sharing, introducing new techniques to build more efficient computation for general circuits. Specifically, we introduce a new approach for handling \\emph{linear attacks} that arise in protocols using packed secret-sharing and we propose a method for unpacking shared multiplication triples without increasing the asymptotic costs. Compared with prior work, we avoid the $\\log |C|$ overhead required when generically compiling circuits of size $|C|$ for use in a SIMD computation, and we improve over folklore committee-based'' solutions by a factor of $O(s)$, the statistical security parameter. In practice, our protocol is up to $10X$ faster than any known construction, under a reasonable set of parameters.\n2019\nJOFC\nWe continue the line of work initiated by Katz (Eurocrypt 2007) on using tamper-proof hardware tokens for universally composable secure computation. As our main result, we show an oblivious-transfer (OT) protocol in which two parties each create and transfer a single, stateless token and can then run an unbounded number of OTs. We also show a more efficient protocol, based only on standard symmetric-key primitives (block ciphers and collision-resistant hash functions), that can be used if a bounded number of OTs suffice. Motivated by this result, we investigate the number of stateless tokens needed for universally composable OT. We prove that our protocol is optimal in this regard for constructions making black-box use of the tokens (in a sense we define). We also show that nonblack-box techniques can be used to obtain a construction using only a single stateless token.\n2017\nPKC\n2014\nTCC\n2011\nTCC\n2011\nTCC\n2011\nTCC\n2009\nASIACRYPT\n\nEurocrypt 2022","date":"2022-05-28 02:05:45","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\": 0.6005528569221497, \"perplexity\": 1301.2683206265358}, \"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-2022-21\/segments\/1652663011588.83\/warc\/CC-MAIN-20220528000300-20220528030300-00500.warc.gz\"}"} | null | null |
L'α-cétoglutarate déshydrogénase, ou 2-oxoglutarate déshydrogénase (OGDH), est une oxydoréductase qui fait partie du complexe alpha-cétoglutarate déshydrogénase, dont elle est l'enzyme E1, complexe qui réalise la conversion de l'α-cétoglutarate en succinyl-CoA et :
Plus précisément, l'enzyme E1 catalyse la fixation de l'α-cétoglutarate sur un lipoamide avec l'aide de thiamine pyrophosphate (TPP).
Le mécanisme de cette réaction, qui fait intervenir successivement les enzymes E1, E2 et E3, chacune avec ses cofacteurs, est assez complexe, et peut être résumé par le schéma simplifié ci-dessous :
Notes et références
Articles connexes
Pyruvate déshydrogénase, enzyme E1 du complexe pyruvate déshydrogénase
3-méthyl-2-oxobutanoate déshydrogénase, enzyme E1 du complexe 3-méthyl-2-oxobutanoate déshydrogénase
EC 1.2.4
Cycle de Krebs
Chromosome 7 humain | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,215 |
package com.example.liangjie06.zuche.activity;
import android.os.Bundle;
import android.support.v4.app.Fragment;
import android.support.v4.app.FragmentActivity;
import android.view.KeyEvent;
import android.view.Window;
import android.widget.RadioGroup;
import android.widget.Toast;
import com.example.liangjie06.zuche.R;
import com.example.liangjie06.zuche.base.BaseActivity;
import com.example.liangjie06.zuche.module.mainpager.HomeFragment;
import com.example.liangjie06.zuche.module.mainpager.NearByFragment;
import com.example.liangjie06.zuche.module.mainpager.OrderFragment;
import com.example.liangjie06.zuche.module.mainpager.PersonalFragment;
import com.example.liangjie06.zuche.module.selectcar.adapter.MyHomePagerAdapter;
import com.example.liangjie06.zuche.view.NoCacheViewPager;
import java.util.ArrayList;
import java.util.Timer;
import java.util.TimerTask;
/**
* Created by Jack-Liang on 2016/8/22.
*/
public class MainActivity extends BaseActivity {
private ArrayList<Fragment> fragments;
private NoCacheViewPager mViewPager;
private RadioGroup rgGroup;
@Override
public void onCreate(Bundle savedInstanceState) {
super.onCreate(savedInstanceState);
requestWindowFeature(Window.FEATURE_NO_TITLE);
setContentView(R.layout.fragment_content);
mViewPager = (NoCacheViewPager) findViewById(R.id.vp_content);
//mViewPager.setPagingEnabled(false);
rgGroup = (RadioGroup) findViewById(R.id.rg_group);
fragments = new ArrayList<Fragment>();
fragments.add(new HomeFragment());
fragments.add(new OrderFragment());
fragments.add(new PersonalFragment());
getFragmentManager();
MyHomePagerAdapter adapter = new MyHomePagerAdapter(getSupportFragmentManager(), fragments);
mViewPager.setAdapter(adapter);
mViewPager.setOffscreenPageLimit(0);
rgGroup.setOnCheckedChangeListener(new RadioGroup.OnCheckedChangeListener() {
@Override
public void onCheckedChanged(RadioGroup group, int checkedId) {
switch (checkedId) {
case R.id.rb_home:
mViewPager.setCurrentItem(0, false);
break;
case R.id.rb_smart:
mViewPager.setCurrentItem(1, false);
break;
case R.id.rb_gov:
mViewPager.setCurrentItem(2, false);
break;
default:
break;
}
}
});
}
/**
* 菜单、返回键响应
*/
@Override
public boolean onKeyDown(int keyCode, KeyEvent event) {
// TODO Auto-generated method stub
if (keyCode == KeyEvent.KEYCODE_BACK) {
exitBy2Click(); //调用双击退出函数
}
return false;
}
/**
* 双击退出函数
*/
private static Boolean isExit = false;
private void exitBy2Click() {
Timer tExit = null;
if (isExit == false) {
isExit = true; // 准备退出
Toast.makeText(this, "再按一次退出程序", Toast.LENGTH_SHORT).show();
tExit = new Timer();
tExit.schedule(new TimerTask() {
@Override
public void run() {
isExit = false; // 取消退出
}
}, 2000); // 如果2秒钟内没有按下返回键,则启动定时器取消掉刚才执行的任务
} else {
finish();
System.exit(0);
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,420 |
\section{Introduction}
\label{sec:introduction}
Early-type galaxies are found in virtually all environments ---
from the field, to small groups, to rich clusters (Hubble \& Humason 1931;
Oemler 1974; Dressler 1980). In the highest density environments,
ellipticals and lenticulars are known to dominate the overall fraction of
bright galaxies, $f_{\rm E+S0} \sim 0.4-0.9$, with the precise contribution
depending on local galaxy density and redshift (Smith {et~al.}\ 2004;
Postman {et~al.}\ 2005). In the Virgo Cluster, the rich cluster nearest to
our own Galaxy, $f_{\rm E+S0} \approx 0.44$ for galaxies brighter than
$B\lesssim 13$ or $M_B \lesssim -18.1$ (Julian et~al. 1997).\footnote{Throughout
this paper, we adopt a Virgo distance modulus of $(m-M)_0 = 31.09$ mag
(Tonry {et~al.}\ 2001; Mei {et~al.}\ 2005b).} If one
considers not just giant galaxies, but also the much more common dwarfs,
then the dominance of early-type galaxies is even more pronounced: i.e.,
among the confirmed members of Virgo with unambiguous morphological
classifications, the early-type fraction is $\approx$ 0.8 (Reaves 1983;
Binggeli, Tammann \& Sandage 1987, hereafter BTS87).
It has long been recognized that early-type galaxies, both in Virgo and
elsewhere, often show compact nuclei near their centers. In their landmark study
of the Virgo Cluster, BTS87 carried out a visual search for nuclei
using wide-field, blue-sensitive photographic plates from the 2.5-m
du Pont telescope. Of the 1277 members and 574 probable members in their
Virgo Cluster Catalog (hereafter VCC), a total of 1192 were classified as non-nucleated dwarfs
(dEs or dS0s) while an additional 415 dwarfs (predominantly dE,Ns) were
found to be nucleated. Thus, roughly 25\% of the dwarf
galaxies in Virgo were found by BTS87 to have a discernible nucleus,
although the precise fraction was also found to depend on galaxy
luminosity and position within the cluster (see Figure~8 of Sandage,
Binggeli \& Tammann 1985 and Figure~19 of BTS87, respectively).
Unfortunately, progress toward understanding the nature of these nuclei
has been limited because of several factors: $e.g.$, ground-based studies must
contend with the contamination from the underlying galaxy light, it is
difficult to de-couple the brightness profiles of the
nuclei from those of their host galaxies, and the nuclei are
sufficiently compact that they appear unresolved in even the sharpest
ground-based images (Caldwell 1983, 1987; Durrell 1997).
While the photographic survey of BTS87 remains a landmark study of
nucleated galaxies in the local universe, there are reasons to believe
that a modern survey of the nuclei belonging to early-type galaxies would
be advisable --- preferably one which capitalizes on the high angular
resolution afforded by the {\it Hubble Space Telescope} ({\it HST}).
First, {\it HST} imaging of late-type galaxies has revealed that
50--70\% of these systems have compact stellar clusters at or near
their photocenters (Phillips et~al. 1996; Carollo, Stiavelli \& Mack
1998; Matthews et~al 1999; B\"oker et~al. 2002; B\"oker et~al. 2004).
Second, if the early-type members of the Local Group are
any guide, then one may expect estimates for the fraction of nucleated galaxies to increase
as better imaging becomes available. For instance, in recent years
a number of Local Group dwarfs have been found to contain
previously unrecognized central substructures and/or nuclei, including
Sagittarius (Layden \& Sarajedini 2000; Monaco et~al. 2005),
Ursa Minor (Kleyna et~al. 2003; Palma et~al. 2003),
Andromeda II (McConnachie \& Irwin 2006)
and Fornax (Coleman et~al. 2004). Third, in their WFPC2
survey of dwarfs in the Virgo and Fornax Clusters, Lotz et~al. (2004)
found that six of the 30 ``non-nucleated" dwarf ellipticals in their sample
actually contained nuclei which had gone unnoticed in the ground-based
surveys (Binggeli, Tammann \& Sandage 1985, hereafter BTS85; BTS87;
Ferguson 1989). Very recently, Grant, Kuipers \& Phillipps (2005) used
imaging from the Wide Field Camera on the Isaac Newton Telescope to
show that faint nuclei in Virgo dwarfs were frequently missed in
photographic surveys.
These results suggest that there may be significant incompleteness in our census of
nuclei in early-type galaxies.
Indeed, in their photographic study of Virgo, BTS87 cautioned that ``most nuclei in the
luminous E and S0 galaxies were probably missed due to [the] high surface
brightness [of the underlying galaxy.]" In addition to this surface-brightness
selection effect, BTS87 state explicitly that any nuclei with $B \gtrsim$ 23
($M_B \gtrsim -8$) would fall below their plate detection limits and hence
be missing from their catalog.
The central regions of early-type galaxies have been favorite targets for
{\it HST} since its launch in 1990. For the most part, such surveys
have tended to focus on the core structure of the galaxies. However, several
studies reported the discovery of compact nuclei in (predominantly bright)
samples of early-type galaxies, beginning with pre-refurbishment (WFPC) imaging
(Crane {et~al.}\ 1993; Lauer {et~al.}\ 1995) and continuing with imaging from
WFPC2 (Rest {et~al.}\ 2001; Lauer {et~al.}\ 2005) and NICMOS (Ravindranath {et~al.}\ 2001).
These studies, which are primarily based on single-filter imaging of samples of
(33--77) galaxies with distances between 1 and 100~Mpc, have confirmed that some
early-type galaxies do contain compact nuclei, but there is
disagreement over their overall frequency (with estimates ranging from 13\%
to $\approx$ 50\%), whether or not they are resolved structures, and their
classification as stellar or non-thermal (AGN) sources.
A better understanding of the physical properties of these nuclei is
important since they almost certainly hold clues to the violent processes
that have shaped the central regions of galaxies, which could include star
formation triggered by infalling gas, collisions and mergers of stars and
star clusters, tidal disruption of clusters and the growth of stellar ``cusps"
by central black holes, and the mechanical and radiative feedback from
accreting black holes or intense nuclear starbusts.
This paper presents a homogenous analysis of the nuclei belonging to
a sample of 100 early-type galaxies in the Virgo Cluster. Our images,
taken with the Advanced Camera for Surveys (ACS; Ford et~al. 1998),
form the basis of the ACS Virgo Cluster Survey (ACSVCS; C\^ot\'e {et~al.}\ 2004;
hereafter Paper~I). Other papers in this series have discussed the
data reduction pipeline (Jord\'an et al. 2004a = Paper II),
the connection between low-mass X-ray binaries in M87 (Jord\'an et al.
2004b = Paper III), the measurement and calibration of surface
brightness fluctuation magnitudes (Mei et~al. 2005ab = Papers IV and V),
the morphology, isophotal parameters and surface brightness
profiles for early-type galaxies (Ferrarese et~al. 2006a = Paper VI),
the connection between globular clusters and ultra-compact dwarf
galaxies (Ha\c{s}egan et al. 2005 = Paper VII), the color distributions
of globular clusters (Peng et~al. 2006a = Paper IX), the half light radii
of globular clusters and their use as a distance indicator (Jord\'an et al.
2006 = Paper X) and the discovery of diffuse star clusters in early-type
galaxies (Peng et~al. 2006b = Paper XI).
There are several features of the ACS Virgo Cluster Survey which make it uniquely
suited to the study of nuclei in early-type
galaxies. First, the survey itself targets a large sample of 100 early-type
galaxies lying at a common distance of about 16.5~Mpc so that the
$\approx 0\farcs1$ FWHM
of the ACS point-spread function (PSF) corresponds to a small, and nearly
constant, physical scale of $\approx 8$~pc. This excellent spatial
resolution, coupled with the fine plate scale of 0\farcs049~pix$^{-1}$,
makes it possible to measure
structural parameters for any nuclei larger than a few parsecs in size.
Second, with blue magnitudes in the range $9.31 \lesssim B_T \lesssim 15.97$
($-21.88 \lesssim M_B \lesssim -15.21$),
our program galaxies span a wide range in luminosity
so it is possible to study the
phenomenon of nucleation in giant and dwarf galaxies simultaneously.
Third, the images are sufficiently deep that they reveal not only the nuclei,
but also the many globular clusters belonging to
our program galaxies; thus, the same images which provide information on
the nucleus and host galaxy can also be used to study the associated
globular cluster systems and to explore possible evolutionary links between the
clusters and nuclei. And finally, because multi-band imaging is available
in two widely-separated bandpasses (F475W and F850LP)
for each object in the survey, it is possible to use broadband colors to
place rough constraints on the star formation and chemical enrichment histories
of the nuclei and their host galaxies.
The organization of this article is as follows. \S2 gives a brief summary of
the observational material used in our analysis. A description of the galaxy
brightness profiles and the method of their analysis is presented in \S3.
\S4 contains a discussion of the empirical properties of the
nuclei in our survey, such as their overall numbers, possible displacements
from the galaxy photocenters, luminosities, colors, surface brightnesses and half-light radii.
In \S5 we discuss the implications of our findings for various formation
scenarios. The article concludes with a summary of the main results in \S6.
A future paper in this series will discuss the broader implications of our
findings for models of nucleus formation in early-type galaxies (Merritt {et~al.}\
2006).
\section{Observations and Data Reductions}
\label{sec:obdata}
Our analysis is based on \textit{HST} imaging for 100 early-type galaxies
having morphological types E, S0, dE, dE,N and dS0. All are confirmed members
of the Virgo Cluster based on radial velocity measurements. Images were taken
with the ACS instrument used in Wide Field Channel (WFC) mode
with the F475W and F850LP filter combination, which are roughly equivalent
to the $g$ and $z$ bands, respectively, in the Sloan Digital Sky Survey
photometric system. These images form the basis
of the ACS Virgo Cluster Survey, a complete description of which may be
found in Paper~I. Note that the 26 brightest galaxies in this survey constitute
a complete sample of early-type members of Virgo with $B_T \le 12.15$,
and that the full sample represents 44\% of all early-type members of
Virgo spanning the magnitude range $9.3 \lesssim B_T \lesssim 16$. A
customized data pipeline (described in detail in Paper II) produces
geometrically-corrected, flux-calibrated, cosmic-ray-free images in the
F475W and F850LP bandpasses.
Table~\ref{tab:data} gives some basic information about the target galaxies, tabulated in order
of increasing blue magnitude (decreasing luminosity). An identification number
for each galaxy is given in the first column, followed by the identification from the
VCC (BST85) and other names for the galaxy in the Messier, NGC, UGC or IC catlogs.
Blue magnitudes, $B_T$, from BST85
are presented in column 4, while the fifth column records the
adopted Galactic reddening from Schlegel, Finkbeiner \& Davis (1998). Columns
6 and 7 record the surface brightness of each galaxy, in both the $g$
and $z$ bandpasses, measured via spline interpolation at a geometric mean
radius\footnote{The geometric mean radius is defined as $r \equiv a\sqrt{(1-\epsilon)}$
where $a$ is the semi-major axis and $\epsilon$ is the ellipticity.} of
$r =$~1\arcsec~($\approx$ 80~pc).
This model-independent choice of surface brightness should closely approximate
the galaxy's {\it central} surface brightness, but is measured at a radius
large enough to ensure negligible contamination from any central nucleus.
The remaining columns of Table~\ref{tab:data} will be described below. Coordinates,
morphological classifications and other information on the program galaxies
may be found in Papers I and VI.
\section{Analysis}
\label{sec:analysis}
Our goals in this paper include the measurement of the structural and
photometric properties of the nuclei in our program galaxies,
and an investigation into the relationship between these nuclei
and their host galaxies. Additionally, we
wish to compare the properties of the nuclei to
those of the globular clusters in the program galaxies and,
more generally, to the Virgo ultra-compact dwarf
(UCD) galaxies (e.g., Drinkwater et~al. 1999; Hilker et~al. 1999;
Drinkwater et~al. 2000; Phillipps et~al. 2001)
identified in the course of this
survey and described in Ha\c{s}egan {et~al.}\ (2005;
hereafter Paper~VII) and Ha\c{s}egan {et~al.}\ (2006). A companion
paper in this series (Paper~VI) presents
an analysis of the surface brightness profiles of the
program galaxies along with a tabulation of the best-fit structural
parameters, while two other articles examine the photometric (Paper~IX)
and structural (Paper~X) parameters of the globular clusters.
As we make use of several results from these
studies, the reader is referred to these papers for complete details.
