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\section{Introduction} \label{sec:intro}
Observations of edge-on galaxies now provide detailed structural and kinematic 3-D information. However attempts to use this data to generate mass
distribution have not been completely successful. One problem is that it is difficult to calculate the potential and force vectors for finite disks.
There is a need for fully solved finite disks which can be used for theoretical studies, for benchmarking computer programs, and as basis functions
to directly model 3-D observations.
Mass modeling commonly assumes that the disk mass in contained in an infinite disk. These include the exponential disk \citep{fre70}, the Mestel
disk \citep{mes63}, the Kuzmin-Toomre disk \citep{too63,bin08,eva92,con00::1}, and the Rybicki disk \citep{eva93}.
Only a few finite disks have been solved analytically for all $(R,z)$. \cite{las83} and \cite{vok98} give a solution for a for all $(R,z)$ for a
finite thin disk with constant density. The gravitational attraction approaches infinity at the edge of this disk. The gravitational attraction of
the finite Mestel disk \citep{mes63,lyn78,hun84} and the truncated exponential disk \citep{cas83} are finite at the edge of the disk but these disks
have not been solved in closed form for points off the disk. \cite{hur08,hur05::1} describe a method to approximate the potential of a power law disk
for points both on and off the disk.
The family of finite disks with surface density $ \Sigma_n(R;\alpha) = \left(1- {R^2}/{\alpha^2}\right)^{n-1/2}$ are related to the Maclaurin
spheroid which has been studied since the time of Newton. This family has recently been studied by \cite{gon06} and \cite{ped08}. \cite{gon06} use
the method of \cite{hun63} to obtain the general solution as a sum of Legendre polynomials in elliptical coordinates and give evaluated expressions
for the potentials for the disks 1, 2, and 3. Here we derive complete closed form solutions in cylindrical co-ordinates for the potential and the
gravitational fields of the n=0, 1, and 2 disks. We simplify the integration by moving to the imaginary domain in way which is similar to the
complex-shift method introduced by Appell. See \cite{cio08,cio07} and references therein
The gravitational potential follows Poisson's equation $\nabla^2\Phi = 4\pi\G\rho$. Poissons's equation is linear so that solutions may be added.
That is, if $\nabla^2\Phi_1=4\pi\G\rho_1$ and $\nabla^2\Phi_2=4\pi\G\rho_2$ then $\nabla^2(\Phi_1+\Phi_2)=4\pi\G(\rho_1+\rho_2)$. Similarly,
solutions may be differentiated with respect to a parameter to obtain a new solution. \cite{too63} and \cite{lyn89} exploited this linearity to find
a family of velocity-density pairs by differentiating the \cite{kuz56} model . In the same way, \cite{sat80} derived new models from the
\cite{miy75} set of potential-density pairs by differentiating with respect to a parameter. Here we use a similar approach by integrating a known
solution with respect to a parameter. If the integral is tractable the result is a new potential-density pair.
Maple 11 was used in this work. Maple was invaluable in simplifying the unwieldy intermediate results but needed a good deal of coaxing to handle the
messier equations. The Maple results were cross checked in several ways.
A common notation is used throughout. Cylindrical co-ordinates $(R,z)$ are used; $\Sigma$ is the disk surface density; $\sigma_0$ is the surface
density at $R=0$; and $\alpha$ is the disk radius. $\Phi$ is the potential where $\Phi$ is always negative and $\Phi(\infty) =0$; $F_R$ and $F_z$
are the gravitational field vectors with the standard sign convention, ie: $\vec{F} = - \vec{\nabla} \Phi $~.
The principal values of elementary functions are used so that, for instance, for $z=x+\I y=r \cos(\theta) + \I r \sin(\theta)$;
~~$\sqrt{z}=\sqrt{r}\cos(\theta/2) +\I \sqrt{r}\sin(\theta/2)$, valid for $-\pi<\theta<\pi$. In this way the function $\sqrt{z}$ is unambiguous with
continuous derivatives except on the negative real axis where it is discontinuous.
\section{The n=0 Disk: The Flattened Homeoidal Shell}
A homeoid is a shell of uniform density which is bounded by similar spheroids. Newton was the first to prove that the net force is 0 (i.e., the
potential is constant) within these shells. See \cite{cha87} for historical background. The infinitely thin homeoidal shell is a differential
element of a spheroid. The homeoidal disk is the limiting case for which the minor axis approaches 0. The collapsed density of the thin homeoid is:
\begin{equation} \label{eq:HomSurfDens}
\Sigma_{hom}(R;\alpha) = \begin{cases} {\sigma_0} /{\sqrt{1-R^2/\alpha^2}} & \textrm{for } R<\alpha \\
0 & \textrm{for } R>\alpha
\end{cases}\end{equation}
\cite{lyn89} gives a formula for the external potential of the thin homeoidal shell. Taking the limit $c=(1-e^2)^{1/2} \alpha\rightarrow0$ gives a
solution for the disk which is valid at all $R$ and $z$.
\cite{cud93} gives an expression for the potential of this disk which is simpler than previous solutions:
\begin{equation} \label{eq:CudPhi}
\Phi_{hom}(R,z;\alpha) =
-2\pi\alpha\sigma_0\G\arcsin\left[\frac{2\alpha}{\sqrt{z^2+(R+\alpha)^2} + \sqrt{z^2+(R-\alpha)^2} }\right]
\end{equation}
Equation \ref{eq:CudPhi} can be further simplified. First make the trivial transformation:
\begin{equation} \label{eq:CudPhi1}
\Phi_{hom}(R,z;\alpha) =
-2\pi\alpha\sigma_0\G\arcsin\left[\frac{\sqrt{z^2+(R+\alpha)^2} - \sqrt{z^2+(R-\alpha)^2}}{2R} \right]
\end{equation}
Now use the identity \ref{eq:IdentA1} to obtain
\begin{equation} \label{eq:HomPhi}
\Phi_{hom}(R,z;\alpha) =
-\pi\alpha\sigma_0\G~\left[\arcsin\left(\frac{ \alpha-\I z}{R}\right)+\arcsin\left(\frac{\alpha+\I z}{R}\right) \right]
\end{equation}
Equation \ref{eq:HomPhi} can be integrated over $\alpha$ whereas Equation \ref{eq:CudPhi1} yields an impossible integral. Equation \ref{eq:HomPhi}
must be real valued on physical grounds. This is easy to prove by noting that \arcsin(x) is an odd function of x and so the odd powers of $\I z$ in
the series expansion of equation \ref{eq:HomPhi} cancel, leaving a real valued result.
The gravitational field for the collapsed homeoid is obtained from the potential. Equation \ref{eq:HomPhi} yields particularly simple expressions for
the field vectors $ F_{R,hom}$ and $ F_{z,hom}$:
\begin{align}
F_{R,hom}(R,z;\alpha) &= -\frac{\pi\alpha\sigma_0\G}{R} \left[ \frac{\alpha-\I z}{\sqrt{R^2 - (\alpha- \I z)^2} } + \frac{\alpha+\I z}{\sqrt{R^2 + (\alpha -\I z)^2}
}\right]\\
F_{z,hom}(R,z;\alpha) &= -{\pi\alpha\sigma_0\G} \left[ \frac{\I}{\sqrt{R^2-(\alpha-\I z)^2}} - \frac{\I}{\sqrt{R^2 - (\alpha+\I z)^2} }\right]
\end{align}
These force vectors can be expressed as entirely real functions using identities \ref{eq:IdentA8} and \ref{eq:IdentA9}:
\begin{align}
F_{R,hom}(R,z;\alpha) &= -{\sqrt{2}\pi\alpha\sigma_0\G}~\frac{\alpha\sqrt{f_1 f_2 -f_3} - \abs{z}\sqrt{f_1 f_2 +f_3}}{ R f_1 f_2}\\
F_{z,hom}(R,z;\alpha) &= -{\sqrt{2}\pi\alpha\sigma_0\G}~\frac{\sgn(z)\sqrt{f_1 f_2 +f_3} } {f_1 f_2 }
\end{align}
Where
\begin{align}\label{eq:Define-f1-f2-f3}
f_1 &= \sqrt{z^2+(R +\alpha)^2} \nonumber \\
f_2 &= \sqrt{z^2+(R -\alpha)^2} \\
f_3 &= \alpha^2 -R^2 -z^2 \nonumber \end{align}
\section{The n=1 Disk: The Maclaurin disk } \label{sec:MaclaurinDisk}
Beginning in the early 18'th century Colin Maclaurin, along with James Ivory and many others, studied the properties of elliptical bodies.
\cite{cha87} includes a very good historical summary. See also \cite{bin08,ber00,sch56,mih68,kal71,kal72}.
The homogeneous oblate spheroid is the simplest case of a spinning body for which the gravitational attraction balances the centrifugal force. The
Maclaurin disk, also known as the Kalnajs disk \citep{kal72}, is a limiting case for which minor axis is 0. The Maclaurin disk is defined by the
surface density:
\begin{equation}\label{eq:MacSurfDens}
\Sigma_{Mac}(R;\alpha) = \begin{cases} \sigma_0 \sqrt{1-R^2/\alpha^2} & \textrm{for } R<\alpha \\
0 & \textrm{for } R>\alpha
\end{cases}\end{equation}
There are a few solutions for the potential of the Maclaurin disk in the literature. \cite{mih68} gives an expression for the potential of an oblate
homogeneous spheroid based on the derivation in \cite{sch56}. The potential of the Maclaurin disk can be found by letting the eccentricity $ e
\rightarrow 1$ while holding the mass constant. \cite{hun63} gives the solution for the Maclaurin disk as a series of Legendre polynomials in
elliptical coordinates. \cite{neu95,mei01,gon06} give a closed form solution for the potential of the Maclaurin disk in elliptic coordinates.
The starting point here is the potential-density pair for the n=0 disk, the flattened homeoid for which
$\Sigma_{hom}(R;\alpha)=\sigma_0/\sqrt{1-R^2/\alpha^2}$. The surface mass density of the Maclaurin disk is found from the transformation:
\begin{equation}
\Sigma_{Mac}(R;\alpha) = \frac{1}{\alpha} \int^\alpha_0{ \Sigma_{hom}(R; \hat{\alpha} ) ~d\hat{\alpha}}
= \frac{1}{\alpha} \int^\alpha_0{\frac{\sigma_0}{\sqrt{1-R^2/\hat{\alpha}^2} } ~d\hat{\alpha}}= \sigma_0\sqrt{1-R^2/\alpha^2}
\end{equation}
The corresponding potential is:
\begin{equation}
\Phi_{Mac}(R,z;\alpha) = \frac{1}{\alpha} \int^{\alpha}_0{ \Phi_{hom}(R;\hat{\alpha}) ~d\hat{\alpha}}
= \frac{-\pi \sigma_0 \G }{\alpha} \int^{\alpha}_0 { \hat{\alpha} \left[\arcsin\left(\frac{\hat{\alpha}- \I z }{R}\right)+\arcsin\left(\frac{\hat{\alpha}+ \I z }{R}\right) \right] d\hat{\alpha}}
\end{equation}
Where the expression for $\Phi_{hom}$ is given by equation \ref{eq:HomPhi}~ above. Use integral 2.813 and 2.833 from \cite{gra94} to obtain:
\begin{equation}\label{eq:MacPhi-Im}
\begin{split}
\Phi_{Mac}(R,z;\alpha)&= -\frac{\pi\sigma_0 \G}{4\alpha}\bigg[
( 2\alpha^2 -R^2 +2z^2 )\left(\arcsin\left(\frac{\alpha+\I z}{R}\right)+\arcsin\left(\frac{\alpha-\I z}{R}\right)\right)\\
&+\alpha \left( \sqrt{R^2-(\alpha-\I z)^2} +\sqrt{R^2-(\alpha+\I z)^2} \right)\\
&-3\ z \left( \I\sqrt{R^2-(\alpha-\I z)^2} -\I\sqrt{R^2-(\alpha+\I z)^2} \right)\bigg]
\end{split}
\end{equation}
Equation \ref{eq:MacPhi-Im} can be converted to an entirely real expression by using the identities \ref{eq:IdentA1}, \ref{eq:IdentA6} and
\ref{eq:IdentA7}:
\begin{equation} \label{eq:MacPhi-Re}
\begin{split}
\Phi_{Mac}(R,z;\alpha) =-\frac{\pi\sigma_0 \G}{4\alpha}\bigg[&
2( 2\alpha^2 -R^2 +2z^2)\arcsin\left( \frac{f_1-f_2}{2 R} \right)\\
&+\sqrt{2}\alpha \sqrt{f_1 f_2 -f_3}
-3\sqrt{2}\abs{z} \sqrt{f_1 f_2 +f_3} \bigg]
\end{split}
\end{equation}
where $f_1, f_2, f_3$ are given by equation \ref{eq:Define-f1-f2-f3} above.
The gravitational field of the Maclaurin disk is found from the potential using $\Phi$ as given by equation \ref{eq:MacPhi-Im}. The resulting
expressions were expressed as entirely real functions by using the identities \ref{eq:IdentA1}, \ref{eq:IdentA8}, and \ref{eq:IdentA9} :
\begin{equation}\label{eq:MacFR-Re}\begin{split}
F_{R,Mac}(R,z;\alpha)=& -\frac{\pi\sigma_0\G}{2 R \alpha} \Bigg[
2 R^2\arcsin\left( \frac{f_1-f_2}{2 R} \right)\\
& +\sqrt{2}\alpha ( \alpha^2 -R^2 +z^2) \frac{\sqrt{f_1 f_2 -f_3}}{f_1 f_2}\\
& -\sqrt{2}\abs{z} ( \alpha^2 +R^2 +z^2) \frac{\sqrt{f_1 f_2 +f_3}}{f_1 f_2}
\Bigg] \end{split}
\end{equation}
\begin{equation} \label{eq:MacFz-Re}\begin{split}
F_{z,Mac}(R,z;\alpha) = & -\frac{\pi\sigma_0 \G}{ \alpha} \Bigg[
-2z\arcsin\left( \frac{f_1-f_2}{2 R} \right) \\
& + 2 \sqrt{2} \alpha z \frac{ \sqrt{f_1 f_2 -f_3}}{f_1 f_2} \\
& +\sqrt{2}~\sgn(z)~( \alpha^2 -R^2 -z^2) \frac{ \sqrt{f_1 f_2 +f_3}}{f_1 f_2}
\Bigg] \end{split}\end{equation}
where $f_1, f_2, f_3$ are given by equation \ref{eq:Define-f1-f2-f3} above.
The potential on the $z$ axis and on the $z=0$ plane can be found by by taking limits of equation \ref{eq:MacPhi-Re}:
\begin{equation}\label{eq:MacPhizaxis}
\Phi_{Mac}( 0,z;\alpha) = - \frac{\pi\sigma_0\G}{\alpha}\left[ \left( \alpha^2+ z^2 \right)\arcsin\left( \frac{\alpha}{\sqrt{\alpha^2+z^2}} \right)
-\alpha \abs{z} ~\right]
\end{equation}
\begin{equation}\label{eq:MacPhiOnDisk}
\Phi_{Mac}(R, 0;\alpha) = \begin{cases}-\frac{\pi^2\sigma_0\G}{4\alpha} \left( 2\alpha^2-R^2\right) & \textrm{for} R \leq \alpha \\
-\frac{\pi \sigma_0\G}{2\alpha} \left[ (2\alpha^2-R^2)\arcsin(\frac{\alpha}{R}) +\alpha\sqrt{R^2-\alpha^2}\right] &\textrm{for} R \geq \alpha
\end{cases}
\end{equation}
The radial force vector in the $z=0$ plane is found by taking the limit of equation \ref{eq:MacFR-Re} or by differentiating equation
\ref{eq:MacPhiOnDisk} with respect to $R$.
\begin{equation}\label{FRMac_in_plane}
F_{R,Mac}(R, 0;\alpha) =
\begin{cases} - \frac{\pi^2 R\sigma_0\G} {2 \alpha}& \textrm{for} R\leq\alpha\\
- \frac{\pi\sigma_0\G} {\alpha}\left[ {R\arcsin({\alpha}/{R}} )-\alpha\sqrt{1-\alpha^2/R^2}\right] & \textrm{for} R\geq\alpha
\end{cases}
\end{equation}
The axial force vector on the $z$ axis is found by taking the limit of equation \ref{eq:MacFR-Re} or differentiating equation \ref{eq:MacPhizaxis}
with respect to $z$.
\begin{equation}\label{FzMac_in_plane}
F_{z,Mac}( 0,z ;\alpha) = -\frac{2\pi\sigma_0\G} {\alpha}\left[z\arcsin\left( \frac{\alpha}{\sqrt{\alpha^2+z^2}}\right) -\alpha~\sgn(z)\right]
\end{equation}
\section{The n=2 Disk } \label{sec:NewDisk}
\cite{gon06} give a closed form solution for the potential of the n=2 disk in elliptic coordinates.
The disk surface density of the n=2 disk is
\begin{equation} \label{eq:D2SurfDens}
\Sigma_{D2}(R;\alpha) = \begin{cases} {\sigma_0} { (1-R^2/\alpha^2)^{3/2}} & \textrm{for } R<\alpha \\
0 & \textrm{for } R>\alpha
\end{cases}\end{equation}
This mass distribution can be obtained from equation \ref{eq:MacSurfDens}, the disk surface density of the n=1 disk, with the transformation:
\begin{equation} \Sigma_{D2}(R;\alpha) =
\frac{3}{\alpha^3} \int^\alpha_0{ \hat{\alpha}^2 \Sigma_{Mac}(R; \hat{\alpha} ) d\hat{\alpha}}
\end{equation}
The corresponding potential is:
\begin{equation} \label{eq:D2PhiTransform}
\Phi_{D2}(R,z;\alpha) = \frac{3}{\alpha^3} \int^{\alpha}_0{ \hat{\alpha}^2 \Phi_{Mac}(R,z;\hat{\alpha}) ~d\hat{\alpha}}
\end{equation}
Where the expression for $\Phi_{Mac}$ is given by equation \ref{eq:MacPhi-Im} above. Equation \ref{eq:D2PhiTransform} contains terms which after
change of variable have with the form $ \int{x^n\arcsin(x) dx }$ and $\int{x^n\sqrt{1\pm x^2} dx}$ and can be solved using \cite{gra94} 2.262,
2.813, and 2.833. After collecting terms, the result is reasonably compact:
\begin{small}
\begin{equation}\label{eq:D2Phi-Im} \begin{split}
\Phi_{D2}(R,z;\alpha) &= -\frac{\pi\sigma_0\G}{64\alpha^3}\Bigg[
3( 8{\alpha}^4 -8{\alpha}^2{R}^2 +16\alpha^2{z}^2 +3{R}^4 -24{R}^2{z}^2 +8{z}^4 ) \left(\arcsin\left(\frac{\alpha+\I z}{R}\right)+\arcsin\left(\frac{\alpha-\I z}{R}\right)\right) \\
&+\alpha ( 18\alpha^2 -9{R}^2 +26{z}^2 ) \left( {\sqrt{R^2-(\alpha+\I z)^2}} + {\sqrt{R^2-(\alpha-\I z)^2}}\right) \\
&-z(58\alpha^2 -55{R}^2 +50{z}^2 ) \left( {\I}{\sqrt{R^2-(\alpha+\I z)^2}} - {\I}{\sqrt{R^2-(\alpha-\I z)^2}}\right)
\bigg]
\end{split} \end{equation} \end{small}
Equation \ref{eq:D2Phi-Im} can be converted to an entirely real expression by using the identities \ref{eq:IdentA1}, \ref{eq:IdentA6} and
\ref{eq:IdentA7}:
\begin{equation}\label{eq:D2Phi-Re}\begin{split}
\Phi_{D2}(R,z;\alpha) =&-\frac{\pi\sigma_0\G}{64\alpha^3}\Bigg[
6( 8{\alpha}^4 -8{\alpha}^2{R}^2 +16\alpha^2{z}^2 +3{R}^4 -24{R}^2{z}^2 +8{z}^4 )\arcsin\left( \frac{f_1-f_2}{2 R} \right)\\
&+\sqrt{2}\alpha ( 18\alpha^2 -9{R}^2 +26{z}^2 ) \sqrt{f_1 f_2 -f_3}\\
&-\sqrt{2}\abs{z} (58\alpha^2 -55{R}^2 +50{z}^2 ) \sqrt{f_1 f_2 +f_3}
\Bigg]
\end{split} \end{equation}
where $f_1, f_2, f_3$ are given by equation \ref{eq:Define-f1-f2-f3} above.
The gravitational field of the n=2 disk is the gradient of the potential using $\Phi$ as given by equation \ref{eq:D2Phi-Im}. The resulting
expressions were expressed as entirely real functions by using the identities by using identities \ref{eq:IdentA1}, \ref{eq:IdentA8}, and
\ref{eq:IdentA9} :
\begin{small}
\begin{equation}\label{eq:D2FR-Re} \begin{split}
F_{R,D2}(R,z;\alpha) &= - \frac{3\pi\sigma_0\G}{16 R \alpha^3} \Bigg[
2R^2 (4\alpha^2R^2 -3R^4 +12 R^2z^2 )\arcsin\left( \frac{f_1-f_2}{2 R} \right) \\
&+ \sqrt{2}\alpha( 2\alpha^4 -5\alpha^2R^2 +4\alpha^2z^2 +3R^4 -25R^2z^2 +2z^4 ) \frac{ \sqrt{f_1 f_2 -f_3}}{f_1 f_2} \\
&+ \sqrt{2}\abs{z}( 2\alpha^4 +9\alpha^2R^2 +4\alpha^2z^2 -13R^4 -11R^2z^2 +2z^4 ) \frac{ \sqrt{f_1 f_2 +f_3}}{f_1 f_2}
\Bigg]
\end{split}\end{equation} \end{small}
\begin{small}
\begin{equation}\label{eq:D2Fz-Re} \begin{split}
F_{z,D2}(R,z;\alpha) & = - \frac{\pi\sigma_0\G}{4 \alpha^3} \Bigg[
-6z( 2\alpha^2 -3R^2 +2z^2 )\arcsin\left( \frac{f_1-f_2}{2 R} \right) \\
& + \sqrt{2} \alpha z ( 13\alpha^2 -13R^2 +17z^2 ) \frac{ \sqrt{f_1 f_2 -f_3}}{f_1 f_2} \\
& + \sqrt{2} ~\sgn(z) (( 4\alpha^4 -8\alpha^2R^2 -3\alpha^2z^2 +4R^4 -7R^2z^2 -11z^4 ) \frac{ \sqrt{f_1 f_2 +f_3} } {f_1 f_2}
\Bigg]
\end{split}\end{equation} \end{small}
where $f_1, f_2, f_3$ are given by equation \ref{eq:Define-f1-f2-f3} above.
The potential on the $z$ axis and on the $z=0$ plane can be found by by taking the limit of equation \ref{eq:D2Phi-Re}:
\begin{equation}\label{eq:D2Phizaxis}
\Phi_{D2}( 0,z;\alpha) = - \frac{\pi\sigma_0\G}{4\alpha^3}\left[ 3\left( \alpha^4+2\alpha^2 z^2 + z^4 \right)\arcsin\left(\frac{\alpha}{\sqrt{\alpha^2+z^2}}\right)
-\alpha \abs{z}(5\alpha^2+3z^2) ~\right]
\end{equation}
\begin{equation} \label{eq:D2PhiOnDisk}
\Phi_{D2}(R,0;\alpha) = \begin{cases}-\frac{3\pi^2\sigma_0\G}{64\alpha^3} \left( 8\alpha^4 -8\alpha^2R^2+3R^4\right) & \textrm{for} R \leq \alpha \\
-\frac{3\pi\sigma_0\G}{32\alpha} \left[ ( 8\alpha^4 -8\alpha^2R^2+3R^4)\arcsin(\frac{\alpha}{R})
+3\alpha(2\alpha^2-R^2)\sqrt{R^2-\alpha^2}\right] &\textrm{for} R \geq \alpha
\end{cases}
\end{equation}
The radial force vector in the $z=0$ plane is found by taking the limit of equation \ref{eq:D2FR-Re} or differentiating equation \ref{eq:D2PhiOnDisk}
with respect to $R$.
\begin{equation}\label{FRD2_in_plane}
F_{R,D2}(R, 0;\alpha) =
\begin{cases} -\frac{3\pi^2 R\sigma_0\G} {16\alpha^3}(4\alpha^2-3R^2)& \textrm{for} R\leq\alpha\\
-\frac{3\pi\sigma_0\G} {8\alpha^3}\left[ R(4\alpha^2-3R^2)\arcsin({\alpha}/{R} )
-\alpha(2\alpha^2-3R^2)\sqrt{1-\alpha^2/R^2}\right] & \textrm{for} R\geq\alpha
\end{cases}
\end{equation}
The axial force vector on the $z$ axis is found by taking the limit of equation \ref{eq:D2Fz-Re} or differentiating equation \ref{eq:D2Phizaxis} with
respect to $z$.
\begin{equation} \begin{split}\label{eq:FzD2_OnZaxis}
F_{z,D2}( 0,z ;\alpha) = - \frac{\pi\sigma_0\G}{2 \alpha^3( \alpha^{2}+{z}^{2}) } \bigg[
& -6(\alpha^2+z^2)^2 z \arcsin\left(\frac{\alpha}{\sqrt{\alpha^2+z^2}}\right)\\
& +\alpha z^2(13\alpha^2+17z^2)
+\alpha\,\sgn(z)(4\alpha^4-3\alpha^2z^2-11z^4)
\bigg] \end{split}\end{equation}
\subsection{Comparison of the Maclaurin disk and the n=2 disk}
\begin{figure*}[t]
\plottwo{fig1a.eps}{fig1b.eps}
\caption{Compare the surface mass density and circular velocity, $V_c=\sqrt{(-V F_R(R,z)}$, for the Maclaurin disk and the n=2 disk. Reduced units are used so that
the masses of both disks and the point mass are 1.0 }
\label{fig:Sigma+Velocity}
\end{figure*}
Table 1 compares important properties of the two disks. Figure \ref{fig:Sigma+Velocity}a compares the surface mass density and potential. Figure
\ref{fig:Sigma+Velocity}b compares rotational velocity in the disks. The n=2 disk is more centrally concentrated than the Maclaurin disk. The
rotational velocity increases more quickly in the inner disk and begins to fall before reaching the edge of the disk. As is apparent from figure
\ref{fig:Sigma+Velocity}b, the derivative of the circular velocity of the Maclaurin disk is discontinuous at the edge of the disk whereas the n=2
disk is better behaved.
\renewcommand{\arraystretch}{1.5}
\begin{tabular} { l r l l} \\
\multicolumn{3}{c}{ \bfseries Table 1 comparison of the Maclaurin and the n=2 disks}\\
\bfseries ~~Property&&\bfseries ~~Maclaurin disk~~~&\bfseries ~~n=2 Disk\\
\hline
Surface density &$ \Sigma(R) =$&$ \sigma_0\sqrt{1 - {R^2}/{\alpha^2}} $&$= \sigma_0 (1 - {R^2}/{\alpha^2})^{3/2} $\\
Total mass &$ M =$&$ \frac23 \pi\alpha^2\sigma_0 $&$= \frac25\pi \alpha^2\sigma_0 $\\
Circular velocity &$ V_c^2(R,0) =$&$ \dfrac{\pi^2 R^2\sigma_0\G} {2\alpha} $&$= \dfrac{3\pi^2 R^2\sigma_0\G (4\alpha^2 -3 R^2) } {16\alpha^3} $\\
&= &$ \dfrac{3\pi R^2 M\G}{4\alpha^3} $&$= \dfrac{15\pi R^2 M \G(4\alpha^2 -3 R^2)}{32\alpha^3} $\\
Disk edge velocity &$ V_c^2(\alpha,0) =$&$ \dfrac{\pi^2 \alpha \sigma_0\G} {2} $&$= \dfrac{3\pi^2 \alpha \sigma_0\G} {16} $\\
&= &$ \dfrac{3 \pi M\G}{4\alpha} $&$= \dfrac{15\pi M\G}{32\alpha} $\\
\hline
\end{tabular}
\section{Example: The force field of a simple galaxy model}\label{sec:apps}
\begin{figure*}[t]
\epsscale{.50}
\plotone{fig2.eps}
\caption{ Circular velocity, $V_c=\sqrt{(-V F_R(R,z)}$, of a simple galaxy model at z=0, z=2.6~\kpc, and z=3.9~\kpc\, above the disk. The model consists of an n=2 disk and a
point mass to represent the core + bulge. The masses of the two components are $M_{disk}=3.5\dex{10} \Msun$ and
$M_{Point Mass} = 4\dex{10} \Msun$. The radius of the disk is $\alpha=19~\kpc$. }
\label{fig:NGC891}
\end{figure*}
Three-dimensional problems such as that of the structure and kinematics of the extra-planar gas will benefit from the use of the new
density-potential pairs. A simple galaxy model was constructed for illustration. The model consists of a an n=2 disk and a core/bulge region
modeled as a point mass. This model is defined by three parameters: the mass of the disk, the mass of the core/bulge region, and the diameter of the
disk. As shown in figure \ref{fig:NGC891}, the circular velocity, calculated as $V_c=\sqrt{(-V F_R(R,z)}$, is nearly constant over much of the disk.
Also, the derivative of the circular velocity with $z$ is nearly linear over a wide range of both $R$ and $z$.
Figure \ref{fig:NGC891} agrees surprisingly well with figure 5 of \cite{fra05::1} which shows that the measured velocity of HI for NGC891 decreases
linearly with the height above the disk. See also \cite{ran97,swa97,kam07,oos07,fra06::3,bar06}. Further work is planned on this topic.
\section{Summary and conclusion}
We have presented new solutions for a family of finite disks. Closed form expressions in cylindrical coordinates using elementary functions are
given for the potential and gravitational force for the disks with surface density $\Sigma_n=\sigma_0 (1-R^2/\alpha^2)^{n-1/2} \textrm{ with } n=0,
1, 2$. Expressions are also given for the limiting cases of $R=0$ and $z=0$.
These solutions fill a need and should make it easier to model 3-D gravitational phenomenon involving disk galaxies. This is particularly important
due to the recent availability of detailed kinematic data above the plane of the disk.
\acknowledgements
I am grateful to the anonymous referee for useful suggestions and comments which improved the presentation.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,640 |
\section{Introduction}
Recently there have been much activity in trying to understand the
nature of 1/4 BPS dyonic configurations in N=4 supersymmetric Yang-Mills
theories~\cite{yi,hash,bak}. The N=4 supersymmetric theories arise as an
effective low energy theory of parallel D3 branes in the type IIB string
theory~\cite{witten}. The expectation values of the six Higgs fields
are the coordinates of these D3 branes in the transverse six
space. When D3 branes lie on a line, or all Higgs expectation
values are aligned in Lie algebra, there can be only 1/2 BPS
configurations. Most general 1/2 BPS states are collections of 1/2
BPS dyons whose electric charges are all proportional to the
magnetic charge, individually as well as collectively. When the
expectation values of the Higgs fields are not aligned, on the other
hand, there can be 1/4 BPS configurations, which have a nice string
interpretation as multi-pronged strings~\cite{bergman}. As a field
theoretic solution, a 1/4 BPS configuration can be thought of as more
than one 1/2 BPS dyons at rest with respect to each other; the positions
of the component dyons are determined by a delicate balance of the
electromagnetic Coulomb and Higgs forces~\cite{yi}. Because of
this, the relative electric charges of distinct-type dyons are
functions of their relative positions.
The low energy dynamics of 1/2 BPS monopoles has been explored before
in many directions, but only in the context of aligned vacua where no
static forces among monopoles are possible. The 1/2 BPS
configurations are characterized by their moduli parameters, and
Manton proposed that the low energy motion of 1/2 BPS monopoles be
treated as the geodesic motion on the moduli space\cite{manton}. There are
several explicitly known moduli space metrics. For a pair of identical
monopoles in SU(2) gauge theory, the moduli space is the
Atiyah-Hitchin manifold\cite{atiyah}. For a pair of distinct
monopoles in SU(3) theory, it is the Taub-NUT manifold~\cite{connell},
and so on.
On the other hand, the low energy dynamics of BPS solitons in
misaligned vacua has been more problematic. In the simplest
example of SU(3) gauge theory, a 1/4 BPS configuration is known to
consist of two 1/2 BPS dyons that are separated by a fixed distance,
at which the Coulomb repulsion due to the relative electric charge is
exactly balanced against a static attraction induced by Higgs
interaction. The explicit form of the potential for the Higgs force,
however, has not been well understood.
Recall that the low energy dynamics of BPS solitons explores physics
that deviates a little bit from the BPS bound. Thus, in certain limit
where 1/4 BPS states are almost 1/2 BPS, it should be possible to
rediscover physics of 1/4 BPS configurations from the dynamics of 1/2
BPS states. Since we have static forces between 1/2 BPS solitons in
misaligned vacua, the simplest possibility is to add a potential term
to the moduli space dynamics. In this paper, we find such a low energy
effective action that describes both the 1/2 BPS and the 1/4 BPS
configurations in misaligned vacua.
The BPS equations for the 1/4 BPS configurations can be grouped into
two sets of equations~\cite{yi}; the first is the old 1/2 BPS
equations that produce purely magnetic monopoles, and the second
solves for the unbroken global gauge modes in this magnetic
background. The solution of the second BPS equation is guaranteed and
determines electric charges carried by monopoles of the first BPS
equation. Thus, the 1/4 BPS dyons are constructed by dressing 1/2 BPS
monopoles electrically, where the amount of the relative electric
charge depends on the monopole moduli parameters. A crucial
consequence is that the moduli space of monopoles also parameterizes
the classical 1/4 BPS dyons but with a twist that some of the
parameters characterizes electric charges.
These observations tell us that there are two different ways of
constructing 1/4 BPS configurations. The first is to obtain an exact
field theoretic classical solution. The energy of such a configuration
would saturate the classical BPS bound exactly. On the other hand,
since 1/4 BPS dyons are all dressed versions of purely magnetic
monopoles, one also should be able to find them as excited charged
configurations on the moduli space dynamics. They should also
saturate a BPS bound of the low energy effective dynamics. In the
limit where the moduli space approximation is good, one then
identifies these two BPS bounds, which should constrain the unknown
potential term. As a matter of fact, this procedure turns out to be
enough to fix the potential completely.
We will first consider the interaction of two distinct dyons in the
SU(3) case. We find the exact form of the potential, using the idea
outlined above. Furthermore, the resulting Hamiltonian of the low
energy dynamics is shown to have a BPS bound. As a consistency check,
we show that the configurations that saturate this low energy BPS
bound describe the identical physics as the field theoretic 1/4 BPS
configurations. We also show that the $1/r$ piece of the potential at
large separation is consistent with the results from the study of the
interaction of two point-like dyons in large separation.
We then generalize this discussion to any combination of magnetic
monopoles in arbitrary gauge group. The form of the potential will
turn out to be half the norm of of certain triholomorphic Killing
vector field on the moduli
space. Here a recent work by D. Tong \cite{tong} plays a crucial role.
This way, the effective Lagrangian is again determined by the geometry
of the moduli space alone. This particular form of potential is known
to admit supersymmetric extension, which we also explore.
The plan of the paper is as follows. In Sec.2, we review briefly the
1/4 BPS configuration of two distinct dyons in SU(3) gauge theory and
the moduli space metric of these dyons in the 1/2 BPS limit. In Sec.3,
we obtain the exact potential for the simplest case of $SU(3)$ when
the deviation from the 1/2 BPS case is small, that is, when the D3
branes are almost on a straight line. We show that the BPS
configurations of the low energy dynamics are identical to the BPS
configuration of the field theoretic ones. In Sec.4, we generalize
this result to arbitrary monopole configurations in arbitrary gauge
group with more emphasis on the geometrical character of the low
energy effective action. The underlying supersymmetric Lagrangian and
supercharge, as well as the BPS conditions, are found in Sec. 5. In
Sec. 6 we conclude with some remarks.
\section{Two Distinct Monopoles in the SU(3) Gauge Theory}
The SU(3) gauge group appears in the low energy dynamics of three
parallel D3 branes. (See Ref.~\cite{yi} for details.) The positions
of D3 branes on a plane of the six-dimensional transverse direction
are dictated by the expectation values of two Higgs fields:
\beqn
&& \hat b\cdot\phi(\infty)= {\rm diag} (h_1,h_2,h_3), \label{basy} \\
&& \hat a\cdot\phi(\infty)= \xi\; {\rm diag} (h_1,h_2,h_3)+ \eta\;
{\rm diag} (\mu_2,-\mu_1-\mu_2,\mu_1) , \label{aasy}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where $h_1<h_2<h_3$, $h_1+h_2+h_3=0$ and $\mu_1=h_2-h_1,
\mu_2=h_3-h_2$. (Here the string tension is multiplied to the positions
of D3 branes so that they acquire the mass dimension.) Two Higgs field
expectation values are the coordinates of D3 branes along two
orthogonal directions ${\hat b} = (1,0)$ and ${\hat a}= (0,1)$. The
relative position vector of the second D3 brane with respect to the
first D3 brane on the plane is
\begin{equation}
{\vec R}_1 = (\mu_1, \xi \mu_1 - \eta(\mu_1+2\mu_2)),
\label{R1}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
while the relative position of the third D3 brane with respect to the
second D3 brane is
\begin{equation}
{\vec R}_2 = (\mu_2, \xi\mu_2+\eta(2\mu_1+\mu_2)).
\label{R2}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
Two simple roots ${\mbox{\boldmath $\alpha$}}$ and ${\mbox{\boldmath $\beta$}}$ of the $SU(3)$ group are
chosen in accordance with the convention ${\mbox{\boldmath $\alpha$}}^2={\mbox{\boldmath $\beta$}}^2=1$,
${\mbox{\boldmath $\alpha$}}\cdot {\mbox{\boldmath $\beta$}} = -1/2$. With ${\mbox{\boldmath $\alpha$}}\cdot {\bf H} =
\frac{1}{2} {\rm diag} (-1,1,0)$ and ${\mbox{\boldmath $\beta$}}\cdot{\bf H} =
\frac{1}{2}{\rm diag}(0,-1,1)$, the masses of isolated ${\mbox{\boldmath $\alpha$}}$ and
${\mbox{\boldmath $\beta$}}$ monopoles are
\beqn
&& m_1 = g |{\vec R}_1| ,\\
&& m_2 = g|{\vec R}_2| ,
\end{eqnarray}} \newcommand{\bde}{{\bf e}
respectively, where $g = 4\pi$ is the charge of magnetic monopole with
$e=1$ assumed for convenience. Notice that when $\xi$ and $\eta$ are very
small,
\beqn
&& m_1 = g\mu_1 + \frac{g}{2\mu_1}(\xi \mu_1 -\eta(\mu_1+2\mu_2))^2, \\
&& m_2 = g\mu_2 + \frac{g}{2\mu_2}(\xi \mu_2 + \eta(2\mu_1+\mu_2))^2.
\end{eqnarray}} \newcommand{\bde}{{\bf e}
There is a third monopole corresponding to the third positive root
${\mbox{\boldmath $\alpha$}} + {\mbox{\boldmath $\beta$}}$ and its mass is $m_3 = g|\vec{R}_1+
\vec{R}_2|$. In contrast to the case when the Higgs vacuum values are
aligned, generically there is no distinction between fundamental or
composite monopoles. (For example, consider the case where three D3
branes lie on the corners of an equitriangle.) However, when the
Higgs vacuum values are almost aligned as in the case we study, we can
still distinguish between fundamental and composite monopoles. Thus,
${\mbox{\boldmath $\alpha$}}$ and ${\mbox{\boldmath $\gamma$}}$ monopoles are fundamental and
${\mbox{\boldmath $\alpha$}}+{\mbox{\boldmath $\gamma$}}$ monopoles are composite.
The dyonic configuration we consider is made of one ${\mbox{\boldmath $\alpha$}}$
and one ${\mbox{\boldmath $\beta$}}$ monopoles, with electric charges $q_1$ and $q_2$,
respectively. Thus the asymptotic forms of the Higgs fields are
\beqn
&& \hat b\cdot\phi = \hat b\cdot\phi(\infty) - \frac{1}{4\pi r} \, g
({\mbox{\boldmath $\alpha$}} +{\mbox{\boldmath $\beta$}})\cdot {\bf H} ,
\label{asymp1} \\
&& \hat a\cdot\phi = \hat a\cdot\phi(\infty) - \frac{1}{4\pi r}\, (q_1
{\mbox{\boldmath $\alpha$}} +q_2{\mbox{\boldmath $\beta$}}) \cdot {\bf H} .\label{asymp2}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
For the given asymptotic (\ref{asymp1}), the solution of the
first BPS equation
\begin{equation}
B_i = D_i \hat b\cdot \phi
\label{bpse1}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
is uniquely characterized by the relative distance, $L$, between two
monopoles. Once the first BPS solution is found, the solution of
the second BPS equation
\begin{equation}
D_i^2 \hat a\cdot\phi - [\hat b\cdot\phi, [\hat b\cdot\phi,\hat
a\cdot\phi]] =0
\label{bpse2}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
is found to be unique for a given asymptotic (\ref{asymp2}). From
this solution, we can read electric
charges
\beqn
&& q_1 = g (\xi +\eta p_1) ,\\
&& q_2 = g (\xi+\eta p_2) ,
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where
\beqn
&& p_1 = \frac{\mu_1 -\mu_2-2(\mu_1+2\mu_2)\mu_2 L}{\mu_1+\mu_2+
2\mu_1\mu_2 L}, \\
&& p_2 = \frac{\mu_1-\mu_2+2(2\mu_1+\mu_2)\mu_1
L}{\mu_1+\mu_2+2\mu_1\mu_2 L}.
\end{eqnarray}} \newcommand{\bde}{{\bf e}
As discussed in the introduction, one of the interesting things about
these BPS solutions is that we
may treat the 1/4 BPS configurations as if they are made of 1/2 BPS
monopoles with some electric
dress. The first BPS equation can be regarded as the 1/2 BPS
equation. Its solution describes a collection of
1/2 BPS monopoles and is characterized uniquely by the moduli space
coordinates.
Here the constituent monopoles carry mass characterized by the Higgs
asymptotic value $\hat b\cdot\phi(\infty)$. Then we solve the second
BPS equation, which is identical to the gauge zero mode equation of the
first BPS equation. The solutions of the second BPS equation is
determined uniquely by the moduli parameters of the solution of the
first BPS equation and by the value $\hat{a}\cdot \phi(\infty)$. One of the
key point of this paper is to take this view further and to regard
the low energy dynamics of 1/4 BPS dyons as that of 1/2 BPS monopoles
described by the first BPS equation.
Thus, instead of $m_1$ and $m_2$, we regard $g\mu_1$ and $g\mu_2$ as
the real mass of monopoles. We
can define the total and relative charges with respect to the
masses $g\mu_1$ and $g\mu_2$, that is,
\beqn
&& q_{\rm tot} = \frac{\mu_1q_1+\mu_2q_2}{\mu_1+\mu_2}= g \left( \xi +
\eta \frac{\mu_1-\mu_2}{\mu_1+\mu_2} \right) \label{qtot}, \\
&& \Delta q(L) = \frac{q_2-q_1}{2} = \frac{\Delta q_c}{1+
\frac{1}{2\mu L}},
\label{dqr}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where the critical relative charge is defined as
\begin{equation}
\Delta q_c \equiv g\eta \frac{\mu_1^2+\mu_1\mu_2 +\mu_2^2}{\mu_1\mu_2},
\label{dqc}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
and the pseudo relative mass is
\begin{equation}
\mu = \frac{\mu_1\mu_2}{\mu_1+\mu_2}.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
Notice that the relative charge vanishes when $L=0$ and
tends to $\Delta q_c$ at $L=\infty$. (When three D3 branes lie on a line
and so $\eta=0$, the relative charge vanishes.) We further
note that the mass difference, $\sum_i(m_i- g\mu_i)$, may be
written, in terms of charges, as
\begin{equation}
m_1+m_2-g(\mu_1+\mu_2)= {\mu_1+\mu_2\over 2g}(q_{\rm tot})^2 +
{2\mu\over g} (\Delta q_c)^2.
\label{difference}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The difference between
$m_i$ and $g\mu_i$ can be considered as a result of the interaction
between monopoles, which even includes a constant potential.
\subsection{BPS Energy}
In the BPS energy bound, it is natural to introduce
the two-dimensional magnetic and electric charge vectors,
\beqn
&& {\vec Q}^M = g( {\vec R}_1 + {\vec R}_2), \\
&& {\vec Q}^E = q_1 {\vec R}_1 + q_2 {\vec R}_2 .
\end{eqnarray}} \newcommand{\bde}{{\bf e}
The central charge of the N=4 supersymmetric algebra, which gives the
BPS energy bound, is
\begin{equation}
Z^2_\pm = {\rm max} \left( {\vec Q}_M^2 + {\vec Q}_E^2 \pm 2{\vec Q}_E\times
{\vec Q}_M \right) .
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
When the $BPS$ equations are satisfied so that the BPS energy bound
is saturated, there is another expression for
the central term $Z$. Since ${\hat a}\cdot {\vec Q}^M = {\hat b}\cdot
{\vec Q}^E$, the energy is
\beqn
Z_+ &=& {\hat b}\cdot {\vec Q}^M + {\hat a}\cdot {\vec Q}^E
\nonumber \\
&=& g(\mu_1+\mu_2) +\frac{\mu_1+\mu_2}{g} (q_{\rm tot})^2
+ \frac{4\mu }{g} \Delta q(L) \Delta q_c,
\label{bps0}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
which is exact. As ${\hat b}\cdot {\vec Q}^M= g(\mu_1+\mu_2) $ is
just the sum of constituent monopole masses, we regard the rest of
the contribution, ${\hat a}\cdot {\vec Q}^E$, to arise from the
dynamics of monopoles.
The low energy dynamics means that the dynamical energy contribution
to the rest mass is small. From Eqs.~(\ref{qtot}) and (\ref{dqr}), we
see that the low energy approximation may hold if
\begin{equation} |\xi|,\;\; |\eta|\; << 1 . \end{equation}} \newcommand{\beqn}{\begin{eqnarray}
{}From Eqs.~(\ref{basy}) and (\ref{aasy}), we see that three D3 branes
are almost collinear, for the low energy approximation to hold. The
above condition also implies that the magnitude of $q_{\rm tot},
\Delta q$ of 1/4 BPS configurations should be very small compared with
the magnetic charge $g$. {}From our point of view, ${\hat a}\cdot
{\vec Q}^E$ in the central charge arises from the low energy dynamics
of monopoles. As we will see, it has the contributions from both
kinetic and potential energies.
\subsection{Moduli Space Metric of 1/2 BPS Monopoles}
When three D3 branes are collinear, the low energy dynamics of two
monopoles can be described by the moduli space or collective
coordinate dynamics. There are four zero modes for each monopole,
three for its position and one for the U(1) phase. We call their
positions and phases to be ${\bf x}_i,\psi_i, i=1,2$ for ${\mbox{\boldmath $\alpha$}}$ and
${\mbox{\boldmath $\beta$}}$ monopoles, respectively. The exact nonrelativistic effective
Lagrangian has been found to be a sum of the Lagrangians for the
center of mass and the relative motion~\cite{connell}. As there is no
external force, the center of mass Lagrangian is free one,
\begin{equation}
{\cal L}_{\rm cm} = \frac{g(\mu_1+\mu_2)}{2} \dot{{\bf X}}^2 +
\frac{g}{2(\mu_1+\mu_2)} \dot{\chi}^2 ,
\label{cmL}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
where the center of mass position is ${\bf X} = ( \mu_1 {\bf x}_1 +
\mu_2 {\bf x}_2)/(\mu_1+\mu_2)$ and the center of mass phase is $\chi
=\psi_1+\psi_2 $.
The relative motion between them is more complicated and described
by the Taub-NUT metric,
\begin{equation}
{\cal L}_{\rm rel}
= \frac{g\mu }{2} \left( ( 1+ \frac{1}{2\mu r}) \dot{\bf r}^2
+ \frac{1}{4\mu^2(1+ \frac{1}{2\mu r})}
(\dot{\psi} + {\bf w}({\bf r})\cdot \dot{\bf r})^2 \right),
\label{relL}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
where the relative position is ${\bf r} = {\bf x}_2-{\bf x}_1$, the
relative phase is $\psi =2(\mu_1\psi_2-\mu_2\psi_1)/(\mu_1+\mu_2)$,
and ${\bf w}({\bf r}) $ is the Dirac potential such that $\nabla
\times {\bf w}({\bf r}) = -{\bf r}/r^3$. The range of $\psi$ is
$[0,4\pi]$. The relative moduli space ${\cal M}_0$ is the Taub-NUT
manifold with the metric given from the above Lagrangian.
The eight-dimensional moduli space is then given by
\begin{equation}
{\cal M} = R^3 \times \frac{ S^1 \times {\cal M}_0}{Z},
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
where $Z$ is the identification map
\begin{equation}
(\chi, \psi) = (\chi + 2\pi, \psi + \frac{4\pi \mu_2}{\mu_1+\mu_2}).
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
One way of obtaining this moduli space metric is by exploring the long
range interaction between two dyons\cite{manton1}. As for two dyons
of charge $(g, \pi_1){\mbox{\boldmath $\alpha$}}$ and $(g,\pi_2){\mbox{\boldmath $\beta$}}$, there are several
interactions between them. Obvious ones are electric and magnetic
Coulomb potentials. Besides, there exists a potential due to Higgs
force between them.. In addition, when they move, there is a time delay
effect which appears in the form of Lienard-Wiechert potential. When
they are put together, we get the Routhian obtained from the Legendre
transformation of phase variables to charges,
\begin{equation}
{\cal R}= \frac{g(\mu_1+\mu_2)}{2} \biggl(\dot{\bf X}^2 -
\frac{(\pi_{\rm t})^2}{g^2} \biggr)
+ \frac{g\mu}{2} \bigl( 1+ \frac{1}{2\mu r}\bigr) \biggl(\dot{\bf r}^2 -
\frac{4(\pi_{\rm r})^2}{g^2} \biggr) +\pi_{\rm r}
{\bf w}({\bf r})\cdot \dot{\bf r},
\label{routhian}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
where the relation between $\pi_{\rm t}, \pi_{\rm r}$ and
$\pi_1$ and $\pi_2$ are given by similar equations as in
Eqs.~(\ref{qtot}) and (\ref{dqr}). We emphasize
here that total charge $\pi_{\rm t}$ and relative charge $\pi_{\rm r}$
are conjugate momenta without any fixed value.
\section{Case of $SU(3)$}
To find the potentials between two dyons in every relative separation,
we explore first the case $q_{tot}=\pi_{\rm t}=0$. Suppose that the
exact potential for the relative motion is ${\cal U}_{\rm rel}(r)$, and
so the effective potential for two dyons is
\begin{equation}
U_{\rm eff}(r) = \frac{2\mu}{g}\left( 1+ \frac{1}{2\mu r}\right)
(\pi_{\rm r})^2 + {\cal U}_{\rm rel}(r)
\label{ueff0}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The first part of the effective potential comes from the charge
kinetic energy. This effective potential should have a minimum at $r$
if $\pi_{\rm r} = \Delta q(r)$ for $\Delta q(r)$ specified by
Eq.~(\ref{dqr}) and the energy at the minimum should be identical to
the BPS energy (\ref{bps0}) once we add the mass of monopoles. Then we
realize that the potential energy is exactly one half of the excessive
BPS energy associated with the relative charge. Using this, we find
the piece of the potential,
\beqn
{\cal U}_{\rm rel}(r) &=& \frac{2\mu}{g}
\frac{(\Delta q_c)^2}{ 1 + \frac{1}{2\mu r}} \nonumber \\
&=& \frac{2\mu}{g}( 1 + \frac{1}{2\mu r})(\Delta q(r))^2
\label{pote0}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where $\Delta q (r)$ is to be understood as function of $r$ given in
Eq.~(\ref{dqr}). However, we have not quite found the potential $\cal
U$. We actually found only the piece that depends on the distance
$r$. By including the center of mass motion, it is not difficult to
guess that the actual potential must be of the form,
\begin{equation}
{\cal U}(r)=\frac{\mu_1+\mu_2 }{2g}q_{\rm tot}^2 + \frac{2\mu}{g}
\frac{(\Delta q_c)^2}{ 1 + \frac{1}{2\mu r}}
\label{pote}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
This guess will be justified below momentarily.
As we have seen, this identification can be made for all possible classical
dyons, which translates to all possible values of $r$. In effect,
we have found the potential $\cal U$ throughout the moduli space.
In the next section, we will discover that this method is trivially
generalized to arbitrary configurations of monopoles and dyons in
arbitrary gauge group, and that it produces a rather special kind
of potential, which, as one may expect, admits supersymmetric extension.
Let us go back to the matter concerning total charge and its
associated constant potential energy. If we minimize the effective
potential (\ref{ueff0}), we get $\pi_{\rm r}$ to be $\Delta q(r) =
\Delta q_c/(1+ \frac{1}{2\mu r})$. However, we could not fix $\pi_{\rm
t} $ to be $q_{\rm tot}$ in this way. Hamilton's equations of motion
imply that $\pi_{\rm t}$ to be constant in time. Thus, the naive
equilibrium condition does not fix the total electric charge, even
though it fixes the relative charge. To understand this, let us now
collect the full low energy Lagrangian for 1/4 BPS dyons, which is the
sum of the kinetic energies (\ref{cmL}), (\ref{relL}) on the moduli
space and the potential energy (\ref{pote})
\begin{equation}
{\cal L}_{\rm low} = {\cal L}_{\rm cm} + {\cal L}_{\rm rel} - {\cal U}(r).
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
In terms of the conjugate momenta, ${\bf P}_{\rm t} = g(\mu_1 + \mu_2)
\dot{\bf X}$, ${\bf p} = g\mu (1+ \frac{1}{2\mu r})\dot{\bf r}
+\pi_{\rm r} {\bf w}({\bf r})$, $\pi_{\rm t}$, and $\pi_{\rm
r}$, the Hamiltonian is
\beqn
{\cal H}&= & \frac{1}{2g(\mu_1+\mu_2)} {\bf P}^2 +
\frac{1}{2g\mu}\frac{1}{1+\frac{1}{2\mu r}} ({\bf p}- \pi_{\rm
r}{\bf w}({\bf r}))^2 \nonumber \\
& & +\frac{\mu_1+\mu_2}{2g} (\pi_{\rm t})^2+
\frac{2\mu}{g}(1+ \frac{1}{2\mu r}) \pi_{\rm r}^2 + {\cal U}(r) .
\end{eqnarray}} \newcommand{\bde}{{\bf e}
The Hamiltonian can be expressed as follows:
\beqn
{\cal H} &= & \frac{1}{2g(\mu_1+\mu_2)} {\bf P}^2 +
\frac{1}{2g\mu}\frac{1}{1+\frac{1}{2\mu r}} ({\bf p}- \pi_{\rm
r}{\bf w}({\bf r}))^2 \nonumber \\
& & + \frac{\mu_1+\mu_2}{2g} (\pi_{\rm t} \mp q_{\rm tot})^2+
\frac{2\mu}{g}(1+ \frac{1}{2\mu r}) (\pi_{\rm r}\mp \Delta q(r))^2 \pm
{\cal Z} ,
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where the new central term is
\begin{equation}
{\cal Z} = \frac{\mu_1+\mu_2}{g} q_{\rm tot} \pi_{\rm t}
+ \frac{4\mu}{g} \Delta q_c \pi_{\rm r}.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The central term is linear in the conjugate momenta $\pi_{\rm t}$
and $\pi_{\rm r}$.
Clearly there is a classical BPS bound on the mechanical energy
\begin{equation}
{\cal H} \ge |{\cal Z}|,
\label{cbps}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
which is saturated for any ${\bf X}$ and ${\bf r}$ when $\pi_{\rm t} =
q_{\rm tot}$ and $\pi_{\rm r} =
\Delta q(r)$ and $\dot{\bf X}=\dot{\bf r} = 0$. Thus the nonrelativistic
BPS configuration matches exactly the field theoretic BPS
configuration. In this case, the sum of the rest mass plus this BPS
energy is exactly the 1/4 BPS energy bound (\ref{bps0}). Thus our 1/4
BPS field configuration corresponds to not the lowest energy
configuration, but a BPS saturated configuration of the nonrelativistic
Hamiltonian.
As an independent check of the above potential, let us consider the
leading term in the large distance. When three D3 branes are not collinear,
two Higgs fields are involved nontrivially and so we expect an additional
Higgs interaction~\cite{fraser}. To see this, let us go back to the
old derivation of the Higgs interaction. We put the resting ${\mbox{\boldmath $\alpha$}}$
dyon at ${\bf x}_1$. The Higgs field far from this dyon is
\begin{equation}
\phi_I({\bf x}) = \phi_I (\infty) - \frac{\hat{R}_{1I}}{4\pi
|{\bf x}-{\bf x}_1|} {\mbox{\boldmath $\alpha$}} \cdot {\bf H} \sqrt{g^2 + \pi_1^2},
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
in the unitary gauge. Note that ${\mbox{\boldmath $\alpha$}} \cdot \phi_I(\infty) = \vec{
R}_{1I}$. The ${\mbox{\boldmath $\beta$}}$ monopole at ${\bf x}={\bf x}_2$ would feel
this Higgs
field as an effective mass
\begin{equation}
-m_{\rm eff}= - \sqrt{g^2+\pi_2^2}\: \biggl| {\vec R}_2 - {\hat R}_1
\sqrt{g^2+ \pi_1^2} \;\frac{{\mbox{\boldmath $\alpha$}} \cdot {\mbox{\boldmath $\beta$}} }{4\pi r} \biggr|,
\label{meff0}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
where $r=|{\bf x}_2-{\bf x}_1|.$
We expand the effective mass to order $1/r$ and to quadratic terms in $\pi_1$
and $\pi_2$ to get
\begin{equation}
-m_{\rm eff} = -\biggl(1+ \frac{\pi_2^2}{2g^2}\biggr) m_2
- \frac{g^2}{8\pi r} {\hat R}_1 \cdot {\hat R}_2 \biggl( 1+
\frac{\pi_1^2}{2g^2}\biggr)\biggl( 1+ \frac{\pi_2^2}{2g^2} \biggr).
\label{meff}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
If three D3 branes are collinear, ${\hat R}_1 = {\hat R}_2$ and
we get the old result. When we include the velocity dependent terms
and electromagnetic forces and keep everything in quadratic order, we
get the previous Routhian (\ref{routhian}).
However, when they are not collinear, there is an additional $1/r$
correction to the old result. As we assume that the deviation from the
straight line is very small, or $\eta << 1 $,
\begin{equation}
{\hat R}_1 \cdot {\hat R}_2 = \cos \theta \approx 1 -
\frac{\theta^2}{2}.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
{}From Eqs. (\ref{R1}) and (\ref{R2}), one can show easily that
\beqn
\theta &=& 2 \eta \frac{\mu_1^2+\mu_1\mu_2+\mu_2^2}{\mu_1\mu_2} +{\cal
O}(\eta^2) \nonumber \\
&=& \frac{2\Delta q_c}{g}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
with $\Delta q_c$ in Eq.~(\ref{dqc}).
The small deviation obtained from Eq.~(\ref{meff}) is then
\beqn
\Delta L &=& -m_1-m_2 +g(\mu_1+\mu_2)+
\frac{g^2}{8\pi r} (1- \hat{ R}_1 \cdot \hat{R}_2) \nonumber
\\ &=& -{\mu_1+\mu_2\over 2g}(q_{\rm tot})^2 -
{2\mu\over g} (\Delta q_c)^2 +
\frac{ (\Delta q_c)^2}{ 4\pi r},
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where we have included the additional constant terms from masses that
are given explicitly in (\ref{difference}). This is the additional
attractive potential which exists since three D3 branes are not
collinear. It is a matter of simple algebra to check that the exact
potential (\ref{pote}) we found above, do contain this term to the
leading $r$-dependent piece at large $r$. One may wonder about terms
of order $1/r^2$. These terms do not come out to be symmetric under
the exchange of ${\mbox{\boldmath $\alpha$}}$ and ${\mbox{\boldmath $\beta$}}$ monopoles and depend on the
monopole core size. It needs a further investigation, which we do not
attempt here.
In the above derivation we have dropped terms of order $(\Delta q_c)^2
(\pi_i)^2$ and $(\Delta q_c)^4$. Terms of order $(\Delta q_c)^2
(\pi_i)^2$ can be regarded as the correction to the kinetic term which
exists since D3 branes are not collinear. Corrections of such order is
expected from the naive moduli space metric as there will be
additional term such as $\int d^3x {\rm tr}\,} \newcommand{\beq}{\begin{equation} (\hat{a}\cdot \dot{\phi})^2$.
Terms of order $(\Delta q_c)^4$ can be regarded as the correction to
the potential energy and is negligible compare with terms of order
$(\Delta q_c)^2$.
The low energy approximation we found holds when the kinetic and
potential energies are much smaller than the rest mass. In terms of
generic velocity $v$ and the coefficient, $\epsilon$, of
$\hat{a}\dot{\phi}(\infty)$, the low enegy approximation holds when
\begin{equation}
v<< 1, \;\;\, \epsilon << 1.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The order of the low energy Lagrangian is then
\begin{equation}
L \sim v^2 + \epsilon^2
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The corrections we have neglected in the above expansion is then of order
\begin{equation}
v^4, \;\;\; v^2 \epsilon^2, \;\;\; \epsilon^4.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
\section{Generalization}
The crucial ingredient of the preceding section came from a rather
simple observation: we have two different ways of determining the 1/4
BPS dyon mass. The field theoretic one gives an exact expression,
while the other is derived from approximate low energy dynamics on the
moduli space of monopoles. This is possible because the moduli space
dynamics already incorporates various internal excitations that induce
relative electric charges. By comparing the two, one obtains an
approximate form of the inter-monopole potential that is arbitrarily
accurate as the Higgs expectations become collinear. In this section,
we will utilize this simple idea to fix the bosonic part of low energy
effective Lagrangian for all 1/2 BPS and 1/4 BPS states in maximally
supersymmetric Yang-Mills theory.
First consider a 1/2 BPS multi-monopole configuration, when an
arbitrary gauge group $G$ is broken to $U(1)^r$ by a single Higgs
field expectation value $\langle {\hat b}\cdot \phi\rangle={\bf h}$ in
the root space. There are always a set of simple roots ${\mbox{\boldmath $\beta$}}_A,
A=1,...r$ such that ${\bf h}\cdot {\mbox{\boldmath $\beta$}}_A>0$. Any BPS magnetic
monopole configuration would have the magnetic charge
\begin{equation}
{\bf g} = g(n_1 {\mbox{\boldmath $\beta$}}_1 + n_2 {\mbox{\boldmath $\beta$}}_2+... + n_r {\mbox{\boldmath $\beta$}}_r),
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
with nonnegative $n_A's$. Without loss of generality, we
assume that it is irreducible so that
all $n_A$ are positive. In this case all monopoles are interacting
with each other directly or indirectly and are not divided into
noninteracting subclusters.
Each monopole of ${\mbox{\boldmath $\beta$}}_A$ type has mass $g\mu_A=g {\bf h}\cdot
{\mbox{\boldmath $\beta$}}_A$ and four zero modes. The total energy is then $g\sum_A n_A
\mu_A = {\bf g}\cdot {\bf h} $ and the total number of zero mode is
$4N=4\sum_A n_A$~\cite{ejw}. The low energy dynamics of this
configuration can be described by the dynamics in the moduli space of
dimension $4N$ with the metric $ds^2 = g_{\mu\nu} dz^\mu dz^\nu$. This
moduli space is known to be hyperk\"ahler and thus are equipped with
three covariantly constant complex structures~\cite{atiyah, blum}. The
low energy Lagrangian is then
\begin{equation}
{\cal L}_1 = \frac{1}{2}\; g_{\mu\nu}\dot{z}^\mu \dot{z}^\nu .
\label{calL}
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The unbroken $U(1)^r$ symmetries act on the moduli space as
translational isometries, and generate cyclic $U(1)$ coordinates
$\psi^A$, whose conjugate momenta $q_A$ are conserved electric
charges. We divide the moduli space coordinates $z^\mu$ into $r$
$\psi^A$'s and remaining $(4N-r)$ $y^i$'s. Up to gauge
transformations, the solution of the first BPS equation (\ref{bpse1})
is uniquely characterized by the values of the coordinates $y^i$. The
Lagrangian (\ref{calL}) can be rewritten as
\begin{equation}
{\cal L}_1 = \frac{1}{2} \; h_{ij} \dot{y}^i \dot{y}^j + \frac{1}{2}
L_{AB} (\dot{\psi}^A + w^A_{\; i} \dot{y}^i) (\dot{\psi}^B+ w^B_{\;
j}\dot{y}^j).
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
Due to the cyclic properties of $\psi^A$'s, the metric components
$h_{ij}, L_{AB}$ and $w^A_{\; i}$ are functions of $y^i$ only.
Since the kinetic energy should be positive, the metric $h_{ij}$ and
$L_{AB}$ are positive definite. Each $U(1)$ generators are associated
with the vector field $K_A=\partial/\partial{\psi^A} = \delta_A^\mu
\partial/\partial z^\mu$,
which is a (triholomorphic) Killing vector field. Finally,
note that $L_{AB}(y) = g_{\mu\nu}K_A^\mu K_B^\nu$.
Denoting the conjugate momenta as
\beqn
&& q_A = L_{AB}(\dot{\psi}^B+ w^B_{\; j} \dot{y}^j), \\
&& p_i = h_{ij} \dot{y}^j + q_A w^A_{\; i},
\end{eqnarray}} \newcommand{\bde}{{\bf e}
the electric charge of the whole configuration is expressed in terms of
the $q_A$'s as
\begin{equation}
{\bf q} = q_1 {\mbox{\boldmath $\beta$}}_1 +q_2 {\mbox{\boldmath $\beta$}}_2+...+q_r {\mbox{\boldmath $\beta$}}_r .
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The Hamiltonian ${\cal H}_1 = \frac{1}{2} g^{\mu\nu}p_\mu p_\nu$ is
\begin{equation}
{\cal H}_1 = \frac{1}{2} h^{ij}(p_i - q_A w^A_{\; i}) (p_j - q_B w^B_{\; j})
+ \frac{1}{2} (L^{-1})^{AB} q_A q_B.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
Since $L^{-1}$ depends on the $y^i$'s, this shows that electric
charge excitations will typically induce a long range potential.
Because of this, 1/4 BPS configurations of dyons with relative charge
cannot be allowed if no other static forces are present,
as would be the case if Higgs expectations were all aligned.
Suppose that the other Higgs field $\hat a\cdot \phi$ acquires an
expectation value which is misaligned with respect to $\langle \hat
b\cdot \phi\rangle$. Then there is an attractive static force between
monopoles. If the misalignment is small enough so that the static
attractive potential is much smaller than the monopole mass scale, we
may incorporate this potential into the above moduli space
dynamics. (The part of Higgs expectation $\hat a\cdot\phi$ that is
proportional to that of $\hat b\cdot \phi$ is not associated with any
static force. Rather, its effect on the BPS configuration is to fix
the total charge to a certain value and to give an additional
contribution to the energy besides the energy carried by the
charge. This can be understood as the correction to the bare mass of
magnetic monopoles.)
Adding a potential to the Hamiltonian and considering the static
configuration with $\dot{y}_i=0$, we get an effective potential
\begin{equation}
{ U}_{\rm eff} = \frac{1}{2} q_A (L^{-1}(y))^{AB} q_B + {\cal U}(y).
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
For 1/4 BPS configurations, $q_A$ should be determined by the
moduli parameters $y^i$, or reversely, given values of $q_A$
restrict the equilibrium positions $y^i$.
On the other hand, an exact expression of the 1/4 BPS configuration
is known. The additional energy due to the electric charge is given
by a simple expression,
\begin{equation}
E_Q = {\hat a}\cdot {\vec Q}^E = a^A \bar{q}_A(y).
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The second equality defines the projected Higgs expectation values $a^A$.
The electric charge
are fixed by the positions of magnetic monopoles, which we emphasize
by introducing new notation $\bar{q}_A(y)$. A recent work by
D. Tong~\cite{tong} brought some light on this quantity. He expressed
$\bar{q}_A(y)$ in
terms of quantities on moduli space and found
\begin{equation}
\bar{q}_A (y) \equiv L_{AB}(y)\, a^B.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
This equation can be viewed in two equivalent ways.
One is as a restriction on the
possible equilibrium positions $y^i$ when the electric charges are given.
Or equivalently, as an expression for electric charges in terms of
the equilibrium positions $y^i$. Either way, the excess energy due to the
electric charge excitation is
\begin{equation}
a^A \bar{q}_A (y) = a^A L_{AB}(y) \,a^B = \bar{q}_A(y)\,
(L^{-1}(y))^{AB} \bar{q}_B(y) .
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
Note that this happens to be twice the charge kinetic energy if we put
$q_A= \bar{q}_A(y)$. Again we demand the potential ${\cal U}(y)$ to
be identical to one half of $E_Q$, that is, when expressed as a
function of $y^i$,
\begin{equation}
{\cal U}(y) = \frac{1}{2}\, a^A L_{AB}(y) \,a^B .
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
We need to pause here for a moment, and explain how it was possible
that we obtained the potential ${\cal U}$ by considering only the
classical minimum energy configurations of ${U}_{\rm eff}$. It may
seem that we made an extrapolation of some kind. However, it is not
the case. We actually have derived the exact potential, as we will
explain below.
{}From Ref.~\cite{yi}, we know that solutions to the
first BPS equation (\ref{bpse1}) are characterized by moduli
parameters $y^i$'s. That is, any given set of $y^i$, a purely magnetic
solution exists. Then one solves the second BPS equation
(\ref{bpse2}), which leads to 1/4 BPS dyons whose configuration
satisfies the relation $q_A= \bar{q}_A(y) = L_{AB}(y) a^B$. In other
words, there are 1/4 BPS configurations for any generic values of
$y^i$'s. Therefore, while we identified the value of potential $\cal
U$ for individual 1/4 BPS states, we can learn the values of $\cal
U$ for all $y^i$ by considering all possible classical 1/4 BPS dyons.
Here, we need to make one final consistency check. Not only the value
of the potential ${\cal U}(y)$ at the `minimum' of the effective potential
$U_{\rm eff}(y)$ should yield the right value, which led to the above
identification, but also it should have the `minimum' at the right
value. In other words, we must recover the central relationship $q_A=
\bar{q}_A(y) = L_{AB}(y) a^B$, which is of field theory origin, by minimizing
the low energy dynamics. The final Hamiltonian with the potential is
\begin{equation}
{\cal H} =
\frac{1}{2} h_{ij}(y)\dot{y}^i \dot{y}^j
+ \frac{1}{2} (L^{-1})^{AB}(y) q_A q_B
+ \frac{1}{2} a^A L_{AB}(y) a^B,
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
where $\dot{y}^i = h^{ij} (p_j - q_A w^A_{\; j})$.
As in the previous section, we can reexpress this as
\begin{equation}
{\cal H} = \frac{1}{2} h_{ij}(y) \dot{y}^i \dot{y}^j
+ \frac{1}{2}(L^{-1})^{AB}(y) (q_A \mp \bar{q}_A(y))(q_B\mp \bar{q}_B(y))
\pm {\cal Z} ,
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
where the central charge is
\begin{equation}
{\cal Z} = \bar{q}_A(y)\,(L^{-1})^{AB}(y)\, q_B = a^A \,q_A.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
The BPS bound is saturated when $\dot{y}^i=0$ and $q_A = \bar{q}_A(y)$,
exactly as we have hoped for. This completes the derivation of the
potential $\cal U$.
The effective Lagrangian for the low energy is the sum of the usual kinetic
term on the moduli space and the potential $\cal U$,
\begin{equation}
{\cal L} = \frac{1}{2} \; h_{ij} \dot{y}^i \dot{y}^j + \frac{1}{2}
L_{AB} (\dot{\psi}^A + w^A_{\; i} \dot{y}^i) (\dot{\psi}^B+ w^B_{\;
j}\dot{y}^j) - \frac{1}{2} a^A L_{AB}(y) a^B,
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
or, more compactly,
\begin{equation}
{\cal L} = \frac{1}{2}\, g_{\mu\nu} \dot{z}^\mu \dot{z}^\nu -
\frac{1}{2}\, g_{\mu\nu} (a\cdot K)^\mu (a\cdot K)^\nu.
\end{equation}} \newcommand{\beqn}{\begin{eqnarray}
Note that, given the Higgs expectation values, the potential is completely
determined in terms of geometry of the monopole moduli space. As showed
by Alvarez-Gaume and Freedman~\cite{alvarez}, and as pointed out by
D. Tong~\cite{tong} very recently,
this form of potential is exactly what one needs to extend the dynamics
to a supersymmetric one. In particular, the triholomorphicity of $a\cdot K$
that follows from gauge invariance, guarantees that the low energy dynamics
will have 8 real supercharges. In this sense the dynamics itself is 1/2 BPS
with respect to the Yang-Mills field theory. The quantum counterpart
of the classical 1/4 BPS dyons should break additional half of these remaining
8 supercharges, and is realized as finite energy BPS states of this low
energy theory itself. In the next section we will explore this supersymmetric
dynamics in some detail.
\section{Supersymmetry and BPS Bound}
We begin with the N=4 supersymmetric quantum extension of the above
effective action~\cite{alvarez}. Its form is similar to the usual
sigma model action but modified by a coupling to the triholomorphic
Killing vector $G\equiv a\cdot K$. The supersymmetric Lagrangian
written with real fermions
is
\beqn
{\cal L}&=&{1\over 2} \biggl( g_{\mu\nu} \dot{z}^\mu \dot{ z}^\nu +
ig_{\mu\nu} \bar\psi^\mu \gamma^0 D_t \psi^\nu + {1\over 6}
R_{\mu\nu\rho\sigma}\bar\psi^\mu \psi^\rho \bar\psi^\nu \psi^\sigma
\biggr.
\nonumber\\
&& \biggl. - g^{\mu\nu} G_\mu G_\nu - D_\mu G_\nu \bar\psi^\mu
\gamma_5\psi^\nu \biggr),
\label{action}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where $\psi^\mu$ is a two-component anticommuting Majorana spinor
and $\gamma^0= \sigma_2$, $\gamma_5= \sigma_3$, and $\bar\psi=\psi^T
\gamma^0$.
The metric here is hyperk\"ahler, endowed with three complex
structures $f^{(a)\mu}\,_\nu (a=1,2,3)$ that satisfy
\beqn
&&f^{(a)}f^{(b)} = - \delta^{ab} +\epsilon^{abc} f^{(c)}, \\
&& D_\mu f^{(a)\nu}\,_\rho =0\, ,
\label{complex}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
and the Killing vector $G^\mu$ is triholomorphic, i.e., its action
preserves the three complex structures. From now on, we
will use $f$ to denote any one of the three complex structures,
unless noted otherwise.
With vielbein $e_\mu^A$ and the spinors $\psi^A = e_\mu^A \psi^\mu$,
we define supercovariant momenta by
\begin{eqnarray}
&& \pi_\mu \equiv p_\mu -{i\over 2}\omega_{AB\mu}
\bar\psi^A \gamma^0 \psi^B
\label{cov}
\end{eqnarray}
where the $p$'s are canonical momenta of coordinate $z$'s,
and $\omega^A\,_{B,\mu}$ is the spin connection. The canonical
commutation relations are $[z^\mu, p_\nu ] = i\delta^\mu_\nu $ and $\{
\psi^A_\alpha, \psi^B_\beta\} = \delta^{AB}\delta_{\alpha\beta}$.
SUSY generators in real form are:
\begin{eqnarray}
&&Q_\alpha = \psi^\mu_\alpha \pi_\mu
+i(\gamma^0\gamma_5 \psi^\mu)_\alpha G_\mu ,\\
&&Q^f_\alpha = f^\mu\,_\nu \psi^\nu_\alpha \pi_\mu
+i(\gamma^0\gamma_5 f^\mu\,_\nu\psi^\nu)_\alpha G_\mu,
\label{generator}
\end{eqnarray}
which satisfy the following SUSY algebra:
\beqn
&&\{Q_\alpha,Q_\beta \} =\{Q^f_\alpha,Q^f_\beta\}=2
\delta_{\alpha\beta} \; {\cal H} +
2 i(\gamma^0\gamma_5)_{\alpha\beta} \; {\cal Z} \\
&& \{Q_\alpha,Q^f_\beta \} =0
\label{algebra}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
Similarly, supercharges associated with different complex
structures $f^{(a)}$ anticommute.
The Hamiltonian $\cal H$ and the central charge $\cal Z$ read
\beqn
&&{\cal H}=
{1\over 2} \biggl( {1\over \sqrt{g}}\pi_\mu \sqrt{g }g^{\mu\nu}\pi_\nu
+ G_\mu G^\mu -{1\over 4}R_{\mu\nu\rho\sigma}\bar\psi^\mu
\gamma^0 \psi^\nu \bar\psi^\rho
\gamma^0 \psi^\sigma + D_\mu G_\nu \bar\psi^\mu \gamma_5\psi^\nu \biggr)\\
&& {\cal Z}= G^\mu \pi_\mu -{i\over 2} (D_\mu G_\nu) \bar\psi^\mu
\gamma^0\psi^\nu
\label{hamiltonian}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
It is easily checked that the central charge $\cal Z$ indeed commutes with
all SUSY generators.
For spectrum analysis, SUSY generators in complex form are more useful.
Introducing $\varphi\equiv {1\over \sqrt{2}} (\psi_1^\mu -i\psi_2^\mu)$,
and defining $ Q\equiv {1\over \sqrt{2}} (Q_1-iQ_2)$,
one finds
\beqn
&&Q = \varphi^\mu \pi_\mu
+ i \varphi^{*\mu} G_\mu,\\
&& Q^{\dagger}=\varphi^{*\mu} \pi_\mu
- i \varphi^{\mu} G_\mu,
\label{comgenerator1}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
which generates the following simple algebra:
\begin{eqnarray}
&&\{Q,Q^{\dagger}\}=\{Q^f,{Q^f}^{\dagger}\} =2 {\cal H},\\
&&\{Q,Q\} = \{Q^f,Q^f\}=-\{Q^{\dagger},Q^{\dagger}\} =
-\{{Q^f}^{\dagger},{Q^f}^{\dagger}\}=
2i{\cal Z},\\
&& \{Q,Q^f\}=\{Q^{\dagger}, Q^f\}=0.
\label{algebra2}
\end{eqnarray}
Again, supercharges associated with different complex
structures $f^{(a)}$ anticommute.
It is easy to read off the BPS condition for quantum states that preserves
half of supersymmetries. Depending on the sign of central charge, we find
\begin{eqnarray}
&&(Q\mp iQ^\dagger)|\Phi\rangle =0,
\label{comgenerator2}
\end{eqnarray}
so that the given state may saturate the condition ${\cal H}=\pm {\cal Z}$.
We can express this BPS condition in more geometrical fashion by transcribing
the wavefunction to differential forms on the moduli
space~\cite{witten1}. Note that
\beqn
&& [i\pi_\mu,\varphi^\nu]= -\Gamma^\nu_{\mu\rho} \varphi^{\rho},\\
&& [i\pi_\mu, \varphi^*_\nu]=
\Gamma^\rho_{\mu\nu} \varphi^*_{\rho},\\
&& \{ \varphi^\mu,\varphi^{*}_\nu\}=\delta_\nu^\mu.
\label{comgenerator3}
\end{eqnarray}
Furthermore, the wavefunction has the following general form,
\beqn
&& |\Phi \rangle =
\sum_p \frac{1}{p!}\,\Omega_{\mu_1 \cdots \mu_p}(z^\mu)
\varphi^{\mu_1}\cdots \varphi^{\mu_p}|0\rangle \\
&& \varphi^{*\mu}|0\rangle =0.
\label{comgenerator}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
The coefficients $\Omega_{\mu_1 \cdots \mu_p}$ are completely antisymmetric
and may be regarded as those of a $p$-form. In this language where we
interpret $\varphi^\mu$ and $\varphi^*_\mu$ as
a natural cobasis $dz^\mu$ and a natural basis $\partial\over \partial
z^\mu$, one finds that the following replacement can be made:
\beqn
&& i\varphi^\mu \pi_\mu \rightarrow d \,,
\ \ i\varphi^{*\mu} \pi_\mu \rightarrow -\delta,\\
&& \varphi^{*\mu} G_\mu \rightarrow i_G\,, \ \
i{\cal Z} \rightarrow {\cal L}_G\equiv di_G +i_G d,
\label{comgenerator4}
\end{eqnarray}} \newcommand{\bde}{{\bf e}
where $i_G$ denotes the natural contraction of the vector field $G$
with a differential form, and $\delta$ is the Hermitian conjugate
of $d$. The BPS equation now becomes
\begin{eqnarray}
(d - i_G)\, \Omega = \mp\, i(\delta -G\wedge)\Omega
\label{comgenerator5}
\end{eqnarray}
Solving this first order system,
we should recover all 1/2 BPS and 1/4 BPS states of the underlying
Yang-Mills field theory. Work is currently in progress to solve this
equation in the simplest case of $SU(3)$ \cite{piljin}.
\section{Conclusion}
We have found the low energy effective Lagrangian of 1/2 BPS monopoles
in vacua of misaligned Higgs expectation values. This low energy
effective theory produces 1/4 BPS dyons as BPS configurations of the
nonrelativitic Hamiltonian. The kinetic term is given by the usual
moduli space metric of 1/2 BPS monopoles, while the potential term is
also determined by the same geometrical data. Its precise form is
given by one half of the norm of certain Killing vector field, which
allows a supersymmetric extension.
There are several directions to go from here. Our derivation relies
heavily on the established properties of 1/4 BPS dyons and monopole
moduli space. While there is little doubt that this is a valid
derivation, it may be worthwhile to rederive our exact result from a
different perspective. For instance, one may imagine deriving the
exact bosonic potential from a point particle point of view. Another
venue would be to find the supersymmetric low effective energy Lagrangian
directly from the field theory, along the line of Gauntlett and
Blum~\cite{blum}. Naturally, we expect to recover
the 1/4 BPS dyon spectra as quantum BPS states of this low energy
dynamics. The actual form of the wavefunction is currently under
investigation for the minimal case of $SU(3)$ \cite{piljin}.
Another interesting question concerns monopoles when the symmetry
breaking is not maximal~\cite{ejw3}. The gauge symmetry breaking is
determined by both $\hat{b}\cdot\phi(\infty)$ and $\hat{a}\cdot
\phi(\infty)$. If there is unbroken nonabelian gauge symmetry by
$\hat{b}\cdot\phi(\infty)$, some of magnetic monopoles become
massless. The moduli space acquires an enhanced isometry,
corresponding to unbroken gauge groups. By $\hat{a}\cdot\phi(\infty)$,
the unbroken gauge symmetry could remain unbroken or gets
broken~\cite{kml}. In the former case, the massless monopole clouds
screen the color magnetic charge of massive monopoles, and the
strength of static monopole-monopole force, say in the singlet
channel, may be different from naive expectation. Such a deviation
has been observed in the large $N$ context quite
recently~\cite{rey}. It would be quite interesting to quantize
massless monopole motion and find the resulting quantum effective
potential between massive monopoles. However, we should also point out
that it is still unclear whether the moduli space dynamics is a valid
approximation in the case of nonmaximal symmetry breaking.
\vspace{3mm}
\centerline{\bf Acknowledgments}
We thank Sangmin Lee for drawing our attention to works related to
Ref.~\cite{alvarez}. D.B. is supported in part by Ministry of
Education Grant 98-015-D00061. C.L. and K.L. are supported in part
by the SRC program of the SNU-CTP and the Basic Science and Research
Program under BRSI-98-2418. D.B. and K.L. are also supported in part
by KOSEF 1998 Interdisciplinary Research Grant 98-07-02-07-01-5, and
C.L. by KOSEF Grant 97-07-02-02-01-3.
\vspace{1cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,193 |
Petiolaris is a Latin name often found in taxonomy to refer to a petiole or leaf stalk.:
Species Latin names abbreviations
F. petiolaris
Ficus petiolaris, a species in the genus Ficus
Varieties
the sundew "petiolaris complex", a group in the genus Drosera, containing tropical Australian species which live in constantly warm but irregularly wet conditions
Senna artemisioides ssp. petiolaris, Randell, the woody cassia
Tilia × petiolaris, a variety in the genus Tilia
See also
Drosera dilatato-petiolaris, a plant species in the genus Drosera | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,623 |
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POLITICO EU Confidential: TV star to run Slovenia — Banned from Schengen — Summer time feedback overload
By Ryan Heath and Laurens Cerulus
August 18, 2018 10:00 am
By RYAN HEATH and LAURENS CERULUS
Send us your "Dear POLITICO" questions and dilemmas | Subscribe on Apple Podcasts | Subscribe on Soundcloud | Subscribe on Stitcher | View in your browser
Listen now to this week's EU Confidential podcast here on SoundCloud | Apple iTunes | Stitcher.
This week we get POLITICO reporters to explain how they found and developed their best stories of the summer: from Ginger Hervey and Jillian Deutsch explaining the EU's dilemma over 4,000 unregulated chemicals in tattoo ink, to how emigration has changed Carmen Paun's childhood village in Romania.
European Commission President Jean-Claude Juncker has requested that European flags and Italian flags today fly at half-mast outside Commission buildings across Europe, in memory of victims of the Genoa bridge collapse.
SLOVENIA GETS A NEW — COMEDIAN-LED — GOVERNMENT
Most countries with a caretaker government would punt the hard decisions until the end of August. Not Slovenia. The country's parliament on Friday approved a minority government led by former comedian Marjan Šarec, ending more than two months of political stalemate since an inconclusive election. Šarec's center-left LMS party struck a coalition deal with the Social Democrats, the Modern Centre Party, pensioners' party Desus, the centrist SAB and the Left party (the latter will not formally participate in the government).
Together they will have 43 seats in the 90-seat parliament (that number rises to 52 with the support of the Left party). It means Slovenia will have a former TV star as prime minister and a former model (Borut Pahor) as president.
The knock-on effect in Brussels is that the liberal ALDE group loses its spot as the joint-biggest group at European Council summits. The loss of Liberal Miro Cerar as Slovenian prime minister brings ALDE back down to seven heads of government, with the European People's Party's on eight (Šarec is so far unaligned at the EU level). At age 40, Šarec becomes the latest member of the youth squad at the European Council. The average age of European leaders is now 50.
Lyudmyla Kozlovska, president of the Open Dialog Foundation (and banned from the Schengen zone)
Ukrainian human rights activist Lyudmyla Kozlovska on Monday flew into Brussels from Kiev, only to find that she had been flagged in the Schengen Information System (SIS) as a person of danger to Polish national security. Kozlovska and her husband Bartosz Kramek have led several campaigns criticizing the Polish government for its record on freedom of expression and the rule of law.
The Polish government won't talk to POLITICO about the matter. Jakub Dudziak, spokesperson for the Polish Foreign Office, said only that the department chief makes decisions on whether to deny entry to Schengen.
Kozlovska told Laurens Cerulus her side of the story.
"I was asked by the border police officer of the Brussels airport what kind of crime I did in Poland. I said I didn't do anything. But they said: 'Poland is looking for you on a high alert. There is a ban to enter the Schengen countries.' I told the police officer that I recognize the case as one of political prosecution of my husband and Polish authorities want to punish him," she said.
Belgian authorities confirmed to EU Confidential that Kozlovska was stopped late Monday night and sent back to Kiev on Tuesday.
In the past year, Kozlovska and her husband have suffered at the hands of Polish authorities, she said. "They wanted to discredit our organization. They wanted to forcibly change the management board of our organization, the minister of foreign affairs personally requested to the court to basically fire me."
"When I moved to Poland it was like the land of opportunities. I was able to establish a foundation, even if I didn't know the language. I was able to come to the European Parliament, to speak to parliamentarians," she said, but now "I lost my Poland, which I treated as my second home."
Kozlovska sees her personal Schengen punishment as a symptom of a deeper trouble stirring in the EU. "When the Schengen system was created the EU was not expecting some member countries would use this instrument as a political tool to punish someone. There is also no possibility for another country to exclude me from it. I am like a hostage."
Would she consider seeking asylum in an EU country? "Asylum is when you're prosecuted in your native country. I'm Ukrainian, but it isn't Ukraine that's prosecuting me, it's Poland."
MEP'S FB RANT
Austrian EPP MEP Claudia Schmidt is in very hot water after posting on Facebook (next to an EPP logo) that "African and Muslim culture are not compatible with our culture." Schmidt went on: "Africans do not want to think and work like Europeans, but just want to live like Europeans" in addition to their "inclination to violence." Der Standard has more details of the fallout. Liberals and Socialists called on her to step down, and her own party colleagues demanded she revoke her statement. Angelika Mlinar, an Austrian MEP who is vice president of the liberal ALDE group, said: "The biggest Group in the EP cannot ignore and be silent about racism in its own ranks."
On Friday, Schmidt said on Facebook that she had deleted the posts, adding "it is and was not my intention to offend anyone. I apologize to anyone offended by my posting."
Speaking of Austrians upsetting people, the country's foreign minister, Karin Kneissl, triggered outrage by inviting Vladimir Putin to her wedding. The Russian president is due to attend the wedding on Saturday afternoon at a vineyard in the Styria region, before flying to Berlin for talks with Angela Merkel. Critics say the invitation undermines the EU's stance against Russia over Ukraine.
THE PEOPLE HAVE SPOKEN, A LOT
The European Commission received a record 4.6 million responses to a consultation on whether Europe should ax daylight savings rules and move to permanent summer time or permanent winter time. That's 108,000 responses a day during the six weeks of the consultation: half of them summer holiday weeks (though if you're the sort of person who'd vote for permanent winter time, you probably didn't take a summer holiday).
One of the supposed benefits of ending daylight saving rules would be eliminating the need to change clocks twice a year. That's a problem that no longer exists for people who use their smartphone and computer as clocks. The Commission can't yet tell us what the public have said. They'll need to spend a long time going through the feedback.
Do you know more about who helped organize the 4.6 million responses? If nothing else the European Parliament would probably love to hire them to organize a get-out-the-vote drive for the 2019 election. The previous record-holder in terms of feedback was the Directive on Birds and Habitat, a consultation that garnered a mere 550,000 responses.
CATCHING OUR EYE
Spot the mouse roaming in what is affectionately known as the European Parliament's "Mickey Mouse" bar in Brussels. H/t Marten Walker
1,080 — Number of U.S. newspaper sites blocked for European users: The list includes big journalism names such as the Los Angeles Times and New York Daily News, and it's a product of the newspapers either not wanting to comply with the EU's new General Data Protection Regulation (GDPR) or being subject to over-zealous lawyers wanting to protect them from GDPR-related lawsuits. See the full list here.
TOP POLITICO READS
Europe's freak weather, explained (Stefan Rahmstorf)
Newspapers just played right into Trump's hands (Jack Shafer)
Poland is Europe's future —but which one? (Matthew Kaminski)
TOP WEEKEND READS FROM ELSEWHERE
The British bluffocracy (Spectator)
The biggest life regrets of people in their 90s (Today)
How millenials killed mayonnaise (Philly.com)
Said something you'd like to forget? CNN's Andrew Kaczynski won't let it go (Washington Post)
Omarosa laid a series of traps for the White House, and they're starting to go off (Slate)
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,363 |
Recording Guitar, July 1997
TIPS FROM FIVE PROS
Mix Staff ⋅ May 17, 2004
It is almost impossible to overstate how important guitar has beenin the world of popular music, whether it is rock, R&B, folk,country, blues, whatever. Sure, drummers, bassists and keyboardists canall lay rightful claim to their essential roles, but somehow guitaristsget the lion's share of the so-called hero-worship, whetherit's quiet acoustic renderings, funky chicken pickin', orpure jack hammer, amp-blowing death rock.
The brilliance may start with the player, but behind the scenes, inthe studio, is usually an attentive, resourceful engineer or producer,trying to catch sparks on tape.
Mix spoke to a handful of notable producers, engineers andplayers who have excelled at capturing great guitar sounds. Thanks toAdrian Belew, John Jennings, Skidd Mills, Michael Wagener and David Zfor their fine input.
MICHAEL WAGENER
Since the late '70s, Michael Wagener has earned a reputation asone of the masters of great-sounding rock recordings. Wagener, who hasamassed more than 100 album credits, may be well-known for his workproducing, engineering and/or mixing aggressive hard rock projects likeSkid Row, Extreme, Ozzy Osbourne, Metallica, Megadeth and Alice Cooper,but he also has credits that range from Janet Jackson and Queen to ThePlasmatics. Wagener is currently producing a video called How to ReallyRecord Guitar.
"There is an important relationship between amp output andspeaker wattage. I subscribe to the theory that you have to push air toget your point across. That means I will always try to use an amp withmore power reserves than the RMS wattage of the speaker cab. Of courseyou have to be careful not to blow the cab to pieces. A tube amp ofabout 100 watts can have peaks around 250 watts, so make sure yourcabinet can stand that occasional peak. Also, if you use a tube amp,that peak is liable to come smoother than or not as sudden as you wouldget from a transistor power amp. A tube power amp will probably giveyou a fatter, saturated sound, whereas a transistor amp will be cleanerwith a bit of a harder attack.
"Another very important part of the power amp is the outputtransformer. The output transformer can make or break the sound of anamplifier. Once, I had to exchange a blown output transformer of agreat-sounding Marshall 100-watt top. I never got the original soundback.
"The distortion doesn't always have to be generated inthe preamp. Sometimes it's better to keep the preamp sectionfairly clean and get the distortion out of the power amp or thespeaker. Speaker distortion is the smoothest distortion you can get.Unfortunately, because of the high volume, it also involves having avery good isolated studio, so the neighbors won't get distortedas well.
"When you pick a speaker cabinet, there are a fewconsiderations to be made. What kind of sound do you want to achieve?Are you looking for a clean sound or a distorted sound? Is theinstrument going to be in the front or the back of the mix? Is it goingto be doubled? Are you playing single notes or chords or both? Howpowerful is your amp? Can your speaker cabinet withstand the poweroutput from the amp? Is your speaker cabinet too 'big' forthe amp, so it won't push enough air? For example, a 4×30-wattcab would be a great, powerful cab for a 100-watt amp if you arelooking for a fat, distorted sound. If you are going for a cleanersound, you might want to try a 4×75-watt cabinet on the same amp. Makesure that the impedance of the cabinet and the amp match.
"Make sure not to download the guitar output by hooking up abunch of amps without a splitter. If you combine amps, it is importantto look at the amp input as a resistor or load on your guitar. When youput two resistors in parallel, their value halves—think about two8-ohm speakers switched parallel, resulting in 4-ohms. The smaller theresistor value the more current [or power] gets drawn by it. Yourguitar only has a very tiny amount of power available on its output, soif you simply Y-cord the guitar into two amps, you are liable to losesome of the pickup power of the guitar to the load of the two parallelamp inputs. The most noticeable side effect is probably a loss of highend or overall crunch.
"The input impedance of a normal tube amp is around 1 millionohms, and the output impedance of a guitar is normally around 250,000ohms. That is a pretty healthy relationship. If you combine two ampinputs, the input impedance goes down to about 500,000 ohms, which is amuch higher load on your guitar output.
"Sometimes, for creating sound options, it might be good toset up a few different amps and cabinets in different rooms—hardand soft, open and dampened. It also works well to have a certain ampjust produce the upper frequencies and another one just for the lowend. That way you can decide on the mix between the two from inside thecontrol room. If you have enough tracks available, record them bothseparately and mix them later when you have a better idea about thewhole sound of the song. If you record the [almost] same signal twice,you have to be careful not to get phase distortion."
SKIDD MILLS
Producer-engineer Skidd Mills has worked on projects ranging from ZZTop's hard guitar skronk to Robert Cray's lyrical bluesstylings. Mills, who primarily works out of the legendary Memphisstudio Ardent Recording, has also worked with pop-rock wonders BigStar, Killjoys and Joe, Marc's Brother, as well as Spin Doctorsand hard rockers Skillet and Audio Adrenilin.
"First off, to me the most important element is the player.That is where most of the tone comes from. As far as amps go, I reallylike Matchless amps. I think they are really cool. I have recorded thema few times, and they've turned out really cool. Some of myfavorites are also old Marshalls, Hiwatts and old Fenders.
"I almost always mike amps the same. I usually use two [Shure]57s on a cabinet, a little off-center from the cone, right up againstthe grille. Sometimes, I will use a [Sennheiser] 421 or a [AKG] 451with a 57.
"I usually don't like to EQ my mics, especiallyseparately, because when you're EQ'ing separate guitarmics, you can get weird phase problems happening. If I'm going todo compression, EQ or anything like that, it's almost alwaysafter the fact.
"I rely more on the actual sound. I will stand out by the ampbefore I start to EQ anything on the board. I'll go out and standby the amp and just make sure that it sounds good. If I do any EQadjustments, I start first on the guitar amp itself. I won't addboard EQ while I'm going to tape, because I really just want toget the sound of the amp. Sometimes I'll compress the guitar totape, if I'm looking for a real heavy sound. One of my favoritesis the Valley People 440. It has a lot of versatility to it.
"For the most part, I don't like to slam guitars. WhenI'm standing in front of my monitors, I like to have the feellike I'm standing in front of the speaker cabinet. In otherwords, I'm pushing a lot of air.
"You have to be careful with compression because you cansqueeze the life out of a guitar sound until it sounds paper-thin. Atthe same time, you don't want to have the guitarist juststrumming along and have one section come bursting out at you. WhenI'm mixing, I would say that my all-time favorite guitarcompressor is the SSL compressor that is sitting in the board.
"Initially, I work with the sound of the player and amp. I getall of that together before I start thinking about what mics and whatcompression I want to use. I listen to the playing and see if theguitar and amp are most complementary to that player's style.Experimenting with different amps, guitars and even picks can make abig difference. I usually like to have a lot of toys lying around, likea box full of distortion boxes and old vintage stuff. I like doingthese things to achieve the best complementary tone for theplayer's style and the type of music, instead of having theguitarist merely plug in and mike it up and sit at the boardEQ'ing all day till I'm blue in the face."
David Z is one of those producer/engineers who has had the good fortuneof being able to successfully defy pigeonholing. Z's creditsinclude dance music divas Jody Watley and Nenah Cherry, as well as workwith Prince. He's also done blues rock up-and-comers Kenny WayneShepard and Kid Johnny Lang. Other credits include Fine Young Cannibalsand the Freddy Jones Band. Most recently, Z has been working with thealternative insurgent country-rock scene, an arena in which he is verycomfortable. After all, he was a friend of and co-songwriter with thatmovement's late icon, Gram Parsons.
"The role of a funk guitar is almost like that of anotherpercussion instrument. It's playing a polyrhythm. Basically, infunk music, everything is a little more percussive. Everything is morea function of the beat than in many other styles of music.
"A lot of times, funk guitars are very clean, bright and oftenintensely compressed, because the way funk is played is like aslapping, hard picking technique to make it bite. It's usually aFender guitar, because Fenders have a good short tone, meaning theyhave a quick attack and quick release on a note, as opposed to asmooth, long tone, like an acoustic or a Gibson or something.
"With Prince, we used a Hohner, which sounded like a Tele, butit was 20,000 times brighter. It would come off with that'skanky' sound. There is also that Gibson or 335 sound fordarker, funky chord sounds. Those are usually recorded pretty straight,with maybe a little chorus. They aren't real elaborate.
"For compressors, I love the LA-2As or ones that grab you alittle bit more, such as a dbx OverEasy or an Inovonics. Those grabhard. Sometimes that is what you want. Usually, I will have it set witha slow attack, to get the head of the note, and then slam it. Then Ihave a fast release. I usually have it set at a 4-to-1 ratio, but itdepends. It's totally by ear. That's just a usual setting Imight use.
"Guitar amplifiers add some power, but they aren't a bigpart of the actual tone of funk guitars. You want to get that speakertone, but the attack is a pretty clean tone. We are not looking fordistortion. Recording blues guitar, on the other hand, is more afunction of the guitar and the amplifier together because of thedistortion factor. A lot of times, I will use a ribbon mic, like aColes ribbon mic. A lot of times with blues guitar and also acoustic, Iwould take what I would call 'multiple sources.' Forexample, on the Big Head Todd & the Monsters record that I did, wehad a lot of multiple sources. We ran through a Leslie and we ranthrough a little Marshall. We miked the strings and then out of hisregular amp all at the same time. We then had four different soundsgoing for the same part. Depending on what you pick and choose, you canget some pretty cool textures doing that.
"Sometimes I will put what I call a 'kamikaze'microphone focused on the bridge of the electric guitar. SometimesI'll put that mic on a stand, or hang it from a stand, placed asclose as you can get it. I mainly use an ECM-50 or ECM-150 lavaliermic, or the kind of Sony that newscasters wear on their ties. I mightuse a 452, or [Neumann] KM84, a bright condenser mic, just to pick upthe zing of the pick hitting the strings. You've got to roll offthe bottom end. You're just trying to get some sort of high-endthing. Obviously, you have to put the amp in another room from theplayer, or you won't get anything worth using.
"If you mix a little bit of that in with what he'splaying, that adds a third dimension to it. You bring the sound into aneven bigger arena, and you can spread it out. I like to do that,because in that way, you can actually make the guitar itself becomemuch bigger-sounding. I may not use some of those elements, but I willusually try to take multiple sources.
"For Leo Kottke, we did a lot of multiple sourcing. We used acouple of mics, and we took a direct out of his pickup. We also usedthis guitar synthesizer that he had, a Roland VG-8. It added stringsounds or other textures that played way underneath what he wasplaying. It gave the music a real eerie quality.
"On Leo, I used the DI to get a little support and clarity. Ihad one signal running through a little Fender Champ in another room. Imiked him with two mics, a 452 up on the neck and a 49 over the hole.Both were placed two or three inches away.
"Acoustic guitars have some sort of a buildup in the lower-endareas, and it can really overwhelm you. I think the buildup is oftenaround 150 Hz. You have to be kind of careful with compression and micplacement. A little roll-off and distancing of the mic helps. I tendnot to compress very much.
"Actually, the big acoustic guitars can be deceiving becausethey can be great-sounding live, but then the microphone picks up allof this boom and it gets all screwed up. As a result, smaller guitarsare sometimes the best. The player obviously can make a bigdifference."
Since the '70s, Adrian Belew has earned the distinction of beingone of the guitar world's most inventive practitioners. He hasappeared on albums by Laurie Anderson, Joan Armatrading, David Bowie,Herbie Hancock, Mike Oldfield, Robert Palmer, Paul Simon, Talking Headsand Frank Zappa. Belew has been a member of King Crimson, one ofrock's most adventurous ensembles, since the early '80s.During the last half of the '80s, Belew has recorded twisted poprock with his group, The Bears. Belew has also recorded 11 solo albumprojects that have ranged from Beatles-influenced melodic pop-rock tothe ambitious 1996 release Experimental Guitar Series #1—TheGuitar as Orchestra, in which he created an orchestra throughsounds designed and executed through guitar and velocity-sensitiveguitar synths.
"Before recording, I try to program most of my sounds into themulti-effects units the way that I want them heard, so there is littleneed for extra things to be done from the console, in terms of dynamicsignal processing or EQ. Of course, there's always a certainamount of EQ'ing that you will do.
"There are always happy accidents or things that occur that Ididn't plan on happening while recording. I always welcome thosethings, but most of the time it's important that I scientificallydevelop the sounds that I really want to use in a song in a way thatallows me to reproduce them again live. I really concentrate more on myguitar setup and its abilities to generate those.
"I like to build a single guitar sound out of severaldifferent guitar sounds. I may overdub three different guitars that areplaying exactly the same thing, but have different variations ofsounds. It's important to me to create clean arrangements. Interms of sound, fewer parts are better.
"I have several choices of amplifiers that I use in severaldifferent rooms of my home studio. I use a DC-30 Matchless amp, whichhas an incredibly good tube sound. I keep it in my studio's maplefloor room. I also have some other amps, like a Fender Twin, a coupleof Jazz amps and a Roland Jazz Chorus 120.
"I mainly like to play through 12-inch speakers. I'llput up a couple of AKG 414s on them and maybe have a room mic, like aC-24, so there is a combination of close mic and room sounds to choosefrom. It just depends on what kind of sounds I'm going for.Sometimes, I'll just plug into the board and play straight intothe console. Most of the time, I like to go through speakers.
"If I'm recording guitar synthesizer stuff, Idon't find that those sounds come out any better coming through aspeaker and a microphone, so I generally just take the signal direct. Imight go through a Tube-Tech to try to warm things up a little bit, ifpossible, and try to get the cleanest signal going right in to theboard. If I'm going to go direct with a guitar, I particularlylike the Eventide Harmonizers, because they have so many sounds.
"I have four different synthesizers. In my rack, I have twothat I use for all of my live sounds—they're the RolandGR-1 and the Roland GR-50. I also have the older GR-700, which has alot of really nice analog-based sounds. I probably have designed about200 sounds with that unit. It's a little hard to give it up, so Ileave it in the studio. I also leave a newer model in the studio calledthe VG-8. It's not actually a guitar synthesizer, but it'syet another thing that I find works better for me in the studio than ina live application.
"The VG-8 has some really nice properties. In particular, itallows you to use altered tunings. You can write in altered tunings andthe guitar sound is very realistic. There are many available guitarsounds, and you can play harmonics and get string noises, and you canreally think that you are playing through a pickup, but you areactually not. In fact, you could use a guitar that has no pickups onit, as long as you're using the MIDI controller, and you wouldnever know that you're not playing through pickups. Again,it's an excellent way to utilize a lot of different tunings, andthat is one of the things that I mainly use it for."
As a producer, guitarist and songwriter, John Jennings has worked withsuch top artists as Mary Chapin Carpenter—with whom he has madeseveral albums that have yielded 11 Top 10 singles, 13 Grammynominations and five Grammy Awards—Indigo Girls, Iris DeMent,Janis Ian, Lyle Lovett and Bill Morrissey. Jennings' most recentsolo album is Buddy, recorded at Bias Recording in Springfield, Va.,where he has cut all of Chapin Carpenter's records.
"For better or worse, I do have several 'default'locations for placing mics. I like to think of them as good startingpoints, rather than rules. They work for me and may not work foryou.
"Go out on the floor and listen to what you're going torecord. Don't just throw up a couple of mics and do yourinspection from the control room. Mics and monitors can lie to you. Ifyou're recording an acoustic guitar, listen with your faceparallel to the face of the instrument. You'll want to be a fewfeet back from the guitar, and you'll want to move around a bit,mostly from side to side. You'll find the 'sweet'spot, where all the elements of the sound are apparent and fairlywell-balanced. Regardless of whether you're recording in stereoor mono, this is the 'zone' you want to try to capture.
"Once you have found a sound that you like, walk around theroom a bit. Listen from behind the guitarist, from the side, and allover. There might also be another place you can add a mic that willhelp the sound overall. Sometimes you have to try fairly unconventionalthings to compensate for an instrument that is lacking in a particulararea, or to find a sound that fits a particular track. There are folkswho will try to convince you, before you even try, that trying someunusual mic placement may not work. Having been guilty of this a timeor two myself, I have reformed. I now say, 'Whatever!' Itonly takes a few minutes to find out.
"I personally prefer to record acoustic guitars in stereo, asI like wide images. I like to use matching pairs of mics and have aparticular fondness for KM84s. Point one toward the middle of the lowerbout and the other at the 15th (or so) fret. Put them a foot or so fromthe guitar, with the capsule roughly parallel to the face, and adjustthe distance to taste. You get that nice bottom end from the bridge andthe articulation from the neck.
"If you're recording direct, try to have a few optionsfor DIs. It's always best to be able to tailor a sound to aparticular track. As for recording electric guitars, I'm alwayssearching for better ways to do it. I've become a proponent ofthe 'multiple mic' method. I really like to try severaldifferent mics on different speakers and move them around a good bit.Do yourself a favor: Buy a Sennheiser MD409 and use it in conjunctionwith an SM57. I place the 409 about a foot from one of the speakers andpoint it toward the outer edge of the cone. I find the 57 useful inadding definition to the sound if the 409 seems a bit too'soft.' Nevertheless, there are many mics; try as many asyou can. There are really useful microphones that are not veryexpensive, like Radio Shack PZMs.
"When recording electric guitars, listen to the amp close upand at different points in the room. If the amp has multiple speakers,each may have its own character, no matter how subtle. Ask theguitarist where the spot is in the room that sounds good to him or her.If the guitarist is standing and has dialed in a tone that works fromhead height, try to make a provision for that. In other words, put upanother mic!"
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CRAS Awards Two Local Arizona High School Student Contest Winners with Live Sound & Recording Sessions | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,075 |
\section{Introduction}
Einstein's general theory of relativity predicts the existence of
gravitational waves (GWs), oscillations in the space-time metric that
propagate at the speed of light. The Laser Interferometer
Gravitational-Wave Observatory, LIGO, is designed to detect and study
astrophysical GWs, with the promise of studying qualitatively new
physics and astrophysics.~\citep{Abramovici:1992p2546} In particular,
the direct detection of GWs will provide information about systems in
which strong-field gravitation dominates, a virtually untested regime
in which space-time curvature self-interacts. Such GW sources include
compact binary coalescences in which a neutron star or black hole
binary system inspirals together, coalescing to form a black hole; the
stellar core collapse thought to power Type II supernovae; rapidly
rotating asymmetric neutron stars; and possibly cosmic-scale processes that
produce a stochastic background of GWs.~\citep{Cutler:2002p7832}
In the past few years, the first generation of long-baseline
gravitational wave detectors has successfully operated at or near
design sensitivity. In collaboration with the Virgo 3~km and the GEO
600~m interferometers,~\citep{Acernese:2008p9535, Willke:2002p3770}
LIGO anchors a worldwide network of instruments in search of the first
direct detection of gravitational waves. The LIGO detectors operated
from November 2005 to October 2007, with joint data taking with Virgo
starting in May 2007. The data is currently being analyzed for GW
signals from inspiraling binary systems, burst sources, a stochastic
GW background, and rapidly rotating neutron stars. The status of
these searches and their astrophysical importance is discussed by
other authors in these proceedings.
This article focuses on the next generation of LIGO interferometers,
in particular Advanced LIGO, currently being designed and assembled at
two sites in the United States. The Livingston Parish, Louisiana
observatory will operate a single interferometer, L1, with 4~km long
arms while the Hanford, Washington observatory will operate two 4~km
interferometers within a common vacuum envelope, H1 and H2. The
second generation Advanced LIGO detectors will improve the sensitivity
of ground-based gravitational wave detectors by an order of magnitude
over current detectors. A preliminary Advanced LIGO design was
described in Ref.~\citep{Fritschel:2003p9356}, here we provide an
overview of the final design as construction begins. We first
describe the Advanced LIGO optical configuration, then follow with a
description of the dominant noise terms and anticipated sensitivities.
Finally, we conclude with comments on the initial tests of Advanced
LIGO and progress towards the first science runs.
\section{Advanced LIGO Optical Design}
\label{sec:advanced-ligo-design}
From a detector perspective, gravitational waves can be thought of as
a quadrupole strain of space, $h = \delta L / L$, which can be probed
by monitoring the relative positions of inertial test masses with
light. Equivalently, in a fixed Lorentzian frame, gravitational waves
create a tidal force, a force proportional to the distance from a
chosen origin. As with electromagnetic waves, GWs are transverse
waves that travel at the speed of light. Unlike electromagnetism, GWs
are constrained by mass and momentum conservation to be quadrupolar:
the strain (or tidal forces) contracts along one transverse dimension
while expanding the orthogonal dimension. Also unlike
electromagnetism, GWs are very weak and interact very weakly as they
propagate through space; detectable GWs are generated only by the
coherent acceleration of stellar masses at relativistic velocities. The
strongest nearby sources will produce strains on Earth no larger than
$h \approx 10^{-21}$. Finally, it's worth noting that GW detectors
measure the amplitude of a GW (as opposed to the power) so that the
observed volume of space scales cubicly with the detector sensitivity.