\subsection{Parameterization of the Surface Brightness Profiles}
Because the nuclei are always superimposed on the light of the underlying
galaxy, measuring their photometric and structural properties requires
a model for the galaxy surface brightness profile. For each galaxy
in our survey, $g$- and $z$-band azimuthally-averaged radial surface
brightness profiles are available from Paper~VI. These profiles were
derived by fitting the isophotes with the ELLIPSE task in IRAF which,
in turn, is based on the algorithm of Jedrzejewski (1987). The $g$- and $z$-band
brightness profiles were parameterized with a standard S\'{e}rsic (1968) model,
\begin{equation}
\begin{array}{rrrrr}
$$I_{g}(r) & = & I_0\exp[-b_n(r/r_e)^{1/n})],$$
\end{array}
\label{eq1}
\end{equation}
where $I_0$ is the central intensity and
$n$ is a shape parameter which yields an $R^{1/4}$-law profile for $n =4$
(de Vaucouleurs 1948) and an exponential profile for $n = 1$.
The parameter $b_n$ is defined such that
$\Gamma(2n) = 2\Gamma_1(2n,b_n)$, where $\Gamma$ and $\Gamma_1$ are
the complete and incomplete gamma functions, respectively (e.g.,
Graham \& Driver 2005).
As shown by Caon, Capaccioli \& D'Onofrio (1993), a convenient
approximation relating $b_n$ to the shape parameter $n$ is
$b_n \approx 1.9992n - 0.3271$ for $1 \lesssim n \lesssim 10$. Given
this definition of $b_n$, $r_e$ is the effective radius of the galaxy.
The $g$- and $z$-band brightness profiles for each galaxy
were also fit with a ``core-S\'{e}rsic'' model,
\begin{equation}
\begin{array}{rrrrr}
I(r) = I^{\prime} \biggl [1 + \biggl ( {r_b \over r} \biggr ) ^{\alpha} \biggl ]^{\gamma / \alpha} \exp \biggl [-b_n \biggl ( {r^{\alpha} + r_b^{\alpha} \over r_e^{\alpha}} \biggr ) ^{1/(\alpha n)} \biggr ],
\end{array}
\label{eq2}
\end{equation}
where
\begin{equation}
\begin{array}{rrrrr}
I^{\prime} = I_b2^{-\gamma / \alpha} \exp \biggl [b_n \biggl (2^{1/\alpha}r_b/r_e \biggl )^{1/n} \biggr ]
\label{eq3}
\end{array}
\end{equation}
This model, which was first proposed by Graham {et~al.}\ (2003),
consists of a power-law
component in the inner region of a galaxy, which ``breaks'' to a
traditional S\`{e}rsic profile beyond some radius, $r_{b}$.
The model has a total of six free parameters: the
logarithmic slope of the inner power-law ($\gamma$);
the shape of the S\'{e}rsic function ($n$); the break radius ($r_b$);
the effective half-light radius of the S\'{e}rsic profile ($r_e$);
the intensity at the break radius ($I_b$) and
a parameter ($\alpha$) which governs the sharpness of the
transition between the inner power law and the outer S\'{e}rsic function. After
some experimentation, it was decided to use the modified parametrization of
Trujillo {et~al.}\ (2004),
$$
I_{g}(r) = I_b \biggl [(r_b/r)^{\gamma}u(r_b-r) +
$$
\begin{equation}
+e^{b_n(r_b/r_e)^{1/n}}e^{-b_n(r/r_e)^{1/n}}u(r-r_b) \biggr ]
\label{eq4}
\end{equation}
in which $\alpha \rightarrow \infty$ and $u(x-z)$ is the Heaviside step function.
This
model produced more stable fits, with better consistency between
the five remaining parameters ($I_b$, $\gamma$, $n$, $r_e$ and $r_b$)
measured in the $g$ and $z$ bandpasses.
Equations (1) and (4) are intended to describe the profiles of
galaxies which have no central nucleus. However, it is obvious that many
galaxies in our sample do indeed have compact sources at or near their centers.
For such nucleated galaxies, a single-component King model (Michie 1963;
King 1966) was used to represent this central component. This introduces three
additional parameters to the fit: the total intensity of the nucleus ($I$);
the projected half-light radius ($r_h$); and the King concentration index
($c$). In other words,
for {\it nucleated
galaxies}, the fitted model, $I(r)$, takes the form
\begin{equation}
\begin{array}{rrrrr}
$$I(r) & = & I_{g}(r) + I_k(r),$$
\end{array}
\label{eq5}
\end{equation}
where $I_{g}(r)$ is either a pure S\'ersic model (Equation 1)
or a core-S\'ersic model (Equation 4), depending on the galaxy in question,
and $I_k(r)$ is the central King model component.
For non-nucleated galaxies, the profiles are fit simply with models
of the form of Equations (1) or (4).
A detailed justification for the choice of galaxy model (i.e.,
S\'{e}rsic vs. core-S\'{e}rsic) is given on a case-by-case basis
in Paper~VI. We adopt these classifications verbatim,
with the exception of three intermediate-luminosity galaxies: VCC543
(UGC7436), VCC1528 (IC3501) and VCC1695 (IC3586).\footnote{Note that
$r_b >> r_h$ for all nucleated (Type~Ia) core-S\'{e}rsic galaxies;
$\langle r_b/r_h\rangle = 74$ for the four galaxies in this category.}
While the {\it global} brightness profiles of
these galaxies are adequately represented by S\'ersic models, such
models overpredict the amount of galaxy light on subarcsecond
scales. For the purposes of measuring photometric and structural
properties for the nuclei in these galaxies, we parameterize the
galaxy profiles with core-S\'ersic models in all three cases.
We note that the definition of a ``nucleus" invariably
hinges on some assumption --- explicit or otherwise --- about the intrinsic
brightness profile of the host galaxy. Our study is no exception in
this regard. Choices for the galaxy profiles made by previous workers have
included King models (Binggeli \& Cameron 1993), pure exponentials (Binggeli
\& Cameron 1993; Stiavelli {et~al.}\ 2001), Nuker laws (Rest {et~al.}\ 2001;
Ravindranath {et~al.}\ 2001; Lauer {et~al.}\ 2005) and Sersic profiles (Durrell
1997; Stiavelli {et~al.}\ 2001). After considerable experimentation (Paper IV),
we opted to use the family of models represented by Equations~(1) and
(2) because they have the great advantage they are flexible
enough to provide accurate fits to the brightness profiles of both
giant and dwarf galaxies (see Paper~VI).
The use of a single
(S\'{e}rsic) model to describe the full sample of galaxies also seems
advisable in light of mounting evidence that, at least in terms of their
{\it structural parameters}, the longstanding perception of a
fundamental dichotomy between giant and dwarf ellipticals
(e.g., Kormendy 1985) may be incorrect (see, e.g., Jerjen \&
Binggeli 1997; Graham \& Guzman 2003; Paper~VI). From a theoretical perspective, the
choice of Equations~(1) and (4) also seems reasonable given recent
findings that the S\'{e}rsic law provides an accurate representation
of the spatial and surface density profiles of dark matter halos
in high-resolution $\Lambda$CDM simulations (Navarro et~al. 2004;
Merritt et~al. 2005).
At the same time, our decision to parameterize the central nuclei with King models is motivated
by high-resolution observations of the nuclei in nearby galaxies. In nucleated
Local Group galaxies such as NGC205 and Sagittarius,
King models are found to provide accurate representations of the central
components (e.g., Djorgovski {et~al.}\ 1992; Butler \& Mart\'inez-Delgado 2005;
Monaco {et~al.}\ 2005). Nevertheless, for galaxies at the distance of the Virgo
Cluster, we are working close to the limits of resolvability, so we caution that
our choice of King models to parameterize the central components may not be unique,
particularly for faint nuclei in the highest surface brightness galaxies.
Alternative parameterizations of the central brightness ``cusps" in our sample
galaxies will be explored in a future paper in this series (Merritt {et~al.}\ 2006).
\subsection{Choice of Drizzling Kernel, PSF Determination and Fitting Procedure}
As described in Paper~II, our analysis of the nuclei, brightness profiles,
and isophotal structure of the galaxies is based on F475W and F850LP images in which
a {\it Gaussian} kernel is used to
distribute flux onto the output (drizzled) images. This choice of kernel
has the advantage that, relative to {\it Lanczos3} kernel, bad pixels can be repaired
more effectively, albeit with the penalty of a slight reduction in angular
resolution.\footnote{Using the {\it Lanczos3} kernel produces images with
better noise characteristics and a somewhat sharper PSF (0\farcs09 versus 0\farcs1),
so this kernel was used for both the surface brightness fluctuation measurements
and the determination of the globular cluster photometric and structural parameters.}
Due to the compact
nature of the nuclei (even the most extended objects have effective radii
$\lesssim 1$\arcsec), it is important that the effects of the PSF
are taken into account when fitting models to the observed brightness profiles.
PSFs in the F475W and F850LP filters, varying quadratically
with CCD position, were derived using DAOPHOT II as described in Paper~II. Briefly,
archival images of the Galactic globular cluster NGC104 (47~Tucanae) taken during
programs G0-9656 and GO-9018 were used to construct empirical PSFs in the two
bandpasses. These archival images were drizzled in the same manner as the
images for the program galaxies. A total of $\approx$~200 stars in each filter were used
to construct the PSFs, which extend to a radius of 0\farcs5 in both bandpasses. To
follow the behavior of the PSFs to still larger radii, we matched our
empirical PSFs at a radius of 0\farcs3 to those measured for high-S/N composite stars
by Sirianni {et~al.}\ (2005). These latter PSFs extend to radii of 3\arcsec, and
were constructed from images of 47~Tuc fields taken as part of the
photometric calibration of ACS. Figure~\ref{fig01} shows azimuthally averaged
PSFs for the F475W and F850LP filters measured at the position of the nucleus in
VCC1303 (NGC4483) --- the program galaxy whose center is nearest to the mean
position for the full sample of program galaxies.
A $\chi^2$ minimization scheme was used to find the models which best fit the
azimuthally-averaged, one-dimensional intensity profiles for each galaxy.
Minimizations were carried out using the {\tt Minuit} package in the CERN program
library; initial determinations of the minima, obtained using
a Simplex minimization algorithm (Nelder \& Mead 1965), were later refined using a variable
metric method with inexact line search (MIGRAD).
Following Byun et al. (1996), all points in the profile were assigned equal weight.
For both nucleated and non-nucleated galaxies, the
PSFs at the location of the galaxy's center were convolved with the models
before fitting to the intensity profiles.
Customized PSFs were created for each galaxy in the survey, centered at the exact
(sub-pixel) location of the nucleus.
While, in practice, the PSF convolution has little impact on the fitted S\'ersic or
core-S\'{e}rsic model parameters, with the exception of $\gamma$, this step
is critically important when evaluating accurate structural parameters
for the central nucleus.
Profile fits are carried out independently in the two
bandpasses, with the exception of the 11 nucleated galaxies brighter
than $B_T = 13.5$ (i.e., Type~Ia galaxies; see \S\ref{sec:results}). Our
numerical experiments suggest that in this
high surface brightness regime, the profile of the underlying galaxy
makes the measurement of nuclei half-light radii and total magnitudes
extremely challenging (see Appendix~A). For these
galaxies, the composite $g$- and $z$-band profiles were first fitted
simultaneously and the individual
fits constrained so that the galaxy shape index parameter,
nucleus concentration index and half-light radius were the same in
the two bandpasses. When dust is present (see below), the models
are fitted to the dust-corrected surface brightness profiles if
$\ge$ 50\% of the points along a given isophote are affected;
otherwise the dust affected regions are masked. More details on the
correction for dust obscuration are given in Paper~VI.
Sufficiently compact nuclei will appear unresolved even in our ACS images. To estimate
the resolution limit of our observations, we constructed brightness profiles
for a number of likely stars which appear in our images. These candidate stars
were classified as unresolved
in the object catalogs produced by KINGPHOT, the reduction package used to measure
structural and photometric parameters for the globular clusters in these
fields (see Papers II and X). Fitting King models to the brightness profiles
of these objects gives median half-light radii of 0\farcs011$\pm$0\farcs004 and
0\farcs018$\pm$0\farcs005 in the F475W and F850LP bandpasses, respectively.
As an additional test, we may make use of the fact that VCC1316 (M87 = NGC4486), one of
the AGN galaxies in our survey (see below), contains a prominent non-thermal
central point source. Although this source is saturated in our F475W images,
a King model fitted to the central source in the $z$-band brightness profile
gives $r_h = 0\farcs021$. In what follows, we
adopt a conservative upper limit of 0\farcs025 $\approx$ 2~pc for
the resolution limit in both bandpasses.
Before proceeding, we pause to demonstrate that the vast majority of the nuclei
belonging to our program galaxies are indeed more extended than point sources. In
Figure~\ref{fig02}, we show $g$-band surface brightness profiles
for a representative sample of nine nucleated galaxies, chosen to span the full
range in fitted half-light radius (with $\langle{r_h}\rangle$ decreasing from left
to right and from top to bottom). In each panel, the red curves show the results of
fitting the nuclear component with a King model, while the blue curves show the
results of fitting a central point source; residuals from both fits are shown in the
lower panel. With the exception of VCC1528 (IC3501), the central nucleus
is resolved for all of the galaxies in Figure~\ref{fig02}. In total, six
galaxies in our sample --- VCC1883 (NGC4612), VCC140 (IC3065), VCC1528,
VCC1695 (IC3586), VCC1895 (UGC7854) and VCC1826 (IC3633) --- have
best-fit half-light radii, measured in at least one bandpasses, that fall below
our nominal resolution limit of 0\farcs025. These half-light radii are given
in parantheses in Table~\ref{tab:data}. They have been included in the
following analysis, but we caution that they are formally unresolved in
our ACS images. We shall return to the issue of these compact nuclei in
\S\ref{sec:discussion}.
Additional tests on the resolution
limits, possible biases in the derived photometric and structural
parameters, and a discussion of measurement errors, are given in
\S4.1 and Appendix~A.
\clearpage
\section{Results}
\label{sec:results}
As many as eighteen of the 100 galaxies in Table~\ref{tab:data} show evidence for
dust --- either as isolated patches and filaments, or in the form of disks having
varying degrees of regularity (see Paper~VI). For the most part, this
dust has no impact on the identification of possible nuclei. However,
for four galaxies in our sample (i.e., VCC1535 = NGC4526, VCC1030 =
NGC4435, VCC685 = NGC4350 and VCC571) the central dust obscuration is
severe enough to make a reliable classification of these galaxies as
nucleated or non-nucleated impossible. Moreover, for VCC1535 and VCC1030, both
of which harbour massive, kpc-scale dust disks, the surface brightness
profiles are themselves so limited that it is not even possible to
place the galaxies in the appropriate S\'ersic or core-S\'ersic
categories.
In general, the census of active galactic nuclei (AGN) in intrinsically
faint galaxies --- and in the ACS Virgo Cluster Survey galaxies
in particular --- is far from complete. However, two of the brighter galaxies in our sample
(VCC1316 = NGC4486, M87, 3C 274 and VCC763 = NGC4374, M84, 3C 272.1) are
known to host AGNs with strongly non-thermal spectral energy distributions
(e.g., Wrobel 1991; Ho 1999; Chiaberge, Capetti \& Celotti 1999).
In both cases, the
unresolved non-thermal nucleus is clearly seen in the ACS images;
in neither instance, however, does there appear to be a resolved
stellar nucleus. A third galaxy (VCC1619 = NGC4550), is classified
as a LINER by Ho {et~al.}\ (1997). This galaxy contains some dust within the
central $\sim$ 25\arcsec, but there is clear evidence for a resolved
stellar nucleus.
Wrobel (1991) detected nuclear radio emission in three other galaxies
in our survey (VCC1226 = NGC4472, M49; VCC1632 = NGC4552, M89; and VCC1978 =
NGC4649, M60). In both VCC1226
and VCC1632, the innermost $\sim$ 1\arcsec~are slightly obscured by dust
(see Paper VI), but once a correction for dust obscuration is performed,
there is no evidence of a stellar nucleus in either case. We see no sign
of a nucleus in VCC1978.
A search for low-level AGN in
our program galaxies is now underway using low- to intermediate-resolution
ground-based optical spectra, the results of which will be presented in a
future paper in this series. These spectroscopic data will be useful in
establishing the extent to which non-thermal sources are responsible for,
or contribute to, the central luminosity excesses observed in a number of
these galaxies. For the time being,
Table~\ref{tab:class} summarizes our classifications for the program galaxies, as
discussed in the next section. We begin by defining a class of galaxies
(Type 0) in which dust obscuration (four galaxies) or AGN emission (two galaxies)
renders a reliable classification as nucleated or non-nucleated impossible. In what
follows, we shall limit our analysis to the remaining 94 galaxies.
\subsection{Identification and Classification of the Nuclei}
As a first step in the identification of nuclei in our program galaxies,
the $g$- and $z$-band surface brightness profiles were each fitted with the
appropriate galaxy model (i.e., either a pure S\'ersic or core-S\'ersic model)
outside a geometric mean radius of 0\farcs5. Those galaxies with brightness profiles
which lay systematically above the inward extrapolation of fitted model for
$\lesssim 0\farcs5$ were considered to be nucleated. Because
many of the nuclei are somewhat bluer than the underlying galaxies, a
central excess was often more apparent in the $g$-band profile than in
the redder bandpass. In addition to classifying the galaxies on the
basis of their brightness profiles, the F475W and F850LP images for each
galaxy were carefully inspected for the presence of a distinct central excess. Using
these two criteria, a total of 62 galaxies were found to show clear
evidence for a central nucleus; such galaxies are classified
as Type Ia or Ib.
Unfortunately, for 11 of these 62 galaxies, although the presence of a
faint central component could be established from the images themselves or
from the brightness profiles, the nucleus itself was too faint
to allow us to recover trustworthy photometric or structural parameters
from the surface brightness profiles. Such galaxies are
referred to as Type Ib in Tables~\ref{tab:data} and \ref{tab:class}. Our analysis of the structural and
photometric properties of the nuclei is therefore based on the subset of
51 nucleated galaxies for which it was possible to obtain a reliable
fit to the central brightness profiles: i.e., Type~Ia galaxies.
The Type~Ib galaxies are classified as
nucleated for the purposes of computing the overall frequency of
nucleation, but their nuclei are omitted from the analysis in \S4.4 to 4.8.
Of the remaining 94--62 = 32 galaxies, five may have nuclei which
are offset by $\approx 0\farcs5$ or more from the centers of the
isophotes (Type~Ie galaxies). We consider these five galaxies in
more detail in \S4.3. The remaining 94--62--5 = 27 objects consist of
galaxies which are either unquestionably non-nucleated, or galaxies with
uncertain classifications.
As described in Appendix~A, we have carried out a series of experiments
in which simulated nuclei having sizes and
luminosities that obey the empirical scaling relations found in
\S\ref{sec:results}, are added to --- and removed from --- the observed
brightness profiles. By re-fitting the brightness profiles obtained
in this way, we aim to refine the nuclear classifications of these galaxies.
To summarize our conclusions from these simulations, we
classify 12 of these 27 galaxies as
{\it certainly non-nucleated} (Type~II),
11 as {\it possibly nucleated} (Type~Id)
and four others as {\it likely nucleated} (Type~Ic).