The Advanced LIGO optical design consists of a Michelson
interferometer with Fabry-Perot arm cavities, a power recycling cavity
and a signal recycling cavity as shown in Fig.~\ref{fig:layout}. The
Michelson topology is well matched to the quadrupole strain: a
properly oriented, linearly polarized GW propagating normal to the
interferometer plane generates a positive strain along one arm, a
negative strain along the other and vice versa, oscillating in time.
In the Advanced LIGO configuration, the arm lengths are controlled so
that Michelson interferometer reflects the input laser beam back
towards the laser while the anti-symmetric port is dark. The
differential motions of the two arms -- the GW signal --
constructively interfere at the beam splitter's Anti-Symmetric port,
labeled ``AS'' in Fig.~\ref{fig:layout}. The common mode signals
generated by common mode motion of the end mirrors, by laser frequency
noise, and by laser intensity noise constructively interfere at the
beam splitter's Symmetric port; to first order, this configuration
eliminates laser technical noise couplings to the GW signals.
Fabry-Perot cavities in the interferometer arms defined by the
partially reflecting Input Test Mass (ITM) and high reflectance End
Test Mass (ETM) resonate the input laser light to increase the power
in the arms. Similarly, the partially reflecting Power Recycling
Mirror (PRM) resonates the light that returns toward the laser from
the beam splitter's symmetric port. Together the arm and power
recycling cavities build up the laser power in the arms by a factor of
$\simeq 6000$. The power recycled Fabry-Perot Michelson topology is
identical to Initial LIGO and has been described in detail in
Ref.~\citep{Abbott:2009p9436}.
\begin{figure}[tb]
\centering
\includegraphics[height=4in]{IFO_diagram_modified.pdf}
\caption{The Advanced LIGO optical layout. The triangular suspended
input mode cleaner filters frequency and amplitude noise from the
laser (not shown) and provides a stable input beam. The Faraday
Isolator (FI) isolates the laser from the interferometer reflected
beam (REFL) used to control the laser frequency. The 4~km long
Fabry-Perot arm cavities are formed between the ITM and ETM test
masses. The Power Recycling Mirror (PRM) and Signal Recycling
Mirrors (SRM) form folded cavities discussed in the text. The GW
signals are carried by the light transmitted through the Output
Mode Cleaner at the Anti-Symmetric (AS) port.}
\label{fig:layout}
\end{figure}
Advanced LIGO has several significant changes in the optics relative
to Initial LIGO: the Signal Recycling Mirror (SRM) is added to the AS
port of the interferometer to form a signal recycling cavity, the
power and signal recycling cavities have a stable geometry, the GW
signal is detected using DC readout, and the laser power is increased.
The SRM forms a resonant cavity for the differential mode signal,
altering the interferometer
dynamics.~\citep{Meers:1988p8021,Mizuno:1993p10166} The impact on the
interferometer quantum noise is discussed further in the
\S\ref{sec:quantum} below.
In Initial LIGO, the $\approx 10$~m long power recycling cavity was
formed by mirrors having $\ge 10$~km radii of curvature, effectively a
flat-flat resonator geometry. That configuration was extremely
sensitive to changes in the curvatures caused by unavoidable thermal
lensing. To reduce the thermal sensitivity, the Advanced LIGO signal
and power recycling cavities are each formed by a folded chain of
three curved mirrors. In effect, the Advanced LIGO recycling cavities
incorporate beam expanding telescopes that reduce the sensitivity to
thermal lenses in the ITMs. In a similar change, the Fabry-Perot arm
cavities have a near concentric configuration motivated by the reduced
coating thermal noise from large spot sizes, discussed in
\S\ref{sec:therm}, and from the improved response to optical torques
for near concentric cavities, discussed in
Ref.~\citep{Sidles:2006p4148}.
The Advanced LIGO differential arm length will be detected in a
homodyne scheme known as DC readout. In DC readout, the GW signals are
measured directly as amplitude modulations on a static field at the AS
port. The static field is created by a small offset in the
differential arm length; GWs make oscillations around the offset,
modulating the output. However, many fields are present at the AS
port that don't carry GW information such as auxiliary control fields
and scattered non-resonant light. Advanced LIGO incorporates an
Output Mode Cleaner (OMC) at the AS port to select only those fields
containing GW signal. The OMC is a $\sim1~$m long optical cavity
which filters the interferometer output before detection, transmitting
only light from the arm cavity.
Finally, the Initial LIGO laser will be upgraded from a 10~W Master
Oscillator/Power Amplifier (MOPA) to a 180~W MOPA for Advanced
LIGO.~\citep{Willke:2008p10173} The input optics are upgraded to match
the laser: high power versions of electro-optic phase modulators,
photodetectors, and Faraday isolators replace conventional
components. With these changes, the maximum Advanced LIGO arm power
approaches 800~kW, improving the shot noise limited sensitivity by a
factor of $\approx 6$ with respect to Initial LIGO.
\section{Advanced LIGO Noise Contributions}
\label{sec:noiselimits}
The Advanced LIGO sensitivity limits are estimated from calculations of
technical and fundamental noises; many of these have been studied with
Initial LIGO and other dedicated experiments.\footnote{See
Ref.~\citep{Abbott:2009p9436} and references therein.} Below 10~Hz,
the sensitivity is limited by the seismic motion of the earth, at
intermediate frequencies thermal noise dominates, and at the highest
frequencies photon shot noise limits the sensitivity. The
interferometer noise contributions, modeled with the GWINC-v2 software
package and plotted in Fig.~\ref{fig:noise_budget}, are described in
the following sections.
\begin{figure}[tb]
\centering
\includegraphics[height=4in]{NoiseTermsPlot-crop.pdf}
\caption{The modeled noise budget for an Advanced LIGO
interferometer with $\phi_{SRM}= 0$ and 125~W input
power. The total noise (grey) is the incoherent sum of each of
the listed noise terms, described in detail below.}
\label{fig:noise_budget}
\end{figure}
\subsection{Acoustic and Seismic Isolation}
\label{sec:isolation}
The Advanced LIGO detector requires a residual RMS differential arm
motion of $\delta x \leq 10^{-15}$~m to maintain the arm power buildup
and to minimize the coupling of laser noise into the GW signal. In
the GW detection band, the required test mass displacement is $\delta
x \leq 10^{-19}~m/\sqrt{Hz}$ at 10 Hz and $\delta x \leq 2 \times
10^{-20}~m/\sqrt{Hz}$ at 100 Hz. Here, $\delta x$ refers only to the
differential arm motion; the common arm motion and the motion of the
other length degrees of freedom may be somewhat larger. To meet these
requirements, the interferometer is be isolated from environmental
influences such as acoustic noise, gas produced phase noise, and
seismic noise. Thus, the interferometer optics are enclosed in an
ultra-high vacuum system. The facility specifications have been
determined by the standard quantum sensitivity limit for a future
interferometer with 1 ton test masses. The $\simeq 10^{-9}$~torr
vacuum reduces the gas produced phase noise to an equivalent strain of
$h \approx 10^{-25}$, well below the anticipated Advanced LIGO
sensitivity. In addition, the interferometer detection beam paths,
including the photodetectors, are enclosed within the vacuum on the
seismic isolation platforms to eliminate acoustic coupling and reduce
the motion of light scatterers.
Isolating the interferometer optics from ground motion is a task
divided into several stages, with each stage providing isolation for
the following stage. The effect of each stage of isolation can be
seen in Fig.~\ref{fig:seismic}. At the lowest frequencies, an active,
6 degree of freedom, hydraulic, external pre-isolator (HEPI) reduces
the motion between 0.1 and 5~Hz by a factor of $\simeq
10$.~\citep{Fritschel:2004p9420} A two stage, active, internal seismic
isolation (ISI) system enclosed within the vacuum reduces ground
motion by a further factor of $\simeq 300$ at 1~Hz and $\simeq 3000$
at 10~Hz.~\citep{Abbott:2002p4431} Both active isolation systems consist
of a spring mounted, actuated platform outfitted with a suite of
motion sensors. The sensors measure the platform motion and feedback
to the actuators, thereby stabilizing the platform to a level limited by the
sensor noise floors and mechanical cross coupling.
\begin{figure}[tb]
\centering
\includegraphics[height=3in]{SeismicSpectra-crop.pdf}
\includegraphics[height=3in]{SuspensionSketch.pdf}
\caption{The Advanced LIGO seismic isolation consists of several
stacked stages of active isolation shown schematically on the
right, with colors matching the curve and the test mass shaded
gray. The reference ground motion (blue) is the average ground
displacement at the Livingston observatory. The HEPI curve (green)
is the motion atop the hydraulic pre-isolator located outside the
vacuum system. The anticipated ISI payload motion (red) is
calculated from models of the two stage active isolator
performance with the HEPI motion as the input spectrum. The
motion of the test mass at the end of the four-stage suspension is
calculated from a model and shown in the Quad curve (black).}
\label{fig:seismic}
\end{figure}
The interferometer optics are suspended from the ISI platforms using a
coupled pendulum system based on the GEO-600 three-stage
suspensions.~\citep{Plissi:2000p9494} The beam splitter, power
recycling cavity and signal recycling cavity optics are hung from
three-stage suspensions consisting of two metal masses linked with
steel wire, followed by the optic itself. Vertical isolation is
provided by cantilevered blade springs mounted to the metal masses.
Above the pendulum resonances, the triple suspension provides
isolation proportional to $1/f^6$. The ITMs and ETMs are suspended
from four-stages consisting of two metal masses with vertical springs,
a fused silica intermediate mass, and the test mass. Welded fused
silica fibers join the test mass to the fused silica mass
above.~\citep{Robertson:2002p9490} The additional isolation stage is
necessary to meet the displacement noise goals at 10~Hz, and the fused
silica fibers reduce the thermal noise as discussed below.
Altogether, the Advanced LIGO isolation systems reduce the
seismic-induced test mass motion by 10 orders of magnitude to $\delta
x \simeq 10^{-20}~m/\sqrt{Hz}$ at 10 Hz, opening the frequency band
from 10 to 40~Hz for gravitational wave searches.
\subsection{Thermal Noise Sources}
\label{sec:therm}
Between 10 and 200 Hz, thermal noise sources limit the interferometer
sensitivity. An unavoidable consequence of energy dissipation, thermal
noise is modeled by applying the fluctuation-dissipation theorem to
all aspects of the system which influence the motion or measurement of
the arm cavity test masses. The fluctuation dissipation theorem is
closely related to Brownian motion, hence thermal noise associated
with mechanical motion is often called Brownian noise. Three sources
of dissipation dominate the Advanced LIGO thermal noise: mechanical
loss in each test mass leads to fluctuations in the mirrored surface
of the test mass; dissipation within the suspension fibers generates a
fluctuating force on the test masses; and losses within the mirror's
dielectric coating generate a fluctuating phase shift of the reflected
light.
The test mass mechanical loss determines the level of substrate
Brownian noise that causes fluctuations of the surface with respect to
the center of mass. Early Advanced LIGO designs considered sapphire
test masses for their superior mechanical and thermal
properties. ~\citep{Rowan:2000p10031} Since then, fused silica test
masses have demonstrated sufficiently low loss and have been adopted
as the substrate material. The Advanced LIGO test masses are 40~kg,
high purity, low-inclusion fused silica cylinders 34~cm in diameter.
To maintain the substrates' excellent mechanical properties, no lossy
materials contact the test mass (eg. magnets). The masses are
suspended via fused silica mounting blocks hydroxy-catalysis bonded to
each side.~\citep{Rowan:1998p10037} Fused silica fibers are welded to
the blocks and connect to the upper fused silica mass in a similar
fashion. The test masses are actuated using a non-contact, low-force
and low-noise electrostatic drive. As a result of these measures, the
test masses have very low loss (mechanical quality factors
exceeding $10^7$), and correspondingly low fluctuations, contributing
to the strain noise at $h \approx 3 \times 10^{-24} f^{-1/2}\;
Hz^{-1/2}$.
Loss in the mechanical structure supporting the test masses generates
fluctuating forces at the masses' suspension
points.~\citep{Gonzalez:2000p3745} The extremely low mechanical loss
needed to limit the fluctuations motivates the fused silica fiber
stage of the four-stage suspension. Since the loss is dominated by
the bending regions near the fiber ends, the cylindrical fibers are
laser polished and drawn from fused silica rod with a carefully
controlled, variable diameter. At the ends where the fiber is welded
to the test mass, the fibers have a large diameter to reduce flexing
of the (potentially higher loss) welded joints. The fibers then taper
with an optimized profile so that the bending occurs predominantly in
a low loss region of the fiber. Suspension thermal noise contributes
to the detector noise below 30~Hz, limiting the low-frequency
sensitivity to $h \approx 2\times 10^{-21}f^{-2}\; Hz^{-1/2}$.
Finally, thermal noise from the dielectric mirror coatings limits the
detector noise between 40 and 140~Hz, the most sensitive region. The
coatings are alternating layers of SiO$_2$ and titanium-doped
TaO$_2$.\citep{Harry:2007p9489} Both thermo-optic and mechanical loss
contribute to the thermal noise. The thermo-optic noises include
thermo-refractive fluctuations in the layers' index of refraction as
well as the thermo-elastic fluctuations that modify the layer
thickness and hence the magnitude and phase of the reflected field.
The coating mechanical loss dominates the thermal noise by an order of
magnitude, primarily in the thick ETM high reflector. Because the
thin film coatings have a high mechanical loss relative to the
substrate, the coating Brownian noise exceeds that of the substrate by
nearly an order of magnitude. The coating noise is inversely
proportional to the beam diameter, motivating large spot sizes on the
optics as described in \S\ref{sec:advanced-ligo-design}. Coating
thermal noise limits the detector sensitivity to $h\approx 2.5 \times
10^{-23} f^{-1/2}\; Hz^{-1/2}$.
\subsection{Quantum Noise}
\label{sec:quantum}
Quantum mechanics limits the precision at which the test mass
positions can be determined. At high frequencies, photon shot noise
limits the sensitivity to $h \propto \sqrt{ f / P}$, while at low
frequencies radiation pressure limits the sensitivity to $h \propto
\sqrt{P}/f^2$. The Advanced LIGO interferometer is a realization of a
Heisenberg microscope: the high laser power required to determine the
position of the test masses exerts a fluctuating radiation pressure
which perturbs the test mass positions. In the absence of
position-momentum correlations, the Advanced LIGO strain sensitivity
is limited by the Standard Quantum Limit (SQL) $h_{SQL} = 1.8 \times
10^{-22}/f\; Hz^{-1/2}$. Because the signal recycling cavity couples
the test mass position and momentum, sub-SQL sensitivity is possible
over a frequency bandwidth $\Delta f \sim f$ at the expense of the
sensitivity at other wavelengths.~\citep{Buonanno:2001p2} The primary
difference between the three curves shown in
Fig.~\ref{fig:noise_curves} is the degree of correlation between the
test mass position and momentum as determined by the SRM reflectivity
and the signal recycling cavity length tuning.\footnote{These noise
curves can be found at
https://dcc.ligo.org/cgi-bin/private/DocDB/ShowDocument?docid=2974
and the documents therein.} The noise curves include the thermal
and seismic noises calculated by GWINC-v2.
\begin{figure}[tb]
\centering
\includegraphics[height=3.5in]{AnticipatedSensitivity-crop.pdf}
\caption{Amplitude spectral densities of the anticipated
sensitivities of the Advanced LIGO interferometers as a function
of the tuning of the signal recycling phase, $\phi_{SRM}$. \textbf{No
SRM} is a potential initial interferometer configuration with no
signal recycling mirror with modest sensitivity; the
\textbf{Broadband} configuration has good sensitivity at all
frequencies; and \textbf{NS/NS} is optimized for the detection of
two coalescing 1.4~M$_\odot$ neutron stars. }
\label{fig:noise_curves}
\end{figure}
\section{Advanced LIGO Progress}
\label{sec:advanc-ligo-progr}
Some Advanced LIGO hardware has already been installed on the Initial LIGO
interferometers. These components include: a 35~W laser
Master-Oscillator/Power-Amplifier; a high-power, in vacuum Faraday
Isolator; a single stage, in-vacuum seismic isolation system; and DC
readout using an in-vacuum Output Mode Cleaner. With these systems,
LIGO has begun another science run, the sixth, with significantly
improved high frequency performance.
The Advanced LIGO project began in 2008, with plans for the first
in-vacuum hardware installation in early 2011. To evaluate the
greatly increased chances for direct GW detection, we consider compact
binary coalescences for which the source rate can be estimated from
observed binary pulsar systems. Once the instruments reach the
anticipated sensitivities, we can expect to detect between 1 and 1,000
compact binary coalescences per year. As installation and
commissioning progress, Advanced LIGO will transform the field from
searching for the first direct GW detection to exploring the rich
phenomena of GW astrophysics.
\section*{Acknowledgments}
We gratefully acknowledge the support of the United States National
Science Foundation for the construction and operation of the LIGO
Laboratory and the Science and Technology Facilities Council of the
United Kingdom, the Max-Planck-Society, and the State of
Niedersachsen/ Germany for support of the construction and operation
of the GEO600 detector. We also gratefully acknowledge the support of
the research by these agencies and by the Australian Research Council,
the Council of Scientific and Industrial Research of India, the
Istituto Nazionale di Fisica Nucleare of Italy, the Spanish Ministerio
de Educacion y Ciencia, the Conselleria d'Economia Hisenda i Innovacio
of the Govern de les Illes Balears, the Scottish Funding Council, the
Scottish Universities Physics Alliance, The National Aeronautics and
Space Administration, the Carnegie Trust, the Leverhulme Trust, the
David and Lucile Packard Foundation, the Research Corporation and the
Alfred P Sloan Foundation.
\bibliographystyle{unsrt}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,467 |
\section{Introduction}
\label{sec:intro}
\input{introduction}
\section{Related Work}
\input{related}
\section{Approach}
\input{approach}
\subsection{Problem Definition}
\label{sec:problem_definition}
\input{problem_definition}
\subsection{Network Structure}
\label{sec:network_structure}
\input{network_structure}
\subsection{Training Process}
\label{sec:training_process}
\input{training_process}
\section{Experiments}
\input{experiment}
\input{dataset}
\input{metrics}
\input{baselines}
\input{implementation}
\subsection{Network output accuracy and computational cost}
\input{computational_cost}
\subsection{Object detection and proposal accuracy}
\input{pascal_result}
\section{Conclusion}
\input{conclusion}
{\small
\section{Spherical Convolution Network Structure}
Fig.~\ref{fig:sphconv_supp} shows how the proposed spherical convolutional network differs from an ordinary convolutional neural network (CNN).
In a CNN, each kernel convolves over the entire 2D map to generate a 2D output.
Alternatively, it can be considered as a neural network with a tied weight constraint,
where the weights are shared across all rows and columns.
In contrast,
spherical convolution only ties the weights along each row.
It learns a kernel for each row,
and the kernel only convolves along the row to generate 1D output.
Also, the kernel size may differ at different rows and layers,
and it expands near the top and bottom of the image.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{./Figures/sphconv_illustration.pdf}
\caption{
Spherical convolution illustration.
The kernel weights at different rows of the image are untied,
and each kernel convolves over one row to generate 1D output.
The kernel size also differs at different rows and layers.
}
\label{fig:sphconv_supp}
\end{figure}
\section{Additional Implementation Details}
We train the network using ADAM~\cite{kingma2014adam}.
For pre-training, we use the batch size of 256 and initialize the learning rate to 0.01.
For layers without batch normalization,
we train the kernel for 16,000 iterations and decrease the learning rate by 10 every 4,000 iterations.
For layers with batch normalization,
we train for 4,000 iterations and decrease the learning rate every 1,000 iterations.
For fine-tuning,
we first fine-tune the network on conv3\_3 for 12,000 iterations with batch size of 1.
The learning rate is set to 1e-5 and is divided by 10 after 6,000 iterations.
We then fine-tune the network on conv5\_3 for 2,048 iterations.
The learning rate is initialized to 1e-4 and is divided by 10 after 1,024 iterations.
We do not insert batch normalization in conv1\_2 to conv3\_3 because we empirically find that it increases the training error.
\section{Data Preparation}
This section provides more details about the dataset splits and sampling procedures.
\paragraph{Pano2Vid}
For the \textbf{Pano2Vid} dataset,
we discard videos with resolution $W{\ne}2 H$ and sample frames at 0.05fps.
We use ``Mountain Climbing'' for testing because it contains the smallest number of frames.
Note that the training data contains no instances of ``Mountain Climbing'', such that our network is forced to generalize across semantic content.
We sample at a low frame rate in order to reduce temporal redundancy in both training and testing splits.
For kernel-wise pre-training and testing,
we sample the output on 40 pixels per row uniformly to reduce spatial redundancy.
Our preliminary experiments show that a denser sample for training does not improve the performance.
\paragraph{PASCAL VOC 2007}
As discussed in the main paper, we transform the 2D PASCAL images into equirectangular projected $360\degree$ data in order to test object detection in omnidirectional data while still being able to rely on an existing ground truthed dataset.
For each bounding box,
we resize the image so the short side of the bounding box matches the target scale.
The image is backprojected to the unit sphere using $\mathcal{P}^{-1}$,
where the center of the bounding box lies on $\hat{n}$.
The unit sphere is unwrapped into equirectangular projection as the test data.
We resize the bounding box to three target scales $\{112, 224, 336\}$ corresponding to $\{0.5R, 1.0R, 1.5R\}$,
where $R$ is the Rf of $N_{p}$.
Each bounding box is projected to 5 tangent planes with $\phi=180\degree$ and $\theta \in \{36\degree, 72\degree, 108\degree, 144\degree, 180\degree\}$.
By sampling the boxes across a range of scales and tangent plane angles, we systematically test the approach in these varying conditions.
\section{Complete Experimental Results}
This section contains additional experimental results that do not fit in the main paper.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{./Figures/pano_err_full.pdf}
\caption{
Network output error.
}
\label{fig:output_error_full}
\end{figure}
Fig.~\ref{fig:output_error_full} shows the error of each meta layer in the VGG architecture.
This is the complete version of Fig.~4a in the main paper.
It becomes more clear to what extent the error of \textsc{SphConv} increases as we go deeper in the network as well as how the error of \textsc{Interp} decreases.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{./Figures/proposal_iou_full.pdf}
\caption{
Proposal network accuracy (IoU).
}
\label{fig:proposal_full}
\end{figure}
Fig.~\ref{fig:proposal_full} shows the proposal network accuracy for all three object scales.
This is the complete version of Fig.~6b in the main paper.
The performance of all methods improves at larger object scales,
but \textsc{Perspective} still performs poorly near the equator.
\section{Additional Object Detection Examples}
Figures~\ref{fig:detection_supp1}, \ref{fig:detection_supp2} and \ref{fig:detection_supp3} show example detection results for \textsc{SphConv-Pre} on the $360\degree$ version of PASCAL VOC 2007.
Note that the large black areas are undefined pixels; they exist because the original PASCAL test images are not $360\degree$ data, and the content occupies only a portion of the viewing sphere.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf112tilt32/000846-bbox002.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/000084-bbox003.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/002160-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/006642-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/005096-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/005886-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/001868-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/006388-bbox000.jpg}
\end{subfigure}
\caption{
Object detection results on PASCAL VOC 2007 test images transformed to equirectangular projected inputs at different polar angles $\theta$.
Black areas indicate regions outside of the narrow field of view (FOV) PASCAL images, i.e., undefined pixels.
The polar angle $\theta = 18\degree, \, 36\degree, \, 54\degree, \, 72\degree$ from top to bottom. Our approach successfully learns to translate a 2D object detector trained on perspective images to $360\degree$ inputs.
}
\label{fig:detection_supp1}
\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/002922-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/008344-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/001829-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/003006-bbox002.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/004740-bbox003.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/005008-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/000434-bbox001.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/001412-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/002406-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/002474-bbox002.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/004844-bbox003.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Detections/008834-bbox000.jpg}
\end{subfigure}
\caption{
Object detection results on PASCAL VOC 2007 test images transformed to equirectangular projected inputs at $\theta = 36\degree$.
}
\label{fig:detection_supp2}
\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/002060-bbox001.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/002360-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/003394-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/003526-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/005976-bbox002.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/006044-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/006982-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/009258-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/009736-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Rf336tilt32/009750-bbox004.jpg}
\end{subfigure}
\caption{
Object detection results on PASCAL VOC 2007 test images transformed to equirectangular projected inputs at $\theta = 18\degree$.
}
\label{fig:detection_supp3}
\end{figure}
Fig.~\ref{fig:detector_failure} shows examples where the proposal network generate a tight bounding box while the detector network fails to predict the correct object category.
While the distortion is not as severe as some of the success cases,
it makes the confusing cases more difficult.
Fig.~\ref{fig:proposal_failure} shows examples where the proposal network fails to generate tight bounding box.
The bounding box is the one with the best intersection over union (IoU),
which is less than 0.5 in both examples.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Failures/000176-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Failures/000894-bbox000.jpg}
\end{subfigure}
\\
\vspace{2pt}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Failures/005474-bbox000.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Examples/Failures/005846-bbox000.jpg}
\end{subfigure}
\caption{
Failure cases of the detector network.
}
\label{fig:detector_failure}
\end{figure}
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Figures/000166-bbox001.jpg}
\end{subfigure}
~
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{./Figures/000668-bbox000.jpg}
\end{subfigure}
\caption{
Failure cases of the proposal network.
}
\label{fig:proposal_failure}
\end{figure}
\clearpage
\section{Visualizing Kernels in Spherical Convolution}
Fig.~\ref{fig:kernels_supp} shows the target kernels in the AlexNet~\cite{alexnet} model and the corresponding kernels learned by our approach at different polar angles $\theta \in \{9\degree, 18\degree, 36\degree, 72\degree\}$.
This is the complete list for Fig.~5 in the main paper.
Here we see how each kernel stretches according to the polar angle,
and it is clear that some of the kernels in spherical convolution have larger weights than the original kernels. As discussed in the main paper, these examples are for visualization only. As we show, the first layer is amenable to an analytic solution, and only layers $l > 1$ are learned by our method.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth]{./Examples/kernels.png}
\caption{Learned conv1 kernels in AlexNet (full). Each square patch is an AlexNet kernel in perpsective projection. The four rectangular kernels beside it are the kernels learned in our network to achieve the same features when applied to an equirectangular projection of the $360\degree$ viewing sphere.}
\label{fig:kernels_supp}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,371 |
About Simon Sinek
Simon Sinek is an optimist. He teaches leaders and organizations how to inspire people. From members of Congress to foreign ambassadors, from small businesses to corporations like Microsoft and 3M, from Hollywood to the Pentagon, he has presented his ideas about the power of why. He has written two books, Leaders Eat Last and Start With Why and is quoted frequently by national publications. Sinek also regularly shares 140 characters of inspiration on Twitter (@simonsinek).
Jocko Willink
Patrick M. Lencioni
Leif Babin
Michael Bungay Stanier
Brené Brown
Getting a Better Job
There is an entire section in our bookstores called "self-help." What we really need is a section called "help others."
To get a better job, you may not need to switch jobs at all.
There is one thing you can do that will not only help you find greater fulfillment, meaning and satisfaction in your career — you will also be helping the people around you create the same.
We feel our greatest achievements of success not when we accomplish something for
3 years ago Read more
Can Theresa May Lead the UK?: 4 Essential Tips for New Leaders Stepping Into A "Damned-If-You-Do-Damned-If-You-Don't" Job
As Theresa May steps in as the new Prime Minister of Britain, she will be faced with the daunting task of unifying a divided nation. It's an unenviable position for any leader; a thankless damned-if-you-do-damned-if-you-don't job. In Theresa May's case, if she goes through with the Brexit, she will infuriate half the population that thinks that's a terrible idea and if she decides not to go through with it she will infuriate the other half that won a popular referendum. What is she to do?
Explaining the Rise of Donald Trump
Whenever I read or hear a news story about the perplexing rise of Donald Trump, I myself am confused. Confused, not by Trump's rise, but how predictable his rise was given the politics of the past decade.
The Republican Party did such an expert marketing job for the past 8 years that it actually worked. Being a "Washington insider," for some reason, became a dirty thing. "Outsiders" were hailed as saviors. This gave rise to the Tea Party movement and laid the foundation for Donald Tru
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The world is a bell curve. Class room test scores, employee performance in a company or how many people really, really like you. No matter the population you're studying, they always fit neatly across the standard deviations of the famous bell curve. The most important thing about the bell curve is that it is always balanced. If there are high performers on one side, there are low performers on the other. It is always balanced. Which brings me to the topic of this blog
To Be Or Not To Be...Inspired
Post by David Mead
On a recent flight, as I often do after I board a plane, I was looking out the window as the ground crew was hussling to get the flight ready for take off. I happened to zero in on the guy who was hurling bags onto the conveyor belt, which led to the belly of the plane.
As I looked at him and tried to imagine what his work day must be like, I couldn't help wondering if he would even care if the airline he works for had a Why - a higher cause or purpose m
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Imagine if your boss walked into your office after a meeting about a particular initiative you were heading up and said, "You've been working on this project for a couple of weeks now and from the report you gave at the meeting, it's already behind schedule. What's going wrong?"
How does that feel?
Let's change your boss's comment, just slightly. "You've been working on this project for a couple of weeks now and from the report
What makes us amazing as human beings is our ability to form communities and groups. Our survival and our progress has depended on working together. And yet, as it turns out, most people seem to miss the connection between success and progress in a business environment and the way we naturally form groups and communities.
Think of it this way - we gladly sacrifice time, energy, money and other resources to join groups, clubs, political parties and associ
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It's not about hiring people who are just like us
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I was in Providence, Rhode Island recently on a speaking assignment. I was sitting in my hotel room when I heard some shouting and chanting outside. I went to the window and noticed a small group of people walking in a circle in front of the hotel, holding picket signs chanting something in Spanish.
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Fifty feet up the trail and already I knew things were going to be rough. My dad turned to me and said, 'Wow, I didn't think I'd be out of breath this quickly. I need to stop for a minute.'
We were about one minute in to a 2.5 mile hike up the very steep, varying terrain to Preikestolen, or Pulpit Rock, which juts out 2000 feet above the Ryfylke fjord in Norway. At 77 years old, with diabetes and a growing problem with balance, this was not going
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The deal of the day was from an auto center offering headlight restora
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Whenever I can, I enjoy mountain or 'fell' running, occasionally entering a mountain marathon. It's great exercise and one can get a completely different perspective of the world from the summit of a mountain.
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Click here to read a
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But recently, a Fairway Market moved in less than a block away from the Food Emporium. Fairway is
The Night The Lighthouse Went Out
Today we've lost a lighthouse. Steve Jobs, the man who stood for so much more than sleek design or innovation, the man who stood for people, is gone.
He was singularly devoted, not to technology, but how people interacted with technology. It wasn't for us to fit into a world of computers, it was for the computers to fit into a world of people. And that's what made Jobs different. That's what made Jobs special.
There are many great CEOs and there are great innovators, but we wo
The Power of the Young
Jill is an entrepreneur. She is a big thinker with big ideas. She is also an idealist. She imagines a world in which companies make their impact on society their primary bottom line and the financial results that follow as their second bottom line. She is smart and articulate and her ideas are really good, but she's struggling to get anyone to take her seriously.
According to the companies that close the door on her, it's because she's only 24.
Obligate Ram Ventilation
Careers are like a shark, to be successful we have to keep moving forwards.
In order to breathe, most sharks have to keep moving at all times. The forward motion allows the oxygen rich water to pass into their mouths over their gills and out through the gill slits. If they stop moving forwards, their gills won't work, they won't be able to breath and they will sink to the bottom and die. This system is called "obligate ram ventilation."
Obligate - as in obligatory,
The Advantage in Disadvantage
The young artist who was told by their high school math teacher that they were lazy.
The son who was told he's going to screw up his life because he didn't want to become an accountant like his dad.
The employee who was passed over for a raise or a promotion and was, instead, given a growth plan to help them overcome their weaknesses in order to make it to the next level.
Even the disciplines of psychology and psychiatry – almost everything in this world is gear
Money Can Buy Happiness
We're told all our whole lives that money can't buy happiness. The fact is, that's simply not true. Money can buy happiness. What money can't buy is lasting happiness. What money can't buy is fulfullment. That takes something else.
Here is the second part of a videoed conversation I had about where fulfillment comes from and how we can find lasting happiness.
So you don't have to look it up, here is Part 1.
The Pictures You'll Never See
I returned from Afghanistan this week. I went as a guest of the Air Force to experience the remarkable work they do, first hand. "The grand ballet," the Commander of Air Mobility Command calls it. I flew on various types of aircraft on various kinds of missions. I watched an air drop, supplying soldiers on the ground with fuel, water and ammunition. I rode on an aeromedical evacuation as 37 wounded soldiers and marines were brought home, one in critical condition. There was one more
The Power of Rachel Beckwith
A few days ago, a 9-year-old girl named Rachel Beckwith died in a pileup on I-90 in Washington state. Her spinal chord was severed and there was nothing the doctors could do to save her life.
June 12th was her last birthday and for her birthday Rachel told everyone that she didn't want any presents and she didn't want a party. Instead, she wanted her friends to donate $9 to Charity Water." Her big crazy goal," said her pastor, "was to raise $300 so that 15 kids in Afri
Fulfillment Is A Right, Not A Luxury
The discovery of Why taught me a valuable lesson: fulfillment, that sense of prolonged happiness in our lives, is not a luxury. In fact it is our right.
I am undertaking a little project to write a manifesto - a statement of belief - to help capture this movement that we are a part of. To put into words the world we imagine and a roadmap to get there.
I sat down with some of the folks on my team, people devoted to this cause, to tell them what I'm up to and wha
Outside vs Inside
Ah...the spa. A place of serenity, calm and oneness with the universe. A place where we can go to escape the hustle and bustle and be treated as human beings. A place where looking after your mind, body and spirit is of the utmost importance. Well...for paying customers, anyway.
It turns out that all that Zen idealism doesn't apply to all living things. There's a high-end spa I know in a major metropolis that doesn't treat its employees very well. I was talking to one of the ma
Books By Simon Sinek
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Start with Why: How Great Leaders Inspire Everyone to Take Action Sep 23, 2009
by Simon Sinek
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The Infinite Game Oct 15, 2019
Leaders Eat Last: Why Some Teams Pull Together and Others Don't Jan 7, 2014
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Find Your Why: A Practical Guide for Discovering Purpose for You and Your Team Sep 5, 2017
by Simon Sinek , David Mead , Peter Docker
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Together Is Better: A Little Book of Inspiration Sep 13, 2016
Empieza con el porque: Cómo los grandes líderes motivan a actuar (Gestión del conocimiento) (Spanish Edition) May 7, 2018
by Simon Sinek , Martín Rodríguez-Courel Ginzo
Los líderes comen al final (edición revisada): Por qué algunos equipos funcionan bien y otros no (Gestión del conocimiento) (Spanish Edition) Jul 10, 2017
Encuentra tu porqué (Gestión del conocimiento) (Spanish Edition) May 7, 2018
by Simon Sinek , Sergio Bulat Barreiro
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Juntos es mejor (Spanish Edition) May 31, 2017
Only 19 left in stock (more on the way).
Start with Why by Simon Sinek Jan 1, 2019
Il gioco infinito (Italian Edition) Oct 24, 2019
The Topline Summary of: Simon Sinek's Start with Why - Be a Great Leader and Inspire Other People to Take Action (Topline Summaries) Jul 2, 2014
by Gareth F. Baines , Brevity Books
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#10 in Books > Business & Money
#21 in Kindle eBooks > Business & Money
#76 in Kindle eBooks > Health, Fitness & Dieting | {
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} | 5,763 |
Correct foot positioning helps players such as Houston's Jeremy Lin shoot accurate jumpers.
You shoot a basketball with your arm, but unless you're dunking or shooting from very close range, your legs are just as important as that arm. Positioning your feet properly when you take jump shots helps you square your body to the hoop and maintain your balance, so you can shoot accurately. Additionally, correct foot positioning on long shots helps you produce the leg drive you need to hit the perimeter shot. So if you want to improve your jump shot, start from the bottom up by developing good footwork.
Set your feet shoulder-width apart.
Shift your weight to the balls of your feet.
Position your dominant foot a few inches forward of the other foot. For instance, if you're right-handed you'd put your right foot in front.
Point your toes in the general direction the basket.
Set your feet directly under your hips to provide maximum lift when you jump, particularly when you're shooting from long range.
Your foot positioning varies depending on the type of shot you take, particularly when you're close to the hoop. On dunks and layups, for example, your feet will typically be in the air before you begin your shooting motion. On hook shots you'll lift one foot as you push off the court with the other. On turnaround jumpers your heels will face the basket before you jump and spin in the air.
Once you learn proper shooting mechanics, tweak your foot positioning in practice to see what works best for you. Once you find your ideal positioning, use the same technique, consistently, for every jump shot.
What Is a Stride Stop in Basketball?
How Does Muscular Strength Improve From Playing Basketball? | {
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} | 7,249 |
Just in time for fall weather and the upcoming holiday season, we're featuring the Anna's Garden Poncho Kit. One of our most popular garments, the poncho is a classic piece, fit for most any occasion.
Originally featured in Alabama Studio Sewing + Design, the poncho is seen here in our 100% organic cotton jersey fabric in Black with Forest stitched in negative reverse appliqué. We constructed our garment using black Button Craft thread tied with knots on the outside.
This one-size-fits-all DIY kit is ideal for novice and experienced sewers alike. For added complexity, work your kit in a reverse appliqué technique or add beads to your order to embellish the stencil work.
Visit our website to choose your own color ways and additional notions. | {
"redpajama_set_name": "RedPajamaC4"
} | 106 |
« Obituary: Micol Fontana, Fashion Designer (1913-2015) | Main | Fashion is Absolutely Tacky: Moschino Men's S/S 2016 »
That Memphis Milano Obsession That Won't Go Away: Arthur Arbesser Resort 2016
It all started in a fashion galaxy far away (time, you see, goes very fast in the fashion universe...) when Bill Gaytten took inspiration for Dior's Autumn/Winter 2011-12 Haute Couture collection from the Memphis Milano movement. The same thread continued in Sergio Rossi's Spring/Summer 2013 collection and later on took a new incarnation with Prada's A/W 2015 designs, with their pastel shades that looked entirely lifted from Michele De Lucchi's Girmi prototypes.
So, in a way, seeing yesterday the 1981 "Suvretta" bookcase by Ettore Sottsass at the top of Arthur Arbesser runway inside the Stazione Leopolda in Florence, quite a few fashion and interior design connoisseurs must have heard "Stuck Inside of Mobile with the Memphis Blues Again", Bob Dylan's song that inspired the Memphis Milano designers, eerily echoing in their minds.
Further pieces scattered along the runway (such as Ettore Sottsass's "Beverly" sideboard, another iconic design from 1981) in a display designed by architect Luca Cipelletti, provided more clues to the main inspirations behind this collection.
Arthur Arbesser's Resort 2016 collection, showcased as part of the Florence-based Pitti event, displayed a strong link with interior design and in particular with revolutionary creators like the Memphis Milano group and Ettore Sottsass, one of Arbesser's design heroes.
Arbesser shares something with Sottsass, first and foremost his origins: the latter was born in Innsbruck, Austria, in 1917, and then studied Architecture at the Politecnico of Turin in 1939, setting his own studio in Milan in 1947 and starting to work as design consultant for Olivetti in 1948.
Arbesser was born and raised in Vienna, studied at London's Central Saint Martins and then moved to Milan where he worked for seven years at Giorgio Armani, and where he recently started focusing on his own collections (he was among the finalists of the LVMH Prize 2015 won by design duo Marques'Almeida).
Arbesser did his own research and tried to come up with less exploited Memphis Milano aspects, reshifting his attention on Ettore Sottsass's collaboration (that lasted roughly 30 years) with the Montelupo Fiorentino-based company Manifattura Ceramica Bitossi.
Sottsass created for them several cercamic pieces, and, in 1959, he came up with an ambitious mural. Originally installed in 1960 in the main entrance of the Triennale di Milano, this monumetal ceramic painting in three parts featured a series of geometrical elements in brilliant colours.
Some of the patterns included in this work seem to have trickled down into Arbesser's boldly geometric prints and intarsia knits (made by a company based just outside Florence), that were definitely among the most vivid pieces of the collection.
The yellow, black and white palette of some of Sottsass's ceramic totems returned in the ample coats, trousers and shirts, but also in some of the knitwear, while orange and black shades called to mind Sottsass' geometrical vases, his Freemont cupboard or his iconic "Valentine" typewriter for Olivetti.
Further references to Bitossi's archival pieces (not necessarily made by Sottsass) could be spotted in one of the colours of the bright palette (a celebration of the main theme for this season's Pitti - "That's Pitticolor!"), an intense blue that called to mind the shades of the "Rimini Blu" ceramics series.
The minimalist elaborations of the ceramic bowls from the '60s found correspondences on the muted surface eleborations, while technical textures such as waxed effect denim pieces were maybe nods to the plastics by Abet Laminati favoured by Memphis Milano.
Shapes were simple and basic for both men and womenswear: the collection included six men's looks as a reference to Pitti Uomo, but it was mainly "agender" (or, as we called it in the '70s, unisex...), with lean dresses and wide pants, ample blouses and shirts.
There were also subtle clinical and medical moods in the nylon tops, laboratory coats and hospital operation uniforms in blue or bright orange, while light padded garments may have pointed not at madness, padded cells and straightjackets, but at a Japanese vision of samurais (an influence visible in some of the skirt shapes and fastenings). All the looks were accessorised with flat androgynous shoes designed in collaboration with Sergio Rossi.
Parallelisms could be made between these designs and Arthur Arbesser's Autumn/Winter 2015 collection: the latter was presented in Milan in February with a sort of installation format with paintings by contemporary artist Hermann Nitsch on the walls and models seated on Austrian-designed chairs listening to a Schubert composition played live on the piano, and featured mesmerising multi-coloured prints inspired by Vienna's Wiener Werkstätte.
This collection was showcased in a sort of arty environment and again featured very colourful graphic patterns, though Arbesser is clearly trying to combine in a stronger way feminine and masculine elements into his pieces.
Other designers such as Alessandro Michele at Gucci and J.W. Anderson are doing the same thing in their collections (though in both of them the feminine prevails over the masculine), but Arbesser is doing so via industrial elements, maybe following Miuccia Prada's lesson.
The display at the Stazione Leopolda was indeed slightly reminiscent of Prada's presentation for its Autumn/Winter 2013 menswear collection, but, rather than proving that those invisible yet tangible bridges between interior design and fashion can definitely be built, this collection was to be considered as a tribute to those industrial skills that Italy lost in the last 20 years or so.
What was missing? As stated in previous posts, Memphis was a reaction to years of rationalism, and a final shift from pure and clean lines to ebullient trends, it was a visual punch in the eye that made you instantly feel optimistic and brave about the present and the future.
At times you genuinely feel that Memphis Milano may turn into the albatross hanging on the neck of the fashion world: so far we have seen fashion mentioning, referencing, imitating and plagiarising the work of this group or infusing, as Arbesser did, some of its semantics into wearable (though not desperately new) garments.
Yet the revolutionary and radical message and the optimistic playfulness weren't certainly there and you were left wishing that, rather than paying tribute to or moving from Memphis Milano & Co, there will be one day a fashion designer interested in writing a manifesto with some of the original people involved in this group.
It would indeed be intriguing to hear from people like Michele De Lucchi if and in case how we could use fashion to criticise design itself, but also society, politics and modern anxieties and maybe build through it not a metaphor for a new existence, but a new life.
There are hopes, though, that we will get there: Arbesser, who self-financed his fashion house, hopes to grow little by little and step by step and develop his dialogue between art, fashion and architecture in a consistent way (there were no boundaries between them for people like his hero Sottsass, who used to see differences only in the techniques and not in the contents of these disciplines), while keeping a close eye on each and every detail of his designs, as Armani taught him to do.
Arbesser was a bit of a change for the Pitti organisers: while they seem to have spent the last few years desperately running after the next trendy thing to look more appealing to the high profile condescending bloggers they invited, they may have finally found something with more substance than uselessly grand or chaotic displays and presentations that generated media revenues and no sales. Have they learnt their lessons? Time will tell, but at the moment they are all getting excited about Katy Perry and other assorted celebrities at Moschino's menswear catwalk show tonight and that's definitely not a good sign...
Academy of Art University Special 3: Spotlight on Amanda Manashi
Over-Embellished Couture Style Details in Modern Ballet Costumes
Book Alert: Louise Brooks, Detective by Rick Geary (NBM Publishing)
Shoes: Pleasure & Pain @ The Victoria and Albert Museum, London
Academy of Art University Special 1: Spotlight on Ye Kuang and Candy Hsinyu Chu | {
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Working for multi-client geoscience data company, TGS, Fugro continues the hunt for hydrocarbon seeps, this time offshore Brazil. Two modern, purpose-built vessels – Fugro Brasilis and Fugro Searcher – have been deployed to acquire high-resolution multibeam echosounder and sub-bottom profiler data in the Campos and Santos Basins.
The survey is designed to mirror TGS's successful 2016-2017 Gigante and Otos projects in the Gulf of Mexico in which Fugro provided similar services. Covering an area of approximately 200,000 square kilometres, Fugro will use these data to identify and recommend the most prospective locations to target for geochemical sampling.
"We have been operating in Brazil for more than two decades, providing a wide variety of geotechnical, survey, subsea and geoscience services, principally for the deepwater oil and gas industry," he added. For much of this time Fugro has been at the forefront of modern marine seep-hunting operations in the energy sector and has led the market in integrated multibeam and geochemical coring campaigns. | {
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Q: Jest - Getting Started lesson not completely I'm using w/ VScode on a mac. I followed the instructions for the first test on https://jestjs.io/docs/en/getting-started.
I'm getting a jest prompt - and it's just hanging. The test is not completing.
Jest is installed and my package.json is set up. What's is going on? I'd love to start doing TDD.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,187 |
Policy and Research Needs to Maximize Independence and Support Community Living: Workshop Summary (2016)
Chapter: 2 The Importance of Community Living and Maximizing Independence to Individuals and Society
Suggested Citation:"2 The Importance of Community Living and Maximizing Independence to Individuals and Society." National Academies of Sciences, Engineering, and Medicine. 2016. Policy and Research Needs to Maximize Independence and Support Community Living: Workshop Summary. Washington, DC: The National Academies Press. doi: 10.17226/21893.
The Importance of Community Living and Maximizing Independence to Individuals and Society
To provide a framework for the workshop's discussions, two keynote speakers talked about what maximizing independence and community living mean, the successes that have already been achieved, the challenges that lie ahead, and suggestions for how to meet those challenges. Kathy Greenlee, the Administrator of the Administration for Community Living (ACL) and the Assistant Secretary for Aging in the U.S. Department of Health and Human Services (HHS), gave a keynote presentation on the history and current state of the federal government's efforts to support community living and independence. Gretchen Alkema, the Vice President for Policy and Communications at The SCAN Foundation, then discussed the contrast between how Americans perceive issues of aging, disability, and independence and the realities of these issues.
FEDERAL EFFORTS TO SUPPORT COMMUNITY LIVING AND INDEPENDENCE
Kathy Greenlee
Administrator, Administration for Community Living
Assistant Secretary for Aging,
Kathy Greenlee of ACL and HHS began her presentation by recounting how when she first began working as the Assistant Secretary for Aging in 2009, HHS was putting together working groups to prepare for what
would eventually become known as the Patient Protection and Affordable Care Act1 (ACA). She recalled how she and Henry Claypool, who was then Secretary Kathleen Sebelius's advisor on disability policy, would receive the lists of those who were to be members of the working groups, and neither one of them would be included. "That was troubling to us and also an early warning that health [care] reform was not long-term care reform," Greenlee said. Although long-term care reform is occurring, she added, it was not organized that way, nor was there recognition of the connection to community-based work on aging and disability. It was for this reason that Greenlee, Claypool, and Sharon Lewis—then the Administrator of the Administration on Intellectual and Developmental Disabilities—worked to create ACL. They envisioned ACL as the agency at HHS where the voices of older adults and people with disabilities would be heard in the health reform conversations.
As the Obama Administration comes to a close, Greenlee and her colleagues at HHS have been working on a proposed budget for 2017. Even though the budget is unlikely to be passed by Congress before the next Presidential election, Greenlee said she and her colleagues have decided to use this budget-planning time as an opportunity to educate as many people as possible within the federal government about the need to integrate community living and community-based supports with programs that provide long-term services and supports (LTSS) and acute care for older adults and people with disabilities. One question the Office of Management and Budget has asked was why the Centers for Medicare & Medicaid Services (CMS) is not doing this work. This question demonstrates how many people incorrectly assume that CMS oversees all things related to aging and disability—in fact, it does not—and it highlights the need to continue to educate both those inside and outside of the federal government on the importance of developing policies and providing funding to integrate services in a way that supports community living and independence.
CMS, Greenlee explained, interfaces with state Medicaid agencies, which are essentially insurance finance companies and not program agencies. Although some at CMS know what work ACL does at the community level, those involved with Medicare or the Center for Medicare & Medicaid Innovation generally do not. What is disheartening, although not surprising, Greenlee said, is that 6 years into the era of health care reform there is still a disconnect between the need to have community-based services so older adults and those with disabilities can stay in their
1 Patient Protection and Affordable Care Act, Public Law 111-148, 111th Cong., 2nd sess. (March 23, 2010).
homes and the need to have federal health care policies to enable and support the delivery of those services.
Over the past 6 years, Greenlee and her colleagues have made what she called tremendous progress on individual pieces of this problem and on demonstrating what is possible when the right policies and funding are in place. States are beginning to replicate some of the demonstration programs that ACL has funded, and researchers are starting to develop approaches for integrating these individual pieces into a long-term care strategy. One big issue still remaining, Greenlee said, is how to reform long-term care financing and preserve the value base of community-based services.
Two specific areas in which ACL and its partners have made what Greenlee characterized as significant strides forward are in business acumen and in the coordination of LTSS.
The ACL Business Acumen Learning Collaborative2 is a community-level effort to improve the business acumen of the organizations and agencies that provide community-based services. Greenlee noted the tremendous support that The John A. Hartford Foundation and The SCAN Foundation have given as ACL has worked to change the network of services it supports and to make previously free services more sustainable in an uncertain funding climate. "We are taking a national network of nonprofits—the area agencies on aging [and the] centers for independent living—and changing their fundamental business structure," Greenlee said. "Not their mission, not their value[s], not their contribution[s], [but] their structure." These are primarily grant-based organizations that receive money from the federal government via the states and the Older Americans Act, she explained. They are learning to become providers, to demonstrate their economic value, and to bill for their services. Possible opportunities include establishing meal providers that hospitals can contract with to deliver meals to the home of someone recently released from the hospital. Other services that might be able to generate fees rather than subsist on grants include community-based transportation networks and case management. There is still a role for services that are provided free of charge, Greenlee said. The challenge is that there is an assumption that all of the free community services are plentiful and will continue to be available for the clients of large long-term care providers, managed-care organizations, and hospital systems, even in an uncertain funding climate.
2 For more information, see http://www.acl.gov/Programs/CIP/OICI/BusinessAcumen/index.aspx (accessed January 28, 2016).
Coordination of LTSS
The other significant stride forward that ACL has made, with strong support from CMS and the U.S. Department of Veterans Affairs, is a state-level effort to help improve the coordination of LTSS. ACL has taken the core principles of what CMS has done with its "no wrong door" approach for beneficiaries to gain access to CMS services and is working to extend a similar model to the entire range of aging and disability services. One goal of this work, Greenlee said, is to bring all potential beneficiaries to the table—not just those who have spent down all of their savings, but also those who still have savings and who, with some assistance, might be able to avoid spending down their savings and having to enroll in Medicaid to pay for their LTSS needs. She cited the collaboration between the Partners in Care Foundation and Blue Shield of California as one example of how states are designing programs so that they can provide a modest amount of help to individuals earlier, which may delay or prevent individuals from ever needing to enter the Medicaid system.
Key Issues for Moving Forward
During the course of ACL's work on improving business acumen and the coordination of LTSS, four key issues have emerged that should be kept in mind as this work moves forward, Greenlee said. The first is a focus on quality and quality outcomes. "If we monetize [services], people will have to know that they are buying something that has an outcome that they want," she said. "I am convinced that this particular pathway will secure the future of community-based organizations once they can prove it." For example, quantifying the value of providing home-delivered meals as a form of health support after a hospital discharge would require research demonstrating the associated positive health outcomes and health care savings. However, she said, focusing on quality outcomes is hard for the aging services network because of its historic emphasis on outputs, not outcomes. For example, when ACL instituted a requirement that states and area agencies could only spend Older Americans Act funds on evidence-based programs, there was some resistance. This requirement was a signal that an exercise program at a senior center could not be created just because someone volunteered to run it. The exercise program must have fidelity and be replicated with outcomes that are proven, Greenlee explained. Toward this end, HHS has asked the National Quality Forum (NQF) to identify the domains needed for quality measures of LTSS. NQF has already identified person-centeredness as one of the necessary domains. The next step, Greenlee said, is to invest in research to populate these domains with data. Although some data are
available at the local level, the research for home- and community-based services needs to be taken to a much larger scale.
The second key issue is to continue the conversation about delivery system reform. Greenlee said that much of the discussion has been about delivery system payment reform and has focused on how doctors and hospitals are paid. However, delivery system reform goes beyond the boundaries of the hospital system to include how community-based services are delivered.
The third key issue relates to technology and, in particular, the development of interoperable electronic health record (EHR) systems. Greenlee pointed to research on care transitions, funded by the Center for Medicare and Medicaid Innovation, as evidence that the lack of access that community-based organizations have to EHRs is a fundamental barrier to these organizations successfully moving forward. In some cases local organizations, such as the Area Agency on Aging in San Diego County—an aging and disability resource center—have gone so far as to build their own systems to interface with regional hospitals. And yet, "no one is investing in this. This is not part of meaningful use proposals for this country. It's not part of what the Office of the National Coordinator does," said Greenlee. "Technology will keep the aging network and disability programs and local communities . . . at the starting gate if we don't talk about it, because no one is investing in [it]," said Greenlee.
The fourth key issue, Greenlee said, is basic education about the disconnects, the opportunities, and the good, effective work that is already taking place in home- and community-based services. "Where is the academic world? Where is the policy and research world? Is anyone writing about this?" asked Greenlee. She asked the workshop audience to continue to increase the visibility of these topics because "if this is only an internal conversation within the federal government, we can't solve the connectivity that we are all looking for."
AMERICAN PERCEPTIONS AND THE REALITIES OF AGING, DISABILITY, AND INDEPENDENCE
Gretchen Alkema
Vice President of Policy and Communications, The SCAN Foundation
How Americans think about aging and disability matters, Gretchen Alkema of The SCAN Foundation told the workshop audience. It affects the conversations that those working in the field have with each other, with policy makers at the state and local levels, and with the general public. It also affects how the media portray the issue. All of this led Alkema to question what should be done to bring the issues of aging,
disability, and independence out of the recesses of people's minds and out of the bedroom conversations of "What are we going to do about mom?" and into the open to start generating change in the United States. Although Americans are beginning to recognize that these issues are part of their lives, they are still not entirely clear about what actions to take, and they sometimes wonder whether anyone else is struggling with the same challenges.
Since 2013, The SCAN Foundation, which is interested in aging and dignity with independence, has commissioned the Associated Press–NORC Center for Public Affairs Research to conduct an annual poll of more than 1,700 individuals who are 40 years of age or older to examine Americans' awareness and perceptions of aging and long-term care.3,4 According to the 2015 poll, 47 percent of those surveyed believe it not too likely or not very likely that they have an aging family member or close friend who will need any level of ongoing living assistance in the next 5 years, a perception that Alkema characterized as not being based on reality. Even more surprising, she said, is that the percentage of poll respondents who held this belief increased substantially between the 2013 poll and the 2015 poll. One possible reason for this concerning trend, she suggested, is that it is driven by those who are 40 to 45 years old and are just beginning to experience the consequences of aging in their own families. Another possible reason, based on anecdotal observations from other health-related polls, is that these results have been affected by the economic recovery in the United States. Regardless of the reason, Alkema said, these results represent an enormous problem regarding Americans' awareness of the impending impact of an aging population.
Given these results, it was not surprising that the poll found that only 28 percent of Americans are very prepared or extremely prepared to help their loved ones. For the approximately half of those polled who were somewhat prepared to help their loved ones, many reported that their preparation might have consisted only of a conversation about funeral planning or other related end-of-life decisions.
The survey also found that approximately 76 percent of respondents said it was not at all likely, not too likely, or only somewhat likely that they themselves would ever need assistance. A 2015 brief from the Office of the Assistant Secretary for Planning and Evaluation (ASPE) at HHS
3 Results from each year of the poll can be found at longtermcarepoll.org (accessed January 30, 2016).
4 Alkema noted that the poll uses the phrase "ongoing living assistance" rather than "long-term care" or "long-term services and supports." For the purposes of the poll, "ongoing living assistance" was defined as help with tasks such as shopping, transportation, meal preparation, or money management. Activities of daily living were not included in this definition.
showed a different reality: The brief projected that approximately 52 percent of adults ages 65 years and older will have a severe long-term care5 need at some point in their lives, where "severe" means having impairment in two or more activities of daily living (ADLs)6 or severe cognitive impairment (Favreault and Dey, 2015). These two studies demonstrate a stark contrast between what the American population thinks and what the reality is. Furthermore, the care needed by 52 percent of the population who reach age 65 will have a tremendous cost. The ASPE brief included projections that the average individual lifetime care cost for the 52 percent of individuals with severe needs will be approximately $138,000 each. This amount is based on the cost of care in 2015, not on a future cost that accounts for inflation. Importantly, this cost does not include the cost of housing, food, clothing, or any other basic needs. These data underscore the importance of building a long-term services delivery system capable of meeting these projected demands through partnerships between community-based organizations and the health care delivery system. Alkema noted that these data are for individuals ages 65 years and older because there are very few, if any, data available for those under 65 years of age or those who have disabilities. She said this is a serious deficiency and an area in much need of research.
In summarizing her presentation, Alkema said that she disagreed with the metaphor many use of describing the United States as facing a tsunami of need, both for aging adults and those with disabilities. "I don't actually believe it's a tsunami at all because people don't experience it that way. People experience it as their own personal crisis," she said. "I would offer that . . . it's really a set of personal realignments when the myth of our life and the reality of our life collide."
When 100 million Americans are undergoing individual personal realignments at the same time, the opportunity arises to create a societal movement for change. The nation will not solve the challenges created by the discontinuity between the myth and reality of aging and disability using current tools for individual personal planning. Although the delivery and financing systems for long-term care should be considered as separate and unique, as also mentioned by Greenlee, they are still connected. Tools that can exist in both systems are needed to support the systems in meaningful ways for the benefit of the public.
Individuals cannot plan their way out of these challenges by them-
5Favreault and Dey (2015) define care needs to include "a range of services and supports individuals may need to meet their health or personal needs over a long period of time. Most [long-term care services and supports] is not medical care, but rather assistance with the basic personal tasks of everyday life, sometimes called 'Activities of Daily Living' (or ADLs)."
6 Activities of daily living (ADLs) are routine, everyday tasks such as bathing, dressing, eating, using the toilet, walking, and transferring (e.g., from a chair to a bed).
selves. "We know that from an economic perspective," Alkema said. "We know that from a delivery system perspective. We know that from a family perspective. So I strongly suggest that we just erase that myth." However, developing a more robust awareness will not be sufficient to address the disconnect between myth and reality. Alkema asked the workshop participants to create and implement new tools, systems of care, and policies to address the growing needs of older adults and those with disabilities. She cautioned not to focus on the negative elements of these challenges but rather to focus on the positive side, which is that this is an opportunity to create a new reality with better systems of care, better policies, and better programs.
An open discussion followed the keynote presentations. Workshop participants were invited to ask questions of and offer comments to the keynote speakers. The following section summarizes the discussion.
Greenlee began the discussion by offering a possible explanation for the concerns Alkema presented that highlighted the disconnect between what Americans believe and the realities of aging and disability. Greenlee suggested that the disconnect results from individuals' feelings about the concept of independence. Most people do not want to admit that they might lose their independence. If admitting that they need care means that they will lose their independence, then they will not admit that they need care. However, if the question of independence and needing care is reframed by asking people what services they require to remain independent, then they make every effort to maintain their independence. She noted that the World Health Organization released a report that frames health as an issue of functional ability (WHO, 2015) because of the way functional ability is linked to independence. "It's important for those of us who work in the health space to continue to connect our health care supports and our health-related supports to functional ability," Greenlee said, "because that is what people need to maintain." As an example, she said that diabetes itself does not affect independence so much, but rather it is the loss of a foot because of diabetes that impacts functional ability and thus independence. Similarly, hypertension itself does not affect independence, but having a stroke that results in paralysis will affect functional ability and independence.
Alkema added, "We have a very rigid idea of independence in this country about I gotta do it . . . and if I am not doing it, then I'm not independent." However, she said humans are dependent on each other for many things; very few grow their own food, sew their own clothes, and generate their own electricity. "There is a way in which we can use lan-
guage to basically reframe the concept of independence and identity and interest and engagement as opposed to executional capacity," she said.
Margaret Campbell of the National Institute on Disability, Independent Living, and Rehabilitation Research asked how to go about building and nurturing a bridge between the fields of aging and disability—two bodies of knowledge and policy streams that have been separate historically—so that there is the potential to create new partnerships and alliances that can help maximize independence equally among older adults and those with disabilities. In particular, she noted the absurdity of large amounts of money having been invested in research collecting data on disability, functional status, and the use of assistive technology when the research tends to study people only up until they reach age 65 or else the study population only begins at age 65.
Greenlee replied that she refers to this as multicultural work. She said that to do this work effectively, it is necessary to be able to live with dissonance. Disability and aging, for example, both have a history of stigma and discrimination, but they are not the same history, and civil rights factors into the field of disability in a way that it does not in aging. Furthermore, the two fields do not use the same terminology definitions. Therefore, bridging the fields of aging and disability requires patience, good intentions, and a lot of discussion about fundamental issues and clear communication about meaning. "The goal is not to think the same way or to be the same at all, but to have and articulate the same objectives," Greenlee said. ACL has a shared vision of what it means to be an autonomous person living in the community with a need for both prevention services and supports. Greenlee cautioned that it is possible to get lost in the complexities of distinguishing between the different fields but to remain respectful and comfortable living with the differences. She said that ACL has changed some of the language it uses to accommodate the differences in how the aging and disability communities think and talk about "choice," living in community "for a lifetime," and end-of-life care. She said that disability and aging work will not be the same, but there are opportunities because they share common threats and common goals, such as the threat of the medical community dictating how individuals should live their lives and the goal of individuals living their lives how they want them to be.
Next: 3 Home and Community Settings: Services and Supports for Community Living and Participation »
Policy and Research Needs to Maximize Independence and Support Community Living: Workshop Summary Get This Book
Living independently and participating in one's community are priorities for many people. In many regions across the United States, there are programs that support and enable people with disabilities and older adults to live where they choose and with whom they choose and to participate fully in their communities. Tremendous progress has been made. However, in many cases, the programs themselves – and access to them – vary not only between states but also within states. Many programs are small, and even when they prove to be successful they are still not scaled up to meet the needs of the many people who would benefit from them. The challenges can include insufficient workforce, insufficient funding, and lack of evidence demonstrating effectiveness or value.
To get a better understanding of the policies needed to maximize independence and support community living and of the research needed to support implementation of those policies, the National Academies of Sciences, Engineering, and Medicine convened a public workshop in October 2015. Participants explored policies in place that promote independence and community living for older adults and people with physical disabilities, and identified policies and gaps in policies that can be barriers to independence and the research needed to support changing those policies. This report summarizes the presentations and discussions from the workshop.
2 The Importance of Community Living and Maximizing Independence to Individuals and Society 7–16
3 Home and Community Settings: Services and Supports for Community Living and Participation 17–32
4 Workforce Needs to Support Community Living 33–54
5 Financing to Support Community Living 55–68
6 Technology to Support Independence 69–80
7 Closing Remarks 81–84
Appendix A: Workshop Agenda 87–92
Appendix B: Biographical Sketches of Workshop Speakers and Moderators 93–104 | {
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HomeUAE recognizes Israel, which will stop annexation in Trump-brokered deal
UAE recognizes Israel, which will stop annexation in Trump-brokered deal
Noa Landau/HaaretzAugust 13, 2020Image by Sarah Silbiger/Getty Imag...
Image by Sarah Silbiger/Getty Imag...
Read this article in Yiddish
Israel and the United Arab Emirates reached a historic peace deal on Thursday that will lead to a full normalization of diplomatic relations between the two Middle Eastern nations in an agreement that U.S. President Donald Trump helped broker.
Under the agreement, Israel has agreed to suspend applying sovereignty to areas of the West Bank that it has been discussing annexing, a joint statement by all three parties to the agreement said.
Trump, in a tweet, called the agreement a "HUGE breakthrough," describing it as a "historic peace agreement between our two GREAT friends."
Prime Minister Benjamin Netanyahu called it "a historic day" on Twitter, and will deliver a speech the agreement on Thursday night.
The joint statement said the three leaders had "agreed to the full normalization of relations between Israel and the United Arab Emirates."
"This historic diplomatic breakthrough will advance peace in the Middle East region and is a testament to the bold diplomacy and vision of the three leaders and the courage of the United Arab Emirates and Israel to chart a new path that will unlock the great potential in the region," the statement said.
The peace deal was the product of lengthy discussions between Israel, the UAE and the United States that accelerated recently, White House officials said.
The agreement was sealed in a phone call on Thursday between Trump, Netanyahu and Sheikh Mohammed Bin Zayed, crown prince of Abu Dhabi.
The officials described the agreement, to be known as the Abraham Accords, as the first of its kind since Israel and Jordan signed a peace treaty in 1994. It also gives Trump a foreign policy success as he seeks re-election on November 3.
'A win for diplomacy'
UAE Ambassador to the United States Yousef Al Otaiba released a statement calling the agreement "a win for diplomacy and for the region" and "a significant advance in Arab-Israeli relations that lowers tensions and creates new energy for positive change."
He added that the move "immediately stops annexation and the potential for violent escalation. It maintains the viability of a two-state solution as endorsed by the Arab League and international community. It creates new dynamics and possibilities in the peace process."
Al Otaiba's statement concludes with a reassurance that "The UAE will remain a strong supporter of the Palestinian people – for their dignity, their rights and their own sovereign state. They must benefit from normalization. We will forcefully advocate for these ends, now directly and bolstered with stronger incentives, policy options and diplomatic tools."
Annexation on hold
The statement said that as "a result of this diplomatic breakthrough and at the request of President Trump with the support of the United Arab Emirates, Israel will suspend declaring sovereignty" over areas of the West Bank that were envisioned in the U.S. Middle East plan unveiled by Trump in January.
"Israel for the foreseeable future will be focused on building this relationship and pursuing all the advantages that can come from having this new relationship with this country, and we also breaks the ice for doing more normalizations and peace agreements with other regional players as well," one White House official told Reuters.
Sources involved said that Netanyahu discussed the agreement with Alternate Prime Minister Benny Gantz and Foreign Minister Gabi Ashkenazi only after the agreement was formally announced by Washington. Both Gantz and Ashkenazi deny that they were previously unaware of the agreement.
A senior political source told Haaretz that annexation is still on the agenda, and that Israel is committed to it. "The Trump administration asked that we temporarily postpone declaring [sovereignty over parts of the West Bank] in order to achieve the beginning of this historic peace agreement with the Emirates."
A senior Likud official praised the agreement, saying: "The Israeli and international left always said that it is impossible to bring peace with Arab nations without peace with the Palestinian people. That there is no other way except from withdrawing from the '67 borders, clearing out the settlements, dividing Jerusalem and establishing a Palestinian state. This is the first time in History that Prime Minister Netanyahu broke the paradigm of 'land for peace' and brought 'peace in exchange for peace.'"
Meanwhile, PLO Executive Committee member Dr. Hanan Ashrawi condemned the agreement and its terms on Twitter. "Israel got rewarded for not declaring openly what it's been doing to Palestine illegally and persistently since the beginning of the occupation," she wrote, referring to the annexation agreement. "The UAE has come out in the open on its secret dealings/normalization with Israel. Please don't do us a favor. We are nobody's fig leaf!"
White House officials said Trump senior adviser Jared Kushner, U.S. Ambassador to Israel David Friedman and Middle East envoy Avi Berkowitz were deeply involved in negotiating the deal, as well as Secretary of State Mike Pompeo and White House national security adviser Robert O'Brien.
Pompeo congratulated Israel and the UAE in both Arabic and Hebrew in a tweet, calling the agreement "a significant step forward for peace in the Middle East."
In a statement, the U.S. secretary of state called the agreement "a remarkable achievement for two of the world's most forward leaning, technologically advanced states, and reflects their shared regional vision of an economically integrated region. It also illustrates their commitment to confronting common threats, as small – but strong – nations.
The United States hopes that this brave step will be the first in a series of agreements that ends 72 years of hostilities in the region. Although the peace treaties between Israel and Egypt and Jordan have not yet fulfilled their full potential, since the 1978 Camp David Accords and the 1994 Wadi Arava Agreement, we have witnessed significant economic development in Egypt and Jordan, an unmistakable dividend of peace."
Delegations from Israel and the United Arab Emirates will meet in the coming weeks to sign bilateral agreements regarding investment, tourism, direct flights, security, telecommunications and other issues, the statement said.
The two countries are expected soon to exchange ambassadors and embassies.
The agreement envisions giving Muslims greater access to the Al-Aqsa Mosque in the Old City of Jerusalem by allowing them to fly from Abu Dhabi to Tel Aviv, White House officials said.
The joint statement said the United Arab Emirates and Israel will immediately expand and accelerate cooperation regarding the treatment of and the development of a vaccine for the novel coronavirus amid the pandemic.
Noa Landau/Haaretz
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Trump deal: Israel stops annexation, UAE recognizes it
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Koncentracijski logor Kampor (Rab) bio je talijanski fašistički koncentracijski logor u Drugom svjetskom ratu. Osnovala ga je talijanska okupatorska vlast na otoku Rabu srpnju 1942. u blizini Kampora.
Logor je raspušten u rujnu 1943. nakon kapitulacije Italije.
U koncentracijskim logoru nalazilo se približno 15.000 zatočenika, i to uglavnom Hrvata iz Gorskog kotara, Židova i Slovenaca. Od gladi i teških uvjeta života, posebice u zimskom i ljetnom razdoblju umrlo je više od 4000 ljudi (od kojih je imenom i prezimenom poznato tek 1490), uključujući više od 150 djece ispod 15 godina. Daljnih 800 osoba je umrlo pri kasnijim transportu u druge koncentracijske logore u Italiji, kao primjerice u Koncentracijski logor Gonars i Padovu.
Od 15.000 zatočenika život je izgubilo ukupno 20 posto.
Godine 1953. na području logora izgrađen je spomen park, kojeg su izgradili politički zatvorenici na prisilnim radu u jugokomunističkom koncentracijskom logoru na Golom otoku. Projekt za izgradnju spomen groblja napravio je slovenski arhitekt Edvard Ravnikar.
Postojanje tog koncentracijskog logora bilo je malo poznato izvan Hrvatske i Slovenije. Godine 2003., talijanski premijer Silvio Berlusconi izazvao je burne reakcije s izjavom da "talijanski fašisti protivnike svojeg režima nisu slali u koncentracijske logore, nego samo na interni egzil".
Povezani članci
Koncentracijski logor Gonars
fašizam
Izvori
Vanjske poveznice
Webstranica spomen muzeja u Kamporu
spomen muzeja u Kamporu
International Herald Tribune: Survivors of war camp lament Italy's amnesia
http://www.ua-rab.hr/index.php/talijanski-fasisticki-logor-kampor
http://www.cro-eu.com/forum/index.php?action=printpage;topic=1981.0
Kampor
Rab
Hrvatska u Drugom svjetskom ratu | {
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{"url":"http:\/\/libros.duhnnae.com\/2017\/aug7\/150323656947-Implementation-of-superconductor-ferromagnet-superconductor-pi-shifters-in-superconducting-digital-and-quantum-circuits-Condensed-Matter-Supercond.php","text":"# Implementation of superconductor-ferromagnet-superconductor pi-shifters in superconducting digital and quantum circuits - Condensed Matter > Superconductivity\n\nImplementation of superconductor-ferromagnet-superconductor pi-shifters in superconducting digital and quantum circuits - Condensed Matter > Superconductivity - Descarga este documento en PDF. Documentaci\u00f3n en PDF para descargar gratis. Disponible tambi\u00e9n para leer online.\n\nAbstract: The difference between the phases of superconducting order parameter plays insuperconducting circuits the role similar to that played by the electrostaticpotential difference required to drive a current in conventional circuits. Thisfundamental property can be altered by inserting in a superconducting circuit aparticular type of weak link, the so-called Josephson $\\pi$-junction havinginverted current-phase relation and enabling a shift of the phase by $\\pi$. Wedemonstrate the operation of three superconducting circuits - two of them areclassical and one quantum - which all utilize such $\\pi$-phase shiftersrealized using superconductor-ferromagnet-superconductor sandwich technology.The classical circuits are based on single-flux-quantum cells, which are shownto be scalable and compatible with conventional niobium-based superconductingelectronics. The quantum circuit is a $\\pi$-phase biased qubit, for which weobserve coherent Rabi oscillations and compare the measured coherence time withthat of conventional superconducting phase qubits.\n\nAutor: A. K. Feofanov, V. A. Oboznov, V. V. Bol'ginov, J. Lisenfeld, S. Poletto, V. V. Ryazanov, A. N. Rossolenko, M. Khabipov, D.\n\nFuente: https:\/\/arxiv.org\/","date":"2018-05-21 21:20:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.379329115152359, \"perplexity\": 10484.988083028078}, \"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-2018-22\/segments\/1526794864544.25\/warc\/CC-MAIN-20180521200606-20180521220606-00315.warc.gz\"}"} | null | null |
Hafez
(1325/26–1389/90)
Contents
The Life and Poetry of Hafez
BRIEF INTRODUCTION: HAFEZ
PERSIAN LYRICS
THE DIVAN
POEMS FROM THE DIVAN OF HAFIZ
SONNETS FROM HAFEZ AND OTHER VERSES
The Biography
INTRODUCTION TO HAFEZ by Getrude Lowthian Bell
The Delphi Classics Catalogue
© Delphi Classics 2017
Version 1
Hafez
By Delphi Classics, 2017
## COPYRIGHT
Hafez - Delphi Poets Series
First published in the United Kingdom in 2017 by Delphi Classics.
© Delphi Classics, 2017.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of the publisher, nor be otherwise circulated in any form other than that in which it is published.
ISBN: 978 1 78656 210 4
Delphi Classics
is an imprint of
Delphi Publishing Ltd
Hastings, East Sussex
United Kingdom
Contact: sales@delphiclassics.com
www.delphiclassics.com
## NOTE
When reading poetry on an eReader, it is advisable to use a small font size and landscape mode, which will allow the lines of poetry to display correctly.
# The Life and Poetry of Hafez
Shiraz, the sixth most populous city of Iran and one of the oldest cities of ancient Persia — Hafez' birthplace
Shiraz in 1681
# BRIEF INTRODUCTION: HAFEZ
Khwāja Shams-ud-Dīn Muḥammad Ḥāfeẓ-e Shīrāzī, known by his pen name 'Hafez', was born in Shiraz, Iran, in c. 1315 and his parents were from Kazerun, Fars Province. Despite his profound effect on Persian life and culture and his enduring popularity and influence, few details of his life have survived. Accounts of his early life rely upon traditional early biographical sketches, which are generally considered unreliable. From an early age, he memorised the Quran and was given the title of Hafez, which he later used as his pen name. The preface of his Divān, an ancient collection of his poems, in which his early life is discussed, was written by an unknown contemporary whose name may have been Moḥammad Golandām.
Hafez was supported by patronage from several successive local regimes; firstly by Shah Abu Ishaq, who came to power while Hafez was in his teens; then Timur at the end of his life. Though his work flourished mostly during the twenty-seven-year rule of Jalal ud-Din Shah Shuja, it is claimed that Hafez briefly fell out of favour with the ruler for mocking inferior poets. Shah Shuja wrote poetry himself and may have taken the comments personally, forcing Hafez to flee from Shiraz, though no tangible historical evidence can support this.
Many semi-miraculous mythical tales were woven around Hafez after his death. According to one tradition, before meeting his patron Hajji Zayn al-Attar, Hafez had been working in a bakery, delivering bread to a wealthy quarter of the town. This was when he first saw Shakh-e Nabat, a woman of great beauty, to whom some of his most celebrated love poems are addressed. Captivated by her beauty, but knowing that his love for her would not be requited, he allegedly held his first mystic vigil in his desire to realise the union. Then he encountered a being of surpassing beauty that identified himself as an angel and his further attempts at union became mystic — a pursuit of spiritual union with the divine. The Western literary parallel of Dante and Beatrice would serve as a good example.
At the age of sixty, Hafez is said to have begun a Chilla-nashini, a forty-day-and-night vigil, conducted by sitting in a circle he had drawn for himself on the ground. On the fortieth day, he once again met with Zayn al-Attar on what is known to be their fortieth anniversary and was offered a cup of wine.
Hafez was acclaimed throughout the Islamic world during his lifetime, with other Persian poets imitating his work, and he received many offers of patronage from as far as Baghdad to India. His poetry was first translated into English in 1771 by William Jones. The verses would make a lasting impression on such Western writers as Thoreau, Goethe and Ralph Waldo Emerson.
Hafez is best known for his poems that can be described as "antinomian" and with the medieval use of the term "theosophical", indicating mystical work by authors inspired by holy books. He wrote primarily in the literary genre of lyric poetry that is the ideal style for expressing the ecstasy of divine inspiration in the mystical form of love poems. His poems are composed in the literary form of the ghazal, comprising rhyming couplets and a refrain, each line sharing the same metre. A ghazal may be understood as a poetic expression of both the pain of loss or separation and the beauty of love in spite of that pain. The form originates in Arabic poetry long before the birth of Islam. It is derived from the Arabian panegyric qasida. The structural requirements of the ghazal are similar in stringency to those of the Petrarchan sonnet. In style and content, it is a genre that has proved capable of an extraordinary variety of expression around its central themes of love and separation.
Themes of Hafez' ghazals include the beloved, faith and exposing hypocrisy. They often deal with love, wine and scenes in the tavern, all presenting ecstasy and freedom from restraint, whether in actual worldly release or in the voice of the lover speaking of divine love. His influence on the lives of Persian speakers is immense and his poems continue to be employed in Persian traditional music, visual art and calligraphy.
Hafez manuscript of the Divan, 1584
Divan of Hafez, Persian miniature, 1585
A collection of ghazals with illuminated headpiece, University of Pennsylvania
Doublures inside a nineteenth century copy of the Divān of Hafez. The front doublure shows Hafez offering his work to a patron.
# PERSIAN LYRICS
OR, SCATTERED POEMS FROM THE DIWAN OF HAFIZ
Translated by John Haddon Hindley, 1800
## CONTENTS
PARAPHRASES IN VERSE.
GAZEL I.
GAZEL II.
GAZEL III.
GAZEL IV.
GAZEL V.
GAZEL VI.
GAZEL VII.
GAZEL VIII.
GAZEL IX.
GAZEL X.
GAZEL XI.
PARAPHRASES IN PROSE.
GAZEL I.
GAZEL II.
GAZEL III.
GAZEL IV.
GAZEL V.
GAZEL VI.
GAZEL VII.
GAZEL VIII.
GAZEL IX.
GAZEL X.
GAZEL XI.
APPENDIX.
ADVERTISEMENT.
SUPPLEMENT.
VARIOUS READINGS OF THESE GAZELS FROM FOUR MANUSCRIPTS OF THE DIWAN-I-HAFIZ.
## PARAPHRASES IN VERSE.
NUNC et ACHÆMENIO
Perfundi nardo IUVAT, et fide Cyllenea
Levare diris pectora solicitudinibus.
HORAT. Ep xiii. 8.
## GAZEL I.
THIS little poem bears strong allusion to the metaphysical theology of the Musselmans. According to the mystical vocabularies on HAFIZ, by wine (mentioned hereafter in one of these stanzas periphrastically as a flaming ruby), the poet invariably means devotion, and, either from contemplating the beauties of nature at sun-rise, or from having been awakened from sleep (there explained to be meditation on the divine perfections), by the fays of the solar light he may here be supposed to be calling on the religious around him to assist in adoring the great Creator. By the breeze, these interpreters say, is meant an illapse of grace; by perfume, the hope of the divine favour; by the tavern or banquet-house, a retired oratory; by its keeper, a sage instructor; by beauty, the perfection of the Supreme Being; and by wantonness, mirth, and ebriety, religious ardour, and disregard of all terrestrial thoughts and objects. (Asiatic Res ii. 02, iii. 170). This Gazel, therefore, may be conceived to open with the poet's impatience not to lose a moment from elevated abstraction on the Deity, and with his invitation to those who are filled with divine love, to regale themselves and imbibe wine or the devotional spirit, and to those who thirst after wisdom, to offer their vows to Heaven and to give themselves up to the religious enjoyments of celestial and angelical love.
It may be here observed, that, deeply versed as our author appears to have been in these mysterious tenets, he is also recorded to have given public lectures on Muhammadan Theology and Jurisprudence, and even to have composed a commentary on the abstruse and doubtful passages of the Koran. Some of his fragments, or marginal notes, are said to be yet extant. It may be remarked also in this place, that from various passages in his poems, he seems to have indulged a great partiality for a secluded and monastic life. Reviski, indeed, supposes him to have been the senior or prefect of some monastery (monasterii alicukus senior vel praecfectus), though he owns he can produce no positive proof of this (Hoc non ausus sim fidenter asserere). Prooem xxi.
It is not perhaps improbable that this may be also descriptive of the morning worship of the Persians in adoration of the sun and its vernal effects upon the vegetable creation. We are informed from good authority, that the ancient Persians worshipped three times each day; most likely, when the sun was rising above,. and sinking beneath the horizon, and at its meridian.
PARAPHRASE.
In roses veil'd the morn displays
Her charms, and blushes as we gaze;
Come, wine, my gay companions, pour
Observant of the morning hour.
See, spangling dew-drops trickling chace,
Adown the tulip's vermeil face;
Then come, your thirst with wine allay,
Attentive to the dawn of day.
Fresh from the garden scents exhale
As sweet as Edens fragrant gale:
Then come, let wine incessant flow
Obedient to our morning vow.
While now beneath the bow'r full-blown
The rose displays her em'rald throne,
Let wine, like rubies sparkling, gleam
Refulgent as morns orient beam.
Come, youths, perform the task assign'd:
What! in the banquet-house confin'd?
Unlock the door; why this delay.
Forgetful of the dawn of day?
Shall guests at this glad season wait?
Come, keeper, open quick the gate:
'Tis strange to let time pass away.
Regardless of the dawn of day.
Ye love-sick youths, come, drain the bowl:
Thirst ye for wisdom? feast the soul;
To heaven your morning homage pay
With hearts that glow like dawn of day.
Kisses more sweet than luscious wine.
Like HAFIZ, sip from cheeks divine,
'Mid smiles as heav'nly Peries bright,
And looks that pierce like orient light.
## GAZEL II.
IN the following lines the poet calls upon his countrymen to join him in celebrating the Nuruz or vernal season, and alludes to that most favourite fable of Eastern poetry, the Loves of the Rose and the Nightingales. The fondness manifested by the bird to the flower, particularly set its first appearance, seems to have given rise to this elegant allegory, the beauty of which, being founded upon local circumstances and local scenery, cannot certainly in the same manner impress the mind of an who has not, like the Asiatic, been accustomed to witness the curious and interesting feet. The candid reader of these poems will prepare himself to make due allowance for the striking difference of Asiatic manners and opinions, and recollect, that many things which may startle him, were not only countenanced by custom, but sanctioned by religion. The concluding stanza alludes to the prostrate mode of salutation among the Asiatics, touching the dust of the ground with their forehead. Our countryman Herbert, in his account of the diversions at this season, says, that, "at the Nuruz, or spring, they send vests to each other: — then also the gardens are opened for all to walk in. The women likewise, for fourteen days, have liberty to appear in public, and, when loose, like birds enfranchised, lose themselves in a labyrinth of wanton sports. The men also, some riding, some sitting, some walking, are all in one tune, drinking, singing, playing, till the bottles prove empty, songs be spent," &c. "In my life, I never saw people more jocund, nor less quarrelsome." — Herbert's Travels, p. 130.
PARAPHRASE.
Hither bring the wine, boy! hither bring the wine, boy!
For the season approaches, the season of joy.
Let us frolic and revel 'midst gardens and bowers.
Since the roses now bud, and the season is ours.
Let the vows of repentance religion has made,
Be forgotten, and broken beneath the cool shade:
Let us warble, like nightingales, through the gay grove,
And, imbedded in roses, here nestle in love.
Come, replenish, replenish the goblet with wine,
For of happiness lo! the sweet rose is the sign:
While she ripens and blows, your enjoyments pursue,
For anon she will wither and bid us adieu.
To the shade then where roses embowering twine,
Come, repair, quick repair, with thy friend, and with wine;
Let oblivious enjoyment there banish distress,
Whilst we warble, like nightingales, 'midst the recess.
'Tis from HAFIZ the rose claims her tribute of praise,
Let him prostrate before her his soul in soft lays,
Let him bow down his head to the dust at her shrine,
And in strains like the nightingale's hail her divine.
ANOTHER, MORE FREE.
Beds of flow'rs of gayest hue
Beckon us to joy anew:
Bring the heart-inspiring wine,
Let the soul its cares resign;
Lo! the vernal zephyr blows,
Scented with the blooming rose.
Borne on pleasure's new-fledg'd wing,
Loud, like nightingales, now sing
'Mid the cool sequester'd shade,
Nestling in sweet flow'r-beds laid:
There, like them, with love repose,
Chanting to the blooming rose.
In the mirth-enliven'd bower
Wine, convivial songsters, pour:
See the garden's flow'ry guest
Comes in happiness full-drest,
Round us joy's perfum'ry throws,
Offspring of the blooming rose.
Hail! sweet flow'r, thy blossom spread,
Here thy welcome fragrance shed;
Let us with our friends be gay,
Mindful of thy transient stay:
Pass the goblet round; who knows
When we lose the blooming rose?
HAFIZ loves, like Philomel,
With the darling rose to dwell:
Let his heart a grateful lay
To her guardian humbly pay,
Let his life with homage close,
To the guardian of the rose.
## GAZEL III.
THE polished Anacreon of Iran now addresses the minstrel or musician. The mirthful and amorous playfulness of this Gazel, is highly characteristic of the gaiety of Asiatic manners, and must be powerfully insinuating to the convivial and voluptuous Persian.
The learned reader will immediately perceive, that the concluding burthen of every stanza totally baffles all attempts at minuteness of version, and may serve to shew the richness of a dialect which can so elegantly adapt the same simple expression to so many varied meanings. He will also notice, that the last stanza is perhaps more dilated than the original will altogether fully authorize; but, we trust, the annexed Paraphrases in Prose, will compensate, in some degree, for these and similar liberties.
PARAPHRASE.
Minstrel, tune some novel lay,
Ever jocund, ever gay;
Call for heart-expanding wine,
Ever sparkling, ever fine.
Sit remov'd from prying eyes;
Love the game, the fair thy prize;
Toying snatch the furtive bliss.
Eager look, and eager kiss;
Fresh and fresh repeat the freak.
Often give, and often take.
Canst thou feed the hung'ring soul
Without drinking of the bowl?
Pour out wine; to her 'tis due:
Love commands thee — Fill anew;
Drink her health, repeat her name,
Often, often do the same.
Frantic love more frantic grows,
Love admits of no repose:
Haste, thou youth with silver feet,
Haste, the goblet bring, be fleet;
Fill again the luscious cup,
Fresh and fresh, come, fill it up.
See, you angel of my heart
Forms for me, with witching art,
Ornaments of varied taste,
Fresh and graceful, fresh and chaste.
Gentle Zephyr, should'st thou roam,
By my lovely charmer's home,
Whisper to my dearest dear,
Whisper, whisper in her ear,
Tales of HAFIZ; which repeat,
Whisper'd soft, and whisper'd sweet;
Whisper tales of love anew,
Whisper'd whisphers oft renew.
## GAZEL IV.
THIS Gazel opens with the artless effusion of an extravagant Amoroso. Fancy pictures to him his mistress passing, as it were, in review before him; and Affection seizes the gratifying moment to turn even defects into charms, and to consider the very minutest thing appertaining to her as invaluable: Nay, he goes so far as to declare, that he would barter away even the renowned Bokhara and Samarcand, the capital cities of and Taimur, were they his, for the mere mole on the cheek of his lovely fair one. His favourite and native Shiraz, its cooling fountains and its rosy bowers, the gay and sprightly damsels that sport within and around it, characterised by the poet's most choice and glowing epithets, who have plundered him of his peace of mind, and whom he compares to Janissaries rushing upon their predatory banquet, seem all to involve him in the happiest of reveries. Yet his powers of praise still fail him. Charms, so all-perfect as these, are too exquisite, too superlative to be described. His love, again, is defective, incomplete, and requires to be ratified by possession. It were just as probable to hope to improve, the finest natural complexion by cosmetics, or the meretricious embellishments of art, as to attempt to heighten such consummate beauty by any thing so feeble as verbal delineation. A change in the tide of his thoughts, therefore, becomes necessary. Accordingly, Epicurean-like, he calls for the minstrel to divert, and for wine to drown his perplexities. He ridicules the casuistry and prophetical folly of prying into the events of futurity, and pronounces it a search always abstruse, presumptuous, and fruitless. Yet all this cannot turn aside the current of his passion: it rather tends to aggravate, than to relieve it; and, by reminding him of a chapter in the Koran, (Joseph, c. 12), brings Zuleikha's case to his recollection, and hints to him, that there did once exist a love, which even overpowered all virtuous considerations. He once more, therefore, cherishes his passion. The beloved object is pathetically conjured by him to attend to the counsels of prudence; to bear in mind, that, in spite of all the suggestions of malice, he still loves her: that he petitions Heaven to preserve her; and that, if she reflects only for a moment on the suavity of her own innate disposition, every expression of malevolence must appear to her unnatural, unbecoming, and detractive from her beauty, as much so as it would be to attribute to her the poison of the scorpion. This thought he seems to prize as sufficiently dazzling to constitute the concluding bead of this melodious string of pearls; and, calling upon himself, in the triumphant pride and rapture of the moment, to sing this Gazel sweetly, the elated and self-applauding bard boasts of his composition as a paragon of harmonious brilliancy, studded and bespangled with poetical beauties, outshining even the Pleiades among the stars of Heaven.
PARAPHRASE.
1. Fair maid of Shiraz, would'st thou take
My heart, and love it for my sake,
For that dark mole my thoughts now trace
On that sweet cheek of that sweet face,
I would Bokhara, as I live,
And Samarcand too, freely give.
2. Empty the flagon, fill the bowl,
With wine to rapture wake the soul:
For, Edens self, however fair,
Has nought to boast that can compare
With thy blest banks, O Rocnabad!
In their enchanting scen'ry clad;
Nor ought in foliage half so gay
As are the bow'rs of Mosellay.
3. Insidious girls with syren eye,
Whose wanton wiles the soul decoy,
By whose bewitching charms beguil'd
Our love-smit town is all run wild,
My stoic heart ye steal away
As Janissaries do their prey!
4. But, ah! no lauréat lovers' praise
The lustre of those charms can raise:
For, vain are all the tricks of art,
Which would to nature ought impart;
To tints, that angelize the face,
Can borrow'd colours add new grace?
Can a fair cheek become more fair
By artificial moles form'd there?
Or, can a neck of mould divine
By perfum'd tresses heighten'd shine?
5. Be wine and music, then, our theme;
Let wizards of the future dream,
Which unsolv'd riddle puzzles still,
And ever did, and ever will.
6. By Joseph's growing beauty mov'd,
Zuleikha look'd, and sigh'd, and lov'd,
'Till headstrong passion shame defy'd,
And virtue's veil was thrown aside.
7. Be thine, my fair, by counsel led,
At wisdom's shrine to bow thy head;
For, lovely maids more lovely shine
Whose hearts to sage advice incline,
Who than their souls more valued prize
The hoary maxims of the wise.
8. But, tell me, Charmer, tell me why
Such cruel words my ears annoy:
Say, is it pleasure to give pain?
Can sland'rous gall thy mouth profane?
Forbid it, Heav'n! it cannot be!
Nought that offends can come from thee:
For, how can scorpion venom drip
From that sweet ruby-colour'd lip,
Which, with good nature overspread,
Can nought but dulcet language shed?
9. THY Gazel-forming pearls are strung,
Come, sweetly, HAFIZ, be they sung:
For, Heav'n show'rs down upon thy lays
Thoughts, which in star-like clusters blaze.
## GAZEL V.
NOT Petrarch himself could approach his favourite Laura with a more extravagant and circumstantial address than this of the idolatrous lover, HAFIZ. He represents his mistress as one of the Temple Idols, loaded with trinkets and brilliant ornaments, and himself under the character of her votary or worshipper, but not without glancing, in the outset, at his past experience of her hard-hearted disposition. He afterwards goes on, painting her as a celestial being kindling his passion into the most flaming and enthusiastic adoration of her personal attributes; and (perhaps in allusion to the peplus or highly-decorated tapestry with which the images are commonly adorned on the great festivals) he wishes himself within the sphere of the fancied nimbus or glory of his belle Idol; or, more directly to meet his idea, equally in possession of her charms with such an embracing veil. Pursuing the same emblematical similitude, he declares his reason and his religion to be lost and absorbed in the divine contemplation of the angelical charms of his Idol; and observes, that no-thing can cure the frailty and infirmity of his love-sick soul, but the gracious aid of her celestial, of her healing love, and compassionate indulgence.
PARAPHRASE.
That Idol with ear-drops so bright,
And whose heart is obdurate as stone,
Of reason has robb'd me outright,
Of myself: for, her captive I'm grown.
No thought the keen glance can pourtray,
Or the mien of my Idol so fair,
No angel such charms can display,
She's an Idol beyond all compare.
Her company breathes soft delight;
Neatly veil'd in a robe she is drest:
The moon cannot shine half so bright;
Love his altar has plac'd in her breast.
Her passion my soul sets on fire,
Thro' my heart I now feel the flame move,
I boil, I boil o'er with desire,
I am all in a ferment of love.
Oh! were she but clasp'd in these arms!
Oh! how happy would then be my case!
No vest, that infolds her rude charms,
Could enjoy, like my heart, the embrace.
Let death close my eyes when it may,
O'er my love she shall still bear controul;
My body may moulder away,
Yet she'll ne'er be forgot by my soul.
Her bosom and shoulders I view —
Yes — again, and again, and again:
My reason then bids me adieu,
My religion grows fruitless and vain.
Religion! — O HAFIZ! how vain!
For thy cure from her mouth thou must sip;
A kiss must relieve thee from pain,
A sweet kiss from her honey-stor'd lip.
## GAZEL VI.
THE sprightly turn of the interrogatory at the conclusion of each distich, contitutes the leading peculiarity of this Gazel, which (for a reason similar to that assigned in Gazel III.), we can hardly hope to imitate with any degree of literal nicety. The poet appears to have quarrelled with the object of his passion; and there seems to have been some interruption to the connection, or at least considerable coolness betwixt them. He apparently offers these effusions as a tributary overture at reconciliaton: and, though he does not stoop to make too great advances, by unbosoming himself over-freely, yet nature speaks, through the veil which his art has thrown over it, sufficiently to shew the full amount of his feelings. He confesses that he has felt the painful anxietude of Love, yet he declines to give a minute description of it: though he owns that his hours have been empoisoned by the effects of absence, yet he is averse to enter into a detail of their influence upon him: even the name of his mistress, the recollection of moments of melting tenderness, soft endearments, goading reproaches, and the afflictive pangs of absence, are circumstances which, however pleasant or painful, seem only to be brought forward in order to evince that they have merely a negative claim to his attention. He, however, sums up his feelings in one word, by declaring that his love has arrived at that pitch of anxiety which it is in vain to ask him to describe.
PARAPHRASE.
Tho' I have felt a lover's woes,
These ask me not to state:
Tho' absence poisons my repose,
This bid me not relate.
Far, far I search'd the world around
For her I love so well.
My charmer's name's a magic sound,
Which ask me not to tell.
My eyes her lovely footsteps trace,
My tears the track bedew;
Ask not the secret of my case,
To whom these tears are due.
No longer since than yesternight,
I heard her tongue declaim,
In accents which, in love's despite,
Oh! ask me not to name.
Why bite that lip? Why hints suggest,
As if I could betray?
A rubied lip, 'tis true, I've prest;
But whose — don't bid me say.
Absent from thee, forlorn, I moan,
Affliction haunts my cot;
But what I bear thus all alone
Ah! prithee ask me not.
HAFIZ, whose heart hath known no woe,
Now feels it in excess;
Ask not his boundless love to know,
'Tis what he can't express.
## GAZEL VII.
HAFIZ, no longer able to endure the painful anxietude occasioned by the absence of his mistress, expatiates upon the effect it has produced upon his mind, and its afflictive operation upon his general feelings. Expostulating on her cold insensibility and total inattention to his just complaint, he describes himself as the victim of her indifference. Despondent, as he has been for some time, he begins entirely to despair, and expects to die of a broken heart.
PARAPHRASE.
Ev'ry moment thy absence I mourn,
But my sighs and my tears are in vain,
Since no zephyr proclaims thy return,
And no zephyr announces my pain.
Night and day I'm abandon'd to grief,
And what truce can extirpate my woes?
Far from thee I can find no relief,
Far from thee can enjoy no repose.
Ah! what else can I do but lament,
When I'm doom'd such affliction to know,
Such that, were I dispos'd to torment,
I should wish to befall my worst foe?
Oh! what sorrow has gush'd from these eyes
Since my fair from my presence has fled!
How my breast has been haunted with sighs!
With what wounds, O my heart! hast thou bled!
When I think of thee, forth the tears start,
From my eye-lashes trickling they fall;
'Tis affection that bids them depart,
It is thoughts which thy image recall.
Say, shall HAFIZ to love fall a prey?
Still shall grief day and night drown his eye?
Shall his soul with despair pine away,
While from thee he obtains not a sigh?
## GAZEL VIII.
THE Poet, in this Gazel, bids the Zephyr bear to the ear of his mistress his complaint of unkind treatment from her, whose coyness and timidity are happily characterised under the form of that delicate and graceful animal the — or Fawn, an image peculiarly tender among the Greek and Roman, as well as among the Asiatic poets. The Rose and Nightingale are here again allegorically alluded to in a manner that, however repeated, still tends to delight the imagination of his Persian readers. He afterwards goes on to hint how much the charms of beauty are heightened and enhanced by a gentle and kind demeanour, and intimates that every being in the creation is delighted with his harmonious strains, except the object of his love; and, that the whole celestial choir, led on by Zorah (the planet Venus), dances in tuneful concert to the melody of his lays.
PARAPHRASE.
O! go, thou kind Zephyr, go, speed thro' the lawn,
And say with a sigh to that diffident fawn,
For her 'tis I wander thro' thicket and grove,
Thro' craggs of steep mountains in quest of her love;
'Tis she that gives charms to the desert so drear,
And makes the rude forest like Eden appear:
Go on still to please, with long life be thou crown'd; —
But, why, thou dear vender of sweetness around,
Ah! why is thy songster thus slighted, O say,
While absent he warbles to thee his soft lay?
One morsel of pity thy parrot O give,
One sigh as a sweetmeat, or else he can't live.
The Rose of her beauty is surely grown vain,
To treat the fond Nightingale thus with disdain!
Charms win by good nature, but not by false glare,
A bird on his guard no decoy can ensnare.
While sipping thy wine thou coquettest so gay,
Think of him who is sighing his hours away!
'Tis strange in such angel-fac'd beauties to find
The heart so obdurate, so fickle the mind!
How perfect, how faultless thy charms would appear,
Were constant thy love, thy affection sincere!
Can HAFIZ be scorn'd? can his lays thy ear tire, When the list'ning planets their sweetness admire; When, by Zorah led on, the celestial train In unison dance to his heavenly strain?
## GAZEL IX.
THE faithful HAFIZ addresses his mistress with the strongest professions of fidelity and constancy — declares himself to have been enamoured of her beauty even from his earliest childhood — asserts the durability and inextinguishable ardency of his passion — and finally concludes by cautioning all mankind against the caprices of the sex, and the dangerous consequences of falling in love, pointing himself out, at the same time, as a striking example to deter others from being duped and driven to the same state of mental distraction.
PARAPHRASE.
Nothing, no; nothing from my heart shall tear
That damsel's image, to my soul so dear;
No, thou most graceful Cypress of the grove,
There grows thy root, deep-planted by my love:
Nor shall stem Fate, in grim misfortune drest,
E'er scare thy lips' memorial from my breasts
In infant life thy locks my passion mov'd,
And something early told me that I lov'd: —
The league, which then with love and them I made,
Shall ne'er by treach'rous mem'ry be betray'd.
With unborn time the innate fondness rose,
And shall with deathless time expiring close:
All but that love may quit my loaded heart,
But that, O! never, never shall depart:
Nought shall destroy it, nought its force controul;
It clings so close united to my soul,
That from this body sever'd were this head,
E'en then my unchang'd love would not be dead.
But, tho' my wounded heart the fair pursues,
Pity my feeble frailty will excuse;
Sick is my soul, and why not seek to find
Some bland restorative to ease my mind?
Whoe'er from wild distraction would be free,
And 'scape the frenzy which thus preys on me,
Let him, by HAFIZ warn'd, avoid his fate,
And shun the sex, lest soon it be too late.
## GAZEL X.
IN this Gazel the Zephyr is again called upon to be the messenger of love, and to waft odours, sighs, and even dust, from the feet of his mistress, to stop the tears of the disconsolate HAFIZ. He considers himself as a mere itinerant outcast, a wandering pilgrim, a poor destitute beggar, craving as a deed of charity that she would return, and strolling about he knows not where, forlorn, as it were, and bewildered in a desert, looking at every shadow for a glimpse of her, at the same time elevated with the hope, and trembling with the fear, of being or of not being successful in his pursuit. He declares, that however callous or insensible her heart may be to his affection, yet such is his extreme regard for her, that he would not even barter away a hair of her head to receive the whole universe in exchange for it. He emphatically concludes by asking, what is the advantage in having a heart emancipated from care, when the soft and tender petition of a poet only tends to make him still more the slave and vassal of her of whom he is enamoured?
PARAPHRASE.
Zephyr, should'st thou chance to rove
By the mansion of my love,
From her locks ambrosial bring
Choicest odours on thy wing.
Could'st thou waft me from her breast
Tender sighs to say I'm blest,
As she lives! my soul would be
Sprinkled o'er with extasy.
But, if Heav'n the boon deny,
Round her stately footsteps fly,
With the dust that thence may rise,
Stop the tears which bathe these eyes.
Lost, poor mendicant! I roam
Begging, craving she would come:
Where shall I thy phantom see,
Where, dear nymph, a glimpse of thee?
Like the wind-tost reed my breast
Fann'd with hope is ne'er at rest,
Throbbing, longing to excess
Her fair figure to caress.
Yes, my charmer, tho' I see
Thy heart courts no love with me,
Not for worlds, could they be mine,
Would I give a hair of thine.
Why, O care! shall I in vain
Strive to shun thy galling chain,
When these strains still fail to save,
And make HAFIZ more a slave.
## GAZEL XI.
THE imagination of the Poet, after dwelling with admiration and enthusiasm on the fine majestic figure and fascinating deportment of his mistress, bursts forth at large into a metaphorical and glowing description of her transcendant beauties. He compares them, according to the style and imagery of the Asiatics, to admired objects in nature, and, with a figurative boldness of expression, delineates their impressive effects upon his senses. He attributes to the magic influence of her omnicreative presence in his mind, all the elegant tints, colouring, embellishments, and peinturesque beauties, with which the flowery repository of his imagination is decorated and stored. After consoling and regaling his mind with the delicious and animating sensations arising from the recollection of her former friendship, he professes his unshaken determination not to give way to reflection, but to risque, at all hazards, the recovery of her society, and never to abandon his project, however peril or despair may thwart him in the pursuit of his object.
PARAPHRASE.
Yes, thy form, my fair nymph, is of elegant mould,
And proportion'd with exquisite grace;
How transporting thy shape, and thy looks to behold,
As sly wantons young Love in thy face.
Like the bloom of the rose, when fresh pluck'd and full-blown,
Sweetly soft is thy nature and air:
Like the beautiful Cypress in Paradise grown,
Thou art ev'ry way charming and fair.
Thy arts so coquettish, thy feigned disdain,
The soft down and sweet mole of thy cheek,
Eyes, and eye-brows, and stature my senses enchain,
While I gaze, not one word can I speak.
When my mind dwells on thee, what a lustre assume
All the objects which fancy presents!
On my memory thy locks leave a grateful perfume,
Far more fragrant than jasmine's sweet scents.
In this wild maze of love is no avenue found
To escape from the torrent of grief,
Yet my heart still emerges, nor fears to be drown'd,
While thy friendship affords it relief.
Should I chance in thy presence to sink and expire,
And before thee to reach my last goal,
Let me look on thy cheek, and in peace I'll retire,
Nor repine when I give up my soul.
Though to roam 'mid the desert and search for thee there,
Nought but hazard and danger proclaim;
Yet HAFIZ shall roam, and tho' mock'd by despair,
Never cease to call out on thy name.
## PARAPHRASES IN PROSE.
## GAZEL I.
[This Gazel is from the History of the Persian Language in the Life of Nader Shah, 8vo. Lond. 1773, p. 179, by the first Orientalist, in point of taste and research, that ever graced any country, the late Sir William Jones, whose numerous and inimitable Translations from Asiatic Authors, pre-eminently entitle him to the following just and characteristic encomium from Ausonius:
Hujus fontis aquas peregrinasferre per urbesy Unum pra reliquis solitus potare Choaspen.
Panegyr. Fontis Burdigal, v. 27.]
1. THE dawn advances, veiled with roses: —
Bring the morning draught, my friends, the morning draught.
2. The dew-drops trickle over the cheek of the tulip:
Bring the wine, my dear companions, bring the wine.
3. A gale of Paradise breathes from the garden: —
Drink, then, incessantly the pure wine.
4. The Rose spreads her emerald throne in the bower;
Reach the liquor that sparkles like a flaming ruby.
5. Are they still shut up in the banquet-house?
Open, O thou keeper of the gate!
6. It is strange, at such a season,
That the door of the tavern should be locked.
7. Oh! hasten, O thou who art in love, drink wine with eagerness;
And ye, who are endued with wisdom, offer your vows to Heaven.
8. Imitate HAFIZ, and drink kisses sweet as wine
From the cheek of a damsel fair as a virgin of Paradise.
## GAZEL II.
1. BOY, bring the wine, — for, the season of Roses is arrived, —
That we may break our vows of repentance again amidst beds of Roses.
2. Jovial, and singing aloud, let us enter the bower;
Like Nightingales let us sink at once into nests of Roses.
3. In the recess of the garden quaff the goblet of wine;
For, the signs of happiness appear also at the command of the Rose. -
4. The Rose is arrived in the garden; be not too confident of the time of her sojourn:
Seek a friend, and wine, and the palace of cultivated rose-bowers.
3. HAFIZ, thou longest after the company of the Rose, like the Nightingales:
Devote thy soul a ransom for the dust of the walk of the Keeper of the Rose-garden.
## GAZEL III.
1. O MINSTREL with a sweet voice! begin an air that is fresh and new:
Call for heart-expanding wine fresh and fresh.
2. Sit down from prying eyes and enjoy thy mistress, as a game, in private:
Snatch eager kisses from her — fresh and fresh.
3. How canst thou eat the bread of life without drinking wine? Quaff wine to her dear remembrance again and again.
4. Ö cup-bearer with legs of silver, I am intoxicated with the love of thy beauty! —
Quick fetch the cup, that I may fill it again and
5. My heart-ravishing angel makes for me Ornaments of various hues and odours afresh and —
6. O! gentle Zephyr, when thou passest by the habitation of my Fairy, Afresh and afresh tell her, in whispers, the tale of HAFIZ.
## GAZEL IV.
[Chiefly from the Persian Grammar by Sir William Jones, p. 129, third Edit. Lond. 1783.]
1. IF that lovely maid of Shiraz would accept my heart, For the black mole on her cheek I would give and Bokhara.
2. Boy, bring me the wine that remains; for, in Paradise thou wilt not find The banks of the fountains of Rocnabad and the rosy bowers of Mosella.
3. Alas! these wanton nymphs, these insidious fair ones, whose beauties raise a tumult in our city, Have borne away the quiet of my heart as Tartars their repast of plunder.
4. Yet the charms of our darlings have no need of our imperfect love:
What occasion has a face naturally lovely for perfumes, paint, moles, or ringlets?
5. Talk to me of minstrels and of wine; and seek not to disclose the secrets of futurity:
No one, however wise, ever has, or ever will discover this enigma.
6. I very well know from that daily increasing beauty which Joseph had, That a resistless love tore away from Zuleikha the veil of her chastity.
7. Attend, O adorable object! to prudent counsels: for, youth of a good disposition Love the advice of the aged better than their own souls.
8. Thou hast spoken ill of me, yet I am not offended: may God forgive thee! — Thou hast spoken well: —
But do bitter words (the answers of the scorpion), become a lip like a ruby, shedding nothing but sweetness, (sugar)?
9. THOU hast composed thy Gazel, and strung thy pearls — Come, sing them sweetly, O HAFIZ!
For, Heaven has sprinkled over thy poetry the clearness and beauty (shining circle) of the Pleiades.
## GAZEL V.
1. THAT Idol with heart of stone and ear-ornaments of silver Hath deprived me of fortitude, power, and reason:
2. For, She is an image of piercing looks, delicate mien, in beauty like a Fairy, A soft companion, bright as the moon, lovely, and robed in the graceful tunick.
3. From the raging fire of her violent love I am continually ebullient (boiling over), like a culinary vessel (pot.)
4. Might I take her in my embraces, like the garment that enfolds her, My heart would be at rest on becoming near her as her nearest vestment (chemise.)
5. Were my very bones even to putrefy, The love I have for her could not be forgotten by my soul.
6. Her bosom and shoulders, her bosom and shoulders, her bosom and shoulders Have deprived me of my heart and religion, my heart and religion:
7; Thy cure, thy cure, O HAFIZ!
Is her honied lip, her honied lip, her honied lip.
## GAZEL VI.
1. I have borne the anguish of love, which ask me not to describe: I have tasted the poison of absence, which ask me not to relate.
2. Far through the world have I roved, and at length I have chosen A sweet creature (a ravisher of hearts), whose name ask me not to disclose.
3. The flowings of my tears bedew her footsteps In such a manner as ask me not to utter.
4. On yesterday night from her own mouth with my own ears I heard Such words, as pray ask me not to repeat.
5. Why dost thou bite thy lip at me? What dost thou not hint ( that I may have told)?
I have devoured a lip like a ruby: but whose ask me not to mention.
6. Absent from thee, and the sole tenant of my cottage, I have endured such tortures, as ask me not to enumerate.
7. Thus am I, HAFIZ, arrived at that pitch (station, experience, or extremity) in the ways of Love, Which, alas! ask me not to explain.
## GAZEL VII.
1. EVERY moment I complain aloud on account of thy absence; But, what if the Zephyr refuses to convey my sighs and complaints to thee?
2. Night and day do I grieve bitterly, and (though I should not grieve), though there should be an interval from grief, When I am thus far from thee, how can my heart be at ease?
3. What can I do but weep, and sob, and lament, When I am reduced to such a state from thy absence, that I should wish an enemy placed in.
4. Since thou hast estranged thyself from my sight, my heart has been consumed with affliction.
Ah! how many are the fountains of blood, that it has opened to me in my eyes!
5. Whenever my poor heart utters its complaints for thy absence A thousand drops of blood trickle down from the root of each eye-lash. —
6. Thus is the distracted HAFIZ immersed in the remembrance of thee day and night: —
Whilst thou art perfectly (free) at ease about thy brokenhearted slave.
## GAZEL VIII.
1. O Zephyr, say with mildness to that delicate Fawn, That she maketh us love to dwell in the hills and desarts.
2. How happens it, that she who dispenses sweetness to all around her (the vender of sugar), — whose life be long! —
Has no sweet morsel of regret for the absence of her poet with a dulcet voice (her parrot feeding on sugar).
3. Perhaps, O Rose, vanity on account of thy beauty will not permit thee To make even a poor inquiry after the fond Nightingale.
4. It is possible to ensnare a prudent heart with softness and gentleness; But a cautious and wise bird is not to be taken with a trap or with a gin.
5. When thou sittest with thy companion, and pourest out the pure wine, Take thought of thy lover, that measurer of the desart (who is lost in the desert of absence).
6. I know not why these damsels have no (tincture) feelings of benevolent sympathy, Damsels, who have black eyes, are tall as Cypresses, and beautiful as the Moon.
7. I can only mention one defect in thy charms; —
Thy fair countenance hath not the hue (disposition) of Love and Constancy.
8. It is not surprising if, in the Heavens, from the strains of HAFIZ, Zorah lead the planets in dance to his melody.
## GAZEL IX.
1. NEVER shall thine image be obliterated from the tablets of my heart and soul:
Never shall that stately moving Cypress (pompatice quit my remembrance:
2. No adverse fortune, nor the angry Fates, shall cause The (imagination) memorial of thy lips to vanish from my distracted brain.
3. From my earliest infancy (eternity without beginning) has my heart been bound in alliance with (the points of) thy ringlets; And 'till my last breath (eternity without end) the contract shall not be broken.
4. Every thing, except the (load of) Love I cherish for thee in this poor heart of mine, May be driven from my affections; but, that shall never go.
5. The love of thee has taken so strong a hold upon my heart and upon my soul, That, though my head were separated from my body, my love for thee would still survive.
6. If my heart does thus pursue the darling sex, it is excuseable, It is sick; and, what can it do but seek for a remedy?
7. Whoever desires not to have his brain turned, like HAFIZ, Let him not give his heart to the fair, nor court their society.
## GAZEL X.
1. ZEPHYR, shouldest thou chance to pass through the region where dwells my mistress, Bring me a profusion of odours from her ambrosial ringlets.
2. By her life! would I sprinkle my soul with sweetness, Wouldest thou but bring me a message from the bosom of my fair one.
3. But, if Heaven should not so far favour thee, Bring dust to these two eyes from the mansion of my beloved.
4. I am miserably destitute, and I am wishing for her arrival —
Alas! bewildered wretch that I am!
Where shall I behold with my eyes the phantom of her countenance?
5. My elevated heart trembles like the reed Through the desire of possessing my fair one, who is like a pine-tree in form and stature.
6. Although this lovely charmer has no esteem for me, I would not exchange a hair of her dear head to receive the whole world in return.
7. Where is the advantage of having his heart (free) emancipated from the pondage of care, When the suaviloquent HAFIZ exists only the slave and vassal of his beloved?
## GAZEL XI.
1. YES, thy whole shape is delicately proportioned; everyplace about thee is exquisite:
My heart is exhilarated with thy sweet and honied blandishments.
2. Like the fresh leaf of the Rose, thy nature is gentleness:
Like the Cypress of the Garden of Paradise (or Eternity) thou art every where (from head to foot) charming.
3. Thy coquetish arts and feigned disdain are sweet; the down and mole of thy cheek are agreeable:
Thy eyes and eyebrows are languishingly brilliant; thy height and stature are lovely.
4. The bower of my ideas is filled by thee with pictures and ornaments:
The odour of my heart becomes fragrant from thy jasmine-scented locks.
5. In the road of Love there is no escape (passage) from the torrent of affliction; But I have (made myself happy) consoled myself by thy friendship.
6. In thy presence I expire: but in that extremity Anguish becomes sweet to me from the smile (cheek) of thy (gracefulness) bright countenance.
7. Though to search for thee in the desert be on all sides dangerous, The despairing HAFIZ proceeds with chearfulness to call upon thy name.
## APPENDIX.
## ADVERTISEMENT.
A Manuscript of the Works of HAFIZ, purchased into the Chetham Library at Manchester after these Papers were prepared for the Press, has given us an opportunity of exemplifying, in part, the remarks which occur in the Notes at pages 15 and 19 of our Introductory Observations; and the Reader, who will only consider that the Persian Gazel is invariably governed by the terminating Rhyme, and that every Poem in the Diwan must fall under the classification of its first final letter, will easily judge of our surprise on comparing the short Catalogue of MENINSKI with this Manuscript. Every step during this comparison gave us occasion to lament, that that truly great and indefatigable Scholar had not left a more adequate solution for our doubts on this subject, and, by the addition of such notices as would have identified each Gazel, enabled us at all times to ascertain its reputed genuineness. This is the more unfortunate, as a Manuscript containing the Shereh of Sudi was constantly used by MENINSKI. We trust, therefore, that the following Catalogue ( the value of which must be proved hereafter on collation with other copies) will not prove, for the present, unacceptable to the Reader of HAFIZ. It must, at all events, prevent his feeling the similar want of a standard reference: and he may remain assured, that it represents with faithfulness the contents of an approved Manuscript of the Diwan, which professes itself to have been transcribed with great care and critical accuracy under the eye of a learned European, and collated with a considerable number of the best Manuscripts in Higher Hindustan, and, under such circumstances, may be expected to aid, in a secondary-view, that appreciation of internal evidence, by which all truly celebrated compositions will ever demonstrate their originality.
NUMBER OF GAZELS IN THE DIWAN, according to the Manuscript MENINSKI, as compared with that of the CHETHAM LIBRARY.
CATALOGUE OF THE DIWAN,
WITH THE NUMBER OF DISTICHS, AND EXTRACT OF THE FIRST HALF BEIT, OF EACH GAZEL, ACCORDING TO
THE ARRANGEMENT
OF
THE CHETHAM MANUSCRIPT.
NOTICE FROM THE CATALOGUE OF THE CHETHAM LIBRARY.
With the additional Odes, Elegies, Chronographs, and other detached pieces. The whole collated with upward of FORTY other Manuscripts by a Gentleman Lucknow."
N. B. The marginal Notices refer to Gazels already printed.
## SUPPLEMENT.
THE following Greek and Latin Versions are added, for the satisfaction of the learned Reader. The Eidyllion by Sir WILLIAM JONES, and the two Odes by Baron REVISKI, are already well known. For the Elegy in Latin Hexameters and Pentameters, the Author is obliged to an old School fellow and College Friend, by whom it was imitated from the English Prose Version, and who has kindly permitted him to produce it as a proof of the Latin language being more fitted than the English, on account of the shackles in rhyme, to express the homotonous repetitions which terminate the stanzas of these Gazels. See the Introductory Remarks.
ODE
Shirazi gracilis puer —
Si votis facilem se dederit meis,
Naevo uni in facie dabo
Samarcandœ opulentam & Bocharee plagam.
Infunde heus! remanens merum,
Non cælo inveniam floriferos aquae
Rucnabadi ego margines,
Musellœque virens perpetuo nemus.
Heuheu! pacem animo meo —
Lull blanditiis quippe procacibus —
Sic totam eripiunt, ferox
Frædatas veluti Turca rapit dapes.
Quid nostrâ face languid!
Augustum pueri contulerim ad decus?
Quid form! facies nitens
Mendaci roseas fuco eget ad genas?
Cantorem memora et merum,
Sed fati ambiguas quaere parum vices,
Quippe aenigmatis istius
Nulla est ingenio aut arte solutio.
Quid possit vehemens amor
Josephe indicio rem teneo tuo,
Te viso male sauciam
Zelicham domitus deserult pudor.
Praebe aurem monitis meis
Attentam, o animae dulce decus meae!
Nam gaudent juvenes probi
Maturis gravium consiliis senum.
Tu convicia quod mihi
Dicis, do veniam, Di tribuant parem,
Nec succenseo, sed reor
Non os verba decere aspera melleum.
Haphyz! quot tibi carmina
Tot gemmae ambrosio desiliunt labro,
Mulce cantibus aethera, ut
Spargant grata tuis pleiades modis.
ELEGIA.
Flammis ah! pereo cœcis, mea cara Ferishta,
Quas nunquam ut referam, cara roga.
Te ploro absentem lachrymis, mea cara
Ferishta, Ne quantum ploro, cara Ferishta, roga.
Erravi totum longe lateque per orbem,
Ne tu, quam propter, cata roga.
Selecta ex aliis passim est mea nympha venusta,
Ne, quam selegi, cara Ferishta, roga.
Oh! forma angelica! oh facies divina puellae!
Cujus ne nomen, cara Ferishta, toga.
Signa pedum lachrymis noster aectatur ocellus: —
At ne, cujusnam, cara Ferishta, roga.
Nocte tuas voces hesternâ his auribus hàusi,
Quas minime ut memorem, cara roga.
Cur torvè labia obmordes? cur suggeris ista?
Quae nunquam lit repetain, cara Ferishta, roga.
Agnosco en! supplex me basia mille dedisse;
Ne cuinam ista dedi, cara Ferishta, roga,
Nostra domus luctu plena est, absente Ferishta,
Ne, cur sic plena est, cara Ferishta, roga.
Infelix Hafiz! quanta infortunia amantis! —
Sed quanta heu! nunquam, cara, roga.
Extremum ah! nequeo verbis depingere amorem,
Ne plura idcirco, cara Ferroga.
ODE
Heus illi tenero dic, Zephyre, hinnulo
Amenti similis quod miser illius
Cæco captus amore
Erro in montibus aviis.
Dulcis venditor heu! pusio sacchari,
Quem servent superi, cur manet immemor?
Nec vel pauca diserto.
Dat bellaria Psittaco?
O quæ punioeo flore nites, rosa!
Estne hæc forma tuæ causa superbiæ?
Flagrantis Philomelas
Quod nil sollicitet te amor.
Incautas volucres retibus et plagis
Venamur; sed enim pulchra animi indoles,
Virtutesque sagaces —
Captivare valent viros.
Si quando eximio cinctus amantium
Coetu, pura libes pocula nectaris,
Esto tunc et amici
Tantillum memor exulis.
Ignoro tereti corpore suavium
Et nigris oculis dulce tuentium,
Cur tam impervia amori
Sint præcordia virginum?
Expertem vitii confiteor tuam
Augustam speciem, sed cuperem magis
Formosi decus oris
Demptâ mente volubili.
Quid mirum! empyreis si tua sedibus
O Haphyz recitent carmina, sique ea,
Lætam aggressa choream
Psallens subsiliat Tenus?
## VARIOUS READINGS OF THESE GAZELS FROM FOUR MANUSCRIPTS OF THE DIWAN-I-HAFIZ.
A Very learned Professor, and great ornament of the University of Oxford, having most obligingly furnished the Author with the following Collations of these Gazels, from three Manuscripts of HAFIZ, in the Bodleian Library, he has been tempted to add the varieties afforded by them to this part of the work, along with those which occur in the Chetham Manuscript. The examination of these readings he leaves to the Persian Scholar, forbearing to encumber a work, already too crouded, with any farther observations.
A. — Represents the Bodleian Manuscripts.
A. 1. Manuscript of Hafiz. Uri CLIV, and Laud, A. 52.
2. Persian text of Hafiz to the letter & with Sururi's. commentary in Turkish. Uri cxxxiv-v, and Laud, B. 38, 39.
B. — The Chetham Manuscript. See Appendix.
# THE DIVAN
Translated by Herman Bicknell
## CONTENTS
NOTE
INTRODUCTION
FRAGMENT BY HБFIZ
I
II
V
VI
XII
XIV
XVI
XVII
XIX
XX
XXII
XXIII
XXV
XXIX
XXXI
XXXV
XLII
XLIII
XLV
LIX
LXIII
LXIX
LXXI
LXXVII
LXXXV
LXXXVI
LXXXVII
LXXXVIII
CI
CXV
CXXIX
CXXXIV
CXXXV
CXLV
CXLVI
CXLIX
CLV
CLVII
CLIX
CLXVI
CLXXIII
CLXXX
ENDNOTES
## NOTE
The reader will be struck with the apparent want of unity in many of the Odes. The Orientals compare each couplet to a single pearl and the entire "Ghazal," or Ode, to a string of pearls. It is the rhyme, not necessarily the sense, which links them together. Hence the single pearls or couplets may often be arranged in various orders without injury to the general effect; and it would probably be impossible to find two manuscripts either containing the same number of Odes, or having the same couplets following each other in the same order.
## INTRODUCTION
We are told in the Persian histories that when Tamerlane, on his victorious progress through the East, had reached Shiraz, he halted before the gates of the city and sent two of his followers to search in the bazar for a certain dervish Muhammad Shams-ad-din, better known to the world by the name of Hбfiz. And when this man of religion, wearing the simple woollen garment of a Sufi, was brought into the presence of the great conqueror, he was nothing abashed at the blaze of silks and jewelry which decorated the pavilion where Tamerlane sat in state. And Tamerlane, meeting the poet with a frown of anger, said, "Art not thou the insolent verse-monger who didst offer my two great cities Samarkand and Bokhara for the black mole upon thy lady's cheek?" "It is true," replied Hбfiz calmly, smiling, "and indeed my munificence has been so great throughout my life, that it has left me destitute, so that I shall be hereafter dependent upon thy generosity for a livelihood." The reply of the poet, as well as his imperturbable self-possession, pleased the Asiatic Alexander, and he dismissed Hбfiz with a liberal present.
This story, we are told, cannot be true, for Tamerlane did not reach Shiraz until after the death of the greatest of Persian lyric poets; but if it is not true in fact, it is true in spirit, and gives the real key to the character of Hбfiz. For we must look upon Hбfiz as one of the few poets in the world who utters an unbroken strain of joy and contentment. His poverty was to him a constant fountain of satisfaction, and he frankly took the natural joys of life as they came, supported under every vicissitude by his religious sense of the goodness and kindliness of the One God, manifested in everything in the world that was sweet and genial, and beautiful to behold. It is strange that we have to go to the literature of Persia to find a poet whose deep religious convictions were fully reconciled with the theory of human existence which was nothing more or less than an optimistic hedonism. There is nothing parallel to this in classic literature. The greatest of Roman Epicureans, the materialist, whose maxim was: enjoy the present for there is no God, and no to-morrow, speaks despairingly of that drop of bitterness, which rises in the fountain of Delight and brings torture, even amid the roses of the feast. It is with mocking irony that Dante places Epicurus in the furnace-tombs of his Inferno amid those heresiarchs who denied the immortality of the soul. Hбfiz was an Epicurean without the atheism or the despair of Epicurus. The roses in his feast are ever fresh and sweet and there is nothing of bitterness in the perennial fountain of his Delight. This unruffled serenity, this joyful acceptance of material existence and its pleasures are not in the Persian poet the result of the carelessness and shallowness of Horace, or the cold-blooded worldliness and sensuality of Martial. The theory of life which Hбfiz entertained was founded upon the relation of the human soul to God. The one God of Sufism was a being of exuberant benignity, from whose creative essence proceeded the human soul, whose experiences on earth were intended to fit it for re-entrance into the circle of light and re-absorption into the primeval fountain of being. In accordance with the beautiful and pathetic imagery of the Mystic, life was merely a journey of many stages, and every manifestation of life which the traveller met on the high road was a manifestation and a gift of God Himself. Every stage on the journey towards God which the soul made in its religious experience was like a wayside inn in which to rest awhile before resuming the onward course. The pleasures of life, all that charmed the eye, all that gratified the senses, every draught that intoxicated, and every fruit that pleased the palate, were, in the pantheistic doctrine of the Sufi considered as equally good, because God was in each of them, and to partake of them was therefore to be united more closely with God. Never was a theology so well calculated to put to rest the stings of doubt or the misgivings of the pleasure-seeker. This theology is of the very essence of Hбfiz's poetry. It is in full reliance on this interpretation of the significance of human existence that Hбfiz faces the fierce Tamerlane with a placid smile, plunges without a qualm into the deepest abysses of pleasure, finds in the love-song of the nightingale the voice of God, and in the bright eyes of women and the beaker brimming with crimson wine the choicest sacraments of life, the holiest and the most sublime intermediaries between divine and human life.
It is this that makes Hбfiz almost the only poet of unadulterated gladsomeness that the world has ever known. There is no shadow in his sky, no discord in his music, no bitterness in his cup. He passes through life like a happy pilgrim, singing all the way, mounting in his own way from strength to strength, sure of a welcome when he reaches the goal, contented with himself, because every manifestation of life of which he is conscious must be the stirrings within him of that divinity of which he is a portion. When we have thus spoken of Hбfiz we have said almost all that is known of the Persian lyric poet, for to know Hбfiz we must read his verses, whose magic charm is as great for Europeans as for Asiatics. The endless variety of his expressions, the deep earnestness of his convictions, the persistent gayety of his tone, are qualities of irresistible attractiveness. Even to this day his tomb is visited as the Mecca of literary pilgrims, and his numbers are cherished in the memory and uttered on the tongue of all educated Persians. The particulars of his life may be briefly epitomized as follows: He was born at Shiraz in the early part of the fourteenth century, dying in the year 1388. The name Hбfiz means, literally, the man who remembers, and was applied to himself by Hбfiz from the fact that he became a professor of the Mohammedan scriptures, and for this purpose had committed to memory the text of the Koran. His manner of life was not approved of by the dervishes of the monastic college in which he taught, and he satirizes his colleagues in revenge for their animadversions. The whole Mohammedan world hailed with delight the lyrics which Hбfiz published to the world, and kings and rulers vied with each other in making offers to him of honors and hospitality. At one time he started for India on the invitation of a great Southern Prince, who sent a vessel to meet him on the way, but the hardships of the sea were too severe for him, and he made his way back to Shiraz without finishing his journey.
His out-and-out pantheism, as well as his manner of life, caused him at his death to be denied burial in consecrated ground. The ecclesiastical authorities were, however, induced to relent in their plan of excommunication at the dictates of a passage from the poet's writings, which was come upon by opening the book at random. The passage ran as follows: "Turn not thy feet from the bier of Hбfiz, for though immersed in sin, he will be admitted into Paradise." And so he rests in the cemetery at Shiraz, where the nightingales are singing and the roses bloom the year through, and the doves gather with low murmurs amid the white stones of the sacred enclosure. The poets of nature, the mystical pantheist, the joyous troubadour of life, Hбfiz, in the naturalness and spontaneity of his poetry, and in the winning sweetness of his imagery, occupies a unique place in the literature of the world, and has no rival in his special domain.
## FRAGMENT BY HБFIZ
In Praise of His Verses.
The beauty of these verses baffles praise:
What guide is needed to the solar blaze?
Extol that artist by whose pencil's aid
The virgin, Thought, so richly is arrayed.
For her no substitute can reason show,
Nor any like her human judgment know.
This verse, a miracle, or magic white —
Brought down some voice from Heaven, or Gabriel bright?
By me as by none else are secrets sung,
No pearls of poesy like mine are strung.
## I
"Alб yб ayyuha's-Sбkн!" — pass round and offer thou the bowl,
For love, which seemed at first so easy, has now brought trouble to my
soul.
With yearning for the pod's aroma, which by the East that lock shall
spread
From that crisp curl of musky odor, how plenteously our hearts have
bled!
Stain with the tinge of wine thy prayer-mat, if thus the aged Magian
bid,
For from the traveller from the Pathway no stage nor usage can be
hid.
Shall my beloved one's house delight me, when issues ever and anon
From the relentless bell the mandate: "'Tis time to bind thy litters
on"?
The waves are wild, the whirlpool dreadful, the shadow of the night
steals o'er,
How can my fate excite compassion in the light-burdened of the shore?
Each action of my froward spirit has won me an opprobrious name;
Can any one conceal the secret which the assembled crowds proclaim?
If Joy be thy desire, O Hбfiz,
From Him far distant never dwell.
"As soon as thou hast found thy Loved one,
Bid to the world a last farewell."
## II
Thou whose features clearly-beaming make the moon of Beauty bright,
Thou whose chin contains a well-pit which to Loveliness gives light.
When, O Lord! shall kindly Fortune, sating my ambition, pair
This my heart of tranquil nature and thy wild and ruffled hair?
Pining for thy sight my spirit trembling on my lip doth wait:
Forth to speed it, back to lead it, speak the sentence of its fate.
Pass me with thy skirt uplifted from the dusty bloody ground:
Many who have been thy victims dead upon this path are found.
How this heart is anguish-wasted let my heart's possessor know:
Friends, your souls and mine contemplate, equal by their common woe.
Aught of good accrues to no one witched by thy Narcissus eye:
Ne'er let braggarts vaunt their virtue, if thy drunken orbs are nigh.
Soon my Fortune sunk in slumber shall her limbs with vigor brace:
Dashed upon her eye is water, sprinkled by thy shining face.
Gather from thy cheek a posy, speed it by the flying East;
Sent be perfume to refresh me from thy garden's dust at least.
Hбfiz offers a petition, listen, and "Amen" reply:
"On thy sugar-dropping rubies let me for life's food rely."
Many a year live on and prosper, Sбkнs of the court of Jem,
E'en though I, to fill my wine-cup, never to your circle come.
East wind, when to Yazd thou wingest, say thou to its sons from me:
"May the head of every ingrate ball-like 'neath your mall-bat be!"
"What though from your dais distant, near it by my wish I seem;
Homage to your Ring I render, and I make your praise my theme."
Shбh of Shahs, of lofty planet, Grant for God what I implore;
Let me, as the sky above thee, Kiss the dust which strews thy floor.
## V
Up, Sбkн! — let the goblet flow;
Strew with dust the head of our earthly woe!
Give me thy cup; that, joy-possessed,
I may tear this azure cowl from my breast,
The wise may deem me lost to shame,
But no care have I for renown or name.
Bring wine! — how many a witless head
By the wind of pride has with dust been spread!
My bosom's fumes, my sighs so warm,
Have inflamed yon crude and unfeeling swarm.
This mad heart's secret, well I know,
Is beyond the thoughts of both high and low.
E'en by that sweetheart charmed am I,
Who once from my heart made sweetness fly.
Who that my Silvern Tree hath seen,
Would regard the cypress that decks the green?
In grief be patient,
Night and day,
Till thy fortune, Hбfiz,
Thy wish obey.
## VI
My heart no longer brooks my hand: sages, aid for God my woe!
Else, alas! my secret-deep soon the curious world must know.
The bark we steer has stranded: O breeze auspicious swell:
We yet may see once more the Friend we love so well.
The ten days' favor of the Sphere — magic is; a tale which lies!
Thou who wouldst befriend thy friends, seize each moment ere it flies.
At night, 'mid wine and flowers, the bulbul tuned his song:
"Bring thou the morning bowl: prepare, ye drunken throng!"
Sikander's mirror, once so famed, is the wine-filled cup: behold
All that haps in Dбrб's realm glassed within its wondrous mould.
O bounteous man, since Heaven sheds o'er thee blessings mild,
Inquire, one day at least, how fares Misfortune's child.
What holds in peace this twofold world, let this twofold sentence show:
"Amity to every friend, courtesy to every foe."
Upon the way of honor, impeded was my range;
If this affect thee, strive my destiny to change.
That bitter, which the Sъfi styled "Mother of all woes that be,"
Seems, with maiden's kisses weighed, better and more sweet to me.
Seek drunkenness and pleasure till times of strait be o'er:
This alchemy of life can make the beggar Kore.
Submit; or burn thou taper-like e'en from jealousy o'er-much:
Adamant no less than wax, melts beneath that charmer's touch.
When fair ones talk in Persian, the streams of life out-well:
This news to pious Pirs, my Sбkн, haste to tell.
Since Hбfiz, not by his own choice,
This his wine-stained cowl did win,
Shaikh, who hast unsullied robes,
Hold me innocent of sin.
Arrayed in youthful splendor, the orchard smiles again;
News of the rose enraptures the bulbul of sweet strain.
Breeze, o'er the meadow's children, when thy fresh fragrance blows,
Salute for me the cypress, the basil, and the rose.
If the young Magian dally with grace so coy and fine,
My eye shall bend their fringes to sweep the house of wine.
O thou whose bat of amber hangs o'er a moon below,
Deal not to me so giddy, the anguish of a blow.
I fear that tribe of mockers who topers' ways impeach,
Will part with their religion the tavern's goal to reach.
To men of God be friendly: in Noah's ark was earth
Which deemed not all the deluge one drop of water worth.
As earth, two handfuls yielding, shall thy last couch supply,
What need to build thy palace, aspiring to the sky?
Flee from the house of Heaven, and ask not for her bread:
Her goblet black shall shortly her every guest strike dead.
To thee, my Moon of Kanaan, the Egyptian throne pertains;
At length has come the moment that thou shouldst quit thy chains.
I know not what dark projects those pointed locks design,
That once again in tangles their musky curls combine.
Be gay, drink wine, and revel;
But not, like others, care,
O Hбfiz, from the Koran
To weave a wily snare!
## XII
Oh! where are deeds of virtue and this frail spirit where?
How wide the space that sunders the bounds of Here and There!
Can toping aught in common with works and worship own?
Where is regard for sermons, where is the rebeck's Tone?
My heart abhors the cloister, and the false cowl its sign:
Where is the Magian's cloister, and where is his pure wine?
'Tis fled: may memory sweetly mind me of Union's days!
Where is that voice of anger, where those coquettish ways?
Can a foe's heart be kindled by the friend's face so bright?
Where is a lamp unlighted, and the clear Day-star's light?
As dust upon thy threshold supplies my eyes with balm,
If I forsake thy presence, where can I hope for calm?
Turn from that chin's fair apple; a pit is on the way.
To what, O heart, aspir'st thou? Whither thus quickly? Say!
Seek not, O friend, in Hбfiz
Patience, nor rest from care:
Patience and rest — what are they?
Where is calm slumber, where?
## XIV
At eve a son of song — his heart be cheerful long! —
Piped on his vocal reed a soul-inflaming lay.
So deeply was I stirred, that melody once heard,
That to my tearful eyes the things of earth grew gray.
With me my Sбkн was, and momently did he
At night the sun of Daп by lock and cheek display.
When he perceived my wish, he filled with wine the bowl;
Then said I to that youth whose track was Fortune's way:
"Sбkн, from Being's prison deliverance did I gain,
When now and now the cup thou lit'st with cheerful ray.
"God guard thee here below from all the haps of woe;
God in the Seat of Bliss reward thee on His day!"
When Hбfiz rapt has grown,
How, at one barleycorn,
Should he appraise the realm,
E'en of Kбъs the Kay?
## XVI
I said: "O Monarch of the lovely, a stranger seeks thy grace this day."
I heard: "The heart's deceitful guidance inclines the stranger from
his way."
Exclaimed I then: "One moment tarry!" "Nay," was the answer, "let me go;
How can the home-bred child be troubled by stories of a stranger's
woe?"
Shall one who, gently nurtured, slumbers with royal ermine for a bed,
"Care if on rocks or thorns reposing the stranger rests his weary head?"
O thou whose locks hold fast on fetters so many a soul known long ago,
How strange that musky mole and charming upon thy cheek of vermil glow!
Strange is that ant-like down's appearance circling the oval of thy
face;
Yet musky shade is not a stranger within the Hall which paintings
grace.
A crimson tint, from wine reflected gleams in that face of moonlight
sheen;
E'en as the bloom of syrtis, strangely, o'er clusters of the pale
Nasrнn.
I said: "O thou, whose lock so night-black is evening in the
stranger's sight,
Be heedful if, at break of morning, the stranger sorrow for his
plight."
"Hбfiz," the answer was, "familiars
Stand in amaze at my renown;
It is no marvel if a stranger
In weariness and grief sit down."
## XVII
'Tis morn; the clouds a ceiling make:
The morn-cup, mates, the morn-cup take!
Drops of dew streak the tulip's cheek;
The wine-bowl, friends, the wine-bowl seek
The greensward breathes a gale divine;
Drink, therefore, always limpid wine.
The Flower her emerald throne displays:
Bring wine that has the ruby's blaze
Again is closed the vintner's store,
"Open, Thou Opener of the door!"
While smiles on us the season's boon,
I marvel that they close so soon.
Thy lips have salt-rights, 'tis confessed,
O'er wounds upon the fire-burnt breast.
Hбfiz, let not
Thy courage fail!
Fortune, thy charmer
Shall unveil.
## XIX
Lo! from thy love's enchanting bowers Rizvбn's bright gardens fresher
grow;
From the fierce heat thine absence kindles, Gehenna's flames intenser
glow.
To thy tall form and cheek resplendent, as to a place of refuge, fleet
Heaven and the Tъbв-tree, and find there— "Happiness — and a fair
retreat."
When nightly the celestial river glides through the garden of the skies,
As my own eye, it sees in slumber, nought but thy drunk narcissus eyes.
Each section of the spring-tide's volume makes a fresh comment on thy
name,
Each portal of the Empyrean murmurs the title of thy fame.
My heart has burned, but to ambition, the aim, still wished for, is
denied:
These tears that tinged with blood are flowing, if I could reach it,
would be dried.
What ample power thy salt-rights give thee (which both thy mouth and
lips can claim),
Over a breast by sorrow wounded, and a heart burnt within its flame!
Oh! think not that the amorous only are drunk with rapture at thy sway:
Hast thou not heard of zealots, also, as reckless and as wrecked as
they?
By thy lips' reign I hold it proven that the bright ruby's sheen is won
By the resplendent light that flashes out of a world-illuming sun.
Fling back thy veil! how long, oh tell me! shall drapery thy beauty
pale?
This drapery, no profit bringing, can only for thy shame avail.
A fire within the rose's bosom was kindled when she saw thy face;
And soon as she inhaled thy fragrance, she grew all rose-dew from
disgrace.
The love thy countenance awakens whelms Hбfiz in misfortune's sea;
Death threatens him! ho there! give help, ere yet that he has ceased
to be!
While life is thine, consent not, Hбfiz,
That it should speed ignobly by;
But strive thou to attain the object
Of thy existence ere thou die.
## XX
I swear — my master's soul bear witness, faith of old times, and
promise leal! —
At early morning, my companion, is prayer for thy unceasing weal.
My tears, a more o'erwhelming deluge than was the flood which Noah
braved,
Have washed not from my bosom's tablet the image which thy love has
graved.
Come deal with me, and strike thy bargain: I have a broken heart to
sell,
Which in its ailing state out-values a hundred thousand which are well.
Be lenient, if thou deem me drunken: on the primeval day divine
Love, who possessed my soul as master, bent my whole nature unto wine.
Strive after truth that for thy solace the Sun may in thy spirit rise;
For the false dawn of earlier morning grows dark of face because it
lies.
O heart, thy friend's exceeding bounty should free thee from unfounded
dread;
This instant, as of love thou vauntest, be ready to devote thy head!
I gained from thee my frantic yearning for mountains and the barren
plain,
Yet loath art thou to yield to pity, and loosen at mid-height my chain.
If the ant casts reproach on Бsaf, with justice does her tongue upbraid,
For when his Highness lost Jem's signet, no effort for the quest he
made.
No constancy — yet grieve not, Hбfiz —
Expect thou from the faithless fair;
What right have we to blame the garden,
Because the plant has withered there?
## XXII
Veiled in my heart my fervent love for him dwells,
And my true eye holds forth a glass to his spells.
Though the two worlds ne'er bowed my head when elate,
Favors as his have bent my neck with their weight.
Thine be the lote, but I Love's stature would reach.
High like his zeal ascends the fancy of each.
Yet who am I that sacred temple to tread?
Still let the East that portal guard in my stead!
Spots on my robe — shall they arouse my complaint?
Nay! the world knows that he at least has no taint.
My turn has come; behold! Majnъn is no more;
Five days shall fly, and each one's turn shall be o'er.
Love's ample realm, sweet joy, and all that is glad,
Save for his bounty I should never have had.
I and my heart — though both should sacrificed be,
Grant my friend's weal, their loss were nothing to me.
Ne'er shall his form within my pupil be dim,
For my eye's cell is but a chamber for him.
All the fresh blooms that on the greensward we view,
Gain but from him their scent and beauty of hue.
Hбfiz seems poor;
But look within, for his breast,
Shrining his love,
With richest treasure is blest.
## XXIII
Prone at my friend's high gates, my Will its head lays still:
Whate'er my head awaits is ordered by that will.
My friend resembles none; in vain I sought to trace,
In glance of moon or sun, the radiance of that face.
Can morning's breeze make known what grief this heart doth hold,
Which as a bud hath grown, compressed by fold on fold?
Not I first drained the jar where rev'lers pass away:
Heads in this work-yard are nought else than wine-jars' clay.
Meseems thy comb has wreathed those locks which amber yield:
The gale has civet breathed, and amber scents the field.
Flowers of verdant nooks be strewn before thy face:
Let cypresses of brooks bear witness to thy grace!
When dumb grow tongues of men that on such love would dwell,
Why should a tongue-cleft pen by babbling strive to tell?
Thy cheek is in my heart; no more will bliss delay;
Glad omens e'er impart news of a gladder day.
Love's fire has dropped its spark
In Hбfiz' heart before:
The wild-grown tulip's mark
Branded of old its core.
## XXV
Breeze of the morn, if hence to the land thou fliest — Of my friend,
Return with a musky breath from the lock so sweet
Of my friend.
Yea, by that life, I swear I would lay down mine in content,
If once I received through thee but a message sent
Of my friend.
But — at that sacred court, if approach be wholly denied,
Convey, for my eyes, the dust that the door supplied
Of my friend.
I — but a beggar mean — can I hope for Union at last?
Ah! would that in sleep I saw but the shadow cast
Of my friend.
Ever my pine-cone heart, as the aspen trembling and shy,
Has yearned for the pine-like shape and the stature high
Of my friend.
Not at the lowest price would my friend to purchase me care;
Yet I, a whole world to win, would not sell one hair
Of my friend.
How should this heart gain aught,
Were its gyves of grief flung aside?
I, Hбfiz, a bondsman, still
Would the slave abide
Of my friend.
## XXIX
Who of a Heaven on earth can tell, pure as the cell — Of dervishes?
If in the highest state you'd dwell, be ever slaves
Of dervishes.
The talisman of magic Might hid in some ruin's lonely site,
Emerges from its ancient night at the wild glance
Of dervishes.
When the proud sun has run his race, and he puts off his crown apace,
He bows before the pomp and place which are the boast
Of dervishes.
The palace portal of the sky, watched by Rizvбn's unsleeping eye,
All gazers can at once descry from the glad haunts
Of dervishes.
When mortal hearts are black and cold, that which transmutes them into
gold
Is the alchemic stone we hold from intercourse
Of dervishes.
When tyranny, from pole to pole, sways o'er the earth with dire control,
We see from first to last unroll the victor-flag
Of dervishes.
There is a wealth which lasts elate, unfearful of decline from fate;
Hear it with joy — this wealth so great, is in the hands
Of dervishes.
Khosrбus, the kiblahs of our prayer have weight to solace our
despair,
But they are potent by their care for the high rank
Of dervishes.
O, vaunter of thy riches' pride! lay all thy vanity aside,
And know that health and wealth abide but by the will
Of dervishes.
Korah lost all his treasured store, which, cursed of Heaven, sinks
daily more,
(Hast thou not heard this tale of yore?) from disregard
Of dervishes,
The smiling face of joy unknown, yet sought by tenants of a throne,
Is only in the mirror shown of the clear face
Of dervishes.
Let but our Бsaf's eye request, I am the slave of his behest,
For though his looks his rank attest, he has the mind
Of dervishes.
Hбfiz, if of the tide thou think, which makes immortal those who drink,
Seek in the dust that fountain's brink, at the cell door
Of dervishes.
Hбfiz, while here on earth, be wise:
He who to empire's rule would rise,
Knows that his upward pathway lies
Through his regard
Of dervishes.
## XXXI
In blossom is the crimson rose, and the rapt bulbul trills his song;
A summons that to revel calls you, O Sъfis, wine-adoring throng!
The fabric of my contrite fervor appeared upon a rock to bide;
Yet see how by a crystal goblet it hath been shattered in its pride.
Bring wine; for to a lofty spirit, should they at its tribunal be,
What were the sentry, what the Sultan, the toper, or the foe of glee?
Forth from this hostel of two portals as finally thou needs must go,
What of the porch and arch of Being be of high span or meanly low?
To bliss' goal we gain not access, if sorrow has been tasted not;
Yea, with Alastu's pact was coupled the sentence of our baleful lot.
At Being and Non-being fret not; but either with calm temper see:
Non-being is the term appointed for the most lovely things that be.
Бsaf's display, the airy courser, the language which the birds employed,
The wind has swept; and their possessor no profit from his wealth
enjoyed.
Oh! fly not from thy pathway upward, for the winged shaft that quits
the bow
A moment to the air has taken, to settle in the dust below.
What words of gratitude, O Hбfiz
Shall thy reed's tongue express anon,
As its choice gems of composition
From hands to other hands pass on?
## XXXV
Now on the rose's palm the cup with limpid wine is brimming,
And with a hundred thousand tongues the bird her praise is hymning.
Ask for a song-book, seek the wild, no time is this for knowledge;
The Comment of the Comments spurn, and learning of the college,
Be it thy rule to shun mankind, and let the Phoenix monish,
For the reports of hermit fame, from Kбf to Kбf astonish.
When yesterday our rector reeled, this sentence he propounded:
"Wine is a scandal; but far worse what men's bequests have founded."
Turbid or clear, though not thy choice, drink thankfully; well knowing
That all which from our Sбkн flows to his free grace is owing.
Each dullard who would share my fame, each rival self-deceiver,
Reminds me that at times the mat seems golden to its weaver.
Cease, Hбfiz! store as ruddy gold
The wit that's in thy ditty:
The stampers of false coin, behold!
Are bankers for the city.
## XLII
'Tis a deep charm which wakes the lover's flame,
Not ruby lip, nor verdant down its name.
Beauty is not the eye, lock, cheek, and mole;
A thousand subtle points the heart control.
## XLIII
Zealot, censure not the toper, guileless though thou keep thy soul:
Certain 'tis that sins of others none shall write upon thy scroll.
Be my deeds or good or evil, look thou to thyself alone;
All men, when their work is ended, reap the harvest they have sown.
Never of Eternal Mercy preach that I must yet despair;
Canst thou pierce the veil, and tell me who is ugly, who is fair?
Every one the Friend solicits, be he sober, quaff he wine;
Every place has love its tenant, be it or the mosque, or shrine.
From the still retreat of virtue not the first am I to roam,
For my father also quitted his eternal Eden home.
See this head, devout submission: bricks at many a vintner's door:
If my foe these words misconstrue— "Bricks and head!" — Say nothing more.
Fair though Paradise's garden, deign to my advice to yield:
Here enjoy the shading willow, and the border of the field.
Lean not on thy store of merits; know'st thou 'gainst thy name for aye
What the Plastic Pen indited, on the Unbeginning Day?
Hбfiz, if thou grasp thy beaker
When the hour of death is nigh,
From the street where stands the tavern
Straight they'll bear thee to the sky.
## XLV
O breeze of morn! where is the place which guards my friend from strife?
Where is the abode of that sly Moon who lovers robs of life?
The night is dark, the Happy Vale in front of me I trace.
Where is the fire of Sinдi, where is the meeting place?
Here jointly are the wine-filled cup, the rose, the minstrel; yet
While we lack love, no bliss is here: where can my Loved be met?
Of the Shaikh's cell my heart has tired, and of the convent bare:
Where is my friend, the Christian's child, the vintner's mansion, where?
Hбfiz, if o'er the glade of earth
The autumn-blast is borne,
Grieve not, but musing ask thyself:
"Where has the rose no thorn?"
## LIX
My Prince, so gracefully thou steppest, that where thy footsteps
fall — I'd die.
My Turk, so gracefully thou glidest, before thy stature tall
I'd die.
"When wilt thou die before me?" — saidst thou. Why thus so eagerly
inquire?
These words of thy desire delight me; forestalling thy desire
I'd die.
I am a lover, drunk, forsaken: Sбkн, that idol, where is he?
Come hither with thy stately bearing! let me thy fair form see,
I'd die.
Should he, apart from whom I've suffered a life-long illness, day by
day,
Bestow on me a glance, one only, beneath that orb dark-gray
I'd die.
"The ruby of my lips," thou saidst, "now bale, now balsam may exhale":
At one time from their healing balsam, at one time from their bale
I'd die.
How trim thy gait! May eye of evil upon thy face be never bent!
There dwells within my head this fancy; that at thy feet content
I'd die.
Though no place has been found for Hбfiz
In Love's retreat, where hid thou art,
For me thine every part has beauty,
Before thine every part —
I'd die.
## LXIII
My heart has of the world grown weary and all that it can lend:
The shrine of my affection holds no Being but my friend.
If e'er for me thy love's sweet garden a fragrant breath exhale,
My heart, expansive in its joy, shall bud-like burst its veil.
Should I upon love's path advise thee, when now a fool I've grown,
'Twould be the story of the fool, the pitcher, and the stone.
Go! say to the secluded zealot: "Withhold thy blame; for know,
I find the arch of the Mihrбb but in an eyebrow's bow."
Between the Ka'bah and the wine-house, no difference I see:
Whate'er the spot my glance surveys, there equally is He.
'Tis not for beard, hair, eyebrow only, Kalandarism should care:
The Kalandar computes the Path by adding hair to hair.
The Kalandar who gives a hair's head,
An easy path doth tread:
The Kalandar of genuine stamp,
As Hбfiz gives his head.
## LXIX
My heart desires the face so fair — Of Farrukh;
It is perturbed as is the hair
Of Farrukh.
No creature but that lock, that Hindъ swart,
Enjoyment from the cheek has sought
Of Farrukh.
A blackamoor by Fortune blest is he,
Placed at the side, and near the knee
Of Farrukh.
Shy as the aspen is the cypress seen,
Awed by the captivating mien
Of Farrukh.
Sбkн, bring syrtis-tinted wine to tell
Of those narcissi, potent spell
Of Farrukh.
Bent as the archer's bow my frame is now,
From woes continuous as the brow
Of Farrukh.
E'en Tartar gales which musky odors whirl,
Faint at the amber-breathing curl
Of Farrukh.
If leans the human heart to any place,
Mine has a yearning to the grace
Of Farrukh.
That lofty soul
Shall have my service true,
That serves, as Hбfiz,
The Hindъ —
Of Farrukh.
## LXXI
When now the rose upon the meadow from Nothing into Being springs,
When at her feet the humble violet with her head low in worship clings,
Take from thy morn-filled cup refreshment while tabors and the harp
inspire,
Nor fail to kiss the chin of Sбkн while the flute warbles and the lyre.
Sit thou with wine, with harp, with charmer, until the rose's bloom be
past;
For as the days of life which passes, is the brief week that she shall
last.
The face of earth, from herbal mansions, is lustrous as the sky; and
shines
With asterisms of happy promise, with stars that are propitious signs.
In gardens let Zoroaster's worship again with all its rites revive,
While now within the tulip's blossoms the fires of Nimrod are alive.
Drink wine, presented by some beauty of Christ-like breath, of cheek
fair-hued;
And banish from thy mind traditions to Бd relating, and Thamъd.
Earth rivals the Immortal Garden during the rose and lily's reign;
But what avails when the immortal is sought for on this earth in vain?
When riding on the windy courser, as Solomon, the rose is found,
And when the Bird, at hour of morning, makes David's melodies resound,
Ask thou, in Solomon's dominion, a goblet to the brim renewed;
Pledge the Vizir, the cycle's Бsaf, the column of the Faith, Mahmud.
O Hбfiz, while his days continue, let joy eternal be thine aim;
And may the shadow of his kindness eternally abide the same!
Bring wine; for Hбfiz, if in trouble,
Will ceaselessly the help implore
Of him who bounty shall aid ever,
As it have aid vouchsafed before.
## LXXVII
Upon the path of Love, O heart, deceit and risk are great!
And fall upon the way shall he who at swift rate
Shall go.
Inflated by the wind of pride, the bubble's head may shine;
But soon its cap of rule shall fall, and merged in wine
Shall go.
O heart, when thou hast aged grown, show airs of grace no more:
Remember that such ways as these when youth is o'er
Shall go.
Has the black book of black locks closed, the album yet shall stay,
Though many a score the extracts be which day by day
Shall go.
## LXXXV
To me love's echo is the sweetest sound
Of all that 'neath this circling Round
Hath stayed.
## LXXXVI
A beggar am I; yet enamoured of one of cypress mould:
One in whose belt the hand bides only with silver and with gold.
Bring wine! let first the hand of Hбfiz
The cheery cup embrace!
Yet only on one condition —
No word beyond this place!
## LXXXVII
When beamed Thy beauty on creation's morn,
The world was set on fire by love new-born.
Thy cheek shone bright, yet angels' hearts were cold:
Then flashed it fire, and turned to Adam's mould.
The lamp of Reason from this flame had burned,
But lightning jealousy the world o'erturned.
The enemy Thy secret sought to gain;
A hand unseen repelled the beast profane.
The die of Fate may render others glad:
My own heart saddens, for its lot is sad.
Thy chin's deep pit allures the lofty mind:
The hand would grasp thy locks in twines entwined,
Hбfiz his love-scroll
To Thyself addressed,
When he had cancelled
What his heart loved best.
## LXXXVIII
The preacher of the town will find my language hard, maybe:
While bent upon deceit and fraud, no Mussulman is he.
Learn drinking and do gracious deeds; the merit is not great
If a mere brute shall taste not wine, and reach not man's estate.
Efficient is the Name Divine; be of good cheer, O heart!
The dнv becomes not Solomon by guile and cunning's art.
The benisons of Heaven are won by purity alone:
Else would not pearl and coral spring from every clod and stone?
## CI
Angels I saw at night knock at the wine-house gate:
They shaped the clay of Adam, flung into moulds its weight.
Spirits of the Unseen World of Purities divine,
With me an earth-bound mortal, poured forth their 'wildering wine.
Heaven, from its heavy trust aspiring to be free,
The duty was allotted, mad as I am, to me.
Thank God my friend and I once more sweet peace have gained!
For this the houris dancing thanksgiving cups have drained.
With Fancy's hundred wisps what wonder that I've strayed,
When Adam in his prudence was by a grain bewrayed?
Excuse the wrangling sects, which number seventy-two:
They knock at Fable's portal, for Truth eludes their view.
No fire is that whose flame the taper laughs to scorn:
True fire consumes to ashes the moth's upgarnered corn.
Blood fills recluses' hearts where Love its dot doth place,
Fine as the mole that glistens upon a charmer's face.
As Hбfiz, none Thought's face
Hath yet unveiled; not e'en
Since for the brides of Language
Combed have their tresses been.
## CXV
Lost Joseph shall return to Kanaan's land — Despair not:
Affliction's cell of gloom with flowers shall bloom:
Despair not
Sad heart, thy state shall mend; repel despondency;
Thy head confused with pain shall sense regain:
Despair not.
When life's fresh spring returns upon the daпs mead,
O night-bird! o'er thy head the rose shall spread:
Despair not,
Hope on, though things unseen may baffle thy research;
Mysterious sports we hail beyond the veil:
Despair not.
Has the revolving Sphere two days opposed thy wish,
Know that the circling Round is changeful found:
Despair not.
If on the Ka'bah bent, thou brave the desert sand,
Though from the acacias thorn thy foot be torn,
Despair not,
Heart, should the flood of death life's fabric sweep away,
Noah shall steer the ark o'er billows dark:
Despair not,
Though perilous the stage, though out of sight the goal,
Whither soe'er we wend, there is an end:
Despair not,
If Love evades our grasp, and rivals press our suit,
God, Lord of every change, surveys the range:
Despair not.
Hбfiz, in thy poor nook —
Alone, the dark night through —
Prayer and the Koran's page
Shall grief assuage —
Despair not.
## CXXIX
Endurance, intellect, and peace have from my bosom flown,
Lured by an idol's silver ear-lobes, and its heart of stone.
An image brisk, of piercing looks, with peris' beauty blest,
Of slender shape, of lunar face, in Turk-like tunic drest!
With a fierce glow within me lit — in amorous frenzy lost —
A culinary pot am I, in ebullition tost.
My nature as a shirt's would be, at all times free from smart,
If like yon tunic garb I pressed the wearer to my heart.
At harshness I have ceased to grieve, for none to light can bring
A rose that is apart from thorns, or honey void of sting.
The framework of this mortal form may rot within the mould,
But in my soul a love exists which never shall grow cold.
My heart and faith, my heart and faith — of old they were unharmed,
Till by yon shoulders and yon breast, yon breast and shoulders charmed.
Hбfiz, a medicine for thy woe,
A medicine must thou sip,
No other than that lip so sweet,
That lip so sweet, that lip.
## CXXXIV
Although upon his moon-like cheek delight and beauty glow,
Nor constancy nor love is there: O Lord! these gifts bestow.
A child makes war against my heart; and he in sport one day
Will put me to a cruel death, and law shall not gainsay.
What seems for my own good is this: my heart from him to guard;
For one who knows not good from ill its guardianship were hard.
Agile and sweet of fourteen years that idol whom I praise:
His ear-rings in her soul retains the moon of fourteen days.
A breath as the sweet smell of milk comes from those sugary lips;
But from those black and roguish eyes behold what blood there drips!
My heart to find that new-born rose has gone upon its way;
But where can it be found, O Lord? I've lost it many a day.
If the young friend who owns my heart my centre thus can break,
The Pasha will command him soon the lifeguard's rank to take.
I'd sacrifice my life in thanks,
If once that pearl of sheen
Would make the shell of Hбfiz' eye
Its place of rest serene.
## CXXXV
I tried my fortune in this city lorn:
From out its whirlpool must my pack be borne.
I gnaw my hand, and, heaving sighs of ire,
I light in my rent frame the rose's fire.
Sweet sang the bulbul at the close of day,
The rose attentive on her leafy spray:
"O heart! be joyful, for thy ruthless Love
Sits down ill-temper'd at the sphere above.
"To make the false, harsh world thyself pass o'er,
Ne'er promise falsely and be harsh no more.
"If beat misfortune's waves upon heaven's roof,
Devout men's fate and gear bide ocean-proof.
"Hбfiz, if lasting
Were enjoyment's day,
Jem's throne would never
Have been swept away."
## CXLV
Breeze of the North, thy news allays my fears:
The hour of meeting with my Loved one nears.
Prospered by Heaven, O carrier pigeon, fly:
Hail to thee, hail to thee, come nigh, come nigh!
How fares our Salmв? What Zъ Salam's state?
Our neighbors there — are they unscathed by Fate?
The once gay banquet-hall is now devoid
Of circling goblets, and of friends who joyed.
Perished the mansion with its lot serene:
Interrogate the mounds where once 'twas seen.
The night of absence has now cast its shade:
What freaks by Fancy's night-gang will be played?
He who has loved relates an endless tale:
Here the most eloquent of tongues must fail.
My Turk's kind glances no one can obtain:
Alas, this pride, this coldness, this disdain!
In perfect beauty did thy wish draw nigh:
God guard thee from Kamбl's malefic eye
Hбfiz, long will last
Patience, love, and pain?
Lovers wail is sweet:
Do thou still complain.
## CXLVI
O thou who hast ravished my heart by thine exquisite grace and thy
shape,
Thou carest for no one, and yet not a soul from thyself can escape.
At times I draw sighs from my heart, and at times, O my life, thy
sharp dart:
Can aught I may say represent all the ills I endure from my heart?
How durst I to rivals commend thy sweet lips by the ruby's tent gemmed,
When words that are vivid in hue by a soul unrefined are contemned?
As strength to thy beauty accrues ev'ry day from the day sped before,
To features consummate as thine, will we liken the night-star no more.
My heart hast thou reft: take my soul! For thine envoy of grief what
pretence?
One perfect in grief as myself with collector as he may dispense.
O Hбfiz, in Love's holy bane,
As thy foot has at last made its way,
Lay hold of his skirt with thy hand,
And with all sever ties from to-day.
## CXLIX
Both worlds, the Transient and Eterne, for Sбkн and the Loved I'd yield:
To me appears Love's satellite the universe's ample field.
Should a new favorite win my place, my ruler shall be still supreme:
It were a sin should I my life more precious than my friend esteem.
## CLV
Last night my tears, a torrent stream, stopped Sleep by force:
I painted, musing on thy down, upon the water-course.
Then, viewing my Beloved one's brow — my cowl burnt up —
In honor of the sacred Arch I drained my flowing cup.
From my dear friend's resplendent brow pure light was shed;
And on that moon there fell from far the kisses that I sped.
The face of Sбkн charmed my eye, the harp my ear:
At once for both mine ear and eye what omens glad were here!
I painted thine ideal face till morning's light,
Upon the studio of my eye, deprived of sleep at night.
My Sбkн took at this sweet strain the wine-bowl up:
I sang to him these verses first; then drank to sparkling cup.
If any of my bird-like thoughts from joy's branch flew,
Back from the springes of thy lock their fleeting wings I drew.
The time of Hбfiz passed in joy:
To friends I brought
For fortune and the days of life
The omens that they sought.
## CLVII
Come, Sъfi, let us from our limbs the dress that's worn for cheat Draw:
Let us a blotting line right through this emblem of deceit
Draw.
The convent's revenues and alms we'd sacrifice for wine awhile,
And through the vintry's fragrant flood this dervish-robe of guile
Draw.
Intoxicated, forth we'll dash, and from our feasting foe's rich stores
Bear off his wine, and then by force his charmer out of doors
Draw.
Fate may conceal her mystery, shut up within her hiding pale,
But we who act as drunken men will from its face the veil
Draw.
Here let us shine by noble deeds, lest we at last ashamed appear,
When starting for the other world, we hence our spirit's gear
Draw.
To-morrow at Rizvбn's green glade, should they refuse to make it ours,
We from their halls will the ghilmбn, the houris from their bowers
Draw.
Where can we see her winking brow, that we, as the new moon of old,
At once may the celestial ball, as with a bat of gold,
Draw?
O Hбfiz! it becomes us not
Our boastful claims thus forth to put:
Beyond the limits of our rug
Why would we fain our foot
Draw?
## CLIX
Aloud I say it, and with heart of glee:
"Love's slave am I, and from both worlds am free."
Can I, the bird of sacred gardens, tell
Into this net of chance how first I fell?
My place the Highest Heaven, an angel born,
I came by Adam to this cloister lorn.
Sweet houris, Tъbв's shade, and Fountain's brink
Fade from my mind when of thy street I think.
Knows no astrologer my star of birth:
Lord, 'neath what plant bore me Mother Earth?
Since with ringed ear I've served Love's house of wine,
Grief's gratulations have each hour been mine.
My eyeball's man drains my heart's blood; 'tis just:
In man's own darling did I place my trust.
My Loved one's Alif-form stamps all my thought:
Save that, what letter has my master taught?
Let Hбfiz' tear-drops
By thy lock be dried,
For fear I perish
In their rushing tide.
## CLXVI
Knowest thou what fortune is?
'Tis Beauty's sight obtaining;
'Tis asking in her lane for alms,
And royal pomp disdaining.
Sev'erance from the wish for life an easy task is ever;
But lose we friends who sweeten life, the tie is hard to sever.
Bud-like with a serried heart I'll to the orchard wander;
The garment of my good repute I'll tear to pieces yonder;
Now, as doth the West-wind, tell deep secrets to the Flower,
Hear now of Love's mysterious sport from bulbuls of the bower.
Kiss thy Beloved one's lips at first while the occasion lingers:
Await thou else disgust at last from biting lip and fingers.
Profit by companionship: this two-doored house forsaken,
No pathway that can thither lead in future time is taken.
Hбfiz from the thought, it seems,
Of Shбh Mansur has fleeted;
O Lord! remind him that the poor
With favor should be treated.
## CLXXIII
With my heart's blood I wrote to one most dear:
"The earth seems doom-struck if thou are not near.
"My eyes a hundred signs of absence show:
These tears are not their only signs of woe."
I gained no boon from her for labor spent:
"Who tries the tried will in the end repent."
I asked how fared she; the physician spake:
"Afar from her is health; but near her ache."
The East-wind from my Moon removed her veil:
At morn shone forth the Sun from vapors pale.
I said: "They'll mock, if I go round thy lane."
By God! no love escapes the mocker's bane.
Grant Hбfiz' prayer:
"One cup, by life so sweet!"
He seeks a goblet
With thy grace replete!
## CLXXX
O thou who art unlearned still, the quest of love essay:
Canst thou who hast not trod the path guide others on the way?
While in the school of Truth thou stay'st, from Master Love to learn,
Endeavor, though a son to-day, the father's grade to earn.
Slumber and food have held thee far from Love's exalted good:
Wouldst thou attain the goal of love, abstain from sleep and food.
If with the rays of love of truth thy heart and soul be clear,
By God! thy beauty shall outshine the sun which lights the sphere.
Wash from the dross of life thy hands, as the Path's men of old,
And winning Love's alchemic power, transmute thyself to gold.
On all thy frame, from head to foot, the light of God shall shine,
If on the Lord of Glory's path nor head nor foot be thine.
An instant plunge into God's sea, nor e'er the truth forget
That the Seven Seas' o'erwhelming tide, no hair of thine shall wet.
If once thy glancing eye repose on the Creator's face.
Thenceforth among the men who glance shall doubtless be thy place.
When that which thy existence frames all upside-down shall be,
Imagine not that up and down shall be the lot of thee.
Hбfiz, if ever in thy head
Dwell Union's wish serene,
Thou must become the threshold's dust
Of men whose sight is keen.
## ENDNOTES
"The traveller of the Pathway" — the Magian, or Shaikh. In former times wine was chiefly sold by Magians, and as the keepers of taverns and caravansaries grew popular, the term Magian was used to designate not only "mine host," but also a wise old man, or spiritual teacher.
An allusion to the dimple and moisture of the chin, considered great beauties by Orientals.
Jem or Jemshнd, an ancient King of Persia. By Jem and his Sбkн are to be understood, in this couplet, the King of Yazd and his courtiers.
By the azure cowl is implied the cloak of deceit and false humility. Hбfiz uses this expression to cast ridicule upon Shaikh Hazan's order of dervishes, who were inimical to the brotherhood of which the poet was a member. The dervishes mentioned wore blue to express their celestial aspirations.
The disciples of Shaikh Hasan. Hбfiz had incurred their displeasure by the levity of his conduct.
In the "Gulistan" of Sa'di a philosopher declares that, of all the trees, the cypress is alone to be called free, because, unlike the others, it is not subject to the vicissitudes of appointed place and season, "but is at all times fresh and green, and this is the condition of the free."
In some MSS. we read: "The mirror of Sikander is the goblet of Jem." King Jem, or Jemshнd, had a talismanic cup: Sikander, or Alexander, had inherited from pre-Adamite times a magic mirror by means of which he was enabled to see into the camp of his enemy Dбrб (Darius). Hбfiz here informs us that the knowledge imputed to either king was obtained by wine.
Referring to wine, which in the Koran is declared to be the
Mother of Vices.
Korah, Kore, or Kбrun, the Dives of his age, was an alchemist. He lived in an excess of luxury and show. At the height of his pride and gluttony he rebelled against Moses, refusing to pay a tithe of his possessions for the public use. The earth then opened and swallowed him up together with the palace in which he dwelt. (See Koran, chap, xxviii, and, for the Bible narrative, The Book of Numbers, chap, xvi.)
It was decreed from all eternity that Hбfiz should drink wine. He had therefore no free agency and could not be justly blamed.
The boy serving at the wine-house.
The curl of hair over a moon-like face is here compared to a curved mall-bat sweeping over a ball.
By "earth" is to be understood Noah himself.
Fate, Fortune, and the Sky, are in Oriental poetry intervertible expressions; and the dome of Heaven is compared to a cup which is full of poison for the unfortunate.
The rebeck is a sort of violin having only three chords.
His locks being black as night and his cheek cheerful as the Sun of Daп or December.
Kai-kбъs, one of the most celebrated monarchs of Persia.
The pictured halls of China, or, in particular, the palace of Arzhang, the dwelling of Manes. Manes lived in the third century of our era, and his palace was famed as the Chinese picture-gallery. Hбfiz compares the bloom upon the cheek of his friend to the works of art executed by Manes, in which dark shadows, like velvety down upon the human face, excite no surprise.
The Nasrнn is the dog-rose.
In Mohammedan countries it is customary to write upon the doors: "O Opener of the gates! open unto us the gates of blessing."
Rizvбn is the gardener and gatekeeper of Paradise.
The lote-tree, known to Arabs as the Tъbв, is a prickly shrub. The Koran says: "To those who believe, and perform good works, appertain welfare and a fair retreat. The men of the right hand — how happy shall be the men of the right hand! — shall dwell among the lote-trees without thorns. Under their feet rivers shall flow in the garden of Delight."
According to Oriental belief, the ruby and all other gems, derive their brilliancy from the action of the sun. By a similar process of Nature, ruby lips obtain their vivid color from the sun above them.
The zodiacal light or faint illumination of the sky which disappears before the light of daybreak.
Бsaf, Solomon's "Vizir," was entrusted with the guardianship of the imperial signet ring, which was possessed of magical properties. While in his care it was stolen. When Solomon granted an audience to animals, and even insects, the ant, it is related, brought as an offering a blade of grass and rebuked Бsaf for having guarded the royal treasure so carelessly. By Бsaf, Hбfiz symbolizes in the present instance his friend or favorite; by the ant is implied a small hair on the face, and by the lost signet of Jem, a beautiful mouth, so small and delicate as to be invisible.
Majnъn, a celebrated lover, maddened by the charms of
Lailа.
This ode may have been written in gratitude for the patronage of a man of rank.
Literally in this toper-consuming shrine (of the world). The second line of the couplet probably means: Other revellers have preceded me, but their heads are now potter's clay in the potter's field of the earth.
The wild tulip of Shiraz has white petals streaked with pink, the inner end of each bearing a deep puce mark. The dark spot formed thus in the centre of the flower is compared to the brand of love, pre-ordained on the Past Day of Eternity to be imprinted on the heart of Hбfiz.
Khosrбu (Cyrus) is the title of several ancient kings of
Persia, and is here used in the plural to denote monarchs in general.
The term "kiblah," fronting-point, signifies the object towards which
the worshipper turns when he prays.
Korah or Kбrun — the miser who disobeyed Moses and was swallowed up with his treasures by the earth. They are said to be still sinking deeper and deeper. (See Numbers, xvi.)
How vain were the glories of Solomon! Бsaf was his minister, the East wind his courser, and the language of birds one of his accomplishments; but the blast of time had swept them away.
The "Comment of the Comments" is a celebrated explanatory treatise on the Koran.
Kбf is a fabulous mountain encircling the world. In this couplet and the following the poet ridicules the ascetics of his time.
The false coiners are inferior poets who endeavor to pass off their own productions as the work of Hбfiz.
Aiman (Happiness) is the valley in which God appeared to
Moses — metaphorically, the abode of the Beloved.
"Mihrбb" — the niche in a mosque, towards which Mohammedans pray.
Kalandars are an order of Mohammedan dervishes who wander about and beg. The worthless sectaries of Kalandarism, Hбfiz says, shave off beard and tonsure, but the true or spiritual Kalandar shapes his path by a scrupulous estimate of duty.
"Farrukh" (auspicious) is doubtless the name of some favorite of the Poet.
"Hindъ" is here equivalent to "slave."
Zerdusht (in Latin, Zoroaster) — the celebrated prophet of the Gulbres, or fire-worshippers. Nimrod is said to have practised a religion, similar to theirs.
Бd and Thamъd were Arab tribes exterminated by God in consequence of their having disobeyed the prophet Sбlih.
By a "grain" is meant a grain of wheat; according to
Mohammedans, the forbidden fruit of Paradise.
Kamбl was an Arab whose glance inflicted death.
"Alif-form," meaning a straight and erect form: the letter Alif being, as it were, of upright stature.
"The men who glance" are lovers. The spiritual or true lover is he who loves God.
# POEMS FROM THE DIVAN OF HAFIZ
Translated by Gertrude Lowthian Bell, 1897
## CONTENTS
TO HAFIZ OF SHIRAZ
I
II
III
IV
V
VI
VII
VIII
IX
X
XI
XII
XIII
XIV
XV
XVI
XVII
XVIII
XIX
XX
XXI
XXII
XXIII
XXIV
XXV
XXVI
XXVII
XXVIII
XXIX
XXX
XXXI
XXXII
XXXIII
XXXIV
XXXV
XXXVI
XXXVII
XXXVIII
XXXIX
XL
XLI
XLII
XLIII
## TO HAFIZ OF SHIRAZ
Thus said the Poet: "When Death comes to you,
All ye whose life-sand through the hour-glass slips,
He lays two fingers on your ears, and two
Upon your eyes he lays, one on your lips,
Whispering: Silence!" Although deaf thine ear,
Thine eye, my Hafiz, suffer Time's eclipse,
The songs thou sangest still all men may hear.
Songs of dead laughter, songs of love once hot,
Songs of a cup once flushed rose-red with wine,
Songs of a rose whose beauty is forgot,
A nightingale that piped hushed lays divine:
And still a graver music runs beneath
The tender love notes of those songs of thine,
Oh, Seeker of the keys of Life and Death!
While thou wert singing, the soft summer wind
That o'er Mosalla's garden blew, the stream
Of Ruknabad flowing where roses twined,
Carried thy voice farther than thou could'st dream.
To Isfahan and Baghdad's Tartar horde,
O'er waste and sea to Yezd and distant Ind;
Yea, to the sun-setting they bore thy word.
Behold we laugh, we warm us at Love's fire,
We thirst and scarce dare tell what wine we crave,
We lift our voices in Grief's dark-robed choir;
Sing thou the wisdom joy and sorrow gave!
If my poor rhymes held aught of the heart's lore,
Fresh wreaths were theirs to lay upon thy grave —
Master and Poet, all was thine before!
## I
ARISE, oh Cup-bearer, rise! and bring
To lips that are thirsting the bowl they praise,
For it seemed that love was an easy thing,
But my feet have fallen on difficult ways.
I have prayed the wind o'er my heart to fling
The fragrance of musk in her hair that sleeps
In the night of her hair-yet no fragrance stays
The tears of my heart's blood my sad heart weeps.
Hear the Tavern-keeper who counsels you:
"With wine, with red wine your prayer carpet dye!"
There was never a traveller like him but knew
The ways of the road and the hostelry.
Where shall I rest, when the still night through,
Beyond thy gateway, oh Heart of my heart,
The bells of the camels lament and cry:
"Bind up thy burden again and depart!"
The waves run high, night is clouded with fears,
And eddying whirlpools clash and roar;
How shall my drowning voice strike their ears
Whose light-freighted vessels have reached the shore?
I sought mine own; the unsparing years
Have brought me mine own, a dishonoured name.
What cloak shall cover my misery o'er
When each jesting mouth has rehearsed my shame!
Oh Hafiz, seeking an end to strife,
Hold fast in thy mind what the wise have writ:
"If at last thou attain the desire of thy life,
Cast the world aside, yea, abandon it!"
Stanza 1. — The first line of this song, the opening poem in the Divan, is borrowed from an Arabic poem by Yezid ibn Moawiyah, the second Khalif of the Ommiad line. This prince was held in abomination by the Persian Shi'ites, both as the head of the Sunnis and because he was the cause of the death of Hussein, the son of Ali, whom the Shi'ites regarded as the rightful successor to the Khalifate. Hafiz was frequently reproached for setting a quotation from the works of the abhorred Yezid at the head of his book, a reproach which he is said to have met with the reply, that it was good policy to steal from the heretics whatsoever they possessed of worth.
"In this country (i.e. North-Eastern China) is found the best musk in the world, and I will tell you how it is produced. There exists in that region a kind of wild animal like a gazelle. It has feet and tail like the gazelle's, a stag's hair of a very coarse kind, but no horns. It has four tusks, two below and two above, about three inches long, and slender in form, one pair growing downwards and the other upwards. It is a very pretty creature. The musk is found in this way: when the creature has been taken, they find at the navel, between the flesh and the skin, something like an imposthume filled with blood, which they cut out and remove, with all the skin attached to it; and the blood inside this imposthume is the musk that produces that powerful perfume. There is an immense number of these beasts in the country we are speaking of. The flesh is very good to eat. Messer Marco brought the dried head and feet of one of these animals to Venice with him." — Travels of Marco Polo.
There is a play of meaning upon the musk which is obtained at the cost of the deer's life-blood and the tears of blood which the lover weeps for his mistress.
Stanza 2. — The title which Hafiz gives to the Tavern-keeper is Pir-i-Maghan-literally, the Old Man of the Magians. The history of this title is an epitome of the history of Persian faiths. It indicated primarily the priest of the first of Persian religions, that of Zoroaster. When the Mahommadans invaded Persia, and the preachers of the Prophet supplanted the priests of Zoroaster, their title fell into disrepute, and was degraded so far that it came to mean only the keeper of a tavern or caravanserai. But in this sense it gradually regained the honourable place from which it had fallen; for the keepers of such places of resort were, for the most part, men well acquainted with the "ways of the road and the hostelry." In their time they may themselves have served travellers upon their journey; they had heard and learnt much from the wayfarers who stopped at their gates, and they were able to guide others upon their journey, sending them forth refreshed and comforted in body. And here the Sufis took up the ancient name and used it to mean that wise old man who supplied weary travellers upon life's road with the spiritual draught of Sufi doctrine which refreshes and comforts the soul.
## II
THE bird of gardens sang unto the rose,
New blown in the clear dawn: "Bow down thy head!
As fair as thou within this garden close,
Many have bloomed and died." She laughed and said
"That I am born to fade grieves not my heart
But never was it a true lover's part
To vex with bitter words his love's repose."
The tavern step shall be thy hostelry,
For Love's diviner breath comes but to those
That suppliant on the dusty threshold lie.
And thou, if thou would'st drink the wine that flows
From Life's bejewelled goblet, ruby red,
Upon thine eyelashes thine eyes shall thread
A thousand tears for this temerity.
Last night when Irem's magic garden slept,
Stirring the hyacinth's purple tresses curled,
The wind of morning through the alleys stept.
"Where is thy cup, the mirror of the world?
Ah, where is Love, thou Throne of Djem?" I cried.
The breezes knew not; but "Alas," they sighed,
"That happiness should sleep so long!" and wept.
Not on the lips of men Love's secret lies,
Remote and unrevealed his dwelling-place.
Oh Saki, come! the idle laughter dies
When thou the feast with heavenly wine dost grace.
Patience and wisdom, Hafiz, in a sea
Of thine own tears are drowned; thy misery
They could not still nor hide from curious eyes.
Stanza 1. — This poem has been expounded to me as a description of the poet's quest for love. In an allegory he shows how he looked for it in vain from that image of earthly devotion, the nightingale; he warns men that it comes not but by humiliation and sorrow; he questions the magic garden, but its breezes cannot answer him; finally, he concludes that love is not that which lies upon the lips of men, and calls upon the Cup-bearer to silence their idle talk with the wine of divine knowledge.
Stanza 2. — The Garden of Irem was planted by the mythical King Shedad, the son of Ad, the grandson of Irem, who was himself the son of Shem. The tribe of Ad settled in the sandy deserts near Aden, where Ad began the building of a great city which his son completed. Round his palace Shedad planted a wonderful garden which was intended to rival in beauty the Garden of Eden. "When it was finished he set out with a great attendance to take a view of it, but when they were come within a day's journey of the place they were all destroyed by a terrible noise from heaven. . . . The city, they tell us, is still standing in the deserts of Aden, being preserved by Providence as a monument of divine justice, though it be invisible, unless very rarely, when God permits it to be seen, a favour one Colabah pretended to have received in the reign of the Khalif Moawiyah, who, sending for him to know the truth of the matter, Colabah related his whole adventure: that, as he was seeking a camel he had lost, he found himself on a sudden at the gates of this city, and entering it, saw not one inhabitant, at which being terrified, he stayed no longer than to take with him some fine stones which he showed the Khalif." — Sale's Koran.
Sudi says that Hafiz composed this poem in a beautiful garden belonging to Shah Shudja, and called by him the Baghi-Irem, after Shedad's legendary Paradise.
"Il y avait jadis en Perse un grand roi nommé Djem ou Djemshid. It régna sept cents ans; je ne saurai vous dire à quelle date au juste, mais 'tant qu'il regna, it n'y eut dans son empire ni mort, ni maladie, ni vicillesse, et tous les hommes marchaient dans la taille de jouvenceaux: de quinze ans; it n'y avait ni chaleur, ni froideur, et jamais ne se desséchaient les eaux ni les plantes.' Mais le pauvre Djem n'avait point la tête solide, et, comme it faisait des immortels, il se crut Dieu et voulut être adoré. Aussitôt, le Fari Yazdan, c'est-à-dire la gloire royale qui vient de Dieu, l'abandonna; un serpent à trois tétes, nommé Zohab, vint de l'Arabie et lui prit son tréne; it s'enfuit dans l'Inde et y resta chaché mille ans durant; puis un beau jour, s'étant aventuré hors de sa retraite, il fut livré au serpent, qui le scia en deux avec une arête de poisson. Entre autres splendeurs, le roi Djemshid, au temps de sa splendeur, possé-dait une coupe magique où il voyait tout l'univers et tout ce qui s'y passe. Certains savants prétendent que cette coupe était le soleil qui voit toute chose; d'autres, que c'était un globe terrestre mis au courant, et il me souvient qu'il y a deux ans, prenant le thé dans un café de Stamboul avec un sage d'Isfahan, nommé Habib, la conversation tomba de la tasse de thé à la coupe de Djemshid, et Habib, me mettant le doigt au front, me dit: Djam-i-Djemshid, dil-i-agah: "la coupe de Djemshid c'est le cœur de l'homme de science." — Darmsteter, "Lettres sur l'Inde."
A few miles from Peshawar, Darmsteter goes on to relate, there is a dried-up pond called the Talab i Djemshid, into which the King is said to have cast his magic cup. The head man of the village told the French traveller that a knife had been discovered there bearing this inscription: "This pond was dug by me, Djemshid, five hundred years before the Hejra." "Elle n'a pas été retrouvée, la coupe de Djemshid," adds Darmsteter, "non plus que la coupe du roi de Thulé, c'est pour ça qu'il n'y a plus parmi les hommes ni science, ni amour."
Djemshid is supposed to have built Persepolis. There is a legend that his cup was found buried in its foundations, and that it was formed of an enormous turquoise. It is said that he was the first to drink wine, and that he recommended it to his subjects as a health-giving beverage. He, too, was the father of chemistry and the possessor of the philosopher's stone.
## III
WIND from the east, oh Lapwing of the day,
I send thee to my Lady, though the way
Is far to Saba, where I bid thee fly;
Lest in the dust thy tameless wings should lie,
Broken with grief, I send thee to thy nest,
Fidelity.
Or far or near there is no halting-place
Upon Love's road-absent, I see thy face,
And in thine car my wind-blown greetings sound,
North winds and east waft them where they are bound,
Each morn and eve convoys of greeting fair
I send to thee.
Unto mine eyes a stranger, thou that art
A comrade ever-present to my heart,
What whispered prayers and what full meed of praise
I send to thee.
Lest Sorrow's army waste thy heart's domain,
I send my life to bring thee peace again,
Dear life thy ransom! From thy singers learn
How one that longs for thee may weep and bum
Sonnets and broken words, sweet notes and songs
I send to thee.
Give me the cup! a voice rings in mine cars
Crying: "Bear patiently the bitter years!
For all thine ills, I send thee heavenly grace.
God the Creator mirrored in thy face
Thine eyes shall see, God's image in the glass
I send to thee.
Hafiz, thy praise alone my comrades sing;
Hasten to us, thou that art sorrowing!
A robe of honour and a harnessed steed
I send to thee."
Stanza 1. — King Solomon sent the lapwing or hoopoe as his messenger to Bilkis, Queen of Sheba. The story is told thus by Al Ta'labi, in his Stories of the Prophets. (The lapwing had already made a journey on his own account, and had brought Solomon news of the great Queen, and told him that she was not a worshipper of the true God.) "Then Solomon wrote a letter saying: From the servant of God, Solomon, son of David, to Bilkis, Queen of Saba, in the name of God the Merciful, the Compassionate, peace be upon him who follows the right road. After which he said :Behave not insolently towards me, but come unto me humbled. And he strewed musk upon it and sealed it with his seal. Then he said to the lapwing: Fly with this letter and deliver it unto them, then turn away, but remain near them and hear what answer they make. And the lapwing took the letter and flew with it to Bilkis. And she was in the land which is called Marib, at a distance of three days' journey, and she had entered into her castle, and the gates of it were shut. For when she slept she was wont to shut the gates and to take the key and lay it beneath her head. So the lapwing came unto her, and she was asleep, lying upon her back; and he laid the letter upon her breast. Wahb ibn Manabbih says that there was a window opposite to the sun so that the sunbeams fell through it at dawn, and when she saw the sun she was wont to bow down and worship it. And the lapwing went to this window and blocked it up with his wings. And the sun rose, but she knew it not. And she thought that the sun was late, and stood up to look for it. Then the lapwing threw a leaf upon her face. And they say that Bilkis took the letter and she was able to read the writing. But when she saw the seal she trembled and bowed down, because of the power of Solomon that was in his seal. For she knew that the power of him who had sent the letter was greater than hers, and she said: Lo, here is a king whose messengers are the birds verily he is a mighty king."
Stanzas 5 and 6. — The accepted explanation of these lines is that by the glass Hafiz means his own heart, which he sends to his mistress that she may see that her own image is reflected in it; but I prefer here (and indeed for the whole poem) a mystical interpretation. The heavenly voice tells him to seek for comfort in Sufiism, and bids him look upon the mirror, for he shall see God himself reflected in it — which is only another way of putting the doctrine that man and God are one. The poet's reputation has gained him admittance into the company of the Sufis, let him hasten to them, for they shall give him that for which he seeks.
A horse and robe is the Eastern gift of honour. Lane in one of his notes to the "Arabian Nights" quotes a significant story concerning these gifts: "A person chancing to look at a register kept by one of the officers of Harun al Rashid, saw in it the following entry: '400,000 pieces of gold, the price of a dress of honour for Jafar ibn Yahya, the Vizir.' A few days after he saw beneath this written: 'Ten kerits, the price of naphtha and reeds for burning the body of Jafar ibn Yahya.' (The kerit of Baghdad was worth a twentieth part of a gold piece.)
Put not your trust in Eastern princes!
## IV
SLEEP on thine eyes, bright as narcissus flowers,
Falls not in vain
And not in vain thy hair's soft radiance showers
Ah, not in vain!
Before the milk upon thy lips was dry,
I said: "Lips where the salt of wit doth lie,
Sweets shall be mingled with thy mockery,
And not in vain!"
Thy mouth the fountain where Life's waters flow,
A dimpled well of tears is set below,
And death lies near to life thy lovers know,
But know in vain!
God send to thee great length of happy days
Lo, not for his own life thy servant prays;
Love's dart in thy bent brows the Archer lays,
Nor shoots in vain.
Art thou with grief afflicted, with the smart
Of absence, and is bitter toil thy part?
Thy lamentations and thy tears, oh Heart,
Are not in vain
Last night the wind from out her village blew,
And wandered all the garden alleys through,
Oh rose, tearing thy bosom's robe in two;
'Twas not in vain!
And Hafiz, though thy heart within thee dies,
Hiding love's agony from curious eyes,
Ah, not in vain thy tears, not vain thy sighs,
Not all in vain!
Verse 3. — The Persians describe the dimple in the chin of their mistress as a dangerous well filled with her lover's tears, into which, when he approaches her mouth, he may fall and be drowned.
Verse 6.— "Oh rose, tearing thy robe in two": that is, bursting into flower beneath the warm breath of the wind that blows from where thou art.
## V
OH Turkish maid of Shiraz! in thy hand
If thou'lt take my heart, for the mole on thy cheek
I would barter Bokhara and Samarkand.
Bring, Cup-bearer, all that is left of thy wine!
In the Garden of Paradise vainly thou'lt seek
The lip of the fountain of Ruknabad,
And the bowers of Mosalla where roses twine.
They have filled the city with blood and broil,
Those soft-voiced Lulis for whom we sigh;
As Turkish robbers fall on the spoil,
They have robbed and plundered the peace of my heart.
Dowered is my mistress, a beggar am I;
What shall I bring her? a beautiful face
Needs nor jewel nor mole nor the tiring-maid's art.
Brave tales of singers and wine relate,
The key to the Hidden 'twere vain to seek;
No wisdom of ours has unlocked that gate,
And locked to our wisdom it still shall be.
But of Joseph's beauty the lute shall speak;
And the minstrel knows that Zuleika came forth,
Love parting the curtains of modesty.
When thou spokest ill of thy servant 'twas well —
God pardon thee! for thy words were sweet;
Not unwelcomed the bitterest answer fell
From lips where the ruby and sugar lay.
But, fair Love, let good counsel direct thy feet;
Far dearer to youth than dear life itself
Are the warnings of one grown wise — and grey!
The song is sung and the pearl is strung
Come hither, oh Hafiz, and sing again!
And the listening Heavens above thee hung
Shall loose o'er thy verse the Pleiades' chain.
Stanza 1.-When the conqueror Timur entered Shiraz it is related that he summoned Hafiz before him and said: "Of all my empire, Bokhara and Samarkand are the fairest jewels; how comes it that in thy song thou hast declared that thou would'st exchange them against the black mole on the cheek of thy mistress?" Hafiz replied: "It is because of such generosity that I am now as poor as thou seest." The Emperor was not to be outdone in repartee: he sent the poet away a richer man by some hundreds of gold pieces.
"Cest du Molière renversé," says Darmsteter of these lines, and quotes
"Si le roi m'avait donné
Paris sa grande ville,
Et qu'il me fallût quitter
L'amour de ma mie,
Je dirais au roi Henri:
Reprenez votre Paris,
J'aime mieux ma mie, ô gué,
J'aime mieux ma mie!"
In the garden of Mosalla, Hafiz lies buried: the stream Ruknabad flows near at hand.
Stanza 2. — The Luli or gipsies, as they were contemptuously called, were a people of the tribe of Keredj, of Indian origin, who inhabited the country between Shiraz and Isfahan. Their young men and maidens were famous for their beauty and musical accomplishments, and furnished minstrels and dancing girls to the wealthy inhabitants of Shiraz. Sir Henry Layard met with a similar tribe near Baghdad. "They bear," he says, "a very bad reputation on the score of morality, and according to general report lead very dissolute lives. The dancing boys and girls who frequent Baghdad, and are notoriously of evil fame, come principally from this district. Whilst we were resting at the caravanserai a party of them came to perform their indecent dances before us, as they were in the habit of doing on the arrival of travellers." — Early Adventures.
In Turkestan there was formerly an institution called the Feast of Plunder. When the pay-day of the soldiers came round, dishes of rice and great quantities of cooked food were prepared and set out on the ground. The soldiers then rode up, armed as if for battle, and carried off the food with mimic violence. Thus they made reparation to their conscience for accepting a pay lawfully earned, and reminded themselves that rapine was their true profession.
Stanza 3. — Joseph is the Oriental type of perfect beauty. The story of his relations with Zuleikha, Potiphar's wife, is one of the famous love stories of the East; Jami made it the theme of a long metaphysical poem. The part played by Zuleikha in Persian tales is far more creditable than that which is assigned to her either in the Bible or the Koran.
Every translator of Hafiz has tried his hand upon this song, which is one of the most famous in the Divan. It is only right to inform the reader that the original is of great beauty.
The whole poem has received a mystical interpretation which seems to me to add but little to its value or to its intelligibility; but in case any one should wish to gather the higher wisdom from it, I may mention that the mole, powder, and paint, of which a beautiful face does not stand in need, represent the ink, colour, dots, and lines of the Koran; and this is the explanation given to the couplet concerning Joseph and Zuleikha by a thorough-going Western mystic: "By reason of that beauty daily increasing that Joseph (the absolute existence, the real beloved, God) had, I (the first day) knew that love for him would bring Zuleikha (us, things possible) forth from the screen of chastity (the pure existence of God)." The learned translator seems to have felt that his version presented some difficulties, and he adds for the use of his weaker brethren the following comment: "In the world of non-existence and possibility, when I beheld the splendour of true beauty with different qualities, I knew for certain that Love would take us out of the ambush." This makes everything clear.
## VI
A FLOWER-TINTED cheek, the flowery close
Of the fair earth, these are enough for me
Enough that in the meadow wanes and grows
The shadow of a graceful cypress-tree.
I am no lover of hypocrisy;
Of all the treasures that the earth can boast,
A brimming cup of wine I prize the most —
This is enough for me!
To them that here renowned for virtue live,
A heavenly palace is the meet reward;
To me, the drunkard and the beggar, give
The temple of the grape with red wine stored!
Beside a river seat thee on the sward;
It floweth past-so flows thy life away,
So sweetly, swiftly, fleets our little day —
Swift, but enough for me!
Look upon all the gold in the world's mart,
On all the tears the world hath shed in vain
Shall they not satisfy thy craving heart?
I have enough of loss, enough of gain;
I have my Love, what more can I obtain?
Mine is the joy of her companionship
Whose healing lip is laid upon my lip —
This is enough for me!
I pray thee send not forth my naked soul
From its poor house to seek for Paradise
Though heaven and earth before me God unroll,
Back to thy village still my spirit flies.
And, Hafiz, at the door of Kismet lies
No just complaint-a mind like water clear,
A song that swells and dies upon the ear,
These are enough for thee!
## VII
FROM the garden of Heaven a western breeze
Blows through the leaves of my garden of earth;
With a love like a huri I'ld take mine ease,
And wine! bring me wine, the giver of mirth!
To-day the beggar may boast him a king,
His banqueting-hall is the ripening field,
And his tent the shadow that soft clouds fling.
A tale of April the meadows unfold —
Ah, foolish for future credit to slave,
And to leave the cash of the present untold!
Build a fort with wine where thy heart may brave
The assault of the world; when thy fortress falls,
The relentless victor shall knead from thy dust
The bricks that repair its crumbling walls.
Trust not the word of that foe in the fight!
Shall the lamp of the synagogue lend its flame
To set thy monastic torches alight?
Drunken am I, yet place not my name
In the Book of Doom, nor pass judgment on it;
Who knows what the secret finger of Fate
Upon his own white forehead has writ!
And when the spirit of Hafiz has fled,
Follow his bier with a tribute of sighs;
Though the ocean of sin has closed o'er his head,
He may find a place in God's Paradise.
Stanza 1. — Those who have seen a Persian garden will not find it difficult to understand why it should play so large a part in Persian poetry. Often enough you may pass with one step out of a barren desert of dust and stones into one of these green and fertile spots, full of violets in the spring, and of roses and lilies in the early summer; and from the blinding glare of a Persian sun into a cool and shadowy retreat planted with great plane-trees. The water which flows in numberless streams through the garden, and leaps in countless fountains, has worked all the miracle. The change from desert to flowery paradise is one of those strong contrasts so common in the East which take hold of the imagination of all who see them.
Stanza 3. — That is, do not attempt to light the torches of a Mahommadan monastery from the lamp of a Jewish synagogue. One of the most famous of the Prophet's sayings is: there is no monasticism in Islam. Nevertheless, from the time of Abu Bekr and Ali onwards, such religious associations grew up and flourished. Nearly all the celebrated doctors of whom the Sufis boast in the first six hundred years after the Hejra belonged to them.
"Verily our messengers write down that which ye deceitfully devise," says the Koran (chap. x.). Two guardian angels attend every man and write down his actions; they are changed daily and a fresh pair takes their place. The books which they have written shall be produced on the Day of judgment.
Stanza 4. — It was this verse which decided the right of Hafiz to receive honourable burial.
## VIII
THE rose has flushed red, the bud has burst,
And drunk with joy is the nightingale
Hail, Sufis! lovers of wine, all hail!
For wine is proclaimed to a world athirst.
Like a rock your repentance seemed to you;
Behold the marvel! of what avail
Was your rock, for a goblet has cleft it in two!
Bring wine for the king and the slave at the gate
Alike for all is the banquet spread,
And drunk and sober are warmed and fed.
When the feast is done and the night grows late,
And the second door of the tavern gapes wide,
The low and. the mighty must bow the head
'Neath the archway of Life, to meet what . . . outside?
Except thy road through affliction pass,
None may reach the halting-station of mirth
God's treaty: Am I not Lord of the earth?
Man sealed with a sigh: Ah yes, alas!
Nor with Is nor Is Not let thy mind contend
Rest assured all perfection of mortal birth
In the great Is Not at the last shall end.
For Assaf's pomp, and the steeds of the wind,
And the speech of birds, down the wind have fled,
And he that was lord of them all is dead;
Of his mastery nothing remains behind.
Shoot not thy feathered arrow astray!
A bow-shot's length through the air it has sped,
And then . . . dropped down in the dusty way.
But to thee, oh Hafiz, to thee, oh Tongue
That speaks through the mouth of the slender reed,
What thanks to thee when thy verses speed
From lip to lip, and the song thou hast sung?
Stanza 3. — When God had created man and made him wiser than the angels, he bound him to himself by a solemn treaty. "Am I not thy Lord who has created thee?" he demanded, and man answered "Yes." But the Arabic word bala, which signifies assent, means also sorrow, and they say that the first of our fathers knew full well what a terrible gift was that life which he had received from his Lord, and sealed the treaty with a seal of grief. Therefore since the earliest day, life and sorrow have gone hand in hand, bound together by the first great pact between God and man.
Stanza 4. — Compare François Villon's rough and powerful treatment of the same theme: —
"Où sont de Vienne et de Grenobles
Le Dauphin, les preux, les senés?
Où de Dijon, Sallin et Dolles,
Les sires et les fils aînés?
Où autant de leurs gens privés,
Hérauts, trompettes, poursuivants?
Ont-ils bien bouté sous le nez? . . .
Autant en emporte le vent!"
Solomon, the type of human greatness, is the King whose mastery has left nothing behind. He harnessed the wind as a steed to his chariot, he spoke with the birds in their own tongue, and the wise and magnificent Assaf was his minister. Upon his seal was engraved the name of God which is unknown to men and before which the Jinn and the Angels must bow down. It was with this seal that he fastened up the bottles in which he imprisoned the Jinn — those bottles which those fishermen in the "Arabian Nights" pull up in their nets.
## IX
OH Cup-bearer, set my glass afire
With the light of wine! oh minstrel, sing:
The world fulfilleth my heart's desire!
Reflected within the goblet's ring
I see the glow of my Love's red cheek,
And scant of wit, ye who fail to seek
The pleasures that wine alone can bring!
Let not the blandishments be checked
That slender beauties lavish on me,
Until in the grace of the cypress decked,
My Love shall come like a ruddy pine-tree
He cannot perish whose heart doth hold
The life love breathes-though my days are told,
In the Book of the World lives my constancy.
But when the Day of Reckoning is here,
I fancy little will be the gain
That accrues to the Sheikh for his lawful cheer,
Or to me for the draught forbidden I drain.
The drunken eyes of my comrades shine,
And I too, stretching my hand to the wine,
On the neck of drunkenness loosen the rein.
Oh wind, if thou passest the garden close
Of my heart's dear master, carry for me
The message I send to him, wind that blows!
"Why hast thou thrust from thy memory
My hapless name?" breathe low in his ear;
"Knowest thou not that the day is near
When nor thou nor any shall think on me?"
If with tears, oh Hafiz, thine eyes are wet,
Scatter them round thee like grain, and snare
The Bird of joy when it comes to thy net.
As the tulip shrinks from the cold night air,
So shrank my heart and quailed in the shade
Oh Song-bird Fortune, the toils are laid,
When shall thy bright wings lie pinioned there?
The heavens' green sea and the bark therein,
The slender bark of the crescent moon,
Are lost in thy bounty's radiant noon,
Vizir and pilgrim, Kawameddin!
Stanza 1. — This poem is addressed to the Vizir of Sultan Oweis of Baghdad, Hadji Kawameddin, who founded a college for Hafiz in Shiraz. With true Persian exaggeration the poet must needs write to his patron much in the same terms in which a lover would write to his mistress; but his words, though they sound strangely to our ears, are nothing more than the Oriental way of saying, "Awake, my St. John!"
The mystical interpretation of the first few lines is said to be: As the wine glows in the cup like the reflection of a ruddy cheek, so in the goblet of my heart I have seen the reflection of God, the true Beloved.
Stanza 6. — It is related that upon a certain occasion when Hafiz was feasting with the Vizir in the latter's garden, a servant handed to him a goblet of wine, and as he took it he saw in it the reflection of the crescent moon overhead. The incident suggested this verse to him. I should say that the anecdote was of doubtful authenticity.
## X
SINGER, sweet Singer, fresh notes strew,
Fresh and afresh and new and new!
Heart-gladdening wine thy lips imbrue,
Fresh and afresh and new and new!
Saki, thy radiant feet I hail;
Flush with red wine the goblets pale,
Flush our pale cheeks to drunken hue,
Fresh and afresh and new and new!
Then with thy love to toy with thee,
Rest thee, ah, rest! where none can see
Seek thy delight, for kisses sue,
Fresh and afresh and new and new!
Here round thy life the vine is twined;
Drink I for elsewhere what wine wilt find?
Drink to her name, to hours that flew,
Hours ever fresh and new and new!
She that has stolen my heart from me,
How does she wield her empery?
Paints and adorns and scents her too,
Fresh and afresh and new and new!
Wind of the dawn that passest by,
Swift to the street of my fairy hie,
Whisper the tale of Hafiz true,
Fresh and afresh and new and new!
This song is not to be found in the best editions of the Divan, and is believed to be spurious; but it is printed in most of the popular editions, and is as widely known as any of the poems which pass with a better right under the name of Hafiz. It is set to a soft and well-nigh tuneless air which sounds like dream music, or the echo of something very beautiful coming from a great distance, the singer ending on an almost whispered repetition of the first exquisite phrase. I have been told that the boatmen on the Ganges sing it as they row, and the monotonous accompaniment of the water under the oars must be even more fitting to the melody than that of the lute strings.
## XI
MIRTH, Spring, to linger in a garden fair,
What more has earth to give? All ye that wait,
Where is the Cup-bearer, the flagon where?
When pleasant hours slip from the hand of Fate,
Reckon each hour as a certain gain;
Who seeks to know the end of mortal care
Shall question his experience in vain.
Thy fettered life hangs on a single thread —
Some comfort for thy present ills devise,
But those that time may bring thou shalt not dread.
Waters of Life and Irem's Paradise —
What meaning do our dreams and pomp convey,
Save that beside a mighty stream, wide-fed,
We sit and sing of wine and go our way!
The modest and the merry shall be seen
To boast their kinship with a single voice;
There are no differences to choose between,
Thou art but flattering thy soul with choice!
Who knows the Curtain's secret? . . . Heaven is mute
And yet with Him who holds the Curtain, e'en
With Him, oh Braggart, thou would'st raise dispute!
Although His thrall shall miss the road and err,
'Tis but to teach him wisdom through distress,
Else Pardon and Compassionate Mercy were
But empty syllables and meaningless.
The Zealot thirsts for draughts of Kausar's wine,
And Hafiz doth an earthly cup prefer —
But what, between the two, is God's design?
Stanza 2. — I have found no explanation of these difficult lines, and, for want of a better, I venture to suggest the following: the Garden of Irem, as has been said in the Note to Poem II., was a mimic Paradise constructed by a certain fabulous King Shedad, who wished to be considered a rival to his Maker by his fellows, for which temerity a swift and sharp judgment fell upon him; the River of Life is one of the many streams which waters the divine Paradise. To my thinking, Hafiz takes the one as a type of the wildest human ambition, the other as a part of the most beautiful vision which the mind of man has conceived. And to what does it all amount? he asks. Only to this: that we are like to one who sits and dreams upon the banks of a mighty and resistless river, fed from many sources, and sings, if he be wise, his song of praise, and so departs.
Stanza 4. — The river Kausar is another of the streams of Paradise; indeed, it is said to be the central spring from whence all the others flow. A part of its waters are led into a great square lake, a month's journey in compass. On the banks of this lake the souls of good Mahommadans rest and find refreshment after they have crossed the terrible bridge, sharper than the edge of a sword, which is laid over the midst of Hell. The waters of the lake are whiter than silver and sweeter than musk. Round it are set as many cups as there are stars in the firmament, and he who has drunk of it shall thirst no more.
## XII
WHERE is my ruined life, and where the fame
Of noble deeds?
Look on my long-drawn road, and whence it came,
And where it leads!
Can drunkenness be linked to piety
And good repute?
Where is the preacher's holy monody,
Where is the lute?
From monkish cell and lying garb released,
Oh heart of mine,
Where is the Tavern fane, the Tavern priest,
Where is the wine?
Past days of meeting, let the memory
Of you be sweet!
Where are those glances fled, and where for me
Reproaches meet?
His friend's bright face warms not the enemy
When love is done —
Where is the extinguished lamp that made night day,
Where is the sun?
Balm to mine eyes the dust, my head I bow
Upon thy stair.
Where shall I go, where from thy presence? thou
Art everywhere.
Look not upon the dimple of her chin,
Danger lurks there!
Where wilt thou hide, oh trembling heart, fleeing in
Such mad haste — where?
To steadfastness and patience, friend, ask not
If Hafiz keep —
Patience and steadfastness I have forgot,
And where is sleep?
## XIII
LADY that hast my heart within thy hand,
Thou heed'st me not; and if thou turn thine ear
Unto the wise, thou shalt not understand —
Behold the fault is thine, our words were clear.
For all the tumult in my drunken brain
Praise God! who trieth not His slave in vain;
Nor this world nor the next shall make me fear!
My weary heart eternal silence keeps —
I know not who has slipped into my heart;
Though I am silent, one within me weeps.
My soul shall rend the painted veil apart.
Where art thou, Minstrel! touch thy saddest strings
Till clothed in music such as sorrow sings,
My mournful story from thy zither sweeps.
Lo, not at any time I lent mine ear
To hearken to the glories of the earth;
Only thy beauty to mine eyes was dear.
Sleep has forsaken me, and from the birth
Of night till day I weave bright dreams of thee;
Drunk with a hundred nights of revelry,
Where is the tavern that sets forth such cheer!
My heart, sad hermit, stains the cloister floor
With drops of blood, the sweat of anguish dire;
Ah, wash me clean, and o'er my body pour
Love's generous wine! the worshippers of fire
Have bowed them down and magnified my name,
For in my heart there burns a living flame,
Transpiercing Death's impenetrable door.
What instrument through last night's silence rang?
My life into his lay the minstrel wove,
And filled my brain with the sweet song he sang.
It was the proclamation of thy love
That shook the strings of Life's most secret lyre,
And still my breast heaves with last night's desire,
For countless echoes from that music sprang.
And ever, since the time that Hafiz heard
His Lady's voice, as from a rocky hill
Reverberates the softly spoken word,
So echoes of desire his bosom fill.
## XIV
THE nightingale with drops of his heart's blood
Had nourished the red rose, then came a wind,
And catching at the boughs in envious mood,
A hundred thorns about his heart entwined.
Like to the parrot crunching sugar, good
Seemed the world to me who could not stay
The wind of Death that swept my hopes away.
Light of mine eyes and harvest of my heart,
And mine at least in changeless memory!
Ah, when he found it easy to depart,
He left the harder pilgrimage to me!
Oh Camel-driver, though the cordage start,
For God's sake help me lift my fallen load,
And Pity be my comrade of the road!
My face is seamed with dust, mine eyes are wet.
Of dust and tears the turquoise firmament
Kneadeth the bricks for joy's abode; and yet . . .
Alas, and weeping yet I make lament!
Because the moon her jealous glances set
Upon the bow-bent eyebrows of my moon,
He sought a lodging in the grave-too soon!
I had not castled, and the time is gone.
What shall I play? Upon the chequered floor
Of Night and Day, Death won the game-forlorn
And careless now, Hafiz can lose no more.
Stanza 1. — Hafiz wrote this poem upon the death of his son.
Stanza 3. — Rosenzweig, in his edition of the Divan, says that the allusion is to the dust and water which God kneaded into the body of Adam, and that, out of derision, Hafiz; calls the human body a house of joy.
The moon, according to Persian superstition, has a baneful influence upon human life.
Stanza 4. — Rosenzweig says that "I had not castled" means that Hafiz; had not taken the precaution of marrying his son, and so securing for himself grandchildren who would have been a consolation to him on their father's death. For that reason he had nothing more to lose, and was indifferent as to what his next move in the game should be.
## XV
RETURN! that to a heart wounded full sore
Valiance and strength may enter in; return!
And Life shall pause at the deserted door,
The cold dead body breathe again and burn.
Oh come! and touch mine eyes, of thy sweet grace,
For I am blind to all but to thy face.
Open the gates and bid me see once more!
Like to a cruel Ethiopian band,
Sorrow despoiled the kingdom of my heart
Return! glad Lord of Rome, and free the land;
Before thine arms the foe shall break and part.
See now, I hold a mirror to mine eyes,
And nought but thy reflection therein lies;
The glass speaks truth to them that understand.
Night is with child, hast thou not heard men say?
"Night is with child! what will she bring to birth?"
I sit and ask the stars when thou'rt away.
Oh come! and when the nightingale of mirth
Pipes in the Spring-awakened garden ground,
In Hafiz' heart shall ring a sweeter sound,
Diviner nightingales attune their lay.
Stanza 3.— "Night is with child" — a Persian proverb extraordinarily suggestive of the clear, deep, Eastern sky. The sight seems to slip through between the stars and penetrate a darkness which is big with possibilities.
## XVI
WHAT is wrought in the forge of the living and life —
All things are nought! Ho! fill me the bowl,
For nought is the gear of the world and the strife!
One passion has quickened the heart and the soul,
The Beloved's presence alone they have sought —
Love at least exists; yet if Love were not,
Heart and soul would sink to the common lot —
All things are nought!
Like an empty cup is the fate of each,
That each must fill from Life's mighty flood;
Nought thy toil, though to Paradise gate thou reach,
If Another has filled up thy cup with blood;
Neither shade from the sweet-fruited trees could be bought
By thy praying-oh Cypress of Truth, dost not see
That Sidreh and Tuba were nought, and to thee
All then were nought!
The span of thy life is as five little days,
Brief hours and swift in this halting-place;
Rest softly, ah rest! while the Shadow delays,
For Time's self is nought and the dial's face.
On the lip of Oblivion we linger, and short
Is the way from the Lip to the Mouth where we pass
While the moment is thine, fill, oh Saki, the glass
Ere all is nought!
Consider the rose that breaks into flower,
Neither repines though she fade and die —
The powers of the world endure for an hour,
But nought shall remain of their majesty.
Be not too sure of your crown, you who thought
That virtue was easy and recompense yours;
From the monastery to the wine-tavern doors
The way is nought
What though I, too, have tasted the salt of my tears,
Though I, too, have burnt in the fires of grief,
Shall I cry aloud to unheeding ears?
Mourn and be silent! nought brings relief.
Thou, Hafiz, art praised for the songs thou hast wrought,
But bearing a stained or an honoured name,
The lovers of wine shall make light of thy fame —
All things are nought!
Stanza 2. — These lines are exceedingly mysterious, as, indeed, is the whole poem. I have looked for an explanation of them in other editions of Hafiz, but have found little more than a bare translation of the Persian words. For the meaning of this stanza, see Introduction, .
Sidreh and Tuba are two trees in the Garden of Paradise. The former is the abode of the angel Gabriel. Concerning the latter Sale says: "They fable that it stands in the palace of Mahommad, though a branch of it will reach to the house of every true believer; that it will be laden with pomegranates, grapes, dates, and other fruits of surprising bigness, and of tastes unknown to mortals. So that if a man desire to eat of any particular kind of fruit, it will immediately be presented to him; or if he choose flesh, birds ready dressed will be set before him, according to his wish. They add that the boughs of this tree will spontaneously bend down to the hand of the person who would gather of its fruits, and that it will supply the blessed not only with food, but also with silken garments and beasts to ride on, ready saddled and bridled and adorned with rich trappings, which will burst forth from its fruits; and that this tree is so large that a person mounted on the fleetest horse would not be able to gallop from one end of its shade to the other in a hundred years." — Introduction to the Koran.
Stanza 4. — He means either facilis descensus Averni, or, more probably, that a great number of those upon whom the orthodox look askance will be found to have equal claim to reward, since the distinction between Sufi and orthodox is in fact nothing.
Stanza 5.— "The lovers of wine" — that is to say the Sufis, who will be equally indifferent whether he comes to them with or without trailing clouds of human approbation, since they will judge of his worth by a different standard.
## XVII
LAY not reproach at the drunkard's door
Oh Fanatic, thou that art pure of soul;
Not thine on the page of life to enrol
The faults of others! Or less or more
I have swerved from my path — keep thou to thine own
For every man when he reaches the goal
Shall reap the harvest his hands have sown.
Leave me the hope of a former grace —
Till the curtain is lifted none can tell
Whether in Heaven or deepest Hell,
Fair or vile, shall appear his face.
Alike the drunk and the strict of fare
For his mistress yearns — in the mosque Love doth dwell
And the church, for his lodging is everywhere.
If without the house of devotion I stand,
I am not the first to throw wide the door
My father opened it long before,
The eternal Paradise slipped from his hand.
All you that misconstrue my words' intent,
I lie on the bricks of the tavern floor,
And a brick shall serve me for argument.
Heaven's garden future treasures may yield —
Ah, make the most of earth's treasury!
The flickering shade of the willow-tree,
And the grass-grown lip of the fruitful field.
Trust not in deeds — the Eternal Day
Shall reveal the Creator's sentence on thee;
But till then, what His finger has writ, who can say.
Bring the cup in thine hand to the Judgment-seat;
Thou shalt rise, oh Hafiz, to Heaven's gate
From the tavern where thou hast tarried late.
And if thou hast worshipped wine, thou shalt meet
The reward that the Faithful attain;
If such thy life, then fear not thy fate,
Thou shalt not have lived and worshipped in vain.
Stanza 3. — The allusion is to the expulsion of Adam from the Garden of Eden.
Stanza 4. — Concerning the Last judgment, a beautiful tradition relates that there are seven degrees of punishment, but eight of blessedness, because God's mercy exceeds His justice.
## XVIII
SLAVES of thy shining eyes are even those
That diadems of might and empire bear;
Drunk with the wine that from thy red lip flows,
Are they that e'en the grape's delight forswear.
Drift, like the wind across a violet bed,
Before thy many lovers, weeping low,
And clad like violets in blue robes of woe,
Who feel thy wind-blown hair and bow the head.
Thy messenger the breath of dawn, and mine
A stream of tears, since lover and beloved
Keep not their secret; through my verses shine,
Though other lays my flower's grace have proved
And countless nightingales have sung thy praise.
When veiled beneath thy curls thou passest, see,
To right and leftward those that welcome thee
Have bartered peace and rest on thee to gaze!
But thou that knowest God by heart, away!
Wine-drunk, love-drunk, we inherit Paradise,
His mercy is for sinners; hence and pray
Where wine thy cheek red as red erghwan dyes,
And leave the cell to faces sinister.
Oh Khizr, whose happy feet bathed in life's fount,
Help one who toils afoot-the horsemen mount
And hasten on their way; I scarce can stir.
Ah, loose me not! ah, set not Hafiz free
From out the bondage of thy gleaming hair!
Safe only those, safe, and at liberty,
That fast enchained in thy linked ringlets are.
But from the image of his dusty cheek
Learn this from Hafiz: proudest heads shall bend,
And dwellers on the threshold of a friend
Be crownèd with the dust that crowns the meek.
Stanza 1. — Blue is the Persian colour of mourning. Hafiz compares the weeping lovers, clad in robes of grief, to a bed of violets, and as the violets bow their heads when the wind passes over them, so they bow down when their mistress passes by with flowing curls.
Stanza 3.— "Erghwan," the Syringa Persica or Persian lilac. In the early spring, before it comes into leaf, it is covered with buds of a beautiful reddish-purple colour.
"Khizr," a prophet whom the Mahommadans confound with Phineas, Elias, and St. George, saying that his soul passed by metempsychosis successively through all three. He discovered the fountain of life and drank of it, thereby making himself immortal. It is said that he guided Alexander to the same fountain, which lay in the Land of Darkness. It was he, too, for whom Moses set out to seek when he had been informed by God that Al Khizr was wiser than he. He found him seated on a rock, at the meeting of the two seas, and followed him for a time, learning wisdom from him, as is related in the eighteenth chapter of the Koran. His name signifies Green; wherever his feet rested, the earth was covered with green herbs.
Hafiz looked upon the prophet Al Khizr as one of his special guardians. About four Persian miles from Shiraz there is a spot called Pir-i-Sabz, the Old Green Man; whosoever should pass forty nights in it without sleeping, on the fortieth night Al Khizr would appear to him and confer upon him the immortal gift of song. Hafiz in his youth fell in love with a beautiful girl of Shiraz called Shakh-i-Nahat, and in order to win her heart he determined to meet Al Khizr and receive from him the art of poetry. For thirty-nine mornings he walked beneath the windows of Shakh-i-Nahat, at noon he ate, then he slept, and at night he kept watch, undismayed by the terrible apparition of a fierce lion which was his nightly companion. At length, on the fortieth morning, Shakh-i-Nahat called him into her house and told him that she was ready to become his wife, for she preferred a man of genius to the son of a king. She would have kept him with her, but Hafiz, though he had gained his original end, was now filled with desire to become a poet, and insisted upon keeping his fortieth vigil. That night an old man dressed in green garments came to him and brought him a cup of the water of immortality.
## XIX
WHAT drunkenness is this that brings me hope —
Who was the Cup-bearer, and whence the wine?
That minstrel singing with full voice divine,
What lay was his? for 'mid the woven rope
Of song, he brought word from my Friend to me
Set to his melody.
The wind itself bore joy to Solomon;
The Lapwing flew from Sheba's garden close,
Bringing good tidings of its queen and rose.
Take thou the cup and go where meadows span
The plain, whither the bird with tuneful throat
Has brought Spring's sweeter note.
Welcome, oh rose, and full-blown eglantine!
The violets their scented gladness fling,
Jasmin breathes purity-art sorrowing
Like an unopened bud, oh heart of mine?
The wind of dawn that sets closed blossoms free
Brings its warm airs to thee.
Saki, thy kiss shall still my bitter cry!
Lift up your grief-bowed heads, all ye that weep,
The Healer brings joy's wine-cup — oh, drink deep!
Disciple of the Tavern-priest am I;
The pious Sheikh may promise future bliss,
He brings me where joy is.
The greedy glances of a Tartar horde
To me seemed kind — my foeman spared me not
Though one poor robe was all that I had got.
But Heaven served Hafiz, as a slave his lord,
And when he fled through regions desolate,
Heaven brought him to thy gate.
Stanza 2. — See Note to Stanza 1 of Poem III.