These classifications are reported on a case-by-case basis in Table~\ref{tab:data},
and summarized for the entire sample in Table~\ref{tab:class}.
Figure~\ref{fig03} shows F475W images for the central
10\arcsec$\times$10\arcsec~regions ($\approx$ 800$\times$800~pc)
of all 100 galaxies in the survey. Each galaxy is labelled according
to the classifications from Table~\ref{tab:data}. Azimuthally-averaged surface
brightness profiles in the $g$-band for all 100 galaxies are
shown in Figure~\ref{fig04}. For the 51 Type Ia galaxies shown
in this figure, the dashed
and dotted curves indicate the best-fit models for the nucleus and galaxy,
while the combined profile is shown by the solid curve. For all remaining
galaxies, the solid curve simply shows the best-fit S\'ersic or
core-S\'ersic model. Note that no fit was possible for either VCC1535
or VCC1030, the two galaxies with the most severe dust obscuration.
Open symbols in Figure~\ref{fig04} denote datapoints that were omitted
when fitting the galaxy profile (e.g., the innermost datapoints for galaxies
which contain nuclei too faint to be fit reliably,
outer datapoints for the close companions of
luminous giant galaxies, and, occasionally, datapoints corresponding to
pronounced rings, shells, or other morphological peculiarities).
\subsection{Errors on Fitted Parameters}
Given that independent fits of the $g$- and $z$-band brightness profiles
are performed for the Type~Ia galaxies, it is natural to
ask how well the photometric and structural parameters of the nuclei measured in the
two bands agree. The first two panels of Figure~\ref{fig05} compare
the King model half-light radii, $r_h$, and total magnitudes,
$g_{\rm AB}$ and $z_{\rm AB}$, measured from the separate profiles (filled
circles). Note that for 11 of these 51 galaxies (i.e., those objects
with $B_T \lesssim 13.5$), the King concentration index and
half-light radii of the nuclei were constrained to be the same in the
two bandpasses; these nuclei are plotted as open stars in the first
panel of Figure~\ref{fig05}. In addition, we include
in this figure the five galaxies with possible offset nuclei,
bearing in mind that in these cases, the $r_h$, $g_{\rm AB}$ and
$z_{\rm AB}$ measurements were carried out in a rather
different way (see \S4.3 for details).
The open circles show the nuclei of these five galaxies.
The third panel of Figure~\ref{fig05} compares two estimates for
the color of the nuclei: i.e., that obtained by integrating the
best-fit $g$- and $z$-band King models, ${\it (g-z)_{AB}}$, and
an aperture color, ${\it (g-z)^a_{AB}}$, obtained using a circular
aperture of radius 4 pixels (0\farcs20 $\approx$ 16~pc) applied to
the nucleus of the galaxy-subtracted image.
The mean difference between the total and aperture colors is 0.018~mag, in
the sense that the aperture colors are slightly redder.
The $rms$ scatter in the measured radii and
colors is found to be $\langle\sigma(r_h)\rangle \sim$ 0\farcs007 and
$\langle\sigma(g-z)\rangle \sim$ 0.059~mag, respectively. Assuming
the latter uncertainty arises equally from errors in the $g$ and $z$ bands,
we find $\langle\sigma(g)\rangle = \langle\sigma(z)\rangle \sim$ 0.041~mag
for the nuclei magnitudes. We adopt these
values for the typical uncertainties on the fitted radii, colors and magnitudes,
bearing in mind that additional systematic errors (e.g., in the photometric
zeropoints or in the construction of the PSFs) may affect the measurements.
In any case, we conclude from Figure~\ref{fig05} that there is
excellent internal agreement between the measured sizes, colors and magnitudes.
\subsection{Frequency of Nucleation}
VCC classifications for our program galaxies are given in Table~\ref{tab:data} where
column (8) reports the classification from BST85: {\tt Y} means nucleated,
{\tt N} means non-nucleated. Our new classifications are given in
column (9). Column 10 indicates which type of model was used to represent the
galaxy: ``{\tt S}" = S\'ersic or ``{\tt cS}" = core-S\'ersic.
The most basic property of the nuclei which might serve as a constraint on
theories for their origin is the overall frequency, $f_n$, with which they
are found in our program galaxies.
Among the 94 galaxies which can be reliably classified as either
nucleated or non-nucleated, we find 62 galaxies, or $f_n = 62/94 \approx 66$\%
of the sample to show clear evidence for a central nucleus (Types~Ia and Ib).
However, we believe
this estimate should be considered a firm lower limit on the frequency. Including
the Type~Ic galaxies (which are very likely to be nucleated but could not be classified
as such unambiguously), gives $f_n = 66/94 \approx 70$\%. If one also includes
the Type~Id galaxies, which {\it may} be nucleated, one then finds
$f_n = 77/94 \approx 82$\%. Finally, if all
five galaxies with possible offset nuclei are included (although we caution
in \S4.3 that the weight of evidence argues against doing so), the
percentage of nucleated galaxies could be as high as $f_n = 82/94 \approx 87$\%.
While the true frequency probably lies between these extremes, it is
nevertheless striking to think that, among our sample of 94 classifiable
galaxies, in only 12 cases can the {\it absence} of a nucleus be established
with any degree of certainty.
\subsubsection{4.3.1 Comparison with Ground-Based Studies}
\label{sec:ground}
Among the 100 elliptical, lenticular and dwarf galaxies in the ACS Virgo
Cluster Survey, 24 dwarf galaxies (dE,Ns) and one E galaxy were classified
as nucleated in the original VCC (BST85; see also Table~\ref{tab:data} of
Paper~I).\footnote{The lone elliptical in our sample which was classified
as nucleated by BST85 is VCC1422 = IC3468 (E1,N:). However, Binggeli \& Cameron (1991) argue
that this galaxy is in fact a misclassified {\it dwarf}. In what
follows, we take the total number of nucleated dwarf galaxies
in our sample, estimated from the BTS87 classifications, to be 25.}
The frequency of nucleation which we derive
here, $f_n \approx$ 66--87\%, is much
higher than the value of $f_n \approx 25\%$ found using the
classifications of BST85, and represents a sharp upward
revision of the nucleation frequency for early-type galaxies
in this luminosity range.
There are several reasons why
such a discrepancy should come as no surprise.
To the best of our knowledge, ours is the first systematic
census of nuclei in early-type galaxies that includes both dwarf and
giant galaxies (spanning a factor of $\sim$ 460 in blue luminosity).
More importantly, the studies of BST85 and BTS87, along with most of the
major subsequent studies of dwarf galaxies and their nuclei (e.g.,
Binggeli \& Cameron 1991; 1993), were based on visual inspection of
photographic plates. As pointed out in \S1 and stressed by
BST85 themselves, the VCC classifications are known to be incomplete
fainter than $B \gtrsim$ 23 ($M_B \gtrsim -8$) and to suffer from
surface brightness selection effects for the luminous E and S0
galaxies. Clearly, selection effects of this sort are less of an
issue for our survey, where the identification of the nuclei is
relatively straightforward thanks to the depth and high spatial
resolution of the ACS images.
In any case, care must be taken when comparing our measurement to
previous estimates since the frequency of nucleation is known to
depend on the luminosities of the galaxies under consideration
(e.g., Sandage, Binggeli \& Tammann 1985).
Figure~\ref{fig06} shows the luminosity functions for our
sample of 62 nucleated galaxies (Types~Ia and Ib) as the double-hatched
histogram; the hatched histogram shows this same sample plus the
15 likely or possibly nucleated galaxies of Types Ic and Id
(i.e., 77 galaxies in total). For comparison,
the 25 galaxies classified as nucleated by BST85 are shown by the filled
histogram, and the open histogram shows the distribution of the 94
classifiable galaxies from the ACS Virgo Cluster
Survey. As expected, the disagreement between our classifications
and that of BST85 is quite dramatic for galaxies brighter than
$B_T \approx 13.7$. This happens to be the approximate dividing
point between dwarf and giant galaxies in the VCC, which strongly
suggests that the disagreement is the result
of selection effects that made it difficult or
impossible for BST85 to identify nuclei in bright, high-surface-brightness
galaxies. For $B_T \gtrsim 13.7$, there is better
agreement although we still find significantly more
nuclei even among these faint galaxies: i.e., we classify 46 of 53 galaxies,
or $87\%$, of this subsample as nucleated, compared to just
25/56 ($\approx$ 47\%) using the BST85 classifications. The
luminosity dependence of $f_n$ is shown explicitly in the lower
panel of Figure~\ref{fig06}.
A vivid demonstration of the importance of surface brightness selection
effects when classifying nuclei
is shown in Figures~\ref{fig07} and \ref{fig08}. The first
of these figures compares the distribution of galaxy surface
brightnesses, measured at a geometric mean radius of 1\arcsec, for the
same four samples shown in the Figure~\ref{fig06}. By contrast,
Figure~\ref{fig08} shows nucleus magnitude as a function of
galaxy surface brightness measured at a geometric mean radius of 1\arcsec.
Filled symbols show the 51 Type~Ia galaxies in our sample, while the
open squares show the 25 galaxies classified as nucleated by BST85.
Open circles in this figure denote the five galaxies with possible offset
nuclei.
Figures~\ref{fig07} and \ref{fig08} leave little doubt that the survey of BST85
preferentially missed nuclei in the bright, high-surface-brightness
galaxies. We further note that a recent survey of 156 Virgo dwarfs with the
Wide Field Camera on the Isaac Newton Telescope uncovered faint nuclei
in 50 galaxies previously classified as non-nucleated, consistent with
our upward revision for frequency of nucleation (Grant, Kuipers \&
Phillipps 2005). Of course, it is conceivable we {\it too} may
be missing faint nuclei in the highest surface brightness
galaxies; it is certainly true that the 11 galaxies for
which we are unable to measure reliable photometric or structural parameters
for the nuclei are among the highest surface brightness galaxies in
our survey. Accordingly, we
stress once again that the estimate of $f_n \approx 66$\% from
\S4.1, which is based on galaxies with unambiguous nuclei, {\it is
certainly a lower limit to the true frequency of nucleation among
our sample of early-type galaxies}.
We shall return to this point in \S5.2 (see also Appendix~A).
Figures~\ref{fig09} and \ref{fig10} illustrate the
importance of {\it HST} imaging for
the identification of nuclei in these galaxies. In
Figure~\ref{fig09} we show a comparison between
the co-added F475W image for VCC2048 (IC3773) --- a Type~Ia galaxy --- with three simulated
ground-based images for this same galaxy. In these three
cases, the co-added F475W frame has been binned $4\times4$ and
convolved with Gaussians having dispersions of 1, 2 and 3 pixels, corresponding
to FWHM of 0\farcs5, 0\farcs9 and 1\farcs4. It is clear that seeing effects
alone make the detection of faint, compact nuclei challenging under normal
conditions of ground-based seeing. This finding is all the more sobering
when one considers that VCC2048, classified as dS0(9) in the
VCC, was thought on the basis of the original BST85 classifications
to be the brightest non-nucleated dwarf galaxy in our sample.
The first two panels of Figure~\ref{fig10} compare the F475W
image for VCC784 (NGC4379) with a $V$-band image taken with the 2.4m Hiltner
MDM telescope on 21 April, 1993 in conditions of 1\farcs14 seeing.
This galaxy, one of the brightest Type Ia galaxies in our survey, was
also classified as non-nucleated in the study of BST85. As the
third panel of Figure~\ref{fig10} demonstrates, there is no hint of
a central nucleus in the ground-based surface brightness profile,
despite the fact that the nucleus, which is clearly visible in the
ACS brightness profile, is among the brightest and largest in our sample.
\subsubsection{4.3.2 Comparison with Previous HST Studies}
\label{sec:previous}
As noted in \S1, a few {\it HST} studies had previously revealed the
presence of compact nuclei in bright early-type galaxies (e.g., Rest {et~al.}\ 2001;
Ravindranath {et~al.}\ 2001; Lauer {et~al.}\ 2005). While these programs
preferentially focussed on distant, high-luminosity ellipticals and
lenticulars --- with 80\% of the galaxies in these respective surveys having
absolute magnitudes brighter than $M_V \sim -20, -20$ and $-20.8$, compared
to $M_V \sim -16$ for the present survey --- there is nevertheless
some overlap with our sample at the bright end due to the large number
of luminous E and S0 galaxies in the Virgo Cluster. In this section, we
compare our nuclear classifications with those reported in these previous
surveys, limiting the comparison to those galaxies in our survey which
have unambiguous classifications (i.e., Types~Ia, Ib and II). For
completeness, we also compare our classifications for three faint galaxies
to those of Lotz {et~al.}\ (2004) who carried out a WFPC2 snapshot survey of
early-type dwarf galaxies in the Virgo and Fornax Clusters.
Table~\ref{tab:comp} summarizes the nuclear classifications for galaxies
in common with the surveys of Rest {et~al.}\ (2001), Ravindranath {et~al.}\ (2001),
Lauer {et~al.}\ (2005) and Lotz {et~al.}\ (2004).
The Rest {et~al.}\ (2001) study presented WFPC2 (F702W $\approx R$) imaging for
67 early-type galaxies between 6 and 54 Mpc, with a mean distance of
$\langle d\rangle = 28\pm$9
Mpc. To minimize spurious detections, Rest {et~al.}\ (2001) adopted rather
conservative criteria in their search for nuclei, identifying nucleated
galaxies as those objects which showed a central excess, along both the major
and minor axes, over the best-fit ``Nuker" model inside a radius of
0\farcs15. Based on these criteria, they identified nuclei in
9 of their 67 galaxies (13\%). No structural and photometric parameters
were measured for the nuclei.
There are six galaxies in common between their survey and ours. We find
reasonable agreement between the two studies, with the exception
of VCC731 (NGC4365): Rest {et~al.}\ (2001) report no nucleus in this galaxy,
whereas we find a small, but definite, central brightness excess.
Accordingly, we classify this galaxy as Type~Ib (i.e., certainly nucleated).
The NICMOS study of Ravindranath {et~al.}\ (2001) was carried out using F160W
($\approx H$) images from the NIC2 (FWHM = 0\farcs17, scale =
0\farcs076) and NIC3 (FWHM = 0\farcs22, scale =
0\farcs2) cameras. For 33 galaxies with distances
in the range 7 to 69~Mpc and $\langle d\rangle = 21\pm$14~Mpc, these authors fitted
two-dimensional, PSF-convolved ``Nuker" models to their NICMOS images. Compact
sources --- consisting of narrow, PSF-convolved Gaussians --- were then included
for those galaxies whose one-dimensional surface brightness profiles
showed evidence for a central excess (14 of their 33 galaxies).
FWHMs and magnitudes for the nuclei were then obtained by $\chi^2$ minimization. We
find good agreement with the Ravindranath {et~al.}\ (2001) classifications.
Specifically, we confirm the absence of nuclei in VCC1226 and VCC881
(M86 = NGC4406). For VCC763 (M84 = NGC4374), which is classified as
nucleated by these authors, we confirm the presence of a
central point source, although the galaxy is classified as Type~0
in Table~\ref{tab:data} due to the presence of strong AGN activity. All of the
nuclei in the study of Ravindranath {et~al.}\ (2001) were found to be
unresolved point sources, although this is probably a consequence of the
relatively poor resolution of their images: i.e., at the mean distance of
their sample galaxies, the NICMOS FWHM corresponds to $\sim$ 20~pc.
The WFPC2 study of Lauer {et~al.}\ (2005) was based on F555W or F606W ($\approx V$)
imaging for 77 galaxies; a little more than half of their galaxies (45) were also
imaged in the F814W ($\approx I$) bandpass. The galaxies have distances
in the range 10 to 97~Mpc, with mean $\langle d\rangle = 33\pm21$, so the
0\farcs07 FWHM for the PC1 CCD corresponds to a physical
scale of 11~pc for the typical galaxy.
Magnitudes and colors for the nuclei in their sample --- identified as an
excess above the ``Nuker" model which best fits the observed brightness
profile --- were derived by direct integration of the model residuals.
In only two of their 25 nucleated galaxies did the nucleus appear resolved.
There are seven ACSVCS galaxies having unambiguous
nuclear classifications which are in common with Lauer {et~al.}\ (2005). The
classifications are in agreement in three cases: VCC1978, VCC731 and
VCC1903 (M59 = NGC4621). For VCC1146 (NGC4458), which Lauer {et~al.}\ (2005)
classify as non-nucleated, we believe the discrepancy may be due to
the highly extended nature of the nucleus. With
$r_h \approx$ 0\farcs8 = 62~pc, it is largest nucleus in our sample, and
would be difficult to distinguish from the underlying galaxy profile
in brightness profiles of limited radial extent; the Lauer {et~al.}\ (2005)
brightness profile for this galaxy covers just the inner 5\arcsec.
The three remaining galaxies --- VCC1226, VCC881 and VCC1632 --- are
listed as nucleated in Lauer {et~al.}\ (2005), but we classify each of
these galaxies as Type~II (non-nucleated). We speculate that the
detection of nuclei in VCC1226 and VCC1632 is
an artifact resulting from the presence of dust in both galaxies,
which partly obscures the innermost $\sim$ 1\arcsec.
Lauer {et~al.}\ (2005)
do not correct their images for dust obscuration; once
such a correction is performed, we find no indication of a central
nucleus in either galaxy (see also Paper~VI).
In the case of VCC881, which is classified as nucleated by Lauer {et~al.}\ (2005),
a faint continuum enhancement is indeed detected in both the $g$ and $z$ bands
at the central location. This feature would certainly be enhanced by the
deconvolution procedure applied by Lauer {et~al.}\ (2005) to their data.
However, it is unclear whether this
corresponds to a stellar nucleus. If one assumes that VCC881 follows the scaling
relation between nucleus and galaxy luminosity obeyed by
the rest of the sample, then the putative nucleus would be
underluminous by a factor of $\sim$ 250. Furthermore, starting around 0\farcs4, the
surface brightness profile of VCC881 {\it decreases} towards the center
(Carollo {et~al.}\ 1997). The origin of this central surface brightness depression
is unclear (Lauer {et~al.}\ 2002; Paper~VI): e.g., an intrinsic decrease in the
luminosity (or mass) density, or perhaps obscuration by gray dust, might be
responsible. Since either processes could produce a modest and localized
continuum enhancement such as the one seen at the nuclear location, we
believe this galaxy is best classifed as non-nucleated (Type~II). We further
note that there is no evidence of nuclear activity in VCC881 from its
optical, radio and X-ray properties (Ho {et~al.}\ 1997; Rangarajan {et~al.}\ 1995;
Fabbiano {et~al.}\ 1989).
Lotz {et~al.}\ (2004) carried out WFPC2 (F555W $\approx V$ and F814W $\approx I$)
imaging for 69 dEs and dE,Ns, mostly belonging to the Virgo and Fornax Clusters.