Stanza 5.— "Narrow-eyedness" is the exact translation of the Persian word for greed, and there is consequently, in the original, a play of meaning between the physical and moral attributes of the Tartars.
It is significant that Hafiz should choose the "narrow-eyed" Tartar robbers as types of cruelty. Just as the Anglo-Saxons prayed to be delivered from the Danes, so a clause in the Persian litany of the thirteenth and fourteenth centuries might have been: "From the power of the Tartars, good Lord, deliver us!" First under Hulagu, and then under Timur, they overran and devastated Persia. The destruction wrought by them was very similar to that wrought by the Arab conquerors in the Roman provinces of North Africa. They rased to the ground great cities; they reduced populous and fertile regions to a barren desert by breaking down the old reservoirs and destroying the irrigating system, completely changing the physical conditions of parts of the country. In the mountains to the north of Tehran, for instance, there are villages bearing names the etymology of which points to their having stood at the outlet of a reservoir of which no other trace remains, and it is said that the country surrounding the town was far more thoroughly irrigated before the Tartar invasion, and supported a larger population. The invaders completely destroyed the ancient city of Rhages, which lay at a distance of about three miles from the modern capital. The same thing happened in North Africa. The ruins of Roman towns are to be found in country which must once have been fertile, but which is now reconquered by the sands of the Sahara.
"One poor robe." The Persian runs "man dervish-i-yek kaba " — i.e. I, a poor man of one robe — dervish signifying in its primary sense, it is hardly necessary to say, poor. I should think that the double meaning is significant. In its mystical sense, the poem describes how Hafiz found consolation in the ecstatic drunkenness of the Sufis, in the minstrel's song, or divine message, which brought him a word from God; and when finally the last shred of his orthodoxy had been tom from him, when in his desperate struggle with existence he was forced to abandon even his dervish robe, Heaven mercifully showed him a safe refuge in the Sufi doctrines.
## XX
FROM out the street of So-and-So,
Oh wind, bring perfumes sweet to me
For I am sick and pale with woe;
Oh bring me rest from misery!
The dust that lies before her door,
Love's long desired elixir, pour
Upon this wasted heart of mine —
Bring me a promise and a sign!
Between the ambush of mine eyes
And my heart's fort there's enmity —
Her eye-brow's bow, the dart that flies,
Beneath her lashes, bring to me!
Sorrow and absence, glances cold,
Before my time have made me old;
A wine-cup from the hand of Youth
Bring me for pity and for ruth!
Then shall all unbelievers taste
A draught or two of that same wine;
But if they like it not, oh haste!
And let joy's flowing cup be mine.
Cup-bearer, seize to-day, nor wait
Until to-morrow! — or from Fate
Some passport to felicity,
Some written surety bring to me!
My heart threw back the veil of woe,
Consoled by Hafiz melody:
From out the street of So-and-So,
Oh wind, bring perfumes sweet to me!
## XXI
NOT all the sum of earthly happiness
Is worth the bowed head of a moment's pain,
And if I sell for wine my dervish dress,
Worth more than what I sell is what I gain!
Land where my Lady dwells, thou holdest me
Enchained; else Fars were but a barren soil,
Not worth the journey over land and sea,
Not worth the toil!
Down in the quarter where they sell red wine,
My holy carpet scarce would fetch a cup
How brave a pledge of piety is mine,
Which is not worth a goblet foaming up!
Mine enemy heaped scorn on me and said
"Forth from the tavern gate!" Why am I thrust
From off the threshold? is my fallen head
Not worth the dust?
Wash white that travel-stained sad robe of thine!
Where word and deed alike one colour bear,
The grape's fair purple garment shall outshine
Thy many-coloured rags and tattered gear.
Full easy seemed the sorrow of the sea
Lightened by hope of gain — hope flew too fast
A hundred pearls were poor indemnity,
Not worth the blast.
The Sultan's crown, with priceless jewels set,
Encircles fear of death and constant dread
It is a head-dress much desired — and yet
Art sure 'tis worth the danger to the head?
'Twere best for thee to hide thy face from those
That long for thee; the Conqueror's reward
Is never worth the army's long-drawn woes,
Worth fire and sword.
Ah, seek the treasure of a mind at rest
And store it in the treasury of Ease;
Not worth a loyal heart, a tranquil breast,
Were all the riches of thy lands and seas!
Ah, scorn, like Hafiz, the delights of earth,
Ask not one grain of favour from the base,
Two hundred sacks of jewels were not worth
Thy soul's disgrace!
Stanza 1. — Sir Henry Layard gives the following account of a party of dervishes with whom he travelled, from which it would appear that the contempt of Hafiz for the dervish habit was not wholly uncalled for: "They were a picturesque and motley crew. One or two of them were what the Persians call luti, young men with well-dyed curls, long garments, and conical caps embroidered in many colours — debauched and dissolute fellows, who, under the guise of poverty, and affecting abstinence and piety, were given to every manner of vice. Others were half-naked savages, with hair hanging down their backs, and the skins of gazelles on their shoulders — barefooted, dirty, and covered with vermin. They carried heavy iron maces, and seemed more disposed to exact than to ask for charity. As they went along they shouted 'Yah Allah! yah Muhammad! yah Ali!' They all had slung from their shoulders the carved cocoa-nut shell, which is indispensable to the dervish, and serves for carrying food and for drinking purposes. Round their necks they wore charms and amulets, with beads and coloured strings and tassels."He goes on to say: "Most Persian dervishes, although they have great pretensions to sanctity, by which they impose upon the people, high and low, are without any religion. They are, however, credited with working miracles, and with being able to give efficacious charms. . . . Although these dervishes are rank impostors, and generally arrant scoundrels, they maintain their influence over the ignorant and superstitious Persians of all classes, who greatly fear, and do not dare to offend them. Consequently no one ventures to refuse them admission into their houses, and even into the women's apartments, where those who go stark-naked, and are looked upon as specialty holy and protected by Allah and Ali, can enter with impunity. Sometimes they will demand a specific sum of money from a rich man, and if he refuses to pay it, will establish themselves in the gateway or porch of his dwelling, or outside close to it, and enclosing a small plot of ground, sow wheat or plant flowers, and remain until what they ask for is paid them, hooting hideously day and night, calling upon Mohammad, Ali, and the Imams, or blowing on a buffalo's horn so as to disturb the whole neighbourhood. The owner and inmates of the house are helpless. They do not dare to remove by force the holy men." — Early Adventures.
Stanza 2. — That is to say, the prayer-carpet of the orthodox Mussulman had not enough value to procure for him so much as one glass of Sufi wine. Nor was he worthy to lay his head even upon the dusty steps of the tavern — the place of instruction in Sufi doctrine.
Stanza 3. — To be clothed in one colour is the Persian idiom for sincerity. He means that the single purple robe of the grape is worth more than the hypocritical garment of the dervish, all torn and patched with long journeying-in the wrong road.
Stanza 5. — So far I have endeavoured to give the mystical interpretation of the poem. There is, however, a story attached to it which turns it into a historical rather than a theological document. It is related that the King of the Deccan, Mahmud Shah Bahmani, had heard of the fame of Hafiz, and having a pretty taste in literature, was desirous of attracting him to his court. Accordingly he ordered his Vizir, Mir Feiz Allah Inju, to send the poet a sufficient sum to pay for his journey from Shiraz. Hafiz resolved to accept the invitation. He wound up his affairs in his native town, using some of the money the Sultan had sent him in paying his debts and in making gifts to his sister's children, and set forth upon his journey. But when he reached the town of Lar he found there an acquaintance in very bad case, having been plundered by robbers and reduced to a state of beggary. Hafiz was moved to compassion and gave him the remainder of the money which Mahmud Shah had sent to him. He was now himself unable to continue his journey for want of means, and perhaps it was bitter experience that taught him that in very fact his prayer-carpet would not fetch him a glass of wine, and that without the necessary silver pieces he would be thrust from out the tavern doors. From these straits he was rescued by two friendly merchants, who were also on their way to India, and who offered to pay his expenses to Hormuz, and there place him on a vessel of Mahmud Shah's which was coming to fetch them. Hafiz accepted the offer, went to Hormuz, and embarked on the ship. But before they had left the port a violent storm arose, and persuaded the poet that no advantages he might reap from the journey would be worth the sorrow of the sea. Under pretext of bidding farewell to some friends, he disembarked, and in all haste made the best of his way back to Shiraz, sending to Feiz Allah this poem as an excuse for failing to keep his engagement. The Vizir read it to Mahmud Shah, who was transported by the beauty of the verses and the philosophic dignity in which Hafiz had cloaked his fears of the dangers of the road and the discomforts of seasickness. With singular generosity he sent the defaulting poet a further present, consisting of some at least of the riches of his lands and seas.
## XXII
THE rose is not fair without the beloved's face,
Nor merry the Spring without the sweet laughter of wine;
The path through the fields, and winds from a flower strewn place,
Without her bright check, which glows like a tulip fine,
Nor winds softly blowing, fields deep in corn, are fair.
And lips like to sugar, grace like a flower that sways,
Are nought without kisses many and dalliance sweet;
If thousands of voices sang not the rose's praise,
The joy of the cypress her opening bud to greet,
Nor dancing of boughs nor blossoming rose were fair.
Though limned by most skilful fingers, no pictures please
Unless the beloved's image is drawn therein;
The garden and flowers, and hair flowing loose on the breeze,
Unless to my Lady's side I may strive and win,
Nor garden, nor flowers, nor loose flying curls are fair.
Hast seen at a marriage-feast, when the mirth runs high,
The revellers scatter gold with a careless hand?
The gold of thy heart, oh Hafiz, despised doth lie,
Not worthy thy love to be cast by a drunken band
At the feet of her who is fairer than all that's fair.
## XXIII
MY lady, that did change this house of mine
Into a heaven when that she dwelt therein,
From head to foot an angel's grace divine
Enwrapped her; pure she was, spotless of sin;
Fair as the moon her countenance, and wise;
Lords of the kind and tender glance, her eyes
With an abounding loveliness did shine.
Then said my heart: Here will I take my rest!
This city breathes her love in every part.
But to a distant bourne was she addressed,
Alas! he knew it not, alas, poor heart!
The influence of some cold malignant star
Has loosed my hand that held her, lone and far
She journeyeth that lay upon my breast.
Not only did she lift my bosom's veil,
Reveal its inmost secret, but her grace
Drew back the curtain from Heaven's mansions pale,
And gave her there an eternal dwelling-place.
The flower-strewn river lip and meadows fair,
The rose herself but fleeting treasures were,
Regret and Winter follow in their trail.
Dear were the days which perished with my friend —
Ah, what is left of life, now she is dead,
All wisdomless and profitless I spend!
The nightingale his own life's blood doth shed,
When, to the kisses of the wind, the morn
Unveils the rose's splendour — with his torn
And jealous breast he dyes her petals red.
Yet pardon her, oh Heart, for poor wert thou,
A humble dervish on the dusty way;
Crowned with the crown of empire was her brow,
And in the realms of beauty she bore sway.
But all the joy that Hafiz' hand might hold,
Lay in the beads that morn and eve he told,
Worn with God's praise; and see! he holds it now.
This poem is said to have been written by Hafiz; upon the death of his wife.
## XXIV
NOT one is filled with madness like to mine
In all the taverns! my soiled robe lies here,
There my neglected book, both pledged for wine.
With dust my heart is thick, that should be clear,
A glass to mirror forth the Great King's face;
One ray of light from out Thy dwelling-place
To pierce my night, oh God! and draw me near.
From out mine eyes unto my garment's hem
A river flows; perchance my cypress-tree
Beside that stream may rear her lofty stem,
Watering her roots with tears. Ah, bring to me
The wine vessel! since my Love's cheek is hid,
A flood of grief comes from my heart unbid,
And turns mine eyes into a bitter sea!
Nay, by the hand that sells me wine, I vow
No more the brimming cup shall touch my lips,
Until my mistress with her radiant brow
Adorns my feast-until Love's secret slips
From her, as from the candle's tongue of flame,
Though I, the singèd moth, for very shame,
Dare not extol Love's light without eclipse.
Red wine I worship, and I worship her —
Speak not to me of anything beside,
For nought but these on earth or heaven I care.
What though the proud narcissus flowers defied
Thy shining eyes to prove themselves more bright,
Yet heed them not! those that are clear of sight
Follow not them to whom all light's denied.
Before the tavern door a Christian sang
To sound of pipe and drum, what time the earth
Awaited the white dawn, and gaily rang
Upon mine ear those harbingers of mirth:
"If the True Faith be such as thou dost say,
Alas! my Hafiz, that this sweet To-day
Should bring unknown To-morrow to the birth!"
Stanza 5. — Shah Shudja, as has been related in the Introduction, was not always on the best of terms with Hafiz, partly because he was jealous of the latter's fame as a poet, and partly because Hafiz had been the protégé of Shah Shudja's former rival, Abu Ishac. Accordingly the King looked about for some means of doing the poet an injury, nor was it long before he found what he sought. He accused Hafiz of denying the Resurrection, basing the accusation upon the last couplet of this poem — the last three lines of the present translation and cited him before the Ulema as an infidel. But Hafiz; was too many for him. Before the day on which he was to answer the charge against himself, he inserted another couplet into the ode, in which he stated that the dangerous lines did not express his own opinion, but that of a heretical Christian. He came off with flying colours; for not only was he entirely cleared, but it was also acknowledged that he had dealt a good blow on behalf of the Mahommadan religion, since he had shown up one of the errors of the infidel.
## XXV
THE days of absence and the bitter nights
Of separation, all are at an end!
Where is the influence of the star that blights
My hope? The omen answers: At an end!
Autumn's abundance, creeping Autumn's mirth,
Are ended and forgot when o'er the earth
The wind of Spring with soft warm feet doth wend.
The Day of Hope, hid beneath Sorrow's veil,
Has shown its face — ah, cry that all may hear:
Come forth! the powers of night no more prevail!
Praise be to God, now that the rose is near
With long-desired and flaming coronet,
The cruel stinging thorns all men forget,
The wind of Winter ends its proud career.
The long confusion of the nights that were,
Anguish that dwelt within my heart, is o'er;
'Neath the protection of my lady's hair
Grief nor disquiet come to me no more.
What though her curls wrought all my misery,
My lady's gracious face can comfort me,
And at the end give what I sorrow for.
Light-hearted to the tavern let me go,
Where laughs the pipe, the merry cymbals kiss;
Under the history of all my woe,
My mistress sets her hand and writes: Finis.
Oh, linger not, nor trust the inconstant days
That promised: Where thou art thy lady stays —
The tale of separation ends with this!
Joy's certain path, oh Saki, thou hast shown —
Long may thy cup be full, thy days be fair!
Trouble and sickness from my breast have flown,
Order and health thy wisdom marshals there.
Not one that numbered Hafiz' name among
The great-unnumbered were his tears, unsung;
Praise him that sets an end to endless care!
Stanza 1. — There are many ways of taking omens which are still practised by the Persians. Concerning astrology and geomancy Mr. Browne questioned a learned Persian, and received the reply that there was positive proof of their truth. The Persian added, however, that the study of these sciences was very difficult, and many who professed to be acquainted with them were mere charlatans. Many dreams also, he said, were capable of interpretation, and might furnish indications to events which were yet to come. Mr. Browne relates that he consulted a geomancer, who, by means of dice, gave him much information as to his future none of which has yet been justified by the event-but on being asked to perform the less difficult task of answering some questions as to his past, turned the conversation into other channels. "I discussed," says the traveller, "the occult sciences with several of my friends, to discover as far as possible the prevailing opinion among them. One of them made use of the following argument to prove their existence: "God," he said," has no bukhl (avarice); it is impossible for Him to withhold from any one a thing for which he strives with sufficient earnestness. Just as if a man devotes all his energies to the pursuit of spiritual knowledge he attains to it, so if he chooses to make occult sciences and magical powers the object of his aspirations they will assuredly not be withheld from him." — A Year Amongst the Persians.
An omen can be taken by opening the Koran or some other well-accredited book (the Divan of Hafiz among the number), pricking a pin into the page, and following whatever directions can be drawn from the verse thus indicated. This method is frequently used before setting out upon a journey. The stars also are consulted in order to select a favourable day for embarking upon any enterprise, certain stars having special influence over men — the influence of the moon, for instance, is dangerous to life, and one of the stars in the constellation of Cassiopea is of evil presage. Besides these omens, divinations are taken from the movements and position of certain animals and birds, and from various passing events. To meet a one-eyed man is of bad omen, especially if he is blind of the left eye, or to hear an unlucky word on setting out from your house of a morning. Lane, in one of his notes to the "Arabian Nights," tells of a Sultan who was setting out on a raid, when one of his standards happening to strike against a cluster (or Pleiades, as they are called in Arabic) of lamps, he regarded this to be of evil import, and was about to abandon the expedition. "Oh our lord!" said one of his officers, "our standards have reached the Pleiades." The Sultan, encouraged by this fortunate suggestion, continued on his way, and returned victorious.
## XXVI
THE secret draught of wine and love repressed
Are joys foundationless — then come whate'er
May come, slave to the grape I stand confessed!
Unloose, oh friend, the knot of thy heart's care,
Despite the warning that the Heavens reveal!
For all his thought, never astronomer
That loosed the knot of Fate those Heavens conceal!
Not all the changes that thy days unfold
Shall rouse thy wonder; Time's revolving sphere
Over a thousand lives like thine has rolled.
That cup within thy fingers, dost not hear
The voices of dead kings speak through the clay
Kobad, Bahman, Djemshid, their dust is here,
"Gently upon me set thy lips!" they say.
What man can tell where Kaus and Kai have gone?
Who knows where even now the restless wind
Scatters the dust of Djem's imperial throne?
And where the tulip, following close behind
The feet of Spring, her scarlet chalice rears,
There Ferhad for the love of Shirin pined,
Dyeing the desert red with his heart's tears.
Bring, bring the cup! drink we while yet we may
To our soul's ruin the forbidden draught
Perhaps a treasure-trove is hid away
Among those ruins where the wine has laughed! —
Perhaps the tulip knows the fickleness
Of Fortune's smile, for on her stalk's green shaft
She bears a wine-cup through the wilderness.
The murmuring stream of Ruknabad, the breeze
That blows from out Mosalla's fair pleasaunce,
Summon me back when I would seek heart's ease,
Travelling afar; what though Love's countenance
Be turned full harsh and sorrowful on me,
I care not so that Time's unfriendly glance
Still from my Lady's beauty turned be.
Like Hafiz, drain the goblet cheerfully
While minstrels touch the lute and sweetly sing,
For all that makes thy heart rejoice in thee
Hangs of Life's single, slender, silken string.
Stanza 2. — For Djemshid, see Note to Stanza 2 of Poem II. He was the fourth king of the First or Pishdadian dynasty, and is supposed to have flourished eight hundred years before the Christian era. Firdusi says he reigned seven hundred years. Kaikobad was the founder of the Second dynasty, the Kayanian. He was set upon the throne by the hero Rustum, son of Zal. It was in his reign that Rustum overcame Afrasiab's arrny, killing his own son in the battle "by the great Oxus stream, the yellow Oxus," a story which all readers of Matthew Arnold know. Kaikobad is said to have reigned one hundred and twenty years. Bahman, another member of the Kayanian house, is better known to the Persians as Ardisher Dirazdast, the Artaxerxes Longimanus of the Greeks. He came to the throne in B.C. 464. He was the grandson of Darius, the Persian Gushtasp. He is supposed to have been the Ahasuerus of Scripture who married Esther. Persian historians ascribe to him also remarkable longevity, and state that he reigned one hundred and twelve years. Kaikaus, mentioned in the next stanza, was the son of Kaikobad, second king of the Kayanian dynasty; Kai may be Kaikhusro, the third king of the same dynasty.
Stanza 3. — The loves of Ferhad and Shirin are famous in Persian legend. Shirin is called by some Mary, and by others Irene. The Greeks describe her as a Roman by birth and a Christian; the Turks and the Persians say that she was a daughter of the Emperor Maurice, and wife of Khusro Parwiz, who came to the Persian throne in A.D. 591. It was Khusro Parwiz who conquered Jerusalem, and carried off, say the Persians, the true Cross, which had been enclosed in a gold box and buried in the ground. He was devotedly attached to his wife Shirin, but she had given her heart to her humble lover Ferhad. He, despairing of ever reaching one whose rank had placed her so far above him, wandered through the deserts and the mountains of Persia calling upon her name, and in order to beguile his weary hours executed the sculptures upon the rock Behistun — so says the legend. At length the King sent to him and told him that if he would cut through the rock and cause a stream upon the other side of the mountains to flow through it, he would relinquish Shirin to him. Ferhad set himself to the task, and had almost accomplished it when Khusro sent him the false news of Shirin's death. On hearing it, Ferhad threw himself from the top of the rock and so died. Shirin's end was scarcely less tragic. Khusro Parwiz was put to a violent death by his son, who proceeded to make proposals of marriage to his father's widow. Shirin promised to marry him if he would allow her to see once more her husband's corpse. She was led to the place where the murdered King lay, and drawing a dagger, she stabbed herself and fell dead across his body.
It is difficult to conceive anything more exquisite than the little scarlet tulip growing upon a barren Persian hillside. On the top of a bleak pass over the mountains between Resht and Tehran, I have seen companies of tiny tulips shining like jewels among the dust and stones.
There is a tradition that this poem was sent to the King of Golconda.
## XXVII
MY friend has fled! alas, my friend has fled,
And left me nought but tears and pain behind!
Like smoke above a flame caught by the wind,
So rose she from my breast and forth she sped.
Drunk with desire, I seized Love's cup divine,
But she that held it poured the bitter wine
Of Separation into it and fled.
The hunter she, and I the helpless prey;
Wounded and sick, round me her toils she drew,
My heart into a sea of sorrow threw,
Bound up her camel loads and fled away.
Fain had I laid an ambush for her soul,
She saw and vanished, and the timid foal,
Good Fortune, slipped the rein and would not stay.
My heart was all too narrow for my woe,
And tears of blood my weeping eyes have shed,
A crimson stream across the desert sped,
Rising from out my sad heart's overflow.
She knew not what Love's meanest slave can tell:
"'Tis sweet to serve!" but threw me a Farewell,
Kissing my threshold, turned, and cried "I go!"
In the clear dawn, before the east was red,
Before the rose had torn her veil in two,
A nightingale through Hafiz' garden flew,
Stayed but to fill its song with tears, and fled.
## XXVIII
HAST thou forgotten when thy stolen glance
Was turned to me, when on my happy face
Clearly thy love was writ, which doth enhance
All happiness? or when my sore disgrace
(Hast thou forgot?) drew from thine eyes reproof,
And made thee hold thy sweet red lips aloof,
Dowered, like Jesus's breath, with healing grace?
Hast thou forgotten how the glorious
Swift nights flew past, the cup of dawn brimmed high?
My love and I alone, God favouring us!
And when she like a waning moon did lie,
And Steep had drawn his coif about her brow,
Hast thou forgot? Heaven's crescent moon would bow
The head, and in her service pace the sky!
Hast thou forgotten, when a sojourner
Within the tavern gates and drunk with wine,
I found Love's passionate wisdom hidden there,
Which in the mosque none even now divine?
The goblet's carbuncle (hast thou forgot?)
Laughed out aloud, and speech flew hot
And fast between thy ruby lips and mine!
Hast thou forgotten when thy cheek's dear torch
Lighted the beacon of desire in me,
And when my heart, like foolish moths that scorch
Their wings and yet return, turned all to thee?
Within the banquet-hall of Good Repute
(Hast thou forgot?) the wine's self-pressed my suit,
And filled the morn with drunken jollity!
Hast thou forgotten when thou laid'st aright
The uncut gems of Hafiz' inmost thought,
And side by side thy sweet care strung the bright
Array of verse on verse-hast thou forgot?
Stanza 1. — According to Oriental belief, Jesus Christ's gift of healing was due to a miraculous quality in His breath.
## XXIX
FROM Canaan Joseph shall return, whose face
A little time was hidden: weep no more —
Oh, weep no more! in sorrow's dwelling-place
The roses yet shall spring from the bare floor!
And heart bowed down beneath a secret pain —
Oh stricken heart! joy shall return again,
Peace to the love-tossed brain — oh, weep no more!
Oh, weep no more! for once again Life's Spring
Shall throne her in the meadows green, and o'er
Her head the minstrel of the night shall fling
A canopy of rose leaves, score on score.
The secret of the world thou shalt not learn,
And yet behind the veil Love's fire may burn —
Weep'st thou? let hope return and weep no more!
To-day may pass, to-morrow pass, before
The turning wheel give me my heart's desire;
Heaven's self shall change, and turn not evermore
The universal wheel of Fate in ire.
Oh Pilgr'm nearing Mecca's holy fane,
The thorny maghilan wounds thee in vain,
The desert blooms again — oh, weep no more!
What though the river of mortality
Round the unstable house of Life doth roar,
Weep not, oh heart, Noah shall pilot thee,
And guide thine ark to the desirèd shore!
The goal lies far, and perilous is thy road,
Yet every path leads to that same abode
Where thou shalt drop thy load — oh, weep no more!
Mine enemies have persecuted me,
My Love has turned and fled from out my door —
God counts our tears and knows our misery;
Ah, weep not! He has heard thy weeping sore.
And chained in poverty and plunged in night,
Oh Hafiz, take thy Koran and recite
Litanies infinite, and weep no more!
Stanza 3. — Maghilan, a thorny shrub which grows on the deserts of Arabia near to Mecca. When the pilgrims see it they know that they have almost reached their goal, and forget the hardships of the journey and the barrenness of the wastes through which their road lies.
## XXX
ALL hail, Shiraz, hail! oh site without peer!
May God be the Watchman before thy gate,
That the feet of Misfortune enter not here!
Lest my Ruknabad be left desolate,
A hundred times, "God forbid!" I pray;
Its limpid stream where the shadows wait
Like the fount of Khizr giveth life for aye.
'Twixt Jafrabad and Mosalla's close
Flies the north wind laden with ambergris —
Oh, come to Shiraz when the north wind blows!
There abideth the angel Gabriel's peace
With him who is lord of its treasures; the fame
Of the sugar of Egypt shall fade and cease,
For the breath of our beauties has put it to shame.
Oh wind that blows from the sun-rising,
What news of the maid with the drunken eyes,
What news of the lovely maid dost thou bring?
Bid me not wake from my dream and arise,
In dreams I have rested my head at her feet —
When stillness unbroken around me lies,
The vision of her makes my solitude sweet.
If for wine the Cup-bearer pour forth my blood,
As the milk from a mother's bosom flows,
At his word let my heart yield its crimson flood.
But, Hafiz, Hafiz! thou art of those
For ever fearing lest absence be near;
For the days when thou held'st the Beloved close,
Why rise not thy thanks so that all may hear?
Stanza 1. — Khizr — see Note to the third stanza of Poem XVIII.
Stanza 2. — The quarter of Jafrabad has ceased to exist. Its position was to the east of the town, opposite to the fields and to the ruined mosque of Mosalla. Between Jaftabad and Mosalla runs the highroad to Isfahan, traversing, at the distance of a mile from Shiraz, the pass of Allahu Akbar.
The angel Gabriel, the Holy Spirit, is the highest of all the angels. It is his duty to write down the decrees of God; through him the Koran was revealed to Mahommad, and it is he who, hovering above the throne of God, shelters it with his wings. Hafiz therefore claims for Shiraz the protection of him who is guardian of the highest place in heaven.
Ibn Batuta, the Arab traveller who visited Shiraz about the year 1340, has left a charming description of the native town of Hafiz and of the manners of his contemporaries. "Shiraz," he says, "is a well-built town of a great size, a wide celebrity, and a high place among cities. It possesses pleasant gardens, far-reaching streams, excellent markets, fine streets, and a numerous population. The town is constructed with taste and admirably arranged; each trade has its own bazaar. The inhabitants are a fine race and well clad. Shiraz lies in a plain; gardens surround it on every side; and five rivers flow through it, amongst them one called Ruknabad, a stream of which the water is excellent to drink, very cold in summer and warm in winter. The principal mosque is called the Old Mosque; it is as spacious and as well built as any one could wish to see. The court of it is vast and paved with marble; in hot weather it is washed with fresh water every night. The wealthy citizens come there every evening to repeat the prayers of sunset and of night. The inhabitants of Shiraz are well-to-do, pious, and chaste; the women in particular are distinguished for their modesty. They go completely veiled, give much in alms, and repair three times a week to the great mosque. Often as many as two thousand are assembled there, sitting with fans in their hands on account of the great heat. Each day in one of the mausoleums the whole Koran is read aloud, and the readers have very beautiful voices. The people bring with them fruits and sweetmeats, and when the congregation has finished eating, the preacher begins his discourse. This takes place between the mid-day and the evening prayers." Ibn Batuta struck up acquaintance with a Sheikh whom he found seated in a small hermitage at the corner of a mosque. The Sheikh was engaged in reading the Koran. In answer to Ibn Batuta's questions, he told him that he had founded the mosque himself, and that the hermitage was to be his tomb. Lifting a carpet, he showed him his grave, covered over with planks. In that box,"he said, pointing to a chest opposite to him, are my winding-sheet, some spices with which my corpse will be perfumed, and a few pieces of money which I earned by digging a well for a pious man. The money will serve to pay for my burial, and what is left over will be distributed among the poor." "I admired his conduct," adds Ibn Batuta. "One of the mausoleums outside the town," he continues, "contains the tomb of Sheikh Sa'di, the first poet of his time. Close at hand is a hermitage built by Sa'di himself, surrounded by a charming garden. It is situated near the source of the Ruknabad. In the garden Sheikh Sa'di constructed a number of basins for the washing of clothes. The citizens of Shiraz make parties of pleasure to this mausoleum; they eat food prepared in the hermitage, wash their garments in the river, and at sunset return to the town. So did I also. May God have mercy on Shiraz!" he concludes piously.
## XXXI
THE breath of Dawn's musk-strewing wind shall blow,
The ancient world shall turn to youth again,
And other wines from out Spring's chalice flow;
Wine-red, the judas-tree shall set before
The pure white jessamine a brimming cup,
And wind flowers lift their scarlet chalice up
For the star-pale narcissus to adore.
The long-drawn tyranny of grief shall pass,
Parting shall end in meeting, the lament
Of the sad bird that sang "Alas, alas!"
Shall reach the rose in her red-curtained tent.
Forth from the mosque! the tavern calls to me!
Would'st hinder us? The preacher's homily
Is long, but life will soon be spent!
Ah, foolish Heart! the pleasures of To-day,
If thou abandon, will To-morrow stand
Thy surety for the gold thou'st thrown away?
In Sha'aban the troops of Grief disband,
And crown the hours with wine's red coronet —
The sun of merriment ere long will set,
And meagre Ramazan is close at hand!
Dear is the rose — now, now her sweets proclaim,
While yet the purple petals blush and blow;
Hither adown the path of Spring she came,
And by the path of Autumn she will go.
Now, while we listen, Minstrel, tune thy lay!
Thyself hast said: "The Present steals away
The Future comes, and bringing — what? Dost know?"
Summoned by thy melody did Hafiz rise
Out of the darkness near thy lips to dwell;
Back to the dark again his pathway lies —
Sing out, sing clear, and singing cry: Farewell!
Stanza 3. — The month of Sha'aban is the eighth month of the Arabic year. It is followed by Ramazan, during which month the Prophet decreed that from two hours before dawn until sunset nothing should pass the lips of his followers. The fast is so strictly observed, especially by the lower orders, that not only do they refrain from eating and drinking, but they will not even smoke until the sunset gun puts an end to the day's abstinence. The night, however, is passed in feasting and revelry, and the richer classes will sleep late in Ramazan and shorten the long hours that must pass before they may breakfast.
## XXXII
UPON a branch of the straight cypress-tree
Once more the patient nightingale doth rest:
"Oh Rose!" he cries, "evil be turned from thee!
I sing thee all men's thanks; thou blossomest
And hope springs up in every joyless heart —
Let not the nightingale lament apart,
Nor with thy proud thorns wound his faithful breast."
I will not mourn my woeful banishment,
He that has hungered for his lady's face
Shall, when she cometh, know a great content.
The Zealot seeks a heavenly dwelling-place,
Huris to welcome him in Paradise;
Here at the tavern gate my heaven lies,
I need no welcome but my lady's grace.
Better to drink red wine than tears, say I,
While the lute sings; and if one bid thee cease,
"God is the merciful!" thou shalt reply.
To some, life brings but joy and endless ease;
Ah, let them laugh although the jest be vain!
For me the source of pleasure lay in pain,
And weeping for my lady I found peace.
Hafiz, why art thou ever telling o'er
The tale of absence and of sorrow's night?
Knowest thou not that parting goes before
All meeting, and from darkness comes the light!
Stanza 3. — According to the popular science of the East, the colouring of precious stones, even of those which are buried deep in the earth, is due to the action of rain and wind and of the rays of the sun.
Stanza 4. — It is a favourite Persian image to describe the hair of the beloved as entangling and entrapping the unfortunate lover. Her long locks are often compared to deadly snakes, and her curls to hooks which catch and tear her lover's heart. One need go no further than the Merchant of Venice to find the same imagery used by a Western poet: "Those crisped snaky golden locks," and again, "A golden mesh to entrap the hearts of men faster than gnats in cobwebs."
## XXXIII
THE jewel of the secret treasury
Is still the same as once it was; the seal
Upon Love's treasure casket, and the key,
Are still what thieves can neither break nor steal;
Still among lovers loyalty is found,
And therefore faithful eyes still strew the ground
With the same pearls that mine once strewed for thee.
Question the wandering winds and thou shalt know
That from the dusk until the dawn doth break,
My consolation is that still they blow
The perfume of thy curls across my cheek.
A dart from thy bent brows has wounded me —
Ah, come! my heart still waiteth helplessly,
Has waited ever, till thou heal its pain.
If seekers after rubies there were none,
Still to the dark mines where the gems had lain
Would pierce, as he was wont, the radiant sun,
Setting the stones ablaze. Would'st hide the stain
Of my heart's blood? Blood-red the ruby glows
(And whence it came my wounded bosom knows)
Upon thy lips to show what thou hast done.
Let not thy curls waylay my pilgrim soul,
As robbers use, and plunder me no more!
Years join dead year, but thine extortionate rule
Is still the same, merciless as before.
Sing, Hafiz, sing again of eyes that weep!
For still the fountain of our tears is deep
As once it was, and still with tears is full.
## XXXIV
LAST night I dreamed that angels stood without
The tavern door, and knocked in vain, and wept;
They took the clay of Adam, and, methought,
Moulded a cup therewith while all men slept.
Oh dwellers in the halls of Chastity!
You brought Love's passionate red wine to me,
Down to the dust I am, your bright feet stept.
For Heaven's self was all too weak to bear
The burden of His love God laid on it,
He turned to seek a messenger elsewhere,
And in the Book of Fate my name was writ.
Between my Lord and me such concord lies.
As makes the Huris glad in Paradise,
With songs of praise through the green glades they flit.
A hundred dreams of Fancy's garnered store
Assail me — Father Adam went astray
Tempted by one poor grain of corn! Wherefore
Absolve and pardon him that turns away
Though the soft breath of Truth reaches his ears,
For two-and-seventy jangling creeds he hears,
And loud-voiced Fable calls him ceaselessly.
That, that is not the flame of Love's true fire
Which makes the torchlight shadows dance in rings,
But where the radiance draws the moth's desire
And send him forth with scorched and drooping wings.
The heart of one who dwells retired shall break,
Rememb'ring a black mole and a red cheek,
And his life ebb, sapped at its secret springs.
Yet since the earliest time that man has sought
To comb the locks of Speech, his goodly bride,
Not one, like Hafiz, from the face of Thought
Has torn the veil of Ignorance aside.
Stanza 1. — The story of the creation of Adam, and of the part played in it by the angels, is told by Mahommad in the following terms: "When thy Lord said unto the angels, I am going to place a substitute on earth; they said, Wilt thou place there one who will do evil therein, and shed blood? but we celebrate thy praise and sanctify thee. God answered, Verily I know that which ye know not; and he taught Adam the names of all things, and then proposed them to the angels, and said, Declare unto me the names of these things if ye say truth. They answered, Praise be unto thee, we have no knowledge but what thou teachest us, for thou art knowing and wise. God said, Oh Adam, tell them their names. And when he had told them their names, God said, Did I not tell you that I know the secrets of heaven and earth, and know that which ye discover and that which ye conceal? And when we said unto the angels, Worship Adam; they all worshipped him, except Eblis, who refused, and was puffed up with pride, and became of the number of unbelievers." — Koran, chap. ii.
Tradition has amplified and adorned this story. It is said that the three archangels, Gabriel, Michael, and Israfil, were each in turn ordered to take from the earth seven handfuls of clay of three different colours, red, white, and yellow, that God might create out of it the races of mankind. But each in turn was moved by the earth's prayer that he would not rob her of her substance, and each returned to heaven empty-handed. The fourth time God sent Azrail, the angel of death, who tore the seven handfuls from the earth, but hearing her lamentations, promised her that when man ceased to live his substance should return to the earth from whence it had been taken. With the clay that Azrail brought him God moulded the figure of man, and when it was finished he left it forty days to dry. The angels came often to gaze upon it, and Eblis, kicking it with his foot, found that it rang hollow. When the figure of clay was dry, God breathed the breath of life into its nostrils, and ordered the angels to submit to the man he had created. But Eblis refused, saying that he had been created of pure fire, and would not serve a hollow mould of clay; for which reason God cast him out of Paradise. The rest of the angels acknowledged the superiority of Adam after God had made him tell them the names of all the creatures of the earth, though they had at first protested that it was not seemly that they should bow down to him, for their love for God was greater than his. It is with this legend in his mind that Hafiz speaks of the angels as standing at the tavern door, where man may enter and receive instruction in God's wisdom, but where they must knock in vain, and as moulding a wine-cup with the despised clay out of which the human body was moulded. I think he means that man himself is the vessel into which divine love and wisdom are poured; and when he says that the angels first brought him wine, he means that by their example they showed him what it was to be intoxicated by the contemplation of God.
Stanza 3.— "Concerning the forbidden fruit," says Sale in a note to the second chapter of the Koran, "the Mohammadans, as well as the Christians, have different opinions. Some say it was an ear of wheat, some will have it to have been a fig-tree, and others a vine."
There are supposed to be seventy-two sects in Islam. Many Mahommadan writers compare them to the seventy-two branches of the family of Noah after the Babylonian confusion of tongues and the dispersal of the children of Adam.
## XXXV
FORGET not when dear friend to friend returned,
Forget not days gone by, forget them not!
My mouth has tasted bitterness, and learned
To drink the envenomed cup of mortal lot;
Forget not when a sweeter draught was mine,
Loud rose the songs of them that drank that wine —
Forget them not!
Forget not loyal lovers long since dead,
Though faith and loyalty should be forgot,
Though the earth cover the enamoured head,
And in the dust wisdom and passion rot.
My friends have thrust me from their memory;
Vainly a thousand thousand times I cry:
Forget me not!
Weary I turn me to my bonds again.
Once there were hands strong to deliver me,
Forget not when they broke a poor slave's chain!
Though from mine eyes tears flow unceasingly,
I think on them whose rose gardens are set
Beside the Zindeh Rud, and I forget
Life's misery.
Sorrow has made her lair in my breast,
And undisturbed she lies — forget them not
That drove her forth like to a hunted beast!
Hafiz, thou and thy tears shall be forgot,
Lock fast the gates of thy sad heart! But those
That held the key to thine unspoken woes —
Forget them not!
Stanza 1. — The second line of this poem is as often quoted as any, perhaps, in the Divan: " Yàd bàd àn ruz-i-gàràn, yàd bàd!" A man will set it upon a letter to an absent friend, even when he is not particularly anxious that days gone by should be preserved from oblivion; and how often must the simple little line have been used by those to whom its very simplicity made it more poignant than pages of sentiment!
Stanza 3. — The Zindeh Rud was a river that flowed past Isfahan. There are unfortunately no longer rose-gardens upon its banks, for it disappeared completely in the terrible earthquake which occurred in the spring of the year 1853. I suspect from internal evidence that this poem was sent to some friends of Hafiz living at Isfahan, upon whom the passionate appeal need refiect no discredit, since it may quite possibly be merely the Oriental way of writing a letter of thanks. At the same time, in spite of this rational explanation, it must be acknowledged that the meaning of the name Zindeh Rud is River of Life. I tremble to think into what a slough of mysticism the innocent little stream might be induced to guide us!
## XXXVI
BELOVED, who has bid thee ask no more
How fares my life? to play the enemy
And ask not where he dwells that was thy friend?
Thou art the breath of mercy passing o'er
The whole wide world, and the offender I
Ah, let the rift my tears have channelled end,
Question the past no more!
If thou would'st know the secret of Love's fire,
It shall be manifest unto thine eyes:
Question the torch flame burning steadfastly,
But ask no more the sweet wind's wayward choir.
Ask me of faith and love that never dies;
Darius, Alexander's sovereignty,
I sing of these no more.
Ask not the monk to give thee Truth's pure gold,
He hides no riches 'neath his lying guise;
And ask not him to teach thee alchemy
Whose treasure-house is bare, his hearth-stone cold.
Ask to what goal the wandering dervish hies,
They knew not his desire who counselled thee:
Question his rags no more!
And in their learned books thou'lt seek in vain
The key to Love's locked gateway; Heart grown wise
In pain and sorrow, ask no remedy!
But when the time of roses comes again,
Take what it gives, oh Hafiz, ere it flies,
And ask not why the hour has brought it thee,
And wherefore ask no more!
Stanza 2.— "Love and Faith," says Rosenzweig, is the name of a well-known Persian story which has been retold by many writers.
## XXXVII
ARISE! and fill a golden goblet up
Until the wine of pleasure overflow,
Before into thy skull's pale empty cup
A grimmer Cup-bearer the dust shall throw.
Yea, to the Vale of Silence we must come;
Yet shall the flagon laugh and Heaven's dome
Thrill with an answering echo ere we go!
Thou knowest that the riches of this field
Make no abiding, let the goblet's fire
Consume the fleeting harvest Earth may yield!
Oh Cypress-tree! green home of Love's sweet choir,
When I unto the dust I am have passed,
Forget thy former wantonness, and cast
Thy shadow o'er the dust of my desire.
Flow, bitter tears, and wash me clean! for they
Whose feet are set upon the road that lies
'Twixt Earth and Heaven Thou shalt be pure," they say,
"Before unto the pure thou lift thine eyes."
Seeing but himself, the Zealot sees but sin;
Grief to the mirror of his soul let in,
Oh Lord, and cloud it with the breath of sighs!
No tainted eye shall gaze upon her face,
No glass but that of an unsullied heart
Shall dare reflect my Lady's perfect grace.
Though like to snakes that from the herbage start,
Thy curling locks have wounded me full sore,
Thy red lips hold the power of the bezoar —
Ah, touch and heat me where I lie apart!
And when from her the wind blows perfume sweet,
Tear, Hafiz, like the rose, thy robe in two,
And cast thy rags beneath her flying feet,
To deck the place thy mistress passes through.
Stanza 4. — See Note to Stanza 4 of Poem XXXIII.
The word bezoar comes from two Arabic roots which signify the annihilator of poison. Murray gives several examples of its use by seventeenth and eighteenth century writers in the sense of an antidote, chiefly to snake bites. Topsell, for instance, in his book on Serpents (1607), remarks that "the juice of apples being drunk, and endive, are the proper Bezoar against the venom of a Phalangie" — whatever that may be. The word was also applied to various substances held as antidotes, especially to a concretion found in the stomach of some animals, formed of concentric layers of animal matter deposited round some foreign substance. This concretion was called the bezoar stone. The original sort was the lapis bezoar orientale obtained from the wild goat of Persia, which was in later times called the bezoar goat; also from various antelopes, &c. The lapis bezoar occidentale, obtained from the llamas of Peru, was less valued. The chamois yielded German bezoar. "The stone," says Frampton, in his "Joyful News," is called the Bezaar, being approved good against Venome" and Hawkins, in his "Voyage to the South Seas," talks about "the becunia and other beasts which breed the beazer stone."
## XXXVIII
I CEASE not from desire till my desire
Is satisfied; or let my mouth attain
My love's red mouth, or let my soul expire,
Sighed from those lips that sought her lips in vain.
Others may find another love as fair;
Upon her threshold I have laid my head,
The dust shall cover me, still lying there,
When from my body life and love have fled.
My soul is on my lips ready to fly,
But grief beats in my heart and will not cease,
Because not once, not once before I die,
Will her sweet lips give all my longing peace.
My breath is narrowed down to one long sigh
For a red mouth that burns my thoughts like fire;
When will that mouth draw near and make reply
To one whose life is straitened with desire?
When I am dead, open my grave and see
The cloud of smoke that rises round thy feet:
In my dead heart the fire still burns for thee;
Yea, the smoke rises from my winding-sheet!
Ah, come, Beloved! for the meadows wait
Thy coming, and the thorn bears flowers instead
Of thorns, the cypress fruit, and desolate
Bare winter from before thy steps has fled.
Hoping within some garden ground to find
A red rose soft and sweet as thy soft cheek,
Through every meadow blows the western wind,
Through every garden he is fain to seek.
Reveal thy face! that the whole world may be
Bewildered by thy radiant loveliness;
The cry of man and woman comes to thee,
Open thy lips and comfort their distress!
Each curling lock of thy luxuriant hair
Breaks into barbèd hooks to catch my heart,
My broken heart is wounded everywhere
With countless wounds from which the red drops start.
Yet when sad lovers meet and tell their sighs,
Not without praise shall Hafiz' name be said,
Not without tears, in those pale companies
Where joy has been forgot and hope has fled.
## XXXIX
CYPRESS and Tulip and sweet Eglantine,
Of these the tale from lip to lip is sent;
Washed by three cups, oh Saki, of thy wine,
My song shall turn upon this argument.
Spring, bride of all the meadows, rises up,
Clothed in her ripest beauty: fill the cup!
Of Spring's handmaidens runs this song of mine.
The sugar-loving birds of distant Ind,
Except a Persian sweetmeat that was brought
To fair Bengal, have found nought to their mind.
See how my song, that in one night was wrought,
Defies the limits set by space and time!
O'er plains and mountain-tops my fearless rhyme,
Child of a night, its year-long road shall find.
And thou whose sense is dimmed with piety,
Thou too shalt learn the magic of her eyes;
Forth comes the caravan of sorcery
When from those gates the blue-veined curtains rise.
And when she walks the flowery meadows through,
Upon the jasmine's shamèd cheek the dew
Gathers like sweat, she is so fair to see!
Ah, swerve not from the path of righteousness
Though the world lure thee! like a wrinkled crone,
Hiding beneath her robe lasciviousness,
She plunders them that pause and heed her moan.
From Sinai Moses brings thee wealth untold;
Bow not thine head before the calf of gold
Like Samir, following after wickedness.
From the Shah's garden blows the wind of Spring,
The tulip in her lifted chalice bears
A dewy wine of Heaven's minist'ring
Until Ghiyasuddin, the Sultan, hears,
Sing, Hafiz, of thy longing for his face.
The breezes whispering round thy dwelling-place
Shall carry thy lament unto the King.
Stanza 1. — It is related that Ghiyasuddin Purabi, who succeeded his father to the throne of Bengal in the year 1367, fell sick. During his illness he was nursed by three faithful handmaidens whose names were Cypress, Tulip, and Rose, and owing to their care he eventually recovered. The rest of the Sultan's ladies were jealous of the gratitude that the three maidens had earned from Ghiyasuddin, and nicknamed them contemptuously "the three bath women," because they had washed the King's body while he was ill. He therefore determined to do them honour by commemorating their devotion in a poem, and to this end he composed the first line of a couplet, and ordered the poets of his court to complete the ode. The line ran thus: "Sàki hadis-i-sarvo gul o làleh miravad" — Cup-bearer, a tale runs of a Cypress, a Rose, and a Tulip. But the poets were unable to perform the task to the King's satisfaction, and at length some one suggested that the line should be sent to Hafiz of Shiraz, the fame of whose great skill had reached Bengal. This was accordingly done, and Hafiz composed the ode here translated, with which the Sultan (whose taste seems to have turned towards the discursive in poetry) was much delighted. The three cups of wine are an allusion to the three maidens who washed the King's body; the parrots of India are the court poets of Ghiyasuddin, and the Persian sweetmeat is the ode that Hafiz sent to Bengal.
Stanza 4. — Samir. Al Samiri belonged, say the Mahommadans, to a certain tribe among the Jews called the Samaritans, whence his name. In this the Mahommadans strangely betray their ignorance of history, for the Samaritans were not formed into a people, nor did they bear that name, until many ages later. Some say that he was a proselyte, but a hypocritical one, and originally of Kerman or some other country. His real name was Musa ibn Dhafar. He was a magician and an alchemist. Pharaoh employed him as a rival to Moses when the latter worked miracles with his hand and his staff, but Al Samiri was unable to show wonders as great as those performed by Moses. It was he and not Aaron, according to Mahommadan tradition, who cast the golden calf. The calf was made of the ornaments of gold and silver and other materials which the Israelites had borrowed from the Egyptians; for Aaron, who commanded in his brother's absence, having ordered Al Samiri to collect those ornaments from the people, who carried on a wicked commerce with them, and to keep them together till the return of Moses, Al Samiri, understanding the founder's art, put them all together into a furnace, to melt them down into one mass, which came out in the form of a calf. The Israelites, accustomed to the Egyptian idolatry, paying a religious worship to this image, Al Samiri went further, and took some dust from the footsteps of the horse of the angel Gabriel, who marched at the head of the people, and threw it into the mouth of the calf, which immediately began to low, and became animated; for such was the virtue of that dust. (Sale, Notes to second and twenty-second chapters of the Koran.) Al Simiri is mentioned by name in the twenty-second chapter of the Koran: "Al Samiri led them astray."
## XL
THE margin of a stream, the willow's shade,
A mind inclined to song, a mistress sweet,
A Cup-bearer whose cheek outshines the rose,
A friend upon whose heart thy heart is laid:
Oh Happy-starred! let not thine hours fleet
Unvalued; may each minute as it goes
Lay tribute of enjoyment at thy feet,
That thou may'st live and know thy life is sweet.
Let every one upon whose heart desire
For a fair face lies like a burden sore,
That all his hopes may reach their goal unchecked,
Throw branches of wild rue upon his fire.
My soul is like a bride, with a rich store
Of maiden thoughts and jewelled fancies decked,
And in Time's gallery I yet may meet
Some picture meant for me, some image sweet.
Give thanks for nights spent in good company,
And take the gifts a tranquil mind may bring;
No heart is dark when the kind moon doth shine,
And grass-grown river-banks are fair to see.
The Saki's radiant eyes, God favouring,
Are like a wine-cup brimming o'er with wine,
And him my drunken sense goes out to greet,
For e'en the pain he leaves behind is sweet.
Hafiz, thy life has sped untouched by care,
With me towards the tavern turn thy feet!
The fairest robbers thou'lt encounter there,
And they will teach thee what to learn is sweet.
Stanza 2. — According to Persian superstition, the smoke of burning rue has the power to avert the evil eye.
## XLI
THE days of Spring are here! the eglantine,
The rose, the tulip from the dust have risen —
And thou, why liest thou beneath the dust?
Like the full clouds of Spring, these eyes of mine
Shall scatter tears upon the grave thy prison,
Till thou too from the earth thine head shalt thrust.
## XLII
TRUE love has vanished from every heart;
What has befallen all lovers fair?
When did the bonds of friendship part? —
What has befallen the friends that were?
Ah, why are the feet of Khizr lingering? —
The waters of life are no longer clear,
The purple rose has turned pale with fear,
And what has befallen the wind of Spring?
None now sayeth: "A love was mine,
Loyal and wise, to dispel my care."
None remembers love's right divine;
What has befallen all lovers fair?
In the midst of the field, to the players' feet,
The ball of God's favour and mercy came,
But none has leapt forth to renew the game —
What has befallen the horsemen fleet?
Roses have bloomed, yet no bird rejoiced,
No vibrating throat has rung with the tale;
What can have silenced the hundred-voiced?
What has befallen the nightingale?
Heaven's music is hushed, and the planets roll
In silence; has Zohra broken her lute?
There is none to press out the vine's ripe fruit,
And what has befallen the foaming bowl?
A city where kings are but lovers crowned,
A land from the dust of which friendship springs —
Who has laid waste that enchanted ground?
What has befallen the city of kings?
Years have passed since a ruby was won
From the mine of manhood; they labour in vain,
The fleet-footed wind and the quickening rain,
And what has befallen the light of the sun?
Hafiz, the secret of God's dread task
No man knoweth, in youth or prime
Or in wisest age; of whom would'st thou ask:
What has befallen the wheels of Time?
Stanza 1. — Khizr. See Note to Stanza 3 of Poem XVIII.
Stanza 3. — Zohra is the planet Venus, the musician of the heavens, and the protector of all musicians and singers upon the earth. Zohra played a part in very ancient mythology. The Mahommadans borrowed and adapted the Magian legends concerning her, and their account runs as follows: Once upon a time the angels fell to marvelling over the wickedness of man and the case with which he was led astray, notwithstanding the warnings sent down to him through the prophets. But God, hearing their words, determined to expose them also to temptation, that they might learn how easy it was to fall. Therefore he appointed two of them, whose names were Harut and Marut, to go down to the earth as judges over man, and he taught them a secret word by the power of which every evening, when their work of judgment was done, they could return to heaven. For some time the two angels accomplished their duties faithfully. But at length a woman called Zohra, more beautiful than any other woman upon earth, came before their judgment-seat demanding redress against her husband, and the two angels conceived a violent passion for her. On the following day, when she returned with the same petition, they drew her aside and declared their love to her. She replied that she would satisfy their desires if they would do three things: destroy her husband, worship the gods she worshipped, and drink wine. Murderers and idolaters the angels could not agree to become, but they consented to drink wine, "not knowing," says the Persian commentator of the Mesnavi of Jelaleddin Rumi, "that wine was the source of sin and the mother of shame." Then said Zohra: "Every night, by the power of a divine word, ye return to heaven. Teach me also that word." The angels confided to her the secret of God, and as soon as she had heard the word she pronounced it in her turn and rose up into heaven, where God changed her form and turned her into a star. The angels attempted to follow her to heaven, but they were refused admittance. On the intercession of a very pious man, however, they were allowed to choose whether they would be punished in this world or the next; they chose the former, and now suffer punishment in the land of Babel — whither, if any man have a mind to learn magic, he may go and learn it of them, for they are masters of all magic arts. Tradition says that Mahommad, whenever he looked upon the planet Venus, was wont to exclaim: "God curse Zohra! for it was she who led the two angels Harut and Marut into sin."
The same story, says Rosenzweig, is to be found in the Talmud, where the two angels are called Asa and Asail. The Talmud relates that the angels, after their sin, were carried into a great mountain and suspended by chains over an abyss. It was they who taught Solomon wisdom.
Stanza 4. — For the superstition concerning the origin of precious stones, see Note to Stanza 3 of poem XXXIII.
## XLIII
WHERE are the tidings of union? that I may arise —
Forth from the dust I will rise up to welcome thee!
My soul, like a homing bird, yearning for Paradise,
Shall arise and soar, from the snares of the world set free.
When the voice of thy love shall call me to be thy slave,
I shall rise to a greater far than the mastery
Of life and the living, time and the mortal span:
Pour down, oh Lord! from the clouds of thy guiding grace.
The rain of a mercy that quickeneth on my grave,
Before, like dust that the wind bears from place to place,
I arise and flee beyond the knowledge of man.
When to my grave thou turnest thy blessed feet,
Wine and the lute thou shalt bring in thine hand to me,
Thy voice shall ring through the folds of my winding-sheet,
And I will arise and dance to thy minstrelsy.
Though I be old, clasp me one night to thy breast,
And I, when the dawn shall come to awaken me,
With the flush of youth on my check from thy bosom will rise.
Rise up! let mine eyes delight in thy stately grace!
Thou art the goal to which all men's endeavour has pressed,
And thou the idol of Hafiz' worship; thy face
From the world and life shall bid him come forth and arise!
This ode is inscribed upon the tomb of Hafiz.
# SONNETS FROM HAFEZ AND OTHER VERSES
Translated by Elizabeth Bridges
## CONTENTS
NOTE
VERSES AFTER HAFEZ
## NOTE
THE last fifteen pieces in this book, which are founded on odes of Hafez, are not translations. Their aim is rather to convey if possible something of the original spirit than to give a faithful rendering of either thought or form; & I have not scrupled to omit, insert, alter or even deliberately to pervert the idea as fancy or feeling dictated. Some of the poems follow the Persian fairly closely (especially nos. 30, 31, 34, & 35); others are merely founded on or suggested by one or two couplets.
No. 4 was suggested by a Persian dialect quatrain by Baba Tahir.
The remainder are original.
E. B.
CHILSWELL, Sept. 1920
TO
A. A. D.
## 1
When sunlight faileth,
& day's glow is gone;
When chill mist traileth
Where warm splendour shone;
When summer's pleasure
Dieth, — dieth too
The transient treasure
That with life up-grew
& none may herit,
Where then wilt thou turn,
O vagrant spirit,
That no home didst earn;
When none replieth
Of thy friends so true,
When sweet Joy crieth
'Adieu, love, adieu!'
## 2
I called to fading day
As o'er the hill she flew,
'Whither, glad light, away?
Take me, O take me too!'
She said, 'O wingless one,
Thou hast thy memoried sun'.
I said to the droop'd rose
Awhile that was so fair,
'Why dost so swiftly lose,
Sweet grace, thy blooming air?'
She said, 'This is my doom;
Cherish thou beauty's tomb'.
I cried to Joy as late
I stood, bidding farewell,
'Must this be too thy fate
Whom I have loved so well?
He said, 'My gift I leave
With her whom I bereave'.
## 3
O youth's young cloudlet, O freshness free,
With heart so light on the winds to fly
Or glisten in spray up-scatter'd, — I
Am sad as the full surgings of the sea;
I gave thee birth, thou shalt return to me.
Thy heart is light as the empty wind
Of barren purposeless change, — but I
Am the thought-burden'd slow-searching mind:
I am the agony to form & find; —
The fluxing travail of eternity.
## 4
Wend I, wander I, past all worlds that be;
Ever have I wander'd or e'er the earth was made;
Urg'd like the álien áir o'er land & sea,
Sleepless as sunlight, joyless as its shade.
Not on your earth travel I; sáy not to mé
'Cease awhile thy wandering, Ó tir'd day!'
Say not, 'O pilgrim, rest thee; comfort thee':
Not hére is my journey's end, Indus nor Cathay.
## 5
ON A HILL
Eyes that o'er the landscape fly,
Over dale & wood & stream
To the cloud-realms of the sky;
Eyes that wander still & dream,
Hopes that ever forward press
Seeking lovelier loveliness;
All the world is yours to roam
— Searching eye & swiftling thought —
Nowhere can ye make your home;
Not where peace so vainly sought
Bower'd in the valley lies,
Nor content's small villages;
Nor can pleasure's garish dress
Tempt you to a mean caress
— Thoughts that will not rest nor stay —
Ever do ye tell her nay;
Still ye wander— 'Where, O say,
Lies our vision'd loveliness?'
## 6
AFTER MUSIC
O what availeth thee thy melting mood,
Thine ecstasy
When once again thy thralldoms o'er thee brood:
& what doth profit thee thy courage high,
& strength so fain;
So soon agen thy coward heart shall fly?
For more & stronger strife our strength shall strain,
Though hope's best good
Be but this hope: to strive, & strive again.
## 7
All things born to break
In meek sacrifice
For another's sake,
All man's striving vain,
Lavish'd as the price
Of the heart's hid pain —
Long, O spirit-bird,
Of thy lonely fear
Hast thou sung unheard
In hope's moon-lit wood,
While no creature near
Knew nor understood.
## 8
If there be any power in passion's prayer —
But no: such ultimate longings have no word:
There is no eloquence in last despair.
Many have voiced their pain & answer heard;
Though 'twere but this, that to give bodied form
To grief, call'd their own heart to combat it:
But not ev'n thus can I pray; — thou strong storm,
All-overpowering, baffling bravest wit,
Wild spirit spurning cage of time or name,
Furious intangible fire, no duteous thought
Can deal with thee, to no calm altar-flame
Confine, nor wish acceptable, — O if aught
From such dumb need can reach aught's hearing ear,
This is it now, O hear, O hear, O hear!
## 9
In love's great ocean, whose calm-shelter'd shore
Must he for ever leave, whose soul is bound
On farthest quest, life's wonders to explore —
That mightiest flood, all-whelming, torment-toss'd,
Wherein must ev'ry lover's self be lost
Ere the Belovéd's lovelier self be found —
Think not, O searcher, in that sea to find
Food for thine earth-born strength & lustful show,
Nor glorious pearl to deck thy worldly mind,
Nor isle of ease; all such doth he forego
Who, recking nought of hurt to pride or limb,
Heark'neth to love's unchallengeable call:
Yea, who would venture, no help is for him
Save whole surrender; health, strength, life & all.
## 10
When sorrow hath outsoar'd our nature's clime,
Leaving it far remote &, like a strong
Eagle lone brooding on her peak sublime,
Graspeth in solitude her towering wrong;
& no more hankereth for petty prey
Nor bleeding victim wherewithal to still
Her hunger of desolate passion, but thus aye
Sitteth, devour'd by her own vital ill,
Motionless, nerveless, where for her no sound
Of life is, only the wind's alien
Moan that meandereth sleeplessly around
The promontory, — what saviour can then
Help helpless sorrow? What shall break that spell
Of icy death in life, that shackling Hell?
## 11
O gentle weariness,
Thine is the power that can all spirits free
From bonding-trouble, thou art a goddess
To all the suffering slaves of misery.
Thy sanctuary
No suppliant vainly seeketh; wheresoe'er
Desperate grief is, then unfailingly
Is thine all-hallowing rest & refuge there.
Our sorrow hath outgrown
Solace, yet still in thine all-mothering hand
Is balm of soft oblivion, who alone
Our never-ending needs dost understand.
## 12
1919
Peace, for whose presence did we erewhile call
With cry sincere, vowing (God knoweth, those
Prótests how passionate were) to love thee all,
Yet when thou camest, pander'd to thy foes
Weaklier than ever, now again the throes
Convulse our being; now, Peace, may'st thou see,
This lust-devoted land is not for thee.
Farewell! Small wonder is it if thou flee
Such faithlessness, yet doth thy memory still
Dwell in each place where thou hast walked with me,
In dawn's fresh mead or by noon's shady rill,
Or when cool evening wafteth, on our hill;
Allwheres that beauty's comfort-laden breath
Sootheth tired sorrow till it slumbereth.
## 13
SONG
Beauty is a waving tree,
Beauty is a flower,
Beauty is a grassy lea
& a shady bower,
Beauty is the verdant Spring
In our hearts awakening.
Beauty is a summer sun
Warming all the land,
Whose full bounty doth o'errun
More than our demand;
Spreadeth Beauty her kind feast
Lavishly for man & beast.
Autumn's quiet hast thou too,
Beauty, who canst feed
Every craving, known or new
Of the spirit's need,
Laying up a lasting store
Of ripe bliss for evermore.
O true Beauty, though joy's vain
Seasons come & go,
Thou a refuge dost remain
From all wintry woe,
Thou art still the perfect clime
Where no transience is nor time.
## 14
Wheresoever beauty flies,
Follow her on eager wings
Beauteous wild imaginings.
Wheresoever she may tread,
Lovely vivid flowers arise,
Springing swift as thoughts unsaid.
Living beauty, more than wise,
Fair art thou to living eyes,
Though less fair than is the dead
Myrtle-wreath that more we prize;
Relic of the one dear head
That for each it garlanded.
## 15
When first to earth thy gentle spirit came
From some soft climate of Elysian field,
Garmented in its own ethereal flame;
When first from heav'n's high peace it enter'd here,
No armour had it then, nor guarding shield,
Nor sword for safety, nor attacking spear,
No pang'd misgivings suffered it, nor fear,
Seeing in every face its own sweet face,
Smiling to treachery with trustful eyes,
Finding in nature its own nature's grace:
— So Adam in his vision'd Paradise
Saw but God's gifts, till taste of bitter ruth
Taught him what earth's creation is in truth: —
Now, O stern angel, none can make relent
Thy steely wrath, thy sword of punishment.
## 16
For sake of these two splendours do the wise
Set store on riches, & for these alone:
For these two glories only do they prize
Power & majesty of kingly throne:
Or this: to succour friendship in distress,
To comfort humble sorrow, nor despise
To cheer the joyless heart of weariness,
To guard & aid whom fortune doth oppress
That he to life's glad kingdom be restor'd
(& thus their monument of thanks they raise
More high than pomp's vain pinnacle of praise),
Or this: to forge therefrom a trenchant sword
Whereat shall poltroon evil cower & fly,
& smite Hell's fiends of foulness that they die.
## 17
She hath not beauty, that ill-fortun'd gem
Wherewith may women dazzle men's meek eyes
Ere they enslave, un-man & slaughter them.
Nor doth she vaunt afar her heart's hid prize,
Nor with wide-lavish'd scent of hope allure
Ere men behold her, nor with rich disguise.
Nor hath she wit, that sword wherewith to smart
Delicate souls, with flashing stroke unsure
Of sharp misprise, wounding some gentle heart.
Yet not unlovely she, my silent rose,
That only may to true love's eyes unclose,
Nor yet doth stintingly her smiles impart;
— But should bold evil venture, O what proud
Pitilessness hath she then, what anger loud!
## 18
When thou art gone, & when are gone all those
That knew thee & that loved thy living grace,
Merged in the formless flood whence all arose,
When thou hast passed, & of thy life no trace
Remaineth, nor remembrancer to say
'Such was he, such his form, his voice, his face',
In that new time shall rise, untouched by thee,
The eddying circles still, & pass away;
Full many a spring shall turn to winter dree,
& morn to nightfall, & life's human day
Shall change from youth's bright hope to darkling pain,
When thy young life in life's hard war is slain,
When thou art gone, & I, & our strong love
Which now Time's change doth but more changeless prove.
## 19
A YOUTH
Play thou on men as on a hárp's stríng:
Though of themselves they lifeless are
In them thy spirit's music shall ring.
Breathe thou in them as through a reed
Thy soul's strong message, through them declare
To the dead world thy living need.
They are to thee as the senseless words
That embody thought's full-felt commands;
Thy passive tools, obedient hands
To serve thy purpose, thy trusty swords:
They shall thy beauteous will express
Though they be graceless & purposeless.
Yea, live thou in men as their life's will,
So long as life is, & good & ill.
## 20
Go, book: go, vessel laden with the mind
That builded thee, go now & bear thy freight
Of man's old messaging to all mankind.
Say first: — Such oft-told things as I relate,
Such pond'rous words strung on unperish'd thought,
Such pearly praises worn by countless queens,
Such temper'd swords wherewith heroes have fought,
Such sceptres that have ruled all earth's demesnes,
Such broider'd robes, such antique jewelry,
Such orbs, such thrones, such treasures (say thou then)
These the rich heritage of poesy
Are as wealth's burden is to untaught men;
Life's costly crowns that but disaster bring,
To who is not in spirit born a king.
## 21
ATTAINMENT
When the strong climber his last mountain-crest
Attaineth, & the point for which he strove
Is reached, & his desire made manifest,
& seating him the topmost heights above
He gazeth on each aspect leisurely,
Considering the path by which he clomb
& which so many attempted, & how he
The first of all his race had strength to come
Unto that eminence, & how this throne
Shall men hereafter to his name recall;
Then more than ever is he strangely lone,
Seeing earth's dwellings spread out far & small;
& more unfathom'd seemeth & more high,
Eternal heaven's unchanged immensity.
## 22
Since neither man's proud pomp & kingly name
Endureth, nor his monumental throne,
Nor honour'd shrine, nor pinnacle of fame;
Since borne in agony to life he came
& in pain too he passeth, one by one
His joys desert him, friendless & alone,
Yea since all lov'd delights their cycles range,
All to their parent elements return,
Air to air, dust to dust, — when Thy Life's breath,
The fire of Thine inscrutable Will them burn,
Scattering, destroying, — yea since finisheth
All things, or death, or marring grief, or change;
Since this is, this must needs Thy purpose be,
Through such dark doors to win Thy works to Thee.
## 23
Pureness of pale moon, loneness of far skies,
Splendour of starry regions & bright sun,
Grandeur of universal vastnesses,
Sweetness of flower, staunchness of stalwart tree,
Delicacy of small verdures ev'ry one,
Loveliness of all lowly forms that be,
Gladness of ev'ry bird that gaily sings,
Ardour of eager beasts that hunt & kill,
Rapture of all life-raptur'd living things;
Yea man's one rapture too & his life's joy,
His strength, his beauty & his living will,
Each soul's will to create & to destroy, —
O thou who visitest these dwellings, why
Thus deignest thou, who art so great & high?
## VERSES AFTER HAFEZ
## 24
DAWN
I saw fair Fortune, one clear morning, touch
Like the bright-sceptred sun's first point of scorn,
With slightest finger my full-ripen'd corn.
I glimps'd her beauty: slender was she, such
As the moon's waning sickle, paled afar,
Or dawn's faint star-sheaves that scarce vision'd are.
I said, 'O my life's crowning queen, for thee
Have I long toiled without repose or rest;
In hope of thee, my harvest heavenly,
Labour'd & waited, still thou lingerest,
Tryest me still' — She turning smil'd & said,
'Though this be, be not thou uncomforted:
Lo now already thy night-ending sun
In world-seen splendour hath his day begun'.
## 25
Come let us drink & deeply drown
In Heav'n's pure wine our sorrowing!
Fling ye earth's faded garlands down,
Scatter away life's flowering!
Though sorrow's myriad armies strive
To subjugate & slay us, we,
O proud cup-bearer, will contrive
To overcome their tyranny.
O earth's sad lover, drink & throw
Unto high heaven thy misery:
So shall perchance bright beauty know
Thy longing need & bend to thee.
Not in this life's sad city grow
Immortal flowers: O friends, arise!
Drink we the wine of truth & go
To deathless joys of Paradise.
## 26
Once more, O happy hill & peaceful plain,
Once more, O kindly meadow, laugh with glee:
Now is all earth's old nature young again.
Once more the lily may her lover see;
Once more the eager tulip lifteth up
Unto the wine-filled rose her golden cup.
O hold thou dear the flowers, that through Spring's door
Enter thy garden: ere thou may'st no more
Behold them, love them who live but for thee.
Greet them, ere they through Autumn's gate depart:
Since for thy pleasuring God made them be,
Gaze on them gladly, — on me too, sweet heart,
Who for thy sake alone live; give me one
Welcome, once smile on me, ere I be gone.
## 27
Tell me not, mournful Preacher, that to prize
Beauty of flower or song or mistress fair,
Is to forgo the sweets of Paradise.
Say not, 'Life's pleasance is a deathly snare:
Shun it, so would'st thou save thy soul alive;
Blind thee, & in drear temple pray & strive'.
Know thou, all gladness is God's house of grace;
All loveliness is thy Belovéd's face;
All beauteous earth is Heav'n's gay garden-ground.
To love the rose, the fair, the gladsome bird,
Life's lovely bliss, wherever it be found,
To love love's truth from whomsoever heard,
This is their faith, who see with seeing eyes,
Their worship & their endless Paradise.
## 28
What madness 'twas, I know not, that thus enchanted me;
What wine, nor who the bringer, nor wherefore enter'd he.
— Lament no more, O full heart, thy love so close confin'd!
O rosebud, ope thy glory, thy beauty nought shall bind!
Behold the flower-fill'd meadows; thou too, O wistful Dear,
Take in thine hand the goblet, & lend to me thine ear!
Lo, for our wintry sorrow I hold the certain cure!
Lo, stern fate kneels before us! Lo, rapture evermore!
— What madness 'twas I know not, that thus entrancéd me;
What wine, nor who the bearer, nor whither vanished he.
## 29
She went. — O whither too, O one true love,
Went my sad heart, thou knowest. Lo my prayer
Followeth thee, & faith that nought may move.
With prayer I came, & now with pleading strong
I leave thee, that my flinchless trust thou share;
So shall God aid us, who to him belong.
Though all earth censure me, by Heaven I swear,
Though tyranny me test with trial untold,
No torments shall enwaver me, nor fear.
Though pleasure her most dazzling joys forth hold,
& luring musics to enravish me,
Thee only see I, thee hear, only thee
I follow: — thou who trav'lest love's long road
Knowest that there no rest is, nor abode.
## 30
I said, 'O heavenly Leader, O truth's day,
Guide thou this wanderer'. He said, 'In quest
Of his own pleasure did thy wand'rer stray'.
I pleaded, 'Bide with me'. He answer'd, 'Nay,
Unmeet were thy cold couch & cheerless rest
For me, soft-nurtur'd: vain is thy request.
The royal-born, how hardly may endure
After pomp's luxury & silk array,
Thy mean provisioning, thy dwelling poor'.
Again I prayed, 'O all life's Sun, O true
Light-giver, to our darkling earth return'.
He said, 'Pray not thus: since not ev'n my few
Dare to behold me, strange 'twere not, nor new,
That a poor wand'rer's wilder'd heart should burn'.
## 31
Where is the pious doer? & I the estray'd one, where?
Behold how far the distance, from his safe home to here!
Dark is the stony desert, trackless & vast & dim,
Where is hope's guiding lantern? Where is faith's star so fair?
My heart fled from the cloister, & chant of monkish hymn,
What can avail me sainthood, fasting & punctual prayer?
What is the truth shall light me to heav'n's strait thoroughfare?
Whither, O heart, thus hastest? Arrest thee & beware!
See what a lone adventure is thine unending quest!
Fraught with what deadly danger! Set with what unseen snare!
Say not, O friend, to Hafez, 'Quiet thee now & rest!'
Calm & content, what are they? Patience & peace, O where?
## 32
I said, 'Thou knowest, O all-knowing Friend,
My trouble for thee'. He said 'Speak not so:
Thy sorrow came; thy sorrow too shall go.'
I said, 'O light of Truth, when wilt thou spend
Thy radiance?' Answer heard I, 'It may be
This will not alway be denied to thee.'
I said, 'O Merciful, when wilt to me
Show mercy?' He replied 'Till that time is
Endure thou patiently my tyrannies.'
I said again, 'All-seeing, who dost see
How long is pain, behold'st thou not then too
How short sweet joy was?' — Answer'd he anew,
'Be not thus comfortless, but comprehend:
Ev'n as thy joys, thy sorrows too shall end.'
## 33
My heart the chamber of His musing is,
Mine eye the mirror of His beauty's face,
My hand the servant of His purposes.
I, who to neither wealth nor worldly place
Incline, nor to religion's promised ease,
Bend low beneath the burden of His grace.
Since all I have is of His bounty given,
So is my poor pride but in His proud name,
His humble service is my hoped-for Heaven.
Nor shun I men's despite & trampling sneer,
Nor heed their slander, nor the infamous fame
Of their blind censure; nay nor do I fear
Death's last defilement: though I pass in shame,
Bright worlds His immortality proclaim.
## 34
Fair is the leisure of life's garden-ground:
Pleasant is friendship's voice & mirth's soft sound.
Sweet are the perfumed flowers; yea, yea, what bliss
Sootheth like hope's fresh scent of loveliness?
Lovely, O nightingale, is thy lament;
Ever to listening love thy plaint is dear;
In the fond thought of love thy life is spent.
Though in this world joy's goal is but a name,
Fair is thy fadeless hope, blest wanderer,
Beauteous its gentle fire & flickering flame.
From the pure lily heard I this clear song:
'Happy their peaceful life who work no wrong;
Sweet idle flowers, whom heav'n's sweet airs do kiss;
No conqu'ring king hath joy more fair than this.'
## 35
Thus spake at dawn to the fresh-open'd rose
The courting-bird, 'Cease thy so empty vaunt:
Comelier than thou full many each day unclose.'
She laughed:— 'In truth I care not, ne'ertheless
Strange lover thou, to use such harsh address:
No gallant vexeth beauty with such taunt.
Or ever thou receive this ruby red,
This wine, first must thy pearl'd disdainfulness
In passion's suppliant sea its jewels shed.'
O vain it were love's chanting voice to chide!
Though may no tongue those burning thoughts expound,
The ardent fire of love no heart can hide.
In his all-whelming tears hath Hafez drowned
Wisdom & patience, yet no peace hath found.
## 36
Though beauty's tress be strayed, 'tis beauteous still:
Though her bright glance should wander, though it err
& wound me, it shall be forgiven her;
Yea, lov'd is the Belovéd though she kill.
Though should love's light'ning ravage & consume
Faith's harvest, & the garner of the wise,
Reproach not nor upbraid her: those bright eyes
Have right all to destroy, that all illume.
Betwixt love's roses should no sharpness be:
Though not uncruel, not unblameworthy
Wast thou, O sweet Love, blame thou only my
Blemish, let not remorse endolour thee.
Yea, censure not afflicting love: thy part
Is but forgiveness, O long-patient heart!
## 37
Arise, O cup-bearer, & bring
Fresh wine for our enrapturing!
O minstrel, of our sorrow sing —
'O joy of whose delight we dreamed,
O love that erst so easy seemed,
What toil is in thy travelling!'
How in the lov'd one's tent can I
Have any rest or gaiety?
Ever anon the horsemen cry,
'O lingering lover, fare thee well!'
Ever I hear the jingling bell
Of waiting steed & harnessry.
O seeker who wouldst surely bring
To happy end thy wandering,
O learner who wouldst truly know,
Let not earth's loves arrest thee. Go!
Mad thee with heaven's pure wine & fling
To those clear skies thy rapturing.
## 38
Our toil is He, & eke our journey's end;
Our life-long ailment & our remedy,
Our foeman & our ever-pitying friend;
Who is more vanquishing than victory,
Fairer than beauty, more belov'd than life;
This is he who our peace is & our strife.
Our strenuous earth is He & eke our Heaven,
The crown of conquest & the armour riven,
The strength, the struggle, yea, the failing even.
Of Him all is, & unto Him also
Doth all return: torment & yearning woe
Surely shall pass, even as pleasures go.
Yea, all have end, beggar & bountied king,
Rapture & tears, resting & wandering.
# The Biography
Isfahan, capital of the Kingdom of Persia, modern day Iran, c. 1725 — in later years it is claimed Hafez briefly fell out of favour with Shah Shuja for mocking inferior poets and was forced to flee from Shiraz to Isfahan.
Isfahan, Naghsh-i Jahan Square, today
# INTRODUCTION TO HAFEZ by Getrude Lowthian Bell
SHEMSUDDIN MAHOMMAD, better known by his poetical surname of Hafiz, was born in Shiraz in the early part of the fourteenth century. His names, being interpreted, signify the Sun of the Faith, the Praiseworthy, and One who can recite the Koran; he is further known to his compatriots under the titles of the Tongue of the Hidden and the Interpreter of Secrets. The better part of his life was spent in Shiraz, and he died in that city towards the close of the century. The exact date either of his birth or of his death is unknown. He fell upon turbulent times. His delicate love-songs were chanted to the rude accompaniment of the clash of arms, and his dreams must have been interrupted often enough by the nip of famine in a beleaguered town, the inrush of conquerors, and the flight of the defeated.
The history of Persia in the fourteenth century is exceedingly confused. Beyond a succession of wars and turmoils, there is little to be learnt concerning the political conditions under which Hafiz lived. Fifty years before the birth of the poet, Hulagu, a grandson of the great Tartar invader Chinghis Khan, had conquered Baghdad, putting to death the last of the Abbaside Khalifs and extinguishing the direct line of the race that had ruled over Persia since 750. For the next 200 years there is indeed a branch of the family of Abbas living in Cairo, members of which were set up as Khalifs by the Mamluk Sultans of Egypt; but they were destitute of any real authority, and their position was that of dependants in the Mamluk court.
The sons and grandsons of Hulagu succeeded him as lords of Persia and Mesopotamia, paying a nominal allegiance to the Great Khan of the Mongols in Cambalec or Pekin, but for all practical purposes independent, and the different provinces of their empire were administered by governors in their name. About the time of the birth of Hafiz, that is to say in the beginning of the fourteenth century, a certain Mahmud Shah Inju was governing the province of Fars, of which Shiraz is the capital, in the name of Abu Said, the last of the direct descendants of Hulagu. On the death of Mahmud Shah, Abu Said appointed Sheikh Hussein ibn Juban to the governorship of Fars, a lucrative and much-coveted post. Sheikh Hussein took the precaution of ordering the three sons of Mahmud Shah to be seized and imprisoned; but while they were passing through the streets of Shiraz in the hands of their captors, their mother, who accompanied them, lifted her veil and made a touching appeal to the people, calling upon them to remember the benefits they had received from their late ruler, the father of the three boys. Her words took instant effect; the inhabitants rose, released her and her sons, and drove Sheikh Hussein into exile. He, however, returned with an army supplied by Abu Said, and induced Shiraz to submit again to his rule. In 1335, a year or two after these events, Abu Said died, and the power of the house of Hulagu crumbled away. There followed a long period of anarchy, which was brought to an end when Oweis, another descendant of Hulagu, seized the throne. He and his son Ahmed reigned in Baghdad until Ahmed was driven out by the invading army of Timur. But during the years of anarchy the authority of the Sultan of Baghdad had been considerably curtailed. On Abu Said's death, Abu Ishac, one of the three sons of Mahmud Shah Inju who had so narrowly escaped from the hands of Sheikh Hussein, took possession of Shiraz and Isfahan, finally ousting his old enemy, while Mahommad ibn Muzaffar, who had earned a name for valour in the service of Abu Said, made himself master of Yezd.
From this time onward the governors of the Persian provinces seem to have given a nominal allegiance now to the Sultan of Baghdad, now to the more distant Khalif. The position of Shiraz between Baghdad and Cairo must have resembled that of Venice between Rome and Constantinople, and, like Venice, she was obedient to neither lord.
Abu Ishac had not steered his bark into quiet waters. In 1340 Shiraz was besieged and taken by a rival Atabeg, and the son of Mahmud Shah was obliged to content himself with Isfahan. But in the following year he returned, captured Shiraz by a stratagem, and again established himself as ruler over all Fars. The remaining years of his reign are chiefly occupied with military expeditions against Yezd, where Mahommad ibn Muzaffar and his sons were building up a formidable power. In 1352, determined to put an end to these attacks, Mahommad marched into Fars and laid siege to Shiraz. Abu Ishac, whose life was one of perpetual dissipation, redoubled his orgies in the face of danger. Uncertain of the fidelity of the people of Shiraz, he put to death all the inhabitants of two quarters of the town, and contemplated insuring himself of a third quarter in a similar manner. But these measures did not lead to the desired results. The chief of the threatened quarter got wind of the King's design, and delivered up the keys of his gate to Shah Shudja, son of Mahommad ibn Muzaffar, and Abu Ishac was obliged to seek refuge a second time in Isfahan. Four years later, in 1357, he was given up to Mahommad, who sent him to Shiraz and, with a fine sense of dramatic fitness, had him beheaded in an open space before the ruins of Persepolis.
The Arab traveller Ibn Batuta, who visited Shiraz between the years 1340 and 1350, has left a description of its ruler: "Abu Ishac," says he, "is one of the best Sultans that can be found" (it must be confessed that the average of Sultans was not very high in Ibn Batuta's time); "he is fair of face, imposing of presence, and his conduct is no less to be admired. His mind is generous, his character remarkable, and he is modest although his power is great and his territories extensive. His army exceeds the number of 30,000 men, Turks and Persians. The most faithful of his subjects are the inhabitants of Isfahan; but he fears the Shirazis, who are a brave people, not to be controlled by kings, and he will not trust them with arms." This view of his relations with the two towns tallies with Abu Ishac's subsequent history, and points to a considerable power of observation on the part of Ibn Batuta. But he relates a tale which would seem to show that Abu Ishac was not unpopular even in Shiraz: on a certain occasion he wished to build a great gate in that city, and hearing of his desire the inhabitants vied with each other in their eagerness to satisfy it; men of all ranks turned out to do the work, putting on their best clothes and digging the foundations with spades of silver. Abu Ishac shared the passion of the age for letters, and was anxious to be accounted a rival to the King of Delhi in his generosity to men of learning; "but," sighs Ibn Batuta, "how far is the earth removed from the Pleiades!" The Persian historian who describes Abu Ishac's execution, quotes a quatrain which the Atabeg is supposed to have written while he was in prison:
"Lay down thine arms when Fortune is thy foe,
'Gainst Heaven's wheel, Wrestler, try not a throw
Drink steadfastly the cup whose name is Death,
Empty the dregs upon the earth, and go."
So perished the first patron of Hafiz.
From 1353 to 1393, when Timur conquered Shiraz for the second and last time, the greater part of Persia was ruled by members of the house of Muzaffar. Scarcely a year passed undisturbed by civil war, scarcely a year in which one of the sons or grandsons of Mahommad did not suffer imprisonment or worse ills at the hands of his brothers. Mahommad himself was the first to fall. Shah Shudja seized his father while he was reading the Koran aloud with a poet of his court, and caused him to be blinded. A few years later the grim life beat itself out against the prison walls of Ka'lah-i-Safid. "Without just cause," sings Hafiz, "the victor of victors suffered imprisonment; guiltless, the mightiest head was laid low. He had overcome Shiraz and Tabriz and Irak; at the last his own hour came. He who, in the eyes of the world, was the light he had kindled (i.e. Mahommad's son, Shah Shudja), through those eyes which had gazed victorious upon the world, thrust the hot iron." A stern and pitiless man was this Mahommad, brave in battle, wise in council, ardent in religion, but hard and cruel beyond measure, a perfidious friend and a relentless enemy. The Persian historian, Lutfallah, relates that on several occasions he had seen criminals brought before Mahommad while the Amir was engaged in reading the Koran. Laying the book aside, he would draw his sword and kill the offenders as they stood, and then return unmoved to his devotions. Shah Shudja once asked his father whether he had killed 1000 men with his own hand. "No," replied Mahommad, "but I think that the number of them that I have slain must reach 800."
After his death, Shah Shudja reigned in Shiraz, and his brother Shah Yahya in Yezd. Shah Shudia was a man of like energy with his father, but it was an energy directed into different channels; the stern religious ardour of the elder man was changed into a spirit of frenzied dissipation in the younger. Whenever he was not engaged in conducting expeditions against his brothers and nephews, he was taking part in. the wildest orgies in Shiraz. He was scarcely less cruel than Mahommad. In a fit of drunkenness he ordered one of his own sons to be blinded, and though, at the instance of his vizir, he repented and sent a second messenger hot foot after the first, it was already too late to save the boy. Before Shah Shudja's death the knell of the house of Muzaffar had sounded — Tamberlain and his Tartar hordes had advanced into Northern Persia. In 1382 Shah Shudja sent a propitiatory embassy to him with gifts — jewels and silks, horses, a scarlet daïs, a royal standard, and a Chinese umbrella; and Timur in return sent the King a robe of honour and a belt studded with jewels.
Worn out before his time with riotous living, Shah Shudja did his utmost to secure the welfare of his family before he died. He sent letters both to Timur and to Sultan Ahmed of Baghdad recommending to their protection his son Zein-el-Abeddin, his brothers, and his nephews. The curtain is drawn aside for a moment from the death-bed of the King, and an anecdote, such as Oriental historians love, reveals to us the fearless and terrible face. Hearing that his brother Ahmed was preparing to dispute the succession with Zein-el-Abeddin, he sent for him in order to persuade him to withdraw his claims. But when Ahmed entered the room where Shah Shudja lay sick to death, both brothers burst into tears, and Ahmed was so much overcome by emotion that he was obliged to withdraw. Thereupon Shah Shudja sent him a letter by the hand of a faithful servant. "The world," he said, "is like unto the shadow of a cloud and a dream of the night; for the one has no resting-place, and when the dreamer awakens there remains to him but a vain memory of the other. I foresee much disturbance in Shiraz; Kerman is the home of our fathers. I have no complaint to lay at your door; but now that I am about to fare upon a long journey, if you were to become a sower of discord, not I alone would reproach you, but God also; and our enemies would rejoice. Go therefore to Kerman and renounce this unhappy city." And Ahmed went.
Shah Shudja died in the odour of sanctity. Ten holy men were with him continually, reading the Koran aloud from end to end each day. He left behind him a name renowned for courage and for liberality. He was a poet, after the fashion of kings, and from boyhood be could repeat the Koran by heart.
The son, whose future he had spent his last hours in assuring, was not to remain for long upon the throne bequeathed to him by his father. During his short reign, Zein-el-Abeddin was engaged in defending himself from the attacks of his cousin Mansur, but in 1388 he was obliged to flee before an enemy more terrible than any he had yet known. Timur, who for several years had been hovering upon the borders of Fars, overran Southern Persia and took Shiraz. Zein-el-Abeddin sought refuge with Mansur, who repaid his confidence by imprisoning and blinding him. It must have been in the year 1388 that the celebrated interview between Hafiz and Timur took place (see note to Poem V.), and not at the time of the second conquest of Shiraz in 1393. The confusion between the two dates has led several writers to doubt the truth of the story, since it is almost certain that the poet had died before 1393. Timur bestowed Shiraz upon Shah Yahya, uncle to Mansur, and some time governor of Yezd; but no sooner was the Tartar army called away by disturbances in the northern parts of the empire than Mansur overthrew his uncle and possessed himself of Shiraz. Hafiz did not live to see the end of the drama, but the end was not far off. In 1393 Timur advanced with 30,000 picked men against Mansur. The Muzaffaride, with Only 3000 or 4000 men, twice charged into the heart of the Tartar force, and at one moment Timur's own life was in danger. Mansur, who was himself fighting in the thickest of the battle, sent a message back to the wings of his army, ordering them to support his desperate charge; but they did not obey his command. He fell fighting beneath the sword of Shah Rukh Mirza, Timur's son, leaving the conqueror to "march in triumph through Persepolis." Courage was a quality in which the descendants of Mahommad ibn Muzaffar were not deficient, but among a race of soldiers Mansur seems to have been distinguished for his reckless bearing. He, too, like the other members of his family, was a patron of learning, and it is related that he used to distribute 200 tomans daily among the poor scholars of Shiraz. Both on account of their popularity and of their bravery, Timur saw that there would be no peace for him in Shiraz while one member of the house of Muzaffar remained alive; Mansur's survivors were put to the sword.
Through all these changes of fortune, Hafiz appears to have played the prudent, if rather unromantic part of the Vicar of Bray. The slender thread of his personal history is made up for the most part of more or less mythical anecdote. He was the son, according to one tradition, of a baker of Shiraz, in which city he was probably educated. The poet Jami says that he does not know under what Sufi doctor Hafiz studied. As a young man, however, he was one of the followers of Sheikh Mahmud Attar, who would seem to have been somewhat of a free-lance among the learned men of Shiraz. Sheikh Mahmud did not give himself up completely to the contemplative life, but combined the functions of a teacher with those of a dealer in fruit and vegetables. "Oh disciple of the tavern!" sings Hafiz, "give me the precious goblet, that I may drink to the Sheikh who has no monastery." Sheikh Mahmud's attitude doubtless brought him under the condemnation of the stricter Sufis, of the disciples of a certain Sheikh Hassan Asrakpush in particular, who, as the title of their master denotes, clad themselves only in blue garments, and declared that their minds were filled with heavenly desires, just as their bodies were clothed in the colour of heaven. Hafiz falls foul of this rival school in several of his poems. "I am the servant," he says, "of all who scatter the dregs of the cup and are clothed in one colour (that is, clothed in sincerity), but not of them whose bodies are clad in blue while black is the colour of their heart." And again: "Give me not the cup until I have torn from my breast the blue robe," by which he means that he cannot receive the teachings of true wisdom until he has divested himself of the errors of the uninitiated. From Sheikh Mahmud, perhaps, he learnt a wholesome philosophy which enabled him to see through the narrow-minded asceticism of other religious teachers, whether Sufi or orthodox, and he was not unmindful of the debt he owed him. "My Grey-Beard," he sings, "who scatters the dregs of the wine, has neither gold nor power, but God has made him both munificent and merciful." And indeed if he succeeded in unchaining the spirit of his disciple from useless prejudice, it may be admitted that the Sheikh went far towards providing him with a good equipment for life. Although he never submitted to any strict monastic rule, Hafiz assumed the dervish habit of which he speaks so contemptuously. We must suppose that he took the precaution, which he himself recommends, of washing it clean in the wine that Sheikh Mahmud provided for him; in other words, that he tempered his orthodoxy with the freer doctrines he had derived from his teacher. He also became a sheikh.
How he first revealed his inimitable gift of song is not known. There is a tradition that upon a certain day one of his uncles was engaged in composing a poem upon Sufiism, and being but a mediocre poetaster, could get no further than the first line. Hafiz took up the sheet in his uncle's absence and completed the verse. The uncle was not a little annoyed; he bade Hafiz finish the poem, and at the same time cursed him and his works. "They shall bring insanity," he declared, "upon all that read them." Men say that the curse still hangs over the Divan, therefore let no one whose reason is not strongly seated venture to study the poet. Whatever were his beginnings, it was not long before the young man rose into high repute. Abu Ishac was his first patron. "By the favour of the victorious standards of a king," says Hafiz, "I was uplifted like a banner among the makers of verse." There is a long poem addressed to Abu Ishac, in which he is called the King under whose feet the garden of his kingdom bursts into flower. "Oh great and holy!" cries the poet, "every man who is a servant of thine is uplifted so high that the stars of Gemini are but as his girdle." Hafiz must have been in Shiraz when Abu Ishac was brought thither, a prisoner, from Isfahan; he may even have witnessed his execution outside Persepolis. "Fate overtook him," he sighs, "all too speedily — alas for the violence and oppression in this world of pitfalls! alas for the grace and the mercy that dwelt among us! Hast thou not heard, oh Hafiz, the laugh of the strutting partridge? Little considered be the clutching talons of the falcon of death."
From the protection of Abu Ishac, Hafiz passed into that of Shah Shudja, but the relations between the two men seem to have been somewhat strained. Shah Shudja may have distrusted the loyalty of one to whom Abu Ishac had been so good a patron; moreover, he nursed a professional jealousy of Hafiz, being himself a writer of occasional verse. The historian Khondamir tells of an interview which cannot have increased the goodwill of either interlocutor towards the other. Shah Shudja reproached Hafiz with the discursiveness of his songs. "In one and the same," he said, "you write of wine, of Sufiism, and of the object of your affections. Now this is contrary to the practice of the eloquent." "That which your Majesty has deigned to speak," replied Hafiz (laying his tongue in his cheek, though Khondamir does not mention the fact), "is the essence of the truth; yet the poems of Hafiz enjoy a wide celebrity, whereas those of some other writers have not passed beyond the gates of Shiraz." But an occasional bandying of sharp speeches, in which the King usually came off second best, did little harm to a friendship which was based upon a marked correspondence in tastes. "Since the hour," declares Hafiz, "that the wine-cup received honour from Shah Shudja, Fortune has put the goblet of joy into the hand of all wine-drinkers"; and in several poems he welcomes Shah Shudja's accession to the throne and the consequent removal of an edict against the drinking of wine: "The daughter of the grape has repented of her retirement; she went to the keeper of the peace (i.e. Shah Shudja) and received permission for her deeds. Forth came she from behind the curtain that she might tell her lovers that she has turned about." Partly out of gratitude, partly with an eye to future favours, Hafiz proclaimed the glory of Shah Shudja, just as he had proclaimed that of the hapless Abu Ishac, and the King was not averse from such good wishes as these from the most famous poet of the age: "May the ball of the heavens be for ever in the crook of thy polo stick, and the whole world be a playing-ground unto thee. The fame of thy goodness has conquered the four quarters of the earth; may it be for all time a guardian unto thee!"
One of Shah Shudja's vizirs, Hadji Kawameddin Hassan, was also a good friend to Hafiz. In the poems he is frequently alluded to as the second Assaf (the first Assaf having been King Solomon's vizir, renowned for his wisdom), while Shah Shudja masquerades under the title of Solomon himself. On his return from a journey, probably to Yezd, Hafiz spent some months in the house of the Vizir-induced thereto by a cogent argument. In one of the poems there is a dialogue between himself and a friend, in which the friend says to him, "When after two years' absence thy destiny has brought thee home, why comest thou not out of thy master's house?" Hafiz replies that the road in which he walks is not of his choosing: "An officer of my judge stands, like a serpent, in ambush upon the path, and whenever I would pass beyond my master's threshold he serves me with a summons and hurries me back into my prison." He goes on to remark that under these painful circumstances he finds his master's house a sure refuge, and the servants of the Vizir useful allies against the officers of the law. "If any one proffers a demand to me there, I call to my aid the strong arm of one of the Vizir's dependants, and with a blow I cause his skull to be cleft in two." A summary manner, one would think, of dealing with the law, and little calculated to incline the heart of his judge towards the offender.
There is another Khawameddin who is frequently mentioned, the Vizir of Sultan Oweis of Baghdad. He founded in Shiraz a college for Hafiz, in which the poet gave lectures on the Koran, and read out his own verses, and whither his fame drew a great number of pupils. We find Hafiz asking his benefactor for money to support this school in the following terms: "Oh discreet friend (my poem), in some retired spot to which even the wind is a stranger, come to the ear of the master, and between jest and earnest place the pointed saying, that his heart may consent unto it; then, of thy kindness, pray his munificence to tell me, if I were to ask for a small stipend, would my request be tolerated?" One cannot but hope that so charming a begging letter, couched in verse withal, was more than tolerated. It was probably this Vizir who sent a robe of honour to Hafiz which, when it came, proved to be too short for him; "but," says the poet politely, "no favour of thine could be too short for any man."
From Oweis himself Hafiz is said to have received kindness, but he does not seem to have been satisfied with the Sultan's conduct towards him: "From my heart," he says, "I am the slave of Sultan Oweis, but he remembers not his servant." The son of Oweis, Sultan Ahmed of Baghdad, whose cruelty caused his subjects to call in the aid of Timur against him, was very anxious to induce Hafiz to visit his court; but Hafiz, perhaps with prudence, declined the invitation, saying that he was content with dry bread eaten at home, and had no desire to taste the honey that pilgrims gather by the roadside. He sent to Ahmed a poem in which he loaded his name with extravagant praise. "On Persian soil," he declared, "the bud of joy has never blown for me. How excellent is the Tigris of Baghdad and the perfumed wine! Oh wind of the dawn, bring unto me the dust from my friend's threshold, that Hafiz may wash bright with it the eyes of his heart."
Once only did he comply with the invitations of foreign kings, and his experience on that occasion was far from encouraging. He visited Shah Yahya, Shah Shudja's brother, at Yezd, but the reward which he received was not commensurate with his expectations. "Long life to thee and thy heart's desire, oh Cup-bearer of Djem's court!" he writes — and the context shows that the allusion is to Shah Yahya— "though while I dwelt with thee my cup was never filled with wine." Moreover, a devoted lover of Shiraz, Hafiz was overcome with homesickness when he was absent from his native town. "Why," he says in a pathetic little poem written while he was at Yezd— "Why should I not return to mine own home? Why should I not lay my dust in the street of mine own beloved? My bosom cannot endure the sorrows of exile; let me return to mine own city, let me be master of my heart's desire." It was after this luckless visit to Shah Yahya that he is said to have remarked, "It seems that Fortune did not intend kings to be wise."
He never again gathered the honey of the roads of pilgrimage. Once, indeed, in answer to the pressing invitation of Shah, Mahmud Purabi, Sultan of Bengal, he set forth for India; but a series of accidents befell him, he lost heart and returned home again. The story is told in a note to Poem XXI.
From the Sultan of Hormuz he received many favours, though he refused to visit him and his pearl fisheries in the Persian Gulf. He compares this Sultan with Shah Yahya, much to the disadvantage of the latter, saying that the King who had never seen him had filled his mouth with pearls, whereas Shah Yahya, to whose court he had journeyed, had sent him empty away.
Shah Shudja was not the only member of the house of Muzaffar who protected Hafiz; the warrior prince Mansur was his staunch friend. He appears to have been absent from Shiraz at the time of Mansur's accession-perhaps he had accompanic imur s retreating army. "The wind has brought me word," he cries, "that the day of sorrow is overpast; I will return to Shiraz through the favour of my friend. On the banners of the Conqueror (i.e. Mansur, of whose name this is the meaning) Hafiz is borne up into heaven; fleeing for refuge, his destiny has set him upon the steps of a throne." Mansur held the poet in high esteem. There is a tradition that when he appointed one of his sons governor over a province, the young man asked his father to give him his vizir, Jelaleddin, as a counsellor, and Hafiz as a teacher. "What!" replied Mansur, "wouldst thou be King even in thy father's lifetime, that thou demandest of him the two wisest men in his realm?"
Hafiz by this time had grown old. Youth had been very pleasant; not without a sigh the greyhaired man relinquished it. "Ah, why has my black hair turned white!" he laments, and tries to warm his old blood with the wine of former days. "Yesterday at dawn I came upon one or two glasses of wine-as sweet as the lip of the Cup-bearer they seemed to my palate. And then, my brain afire, I desired to return to my mistress, Youth, but between us a divorce had been pronounced." And again: "Last night Hafiz strayed into the tavern, and it seemed to him that Youth, his mistress, had come back, and that love and madness had returned to his old head." "Gieb meine Jugend mir zurück!" Other poets besides Hafiz have sung to the same tune. Whether or no he lived to witness the overthrow of the race that had sheltered him, he foresaw the troubles that were coming upon it and upon his beloved Shiraz. There is a short poem full of foreboding which is said to have been written after the entry of Timur: "What tumult I see beneath the moon's orbit, every quarter of the earth is full of evil and wickedness! There is strife among our daughters, and among our mothers contention, and the father is evilly disposed towards his son. Only the foolish are drinking sherbet of rose-water and sugar; the wise are nourished upon their own heart's blood. The Arabian horse is wounded beneath the saddle, and the ass wears a collar of gold about his neck. Master, take the counsel of Hafiz: 'Go and do good!' for I see that this maxim is worth more than a treasure-house of jewels." In several verses he congratulates Mansur upon a victory and a fortunate return to Shiraz, which may perhaps refer to the re-establishment of the Muzaffaride line after Timur's departure. "Give me the cup," he says in one of these, "for the airs of youth blow through my old head, so glad am I to see the King's face again."
The date of his death is variously given as 1388, 1389, 1391, and 1394, but it seems unlikely that he should have been alive as late as 1394. 1389 is the year given in a couplet by an unknown author, which is inscribed upon his tomb: "If thou wouldst know when he sought a home in the dust of Mosalla, seek his date in the dust of Mosalla." The letters of the Persian words Khak-i-Mosalla, dust of Mosalla, give the number 791, that is 1389 of our era. He lies in the garden of Mosalla outside Shiraz, a garden the praises of which he was never tired of singing, and on the banks of the Ruknabad, where he had so often rested under the shade of cypress-trees. When, some sixty years after the poet's death, Sultan Baber conquered Shiraz, he erected a monument over the tomb of Hafiz. An oblong block of stone on which are carved two songs from the Divan, marks the grave. At the head of it is inscribed a sentence in Arabic: "God is the enduring, and all else passes away." The garden contains the tombs of many devout Persians who have desired to rest in the sacred earth which holds the bones of the poet, and his prophecy that his grave should become a place of pilgrimage for all the drunkards of the world has been to a great extent fulfilled. A very ancient cypress, said to be of Hafiz's own planting, stood for many hundreds of years at the head of his grave, and It cast its shadow o'er the dust of his desire."
It is not often that a teacher and the favourite of princes enjoys unmixed popularity, especially when his criticisms of such as disagree with him are as harsh and as often repeated as are those of Hafiz; nor does he seem to have been an exception to the general rule. Moreover, his own conduct gave his enemies sufficient grounds for complaint. His biographers, as biographers will, take a rosy view of his life. Daulat Shah, for instance, states that "he turned always to the company of dervishes and of wise men, and sometimes he attained also to the society of princes; a friend of persons of eminent virtue and perfection, and of noble youths." But such accounts as these are not entirely borne out by other traditions, and his poems do not seem to the unbiased reader to be the works of a man of ascetic temperament. With all due deference to Daulat Shah, I would submit that Abu Ishac, Shah Shudja, and Shah Mansur were none of them persons of eminent virtue; indeed, it is difficult to imagine that a friend and panegyrist of theirs could have renounced all the joys of life. His enemies went so far as to accuse him of heresy and even of atheism, and so strong was popular feeling against him that, on his death, it was debated whether his body might be given the rites of burial. The question was only settled by consulting his poems, which, on being taken at haphazard, opened upon the following verse: "Fear not to follow with pious feet the corpse of Hafiz, for though he was drowned in the ocean of sin, he may find a place in paradise." It is a fortunate age which will allow a man's writings to stand his doubtful reputation in such good stead.
Hafiz was married and he had a son. He laments the death of both wife and child in two poems which are translated in this volume. In spite of all the favours which he received from the great men of his day, he is said to have died poor.
During his lifetime he was too busy "teaching and composing philosophical treatises," says his great Turkish editor, Sudi, "to gather together his songs; he used to recite them in his school, expressing a wish that these pearls might be strung together for the adornment of his contemporaries," This was done after his death by his pupil Sayyed Kasim el Anwar, and the Divan of Hafiz is one of the most popular books in the Persian language. From India to Constantinople his songs are sung and repeated by all who speak the Persian tongue, and the number of his European translators shows that his uncle's curse has a special and peculiar influence in Western countries. Like the Æneid, the Divan of Hafiz is consulted as a guide to future action. There are several stories of famous men who have had recourse to these Sortes Hafizianæ. It is related that Nadir Shah took counsel from Hafiz's book when he was meditating an expedition against Tauris, and opened it at the following verse: "Irak and Fars thou hast conquered with thy songs, oh Hafiz; now it is the turn of Baghdad and the appointed hour of Tabriz." Nadir Shah took this as an encouragement to fresh conquest, and went on his way rejoicing.
It is not only as a maker of exquisite verse but also as a philosopher that Hafiz has gained so wide an esteem in the East. No European who reads his Divan but will be taken captive by the delicious music of his songs, the delicate rhythms, the beat of the refrain, and the charming imagery. Some of them are instinct with the very spirit of youth and love and joy, some have a nobler humanity and cry out across the ages with a voice pitifully like our own; and yet few of us will turn to Hafiz for wisdom and comfort, or choose him as a guide. It is the interminable, the hopeless mysticism, the playing with words that say one thing and mean something totally different, the vagueness of a philosophy that dares not speak out, which repels the European just as much as it attracts the Oriental mind. "Give us a working theory," we demand. "Build us imaginary mansions where our souls, fugitives from the actual, may dream themselves away" — that, it seems to me, is what the Persian asks of his teacher.
Hafiz belonged to the great sect from which so many of the most famous among Persian writers have sprung. Like Sa'di and Jami and Jelaleddin Rumi and a score of others, he was a Sufi. The history of Sufiism has yet to be written, the sources from which it arose are uncertain, and that it should have found a home in Mahommadanism, the least mystical of all religions, is still unexplained. Some have supposed that Sufiism was imported from India after the time of Mahommad; some that it was a development of the doctrines of Zoroaster which the Prophet's successors silenced but did not destroy. In reply to the first theory it has been objected that there is no historic proof of relations between India and Mahommadan countries after the Mahommadan era and before the rise of Sufiism, by which the doctrines of the Indian mystics could have been propagated; and as for the second, it seems improbable that Sufiism, of which the essential doctrine is unity, could have borrowed much from a religion as sharply opposed to it as that of Zoroaster, whose creed is founded upon a dualism. A third theory is that the origins of Sufiism are to be looked for in the philosophy of the Greeks, strangely distorted by the Eastern mind, and in the influence of Christianity; but though the works of Plato are frequently quoted by mystical writers, and though it seems certain that they owe something both to the Neo-Platonic school of Alexandria and to the Christian religion, this would not be enough to account for the great perversion of Mahommad's teaching.
Baron Sylvestre de Sacy suggested the following explanation of the matter. The second century of the Hejira was a time of fermentation and of the rise of sects. This was due in the first place to the introduction of Greek philosophy, and in the second to the rivalry between the partisans of Ali and those of the Ommiad and Abbaside Khalifs. It was among the followers of Ali that the doctrines of the union of God and man, the infusion of the Divinity in the imams, and the allegorical interpretation of religious ceremonies grew up. Daulat Shah in his Biography of the Persian Poets traces back mysticism as far as to Ali himself, though it is probable that he is imputing to the son-in-law of the Prophet beliefs which were of a somewhat later date. By force of circumstances the Alides were placed in opposition to the ruling Khalifs, and were obliged to find a justification for their attitude, and for submitting to the observances enjoined by those whom they refused to recognise as true representatives of Mahommad. They read the Koran by the light of a new creed, and interpreted it in a manner far different from that intended by its author. From the moment when the division between Shi'ite and Sunni sprang into being, the Shi'ites, or followers of Ali, made the eastern provinces of the Khalifate their stronghold. It is not unreasonable to suppose that a mysticism, in every way contrary to the true spirit of the Koran, made in those provinces nearest to India so rapid a progress, because, before the conquest of Persia by the Arabs, Indian mysticism had already struck root there. That is to say, that there had grown up, side by side with Zoroastrianism, a mysticism eminently congenial to the peculiar temper of the Persian mind — so congenial, indeed, that it was not stamped out by the Arab conquerors, but insinuated itself into the stern and practical creed which they forced upon a nation of dreamers and metaphysicians. The author of the Dabistan, a book written in the seventeenth century, containing the description of twelve different faiths, relates that there existed in Persia a sect belonging to the Yekaneh Bina, of those whose eyes are fixed upon One alone: "They say that the world has no external or tangible existence; all that is, is God, and beyond him there is nothing. The intelligences and the souls of men, the angels, the heavens, the stars, the elements, and the three kingdoms of nature exist only in the mind of God and have no existence beyond." "If this Indian doctrine of Maya, or Illusion," adds M. de Sacy, "had been transferred to Persia, there is every reason to believe that mysticism, grounded on the doctrine that all things are an emanation from God and that unto him they shall return, may be traced to the same source."
The keynote of Sufiism is the union, the identification of God and man. It is a doctrine which lies at the root of all spiritual religions, but pushed too far it leads to pantheism, quietism, and eventually to nihilism. The highest good to which the Sufis can attain, is the annihilation of the actual — to forget that they have a separate existence, and to lose themselves in the Divinity as a drop of water is lost in the ocean. In order to obtain this end they recommend ascetic living and solitude; but they do not carry asceticism to the absurd extremes enjoined by the Indian mystics, nor do they approve of artificial aids for the subduing of consciousness, such as opium, or hashish, or the wild physical exertions of the dancing dervishes. The drunkenness of the Sufi poets, say their interpreters, is nothing but an ecstatic frame of mind, in which the spirit is intoxicated with the contemplation of God just as the body is intoxicated with wine. According to the Dabistan there are four stages in the manifestation of the Divinity: in the first the mystic sees God in the form of a corporal being; in the second he sees him in the form of one of his attributes of action, as the Maker or the Preserver of the world; in the third he appears in the form of an attribute which exists in his very essence, as knowledge or life; in the fourth the mystic is no longer conscious of his own existence. To the last he can hope to attain but seldom.