In their analysis, a nucleus was identified as a bright compact object within
1\farcs5 of the galaxy photocenter. While the Lotz {et~al.}\ (2004) survey
tended to focus on fainter galaxies than does our survey (i.e., their program galaxies have
absolute blue magnitudes in the range $-17 \lesssim M_V \lesssim -11.7$ mag,
with mean $\langle M_V\rangle = -14.2$~mag), there are three galaxies which
appear in both studies: VCC9 (IC3019), VCC543 and VCC1948. In the case of VCC9 and
VCC1948, the two studies agree in finding no evidence of a nucleus at the
position of the galaxy photocenter. However, we have identified both of these
galaxies as possible examples of galaxies with offset nuclei (see \S\ref{sec:offset}).
Although Lotz {et~al.}\ (2004) do not comment on a possible offset nucleus in the
case of VCC1948, they state that: ``VCC9 was originally classified
as nucleated by Binggeli {et~al.}\ (1985), but its brightest globular cluster
candidate is 1\farcs8 from its center". For comparison, we measure an
offset of 1\farcs91$\pm$0\farcs07 for this object and, like Lotz {et~al.}\ (2004),
conclude that it is probably a star cluster projected close to the galaxy
photocenter, rather than a {\it bonafide} nucleus. The remaining galaxy, VCC543,
appears in the list of non-nucleated galaxies in Table~3 of Lotz {et~al.}\ (2004),
although we find unmistakable evidence for a nucleus in this object (see
Figures~\ref{fig03} and \ref{fig04}) that is offset by no more than
0\farcs07$\pm$0\farcs12 from the galaxy photocenter.
Finally, we note that two recent papers (de Propris {et~al.}\ 2005; Strader {et~al.}\ 2006)
have examined the properties of nuclei belonging to subsets of the galaxies from
the ACS Virgo Cluster Survey, based on the same observational material used in
this paper. A detailed comparison of the sizes, magnitudes and colors we measure
for the nuclei with those reported by de Propris {et~al.}\ (2005) and
Strader {et~al.}\ (2006) is given in Appendix~B.
\subsection{Possible Offset Nuclei}
\label{sec:offset}
Before proceeding, we pause to consider those galaxies that may have
offset nuclei. Nuclei displaced from the photocenters of their host
galaxies are potentially interesting since they may hold clues to the general processes
which trigger and/or regulate the formation of nuclei in general. For instance, offsets
may arise through the ongoing merging of globular clusters through
dynamical friction (Tremaine, Ostriker \& Spitzer 1975; Miller \& Smith 1992),
the fading of stellar populations in dwarf irregular or blue compact dwarf
galaxies as they evolve into dwarf ellipticals (e.g., Davies \& Phillips 1988),
recoil events following the ejection of a supermassive black hole from the nucleus
(Merritt {et~al.}\ 2004) or counter-streaming instabilities that develop in flat
and/or non-rotating systems (Zang \& Hohl 1978; De Rijcke \& Debattista 2004).
From an observational perspective, the identification of such nuclei is
a complicated problem. They are prone to confusion with globular clusters, foreground
stars or background galaxies --- difficulties that are particularly serious in ground-based
imaging, where the nuclei and contaminants will appear unresolved. The most
ambitious study of offset nuclei undertaken to date is that of Binggeli,
Barazza \& Jerjen (2000), who measured offsets for a sample of
78 nucleated dwarf galaxies in the Virgo Cluster using digitized images of
blue photographic plates obtained in conditions of FWHM $\approx$ 1\farcs2 seeing.
They found offset nuclei to be commonplace, with
$\delta{r_n} \gtrsim 0\farcs5$ in 45 (58\%) and
$\delta{r_n} \gtrsim 1$\arcsec~ in 14 of the objects (18\%).
It is of interest to check these results given the small sizes of the measured
offsets relative to the ground-based seeing disk, the absence of color information
that might be used to identify contaminants,
and the possibility of confusion with Galactic stars, globular
clusters and, to a lesser extent, background galaxies.
We have used our ACS images to measure offsets for the nuclei of the 62 Type~Ia
and Ib galaxies
in our sample. In both the F475W and F850LP images for each galaxy,
we first calculate the centroid of the nucleus and its corresponding uncertainty.
The location of the galaxy photocenter is then found by averaging the centers
of ellipses fitted to the galaxy isophotes over the range
1\arcsec~$\le r \le r_e$ (Paper~VI). The uncertainty on the
position of the photocenter is taken to be the standard deviation
about the mean ellipse center. Adding in quadrature
the uncertainties for the position of the nucleus and photocenter
then yields the uncertainties for the offset. The
results reported in column 17 of Table~\ref{tab:data} are
averages of the offsets measured from the F475W and F850LP images.
Figure~\ref{fig11} shows the measured offsets for the 62 galaxies.
Offsets are shown both in arcseconds (upper panel)
and in units of the effective radius of the galaxy, $\langle r_e\rangle$,
taken from Paper~VI (lower panel). In only three galaxies do we see
evidence for an offset as large as 1\arcsec. Using a less restrictive criterion
of $\delta{r_n} \gtrsim$ 0\farcs5, we find
only five galaxies that may have offset nuclei (i.e., Type~Ie galaxies).
These galaxies, which are shown as the open circles in
Figure~\ref{fig11}, are:
\begin{itemize}
\item[] {\it VCC9}. This very low surface brightness galaxy has
multiple bright sources near its photocenter; it may be a dIrr/dE
transition object and seems to contain a rich population of ``diffuse
star clusters" (Paper~XI). In addition to the
presumed nucleus, there is a second source about two magnitudes
fainter which is located $\approx$ 1\farcs5 from the photocenter (and a
similar distance from the presumed nucleus). Both the color and
the half-light radius of the presumed nucleus are similar to those of
metal-poor globular clusters in our dwarf galaxies. Thus, it is
conceivable that this galaxy has no nucleus at all.
\item[] {\it VCC21 (IC3025)}. More than a dozen bright sources are found in the
inner regions of this very low surface brightness galaxy. Based on its
mottled appearance, this galaxy too should be re-classified as a dIrr/dE
transition object. The presumed nucleus is
located $\approx$ $0\farcs76\pm0\farcs07$ from the galaxy photocenter, the
smallest offset in our sample of five candidates. There are, however,
two fainter sources close by, and given the large number of sources
in this galaxy, its transitional morphology, and the fact that the
presumed nucleus has a very blue color of $(g-z) \approx 0.3$,
we believe the evidence indicates that the ``nucleus" in VCC21 is
probably a young star cluster.
\item[] {\it VCC1779 (IC3612)}. This highly flattened galaxy ($\epsilon \simeq 0.5$)
is noteworthy in that it contains dust filaments --- unusual for
low- and intermediate-luminosity galaxies in our sample (see Paper~VI).
Like VCC9 and VCC21, this galaxy may be a dIrr/dE transition object. The ACS images
reveal four bright sources, all of which may be globular clusters, near
the galaxy center. We identify the brightest
of these sources, which is {\it not} the nearest to the center, as the
putative nucleus. If the nearest (and second brightest) source is instead
identified as the nucleus, then the offset would be $\approx$ 0\farcs4
rather than $\approx$ 0\farcs5.
\item[] {\it VCC1857 (IC3647)}. This galaxy, another very low surface brightness
object, has a very bright source located $\approx$~7\arcsec from its
center. This is by far the largest offset for any galaxy in our
sample, so the identification of this source as a nucleus should
be viewed with some caution. The color and half-light
radius of the presumed nucleus are consistent with those expected
for an otherwise normal (metal-poor) globular cluster.
\item[] {\it VCC1948}. The presumed nucleus in this galaxy, another
very low surface brightness object, is located $\approx 1\farcs4$
from the galaxy photocenter. It is the brightest of several sources in
the inner few arcseconds. There also appears to be a
very faint surface brightness ``excess" that is nearly coincident
with the galaxy photocenter. It is therefore possible that this galaxy
may be a normal (Type Ib) nucleated galaxy, albeit one with an
unusually faint nucleus. If so, then the source identified as the
possible nucleus may be a globular cluster.
\end{itemize}
We conclude that in every case there is considerable uncertainty
regarding the nature of the presumed offset nucleus. It is possible ---
and we consider it likely --- that the ``offset nuclei" in all five of
these galaxies are merely globular clusters residing in non- or weakly-nucleated galaxies.
Are there nuclei with even smaller offsets? The nuclei of four other galaxies
(VCC1539, VCC2019, VCC1895 and VCC1695) have offsets $0\farcs1 \lesssim \delta_{r_n} \lesssim 0\farcs5$ and, in
fractional terms, there are four other galaxies that have nuclei offset by more than 1\% of
the galaxy effective radius (VCC2019, VCC1199, VCC1695 and VCC1895).
We remind the reader, however, that these offsets correspond to just two ACS/WFC
pixels, and should be confirmed using deeper, higher resolution imaging.
Our only secure conclusion is that, {\it at most}, only five of the
nucleated galaxies in our survey,
or $\approx$ 7\% of the sample, have nuclei that are offset
by more than 0\farcs5~from their photocenters --- at least three times
smaller than the value of $\sim$~20\% found by Binggeli {et~al.}\ (2001).
Moreover, we believe that most --- and perhaps {\it all} --- of the
``nuclei" in the Type~Ie galaxies
are probably globular clusters, so this should be considered a
firm upper limit on the percentage of galaxies
with nuclei offset by more than 0\farcs5. The actual percentage may
in fact be zero.
\subsection{The Spatial Distribution of Nucleated and Non-Nucleated Galaxies}
\label{sec:spat}
A key result to emerge from the survey of BTS87 was the discovery of a
spatial segregration between nucleated and non-nucleated dwarf galaxies in
Virgo: the
dE,Ns are more strongly concentrated to the cluster center than the dEs (see,
e.g., Figure~9 of BTS87). A similar trend was
later reported for the Fornax Cluster by Ferguson \& Sandage (1989).
Our discovery of nuclei in many of the galaxies classified as non-nucleated by
BST85 suggests that it is worth reconsidering this important issue.
To facilitate comparison with the BTS87 and BST85 results, we limit our analysis
in this section to those galaxies fainter than $B_T \gtrsim 13.7$, the approximate
dividing point between dwarfs and giants in the VCC. As it happens, this magnitude
also divides the ACS Virgo Cluster Survey equally into two samples of 50 galaxies.
In the left panel of Figure~\ref{fig12}, the heavy solid curve shows the
cumulative distribution of projected distances from the center of the Virgo Cluster
for the 50 galaxies with $B_T \gtrsim 13.7.$\footnote{The cluster center is taken to
be the position of M87: $\alpha$(J2000) = 12:30:49.4 and $\delta$(J2000) = 12:23:28.}
Using the VCC classifications, one finds this sample to be composed predominantly
of dwarfs (i.e., 33 of 50 galaxies according to Table~1 of Paper~I).
The dotted and dashed curves show the corresponding distributions for the nucleated
(23) and non-nucleated (27) galaxies in this sample, once again using the VCC
classifications. A KS test confirms the visual impression from this figure
that the nucleated galaxies in our survey exhibit the same trend noted by BTS87 for
the full sample of nucleated dwarfs in the VCC: i.e., the dE,N galaxies are more
centrally concentrated than their non-nucleated counterparts.
In the right panel of this figure, we show the sample of 49 galaxies
with $B_T \gtrsim 13.7$ which we are able to classify as nucleated or
non-nucleated from our ACS images (heavy solid
curve).\footnote{One galaxy in this magnitude range, VCC571, is
excluded because of an irregular dust lane which obscures
the nucleus and makes a definitive classification impossible.} Excluding
for the moment the five galaxies with possible offset nuclei, whose true
nature is uncertain, we find
40 of 44 remaining galaxies to be nucleated (dotted curve). Given
the preponderance of nucleated galaxies, it is not surprising to see that
a systematic difference in central concentration between the two populations
is no longer apparent. Of course, with just four non-nucleated galaxies in
this regime, the sample falls below the minimum
size needed for statistically reliable results using a KS test, but
our point in showing this comparison is to stress again that the
overwhelming majority of program galaxies with $B_T \le 13.7$ contain nuclei.
A definitive investigation into the spatial distribution of nucleated and
non-nucleated galaxies in the Virgo Cluster would require ACS imaging for
many hundreds of galaxies. Nevertheless, we can speculate on
the origin of the trend noted by BTS87. It has long been known that galaxies
in Virgo show some segregation in terms of luminosity and morphology. Ichikawa
et~al. (1988) noted that the dwarf elliptical galaxies in the central regions of
Virgo appear to be larger and brighter than those in the cluster outskirts. At the
same time, BTS87 showed that the bright early-type galaxies (E+S0) are less
strongly concentrated to the cluster center than the faint (dE) early-type
galaxies (see their Figures~7 and 8). Since central surface brightness
is proportional to total luminosity for early-type galaxies, the BTS87 finding implies
that bright, high-surface-brightness dwarfs (HSBDs) in Virgo are more
spatially extended than low-surface-brightness dwarfs (LSBDs). Because
the original BTS87 classifications suffer from a serious surface brightness
selection effect --- in the sense that nuclei belonging to galaxies with central
surface brightnesses $\mu_g(1\arcsec) \lesssim 20$~mag~arcsec$^{-2}$
will go undetected; see Figure~\ref{fig08} --- the observed
trend may simply be a consequence of this surface brightness selection effect.
To put this claim on a more quantitative footing, we have calculated the density
profiles for HSBD and LSBD early-type galaxies in Virgo, using a surface brightness
of $\mu_g(1\arcsec) \approx 20$~mag~arcsec$^{-2}$ as a dividing point
between the two subgroups. As shown in Figure~\ref{fig08}, BTS87 would
have tended to classify LSBDs as nucleated, while the HSBDs would have
been preferentially classified as non-nucleated.
A least-squares fit to
our sample galaxies gives $\mu_g(1\arcsec) = 1.139B_T + 3.44$, so that
$\mu_g(1\arcsec) \lesssim 20$~mag~arcsec$^{-2}$
corresponds to a total galaxy magnitude of $B_T \approx 14.55$. Restricting
ourselves to the
early-type members of Virgo with $13.7 \lesssim B_T \lesssim 18$, this
leaves us with a total of 448 galaxies. The upper limit of $B_T = 13.7$ represents
the approximate transition between dwarfs and giants in Virgo, while the lower
limit reflects the completeness limit of the BTS87 survey. Among this sample
of 448 galaxies, there are 42 HSBDs with $13.7 \lesssim B_T \lesssim 14.55$,
and 406 LSBDs with $14.55 \lesssim B_T \lesssim 18$.
Figure~\ref{fig13} shows the density profiles, $\Sigma(r)$, for these two
populations. In calculating the profiles, we have excluded galaxies belonging
to the M and W Clouds, and discarded galaxies with declinations less
than 9$^{\circ}$ to guard against contamination from the Virgo B subgroup
centered on VCC1226 (M49). Fitting exponentials of
the form $\Sigma(R) \propto e^{-\alpha R}$ gives scalelengths of
$\alpha = 0.49\pm0.06$ and $0.36\pm0.06$~deg$^{-1}$ for the LSBD
and HSBD populations, respectively. In other words, there is a statistically
significant tendency for the HSBD early-type galaxies --- the same galaxies
which would preferentially be misclassified as non-nucleated in the VCC
because of their high central surface brightnesses --- to be more spatially
extended. This is consistent with the interpretation
that the differing spatial distributions for dE and dE,N galaxies noted by
BTS87 is, in fact, a consequence of their survey's surface brightness limit.
\subsection{The Nucleus-to-Galaxy Luminosity Ratio}
\label{sec:lum}
Lotz {et~al.}\ (2004) and Grant, Kuipers \& Phillipps (2005) found that brighter
galaxies tend to contain brighter nuclei.
The upper panels of Figure~\ref{fig14} plot the magnitudes of the nuclei against
those of the host galaxies; results for the $g$ and $z$ bandpasses are shown in the
left and right panels, respectively. Filled and open symbols show the results for
51 Type~Ia and five Type~Ie galaxies.
The dashed lines in these panels show the least-squares lines of best fit:
$g_n^{\prime} = (0.90\pm0.18)g_g^{\prime} + (7.59\pm2.50)$ and
$z_n^{\prime} = (1.05\pm0.18)z_g^{\prime} + (5.77\pm2.19)$. For comparison,
the solid lines show the best-fit
relations, with (fixed) unity slope:
\begin{equation}
\begin{array}{rrrrrr}
g_n^{\prime} & = & g_g^{\prime} + (6.25\pm0.21) \\
z_n^{\prime} & = & z_g^{\prime} + (6.37\pm0.22) \\
\end{array}
\label{eq6}
\end{equation}
The lower panels of Figure~\ref{fig14} show these same data in a slightly
different form. In these panels, we plot the ratio of nucleus luminosity, ${\cal L}_n$,
to host galaxy luminosity, ${\cal L}_g$,
\begin{equation}
\begin{array}{rrrrr}
$$\eta & = & {\cal L}_n / {\cal L}_g,$$
\end{array}
\label{eq7}
\end{equation}
as a function of galaxy magnitude. Total luminosities for the nuclei were obtained by integrating the brightness
profiles of the best-fit King model components (see \S3). These
magnitudes are recorded in columns (11) and
(12) of Table~\ref{tab:data}. Galaxy luminosities are taken from Paper~VI, in which the
best-fit galaxy model --- either S\'ersic or core-S\'ersic, as specified in
column (10) of Table~\ref{tab:data} --- was integrated over all radii to obtain the total luminosity.
The contribution of the nucleus itself was excluded in the calculation
of~${\cal L}_g$.
The primary conclusion to be drawn from Figure~\ref{fig14} is that the nucleus-to-galaxy
luminosity ratio does not vary with galaxy luminosity, although there is considerable
scatter about the mean value.
In terms of $\eta$, the relations in equation~6 are equivalent to
\begin{equation}
\begin{array}{rrrrr}
\langle \eta_g\rangle & = 0.32\pm0.06~\% \\
\langle \eta_z\rangle & = 0.28\pm0.06~\% \\
\end{array}
\label{eq8}
\end{equation}
for the two bands, where the quoted uncertainties refer to the mean errors.
Our best estimate for the mean nucleus-to-galaxy luminosity ratio is then
\begin{equation}
\begin{array}{rrrrr}
\langle \eta \rangle & = 0.30\pm0.04~\%. \\
\end{array}
\label{eq9}
\end{equation}
This is well below previous
estimates: only 5 of the 51 nucleated galaxies in Figure~9
of Binggeli, Barazza \& Jerjen (2000) have nuclei with
fractional luminosities smaller than this. While the discrepancy
may partly be the result of different choices for the models used
to parameterize the galaxy brightness profiles (e.g., Binggeli et~al. 2000
use King models for the galaxy when calculating the luminosity of the
central excess), it is also
true that the greater depth and sensitivity of the ACS imaging allows
us to identify fainter nuclei than is possible from the ground, while
the high spatial resolution allows a more
accurate subtraction of the underlying galaxy light.
\subsection{Luminosity Functions}
\label{sec:lf}
The luminosity function of nuclei is one of the most
powerful observational constraints on models for their formation.
For instance, one theory involves
the growth of a central nucleus through mergers of globular clusters whose
orbits have decayed because of dynamical friction (Tremaine, Ostriker \&
Spitzer 1975; Tremaine 1976; Lotz {et~al.}\ 2001). While this scenario is consistent
with the well known fact that the brightest nuclei have luminosities
that exceed those of the brightest globular clusters (e.g., Durrell et~al.
1996; Durrell 1997), a reliable measurement for the luminosity function of the nuclei has
been hard to come by due to the lack of high-resolution CCD imaging for
large, homogenous samples of early-type galaxies. The need for {\it HST}
imaging in this instance is clear, since subtle differences in the
subtraction of the galaxy light (particularly the choice of model to
represent the galaxy) can lead to large differences in the inferred
luminosities of the nuclei (see, e.g., section 5 of Binggeli \&
Cameron 1991).
In Figure~\ref{fig15}, we plot the luminosity functions, in $g$ and $z$, for the
sample of 51 Type~Ia nuclei given in Table~\ref{tab:data}. A Gaussian distribution,
\begin{equation}
\begin{array}{rrrrr}
\Phi(m_n^0) & \propto A_ne^{-(m_n^0-\overline{m}_n^0)^2 / 2\sigma_n^2} \\
\end{array}
\label{eq10}
\end{equation}
provides an adequate representation of the luminosity functions,
although there is no physical justification for this particular choice of fitting
function (and it is likely that the luminosity function suffers from some degree
of incompleteness at both the bright and faint ends). Moreover, if the mean luminosity
of nuclei in early-type galaxies is indeed a roughly constant fraction,
$\eta \approx 0.3\%$, of that of their host (\S\ref{sec:lum}), then the distribution
shown in Figure~\ref{fig14} may largely be a reflection of the luminosities of
the program galaxies. With these caveats in mind, we
overlay the best-fitting Gaussian distributions in
Figure~\ref{fig15} as the dotted curves. Fitted parameters and their
uncertainties are recorded in Table~\ref{tab:lf}.
A core objective of the ACS Virgo Cluster Survey is a study of the
globular cluster populations associated with early-type galaxies. Since
many thousands of Virgo globular cluster candidates have been identified
in the course of the survey (e.g., Papers IX, X and XI), it
is possible to compare directly the luminosity
functions of the nuclei with those of the globular clusters.
Figure~\ref{fig15} shows the $g$- and $z$-band luminosity
functions for $\approx$ 11,000 high-probability globular
cluster candidates from the survey. These objects were chosen to
have globular cluster probability indices, ${\cal P}_{\rm gc}$,
in the range $0.5 \le {\cal P}_{\rm gc} \le 1$ (see Jord\'an {et~al.}\ 2006
for details). A complete discussion of the globular cluster luminosity
function is beyond the scope of this paper and will be presented in a
future article. For the time being, we simply note that,
brighter than the 90\% completeness limits of $g_{\rm lim} \sim 26.1$~mag and
$z_{\rm lim} \sim 25.1$~mag, the luminosity functions of the globular
clusters in Virgo (which are dominated by the contributions from the brightest
galaxies) are well
described by Gaussian distributions with $\sigma = 1.3$~mag and
reddening-corrected turnover magnitudes of $g_{\rm to} \approx 23.9$~mag and
$z_{\rm to} \approx 22.8$~mag. These Gaussians are shown as the
upper dotted curves in each panel of Figure~\ref{fig15}.
It is apparent that the luminosity function of the nuclei extends
to higher luminosities than that of the globular clusters and that,
irrespective of the functional form used to parameterize the luminosity
function of the nuclei, their distribution is significantly
broader than that of the globular clusters. In addition, their distribution is
displaced to higher luminosities than that of the globular clusters. We
measure this
offset to be $\Delta{g} = 3.52$~mag and $\Delta{z} = 3.63$~mag
in the two bands. Thus, on average, the nuclei are $\sim$ 25
times brighter than a typical globular cluster.
We shall return to this point in \S5.2.
Also shown in Figure~\ref{fig15} are seven probable UCD galaxies in the
Virgo Cluster, drawn from Paper~VII and from Ha\c{s}egan {et~al.}\ (2006). These
objects were identified on the basis of magnitude and half-light
radius from the same images used to study the nuclei and globular
clusters. Although the UCD sample is limited in size, membership in Virgo
has been established for each object through radial velocity
measurements, surface brightness fluctuation distances, or both.
Furthermore, the mass-to-light ratios presented in Paper VII demonstrate
that at least some of these objects appear genuinely distinct
from globular clusters. Several explanations for their origin
have been proposed; according to
what is probably the leading formation scenario, they
are the surviving nuclei of dwarf galaxies which have been extensively
stripped by gravitational tidal fields in the host cluster (e.g., Bassino
{et~al.}\ 1994; Bekki {et~al.}\ 2001).
The UCDs shown in Figure~\ref{fig15} have luminosities
which coincide with the peak of the nuclei luminosity function, which is
certainly consistent with this ``threshing" scenario. However, it
is important to bear in mind that the luminosities of the UCDs shown in
Figure~\ref{fig15} are entirely due to the construction of the sample: i.e., candidates
from Paper~VII and Ha\c{s}egan {et~al.}\ (2006) were {\it selected} to
have $18 \le g \le 21$~mag and $17 \le z \le 20$~mag.
\subsection{Structural Properties}
\label{sec:structural}
Prior to the launch of {\it HST}, virtually nothing was known about the sizes
of the nuclei in dwarf galaxies. A notable exception was the compact,
low-luminosity nucleus of the Local Group dwarf elliptical galaxy NGC205,
which was measured to have $r_h \sim$~0\farcs4 = 1.4~pc by Djorgovski et~al.
(1992). This early estimate, which was based on deconvolved ground-based
images, is in good agreement with more recent values obtained using
ACS surface brightness profiles (e.g., Merritt {et~al.}\ 2006). However,
measuring half-light radii for nuclei at the distance of Virgo
using ground-based images is impossible (e.g., Caldwell 1983;
Sandage, Binggeli \& Tammann 1985; Caldwell \& Bothun 1987). For instance, using
high-resolution CFHT images for ten Virgo dwarfs, Durrell (1997) was only able
to place an upper limit of $r_h \lesssim$ 0\farcs4--0\farcs5 (30--40~pc) on
the sizes of the nuclei.
Even with the excellent angular resolution and spatial sampling afforded by {\it HST}/ACS,
the measurement of structural parameters for the nuclei is challenging --- more so
than for a typical Virgo globular cluster because the nuclei are observed
on a bright background which is varying rapidly in both the radial and
azimuthal spatial directions.
In their WFPC2 snapshot survey of dwarf galaxies in the Virgo
and Fornax clusters, Stiavelli {et~al.}\ (2001) and Lotz~et~al. (2004) did not
attempt to measure half-light radii for the nuclei, but they noted that
these nuclei have ``sizes" less than 0\farcs13 (10~pc). Working from the same
WFPC2 data for a subset of five nucleated dwarfs, Geha,
Guhathakurta \& van der Marel (2002) derived half-light radii in the
range 9--14~pc (0\farcs11--0\farcs18).
With their greater depth and superior
sampling of the instrumental PSF, our
ACS images are better suited to the measurement of half-light
radii than any previous dataset, including the WFPC2 imaging.
Moreover, images are available in two filters, so it is also possible to carry out
independent size measurements and identify possible systematic errors
arising from uncertainties in the F475W and F850LP PSFs. As shown in
Figure~\ref{fig05}, we have made such a comparison and find
good agreement between the half-light radii
measured in the different bandpasses, with a typical {\it random} measurement
error of $\sigma(r_h) \sim$ 0\farcs007. We note that half-light radii
measured for the nuclei of approximately two dozen of our program galaxies have
recently been reported by de Propris {et~al.}\ (2005) and Strader {et~al.}\ (2006).
Appendix~B presents a comparison of our structural and photometric parameters
with those measured in these studies.
Figure~\ref{fig16} shows the distribution of half-light radii for the nuclei
of Type~Ia galaxies from Table~\ref{tab:data}.
The distribution is evidently quite broad,
with a peak at $r_h \lesssim0\farcs05$ (4~pc) and an extended tail to much
larger radii ($0\farcs83$ $\approx$ 62~pc). The dashed line at 0\farcs025
($\approx$ 2~pc) in each panel shows our estimate for the resolution limit
of the images used to characterize the properties of the nuclei.\footnote{Note
that this resolution limit applies only to those images which were drizzled
with a {\it Gaussian} kernel. In Paper~X, we estimated from numerical simulations that
the half-light radii of globular clusters --- which are measured using the KINGPHOT
software package directly from 2D images
generated with a {\it Lanczos3} kernel --- are largely unbiased for
$r_h \gtrsim$ 0\farcs0125 $\approx$ 1~pc. However, the negative lobes of this
kernel makes it difficult to repair bad pixels, so the {\it Gaussian}
kernel is preferred for the analysis
of the galaxy/nucleus surface brightness profiles.}
In both bandpasses, the median
half-light of the nuclei in our sample is found to be $0\farcs05$ (4~pc).
Clearly, the nuclei have a size distribution that is different from
that of the globular clusters. In the latter case, the distribution
is sharply peaked, with a typical (and nearly constant; Paper~X) half-light radius of
$\langle{r_h}\rangle = 0\farcs033 \approx 2.7$~pc (i.e., $\sim$ 30\% larger than the
resolution limit for the nuclei).
It is clear that
the nuclei are not only {\sl brighter} than typical globular clusters
(\S4.5) but they are also, on average, larger. There is, however,
considerable overlap between the two distributions, and the most
compact nuclei have half-light radii that are indistinguishable from
those of globular clusters. The UCDs, on the other hand,
have half-light radii which resemble those of the nuclei. As
with the luminosity functions, this agreement may be a consequence of the
selection process: i.e., UCD candidates were identified from the sample of
sources with sizes in the range 0\farcs17--0\farcs5 (14--40~pc).
Figure~\ref{fig17} shows that there is a clear correlation between size
and luminosity for the nuclei, in the sense that the brighter objects have
larger half-light radii.
We have fit relations of the form $r_h \propto {\cal L}^{\beta}$
to the data in Figure~\ref{fig17}, excluding both the offset nuclei and
the 5--6 nuclei which fall below
the nominal resolution limit of 0\farcs025 (shown by the dashed lines in the
two panels). The solid lines in the two panels show the relations:
\begin{equation}
\begin{array}{rrrrr}
r_{h,g} & \propto & {\cal L}_g^{0.505\pm0.042} \\
r_{h,z} & \propto & {\cal L}_z^{0.503\pm0.039} \\
\end{array}
\label{eq11}
\end{equation}
This luminosity dependence constitutes another clear point of distinction
between nuclei and globular clusters: the latter, both in the Milky Way
(van den Bergh et~al. 1991; Paper~VII) and in our program galaxies
(Paper~X), have a near-constant size of $\langle r_h\rangle = 2.7$~pc.
This value is indicated by the arrows in Figure~\ref{fig17}. Nuclei
fainter than $g \sim 19$~mag and $z \sim 18$~mag have typical half-light
radii of 0\farcs04 (3.2~pc), or about 20\% larger than a
typical globular cluster; the brightest nuclei are an order
of magnitude larger still. Given their uncertain nature, it is
worth noting that all five of the candidate offset nuclei from \S4.3
have half-light radii close to the mean of the globular clusters.
It is interesting to see that the UCDs --- which in Figure~\ref{fig16} were
found to have half-light radii comparable to those of the largest nuclei ---
are outliers in this size-luminosity plane.
Compared to nuclei of comparable luminosity, the UCDs are nearly
three times larger, with $r_h \approx$ 0\farcs2--0\farcs5.
Alternatively, one might consider the UCDs to be
$\sim$~2~mag underluminous for their size. In any event, the
fact that UCD candidates from Paper VII were chosen to lie within a specific
range of magnitude and half-light radius suggests that a general conclusion
about systematic size differences between the two populations would be premature.
Figure~\ref{fig18} shows that there also exists a difference in surface
brightness between the globular clusters and nuclei. This figure plots
the average surface brightness within the half-light radius,
\begin{equation}
\begin{array}{rrrrr}
\langle \mu_h^{\prime}\rangle & = m^{\prime} + 0.7526 + 2.5\log{(\pi{r_h}^2}),\\
\end{array}
\label{eq12}
\end{equation}
for these two populations.
Because of their near-constant size, the globular clusters fall along
a diagonal swath in this diagram. Note that the dashed line in Figure~\ref{fig18}
is {\it not} a fit to the globular clusters, but simply the expected relation for
clusters which obey Equation~11 and have a constant half-light radius
of $r_h \equiv 2.7$~pc. Although there is sizeable scatter,
The nuclei have a mean surface brightness of
$\langle{\mu_h}\rangle = 16.5$ mag~arcsec$^{-2}$ in $g$ and
$15.2$ mag~arcsec$^{-2}$ in $z$, although there is considerable scatter
($\sigma \approx 1.5$~mag) about these values.
The basic properties for the UCDs and nuclei in Virgo are compared in Table~\ref{tab:global}.
By virtue of their larger radii at fixed luminosity,
the Virgo UCDs have surface brightnesses that are lower than those of comparably
bright nuclei. This is opposite to the claim of de~Propris et~al. (2005)
who argued that Fornax UCDs have {\it higher} surface brightness than the
Virgo nuclei. However, these authors seem to base this conclusion on a
visual comparison of the nuclei brightness profiles with that for
their ``mean UCD". We have calculated the average surface brightness
within the half-light radius for the four Fornax UCDs which have
absolute magnitudes and half-light radii reported in their Table~2.
In doing so, we have transformed their $V$-band magnitudes
into the $g$ and $z$ bandpasses using assumed colors of $(g-V) = 0.48$ and
$(V-z) = 0.76$, which are appropriate for old, intermediate-metallicity
populations (see Table~3
of Paper~III). Their radii have been converted from parsecs to arcseconds
using their adopted Fornax distance modulus of $(m-M)_0$ = 31.39. The
resulting surface brightnesses for these four UCDs are shown as the
open squares in Figure~\ref{fig18}. We find that the Fornax and
Virgo UCDs occupy similar locations in the diagram, and that both populations
have {\it lower} surface brightness (by $\sim$
2.5~mag~arcsec$^{-2}$ in both bandpasses) than comparably bright nuclei,
contrary to the claims of de~Propris et~al. (2005).
Finally, we note that the five candidate offset nuclei are observed to fall along
the diagonal swath defined by the globular clusters in Figure~\ref{fig18}. This
strengthens the conclusion from \S4.3 that these objects are globular clusters,
rather than {\it bonafide} nuclei.
\subsection{Nuclei Colors}
\label{sec:col}
The first comprehensive investigation into the colors of nuclei in dwarf galaxies
was the series of papers by Caldwell (1983; 1987) and Caldwell \& Bothun
(1987). Based on imaging of 30 dwarfs in the Fornax Cluster, Caldwell \& Bothun
(1987) found no evidence for a color difference between the nuclei and their
host galaxies. They did, however, find a correlation between
nuclei luminosity and galaxy color, in the sense that the reddest galaxies
tended to harbor the brightest nuclei. At a given luminosity
the nucleated galaxies were also found to be slightly redder than
their non-nucleated counterparts. A decade later, high-resolution CFHT imaging for
two Virgo dwarfs (Durrell 1997) hinted at an apparent diversity in nuclei
colors: in one galaxy (VCC1254), the nucleus was found to be significantly
bluer than the galaxy, while in the case (VCC1386), the colors
were indistinguishable.
Recently, Lotz et~al. (2004) have carried out aperture photometry for the nuclei
of 45 dE,N galaxies in the Virgo and Fornax Clusters using $VI$ images from
three WFPC2 snapshot programs. They find that: (1) the nuclei are
consistently bluer than the underlying galaxy light, with offset ${\Delta}$($V-I$) = 0.1--0.15~mag;
(2) the nuclei colors correlate with galaxy colors and luminosities, in
the sense that the redder nuclei are found in the redder and brighter galaxies;
(3) and the nuclei are slightly redder
than the globular clusters associated with the host galaxy.
Our examination of the nuclei colors begins with Figure~\ref{fig19}. The left
panel of this figure shows
the color-magnitude diagram for the nuclei of the 51 Type~Ia galaxies
(filled circles) and the five galaxies with possible offset nuclei (open circles).
For the Type~Ia galaxies, the symbol size is proportional to the blue luminosity of the
host galaxy. For reference, 11 galaxies with nuclei redder than
$(g-z)^{\prime}_{AB} = 1.35$ have been labeled.\footnote{These galaxies are
VCC1146, VCC1619, VCC1630, VCC1913, VCC784,
VCC1720, VCC828, VCC1627, VCC1250, VCC1242 and VCC1283.}
Note that one other galaxy, VCC21, has a very blue nucleus with $(g-z)^{\prime}_{AB} \approx 0.30$.
Although it is listed in Table~\ref{tab:data} as a possible example of a galaxy with an offset
nucleus, we have argued in \S4.3, \S4.6 and \S4.7 that such offset ``nuclei"
are likely to be misclassified star clusters. In the case of VCC21, the blue color of the
object points to a young age (i.e., $\le 1$~Gyr for virtually any choice of metallicity;
see Figure~6 of Paper~I). This interpretation is consistent with the galaxy's dIrr/dE
transitional morphology.
There are a number of noteworthy features in the color-magnitude diagram shown in
Figure~\ref{fig19}.
First, both the colors and luminosities of the nuclei are seen to correlate with
host galaxy luminosity, in the sense that the nuclei belonging to the
most luminous galaxies are the brightest and reddest objects in our sample. This finding
is consistent with the trend noted by Lotz et~al. (2004). Even more striking,
though, is the tendency for {\it the nuclei themselves to follow a clear
color-luminosity relation}. To the best of our knowledge, this is the first time
such a trend has been detected. The dashed line in Figure~\ref{fig19} shows the
relation
\begin{equation}
\begin{array}{rrlrr}
(g-z)^{\prime}_{AB} & = & -0.095(\pm0.015)g^{\prime}_{AB} & + & 2.98(\pm0.30), \\
\end{array}
\label{eq13}
\end{equation}
obtained from a least-squares fit to the 37 nuclei belonging to galaxies with $B_T \le 13.5$.
While this relation provides an excellent description of the color-magnitude
relation for the nuclei in faint galaxies, it appears to break down
for brighter galaxies: in this regime, the nuclei not only show considerable scatter,
but they lie systematically to the faint/red side of the extrapolated relation.
These red nuclei cause the histogram of nuclei colors to have secondary peak at
$(g-z)^{\prime}_{AB} \approx$ 1.5 (see the right panel of Figure~\ref{fig19}).