This losing of the soul in God is only a return (and here we come near to such Platonic doctrines as those embodied in the Phædrus) to the conditions which existed before birth into the world. Just as in the Dialogue the immortal steed which is harnessed to the chariot of the soul, longs to return to the plain of birth, and to see again the true justice, beauty, and wisdom of which it has retained an imperfect recollection, so the soul of the Sufi longs to return to God, from whom it has been separated by the mortal veil of the body. But this reunion is pushed much further by the Eastern philosophers than by Plato; it implies, according to them, the complete annihilation of distinct personality, corresponding to the conditions, quite unlike those described by the Platonic Socrates, which they believe to have existed before birth. There is nothing which is not from God and a part of God. In himself he contains both being and not being; when he chooses he casts his reflection upon the void, and that reflection is the universe. There is a fine passage in Jami's Yusuf and Zuleikha in which he sets forth this doctrine of the creation. "Thou art but the glass," the poet concludes, "his is the face reflected in the mirror; nay, if thou lookest steadfastly, thou shalt see that he is the mirror also." In a parable, Jami illustrates the universal presence of God, and the blind searching of man for that by which he is surrounded on every side. There was a frog which sat upon the shores of the ocean, and ceaselessly, day and night he sang its praise. "As far as mine eyes can see," he said, "I behold nothing but thy boundless surface." Some fish swimming in the shallow water heard the frog's song, and were filled with a desire to find that wonderful ocean of which he spoke, but go where they would they could not discover it. At last, in the course of their search, they fell into a fisherman's net, and as soon as they were drawn out of the water they saw beneath them the ocean for which they had been seeking. With a leap they returned into it.
The story of the creation as told in the Koran it is impossible for the Sufis to accept; they are bound to give an outward adhesion to it, but in their hearts they treat it as an allegory. The world is posterior to God only in the nature of its existence and not in time: the Sufis were not far from the doctrine of the eternity of matter, from which they were only withheld by the necessity of conforming with the teaching of the Koran. They content themselves with saying that the world came into existence when it pleased God to manifest himself beyond himself, and will cease when it shall please him to return into himself again. It is more difficult to dispose of the resurrection of the body, which is constantly insisted upon by Mahommad. That the soul, when it has at last attained to complete union with God, should be obliged to return to the prison from whence it has escaped at death, is entirely repugnant to all Sufis nor can they explain satisfactorily the divergence of their opinions from those of the Prophet.
It has been well said that all religious teachers who have honestly tried to construct a working formula, have found that one of their greatest difficulties lay in reconciling the all-powerfulness of God with man's consciousness of his will being free; for on the one hand it is impossible to conceive a God worth the name who shall be less than omnipotent and omniscient, and on the other it is essential to lay upon man some responsibility for his actions. Mahommad more especially, as Count Gobineau points out in his excellent little book, found himself confronted with this difficulty, since his primary object was to exalt the divine personality, and to lift it out of the pantheism into which it had fallen among the pre-Islamitic Arabs; but if he did not succeed in indicating a satisfactory way out of the dilemma, it is at least unjust to accuse him of having failed to recognise it. He insisted that man is responsible for his own salvation: "Whosoever chooseth the life to come, their desire shall be acceptable unto God." There is a tradition that when some of his disciples were disputing over predestination, he said to them: "Why do you not imitate Omar? For when one came to him and asked him, 'What is predestination?' he answered, 'It is a deep sea.' And a second time he replied, 'It is a dark road.' And a third time, 'It is a secret which I will not declare since God has seen fit to conceal it.' "The Sufis were obliged to abandon free will: it was impossible to attach any responsibility to the reflection in the mirror. But here, again, they did not venture to give expression to their real opinions, and their statements are therefore both confused and contradictory. "A man may say," remarks the author of the Dabistan, "that his actions are his own, and with equal truth that they are God's." In the Gulshen-i-Raz, a poem written in the year 1317, and therefore contemporary with Hafiz, it is distinctly laid down that God will take men's actions into account: "After that moment (i.e. the Day of Judgment) he will question them concerning good and evil." But such expressions as these are in direct opposition to the rest of Sufi teaching. There is neither good nor evil, since both alike flow from God, from whom all flows. Some go so far as to prefer Pharaoh to Moses, Nimrod to Abraham, because they say that though Pharaoh and Nimrod were in apparent revolt against the Divinity, in reality they knew their own nothingness and accepted the part that the divine wisdom had imposed upon them. There is neither reward nor punishment; Paradise is the beauty, Hell the glory of God, and when it is said that those in Hell are wretched, it is meant that the dwellers in Heaven would be wretched in their place. And finally, there is no distinction between God and man; the soul is but an emanation from God, and a man is therefore justified in saying with the fanatic Hallaj, "I am God." Though Hallaj paid with his life for venturing to give voice to his opinion, he was only repeating aloud what all Sufis believe to be true. "Is it permitted to a tree to say, 'I am God,'" writes the author of the Gulsheni-Raz (the allusion is to the burning bush that spoke to Moses) why then may not a man say it?" And again: "In God there is no distinction of quality; in his divine majesty I, thou, and we shall not be found. I, thou, we, and he bear the same meaning, for in unity there is no division. Every man who has annihilated the body and is entirely separated from himself, hears within his heart a voice that crieth, 'I am God.'"
The conception of the union and interdependence of all things divine and human is far older than Sufi thought. It goes back to the earliest Indian teaching, and Professor Deussen, in his book on Metaphysics, has pointed out the conclusion which is drawn from it in the Veda. "The gospels," he says, "fix quite correctly as the highest law of morality, Love thy neighbour as thyself. But why should I do so, since by the order of nature I feel pain and pleasure only in myself, not in my neighbour? The answer is not in the Bible (this venerable book being not yet quite free from Semitic realism), but it is in the Veda: You shall love your neighbour as yourselves because you are your neighbour; a mere illusion makes you believe that your neighbour is something different from yourselves. Or in the words of the Bhagaradgitah: He who knows himself in everything and everything in himself, will not injure himself by himself. This is the sum and tenor of all morality, and this is the standpoint of a man knowing himself a Brahman."
The Sufis were forced to pay an exaggerated deference to the Prophet and to Ali in order to keep on good terms with the orthodox, but since they believed God to be the source of all creeds they could not reasonably place one above another; nay more, since they taught that any man who practised a particular religion had failed to free himself from duality and to reach perfect union with God, they must have held Mahommadanism in like contempt with all other faiths. "When thou and I remain not (when man is completely united with God), what matters the Ka'ba and the Synagogue and the Monastery?" That is, what difference is there between the religion of Mahommadan, Jew, and Christian? "One night," says Ferideddin Attar in a beautiful allegory, "the angel Gabriel was seated on the branches of a tree in the Garden of Paradise, and he heard God pronounce a word of assent. 'At this mornent,' thought the angel, 'some man is invoking God. I know not who he is; but this I know, that he must be a notable servant of the Lord, one whose soul is dead to evil and whose spirit lives.' Then Gabriel desired to know who this man could be, but in the seven zones he found him not. He traversed the land and the sea and found him not in mountain or in plain. Therefore he hastened back to the presence of God, and again he heard him give a favourable answer to the same prayers. Again he set forth and sought through the world, yet he saw not the servant of God. 'Oh Lord,' he cried, 'show me the path that leads to him upon whom thy favours fall!' 'Go to the Land of Rome,' God answered, 'and in a certain monastery thou shalt find him.' Thither fled Gabriel, and found him whom he sought, and lo! he was worshipping an idol. When he returned, Gabriel opened his lips and said, 'Oh Master, draw aside for me the veil from this secret: why fulfillest thou the prayers of one who invokes an idol in a monastery?' And God replied, 'His spirit is darkened and he knows not that he has missed the way; but since he errs from ignorance, I pardon his fault: my mercy is extended to him, and I allow him to enter into the highest place.'"
In the language of religious mysticism, God is not only the Creator and Ruler of the world, he is also the Essentially Beautiful and the True Beloved. Love, of which the divine being is at once the source and the object, plays a large part in Sufi writings, a part which it is difficult, and sometimes unwise, to distinguish from an exaggerated expression of the human affections. Jami describes Pure Being, before it had been manifested in Creation, "singing of love unto itself in a wordless melody," and in the same strain Hafiz sings of "the Imperial Beauty which is for ever playing the game of love with itself." Like the echo of a Greek voice falls Jami's doctrine of human love: "Avert not thy face from an earthly beloved, since even this may serve to raise thee to the love of the True." It is almost possible to read in the Persian poem the words of the wise Diotima to Socrates: "He who has been instructed thus far in the things of love, and has learnt to see the Beautiful in true order and succession, when he comes towards the end will suddenly perceive a nature of wonderful beauty, not growing or decaying, waxing or waning . . . he who, under the influence of true love, rising upward from these things begins to see that beauty, is not far from the end."
The Sufis had no difficulty in finding in the Koran texts in support of their teaching. When Mahommad exclaims, "There are times when neither cherubim nor prophet are equal unto me!" the Sufis declare that he alludes to moments of ecstatic union with God; and his account of the victory of Bedr— "Thou didst not slay them, but God slew them, and thou didst not shoot when thou didst shoot, but God shot" — they take as a proof of the Prophet's belief in the essential oneness of God and man. The whole book is twisted after this fashion into agreement with their views.
Beautiful and spiritual as some of these doctrines are, they can hardly be said to form an adequate guide to conduct. The Sufis, however, are regarded in the East as men leading a virtuous and pure life. Even the etymology of their name points to the same conclusion: Sufi comes from an Arabic word signifying wool, and indicates that they were accustomed to clothe themselves in simple woollen garments. They occupy in the East much the same position that Madame Guyon and the Jansenists occupied in the West, and they teach the same doctrine of quietism, which, while it lends to its followers the virtues of exaggerated submission, saps the root of a faith that is manifested in works. So far as the Sufis are striving earnestly after union with God, they are saved from the logical consequences of their doctrines: "Their ear is strained to catch the sounds of the lute, their eyes are fixed upon the cup, their bosoms are filled with the desire of this world and of the world to come." And in the same spirit Hafiz sings: "Though the wind of discord shake the two worlds, mine eyes are fixed upon the road from whence cometh my Friend." The idealism of the Sufis led them to deny the morality of all actions, but they restricted the consequences of their principles to the adepts who had attained to perfect union with God, and even for them the moments of ecstasy are few. Most Sufis are good and religious men, holding it their duty to conform outwardly, and no discredit to use all artifices to conceal from the orthodox the beliefs which they cherish in their heart, but holding also that the practice of the Mahommadan religion, to the rites of which they have attached symbolic meanings, is the only way to the perfection to which they aspire. Nevertheless, Count Gobineau is of opinion that quietism is the great curse of the East. "The dominant characteristic of Sufiism," he says, "is to unite by a weak chain of doctrine, ideas the significance of which is very different, so different that there is in reality but one connecting link between them, and that link is a quietism adapted to them all, a passive disposition of spirit which surrounds with a nimbus of inert sentiment all conceptions of God, of man, and of the universe. It is this quietism, and not Islam, which is the running sore of all Oriental countries."
Unfortunately, as he points out, the conditions of Oriental life are such as to enforce rather than to control a disposition to mysticism. The poets found ready to their hand a mass of vague and beautiful thought eminently suited to imaginative treatment; whether they believed in it or not they used it, and thereby popularised it, delighting, as only an Oriental can, in the necessity of veiling it with exquisite symbolism, and throwing round it a cloud of charming phrases. These phrases caught and held the Oriental ear; and the Oriental mind is faithful to a formula once accepted. Moreover, when a man looked about him and saw the vicissitudes of mortal existence — nowhere more marked than in the East — how conqueror succeeded conqueror and empire empire, how the humble was exalted and the mighty thrown from his seat, how swift was the vengeance of God in sweeping pestilence and resistless famine, and how unsparing the forces of nature, he turned to a philosophy which taught that all earthly things were alike vain-virtue and patriotism and the love of wife and child, power and beauty and the bold part played in a hopeless fight; he remembered what he had learnt from poets and story-tellers— "Behold the world is as the shadow of a cloud and a dream of the night."
How far the Divan of Hafiz can be said to embody these doctrines, each reader must decide for himself, and each will probably arrive at a different conclusion. Between the judgment of Jami, that Hafiz was undoubtedly an eminent Sufi, and that of Von Hammer, who, playing upon his names, declared that the Sun of the Faith gave but an uncertain light, and the Interpreter of Secrets interpreted only the language of pleasure — between these two there is a wide field for differences of opinion. For my part, I cannot agree entirely either with Jami or with Von Hammer. Partly, perhaps, owing to the wise guidance of Sheikh Mahmud Attar, partly to a natural freedom of spirit, Hafiz seems to me to rise above the narrow views of his co-religionists, and to look upon the world from a wider standpoint. The asceticism of Sufi and orthodox he alike condemns: "The ascetic is the serpent of the age!" he cries. I think it was not only to curry favour with a king that he welcomed the accession of Shah Shudja, nor was it only to disarm the criticism of stricter Mohammadans that he described himself as a weary seeker after wisdom, praying God to show him some guiding light by which he might direct his steps. Of the two conclusions that are commonly drawn from the statement that to-morrow we die, Hafiz accepted neither unmodified by the other. "Eat and drink," seemed to him a poor solution of the mysterious purpose of human life, and an unsatisfactory sign-post to happiness; "the abode of pleasure," he says, "was never reached except through pain." On the other hand, he was equally unwilling to despise the good things of this world. "The Garden of Paradise may be pleasant, but forget not the shade of the willow-tree and the fair margin of the fruitful field." "Now, now while the rose is with us, sing her praise; now, while we are here to listen, Minstrel, strike the lute! for the burden of all thy songs has been that the present is all too short, and already the unknown future is upon us." He, too, would have us cut down far reaching hope to the limit of our little day, though he cherished in his heart a more or less elusive conviction that he should find the fire of love burning still, and with a purer flame, behind the veil which his eyes could not pierce.
Be that as it may, one who sings the cool rush of the wind of dawn, the scarlet cup of the tulip uplifted in solitary places, the fleeting shadows of the clouds, and the praise of gardens and fountains and fruitful fields, was not likely to forget that even if the world is no more than an intangible reflection of its Creator, the reflection of eternal beauty is in itself worthy to be admired. I wish I could believe that such innocent delights as these, and a wholehearted desire for truth, had been enough for our poet, but I have a shrewd suspicion that the Cupbearer brought him a wine other than that of divine knowledge, and that his mistress is considerably more than an allegorical figure. How ever willing we may be to submit to the wise men of the East when they tell us that the revelry of the poems is always a spiritual exaltation, it must be admitted that the words of the poet carry a different conviction to Western ears. There is undoubtedly a note of sincerity in his praise of love and wine and boon-companionship, and I am inclined to think that Hafiz was one of those who, like Omar Khayyam, were wont to throw the garment of repentance annually into the fire of Spring. It must be remembered that the morality of his day was not that of our own, and that the manners of the East resemble but vaguely those of the West; and though as a religious teacher Hafiz would have been better advised if he had less frequently loosened the rein of his desires, I doubt whether his songs would have rung for us with the same passionate force. After all, the poems of St. Francis of Assisi are not much read nowadays. Nevertheless, the reader misses a sense of restraint both in the matter and in the manner of the Divan. To many Persians, Hafiz occupies the place that is filled by Shakespeare in the minds of many Englishmen. It may be a national prejudice, but I cannot bring myself to believe that the mental food supplied by the Oriental is as good as the other. But, then, our appetites are not the same.
The tendency in dealing with a mystical poet is to read into him so-called deeper meanings, even when the simple meaning is clear enough and sufficient in itself. Hafiz is one of those who has suffered from this process; it has removed him, in great measure, from the touch of human sympathies which are, when all is said and done, a poet's true kingdom. Of a different age, a different race, and a different civilisation from ours, there are yet snatches in his songs of that melody of human life which is everywhere the same. When he cries, "My beloved is gone and I had not even bidden him farewell!" his words are as poignant now as they were five centuries ago, and they could gain nothing from a mystical interpretation. As simple and as touching is his lament for his son: "Alas! he found it easy to depart, but unto me he left the harder pilgrimage." And for his wife: "Then said my heart, I will rest me in this city which is illumined by her presence; already her feet were bent upon a longer journey, but my poor heart knew it not." Not Shakespeare himself has found a more passionate image for love than: "Open my grave when I am dead, and thou shalt see a cloud of smoke rising out from it; then shalt thou know that the fire still burns in my dead heart-yea, it has set my very winding-sheet alight." Or: "If the scent of her hair were to blow across my dust when I had been dead a hundred years, my mouldering bones would rise and come dancing out of the tomb." And he knows of what he writes when he says, "I have estimated the influence of Reason upon Love and found that it is like that of a raindrop upon the ocean, which makes one little mark upon the water's face and disappears." These are the utterances of a great poet, the imaginative interpreter of the heart of man; they are not of one age, or of another, but for all time. Fitz-Gerald knew it when he declared that Hafiz rang true. "Hafiz is the most Persian of the Persians," he says. "He is the best representative of their character, whether his Saki and wine be real or mystical. Their religion and philosophy is soon seen through, and always seems to me cuckooed over like a borrowed thing, which people once having got do not know how to parade enough. To be sure their roses and nightingales are repeated often enough. But Hafiz and old Omar Khayyam ring like true metal." The criticism and the praise seem to me both just and delicate.
To a certain extent it may be said that the Sufiism of Hafiz is partly due to the natural leaning of the Oriental poet towards a picturesque diction (for all poetry must, to satisfy Eastern readers, be couched in a veiled and enigmatic speech), and has partly been read into the Divan by later ages. But this is not all. With Shah Shudja, I would accuse him of mixing up inextricably wine and love and Sufi teaching, and perhaps more besides. To some at least of the innumerable difficulties which assail every man who turns a thoughtful eye upon life and its conditions, Hafiz seems to have accepted the solution presented to him by Sufiism. He understood and sympathised with the bold heresy of Hallaj, "though fools whom God hath not uplifted know not the meaning of him who said, I am God." Sometimes we find him enunciating one of the abstruser of the Sufi doctrines: "How shall I say that existence is mine when I have no knowledge of myself, or how that I exist not when mine eyes are fixed upon Him?" — a man, that is, can lay claim to no individual existence; all that he knows is that he is a part of the eternally existing. Or, again, he declares that his words are metaphorical, and should receive the full Sufi interpretation, as in the following couplet: "Boon companion, minstrel, and cup-bearer, all these are but names for Him; the image of water and clay (man) is an illusion upon the road of life." But he handles Sufiism in a broad and noble manner, which links it on to the highest codes of morality accepted among the civilised races of mankind. "For all eternity the perfume of love comes not to him who has not swept with his cheek the dust from the tavern threshold"— "Blessed are the poor in spirit," Hafiz is saying in phraseology suited to the ears of those whom he addressed. "If thou desire the jewelled cup of ruby wine," he continues (and it is of the hunger and thirst after wisdom that he speaks), "ah, many tears shall thine eyes thread upon thine eyelashes!" He did not forget that "the Sufi gold is not always without alloy," and he was not one of those who believe that they have discovered the answer to all human demands when their own heart is satisfied. "Since thou canst never leave the palace of thyself," he warns us, "how canst thou hope to reach the village of truth." The song that filled his soul with gladness might strike on other ears to a different measure; and "where is the music to which both the drunk and the sober can dance?" He was, indeed, profoundly sceptical as to the infallibility of any creed, judging men not by the practice, but by the spirit that lay beneath it: "None shall die whose heart has lived with the life love breathed into it; but when the day of reckoning comes, I fancy that the Sheikh will find that he has gained as little by his abstinence as I by my feasting."
Sufiism apart, an undercurrent of mysticism runs through the poems which it is impossible to explain away. If we should attempt to ignore it, many of the odes would have no meaning at all, and most of them would lose a good half of their interest. Take, for instance, such verses as the following: "Heart and soul are fixed upon the desire of the Beloved: this at least is, for if not, heart and soul are nought. Fate is that which comes to the brink without the heart's blood; if not, all thy striving after the Garden of Paradise is nought. Throw thyself not at the foot of its sacred trees hoping for their shade; dost thou not see, oh cypress, that even these are nought unto thee?" Hafiz is engaged in that terrible weighing of possibilities which every man who thinks must know: "Surely the soul which is filled with the desire of God must have some quality which shall be stronger than death? But if this were not so . . . then indeed the soul itself is nought. Surely Fate is like an empty bowl standing upon the edge of the river of life? But if the bowl had been already filled with blood then all your striving to reach the Garden of Paradise shall avail you nothing. For do you not see, you who dare to acknowledge the truth, that you cannot battle against an appointed Destiny, and however grateful may be the shade of the holy trees, they could afford you no protection." Nor can I believe that it is an earthly love of whom he speaks when he says, "Since the Beloved has veiled his face, how comes it that his lovers are reciting his beauties? They can only tell what they imagine to be there." We are all engaged in telling each other — only what we imagine to be there.
It is a curious coincidence (if it be nothing more) that at the time when mystical poetry was taking a recognised place in the literature of Persia and of India, it was also springing into existence in the West. The songs of the Troubadours were avowedly intended to convey a meaning deeper than that which lay upon the surface; the Romance of the Rose comes nearer than any other Western allegory to a full-fledged mysticism worthy of an Oriental poet. St. Francis addresses his Redeemer in terms not very different from those used by Hafiz to express his longing after divine wisdom, and the Beatrice, perhaps of the Vita Nuova, certainly of the Divine Comedy, is no less intangible than the allegorical mistress (when she is allegorical) of the Persian.
Hafiz and Dante, it is interesting to note, were almost contemporaries. At the time when Dante was climbing Can Grande's weary stair, Hafiz was opening his eyes upon a yet more tumultuous world. Both were driven by the confusion around them to look for some solid platform on which to build a theory of existence, but Dante found it in that strenuous personal faith which is for ever impossible to minds of the temper of that of Hafiz. Moreover, the mysticism of Dante stands with its feet planted firmly upon the earth: man and his deeds might be fleeting, but they laid so strong a hold upon the poet's imagination that he welded them into a stepping-stone to that which shall not pass away. His own life was spent in a ceaseless political activity; for all his visionary journeys through heaven and hell, Dante lived as keenly as any of his contemporaries. The fire still burns in the dead heart; the fierce and tender spirit, roused by turns to merciless condemnation and exquisite pity, still glows with a flame removed from mortal conditions, which the chill of death cannot extinguish as long as men shall read and understand. Through him his age lives. The people whom he had met, those of whom he had only heard, the smallest incidents of his time, the sum of all that it knew and of all that it believed, are struck out for ever, hard and sharp, in his vivid lines; and the fortunes of Florence, of one little town in a little corner of the world, loom to us, under the poet's influence, as big and as tragic as they seemed to that most ardent of citizens. To Hafiz, on the contrary, modern instances have no value; contemporary history is too small an episode to occupy his thoughts. During his lifetime the city that he loved, perhaps as dearly as Dante loved Florence, was besieged and taken five or six times; it changed hands even more often. It was drenched with blood by one conqueror, filled with revelry by a second, and subjected to the hard rule of asceticism by a third. One after another Hafiz saw kings and princes rise into power and vanish "like snow upon the desert's dusty face." Pitiful tragedies, great rejoicings, the fall of kingdoms, and the clash of battle-all these he must have seen and heard. But what echo of them is there in his poems? Almost none. An occasional allusion which learned commentators refer to some political event; an exaggerated effusion in praise first of one king, then of another; the celebration of such and such a victory and of the prowess of such and such a royal general-just what any self-respecting court poet would feel it incumbent upon himself to write and no more.
But some of us will feel that the apparent indifference of Hafiz lends to his philosophy a quality which that of Dante does not possess. The Italian is bound down within the limits of his own realism, his theory of the universe is essentially of his own age, and what to him was so acutely real is to many of us merely a beautiful or a terrible image. The picture that Hafiz drew represents a wider landscape, though the immediate foreground may not be so distinct. It is as if his mental eye, endowed with wonderful acuteness of vision, had penetrated into those provinces of thought which we of a later age were destined to inhabit. We can forgive him for leaving to us so indistinct a representation of his own time, and of the life of the individual in it, when we find him formulating ideas as profound as the warning that there is no musician to whose music both the drunk and the sober can dance.
Renan has put into a few luminous sentences his view of the mystical poets of India and Persia. "On sait que dans ces pays," he says, "s'est développée une vaste littérature où l'amour divin et Famour terrestre se croisent d'une façon souvent difficile à démêler. L'origine de se singulier genre de poésie est une question qui n'est pas encore éclaircie. Dans beaucoup de cas les sens mystiques prêtés à certaines poésies érotiques persanes et hindoues n'ont pas plus de réalité que les allégories du Cantique des Cantiques. Pour Hafiz, par exemple, il semble bien que l'explication allégorique est le plus souvent un fruit de la fantaisie des commentateurs, ou des précautions que les admirateurs du poète étaient obligés de prendre pour sauver l'orthodoxie de leur auteur favori. Puis l'magination étant montée sur ce thème, et les esprits étant faussés par une exégèse qui ne voulait voir partout qu'allégories, on en est venu à faire des poèmes réellement à double sens. Comme ceux de Djellaleddin Rumi, de Wali, &c. . . . Dans l'Inde et la Perse ce genre de poésie (érotico-mystique) est le fruit d'un extrème raffinement, d'une imagination vive et portée au quiétisme, d'un certain goût du mystére, et aussi, en Perse du moins, de l'hypocrisie imposée par le fanatisme musulman. C'est, en effet, comme réaction contre la sécheresse de l'Islamisme que le soufisme a fait fortune chez les musulmans non arabes. Il y faut voir une révolte de l'esprit arien contre l'effroyante simplicité de l'esprit sémitique, excluant par la rigueur de sa théologie toute devotion particulière, toute doctrine secrète, toute combinaison religieuse vivante et variée."
Those who have written poems "réellement à double sens" are careful to insist upon the mighty secrets that their words convey. "The things which wise men, who are sometimes called drunkards and sometimes seers," says one of them, "wish to express by the words wine, cup and cup-bearer, musician, magian, and Christian girdle, are so many profound mysteries which sometimes they translate by an enigma and sometimes they reveal." The symbols used by each writer are more or less the same; there is an accepted Sufi code with which the initiated are acquainted. "The nightingale, and none beside, knows the full worth of the rose," sings Hafiz, "for many a one reads the leaf and understands not the meaning thereof." But though we may not all be nightingales, we have some guide to the interpretation of the leaf. Many of the words in the Sufi dictionary have been expounded to the outer world. The tavern, for instance, is the place of instruction or worship, of which the tavern-keeper is the teacher or priest, and the wine the spirit of divine knowledge which is poured out for his disciples the idol is God; beauty is the divine perfection shining locks the expansion of his glory; down on the cheek denotes the cloud of spirits that encircles his throne; and a black mole is the point of indivisible unity. The catalogue might be continued to any extent; almost every word has a vague and somewhat shifting significance in the language of mysticism, which he who has a mind for such exercises may decipher if he choose.
Hafiz is rather the forerunner than the founder of this school of poets. It is equally unsatisfactory to give a completely mystical or a completely material interpretation to his songs. He wrote of the world as he found it. In his experience pleasure and religion were the two most important incentives to human action; he ignored neither the one nor the other. I am very conscious that my appreciation of the poet is that of the Western. Exactly on what grounds he is appreciated in the East it is difficult to determine, and what his compatriots make of his teaching it is perhaps impossible to understand. From our point of view, then, the sum of his philosophy seems to be, that though there is little of which we can be certain, that little must always be the object of all men's desire; each of us will set out upon the search for it along a different road, and if none will find his road easy to follow, each may, if he be wise, discover compensations for his toil by the wayside. And for the rest, "Who knows the secret of the veil?" Like many a good and brave man before his time and since, I think he was content to "faintly trust the larger hope."
## ENDNOTES
For the history of the times of Hafiz, see Defrémery in the Journal Asiatique for 1844 and 1845, Malcolm's "History of Persia," Price's "Mohammedan History," Markham's "History of Persia", For the life of the poet, see V. Hammer; Defrémery in the Journal Asiatique for 1858; Sir Gore Ouseley and Daulat Shah, whose work is mainly a string of anecdote — I have been told that Lutfallah's is little better.
The "Travels of Ibn Batuta," edited by Defrémery and Sanguinetti.
Journal des Savants for 1821 and 1822.
Numberless beautiful images are used to describe the union of God and man. Jelaleddin Rumi points the same moral in the following exquisite apologue: "There came one and knocked at the door of the Beloved. And a voice answered and said, 'Who is there?' The lover replied, 'It is I' 'Go hence,' returned the voice; 'there is no room within for thee and me.' Then came the lover a second time and knocked, and again the voice demanded, 'Who is there?' He answered, 'It is thou.' 'Enter,' said the voice, 'for I am within.'
Dr. Johnson's contribution to this vexed question is perhaps as good as any other: "Sir," said he to Boswell, "we know the will is free, there's an end on't."
Les Religions de l'Asie Centrale.
Cf. St. Paul, who is scarcely more explicit: "Work out your own salvation; for it is God which worketh in you both to will and to work for his good pleasure " (Phil. ii. 12).
Dabistan.
Hallaj lived in the ninth century. He was believed by some to be a sorcerer, and by others a holy worker of miracles. He was condemned to death with horrible tortures by the Khalif of Baghdad in gig, and his ashes were thrown into the Tigris. It is said that a Sufi once asked God why he suffered his servant Hallaj to fall into the Khalifs hands, and was answered, "Thus the revealers of secrets are punished."
Gulshen-i-Raz.
Yusuf and Zuleikha.
"A Year among the Persians." Browne.
Sayyed Ahmed of Isfahan.
Listen to the advice of an Afghan singer who wrote his Ars Poetica in the mountains south of Peshawar about the middle of the seventeenth century: —
"The arrow needs an archer, and poetry a magician.
"He must hold ever in the hand of his mind the weighing scales of metre, rejecting the verse which is too short and that which is too long.
"His mistress, Truth, shall mount her black steed, the veil of allegory drawn across her brow.
"Let her shoot from beneath her eyelashes a hundred glances, challenging and victorious.
"Let the poet place upon her fingers the jewels of the art of many hues, adorn her with the sandal-wood and the saffron of metaphor;
"The bells of alliteration like bangles upon her feet, and on her bosom the necklace of a mysterious rhythm.
Add to these the hidden meaning, like eyes half seen through their lashes, that her whole body may be a perfect mystery."-"Translation of the Kilidi Afghani," by T. C. Plowden.
I fear the outcome of these directions is too often "amphora coepit institui, currente rota cur urceus exit," and perhaps the advice of Horace may be the better of the two "denique sit quod vis, simplex dumtaxat et unum."
Cantique des Cantiques.
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## Series Contents
Series One
Anton Chekhov
Charles Dickens
D.H. Lawrence
Dickensiana Volume I
Edgar Allan Poe
Elizabeth Gaskell
Fyodor Dostoyevsky
George Eliot
H. G. Wells
Henry James
Ivan Turgenev
Jack London
James Joyce
Jane Austen
Joseph Conrad
Leo Tolstoy
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Mark Twain
Oscar Wilde
Robert Louis Stevenson
Sir Arthur Conan Doyle
Sir Walter Scott
The Brontës
Thomas Hardy
Virginia Woolf
Wilkie Collins
William Makepeace Thackeray
Series Two
Alexander Pushkin
Alexandre Dumas (English)
Andrew Lang
Anthony Trollope
Bram Stoker
Christopher Marlowe
Daniel Defoe
Edith Wharton
F. Scott Fitzgerald
G. K. Chesterton
Gustave Flaubert (English)
H. Rider Haggard
Herman Melville
Honoré de Balzac (English)
J. W. von Goethe (English)
Jules Verne
L. Frank Baum
Lewis Carroll
Marcel Proust (English)
Nathaniel Hawthorne
Nikolai Gogol
O. Henry
Rudyard Kipling
Tobias Smollett
Victor Hugo
William Shakespeare
Series Three
Ambrose Bierce
Ann Radcliffe
Ben Jonson
Charles Lever
Émile Zola
Ford Madox Ford
Geoffrey Chaucer
George Gissing
George Orwell
Guy de Maupassant
H. P. Lovecraft
Henrik Ibsen
Henry David Thoreau
Henry Fielding
J. M. Barrie
James Fenimore Cooper
John Buchan
John Galsworthy
Jonathan Swift
Kate Chopin
Katherine Mansfield
L. M. Montgomery
Laurence Sterne
Mary Shelley
Sheridan Le Fanu
Washington Irving
Series Four
Arnold Bennett
Arthur Machen
Beatrix Potter
Bret Harte
Captain Frederick Marryat
Charles Kingsley
Charles Reade
G. A. Henty
Edgar Rice Burroughs
Edgar Wallace
E. M. Forster
E. Nesbit
George Meredith
Harriet Beecher Stowe
Jerome K. Jerome
John Ruskin
Maria Edgeworth
M. E. Braddon
Miguel de Cervantes
M. R. James
R. M. Ballantyne
Robert E. Howard
Samuel Johnson
Stendhal
Stephen Crane
Zane Grey
Series Five
Algernon Blackwood
Anatole France
Beaumont and Fletcher
Charles Darwin
Edward Bulwer-Lytton
Edward Gibbon
E. F. Benson
Frances Hodgson Burnett
Friedrich Nietzsche
George Bernard Shaw
George MacDonald
Hilaire Belloc
John Bunyan
John Webster
Margaret Oliphant
Maxim Gorky
Oliver Goldsmith
Radclyffe Hall
Robert W. Chambers
Samuel Butler
Samuel Richardson
Sir Thomas Malory
Thomas Carlyle
William Harrison Ainsworth
William Dean Howells
William Morris
Series Six
Anthony Hope
Aphra Behn
Arthur Morrison
Baroness Emma Orczy
Captain Mayne Reid
Charlotte M. Yonge
Charlotte Perkins Gilman
E. W. Hornung
Ellen Wood
Frances Burney
Frank Norris
Frank R. Stockton
Hall Caine
Horace Walpole
One Thousand and One Nights
R. Austin Freeman
Rafael Sabatini
Saki
Samuel Pepys
Sir Issac Newton
Stanley J. Weyman
Thomas De Quincey
Thomas Middleton
Voltaire
William Hazlitt
William Hope Hodgson
Series Seven
Adam Smith
Benjamin Disraeli
Confucius
David Hume
E. M. Delafield
E. Phillips Oppenheim
Edmund Burke
Ernest Hemingway
Frances Trollope
Galileo Galilei
Guy Boothby
Hans Christian Andersen
Ian Fleming
Immanuel Kant
Karl Marx
Kenneth Grahame
Lytton Strachey
Mary Wollstonecraft
Michel de Montaigne
René Descartes
Richard Marsh
Sax Rohmer
Sir Richard Burton
Talbot Mundy
Thomas Babington Macaulay
W. W. Jacobs
Series Eight
Anna Katharine Green
Arthur Schopenhauer
The Brothers Grimm
C. S. Lewis
Charles and Mary Lamb
Elizabeth von Arnim
Ernest Bramah
Francis Bacon
Gilbert and Sullivan
Grant Allen
Henryk Sienkiewicz
Hugh Walpole
Jean-Jacques Rousseau
John Locke
John Muir
Joseph Addison
Lafcadio Hearn
Lord Dunsany
Marie Corelli
Niccolò Machiavelli
Ouida
Richard Brinsley Sheridan
Sigmund Freud
Theodore Dreiser
Walter Pater
W. Somerset Maugham
Ancient Classics
Achilles Tatius
Aeschylus
Ammianus Marcellinus
Apollodorus
Appian
Apuleius
Apollonius of Rhodes
Aristophanes
Aristotle
Arrian
Augustine
Aulus Gellius
Bede
Cassius Dio
Cato
Catullus
Cicero
Clement of Alexandria
Cornelius Nepos
Demosthenes
Diodorus Siculus
Diogenes Laërtius
Euripides
Frontius
Herodotus
Hesiod
Hippocrates
Homer
Horace
Isocrates
Josephus
Julius Caesar
Juvenal
Livy
Longus
Lucan
Lucian
Lucretius
Marcus Aurelius
Martial
Nonnus
Ovid
Pausanias
Petronius
Pindar
Plato
Plautus
Pliny the Elder
Pliny the Younger
Plotinus
Plutarch
Polybius
Procopius
Propertius
Ptolemy
Quintus Curtius Rufus
Quintus Smyrnaeus
Sallust
Sappho
Seneca the Younger
Septuagint
Sidonius
Sophocles
Statius
Strabo
Suetonius
Tacitus
Terence
Theocritus
Thucydides
Tibullus
Virgil
Xenophon
Delphi Poets Series
A. E. Housman
Alexander Pope
Alfred, Lord Tennyson
Algernon Charles Swinburne
Andrew Marvell
Beowulf
Charlotte Smith
Christina Rossetti
D. H Lawrence (poetry)
Dante Alighieri (English)
Dante Gabriel Rossetti
Delphi Poetry Anthology
Edgar Allan Poe (poetry)
Edmund Spenser
Edward Lear
Edward Thomas
Edwin Arlington Robinson
Ella Wheeler Wilcox
Elizabeth Barrett Browning
Emily Dickinson
Ezra Pound
Friedrich Schiller (English)
George Chapman
George Herbert
Gerard Manley Hopkins
Gertrude Stein
Hafez
Heinrich Heine
Henry Howard, Earl of Surrey
Henry Wadsworth Longfellow
Isaac Rosenberg
James Russell Lowell
Johan Ludvig Runeberg
John Clare
John Donne
John Dryden
John Keats
John Milton
John Wilmot, Earl of Rochester
Joseph Addison
Leigh Hunt
Lord Byron
Ludovico Ariosto
Luís de Camões
Matthew Arnold
Matthew Prior
Michael Drayton
Nikolai Nekrasov
Percy Bysshe Shelley
Petrarch
Ralph Waldo Emerson
Robert Browning
Robert Burns
Robert Frost
Robert Southey
Rumi
Rupert Brooke
Samuel Taylor Coleridge
Sir Philip Sidney
Sir Thomas Wyatt
Sir Walter Raleigh
Thomas Chatterton
Thomas Gray
Thomas Hardy (poetry)
Thomas Hood
Thomas Moore
Torquato Tasso
T. S. Eliot
W. B. Yeats
Walter Savage Landor
Walt Whitman
Wilfred Owen
William Blake
William Cowper
William Wordsworth
Masters of Art
Albrecht Dürer
Amedeo Modigliani
Artemisia Gentileschi
Camille Pissarro
Canaletto
Caravaggio
Caspar David Friedrich
Claude Lorrain
Claude Monet
Dante Gabriel Rossetti
Diego Velázquez
Edgar Degas
Édouard Manet
Edvard Munch
Eugène Delacroix
Francisco Goya
Giotto
Giovanni Bellini
Gustave Courbet
Gustav Klimt
J. M. W. Turner
Johannes Vermeer
John Constable
Leonardo da Vinci
Michelangelo
Paul Cézanne
Paul Gauguin
Paul Klee
Peter Paul Rubens
Piero della Francesca
Pierre-Auguste Renoir
Pieter Bruegel the Elder
Sandro Botticelli
Raphael
Rembrandt van Rijn
Thomas Gainsborough
Tintoretto
Titian
Vincent van Gogh
Wassily Kandinsky
## Alphabetical List of Titles
A. E. Housman
Achilles Tatius
Adam Smith
Aeschylus
Albrecht Dürer
Alexander Pope
Alexander Pushkin
Alexandre Dumas (English)
Alfred, Lord Tennyson
Algernon Blackwood
Algernon Charles Swinburne
Ambrose Bierce
Amedeo Modigliani
Ammianus Marcellinus
Anatole France
Andrew Lang
Andrew Marvell
Ann Radcliffe
Anna Katharine Green
Anthony Hope
Anthony Trollope
Anton Chekhov
Aphra Behn
Apollodorus
Apollonius of Rhodes
Appian
Apuleius
Aristophanes
Aristotle
Arnold Bennett
Arrian
Artemisia Gentileschi
Arthur Machen
Arthur Morrison
Arthur Schopenhauer
Augustine
Aulus Gellius
Baroness Emma Orczy
Beatrix Potter
Beaumont and Fletcher
Bede
Ben Jonson
Benjamin Disraeli
Beowulf
Bram Stoker
Bret Harte
C. S. Lewis
Camille Pissarro
Canaletto
Captain Frederick Marryat
Captain Mayne Reid
Caravaggio
Caspar David Friedrich
Cassius Dio
Cato
Catullus
Charles and Mary Lamb
Charles Darwin
Charles Dickens
Charles Kingsley
Charles Lever
Charles Reade
Charlotte M. Yonge
Charlotte Perkins Gilman
Charlotte Smith
Christina Rossetti
Christopher Marlowe
Cicero
Claude Lorrain
Claude Monet
Clement of Alexandria
Confucius
Cornelius Nepos
D. H Lawrence (poetry)
D.H. Lawrence
Daniel Defoe
Dante Alighieri (English)
Dante Gabriel Rossetti
Dante Gabriel Rossetti
David Hume
Delphi Poetry Anthology
Demosthenes
Dickensiana Volume I
Diego Velázquez
Diodorus Siculus
Diogenes Laërtius
E. F. Benson
E. M. Delafield
E. M. Forster
E. Nesbit
E. Phillips Oppenheim
E. W. Hornung
Edgar Allan Poe
Edgar Allan Poe (poetry)
Edgar Degas
Edgar Rice Burroughs
Edgar Wallace
Edith Wharton
Edmund Burke
Edmund Spenser
Édouard Manet
Edvard Munch
Edward Bulwer-Lytton
Edward Gibbon
Edward Lear
Edward Thomas
Edwin Arlington Robinson
Elizabeth Barrett Browning
Elizabeth Gaskell
Elizabeth von Arnim
Ella Wheeler Wilcox
Ellen Wood
Émile Zola
Emily Dickinson
Ernest Bramah
Ernest Hemingway
Eugène Delacroix
Euripides
Ezra Pound
F. Scott Fitzgerald
Ford Madox Ford
Frances Burney
Frances Hodgson Burnett
Frances Trollope
Francis Bacon
Francisco Goya
Frank Norris
Frank R. Stockton
Friedrich Nietzsche
Friedrich Schiller (English)
Frontius
Fyodor Dostoyevsky
G. A. Henty
G. K. Chesterton
Galileo Galilei
Geoffrey Chaucer
George Bernard Shaw
George Chapman
George Eliot
George Gissing
George Herbert
George MacDonald
George Meredith
George Orwell
Gerard Manley Hopkins
Gertrude Stein
Gilbert and Sullivan
Giotto
Giovanni Bellini
Grant Allen
Gustav Klimt
Gustave Courbet
Gustave Flaubert (English)
Guy Boothby
Guy de Maupassant
H. G. Wells
H. P. Lovecraft
H. Rider Haggard
Hafez
Hall Caine
Hans Christian Andersen
Harriet Beecher Stowe
Heinrich Heine
Henrik Ibsen
Henry David Thoreau
Henry Fielding
Henry Howard, Earl of Surrey
Henry James
Henry Wadsworth Longfellow
Henryk Sienkiewicz
Herman Melville
Herodotus
Hesiod
Hilaire Belloc
Hippocrates
Homer
Honoré de Balzac (English)
Horace
Horace Walpole
Hugh Walpole
Ian Fleming
Immanuel Kant
Isaac Rosenberg
Isocrates
Ivan Turgenev
J. M. Barrie
J. M. W. Turner
J. W. von Goethe (English)
Jack London
James Fenimore Cooper
James Joyce
James Russell Lowell
Jane Austen
Jean-Jacques Rousseau
Jerome K. Jerome
Johan Ludvig Runeberg
Johannes Vermeer
John Buchan
John Bunyan
John Clare
John Constable
John Donne
John Dryden
John Galsworthy
John Keats
John Locke
John Milton
John Muir
John Ruskin
John Webster
John Wilmot, Earl of Rochester
Jonathan Swift
Joseph Addison
Joseph Addison
Joseph Conrad
Josephus
Jules Verne
Julius Caesar
Juvenal
Karl Marx
Kate Chopin
Katherine Mansfield
Kenneth Grahame
L. Frank Baum
L. M. Montgomery
Lafcadio Hearn
Laurence Sterne
Leigh Hunt
Leo Tolstoy
Leonardo da Vinci
Lewis Carroll
Livy
Longus
Lord Byron
Lord Dunsany
Louisa May Alcott
Lucan
Lucian
Lucretius
Ludovico Ariosto
Luís de Camões
Lytton Strachey
M. E. Braddon
M. R. James
Marcel Proust (English)
Marcus Aurelius
Margaret Oliphant
Maria Edgeworth
Marie Corelli
Mark Twain
Martial
Mary Shelley
Mary Wollstonecraft
Matthew Arnold
Matthew Prior
Maxim Gorky
Michael Drayton
Michel de Montaigne
Michelangelo
Miguel de Cervantes
Nathaniel Hawthorne
Niccolò Machiavelli
Nikolai Gogol
Nikolai Nekrasov
Nonnus
O. Henry
Oliver Goldsmith
One Thousand and One Nights
Oscar Wilde
Ouida
Ovid
Paul Cézanne
Paul Gauguin
Paul Klee
Pausanias
Percy Bysshe Shelley
Peter Paul Rubens
Petrarch
Petronius
Piero della Francesca
Pierre-Auguste Renoir
Pieter Bruegel the Elder
Pindar
Plato
Plautus
Pliny the Elder
Pliny the Younger
Plotinus
Plutarch
Polybius
Procopius
Propertius
Ptolemy
Quintus Curtius Rufus
Quintus Smyrnaeus
R. Austin Freeman
R. M. Ballantyne
Radclyffe Hall
Rafael Sabatini
Ralph Waldo Emerson
Raphael
Rembrandt van Rijn
René Descartes
Richard Brinsley Sheridan
Richard Marsh
Robert Browning
Robert Burns
Robert E. Howard
Robert Frost
Robert Louis Stevenson
Robert Southey
Robert W. Chambers
Rudyard Kipling
Rumi
Rupert Brooke
Saki
Sallust
Samuel Butler
Samuel Johnson
Samuel Pepys
Samuel Richardson
Samuel Taylor Coleridge
Sandro Botticelli
Sappho
Sax Rohmer
Seneca the Younger
Septuagint
Sheridan Le Fanu
Sidonius
Sigmund Freud
Sir Arthur Conan Doyle
Sir Issac Newton
Sir Philip Sidney
Sir Richard Burton
Sir Thomas Malory
Sir Thomas Wyatt
Sir Walter Raleigh
Sir Walter Scott
Sophocles
Stanley J. Weyman
Statius
Stendhal
Stephen Crane
Strabo
Suetonius
T. S. Eliot
Tacitus
Talbot Mundy
Terence
The Brontës
The Brothers Grimm
Theocritus
Theodore Dreiser
Thomas Babington Macaulay
Thomas Carlyle
Thomas Chatterton
Thomas De Quincey
Thomas Gainsborough
Thomas Gray
Thomas Hardy
Thomas Hardy (poetry)
Thomas Hood
Thomas Middleton
Thomas Moore
Thucydides
Tibullus
Tintoretto
Titian
Tobias Smollett
Torquato Tasso
Victor Hugo
Vincent van Gogh
Virgil
Virginia Woolf
Voltaire
W. B. Yeats
W. Somerset Maugham
W. W. Jacobs
Walt Whitman
Walter Pater
Walter Savage Landor
Washington Irving
Wassily Kandinsky
Wilfred Owen
Wilkie Collins
William Blake
William Cowper
William Dean Howells
William Harrison Ainsworth
William Hazlitt
William Hope Hodgson
William Makepeace Thackeray
William Morris
William Shakespeare
William Wordsworth
Xenophon
Zane Grey
www.delphiclassics.com
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Sunset over Shiraz, where Hafez spent his final days
Tomb of Hafez in Shiraz
| {
"redpajama_set_name": "RedPajamaBook"
} | 304 |
//Distributed under 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)
#ifndef UUID_DFFD61ACC72311E6955FFF1F07624D53
#define UUID_DFFD61ACC72311E6955FFF1F07624D53
#include <boost/qvm/inline.hpp>
#include <boost/qvm/assert.hpp>
namespace
boost
{
namespace
qvm
{
template <class>
struct vec_traits;
namespace
qvm_detail
{
template <int I,int N>
struct
vector_w
{
template <class A>
static
BOOST_QVM_INLINE_CRITICAL
typename vec_traits<A>::scalar_type &
write_element_idx( int i, A & a )
{
return I==i?
vec_traits<A>::template write_element<I>(a) :
vector_w<I+1,N>::write_element_idx(i,a);
}
};
template <int N>
struct
vector_w<N,N>
{
template <class A>
static
BOOST_QVM_INLINE_TRIVIAL
typename vec_traits<A>::scalar_type &
write_element_idx( int, A & a )
{
BOOST_QVM_ASSERT(0);
return vec_traits<A>::template write_element<0>(a);
}
};
}
template <class VecType,class ScalarType,int Dim>
struct
vec_traits_defaults
{
typedef VecType vec_type;
typedef ScalarType scalar_type;
static int const dim=Dim;
template <int I>
static
BOOST_QVM_INLINE_CRITICAL
scalar_type
read_element( vec_type const & x )
{
return vec_traits<vec_type>::template write_element<I>(const_cast<vec_type &>(x));
}
static
BOOST_QVM_INLINE_CRITICAL
scalar_type
read_element_idx( int i, vec_type const & x )
{
return vec_traits<vec_type>::write_element_idx(i,const_cast<vec_type &>(x));
}
protected:
static
BOOST_QVM_INLINE_TRIVIAL
scalar_type &
write_element_idx( int i, vec_type & m )
{
return qvm_detail::vector_w<0,vec_traits<vec_type>::dim>::write_element_idx(i,m);
}
};
}
}
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
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Novosibirsk, Russia: October 25, 2010 – LEDAS Ltd today announced the release of version 4.0 of LGS 3D, its powerful geometric constraint solver. This is the core technology for parametric design software, allowing users to express their design intent in 3D parts and assemblies through geometric and dimensional constraints.
Version 4.0 introduces important new functions that customers will find useful for assembly design and direct modeling. It also significantly improves reliability and performance.
LGS 3D v4 introduces (infinite) cone and torus objects. They can be fixed, or be linked with other objects with different constraints: coincidence, tangency, parallelism, perpendicularity, and angular dimensions. In the context of assembly designs, cone and torus objects are used to model spatial and kinematic relationships in mechanisms, such as bearings and variators. In direct modeling applications, they can represent fillets and chamfers, and they adjust their form to satisfy constraints. For example, cones can change their angles and tori change any of their radii. In addition to non-rigid cylindrical and spherical surfaces introduced in previous versions of LGS 3D, cone and tori now allow users to models many kinds of parametrized mechanical parts, such as brackets, bearings, and hinges.
An important function introduced with LGS 3D v4 computes the degrees of freedom (DOF) of parts in mechanisms. Any rigid set in LGS 3D can have up to three translational and three rotational DOFs. When rigid sets are linked by constraints to other fixed objects, the appropriate DOFs are eliminated automatically. A new function returns the number of remaining DOFs for a particular rigid set. Also returned is geometric information, such as directions and axes. LGS-based applications can now represent this data to the end user, a feature that has been highly requested for kinematic analysis.
LGS 3D v4 supports tangency constraints with parametric curves and surfaces that represent non-canonic objects. These features are used to express complex kinematic relations in advanced mechanisms.
Also new in LGS 3D is its ability to deform canonic curves and surfaces when they are moved under constraints. This applies specifically to circles, cylinders, spheres, cones and tori. (In previous versions, the deformation was limited to static constraint solving.) Version 4 also generates the correct diagnostics for models with these non-rigid objects.
A new LGS 3D function returns information about loops in over-defined constraints. (In previous versions, users were only alerted to the condition and were provided no additional information.) Version 4 now allows constraint relaxation to be applied independently of each over-defined loop. This means that it is now faster for users to make their models consistent.
Version 4.0 of LGS 3D contains significant advances in reliability and performance. A new analysis on inconsistent models improves performance three-fold. The technical and algorithmic optimization on large models with thousands constraints increases the performance of LGS 3D by 10-25%.
LGS 3D v4 also contains enhancements to its API that provide better context management and more detailed journaling.
Since the first commercial release in 2004, LGS 2D and 3D software components have been licensed by a dozen CAD/CAM/CAE vendors, who have embedded the components into their applications successfully.
The 2D and 3D geometric constraint solvers are used as parametric engines for 2D sketching and drawing, 3D direct modeling, assembly design, motion analysis, and other applications. LGS 2D and 3D are cross-platform software packages running on 32- and 64-bit versions of Windows, Linux, Mac OS X, *BSD, AIX, HP-UX, and other operating systems. Both solvers have a C-style API for easy integration into a broad range of software applications using a variety of wrappers, such as .NET, Java, and C++.
LGS 2D/3D supports the creation and modification of geometric models through explicit and implicit constraints. Geometric objects that can be constrained include points, lines, circles, ellipses, planes, cylinders, spheres, cones, tori, NURBS, application-defined parametric curves, and surfaces. Objects can be fixed in the absolute coordinate system or relative to each other. Supported constraints include geometric relations between objects (coincidence, parallelism, tangency, and so on) and dimensions that specify the required values for distances, angles, and radii. LGS 2D/3D moves and rotates objects to positions that satisfy all constraints, yet minimizes transformations from initial configurations.
Available as free downloads from the LEDAS web site are "Lege'n'd 2D/3D" example applications. These consist of sample models representing different kinds of 2D sketches and 3D assemblies. The applications can be used to test the functionality, robustness, and performance of LGS 2D/3D. They were created with the Open CASCADE open-source application framework; the source code is available to all licensees.
To learn more about LGS 2D/3D, visit the LEDAS Web site at www.ledas.com/products/lgs3d.
LEDAS Ltd. is an independent software development company founded in 1999. It is based in Novosibirsk Scientific Centre in Akademgorodok, the Siberian Branch of the Russian Academy of Science. A leader in constraint-based technologies, LEDAS is a well-known provider of computational software components for PLM (Product Lifecycle Management) solutions: geometric constraint solvers for CAD/CAM/CAE, optimization engines for project management, work scheduling and meeting planning as well as interval technologies for knowledge-based engineering and collaborative design. The company also provides services for PLM markets: software development, consulting, reselling as well as education and training. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,960 |
{"url":"https:\/\/jacqueskaiser.com\/posts\/2020\/08\/synaptic-learning","text":"# PhD on Synaptic Learning for Neuromorphic Vision\n\nPublished:\n\nAfter over five years of researching in the field of biological learning for robotics, I have successfully completed my PhD. The full dissertation is published on the KIT bibliothek. In this blog post, I introduce the field of neuromorphic computing in layman\u2019s terms, and present some of the results.\n\n## Silicon retina for a silicon brain\n\nThe Dynamic Vision Sensor is a very special type of camera. Conventional cameras take images at a given rate, around 30 times per seconds. Standard computer vision algorithms have to read and process these images, one by one. Modern smartphones are capable to perform such processing, but it consumes a lot of energy. On the other hand, the Dynamic Vision Sensor will only emit an individual event, when the light received by one pixel has changed. An event consists of the address of the pixel which changed, and whether light increased or decreased for this pixel. In other words, when the scene does not change, and the sensor does not move, no events are emitted. A computer vision algorithm can now run more efficiently: instead of processing full images regularly, it only has to process individual events, when the scene changes.\n\nHow the Dynamic Vision Sensor works\n\nUntil now, most computer vision algorithms were tailored to process images, not events. This also includes deep neural networks, which are used today in almost every computer vision problems. A deep neural network consists of neurons connected to each other in layers. Every neuron has a value, which is computed from the neurons in the previous layers and communicated to the neurons in the next layer. The actual computations depends on some values, called weights, which are learned. To process an image, the neurons of the first layer simply take the value of the pixels in the image. The values are used to compute the activation of the second layer of neurons and so on, until the last layer of neuron which is the output. Therefore, for every processed image, all neurons of a layer have to communicate their values synchronously. This is another expensive process (although it fits the GPU very well).\n\nConventional deep learning neuron\n\nSpiking neurons are biologically-plausible version of conventional artificial neurons. They are particularly suited to processing events rather than images. Unlike a conventional neuron, a spiking neuron maintains a dynamical state which is not communicated to other neurons. Communication between neurons only consists of spikes, emitted when the state reaches a threshold. Spikes are spontaneous, stereotypical events: the information resides in the time at which the spike was emitted. Spikes and address events from the Dynamic Vision Sensors are very similar. In fact, to process address events with a spiking network, the first layer of neurons simply emit spikes for every address event.\n\nSpiking neuron\n\nIntuitively, processing a stream of events with a spiking network requires far less computations\/communication than processing a sequence of images with conventional neural networks. The reason is, instead of processing the full information about the visual scene at every time-step again and again, we only process and communicate differences. Less computations implies potentially saving energy while running faster. However, this style of event-based computations do not fit conventional hardware very well: it is a paradigm shift. Taking advantage of these saved computations requires dedicated hardware, called neuromorphic hardware. Designing and manufacturing neuromorphic hardware was initiated in academia (Caltech, Z\u00fcrich, Manchester, Dresden, \u2026). Since then, both Intel and IBM started developing neuromorphic chips.\n\nDeep learning networks are used everywhere today, because they can be easily trained with an algorithm called backpropagation. But how can deep networks of spiking neurons be trained? Are such networks capable of learning directly from the events of the Dynamic Vision Sensor? These are research questions I have investigated as part of my PhD.\n\n## Gradient descent in spiking networks\n\nOver all the methods I have evaluated during my research, approximations of backpropagation turned out to provide the best performance. Not only was it learning more accurately, these approximations also solve some of the biologically implausible aspects of backpropagation, yielding a new theory of learning in the brain.\n\nTo go further, let\u2019s formalize what is learning. Machine learning is a neat technique to solve complex problems, whenever data is available. At its core, learning requires:\n\n\u2022 a parameterized model: some mathematical operations, which is a function of some parameters. In deep learning, the neural network is the model, which is parameterized by the synaptic weights $W$.\n\u2022 a loss function $L: \\mathbb{R}^n \\rightarrow \\mathbb{R}$, a metric assessing the performance of the synaptic weights on the task, using the available data. The goal of learning is to find the weights which minimize this loss function: $W^\\star=argmin_W L(W)$.\n\nTo find the best weights, a naive approach would be to try random weights, evaluate the loss function, and simply keep the ones where the loss function is the smallest. In fact, this is what evolutionary algorithms are doing. However, evaluating the loss function is expensive, and we would like to do it as few times as possible. To this end, the best method to date is gradient descent. The gradient of the loss by the weights $\\frac{\\partial L}{\\partial w_{ij}}$ for all weights $w_{ij} \\in W$ is computed at the same time the loss $L(W)$ is evaluated. By definition, the gradient of the loss by a weight $w_{ij}$ means how an infinitesimal change in $w_{ij}$ affects the loss. Therefore, by taking an (arbitrary small) step in the opposite direction of the gradient, we have good chances that the loss function will decrease. Gradient descent repeats this operation for all weights $w_{ij} \\in W$ until the loss function stop decreasing:\n\n$w_{ij} \\leftarrow w_{ij} - \\alpha \\times \\frac{\\partial L}{\\partial w_{ij}},$\n\nwith $\\alpha \\in \\mathbb{R}$ the learning rate: an arbitrary small step.\n\nNow back to neural networks. Backpropagation is the method used to compute the gradient $\\frac{\\partial L}{\\partial w_{ij}}$ in deep learning. To explain how it works, we need to dive in the equations of artificial neurons. A conventional deep learning neuron starts by summing up the values of its pre-synaptic neurons, weighted by the synaptic weights. It then applies a non-linear function, usually simply nulling-out negative values (ReLU). These are the $u_i$ and the $s_i$ terms in the following neural equations:\n\n\\begin{aligned} u_i &= \\sum_{j\\in \\textit{pre}} w_{ij} \\times s_j \\\\ s_i &= u_i \\: \\text{if} \\: u_i > 0 \\: \\text{else} \\: 0 \\end{aligned}\n\nUsing this definition, the step from a conventional neuron to a spiking neuron is small. In fact, there are only two important differences with spiking neurons:\n\n\u2022 $u_i$ carries a trace of the previous activity (it is a state which needs to be stored and updated). It is called the membrane potential.\n\u2022 $s_i$ is a threshold function (0 or 1, denoting spike and absence of spike, unlike conventional neuron which have output in $\\mathbb{R}$).\n\nMathematically, a spiking neuron can be expressed as:\n\n\\begin{aligned} u_i &= \\sum_{j\\in \\textit{pre}} w_{ij} (\\epsilon \\ast s_j) + \\eta \\ast s_i \\\\ s_i &= 1 \\: \\text{if} \\: u_i > 0 \\: \\text{else} \\: 0 \\end{aligned}\n\nThe $\\ast$ operation denotes a temporal convolution, with kernel $\\epsilon$ and $\\eta$ respectively. In words, this means that $u_i$ also depends on the previous values of $s_j$ and $s_i$. Both dependencies are important. For the first convolution: a spike $s_j$ is instantaneous, so it should leave a trace in $u_i$. For the second convolution: after a neuron spikes, it should stop spiking for some time \u2013 this is called the refractory period. A simple way to model this is simply by having the neuron inhibiting itself, with $\\eta \\ast s_i < 0$.\n\nExample dynamics of a spiking neuron\n\nLet\u2019s calculate the gradient $\\frac{\\partial L}{\\partial w_{ij}}$ for both an analog neuron and a spiking neuron. We can use the chain rule to express the gradient of a loss function $L$ with respect to the weight $w_{ij}$:\n\n$\\frac{\\partial L}{\\partial w_{ij}} = \\frac{\\partial L}{\\partial s_i} \\times \\frac{\\partial s_i}{\\partial u_i} \\times \\frac{\\partial u_i}{\\partial w_{ij}}$\n\nUsing this formulation, we see that the gradient is a multiplication of three terms. Let\u2019s discuss these terms one by one.\n\n### Calculation of $\\dfrac{\\partial u_i}{\\partial w_{ij}}$:\n\nThis term captures how a change in synaptic weight affects the weighted sum of inputs $u_i$. For a conventional deep learning neuron, this term is simple to calculate, it is simply $s_j$. For a spiking neuron, the calculation is more involved, especially because of how a change in weight $w_{ij}$ affects the refractory period. Here comes the first approximation to backpropagation for spiking neuron: we simply ignore this term. Note that if neurons do not spike often (in the brain neurons spike at maximum 60Hz) then refractory periods won\u2019t play a big role in the dynamics. Therefore ignoring it in the gradient computation won\u2019t affect learning too much. Using this approximation we can write $\\frac{\\partial u_i}{\\partial w_{ij}} = \\epsilon \\ast s_j$.\n\n### Calculation of $\\dfrac{\\partial s_i}{\\partial u_{i}}$:\n\nThis term captures how a change the membrane potential $u_i$ affects spikes $s_i$. With conventional deep learning neuron, this is simply the derivative of the ReLU function. For a spiking neuron however, the threshold function is not differentiable (it is undefined at the threshold value). Here comes the second approximation. We simply pretend that the threshold function was a similar, differentiable function \u2013 this is called a surrogate. For example, a function that linearly increases around the threshold value. We therefore have: $\\frac{\\partial s_i}{\\partial u_{i}} = 1$ around the threshold value, 0 otherwise.\n\n### Calculation of $\\dfrac{\\partial L}{\\partial s_i}$:\n\nThis term captures how a change in this neuron activity affects the loss of the whole network. The calculation of this term is the same for conventional neurons and spiking neurons. Note that $s_i$ does not affect the loss directly. Instead, it affects the activity of the neurons in the next layer, which themselves affect the next next layer, and so on until the output layer. Therefore, an exact computation of $\\frac{\\partial L}{\\partial s_i}$ also depends on the synaptic weights in the following layers. In conventional deep learning, this is not a big deal: when the error at the output layer becomes available, it is simply sent backward through the whole network. Hence the name backpropagation. For spiking networks, this is a problem: we can not have signals traveling backward through the same synapses as the forward ones. This is where the third and last approximation jumps in. For spiking networks, we instantiate dedicated feedback synapses with random weights, distributing the network error to all neurons in all layers. This method is called feedback alignment, because the whole network will learn to align with the randomly chosen feedbacks. The actual choice of the loss function, and how feedback alignment is applied, is currently how modern synaptic learning rules for spiking networks differ.\n\n### Putting it all together\n\nInjecting the three terms back into the gradient equation yields the synaptic learning rule for spiking neural networks:\n\n$\\frac{\\partial L}{\\partial w_{ij}} = e_i \\times \\epsilon \\ast s_j \\: \\text{if} \\: u_i \\: \\text{around threshold, 0 otherwise},$\n\nwith $e_i$ the loss-dependent error. Like the dynamics of a spiking neuron, note that this gradient also has a temporal dynamics, due to the presence of the temporal convolution. In practice, this allows the spiking network to learn temporal correlations at no additional cost in memory, since the dynamical state for learning can be shared with the forward dynamics. Additionally, weight updates can be performed online, while sensory input is being streamed inside the network.\n\nThere are multiple instances of synaptic learning rules which relate to this derivation. As part of my PhD, we developed the DECOLLE learning rule [paper, code] together with Prof. Emre Neftci during a stay at the University of California, Irvine, funded by a DAAD scholarship. This learning rule achieves state-of-the-art accuracy on the popular gesture recognition dataset from IBM, DvsGesture.\n\n## Conclusion\n\nThe CPU in your phone or laptop will not be replaced by a neuromorphic chip. Instead, neuromorphic computing opens the door to novel applications where computing speed and energy efficiency matters. Spiking neural networks are capable of learning accurate representations online with gradient descent.","date":"2020-11-27 20:11:51","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\": 3, \"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\": 0.732505202293396, \"perplexity\": 738.9291012966972}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141194171.48\/warc\/CC-MAIN-20201127191451-20201127221451-00515.warc.gz\"}"} | null | null |
Handlebar - Nitto Bullmoose, dull bright FILLET BRAZED - 150mm reach, 190mm quill (B901R)
Home / Handlebars / Handlebar - Nitto Bullmoose, dull bright FILLET BRAZED - 150mm reach, 190mm quill (B901R)
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The Bullmoose Bar
(fillet brazed CrMo, made exclusively for us by Nitto--who used to make 5000 of them per month in the early '80s, Not 500. Not per year.)
It's a handlebar and a stem in one, was invented by Tom Ritchey in 1980, and named by Charlie Kelly. We got Tom's approval to recreate it with the slight modifications noted below, and Charlie, all he wants for our using "his" name on this modified bar, is one of them for himself, and that's easy.
This is the handlebar that graced the early production mountain bikes and defined the breed. It first appeared on the Ritcheys, but by 1982-4 or so, it was on all the Japanese models. It was always an expensive bar to make, and that combined with the onset of the weight-obsessed mtn bike in 1985 killed it.
The Bullmoose bar is still unequaled for rough terrain. You're on a big bike riding fast over rough ground, and the wide bar lets you control the bike like a monkey manipulating a peanut. You wrestle it here, pound it there, lift it over that last-second cow-pie, whatever. The 66cm wide bar gives you the leverage you need, and if you don't need it, cut off an inch or so. But try wide first.
More normal bars are easier to make, offer more adjustability, weigh lessbut represent a caving in to practicality and manufacturer's wishes at the expense of this bold, unique handlebar style. In some tiny way, in the right circumstances, under the wrong influences, some slightly irrational person in the passion of the moment could say that being a bicycle rider in the 21st century without having ridden Bullmoose handlebars is like being a photographer who's never shot film or rewound a roll of it. True, film and rewind levers have been around and more popular than the Bullmoose bar, but this is the bar that really made the mountain bike, back when the mountain bike was tough, rough and tumble, and all business.
There were Bullmoose variants, and the one we offer is yet another. It retains the signature triangle and the expansive width of the originals, but with two subtle differences that make it the best one yet (so we think).
(1) The far end of the triangle is curved. It looks just as good as a flat side, but offers a friendlier place to perch your hands, if you want a change-of grip. These Bullmoose bars offer a good change of pace from the originals, without looking like a whole new bar.
(2) The grip-part sweeps back a bit more, for a more ergo position that feels more normal and slightly improves climbing.
They're CrMo steel, fillet-brazed by Nitto. That's the way it was back then, and there's no way to improve on that part.
The quill is uncharacteristically short for us, because we designed it mainly for the Bombadil, which already has a high head tube; and because there's plenty of built-in rise on the extension. Vaughn here put them on his Bridgestone XO-3, and that bike has never ridden better.
The dull bright finish is an exact match of the dull-bright finish of Nitto bars and racks.
Weight: 30.6 ounces (867.5 grams)
Reach: 150mm
Top of quill to max height line: 3-1/3" (8.5 cm)
1-3/4" (4.5 cm) of vertical adjustment.
Requires brake levers with 22.2mm clamp diameter.
Since this bar is steel, please squirt some boesheild inside the bar and roll the goop around before installing it on your bike. | {
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Allaire is an unincorporated community located within Wall Township in Monmouth County, New Jersey, United States. As of the 2010 United States Census, the ZIP Code Tabulation Area for ZIP Code 07727 had a population of 7,050.
References
Wall Township, New Jersey
Unincorporated communities in Monmouth County, New Jersey
Unincorporated communities in New Jersey | {
"redpajama_set_name": "RedPajamaWikipedia"
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Home / Shop / Hadrian's Wall Operations Manual
Hadrian's Wall Operations Manual
From Construction to World Heritage Site
Hadrian's Wall is the largest, most spectacular historical monument in Britain. Nothing else approaches its vast scale: a land wall running 73 miles from east to west and a sea wall stretching at least 26 miles down the Cumbrian coast.
Some of its forts are as large as Britain's most formidable medieval castles, and with its mile towers, barracks and soldier's leisure facilities, the site allows an astonishingly rich insight into Roman frontier life. Hadrian's Wall Manual looks at the design and construction of the wall, from the initial land survey to its busiest period as Rome's most northern frontier.
Hardback; 275 x 215mm
Hadrian's Wall Operations Manual quantity
SKU: 6428 Categories: All Titles, Cumbrian Books, Cumbrian Outdoors, Hadrian's Wall & Borders, History, Roman and Hadrian's Wall
Britain's First Black Policeman
Ray Greenhow
Old Grasmere and Rydal
Neil Honeyman
Drive and Stroll in Cumbria & The Lake District
Chris Bagshaw
Lake District Fell Farming
Historical and literary perspectives 1750-2017
Terry McCormick
Nella – Lost and Found
Nella Black, Wallace | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,226 |
{"url":"https:\/\/zbmath.org\/?q=an:1221.35213","text":"# zbMATH \u2014 the first resource for mathematics\n\nHarnack estimates for quasi-linear degenerate parabolic differential equations. (English) Zbl\u00a01221.35213\nThe article is devoted to proving Harnack estimates for quasi-linear degenerate parabolic equations of the form $u_t - \\text{div }\\vec A(x,t,u,Du) = B(x,t,u,Du)$ given in the cylindrical domain $$E_t = E\\times (0,T]$$ where $$E$$ is an open set of $$\\mathbb{R}^N$$. Here the measurable functions $$\\vec A:\\, E_T\\times \\mathbb{R}^{N+1} \\to \\mathbb{R}^N$$ and $$B:\\, E_T\\times \\mathbb{R}^{N+1} \\to \\mathbb{R}$$ subject to the constraints \\begin{aligned} & \\vec A(x,t,u,Du)\\cdot Du \\geq C_0| Du| ^p - C^p,\\\\ & | \\vec A(x,t,u,Du)| \\leq C_1| Du| ^{p-1} + C^{p-1}, \\quad \\text{a.e. in } E_t,\\\\ & | B(x,t,Du)| \\leq C| Du| ^{p-1}, \\end{aligned} where $$p > 2$$, $$C_0$$ and $$C_1$$ are given positive constants, and $$C$$ is a non-negative constant. For $$\\varrho > 0$$ let $$K_{\\varrho}$$ be the cube centered at the origin on $$\\mathbb{R}^N$$ with edge $$2\\varrho$$, and for $$y\\in \\mathbb{R}^N$$ let $$K_{\\varrho}(y)$$ denote the homothetic cube centered at $$y$$. For $$\\theta > 0$$ set $Q^-_{\\varrho}(\\theta) = K_{\\varrho} \\times (-\\theta\\varrho^p,0], \\quad Q^+_{\\varrho}(\\theta) = K_{\\varrho} \\times (0,\\theta\\varrho^p],$ and for $$(y,s) \\in \\mathbb{R}^N\\times \\mathbb{R}$$, \\begin{aligned} &(y,s) + Q^-_{\\varrho}(\\theta) = K_{\\varrho}(y) \\times (s-\\theta\\varrho^p,s], \\\\ & (y,s) + Q^+_{\\varrho} (\\theta) = K_{\\varrho}(y) \\times (s,s+\\theta\\varrho^p]. \\end{aligned} Let $$u = C_{\\text{loc}}(0,T;L^2_{\\text{loc}}(E)\\cap L^P_{\\text{loc}}(0,T;W^{1,p}_{\\text{loc}}(E))$$ be a continuous, non-negative weak solution of the above-mentioned equation, fix $$(x_0,t_0)\\in E_T$$ such that $$u(x_0,t_0) > 0$$ and construct the cylinders $(x_0,y_0) + Q^{\\pm}_{4\\varrho}(\\theta), \\quad \\theta = \\left(\\frac{c}{u(x_0,t_0)}\\right)^{p-2},$ where $$c > 0$$. The main results (Intrinsic Harnack Inequality and H\u00f6lder continuity) read: There exist positive constants $$c$$ and $$\\gamma$$ depending only upon the data, such that for all intrinsic cylinders $$(x_0,y_0) + Q^{\\pm}_{4\\varrho}(\\theta)$$ contained in $$E_T$$, either $$u(x_0,t_0) \\leq \\gamma C\\varrho$$ or $u(x_0,t_0) \\leq \\gamma \\inf_{K_{\\varrho}(x_0)}u(x,t_0 + \\theta\\varrho^p).$ Any locally bounded weak solutions of the equation, with no sign restrictions, is locally H\u00f6lder continuous in $$E_T$$.\n\n##### MSC:\n 35K65 Degenerate parabolic equations 35K59 Quasilinear parabolic equations 35B45 A priori estimates in context of PDEs 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35D30 Weak solutions to PDEs\nFull Text:\n##### References:\n [1] Aronson, D. G. & Serrin, J., Local behavior of solutions of quasilinear parabolic equations. Arch. Rational Mech. Anal., 25 (1967), 81\u2013122. \u00b7 Zbl\u00a00154.12001 \u00b7 doi:10.1007\/BF00281291 [2] De Giorgi, E., Sulla differenziabilit\u00e0 e l\u2019analiticit\u00e0 delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat., 3 (1957), 25\u201343. \u00b7 Zbl\u00a00084.31901 [3] DiBenedetto, E., Intrinsic Harnack type inequalities for solutions of certain degenerate parabolic equations. Arch. Rational Mech. Anal., 100 (1988), 129\u2013147. \u00b7 Zbl\u00a00708.35017 \u00b7 doi:10.1007\/BF00282201 [4] \u2013 Harnack estimates in certain function classes. Atti Sem. Mat. Fis. Univ. Modena, 37 (1989), 173\u2013182. \u00b7 Zbl\u00a00698.35023 [5] \u2013 Degenerate Parabolic Equations. Universitext. Springer, New York, 1993. \u00b7 Zbl\u00a00794.35090 [6] DiBenedetto, E. & Friedman, A., H\u00f6lder estimates for nonlinear degenerate parabolic systems. J. Reine Angew. Math., 357 (1985), 1\u201322. \u00b7 Zbl\u00a00549.35061 \u00b7 doi:10.1515\/crll.1985.357.1 [7] DiBenedetto, E., Gianazza, U. & Vespri, V., Local clustering of the non-zero set of functions in W 1,1(E). Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 17 (2006), 223\u2013225. \u00b7 Zbl\u00a01223.46034 \u00b7 doi:10.4171\/RLM\/465 [8] Hadamard, J., Extension \u00e0 l\u2019\u00e9quation de la chaleur d\u2019un th\u00e9or\u00e8me de A. Harnack. Rend. Circ. Mat. Palermo, 3 (1954), 337\u2013346. \u00b7 Zbl\u00a00058.32201 \u00b7 doi:10.1007\/BF02849264 [9] Ladyzhenskaya, O. A., Solonnikov, V. A. & Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type. Nauka, Moscow, 1968 (Russian); English translation: Translations of Mathematical Monographs, 23. Amer. Math. Soc., Providence, RI, 1967. [10] Ladyzhenskaya, O. A. & Uraltseva, N. N., Linear and Quasilinear Elliptic Equations. Nauka, Moscow, 1964 (Russian); English translation: Academic Press, New York, 1968. [11] Moser, J., On Harnack\u2019s theorem for elliptic differential equations. Comm. Pure Appl. Math., 14 (1961), 577\u2013591. \u00b7 Zbl\u00a00111.09302 \u00b7 doi:10.1002\/cpa.3160140329 [12] \u2013 A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math., 17 (1964), 101\u2013134. \u00b7 Zbl\u00a00149.06902 \u00b7 doi:10.1002\/cpa.3160170106 [13] \u2013 On a pointwise estimate for parabolic differential equations. Comm. Pure Appl. Math., 24 (1971), 727\u2013740. \u00b7 Zbl\u00a00227.35016 \u00b7 doi:10.1002\/cpa.3160240507 [14] Nash, J., Continuity of solutions of parabolic and elliptic equations. Amer. J. Math., 80 (1958), 931\u2013954. \u00b7 Zbl\u00a00096.06902 \u00b7 doi:10.2307\/2372841 [15] Pini, B., Sulla soluzione generalizzata di Wiener per il primo problema di valori al contorno nel caso parabolico. Rend. Sem. Mat. Univ. Padova, 23 (1954), 422\u2013434. \u00b7 Zbl\u00a00057.32801 [16] Serrin, J., Local behavior of solutions of quasi-linear equations. Acta Math., 111 (1964), 247\u2013302. \u00b7 Zbl\u00a00128.09101 \u00b7 doi:10.1007\/BF02391014 [17] Trudinger, N. S., Pointwise estimates and quasilinear parabolic equations. Comm. Pure Appl. Math., 21 (1968), 205\u2013226. \u00b7 Zbl\u00a00159.39303 \u00b7 doi:10.1002\/cpa.3160210302\nThis reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.","date":"2021-01-23 18:53:50","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.9350336790084839, \"perplexity\": 1667.1124525295577}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703538226.66\/warc\/CC-MAIN-20210123160717-20210123190717-00415.warc.gz\"}"} | null | null |
Q: How to make buttons cover full screen width in Android Linear Layout? I have following three button in a Linear Layout with width fill_parent.
How can I set the width of these buttons to equally cover the whole screen area?
<Button
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:id="@+id/btnReplyMessage"
android:gravity="left"
android:text="Reply"
/>
<Button
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:id="@+id/btnMarkAsUnread"
android:gravity="left"
android:text="Mark as unread"
/>
<ImageButton
android:id="@+id/btnDeleteMessage"
android:src="@drawable/imgsearch"
android:gravity="right"
android:layout_height="wrap_content"
android:layout_width="wrap_content"
android:layout_alignParentRight="true"
/>
A: You should just specify these attributes for each button:
android:layout_width="fill_parent"
android:layout_weight="1"
So it should be something like that:
<LinearLayout
android:layout_width="fill_parent"
android:layout_height="wrap_content">
<Button
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:layout_weight="1"/>
<Button
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:layout_weight="1"/>
<Button
android:layout_width="fill_parent"
android:layout_height="wrap_content"
android:layout_weight="1"/>
</LinearLayout>
A: Give all buttons the following properties
android:layout_width="fill_parent"
android:layout_weight="1"
fill_parent tells them to consume as much width as possible, and weight determines how that width shall be distributed, when more than one control are competing for the same space. (Try playing around with different values of weight for each button to see how that works)
A: you can use following code:
<Button
android:layout_width="0dp"
android:layout_height="wrap_content"
android:layout_weight="1"/>
You must to do 0dp in width on every button.
| {
"redpajama_set_name": "RedPajamaStackExchange"
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{"url":"https:\/\/www.nature.com\/articles\/s41598-019-55074-1","text":"Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.\n\n# Quantitative phase contrast imaging of a shock-wave with a laser-plasma based X-ray source\n\n### Subjects\n\nAn Author Correction to this article was published on 12 May 2020\n\n## Abstract\n\nX-ray phase contrast imaging (XPCI) is more sensitive to density variations than X-ray absorption radiography, which is a crucial advantage when imaging weakly-absorbing, low-Z materials, or steep density gradients in matter under extreme conditions. Here, we describe the application of a polychromatic X-ray laser-plasma source (duration ~0.5 ps, photon energy >1\u2009keV) to the study of a laser-driven shock travelling in plastic material. The XPCI technique allows for a clear identification of the shock front as well as of small-scale features present during the interaction. Quantitative analysis of the compressed object is achieved using a density map reconstructed from the experimental data.\n\n## Introduction\n\nAdvanced X-ray diagnostics can benefit many areas of physics. These include the study of matter in extreme states, inertial fusion (ICF)1,2, medical physics and biophysics3,4,5 etc. For instance, within the domain of high energy density physics, time-resolved X-ray radiography has been used to follow the dynamics of a laser-driven shock-wave travelling in a target material6,7,8. The nanosecond evolution of the shock-wave requires an X-ray pulse in the picoseconds regime and high photon flux to freeze the phenomenon in a single-shot experiment. The standard technique employed to observe shock-waves is based on the absorption of photons travelling through matter6,7,8. However absorption X-ray radiography has several drawbacks. In particular, it is well known that strong shocks do not allow compressing matter more than a factor of about 4. In many cases, this is not enough to ensure sufficient contrast. Furthermore, when thick or complex targets are used (as in many ICF-related experiments1,2) one must use quite hard X-rays and again this leads to poor X-ray absorption contrast. These drawbacks might be overcome by using X-ray Phase Contrast Imaging (XPCI9). This technique relies upon the phase shift induced by an object on the travelling e.m. waves. Being particularly sensitive to density gradients it allows to probe (poorly absorbing) low-Z targets, as well as to observe subtle features which are invisible with absorption radiography10. In particular, XPCI results in contrast enhancement at the interfaces and it is therefore particularly suited to the study of shock-waves, as demonstrated in recent works11,12,13,14,15,16,17,18,19,20,21.\n\nIn this work, we present the analysis of a laser-driven shock-wave via XPCI. In particular the XPCI simulation code, already employed by Antonelli et al.18 is now explained in details. Moreover a different approach is followed. A density map is reconstructed from the measured image by combining a phase-retrieval code coupled with a tomographic reconstruction algorithm. The experiment was performed at the Petawatt High-Energy Laser for Heavy Ion EXperiments (PHELIX) in GSI (Germany)22, employing the configuration shown in Fig.\u00a01. We used two laser pulses: a long pulse (nanosecond time duration) to launch the shock and a short pulse (half picosecond time duration) to generate the X-ray source. We employed a propagation-based XPCI geometry that does not require optics (crystals) or interferometers (based on transmission gratings)23,24, making it suitable for use on large laser fusion facilities like NIF25 and LMJ26. In propagation-based XPCI experiments, X-rays propagate freely in the vacuum after they have passed through the object of interest. The distance between the object and the detector determines the degree of interference between phase-shifted and unperturbed rays.\n\nOur X-ray source was produced by irradiating a solid target at intensities larger than 1017 W\/cm2, generating hot electrons that cross the backlighter target. These hot electrons can interact with bound electrons in the atoms, generating vacancies that are immediately filled by electrons from higher shells. The photons that are emitted during this bound-bound transition have wavelengths dependent on the transition energy e.g. L-shell to K-shell transitions result in emission. These sources are characterized by subpicosecond time duration27 (shorter than the hydrodynamic time) and a spectrum with few energetic lines (, etc.) sitting on Bremsstrahlung continuum28. The energy conversion efficiency, from laser to X-rays, for on-target laser intensities above 1018 W\/cm2 oscillates between29 1\u2009\u00d7\u200910\u22125 and 3\u2009\u00d7\u200910\u22124. As an example, using 1\u2009J laser pulse, the X-ray photon flux for the copper transition (8.047\u2009keV) is 8\u2009\u00d7\u20091010 photons\/4\u03c0 sr\/pulse. These characteristics make laser-driven X-ray sources attractive candidates for X-ray imaging in experiments at high-power laser facilities.\n\nIn our experiment, we used a 5\u2009m diameter tungsten wire as a backlighter. Using a wire instead of a flat foil limits the spread of electrons in the target and hence limits the source size29. Tungsten wire was used in previous experiment at this facility30, providing good performance in terms of photon flux, energy and, above all, source size. A small source size makes XPCI possible in such a configuration. Tungsten can be manufactured easily down to 5\u2009m diameter wire. Mass limited target results in a better flux compared to pin-holes which limit the flux. The wire was illuminated by a laser pulse with the following parameters: 0.5\u2009ps time duration, maximum pulse energy of 25\u2009J and wavelength of 1.06\u2009m. The focusing optics used for the backlighter pulse was a 45\u2009deg\u2009f\u2009=\u2009400\u2009mm off-axis parabola. The FWHM of the focal spot was ~5\u2009m, the energy encircled in this diameter is ~6\u2009J (1\/4 of the pulse energy), leading to on-target intensities of around 6\u2009\u00d7\u20091019 W\/cm2 for the backlighter beam.\n\nThe X-ray source was used to image a shock launched inside a polystyrene cylinder (300\u2009m diameter and 300\u2009m long) using a second laser pulse with a wavelength of 1.06\u2009m, energy of 25\u2009J and adjustable time duration between 1 and 10\u2009ns. The focusing optic was a 90\u2009deg\u2009f\u2009=\u20091500\u2009mm off-axis parabola leading to a focal spot diameter of 50\u2009m, which corresponds to an intensity on the target of 3\u2009\u00d7\u20091014 W\/cm2. We did not use any smoothing techniques (random phase plate etc.). The interaction face of the plastic cylinder was coated with a 100\u2009nm-thick layer of aluminium in order to increase laser absorption and avoid laser shine-through at early times. The time delay between the two pulses was varied between 1 and 10\u2009ns. We chose a plastic object for two reasons. First, the transmission of polystyrene in the X-ray photon range (6\u20139\u2009keV) is above 70%, allowing for a clear demonstration of the performance of XPCI. Second, we have checked that the EOS used by DUED reproduces with high accuracy the experimental data31 for plastic at pressures in the range 1\u201310\u2009Mbars.\n\n## Result\n\nA real image is given by the convolution of several contributions, summarized by the following formula32,33,34:\n\n$${I}_{real}={I}_{ideal}\\ast PS{F}_{{\\det }}\\ast {\\sigma }_{s}[{R}_{1}\/{R}_{0}]$$\n(1)\n\nwhere Ireal is the \u201creal\u201d observed image, Iideal the image obtained assuming an ideal point-like source and perfect detector, PSFdet is the detector point spread function (PSF), and \u03c3s is the source size. The last parameter has to be rescaled by the ratio R1\/R0 (where R0 is the source-object distance and R1 is the object-detector distance). The blurring effect induced by the source is proportional to R1, meaning that the smaller the value of R1, the lower the negative effects due to the finite source size.\n\nTo help to choose the correct distances, we developed a simulation code based on the Fresnel-Kirchoff formalism35 (cf. Image Simulation in Methods), allowing generating a synthetic XPC-image starting from a density map generated by hydrodynamic codes (in particular we used the code DUED36). The parameters used to simulate the shock are the same as those detailed above (laser and object), while the X-ray source was modelled as a Gaussian beam with a FWHM of 5\u2009m and 7.5\u2009keV photon energy. The X-ray energy is close to the average one for a tungsten Bremsstrahlung backlighter.\n\nFigure\u00a02 shows a preliminary study of the simulated shock Contrast37 versus R0 at different magnifications. For the evaluation of Contrast\u2009=\u2009(Imax\u2009\u2212\u2009Imin)\/(Imax\u2009+\u2009Imin), the maximum (Imax) and minimum (Imin) light intensity in the neighbourhood of the shock-front were considered. From the plot in Fig.\u00a02, it is clear that increasing the Contrast at magnifications higher than 4 requires the distance R0 to be increased as well. This behaviour can be explained by the sensitivity of the propagation-based geometry to source lateral coherence L\u2009=\u2009\u03bbR0\/\u03c3s, where \u03bb and \u03c3s are the source wavelength and size respectively. The spatial coherence requirements for propagation-based XPCI have been studied by Wu et al.38,39,40. Our experimental configuration is represented in Fig.\u00a02 by a red cross, where R0\u2009=\u200925\u2009cm and R1\u2009=\u2009205\u2009cm. This operating point was chosen as a compromise between the geometrical constraints imposed by the vacuum chamber, the layout of the laser optics and the detector resolution. Our detector was an SR-type imaging plate (IP-SR), with 100\u2009m intrinsic resolution41,42. The IP was installed in air, after a 0.5\u2009mm-thick PMMA window and filtered with 13\u2009\u03bcm-thick aluminium foil ensuring that all X-ray photons with energies below 5\u2009keV are suppressed.\n\nOn each shot, the backlighter spectrum was monitored with a HOPG crystal spectrometer43 and a gold knife edge was used to measure the source size. The knife edge (see Fig.\u00a01) was placed 55\u2009mm from the source and 152\u2009cm from the detector (IP-SR). The high (26x) magnification used for the ESF ensured that the source-extension contribution dominated over the detector PSF. In Fig.\u00a03, the source Edge Spread Functions (ESF) along the X and Y axes are shown for a single shot. The experimental curve is fitted with a function obtained by convolving a Gaussian with a Heaviside step function. The source widths are different along the two axes, due to the geometry used to illuminate the tungsten wire and the processes involved in laser-matter interaction. As observed by Park et al.29, the source size is bigger than the laser focal spot due to lateral spreading of hot electrons in cold matter. In our case, the width along the X-axis was constrained by the wire diameter (5\u2009um), which means that the resolution on this axis is limited by the intrinsic detector resolution, corresponding to 10.6\u2009um on the object plane. On the Y-axis, no physical constraints were present besides the wire length. Hence, due to an average source size of 30\u2009um, the resolution was limited to 27\u2009um. The average photon energy was measured to be around 7.5\u2009keV. The source size so measured is later used to simulate the XPC-image, after being rescaled by the source magnification (R1\/R0) as reported in Eq.\u00a01.\n\nIn order to compare experimental and measured images, both are normalized using a flat-field image without the object. During the experiment, it was not possible to simultaneously acquire an object image and a flat-field image. Moreover, the reproducibility of the X-ray source (in terms of flux, source size and spectrum) was not sufficient to allow using a single flat-field measurement for all data. Therefore starting from the acquired image, we estimated the flat-field performing a polynomial fit to each column within the image, excluding any pixels containing the object.\n\nXPC-images of the object without a laser-driven shock are shown in Fig.\u00a04. Experimental images are on the left-hand side (shot #14881) and the simulated versions are positioned on the right (labelled as Sim.). The synthetic image was generated using a density map of the unperturbed object, the measured spectrum and source size (5\u2009\u00d7\u200930\u2009\u03bcm) as source parameters. Profiles taken along the object axis for both images are shown in Fig.\u00a04c. In order to improve the signal-to-noise ratio we performed an average over a 10 pixel width area (shown in red in Fig.\u00a04). The agreement between the two profiles, confirms that the code is able to reproduce both phase-contrast and absorption. The first one is clearly visible as contrast enhancement at object boundary. Pure absorption contribution is visible in the middle of the object.\n\nIn Fig.\u00a05, we present some raw phase contrast images that clearly display the matter-vacuum and compressed-uncompressed matter interfaces in laser-driven shock-waves. Note that the pump laser was not perfectly aligned with the object on the last two shots (#14902, #14896). In Fig.\u00a06, we show the same images but filtered with the so-called \u201cwrong filter\u201d (in Method). As it will be explained in the Method section, this filter allows reducing noise while enhancing the edge-contrast. We have added a dashed line to highlight the position of the shock front and used white arrows to indicate the laser interaction point. The shock front is clearly visible in all the images. In addition other, more subtle features can also be discerned. In shot #14902, for example, one can see a plastic shrapnel blown off during the interaction (see Fig.\u00a06). In shot #14896 instead, the laser illuminated the metal support (white arrow), resulting in a shock-wave driven by the emitted radiation, which propagates parallel to the object axis. In Fig.\u00a05, the images #14886 and #14890 show two perfectly-aligned shots. The images were obtained using identical laser pulse parameters: energy, temporal length and delay between the pump and the probe (8.3\u2009ns). In shot #14886, however, the phase contrast is lower because the X-ray source intensity was 30% lower than for shot #14890, which means a lower signal-to-noise ratio. Moreover the source dimensions were larger on shot #14886 (10% and 20% larger along the X-axis and Y-axis respectively). The finite-source size acts as frequency filter, smoothing and removing the oscillations due to phase-contrast. In Fig.\u00a07c, we compare the profiles taken over an area along the object axis (red box in Fig.\u00a05) with the profiles from the simulated images in Fig.\u00a07a,b. Figure\u00a07a is generated taking in account only the absorption contribution, while in Fig.\u00a07b both effects (absorption and phase-contrast) are taken in account. The Sim. XPC\u2009+\u2009Abs. simulation reproduces the object boundary (arrow 4 in Fig.\u00a07), which is still intact 8.3\u2009ns after the laser interaction. The shock front (arrow 1 in Fig.\u00a07) is also well reproduced: the discrepancy of 13\u2009\u03bcm between shot #14886 and the simulation is commensurate with the imaging resolution (10\u2009\u03bcm) in the object plane. Instead, the XPCI code fails to reproduce structural features between the shock front and the object boundary (between arrows 2 and 3 in Fig.\u00a07). This behaviour can be explained by the presence, of a strong density gradient behind the shock-front, feature not shown by the hydrodynamic simulations. In ref.\u200918 we argued that the discrepancy between the arrows 2 and 3 as a consequence of high intensity spikes inside the laser focal spot (no smoothing techniques were used). To mimic this behaviour we performed several hydrodynamic simulations where the central spike inside the laser spot was reduced from 50\u2009m down to 5\u2009m. In this way, we proved that a spike in laser intensity can effect the resulting XPC-image. The double border observable in Fig.\u00a07b, indicated by the black arrows 3 and 4, is caused by the geometry of the imaging system. The source is not aligned with the object vertical face, producing a double-layer effect. This was also seen for shot #14890 in Figs.\u00a05 and 6.\n\nThe simulated profile from the absorption image (Sim. Abs in Fig.\u00a07c), does not reproduce all the features (arrows 1\u20134) observed in the experimental image. The object boundary (arrow 4) and the shock front (arrow 1) are not clearly identifiable in the absorption image.\n\nBefore one can extract quantitative information from a phase-contrast image, target homogeneity must also be taken into account. The phase-retrieval algorithm requires a homogeneous object, which means that the ratio \u03b4\/\u03b2 should be constant over the entire object for all wavelengths. To prove that this assumption is justified, a simulated shock-wave was used as a test and the ratio \u03b4\/\u03b2 was calculated across the object for all relevant wavelengths. We find that the requirement is fulfilled since emission lines from the source were far from the absorption edges of the elements composing our object44.\n\nIn Fig.\u00a08, we show the density map generated by the hydrodynamic code (blurred according to the source size measured in the experiment) on the left-hand side, and the two maps retrieved from experimental images (shots #14886 and #14890) on the right. The measured density maps are the result of a phase-retrieval code plus a Feldkamp-Davis-Kress (FDK) tomographic reconstruction algorithm45 was used. FDK is an algorithm suitable for point-projection tomography. A tomographic algorithm was chosen instead of Abel inversion for the following. First, the idea is to implement the diagnostic also at facilities where it is possible to acquire multiple projections at different angles. Second, one of the main problems of Abel inversion is its instability with noise fluctuations. The FDK algorithm is more stable because it works in the Fourier space, with a consequent smoothing of the noise fluctuation. Of course, an external Fourier filter could be applied before proceeding with an Abel inversion. The problem would be optimizing the filter magnitude in order to smooth noise without compromising the data.\n\nThe reconstructed slices along the object axis (#14886 and #14890 in Fig.\u00a07) show the advancement of the shock-wave inside the unperturbed object. In Fig.\u00a09a, we compare the density profile along the Y-axis (in front of the shock-front) with the theoretical prediction. We see that the experimental curve follows the theoretical one, noting that modulations on top of the experimental line are artefacts of the reconstruction code. In Fig.\u00a09, dashed lines represent the density profile convolved with a Gaussian function in order to reproduce the imaging system resolution (the values used are 10.6\u2009\u03bcm\u2009\u00d7\u200927\u2009\u03bcm as estimated above). For the Y-profile, there is a good agreement between the experimental profile and the blurred simulated profile on both shots.\n\nFigure\u00a09b instead, shows a significant discrepancy between the simulated density profiles along the object axis and the reconstructed ones. Here, the hydrodynamic code estimates a shock front density of 2.3\u2009g\/cm3, while the density inferred from the experimental images is 1.3\u2009g\/cm3. The dashed line is the simulated profile which is treated with the same Gaussian filter as before. The shape of the shock is roughly reproduced by the simulation, however, the peak density of 1.7\u2009g\/cm3 is still higher than the measured value. Furthermore, the behaviour of the density behind the shock is quite different in the simulation and the experiment. The reconstructed density profiles show a faster decay than the simulated one behind the shock. This discrepancy can be explained by the difficulty of accurately modelling the laser-target alignment and the detailed intensity distribution of laser focal spot in the hydrodynamic simulator DUED36. It was not possible to acquire an image of the focal spot and pin down exactly its position on the object face during the experiment (especially along the horizontal axis). Every shot required to repeat the laser alignment and focusing procedure for both backlighter and object in order to compensate for the differences between each target.\n\n## Discussion\n\nThe use of XPCI for shock-wave imaging and characterisation has been reported in several previous works16,18,19,20,21, which demonstrated how XPCI is superior to X-ray absorption imaging in shock detection. However, only recently has XPCI was demonstrated to work also with for broadband laser-plasma X-ray source18,19, in a configuration similar to the one employed in ICF facilities.\n\nIn this paper, we have presented an analysis of laser-driven shock-waves using a laser-plasma X-ray backlighter. XPCI allowed obtaining higher object definition than absorption radiography6,7,8. We verified that XPCI can reveal structural features that are undetectable using x-ray absorption techniques3,5 (see Fig.\u00a07). Enhanced contrast at density interfaces, that occurs in XPCI, allows for clear identification of the shock front.\n\nXPCI, as absorption radiography6, is also suitable for quantitative analysis of the acquired image. In this work, we have employed two approaches. In the first one we have tried to reproduce the experimental images starting from hydrodynamic simulations. In the second, we have used the acquired images to generate density maps and comparing them to the hydrodynamic simulations. Both ways have shown discrepancies between simulated and experimental data. As explained above, these discrepancies can be attributed to the difficulty of identifying the complex experimental parameters (e.g. laser alignment and laser spot morphology) that are used as inputs to the hydrodynamic simulations.\n\n## Methodology\n\n### Image simulation\n\nThe ideal intensity distribution of a XPC-image (the quantity Iideal in Eq.\u00a01) is given by $${I}_{0}\\parallel \\tilde{u}({P}_{1}){\\parallel }^{2}$$, where I0 is the incoming intensity on the object and $$\\tilde{u}({P}_{1})$$ is the complex-valued wave-field distribution at the detector plane32,34,37:\n\n$$\\tilde{u}({P}_{1})=-\\,\\frac{i}{2\\lambda }r{e}^{-ikr}{\\iint }_{{{\\mathbb{R}}}^{2}}[\\frac{{R}_{0}}{{r}_{0}}+\\frac{{R}_{1}}{{r}_{1}}]\\frac{t(P){e}^{ik({r}_{0}+{r}_{1})}}{{r}_{0}{r}_{1}}\\,dxdy$$\n(2)\n\nwhere r is the length of the vector from the source S to the point P1 on the detector plane, t(P) is the object transfer-matrix, k is the wave number, R0 and R1 are respectively the distance source-object and object-detector, r0 and r1 are the lengths of the vectors SP and PP1 respectively (see Fig.\u00a01 for details). Equation\u00a02 is sufficient to generate a phase-contrast image. Nevertheless, two considerations are necessary. First, the computation of a triple integral is required (t(P) is the result of an integral). To implement Eq.\u00a02 in a code at least six loops (for a 2D image) have to be used, resulting in a large computational time. Second, here the source has been considered point-like. In a real experiment, the source has a finite size producing a blurring effect on the detected image.\n\nHowever, if the waves propagate a long distance after the object and the transverse dimension of the sample itself is small compared to the propagation distance, the Fresnel approximation (or paraxial R0\/r0R1\/r1\u20091) can be used. This requires the inequality Neff\u2009>\u20091 to be fulfilled where NF,eff is the so-called effective Fresnel4 number equal to:\n\n$${N}_{F,eff}={(R\\bar{\\lambda })}^{-1}\\frac{M}{M-1}{\\sigma }_{obj}\\sqrt{{M}^{2}{\\sigma }_{obj}^{2}+{(M-1)}^{2}{\\sigma }_{src}^{2}+PS{F}_{\\det }^{2}}$$\n(3)\n\nwhere R\u2009=\u2009R1\u2009+\u2009R0, $$\\bar{\\lambda }$$ is the weighted sum of the source spectrum wavelength (Eq.\u00a010), M\u2009=\u2009(R0\u2009+\u2009R1)\/R0 is the magnification, \u03c3obj is the width of the smallest object feature, \u03c3src is the source width and PSFdet is the detector PSF. Considering the parameters of the employed imaging system, an average photon energy around 7.5\u2009keV, the average source and detector PSF ranging respectively between 5\u2009\u00f7\u200950\u2009\u03bcm and 100\u2009\u00f7\u2009200\u2009\u03bcm, Neff\u2009>\u20091 is fulfilled.\n\nEquation\u00a02 can then be reduced to a convolution integral and solved using Fourier transform. Applying the Fresnel approximation, Eq.\u00a02 in the Fourier space becomes:\n\n$$\\tilde{U}(u,v)={M}^{2}T(Mu,Mv)\\exp (\\,-\\,\\pi i\\lambda {R}_{1}M({u}^{2}+{v}^{2}))\\exp \\,[2\\pi i\\frac{{R}_{1}}{{R}_{0}}({x}_{0}u+{y}_{0}v)]$$\n(4)\n\nwhere u and v are the spatial frequencies corresponding to the variables x1 and y1, and T(Mu, Mv) is the transform of the object transfer function t(P). This last quantity is a function of (Mu, Mv) because the transform operator is applied with respect to x1. In this form, the equation is suitable for numerical evaluation using the Fast Fourier Transform (FFT) algorithm. The sample is described in the above formalism with the transfer function t(P):\n\n$$t(P)=\\exp [iD(P)-B(P)]$$\n(5)\n\nwhere the two quantities in the argument of the exponential are obtained from the following integrals:\n\n$$D(P)=\\frac{-2\\pi }{\\lambda }\\int dr\\delta (x,y,z);$$\n(6a)\n$$B(P)=\\frac{2\\pi }{\\lambda }\\int dr\\beta (x,y,z)$$\n(6b)\n\nThe two integrals are calculated along the path from the source point S to the detector point P1. Equations\u00a05 and 6 are only valid in the thin sample approximation. A sample can be considered thin as long as the smallest features to be imagined are larger than the quantity $$\\sqrt{\\lambda d}$$, where d is the sample thickness40. At this level, the above equations are coupled with the hydrodynamic code. In particular, the two-dimensional density map generated by a hydrodynamic code in cylindrical geometry is used as an input to Eq.\u00a06. In order to increase the calculation speed, the code uses a dedicated FORTRAN library, written for such a purpose, that performs all the calculations in parallel mode via OpenMP library. The transfer matrix is later used to generate the synthetic image considering all the parameters of the real imaging system, namely polychromaticity, source and detector PSF. Regarding the polychromaticity, for a polychromatic source the final image is a weighted sum of the monochromatic ones32,40:\n\n$${I}_{poly}=\\sum _{\\lambda }\\,w(E){I}_{mono}(\\lambda )$$\n(7)\n\n### Image analysis\n\nThe condition required to use the Eq.\u00a04 together with that of object homogeneity, and an X-ray photon energy below 50\u2009keV allow phase retrieving from the measured image, by using the Transport Intensity Equation (TIE)46,47. In particular, the object project phase is:\n\n$$\\varphi (P)=-\\,\\frac{\\bar{\\delta }}{\\bar{\\mu }}\\,\\mathrm{ln}\\,[{ {\\mathcal F} }^{-1}\\,[\\frac{ {\\mathcal F} ({I}_{rel})}{4{\\pi }^{2}{R}_{eff}\\frac{\\bar{\\delta }}{\\bar{\\mu }}[{u}^{2}+{v}^{2}]{M}^{2}+1}]]$$\n(8)\n\nwhere $${ {\\mathcal F} }^{-1}$$ and $${\\mathcal F}$$ are the inverse and direct Fourier transform and Reff\u2009=\u2009R1\/M is the effective propagation distance. Equation 8 has been implemented in a numerical code via FFT.\n\nAs explained before, as far as the inequality NF,eff\u2009>\u20091 is fulfilled, the above equations can also be applied in the case of a polychromatic source48. The quantities $$\\bar{\\delta }$$, $$\\bar{\\mu }$$, $$\\bar{\\lambda }$$ in Eq.\u00a08 are then averaged over the source spectrum and the detector function response D(\u03bb):\n\n$$\\bar{\\mu },\\bar{\\delta }=\\frac{\\int \\mu (\\lambda ),\\delta (\\lambda )D(\\lambda )I(\\lambda )d\\lambda }{\\int D(\\lambda )I(\\lambda )d\\lambda }$$\n(9)\n$$\\bar{\\lambda }=\\frac{\\int \\lambda D(\\lambda )I(\\lambda )d\\lambda }{\\int D(\\lambda )I(\\lambda )d\\lambda }$$\n(10)\n\nPrior to proceeding with phase retrieval, the image is processed with a Wiener filter49 in the Fourier space:\n\n$$W(u,v)=\\frac{PS{F}^{\\ast }}{|PSF{|}^{2}+K}$$\n(11)\n\nwhere PSF is the total point spread function (PSFsrc*PSFdet) and K is a constant that takes noise variance in account. The filter has two effects: the first one is to partially remove blurring induced by the imaging system and the second is to normalize the noise.\n\nOne consideration can be done on Eq.\u00a08. The term 4\u03c02 on the denominator is required by the FFT used for the numerical evaluation of Irel. Removing such term from the filter will result in a non-physical behaviour at the object border, and in general at all the interfaces. The result is an enhancement of the contrast at the border. We found useful applying such \u201cwrong filter\u201d on the measured images for qualitative analysis.\n\nThe calculated projected phase map can be used to evaluate a three-dimensional map. Since only one projection is available, this requires that the object is cylindrical. The reconstruction is performed with a Feldkamp-Davis-Kress (FDK) algorithm45,50,51. It allows performing a tomographic reconstruction starting from a set of cone-beam projections of the object. In our case, the phase-map projection \u03c6(P) at the object plane, obtained from Eq.\u00a08, is used to generate a three-dimensional spectral averaged50,52:\n\n$$\\varphi {({\\bf{x}})}_{poly}=-\\,\\frac{{R}_{0}^{2}d}{2}{\\int }_{0}^{2\\pi }\\,\\frac{1}{{p}^{2}}{ {\\mathcal F} }_{1}^{-1}\\,[|Mv|{ {\\mathcal F} }_{1}\\,[\\frac{\\varphi (P,\\alpha )}{\\sqrt{{R}_{0}^{2}+|P{|}^{2}}}]]\\,d\\alpha$$\n(12)\n\nwhere x\u2009=\u2009(x, y, z) is a vector describing the object volume, d is the distance R0 minus the radius of the cylinder containing the object, and\n\n$$p=d-x\\,\\cos (\\alpha )-y\\,\\sin (\\alpha )$$\n(13)\n\nThe three-dimensional distribution of $$\\bar{\\delta }({\\bf{x}})=\\varphi {({\\bf{x}})}_{poly}\/k$$ can be reconstructed from the phase, and finally the density distribution is obtained solving the equations\n\n$$\\delta =\\frac{{r}_{e}{N}_{A}{\\lambda }^{2}\\rho }{2\\pi }\\sum _{j}\\,\\frac{{w}_{j}[{Z}_{j}+{f}_{j}^{^{\\prime} }]}{{A}_{j}}$$\n(14)\n\nwhere re is the classical electron radius, NA is the Avogadro number, \u03c1 is the mass density of the compound, Zj and Aj are the atomic number and the atomic weight of jth element of the compound, $${f}_{j}^{^{\\prime} }$$ is the real part of the dispersion correction factor.\n\nThe parameters in Eq.\u00a014 are evaluated using the library Xray-lib described in ref.\u200953.\n\n## Change history\n\n\u2022 ### 12 May 2020\n\nAn amendment to this paper has been published and can be accessed via a link at the top of the paper.\n\n## References\n\n1. 1.\n\nMontgomery, D. S., Nobile, A. & Walsh, P. J. Characterization of National Ignitition Facility Cryogenic Beryllium Capsules Using X-ray Phase Contrast Imaging. Rev. Sci. Instruments 75, 3986\u20133988, https:\/\/doi.org\/10.1063\/1.1790054 (2004).\n\n2. 2.\n\nKozioziemski, B. J. et al. Quantitative characterization of inertial confinement fusion capsules using phase contrast enhanced x-ray imaging. J. Appl. Phys. 97, https:\/\/doi.org\/10.1063\/1.1862764 (2005).\n\n3. 3.\n\nArfelli, F. et al. Low-dose phase contrast x-ray medical imaging. Phys. Medicine Biol. 43, 2845\u20132852, https:\/\/doi.org\/10.1088\/0031-9155\/43\/10\/013 (1998).\n\n4. 4.\n\nGureyev, T. E. et al. Refracting R\u00a8ontgen\u2019s rays: Propagation-based x-ray phase contrast for biomedical imaging. J. Appl. Phys. 105, 102005, https:\/\/doi.org\/10.1063\/1.3115402 (2009).\n\n5. 5.\n\nLundstr\u00f6m, U. et al. X-ray phase contrast for CO2 microangiography. Phys. Medicine Biol. 57, 2603\u20132617, https:\/\/doi.org\/10.1088\/0031-9155\/57\/9\/2603 (2012).\n\n6. 6.\n\nAntonelli, L. et al. Laser-driven shock waves studied by x-ray radiography. Phys. Rev. E 95, 063205, https:\/\/doi.org\/10.1103\/PhysRevE.95.063205 (2017).\n\n7. 7.\n\nRavasio, A. et al. Hard x-ray radiography for density measurement in shock compressed matter. Phys. Plasmas 15, 060701, https:\/\/doi.org\/10.1063\/1.2928156 (2008).\n\n8. 8.\n\nMorace, A. et al. Development of x-ray radiography for high energy density physics. Phys. Plasmas 21, 102712, https:\/\/doi.org\/10.1063\/1.4900867 (2014).\n\n9. 9.\n\nOlivo, A. & Castelli, E. X-ray phase contrast imaging: From synchrotrons to conventional sources. Rivista del Nuovo Cimento 37, 467\u2013508, https:\/\/doi.org\/10.1393\/ncr%2Fi2014-10104-8 (2014).\n\n10. 10.\n\nDavis, T. J., Gao, D., Gureyev, T. E., Stevenson, A. W. & Wilkins, S. W. Phase-contrast imaging of weakly absorbing materials using hard X-rays. Nature 373, 595\u2013598, https:\/\/doi.org\/10.1038\/373595a0 (1995).\n\n11. 11.\n\nYeager, J. et al. High-speed synchrotron x-ray phase contrast imaging for analysis of low-z composite microstructure. Compos. Part A: Appl. Sci. Manuf. 43, 885\u2013892, https:\/\/doi.org\/10.1016\/j.compositesa.2012.01.013 (2012).\n\n12. 12.\n\nJensen, B. et al. Ultrafast, high resolution, phase contrast imaging of impact response with synchrotron radiation. AIP Adv. 2, 012170, https:\/\/doi.org\/10.1063\/1.3696041 (2012).\n\n13. 13.\n\nLuo, S. et al. Gas gun shock experiments with single-pulse x-ray phase contrast imaging and diffraction at the advanced photon source. Rev. Sci. Instruments 83, 073903, https:\/\/doi.org\/10.1063\/1.4733704 (2012).\n\n14. 14.\n\nHudspeth, M. et al. High speed synchrotron x-ray phase contrast imaging of dynamic material response to split Hopkinson bar loading. Rev. Sci. Instruments 84, 025102, https:\/\/doi.org\/10.1063\/1.4789780 (2013).\n\n15. 15.\n\nOlbinado, M. P. et al. Ultra high-speed x-ray imaging of laser-driven shock compression using synchrotron light. J. Phys. D: Appl. Phys. 51, 055601, https:\/\/doi.org\/10.1088\/1361-6463\/aaa2f2 (2018).\n\n16. 16.\n\nWood, J. C. et al. Ultrafast Imaging of Laser Driven Shock Waves using Betatron X-rays from a Laser Wakefield Accelerator. Sci. Reports 8, 11010, https:\/\/doi.org\/10.1038\/s41598-018-29347-0 (2018).\n\n17. 17.\n\nBarbato, F. et al. Propagation-based imaging phase-contrast enhanced imaging setup for single shot acquisition using lasergenerated x-ray sources. J. Instrumentation 14, C03005\u2013C03005, https:\/\/doi.org\/10.1088\/1748-0221\/14\/03\/C03005 (2019).\n\n18. 18.\n\nAntonelli, L. et al. X-ray phase-contrast imaging for laser-induced shock waves. EPL (Europhysics Lett. 125, 35002, https:\/\/doi.org\/10.1209\/0295-5075\/125\/35002 (2019).\n\n19. 19.\n\nKar, A. et al. Simulated refraction-enhanced X-ray radiography of laser-driven shocks. Phys. Plasmas 26, 032705, https:\/\/doi.org\/10.1063\/1.5084968 (2019).\n\n20. 20.\n\nHawreliak, J., Erskine, D., Schropp, A., Galtier, E. C. & Heimann, P. Using phase contrast imaging to measure the properties of shock compressed aerogel. AIP Conf. Proc. 1793, 090006, https:\/\/doi.org\/10.1063\/1.4971625 (2017).\n\n21. 21.\n\nSchropp, A. et al. Imaging Shock Waves in Diamond with Both High Temporal and Spatial Resolution at an XFEL. Sci. Reports 5, 11089, https:\/\/doi.org\/10.1038\/srep11089 (2015).\n\n22. 22.\n\nNeumayer, P. et al. Status of PHELIX laser and first experiments. Laser Part. Beams 23, 385, https:\/\/doi.org\/10.1017\/S0263034605050548 (2005).\n\n23. 23.\n\nSnigirev, A., Snigireva, I., Kohn, V., Kuznetsov, S. & Schelokov, I. On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation. Rev. Sci. Instruments 66, 5486\u20135492, https:\/\/doi.org\/10.1063\/1.1146073 (1995).\n\n24. 24.\n\nWilkins, S. W., Gureyev, T. E., Gao, D., Pogany, A. & Stevenson, A. W. Phase-contrast imaging using polychromatic hard X-rays. Nature 384, 335\u2013338, https:\/\/doi.org\/10.1038\/384335a0 (1996).\n\n25. 25.\n\nMiller, G. H., Moses, E. I. & Wuest, C. R. The National Ignition Facility: enabling fusion ignition for the 21st century. Nucl. Fusion 44, S228\u2013S238, https:\/\/doi.org\/10.1088\/0029-5515\/44\/12\/S14 (2004).\n\n26. 26.\n\nFleurot, N., Cavailler, C. & Bourgade, J. L. The Laser M\u00b4egajoule (LMJ) Project dedicated to inertial confinement fusion: Development and construction status. Fusion Eng. Des. 74, 147\u2013154, https:\/\/doi.org\/10.1016\/j.fusengdes.2005.06.251 (2005).\n\n27. 27.\n\nReich, C., Gibbon, P., Uschmann, I. & F\u00f6rster, E. Yield optimization and time structure of femtosecond laser plasma Ka sources. Phys. Rev. Lett. 84, 4846\u20134849, https:\/\/doi.org\/10.1103\/PhysRevLett.84.4846 (2000).\n\n28. 28.\n\nVon Der Linde, D. et al. Generation and application of ultrashort X-ray pulses. Laser Part. Beams 19, 15\u201322, https:\/\/doi.org\/10.1017\/S0263034601191020 (2001).\n\n29. 29.\n\nPark, H.-S. et al. High-energy Ka radiography using high-intensity, short-pulse lasers. Phys. Plasmas 13, 056309, https:\/\/doi.org\/10.1063\/1.2178775 (2006).\n\n30. 30.\n\nBorm, B., Khaghani, D. & Neumayer, P. Properties of laser-driven hard x-ray sources over a wide range of laser intensities. Phys. Plasmas 26, 023109, https:\/\/doi.org\/10.1063\/1.5081800 (2019).\n\n31. 31.\n\nBarrios, M. A. et al. High-precision measurements of the equation of state of hydrocarbons at 1\u201310 mbar using laser-driven shock waves. Phys. Plasmas 17, 056307, https:\/\/doi.org\/10.1063\/1.3358144 (2010).\n\n32. 32.\n\nOlivo, A. & Speller, R. Experimental validation of a simple model capable of predicting the phase contrast imaging capabilities of any x-ray imaging system. Phys. Medicine Biol. 51, 3015\u20133030, https:\/\/doi.org\/10.1088\/0031-9155\/51\/12\/001 (2006).\n\n33. 33.\n\nGolosio, B. et al. Phase contrast imaging simulation and measurements using polychromatic sources with small sourceobject distances. J. Appl. Phys. 104, 093102, https:\/\/doi.org\/10.1063\/1.3006130 (2008).\n\n34. 34.\n\nPeterzol, A., Olivo, A., Rigon, L., Pani, S. & Dreossi, D. The effects of the imaging system on the validity limits of the ray-optical approach to phase contrast imaging. Med. Phys. 32, 3617\u20133627, https:\/\/doi.org\/10.1118\/1.2126207 (2005).\n\n35. 35.\n\nCowley, J. M. Diffraction physics (Elsevier, 1995).\n\n36. 36.\n\nAtzeni, S. et al. Fluid and kinetic simulation of inertial confinement fusion plasmas. Comput. Phys. Commun. 169, 153\u2013159, https:\/\/doi.org\/10.1016\/j.cpc.2005.03.036 (2005).\n\n37. 37.\n\nBorn, M. & Wolf, E. Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).\n\n38. 38.\n\nWu, X. & Liu, H. Clarification of aspects in in-line phase-sensitive x-ray imaging. Med. Phys. 34, 737\u2013743, https:\/\/doi.org\/10.1118\/1.2431475 (2007).\n\n39. 39.\n\nWu, X. & Liu, H. Phase-space evolution of x-ray coherence in phase-sensitive imaging. Appl. Opt. 47, E44, https:\/\/doi.org\/10.1364\/AO.47.000E44 (2008).\n\n40. 40.\n\nWu, X. & Liu, H. Clinical implementation of x-ray phase-contrast imaging: Theoretical foundations and design considerations. Med. Phys. 30, 2169\u20132179, https:\/\/doi.org\/10.1118\/1.1593836 (2003).\n\n41. 41.\n\nSeely, J. F., Holland, G. E., Hudson, L. T. & Henins, A. X-ray modulation transfer functions of photostimulable phosphor image plates and scanners. Appl. Opt. 47, 5753, https:\/\/doi.org\/10.1364\/AO.47.005753 (2008).\n\n42. 42.\n\nBoutoux, G. et al. Study of imaging plate detector sensitivity to 5-18 MeV electrons. Rev. Sci. Instruments 86, 113304, https:\/\/doi.org\/10.1063\/1.4936141 (2015).\n\n43. 43.\n\nD\u00f6ppner, T. et al. High order reflectivity of highly oriented pyrolytic graphite crystals for x-ray energies up to 22 keV. Rev. Sci. Instruments 79, 10E311, https:\/\/doi.org\/10.1063\/1.2966378 (2008).\n\n44. 44.\n\nGuinier, A. X-ray diffraction in crystals, imperfect crystals, and amorphous bodies (Courier Corporation, 1994).\n\n45. 45.\n\nFeldkamp, L. A., Davis, L. C. & Kress, J. W. Practical cone-beam algorithm. J. Opt. Soc. Am. A 1, 612, https:\/\/doi.org\/10.1364\/JOSAA.1.000612 (1984).\n\n46. 46.\n\nPaganin, D., Mayo, S. C., Gureyev, T. E., Miller, P. R. & Wilkins, S. W. Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object. J. Microsc. 206, 33\u201340, https:\/\/doi.org\/10.1046\/j.1365-2818.2002.01010.x (2002).\n\n47. 47.\n\nBurvall, A., Lundstr\u00f6m, U., Takman, P. A. C., Larsson, D. H. & Hertz, H. M. Phase retrieval in X-ray phase-contrast imaging suitable for tomography. Opt. Express 19, 10359, https:\/\/doi.org\/10.1364\/OE.19.010359 (2011).\n\n48. 48.\n\nArhatari, B. D., Hannah, K., Balaur, E. & Peele, A. G. Phase imaging using a polychromatic x-ray laboratory source. Opt. Express 16, 19950, https:\/\/doi.org\/10.1364\/OE.16.019950 (2008).\n\n49. 49.\n\nOlivo, A. & Speller, R. D. Deconvolution of x-ray phase contrast images as a way to retrieve phase information lost due to insufficient resolution. Phys. Medicine Biol. 54, N347\u2013N354, https:\/\/doi.org\/10.1088\/0031-9155\/54\/15\/N02 (2009).\n\n50. 50.\n\nMyers, G. R., Mayo, S. C., Gureyev, T. E., Paganin, D. M. & Wilkins, S. W. Polychromatic cone-beam phase-contrast tomography. Phys. Rev. A 76, 045804, https:\/\/doi.org\/10.1103\/PhysRevA.76.045804 (2007).\n\n51. 51.\n\nXiao, S., Bresler, Y. & Munson, D. C. J. Fast Feldkamp algorithm for cone-beam computer tomography. Proc. 2003 Int. Conf. on Image Process. (Cat. No. 03CH37429) 2, 3\u20136, https:\/\/doi.org\/10.1109\/ICIP.2003.1246806 (2003).\n\n52. 52.\n\nMiao, H., Zhao, H.-J., Gao, F. & Gong, S.-R. Implementation of FDK Reconstruction Algorithm in Cone-Beam CT Based on the 3D Shepp-Logan Model. In 2009 2nd International Conference on Biomedical Engineering and Informatics, 1\u20135, https:\/\/doi.org\/10.1109\/BMEI.2009.5304987 (2009).\n\n53. 53.\n\nSchoonjans, T. et al. The xraylib library for X-ray\u2013matter interactions. Recent developments. Spectrochimica Acta Part B: At. Spectrosc. 66, 776\u2013784, https:\/\/doi.org\/10.1016\/j.sab.2011.09.011 (2011).\n\n## Acknowledgements\n\nThe authors acknowledge the support of the PHELIX staff at GSI. The authors thank Dr M. Griffa for the useful discussion. This work has been carried out within the framework of the EUROfusion Enabling Research Project: AWP17- ENR-IFE-CEA-01 \u201cPreparation and Realization of Eu- ropean Shock Ignition Experiments\u201d and has received funding from the Euratom research and training pro- gramme 2014\u20132018 under grant agreement No. 633053. The views and opinions expressed herein do not necessar- ily reflect those of the European Commission. The research leading to these results has received funding from LASERLAB-EUROPE (grant agreement No. 654148, Eu- ropean Union\u2019s Horizon 2020 research and innovation pro- gramme). The work was also supported by the Competitiveness Program of NRNU MEPhI, Russia. The authors acknowledge the \u201cLLNL Academic Partnership in ICF\u201d, the Plasma kinetics, pre-heat, and the emergence of strong shocks in laser, EPSRC grant EP\/P026796\/1 and EPSRC CDT in the Science and Technology of Fusion, EPSRC grant EP\/L01663X\/1. The authors acknowledge the Italian MIUR Grant No. PRIN 2012AY5LEL and Sapienza University Grant No. 2015 C26A15YTMA.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nF.B. and L.A. conceived and coordinated the experiment. F.B., L.A., D. Batani and P.N. conceived and developed the experimental set-up. The experiment was carried out by F.B., L.A., G.B., D.M., J.T. and G.Z. The laser was set-up by C.B. and P.N. The data were analysed by F.B., L.A. and J.T., the simulations were performed by F.B., L.A., S.A. and A.S. The manuscript was written by F.B., L.A. and D. Batani. F.B., S.A., D.B., G.B., C.B., P.B., D.M., P.N., A.S., J.T., L.V., G.Z., N.W. and L.A. took part in the discussion and interpretation of the result and reviewed the manuscript. D. Bleiner contributed as FB\u2019s thesis supervisor and by providing funds for the purchase of a few targets.\n\n### Corresponding author\n\nCorrespondence to F. Barbato.\n\n## Ethics declarations\n\n### Competing interests\n\nThe authors declare no competing interests.\n\nPublisher\u2019s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Rights and permissions\n\nReprints and Permissions\n\nBarbato, F., Atzeni, S., Batani, D. et al. Quantitative phase contrast imaging of a shock-wave with a laser-plasma based X-ray source. Sci Rep 9, 18805 (2019). https:\/\/doi.org\/10.1038\/s41598-019-55074-1\n\n\u2022 Accepted:\n\n\u2022 Published:\n\n\u2022 ### Exploring phase contrast imaging with a laser-based K\u03b1 x-ray source up to relativistic laser intensity\n\n\u2022 M. Gambari\n\u2022 , A. Stolidi\n\u2022 , O. Ut\u00e9za\n\u2022 , M. Sentis\n\u2022 \u00a0&\u00a0A. Ferr\u00e9\n\nScientific Reports (2020)","date":"2021-10-19 06:45:01","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\": 2, \"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\": 0.6994170546531677, \"perplexity\": 2491.9144669839698}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585242.44\/warc\/CC-MAIN-20211019043325-20211019073325-00407.warc.gz\"}"} | null | null |
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Built with help from controlGen. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 970 |
Q: How to get the TFS discussion comments from TFS api using c# Am working on the TFS work item to get the comments ,latest added comment date and first added comment date to store in database. Can anybody please assist me on achieving this.
foreach (Revision r in RqInfo.Revisions)
{
foreach (Field f in r.Fields)
{
Console.WriteLine(f.Name, f.Value);
}
}
Have tried above code section, it gives all the Changed Date including for the attached link. I need only for comments from history. Please assist me.