They are found exclusively in high-surface-brightness environments,
which raises the possibility of a bias in the measured colors. However,
the simulations described in Appendix~A --- in which artificial nuclei of known size,
magnitude and color are added to the non-nucleated galaxy VCC1833 and
their properties measured in the same way as the actual nuclei --- show no
evidence for a significant color bias for such bright nuclei. In addition, the
colors for most of these nuclei are actually
redder than the galaxies themselves, by $\sim$~0.1~mag, so it seems unlikely
that contamination from the host galaxy can entirely explain their red colors.
For comparison, the open stars in Figure~\ref{fig19} show the sample of Virgo
UCDs from Paper~VII and Ha\c{s}egan {et~al.}\ (2006).
The agreement with the nuclei is striking: i.e., with a
mean color of $\langle(g-z)^{\prime}_{AB}\rangle$ = 1.03$\pm$0.06~mag, the UCDs
have colors that are virtually identical to those of comparably bright nuclei.
This constitutes yet another piece of evidence for a link between UCDs and the nuclei
of early-type galaxies.
To better visualize how the properties of the host galaxy may affect
the relationship between nuclei color and magnitude,
Figure~\ref{fig20} divides the sample by host galaxy magnitude
into four subgroups. These subsamples are shown in the four separate panels, with the
color-magnitude relation given by Equation (13) repeated in each case
(the dashed line). Also included in each panel are the
globular clusters (small points) belonging to the
galaxies in each of these magnitude intervals; to reduce contamination from
stars and compact galaxies, we plot only those sources
with ``globular cluster probabilities" (see Paper~IX for details) in the
range $0.5 \le {\cal P}_{\rm gc} \le 1$. Note the clear bimodality in the
colors of globular clusters belonging to
these galaxies (Paper~IX). With the exception of the very red nuclei
noted above, we conclude that the nuclei in our Type~Ia
galaxies have colors which fall within the range spanned by the bulk of
the globular clusters in these same galaxies:
$0.7 \lesssim (g-z)^{\prime}_{AB} \lesssim 1.4$~mag.
Comparing the globular cluster colors to those of the UCDs
from Figure~\ref{fig19}, we see that the UCDs
are $\approx$ 0.1--0.2 mag redder than the population of blue globular clusters,
but $\approx$ 0.2--0.3 mag bluer than the red clusters. This may be a point of distinction
with the UCDs in Fornax, which Mieske {et~al.}\ (2006) find to have
colors similar to the red globular clusters.
Figure~\ref{fig21} shows how the colors of the galaxies, nuclei and globular
clusters depend on the galaxy luminosity. Results are shown for the $g$ and $z$
bands in the left and right panels, respectively. A common distance
of 16.5~Mpc has been assumed for all galaxies (Mei {et~al.}\ 2005). Galaxy
colors are taken from Paper~VI and represent the average color in the range
1\arcsec~$\le r \le r_e$, subject to the ACS/WFC field view and
excluding those regions with surface brightnesses 1 mag~arcsec$^{-2}$
or more below the sky.
The majority of our galaxies show no evidence for strong color gradients,
so these colors should accurately reflect their integrated colors.
For the globular clusters, we plots colors for the red and blue subpopulations,
as determined in Paper~IX, along with that of the composite cluster system.
To highlight the subtle trends exhibited by these various samples,
we show mean colors for the nuclei, globular clusters
and galaxies in four broad bins of approximately
equal width in galaxy magnitude ($\sim$~2~mag). Results for the nuclei
are shown for three bins containing an equal number of objects.
This figure reveals a correlation between nucleus color and
galaxy luminosity that is broadly consistent with the finding of Lotz {et~al.}\
(2004) for fainter galaxies. However, the trend is relatively weak and is
in fact due mainly to the $\sim$ one dozen galaxies noted above that have very
red nuclei.
These galaxies make up most of the objects in the bins at $M_g \approx -18.6$,
$(g-z)_{AB}^{\prime} \approx 1.4$ and $M_z \approx -19.6$, $(g-z)_{AB}^{\prime} \approx 1.35$.
In the fainter galaxies, the nuclei colors show
essentially no correlation with galaxy luminosity.
For galaxies fainter than $M_g = -17$, the nuclei have
$\langle(g-z)_{AB}\rangle \approx 1.02$ ---
intermediate in color to the globular clusters and stars in galaxies
in this luminosity regime.
\section{Discussion}
\label{sec:dis}
In the preceding sections we have focussed on the observed properties
of the nuclei found in our program galaxies. We now turn to the broader question of
what these observations may be telling us about the origin and evolution of
galactic nuclei. Before doing so, we pause to briefly review some of the
scenarios that have been proposed as possible explanations for stellar nuclei
in early-type galaxies. A more complete discussion of the theoretical implications
of our findings will be given in Merritt {et~al.}\ (2006).
\subsection{A Review of Formation Models}
\label{sec:review}
Tremaine, Ostriker \& Spitzer (1975) were the first to suggest that
galactic nuclei may be the remains of merged globular clusters, which
were driven inward to the galactic center by dynamical friction
(Chandrasekhar 1943).
According to this ``merger
model", the metallicity and luminosity of the nucleus should be a superposition
of the metallicity and luminosity of the progenitor clusters.
Because dynamical friction causes the orbits of most massive globular
clusters to decay most rapidly, a nucleated galaxy would be expect to show a
selective depletion of {\it bright} globular clusters, at least in the inner
regions where the dynamical friction timescale is short compared to a
Hubble Time. The contribution of globular clusters mergers
to the growth of central black holes and galactic nuclei has
been explored in a series of papers by Capuzzo-Dolcetta and coworkers
(e.g., Capuzzo-Dolcetta 1993; Capuzzo-Dolcetta \& Tesseri 1999).
Recently, Bekki {et~al.}\ (2004) have used numerical simulations
to examine the physical properties (e.g., half-light radii, central
velocity dispersion, mean density) of nuclei that form in such mergers.
Motivated by the discovery that the dE galaxies in Virgo are less centrally
concentrated than the dE,N galaxies (c.f. \S4.4), Oh \& Lin (2000)
revisited the question of how the tidal field from the Virgo cluster
affects the evolution of globular cluster orbits within dE galaxies.
They found that tidal perturbations acting on galaxies near the
center of the cluster tend to be compressive, and have little net
effect on the rate of decay of the globular cluster orbits.
In the outer regions of the Virgo cluster, tidal forces tend
to disrupt galaxies, and the resulting decrease in density
leads to longer time scales for dynamical friction. Thus,
tidal forces favor the formation of nuclei in galaxies which are
located in the cluster core, and suppress the formation in more
distant galaxies. Clearly, this will cause the relative number of
nucleated and non-nucleated galaxies to vary within the cluster,
with the highest fraction of nucleated galaxies in the core.
A second, broad category of models focuses on a dissipational origin for the
nuclei. Noting that galaxies with nuclei are
typically rounder than those without, van den Bergh (1986) speculated
that nuclei form from the gas which collects more easily in the centers
of slowly-rotating galaxies.
Silk, Wyse \& Shields (1987) argued that
dwarf galaxies experience late accretion of cool gas from the
intergalactic medium, leading to star formation and the growth of
compact central nuclei. In a similar vein,
Davies \& Phillips (1988) proposed that early-type dwarfs result from
the fading of stellar populations in dwarf irregular or blue compact
dwarf galaxies. In this scenario, intermittent bursts of central star
formation --- driven by the infalling gas --- continue until the
gas reservoir is depleted. According to this scenario, the final
star-forming event gives rise to the nucleation observed today.
Babul \& Rees (1992) examined the impact of the local intergalactic medium
on the evolution of a low-mass galaxy. They argued that the pressure of the
intergalactic medium acts as a confining agent: in a high-pressure
environment, early-type dwarfs are able to retain more gas and produce
a nucleus from the gas that has been prevented from escaping by the
intergalactic medium. Since the external pressure acting on galaxies
decreases with increasing distance from the cluster center,
some properties of the nuclei (such as their luminosity and color)
should also depend on position within the cluster, with the highest
frequency of nucleation in the central regions of a cluster.
Gas inflow models have also been explored within the context of disk
galaxy mergers. Mihos \& Hernquist (1994) used N-body/hydrodynamical
simulations to show that such mergers are accompanied by gas dissipation
and central star formation which may result in the formation of
a dense stellar core, or the fueling of a pre-existing AGN. Following Weedman
(1983), Mihos \& Hernquist (1994) further note that the dense stellar
core may itself collapse to form a supermassive black hole (SBH). The
observational evidence for a possible link between such SBHs and the
stellar nuclei of early-type galaxies is examined in \S\ref{sec:bh}.
\subsection{Implications for Nucleus Formation}
\label{sec:discussion}
\subsubsection{5.2.1 Connection to Nuclear Star Clusters in Late-Type Galaxies}
\label{sec:boker}
High-resolution {\it HST} imaging for Sa-Sd galaxies has shown that these
objects frequently contain compact nuclear clusters near their photocenters
(e.g. Phillips et~al. 1996; Carollo, Stiavelli \& Mack
1998; Matthews et~al 1999; B\"oker et~al. 2002; B\"oker et~al. 2004).
Figure~\ref{fig22} compares the sample of nuclear clusters from
B\"oker {et~al.}\ (2004) to our sample of early-type nuclei. In the upper panel,
we plot the physical sizes for both samples, where we have assumed a common
distance of 16.5~Mpc for the Virgo galaxies (Tonry {et~al.}\ 2001; Paper V).
It is clear that the nuclear clusters of B\"oker {et~al.}\
(2004) have sizes similar to the early-type nuclei.
The lower panel of Figure~\ref{fig22} compares the absolute magnitudes
of the two samples. Note that the observations of B\"oker {et~al.}\ (2004)
were carried out in the F814W ($I$) bandpass. Comparing the means of the
samples, we find the two populations to be comparably bright, with
$\langle M_g\rangle = -10.9$ and $\langle M_z\rangle = -12.0$
for the early-type nuclei, and $\langle M_I\rangle = -11.7$ for the
nuclear clusters.
B\"oker {et~al.}\ (2002) further report that
59 of 77 late-type spirals in their survey contain a nuclear
cluster close to the galaxy photocenter, giving an overall frequency
of nucleation of $f_n \approx77\%$. For comparison, in \S4.2 we estimated
$66 \lesssim f_n \lesssim 82\%$ for our sample of
early-type galaxies, counting galaxies with possible offset nuclei
as non-nucleated. Thus, in all
these respects, the nuclear clusters found in late-type galaxies are nearly
identical to the nuclei studied here. The lone point of
distinction between the nuclear clusters and the early-type nuclei seems
to be one of age: the majority of the nuclear clusters appear to have
$\tau \lesssim 10^8$ yr (Walcher {et~al.}\ 2005 and references therein),
while the broadband colors rule out such young ages for {\it all} of the
Type~Ia nuclei, irrespective of metallicity (see \S\ref{sec:stellpop} and
Figure~6 of Paper~I). This difference notwithstanding, the similar properties
of the nuclei and nuclear clusters --- and their appearance in galaxies of such
disparate morphology --- clearly points to a rather generic formation
mechanism: e.g., one which is largely independent of local or global environmental
factors, such as the gas content and present-day morphology of the
host galaxy, or the density of neighboring galaxies.
\subsubsection{5.2.2 A Fundamental Division Between S\'{e}rsic and core-S\'{e}rsic Galaxies}
\label{sec:fund}
The above conclusion applies equally well to the luminosity of the host
galaxy: i.e., the nuclei are not confined to just the dwarfs, but are also
found with regularity in many of the giants. In fact,
half (21/42) of the galaxies brighter than $B_T = 13.6$ or $M_B \approx -17.6$
(the approximate division between dwarfs and giants in the VCC) have
classifications of Type~Ia or Ib.\footnote{Excluding
the five Type 0 galaxies in this luminosity range.} The fact that nuclei are
common above and below the canonical dwarf-giant boundary suggests that, at least in terms
of their {\it nuclear} properties, there is no evidence for a fundamental
change in galaxies at this magnitude. This is consistent with
the mounting evidence from photometric scaling relations that the
``dichotomy" between normal and dwarf ellipticals, as originally envisioned
by Kormendy (1985) and others, may not be real (e.g., Jerjen \&
Binggeli 1997; Graham \& Guzm\'an 2003; Paper~VI).
On the other hand, there {\it does} appear to be a fundamental transition at
$M_B \approx -20.5$ in terms of nuclear properties. Brighter than
this, we measure $f_n \sim 0$ and, in almost all cases, the presence
of a nucleus can be ruled out with some confidence (see Appendix A).
Fainter than $M_B \approx -20.5$, the fraction of nucleated galaxies
rises sharply, as shown in the lower panel of Figure~\ref{fig06}.
It has been argued (Graham \&
Guzm\'an 2003; Trujillo {et~al.}\ 2004; Graham 2004; Paper~VI)
that this
magnitude marks a transition from faint, S\'ersic-law
galaxies to bright, core-S\'ersic-law galaxies, whose flat
``cores" are presumed to result from core depletion
by coalescing of supermassive black holes (Ebisuzaki {et~al.}\ 1991; Quinlan \&
Hernquist 1997; Faber {et~al.}\ 1997; Milosavljevi\'c \& Merritt 2001). The absence
of nuclei in galaxies brighter than $M_B \approx -20.5$ is
consistent with this scenario. Of course, it is equally possible that
these ``missing" nuclei are absent in the bright galaxies not because
of the disruptive effects of mergers and black hole coalescence,
but because they failed to form in such galaxies in the first place.
Discriminating between these competing scenarios should prove to be
a fruitful area for future theoretical study.
\subsubsection{5.2.3 Coincidence of Nuclei with Galaxy Photocenters}
In almost all cases, the nuclei are found to be coincident with
the photocenters of their host galaxies. In only five cases does there appear
to be a statistically significant offset of ${\delta}r_n / \langle r_e\rangle \ge 0.04$.
The bulk of the evidence, however, favors the view that these ``nuclei"
are, in actuality, star clusters projected close to the galaxy photocenters.
That is to say, the sizes, surface brightnesses and colors of the five possible
offset nuclei more closely resemble those
of globular clusters than those of the other nuclei in our sample. Interestingly,
all five of the host galaxies show some characteristics of dIrr/dE transition objects,
including blue colors, low central surface brightnesses, the presence of dust,
and a mottled or irregular appearance.
This suggests that if dwarf ellipticals represent an evolutionary
stage that follows gas exhaustion and stellar fading (Davies \& Phillips 1988),
ram pressure stripping (Mori \& Burkert 2000) or harrassment (Moore, Lake \&
Katz 1998) of gas-rich dIrr/disk galaxies, then the formation of a
central nucleus is not an immediate or inevitable consequence. Additional
time seems to be required to ``grow" a central nucleus.
\subsubsection{5.2.4 Nucleus Formation through Globular Clusters Mergers?}
Because the theoretical framework for the globular cluster merger model is at
a more mature stage than for any other model, we now turn our attention to the
question of whether this scenario is consistent with our new observations for
the nuclei. We note that Lotz {et~al.}\ (2001) have previously examined the
viability of the merger hypothesis on the basis of data collected for
nuclei and globular clusters in their WFPC2 snaphot survey of dwarf galaxies. Apart from
identifying a possible depletion of bright clusters in the innermost regions
of the galaxies, they could find no strong evidence for a merger origin
of the nuclei, either from the spatial distribution of the clusters or
from the measured luminosities of the nuclei.
As pointed out in \S4.6, a comparison of the luminosity functions of
nuclei and globular clusters in these galaxies shows that the {\it typical} nucleus
is $\approx$ 3.5 magnitudes brighter than a typical globular cluster. If
cluster mergers are responsible for the formation of a central nucleus, then
one might expect an average of $\sim$ 25 mergers would be needed to assemble
a nucleus from typical clusters. Of course, as Figure~\ref{fig20} shows, a
single number does not tell the whole story. The four panels
of this figure plot the distribution of nuclei and globular clusters in
the color-magnitude diagram. For the brightest galaxies (shown in the first panel),
the nuclei have a median luminosity $\approx$ 125$\times$ that of globular
clusters at the peak of the cluster luminosity function. For the fainter galaxies
(shown in the three remaining panels), the nuclei are brighter than a typical
globular clusters by factors of 29, 15 and 17, respectively.
Are these numbers feasible? In Figure~\ref{fig23} we attempt to answer
this question by plotting the integrated luminosity in globular clusters
against the luminosity of the nucleus for Type~Ia galaxies in our survey. Results are
shown in the upper panels, with measurements made in the $g$ and $z$ bands
given in the left and right panels, respectively. As in Figures~\ref{fig19} and
\ref{fig20}, symbol type indicates the magnitude of the host galaxy. In calculating the
total luminosity in globular clusters for these galaxies, we have simply
summed the luminosities of globular cluster candidates with probabilities in the
range ${\cal P_{\rm gc}} \ge 0.5$. Although this approach will obviously miss
any globular clusters located outside the ACS field, the correction should be
small for the Type~Ia galaxies in our survey which, with $M_B \lesssim -19$, have
$\langle{R_e}\rangle \sim$ 15\arcsec~or less (Paper~VI). The correlations
apparent in these panels are a consequence of the fact that both the total number
of globular clusters, and the luminosity of the nucleus, scale with host
galaxy luminosity.
The lower panels of Figure~\ref{fig23} plot the ratio of globular-to-nucleus
luminosities, $\kappa$, in the two bandpasses. In both cases, the ratio is
near unity. This should perhaps come as no surprise since the mean nucleus-to-galaxy
luminosity ratio, $\eta = 0.30\pm0.04$\%, found in \S4.5 is nearly
identical to the globular
cluster formation efficiency of $\epsilon = 0.26\pm0.05$\% measured by
McLaughlin (1999) for early-type galaxies. This latter measurement is in
turn based on observations of 97 early-type galaxies
and represents the total mass in globular clusters normalized
to the total baryonic (stellar + gas) mass.
While the agreement may be purely coincidental, it is
a remarkable empirical result that the formation of early-type galaxies
results in a nearly constant fraction of the initial
baryonic mass, $\sim$ 0.3\%, being deposited into both globular clusters and, in many
cases, a central nucleus. Of course, this conclusion appears {\it not} to
apply to galaxies brighter than $M_B \approx -20.5$, which lack nuclei
either because they did not form in the first place or because they were
subsequently destroyed by some as-yet-unidentified process.
In any case, galaxies which lie below the dashed line at $\kappa = 1$ in
Figure~\ref{fig23} pose a clear challenge to the merger model for the
obvious reason that they simply have too few clusters
to explain the luminosity of the nucleus. The difficulty is most severe for the
dozen or so red nuclei associated with the brightest Type~Ia galaxies. Of course,
this argument is based on the number of clusters contained by the host galaxy
{\it at the present time}. If the observed clusters are the rare ``survivors"
descended from a much larger initial cluster population, then it may be possible
to circumvent this problem.
An additional test of the merger model is possible. If the mergers were
dissipationless so that star formation and chemical enrichment can be
ignored, then we can use the observed colors of globular
clusters to predict colors for the nuclei. Since both globular cluster
color and the fraction of red
globular clusters are increasing functions of host galaxy
luminosity (Paper~IX), we expect the nuclei in this model to
become progressively redder in brighter galaxies. The heavy
solid curve shown in the four panels of Figure~\ref{fig20} shows the
predicted color magnitude relation for nuclei which grow through
globular cluster mergers. This curve is based on Monte Carlo
experiments in which the color evolution of the nuclei is followed
using the observed colors of the globular clusters in these galaxies.
The thin curves show the 95\% confidence limits on the relation.
Although these simulations do indeed predict redder colors for
the brighter nuclei (which are found preferentially in the brighter
galaxies containing a larger proportion of red clusters),
the predicted scaling is much milder than what is observed.
Bekki {et~al.}\ (2004) have used numerical simulations to investigate the physical
properties of nuclei which form through repeated mergers of globular clusters.
Their predicted scaling relation between half-light radius and luminosity,
$r_h \propto {\cal L}^{0.38}$, is shown by the dotted line in Figure~\ref{fig17}.
The relation has a somewhat milder luminosity dependence than the
observed relation, $r_h \propto {\cal L}^{0.50\pm0.03}$, but is nevertheless
in reasonable agreement. A similar conclusion applies to the luminosity
dependence of surface brightness averaged within the half-light radius. We
find no strong correlation between $\langle\mu_h\rangle$ and $\cal L$, but
the predicted relation, $r_h \propto {\cal L}^{0.23}$ (shown as the dotted line in
Figure~\ref{fig18}), is reasonably consistent with the observations, having
only a weak luminosity dependence.
In the future, it will be of interest to compare the predicted and observed
relationship between luminosity and central velocity dispersion. Spectra for most
of our program galaxies are now in hand and such a comparison will be the subject of a
future paper in this series.
\subsubsection{5.2.5 Stellar Populations in the Nuclei: Clues from Colors}
\label{sec:stellpop}
Ground-based spectroscopy will also be useful for investigating
the history of star formation and chemical enrichment in the nuclei,
although care must be exercised when decoupling the contributions
from the galaxy and nucleus. This separation is more straightforward
in the {\it ACS} imaging, although in this case we are limited in
our ability to measure ages and metallicities because of the well known
age-metallicity degeneracy of broadband colors. The upper
panel of Figure~\ref{fig24} shows linear interpolations of
the [Fe/H]-$(g-z)_{AB}^{\prime}$ relation for simple
stellar populations from the models of Bruzual \& Charlot (2003). The four
relations show color-metallicity relation for ages of $\tau$ = 1, 2, 5 and
10 Gyr, although it is, needless to say, quite unlikely that a single age is
appropriate for all of the nuclei in our sample. For comparison,
the heavy dashed curve in black shows the
color-metallicity relation derived from globular clusters in the Milky Way,
M49 and M87 (Paper~IX). If it is assumed that the nuclei have ages
similar to the globular clusters, then this empirical relation may be
used to derive metallicities for the nuclei.
Converting from colors to metallicities with these relations, we find
the five metallicity distributions shown in the lower panel of
Figure~\ref{fig24}. The results are summarized in Table~\ref{tab:met}.
Not suprisingly, the metallicity distributions derived from the models
depend sensitively on
the assumed age. For $\tau$ = 10 Gyr, the colors of the bluest nuclei, with
$(g-z)_{AB} \sim 0.8$~mag, would require very low
metallicities: i.e., [Fe/H] $\sim -2$ or less. By the same token,
the reddest nuclei in our sample would require
metallicities 1-100$\times$ solar for an assumed age of 1 Gyr.
For ages as young as $\tau \lesssim 10^8$ yr, which is appropriate
for many of the nuclear clusters in late-type galaxies (see \S\ref{sec:boker})
no reasonable choice of metallicity can explain the colors
of $\approx$ 0.8--1.5 that are observed. Thus, to the
extent that the nuclei can be characterized by a single formation epoch, they
show evidence for an old to intermediate age: i.e., $\tau > 1$~Gyr. Using
the globular cluster color metallicity relation gives a mean metallicity of
$\langle{\rm [Fe/H]}\rangle$ = $-0.88\pm0.79$~dex. Firmer conclusions
on the ages and metallicities of the nuclei must await the spectroscopic
analysis.
Spectroscopic constraints on the mix of stellar populations in the nuclei should also
shed some light on what may be the most serious challenge facing the merger model:
the existence of a tight correlation between nucleus luminosity
and color (Figures~\ref{fig19}-\ref{fig20}). Such a correlation
is generally thought to be a signature of self enrichment in stellar systems, and
is reminiscent of the color-magnitude relation for dwarf and giant galaxies
(e.g., Bower, Lucey \& Ellis 1992; Caldwell {et~al.}\ 1992). That the
colors correlate tightly with the luminosities of the nuclei, and less so
with those of the host galaxies, suggests that the chemical enrichment
process was governed primarily by local/internal factors. The existence of a
tight color-magnitude relation for the nuclei is a difficulty for the
merger model as envisioned by Tremaine {et~al.}\ (1975) since the
clusters from which the nuclei are assembled show
no color-magnitude relation themselves, and our Monte-Carlo experiments reveal
the slope of the observed color-magnitude relation is steeper than
that predicted in dissipationless cluster mergers.
We speculate that the merger model in its original form (i.e., ``dry"
mergers of stellar clusters) is an oversimplication of a process that almost
certainly involves some gas dissipation. In fact, if nuclei do indeed
have stellar ages of a few Gyr old or more, then they were
assembled during the earliest, most gas-rich stage of galaxy evolution. It would be
interesting to revisit the merger model with the benefit of numerical simulations that
include the effects of not just dark matter and stars, but also gas, to see
if star formation and chemical enrichment caused by mergers/inflows are
capable of producing a color-magnitude relation consistent with
that shown in Figure~\ref{fig19}. In a number of respects, the dozen or
so bright nuclei labelled in Figure~\ref{fig19} appear to form a
population distinct from their faint counterparts, most notably in
their integrated colors (which appear redder than the galaxies themselves).
These nuclei may be candidates for the ``dense stellar cores"
which form in numerical simulations (Mihos \& Hernquist 1994)
when (chemically-enriched) gas is driven inward, perhaps as a result of mergers.
\subsubsection{5.2.6 Ultra-Compact Dwarfs, Nuclei and Galaxy Threshing}
Our ACS observations may also provide some clues to the origin of
UCD galaxies. In terms of color, luminosity and size, the UCDs from Paper~VII
bear a strong resemblance to many of the nuclei studied
here, leading credence to the galaxy threshing scenario
(Bassino {et~al.}\ 1994; Bekki {et~al.}\ 2001).
However, these UCDs were selected for study on the basis of luminosity
and half-light radius, so it is unclear to what extent these conclusions
may apply to the population of UCDs as a whole. An unbiased survey of the
UCD population in Virgo should be undertaken to clarify this issue,
although this will be a difficult and time-consuming task as it requires
high-resolution spectroscopy and HST imaging to distinguish true UCDs from bright
globular clusters (see \S7 of Paper~VII). Jones {et~al.}\ (2006) have recently
reported the first results from a program to search for UCDs
in Virgo using radial velocities for $\sim$ 1300 faint,
star-like sources in the direction of the cluster. However, lacking structural
and internal dynamical information for UCD candidates found in this way,
it is impossible to know to what extent their sample has been ``polluted" by
globular clusters: either those associated
with galaxies or intergalactic in nature (e.g., West {et~al.}\ 1995).
For the time being, we may use the existing sample of Virgo UCDs from Paper~VII
and Ha\c{s}egan {et~al.}\ (2006) to re-examine
the threshing model in light of our findings for the Virgo nuclei.
Specifically, we may estimate the luminosities of the
UCD progenitor galaxies within the context of the threshing model:
for the
typical ratio of nucleus-to-galaxy luminosity found in \S4.5,
$\langle\eta\rangle \approx 0.3$\%, we expect the progenitors
to have $-18.2 \lesssim M_B \lesssim -16.6$, with a
value mean of $\langle M_B\rangle = -17.3\pm0.6$.
If the threshing
scenario is correct, then we might expect the surviving analogs of the UCD
progenitors to resemble galaxies \#40--69 in Table~\ref{tab:data}. It is
interesting to note that only about half (16/30) of these galaxies were
classified as dwarfs by BST85, meaning that a significant fraction of
{\it bright} galaxies may need to have been ``threshed" to explain the UCD
luminosity function within the context of this model. Photometric, dynamical
and structural parameters
for these candidate UCD progenitor galaxies may serve as useful constraints
for self-consistent numerical simulations of galaxy threshing and UCD formation.
\subsubsection{5.2.7 Connection to Supermassive Black Holes}
\label{sec:bh}
A large body of literature now exists on the SBHs that reside in the
centers of many galaxies (see, e.g., the review of Ferrarese \& Ford 2005).
While it had been known for some time that SBH masses, $\cal M_{\bullet}$,
correlate with the bulge masses, ${\cal M}_{gal}$, of their host galaxies
(Kormendy \& Richstone 1995), it was only after the discovery of a tight relation
between $\cal M_{\bullet}$ and bulge velocity dispersion (Ferrarese \&
Merritt 2001; Gebhardt {et~al.}\ 2001) that
Merritt \& Ferrarese (2001) were able to show that the frequency function for
galaxies with SBHs
has a roughly normal distribution in $\log_{10} {( {\cal M}_{\bullet} / {\cal M}_{gal} )}$. Fitting to the data available at that time,
Merritt \& Ferrarese (2001) found a mean of value of $-2.90$ (0.13\%)
and a standard deviation of 0.45~dex.
Remarkably, this mean value is, to within
a factor of $\approx$ two, identical to the
mean fractional luminosity contributed by nuclei to their host galaxies (\S\ref{sec:lum}).
In fact, the nuclei and SBHs share a number of other key similarities that are
highly suggestive of a direct connection: e.g., they
share a common location at the bottom of the gravitational potential wells
defined by their parent galaxies and dark matter halos, and both are probably
old components that formed during the earliest stages of galaxy evolution
(\S\ref{sec:stellpop}; Haehnelt {et~al.}\ 1998; Silk \& Rees 1998; Wyithe \&
Loeb 2002). Could it be that the compact nuclei which are found
so frequently in the low- and intermediate-luminosity early-type galaxies are
related in some way to
SBHs dectected in the brighter galaxies?
Figure~\ref{fig25} examines the connection between nuclei and SBHs in more
detail. In the upper panel, we show the distribution of absolute blue
magnitudes, $M_B$, for the 51 galaxies in our survey that contain Type~Ia nuclei
(solid histogram).
This distribution should be compared to
that for the early-type galaxies having SBH detections (open squares and dotted
histogram). This latter sample is based on data from Table~II of Ferrarese \&
Ford (2005), which reports SBH mass measurements from
resolved dynamical studies for 30 galaxies. Among this sample, there are 23
early-type galaxies with measured SBH masses (all based on stellar and/or gas dynamical
methods). It is clear from Figure~\ref{fig25}
that the two samples have very different distributions. With the exception
of M32 (with $M_B = -15.76$~mag and ${\cal M}_{\bullet} = 2.5\times10^6$ solar masses),
the SBH galaxies are all brighter than
$M_B \approx -18$, a cutoff that is thought to reflect the
formidable technical challenges involved in detecting smaller SBHs in fainter early-type
galaxies.
By contrast, the Type Ia galaxies have $M_B \gtrsim -18.9$~mag. Note that this does
{\it not} reflect the true upper boundary for nucleated galaxies,
since nuclei definitely exist in galaxies brighter than this --- Table~\ref{tab:data}
lists 14 galaxies with certain or suspected nuclei (i.e., Types Ib, Ic or Id)
having $M_B \lesssim -18.9$~mag ---
but the high surface brightness of the host galaxies do not allow a reliable
measurement of the nuclei luminosities or sizes. As we have argued in
\S\ref{sec:fund}, the more fundamental cutoff seems to occur at $M_B \sim -20.5$~mag
since we find no nucleated galaxies brighter than this.
Before moving on, we note that four of the galaxies with SBH masses in Table~II
of Ferrarese \& Ford (2005) appear in our survey. In two cases --- VCC1978 (NGC4649)
and VCC1231 (NGC4473) --- there is no evidence for a nucleus so we classify the
galaxies as Type II. In a third case,
VCC763 (NGC4374), the center of the galaxy is partly obscured by an AGN (Type~O)
but we see no evidence of a resolved stellar nucleus (\S4). The fourth and final
galaxy, VCC1664 (NGC4564), has a reported SBH mass of 5.6$\times10^7{\cal M}_{\odot}$
(Gebhardt {et~al.}\ 2003). We classify this object as Type~Ic, meaning that
we see evidence for a resolved nucleus but are unable to measure its properties
owing to the high surface of the galaxy.
In the lower panel of Figure~\ref{fig25}, we compare the frequency functions
of SBHs and Type~Ia nuclei. Bulge masses for the SBH galaxies were found by
assuming a constant bulge color of $(B-V)=0.9$~mag and combining the
magnitudes from Ferrarese \& Ford (2005) with the mass-to-light ratio
relation $\Upsilon_V = 6.3(L_V/10^{11})^{0.3}$ from Paper VII.
For the SBH sample, we find
\begin{equation}
\begin{array}{rrlll}
\langle \log_{10}({\cal M}_{\bullet} / {\cal M}_{gal})\rangle & = &
-2.61\pm0.07~{\rm dex} & \\
& = & \phantom{-}0.25\pm0.04~\% \\
\sigma(\log_{10}{\cal M}_{\bullet} / {\cal M}_{gal}) & = & \phantom{-}0.45\pm0.09~{\rm dex} \\
\end{array}
\label{eq14}
\end{equation}
whereas for the nuclei, we find
\begin{equation}
\begin{array}{rrrrr}
\langle \log_{10}{\eta}\rangle & = & -2.49\pm0.09~{\rm dex} & {\rm (= 0.32\pm0.07\%)} \\
\sigma(\log_{10}{\eta}) & = & 0.59\pm0.10~{\rm dex} \\
\end{array}
\label{eq15}
\end{equation}
For comparison, Gaussian distributions with these parameters are shown in the lower panel
of Figure~\ref{fig25}.
In our view, the available evidence favors the view that the compact stellar nuclei
found in many of the fainter galaxies may indeed be the counterparts of SBHs in the
brighter galaxies. If this speculation is correct, then it might be more
appropriate to think in terms of {\it Central Massive
Objects} --- either SBHs or compact stellar nuclei --- that accompany the formation
of almost all early-type galaxies and contain a mean fraction $\approx$ 0.3\%
of the total bulge mass. We note that a similar conclusion has been reached
independently by Wehner \& Harris (2006).
Models for the formation of SBHs in massive galaxies would
then be confronted with the additional challenge of explaining the different
manifestations of Central Massive Object formation along the galaxy luminosity
function, with an apparent preference for SBH formation above
$-20.5 \lesssim M_B \sim -18$~mag. These issues are explored in
more detail in Ferrarese {et~al.}\ (2006b).
\section{Summary}
Using multi-color ACS imaging from the {\it Hubble Space Telescope}, we have
carefully examined the central structure of the 100 early-type galaxies which
make up the ACS Virgo Cluster Survey. In this paper, we have focussed
on the compact nuclei which are commonly found at, or close to, the photocenters
of many of the galaxies. Our main conclusions are as follows:
\begin{itemize}
\item[1.] Nuclei in early-type galaxies are more common than previously
believed. Excluding the six galaxies for which the presence of a nucleus
could not be established, either because of dust obscuration or the
presence of an AGN, and counting the five galaxies with possible offset
nuclei as non-nucleated, we find the
frequency of nucleation to fall in the range $66 \% \lesssim f_n \lesssim 82$\% for
early-type galaxies brighter than $M_B \approx -15$.
\item[2.] Nuclei are found in galaxies both above and below the canonical dividing
line for dwarfs and giants ($M_B \approx -17.6$). Half (21/42) of the galaxies brighter
than $M_B \approx -17.6$ are found to contain nuclei. This is almost
certainly a lower limit on the true frequency of nucleation because
of incompleteness and surface brightness selection effects in these bright galaxies.
\item[3.] On the other hand, galaxies brighter than $M_B \approx -20.5$ appear to
be distinct in that they do {\it not} contain nuclei --- at least, not those of the
kind expected from upward extrapolations of the scaling relations obeyed by nuclei
in fainter galaxies. Whether this means that nuclei
never formed in these ``core-S\'{e}rsic" galaxies (see Paper VI and references therein), or
were instead subsequently destroyed by violent mergers and core evacuation due to
coalescing supermassive black holes, is unclear. The absence of nuclei in galaxies
brighter than $M_B \approx -20.5$ adds to the mounting evidence for a fundamental
transition in the structural properties of early-type galaxies at this magnitude.
\item[4.] Few, if any, of the nuclei are found to be significantly offset from the
photocenters of their host galaxies. In only five galaxies do we measure offsets
$\delta{r_n} \gtrsim 0\farcs5$ or $\delta{r_n}/\langle{r_e}\rangle \gtrsim 0.04$.
In all fives cases, however, the available evidence (i.e., magnitudes, colors,
half-light radii and surface brightness measurements) suggests that such ``nuclei"
are probably star clusters projected close to the galaxy photocenters.
\item[5.] The central nuclei are {\it resolved}, with a few notable exceptions: such as
the two AGN galaxies, M87 and M84, which have prominent non-thermal nuclei, and
a half dozen of the faintest galaxies with very compact, but presumably stellar, nuclei.
This observation rules out low-level AGN as an explanation
for the central luminosity excess in the vast majority of the galaxies. Excluding
those galaxies with faint, unresolved nuclei, we find the half-light
radii of the nuclei to vary with luminosity according to the relation
$r_h \propto {\cal L}^{0.50\pm0.03}$.
\item[6.] A Gaussian distribution provides an adequate, though by no means unique, description
of the luminosity function for the nuclei. The peak of the luminosity function occurs at
$\langle{M_g}\rangle = -11.2\pm0.6$ and $\langle{M_z}\rangle = -12.2\pm0.6$, or
roughly 25$\times$ brighter than the peak of the globular cluster luminosity functions
in these same galaxies. We find the ratio of nucleus-to-galaxy luminosities to
be $\eta \approx $ 0.3\%, independent of galaxy luminosity but with significant
scatter.
\item[7.] A comparison of the nuclei to the nuclear star clusters in
late-type galaxies shows a remarkable similarity in luminosity, size
and overall frequency. This suggests that a quite generic formation mechanism
is required to explain the nuclei: one which is largely independent of local
or global environmental factors, such as the gas content and present-day morphology
of the host galaxy, or the density of neighboring galaxies.