This Microsoft.TeamFoundation.Discussion.Client namespace used but getting the comments for particular work item is not able find.
A: You should check only the System.History field. That was considered here: Fetching the comment history for a work item in TFS
Here is an example:
static void Main(string[] args)
{
string historyField = "System.History";
string changedDateField = "System.ChangedDate";
string changedByField = "System.ChangedBy";
WorkItemStore store = new WorkItemStore("<your_server_url>/tfs/DefaultCollection");
WorkItem wi = store.GetWorkItem(your_id);
foreach (Revision rev in wi.Revisions)
{
if (rev.Fields[historyField].Value.ToString() != "")
{
Console.WriteLine("{0}: {1} says: \n{2}",
rev.Fields[changedDateField].Value,
rev.Fields[changedByField].Value,
rev.Fields[historyField].Value);
}
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,263 |
{"url":"https:\/\/www.physicsforums.com\/threads\/space-elevator-feasibility-split-from-nasa-announces-new-launch-vehicle-and-cev.90419\/","text":"# Space elevator feasibility; split from: NASA Announces New Launch Vehicle and CEV\n\n\u2022 NASA\nLunchBox\nWhat I wish is that lots of money would be dumped into a ribbon-type space elevator or a space fountain That's the real way to get to space cheaply and safely.\n\nYeah, they'll have those right after they have satellites that can scratch your ass with a laser beam from space.\n\nCheers...\n\nSkepticJ\nLunchBox said:\nYeah, they'll have those right after they have satellites that can scratch your ass with a laser beam from space.\n\nCheers...\n\nWhile I admire your skepticism, you seem to have never heard of carbon nanotubes have you? They're already strong enough to create such a ribbon, give it a few decades. Remember, we'll never break the speed of sound, land on the moon or split the atom; don't be absurd.\n\nStaff Emeritus\nGold Member\nI'm afraid Lunchbox is right on this one...\n\nIt's possible... (possibly), but anyone who says that building it is anything short of the most complicated engineering feat ever attempted has their head in the clouds and their pie in the sky.\n\nBetween grounding out the Van Allen Belts and the slightest gravitational, atmospheric, or solar pressure perturbations exciting 1047th natural mode of the ribbon and causing it to oscillate out of control... I just don't think it's possible in our lifetime, nor the lifetimes of our children or our children's children's children.\n\nSkepticJ\nenigma said:\nI'm afraid Lunchbox is right on this one...\n\nIt's possible... (possibly), but anyone who says that building it is anything short of the most complicated engineering feat ever attempted has their head in the clouds and their pie in the sky.\n\nBetween grounding out the Van Allen Belts and the slightest gravitational, atmospheric, or solar pressure perturbations exciting 1047th natural mode of the ribbon and causing it to oscillate out of control... I just don't think it's possible in our lifetime, nor the lifetimes of our children or our children's children's children.\n\nWould you please give links etc. to such data? I've seen \"show stopping\" problems(at least they appear to be before rebuttals are given) with space elevators before, and I'm not persuaded they aren't feasible yet. I'm not saying one will be built within thirty years, but not within my great great grandchildren's lives? Come on. For one thing I'm only twenty, and for another, in a smaller span of \"grands\" we've gone from http:\/\/quest.arc.nasa.gov\/aero\/wright\/background\/otto.jpg\n\nLast edited by a moderator:\nStaff Emeritus\nGold Member\nSkepticJ said:\n\nA few semesters of stuctures vibrations courses is my only source. Damping out a string under tension with one fixed end and one free end will be damn near impossible.\n\nThere was a shuttle experiment where they extended wires a hundred meters or so radially and the current produced destroyed the experiment, and the shuttle wasn't even in the VA belt. A space elevator is thousands of kilometers long. Even if the ribbon is built strong enough to withstand the current, due to IXB, you'll get a time-varying tangential force on top of the floating string in tension model which now also needs to be damped out.\n\nI did a bunch of research a few years back and read reports by David Smitherman at NASA which said that it will be feasible in the next century, but I am highly skeptical that his analysis was thorough.\n\nAgain, I'm the first person to say guffaw to anyone who says it will never happen... but I don't think this one will be attainable anytime soon.\n\nSkepticJ\nenigma said:\nA few semesters of stuctures vibrations courses is my only source. Damping out a string under tension with one fixed end and one free end will be damn near impossible.\n\nThere was a shuttle experiment where they extended wires a hundred meters or so radially and the current produced destroyed the experiment, and the shuttle wasn't even in the VA belt. A space elevator is thousands of kilometers long. Even if the ribbon is built strong enough to withstand the current, due to IXB, you'll get a time-varying tangential force on top of the floating string in tension model which now also needs to be damped out.\n\nI did a bunch of research a few years back and read reports by David Smitherman at NASA which said that it will be feasible in the next century, but I am highly skeptical that his analysis was thorough.\n\nAgain, I'm the first person to say guffaw to anyone who says it will never happen... but I don't think this one will be attainable anytime soon.\n\nAh, but this wouldn't be a problem to a space fountain which I linked. A space fountain isn't under tension, it's under compression. It's held up using mass beams (steams of mass pellets shot out of an auto fire magnetic accelerator gun) Another advantage to space fountains is they need not be built on the equator. One could build one at the North Pole if they wanted to, it would be stupid and fuel wasting, but could be done. The best place for a space fountain would be at the equator, to give the cars traveling up it that added boost from the spin of the Earth; just like rockets make use of.\n\nKi Man\n20 mile high elevator.\n\nwhat the heck is the top of it going to lead to? what's the point? is it going to be like the cn tower but blown up in size 20x? threes no point in going up if there's nothing to do when your up there.\n\ni feel sorry for the \"astronaut\" who has to be first to ride it.\n\nnot impossible idea but not very plausible. your going to make a building\/stucture partially leave the atmoshere while still tethered tot he ground.\n\nmaybe it can happen. right after pigs take over england and i can go around the world with the press of a button while having my but scratched by a satalite guided laser while being served fruit that came from our new martian neighbors.\n\nSkepticJ\nKi Man said:\n20 mile high elevator.\n\nwhat the heck is the top of it going to lead to? what's the point? is it going to be like the cn tower but blown up in size 20x? threes no point in going up if there's nothing to do when your up there.\n\ni feel sorry for the \"astronaut\" who has to be first to ride it.\n\nnot impossible idea but not very plausible. your going to make a building\/stucture partially leave the atmoshere while still tethered tot he ground.\n\nmaybe it can happen. right after pigs take over england and i can go around the world with the press of a button while having my but scratched by a satalite guided laser while being served fruit that came from our new martian neighbors.\n\nOne would think we could dispense with strawman arguments and engage critical thinking at a science forum. :yuck:\n\nIt'd be far taller than 20 miles. I space fountain could be 200 mi. high to provive service to low Earth orbit; or as high as you want to go. They have no limit in height. You should read up on them at the link I gave, or at this link before you strawman and mock them.\nA \"beanstalk\" ribbon type elevator, which has the problem noted in this thread, would go up about 60,000 miles into the sky.\n\nOh I don't know, how about a space station or something? What do we sent those Russian craft or the Shuttle into space for?\nThe idea is that craft ride up into orbit and then detach from the elevator at the height they want to go to; or go on to the end.\n\nWhy? If mag lev rails went up the side of the space fountain tower the elevator car could reach the top in less than a few hours. The ribbon type's transit time might be up to a few weeks from bottom to top. As long as some good Led Zeppelin music was playing over the speakers, Internet link with the ground, TV beamed up and perhaps love mate(s) as well the ride up would be great.\n\nThis matters why? What, you're talking about making a metal tube soar through the air using metal \"wings\" and loud tube-like things with spinning \"turbines\" to propell them? Carry four-hundred people as well? At 30,000 ft.? At 600mph? Across an entire ocean on one tank of fuel? :rofl: Arguments from Incredulity aren't valid arguments.\n\n*Sigh* I won't dignify this pile of BS hyperbole with a response longer than this.\n\nLast edited:\nKi Man\nthats different. planes have good physics backing them up. this is just... whoa.\n\nthink about it. even if we had a design that would stand the test of time, altitude, weather, and earthquakes, along with hunderds of other things, how the heck are we planning on building this thing? AND how much will this cost? AND what are we going to do when its up there? were not going to be able to leave what ever is at the top of the elevator in any way other than going back down the elevator. we're going to be limited to spacewalks unless we have a spacecraft taken up there, but if we take a spacecraft up there its probably going to be using rocketsto get there so then there's no point in an elevator in the first place. and there's always the danger of being hit by lowflying aircraft AND satalites. if a b-29 hit the empire state building, somethings going to hit this baby sometime.\n\nIntuitive\nI remember reading about an article that a dirigible made of rigid nano tubes and if the area of displacement was as large as a football field, You could lift 800 lbs of mass, The Rigid dirigible was an evacuated Blimp and contain a football field size vacuum of displacement, You can could put more into space this way and it's reusable, If you have efficient vacuum pumps and a super material like nanotubes, If I had the money to make it I bet I could win some X prize, The rigid evacuated dirigible will seek an equal equalibrium of space.\n\nIf you can created a super hard vacuum in a rigid dirigible it would be pushed up to space.\n\nI bet if we had a good engineer team work on this we could have a blimp in space, work out all the bugs. (Specific Gravity is our friend, Buddy up!)\n\nSkepticJ\nKi Man said:\nthats different. planes have good physics backing them up. this is just... whoa.\n\nthink about it. even if we had a design that would stand the test of time, altitude, weather, and earthquakes, along with hunderds of other things, how the heck are we planning on building this thing? AND how much will this cost? AND what are we going to do when its up there? were not going to be able to leave what ever is at the top of the elevator in any way other than going back down the elevator. we're going to be limited to spacewalks unless we have a spacecraft taken up there, but if we take a spacecraft up there its probably going to be using rocketsto get there so then there's no point in an elevator in the first place. and there's always the danger of being hit by lowflying aircraft AND satalites. if a b-29 hit the empire state building, somethings going to hit this baby sometime.\n\nSpace fountains don't violate known physics. If they did I wouldn't suggest them as a possible way to space. If you want to keep talking about them it's best you actually read how they'd work at the links I gave.\n\nWould you care to explain what negative effect altitude above sea level would have on metals? Ummm, I have; please don't be condescending with me. Read the links, they explain how they'd be made.\nCan you read? I said the stuff that goes up detaches from the tower. Nothing is \"stuck\" any more than payloads are stuck to the Space Shuttle or other rockets that take them to orbit or beyond. If you're not going to stop being stupid with me I'm not going to spend time conversing with you any longer; your choice what happens.\nNo, the most energy use is getting from the ground up to orbit; which is what the elevator does. Once you're in orbit rockets can be much smaller to get where you want to go. Air space around such a tower would be as tight as the space around a government building. Idiots or terrorists would be shot down long before they could hit the tower. We have a thing called RADAR for tracking things in orbit. How do you think we can keep the ISS or Shuttle from impacting with objects? Because we know where they are. A tower won't be built in the orbital paths of satellites.\nThe B-29 that you speak of hit the Empire State Building in heavy fog, before the days of RADAR and it still didn't make the tower fall. Name a modern event where a plane hit a building by accident and you'd have a better point. Such a tower would, as a last resort, have guns similar to the missile defence guns on Navy ships that shred enemy missles into tiny, tiny bits several miles away. Such a tower could have missles, and would have protective ocean and aircraft that could take out threats.\n\nKi Man\nWhat I wish is that lots of money would be dumped into a ribbon-type space elevator or a space fountain That's the real way to get to space cheaply and safely.\n\nsince when is building an elevator many many miles high cheap\n\nI never said its impossible, but there's a whole lot of hurdles we need to jump before we can start taking this to the next level.\n\n*sigh* everything i say always comes out wrong. i need to work on my charisma. your not going to hear anything from me for a while until i get better at speaking.\n\nLast edited:\nSkepticJ\nKi Man said:\nsince when is building an elevator many many miles high cheap\n\nI never said its impossible, but there's a whole lot of hurdles we need to jump before we can start taking this to the next level.\n\n*sigh* everything i say always comes out wrong. i need to work on my charisma. your not going to hear anything from me for a while until i get better at speaking.\n\nIt's not cheap, at first. I don't know if you know this or not, but every time the US's Space Shuttle goes into orbit, the effort that went into that feat costs the tax paying public about 1,000,000,000 US Dollars. This breaks down to about $10,000 dollars per pound to get a payload into space. Why did Dennis Tito, the first space tourist, have to pay$20,000,000? Because of his weight and the weight of the air, water, food and fuel that was needed to keep him alive. Plus a tiny bit of profit to the Russians for their trouble, I guess. I'm all for NASA and the Russian's space program. In fact I wish they got about $0.05 per tax dollar instead of the$0.01 or less they get. During the Apollo program NASA got about \\$0.04 per tax dollar, and we had missions to the effing moon! But I digress. A space elevator would cost a lot up front, but the cost after that would be very low; the level of low that would let you take a tourist trip to space, if millions of other people weren't on the waiting list cash in hand, that is.\n\nYes, there is things that need to be developed to make the idea even more economical. Such as higher temp superconductor materials to cut down on the energy lost as heat from slowing down the mass beams and speeding them up again. And better magnetic guns.\nTalk to you again on Tuesday, have to go until then.\n\nKi Man\nthe quote i was referring too said that the ribbon would be a cheap and safe alternative. safe, maybe. cheap? definitely not\n\nmoney money money money\n\ngoing to space costs billions into begin with.\n\nwhat about 2 or 3 stage jet\/rocket aircraft. definitely cheaper, re-usable, and reasonable safe.\n\nStaff Emeritus\nGold Member\ncheaper like the shuttle is cheaper?\n\nRe-usable spacecraft are some of the most expensive things on the planet.\n\nKi Man\ncheaper as in a small craft being carried up on something kind of like a boeing 747 and then shooting off when its high enough, then entering another stage, wheich means it will take less than a giant red tank of fuel 300 feet large and 2 boosters to get up. just enough for an airliner and some small rockets.\n\nStaff Emeritus\nGold Member\nSo instead of having one single system to get to 7.75 kilometers per SECOND and 250 kilometers high, you propose that designing all of the interlinkages and safety systems so you can launch it from 0.25 kilometers per second and 10 kilometers up?\n\nAll that and you now need to size the rocket small enough to be able to be carried by a 747. I'm sorry... if it was more feasible to launch from a plane for larger rockets, they'd do it.\n\nLunchBox\nI gots some numbers\n\nI just tried to post this and my browser crapped out on me... f'ing FireFox... anywho... here it is in condensed form because I don't feel like retyping it all...\n\nSkepticJ... your 'space fountain' is crap. Your grandchildren's grandchildren's grandchildren won't see it. Here's why:\n\n1.) Bending. This thing will sway like a drunken frat whore at Mardi Gras. For comparison, the Sears Tower, at 1450 ft, sways an average of 6 into either side. Comparing that ratio with the 656000 ft 'space fountain' yields a sway of 226 ft. Now, that's to BOTH SIDES. So the magnetic catch on the redirector needs to be 500 FEET IN DIAMETER! And that's for AVERAGE SWAY. This thing will also be cutting through the jet stream so that 226 ft mean sway is so conservative, W is telling it to back off.\n\n2.) Torsion. Everything from 1.) applies. Oops... just increased the necessary magnetic catch diameter.\n\n3.) Projectiles. These things will require a TREMENDOUS amount of energy just to get to the top with NO residual energy. Oh, and you need residual energy to keep the structure from falling down like a lightweight frat pledge at initiation. (Wow... two drunken references in an analysis... new record.) The amount of energy required just to get the projectile to the top is 2 MJ\/kg... yes 2 MILLION Joules per kilogram. Even assuming you have a rail launcher that is 1 km long, that is a required initial velocity of 2 km\/s. Oh, and all of those numbers are excluding aerodynamic drag which will be substantial on a projectile leaving an accelerator at SIX TIMES THE SPEED OF SOUND!\n\nI could go on... but... well... [URL=[URL=[PLAIN]http:\/\/groups.msn.com\/_Secure\/0TwDtAuIY!vxCf!LqsXBkoLXuJ0MS!*SqpH8VwXdtWpv4XVx7NBucxjJh2umHmm2c83SmaHcre6HAhkr33eDqi82b2CUpiYc1WH7nGFtSe74!5sVUE*1!vg\/beating-a-dead-horse.gif[\/URL] [Broken] pretty much sums up what I'm already doing.\n\nCheers...\n\nLast edited by a moderator:\nSkepticJ said:\nAh, but this wouldn't be a problem to a space fountain which I linked.\nYou provided no link that I can see. I am a huge skeptic of the elevator notion but I would like to see something on the fountain idea before I start with my views on the subject.\n\nStaff Emeritus\nGold Member\nFredGarvin said:\nYou provided no link that I can see. I am a huge skeptic of the elevator notion but I would like to see something on the fountain idea before I start with my views on the subject.\n\nHe did, but it got trimmed when I split the threads.\n\nWikipedia knows all\n\nKi Man\ni'm thinking more like, large version of spaceship one, maybe not as huge as i said with the 747\n\nSkepticJ\nLunchBox said:\nSkepticJ... your 'space fountain' is crap. Your grandchildren's grandchildren's grandchildren won't see it. Here's why:\n\n1.) Bending. This thing will sway like a drunken frat whore at Mardi Gras. For comparison, the Sears Tower, at 1450 ft, sways an average of 6 into either side. Comparing that ratio with the 656000 ft 'space fountain' yields a sway of 226 ft. Now, that's to BOTH SIDES. So the magnetic catch on the redirector needs to be 500 FEET IN DIAMETER! And that's for AVERAGE SWAY. This thing will also be cutting through the jet stream so that 226 ft mean sway is so conservative, W is telling it to back off.\n\n2.) Torsion. Everything from 1.) applies. Oops... just increased the necessary magnetic catch diameter.\n\n3.) Projectiles. These things will require a TREMENDOUS amount of energy just to get to the top with NO residual energy. Oh, and you need residual energy to keep the structure from falling down like a lightweight frat pledge at initiation. (Wow... two drunken references in an analysis... new record.) The amount of energy required just to get the projectile to the top is 2 MJ\/kg... yes 2 MILLION Joules per kilogram. Even assuming you have a rail launcher that is 1 km long, that is a required initial velocity of 2 km\/s. Oh, and all of those numbers are excluding aerodynamic drag which will be substantial on a projectile leaving an accelerator at SIX TIMES THE SPEED OF SOUND!\n\nI could go on... but... well... [URL=[URL=[PLAIN]http:\/\/groups.msn.com\/_Secure\/0TwDtAuIY!vxCf!LqsXBkoLXuJ0MS!*SqpH8VwXdtWpv4XVx7NBucxjJh2umHmm2c83SmaHcre6HAhkr33eDqi82b2CUpiYc1WH7nGFtSe74!5sVUE*1!vg\/beating-a-dead-horse.gif[\/URL] [Broken] pretty much sums up what I'm already doing.\n\nCheers...\n\nYep, total crap. Those people at Lawrence Livermore National Laboratory are time wasting idiots; and so was Robert Lull Forward.\n\n1. Amazing this can happen, since much of the tower is above atmosphere of any appreciable thickness. The thinner the air, less less the force of the wind per the wind's speed. Even if magnets of this size would be needed I see no problem. Particle accelerators, km in length, have been around longer than I have.\nWe know where the jet streams are. We could build it where the streams never go, if they'd be a problem.\n\n2. From what? You'll have to explain in detail.\n\nYou lack the ability to read what the links I give say I see. You're \"Oh, mass pellets being shot up, energy lost, debunked!\" without even looking at the math these people did. Sloppy. So? If you think 2km per sec. gun velocity is current science fiction you'd be wrong. There are rail guns that can shoot spike-like rounds at 7+km per second already. IIRC the finned spike projectiles have a mass of about a kilogram. The rail gun is less than 20 meters long IIRC. Particle accelerators get particles, with mass, up to just below the speed of light, currently.\n\nI'd rather you did, because your link doesn't come up for me.\n\nLast edited by a moderator:\nLunchBox\nThose people at Lawrence Livermore National Laboratory\n...\n\n(Psst... just because the people work for a place who's name ends in \"national laboratory\" doesn't mean they have a direct pipeline to the wisdom of the ages. In fact, they are just as likely to be wrong as anyone else... I know... I work with these people from time to time.)\n\n1. Amazing this can happen\n...\n\nIsn't it? I've always found vibrations to be an interesting topic. The numbers were for comparison. Now, in the 'articles' you link, the authors stress how only a spindly, lightweight structure will be needed. Spindly, lightweight structures have extremely small bending (EI) stiffnesses. The Sears Tower is practically rigid compared with a 'space fountain' structure. Pick your favorite 'space fountain' design (as the articles you linked surprisingly had no structure sizing) and perform a sinusoidal gust loading along the height... I imagine the tip displacement value will astonish you.\n\n2. From what? You'll have to explain in detail.\n...\n\nOk... I'll explain. The gust loading will not be even over the side area of the structure facing the wind. This difference in loading along the windward face will create a torque on the structure. And spindly structures have even lower torsional (GJ) stiffnesses than bending stiffnesses.\n\nYou lack the ability to read what the links I give say I see.\n\nNo... they just fail to say anything useful...\n\nYou're \"Oh, mass pellets being shot up, energy lost, debunked!\" without even looking at the math these people did.\n\nI'll assume you meant 'your' and not 'you are'. I saw no 'math' in your Wikipedia article and the only technical publication linked from there was on a 'launch loop', not a 'space fountain'. The wonderful catch-all in the Wikipedia article\nbut Roderick Hyde worked out all the engineering design details for a Space Fountain and showed that there were no show-stoppers.\nreally means that the problems were manageable within the scope of the universe we occupy, not that the problems were practically solvable, or that they were economically practical. I would ABSOLUTELY LOVE to see that analysis done by Roderick Hyde. If you have it, please send me a link or a pdf. Also, always remember Akin's Law of Spacecraft Design Number 17\n\nSo? If you think 2km per sec. gun velocity is current science fiction you'd be wrong. There are rail guns that can shoot spike-like rounds at 7+km per second already.\n\nNo... I didn't say that they were science fiction. In fact, I am a huge proponent of rail gun technology. As soon as it becomes economically feasible, I think the military should put those bad boys on EVERYTHING. However, what I was referring to was the atmospheric drag that will suck momentum from these projectiles. Not only that, but the projectiles will need to be ferromagnetic in order to be redirected at the top and bottom of the tower. The atmospheric drag on a projectile traveling at Mach 6+ will cause tremendous heating and could exceed the Curie temperature of the material making it no longer ferromagnetic. Also, the velocity of the returning projectiles will will be limited by the terminal velocity of the projectile profile. All these losses will add up to necessitate a tremendous energy expenditure to bring the projectiles back up to speed at the bottom of the 'space fountain'.\n\nNow look, maybe I started a little harsh, but I'm sick and tired of people thinking you can get to space easily by climbing successively taller trees. My goal it not to stifle creativity... far from it. However, I think a little realism and practicality needs to be brought into every discussion. Oh and I was serious about wanting to see that analysis...\n\nCheers...\n\nMentor\nKi Man said:\ni'm thinking more like, large version of spaceship one, maybe not as huge as i said with the 747\nThe important thing to know about SpaceShip One is that it is not a space ship. It's just the most expensive amusement park ride ever created. Ok, so I'm not sure if there is a formal definition of the term, but my point is that Spaceship One does not come anywhere close to achieving orbit, which is what it must be able to do have any real use. So scaling it up would accomplish little of value.\n\nAlso, while SpaceShip One had a number of unique design features, the overall concept is an old one, the same as the X-15. The performance is somewhat less than what the X-15 achieved 50 years ago.\n\nLast edited:\nGold Member\nKi Man said:\nthats different. planes have good physics backing them up. this is just... whoa.\nI was unaware that there was \"good\" physics and \"bad\" physics.\nAs SkepticJ says: \"Space fountains don't violate known physics.\"\n\nPerhaps what you meant was that there are a number of technical hurdles to overcome. To which I think most of us agree. We only disagree on how much, how long and how costly.\n\nKi Man said:\nsince when is building an elevator many many miles high cheap\n\nWhen? Why, by trip #2!\n\nThe whole point of these devices is that, utterly unlike any kind of rocket, you only put out the expense once, not every time.\n\nGold Member\nenigma, re: your technical problems mentioned in post #5, would any or all of these also apply to a skyhook?\n\nSkepticJ\nLunchBox said:\n...\n\n(Psst... just because the people work for a place who's name ends in \"national laboratory\" doesn't mean they have a direct pipeline to the wisdom of the ages. In fact, they are just as likely to be wrong as anyone else... I know... I work with these people from time to time.)\n\n...\n\nIsn't it? I've always found vibrations to be an interesting topic. The numbers were for comparison. Now, in the 'articles' you link, the authors stress how only a spindly, lightweight structure will be needed. Spindly, lightweight structures have extremely small bending (EI) stiffnesses. The Sears Tower is practically rigid compared with a 'space fountain' structure. Pick your favorite 'space fountain' design (as the articles you linked surprisingly had no structure sizing) and perform a sinusoidal gust loading along the height... I imagine the tip displacement value will astonish you.\n\n...\n\nOk... I'll explain. The gust loading will not be even over the side area of the structure facing the wind. This difference in loading along the windward face will create a torque on the structure. And spindly structures have even lower torsional (GJ) stiffnesses than bending stiffnesses.\n\nNo... they just fail to say anything useful...\n\nI'll assume you meant 'your' and not 'you are'. I saw no 'math' in your Wikipedia article and the only technical publication linked from there was on a 'launch loop', not a 'space fountain'. The wonderful catch-all in the Wikipedia article really means that the problems were manageable within the scope of the universe we occupy, not that the problems were practically solvable, or that they were economically practical. I would ABSOLUTELY LOVE to see that analysis done by Roderick Hyde. If you have it, please send me a link or a pdf. Also, always remember Akin's Law of Spacecraft Design Number 17\n\nNo... I didn't say that they were science fiction. In fact, I am a huge proponent of rail gun technology. As soon as it becomes economically feasible, I think the military should put those bad boys on EVERYTHING. However, what I was referring to was the atmospheric drag that will suck momentum from these projectiles. Not only that, but the projectiles will need to be ferromagnetic in order to be redirected at the top and bottom of the tower. The atmospheric drag on a projectile traveling at Mach 6+ will cause tremendous heating and could exceed the Curie temperature of the material making it no longer ferromagnetic. Also, the velocity of the returning projectiles will will be limited by the terminal velocity of the projectile profile. All these losses will add up to necessitate a tremendous energy expenditure to bring the projectiles back up to speed at the bottom of the 'space fountain'.\n\nNow look, maybe I started a little harsh, but I'm sick and tired of people thinking you can get to space easily by climbing successively taller trees. My goal it not to stifle creativity... far from it. However, I think a little realism and practicality needs to be brought into every discussion. Oh and I was serious about wanting to see that analysis...\n\nCheers...\n\nGovernment Labs don't usually write stuff that is BS though. If it involves physics breaking stuff they have to show that physics, as it was understood up to that point, isn't correct. This, as far as I know, hasn't happened yet.\n\nRight.\n\nThat's because it's a general concept. The towers could be whatever diameter, mass etc. is wanted\/needed.\n\nThat's better.\n\nNeither did your non-functioning \"link\". You might want to go back and fix that, because I can't be privy to the \"debunking\" of \"crap\" concept space fountains.\n\nIt's very ironic your critiquing of my typing flaw when you, while using a sock puppet, made this typing error: \"maybe it can happen. right after pigs take over england and i can go around the world with the press of a button while having my but scratched by a satalite guided laser while being served fruit that came from our new martian neighbors.\"\nI think you and Ki Man are one and the same because of these two quotes:\n\n\"having my but scratched by a satalite guided laser\"-Ki Man\n\n\"Yeah, they'll have those right after they have satellites that can scratch your ass with a laser beam from space.\"-LunchBox\n\nEven if you're not one and the same, this should do: \"Also, the velocity of the returning projectiles will will be limited by...\"\n\nIf you're going to debate with me, do it with one account, mmmmkay?\n\nIf you're not one and the same person, a little advice, don't correct people's typing in a debate and in a pissy manner. Especially in a debate with someone that is anal retentive about doing correct spelling in his own work.\nBe nice to me and I'll be nice to you in return.\n\nVacuum pipes, which the streams travel through, make this a non-issue. No air, no drag. *sigh* I guess I'll have to say everything that the linked papers say before you get it. Why, oh, why did I waste my time? What's the difference between this \"manageable within the scope of the universe we occupy\" and this \"not that the problems were practically solvable\"? http:\/\/www.answers.com\/practical&r=67 [Broken] seems to fit both. Economic is another matter though.\n\nI'll probably have to read through Dr. Robert Forward's Indistinguishable from Magic to find the calculations\/analysis. Said book has a lot of information on Space Fountains in it; plus other stuff.\n\nLast edited by a moderator:\nKi Man\nmeh, ignore what i said before i was drunk or something (not literally)\n\nif this is about 200 miles high, how quickly will we be traveling to get up this thing. is it going to be like a 5 hour take your time kind of thing as you go straight up into the atmosphere or is it going to be somekind of accelerated trip to make you get there quickly. i understand what's backing this up now, i can have a concept design done some time this week.\n\nhow fast are we talking? how high is it going to be (estimate) and how big is the elevator going to have to be (like how wide is it going to need to be at most and what kind of things are going to need to be carried up by the elevator carts.)?\n\nand no this is not lunchbox! i doubt he is a 13 year old living in south california. i just thought using his comment in my own would be a good idea. outside of the forum i am not associated with him.\n\ni am not him. he's not smart enough to be me lol :rofl:\n\nokay there ARE similarities in our posts but if a mod checked out our IP adresses he'd see that they are different (unless lunchbox is really my sister which i really really doubt). can i get a mods help here.\n\nLast edited:\nSpace-elevator at Yahoo\n\nKi Man said:\nif this is about 200 miles high\nIt goes to geostationary orbit, which is about 22,200 miles high.\n\nKI Man said:\nhow quickly will we be traveling to get up this thing.\nThere exists a variety of space-elevator conceptions. There are speculation links at the bottom of this page...\nen.wikipedia.org\/wiki\/Space_elevator#See_also\n\n...and a good space-elevator discussion group here:\ngroups.yahoo.com\/group\/space-elevator\n\nStaff Emeritus\nGold Member\nSkepticJ said:\nGovernment Labs don't usually write stuff that is BS though.\n\nUsually, no. Occasionally? Yes. Just because you have PhD after your name does not mean you can't make mistakes or overlook certain aspects of a problem. That's why designs are always done in groups. In this case, the idea has so many critical engineering details neglected as to be worthless. Where is the report? All I've seen is a non-technical Wikipedia article.\n\nNeither did your non-functioning \"link\". You might want to go back and fix that, because I can't be privy to the \"debunking\" of \"crap\" concept space fountains.\n\nLink works fine for me. Links to the website of University of Maryland's Space Systems lab director Dr. Dave Akin. Law #17 states:\n\nDave Akin said:\n17. The fact that an analysis appears in print has no relationship to the likelihood of its being correct.\n\nSomething which is absolutely true. I have read many technical journals with blatant errors that got overlooked.\n\nEven if you're not one and the same,\n\nLunchbox and Ki Man are not the same person.\n\nVacuum pipes, which the streams travel through, make this a non-issue. No air, no drag.\n\nIt is proposing a pump which drains the air (and keeps the air drained) out of a pipe _200 miles long_?!? Amazing what is trivially attainable when you just wave the magic \"engineers can do anything... and easily\" wand.\n\nSteel is unable to sustain its own weight in the Earth's gravity field if it's on the order of 10 miles long. Stresses inside the structure due to the gravity loading will cause it to fail, even if it's being pushed up at the end. The failures would occur in the middle anyways.\n\nUntil I see an in-depth technical analysis which covers structure vibrations and stress analyses, I'm going to say it's not feasible as well, just from back-of-the-envelope calculations.\n\nI guess I'll have to say everything that the linked papers say before you get it\n\nI see a grand total of zero linked technical papers in this thread, or the one which I separated this discussion out of.\n\nLast edited:\nSkepticJ said:\nAnother advantage to space fountains is they need not be built on the equator. One could build one at the North Pole if they wanted to\nRibbon-type space elevators do not need to be anchored on the equator. They can be anchored anywhere, including the poles.\n\nLast edited:\nblimkie\nKi Man said:\nmeh, ignore what i said before i was drunk or something (not literally)\n\nhow fast are we talking? how high is it going to be (estimate) and how big is the elevator going to have to be (like how wide is it going to need to be at most and what kind of things are going to need to be carried up by the elevator carts.)?\n\nKi man the space elevator is not an actualy elevator as u have in the CN tower empire state etc.\nIts not gonne be built as if it were a massive cn tower reaching into space. Keyword \"ribbons\" made from carbon nano tubes its flexible.\n\nIM NOT saying that it is 100% possible because i don't know all the complicated physics involved exceeding high school. The idea was for a spacecraft to latch onto the ribbon and drive itself into space at a slower velocity than rocket ships and supposedly be more safer. A company called lift port has smalled scale models suspended by balloons. The question is can they create one much larger and extend it into space.\nI watched a small piece about it discovery channel last night and was quite suprised but beleives they can make this work by 2018. http:\/\/liftport.com\/research1.php [Broken]\n\nLast edited by a moderator:\nSkepticJ\nRibbon-type space elevators do not need to be anchored on the equator. They can be anchored anywhere, including the poles.\n\nUntil the asteroid falls out of the sky blowing the Arctic ice sheet or Antarctica a new crater. It's the centripital force of the asteroid on the end that holds the ribbon tight. Other places as well have a problem, because geosynchronous orbit can only happen above the equator.\n\nSkepticJ\nenigma said:\nUsually, no. Occasionally? Yes. Just because you have PhD after your name does not mean you can't make mistakes or overlook certain aspects of a problem. That's why designs are always done in groups. In this case, the idea has so many critical engineering details neglected as to be worthless. Where is the report? All I've seen is a non-technical Wikipedia article.\n\nLink works fine for me. Links to the website of University of Maryland's Space Systems lab director Dr. Dave Akin. Law #17 states:...\n\nIt is proposing a pump which drains the air (and keeps the air drained) out of a pipe _200 miles long_?!? Amazing what is trivially attainable when you just wave the magic \"engineers can do anything... and easily\" wand.\n\nGood point. I'll see if I can find that technical paper. Google searches aren't finding anything, but I'll keep trying to get it.\n\nNo, not that link; the link in post #18's bottom. It doesn't work for me.\n\nShouldn't be hard at all. For one thing, the pipe could be very narrow in diameter; which means less air inside. For another thing, the vacuum pump isn't fighting against gravity. If you had a pipe as tall as this, connected even just a small vacuum pump, it could eventually suck all the air out. \"Eventually\" wouldn't do, so something like a 777's jet engine powered vacuum pump would do nicely. It'd probably be able to suck the pipe to a vacuum in several hours to a day. Since the top of the pipe is above the atmosphere no more, or very, very little and slowly, will get in again that way. Going through the pipe material itself, hydrogen and helium are the only gases I know of that can squeeze through metal. There's not much of either gas in Earth's atmosphere, because they rise to the top of the atmosphere and are blown away by Sol's solar wind.","date":"2023-02-04 17:48:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.4111977815628052, \"perplexity\": 1561.747998536815}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764500151.93\/warc\/CC-MAIN-20230204173912-20230204203912-00789.warc.gz\"}"} | null | null |
{"url":"http:\/\/zbmath.org\/?q=an:1142.34367","text":"# zbMATH \u2014 the first resource for mathematics\n\n##### Examples\n Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. \"Topological group\" Phrases (multi-words) should be set in \"straight quotation marks\". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. \"Quasi* map*\" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. \"Partial diff* eq*\" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.\n\n##### Operators\n a & b logic and a | b logic or !ab logic not abc* right wildcard \"ab c\" phrase (ab c) parentheses\n##### Fields\n any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)\nAlmost periodic solutions of neutral functional differential equations. (English) Zbl\u00a01142.34367\nSummary: We study a non-autonomous neutral functional differential equation in a Banach space. Applying the theory of semigroups of operators to evolution equations and Krasnoselskii\u2019s fixed point theorem we establish the existence and uniqueness of a mild almost periodic solution of the problem under consideration.\n##### MSC:\n 34K14 Almost and pseudo-periodic solutions of functional differential equations 34K40 Neutral functional-differential equations\n##### References:\n [1] B.M. Levitan, V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow, 1978 \u00b7 Zbl\u00a00414.43008 [2] Besicovich, A. S.: Almost periodic functions, (1958) [3] Amerio, L.; Prouse, G.: Almost periodic functions and functional equations, (1971) \u00b7 Zbl\u00a00215.15701 [4] Rubanik, V. P.: Oscillations of quasilinear systems with retardation, (1969) [5] Hale, J. K.; Lunel, V. M.: Introduction to functional differential equations, Appl. math. Sci. 99 (1993) \u00b7 Zbl\u00a00787.34002 [6] Gurtin, M. E.; Pipkin, A. C.: A general theory of heat conduction with finite wave speed, Arch. ration. Mech. anal. 31, 113-126 (1968) \u00b7 Zbl\u00a00164.12901 \u00b7 doi:10.1007\/BF00281373 [7] Zaidman, S.: Abstract differential equations, (1979) [8] Zaidman, S.: Topics in abstract differential equations 1, Pitman research notes in mathematics 304 (1994) \u00b7 Zbl\u00a00806.34001 [9] Zaidman, S.: Topics in abstract differential equations 2, Pitman research notes in mathematics 321 (1995) \u00b7 Zbl\u00a00834.34001 [10] Islam, Mohammad N.; Raffoul, Youssef N.: Periodic solutions of neutral nonlinear system of differential equations with functional delay, J. math. Anal. appl. 331, 1175-1186 (2007) \u00b7 Zbl\u00a01118.34057 \u00b7 doi:10.1016\/j.jmaa.2006.09.030 [11] Acquistapace, P.; Terreni, B.: A unified approach to abstract linear parabolic equations, Rend. sem. Math. uni. Padova 78, 47-107 (1987) \u00b7 Zbl\u00a00646.34006 \u00b7 doi:numdam:RSMUP_1987__78__47_0 [12] Acquistapace, P.: Evolution operators and strong solution of abstract linear parabolic equations, Differential integral equations 1, 433-457 (1988) \u00b7 Zbl\u00a00723.34046","date":"2014-04-16 16:41:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"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\": 0.71623694896698, \"perplexity\": 9450.42314871619}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-15\/segments\/1397609524259.30\/warc\/CC-MAIN-20140416005204-00133-ip-10-147-4-33.ec2.internal.warc.gz\"}"} | null | null |
Petty Officer 2nd Class Matthew G. Axelson
29 years old from Cupertino, California
Petty Officer 2nd Class Matthew G. Axelson was part of a four-man unit that was attacked June 28, 2005 in a wooded, mountainous area of Kunar province, which is along the Pakistani border. The commandos were on a reconnaissance mission when they were ambushed and called for help, said Lt. Taylor Clark, public affairs officer for Naval Special Forces Command. He was a member of the SEAL Delivery Vehicle Team One, based in Pearl Harbor, Hawaii, which trains with submersible vehicles. The Monta Vista High School graduate had earned a degree in political science at Chico State University and joined the Navy in 2000. He was awarded military honors, including the Silver Star and the Purple Heart.
Ray Garland, a neighbor who watched him grow up in their Cupertino neighborhood, said Axelson had been "a delightful, fun-loving young boy who loved riding his bike and going on skateboard ramps on our cul-de-sac."
Axelson grew into a young man who "had a very calm, unflappable personality ... who had the ability to defuse tension wherever he found it," Garland said. "He just had a very easy, comfortable, enjoyable way about him."
These brave men and women have given their lives so that others may enjoy the freedoms we get to enjoy everyday. For that, I am proud to call them Hero.
If you have a loved who has served or is currently serving in the armed forces, and would like for me to profile them as a Wednesday Hero, I would be proud to. Just email me with some info and a photo.
We Have Every Right To Dream Heroic Dreams.
Those Who Say That We're In A Time When There Are No Heroes, They Just Don't Know Where To Look
Filed under Wednesday Heroes
Categories: United States Military, War On Terror, Wednesday Hero
Immigration Plan
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Right-Wing & Right Minded Poll
He Does It Again | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,808 |
require 'spec_helper'
require 'pry-byebug'
RSpec.describe Molecular::RecipientsController, type: :controller do
let(:campaign) { create(:campaign) }
let(:users) { create_list(:user, 3) }
before do
allow_any_instance_of(Molecular::Campaign).to receive(:recipients).
and_return(User.where(id: users))
end
describe "GET #index" do
it "returns a 200 OK status" do
get :index, campaign_id: campaign.id
expect(response).to have_http_status(:ok)
end
it "assigns the requested campaign as @campaign" do
get :index, campaign_id: campaign.id
expect(assigns(:campaign)).to eq(campaign)
end
it "assigns campaign recipients as @recipients" do
get :index, campaign_id: campaign.id
expect(assigns(:recipients)).to eq(users)
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,310 |
\section{Introduction}
Estimating the state of the atmosphere and the ocean has long been one
of the main goals of modern science. Data assimilation, which consists
of combining data and dynamical models to determine the best possible
estimate of the state of a system, is now recognized as the best
approach to tackle this problem \citep{Ghil1991-data}. The strongly
nonlinear character of the atmospheric and oceanic models, combined
with their important computational burden, makes data assimilation in
these systems quite challenging.
Based on the Bayesian estimation theory, the optimal solution of the
nonlinear data assimilation problem can be obtained from the optimal
nonlinear filter (ONF) \citep{Doucet2001-sequential}. This involves the
estimation of the conditional probability distribution function ({\it
pdf}) (not necessarily Gaussian) of the system state given all
available measurements up to the estimation time. Knowledge of the
state {\it pdf} allows determining different estimates of the state,
such as the minimum variance estimate or the maximum a posteriori
estimate \citep{Todling1999-estimation}. The ONF recursively operates
as a succession of a correction (or analysis) step at measurement
times to correct the state (predictive) {\it pdf} using the Bayes'
rule, and a prediction step to propagate the state (analysis) {\it
pdf} to the time of the next available observation. Although
conceptually simple, the numerical implementation of the optimal
nonlinear filter can be computationally prohibitive, even for systems
with few dimensions \citep{Doucet2001-sequential}. Its use with atmospheric and
oceanic data assimilation problems is therefore not possible because
of the huge dimension of these systems.
In recent years, two approximation schemes of the ONF have attracted
the attention of researchers for their potentials to tackle nonlinear
and non-Gaussian data assimilation problems. One is based on the
point-mass representation (mixture of Dirac functions) of the state
{\it pdf}, and leads to the celebrated particle filter (PF)
\citep{Doucet2001-sequential,Pham2001,Nakano2007-merging,vanLeeuwen-variance,VanLeeuwen2009}. The
other is based on the Gaussian mixture representation of the state
{\it pdf}, and results in a filter that is in between the Kalman
filter and the particle filter
\citep{Anderson-Monte,Bengtsson2003,Chen2000-mixture,Hoteit2008,Luo2008-spgsf1,Sorenson-recursive},
as to be shown later. For this reason, we refer to this filter as the
particle Kalman filter (PKF).
In terms of computational efficiency, the particle filter needs to
generate large samples for a good approximation of the state {\it
pdf}. In certain circumstances, in order to avoid weights collapse,
the number of samples needs to scale exponentially with the dimension
of the system in assimilation \citep{Bengtsson2008}, which may be
infeasible for high-dimensional systems \citep{Snyder2008}. On the
other hand, in some comparison studies
\citep{Han-evaluation,Nakano2007-merging}, it has been reported that
the ensemble Kalman filter (EnKF) and its variants
\citep{Anderson-ensemble,Bishop-adaptive,Burgers-analysis,Evensen-sequential,Evensen-assimilation,Houtekamer1998,Whitaker-ensemble}
can achieve lower estimation errors than the particle filter given a
small ensemble size. To save space, in this paper we confine ourselves
to the PKF, and make performance comparison only between the PKF and
the EnKF.
Using a Gaussian mixture representation of the state {\it pdf}, the
resulting PKF consists of an ensemble of parallel nonlinear Kalman
filters \citep{Hoteit2008,Luo2008-spgsf1}. Different variants of the
Kalman filter (KF), including the extended Kalman filter
\citep{Chen2000-mixture,Sorenson-recursive}, the reduced-rank Kalman
filter \citep{Hoteit2008,Luo2008-spgsf1}, the EnKF
\citep{Anderson-Monte,Bengtsson2003}, can be used to construct the
PKF. The focus of this paper is to investigate the PKF that is
constructed by an ensemble of parallel EnKFs. Common to all the
implementations of the PKF, the mixture of normal distributions (MON)
-- a more general {\it pdf} representation than the single Gaussian
{\it pdf} approximation in the EnKF -- can be used to tackle
nonlinearity and non-Gaussianity in data assimilation. On the other
hand, choosing the EnKF to construct the PKF is based on the
consideration of computational efficiency, since the EnKF itself is a
very efficient algorithm for data assimilation in high dimensional
systems. In this regard, this work is very similar to the earlier
works of \citet{Anderson-Monte} and \citet{Bengtsson2003}, but is
different from them mainly in the following aspect.
In \citet{Anderson-Monte} and \citet{Bengtsson2003}, the PKF was
constructed without a re-sampling step. As a result, the PKF may
suffer from weights collapse as in the particle filter. To overcome
this problem, \cite{Bengtsson2003} considered a hybrid of the EnKF and
the PKF, which, however, involves the computation of the inverses of
sample covariance matrices in the ``global-to-local'' adjustments. In
doing so, it is not only computationally intensive, but also
encounters singularities in computing the inverses when the ensemble
size is smaller than the system dimension, such that the sample
covariances themselves are rank deficient. Therefore, it is not clear
how the hybrid scheme in \cite{Bengtsson2003} can be applied to the
scenario with the ensemble size smaller than the system dimension. For
the implementation of the PKF scheme in this work, we introduce a
re-sampling step similar to those in \cite{Musso-improving-2001} and
\citet{Stavropoulos-improved-2001} to tackle weights collapse. Our
experience shows that, with this re-sampling step, the PKF becomes
much more stable and can conduct data assimilation in the small
ensemble scenario, as to be demonstrated through the numerical
experiments presented in this work.
As may be of particular interest for the ensemble filtering community,
we will show that different EnKFs can be considered as special cases
of the PEnKF following our implementation. This point of view allows
for a better understanding of the EnKFs' behaviors and/or their
differences.
The paper is organized as follows. The optimal nonlinear filter is
first described in section \ref{optfilter}. The PKF and its ensemble
implementation are discussed in section \ref{pkf}. Results of
numerical experiments with the Lorenz-$96$ model are presented in
section \ref{experiments}. A summary of the main results and a general
discussion on the potential of the PEnKF for tackling realistic
atmospheric and oceanic data assimilation problems concludes the paper
in section \ref{discussion}.
\section{The Optimal Nonlinear Filter}
\label{optfilter}
Starting from a random initial condition with a known probability
density function, the optimal nonlinear filter provides the
conditional density function of the system state given all available
measurements up to the estimation time. To describe the algorithm of
the optimal nonlinear filter, consider the nonlinear stochastic
discrete-time dynamical system
\begin{eqnarray}
\label{model}
\mbox{\boldmath$x$}_k &=& \mbox{\boldmath$M$}_k(\mbox{\boldmath$x$}_{k-1}) + \mbox{\boldmath$\eta$}_k,\\
\label{obs}
\mbox{\boldmath$y$}_k &=& \mbox{\boldmath$H$}_k(\mbox{\boldmath$x$}_k) + \mbox{\boldmath$\epsilon$}_k,
\end{eqnarray}
where $\mbox{\boldmath$x$}_k$ is the state vector (to be estimated), of dimension $n$,
$\mbox{\boldmath$y$}_k$ is the observation vector, of dimension $p$, $\mbox{\boldmath$M$}_k$ and
$\mbox{\boldmath$H$}_k$ are two continuously differentiable maps from ${\rm I\kern-.15em R}^n$ to
${\rm I\kern-.15em R}^n$ and from ${\rm I\kern-.15em R}^n$ to ${\rm I\kern-.15em R}^p$ respectively representing the
transition and the observational operators, and $\mbox{\boldmath$\eta$}_k$ and
$\mbox{\boldmath$\epsilon$}_k$ denote the dynamical and the observational noise,
respectively. We assume that $\mbox{\boldmath$\eta$}_k$ and $\mbox{\boldmath$\epsilon$}_k$ are Gaussian
with zero mean and non-singular covariance matrices $\mbox{\boldmath$Q$}_k$ and
$\mbox{\boldmath$R$}_k$, respectively, and are independent of the system state at any
time instant. Under this setting, the dynamical system
Eq.~(\ref{model}) is Markovian.
The optimal nonlinear filter recursively operates with a succession of
prediction and correction steps as summarized below. The reader is
referred to \cite{Doucet2001-sequential} for an extensive description of the
filter. To simplify the notation, $\mbox{\boldmath$y$}_{1:k}$ is defined as a
shorthand for the set of all observations $\mbox{\boldmath$y$}_1, \dots, \mbox{\boldmath$y$}_k$ up to
and including time $t_k$. Let $p^f_k(~\cdot \mid\mbox{\boldmath$y$}_{1:k-1})$ be the
conditional (predictive) {\it pdf} of $\mbox{\boldmath$x$}_k$ given $\mbox{\boldmath$y$}_{1:k-1}$ and
$p^a_k(~\cdot \mid \mbox{\boldmath$y$}_{1:k})$ be the conditional (analysis) {\it pdf}
of $\mbox{\boldmath$x$}_k$ given $\mbox{\boldmath$y$}_{1:k}$, both determined at time $t_k$. The
filter steps are described as follows.
\begin{enumerate}
\item[$\bullet$] {\underline {\it Prediction step}:} Given the
analysis {\it pdf} $p^a_{k-1}(~\cdot \mid \mbox{\boldmath$y$}_{1:k-1})$ at time
$t_{k-1}$, the predictive {\it pdf} $p^f_k(~\cdot \mid \mbox{\boldmath$y$}_{1:k-1})$ is
obtained by integrating $p^a_{k-1}(~\cdot \mid \mbox{\boldmath$y$}_{1:k-1})$ with the
model (\ref{model}) to the time of the next available observation
$t_k$. Under the assumptions made on the model noise $\mbox{\boldmath$\eta$}_k$, the
likelihood function for the state vector $\mbox{\boldmath$x$}_{k-1}$ to transit to $\mbox{\boldmath$x$}_{k}$ at the next time instant is described by the Gaussian {\it pdf} $N \left(\mbox{\boldmath$x$}_{k}: \mbox{\boldmath$M$}_k(\mbox{\boldmath$x$}_{k-1}), \mbox{\boldmath$Q$}_k\right)$, where $N \left(\mbox{\boldmath$x$}: \mathbf{\mu}, \mbox{\boldmath$\Sigma$}\right)$ denotes the Gaussian {\it pdf} with mean $\mathbf{\mu}$ and covariance $\mbox{\boldmath$\Sigma$}$. Thus,
\begin{eqnarray}
\label{e:predicden}
p^f_k(\mbox{\boldmath$x$}_{k} \mid \mbox{\boldmath$y$}_{1:k-1}) &=& \int_{{\rm I\kern-.15em R}^n} N \left(\mbox{\boldmath$x$}_{k}:
\mbox{\boldmath$M$}_k(\mbox{\boldmath$x$}_{k-1}) , \mbox{\boldmath$Q$}_k\right) p^a_{k-1}(\mbox{\boldmath$x$}_{k-1} \mid \mbox{\boldmath$y$}_{1:k-1}) d\mbox{\boldmath$x$}_{k-1}.
\end{eqnarray}
\item[$\bullet$] {\underline {\it Correction step}:} After a new
observation $\mbox{\boldmath$y$}_k$ has been made, the analysis {\it pdf} $p_k^a (~\cdot
\mid \mbox{\boldmath$y$}_{1:k})$ at time $t_k$ is updated from $p_k^f (~\cdot
\mid \mbox{\boldmath$y$}_{1:k-1})$ using Bayes' rule, i.e.,
\begin{eqnarray}
\label{e:conden}
p^a_k(\mbox{\boldmath$x$}_{k} \mid \mbox{\boldmath$y$}_{1:k}) &=& \frac{1}{b_k}p^f_k(\mbox{\boldmath$x$}_{k} \mid
\mbox{\boldmath$y$}_{1:k-1}) N \left(\mbox{\boldmath$y$}_k: \mbox{\boldmath$H$}_k(\mbox{\boldmath$x$}_{k}), \mbox{\boldmath$R$}_k\right).
\end{eqnarray}
The analysis {\it pdf} is therefore obtained by multiplying the
predictive {\it pdf} by the observation likelihood function $N \left(\mbox{\boldmath$y$}_k: \mbox{\boldmath$H$}_k(\mbox{\boldmath$x$}_{k}), \mbox{\boldmath$R$}_k\right)$, and then being normalized by $b_k
= \int_{{\rm I\kern-.15em R}^n} p^f_k(\mbox{\boldmath$x$}_{k} \mid \mbox{\boldmath$y$}_{1:k-1}) N\left(\mbox{\boldmath$y$}_k:
H_k(\mbox{\boldmath$x$}_{k}) , R_k\right) d\mbox{\boldmath$x$}_{k}$.
\end{enumerate}
While the expressions of the state {\it pdf}s can be obtained
conceptually, determining the exact values of them at each point of
the state space is practically infeasible in high dimensional systems
\citep{Doucet2001-sequential}. For instance, the determination of the predictive
{\it pdf} requires the evaluation of the model $\mbox{\boldmath$M$}_k(\mbox{\boldmath$x$})$ for a
prohibitively large number of $\mbox{\boldmath$x$}$, given that one single evaluation
might already be computationally very expensive in realistic
atmospheric and oceanic applications.
\section{The Particle Ensemble Kalman Filter}
\label{pkf}
\subsection{Particle Kalman Filtering and Its Ensemble Implementation}
Given $N$ independent samples $\mbox{\boldmath$x$}^1, \ldots, \mbox{\boldmath$x$}^N$ from a
(multivariate) density $p$, an estimator $\hat{p}$ of $p$ can be
obtained by the kernel density estimation method
\citep{Silverman1986}, in the form of a mixture of $N$ Gaussian {\it
pdf}s:
\begin{eqnarray}
\hat{p}(\mbox{\boldmath$x$}) &=& \frac{1}{N}\sum_{i=1}^N N(\mbox{\boldmath$x$}: \mbox{\boldmath$x$}^i , \mbox{\boldmath$P$}),
\end{eqnarray}
where $\mbox{\boldmath$P$}$ is a positive definite matrix. Inspired from this
estimator, the particle Kalman filter (PKF) approximates the
conditional state {\it pdf}s in the optimal nonlinear filter by
mixtures of $N$ Gaussian densities of the form
\begin{eqnarray}
\label{approxmix}
p^s_k(\mbox{\boldmath$x$}_k \mid \mbox{\boldmath$y$}_{1:k}) &=& \sum_{i=1}^N w^i_k N(\mbox{\boldmath$x$}_k: \mbox{\boldmath$x$}^{s,i}_k, \mbox{\boldmath$P$}^{s,i}_k).
\end{eqnarray}
The subscript $s$ replaces $a$ at the analysis time and $f$ at the
prediction time. The parameters of the mixture are the weights
$w^i_k$, the centers of the distributions $\mbox{\boldmath$x$}^{s,i}_k$, and the
covariance matrices $\mbox{\boldmath$P$}^{s,i}_k$. In particular, if $N=1$,
$p^s_k(\mbox{\boldmath$x$}_k \mid \mbox{\boldmath$y$}_{1:k})$ reduces to a single Gaussian {\it pdf},
so that the PKF reduces to the Kalman filter (KF) or its variants
trivially (a non-trivial simplification will also be discussed
below). Consequently, the KF and its variants can be considered
special cases of the PKF.
Two special cases of Eq.~(\ref{approxmix}) may be of particular
interest. In the first case, $\mbox{\boldmath$P$}^{s,i}_k \rightarrow \mathbf{0}$,
such that the Gaussian {\it pdf}s $N(\mbox{\boldmath$x$}_k: \mbox{\boldmath$x$}^{s,i}_k, \mbox{\boldmath$P$}^{s,i}_k)$
tend to a set of Dirac functions $\delta(\mbox{\boldmath$x$}^{s,i}_k)$, with the mass
points at $\mbox{\boldmath$x$}^{s,i}_k$. In this case, the Gaussian mixture
Eq.~(\ref{approxmix}) reduces to the Monte Carlo approximation used in
the particle filter \citep{Doucet2001-sequential}. In the second case, all
Gaussian {\it pdf}s $N(\mbox{\boldmath$x$}_k: \mbox{\boldmath$x$}^{s,i}_k, \mbox{\boldmath$P$}^{s,i}_k)$ have (almost)
identical centers and covariances, such that the Gaussian mixture
Eq.~(\ref{approxmix}) tends to a (single) Gaussian approximation, an
assumption often used in various nonlinear Kalman filters (including
the EnKF). In this sense, the PKF can be considered as a filter in
between the Kalman filter and the particle filter
\citep{Hoteit2008,Luo2008-spgsf1}.
The main procedures of the PKF are summarized as follows. Without loss
of generality, suppose that at time instant $k-1$, the analysis {\it
pdf}, after a re-sampling step, is given by
$\tilde{p}_{k-1}(\mbox{\boldmath$x$}_{k-1} \mid \mbox{\boldmath$y$}_{1:k-1}) = \sum_{i=1}^N
\tilde{w}^i_{k-1} N(\mbox{\boldmath$x$}_{k-1}: \mathbf{\theta}^{i}_{k-1},
\mathbf{\Phi}^{i}_{k-1})$. Then by applying Eq.~(\ref{e:predicden}) at
the prediction step, one obtains the background {\it pdf}, in terms of
a new MON
\begin{eqnarray}
\label{approxpred}
p^f_k(\mbox{\boldmath$x$}_k \mid \mbox{\boldmath$y$}_{1:k-1}) &\approx& \sum_{i=1}^N \tilde{w}^i_{k-1}
N \left(\mbox{\boldmath$x$}_k: \hat{\mbox{\boldmath$x$}}^{f,i}_{k} , \hat{\mbox{\boldmath$P$}}^{f,i}_{k}\right),
\end{eqnarray}
where $\hat{\mbox{\boldmath$x$}}^{f,i}_{k}$ and $\hat{\mbox{\boldmath$P$}}^{f,i}_{k}$ are the
propagations of the mean $\mathbf{\theta}^{i}_{k-1}$ and the
covariance $\mathbf{\Phi}^{i}_{k-1}$ of the Gaussian component
$N(\mbox{\boldmath$x$}_{k-1}: \mathbf{\theta}^{i}_{k-1}, \mathbf{\Phi}^{i}_{k-1})$
through the system model Eq.~(\ref{model}), respectively.
Given an incoming observation $\mbox{\boldmath$y$}_k$, one applies
Eq.~(\ref{e:conden}) to update $p^f_k(\mbox{\boldmath$x$} \mid \mbox{\boldmath$y$}_{1:k-1})$ to the
analysis {\it pdf}, also in the form of an MON
\begin{eqnarray}
\label{densmix}
p^a_k(\mbox{\boldmath$x$}_k \mid \mbox{\boldmath$y$}_{1:k}) &=& \sum_{i=1}^N w^i_{k} N \big(\mbox{\boldmath$x$}_k:
\hat{\mbox{\boldmath$x$}}^{a,i}_{k}, \hat{\mbox{\boldmath$P$}}^{a,i}_{k}\big),
\end{eqnarray}
where $\hat{\mbox{\boldmath$x$}}^{a,i}_{k}$ and $\hat{\mbox{\boldmath$P$}}^{a,i}_{k}$ are updated from
$\hat{\mbox{\boldmath$x$}}^{f,i}_{k}$ and $\hat{\mbox{\boldmath$P$}}^{f,i}_{k}$ through the Kalman
filter or its variants, and the new weights
\begin{eqnarray}
\label{corrp}
w^i_k &=& \frac{\tilde{w}^i_{k-1} N \big(\mbox{\boldmath$y$}_k: \mbox{\boldmath$H$}_k(\hat{\mbox{\boldmath$x$}}^{f,i}_k) ,
\mbox{\boldmath$\Sigma$}^i_k \big)}{\sum_{j=1}^N \tilde{w}^i_{k-1} N \big(\mbox{\boldmath$y$}_k: \mbox{\boldmath$H$}_k(\hat{\mbox{\boldmath$x$}}^{f,i}_k), \mbox{\boldmath$\Sigma$}^i_k \big)} \, ,
\end{eqnarray}
where $\mbox{\boldmath$\Sigma$}^i_k$ is the innovation matrix. If evaluated through the
extended Kalman filter, $\mbox{\boldmath$\Sigma$}^i_k = {\bf H}^i_k \hat{\mbox{\boldmath$P$}}^{f,i}_k
({\bf H}_k^{i})^T + \mbox{\boldmath$R$}_k$, with ${\bf H}^i_k$ being the gradient of
$\mbox{\boldmath$H$}_k$ evaluated at $\hat{\mbox{\boldmath$x$}}^{f,i}_k$. Alternatively, if evaluated
in the context of the EnKF, $\mbox{\boldmath$\Sigma$}^i_k$ can be expressed as the
covariance of the projected background ensemble onto the observation
space plus the observation covariance $\mbox{\boldmath$R$}_k$
\citep{Evensen-sequential,Whitaker-ensemble}. Finally, a re-sampling
step can be introduced to improve the performance of the PKF
\citep{Hoteit2008,Luo2008-spgsf1}, so that the analysis {\it pdf}
becomes $\tilde{p}_{k}(\mbox{\boldmath$x$}_{k} \mid \mbox{\boldmath$y$}_{1:k}) = \sum_{i=1}^N
\tilde{w}^i_k N(\mbox{\boldmath$x$}_{k}: \mathbf{\theta}^{i}_{k},
\mathbf{\Phi}^{i}_{k})$. Such a re-sampling algorithm is presented in
the next section.
The PKF correction step can be interpreted as composed of two types of
corrections: a {\it Kalman-type correction} used to update
$\hat{\mbox{\boldmath$x$}}^{f,i}_{k}$ and $\hat{\mbox{\boldmath$P$}}^{f,i}_{k}$ to
$\hat{\mbox{\boldmath$x$}}^{a,i}_{k}$ and $\hat{\mbox{\boldmath$P$}}^{a,i}_{k}$, and a {\it
particle-type correction} used to update the weights
$\tilde{w}^i_{k-1}$ to $w^i_{k}$. In the PKF, the Kalman correction
reduces the risk of weights collapse by allocating the estimates
$\hat{\mbox{\boldmath$x$}}^{f,i}_k$ (whose projections onto the observation space) far
away from the observation $\mbox{\boldmath$y$}_k$ relatively more weights than in the
particle filter \citep{Hoteit2008,VanLeeuwen2009}. Indeed,
Eq. (\ref{corrp}) has the same form as in the PF \citep{Doucet2001-sequential},
but uses the innovation matrices $\mbox{\boldmath$\Sigma$}^i_k$ to normalize the
model-data misfit, rather than $\mbox{\boldmath$R$}_k$. As $\mbox{\boldmath$\Sigma$}^i_k$ are always
greater than $\mbox{\boldmath$R$}_k$, the estimates that are close to the observation
will receive relatively less weights than in the PF, while those far
from the observation will receive relatively more weights. This means
that the support of the local predictive {\it pdf} and the observation
likelihood function will be more coherent than in the PF. Re-sampling
will therefore be needed less often, so that Monte Carlo fluctuations
are reduced.
The main issue with the PKF is the prohibitive computational burden
associated with running an ensemble of KFs, knowing that running a
Kalman filter (KF) or an extended KF in high dimensional systems is
already a challenge. To reduce computational cost, we use an ensemble
of EnKFs, rather than the KF or the extended KF, to construct the
PKF. We refer to this approach as the Particle Ensemble Kalman Filter
(PEnKF). In the PEnKF, the (analysis) ensembles representing the
Gaussian components are propagated forward in time to obtain a set of
background ensembles at the next assimilation cycle. Then for each
background ensemble, a stochastic or deterministic EnKF is used to
update the background ensemble to its analysis counterpart. This
amounts to simultaneously running a weighted ensemble of EnKFs, and
the final state estimate is the weighted average of all the EnKFs
solutions.
\subsection{A Re-sampling Algorithm}
\label{sec:re-sampling}
We adopt a re-sampling algorithm that combines those in
\citet{Hoteit2008,Luo2008-spgsf1,Pham2001}. The main idea is as
follows: Given a MON, we first employ an information-theoretic
criterion used in \citet{Hoteit2008} and \citet{Pham2001} to check if
it needs to conduct re-sampling. If there is such a need, we then
re-approximate the MON by a new MON, based on the criterion that the mean and covariance of the new MON match those of the original MON as far as possible \citet{Luo2008-spgsf1}.
More concretely, let $p \left( \mathbf{x} \right)$ be the {\it pdf} of
the $n$-dimensional random vector $\mathbf{x}$, expressed in terms of
an MON with $N$ Gaussian {\it pdf}s so that
\begin{equation} \label{eq:original_GMM}
p \left( \mathbf{x} \right) = \sum\limits_{i=1}^{N} w_i N \left(\mathbf{x}: \mathbf{\mu}_i, \mathbf{\Sigma}_i\right) \, ,
\end{equation}
where $w_i$ are the set of normalized weights of the Gaussian {\it
pdf}s $N \left(\mathbf{x}: \mathbf{\mu}_i, \mathbf{\Sigma}_i\right)$
with mean $\mathbf{\mu}_i$ and covariance $\mathbf{\Sigma}_i$,
satisfying $w_i \geq 0$ for $i=1,\dotsb,N$ and $\sum_{i=1}^{N}
w_i=1$. To decide whether to conduct re-sampling or not, the entropy
$E_w$ of the weights $w_i$ is computed, which reads
\citep{Hoteit2008,Pham2001}
\begin{equation}
E_w = - \sum\limits_{i=1}^{N} w_i \text{log} w_i \, .
\end{equation}
Ideally, when the distribution of the weights $w_i$ is uniform, which
yields the maximum weight entropy $E_w^{u}=\text{log} N$, there is no
need to conduct re-sampling. Thus, as a criterion, if $E_w$ is within
a certain distance $d$ to $E_w^u$, i.e.,
\begin{equation} \label{eq:resampling_criterion}
E_w^u - E_w = \text{log} N + \sum\limits_{i=1}^{N} w_i \text{log} w_i \leq d \, ,
\end{equation}
where $d$ is a user-defined threshold, then we choose not to conduct
re-sampling. In this work we set the threshold $d=0.25$ following
\citet{Hoteit2008}.
In case that there is a need to conduct re-sampling, we follow the
procedure similar to that in \citet{Luo2008-spgsf1}. Here the idea is
to treat re-sampling as a {\it pdf} approximation problem, in which we
seek a new MON
\begin{equation} \label{eq:new_GMM}
\tilde{p} \left( \mathbf{x} \right) = \dfrac{1}{q} \sum\limits_{i=1}^{q} N \left(\mathbf{x}: \mathbf{\theta}_i, \mathbf{\Phi}_i\right) \, ,
\end{equation}
with $q$ equally weighted Gaussian {\it pdf}s, to approximate the
original $p \left( \mathbf{x} \right)$ in
Eq.~(\ref{eq:original_GMM}). In approximation, we require that the
mean and covariance of $\tilde{p} \left( \mathbf{x} \right)$ be as
close as possible to those of $p \left( \mathbf{x} \right)$. To this
end, we need to choose proper values of $\mathbf{\theta}_i$ and
$\mathbf{\Phi}_i$ in order to achieve this objective.
The means and covariances of $p \left( \mathbf{x} \right)$ and
$\tilde{p} \left( \mathbf{x} \right)$, denoted by $\bar{\mathbf{x}}$
and $\bar{\mathbf{P}}$, and $\tilde{\mathbf{x}}$ and
$\tilde{\mathbf{P}}$, respectively, are given by
\begin{subequations} \label{eq:stat_original_pdf}
\begin{align}
\bar{\mathbf{x}} &= \sum_{i=1}^{N} w_i \mathbf{\mu}_i \, , ~\text{and}~
\bar{\mathbf{P}} = \sum_{s=1}^{N} w_i \left(\mathbf{\Sigma}_i + \left( \mathbf{\mu}_i - \bar{\mathbf{x}} \right) \left( \mathbf{\mu}_i - \bar{\mathbf{x}} \right)^T \right) \, ,\\
\tilde{\mathbf{x}} &= \dfrac{1}{q} \sum_{i=1}^{q} \mathbf{\theta}_i \, , ~\text{and}~
\tilde{\mathbf{P}} = \dfrac{1}{q} \sum_{i=1}^{q} \left(\mathbf{\Phi}_i + \left( \mathbf{\theta}_i - \tilde{\mathbf{x}} \right) \left( \mathbf{\theta}_i - \tilde{\mathbf{x}} \right)^T \right) \, .
\end{align}
\end{subequations}
Thus our objective is equivalent to balancing the above equation such that
\begin{equation} \label{eq:auxiliary_objective}
\tilde{\mathbf{x}} = \bar{\mathbf{x}} \, , ~\text{and}~ \tilde{\mathbf{P}} \approx \bar{\mathbf{P}} \, .
\end{equation}
In the trivial case with $q = N = 1$,
Eq.~(\ref{eq:auxiliary_objective}) can be satisfied by letting
$\mathbf{\theta}_1 = \mathbf{\mu}_1$ and $\mathbf{\Phi}_1 =
\mathbf{\Sigma}_1$, and the PEnKF reduces to an EnKF. In non-trivial
cases, for simplicity in solving Eq.~(\ref{eq:auxiliary_objective})
and reducing computational cost (as to be shown later), one may choose
the covariances $\mathbf{\Phi}_i$ to be constant, say
$\mathbf{\Phi}_i= \mathbf{\Phi}$, for $i=1,\dotsb,q$, so that
\begin{equation} \label{eq:objective_expression}
\dfrac{1}{q} \sum_{i=1}^{q} \mathbf{\theta}_i = \bar{\mathbf{x}} \, , ~\text{and}~
\mathbf{\Phi} + \dfrac{1}{q} \sum_{i=1}^{q} \left( \mathbf{\theta}_i - \bar{\mathbf{x}} \right) \left( \mathbf{\theta}_i - \bar{\mathbf{x}} \right)^T \approx \bar{\mathbf{P}} \, .
\end{equation}
When an EnKF is used to construct the PKF, one needs to represent the
solution of Eq.~(\ref{eq:objective_expression}) in terms of some
ensembles $\{ \mathbf{X}_{en}^i, i=1,\dotsb,q\}$, where
$\mathbf{X}_{en}^i$ is a matrix containing the (analysis) ensemble of
the $i$th Gaussian component in Eq.~(\ref{eq:new_GMM}), with mean
$\mathbf{\theta}_i$ and covariance $\mathbf{\Phi}$. For simplicity, we
assume that $\mathbf{X}_{en}^i$ are all of dimension $n \times m$,
with the ensemble size $m$ for each $i$. Similar results can be easily
obtained in the case with non-uniform ensemble sizes.
We then define a constant $c$, called \emph{fraction coefficient}
hereafter, which satisfies that $0 \leq c \leq 1$. We let
$\mathbf{\Phi} \approx c^2 \bar{\mathbf{P}}$, so that
Eq.~(\ref{eq:objective_expression}) is reduced to
\begin{equation} \label{eq:reduced_objective_expression}
\dfrac{1}{q} \sum_{i=1}^{q} \mathbf{\theta}_i = \bar{\mathbf{x}} \, , ~\text{and}~
\dfrac{1}{q} \sum_{i=1}^{q} \left( \mathbf{\theta}_i - \bar{\mathbf{x}} \right) \left( \mathbf{\theta}_i - \bar{\mathbf{x}} \right)^T \approx (1-c^2) \bar{\mathbf{P}} \, .
\end{equation}
In other words, the centers $\{ \mathbf{\theta}_i, i = 1, \dotsb, q
\}$ can be generated as a set of state vectors whose sample mean and
covariance are $\bar{\mathbf{x}}$ and $(1-c^2) \bar{\mathbf{P}}$,
respectively. After obtaining $\mathbf{\theta}_i$, one can generate
the corresponding ensembles $\mathbf{X}_{en}^i$, with the sample means
and covariances being $\mathbf{\theta}_i$ and $\mathbf{\Phi} \approx
c^2 \bar{\mathbf{P}}$, respectively. How $ \mathbf{\theta}_i$ and
$\mathbf{X}_{en}^i$ can be generated is discussed with more details in
the support material.
From the above discussion, we see that $c$ is a coefficient that
decides how to divide $\bar{\mathbf{P}}$ among $\mathbf{\Phi}$ and
$\dfrac{1}{q} \sum_{i=1}^{q} \left( \mathbf{\theta}_i -
\bar{\mathbf{x}} \right) \left( \mathbf{\theta}_i - \bar{\mathbf{x}}
\right)^T$, so that the constraints in
Eq.~(\ref{eq:objective_expression}) are satisfied. When $c \rightarrow
0$, we have $\mathbf{\Phi} \rightarrow \mathbf{0}$ so that $\tilde{p}
\left( \mathbf{x} \right)$ in Eq.~(\ref{eq:new_GMM}) approaches the
Monte Carlo approximation in the particle filter, with the mass points
equal to $\mathbf{\theta}_i$. On the other hand, when $c \rightarrow
1$, we have $\dfrac{1}{q} \sum_{i=1}^{q} \left( \mathbf{\theta}_i -
\bar{\mathbf{x}} \right) \left( \mathbf{\theta}_i - \bar{\mathbf{x}}
\right)^T \rightarrow \mathbf{0}$, so that all $\mathbf{\theta}_i$
approach $\bar{\mathbf{x}}$ and $\mathbf{\Phi}$ approaches
$\bar{\mathbf{P}}$. As a result, $\tilde{p} \left( \mathbf{x} \right)$
in Eq.~(\ref{eq:new_GMM}) approaches the Gaussian {\it pdf}
$N(\mathbf{x}: \bar{\mathbf{x}},\bar{\mathbf{P}})$, which is
essentially the assumption used in the EnKF. In this sense, when
equipped with the re-sampling algorithm, the PEnKF is a filter in
between the particle filter and the EnKF, with an adjustable parameter
$c$ that influences its behavior.
We note that, when $c \rightarrow 0$, under the constraint of matching the first two moments, our re-sampling scheme is very close to the posterior Gaussian re-sampling strategy used in the Gaussian particle filter \citep{Kotecha-signal-2003,Xiong-note-2006}, in which one generates particles from a Gaussian distribution with mean and covariance equal to those of the posterior {\it pdf} of the system states. As a result, there is no guarantee that higher order moments of the new MON match those of the original MON in our re-sampling scheme. If matching higher-order moments is a concern, one may adopt alternative criteria, for instance, the one that aims to minimize the distance (in certain metric) between the new MON and the original one, so that the re-sampling procedure is recast as an optimization problem, in which one aims to choose appropriate parameters, i.e., means and covariances of the new MON, that satisfy the chosen criterion as far as possible. In principle, this type of parameter estimation problem may be solved by the expectation-maximization (EM) algorithm \citep{Redner-mixture-1984,Smith-cluster}. But in practice, it is often computationally very intensive in doing so, due to the slow convergence rate of the EM algorithm and the high dimensionality of the parameter space in constructing the new MON. Therefore we do not consider this type of more sophisticated re-sampling strategy in this study.
For the purpose of pdf re-approximation, it is clear that the MON is not the only choice. A few alternatives are developed in the context of kernel density estimation (KDE) \citep{Silverman1986}, and in principle all of them can be applied for pdf re-approximation. For instance, KDE is adopted at the re-sampling step in the regularized
particle filter (RPF) \citep{Musso-improving-2001,Stavropoulos-improved-2001} to construct a continuous pdf with respect the particles before re-sampling, and to draw a number of new particles from the continuous pdf afterwards. In this regard, the PEnKF is similar to the RPF, especially if the Gaussian kernel is adopted in the RPF for density estimation. However, there also exist differences. We list some of them as follows.
\begin{itemize}
\item The RPF first constructs a continuous pdf, and then draws a number of new particles with equal weights from the resulting pdf. In contrast, the PEnKF aims to directly approximate a MON by a new MON with equal weights.
\item In the RPF, various kernels can be adopted for the purpose of constructing the continuous pdf. However, in the PEnKF, we are confined to use the MON, since we aim to build the PEnKF consisting of a set of parallel EnKFs.
\item The pdf re-approximation criterion used in the PEnKF only captures the first two moments of the underlying pdf. In contrast, KDE used in the RPF in principle can yield a very good pdf estimate, provided that there are sufficient particles. In certain circumstances, though, the number of required particles may also suffer from the ``curse-of-dimensionality'' \citep[ch.~4]{Silverman1986}.
\end{itemize}
\subsection{Outline of the PEnKF Algorithm}
To facilitate the comprehension of the PEnKF, here we provide an
outline of the main steps of its algorithm. To avoid distraction, we
will discuss the initialization of the PEnKF in the next
section. Throughout this paper, we assume that the number $q$ of
Gaussian components at the re-sampling step and the number $N$ of
Gaussian components at the prediction and correction steps are time
invariant. This implies the choice $q = N$.
Without loss of generality, we also assume that at time instant $k-1$,
the posterior pdf $p^a_{k-1} (\mathbf{x}_{k-1} \mid \mbox{\boldmath$y$}_{1:k-1})$ is
re-approximated, through the re-sampling step, by a mixture model
\[
\tilde{p}_{k-1} (\mathbf{x}_{k-1} \mid \mbox{\boldmath$y$}_{1:k-1}) =
\sum\limits_{i=1}^{q} \tilde{w}_{k-1}^i N \left(\mathbf{x}_{k-1}:
\mathbf{\theta}_{k-1,i}, \mathbf{\Phi}_{k-1} \right) \, .
\]
Moreover, the re-approximated analysis ensembles $\{
\mathbf{X}_{approx}^{k-1,i}, i=1,\dotsb,q\}$ representing the Gaussian
components $N \left(\mathbf{x}_{k-1}: \mathbf{\theta}_{k-1,i},
\mathbf{\Phi}_{k-1} \right)$ are also generated. The procedures at
the next assimilation cycle are outlined as follows.
\begin{enumerate}
\item[$\bullet$] {\underline {\it Prediction step}:} For
$i=1,\dotsb,q$, propagate the ensembles
$\mathbf{X}_{approx}^{k-1,i}$ forward through Eq.~(\ref{model}) to
obtain the corresponding background ensembles
$\mathbf{X}_{bg}^{k,i}$ at instant $k$. Accordingly, the background
{\it pdf} becomes
\[
p^b_{k} (\mathbf{x}_k \mid \mbox{\boldmath$y$}_{1:k-1}) = \sum\limits_{i=1}^{q}
\tilde{w}_{k-1}^i N \left(\mathbf{x}_k: \hat{\mathbf{x}}_{k,i}^b,
\hat{\mathbf{P}}_{k,i}^b \right) \, ,
\]
with $\hat{\mathbf{x}}_{k,i}^b$ and $\hat{\mathbf{P}}_{k,i}^b$ being
the sample mean and covariance of the ensemble
$\mathbf{X}_{bg}^{k,i}$, respectively.
\item[$\bullet$] {\underline {\it Correction step}:} With an incoming
observation $\mbox{\boldmath$y$}_k$, for each background ensemble
$\mathbf{X}_{bg}^{k,i}$, $i=1,\dotsb,q$, apply an EnKF to obtain the
analysis mean $\hat{\mathbf{x}}_{k,i}^a$ and the analysis ensemble
$\mathbf{X}_{ana}^{k,i}$. During the correction, covariance
inflation and localization (cf. \S~\ref{sec:inflation and
localization}) can be conducted on the EnKF. In addition, update
the associated weights $\tilde{w}_{k-1}^i$ to $w_k^i$ according to
Eq~(\ref{corrp}). After the corrections, the analysis {\it pdf}
becomes
\[
p^a_{k} (\mathbf{x}_k \mid \mbox{\boldmath$y$}_{1:k}) = \sum\limits_{i=1}^{q} w_k^i N
\left(\mathbf{x}_k: \hat{\mathbf{x}}_{k,i}^a, \hat{\mathbf{P}}_{k,i}^a
\right) \, ,
\]
where $w_k^i$ are computed according to Eq.~(\ref{corrp}) in the
context of the EnKF, and $\hat{\mathbf{P}}_{k,i}^a$ are the sample
covariances of $\mathbf{X}_{ana}^{k,i}$.
\item[$\bullet$] {\underline {\it Re-sampling step}:} Use the
criterion in (\ref{eq:resampling_criterion}) to determine whether to
conduct re-sampling or not.
\begin{enumerate}
\item[$(1)$] If there is no need for re-sampling, then assign
$\tilde{p}_{k} (\mathbf{x}_k \mid \mbox{\boldmath$y$}_{1:k}) = p^a_{k} (\mathbf{x}_k
\mid \mbox{\boldmath$y$}_{1:k})$, and $\mathbf{X}_{approx}^{k,i} =
\mathbf{X}_{ana}^{k,i}$ for $i = 1, \dotsb, q$;
\item[$(2)$] Otherwise, $\tilde{p}_{k} (\mathbf{x}_k \mid \mbox{\boldmath$y$}_{1:k}) =
\dfrac{1}{q} \sum\limits_{i=1}^{q} N \left(\mathbf{x}_k:
\mathbf{\theta}_{k,i}, \mathbf{\Phi}_{k} \right)$, where
parameters $\mathbf{\theta}_{k,i}$ and $\mathbf{\Phi}_{k}$ are
computed following the method in \S~\ref{sec:re-sampling}, and the
associated weights become $1/q$. The ensembles
$\mathbf{X}_{approx}^{k,i}$ are produced accordingly.
\end{enumerate}
\end{enumerate}
\section{Numerical Experiments}
\label{experiments}
\subsection{Experiment Design}
\label{exp:design}
In the present work, we focus on two different implementations of the
PEnKF: the first is based on the stochastic EnKF (SEnKF) of
\cite{Evensen-sequential} and the second based on the ensemble
transform Kalman filter (ETKF) of \cite{Bishop-adaptive}. These two
implementations are referred to as the PSEnKF and the PETKF,
respectively.
The strongly nonlinear $40$-dimensional system model due to
\cite{Lorenz-optimal} (LE98 model hereafter) was chosen as the testbed
to evaluate and study the performance of these two filters. This model
mimics the time-evolution of a scalar atmospheric quantity. It is
governed by the following set of equations:
\begin{equation} \label{LE98}
\frac{dx_i}{dt} = \left( x_{i+1} -
x_{i-2} \right) x_{i-1} - x_i + 8, \, i=1, \dotsb, 40,
\end{equation}
where the nonlinear quadratic terms simulate advection and the linear
term represents dissipation. Boundary conditions are cyclic, i.e. we
define $x_{-1}=x_{39}$, $x_{0}=x_{40}$, and $x_{41}=x_{1}$. The model
was numerically integrated using the Runge-Kutta fourth order scheme
from time $t = 0$ to $t = 35$ with a constant time step $\Delta t =
0.05$ (which corresponds to $6$ hours in real time). To eliminate the
impact of transition states, the model trajectory between times $t =
0$ and $t = 25$ was discarded. The assimilation experiments were
carried out during the period $t = 25.05$ to $t = 35$ where the model
trajectory was considered to be the 'truth'. Reference states were
then sampled from the true trajectory and a filter performance is
evaluated by how well it is able to estimate the reference states
using a perturbed model and assimilating a set of (perturbed)
observations that was extracted from the reference states.
In this work we consider two scenarios: one with a linear observation
operator and the other with a nonlinear operator. The concrete forms
of these two observational operators will be given in the relevant
sections below.
The time-averaged root mean squared error (rmse for short) is used to
evaluate the performance of a filter. Given a set of $n$-dimensional
state vectors $\{\mathbf{x}_k: \mathbf{x}_k =
(x_{k,1},\dotsb,x_{k,n})^T, k=0,\dotsb, k_{max} \}$, with $k_{max}$
being the maximum time index ($k_{max}=199$ in our experiments), then
the rmse $\hat{e}$ is defined as
\begin{equation}
\hat{e} = \dfrac{1}{k_{max}+1} \sum\limits_{k=0}^{k_{max}} \sqrt{\dfrac{1}{n} \sum\limits_{i=1}^{n}
(\hat{x}_{k,i}^a - x_{k,i})^2 }\, ,
\end{equation}
where $\hat{\mathbf{x}}_k^a =
(\hat{x}_{k,1}^a,\dotsb,\hat{x}_{k,n}^a)^T$ is the analysis state of
$\mathbf{x}_k$.
A possible problem in directly using $\hat{e}$ as the performance
measure is that $\hat{e}$ itself may depend on some intrinsic
parameters of the filters, for instance, the covariance inflation
factor and localization length scale as to be discussed later. This
may lead to inconsistent conclusions at different parameter values. To
avoid this problem, we adopted the following strategy: we relate a
filter's best possible performance to the minimum rmse
$\hat{e}_{min}$, which is the minimum value of $\hat{e}$ that the
filter can achieve within the chosen ranges of the filter's intrinsic
parameters. In performance comparison, if the minimum rmse
$\hat{e}_{min}^A$ of filter $A$ is less than the minimum rmse
$\hat{e}_{min}^B$ of filter $B$, filter $A$ is said to perform better
than filter $B$.
\subsection{Implementation Details}
\label{exp:ini}
\subsubsection{Filter Initialization}
To initialize the PEnKF, we first estimate the mean and covariance of
the LE98 model over some time interval following
\citet{Hoteit2008}. These statistics are then used to produce the {\it
pdf} $p_0^f(\mathbf{x}_{0})$ of the background at the first
assimilation cycle as a MON.
Concretely, the LE98 model was first integrated for a long period
(between $t=0$ and $t=1000$) starting from an initial state that has
been drawn at random. The trajectory that falls between $t=50.05$ and
$t=1000$ was used to estimate the mean $\hat{\mathbf{x}}_{ds}$ and
covariance $\hat{\mathbf{P}}_{ds}$ of the dynamical system. To
initialize $p_0^f(\mathbf{x}_{0})$ as a mixture of $N$ Gaussian
distributions
\begin{equation}
p_0^f(\mathbf{x}_{0}) = \dfrac{1}{N} \sum\limits_{i=1}^{N} N(\mathbf{x}_{0}:\mathbf{x}_{0}^{f,i} \, , \mathbf{P}_{com}) \, ,
\end{equation}
where $\mathbf{x}_{0}^{f,i}$ are the means, and $\mathbf{P}_{com}$ the
common covariance matrix of the Gaussian distributions in the mixture,
we draw $N$ samples $\mathbf{x}_{0}^{f,i}$ from the Gaussian
distribution $N(\mathbf{x}_{0}:\hat{\mathbf{x}}_{ds},
\hat{\mathbf{P}}_{ds})$, and set $\mathbf{P}_{com} =
\hat{\mathbf{P}}_{ds}$. If $\hat{\mathbf{x}}_{0}^{f} = \dfrac{1}{N}
\sum\limits_{i=1}^{N} \mathbf{x}_{0}^{f,i}$ denotes the sample mean of
$\mathbf{x}_{0}^{f,i}$, then the covariance $\mathbf{P}_{0}^f$ of
$p_0^f(\mathbf{x}_{0})$ is given by
\begin{equation}
\mathbf{P}_{0}^f = \hat{\mathbf{P}}_{ds} + \dfrac{1}{N} \sum\limits_{i=1}^{N} (\mathbf{x}_{0}^{f,i} - \hat{\mathbf{x}}_{0}^{f} )(\mathbf{x}_{0}^{f,i} - \hat{\mathbf{x}}_{0}^{f} )^T \, ,
\end{equation}
which is always larger than $\hat{\mathbf{P}}_{ds}$. The rationale
behind this choice is not far from the covariance inflation technique
\citep{Anderson-Monte,Whitaker-ensemble}. In practice, a data
assimilation system is often subject to various errors, such as poorly
known model and observational errors, sampling errors, etc. In such
circumstances, an inflated background covariance would allocate more
weights to the incoming observation when updating the background to
the analysis, making the filter more robust
\citep{Jazwinski1970,Simon2006}.
\subsubsection{Covariance Inflation and Localization} \label{sec:inflation and localization}
Covariance inflation \citep{Anderson-Monte,Whitaker-ensemble} and
localization \citep{Hamill-distance} are two popular techniques that
are used to improve the stability and performance of the EnKF
\citep{Hamill2009,VanLeeuwen2009}, especially in the small ensemble
scenario. In our experiments, these two techniques are implemented for
each EnKF in the PEnKF.
More concretely, to introduce covariance inflation to the $i$th EnKF
at instant $k$, we multiply the analysis covariance
$\hat{\mathbf{P}}_{k,i}^a$ (before the re-sampling step) by a factor
$(1+\delta)^2$, where the scalar $\delta \geq 0$, called {\it
covariance inflation factor}, is introduced as an intrinsic
parameter of the EnKF. On the other hand, we follow the method in
\citet{Hamill-distance} to conduct covariance localization on the
background covariance and its projection onto the observation space,
with the tapering function (for smoothing out spuriously large values
in covariance matrices) being the fifth order function defined in
Eq.~(4.10) of \citet{Gaspari1999}. In doing so, another intrinsic
scalar parameter $l_c > 0$, called {\it length scale}
\citep{Hamill-distance}, is introduced to the EnKF. Roughly speaking,
$l_c$ is a parameter that determines the critical distance beyond
which the tapering function becomes zero.
\subsection{Experiments Results with a Linear Observation Operator}
\label{exp_linear:results}
In the first scenario, we let the (synthetic) observations be generated every day ($4$ model time
steps) from the reference states using the following linear
observation system
\begin{equation} \label{sim:lin_observer}
\mathbf{y}_k = (x_{k,1},x_{k,3},\dotsb,x_{k,39})^T + \mathbf{v}_k\, ,
\end{equation}
where only the odd state variables $x_{k,i}$ ($i=1,3,\dotsb,39$) of
the system state $\mathbf{x}_k \equiv (x_{k,1},\dotsb,x_{k,40})^T$ at
time index $k$ are observed. The observation noise $\mathbf{v}_k$
follows the $20$-dimensional Gaussian distribution $N(\mathbf{v}_k:
\mathbf{0}, \mathbf{I}_{20})$ with $\mathbf{I}_{20}$ being the $20
\times 20$ identity matrix.
\subsubsection{Effect of the Number of Gaussian Distributions}
\label{sec:lin_exp_c_vs_q}
In the first experiment we examine the effect of the number of
Gaussian distributions on the performance of the PSEnKF and the
PETKF. The experiment settings are as follows.
We initialize the pdf $p_0^f(\mathbf{x}_{0})$ with $N$ Gaussian {\it
pdf}s. In our experiments we let $N$ take values between $1$ and
$60$. Since it is costly to carry out the computation for each integer
in this interval, we choose to let $N$ increase from $1$ to $10$, with
an even increment of $1$ each time, and then increase it from $15$ to
$60$, with a larger increment of $5$ each time, as $N$ becomes
larger. For convenience, we denote this choice by $N \in
\{1:1:10,15:5:60\}$, where the notation $v_{min}:v_{inc}:v_{max}$
represents a set of values increasing from $v_{min}$ to $v_{max}$,
with an even increment of $v_{inc}$ each time. If there is a need to
conduct re-sampling, we re-approximate the analysis MON by a new MON
with equal weights and with the same number of normal distributions.
In doing so, we introduce a new parameter, i.e., the
fraction coefficient $c$ defined in \S~ \ref{sec:re-sampling}, to the
PSEnKF/PETKF. To examine its effect on the performance of the filter,
we let $c \in \{0.05:0.1:0.95\}$. The ensemble size is set to $m=20$
in each SEnKF/ETKF, which is relatively small compared to the system
dimension $40$. In this case, it is customary to conduct covariance
inflation \citep{Anderson-Monte,Whitaker-ensemble} and localization
\citep{Hamill-distance} to improve the robustness and performance of
the filters \citep{Hamill2009,VanLeeuwen2009}. The impacts of
covariance inflation and localization on the performance of the EnKF
have been examined in many works, see, for example,
\citet{Whitaker-ensemble}. In our experiments we let the covariance
inflation factor $\delta=0.02$. We follow the settings in
\citet[\S~7.2.3]{Luo2008-spgsf1} to conduct covariance localization
and choose the length scale $l_c=50$. To reduce statistical
fluctuations, we repeat the experiments for $20$ times, each time with
a randomly drawn initial background ensemble, but the same true
trajectory and the corresponding observations. The same repetition
setting is adopted in all the other experiments.
In Fig.~\ref{fig_lin:GMM_contour_fraction_vs_nGMM_grayColor} we show
the rms errors of both the PSEnKF and PETKF as functions of the
fraction coefficient $c$ and the number $N$ of Gaussian {\it
pdf}s. First, we examine how the rmse of the PSEnKF changes with $c$ when $N$ is fixed. In Fig.~\ref{fig_lin:sEnKF_based_GMM_contour_fraction_vs_nGMM_grayColor}, if $N$ is relatively small (say $N<40$), the rmse tends to decrease as $c$ increases. For larger $N$ (say $N=55$), the rmse of the filter exhibits the bell-shape behavior: at the beginning it increases when $c$ grows from $0$; after $c$ becomes relatively large (say $c=0.4$), further increasing $c$ reduces the rmse instead. Next, we examine the behavior of the rmse of the PSEnKF with respect to $N$ when $c$ is fixed. When $c$ is relatively small (say $c=0.1$), the rmse exhibits the U-turn behavior: at the beginning it intends to decrease as $N$ grows; after $N$ becomes relatively large (say $N=45$), further increasing $N$ increases the rmse instead. When $c$ is larger, say, $c=0.6$, the rmse appears less sensitive to the change of $N$. However, for even larger values of $c$, say, $c=0.9$, the rmse appears to monotonically decrease with $N$.
The behavior of the PETKF (cf. Fig.~\ref{fig_lin:ETKF_based_GMM_contour_fraction_vs_nGMM_grayColor}) with respect to the changes of $N$ and $c$ is similar to that of the PSEnKF. Therefore we do not repeat its description here.
To examine the minimum rms errors $\hat{e}_{min}$ of the PSEnKF and
the PETKF within the tested values of $c$ and $N$, we plot
$\hat{e}_{min}$ of both filters as functions of $N$ in
Fig.~\ref{fig_lin:sEnKF_ETKF_based_GMM_minRMSE_vs_nGMM_grayColor}. The
$\hat{e}_{min}$ of both filters tends to decrease as the number $N$ of
Gaussian distributions increases, though there also exhibit certain local minima. The PSEnKF achieves its lowest $\hat{e}_{min}$ at $N=60$, while the PETKF at $N=50$. As $N$ grows, both the PSEnKF
and the PETKF tend to have lower $\hat{e}_{min}$ than their corresponding base
filters, the SEnKF and the ETKF (corresponding to the PSEnKF and the
PETKF with $N=1$, as discussed in \S~\ref{sec:re-sampling}),
respectively. This confirms the benefit of accuracy improvement by
using the PEnKF instead of an EnKF. A comparison between the PSEnKF
and the PETKF shows that the PETKF performs better than the PSEnKF
when the number $N$ of Gaussian distributions is relatively small
(say, $N \leq 7$). However, as $N$ becomes larger, the PSEnKF
outperforms its ETKF-based counterpart instead. Similar phenomena can
also be observed in other experiments, as to be shown later.
\subsubsection{Effect of the Ensemble Size} \label{sec:lin_exp_n_vs_c}
In the second experiment we examine the effect of the ensemble size of
each SEnKF/ETKF in the PEnKF, on the performance of the
PSEnKF/PETKF. For reference, we also examine the performance of the
SEnKF and the ETKF under various ensemble sizes. The experiment
settings are as follows. For the PSEnKF and the PETKF, we let the
ensemble size $m$ of each EnKF take values from the set
$\{20,40,80,100,200,400,800,1000\}$. For a single SEnKF/ETKF, we
let $m \in \{20,40,60,80,100,200,400,600,800,1000\}$, with two more values at $60$ and $600$.
In the PSEnKF and the PETKF, we also vary the fraction coefficient $c$
such that $c \in \{0.05:0.1:0.95\}$. We fix the number $N$ of Gaussian
{\it pdf}s, i.e., the number of ensemble filters, to be $3$. To
conduct covariance inflation, we let the inflation factor
$\delta=0.02$. We choose to conduct covariance localization, and set
the length scale $l_c=50$, only if the ensemble size $m$ is not larger
than the dimension $40$ of the LE98 model. No covariance localization
was conducted if $m>40$. Our experience shows that, for $m>40$, the
benefit of conducting localization is not significant even if the
length scale $l_c$ is properly chosen, while an improper value of
$l_c$ is more likely to deteriorate the filter performance. To reduce
statistical fluctuations, the experiments are again repeated for $20$
times.
In Fig.~\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor} we show the rms
errors of the SEnKF and the ETKF as functions of the ensemble size
$m$. The rmse of the ETKF exhibits a U-turn behavior. The rmse of the
ETKF monotonically decreases as long as $m < 100$. Beyond that, the
rmse monotonically increases instead as $m$ increases. On the other
hand, the SEnKF exhibits a different behavior. Its rmse decreases for
$m \leq 200$, and then reaches a plateau where the rmse remains almost
unchanged as $m$ further increases.
Fig.~\ref{fig_lin:GMM_contour_fraction_vs_nEn_grayColor} plots the rms
errors of the PSEnKF and the PETKF as functions of the fraction
coefficient $c$, and the ensemble size $m$ in the SEnKF and the ETKF
used to construct the corresponding PEnKFs. The rms errors, as
functions of the ensemble size $m$ (with fixed $c$), are consistent
with our observations in
Fig.~\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor}. On the other
hand, for both PEnKFs, their rms errors tend to decrease as the
fraction coefficient $c$ increases.
Per analogy to the first experiment,
Fig.~\ref{fig_lin:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} plots
the minimum rms errors $\hat{e}_{min}$ of the PSEnKF and the PETKF
within the tested fraction coefficient $c$ and the ensemble size
$m$. A comparison between
Figs.~\ref{fig_lin:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} and
\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor} shows that, the minimum
rms errors $\hat{e}_{min}$ of the PEnKFs behave very similarly to the
rms errors of their corresponding EnKFs in
Fig.~\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor}. Moreover, the values of $\hat{e}_{min}$ in
Fig.~\ref{fig_lin:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} tends to be
lower than the corresponding rms errors in
Fig.~\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor}, indicating the
benefit of accuracy improvement in using the PEnKFs. Again, a
comparison between the PSEnKF and the PETKF shows that the PETKF
performs better than the PSEnKF when the ensemble size $m$ is
relatively small (say, $m \leq 40$). However, as $m$ becomes larger,
the PSEnKF outperforms the PETKF instead.
\subsection{Experiments Results with a Nonlinear Observation Operator}
\label{exp_nln:results}
In the second scenario, we introduce nonlinearity to the
observation system. To this end, we let the observations be generated
by the following nonlinear process
\begin{equation} \label{sim:nln_observer}
\mathbf{y}_k = 0.05(x_{k,1}^2,\dotsb,x_{k,39}^2)^T + \mathbf{v}_k\,
\end{equation}
for every $4$ model time steps. In Eq.~(\ref{sim:nln_observer}), again
only the odd state variables $x_{k,i}$ ($i=1,3,\dotsb,39$) of the
system state $\mathbf{x}_k \equiv (x_{k,1},\dotsb,x_{k,40})^T$ at time
index $k$ are observed. The observation noise $\mathbf{v}_k$ also
follows the $20$-dimensional Gaussian distribution
$N(\mathbf{v}_k:\mathbf{0}, \mathbf{I}_{20})$. We conduct the same
experiments as those in the case of linear observation operator.
\subsubsection{Effect of the Number of Gaussian Distributions}
\label{sec:nin_exp_c_vs_q}
We first examine the effect of the number of Gaussian
distributions. The experiment settings are the same as those in
\S~\ref{sec:lin_exp_c_vs_q}. Concretely, For either the PSEnKF or the
PETKF, the number of Gaussian distributions $N \in
\{1:1:10,15:5:60\}$, the fraction coefficient $c \in
\{0.05:0.1:0.95\}$. For each individual SEnKF/ETKF in the PEnKF, the
ensemble size $m=20$, the covariance inflation factor $\delta=0.02$
and the length scale $l_c=50$ for covariance localization. As before,
the experiments are repeated for $20$ times to reduce statistical
fluctuations.
Fig.~\ref{fig:GMM_contour_fraction_vs_nGMM_grayColor} plots the rms
errors of both the PSEnKF and the PETKF as functions of the fraction
coefficient $c$ and the number $N$ of Gaussian {\it pdf}s. When $c$ and $N$ changes, both the PSEnKF and the PETKF behave very similar to their counterparts in the linear case. The rms errors of the filters tend to decrease as $N$ increases, meaning that the PSEnKF/PETKF with $N>1$ in general performs better than the stochastic EnKF /ETKF
(corresponding to the case $N=1$ in the PEnKF), consistent with the results obtained in the linear observer case.
We also examine the minimum rms errors $\hat{e}_{min}$ of the PSEnKF
and the PETKF within the tested values of $c$ and
$N$. Fig.~\ref{fig:sEnKF_ETKF_based_GMM_minRMSE_vs_nGMM_grayColor}
plots $\hat{e}_{min}$ as functions of $N$. For the PSEnKF, the lowest
$\hat{e}_{min}$ is achieved at $N=50$.
And for the PETKF, its $\hat{e}_{min}$ tends to decrease within
the tested range of $N$, and achieves its minimum at $N=60$. The PEnKF with more than
one Gaussian distributions ($N>1$) performs better than the
corresponding EnKF ($N=1$). In addition, a comparison between the
PSEnKF and the PETKF shows again that the PETKF performs better than
the PSEnKF when the number $N$ of Gaussian distributions is relatively
small, but tends to become worse as $N$ increases.
A comparison between Figs.~\ref{fig_lin:sEnKF_ETKF_based_GMM_minRMSE_vs_nGMM_grayColor} and \ref{fig:sEnKF_ETKF_based_GMM_minRMSE_vs_nGMM_grayColor} shows that the rmse of a filter (e.g. the PSEnKF at $N=2$) with a nonlinear observer sometimes may be lower than that of the same filter with a linear observer \footnote{The result of comparison would also depend on the filter in use, its configuration, the system in assimilation, and so on, and therefore may change from case to case.}. This seemingly counter-intuitive result happens possibly because in such situations, the effect of sampling error due to the relatively small ensemble size dominates the effect of nonlinearity in the observation system. However, as the number $N$ of Gaussian distributions increases, the effect of nonlinearity becomes more prominent so that the rmse with a nonlinear observer tends to be higher than that with a linear one. Similar phenomenon can also be found by comparing Figs.~\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor} and \ref{fig_lin:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} with Figs.~\ref{fig:sEnKF_ETKF_RMSE_vs_nEn_grayColor} and \ref{fig:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} (to be shown below), respectively, at different ensemble sizes.
\subsubsection{Effect of the Ensemble Size}
In the second experiment we examine the effect of the ensemble size in
each ensemble filter on the performance of the corresponding
PEnKF. For reference, we also examine the performance of the SEnKF and
the ETKF under various ensemble sizes. The experiment settings are the
same as those in \S~\ref{sec:lin_exp_n_vs_c}. In the PSEnKF and PETKF,
we choose the fraction coefficient $c \in \{0.05:0.1:0.95\}$. We also
choose the number of ensemble filters in each PEnKF to be $3$. For
each individual EnKF in the corresponding PEnKF, we let the ensemble
size $m$ take values from the set
$\{20,40,80,100,200,400,800,1000\}$, and for the experiments on the
single EnKF, we let $m \in
\{20,40,60,80,100,200,400,600,800,1000\}$. To conduct covariance
inflation and localization in each individual EnKF, we choose the
inflation factor $\delta=0.02$, and the length scale $l_c=50$. As in
\S~\ref{sec:lin_exp_n_vs_c}, covariance localization is conducted only
if the ensemble size $m$ is no larger than the dimension $40$.
Fig.~\ref{fig:sEnKF_ETKF_RMSE_vs_nEn_grayColor} shows the rms errors
of the SEnKF and the ETKF as functions of the ensemble size $m$. For
both filters, their rms errors decrease as the ensemble size $m$
increases. The ETKF performs better than the SEnKF in the small sample
scenario with $m=20$. But as $m$ increases, the SEnKF outperforms the
ETKF instead. In particular, divergence in the ETKF occurs if $m>400$,
which did not happen in the linear observer case
(cf. Fig.~\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor}). On the
other hand, the rmse of the SEnKF appears to reach a plateau for
$m>400$, similar to the linear observer case. Comparing
Fig.~\ref{fig:sEnKF_ETKF_RMSE_vs_nEn_grayColor} with
Fig.~\ref{fig_lin:sEnKF_ETKF_RMSE_vs_nEn_grayColor}, it is easy to see
that, except for the stochastic EnKF at $m=20$, the presence of nonlinearity in the observer
deteriorates the performance of the ensemble filters.
Fig.~\ref{fig:GMM_contour_fraction_vs_nEn_grayColor} plots the rms
errors of the PSEnKF and the PETKF as functions of the fraction
coefficient $c$, and the ensemble size $m$ in the corresponding SEnKF
and the ETKF, respectively. In the PSEnKF
(cf. Fig.~\ref{sEnKF_based_GMM_contour_fraction_vs_nEn_grayColor}),
the rmse tends to decrease as both $c$ and $m$ increases when the
ensemble size $m \leq 800$. However, when $m>800$, the impact of $m$
on the filter performance is not significant, which is consistent with
the results in Fig.~\ref{fig:sEnKF_ETKF_RMSE_vs_nEn_grayColor}. On the
other hand, in the PETKF
(cf. Fig.~\ref{fig:ETKF_based_GMM_contour_fraction_vs_nEn_grayColor}),
filter divergence occurs for $m>200$, which is why we only report its
rmse with $m \leq 200$ in
Fig.~\ref{fig:ETKF_based_GMM_contour_fraction_vs_nEn_grayColor}, where
the rmse of the PETKF appears to be a monotonically decreasing
function of $m$ and $c$.
In analogy to the first experiment,
Fig.~\ref{fig:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} plots the
minimum rms errors $\hat{e}_{min}$ of the PSEnKF and the PETKF within
the tested fraction coefficient $c$ and ensemble size $m$. One may
observe that, similar to the SEnKF and the ETKF themselves, the
$\hat{e}_{min}$ of both the PSEnKF and the PETKF decrease as $m$
increases. However, for the PETKF, divergence occurs if $m>200$,
rather than $m>400$ as in
Fig.~\ref{fig:sEnKF_ETKF_RMSE_vs_nEn_grayColor}, but overall its rmse
is closer to that obtained in the PSEnKF. Meanwhile, a comparison
between Fig.~\ref{fig:sEnKF_ETKF_RMSE_vs_nEn_grayColor} and
Fig.~\ref{fig:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} shows that the PEnKFs perform better than the corresponding EnKFs. Also, a comparison between Fig. \ref{fig_lin:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} and \ref{fig:sEnKF_ETKF_based_GMM_minRMSE_vs_nEn_grayColor} shows that, except for the PSEnKF at $m=20$, the nonlinearity in the observer again deteriorates the performance of the ensemble filters.
\section{Discussion}
\label{discussion}
This paper presented a discrete solution of the optimal nonlinear
filter, called the particle Kalman filter (PKF), based on the Gaussian
mixture representation of the state {\it pdf} given the
observations. The PKF solves the nonlinear Bayesian correction step by
complementing the Kalman filter-like correction step of the particles
with a particle filter-like correction step of the weights. The PKF
simultaneously runs a weighted ensemble of the Kalman filters in
parallel. This is far beyond our computing capabilities when dealing
with computationally demanding systems, as the atmospheric and oceanic
models. Therefore, to reduce computational cost, one may instead
consider a low-rank parametrization of the Gaussian mixture covariance
matrices of the state {\it pdf}s. An efficient way to do that is to
resort to the ensemble Kalman filter (EnKF) and use an EnKF-like
method to update each component of the Gaussian mixture {\it
pdf}s. This amounts to running a weighted ensemble of the EnKFs. In
this work, the PKF was implemented using the stochastic EnKF and a
deterministic EnKF, the ensemble transform Kalman filter (ETKF). We
call this type of implementation the particle ensemble Kalman filter
(PEnKF).
The PEnKF sets a nonlinear Bayesian filtering framework that
encompasses the EnKF methods as a special case. As in the EnKF, the
Kalman correction in the PEnKF attenuates the degeneracy of the
ensemble by allocating the ensemble members far away from the incoming
observation relatively more weights than in the particle filter, so
that the filter can operate with reasonable size ensembles. To further
improve the performance of the PEnKF, we also introduced to the PEnKF
a re-sampling step similar to that used in the regularized particle
filter \citep{Musso-improving-2001,Stavropoulos-improved-2001}.
The stochastic EnKF and ETKF-based PEnKFs, called the PSEnKF and the
PETKF, respectively, were implemented and their performance was
investigated with the strongly nonlinear Lorenz-96 model. These
filters were tested with both linear and nonlinear observation
operators. Experiments results suggest that the PSEnKF and the PETKF
outperform their corresponding EnKFs. It was also found that the ETKF
outperforms the stochastic EnKF for small size ensembles while the
stochastic EnKF exhibits better performance for large size
ensembles. We argued that this happens because the EnKF endures less
observational sampling errors when the ensemble size is large. Another
reason would also be the better approximation of the PEnKF
distributions provided by the stochastic EnKF compared to the
ETKF. This was also true for their PEnKF counterparts. Overall, the
conclusions from the numerical results obtained with the linear and
nonlinear observation operators were not fundamentally different,
except that in general better estimation accuracy was achieved with the linear
observer when the sampling error is not the dominant factor.
The results also suggest that the PEnKFs could more benefit from the use of more components in the
mixture of normals (MON) and larger ensembles in the EnKFs in the
nonlinear observations case.
Future work will focus on developing and testing new variants of the
PEnKF that applies more efficient approximations, in term of
computational cost, to update the mixture covariance matrices. Another
direction for improvement would be also to work on localizing the
correction step of the particle weights \citep{VanLeeuwen2009}. Our
final goal is to develop a set of computationally feasible suboptimal
PEnKFs that can outperform the EnKF methods at reasonable
computational cost. As stated by \citet{Anderson2003-local},
developing filters in the context of the optimal nonlinear filtering
problem, rather than starting from the Kalman filter, should lead to a
more straightforward understanding of their capabilities.
The paper further discussed how the PEnKF can also be used as a
general framework to simultaneously run several assimilation
systems. We believe that this approach provides a framework to merge
the solutions of different EnKFs, or to develop hybrid
EnKF-variational methods. Work in this direction is under
investigation.
\section*{Acknowledge}
We would like to thank the three anonymous reviewers for their
valuable comments and suggestions. Ibrahim Hoteit was partially
supported by ONR grant N00014-08-1-0554.
\bibliographystyle{ametsoc}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,048 |
The common name "groundfish" refers to any fish that lives near or on the floor of the ocean.
The fishery includes some well-known fish like flounder, halibut, and sole – along with more than 90 different species of flatfish, rockfish, and roundfish. Groundfish can be categorized as midwater, deep slope, shelf, slope rockfish, nearshore rockfish, and nearshore flatfish. On the West Coast, some of the better-known species are Dover Sole, English Sole, Petrale Sole, Pacific Sanddab, Starry Flounder, various Rockfish, Longspine and Shortspine Thornyhead, Lingcod, and Longnose Skate.
Near the ocean floor, groundfish live in varying habitats, some rich with coral and grasses, and others on sandy bottom. For the fishermen in the Fort Bragg Groundfish Assocation, over 95% of their fish are caught from sandy bottom. Groundfish typically feed on a diet of krill, shrimp, smaller fish, and other ocean invertebrates. To adapt to their habitats, many species have developed unique characteristics – sometimes strange or surprising! For example, many flatfish have downward- or upward-facing mouths, depending on the usual direction that they catch their food. The flat-shaped halibut has a "migrating" eye that slowly, as the fish matures, moves from the bottom to top of the body, so both eyes eventually look upward. All species have certain features – shape, color, texture – that help them to camouflage along the ocean's floor.
On the West Coast, groundfish have been part of the human diet since the earliest natives inhabited this area.
Over the years, groundfish have been caught with trawl, troll, longline, hook and line, pots, gillnets, and other gear. The commercial market for groundfish was first established by settlers in the early to mid-1800s. This market shifted and grew over the 1900s, as economic markets and fishing policies shifted. After significant growth, the groundfish fishery hit "bottom" at the end of the 20th century, when a combination of longterm factors led to an overfished and heavily impacted fish population. Since then, enormous efforts have been made towards environmental sustainability. Today, the groundfish fishery has evolved into a sustainability success story, as it continues a remarkable recovery from its near collapse in 2000.
In 2014, the Monterey Bay Aquarium's Seafood Watch program rated 21 species as "Best Choice (green)" or "Good alternative (yellow)" – a significant move for many species that had been labeled "Avoid (red)" just years earlier. The fishery's newfound success is thanks to collaboration at the federal, state, and regional levels – as well as the hard work and dedication of forward-thinking fishermen. All are invested in the longterm sustainability and availability of groundfish for years to come.
Today, groundfish are a delicious and abundant, if too often overlooked, option at the local fish market.
Available in great abundance, groundfish are a top-quality resource – both in flavor and nutrition. Like many fish, they are a great value for a lean, healthy source of protein. Their mild flavor and flaky texture are a favorite of both professional chefs and home cooks. The ongoing consumers helps the fishery continue to make its economic recovery in tandem with its environmental work. On the West Coast, many seafood lovers and environmentalists alike are coming out in support and celebration of the groundfish.
Want to learn more? Read more details about our seafood. | {
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Abner ist im Alten Testament der Heerführer König Sauls. Im 1. und 2. Buch Samuel wird von ihm berichtet.
Etymologie
Der hebräische Personenname "Abner" ist ein Nominalsatzname, bestehend aus Subjekt ud Prädikat. Subjekt (und zugleich theophores Element) ist der Gottesname , Prädikat ist das Substantiv . Der Name lässt sich daher als "Vater ist Leuchte" übersetzen. Eine Nebenform des Namens mit identischer Bedeutung, wie sie nur in vorkommt, lautet . Die Septuaginta gibt Name mit Αβεννηρ abennēr wieder, die Vulgata mit Abner.
Biblische Erzählung
In der biblischen Erzählung unterstützte Abner nach dem Tod Sauls bei der Schlacht von Gilboa den jüngsten und einzig übriggebliebenen Sohn Sauls, Isch-Boschet, bei seinen Thronbestrebungen gegen König David. Nach der sich abzeichnenden Niederlage von Isch-Boschets Kräften lief Abner zu David über. Er wurde aber von Joab, dem Heerführer Davids, ermordet, da er zuvor dessen Bruder Asaël in einer Schlacht getötet hatte. Abner wurde in Hebron beerdigt. Sein Grab ist bis heute dort erhalten.
Literatur
Weblinks
Bilder von Abners Grab in Hebron
Einzelnachweise
Person im 1. Buch Samuel
Person im 2. Buch Samuel
Militärperson (jüdisches Altertum)
Saul
ca:Llista de personatges bíblics#Abner | {
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{"url":"http:\/\/scylla.ceas.uwm.edu\/465\/summary\/new\/sum_mar_13.html","text":"Optical filters II Mar. 13, 2018\n\n\u2022 Acousto-optics tunable filters (pp. 143-146 2nd Ed, pp. 149-152 3rd Ed): Use RF signal to induce acoustic oscillation -> Use RF signal to induce generate a Bragg grating that diffracts light.\nAcoustic wave parameters - freq f_a, velocity v_a and wavelength Lambda = v_a \/f_a\nRF signal induces strain s(x,t) = S_o cos ( Omega t - qx) where q = (2pi)\/Lambda and Omega = 2 pi f_a\nacousto-optical effect causes refractive index change Delta n = -(p n^3)\/2 s(x,t) where p is the photoelastic constant.\nSelect wavelength lambda = Delta n Lambda\nResponse time D \/ v_a where D is the optical beam diameter.\nOperation - Use birefringent material to make a polarizing beam splitter. The resulting TE wave will be converted into TM wave if grating condition lambda = |n_(TE) - n_(TM) | Lambda = Delta n Lambda is satisfied. The polarizing beam splitter selects only TM waves.\nFreq response T(lambda) = (sin^2 (pi\/2 sqrt{1+X^2}))\/(1+X^2) where X = (2 Delta lambda)\/(Delta_(AO)), Delta lambda = lambda - lambda_B and Delta_(AO) = lambda_B^2\/(D Delta n)\n\u2022 Mach-Zehnder (MZ) filter (pp. 135-139 2nd Ed, pp. 141-145 3rd Ed): Similar to Mach-Zehnder interferometer; replace beam splitter with directional coupler.\nDirectional coupler is a 2 ports device (pp. 108-112 2nd Ed, pp. 114-118 3rd Ed).\nIts transfer function for straight thru (e.g. input 1 -> output 1) is T_{11} = cos^2 ( kappa l) and for cross over (e.g. input 1 -> output 2) is T_{12} = sin^2 (kappa l) where kappa is the coupling coefficient and l is the coupling length..\nBy choosing appropriate l , we can construct a 3dB coupler (i.e. 50%\/50% splitter).\nFor 3dB coupler, signal from each input port is split across 2 output ports. Phase shift of pi\/2 occurs when signal crosses ports, e.g. input port 1 to output port 2 or input port 2 to output port 1.\nTwo input ports separate freq depending on the phase difference between the two paths for input signals, e.g. signal at output port 1 satisfies Delta phi_1 = (2 m -1 ) pi for lambda_1 ;\nsignal at output port 2 satisfies Delta phi_2 = 2 m pi for lambda_2\nChannel frequency spacing f_2 - f_1 = Delta f = c \/ (2n Delta L)\nTunable output freq by RF, thermal or mechanical controls.\nCascading MZ to get more than 2 outputs.","date":"2018-07-22 16:10:22","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.784955620765686, \"perplexity\": 6852.625088529664}, \"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-2018-30\/segments\/1531676593378.85\/warc\/CC-MAIN-20180722155052-20180722175052-00285.warc.gz\"}"} | null | null |
JEDBURGH'S FIRST AND LONGEST ESTABLISHED FIVE STARS BED AND BREAKFAST IN THE SCOTTISH BORDERS. AWARDED VISIT SCOTLAND'S PRESTIGIOUS THISTLE AWARD IN 2005 AND THEIR GOLD AWARD IN SEPTEMBER 2009.
The School House Bed and Breakfast were the winners of Visit Scotland's Thistle Award for Customer Care in 2005 and a Finalist in 2004. These awards are open to all members of VisitScotland's Quality Assurance scheme and, no matter how large or small the business, it's the size of the welcome that counts!
The School House at Edgerston lies in the heart of the historic Scottish Borders. Built in 1845 by the owners of the Edgerston estate the school served the local community until 1945. This listed historic structure has now been carefully modified and tastefully furnished to enhance the period features of the building and to create a very high standard of comfortable and friendly accommodation for up to four people.
Each room has a unique theme and has been furnished with antique and period furnishings. The bedroom on the ground floor has a double bed and an ensuite shower room. The bedroom on the first floor has twin beds and a private bathroom. The guest lounge has an open fire, TV, video, library, maps and touring information.
Breakfasts are served in our farmhouse style kitchen and for those taking dinner there is our romantic minstrel gallery dining room.
Guests are offered a superb range of traditional Scottish cuisine such as local venison, salmon or Border lamb.
Catering for any special diet is also available by prior arrangement.
Guests can also relax and enjoy the rural setting from one of the seated areas in the large garden or take one of the walks in the woods and farmlands.
There are many towns and places of historical and scenic interest in easy reach of Edgerston. We can help you plan your tours or activities in the Borders and can provide you with any detailed maps.
Golf, fishing, riding, cycling and hiking is available by prior arrangement.
**Please note unfortunately we are unable to accept payment by credit card.
The School House B&B at Edgerston The School House B&B at Edgerston Twin Room, upstairs, with private bathroom Breakfasts are served in our country style kitchen Double Room, downstairs, with en-suite shower room Rockery and water feature Dining Area in our minstrel gallery Overnight use of our garage is appreciated by bikers & cyclists. | {
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\section{Introduction}
Exclusive $\bar{B}\to D^{(*)} \ell \bar\nu$ decays have become precision probes
of the semileptonic parton-level transitions $b\to c \ell \bar\nu$. As such,
they provide excellent means for the determination of the corresponding
Cabibbo-Kobayashi-Maskawa (CKM) matrix element $|V_{cb}|$ of the Standard
Model~(SM). The combination of good experimental and theoretical control renders
them also sensitive probes of beyond-the-SM (BSM) physics that potentially
modifies both the normalization and the angular distribution of these modes.
In the SM, the lepton-flavour universal (LFU) nature of the underlying
$W^\pm$-boson exchange allows for precision predictions of LFU ratios that are
almost free of hadronic uncertainties. Measurements of the three different
lepton modes $\ell=e, \mu, \tau$ then allow to test SM paradigms such as
CKM unitarity and LFU.
Improved LFU tests are especially important in light of the recent
indications for LFU violation in the so-called \emph{B anomalies}, concerning
$b\to c\tau\bar \nu$ and $b\to s\ell^+\ell^-$ ($\ell=e,\mu$) transitions.
Further motivation for precision analyses of $\bar{B}\to D^*\ell\bar\nu$
decays is provided by the persisting $V_{cb}$ puzzle, \textit{i.e.} a tension between the
inclusive and exclusive determinations of this CKM element.\\[-0.3cm]
This work is triggered by three recent developments:
\begin{description}
\item[Availability of experimental data] Starting with the 2015 analysis of
$\bar B\to D\ell\bar\nu$ decays by Belle~\cite{Glattauer:2015teq},
experimental collaborations made their data on $b\to c\ell\bar\nu$ transitions
available in a model-independent way \cite{Glattauer:2015teq,Abdesselam:2017kjf,
Waheed:2018djm,Aaij:2020hsi,Aaij:2020xjy}, thereby making phenomenological
analyses possible that vary the form-factor parametrizations and BSM scenarios.
In particular, a recent Belle analysis \cite{Waheed:2018djm} presents for the
first time four single-differential distributions of $\bar B\to D^*\ell\bar\nu$
decays for both $\ell=e,\mu$ including their full correlation matrices.
\item[Improved form-factor determinations] There has been significant progress
in the theoretical determination of hadronic $\bar B\to D^{(*)}$ form factors,
both from lattice QCD computations \cite{Lattice:2015rga,Na:2015kha,
Harrison:2017fmw,Bailey:2014tva} and from light-cone sum rules
\cite{Gubernari:2018wyi}. These determinations allow for precise predictions
of the complete set of form factors in $\bar B\to D^*\ell\bar\nu$ in the
whole phase space~\cite{Bordone:2019guc, Bordone:2019vic}. These predictions
are using the heavy-quark expansion and account for contributions up to and
including $\order{1/m_c^2}$. They are a prerequisite for a general BSM
analysis of these modes.
\item[Impending progress in experimental and theoretical precision]
Both the experimental and the theoretical precision are expected to improve
significantly: the ongoing Belle II and LHCb upgrade experiments are bound
to deliver $\bar{B}\to D^{(*)}\ell\bar\nu$ results based on multiples of the
current datasets~\cite{Kou:2018nap, Cerri:2018ypt, Bediaga:2018lhg},
and updated lattice QCD results for several $\bar{B}\to D^*$ form factors
beyond zero recoil are upcoming~\cite{Kaneko:2019vkx, Vaquero:2019ary,
Bhattacharya:2020xyb}, see also the discussions in Refs.~\cite{Lehner:2019wvv,
Gambino:2020jvv}. This renders the discussion of presently negligible
effects important for the full phenomenological exploitation of the
upcoming experimental and theoretical results.
\end{description}
The discussions resulting from the first two items significantly improve our
understanding of these modes, and their sensitivity to the adopted form-factor
parametrization. Recent phenomenological analyses have also shown that the
$V_{cb}$ puzzle is significantly reduced, albeit not yet fully resolved
\cite{Bigi:2016mdz, Bigi:2017njr, Bernlochner:2017xyx, Bigi:2017jbd,
Bernlochner:2017jka, Gambino:2019sif, Bordone:2019guc, Bordone:2019vic,
Bernlochner:2019ldg, Jaiswal:2020wer}. We pose the following questions that
affect existing and future angular analyses of $\bar B\to D^*\ell\bar\nu$ data:
\begin{enumerate}
\item What is the amount of LFU violation in the SM induced by the muon mass? Is the muon mass still negligible given the achieved experimental and theoretical precision?
\item What amount of information can be extracted from the available
single-differential distributions in comparison to a fully-differential
angular analysis of $\bar B\to D^*\ell\bar\nu$? Is it possible to increase
the sensitivity to BSM physics with available data by modifying the
analysis strategy?
\item What are the limits on BSM physics from existing
$\bar B\to D^*\ell\bar\nu$ data? Which effective operators could resolve
a potential tension with the SM and what would be their implications
on so far unmeasured observables?
\end{enumerate}
In order to answer these questions, we proceed as follows: We begin by describing
the general properties of the $\bar{B}\to D^* \ell\bar\nu$ angular distribution
and the BSM physics reach of the angular observables arising from this distribution
in \refsec{fit-model}. In \refsec{data} we prepare a full angular analysis on
the basis of the Belle data published in Ref.~\cite{Waheed:2018djm}. In doing
so, we identify two obstacles to the full use of these data. In \refsec{fit} we
carry out a fit of the full angular distribution to the Belle data, and discuss
the compatibility with SM predictions. In light of an observed tension, we
further discuss possible BSM interpretations of our results.
We conclude in \refsec{conclusions}.
\section{Full angular distribution and its BSM reach}
\label{sec:fit-model}
The four-fold differential distribution of $\bar B\to D^{*}
(\to D\pi) \ell \bar\nu$ decays constitutes a powerful tool for assessing SM
as well as BSM physics. It is given as
\begin{align}\label{eq:d4Gamma}
\frac{d^4 \Gamma^{(\ell)}}{d q^2\, d\!\cos{\theta_\ell}\, d\!\cos{\theta_{D}}\, d\chi} &
= \frac{3}{8 \pi} \sum_i J_i^{(\ell)}(q^2) \;
f_i(\cos{\theta_\ell},\, \cos{\theta_{D}},\, \chi) \,.
\end{align}
Assuming a purely P-wave $D\pi$ final state, this distribution is fully
described by twelve angular observables $J_i^{(\ell)}$ and their respective
angular coefficient functions $f_i$. The dependence of the functions $f_i$
on the three angles $\cos{\theta_\ell}$, $\cos{\theta_{D}}$ and~$\chi$, given in
\refeq{ang-dist} in \refapp{AngularDistribution}, is
lepton-flavour universal and completely determined by conservation of
angular momentum.\\[-0.3cm]
The angular observables $J_i^{(\ell)}$ depend on the momentum
transfer $q^2$, or equivalently the hadronic recoil $w$. Their calculation
involves the lepton-flavour-universal hadronic form factors, as well as the
short-distance coefficients of the low-energy effective theory. The latter
encode short-distance SM effects (which are again lepton-flavour universal)
as well as potential BSM effects (which are in general non-universal). These
dependencies are listed in \reftab{J_i-C_a}. Additional sources of
lepton-flavour non-universality are known kinematic phase-space effects
$\sim m_\ell/\sqrt{q^2}$, which are most pronounced for $\ell = \tau$.
Under the assumption that the short-distance behaviour corresponds to the SM
expectation, the angular observables $J_i^{(\ell)}$ can be used to extract
information on the hadronic form factors. When lifting this assumption in
BSM scenarios, the BSM short-distance coefficients cannot be fully disentangled
from the form factors, making theory input for
the $q^2$-dependence of the form factors and their ratios
indispensable. Below
we discuss the necessary amount of experimental information on the angular
observables $J_i^{(\ell)}$ for a reliable determination of BSM contributions.
Details on the definitions of the angular observables are given in
\refapp{AngularDistribution}.\\
The complete dependence of the angular distribution on BSM contributions
in terms of the BSM couplings has been given for the first time in
Ref.~\cite{Duraisamy:2014sna}, see also Ref.~\cite{Ivanov:2016qtw},
with previous partial results throughout the literature \cite{Tanaka:1994ay,
Biancofiore:2013ki,Duraisamy:2013pia,
Fajfer:2012vx,
Tanaka:2012nw,
Korner:1989qb,Hagiwara:1989gza}.
We use the conventions/notation provided in \refapp{AngularDistribution}.
The sensitivity to various BSM couplings and lepton-mass effects have been
studied in detail \cite{Alguero:2020ukk} based on helicity amplitudes.\\[-0.3cm]
Here we would like to address properties that are not mentioned previously,
or that are particularly important for our work.
An important observation in charged-current semileptonic decays is that to
extremely good approximation no CP-conserving scattering phases appear in the
$J_i^{(\ell)}$.\footnote{%
Such CP-conserving phases are strongly suppressed in $\bar{B}\to D^* \ell \bar\nu$
and can arise, \emph{e.g.} at the level of dimension eight in the
low-energy EFT or due to radiative QED corrections.
}
This simplifies their properties under CP conjugation, rendering them simply
even (for $i\in\{1c,1s,2c,2s,3,4,5,6c,6s\}$) or odd (for $i\in\{7,8,9\}$).
As a consequence, the numerators in the combinations
\begin{align}
\label{eq:def-S_i-A_i}
\braket{S_i^{(\ell)}} &
\equiv \frac{\braket{J_i^{(\ell)}} + \braket{\bar{J}_i^{(\ell)}}}
{\Gamma^{(\ell)} + \bar{\Gamma}^{(\ell)}} \,,
&
\braket{A_i^{(\ell)}} &
\equiv \frac{\braket{J_i^{(\ell)}} - \braket{\bar{J}_i^{(\ell)}}}
{\Gamma^{(\ell)} + \bar{\Gamma}^{(\ell)}} \,,
\end{align}
either vanish or are given by $2\langle J_i^{(\ell)}\rangle$. Here the notation
$\braket{\ldots}$ denotes integration over the full range of the dilepton-invariant
mass as defined in \refeq{def-braket}.
The experimental determination of the fully differential rate is rather
involved. Many analyses therefore present only results for the partially
or fully integrated rate, typically CP-averaged. Doing so simplifies the
experimental analysis, but the sensitivity to some of the angular observables
is lost, which can render the determination of some parameters of interest
impossible. The two recent Belle analyses for instance \cite{Abdesselam:2017kjf,
Waheed:2018djm} provide binned CP-averaged measurements of the four
single-differential distributions
\begin{align}
\label{eq:dG:dw}
\frac{d \widehat{\Gamma}^{(\ell)}}{dw} & \equiv
\frac{1}{2} \frac{d(\Gamma^{(\ell)} + \bar{\Gamma}^{(\ell)})}{dw} \,,
\\[0.2cm]
\label{eq:dG:dcosthL}
\frac{1}{\widehat{\Gamma}^{(\ell)}}
\frac{d \widehat{\Gamma}^{(\ell)}}{d\!\cos{\theta_\ell}} & =
\frac{1}{2}
+ \braket{A_\text{FB}^{(\ell)}} \cos{\theta_\ell}
+ \frac{1}{4} \left(1 - 3 \braket{\wT{F}^{(\ell)}_L} \right)
\frac{3 \cos^2 {\theta_\ell} - 1}{2} \,,
\\[0.2cm]
\label{eq:dG:dcosthD}
\frac{1}{\widehat{\Gamma}^{(\ell)}}
\frac{d \widehat{\Gamma}^{(\ell)}}{d\!\cos{\theta_{D}}} & =
\frac{3}{4} \left(1 - \braket{F_L^{(\ell)}}\right) \sin^2\!{\theta_{D}}
+ \frac{3}{2} \braket{F_L^{(\ell)}} \cos^2\!{\theta_{D}} \,,
\\[0.2cm]
\label{eq:dG:dchi}
\frac{1}{\widehat{\Gamma}^{(\ell)}}
\frac{d \widehat{\Gamma}^{(\ell)}}{d\chi} & =
\frac{1}{2\pi}
+ \frac{2}{3\pi} \braket{S_3^{(\ell)}} \cos 2\chi
+ \frac{2}{3\pi} \braket{S_9^{(\ell)}} \sin 2\chi \,,
\end{align}
where $\widehat{\Gamma}^{(\ell)}$ denotes the CP-averaged decay rate.
In particular, in Ref.~\cite{Waheed:2018djm} the authors separate the data
by the light lepton flavours $\ell=e$ and $\ell=\mu$. The three CP-averaged
single-angular distributions depend on only five out of the twelve angular
observables defined in \refeq{d4Gamma}. Out of these five observables,
the CP-averaged $\braket{S_9^{(\ell)}}$ vanishes independently of the BSM
scenario, as discussed above \refeq{def-S_i-A_i}, and is thus not relevant
for our analysis. This leaves the $D^*$-longitudinal polarization fraction
$\braket{F^{(\ell)}_L}$, the lepton forward-backward asymmetry
$\braket{A_\text{FB}^{(\ell)}}$, and two further angular observables
$\braket{\wT{F}^{(\ell)}_L}$ and $\braket{S_3^{(\ell)}}$ as independent
observables in the distributions. Within the SM, $\braket{F^{(\ell)}_L}$
and $\braket{\wT{F}^{(\ell)}_L}$ differ by lepton-mass suppressed terms, only.
In a generic BSM scenario, the two observables can further differ due to
contributions from pseudoscalar and tensor operators, see \reftab{J_i-C_a}.
For more details see \refapp{AngularDistribution}.\\[-0.3cm]
The presentation of the data in terms of single-differential distributions
implies that all angular observables are integrated over the full $q^2$ range.
By binning in $q^2$, the data will provide more information about
the BSM couplings through the $q^2$ shape of the angular observables.
In particular, the binned angular observables yield access to more
and independent bilinear combinations of the BSM couplings than the
$q^2$-integrated ones do. Hence, binning the angular observables will constitute
a powerful tool to discriminate between BSM scenarios, as discussed in more
detail below.
The CP asymmetries of the single-differential rates \eqs{eq:dG:dw}{eq:dG:dcosthD}
vanish independently of the BSM scenario. This can be used to validate
the experimental analyses. The CP asymmetry of the $\chi$-dependent rate in
\refeq{dG:dchi} is fully described by the angular observable $A_9^{(\ell)}$.
A measurement of this CP asymmetry could be accomplished with existing datasets
and would provide important information about potential CP-violating BSM effects.
\subsection{Parametrization of BSM Physics}
BSM physics in $\bar{B}\to D^* \ell \bar\nu$ decays
has been investigated, usually based on the assumption of
three light left-handed neutrino flavours below the electroweak scale.
The corresponding most general low-energy effective theory at dimension
six~\cite{Goldberger:1999yh} can be written as~\cite{Jung:2018lfu}
\begin{equation}
\label{eq:NPlagrangian}
\mathcal{L}(b\to c\ell\bar\nu
= \frac{4 G_F}{\sqrt{2}}V_{cb} \;
\sum_i \sum_{\ell'} C_i^{\ell\ell^\prime} \mathcal{O}_i^{\ell\ell^\prime}
+ \text{h.c.} \,.
\end{equation}
Here the operators are constructed out of SM fermion fields
and read
\begin{equation}
\label{eq:NP_op_basis}
\begin{aligned}
\mathcal{O}_{V_L}^{\ell\ell^\prime} &
=(\bar{c}\gamma^\mu P_L b)(\bar{\ell}\gamma_\mu P_L \nu_{\ell^\prime})\,,
\qquad
&
\mathcal{O}_{S_L}^{\ell\ell^\prime} &
= (\bar{c} P_L b)(\bar{\ell} P_L \nu_{\ell^\prime})\,,
\qquad
& \mathcal{O}_T^{\ell\ell^\prime} &
= (\bar{c}\sigma^{\mu\nu} P_L b)(\bar{\ell}\sigma_{\mu\nu} P_L \nu_{\ell^\prime})\,,
\\[0.2cm]
\mathcal{O}_{V_R}^{\ell\ell^\prime} &
= (\bar{c}\gamma^\mu P_R b)(\bar{\ell}\gamma_\mu P_L \nu_{\ell^\prime})\,,
&
\mathcal{O}_{S_R}^{\ell\ell^\prime} &
= (\bar{c} P_R b)(\bar{\ell} P_L \nu_{\ell^\prime})\,.
\end{aligned}
\end{equation}
They account for lepton-flavour violation (LFV) by $\ell\neq\ell'$.\\
The observables in $\bar{B}\to D^* \ell \bar\nu$ depend only on four
combinations of Wilson coefficients:
\begin{align}
C_V^{\ell\ell^\prime} & = C_{V_R}^{\ell\ell^\prime} + C_{V_L}^{\ell\ell^\prime} , &
C_A^{\ell\ell^\prime} & = C_{V_R}^{\ell\ell^\prime} - C_{V_L}^{\ell\ell^\prime} , &
C_P^{\ell\ell^\prime} & = C_{S_R}^{\ell\ell^\prime} - C_{S_L}^{\ell\ell^\prime} , &
\end{align}%
together with $C_T^{\ell\ell^\prime}$, whereas the combination
$C_S^{\ell\ell^\prime} = C_{S_R}^{\ell\ell^\prime} + C_{S_L}^{\ell\ell^\prime}$
enters only in $\bar{B}\to D \ell \bar\nu$.
Since the neutrino flavour~$\ell'$ is not detectable, it must be summed over
in every observable.
\begin{table}
\centering
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|c|cccc|ccc|cc|c|}
\hline
Observable & $|C_A|^2$ & $|C_V|^2$ & $|C_P|^2$ & $|C_T|^2$
& $\re(C_A^{} C_V^*)$ & $\re(C_A^{} C_P^*)$ & $\re(C_A^{} C_T^*)$
& $\re(C_V^{} C_P^*)$ & $\re(C_V^{} C_T^*)$
& $\re(C_P^{} C_T^*)$
\\
\hline\hline
$J_{1c} = V_1^0$ & $\checkmark$ & -- & $\checkmark$ & $\checkmark$ & -- & $(m)$ & $(m)$ & -- & -- & --\\
$J_{1s} = V_1^T$ & $\checkmark$ & $\checkmark$& -- & $\checkmark$ & -- & -- & $(m)$ & -- & $(m)$ & --\\
$J_{2c} = V_2^0$ & $\checkmark$ & -- & -- & $\checkmark$ & -- & -- & -- & -- & -- & --\\
$J_{2s} = V_2^T$ & $\checkmark$ & $\checkmark$& -- & $\checkmark$ & -- & -- & -- & -- & -- & --\\
$J_{3} = V_4^T$ & $\checkmark$ & $\checkmark$& -- & $\checkmark$ & -- & -- & -- & -- & -- & --\\
$J_{4} = V_1^{0T}$ & $\checkmark$ & -- & -- & $\checkmark$ & -- & -- & -- & -- & -- & --\\
$J_{5} = V_2^{0T}$ & $(m^2)$ & -- & -- & $(m^2)$ & $\checkmark$ & $(m)$ & $(m)$ & -- & $(m)$ & $\checkmark$ \\
$J_{6c} = V_3^0$ & $(m^2)$ & -- & -- & -- & -- & $(m)$ & $(m)$ & -- & -- & $\checkmark$ \\
$J_{6s} = V_3^T$ & -- & -- & -- & $(m^2)$ & $\checkmark$ & -- & $(m)$ & -- & $(m)$ & -- \\
\hline
$d\Gamma/dq^2$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & -- & $(m)$ & $(m)$ & -- & $(m)$ & -- \\
num($A_\text{FB}$) & $(m^2)$ & -- & -- & $(m^2)$ & $\checkmark$ & $(m)$ & $(m)$ & -- & $(m)$ & $\checkmark$ \\
num($F_L$) & $\checkmark$ & -- & $\checkmark$ & $\checkmark$ & -- & $(m)$ & $(m)$ & -- & -- & --\\
num($F_L$-1/3) & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & -- & $(m)$ & $(m)$ & -- & $(m)$ & --\\
num($\wT F_L$) & $\checkmark$ & $(m^2)$ & $\checkmark$ & $\checkmark$ & -- & $(m)$ & $(m)$ & -- & $(m)$ & --\\
num($\wT F_L$-1/3) & $\checkmark$ & $\checkmark$ & -- & $\checkmark$ & -- & -- & -- & -- & -- & --\\
num($S_3$) & $\checkmark$ & $\checkmark$ & -- & $\checkmark$ & -- & -- & -- & -- & -- & --\\
\hline\hline
Observable & -- & -- & -- & --
& $\im(C_AC_V^*)$ & $\im(C_AC_P^*)$ & $\im(C_AC_T^*)$
& $\im(C_VC_P^*)$ & $\im(C_VC_T^*)$
& $\im(C_PC_T^*)$
\\
\hline
$J_{7} = V_3^{0T}$ & & & & & $(m^2)$ & -- & $(m)$ & $(m)$ & -- & $\checkmark$\\
$J_{8} = V_4^{0T}$ & & & & & $\checkmark$ & -- & -- & -- & -- & -- \\
$J_{9} = V_5^T$ & & & & & $\checkmark$ & -- & -- & -- & -- & -- \\
\hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\caption{The dependence of angular observables on combinations of Wilson
coefficients. An entry of $\checkmark$ denotes the presence of this combination.
An entry of $m^n$ denotes the presence of this term, but with kinematic lepton-mass
suppression $\propto (m_\ell/\sqrt{q^2})^n$ ($n=1,2$). The
``num($\cdot$)'' indicates that only the dependence of the numerator of
this observable is given. The $V_i^a$ have been introduced in Ref.~\cite{Duraisamy:2014sna}.
}
\label{tab:J_i-C_a}
\end{table}
We determine the minimal number of parameters and their ranges necessary
to parametrize these BSM coefficients for different cases.
Starting from the lepton-flavour conserving case, \refeq{NPlagrangian}
contains five complex parameters $C_i^\ell\equiv C_i^{\ell\ell}$ per
charged-lepton species $\ell$.
In the context of BSM analyses of $\bar{B} \to D^*\ell\bar\nu$, the fact that
matrix elements of the scalar $\bar{c}b$ currents vanish implies that one
can maximally determine four linear combinations out of the five Wilson coefficients.
These four complex coefficients can be parametrized by seven real parameters,
since an overall phase is unobservable, \emph{i.e.} all observables are invariant under a
joint phase rotation $C_i^\ell\to \exp(i\phi^\ell)C_i^\ell$.
For instance, one of the complex coefficients can be chosen real and positive,
which leaves four real and three imaginary parts or four absolute values and three relative phases
as free parameters.
The Lagrangian \refeq{NPlagrangian} is conveniently normalized to $G_F\, V_{cb}$
to ensure that in the SM $C_{V_L} = 1$ at tree-level.
In general, these factors cannot
be separated from the BSM Wilson coefficients since only their products
enter observables.
Hence, they do not count as additional parameters.
The set of seven real parameters is therefore the maximal information
we can hope to extract from $\bar B\to D^*\ell\bar\nu$ decays for a given $\ell$ without LFV.\\[-0.3cm]
All CP-averaged observables depend on these seven parameters through
the combinations
\begin{equation}
\label{eq:LFC-comb-1}
\begin{aligned}
|C_i^\ell|^2
& =\re^2(C_i^\ell) + \im^2(C_i^\ell)\,,\\
\re(C_i^\ell C_j^{\ell*})
& = \re(C_i^\ell)\re(C_j^\ell) - \im(C_i^\ell)\im(C_j^\ell)\,.
\end{aligned}
\end{equation}
These combinations, however, are invariant under the discrete symmetry transformation
$\im(C_i^\ell) \to -\im(C_i^\ell) \;\forall\; i$.
The combinations
\begin{equation}
\label{eq:LFC-comb-2}
\im(C_i^\ell C_j^{\ell*})
= \im(C_i^\ell)\re(C_j^\ell) - \re(C_i^\ell)\im(C_j^\ell)\,,
\end{equation}
can therefore still be determined from CP-averaged observables, albeit only
up to an overall sign.
One is free to choose one of these signs freely in the fit, since the second
solution can always be obtained by inverting the signs of the imaginary
parts.\\[-0.3cm]
In the limit of a massless lepton, the two classes of Wilson coefficients
$C_{A,V}^\ell$ and $C_{P,T}^\ell$ decouple in the observables, since their
interference is $m_\ell$ suppressed, as shown in \reftab{J_i-C_a}.
As we will see below, this applies only to electrons, since in precision
analyses the muon mass cannot be neglected anymore. This implies a separate
symmetry for each class, $C_{V,A}^\ell\to \exp(i\phi^\ell)C_{V,A}^\ell$
and $C_{P,T}^\ell\to \exp(i\varphi^{\ell})C_{P,T}^\ell$.
Therefore another phase cannot be determined from any $\bar{B}\to D^*\ell^-\bar\nu$
observable in this limit.
In fact, it can be eliminated altogether from the parametrization, leaving
maximally six parameters to be determined from $\bar B\to D^*\ell\bar \nu$
for massless charged leptons. In this case also the discrete
symmetry for the imaginary parts holds separately for each class, allowing
to choose another sign freely. Hence, the most general parametrization of
CP-averaged $\bar B\to D^* e\bar\nu$ data within the weak effective theory
and when neglecting LFV requires only six parameters, four of which can be
chosen positive. Taking into account lepton-mass effects
requires a seventh parameter, and only two of these parameters can be chosen positive.\\[-0.3cm]
Note that in the counting above we have assumed the couplings for the different
lepton flavours to be completely independent, allowing in particular for
independent phase rotations.
Such an assumption does not hold in all BSM scenarios; in particular
it does not hold in the
Standard Model Effective Field Theory (SMEFT) at mass dimension six.
In the matching of \refeq{NPlagrangian} to the SMEFT, the coefficient $C_{V_R}^{\ell\ell'}$
is lepton-flavour universal, a property inherited from the SM gauge group \cite{Cata:2015lta, Cirigliano:2009wk}.
This universality couples the different sectors and
consequently the phase rotations cannot be performed independently anymore.
This gives rise to
an additional measurable phase in this scenario, and therefore necessitates a new corresponding
parameter.
For instance, for the common and convenient choice of a real and positive
$C_{V_L}^\ell$, the coefficients $C_{V_R}^{\ell}$
cannot be trivially identified with each other. Instead they fulfill
\begin{align}\label{eq::CVRuni}
C_{V_R}^e & = \exp(i\phi_L) \, C_{V_R}^\mu\,,
&
\phi_L & = \phi_{V_L}^e-\phi_{V_L}^\mu\,,
\end{align}
and similarly for $\ell=\tau$.
The relative phase between the two Wilson coefficients $C_{V_L}^e$ and
$C_{V_L}^\mu$ appears explicitly, while it can be absorbed everywhere else.
This implies that although two real parameters are removed (one of the complex
$C_{V_R}^\ell$ coefficients), one is added (the relative phase),
and hence the overall number of parameters is reduced only by one.
Generalizing the above observations in the presence of lepton-flavour-violating
interactions, $\ell\neq \ell'$, is straight-forward insofar as the
contributions with different neutrino flavours do not interfere. Hence all
expressions in \eqs{eq:LFC-comb-1}{eq:LFC-comb-2} remain valid with the
generalizations
\begin{align}
\label{eq:LFV-generalization}
\left|C_i^\ell\right|^2 &
\to \sum_{\ell'} \left|C_i^{\ell\ell'}\right|^2 ,
&
\re\left(C_i^\ell C_j^{\ell*}\right) &
\to \sum_{\ell'} \re\left(C_i^{\ell\ell'} C_j^{\ell\ell'*}\right) ,
&
\im\left(C_i^\ell C_j^{\ell*}\right) &
\to \sum_{\ell'} \im\left(C_i^{\ell\ell'} C_j^{\ell\ell'*}\right) .
\end{align}
The symmetry considerations hold for each neutrino flavour separately.
Naively the number of parameters simply triples compared to
the lepton-flavour-conserving case above. The situation is nevertheless
significantly different from the lepton-flavour conserving case, for which the number
of parameters is smaller than the number of combinations of Wilson coefficients
appearing in the description of the decay.
This implies (non-linear) relations
between these combinations in the lepton-flavour conserving case, for instance,
\begin{equation}
\label{eq:LFV-relation}
\im^2(C_i^\ell C_j^{\ell*})
= |C_i^\ell|^2 |C_j^{\ell}|^2-\re^2(C_i^\ell C_j^{\ell*})\,.
\end{equation}
With the generalizations in \refeq{LFV-generalization}, the number of
BSM parameters is larger than the number of combinations of Wilson
coefficients. Hence, the latter determine the maximal number of parameters
(parameter combinations) that can be extracted.
This implies that relations such as \refeq{LFV-relation} do not hold anymore in the presence of lepton-flavour violation and can be used instead
to test for LFV in charged-current decays without the need to identify
the neutrino flavour experimentally.\\[-0.3cm]
In the presence of light right-handed neutrinos, similar considerations
as for the LFV case apply, since also here more BSM parameters are
introduced and the corresponding contributions do not interfere.
The generalization to light right-handed neutrinos is therefore analogous to \refeq{LFV-generalization}
and similar comments apply for the determination of the corresponding
parameters.
\subsection{\boldmath BSM reach in
\texorpdfstring{$\bar{B}\to D^*\ell \bar\nu$}{B -> D*+ l v}}
We now turn to the determination of the discussed parameters from the
differential distributions. Each fully $q^2$-integrated angular observable provides
one linear combination of the combinations of Wilson coefficients only, as indicated
in \reftab{J_i-C_a}. The measurement of their $q^2$ dependence
allows further to separate different BSM contributions to the same angular observable,
if their $q^2$ dependence \cite{Alguero:2020ukk} is different. For instance,
the $q^2$-differential
rate allows to determine all four absolute values of the BSM parameters.
The question is what amount of
experimental information is necessary to determine the maximal amount
of parameters in a given scenario. \reftab{BSMinfo} shows the situation
in a few scenarios for different sets of experimental measurements.\\[-0.3cm]
\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{l c c c c l}
\hline\hline
& \multicolumn{2}{c}{no LFV} & \multicolumn{2}{c}{LFV} & \\
Measurement & $m_\ell\to0 \quad$ & $m_\ell>0 \quad$ & $m_\ell\to0 \quad$ & $m_\ell>0 \quad$ & Comments \\
\hline
4-fold differential, S+A & 6 & 6+1 & 8 & 8+5 & Maximum achievable\\
4-fold differential, S only & 6 & 6+1 & 6 & 6+3 & $\sgn[\im(C_i^\ell C_j^{\ell *})]$ not resolved\\
$4\times1$-fold differential, S+A & 6 & 6+1 & 6 & 6+3 & 2-fold ambiguity in $m_\ell\to0$\\
$4\times1$-fold differential, S only& 5 & 5+2 & 5 & 5+3 & Insufficient for $m_\ell\to 0$,\\
& & & & & $\sgn[\im(C_i^\ell C_j^{\ell*})]$ not resolved\\
\hline\hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\caption{Amount of BSM physics information that can be extracted in
different scenarios, see also text. Here S and A denote the measurement
of the CP average and the CP asymmetry of the respective differential rate.
The first and second number corresponds to the number of parameters that can be
extracted without and with mass suppression, respectively.}
\label{tab:BSMinfo}
\end{table}
A few general comments are in order:
\begin{itemize}
\item It is necessary to consider the CP-conjugated modes separately
if the sign ambiguity for the imaginary parts is to be resolved.
Since the lepton charge tags the $B$ meson flavour, this is not difficult
to achieve experimentally.
\item The interference between the two classes of BSM coefficients
$C_{A,V}^\ell$ and $C_{P,T}^\ell$ is always lepton-mass suppressed, see
\reftab{J_i-C_a}. Hence its determination requires high statistical power,
as expected from the upcoming datasets at Belle 2 and the LHC experiments.
\item While for $\ell=\mu$ there is some sensitivity to additional combinations
of Wilson coefficients, these combinations are still strongly suppressed.
The corresponding parameters will therefore be determined
comparatively poorly. Generally the best chance to determine them is to
consider rather low values of $q^2$, given the suppression by
powers of $m_\ell/\sqrt{q^2}$, both for the angular
observables and the $q^2$-differential rate. Probing different bins in~$q^2$
can also improve the sensitivity to other BSM coefficients. Tensor
interactions for instance can be probed particularly well at low $q^2$
in $d\widehat\Gamma_T/dq^2\sim 3S_{1s}-S_{2s}$, since the SM
contributions vanish for $q^2\to 0$, while the tensor contributions
remain finite \cite{Jung:2018lfu}, see also Ref.~\cite{Bhattacharya:2015ida}.
\end{itemize}
Considering some of the scenarios in more detail, we make the following
observations:
\begin{itemize}
\item It is impossible to determine the full set of physical BSM
parameters for $m_\ell\to0$ (\emph{e.g.} $\ell=e$) from the CP-averaged
single-differential rates alone, even disregarding ambiguities
in the signs of imaginary parts. The reason is that in this case only
$\braket{A_\text{FB}^{(\ell)}}$ is sensitive to the relative
phases between the coefficients.
Since there are two observable relative phases (one between $C_A^\ell$
and $C_V^\ell$, one between $C_P^\ell$ and $C_T^\ell$), they cannot
both be determined from this single observable.
\item Assuming the flavour-conserving case, the extraction of all seven parameters
is possible for finite $m_\ell$ from the CP-averaged single-differential rates,
modulo discrete ambiguities. However, one relative phase can only be obtained from
lepton-mass-suppressed contributions, even though in more sophisticated
measurements it would be accessible without lepton-mass suppression.
\item Beyond the lepton-flavour-conserving case, it becomes
clearer how much more information is contained in a fully $q^2$-differential
measurement. Strictly speaking, such a measurement is not necessary when
assuming lepton-flavour conservation. However, also in this case there are
additional crosschecks possible and additional form-factor information
can be extracted together with the BSM parameters.
\end{itemize}
These observations apply fully to the recent Belle measurements \cite{Waheed:2018djm}.\\[-0.3cm]
Considering the determination of the full BSM information in the
lepton-flavour-conserving case as an important intermediate goal, there are
several ways this could be achieved with existing data, extending the experimental
analyses only slightly:
\begin{enumerate}
\item Measuring $A_\text{FB}^{(\ell)}$ in at least two $q^2$ bins. This disentangles
$S_{6s}^{(\ell)}$ from $S_{6c}^{(\ell)}$ entering this observable. Given that the
$\cos\theta_\ell$-differential distribution \eqref{eq:dG:dcosthL}
has been measured in 10 bins in
Refs.~\cite{Waheed:2018djm, Abdesselam:2017kjf}, but contains only two
angular observables, this seems feasible by reducing the number of
$\cos\theta_\ell$ bins and providing the observables in two or
three $q^2$ bins instead.
This would give access to all BSM parameters, leaving only two signs of
imaginary parts undetermined.
\item Measuring $d\Gamma/d\chi$ separately for the two lepton charges.
This would give access to $A_9^{(\ell)}$, and thereby to
$\im(C_A^\ell C_V^{\ell *})$. This in turn would determine also $\re(C_A^\ell
C_V^{\ell *})$ up to a sign, and thereby allow to access $\re(C_P^\ell
C_T^{\ell *})$ from $\braket{A_\text{FB}^{(\ell)}}$ up to a two-fold ambiguity.
Each of the solutions would still have a two-fold sign ambiguity for the
corresponding imaginary part. Together with the first option, this measurement
would resolve the sign ambiguity in $\im(C_A^\ell C_V^{\ell *})$, leaving
only the one in $\im(C_P^\ell C_T^{\ell *})$ (should this parameter combination
be found to be different from zero).
\item Assessing $S_5^{(\ell)}$, $A_7^{(\ell)}$ and/or $A_8^{(\ell)}$.
The measurement of each of these requires a different binning
scheme, since these observables do not enter the single-differential
rates. The latter two further require tagging by the lepton charge.
Of particular interest is $A_7^{(\ell)}$, since contributions linear in BSM
parameters are additionally lepton-mass suppressed, rendering the quadratic BSM
contributions potentially dominant.
A similar statement holds for~$S_{6c}^{(\ell)}$.
\end{enumerate}
\section{Available Experimental Data}
\label{sec:data}
Semileptonic $\bar{B}\to D^{(*)}\ell\bar\nu$ decays have been of key interest for many years,
see Ref.~\cite{Amhis:2019ckw} for a list of analyses over the last
$\sim 25$~years. However, until recently, almost all experimental analyses have been
tied to a specific form-factor parametrization, specifically the so-called
CLN~parametrization \cite{Caprini:1997mu}. This parametrization involves
assumptions that are not adequate anymore for precision analyses. Applying
instead the underlying formalism of a heavy-quark expansion more consistently
\cite{Bernlochner:2017jka} and extending it to include $1/m_c^2$ contributions
\cite{Bordone:2019guc, Bordone:2019vic}, allows for a consistent description
of the available experimental data and form factor results. However, since
experimental analyses presented in most cases \emph{only} parametrization-specific
results, a model-independent reanalysis under different theory assumptions
of the underlying experimental data is impossible.\footnote{The total branching
ratio results have been found to be approximately parametrization-independent,
see, \emph{e.g.} Refs.~\cite{Bigi:2017jbd, Waheed:2018djm}, but might still
suffer from underestimated uncertainties to some extent.} Unfortunately, this
problem persists in the most recent BaBar analysis~\cite{Dey:2019bgc}, which
includes a second form factor parametrization, but still does not allow for an
independent analysis of the data. Furthermore, in many cases electron and muon
data have been averaged without presenting separate results, rendering them of
limited use for the analysis of LFU. A notable exception among these past studies
is the 2010 untagged Belle analysis \cite{Dungel:2010uk}, which presented
lepton-specific differential rates separately for longitudinal and transverse
$D^*$ polarizations, but lacked the necessary correlations.\\[-0.3cm]
More recently, the Belle analysis of $\bar{B}\to D\ell\bar\nu$~\cite{Glattauer:2015teq}
presented for the first time lepton-specific differential rates including
their full correlations, which made possible precision studies with arbitrary
form-factor parametrizations for the first time, initiating an intense ongoing
discussion regarding the best way to analyze these and similar data.
Similar comments apply to the preliminary $\bar{B}\to D^*\ell\bar\nu$ data with hadronic tag in
Ref.~\cite{Abdesselam:2017kjf}, which were however again lepton-flavour averaged
and are presently reanalyzed, and the 2018 untagged analysis \cite{Waheed:2018djm},
superseding the results of Ref.~\cite{Dungel:2010uk}, which we discuss in detail in the following.
\subsection{Belle's 2018 untagged analysis}
\label{sec:BelleData}
The dataset for the angular distribution provided by Belle~\cite{Waheed:2018djm}
is the first analysis that separates the electron mode from the muon mode in both
the bin contents and the statistical covariance matrix, and also the systematic
covariance matrix can be reconstructed for both lepton species
separately.\footnote{%
Note that the arXiv version v3 of Ref.~\cite{Waheed:2018djm}
contains erroneous statistical and systematic correlation matrices. The journal
version of Ref.~\cite{Waheed:2018djm} contains the correct statistical correlation matrix,
but still an erroneous systematic one: both off-diagonal $20\times 20$ blocks
of the $40\times 40$ matrix should be transposed.
}
Unfortunately the correlations between the electron and muon modes are not given
explicitly. Yet Belle has used these data for a high-precision LFU test
that compares
the branching fractions to electrons and muons integrated over the entire
phase space. They found the ratio to be in agreement with lepton flavour universality,
$R_{e/\mu} = 1.01 \pm 0.01\mbox{(stat.)} \pm 0.03\mbox{(sys.)}$.
In our study we aim to extend the study of LFU to the angular observables using the same Belle data. For
this purpose we need to construct a combined correlation matrix for the full dataset, including correlations between electrons and muons.\\[-0.3cm]
Before going into these details, however, we comment on an issue present in the
statistical correlation matrix. Belle provides the number of (background-subtracted)
events before unfolding in bins of the four aforementioned single-differential
distributions. These are the distribution in
\begin{align}
w & = \frac{m_B^2 + m_{D^*}^2 - q^2 }{2 m_B m_{D^*}} ,
\end{align}
and the three angular distributions Eqs.~\eqref{eq:dG:dcosthL}--\eqref{eq:dG:dchi}.
There are 10 equidistant bins for each distribution, resulting in 40 bins per
lepton flavour (LF). The events in the 10 bins for each of the four distributions
sum up to the same number for each lepton flavour:
\begin{equation}
\label{eq:obs-norm}
\sum_{i= 1}^{10} N^\text{obs}_{i,\ell}
= \sum_{i=11}^{20} N^\text{obs}_{i,\ell}
= \sum_{i=21}^{30} N^\text{obs}_{i,\ell}
= \sum_{i=31}^{40} N^\text{obs}_{i,\ell}
= \left\{ \begin{array}{cc} 90743.4 & \ell = e \\[0.2cm]
89087.0 & \ell = \mu
\end{array} \right. ,
\end{equation}
\emph{i.e.} the same signal candidates have been histogrammed in four different ways
in the four single-differential distributions.
These relations imply that for both electrons and muons only 37 of the measured
bins are independent, since the content of 3 bins can be calculated as the total
yield minus the yields of the other 9 bins of the corresponding distributions.
This in turn implies that the corresponding statistical correlation matrices have
to be singular; each of the $40\times 40$ matrices should exhibit three vanishing
eigenvalues. This is, however, not the case: the determinant of both matrices is
rather large and all eigenvalues of both statistical correlation matrices are
$\mathcal O(1)$. It remains unclear why the statistical correlation matrices do
not reflect the linear dependence of the 3 bins, which should by construction
be a result of the description of 10 bins per single-differential distribution
used by the Belle collaboration. Note that the issue of the linearly dependent
bins affects the determination of $V_{cb}$ from these data:\footnote{%
Depending on
the source of this problem, it might not affect the determination of $V_{cb}$
directly from the electron and muon event samples in Ref.~\cite{Waheed:2018djm}.
}
if the sum over each set of 10 bins is identical, no information is added to the
determination of the total rate by having the four binnings. However, if the
correlations are such that these sums become effectively independent, the total
rate is more precisely determined by considering all four binnings than by
considering only a single one, leading to an underestimation of the uncertainty
of the total rate (and hence $V_{cb}$).
The effect is not large with the given data, but it is non-vanishing: the determination
of the total rate is a couple of per mil better than from each individual distribution.
It is important to note that this small numerical impact is \emph{not} an
indication that a correct extraction of the statistical correlation matrices
will lead to small corrections in the analysis. Since there is an unknown problem
in the extraction of the statistical correlation matrices, there is no way of
knowing what the effect of its resolution will be. Given this numerical smallness
within our analysis, however, we will work below with $40 \times 40$ matrices.
In LF-specific fits with a $37\times 37$ matrix the result varies very slightly,
depending on the choice of the three discarded bins, and any specific choice
would be arbitrary. We have checked that our numerical results below remain
essentially unaffected. The issue with the statistical correlation matrices
must be kept in mind when interpreting \emph{any} results obtained from the
data from Ref.~\cite{Waheed:2018djm}.\\[-0.3cm]
In the remainder of this section, we describe the construction of a combined
electron-muon $80\times 80$ covariance matrix based on Ref.~\cite{Waheed:2018djm},
with only one mild additional assumption. According to Ref.~\cite{Waheed:2018djm},
the only source of systematic uncertainties that is different for $\ell=e$ and
$\ell=\mu$ is the procedure of lepton identification (Lepton ID). Given the
statistical independence of electron and muon samples, this implies the
following form for the total covariance matrix:
\begin{align}
\text{Cov}^\text{total}_{80\times 80} &
= \text{Cov}^\text{stat}_{80\times 80}+\text{Cov}^\text{sys}_{80\times 80}
\\[0.2cm] &
= \begin{pmatrix}
\text{Cov}^{\text{stat},e}_{40\times 40} & 0_{40\times 40}
\\[0.2cm]
0_{40\times 40} & \text{Cov}^{\text{stat},\mu}_{40\times 40}
\end{pmatrix}
+ \begin{pmatrix}
\text{Cov}^{\text{sys,uni}}_{40\times 40} & \text{Cov}^{\text{sys,uni}}_{40\times 40}
\\[0.2cm]
\text{Cov}^{\text{sys,uni}}_{40\times 40} & \text{Cov}^{\text{sys,uni}}_{40\times 40}
\end{pmatrix}
+ \text{Cov}^\text{sys,lep-ID}_{80\times 80}\,.
\end{align}
The lepton-ID systematic uncertainties are provided individually for both
lepton flavours, but also for the ``LF-combined'' which enter the
systematic correlation matrix given explicitly in the article. We therefore
have
\begin{align}
\text{Cov}^\text{sys,uni}_{40\times 40} &
= \text{Cov}^\text{sys,LF-comb}_{40\times 40}
- \text{Cov}^\text{sys,lep-ID-comb}_{40\times 40} \,.
\end{align}
Together with the information that the Lepton-ID systematic uncertainties
are 100\% positively correlated throughout all bins \cite{Waheed-PhD},
$\text{Cov}^{\text{sys,lep-ID-comb}}_{ij} = \sigma_i^{\text{lep-ID-comb}}
\sigma_j^{\text{lep-ID-comb}}$, where $\sigma_i$ are systematic uncertainties
of the $i$th bin taken from tables XI--XIV \cite{Waheed:2018djm}, the ``LF
combination'' can thus be undone for the systematic correlations. We compute
the LF-specific systematic covariances
($\text{Cov}^{\text{sys},\ell}$) from the ``LF-combined'' ones
($\text{Cov}^\text{sys,LF-comb}$) of \cite{Waheed:2018djm} consequently as
\begin{align}
\text{Cov}^{\text{sys},\ell}_{ij} &
= \text{Cov}^\text{sys,LF-comb}_{ij}
- \sigma_i\sigma_j |^{\text{lep-ID-comb}}
+ \sigma_i\sigma_j |^{\text{lep-ID,}\ell} ,
&
i,j & = 1,\ldots, 40\,.
\end{align}
LF-specific analyses can be performed with these LF-specific $40 \times 40$
statistical and systematic correlation matrices at hand.
The \emph{only} assumption we make for the construction of the full
$80\times 80$ covariance matrix is that the lepton-ID uncertainties
for electrons and muons are uncorrelated:
\begin{align}
\text{Cov}^\text{sys,lep-ID}_{80\times 80}
= \begin{pmatrix}
\text{Cov}^{\text{sys,lep-ID},e}_{40\times 40} & 0_{40\times 40}
\\[0.1cm]
0_{40\times 40} & \text{Cov}^{\text{sys,lep-ID},\mu}_{40\times 40}
\end{pmatrix}\,.
\end{align}
This is plausible (as confirmed by Belle collaboration members \cite{PrivateCommunications}), given
they concern different detector parts, but not fully guaranteed.
We consider this assumption to be at a comparable level to the assertion
in Ref.~\cite{Waheed:2018djm} that the lepton ID constitutes the only
non-universal contribution to the systematic uncertainty. Note that this
is an approximation that might not hold well enough to analyze LFU.
In that case the systematic uncertainty given in \cite{Waheed:2018djm}
for the LFU ratio $R_{e/\mu}$ would be underestimated, as would be our $e-\mu$
covariance. However, we perform below an extremely conservative check
that our observation of a tension with the SM does not depend on this
assumption.
\section{\boldmath Fits to
\texorpdfstring{$\bar{B}\to D^* (e, \mu)\bar\nu$}{B-> D*+ (e, mu) v}
Data and Discussion}
\label{sec:fit}
We analyze the data from the Belle analysis \cite{Waheed:2018djm} in detail,
based on the general analysis in \refsec{fit-model} and the covariance
matrix derived in \refsec{data}.
\subsection{Angular analysis and comparison with the SM}
In the first step our fit is completely model-independent: we use the observation
made in \refsec{fit-model} that the three single-differential CP-averaged
angular distributions can be fully described by only four angular observables
\begin{align}
\label{eq:obs-set-1}
\braket{A_\text{FB}^{(\ell)}}, \quad
\braket{F_L^{(\ell)}}, \quad
\braket{\wT F_L^{(\ell)}}, \quad
\braket{S_3^{(\ell)}} ,
\end{align}
retaining all information. Further, we parametrize the 10 bins of
the $w$-distribution again in full generality as the
total decay rate and nine independent bins of the normalized $w$-differential rate:
\begin{align}
\label{eq:obs-set-2}
\widehat{\Gamma}^{(\ell)}, &
&
x_i^{(\ell)} & \equiv \frac{1}{\widehat{\Gamma}^{(\ell)}}
\int_{w_{i-1}}^{w_i} \!\! dw \, \frac{d \widehat{\Gamma}^{(\ell)}(w)}{dw} ,
&
w_i & = 1 + i \frac{(w_\text{max}-1)}{10}
& (i & = 2, \ldots 10)\,.
\end{align}
Here $w_\text{max} = 1.5$ to comply with the choice in \cite{Waheed:2018djm},
which excludes a tiny part of the low-$q^2$ phase space.
From this parametrization we calculate the bin contents $N_{i,\ell}^\text{obs}$
by integrating over the relevant angle intervals where necessary, and folding
these predictions with the corresponding response matrices and efficiencies provided
by the Belle collaboration for each lepton flavour separately, as described in
\cite{Waheed:2018djm}. We thus arrive at a description of the 40 bins per lepton
flavour given in \cite{Waheed:2018djm} in terms of only $10 + 4 = 14$ observables
in \eqs{eq:obs-set-1}{eq:obs-set-2}.
We emphasize that our fit parameters appear up to the common normalization factor
linearly, assuring a unique minimum and no distortion of their distributions
from a multivariate gaussian shape.\\[-0.3cm]
The conversion of number of events to decay rate involves the following
numerical input:
\begin{equation}
\begin{aligned}
N_{B\bar{B}} & = (772 \pm 11) \cdot 10^6 , \qquad\qquad
&
\mathcal{B}(D^{*+} \to D^0 \pi^+) & = (67.7 \pm 0.5)\, \% ,
\\
f_{00} & = 0.486 \pm 0.006 ,
&
\mathcal{B}(D^0 \to K^- \pi^+) & = (3.950 \pm 0.031)\, \% ,
\\
\tau_{B^0} & = (1.519 \pm 0.004) \cdot 10^{-12} \,\text{s} ,
\end{aligned}
\end{equation}
with $N_{B\bar{B}}$ from \cite{Natkaniec:2006rv}, $f_{00}$ and from the
$B^0$~lifetime from \cite{Amhis:2019ckw} (see also the discussion on
$f_{00}$ in \cite{Jung:2015yma}), and the latest values of the branching
fractions from \cite{Zyla:2020zbs}. Note that the value for $\mathcal{B}(D^0 \to K^- \pi^+)$
was updated w.r.t the value used in Ref.~\cite{Waheed:2018djm, Ferlewicz:2020lxm},
which slightly
impacts the determination of $V_{cb}$. The corresponding uncertainties
cancel in all ratios and hence affect only the total decay rate, for
which they are included in the systematic uncertainties provided by the
Belle collaboration \cite{Waheed:2018djm}.\\[-0.3cm]
We further introduce the averages and differences of LF-specific observables
\begin{align}
\label{eq:def-LF-ave-obs}
\Sigma X & \equiv \frac{X^{(\mu)} + X^{(e)}}{2} \,,
&
\Delta X & \equiv X^{(\mu)} - X^{(e)} \,,
\end{align}
for later convenience in the study of LFU violation where $X^{(\ell)}$ stands
for any of the considered observables.\\[-0.3cm]
\begin{table}
\centering
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{|c|cc|cc|}
\hline
& \multicolumn{2}{c|}{SM} & \multicolumn{2}{c|}{Fit Belle data}
\\
\hline
observable & $\ell = e$ & $\ell = \mu$
& $\ell = e$ & $\ell = \mu$
\\
\hline\hline
$10^{14} \cdot \widehat{\Gamma}^{(\ell)}$ [GeV] &
$(1.469 \pm 0.079)\cdot 10^{3}\cdot|V_{cb}|^2$ &
$(1.465 \pm 0.078)\cdot 10^{3}\cdot|V_{cb}|^2$ &
$ 2.1840 \pm 0.0682$ &
$ 2.1656 \pm 0.0719$
\\
$x_2^{(\ell)}$ &
$0.0920 \pm 0.0034$ & $0.0923 \pm 0.0034$ &
$0.0956 \pm 0.0022$ & $0.0915 \pm 0.0023$
\\
$x_3^{(\ell)}$ &
$0.1085 \pm 0.0030$ & $0.1088 \pm 0.0030$ &
$0.1146 \pm 0.0024$ & $0.1120 \pm 0.0025$
\\
$x_4^{(\ell)}$ &
$0.1161 \pm 0.0020$ & $0.1164 \pm 0.0020$ &
$0.1184 \pm 0.0027$ & $0.1208 \pm 0.0029$
\\
$x_5^{(\ell)}$ &
$0.1180 \pm 0.0010$ & $0.1184 \pm 0.0010$ &
$0.1198 \pm 0.0033$ & $0.1241 \pm 0.0033$
\\
$x_6^{(\ell)}$ &
$0.1160 \pm 0.0006$ & $0.1163 \pm 0.0006$ &
$0.1169 \pm 0.0037$ & $0.1134 \pm 0.0037$
\\
$x_7^{(\ell)}$ &
$0.1108 \pm 0.0015$ & $0.1110 \pm 0.0015$ &
$0.1079 \pm 0.0040$ & $0.1145 \pm 0.0040$
\\
$x_8^{(\ell)}$ &
$0.1031 \pm 0.0025$ & $0.1033 \pm 0.0025$ &
$0.1045 \pm 0.0039$ & $0.0942 \pm 0.0040$
\\
$x_9^{(\ell)}$ &
$0.0934 \pm 0.0033$ & $0.0936 \pm 0.0033$ &
$0.0923 \pm 0.0036$ & $0.0923 \pm 0.0037$
\\
$x_{10}^{(\ell)}$ &
$0.0820 \pm 0.0040$ & $0.0818 \pm 0.0040$ &
$0.0758 \pm 0.0033$ & $0.0777 \pm 0.0034$
\\
$\braket{F_L^{(\ell)}}$ &
$0.541 \pm 0.011$ & $0.542 \pm 0.012$ &
$0.5336 \pm 0.0045$ & $0.5271 \pm 0.0046$
\\
$\braket{A_\text{FB}^{(\ell)}}$ &
$0.204 \pm 0.012$ & $0.198 \pm 0.012$ &
$0.1951 \pm 0.0069$ & $0.2300 \pm 0.0059$
\\
$\braket{\wT F_L^{(\ell)}}$ &
$0.541 \pm 0.011$ & $0.536 \pm 0.011$ &
$0.5491 \pm 0.0102$ & $0.5384 \pm 0.0103$
\\
$- \braket{S_3^{(\ell)}}$ &
$0.1350 \pm 0.0036$ & $0.1344 \pm 0.0036$ &
$0.1325 \pm 0.0076$ & $0.1452 \pm 0.0079$
\\
\hline
\hline
& \multicolumn{2}{c|}{SM} & \multicolumn{2}{c|}{Fit Belle data}
\\
\hline
observable & $\Sigma X$ & $\Delta X$
& $\Sigma X$ & $\Delta X$
\\
\hline\hline
$10^{14} \cdot \widehat{\Gamma}^{(\ell)}$ [GeV] &
$(1.467 \pm 0.078)\cdot 10^{3}\cdot|V_{cb}|^2$ & $(-3.80 \pm 0.24)\cdot|V_{cb}|^2$ &
$2.1748 \pm 0.0647$ & $-0.0184 \pm 0.0539$
\\
$x_2^{(\ell)}$ &
$0.0922 \pm 0.0034$ & $(2.272 \pm 0.049)\cdot 10^{-4}$ &
$0.0936 \pm 0.0017$ & $-0.0040 \pm 0.0029$
\\
$x_3^{(\ell)}$ &
$0.1086 \pm 0.0030$ & $(3.119 \pm 0.056)\cdot 10^{-4}$ &
$0.1133 \pm 0.0018$ & $-0.0025 \pm 0.0033$
\\
$x_4^{(\ell)}$ &
$0.1162 \pm 0.0020$ & $(3.211 \pm 0.087)\cdot 10^{-4}$ &
$0.1196 \pm 0.0021$ & $+0.0024 \pm 0.0038$
\\
$x_5^{(\ell)}$ &
$0.1182 \pm 0.0010$ & $(3.10 \pm 0.13)\cdot 10^{-4}$ &
$0.1220 \pm 0.0024$ & $+0.0043 \pm 0.0046$
\\
$x_6^{(\ell)}$ &
$0.1161 \pm 0.0006$ & $(2.84 \pm 0.17)\cdot 10^{-4}$ &
$0.1151 \pm 0.0027$ & $-0.0035 \pm 0.0052$
\\
$x_7^{(\ell)}$ &
$0.1109 \pm 0.0015$ & $(2.47 \pm 0.21)\cdot 10^{-4}$ &
$0.1112 \pm 0.0029$ & $+0.0066 \pm 0.0056$
\\
$x_8^{(\ell)}$ &
$0.1032 \pm 0.0025$ & $(1.99 \pm 0.24)\cdot 10^{-4}$ &
$0.0993 \pm 0.0029$ & $-0.0103 \pm 0.0054$
\\
$x_9^{(\ell)}$ &
$0.0935 \pm 0.0033$ & $(1.37 \pm 0.27)\cdot 10^{-4}$ &
$0.0923 \pm 0.0026$ & $-0.0000 \pm 0.0052$
\\
$x_{10}^{(\ell)}$ &
$0.0819 \pm 0.0040$ & $(2.21 \pm 0.15)\cdot 10^{-4}$ &
$0.0768 \pm 0.0025$ & $+0.0019 \pm 0.0044$
\\
$\braket{F_L^{(\ell)}}$ &
$0.541 \pm 0.011$ & $(5.43 \pm 0.36)\cdot 10^{-4}$ &
$0.5303 \pm 0.0035$ & $-0.0065 \pm 0.0059$
\\
$\braket{A_\text{FB}^{(\ell)}}$ &
$0.201 \pm 0.012$ & $(-5.33 \pm 0.24)\cdot 10^{-3}$ &
$0.2125 \pm 0.0047$ & $+0.0349 \pm 0.0089$
\\
$\braket{\wT F_L^{(\ell)}}$ &
$0.539 \pm 0.011$ & $(-5.20 \pm 0.30)\cdot 10^{-3}$ &
$0.5437 \pm 0.0074$ & $-0.0107 \pm 0.0142$
\\
$- \braket{S_3^{(\ell)}}$ &
$0.1347 \pm 0.0036$ & $(5.81 \pm 0.22)\cdot 10^{-4}$ &
$0.1388 \pm 0.0055$ & $+0.0127 \pm 0.0109$
\\
\hline
\end{tabular}
\renewcommand{\arraystretch}{1.0}
\caption{The SM predictions of observables for $\ell = e$ and $\ell = \mu$,
using the form-factor results \cite{Bordone:2019guc}, together with
their values obtained from our fit to the Belle data \cite{Waheed:2018djm}.
For the prediction of the total rate, we leave the value of $|V_{cb}|$
unspecified.
}
\label{tab:SM-predictions}
\end{table}
\begin{figure}
\includegraphics[width=.32\textwidth]{plots/Belle_vs_SM-electron_x.pdf}
\includegraphics[width=.32\textwidth]{plots/Belle_vs_SM-muon_x.pdf}
\includegraphics[width=.32\textwidth]{plots/Belle_vs_SM-Sigma_x.pdf}
\\[0.2cm]
\hskip 0.01\textwidth
\includegraphics[width=.33\textwidth]{plots/Belle_vs_SM-ang_FL.pdf}
\hskip 0.18\textwidth
\includegraphics[width=.33\textwidth]{plots/Belle_vs_SM-ang_AfbS3.pdf}
\\[0.2cm]
\includegraphics[width=.34\textwidth]{plots/Belle_vs_SM-Delta_x.pdf}
\hskip 0.18\textwidth
\includegraphics[width=.34\textwidth]{plots/Belle_vs_SM-Delta_ang.pdf}
\caption{\label{fig:x} The measured $x_i^{(\ell)}$ from Belle versus
the SM predictions for $\ell = e$ [left], $\ell = \mu$ [middle]
and the lepton-flavour averaged $\Sigma x_i$ [right], and
also shown the differences $\Delta x_i$ [lower]. The numbers
are collected in \reftab{SM-predictions}.
}
\end{figure}
We perform two types of fits with our approach to test the stability of
the results:
\begin{enumerate}
\item
a simple $\chi^2$ fit,
\item
a fit using pseudo-Monte Carlo techniques, following the procedure
described in Ref.~\cite{Ferlewicz:2020lxm},
\end{enumerate}
both using the full $80\times 80$ covariance matrix. In addition, we have
applied a correction to the systematic correlations for d'Agostini bias
\cite{DAgostini:1993arp}, following the procedure described in Ref.~\cite{Jung:2018lfu}.
We find the results of the two fits to be virtually identical.
In Ref.~\cite{Ferlewicz:2020lxm} the authors observe that in their
joint fit of $V_{cb}$ and form-factor parameters the two procedures
produce markedly different results. They conclude that this difference is due
to the large correlations present in the experimental data and that
the usage of the pseudo-Monte Carlo technique is mandatory for phenomenological analyses.
Our findings are in stark contrast to this conclusion and indicate instead
that large correlations alone are not the cause for this difference.
Our
interpretation is that the observed difference is related to the form-factor
parameters entering non-linearly in the fit of Ref.~\cite{Ferlewicz:2020lxm},
while our angular observables and $x_i^{(\ell)}$ parameters enter bilinearly.
It is worth emphasizing in this context that
\begin{itemize}
\item
our fit results are extremely well described by Gaussian distributions;
and that
\item
the correlations between our fit parameters are much smaller than the
ones present in the $80\times 80$ matrix describing the bin contents.
\end{itemize}
As a consequence, we do not distinguish between the results from the
two fit procedures in the following.\\[-0.3cm]
The fit results for our parameters as defined in \eqs{eq:obs-set-1}{eq:obs-set-2}
are listed in \reftab{SM-predictions} and shown in \reffig{x}. At the best-fit point we
find $\chi^2 = 48.9$ for $80 - 2\times 14 = 52$ degrees of freedom (dof),
indicating a good fit.\footnote{The dof would be $46$ when considering that
3~bins should be linearly dependent among the 40~bins per lepton flavour,
still indicating a good fit.} This suggests that the assumption
of a pure $P$-wave $D\pi$ final state is well justified.\\[-0.3cm]
In both \reftab{SM-predictions} and \reffig{x} we juxtapose the fit results
with their corresponding SM predictions. The latter depend on the $\bar{B}\to D^*$
form factors. Here, we use the form-factor determinations from Refs.~\cite{Bordone:2019guc, Bordone:2019vic}.
All SM predictions are obtained using the EOS software \cite{EOS}. The EOS code for the computation
of $\bar{B}\to D^*\ell\bar\nu$ observables has been independently checked.
We also predict the ratio $R_{e/\mu}$ in the SM and obtain:
\begin{equation}
R_{e/\mu} = 1.0026 \pm 0.0001\,,
\end{equation}
which does \emph{not} include possible structure-dependent QED corrections.\\[-0.3cm]
We emphasize that the predictions \cite{Bordone:2019guc, Bordone:2019vic} of the
$\bar{B}\to D^*$ form factors are conservative in that the corresponding uncertainties
include higher-order contributions in the heavy-quark expansion. Furthermore
they rely only on theory input from various sources,
\emph{i.e.} no experimental input has been used for their determination.
Note that $|V_{cb}|$ cancels in the predictions for the normalized bins
$x_i^{(\ell)}$ as well as in the angular observables; only the total
decay rate is proportional to $|V_{cb}|^2$. Moreover, theoretical uncertainties
of the normalization of the leading hadronic $B\to D^*$ form factor cancel
in the normalized observables. However, we do not include structure-dependent
electromagnetic corrections to the angular distribution. Given the expected precision
of the experimental data and the impact of muon-mass effects as discussed in this
work, we expect that including these effects will become mandatory soon.
\\[-0.3cm]
Before comparing to our numerical SM predictions, we test the qualitative
expectation of approximate lepton-flavour-universality, \emph{i.e.}
$\Delta X\equiv 0$, which does not require a specific form-factor parametrization.
We find that most quantities are
well compatible with lepton-flavour universality, with the exception of
$\braket{A_\text{FB}^{(\ell)}}$, which shows a deviation from exact universality
at the $3.9\sigma$ level, to be discussed below.
This strong violation is not readily observable in the 80~bins provided by
the Belle collaboration, but becomes obvious in the results of the fit of the
non-redundant set of angular observables to the underlying angular distributions, see \reffig{x}.
The violation is further hidden by the fact that the lepton-flavour averaged
data are compatible with the SM expectation.\\[-0.3cm]
In the comparison of our SM predictions with the fit results we find:
\begin{enumerate}
\item
As expected, the precision for most normalized quantities is better
than that for the total rate, typically at the level of a few percent.
This is true for both the SM predictions and the fit results.
\item
Overall we find very good agreement of the fit results with our SM
predictions, as can be seen in \reffig{x}, especially when considering
the individual lepton species. There are a few smaller differences of roughly
$1\sigma$, only $\langleA_\text{FB}^{(\mu)}\rangle$ shows a tension above
the $2\sigma$ level.
\item
The differences of the lepton-flavour-specific observables, $\Delta X$,
are predicted
with very small absolute uncertainties due to the muon-mass suppression.
Their predictions have similar relative uncertainties as the ones for the
angular observables themselves. Their absolute values are also very small, with
$\Delta X/\Sigma X=\mathcal O(\permil)$ in most cases. This can be readily
understood, since these observables receive only corrections of
$\mathcal O(m_\mu^2)$ in the SM. The only sizable central values are those
of $\DeltaA_\text{FB}$ and $\Delta \wT F_L$, which are slightly enhanced by numerical
factors. Most importantly, we find that the latter shifts are still small,
but already comparable to the corresponding experimental uncertainties,
see \reftab{SM-predictions}. This implies that the muon mass cannot be
neglected anymore in precision analyses.
\item
The pattern of the shifts in $\Delta x_i$ is surprising at first sight,
since $|\Delta x_i|/\Sigma x_i$ is almost constant over the whole range
of $w$ (or $q^2$), while we argued that the effect scales like
$(m_\mu/\sqrt{q^2})^2$. This can be understood from the normalization to
the total rate. The shifts in $\Delta(\Delta \Gamma_i)/\Sigma(\Delta\Gamma_i)$
scale as expected, from significantly less than $1\permil$ at $w\sim 1$
(high $q^2$) to $-5\permil$ in the bin with maximal $w$ (lowest $q^2$).
The shift in the total rate is about $-3\permil$, so normalizing yields
shifts in $\Delta x_i/\Sigma x_i$ to the range $[-3\permil, 3\permil]$.
\item
For LFU observables we still find mostly excellent agreement between
experiment and our SM predictions.
However, the aforementioned difference between the measurements of
$A_\text{FB}^{(\mu)}$ and $A_\text{FB}^{(e)}$ becomes more significant,
given the smaller absolute
uncertainty in $\DeltaA_\text{FB}$ and the fact that the relatively large SM
prediction carries the opposite sign from the one determined in the fit.
This quantity differs therefore by approximately $4\sigma$ from its SM
prediction. In \reffig{DeltaAFBDeltaFLtilde} we show the pair-wise
2-dimensional best-fit regions of $\Delta A_\text{FB}$ with $\Delta F_L$,
$\Delta \wT F_L$, $\Delta S_3$, and $\Sigma A_\text{FB}$. The discrepancy with the
predictions reaches the $4\,\sigma$ level, compatible with similar levels seen for the 1-dimensional discrepancy for $\Delta A_\text{FB}$ in \reftab{SM-predictions}.
\end{enumerate}
\begin{figure}
\includegraphics[width=.45\textwidth]{plots/DeltaAFB-DeltaFL.pdf}
\includegraphics[width=.45\textwidth]{plots/DeltaAFB-DeltaFtildeL.pdf}\\
\includegraphics[width=.45\textwidth]{plots/DeltaAFB-DeltaS3.pdf}
\includegraphics[width=.45\textwidth]{plots/DeltaAFB-AFBbar.pdf}
\caption{\label{fig:DeltaAFBDeltaFLtilde}
Fit to the Belle data in the planes of $\Delta A_\text{FB} = A_\text{FB}^{(\mu)} - A_\text{FB}^{(e)}$
vs. $\Delta F_L=F_L^{(\mu)} - F_L^{(e)}$ (top left),
$\Delta A_\text{FB}$ vs. $\Delta\wT F_L=\wT F_L^{(\mu)} - \wT F_L^{(e)}$ (top right),
$\Delta A_\text{FB}$ vs. $\Delta S_3 = S_3^{(\mu)} - S_3^{(e)}$ (bottom left),
and $\Delta A_\text{FB}$ vs. $\Sigma A_\text{FB} = (A_\text{FB}^{(\mu)} + A_\text{FB}^{(e)}) / 2$ (bottom right).
Contours correspond to $68\%$, $95\%$ $99.7\%$, and $99.99\%$ probability, respectively.
The ragged outermost contours are artefacts due to lack of samples
so far in the periphery of the best-fit point.
The SM predictions based on the form factors
obtained in Refs.~\cite{Bordone:2019guc,Bordone:2019vic} are shown as blue crosses.
The SM uncertainties are found to be much smaller than $10^{-2}$ and
hence negligible, with the exception of the last panel.
The uncertainty in the $\DeltaA_\text{FB}$--$\Sigma A_\text{FB}$ plane
is shown as a (highly degenerate) ellipse at the $68\%$ probability level.
}
\end{figure}
These observations mildly depend on the covariance matrix used in the fit.
As stated above, we consider our construction of the $80\times 80$ covariance
matrix reliable to the extent that the data in Ref.~\cite{Waheed:2018djm} are
correct. To make absolutely sure that our assumption regarding the $e-\mu$
correlations is not the reason for the observed discrepancy, we adopt the
following alternative procedure: We determine the $A_\text{FB}^{(e)}$ and
$A_\text{FB}^{(\mu)}$ with separate statistical and systematic uncertainties
in two separate fits to the lepton-specific data, using the corresponding
$40\times40$ covariance matrices for which we do not have to rely on our
assumption. We then \emph{minimize} the discrepancy with respect to our
(strongly correlated) SM predictions by assuming a diagonal $2\times2$
statistical correlation matrix for $A_\text{FB}^{(e)}$ and $A_\text{FB}^{(\mu)}$, but
allowing for an arbitrary correlation $\rho \in[-1, 1]$ between the
systematic uncertainties.
We find that the minimal tension with respect to
the SM for the combined $A_\text{FB}^{(e)}$, $A_\text{FB}^{(\mu)}$ occurs for maximal
anti-correlation ($\rho = -1$), which is not a realistic value.
The correlation determined in the fit to the $80\times80$ covariance matrix is
actually very small.
Adopting nevertheless this most conservative choice of $\rho = -1$ still leads to a tension of $3.6\sigma$.
We emphasize again that this result is not changed by employing the
pseudo-Monte Carlo approach with Cholesky decomposition for the fit as done
in \cite{Ferlewicz:2020lxm}, nor by the d'Agostini effect (the plots shown in \reffig{x}
include the corresponding shifts). Therefore, even adopting this maximally
conservative procedure, our results amount to evidence for $\mu$-$e$-non-universality
beyond the SM in charged-current $b\to c\ell\nu$ transitions. However, our finding
hinges on the approximate validity of the data and specifically the
correlation matrices given in Ref.~\cite{Waheed:2018djm}.
We also perform a full SM fit to the $2\times 14$ observables in
\reftab{SM-predictions}, including their correlations given in ancillary files attached to the arXiv preprint of this article. Starting from a fit of form-factor parameters from
theory input, only \cite{Bordone:2019guc, Bordone:2019vic}, the inclusion
of the experimental information on these 28 observables increases the
minimal $\chi^2$ by $68.5$, while only $|V_{cb}|$ is introduced as an
additional parameter in the fit. This does indicate a bad fit, with a
$p$ value of $2\times 10^{-5}$, or a tension at the $4.3\sigma$ level. The discrepancy remains
driven by a $\sim 4\sigma$ tension in $\braket{A_\text{FB}^{(\mu)}}$ and a $\sim 2\sigma$ tension
in $\braket{A_\text{FB}^{(e)}}$. The experimental and theoretical correlations
with other observables play a minor role, see also \reffig{DeltaAFBDeltaFLtilde}.
We note in passing that S-P wave interference cannot affect the numerator of $A_\text{FB}$,
and can only decrease the magnitude of $A_\text{FB}$ by a coherent contribution to the
denominator \cite{Matias:2012qz}.
We refrain from providing the value of $|V_{cb}|$ from either lepton mode,
which would be compatible with the values obtained from the lepton-flavour
average in Refs.~\cite{Waheed:2018djm, Ferlewicz:2020lxm} and continue to exhibit a substantial
tension with respect to the inclusive determination $|V_{cb}|_{B\to X_c} = (42.00 \pm 0.64)
\cdot 10^{-3}$~\cite{Gambino:2016jkc}.
Given the incompatibility of the data with the SM prediction, we consider it misleading
to use it to extract $|V_{cb}|$.
To summarize, we find in our fits a discrepancy between data and the SM of $\sim 4\sigma$.
This result is stable with respect to the treatment of the d'Agostini bias,
the type of fit we are performing ($\chi^2$ fit vs.~pseudo-Monte Carlo
techniques), and importantly also the precise treatment of the correlations
of the systematic uncertainties between electrons and muons. We reiterate,
however, the concerns discussed in \refsec{BelleData}: the statistical correlation
matrices given in~\cite{Waheed:2018djm} do not seem to be correct, since they
are not singular as they should be, given the performed redistribution of
events to obtain the different single-differential rates. Bearing this caveat
in mind, we still investigate in the following the possibility that the observed
discrepancy is an effect of BSM physics.
\subsection{Possible BSM interpretation}
We consider the possibility that the observed discrepancy is due to BSM physics.
To that aim, we investigate the Lagrangian \refeq{NPlagrangian} in the limit of
lepton-flavour conservation $\ell=\ell'$. From our general analysis in
\refsec{fit-model} we have seen that $\braket{A_\text{FB}^{(\ell)}}$ is special in that
it is determined to $\mathcal O(m_\mu)$ only by interference contributions
$\sim \re(C_i^\ell C_j^{\ell*})$, and is the only observable in the
single-differential distributions to which interference terms contribute in the
massless limit. Given the size of the observed effect, $\DeltaA_\text{FB} /
\SigmaA_\text{FB}\sim\mathcal O(10\%)$, a muon-mass suppressed contribution does
not seem likely as its source. This suggests that in order to accommodate
$\DeltaA_\text{FB}$, the first options to consider are BSM contributions to right-handed
vector operators, to both pseudoscalar and tensor operators, or to left-handed
vector operators. Notably, the first two options correspond to second-order BSM
contributions: for the interference between pseudoscalar and tensor operators
this is obvious. For the right-handed vector operator the interference term
$\re(C_V^\ell C_A^{\ell*}) = |C_{V_R}^\ell|^2-|C_{V_L}^\ell|^2$ is manifestly
second order in the $C_{V_R}^\ell$. For the BSM contributions to the left-handed
vector operator only, the discussion is more involved. The interference terms
$\re(C_V^\ell C_A^{\ell*}) = |C_{V_R}^\ell|^2-|C_{V_L}^\ell|^2$ contain in
principle a linear contribution in $|C_{V_L}^\ell|^2 = |1 + \Delta
C_{V_L}^{\ell,\text{BSM}}|^2$, wherein the $1$ stands for the SM contribution.
However, if $C_{V_L}^\ell$ were the only BSM contribution it would cancel in
all normalized observables. This is not true for the contribution from
right-handed vector operators, the real parts of which, however, enter linearly
in $|C_{A,V}^\ell|^2$. Given the compatibility of all other observables with
the SM, this scenario would therefore require the main contribution to either
have a sizable imaginary part, or specific cancellations with other BSM
contributions, in order not to upset this agreement.\\[-0.3cm]
Taking here the Belle data at face value, we perform fits analogous to the ones
described above, including different sets of BSM contributions. Note that we
keep our description qualitative, since numerical statements are likely to be
upset by an eventual correction of the Belle dataset~\cite{Waheed:2018djm}.
For the same reason we do not perform a combined fit with other
$b\to c\ell\bar\nu$ modes, which would of course be required to confirm the
viability of potential BSM scenarios that resolve the tension in this dataset. \\[-0.3cm]
We find that either contributions from right-handed vector operators, or from
both pseudoscalar and tensor operators are necessary to accommodate the observed
$\Delta A_\text{FB}$, confirming our previous considerations. In order to describe the
dataset well with real BSM Wilson coefficients, only, LFUV contributions to both
the right- \emph{and} left-handed vector operators are required.
The three minimal BSM scenarios that fit the present Belle
$\bar B\to D^*\ell\bar\nu$ data~\cite{Waheed:2018djm} can be summarized as follows:
\begin{enumerate}
\item
$C_{V_R}^\ell\neq 0$: This scenario does require a sizable imaginary part
(as anticipated above) and LFU violation. The latter fact is interesting,
since it might point to BSM physics beyond SMEFT \cite{Cata:2015lta}.
The imaginary part of $C_{V_R}^\ell$ implies that $\braket{A_8^{(\ell)}}$
and $\braket{A_9^{(\ell)}}$ are sizable. We strongly encourage an
experimental measurement of these observables. \\[-0.5cm]
\item
$C_{V_R}^\ell\neq 0$ and $C_{V_L}^\ell\neq 1$: This scenario can obviously
describe the data well, given that in principle already $C_{V_R}^\ell\neq 0$
suffices. However, to our surprise it is also compatible with an LFU BSM
contribution to $C_{V_R}^\ell$, which is required in a SMEFT scenario.
Enforcing this flavour-universal $C_{V_R}^\ell$, \emph{i.e.,} $C_{V_R}^e =
C_{V_R}^\mu$, results in significantly different absolute values and a
sizable phase difference between $C_{V_L}^e$ and $C_{V_L}^\mu$. Sizable
$\braket{A_{8,9}^{(\ell)}}$ are also likely in this case, although not strictly
necessary. It is possible to have all BSM coefficients real, and hence
$\braket{A_{8,9}^{(\ell)}} = 0$, but only with a phase between the left-handed coefficients
$\phi_L=\pi$. This corresponds to a BSM contribution of about twice the
SM one and is therefore highly fine-tuned.\\[-0.5cm]
\item
$C_{P}^\ell\neq0$ and $C_{T}^\ell\neq0$: Also this scenario provides a
good fit to the data, both for complex and real-valued Wilson coefficients.
The fact that both $C_T^\ell$ and $C_P^\ell$
are required means that this scenario can be tested by measuring
$\braket{S_{6c}^{(\ell)}}$ and $\braket{A_7^{(\ell)}}$, at least one of which is
expected to show significant differences relative to their SM predictions, which
are small for $\braket{S_{6c}^{(\ell)}}$ and zero for $\braket{A_7^{(\ell)}}$.
\end{enumerate}
While we do not attempt to include additional datasets as explained above and
therefore cannot quantitatively test specific BSM scenarios, we still observe
a few general features of a possible BSM explanation in the context of the
$B$ anomalies, especially in $b\to c\tau\bar\nu$ transitions:
\begin{enumerate}
\item
While moderate shifts in one or several Wilson coefficients are required
to fit the present Belle data \cite{Waheed:2018djm}, the total rates are
not strongly affected. Hence it is not possible to explain the discrepancy
in $R(D^{*})$ with these shifts, \emph{i.e.} additional new contributions
in $b\to c\tau\bar\nu$ coefficients are required to explain
the deviations of LFU ratios involving $\ell = \tau$ from SM predictions.
\item
If the observations made here based on the Belle data persist after future
updates or corrections, they would have strong implications for scenarios
addressing the $B$ anomalies: Scenarios that only shift $C_{V_L}^\ell$
would be ruled out, which are currently favoured as simultaneous explanations
of the $b\to c\tau\bar\nu$ and $b\to s\ell^+\ell^-$ anomalies.
\item
Based on the picture provided by the observables, one would naively expect
a hierarchy $\Delta_\mu > \Delta_e$. In light of the more substantial
deviations in $b\to c\tau\bar\nu$, this could be extended to $\Delta_\tau >
\Delta_\mu$, which is quite natural in scenarios addressing both $B$ anomalies.
However, we find that $\Delta_\mu > \Delta_e$ is far from being established
in our fits at the level of the Wilson coefficients.
\end{enumerate}
There will therefore be far-reaching consequences for the field of particle
physics, should this discrepancy be confirmed.
\section{Conclusions}
\label{sec:conclusions}
In this article we pave the way for precision analyses of $b\to c\ell\bar\nu$
processes beyond the assumption of $e-\mu$ universality. This endeavour is
important for the determination of $V_{cb}$ in the Standard Model, a complete
understanding of the weak effective theory (WET) beyond the SM (BSM), and also
to gain new insights into the persistent $b\to c\tau\bar\nu$ anomaly.
We focus on the angular distribution in $\bar{B}\to D^* \ell\bar\nu$ with light
leptons $\ell = e, \mu$ and highlight strategies for improved experimental
analyses.\\[-0.3cm]
We discuss the complete set of CP-even and CP-odd angular observables that arise
from the fully-differential angular distribution of $\bar{B}\to D^* (\to D \pi)\,
\ell\bar\nu$. In particular we discuss the influence of a finite mass of the
charged lepton on these observables in and beyond the SM. We consider in detail
the specific case of four single-differential CP-averaged rates that have been
experimentally analyzed in Refs.~\cite{Waheed:2018djm, Abdesselam:2017kjf}.
We find that only four flavour-specific angular observables per lepton flavour
are sufficient to describe the three single-differential CP-averaged angular
distributions including arbitrary BSM contributions: the lepton-forward-backward
asymmetry $A_\text{FB}^{(\ell)}$, the longitudinal $D^*$-polarization $F_L^{(\ell)}$,
and two further observables $\wT F_L^{(\ell)}$ and $S_3^{(\ell)}$. However, we
find that it is principally not possible to extract the full information on the
BSM contributions to the WET Wilson coefficients for the electron mode when using
only the single-differential CP-averaged rates. For the muon mode, part of that
information enters only muon-mass suppressed, although it can be extracted
without that suppression when considering a different presentation of the data.
We further emphasize the existence non-linear relations between the Wilson
coefficients that allow to test for lepton-flavour violation (LFV) and
right-handed neutrinos.\\[-0.3cm]
The most precise lepton-flavour-specific analysis to date \cite{Waheed:2018djm}
presents the three CP-averaged single-differential angular distributions for
electron and muon flavours separately. Since they depend on only four angular
observables per lepton flavour, the chosen number of kinematic bins is much
larger than necessary. We show that this redundant presentation accidentally
hides tensions between SM predictions and data. We encounter an issue with the
statistical correlation matrices that can only be clarified by the Belle
collaboration. We describe our approach to the combination of statistical and
systematic correlations for the electron and muon datasets and extract the
non-redundant lepton-flavour specific CP-averaged angular observables from the
Belle data. For most of the angular observables we find good agreement with our
up-to-date SM predictions, except for $A_\text{FB}^{(\mu)}$.
The observed tension with the SM predictions is even more pronounced for the
observable $\Delta A_\text{FB} \equiv A_\text{FB}^{(\mu)} - A_\text{FB}^{(e)}$ in which the
correlations of form factors lead to a strong cancellation of uncertainties,
reaching the $4\,\sigma$ level.
We perform numerous checks that this tension is not a result of our specific
treatment of the data. In particular, even when allowing for arbitrary
systematic correlations between the electron and muon data, we find that this
tension does not drop below $3.6\,\sigma$.
This constitutes evidence for lepton-flavour universality violation.\\[-0.3cm]
We continue by investigating in a qualitative manner the most economic BSM
scenarios that can potentially explain the observed tensions. To this end,
we assume lepton-flavour conservation, but allow for lepton-flavour non-universality
in the WET description. We find that either right-handed vector operators
or both pseudoscalar and tensor operators are necessary to accomodate the
observed tension. If only right-handed vector operators are present, large
imaginary parts in the Wilson coefficients are necessary. As a consequence,
the CP-odd angular observables $A_{8,9}^{(\ell)}$ would be expected to deviate
sizably from their SM predictions. A solution with purely real-valued Wilson
coefficients appears only as a highly fine-tuned solution in a combined scenario
with left- and right-handed vector operators. For the combination of pseudoscalar
and tensor operators, we do not find the necessity of sizable imaginary parts.
In this case, $\braket{S_{6c}^{(\ell)}}$ or $\braket{A_7^{(\ell)}}$
are expected to show significant differences relative to their SM predictions.
None of these three scenarios coincides with the preferred explanation of
the $b\to c \tau\bar\nu$ anomaly.\\[-0.3cm]
Given the far-reaching consequences of our findings, we consider it essential
that the Belle collaboration reviews --- and if need-be corrects --- the
published dataset from Ref.~\cite{Waheed:2018djm}. Without such scrutiny, we
cannot determine the impact of the identified issues on results inferred from
the data. We strongly recommend that
future measurements separate between the two light-lepton flavours in a transparent
way. This is also important for the comparison with existing and upcoming
LHCb analyses, which focus on the muon mode, only.
\acknowledgments
We are very grateful to David Straub for early discussions of the Belle data and
their possible interpretation in a BSM fit. We also thank Paolo Gambino for
useful discussions. We are grateful to a number of
members of the Belle collaboration for help with the interpretation of the Belle
data, especially Eiasha Waheed, but also Florian Bernlochner, Daniel Ferlewicz,
Daniel Greenwald, Thomas Kuhr, Christoph Schwanda, and Phillip Urquijo.
The work of CB is supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under grant BO-4535/1-1.
The work of MB and MJ is supported by the Italian Ministry of Research (MIUR) under
grant PRIN 20172LNEEZ.
The work of NG is partially supported by DFG under grant 396021762 -- TRR 257
``Particle Physics Phenomenology after the Higgs Discovery''.
The work of DvD is supported by the DFG within the Emmy Noether Programme
under grant DY130/1-1 and by the NSFC and the DFG through the funds provided
to the Sino-German Collaborative
Research Center TRR110 ``Symmetries and the Emergence of Structure in QCD''
(NSFC Grant No. 12070131001, DFG Project-ID 196253076 - TRR 110).
This research was supported by the Cluster of Excellence ``ORIGINS'' and the
Munich Institute for Astro- and Particle Physics (MIAPP) which are funded by
the DFG under Germany's Excellence Strategy -- EXC-2094 -- 390783311.
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{"url":"https:\/\/www.thejournal.club\/c\/paper\/189\/","text":"2-State 3-Symbol Universal Turing Machines Do Not Exist\n\nCraig Alan Feinstein\n\nIn this brief note, we give a simple information-theoretic proof that 2-state 3-symbol universal Turing machines cannot possibly exist, unless one loosens the definition of \"universal\".\n\narrow_drop_up","date":"2022-01-19 05:50:41","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8502888679504395, \"perplexity\": 1982.9025632005598}, \"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-2022-05\/segments\/1642320301263.50\/warc\/CC-MAIN-20220119033421-20220119063421-00692.warc.gz\"}"} | null | null |
import { PureComponent } from 'react';
export default { create };
function create(coroutine) {
class Coroutine extends PureComponent {
constructor(props) {
super(props);
this.state = { view: null };
this.iterator = null;
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iterate(props) {
let target = coroutine(props);
this.iterator = target;
let shouldStop = () => this.iterator !== target;
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if (target && typeof target.then === 'function') {
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componentDidUpdate(prevProps) {
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return this.state.view;
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Coroutine.displayName = coroutine.name || coroutine.displayName || 'Anonymous';
return Coroutine;
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function resolveAsyncIterator(iterator, done, shouldStop) {
return iterator.next().then(data => {
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for (let k in a) {
if (a.hasOwnProperty(k)) {
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Whether you're learning to play the piano again, after a decade's long hiatus; or you're trying to teach a young child according to best practices, you may have been led astray by the following 3 myths about what it takes to become an adept (or even a passable) pianist.
Myth #1: You can learn to play by watching online tutorials and by supplementing with "computer based" practice, reading about music theory on your iPad, etc.
Reality: There is no substitute for real world practice, guided by a live instructor.
Playing piano is a physical activity. You would never expect someone to learn how to ice skate correctly by studying ice-skating on the computer. You need a real piano – maybe a Yamaha upright piano – and an experienced teacher to show you correct hand positioning, instruct you on the dynamic features of the instrument, etc. This isn't to say that virtual coaching or computer-assisted learning can't supplement and support your quest.
Myth #2: You can find special systems to transform you into a brilliant pianist quickly.