\item[8.] In terms of color, luminosity and size, the UCDs from Paper~VII bear
a strong resemblance to the compact nuclei in a number of these galaxies,
leading credence to the ``threshing" scenario in which UCDs are assumed to be the
tidally stripped remains of nucleated galaxies. If this model is correct, then
we argue that the UCD progenitor galaxies would --- if they avoided ``threshing" ---
now resemble galaxies with magnitudes in the range $-18.2 \lesssim M_B \lesssim -16.6$.
Simulations to test the viability of the threshing mechanism for such galaxies are advisable.
\item[9.] The colors of the nuclei are tighly correlated with their total luminosities,
but only weakly with those of their host galaxies. This may suggest that the history
of chemical enrichment in the nuclei was governed by local or internal factors.
\item[10.] The mean of the frequency function for the nucleus-to-galaxy luminosity ratio
in our nucleated galaxies, $\langle\log_{10}\eta\rangle = -2.49\pm0.09$~dex ($\sigma = 0.59\pm0.10$),
is indistinguishable from that of the SBH-to-bulge mass ratio,
$\langle \log_{10} {( {\cal M}_{\bullet} / {\cal M}_{gal} )} \rangle = -2.61\pm0.07$~dex
($\sigma = 0.45\pm0.09$),
calculated in 23 early-type galaxies with detected SBHs.
\item[11.] We argue that the compact stellar nuclei
found in many of our program galaxies may be the low-mass counterparts of
SBHs detected in the bright galaxies, a conclusion reached
independently by Wehner \& Harris (2006).
If this view is correct, then one should
think in terms of {\it Central Massive Objects (CMOs)} --- either SBHs or compact
stellar nuclei --- that accompany the formation of almost all early-type galaxies and
contain a mean fraction $\approx$ 0.3\% of the total bulge mass. In
this view, SBHs would be the dominant mode of CMO formation above $M_B \approx -20.5$.
\end{itemize}
As the nearest large collection of early-type galaxies, the Virgo cluster is
likely to remain, for the forseeable future, at the center of efforts to understand the physical
processes that drive nucleus formation. Unfortunately, exploring the stellar
dynamics of the most compact nuclei --- and modeling the mass distribution within
the central few parsecs of the host galaxies --- requires integrated-light spectra with an angular
resolution of $\sim$~0\farcs1 or better. Thus, the Virgo nuclei are obvious
targets for diffraction-limited, near-IR spectrographs on 8m-class
ground-based telescopes, particularly since the demise of the {\it Space
Telescope Imaging Spectrograph} on {\it HST}. For the time being, though, ACS imaging of the nuclei should
serve to guide models of their formation and evolution. This will be the
subject of a future paper in this series, in which we will examine the
implications of these observations for theories of nucleus formation
(Merritt {et~al.}\ 2006).
\acknowledgments
We thank Peter Stetson for assistance with the construction of the PSFs used
in this study. Support for program GO-9401 was provided through a grant from
the Space Telescope Science Institute, which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract
NAS5-26555.
P.C. acknowledges additional support provided by NASA LTSA grant NAG5-11714.
M.M. acknowledges additional financial support provided by the Sherman
M. Fairchild foundation. D.M. is supported by NSF grant AST-020631,
NASA grant NAG5-9046, and grant HST-AR-09519.01-A from STScI.
M.J.W. acknowledges support through NSF grant AST-0205960.
This research has made use of the NASA/IPAC Extragalactic Database (NED)
which is operated by the Jet Propulsion Laboratory, California Institute
of Technology, under contract with NASA.
\begin{appendix}
\section{Tests for Completeness, Resolvability and Bias}
The approach used to classify galaxies according to the presence or absence of
a central nucleus has been described in \S4. Briefly, the classification procedure
relies on both a visual inspection of the ACS images and the detection of
a central ``excess" in the brightness profile relative to the fitted S\'{e}rsic
or core-S\'{e}rsic galaxy model. The results are summarized in Table~\ref{tab:class}.
We find a total of 62 galaxies in which the presence of a nucleus could be
established with a high level of confidence (i.e., the Type~Ia and Ib galaxies).
Five more galaxies (Type~Ie) {\it may} contain an offset nucleus but,
as we have argued above, the weight of evidence favors the view that these
``nuclei" are actually globular clusters. Six other galaxies (Type~0) contain
either an AGN or dust at the photocenters, making the identification of a nucleus
difficult or impossible.
This leaves us with a sample of 100 -- 62 -- 5 -- 6 = 27 galaxies which
may be classified provisionally as non-nucleated. Of course,
the faintest, most extended nuclei will go undetected in any survey, especially
when superimposed on a bright galaxy background. It therefore seems
likely that at least some of these galaxies may, in fact, be nucleated. In this
Appendix, we attempt to elucidate the nature of these
galaxies with the aid of numerical simulations guided by our findings from \S4.
For the 27 galaxies in question, Figure~\ref{fig26} plots residuals, over the
innermost 10\arcsec, between the {\it observed} brightness profile and the fitted
models shown in Figure~\ref{fig04}. Because a few of
these galaxies contain multiple components (e.g., rings, bars or shells), or have
outer brightness profiles that are contaminated by the light of nearby giant galaxies,
the profiles were sometimes fit over a restricted range in radius. In two cases
where this outer fitting radius is $\le 10$\arcsec~(VCC778 = NGC4377 and
VCC575 = NGC4318), an upward arrow shows the adopted limit. Likewise, six galaxies
(VCC1664 = NGC4564, VCC944 = NGC4417, VCC1279 = NGC4478, VCC355 = NGC4262,
VCC1025 = NGC4434 and VCC575) in which the presence of
a faint central nucleus was suspected on the basis of an upturn in the central
brightness profile, an upward arrow at 0\farcs2--0\farcs3 shows the
inner limit used to avoid biasing the fitted galaxy parameters. Note that in
most cases, the fitted S\'{e}rsic or core-S\'{e}rsic model provides a reasonably
accurate match to the central brightness profile, meaning that any nuclei which
may be hiding in these galaxies have had only a minor impact on the observed profiles.
Of course, two possibilities exist for any given galaxy: either it contains a
nucleus or it does not. To test the first possibility, we use the scaling relation
from \S4 which links the luminosity of the galaxy to that of its nucleus
(Equation 6). Meanwhile, Equation 11 provides a link between galaxy luminosity
and nucleus size (half-light radius). For each of the 27 galaxies
in Figure~\ref{fig26}, we then subtract a nucleus of the expected
size and luminosity based on the magnitude of the galaxy itself.
If the nucleus-subtracted profile of the galaxy is better represented by a
Sersic or core-S\'{e}rsic model than the original profile, this would be (circumstantial)
evidence for the presence of a faint nucleus.
The alternative possibility is that the galaxy is truly non-nucleated. In this
case, we can {\it add} a
simulated nucleus of the appropriate size and magnitude to see if it would be detectable
from the images and/or the surface brightness profile. Taken together, these two experiments
allow us to crudely estimate the overall completeness of our survey and to refine the
provisional classifications for these 27 galaxies. We caution, however, that
the approach of adding and subtracting nuclei not only assumes that the scaling relations
found in \S4 are valid for all galaxies in the survey, but it ignores the significant
scatter about the fitted relations.
With these caveats in mind, we present the results of this exercise in Figure~\ref{fig26}.
In each panel, the small blue squares show the residuals profile obtained after adding
a simulated nucleus to the image and recalculating the brightness profile. Small red squares
show the profile obtained if this nucleus is instead subtracted. For four galaxies
(VCC1692, 1664, 944 and 1025), the best-fit S\'{e}rsic/core-S\'{e}rsic model for
the subtracted profile provides some improvement over the original fit.
We therefore classify these four as galaxies as Type~Ic systems.
At the same time, we identify 12 other galaxies which can be classified
unambiguously as non-nucleated (Type~II). In these galaxies, the subtracted brightness profiles
show strong inward gradients in their central regions: an obviously
non-physical result. Interestingly, these 12 galaxies fall into two rather
distinct categories: (1) {\it bright giants} which have shallow ``cores" in the
central few arcseconds and thus are best fit with core-S\'{e}rsic models; and (2)
{\it faint dwarfs} which are best
fit with pure S\'{e}rsic models. The common feature linking these two
populations is the presence of a low-surface brightness core that facilitates
the detection of a central nucleus. For this reason, we can say with some confidence
that these 12 galaxies do {\it not} contain nuclei that follow the scaling
relations observed in \S4 for the Type~Ia galaxies.
The final 11 galaxies remain elusive since we can neither confirm
nor rule out the presence of a nucleus in these cases. We classify these objects as
possibly nucleated (Type~Id).
Figures~\ref{fig06}-\ref{fig07} clearly demonstrate that it is possible to detect
nuclei in galaxies that span a wide range in luminosity and central surface brightness.
But to what extent is our ability to detect nuclei --- and to measure their sizes and
magnitudes ---
affected by the surface brightness of the underlying galaxy and their own
luminosity or size? Needless to say, a complete characterization of the biases and incompleteness
affecting the nuclei requires {\it a priori} knowledge of their intrinsic properties:
information that we are
obviously lacking. Nevertheless, we may take a first step towards answering these questions
by adding simulated nuclei of known size and
magnitude to the center of a non-nucleated galaxy. For these experiments,
we focus on a single non-nucleated galaxy, VCC1833, which, as a S\'{e}rsic-law galaxy with a central
surface brightness of $\mu_g(1\arcsec) \approx 19.3$ and
$\mu_z(1\arcsec) \approx 18.1$~mag~arcsec$^{-2}$, is representative of the Type Ia galaxies
in our survey.
Nuclei that span a range in both magnitude and size were added to the
galaxy photocenter. Input magnitudes covered the intervals $16 \le g \le 25$
and $16 \le z \le 24$ in 1~mag increments; at each magnitude, nuclei were added
with half-light radii of 0\farcs00, 0\farcs02, 0\farcs03, 0\farcs04, 0\farcs1,
0\farcs05, 0\farcs1, 0\farcs15 and 0\farcs2.
Simulations were carried out independently for the
F475W and F850LP images, and for each simulation, the surface brightness profile was
measured from the artificial image using the same procedure as for the real galaxy.
A nucleated S\'{e}rsic model was then fitted to the profile of the simulated
galaxy+nucleus and the best-fit parameters for the nucleus recorded.
The results of these simulations are shown in Figure~\ref{fig27}. The upper panel
of this figure shows the difference between the recovered and input half-light
radius, $\Delta{r_h}$, as a function of input radius. The lower panel plots the difference
between the recovered and input magnitude, $\Delta{m}$, as a function of input magnitude.
In both panels, results are shown for the separate F475W and F850LP bandpasses (blue
and red squares, respectively). The symbol size is proportional to either input magnitude
(as in the upper panel, where larger symbols correspond to brighter nuclei) or input
radius (as in the lower panel, where larger symbols correspond to more compact nuclei).
There are several conclusions which may be drawn from this figure, although
sweeping claims must be avoided because the results of the simulations
will almost certainly depend on the central surface brightness of the galaxy,
the steepness of its brightness profile, etc, so the findings are
not generalizable to the other program galaxies in any straightforward way.
With these caveats in mind, we note that nuclei brighter than $g \approx 19$~mag
or $z \approx 18$~mag in this particular galaxy
would be detected for any choice of $r_h$ in the range 0-0\farcs2. Conversely,
nuclei fainter than $g \approx 23$~mag or $z \approx 24$~mag would never be
detected. There appears to be no serious bias affecting the $r_h$ measurements
for nuclei with $r_h \le 0\farcs05$, at least for nuclei brighter than
$g \sim$ 20--21~mag. In this size regime --- a range which
encompasses half of the nuclei in Table~\ref{tab:data} --- the input half-light
radii are recovered to a precision of $\sim$ 15\% or better. For larger nuclei,
with $r_h \gtrsim 0\farcs05$, there is a bias which ranges from $\lesssim$ 10\%
for the brightest nuclei, to nearly a factor of two for the faintest detectable
nuclei, in the sense that the recovered nuclei are smaller. Unfortunately, the intrinsic
distribution of nuclei sizes is unknown, so it is not possible to apply an {\it a posteriori}
correction to the measured sizes. In any case, we note that the result from
\S\ref{sec:results} that would be most directly affected by a bias of this sort
is the observed scaling between nucleus and luminosity (Figure~\ref{fig17}),
where it was found ${\cal L} \propto r^{\beta}$ with $\beta = 0.50\pm0.03$.
If we make the admittedly dubious assumption that the luminosity dependence of the
bias found in the case of VCC1833 is representative of the full sample of
Type~Ia galaxies, then we would expect the exponent in Equation~11 to fall to
$\beta \sim 0.4$.
For the faintest nuclei, the simulations reveal that completeness is a function
of surface brightness, in the expected sense that, at fixed luminosity, more
compact nuclei (i.e., higher surface brightness) are more likely to be detected.
As the lower panel of Figure~\ref{fig27} shows, there is also a tendency to
measure fainter magnitudes for the simulated nuclei, at least in this galaxy. Not
surprisingly, the importance of this bias depends sensitively on the input magnitude;
for the brightest nuclei, the bias is less than $0.1$~mag in both filters,
irrespective of the input $r_h$. For the fainter nuclei, the bias can be as large as
$\sim$ 0.5~mag, and is slightly worse in the F850LP filter. To the
extent that the simulations for VCC1833 are applicable to other galaxies in
the survey, this means that the faintest Type~Ia nuclei may have
measured colors that are systematically too blue by $\sim$ 0.1~mag.
\section{Comparison with de Propris {et~al.}\ (2005) and Strader {et~al.}\ (2006)}
Two recent papers, de~Propris~et~al.~(2005) and Strader {et~al.}\ (2006), have presented
magnitudes, colors and half-light radii for the nuclei belonging to a subset of our
program galaxies. Since the same observational material that forms the basis of
our analysis was used in each of these studies, it is of interest to compare the various
measurements.
Based on the VCC classifications that were available when the ACS Virgo Cluster Survey
was begun, 25 of the 100 program galaxies were thought to contain nuclei
(see Table~1 of Paper~I). As we have shown in \S4, the actual number of nucleated
galaxies in our survey is about three times larger than this, although in a number
of cases the nuclei were too faint to determine reliable photometric or
structural parameters; in the final analysis, magnitudes, colors and sizes could
be measured for 51 (Type~Ia) nuclei.
We first consider the results of de~Propris~et~al.~(2005), who studied 18 of the
25 galaxies originally classified as nucleated dwarfs. These authors parameterized
the underlying galaxies as S\'{e}rsic models. After subtracting this S\'{e}rsic component,
colors and magnitudes for the nuclei were determined by summing the light within
a 1\arcsec~aperture, while half-light radii for the nuclei were measured with the
ISHAPE software package (Larsen 1999) for a circular Plummer profile and Tiny Tim
PSF.
We have transformed the de Propris {et~al.}\ (2005) VEGAMAG photometry onto
the AB system using the zeropoints given in Table~11 of Sirianni {et~al.}\
(2005). Their half-light radii were converted from parsecs to
arcseconds using their adopted Virgo distance of 15.3~Mpc. Extinction
corrections, which in both studies are based on the DIRBE maps of Schlegel
{et~al.}\ (1998), were applied to our photometry as described in Paper~II.
The two upper panels of Figure~\ref{fig28} compare our magnitudes
with those of de~Propris~et~al. (2005) (filled circles), where the dashed lines show
the one-to-one relations. There is only fair agreement between the measured
magnitudes (the $rms$ scatter is $\approx$ 0.30~mag in both bands).
In the lower left panel of Figure~\ref{fig28}, we compare our two estimates for the nuclei colors
with those of de~Propris~et~al. (2005). Whether one uses integrated or aperture
colors, the agreement is fair at best ($rms$ scatter $\approx$ 0.17~mag in either case).
As discussed in \S4.1, an internal comparison of our color measurements
shows a an $rms$ scatter of 0.059 mag between the integrated and aperture
colors. In any case, the scatter in the comparison with the de~Propris~et~al.
(2005) colors is largely driven by three galaxies --- VCC200, VCC1826 and
VCC2050 (IC3779) --- which de~Propris~et~al.
(2005) find to host exeptionally blue nuclei, $(g-z)_{AB} \lesssim 0.75$~mag. For
single burst stellar populations, such colors would require ages
$\lesssim 3$~Gyr for virtually any choice of metallicity (see Figure~6
of Paper~I). By contrast, we measure colors in the range 0.8--1.2 for
these three nuclei.
In addition, we find poor agreement ($rms$ scatter = 0\farcs056)
between the half-light radii measured in the two studies. In the lower
right panel of Figure~\ref{fig28}, we plot the de~Propris~et~al.~(2005)
half-light radii against the mean of our
measurements in the $g$ and $z$ bandpasses. An internal comparison of
our $g-$ and $z$-band measurements shows good agreement, with an $rms$ scatter of
$\sim$ 0\farcs01 (\S4.1). Unfortunately, an internal comparison of the
de~Propris~et~al.~(2005) is not possible since they report a single value of
the half-light radius for each nucleus, and it is not clear
if this value refers to a measurement made in a single bandpass, or the average
of measurements in the $g$ and $z$ bandpasses.
Figure~\ref{fig28} also shows a comparison of our magnitudes, colors and
half-light radii for 25 nuclei to those of Strader {et~al.}\ (2006) (open squares).
The Strader {et~al.}\ (2006) measurements were also determined using
the ISHAPE package (Larsen 1999), although these authors used an empirical
PSF and assumed a King model
nucleus of fixed concentration index $c \equiv \log{(r_t/r_c)} = 1.477$.
Although there is no discussion of how the contribution from the underlying galaxy
was modeled in their analysis of the nuclei,
the authors do state that photometry and size measurements for the nuclei
were carried out using procedures identical to the globular
clusters, in which the background is usually modeled as a constant
or a plane. However, near the photocenter where the nuclei are found, the galaxy
light is varying rapidly in both the radial and azimuthal directions, and since
the galaxy brightness profiles nearly always exhibit an inward
rise, this procedure will lead to overestimates of the nuclei
luminosities and sizes.
From the upper panels of Figure~\ref{fig28}, we see
that the Strader {et~al.}\ (2006) magnitudes are, on average, $\sim$ 0.4~mag
brighter than ours. In addition, the discrepancy rises to $\gtrsim$~1~mag
for the faintest nuclei --- those which should be most prone to errors in
modeling the underlying galaxy light.
There is better agreement between the measured
colors from the two studies, as the lower left panel shows ($rms$ scatter
= 0.098 and 0.072~mag for the integrated and aperture colors, respectively).
At the same time, however, there is poor agreement between the measured half-light
radii ($rms$ scatter = 0\farcs055), where their radii
are $\sim$ 80\% larger than ours. Unfortunately, Strader {et~al.}\ (2006) tabulate a single
value of the radius for each nucleus, so no internal comparison of their
size measurements is possible.
\end{appendix}
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