To gain expertise on the instrument, you need to put in hours of hard practice. Your body and mind (and some might even say soul) must grow to understand the instrument on a deep level.
Myth #3: All practice is good practice.
Reality: Incorrect practice may be worse than no practice at all. You need correct practice.
Correct, deliberate practice is a very specific kind of practice.
Yes, you need time to "mess around" on your piano — to noodle around in an unstructured way to develop your musicality and just "have fun." However, you need to lay in a solid foundation of musical and piano knowledge. You need to learn basic music theory; scales; good hand technique; good rhythm; and general musicianship and sensitivity.
Of course, if you don't have a piano, your quest to learn the instrument is going to be a lot harder!
If you live in or around the Northern California area, come see us here at Pianos Plus. We're a family run, Bay Area piano store that's been serving budding musicians like you for four decades with excellent service.
This entry was posted in Frequently Asked Questions, Pianos How to's and Tips and tagged Bay Area pianos, piano lessons, yamaha by Pianos Plus. Bookmark the permalink.
Good deal, although I can't really get behind a couple things (more on this later but first the agreements!) I tell folks all the time that they can try to find free crap online but they should note my site just in case it doesn't work. They'll be back! I love the comparison about ice skating – that is sheer genius! Right!, playing at any level is purely physical (although you do have to engage the mind right at the start!) Now on to the few things above I disagree with. 1. You seem very focused on the acoustic piano. Why? 2. Not everyone wants to be really, really good – most of the folks I've taught (1000's in 25+ years) wanted only a nice musical hobby. | {
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Business and Society 2.0: Why do we need to invest in knowledge management?
art information processing and intellectual assets in a sustainable value for customers and employees - Pricewaterhouse Coopers; Read more Why do we need to invest in knowledge management?
Most companies do not consider the management of the knowledge inside and across the organization, which is created and refined by their professionals. The knowledge is them whitin the heads of each emplyeee, it is not shared, will not be possible to improve this knowledge and contribute to improve the overall performance of the companies. Sharing the knowledge and creating a metholology to organize this knowledge is a key and strategic issue. | {
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" Frank Schaeffer: I'm Now a 'Liberal' Because I'm a Conservative "
Frank Schaeffer: I'm Now a 'Liberal' Because I'm a Conservative
By Frank Schaeffer on 11/4/2009, 11:39am PT
I grew up in a fundamentalist missionary family that in the 1970s and 80s morphed into my father's activity as one of the founders of the Religious Right. We would hobnob with Republican leaders from Ronald Reagan to Gerald Ford and the Bush family, Jack Kemp and many others. One day it dawned on me that the far right of the Republican Party --- in other words its base --- actually hates America.
The Religious Right reveled in rising crime statistics, "family breakdown" statistics, failing public schools and so forth. As I explain in my book Patience with God: Faith for People Who Don't Like Religion (or Atheism) , if crime started going down, or public school test results started going up --- without the country "turning back to Jesus" --- then that would prove that somehow "we" were wrong.
We wanted our country to fail because it had "turned away" from what we believed to be true.
Combined with the fact that we began to lose parts of the culture war, when it came to other Americans beginning to recognize gay rights, expanding women's rights, abortion rights and such, the Religious Right and the Republican Party infected gun-toting America with a chip on its shoulder about a mile wide. This led to the myth that "they" (fill in the blank, gays, Jews, blacks, liberals --- whatever) are "taking away our country from 'us'"...
"Conservative" means that you believe it's right to legalize torture, but reject health care for all.
These days to be a conservative means that you hate the United States government elected by the people; believe that if millions of citizens are out of work that it's their own fault and that the rest of the community should not help them by spending tax dollars; think that Sarah-believes-in-casting-out-demons-before-she-ran-for-governorship-Palin speaks for you. To be a conservative means you believe that healthcare reform will lead to "death panels"; that the president of the United States is not a "real American"; that a university education is a dangerous thing; that Americans who live in big cities are less American than those who live in small towns; that brown people, blacks, progressive whites, gays, public school teachers, Hispanics, immigrants, are somehow conspiring to subvert the "real America" with a "gay agenda" or a "Muslim agenda" or at least the browning of "our" white America.
In other words to be a conservative today is to be an anti-American, nihilistic libertarian know-nothing who believes in unregulated consumerism and the theology of dominion, and the Rapture that many conservatives also subscribe to along with such "facts" as that Obama is the --- literal! --- Antichrist.
Other than trying to stop women from having abortions and fighting the whole world, our "terrorist enemies", in other words everyone "not like us", conservatism today is nothing more than a pent up reaction against everything "we" don't understand --- like art, literature, government, history, geography, diversity, how people get to be gay, black or female... things like that.
Conservatism today is actually not for anything. It is just against everyone but "us" and a few like us bound together by an alternative reality, otherwise known as Fox/NRA/Beck/Palin/Jesus's Return--"News."
The irony is that conservatives used to wrap themselves in the American flag and belonging to a cause built on higher ideals than pure selfishness and individual choice. Patriotism was based on principle, not fear and anger. Conservatism led by people such as the late William F. Buckley, Barry Goldwater and others had its feet firmly planted in what it regarded as the reality-based community as opposed to liberal wishful thinking about progress coming from government, human nature, etc.
The problem for the conservative movement --- hence the Republican Party --- is that the us in "us" was never more narrowly defined.
No one said it openly, in fact it was denied, but it really amounted to we Real Americans boiling down to mostly uneducated white people, dumb enough to believe things such as Sarah Palin's barefaced lies about Obama consorting with terrorists, and/or, post the Obama election, conspiring to unleash "death panels" on unsuspecting elderly and/or handicapped Americans while turning us into a "Communist state" as everyone knows Hitler did to Germany, that other "communist country" famous right up there with Canada and the UK for killing its sick, tired and poor.
What is the conservative movement today, and/or the Republican Party?
It's about as far away from conservatism as it can get. It is a party ready to trash its own country in support of nihilistic, selfish market-driven "values" the very opposite of conservative values of family, community and stability. It is in fact what conservatives of the 60s said the hippies were: selfish brats with no sense of responsibility to anyone. It's also a party of armed revolution not so subtly egging on its lunatic fringe to commit violence. It applauds white rubes who show up at public meetings carrying loaded assault weapons "to make a point" and signs reminiscent of Timothy McVeigh and his famous T-shirt; "the tree of Liberty must be watered with the blood of tyrants" and the like are held up by Murdoch/Beck/Fox and company --- those profiteers off the unregulated market --- as paragons of good sense and free enterprise and gun rights.
To be an actual conservative today is to be a progressive Democrat.
An actual conservative believes in community and accountability to a moral tradition that puts the greater good of others ahead of oneself. Take a look at the way the very conservative communities of New England's Puritan towns were arranged around the village green known as "the commons."
Shared public spaces were owned by the community, for instance grazing land, and town meetinghouses. People were obliged to show up and participate in the fledgling democracy and vote. Taxes were dispensed by committees for charitable purposes. A duty to government and obligations placed on citizens by other citizens --- when it came to putting the life of the community ahead of the self --- were the norm. The free-market and individual enterprise were strictly curtailed based on not just the needs of the community but, when it came to things like banking and lending, the Old Testament teachings that frowned on "usury" --- in other words banks making more money than they should from ordinary people-- were upheld.
President Obama is a conservative. He believes in the brotherhood of all people. He believes in the freedom of the individual to make moral decisions. He believes that sexuality, religion and skin color should not define us but the content of our characters should define us. He believes that we are our brother's keeper. He believes in loyalty to community and country --- in other words patriotism, whether that's the honor of serving in the military or the honor of paying taxes to support not just national defense but how we treat what the Bible calls the least amongst us.
People ask me why I'm a progressive these days and "changed sides" from being a conservative. I didn't change sides.
What changed --- ironically with my father's and my nefarious "help!" --- was a conservative movement that became an enclave for hate-filled ignorance, anti-American sentiment and nihilistic individualism. What changed was my bare faced self deception as I profited from the God business and the far right even though I knew better. Today I am an independent voter, and an Obama supporter, and a progressive because I am a conservative.
Frank Schaeffer is the author of Crazy for God: How I Grew Up as One of the Elect, Helped Found the Religious Right, and Lived to Take All (or Almost All) of It Back and the forthcoming Patience with God: Faith for People Who Don't Like Religion (or Atheism) .
He will be making appearances across the country over the next several months to promote his new book. His current tour schedule is here. Please stop by to say hello if he comes to your home town, and tell him you read him on The BRAD BLOG!
Previously from Frank Schaeffer at The BRAD BLOG:
• 10/22/09: "A Letter to Rupert Murdoch"
• 10/23/09: Brad Interviews Schaeffer on Green960
• 11/2/09: "The Loony Right Eats Its Own"
Article Categories: Fox 'News', Rights And Freedoms, Accountability, Barack Obama, Health Care, Glenn Beck, Republicans, Rush Limbaugh, Marriage Equality, Torture, Sarah Palin, Bush Legacy
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"Frank Schaeffer: I'm Now a 'Liberal' Because I'm a Conservative"
... jhimmi said on 11/4/2009 @ 1:41 pm PT...
Funny, I guess I'm now a conservative, BECAUSE I'M LIBERAL. I think a huge, totalitarian federal government running a brave-new-world society will put a crimp in my free-spiritedness.
... Joshua said on 11/4/2009 @ 3:30 pm PT...
Thoughtful article Mr. Schaeffer. I was reared in a more liberal Christian community in the northern San Francisco bay area; however, it was the early to mid eighties and the fervor of divine right to all of America and the world by the Christian community was somewhat disgusting. I am an atheist now. Much simpler.
... don knuttel said on 11/4/2009 @ 4:16 pm PT...
Mr. Schaeffer,I think you are desperately needed in the Congress. You surly have my vote. Corporations have taken over the Republican Party and going hard after the Democrats.
How many of your past associates really think as you do but can't get out, because the Money is good?
... Lora said on 11/4/2009 @ 4:46 pm PT...
Frank Schaeffer wrote:
It is a party ready to trash its own country in support of nihilistic, selfish market-driven "values" the very opposite of conservative values of family, community and stability.
Indeed. Yes, indeed. Those nihilistic, selfish, market driven corporate moguls are using the religious right to advance their own nefarious purposes: gather as much money and power for themselves as possible. They are conducting as brilliant a psy-op as I could imagine.
... Soul Rebel said on 11/4/2009 @ 7:28 pm PT...
Frank - I read your article on Alternet yesterday, and I'll leave the same comment here as I did there (granted, your writing here is not the same, but what was posted there makes me not want to read your post here - I will, and maybe I'll comment on it later in the thread):
WHAT A STUPID ARTICLE
No, of course "New Atheists" are not as dangerous as fundamentalist religious kooks. For one, we are not likely to kill someone over whether or not God exists. Why is it not OK to view the entire concept of 'faith' as idiocy? It certainly has a historical precedent as such. Seriously, what is so virtuous about believing in something with no evidence? (No, the bible does NOT count as evidence, not in any meaningful way.)
People are welcome to 'believe' whatever LooneyTunes ideas they want. They are all as LooneyTunes as each other - virgin birth, 72 virgins upon martyrdom, resurrection after 3 days, jesus came among the native americans - take your pick, all equally loopy...on par with the actual belief in a god, which is no more or less nutty than any of the surrounding myths. I would just like to see us get to a point where believing in god is something people grow out of, like the tooth fairy or the easter bunny or santa claus.
I don't have to respect "belief". I can accept that people "believe" many strange things, but I don't have to respect it. Any real moral ideas (the Golden Rule being the big one) do not need a god for validity.
Aside from writing articles on liberal blogs (no, I won't use the word conservative to describe myself ever) and writing books that seem to pander to a liberal audience (no, I haven't read any of them, but if your aforementioned Alternet piece is any kind of representation of your work, I won't be reading them) what have you done, Mr. Schaeffer, to repent for your time spent being part of the 'haters.' It's all well and good to throw mea culpae towards a receptive audience (oh goody, he's one of us now), but you were at one point a powerful cog in a destructive and traitorous machine, one that justified the death of kids like Matthew Shepard. I'm certainly glad that you are no longer a part of that machine, but what are you really doing to undo the 'sins' of your past? How are you taking on the current Religious Right directly? Have you had a conversation with Fred Phelps about how you were wrong? What about other more general haters like George Bush (apparently you have/had access)? Glenn Beck? Michael Savage? Lou Dobbs?
Preaching to the choir, telling us all here things about conservatism that we already knew, sorry it don't impress me much.
BTW - I did read the letter to Rupert Murdoch. While it was quite a good letter, I have a feeling he didn't read it. I have a feeling it was really for us.
... colinjames said on 11/5/2009 @ 12:05 pm PT...
Maybe you should change your name to "Rebel Without a Soul"... Negative much? Sheesh! What do you expect the guy to do get down on his knees and beg your forgiveness? If his book makes even one so-called conservative see the light regarding the hypocrisy of the religious right, he'll have gone a long way towards atoning for his previous crimes against community. Being intolerant of people of faith and trashing their belief just makes you guilty of the same kind of behavior you seem to be so against. NOT progressive in my book.
By the way, where did I say I was "intolerant" of people of faith? I fully support everyone 1st Amendment guarantee to freedom of religion. That I think "faith" is NOT a virtue, nor is it rational, and that I also think religion is the most destructive force on the planet does not render me intolerant.
What I AM intolerant of is people making idiotic statements like Schaeffer's "New Atheists are as dangerous as religious fundamentalists." Not true, not even close to being true. Atheists have been a persecuted minority since the dawn of religion (not like gays, or blacks in contemporary history, but certainly in the histiry of mankind. Recent polls have said that the least likely demographic to be elected to public office is Atheist/Nonbeliever. THAT is a ridiculous and troubling commentary on the state of our society!) Now that we are becoming a vocal minority that is actually standing up for ourselves, everyone starts pissing in their pants.
So I guess people will always read into things what they want. And my moniker should be interpreted exactly as you have indicated (at least that is one of my intents) - I Rebel against the very notion of a Soul.
... David Ashton said on 11/6/2009 @ 12:16 am PT...
Dear Frank
Having read some of your father's books I find it hard to see him as a Fundamentalist, especially of the American variety.
He did not seem to be a ranter or bigoted in his opinions and ideas. To rank him with people like Jerry Falwell, Oral Roberts, and the Bush family seems unfair if not unreal.
I agree that the christian right seems to hate America, or at least democracy. They are your equivalent of the Taliban and others like them, and it seems that it will not take much for them to turn to violence to get what they want.
I think in some ways that might not be a bad thing, since it seems the only way some countries can only rid themselves of the sort of poison that permeates your country.
I am continually disappointed that President Obama does not allow the Justice department to pursue Bush and co for their crimes (they can't be called anything else), especially their incompetence or deliberate willful neglect of the prospect of the 9/11 attack, and their callous use of it and the trust of the American people in going to war against Iraq.
... tim said on 11/11/2009 @ 11:02 pm PT...
Most Christians have an experience similar to that Palin-demon-smear, you mention. Anybody who converts to Catholicism has that within their Rite. You twist and conflate in a hateful way.
Now I've heard it all, "nihilistic individualism". And these so-labeled nihilists are the family people, church-goers, taxpayers and producers in society, it seems. So you disagree with them. Fine. That justifies this anti-intellectual smear?
Obama is many things, and "conservative" isn't one. Again, twisting a definition, or revising facts to fit one is off-the-wall.
Have conservatives changed? Or has society's expectation of what they must DO, ACCEPT & BELIEVE changed? They don't want to go along with the changes proposed, so you change definitions and smear them.
They are the same people they were 30-40 years ago, but won't buy in to what you want. They resist and you hate them for it.
Hint: Leave them alone, and all the fireworks go away, Mr. Social Engineer.
I'd hate to see what drives your demagogic rant...it could even be nihilism in your psyche.
... Brian said on 11/13/2009 @ 8:05 pm PT...
What a completely ridiculous article but certainly a window into a demented delusional mind that takes truth and twists it to such a degree as to be unrecognizble. | {
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{"url":"https:\/\/softwareengineering.stackexchange.com\/questions\/86254\/what-are-the-consequences-of-immutable-classes-with-references-to-mutable-classe?answertab=active","text":"# What are the consequences of immutable classes with references to mutable classes?\n\nI've recently begun adopting the best practice of designing my classes to be immutable per Effective Java [Bloch2008]. I have a series of interrelated questions about degrees of mutability and their consequences.\n\nI have run into situations where a (Java) class I implemented is only \"internally immutable\" because it is immutable except that it uses (immutable) references to other mutable classes. To clarify, if the object were constructed with its references set to immutable classes, the class would be immutable.\n\n1. Do any of the benefits (see below) of immutable classes hold true even by only \"internally immutable\" classes?\n\n2. Is there an accepted term for the aforementioned \"internal mutability\"? Wikipedia's immutable object page uses the unsourced term \"deep immutability\" to describe an object whose references are also immutable.\n\n3. Is the distinction between mutability and side-effect-ness\/state important?\n\nThe \"Java Practices\" site lists the following benefits of immutable classes:\n\n\u2022 are simple to construct, test, and use\n\n\u2022 are automatically thread-safe and have no synchronization issues\n\n\u2022 do not need a copy constructor\n\n\u2022 do not need an implementation of clone\n\n\u2022 allow hashCode to use lazy initialization, and to cache its return value\n\n\u2022 do not need to be copied defensively when used as a field\n\n\u2022 make good Map keys and Set elements (these objects must not change state while in the collection)\n\n\u2022 have their class invariant established once upon construction, and it never needs to be checked again\n\n\u2022 always have \"failure atomicity\" (a term used by Joshua Bloch) : if an immutable object throws an exception, it's never left in an undesirable or indeterminate state\n\nWell, simply put; can your state change?\n\nThis is not just through your objects API, but in every way an object can change state.\n\n\u2022 You can't leak mutable references.\n\u2022 You can't inject mutable references.\n\u2022 You can't change state on any references, not even if the object is completely self contained.\n\nYou can initialize a mutable object, as long as no calls are ever made that change state, but you can still query it. So, if internal mutability is not what I described here, then no, you can't have that.\n\nWell, the concept is reading the JLS and understanding it. In this case, the JLS says:\n\nfinal fields also allow programmers to implement thread-safe immutable objects without synchronization. A thread-safe immutable object is seen as immutable by all threads, even if a data race is used to pass references to the immutable object between threads. This can provide safety guarantees against misuse of an immutable class by incorrect or malicious code. final fields must be used correctly to provide a guarantee of immutability.\n\nThe usage model for final fields is a simple one: Set the final fields for an object in that object's constructor; and do not write a reference to the object being constructed in a place where another thread can see it before the object's constructor is finished. If this is followed, then when the object is seen by another thread, that thread will always see the correctly constructed version of that object's final fields. It will also see versions of any object or array referenced by those final fields that are at least as up-to-date as the final fields are. So you need to:\n\nMake address both final and private. Prevent any mutable reference to that object from being seen externally to your immutable object. In this case, #2 probably means you can't return a reference to Address like you have with getAddress(). And you have to make a defensive copy. Make a copy of the object, and store that copy in Employee. If you can't make a defensive copy, there's really no way to make Employee immutable.\n\npublic final class Employee{\nprivate final int id;\n{\nthis.id = id;\n}\npulbic int getId(){\nreturn id;\n}\n}\n\n\nImplementing clone() or something similar (a copy ctor) would make creating defensive objects easier for complicated classes. However, the best recommendation I think would be to make Address immutable. Once you do that you can freely pass around its reference without any thread-safety issues.\n\nIn this example, notice I do NOT have to copy the value of street. Street is a String, and strings are immutable. If street consisted of mutable fields (integer street number for example) then I would have to make a copy of street also, and so on ad infinitum. This is why immutable objects are so valuable, they break the \"infinite copy\" chain.\n\nIt doesn't really matter as long as the class you say is immutable cannot be observed to be mutable. E.g., memoisation uses mutation but a memoised value can be treated as immutable for all intents.\n\nIf I can't tell that your object uses mutable variables internally then it doesn't compromise my program and I can compose it with other objects as usual.\n\nImmutability is useful when designing concurrent applications because of the object state guarantees it provides. It's also quite useful for Sets and as Map keys... As far as i am concerned there's no choice between immutability and mutability. As an example, in a multithreaded Bank application, an Account object cannot be made immutable because its a dynamic entity which changes overtime and its unique to one holder. However a Discount or APR can be designed as an Immutable object because they do not vary beyond definition. In short your problem domain will decide how you model your objects. Too many times Java programmers write code for \"Gang Of Four\" as opposed to coding to customer's requirement.\n\nA Class isn't truly immutable if any of this children references aren't immutable as well. If your root Class has final references to all its instance variables and all those instance variables are immutable as well as all their children following this same restriction then you can say the root Class is immutable.\n\nIf any of the children references are non-final or any of the children instances are mutable then the containing Class is not immutable.\n\nIt doesn't matter about the internal immutablity whatever that means is anyone's guess, all that matters is that the public contract to the class immutability is kept. You can't have partial immutability anymore than you can be partially dead. Your class is either immutable or it isn't. And it isn't if any of its referred to classes or their classes, etc. are mutable.\n\nIf you have a graph and any of the members of the graph are mutable you can't deterministically say that you have an immutable state. Concurrency guarantees go out the window, .equals() and .hashCode() become non-deterministic and simple to test cloning and serializing go out the window as well.\n\nIf anything breaks this immutability contract you lose all the benefits of trying to maintain this immutability contract.\n\nSimpler concurrency issues is a main motivational force of immutability.\n\nHaving side effect free code helps with predictability and maintainability. Since you don't have to wonder where things are getting mutated in the call tree because you know they can't be mutated.\n\nPerformance is another less important factor, in Java the JVM can make some highly optimized decisions about caching and other factors if it knows data can't change state. It does provide important hints to the compiler at compile time and the JIT algorithms at runtime.\n\n\u2022 Re: \"A Class isn't truly immutable if any of this children references aren't immutable as well\", I don't think that's quite true. It depends on the semantics of the class and how it actually uses the non-immutable object. It may be OK for an \"immutable\" DocumentResultSet to have references to mutable Document objects, as long as those references are final. \u2013\u00a0Darien Jun 22 '11 at 22:09\n\u2022 This is a good answer, but why do you have to bold every single instance of mutable and immutable? Really makes it harder to read. \u2013\u00a0Aaronaught Jun 23 '11 at 3:05\n\u2022 @Darien I sort of agree. For example, I like to implement immutable lazy lists (handy for parsing) using an Iterator (the source), an ArrayList (elements that have been read from the source), and a position. In a sense the class is mutable because it references two mutable objects, but the mutability is completely encapsulated; the class fulfills the contract of an immutable structure. \u2013\u00a0Daniel Lubarov Jun 23 '11 at 4:01\n\u2022 please correct your answer to reflect the fact that an immutable object can reference mutable objects as long as: 1. The references to its are not leaked; and 2. the immutable object cannot modify the mutable objects after construction. (Java Concurrency in Practice, Section 3.4 Immutability, Goetz2006) \u2013\u00a0glenviewjeff Jun 29 '11 at 14:23\n\u2022 A class can hold references to mutable objects and still be considered immutable if the purpose of the references is to IDENTIFY rather than CONTAIN the objects referred to thereby. For example, an immutable list of cars would be one that always refers to the same automobiles. If the fifth car on the list is a red car with VIN #24601, it might tomorrow be a blue car with VIN #24601, but it won't have any other VIN#. \u2013\u00a0supercat Jul 17 '12 at 21:01\n\n## protected by gnatSep 12 '17 at 7:07\n\nThank you for your interest in this question. 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Herr Jesu Christ, wahr' Mensch und Gott (BWV 127) ist eine Kirchenkantate von Johann Sebastian Bach. Er komponierte die Choralkantate in Leipzig für den Sonntag Estomihi und führte sie am 11. Februar 1725 erstmals auf.
Geschichte und Worte
Bach schrieb die Choralkantate in seinem zweiten Amtsjahr in Leipzig für den Sonntag Estomihi, den Sonntag vor Aschermittwoch. Nach diesem Sonntag wurde in Leipzig während der Passionszeit tempus clausum eingehalten, das heißt, es wurden keine Kantaten musiziert außer am Fest Mariä Verkündigung. Im Jahr 1723 hatte Bach in Leipzig zu Estomihi wahrscheinlich zwei Kantaten aufgeführt, Du wahrer Gott und Davids Sohn, komponiert bereits in Köthen, und Jesus nahm zu sich die Zwölfe, beide als Probestücke für das Amt des Thomaskantors.
Die vorgeschriebenen Lesungen für den Sonntag waren und , die Heilung eines Blinden und damit verbunden die Ankündigung des Leidens in Jerusalem. Der Kantatentext basiert auf dem Sterbelied in acht Strophen von Paul Eber (1562). Das Lied passt zum Evangelium, da es die Passion betont und der Ruf am Ende der ersten Strophe "Du wollst mir Sünder gnädig sein" dem Anruf des Blinden ähnelt. Das Lied betrachtet den Weg Jesu nach Jerusalem als Vorbild für den Weg des Glaubenden zu seinem eigenen erlösten Ende. Ein unbekannter Dichter behielt die erste und die letzte Strophe des Liedes im Wortlaut bei und dichtete die Binnenstrophen zu einer Folge von Rezitativen und Arien um, wobei er bedeutungsvoll einige Zeilen wörtlich einbezog. Die Strophen 2 und 3 wurden zu einem Rezitativ, Strophe 4 zu einer Arie, Strophe 5 zum Rezitativ, Strophen 6 und 7 zu einer weiteren Arie.
Bach führte die Kantate am 11. Februar 1725 erstmals auf. Sie ist die vorletzte Choralkantate seines 2. Kantatenzyklus, vor Wie schön leuchtet der Morgenstern, BWV 1, für Mariä Verkündigung.
Besetzung und Aufbau
Die Kantate ist besetzt mit drei Solisten, Sopran, Tenor, Bass, vierstimmigem Chor, Trompete, zwei Blockflöten, zwei Oboen, zwei Violinen, Viola und Basso continuo.
Choral: Herr Jesu Christ, wahr' Mensch und Gott
Recitativo (Tenor): Wenn alles sich zur letzten Zeit entsetzet
Aria (Sopran): Die Seele ruht in Jesu Händen
Recitativo e aria (Bass): Wenn einstens die Posaunen schallen – Fürwahr, fürwahr, euch sage ich
Choral: Ach, Herr, vergib all unsre Schuld
Musik
Der Eingangschoral ist durch ein ausgedehntes Vorspiel und Zwischenspiele gegliedert. Sie beruhen auf einem Motiv, das aus der ersten Zeile des Chorals abgeleitet ist. Sie enthalten außerdem den cantus firmus eines weiteren Chorals, des lutherischen Agnus Dei "Christe, du Lamm Gottes", der zuerst in den Streichern, dann in den Oboen und Blockflöten erscheint. Dieser Choral wird ähnlich eingesetzt wie im Eingangschor von Bachs Matthäus-Passion "O Lamm Gottes, unschuldig". Seine Bitte "erbarm dich unser" entspricht der des Blinden im Evangelium. Als ein dritter Choral erscheint mehrfach im Continuo der Beginn des Passionsliedes "O Haupt voll Blut und Wunden". Christoph Wolff merkt an, dass Bach wenig später am Karfreitag des Jahres 1725 die zweite Fassung seiner Johannes-Passion aufführte, in der er Eingangschor und Schlusschor durch Choräle ersetzte, "O Mensch, bewein dein Sünde groß", das er später zum Abschluss des ersten Teils seiner Matthäus-Passion umformte, und erneut "Christe, du Lamm Gottes".
Für die erste Arie wählte Bach eine ausgefallene Instrumentierung. Die Oboe spielt die Melodie, unterstützt von kurzen Akkorden in den Blockflöten. Im Mittelteil werden die "Sterbeglocken" durch pizzicato der Streicher dargestellt. Satz 4 illustriert das Jüngste Gericht. Zum Text "Wenn einstens die Posaunen schallen" tritt erstmals die Trompete auf. Der ungewöhnliche Satz kombiniert ein accompagnato-Rezitativ mit einer Arie. Er kontrastiert den Weltuntergang mit der Sicherheit der Glaubenden, die in Worten und Musik des Chorals ausgedrückt wird. John Eliot Gardiner vergleicht den Satz mit dem Doppelchor aus der Matthäus-Passion "Sind Blitze, sind Donner in Wolken verschwunden". Der Schlusschoral ist ein vierstimmiger Satz, der subtil auf den Text eingeht, zum Beispiel "auch unser Glaub stets wacker sei" durch Bewegung in den Unterstimmen verdeutlicht und "bis wir einschlafen seliglich" kunstvoll harmonisiert.
Einspielungen
J.S. Bach: Cantatas BWV 67, 108 & 127, Karl Richter, Münchener Bach-Chor, Bayerisches Staatsorchester, Antonia Fahberg, Peter Pears, Kieth Engen, Teldec 1958
J.S. Bach: Cantatas BWV 127 & BWV 171, Wolfgang Gönnenwein, Süddeutscher Madrigalchor, Südwestdeutsches Kammerorchester, Herrad Wehrung, Georg Jelden, Jakob Stämpfli, Cantate 1961
Die Bach Kantate Vol. 40, Helmuth Rilling, Gächinger Kantorei, Bach-Collegium Stuttgart, Arleen Augér, Lutz-Michael Harder, Wolfgang Schöne, Hänssler 1980
J.S. Bach: Das Kantatenwerk · Complete Cantatas · Les Cantates, Folge / Vol. 31, Gustav Leonhardt, Knabenchor Hannover, Collegium Vocale Gent, Leonhardt-Consort, Solist des Knabenchor Hannover, Kurt Equiluz, Max van Egmond, Teldec 1982
J.S. Bach: Complete Cantatas Vol. 11, Ton Koopman, Amsterdam Baroque Orchestra & Choir, Sibylla Rubens, Christoph Prégardien, Klaus Mertens, Antoine Marchand 1999
Bach Cantatas Vol. 21: Cambridge/Walpole St Peter / For Quinquagesima Sunday (Estomihi) / For Annunciation / Palm Sunday / Oculi, John Eliot Gardiner, Monteverdi Choir, Choir of Clare College, Cambridge & The Choir of Trinity College, Cambridge, English Baroque Soloists, Ruth Holton, James Oxley, Stephan Loges, Soli Deo Gloria 2000
Bach Edition Vol. 20 – Cantatas Vol. 11, Pieter Jan Leusink, Holland Boys Choir, Netherlands Bach Collegium, Ruth Holton, Nico van der Meel, Bas Ramselaar, Brilliant Classics 2000
J.S. Bach: Cantatas Vol. 34 – (Cantatas from Leipzig 1725), Masaaki Suzuki, Bach Collegium Japan, Carolyn Sampson, Gerd Türk, Peter Kooij, BIS 2005
J.S. Bach: Jesus, deine Passion – Cantates BWV 22, 23, 127 & 159, Philippe Herreweghe, Collegium Vocale Gent, Dorothee Mields, Jan Kobow, Peter Kooij, Harmonia Mundi France 2007
Literatur
Alfred Dürr: Johann Sebastian Bach: Die Kantaten. Bärenreiter, Kassel 1999, ISBN 3-7618-1476-3 und Deutscher Taschenbuchverlag, München 1995, ISBN 3-423-04431-4.
Werner Neumann: Handbuch der Kantaten J.S.Bachs. 1947. 5. Auflage: 1984, ISBN 3-7651-0054-4
Hans-Joachim Schulze: Die Bach-Kantaten: Einführungen zu sämtlichen Kantaten Johann Sebastian Bachs. Evangelische Verlags-Anstalt, Leipzig 2006, ISBN 3-374-02390-8; Carus-Verlag, Stuttgart 2006, ISBN 3-89948-073-2
Christoph Wolff, Ton Koopman: Die Welt der Bach-Kantaten Verlag J.B. Metzler, Stuttgart / Weimar 2006, ISBN 3-476-02127-0
Weblinks
Herr Jesu Christ, wahr' Mensch und Gott auf der Bach.de-Website
Text, Aufbau und Besetzung auf der persönlichen Homepage von Walter F. Bischof bei der University of Alberta
Einzelnachweise
Bachkantate
Musik 1725
Chorwerk aus dem 18. Jahrhundert | {
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Q: Combinatorics (discrete math course) help problem 1:
you have 4 balls with different weights and 6 drawers stacked on top of each other.
how many ways are there to organize the balls such that the top drawer will have exactly 1 ball and the bottom drawer will have at least 1 ball?
problem 2:
from a group of 5 women and 6 men you need to choose a council made of a ceo, vp, cto, cfo.
in how many ways can you do that such that the ceo is a women and the oldest of the men will be in the council?
problem 3:
given a triangle on the plane; on each of its edges are 5 points that are not on the vertices.
how many triangles can be created when each of their vertices are on the points stated?
thx for your help,
A: 1) see the answer of @Rajkumar.
2) $5\times 3\times 9\times 8$.
Factor $5$ corresponds with the number of choices for the ceo. Factor $3$ with the number of duties that can be given to the oldest man. Then $9$ persons are left for $2$ duties, leading to $9\times 8$ possibilities.
3) $5^3+3\times\binom{5}{2}\times10$.
If the $3$ vertices are on $3$ different edges then there are $5^3$ possibilities.
Looking at triangles having $2$ vertices on the same edge there are $3\times\binom{5}{2}\times10$ possibilities. Factor $3$ corresponds with the choice of the edge with $2$ vertices. Factor $\binom{5}{2}$ with the choice of these $2$ vertices, and factor
$10$ with the choice of the third vertex.
A: Problem 1:
One ball should be in top drawer, so choosing one ball from 4 in 4 ways.
The other 3 calls can be arranged in 5 drawers in 3^5 ways.
But bottom should posses atleast one ball. So from 3^5 remove (number of ways 3 balls can be arranged in middle 4 drawers).
i.e. 4 * (3^5 - 3^4)
[Modified based on the inputs from drhab]
Problem 2:
Choosing CEO ( one woman from a group of 5 ) in 5 ways.
Oldest among men will be in the council in 1 way.
Remaining 2 could be from other 4 women & 5 men, i.e. from 9 in 9C2 ways.
So ans is = 5 * 1 * 9C2.
| {
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} | 795 |
Our warmly modern, residentially styled luxury New York hotel standard rooms are designed with the modern day traveler in mind. The layout consists of a palette of warm tones with an artful balance of form and function – from the spacious slate-tile bathrooms to the flat screen TVs. Oversized windows provide an abundance of natural light and city views. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,427 |
{"url":"http:\/\/mathhelpforum.com\/algebra\/277018-logarithm.html","text":"1. ## Logarithm\n\nSmall logarithm question:\n\n(2x-1) * ln2 = (3x-2) * ln3\n\n2. ## Re: Logarithm\n\nDistribute, bring all of the $x$'s to one side and everything else to the right side. Divide both sides by the coefficient of $x$.\n\n3. ## Re: Logarithm\n\nYou titled this \"logarithm\" as if the logarithms here are the important part of the problem- they are not.\n\nln(2) and ln(3) are simply numbers. How would you solve A(2x- 1)= B(3x- 2) for numbers A and B?","date":"2017-11-18 01:44:40","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9257577061653137, \"perplexity\": 2376.0418767728097}, \"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-2017-47\/segments\/1510934804125.49\/warc\/CC-MAIN-20171118002717-20171118022717-00416.warc.gz\"}"} | null | null |
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Tear Down Mount Davis!
The first and only time I ever attended a ballgame at the Oakland Coliseum was on the evening of August 5, 1977, when I saw Luis Tiant of the Red Sox pitch a five-hit shutout against an A's team that had Manny Sanguillen (who collected three of those hits) in the lineup but little else to recommend it. 1977 was the beginning of the A's dark post-dynasty days — owner Charlie Finley was too distracted and discouraged by the onset of free agency to sink another penny into his team — and only a little more than 7,000 fans showed up that night to cheer on a team that was already more than 20 games below the .500 mark. It was no trouble at all for my family to get tickets in the section behind the A's dugout, and I remember getting "designated runner" Larry Lintz's autograph before the game — the first time I'd ever gotten an autograph at the ballpark.
Even cooler than that, though, at least to my eleven-year-old eyes, was the Oakland Coliseum itself, which looked pretty much the same then as it does in the 1980 photo above. I loved that the stadium had been home to three straight World Champion ballclubs, but I loved its sleek, modern design even more. I especially loved the green slope above the outfield seats, which gave the futuristic structure a touch of pastoral charm, kind of like the way Bruce Dern's gardens in Silent Running mitigated the harsh utilitarian feel of his space station.
That alluring slope is long gone, of course, replaced by "Mount Davis," Raiders owner Al Davis's multi-tiered monument to greed, which completely ruined the balanced, open feel of the Coliseum. But the "Mausoleum" still stands, and remains true to its original multi-purpose purpose — which, in a weird way, is something to celebrate. Its continued existence inspired "The Last of the Concrete Donuts," a piece that I recently penned for the new Sphere site, which in turn has inspired someone to start a new Facebook page called Tear Down Mount Davis, which is a sentiment I can definitely get behind. I mean, just LOOK at this hideousness...
While nostalgia for the "ashtray stadiums" of yore might seem a bit silly, I remember thinking as a kid that no one would ever be nostalgic for 1970s music or clothing, either. Funny how the passing of a few decades — and the emergence of new, previously unimagined monstrosities — can put things in perspective. I'm not an A's fan, and I have no skin in this game; but the A's fanbase deserves a ballpark that's as funky as they are, and a Coliseum restored to its 1968 glory would be just the ticket.
Posted at 06:10 PM in Ballparks | Permalink
I attended my first game at the Coliseum in August 1972 when I was 10 years old. I have great fondness for this ballpark, not so much for its architecture, but for the many wonderful memories I've developed over the last forty plus years.
The "renovation" of the Coliseum and the erection of Mount Davis is the price the city and county paid to bring back the Raiders after their sojourn to Los Angeles. It was a financial boondoggle from the beginning, one the taxpayers are still paying for, and not just from an aesthetics point of view.
Posted by: Acbeam | 08/21/2014 at 07:02 PM | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,355 |
HomeTexasRise Against performing at the Austin360 Amphitheater in Austin, Texas
Rise Against performing at the Austin360 Amphitheater in Austin, Texas
Rise Against performing at the Austin360 Amphitheater in Austin, Texas on June 28, 2017, with Tim, Joe, Brandon, and Zach.
Chicago-based band Rise Against's most recent album is The Black Market, which hit No. 1 on the Alternative, Top Rock and Hard Rock albums charts. Alternative Press awarded The Black Market four stars, noting, "Whatever meaning you take from them [the songs]…one thing remains solid—the sheer heart with which they were forged and the very visceral emotion they contain" and Kerrang praised the album as "a magnificent wake-up call." Rise Against is Tim McIlrath (lead vocals/rhythm guitar), Zach Blair (lead guitar), Joe Principe (bass) and Brandon Barnes (drums). (Source: AllMusic)
The setlist for the show at the Austin360 Amphitheater:
Ready to Fall
Re-Education (Through Labor)
The Good Left Undone
People Live Here
Make It Stop (September's Children)
Prayer of the Refugee
Rise Against and Deftones were the main act at the Austin360 Amphitheater with guests Thrice, and Frank Iero and the Patience. The tour continues across the United States, Europe, and Canada ending at Halle622 in Zurich, Switzerland on November 21, 2017.
Rise Against contact: | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,417 |
/*
* To change this license header, choose License Headers in Project Properties.
* To change this template file, choose Tools | Templates
* and open the template in the editor.
*/
package be.fror.projecteuler.problem;
import static java.lang.Math.ceil;
import static java.lang.Math.log10;
import static java.lang.Math.sqrt;
/**
*
* @author Olivier Grégoire
*/
public class Problem25 extends AbstractProblem {
@Override
public Object solve() {
int N = 1000;
double phi = (1.0 + sqrt(5.0)) / 2.0;
return (int) ceil((N + log10(5.0) / 2.0 - 1.0) / log10(phi));
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,900 |
Q: Swift map function variability I've got this properly working map function:
let movingImages = (1...71).map { UIImage(named: "Animation.\($0)")! }
Except there is no Animation.1 file - it's Animation.01
Rather than renaming the files, how can I code it to add a 0 to the numbers before 10?
Edit: this question differs from others because the answer provided elsewhere about adding a 0 to single digit number in Swift needs additional context in order to be used within a mapping that needs to return a value
A: You can use the String(format: ...) with the %02d for this:
let movingImages = (1...71).map { index -> UIImage? in
let imageNumber = String(format: "%02d", index)
return UIImage(named: "Animation.\(imageNumber)")!
}
Also, on the side note, avoid using force unwrapping (!).
A more safe way:
Swift 4:
let movingImages = (1...71).flatMap { index -> UIImage? in
let imageNumber = String(format: "%02d", index)
return UIImage(named: "Animation.\(imageNumber)")
}
Swift 4.1:
let movingImages = (1...71).compactMap { index -> UIImage? in
let imageNumber = String(format: "%02d", index)
return UIImage(named: "Animation.\(imageNumber)")
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,025 |
iMend run one of the most trusted and recognised training schools in the UK, administering industry-changing programmes for trainees of various skill sets. We run the only government-listed Mobile Phone repairs school.
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Our Training Centre is situated in the heart of the country just, just off junction 11 of the M6. Our central location is perfect for travelling trainees with access to all major motorways. There are local hotels and restaurants located in town which are easy to travel between. If you are in need of any help, we have plenty of recommendations and contacts.
Over recent months, our Training Centre has undergone some serious changes. Now deemed as one of the most vibrant and state-of-the-art hubs in the UK, our training programmes have just gone from incredible to impeccable.
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For those that are new to Mobile Phone and Tablet Repairs, we recommend our Level 1 – New Technician Programme. If you are looking to get to grips with the basics in repairing both iPhone, Samsung and other devices, this course is perfect for you. During this 3 day course, trainees will master Screen Repairs, Battery Replacements and other Small Parts Repairs.
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If you are already savvy with Mobile Phones and Tablet Repairs but are looking to brush up on some new techniques, our Level 3 – Advanced Technician is the perfect course for you. Our 3-day course covers the highest calibre of repairs including Touch, Charge & Power IC Replacement, Charging Port Repairs and BGA Rework.
By far the most popular of our courses is the Fast Track 1,2,3 Training Programme. The course is a combination of all three courses stated above, starting from the very essence of repairs all the way to the most difficult techniques. This 6 day programme is perfect for someone who has little to no real experience and is keen on becoming a confident and skilled Technician across all major repairs and device models.
You will be trained by Mobile and Tablet Experts, who have over 12 years experience in the industry. Our experts are able to train technicians of all skill levels. Whether you are interested in learning how to repair the latest iPhone Mobile or wanting to start a new career in phone repairs, our experts provide you with the skills you desire.
After completing your chosen course, there are plenty of post-training benefits. All trainees have the opportunity to join the iMend Network, where you will work as a call-out technician repairing devices within your local area. It's a great way to reinvest your time and money spent on our training programme. It is a very popular choice for those looking to kickstart a new career in the Mobile Phone and Tablet Repairs Industry.
Even if you have other plans once the course is complete, we guarantee life-time tech support. If you have a question or query, our team are on hand to help. All trainees receive an iMend Approved Technician Certificate after completing the chosen , a qualification which is highly regarded across the whole of the industry.
Are you interested in joining one of our many training programmes, click here to enquire today. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,223 |
Q: How can I make human readable a google chart line with large variations? The image graph represents the number of users per day over two years (730 days). At the moment it is far from being human readable. How can I make it more human readable?
The options on the chart at this point are:
options = {
colors: elevenColors,
vAxis: { minValue: 0, format: '0', maxTextLines: 1 },
hAxis: { textPosition: 'none' },
legend: { position: 'none' },
dataOpacity: 0.7,
chartArea: {'width': '90%', 'height': '80%'},
}
I've already tried increasing the maxValue, and changing the graph to scatter type (I'm open to changing the graph type as long as it's more readable), but nothing seems to work since the sample seems to be too big.
A: My suggestion would be to change the data structure on the x axis.
This could be achieved through grouping the results eg. monthly users or by shortening the time scale (and probably adding a time filter to change the results depending on the selected time range).
Posting the data the graph is drawn from would probably help this question.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,925 |
<?xml version="1.0" encoding="utf-8"?>
<selector xmlns:android="http://schemas.android.com/apk/res/android">
<!-- Non focused states -->
<item android:drawable="@color/share" android:state_focused="false" android:state_pressed="false" android:state_selected="false"/>
<item android:drawable="@color/share_pressed" android:state_focused="false" android:state_pressed="false" android:state_selected="true"/>
<!-- Focused states -->
<item android:drawable="@color/share_pressed" android:state_focused="true" android:state_pressed="false" android:state_selected="true"/>
<!-- Pressed state -->
<item android:drawable="@color/share_pressed" android:state_pressed="true"/>
</selector> | {
"redpajama_set_name": "RedPajamaGithub"
} | 735 |
I'm pleased to welcome back SaneBox as this week's sponsor. I've been using Sanebox for years and it still saves my bacon every day. There are so many great benefits to Sanebox. First and foremost is filtering. Sanebox looks at all of my incoming email and filters it for me to appropriate mailboxes. That way, the first thing in the morning I see are only those emails that are most important.
However, Sanebox can do so much more. I used to make fun of email deferment until I started using it with Sanebox. Now I use it all the time. When an email comes in that I don't need to deal with for a couple days, rather than giving myself the extra work of scheduling a response in OmniFocus, I just defer the email two days. It disappears and then shows up again in my inbox in two days and I can deal with it then. Sanebox is highly customizable so you can set defer periods in hours, days, and weeks.
Because Sanebox is a cloud based service, it works in the email client of your choice, including Apple Mail. There is also an ingenious reminder system where you can blind copy an email to 1week@sanebox.com and the service will remind you if you don't get a response to the email within one week.
There is a lot more to SaneBox. If you get a lot of email, the service can really help. You can learn more at SaneBox.com. Note the links in this post will get you a discount when you subscribe. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,619 |
import { NgModule } from '@angular/core';
import { BrowserModule } from '@angular/platform-browser';
import { ServerModule } from '@angular/platform-server';
import { NoopAnimationsModule } from '@angular/platform-browser/animations';
// libs
import { ServerStateTransferModule, StateTransferService } from '@ngx-universal/state-transfer';
import { CACHE, CacheService, STORAGE } from '@ngx-cache/core';
import { FsCacheService, ServerCacheModule } from '@ngx-cache/platform-server';
import { fsStorageFactory, FsStorageLoader, FsStorageService } from '@ngx-cache/fs-storage';
// modules & components
import { AppModule } from '../../client/app/app.module';
import { AppComponent } from '../../client/app/app.component';
@NgModule({
bootstrap: [AppComponent],
imports: [
NoopAnimationsModule,
BrowserModule.withServerTransition({
appId: 'cd'
}),
ServerModule,
ServerStateTransferModule.forRoot(),
ServerCacheModule.forRoot([
{
provide: CACHE,
useClass: FsCacheService
},
{
provide: STORAGE,
useClass: FsStorageService
},
{
provide: FsStorageLoader,
useFactory: (fsStorageFactory)
}
]),
AppModule
]
})
export class AppServerModule {
constructor(private readonly stateTransfer: StateTransferService,
private readonly cache: CacheService) {
}
ngOnBootstrap = () => {
this.stateTransfer.set(this.cache.key, JSON.stringify(this.cache.dehydrate()));
this.stateTransfer.inject();
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,203 |
'use strict';
/* jshint ignore:start */
/**
* This code was generated by
* \ / _ _ _| _ _
* | (_)\/(_)(_|\/| |(/_ v1.0.0
* / /
*/
/* jshint ignore:end */
var Q = require('q'); /* jshint ignore:line */
var _ = require('lodash'); /* jshint ignore:line */
var util = require('util'); /* jshint ignore:line */
var Page = require('../../../../../base/Page'); /* jshint ignore:line */
var deserialize = require(
'../../../../../base/deserialize'); /* jshint ignore:line */
var values = require('../../../../../base/values'); /* jshint ignore:line */
var UserChannelList;
var UserChannelPage;
var UserChannelInstance;
/* jshint ignore:start */
/**
* Initialize the UserChannelList
*
* @constructor Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelList
*
* @param {Twilio.IpMessaging.V1} version - Version of the resource
* @param {string} serviceSid - The service_sid
* @param {string} userSid - The sid
*/
/* jshint ignore:end */
UserChannelList = function UserChannelList(version, serviceSid, userSid) {
/* jshint ignore:start */
/**
* @function userChannels
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext#
*
* @param {string} sid - sid of instance
*
* @returns {Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelContext}
*/
/* jshint ignore:end */
function UserChannelListInstance(sid) {
return UserChannelListInstance.get(sid);
}
UserChannelListInstance._version = version;
// Path Solution
UserChannelListInstance._solution = {serviceSid: serviceSid, userSid: userSid};
UserChannelListInstance._uri = `/Services/${serviceSid}/Users/${userSid}/Channels`;
/* jshint ignore:start */
/**
* Streams UserChannelInstance records from the API.
*
* This operation lazily loads records as efficiently as possible until the limit
* is reached.
*
* The results are passed into the callback function, so this operation is memory
* efficient.
*
* If a function is passed as the first argument, it will be used as the callback
* function.
*
* @function each
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelList#
*
* @param {object} [opts] - Options for request
* @param {number} [opts.limit] -
* Upper limit for the number of records to return.
* each() guarantees never to return more than limit.
* Default is no limit
* @param {number} [opts.pageSize] -
* Number of records to fetch per request,
* when not set will use the default value of 50 records.
* If no pageSize is defined but a limit is defined,
* each() will attempt to read the limit with the most efficient
* page size, i.e. min(limit, 1000)
* @param {Function} [opts.callback] -
* Function to process each record. If this and a positional
* callback are passed, this one will be used
* @param {Function} [opts.done] -
* Function to be called upon completion of streaming
* @param {Function} [callback] - Function to process each record
*/
/* jshint ignore:end */
UserChannelListInstance.each = function each(opts, callback) {
if (_.isFunction(opts)) {
callback = opts;
opts = {};
}
opts = opts || {};
if (opts.callback) {
callback = opts.callback;
}
if (_.isUndefined(callback)) {
throw new Error('Callback function must be provided');
}
var done = false;
var currentPage = 1;
var currentResource = 0;
var limits = this._version.readLimits({
limit: opts.limit,
pageSize: opts.pageSize
});
function onComplete(error) {
done = true;
if (_.isFunction(opts.done)) {
opts.done(error);
}
}
function fetchNextPage(fn) {
var promise = fn();
if (_.isUndefined(promise)) {
onComplete();
return;
}
promise.then(function(page) {
_.each(page.instances, function(instance) {
if (done || (!_.isUndefined(opts.limit) && currentResource >= opts.limit)) {
done = true;
return false;
}
currentResource++;
callback(instance, onComplete);
});
if (!done) {
currentPage++;
fetchNextPage(_.bind(page.nextPage, page));
} else {
onComplete();
}
});
promise.catch(onComplete);
}
fetchNextPage(_.bind(this.page, this, _.merge(opts, limits)));
};
/* jshint ignore:start */
/**
* Lists UserChannelInstance records from the API as a list.
*
* If a function is passed as the first argument, it will be used as the callback
* function.
*
* @function list
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelList#
*
* @param {object} [opts] - Options for request
* @param {number} [opts.limit] -
* Upper limit for the number of records to return.
* list() guarantees never to return more than limit.
* Default is no limit
* @param {number} [opts.pageSize] -
* Number of records to fetch per request,
* when not set will use the default value of 50 records.
* If no page_size is defined but a limit is defined,
* list() will attempt to read the limit with the most
* efficient page size, i.e. min(limit, 1000)
* @param {function} [callback] - Callback to handle list of records
*
* @returns {Promise} Resolves to a list of records
*/
/* jshint ignore:end */
UserChannelListInstance.list = function list(opts, callback) {
if (_.isFunction(opts)) {
callback = opts;
opts = {};
}
opts = opts || {};
var deferred = Q.defer();
var allResources = [];
opts.callback = function(resource, done) {
allResources.push(resource);
if (!_.isUndefined(opts.limit) && allResources.length === opts.limit) {
done();
}
};
opts.done = function(error) {
if (_.isUndefined(error)) {
deferred.resolve(allResources);
} else {
deferred.reject(error);
}
};
if (_.isFunction(callback)) {
deferred.promise.nodeify(callback);
}
this.each(opts);
return deferred.promise;
};
/* jshint ignore:start */
/**
* Retrieve a single page of UserChannelInstance records from the API.
*
* The request is executed immediately.
*
* If a function is passed as the first argument, it will be used as the callback
* function.
*
* @function page
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelList#
*
* @param {object} [opts] - Options for request
* @param {string} [opts.pageToken] - PageToken provided by the API
* @param {number} [opts.pageNumber] -
* Page Number, this value is simply for client state
* @param {number} [opts.pageSize] - Number of records to return, defaults to 50
* @param {function} [callback] - Callback to handle list of records
*
* @returns {Promise} Resolves to a list of records
*/
/* jshint ignore:end */
UserChannelListInstance.page = function page(opts, callback) {
if (_.isFunction(opts)) {
callback = opts;
opts = {};
}
opts = opts || {};
var deferred = Q.defer();
var data = values.of({
'PageToken': opts.pageToken,
'Page': opts.pageNumber,
'PageSize': opts.pageSize
});
var promise = this._version.page({uri: this._uri, method: 'GET', params: data});
promise = promise.then(function(payload) {
deferred.resolve(new UserChannelPage(this._version, payload, this._solution));
}.bind(this));
promise.catch(function(error) {
deferred.reject(error);
});
if (_.isFunction(callback)) {
deferred.promise.nodeify(callback);
}
return deferred.promise;
};
/* jshint ignore:start */
/**
* Retrieve a single target page of UserChannelInstance records from the API.
*
* The request is executed immediately.
*
* If a function is passed as the first argument, it will be used as the callback
* function.
*
* @function getPage
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelList#
*
* @param {string} [targetUrl] - API-generated URL for the requested results page
* @param {function} [callback] - Callback to handle list of records
*
* @returns {Promise} Resolves to a list of records
*/
/* jshint ignore:end */
UserChannelListInstance.getPage = function getPage(targetUrl, callback) {
var deferred = Q.defer();
var promise = this._version._domain.twilio.request({method: 'GET', uri: targetUrl});
promise = promise.then(function(payload) {
deferred.resolve(new UserChannelPage(this._version, payload, this._solution));
}.bind(this));
promise.catch(function(error) {
deferred.reject(error);
});
if (_.isFunction(callback)) {
deferred.promise.nodeify(callback);
}
return deferred.promise;
};
/* jshint ignore:start */
/**
* Provide a user-friendly representation
*
* @function toJSON
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelList#
*
* @returns Object
*/
/* jshint ignore:end */
UserChannelListInstance.toJSON = function toJSON() {
return this._solution;
};
UserChannelListInstance[util.inspect.custom] = function inspect(depth, options)
{
return util.inspect(this.toJSON(), options);
};
return UserChannelListInstance;
};
/* jshint ignore:start */
/**
* Initialize the UserChannelPage
*
* @constructor Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelPage
*
* @param {V1} version - Version of the resource
* @param {Response<string>} response - Response from the API
* @param {UserChannelSolution} solution - Path solution
*
* @returns UserChannelPage
*/
/* jshint ignore:end */
UserChannelPage = function UserChannelPage(version, response, solution) {
// Path Solution
this._solution = solution;
Page.prototype.constructor.call(this, version, response, this._solution);
};
_.extend(UserChannelPage.prototype, Page.prototype);
UserChannelPage.prototype.constructor = UserChannelPage;
/* jshint ignore:start */
/**
* Build an instance of UserChannelInstance
*
* @function getInstance
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelPage#
*
* @param {UserChannelPayload} payload - Payload response from the API
*
* @returns UserChannelInstance
*/
/* jshint ignore:end */
UserChannelPage.prototype.getInstance = function getInstance(payload) {
return new UserChannelInstance(
this._version,
payload,
this._solution.serviceSid,
this._solution.userSid
);
};
/* jshint ignore:start */
/**
* Provide a user-friendly representation
*
* @function toJSON
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelPage#
*
* @returns Object
*/
/* jshint ignore:end */
UserChannelPage.prototype.toJSON = function toJSON() {
let clone = {};
_.forOwn(this, function(value, key) {
if (!_.startsWith(key, '_') && ! _.isFunction(value)) {
clone[key] = value;
}
});
return clone;
};
UserChannelPage.prototype[util.inspect.custom] = function inspect(depth,
options) {
return util.inspect(this.toJSON(), options);
};
/* jshint ignore:start */
/**
* Initialize the UserChannelContext
*
* @constructor Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelInstance
*
* @property {string} accountSid - The account_sid
* @property {string} serviceSid - The service_sid
* @property {string} channelSid - The channel_sid
* @property {string} memberSid - The member_sid
* @property {user_channel.channel_status} status - The status
* @property {number} lastConsumedMessageIndex - The last_consumed_message_index
* @property {number} unreadMessagesCount - The unread_messages_count
* @property {string} links - The links
*
* @param {V1} version - Version of the resource
* @param {UserChannelPayload} payload - The instance payload
* @param {sid} serviceSid - The service_sid
* @param {sid} userSid - The sid
*/
/* jshint ignore:end */
UserChannelInstance = function UserChannelInstance(version, payload, serviceSid,
userSid) {
this._version = version;
// Marshaled Properties
this.accountSid = payload.account_sid; // jshint ignore:line
this.serviceSid = payload.service_sid; // jshint ignore:line
this.channelSid = payload.channel_sid; // jshint ignore:line
this.memberSid = payload.member_sid; // jshint ignore:line
this.status = payload.status; // jshint ignore:line
this.lastConsumedMessageIndex = deserialize.integer(payload.last_consumed_message_index); // jshint ignore:line
this.unreadMessagesCount = deserialize.integer(payload.unread_messages_count); // jshint ignore:line
this.links = payload.links; // jshint ignore:line
// Context
this._context = undefined;
this._solution = {serviceSid: serviceSid, userSid: userSid, };
};
/* jshint ignore:start */
/**
* Provide a user-friendly representation
*
* @function toJSON
* @memberof Twilio.IpMessaging.V1.ServiceContext.UserContext.UserChannelInstance#
*
* @returns Object
*/
/* jshint ignore:end */
UserChannelInstance.prototype.toJSON = function toJSON() {
let clone = {};
_.forOwn(this, function(value, key) {
if (!_.startsWith(key, '_') && ! _.isFunction(value)) {
clone[key] = value;
}
});
return clone;
};
UserChannelInstance.prototype[util.inspect.custom] = function inspect(depth,
options) {
return util.inspect(this.toJSON(), options);
};
module.exports = {
UserChannelList: UserChannelList,
UserChannelPage: UserChannelPage,
UserChannelInstance: UserChannelInstance
};
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,004 |
# Copyright
Copyright © 2017 by CN Lester
Hachette Book Group supports the right to free expression and the value of copyright. The purpose of copyright is to encourage writers and artists to produce the creative works that enrich our culture.
The scanning, uploading, and distribution of this book without permission is a theft of the author's intellectual property. If you would like permission to use material from the book (other than for review purposes), please contact permissions@hbgusa.com. Thank you for your support of the author's rights.
Seal Press
Hachette Book Group
1290 Avenue of the Americas, New York, NY 10104
www.sealpress.com
Originally published in trade paperback and ebook by Virago in May 2017
First US Edition: June 2018
Published by Seal Press, an imprint of Perseus Books, LLC, a subsidiary of Hachette Book Group, Inc. The Seal Press name and logo is a trademark of the Hachette Book Group.
The publisher is not responsible for websites (or their content) that are not owned by the publisher.
Library of Congress Cataloging-in-Publication Data
Names: Lester, CN, author.
Title: Trans like me: conversations for all of us / CN Lester. Description: First US edition. | New York, NY: Seal Press, [2018] | "Originally published in trade paperback and ebook by Virago in May 2017."
Identifiers: LCCN 2017051262 | ISBN 9781580057851 (pbk.) | ISBN 9781580057844 (ebook)
Subjects: LCSH: Lester, CN | Transgender people—Biography. | Transgender people—Identity. | Transgender people—Conduct of life.
Classification: LCC HQ77.8.L48 A3 2018 | DDC 306.76/8—dc23
LC record available at https://lccn.loc.gov/2017051262
ISBNs: 978-1-58005-785-1 (paperback), 978-1-58005-784-4 (ebook)
E3-20180523-JV-PC
# Contents
1. Cover
2. Title Page
3. Copyright
4. Prologue
5. 1 The Production of Ignorance
6. 2 "Call Me Caitlyn"
7. 3 Finding My Voice
8. 4 Couldn't You Just... Not Be?
9. 5 What About Sex?
10. 6 Think of the Children
11. 7 Delusional and Disturbed
12. 8 A Different Approach
13. 9 Trans/Love
14. 10 Are Trans People Real?
15. 11 The Denial of History
16. 12 Beyond Binaries
17. 13 The T from the LGB
18. 14 Trans Feminisms
19. 15 Futures
20. Acknowledgments
21. About the Author
22. Further Resources
23. Endnotes
# _Prologue_
W **hat does the word "trans" mean to you? A term** picked up from magazines, newspapers, the internet? A way of describing a colleague—maybe even a friend or family member? A phenomenon affecting other people? Or, perhaps, a facet of who you think you might be?
When I work with organizations to provide trans diversity training, I always begin by asking for other people's definitions. Inevitably, writing them up, I run out of space on the whiteboard. Suggestions for what trans means includes people who were assigned one sex at birth (male or female) who medically transition to the other, people who live as the opposite sex without surgery or hormones, people who combine or blur sex and gender categories, and people who cross-dress. Participants suggest words and meanings: transsexual to mean someone who has or wants to transition; transgender to indicate a wide variety of unusual gendered experiences; and genderqueer, gender-neutral, and non-binary to mean someone who complicates the simple division of female and male, combining or denying these categories.
All of these ways of being can be included in the word "trans." Any person who has had to challenge or change the sexed and gendered labels placed upon them at birth to honor their true selves can, by their own or others' volition, find themselves under this trans umbrella. The category of trans can be an uncomfortable place to be, filled with the fears society has about disruption, fitting in, danger, and change.
It is also a place filled with unlooked-for knowledge, new experiences, new perspectives, and unanticipated joy.
This is a book about gender—how we all live it, how we think about it, why we do and think what we do. It is also about how some of those things have changed and will change. It is a book about what it is to be trans, to be a woman or a man or a person who is neither, in different times and in different places. It is about our gendered words, and how those words lead to the concepts, laws, and categories that shape our frequently conflicting realities: our families, our love lives, our place in the world. It is a book about my journey through gender, sex, selfhood—and, if you want it to be, about yours.
Throughout this book, I'm going to break down the most common myths about trans lives: panic about "sex changes" for young children, the fear of trans people in bathrooms and changing rooms, and the assumption that being trans is a faddish new trend. A little knowledge is a dangerous thing; luckily, we have a wealth of information about what it is to be trans, and how the world is changing and growing in response to the challenge of trans lives.
Inevitably, when we talk about being trans we have to talk about transphobia. It is hard to live it; it can be hard to read it. It is so tempting to become defensive, to shut down and try to shut it out, to ignore it until it goes away.
But we—all of us—cannot fix this problem until we look it square in the face. And what it is that we gain from embracing the truth of trans existence is so great, so important, that this is a challenge we must take on. We cannot have the good without tackling the bad—and the good is better than we have ever been taught to expect.
I came out as trans half a lifetime ago. In all those intervening years, I have had my share of hardships, but I have never regretted the decision to tell the truth about my gender. I cannot put a price on the wonder of being at peace with who I am, and of living my life as my whole, authentic self. I believe we all deserve a chance at that happiness.
Opening ourselves up to the trans experience—whether or not we ourselves are trans—means embracing a new way of seeing the world. It requires trust, empathy, and imagination. It is not a one-way process, but a journey we are all on together.
This is one book, and one story, out of millions. This is a work in progress.
#
# The Production of Ignorance
_New York Post_ , 2015: "Uncle Sam's Insane Push for TransGender Rights in School Locker Rooms."
_Courier Mail_ , October 2014: "Monster Chef and the She Male."
_Fox News_ , July 2017: "Pediatrician: Transgender Ideology causing child abuse."
_Wall Street Journal_ , 2014: "Transgender Surgery Isn't the Solution."
W **e're often the butt of the joke:the _Sun'_ s 2011 game** "Tran or woman?" There's an air of the freak show about us, an invitation to peer into the bizarre realities of our lives: "Transsexual, 44, Elects to Die by Euthanasia After Botched Sex-Change Operation Turned Him into a 'Monster.'" Even when the intent is celebratory, we are marked out as different and strange: CNN's list of the most influential people of 2014 described actress and advocate Laverne Cox as "The Gender Bender."
Here are a few of the things the media shows, and has shown, trans people to be: confused, deceitful, delusional, damaged, predatory, brave (sometimes), pitiable, pathetic. A punch line, a warning, a mistake.
Here are a few of the things I am: a singer, a teacher of music, a good (if forgetful) friend, a loving child and grandchild, a loved and loving partner. I am a doctoral student, a decent cook, too ambitious, too anxious, a composer of all kinds, and someone who tries, at least, to be better than my worries would have me be. And I'm also transgender.
Rarely has that disconnect between trans reality and its interpretation been so clearly shown as with the publication of British journalist Richard Littlejohn's 2012 "character assassination" of primary schoolteacher Lucy Meadows. Meadows was not a public figure. She hadn't contacted the press to sell her story, hadn't issued a press release, hadn't promoted herself via social media. It didn't matter. "He's Not Only in the Wrong Body... He's in the Wrong Job," the headline announced, accompanied by a photo of Meadows on her wedding day, back before she had transitioned. Referring to her as "he," Littlejohn warned that Meadows's mere presence would have a "devastating effect" on her young pupils. "[Meadow's previous name] is entitled to his gender reassignment surgery, but he isn't entitled to project his personal problems on to impressionable young children."
As a teacher of students ranging in age from five years old to fifty, I had never before heard about projecting my personal issues onto my pupils. My lesson plans focus mostly on technique, creativity, and personal growth, with a side order of self-confidence boosting and chatting about musical history. My private life doesn't come into it. I don't hide who I am with my students and their families, and neither do I dwell on it; the fact that I am trans is as fundamental as any other part of me, but far less important in this context than my knowledge of vocal production and how best to play staccato. I teach for many reasons, but most of all because my own music teachers gave me so much, and I have a debt to repay through sharing the joy of making music. It makes no sense to me that my support of my young students would somehow rob them of their innocence, nor can I interpret the work I put into others' learning as "selfish."
In the end, Littlejohn need not have worried. Lucy Meadows did not remain in her job for long. Three months after transitioning, in March 2013, she killed herself.
Ascribing a single cause to any suicide is both dangerous and disingenuous. Lucy Meadows, like many of us, struggled under multiple burdens. But it is also true that the coroner who investigated her death, Michael Singleton, believed that the press had played a role. Speaking at the inquest, he said: "Lucy Meadows was not somebody who had thrust herself into the public limelight. She was not a celebrity. She had done nothing wrong. Her only crime was to be different. Not by choice but by some trick of nature. And yet the press saw fit to treat her in the way that they did." Finishing his statement, he turned to the gathered reporters and said: "And to you the press, I say shame, shame on all of you."
How can it be that both trans people and the journalists who write about us believe ourselves to be talking about the same subject, and yet have such wildly different beliefs, words, and ways of speaking? Differences so vast that the same life can be deemed both worthy of respect and worthy of public ridicule, an inspiration and also a disgusting threat? Differences that play out not just in the media, but in how wider society treats trans people?
These writers are recording the trans "debate" in one language, and trans people like me are speaking the realities of our lives in a totally different tongue.
How are we meant to reach the people who are not trans, when they are primed to believe the opposite of how trans people live our actual lives? How much longer must we misunderstand each other, trapped in the falsehoods created by the production of ignorance?
I WAS FIRST introduced to the concept of "the production of ignorance" at an early music conference in 2015. The focus of the day was on women composers and music-makers in Western history, and of particular interest to me was the question why, after celebration and acceptance in their own eras, after decades of careful research, reevaluation and performance, so many people, even musicians, believe that there were no women composers before the twentieth century. In one of the question-and-answer sessions, musicologist Melanie Marshall put forward an answer that clarified not only this problem, but which also explained to me so much about popular knowledge and general confusion over gender issues. Referencing the work of Nancy Tuana, Marshall described the concept and process of the production of ignorance: it is not just the absence of knowledge that keeps a truth from being widely known and accepted; it is also the active production of ignorance that suppresses that truth. It is not only that we are unaware of the many women composers throughout history: we are actively taught that there were none, or certainly none worth bothering about.
Similarly, it is not that trans people are ignored entirely, but that what we are taught as fact can often obscure and distort the truth in a way that even silence could not.
Not that silence is the solution, even if it were still possible. Much has been made of the "trans tipping point," from the front cover of _Time_ magazine to the daily, twice daily, articles in English-language media throughout the world. It is undeniable that the media is having a trans moment. Interviews with trans adults, features on trans children, possible changes in legislation that would help trans people, definite changes in legislation that will hurt us, another trans death in prison, another trans person in custody, a gender-fluid celebrity, a charity campaign. Some of this content is incredibly good, and some is just incredible. In an age of declining sales of offline media, the end of physical newsprint, and the importance of clickbait ad revenue, there's a particular winning formula when it comes to trans issues: anti-trans opinion piece (as shocking as possible), report on the hurt caused by said piece (search Twitter), and pro-trans rebuttal (in the same paper). Rinse and repeat on a regular basis. It doesn't matter why people are reading—agreement, rage, or the hope of titillation—so long as it sells. And, right now, trans sells.
When we apply the concept of the production of ignorance to this cycle, we can see why and how it plays out.
To learn how to learn about trans people, about the ways in which what we know about gender is shifting and growing, we first must unlearn.
THE QUESTION I am most often asked about being trans—on the internet, in a bar, on the bus, at work—is the one I most dread answering. Sometimes it's delivered through euphemism, sometimes crudely, and worst of all by a groping, uninvited hand. "Have you... you know?" or "So you're... post- or pre-op?" It's colleagues I barely know asking me what kind of genitals I have and whether I'm going to change them—and, if so, how—and strangers recoiling in horror, because "I don't know what you have down there." This, for them, is _the_ defining point of being trans: the "sex change," the "op." Never mind that there are many different kinds of medical treatments that trans people may undergo, if they are right for us, if we have the money, if our country's medical systems allow it. In the popular imagination there's a singular operation, and a violent, last-option one at that. Various forms of detailed, sensitive reconstruction work become "lop your tits off" and "cut your cock off." It's the supposed proof of being trans and, more than that, it's everybody else's business.
What we experience on a day-to-day basis we see modeled by the media: documentaries, interviews, movies, TV shows. It's the triumphant finale of the 2005 film _TransAmerica_ : Felicity Huffman sliding her hands between her legs in relief at the absence of her (much publicized prosthetic) penis. In the 2015 documentary _Girls to Men_ , the filmmakers framed the stories of their young trans masculine protagonists in terms of their journey toward genital surgery. Gory surgical footage and close-up cock shots: that the audience should become a voyeur is a given, because they, somehow, have not only the right to know but the right to gape. Even when a trans person has not volunteered the information, the topic is considered fair game—more than that, essential. Watch the 2014 interview of Carmen Carrera and Laverne Cox by Katie Couric: the ease with which Couric asks about her interviewees' genitals, and her confusion at being denied an answer.
For many of the people who ask, the fact that a ready answer might not be forthcoming is baffling. After all, isn't that how being trans is meant to work? Someone realizes that they're "trapped in the wrong body," then gets that body overhauled and emerges a new person. It's everything we've been taught from the earliest age: women have vaginas and men have penises. If we, trans people, want public acknowledgment of who we are then, the argument goes, we should accept the public judgment of our genitals.
If we were to take another example, and apply the same rules, it becomes obvious just how inappropriate and harmful this trope is. For some (not all) trans people, one element of being trans is the physical process of transition. It can be joyful, it can be painful, it can be messy, and it can involve surgery. The same could be said of parenthood. Conception, pregnancy, and childbirth are necessary parts of making a family for the majority of people. Like medical transition, it is vital that we're educated about these processes if there's a chance we'll find ourselves personally affected. And luckily, in both of these cases, the medical information is freely and easily available online, through public health initiatives, in libraries, and from the relevant medical authorities.
But it would _never_ be appropriate to approach a new mother in a café and say, "So, did you rip your vagina giving birth to that one?" When greeting a colleague returning to the office after maternity leave, we don't ask if we can examine the stretch marks and possible scars, or ask about hemorrhaging and post-natal incontinence. If we're close friends or family, we might well talk about the most personal physical aspects of creating and delivering a baby—the same is true of transition. But the need to be honest and close with our loved ones doesn't make the intrusion of strangers okay.
The second problem is that of language. Obvious transphobic language in the media—and in the wider world—is hard to ignore. Even those people who are themselves transphobic could hardly pretend that Julie Burchill's infamous 2013 column for _The Observer_ was inoffensive, with her descriptions of trans women as "bedwetters in bad wigs" and "dicks in chicks' clothing." You don't need to know anything about trans people to know that referring to us with insults is cruel.
What worries me more is the trend to describe all trans-related language as somehow "made up," difficult, and too PC to be allowed.
When I'm asked to give a talk, write an article, or deliver training on trans issues, I'm well aware of the fact that the words I use won't be familiar to everyone, and am happy to explain. "Trans" is the word I favor, as it has the broadest and most flexible definition: any person who, in some way or combinations of ways, has found that how they experience their gendered self does not fit with the gender and sex they were assigned at birth. "Cis" is the antonym of trans; just as we cannot describe being gay without having a word for straight, we need a word to describe experiences which are _not_ trans, as well as experiences which are. These words are blunt instruments, designed to give a rough understanding of the ever-changing world we find ourselves in; tools to help us to understand and challenge the ignorance and prejudice between us. They will change with time, and new words will take their place: humans are quite remarkable in their capacity to learn new words. For example, we now use the word "you" for both the singular and the plural: not so in Early Modern English. In the past twenty years the word "internet" and all its related terms and add-ons (including the term "add-on") have entered into daily, unremarked usage. As a teacher, I'm constantly introducing words that are new to my students: rubato, cantabile, légèrement. When new words can bring us closer to something we want to say, then we are all too happy to learn them. And this is why I'm suspicious of the claim that trans-related words are too much, too hard, and of no use.
Even when a word has been in usage for a long time, those who are suspicious of what that means in terms of gender are quick to claim that the change is too fast. "They" has been used as a singular pronoun in English for hundreds of years; we find examples of the singular "they" in the works of of the singular "they" in the works of Shakespeare, Austen, and Swift. But trans people like me, who use the pronoun "they" as a gender-neutral alternative to "he" or "she," are often mislabeled in the media by editors who struggle with its usage. By implying that trans people are faddish and difficult about words, writers can cast aspersions on the validity of our language—and of our selves. By claiming that our words are too hard to understand, the media perpetuates the idea that _we_ are too hard to understand, and suggests that there's no point in trying.
Learning how to talk about trans people is not difficult, and doesn't require any specialist knowledge. Just as you would in any other situation, you have to reflect back the words a person uses about themselves. Wanting to be referred to in an accurate and respectful way isn't a trans-specific thing, but a cornerstone of polite society. I don't call my Jewish friends Buddhist. It's the same with trans people. Use the right names, use the right pronouns, and don't fall for the line that we're too difficult for our own good. I know many cis people who are so nervous about getting it wrong that they're scared to try to get it right, but it's okay to ask. I would far rather someone ask me what pronoun I use than try, out of embarrassment, to guess, and get it wrong.
The final problem of the framing of trans lives so often recycled by the media is perhaps the hardest one to see. So often it is the only way in which trans people are included in the media at all. Less obviously pernicious, but still dangerous, is the way in which trans people are only featured when _being trans_ is the story.
The most obvious, and most egregious, example in recent years must surely be in the press treatment of scientist Kate Stone. Dr. Stone was gored by a stag in a freak accident in late 2013; as someone who had not sold her story, who was not in the public eye, she had no reason to suspect that her accident would hit the news. And yet she, her family, and her friends were confronted with headlines such as "Sex Swap Scientist in Fight for Life" and "Deer Spears Sex-Swap Kate." Speaking to the _Guardian_ , Stone explained: "I have no regrets about the accident. I have never for one moment thought, 'Why me?' But some of the reporting was horrendous. The media door-stepped my family, my friends and colleagues. On radio, one 'expert' was asked, 'Was Kate gored by a stag because she was transgender?'"
This is an extreme example, for sure. Most of us will never experience this kind of treatment, although more trans people have experienced door-stepping than you might expect. Stone sought help from the Press Complaints Commission and, eventually, the intrusive stories were withdrawn. But the broader point—that being trans is, in its own right, newsworthy—impacts on the way all trans rights are framed.
When I was first starting out as a performer, I was shocked by the number of people in the media who were more than happy to write about me, but not as a musician, only as a trans sob story. I refused to provide "before" and "after" pictures, to give away the personal details of my life; most of the press interest disappeared. We're forced into a double bind; if we're to speak honestly about who we are, then we must have the freedom to talk about being trans, but we cannot be reported honestly if being trans is the _only_ aspect of our lives discussed.
I know many trans people who have spoken to the media about what it is that they _do—_ their professional expertise, their artistic ventures, their latest projects—and are later confronted with a final copy that cuts out all of that detail for a clichéd trans narrative that has nothing to do with the actual life of the person featured. Through this framing we are made to look like attention seekers and oddities. If we don't mention being trans, we risk one of two options. If, like me, we are visibly different, then we are usually pressed to talk about it. If we are not seen as trans, we run the risk of accusations of deception, of a scandalous "reveal," if we don't announce that we are trans from the get-go.
As in the media, as in everyday life. Without being able to talk about being trans, I can't speak about how I have been made to suffer for it, and also what I have learned through those experiences. I can't make things better by being silent. But neither can I speak about every other part of my life—live every other part of my life—if other people focus only on my transness as something shocking and different.
It has to be our choice to talk or not talk about being trans, and—whether we talk about it or not—we still need to be recognized as whole, complex people. Our lives are truncated when we are seen only through the stereotypes of others, and we waste so much time struggling against those constraints. Whether it's on the front pages or in the workplace, "being trans" is never the most interesting thing about us. Accept it as one crucial part and then, please, keep listening.
IF THESE AREN'T the parts of trans life trans people themselves would like you to know, then how have these stories become so prevalent?
Because we, as trans people, are not the ones in control of the trans news story.
In 2014, American scholar Jamie Colette Capuzza published a study analyzing sourcing patterns of trans stories in the US media. Looking at data from the preceding four years, Capuzza found evidence to support what has long been noted within our trans communities: trans people are far more likely to be written _about_ as an "issue" than we are to be recording our experiences and insights as equal participants. Just as often as not, the cis journalists writing an article or putting together a news segment would fail to include even a single quote from a trans person. Of the trans people who were quoted, the vast majority were white, the vast majority were trans women, and trans people who don't fit into the gender binary were hardly present at all.
Beyond that, Capuzza found a distinct skewering of focus: trans people were far more likely to be written and talked about in the entertainment, beauty, and lifestyle sections of the media than in the "hard news" categories of political, legal, economic, and medical reporting.
Trans people are not always—not even often—approached by the press for comment or explanation when trans topics come up. When we are allowed to speak for ourselves, our answers are usually trimmed to fit a script written by others. And when that script is offered up as the truth of what trans people are and used as the foundation for future scriptwriters, then we end up with a trans "reality" created and maintained by those who aren't: a perfect trans chimera that mutates into the snake swallowing its own tail.
This isn't just a trans thing, of course; all kinds of people and subjects are distorted by reporting. The "news" is a funny combination of playing to a known audience, keeping advertisers happy, trying to attract attention in a crowded marketplace, and appealing to the political sensibilities of editors and stakeholders. Sometimes there's some great journalism thrown in as well, if we're lucky. But when not all that many cis people know a trans person in real life—or don't think that they do—that understanding of the margin of difference between media spin and everyday reality can slip down to nothing at all.
WE CAN BE misinterpreted through lack of representation—but also through the particular prejudices of popular writers. The denial of reality, the cutting of a story to fit a particular narrative, and presenting uninformed opinion as fact: on a weekly basis, these are the ways in which trans people are represented to the wider world by those who know nothing about our lives.
What would someone who has never met a trans person—never worked alongside a trans colleague, had a beer with a trans friend, watched a movie with a trans sibling—think of Jeremy Clarkson's recent op-ed in _The Times_ , "Transgender Issues Are Driving Me Nuts"? If you know Clarkson, you'll know that this is the kind of piece he made his fortune with: reactionary, cutting, the kind of thing described as "not politically correct." He writes: "[Children] dream impossible dreams. You don't actually take them seriously. You don't take them to a hospital when they're 10 and say, 'He wants to be a girl, so can you lop his todger off?'" Anyone who knows anything about medical transition and the treatment of trans youth knows that genital reconstruction surgery is only available for adults. But those people are not who this article is written for—and the people for whom it _is_ written now have another piece of proof that trans people are deranged, delusional, and not to be trusted with children.
On a much subtler note is _New York_ magazine's article on the removal of Kenneth Zucker from the Child Youth and Family Gender Identity Clinic in Toronto. A long-form read on the debate around the treatment of gender non-conforming children and teenagers, the reporting appears unbiased, nuanced—that suspect word, "objective"—the kind of piece that requires time and attention from a reader, and rewards you for your efforts with the sense that you have learned something concrete. But the article suffers from a number of omissions that would have given a more balanced picture of the current debate around treatment options for young people. Only a fraction of the research that contradicts Zucker's approach is mentioned: notable absences include an American Psychological Association award-winning paper from Y. Gavriel Ansara and Peter Hegarty. The World Professional Association for Transgender Health, the global leader in standards of trans-related health care, has condemned reparative therapy—the kind of therapy Zucker is alleged to have practiced—as unethical. This vital detail is missing from _New York_ magazine's article. The leading trans researcher consulted by the author on the historical, psychological, and academic context for Zucker's work was not quoted, and more time is given to the trans activists who have protested Zucker's work than to the trans psychologists and researchers who have criticized his methods from within the field. The risk with articles such as this is that readers may be left with the impression that trans people are hypersensitive fanatics unable to function in the "real world," most particularly that most logical of real worlds, that of scientific research and development. Researching the full detail of all the issues would take a full day and access to an academic library. It's heartbreaking.
And the most popular error of framing is, as ever, asking a non-expert to weigh in on a sensitive issue requiring expert knowledge. Before the publication of the results of the first-ever UK Trans Inquiry, a cross-party parliamentary investigation into the current state of trans rights and experiences in the UK, the _Evening Standard_ published a piece entitled "Changing Sex Is Not to Be Done Just on a Whim." Written with the kind of hyperbolic humor frequently found in newsprint editorials, the arguments contained within went beyond opinion and into the realm of misinformation. Following decades' worth of campaigning from trans activists and extensive consultation from experts and laypeople from across the country, the inquiry recommended reforming the current confusing, time-consuming process of legal gender recognition, instead allowing UK trans people to update their documents with a simple online form. That update has already enjoyed great success in Ireland, with no sign of complications or dire societal fallout. But instead of focusing on these facts, and the genuine debate around them, the author instead weighed in against "gender as a choice issue" and misrepresented the concerns of trans campaigners and our supporters as displaying "a worrying indifference to a basic question of what makes us ourselves." It is not a furious or hateful piece, but it mattered. The _Evening Standard_ , given away free every evening throughout London, is impossible to escape. Its message carries. And that line—that trans people "change sex on a whim"—was one that I had heard again and again in political discussions leading up the publication of said inquiry. Despite its lack of foundation, it is used as an excuse by lawmakers, civil servants, and politicians to reject calls for a simplified change of legal gender. Even when the person spouting it claims that they don't personally believe it, they put that phrase in the mouth of "the public," "the electorate," and use it as a reason why trans people cannot be allowed to have equal rights. And while I might have read four or five excellent takedowns of this piece, of that idea, online, I'd be willing to bet that the people who were already primed to believe that trans people are fickle and confused read only that piece—and agreed with it.
This is the reason why it's insufficient to respond with accusations of being "offended," to say that anyone who disagrees with these pieces is not obliged to read them and can take their support elsewhere. Trans people may choose not to consume transphobic media; we have no choice about living in a world shaped by this misinformation.
A study from the University of Saskatchewan, in Canada, published in 2014, showed the real-world impact of such media. Looking at personally and culturally endorsed stereotypes of trans people and behaviors, the researchers found that, in the absence of real-life experience, cis people fell back on what they had learned through the media. Overwhelmingly, what they had "learned" was that trans people are "confused." Respondents, relying on images of trans people in films, in the news, on TV, described trans women as wig-wearing caricatures of femininity, who most likely had no ambitions beyond looking pretty and finding stereotypical feminine employment and male approval. Trans men, on the other hand, probably wanted to pursue typically masculine careers and hobbies—working on an oil rig, playing team sports—but were let down by their smaller, weaker bodies and inability to be accepted by "real" men. Some choice words used to describe trans people were "odd," "gross," and "freaks."
FOR ALL OF these reasons, large numbers of trans people refuse to have anything to do with the mainstream media. We do have alternatives, after all. I'm a member of the first generation to have had internet access at home. Being able to research trans people, trans history, surgical options, was a lifeline for me, although it took some getting used to. Compared with how trans youth use the internet now, my experiences are already archaic. I still had to learn the language from the mainstream media, before I had enough information, and the courage, to enter those words into a Yahoo! search. What I found was limited to a handful of forums, a small list of books, and an overwhelming amount of vitriol and anti-trans hate. For many teenagers now, mainstream media is no longer the first place they hear the words "transgender"; they already have the entire trans world, or an approximation of it, at their fingertips. They can follow the real-time transitions of popular vloggers, swap advice and support on Tumblr, learn the theory and practice of social justice through reading blog posts and online articles, and catalogue their own transitions on Instagram. For trans adults, too, all that we can't find in a newspaper, on TV, in a cinema, is available for us in the comfort of our own homes. I recently binge-watched _Her Story_ , an Emmy-nominated web series created by trans and queer women, starring trans and queer women. I was in heaven just watching trans people—actual trans people like me and my friends, but with better outfits—navigating questions of friendship, love, societal pressures, and internal doubts. I put my money in for the crowdfunding of _Happy Birthday, Marsha!—_ a movie about the history of the Stonewall riots that is everything Roland Emmerich's spectacular flop _Stonewall_ was not. There are trans speakers on TED, funny but serious lists on Cracked and BuzzFeed, and a host of well-informed, well-researched bloggers and academics to follow. Complexity, nuance, a basic level of humanity: trans people speaking for ourselves.
The only problem is that the people we need to reach are not always—not even often—there with us. As much as we can assume that private internet access is universal, it is not. When we rely entirely on alternative media, we cut ourselves off from those who cannot afford to join us. Even with that access, there are hurdles to overcome. You need a magic word to get in—a search word, a recommendation, a click-through— _something_ that will open up your media options from the traditional to the new. You also need the will to seek out something different, the need to find something better. What I have found, in outreach and consultation and most specifically in political work, is that the people who have the greatest power to affect trans people's lives through legislation, employment, housing, and environment are also the people most likely to accept traditional media portrayals of what trans people are, and not understand the need to look elsewhere for representation.
So, as much as I would like to say "enough"—to withdraw from the mainstream media altogether, as consumer, as spectacle, and as participant—some of us stay. We stay in the hopes that, if at least one of our quotes makes it into a finished piece, then that quote will change one mind. We offer an interview, knowing that it will be cut and edited in ways we would not like, because we think of a young person without internet access who might just pick up a paper copy because it's there. We go on television, then try to protect ourselves from the inevitable abuse that follows, because we hope to be heard by the people who would never dream of watching an online news show produced by trans people. And we become complicit in the machine, knowing that if other people had not done the same for us, we would not be here today to keep the fight going.
But we can still expect more, work for more, and ask for better.
#
# "Call Me Caitlyn"
I **do not know Caitlyn Jenner. I do not know anybody who** knows Caitlyn Jenner, nor anyone who has worked with Caitlyn Jenner and, to be honest, Caitlyn Jenner does not take up a great deal of my time. I say this to ward off the inevitable questions, because being trans in a post–Caitlyn Jenner world is a little bit like being British and visiting small-town America: no, I do not know your cousin Gary who lives in Brighton; no, I do not drink tea every day at 4 p.m.; I have never met the Queen; and I'm sorry, but I don't miss Princess Diana.
Caitlyn Jenner might well be the most famous trans person in the world, but I don't know a single trans person who would call her a trans icon. I don't say this to be cruel, but because I am confused as to why the mainstream press—the _Washington Post,_ NBC, MSN—would give her that title without the support of trans people. Mey Rude, trans editor for the online magazine _Autostraddle_ , sums up everything I've heard from my own London-based community here:
Apart from being a big moment, this is also a pretty strange one. It's strange that a trans woman who, so far, hasn't done any work in the trans community has been crowned our queen. It's strange that a trans woman who is famous, rich, white and conservative, four things that do not describe most trans women, is now the face most cis people think of when they hear the word "transgender." It's weird that people are saying that a famous, rich, white, conservative and conventionally attractive trans woman is humanizing trans people to a whole new group of people. Why didn't Janet Mock or Laverne Cox do that for them? Why didn't CeCe McDonald? Why didn't Islan Nettles?
But look at this phenomenon in a historical context and I guess it isn't so surprising. The Caitlyn Jenner story, the Caitlyn Jenner brand is, for all the newness, all the talk of being game-changing, following a path that has been trodden before. Christine Jorgensen, labeled the "GI Joe turned blonde bombshell," hit the headlines in the early 1950s and stayed there for nearly twenty years. Born in New York in 1926, Jorgensen traveled to Denmark to pursue medical transition after a brief spell in the army. She left the States a "shy, miserable person"—she returned to see her story on the front page of the _New York Daily News_. Coverage of her transition garnered more press attention than the news of the polio vaccine and, in her role as glamorous media darling, Jorgensen's fortune was made.
An examination of the trans success stories of the past seven decades, as measured by media celebration, attention, and payout, reveals a well-established pattern of which Caitlyn Jenner is simply the most recent example.
The first ingredient is the "can you believe it?" element. A dramatic shift from hyper-masculine to hyper-feminine, or vice versa, is more shocking, more titillating and makes for a better scoop. Even if this sudden change is at odds with the actual story—how the trans person in question experienced their growing self-knowledge and transition—it can be used as a frame and a guide. "Before" and "after" photos are necessary props, preferably as stereotypically masculine and feminine as possible.
Next comes the Cinderella moment. Nobody wants to hear about waiting years for a doctor's appointment, waiting to see what dose of hormones works best, trying to unpick gendered expectations, trying to make sense of a frequently hostile world. Instead, what works best is an almost overnight "sex change"; not an ordinary life lived in stages and negotiated with compromises and setbacks, but a magical transformation, granted with the wave of a wand.
And finally, in this most popular trans media script, there is the focus on simplification and exceptionalism. Instead of focusing on trans communities, and the most common and the most pressing issues faced by trans people, these stories focus on personal fulfillment and the desire to fit in. Transphobia is relegated to personal slights suffered, rather than a deeply entrenched, multifaceted problem. The trans subject becomes one of the "good" trans people: not angry, ground down, and difficult, but seemingly eager to jump through the hoops set for them by the cis majority.
This is not a personal attack on Caitlyn Jenner. I don't know her. I know that she has raised money for trans causes, and that she has raised awareness of young trans people contemplating suicide. I know that we have very different ideas about what it is to be a woman or a man or any other gender. I know that her political choices are not my choices. I know that I don't know anything of her private life, her character, the struggles that she must bear, as we all bear our own struggles. I know that she is not my icon, and not my representative, though I wish her every happiness.
I know that this conversation is so much bigger than the public figure of Caitlyn Jenner.
And I know that this historical link—between Caitlyn Jenner and Christine Jorgensen—extends to the trans communities impacted by this double-edged sword of media success and successive waves of visibility. And that we have a chance to do things better the second time around.
I BELIEVE THAT the reason why so many trans people today feel ambivalent toward Caitlyn Jenner is because she is our Christine Jorgensen. And because, half a century later, the members of our community who are suffering the most are still being ignored in favor of a glamorous makeover and a tell-all story.
During the years when Christine Jorgensen was working on the movie adaption of her autobiography, there were many other trans people in the United States making waves of their own. Unlike Jorgensen, the media paid them little attention. Unlike Jorgensen, they didn't get compensated for their work. _The Christine Jorgensen Story_ was proposed in 1960 and hit cinemas in 1970, complete with lurid marketing and the tagline: "Did the surgeon's knife make me a woman or a freak?" For Jorgensen, just another aspect of the media machine; for those without her privileges, a life-or-death question, most often answered with hatred and violence. It was not a situation that could hold for long.
Nineteen sixty-five saw one of the first in a line of protests that would lead to the famous Stonewall riots, considered by many to be the birth of the LGBT movement we know today. Dewey's Lunch Counter in Philadelphia had been refusing to serve gender non-conforming customers: queer and trans people who didn't look the way society thought they should, who didn't act in a so-called appropriate manner. So those same customers—more than one hundred and fifty of them—turned up, sat down, and wouldn't leave. They handed out leaflets, and waited for the management to back down—which they did. The majority of the protesters at Dewey's Lunch Counter were black. They had taken some of the tactics from the civil rights movement, tried them in another setting—and won.
Another major protest followed the next year, at Compton's Cafeteria in San Francisco. In her groundbreaking 2008 work _Transgender History_ , historian Susan Stryker recounts the details: that, again, it was the most vulnerable queer and trans people who came under attack, who were pushed too far, who fought back. What began with the usual police abuse finished in an all-out street fight. No date is available for the Compton's Cafeteria riot because recording it wasn't thought of as important: there was no press coverage, the police report disappeared, and when the surviving participants were interviewed, years later, they couldn't remember the exact day.
The reasons why it all kicked off are depressingly familiar to many trans people, and our wider communities, today: racism, classism, the pressures of gentrification, and the abuse of trans women by the police. Stryker explains:
The police could be especially vicious to "street queens," whom they considered bottom-of-the-barrel sex workers, and who were the least able to complain about mistreatment. Transgender women working the streets were often arrested on suspicion of prostitution even if they were just going to the corner store or talking to friends; they might be driven around in squad cars for hours, forced to perform oral sex, strip-searched, or, after arriving at the jail, humiliated in front of other prisoners. Transgender women in jail often would have their heads forcibly shaved, or if they resisted, be placed in solitary confinement in "the hole." And because they were legally men... they would be placed in the men's jail, where their femininity made them especially vulnerable to sexual assault, rape, and murder.
The same people who paid money to gawk at Jorgensen, who might even call her "brave," were all too often complicit in making and maintaining a world savage in its punishment of her trans siblings.
Which is not to say that this was Christine Jorgensen's fault, or that she did not do a great deal of good in the world—clearly, demonstrably, she did. She was a lifeline to hundreds of thousands of people, and her openness inspired compassion and hope in many. Her story, and her willingness to tell it, spread knowledge and spread ignorance, enlightened and erased. Both of these things can be true.
OF COURSE, THERE are worse things than being ignored.
The last time I saw a trans woman on television—or, more correctly, the _character_ of a trans woman on television—was during the latest series of _The X-Files_. I couldn't resist the pull of nostalgia, but wished I had with the introduction of "Annabelle." A minor background character, she appears only to flesh out the monster-of-the-week story. A black woman, a sex worker, and, as is quickly explained, a trans woman who is also a drug addict. Who's had "the surgery"—you know, on her genitals.
That's it. That's the joke. That's the reason for the character to exist: to be a black trans woman, who is a sex worker and a drug addict, because being that kind of woman, combining those particular traits, is apparently intrinsically funny. In a show designed to stretch credibility in all kinds of glorious ways, credibility disappears in an instant as Annabelle reveals her vulnerabilities to the FBI agents capable of arresting her and holding her in a men's prison—all for the sake of a punch line.
The inclusion of marginalized trans women as jokes or victims only is not a new trend. The long-running series _CSI: Crime Scene Investigation_ has found a particular niche in sensationalizing vulnerable trans women for ratings, offering us a psychotic trans murderer, numerous dead trans women (complete with genitalia jokes), and a "delusional" trans woman stabbing herself to death.
Even an actress as gifted and groundbreaking as Laverne Cox has played these roles, early in her career. Before her breakthrough role in _Orange Is the New Black_ , Cox appeared in _Law & Order_, in _Law & Order: Special Victims Unit_, and in the role of "Transsexual Prostitute" in _Bored to Death_.
As noted by _Bitch_ magazine: "In a 2012 survey, GLAAD found that transgender characters were victims in 40 percent of appearances. Additionally, 20 percent were sex workers. These are the roles people associate with transgender people, especially women."
This is not about only wanting "respectable" trans people to be portrayed.
This is about asking why these facts—that many trans people are sex workers, that many trans people turn to drugs and alcohol, that many trans people suffer violence both at the hands of those they know and at the hands of strangers, and that trans people who suffer from the effects of racism are most likely to suffer further violence and abuse—are suitable fodder for light entertainment, but not for an urgent and sincere investigation into the oppressions which are killing the most marginalized members of the trans community.
Instead of reporting on the _whys_ of all of this—the scandals that are endemic racism, endemic transphobia, the particular hatred of trans femininity and womanhood that is transmisogyny, the daily ways in which it is decided that some people are not as worthy of protection, of life, as others—instead, the lives of marginalized trans women are used as fodder for schlocky drama series, the background hum of an oversaturated media machine.
In the rare instances where real-life reporting does take place, the narrative tends to follow the fictionalized example. Scandal and titillation, the wrong names and the wrong pronouns, lurid accounts of violence committed and theories as to why (it's hinted) that violence could be justified. It can seem as though the journalists writing these reports forget the reality of what it is they're covering—very often the ending of a human life, the final, desperate moments of another human being—and instead write as though they're recapping an episode of the latest crime serial.
When leading human rights lawyer Sonia Burgess was pushed to her death on the London tube tracks in 2010, the mainstream media coverage was, in the opinion of many trans people and our supporters, depressingly lacking in sensitivity. Some reporters focused on unconnected and salacious details of the victim's sex life; most misgendered both the deceased and the accused, a fellow transgender person. Headlines such as "Sex Change 'Woman' Accused of Killing Cross-Dress Lawyer on Tube" felt like a trivialization of a tragedy. The behavior of the media was so upsetting that it led to the founding of All About Trans, a UK charity devoted to ending transphobia in the press.
Five years later, Senthooran/Nina Kanagasingham, the person responsible for Sonia Burgess's death, was found dead in their police cell. They had suffocated to death with their hands bound and a plastic bag over their head—fellow prisoners reported hearing them shouting "Help me" before their body was discovered. None of the newspapers that had so readily reported on Kanagasingham's appearance and demeanor when on trial could be bothered to record the result of the inquest into their death: "self-inflicted," according to the National Offender Management Service Deaths in Custody Database.
And it's not only this exploitation of the suffering and deaths of the most marginalized trans people that does so much damage, but the ways in which it is _only_ this suffering, these deaths, which are considered newsworthy. In all the trans-related media I've consumed, I could count on one hand the mainstream outlets that have covered the lives of marginalized trans people in an accurate and respectful way, rather than simply capitalizing on their pain. A "sensational" murder may be worth a headline, but nothing of the artists, lawyers, advocates, protesters, activists, everyday and extraordinary people resisting and fighting and thriving. The 2015 film _Tangerine_ , following a day in the life of two trans women of color, received widespread critical praise—but has nothing like the reach of a popular TV show or tabloid newspaper. There's a reason why so many trans people are such fans of prison dramedy _Orange Is the New Black_ : where else do we get to see such a realistic, respectfully written trans character, beautifully portrayed by a trans actress, highlighting real issues faced by trans people? Activists who flip the script on who gets to be the subject of the news, who gets to write the story and what that story contains—women like Monica Jones, Miss Major, and Lourdes Hunter—are rightly praised by online LGBT outlets, but I have yet to see their groundbreaking work covered by the mainstream press. There is a strict dichotomy of trans narratives which are deemed "worthy" and those that are not. There is endless space available to speculate on whether Caitlyn Jenner will detransition, or to number trans women of color as misnamed, misgendered victims of violent crime. For documenting the richness of the lives of those challenging this system? Not so much.
THESE ARE THE two sides of the coin, the two ways in which trans people—overwhelmingly trans women—are portrayed. On one the glamorous "sex change" and, on the other, the victim, the freak, the joke, the threat. For one, power (of a kind), and for the other, none.
And maybe this is why, after all, I do have something to say about Caitlyn Jenner. Because, despite her wealth, despite her whiteness, her prestige, and her celebrity, huge numbers of people still laugh at her. They call her a man in social media memes, and journalists are paid to write op-eds in which they deny her the right to define her own gender. You can purchase a Caitlyn Jenner Halloween costume, complete with lingerie and mask, and you can rip her apart in a comedy routine. You can make her pronouns conditional; watch the liberal "she" turn into "he" when she says something out of line. She is protected from the worst of what trans people go through, but the received truth of her gender is still predicated on the goodwill of the cis majority. The received truths of all our genders are too often predicated on the goodwill of the cis majority. One side of the coin is far safer than the other, but that safety is not guaranteed, and not often ours to secure. There is a spectrum of experience, from danger and prejudice to safety and success: Caitlyn Jenner is further along that spectrum than the majority of trans people, but cannot escape it. We are all still linked.
And this, then, is the reason why, despite not knowing Caitlyn Jenner, I can feel let down by her actions. Because one of the reasons for her popularity is that she's seemingly cool with this media narrative, seemingly okay with the assumptions and limitations—at least in public. Instead of using the advantage she has to tackle a hatred that hurts us all, she plays into the stereotypes, laughs along with the jokes, positions herself against the less palatable (more "political") members of our community, and aligns herself with those who are doing the most to hurt us. One of the saddest things I've learned, consulting with members of the mainstream media on trans coverage, is that there are plenty of cis people in the industry who don't like what they usually hear from trans activists: please stop hurting us, please stop misrepresenting us, please stop demeaning us. If Caitlyn Jenner says it's not that bad, then they would rather believe her. If Caitlyn Jenner's made it, why haven't you?
It is, perhaps, old-fashioned to believe in the idea of moral debt and moral duty. It's certainly not something I could impose upon anyone else, and I understand the opposition to it: Why should Caitlyn Jenner be expected to do more for trans people, simply because she's trans herself? Why should trans people have to work harder, be less selfish, than everyone else? But when I think of how much has been given for so little reward by all the trans people before us—when I realize that neither I nor Caitlyn Jenner would exist as we are without the suffering and sacrifices of our shared communities—then I can't pretend that that debt isn't there, acknowledged or not.
I'll say it one more time—I don't know Caitlyn Jenner. I don't know if the insults hurled against her, the denial of her reality, hurt her—they would certainly hurt me. But the real-life implication of the attitudes she lets slide—that trans woman are actually men, that the underlying issues of inequality and prejudice matter less than looking right and fitting in, that you can't expect cis people to do better—those attitudes feed the foundation of apathy, ignorance, and cruelty that harms trans people without her protections.
Caitlyn Jenner is, most likely, the most powerful trans woman in the world. Think of what she could do, if she wanted to.
#
# Finding My Voice
B **efore I learned that there were words for people like** me, I knew what it was I was looking for. I just didn't know how to capture that in a way I could fit into my world and hold on to, to put my feelings into language. Without language, those feelings couldn't solidify. Instead of a stable narrative, my memories of growing up to be what I would later call genderqueer are little flashes of recognition and fascination, sunk back down into what I had been taught I should be.
I spent a lot of time staring at famous faces in absolute wonder: Grace Jones, David Bowie, Lou Reed, Tracy Chapman, Brian Molko, Patti Smith. I had no idea what these people's genders were, but I knew that I couldn't tell by looking; that when I looked at them I felt full of possibilities and longing. I felt the same way about expressions of androgyny in classical art, in movies, in poetry. Where I couldn't find them, I made them up; there were so many experiments with clothes and hair and makeup behind my bedroom door. I had murky, childish fantasies of being able to change my sex in any way I liked just by pressing a button on a magical machine. One of my most persistent daydreams was that, as an adult, nobody would be able to tell what my gender was, but that I would be so talented and so charming that nobody would ever try to find out.
But I can't overstate how hard it was to recognize myself as neither/nor when the whole world seemed to run as one or the other. There seemed to be a little gender leeway allowed for some kinds of gay people—for gay men who did drag and butch lesbians—but both of those categories made me feel like an outsider looking in. The model of transsexuality I'd heard whispers of was similarly fraught. "Transsexual," in this definition, meant a man who wanted to be a woman, or a woman who wanted to be a man. In the ways in which they were discussed by others, these people were still trapped by other people's definitions, and the best they could hope for was to exchange one set for another, if they accepted being treated as a freak show. Beside which, there was the question of surety. In the newspapers, transsexual women always said "I knew I was a girl," "I was a woman trapped in the body of a man," but I barely knew what it felt like to be me, and didn't know how I was meant to decide whether that internal experience fit into a boy or a girl camp. I felt trapped in my body a great deal of the time, but not all parts of my body: not my hands and not my voice. They always felt real to me. I could know that I needed my body to be different and, almost in the same moment, lose that knowledge from fear and from the impossibility of realizing those differences in an honest way.
It came to a head when I came out at the age of fifteen. I wasn't sure of what I was meant to call my gender or do about my body, but I needed to be honest about my sexuality, so I told my friends, classmates, family, that I was bisexual (queer for preference), and then added the rejoinder "gender-blind," because it was the best term I could come up with on my own. I mostly settled on the androgynous style that made me feel most like myself, and the binding that paradoxically left me able to breathe, but still felt enormous pressure to present myself the way other people wanted me to be—to be found attractive.
People had commented on my gendered appearance and behavior before; now, it was a subject for public debate. Some people said that they knew I had to be one way, because I was so "forceful," "dominant," and "like a man." There were as many saying that I had to be the other, and for the opposite reasons—femininity and prettiness. Girls excused their crushes on me by saying I was like a boy, and boys got angry with me for their crushes, because why couldn't I be more like a girl? All of that pressure pushed me into having to find an answer—but still I had no language.
After a year of reading absolutely everything I could find on being queer, I started noticing the breadcrumb trail left in the margins, in the footnotes. Alison Bechdel, author of _Dykes to Watch Out For_ , was a godsend: the background detail of her comic strip often included the names of influential LGBT works and authors. I discovered Kate Bornstein, and ordered a copy of _My Gender Workbook_ from America. I felt as nervous as if I had ordered porn through the mail. When it arrived, I could barely stand to open it, despite how desperate I was to learn what was inside. It was the old naming magic: I knew that, once I had the words, I wouldn't be able to escape the fact of what I was. Bornstein used the term "transgender," and in her broad, evolving description—the contrasting descriptions of other trans people which she included alongside her own—I knew I had an answer. Not that I had a category I could slot myself into, but that I finally had the key to unlock all that I needed to tell about myself, and a tool with which to craft my future. I found a T-shirt that said "gender free" and wore it with great pride, alongside my Doc Martens, black suits, and heavy eyeliner. I grew up, went to university, grew up some more, got better at explaining myself, set my heart on medical transition, had my plans changed by bad luck, got there in the end, found my place. I'm sure I'll collect more words in the end, and look forward to watching them change and evolve in turn.
A QUESTION I am often asked is why, as someone who wants to subvert gender norms, I would want or need an additional gendered label. Couldn't I simply refuse all descriptors? Or, failing that, call myself a feminine man or a masculine woman?
What this question misses is the twofold job of words like transgender, genderqueer, gender-fuck, androgynous. They do duty as the personal language for who I am but, far more importantly, their usage helps to develop a cultural language of greater gender plurality and nuance.
It's not about absolutes, or strict lines of demarcation between degrees of gendered expression. There are women in the world far more masculine than I am, and men far more feminine. I don't want to force them to use my words, nor do I have a desire to steal theirs. This isn't about creating more fixed categories, with more rules about who can and can't be admitted.
It's about understanding that there needs to be space in our shared cultural language for every individual iteration of selfhood to be communicated accurately. We find our own personal language within the broader terms, and accept the universality of the broader terms within our own personal experience.
What is right for me is my own use of transgender, androgynous, genderqueer. It is in my appearance, in my crafting of my sexed body, in how I see myself in the world: a denial of being either a man or a woman, and the embracing of many wonderful things in both of those categories. It is only one way to live outside the gender binary: there are millions more. Some of these ways will seem obvious to an outsider, and some ways won't, but they are all of them valid, and each word used is worth learning. Every time each one of us uses these gendered words, and others, we are enriching the total meaning of every term. The more we expand our definitions, the more space there is for everyone.
We push for the inclusion of these words into our common lexicon because, without them, it is so much easier to pretend that we're too impossible to exist. Being genderqueer can be a daily fight against being made invisible. My words are a challenge to the people who would strip my experiences from me. What can be described can be communicated and made real, becomes a shield against that invisibility and dissolution.
Trying to take away someone's language is usually the first step in trying to change them. When I was twenty-one I worked as volunteer at the London Lesbian and Gay Switchboard, a phone advice and listening service that was meant to serve the entire LGBT community. While explaining who I was, that I was transgender, one of the trainers shot me down: I couldn't be transgender, she said, because I didn't look transgender. Obviously, I was "just" a drag king. That's what she wanted me to be, so that's what she described. My ordering of my experiences into my own descriptors is a challenge to that kind of erasure.
Pronouns are a vital case in point. I do not doubt that the recent surge in the visibility of people like me is linked to the increased use of the singular "they" as a gender-neutral pronoun. When I was coming out, even that basic building block was lacking. Every time I was referred to with the wrong pronoun, a fundamental part of me was spoken away. The proposed pronouns I had read about—sie and hir—didn't translate into daily speech, at least not for the people around me. Now, through the use of they, I can feel the daily difference made by a language that supports the validity of my self. Each correct usage solidifies the awareness that people like me are just as much a part of the world as any other person.
HAVING FOUND MY own voice, my own language, was part of the battle. The other part, far from over, is in trying to be understood by other people not prepared to understand.
The majority of strangers still try to place me as male or female. On any given day, I'll collect a handful of gendered labels: "sir," "miss," "madam," "mate," and (my favorite) "my good man." In the face of disagreement, some people can double down on their need to enforce what they see. One time, ordering a coffee to go, I listened to two baristas argue over what to call me. They kept correcting each other's language: "Did she want a small or a regular?" "No, _he_ wanted a large, no cream." I joke that the difference between being a man or a woman is half an inch more on my undercut but, genuinely, how other people decide me is frequently confusing. I've been called sir with my mascara running, and miss in a three-piece suit. Being a semi-public figure, I find appraisals of myself in odd places: an academic text on the meaning of gendered identities was the strangest. The author was at pains to dissect my haircut along gendered lines, describing the shaved back and sides as masculine, the "shock" of hair in the front as feminine. If they had asked me, I could have told them that the reason I've sported a standard-issue hipster undercut for the past five years is because I live in central London and I found the temptation too great to resist. Once, in a master class, the famous singer giving instruction to a room full of young musicians waved her hand to encompass the whole of me and said, "I don't know what _this_ is." She couldn't even name me. I would be scattered away into pieces if I let other people decide me in their own words.
What the people who fail to understand me as I am might be surprised to learn is that their way of reacting to me, while common, is not the only way. Even when we are confused about someone's gender, and don't have a greater awareness of what it means to be trans, we have a choice to respond with kindness rather than cruelty. I've had some very special conversations with my nana about appearance, gendered norms, and being yourself; with her usual talent for appropriate understatement, she says, "Where would we be if we were all the same?" I've had similar conversations with my partner's grandmother over coffee after dinner, about the gendered pressures we all feel, the compromises we all make between who we want to be and how the world wants us to behave. I teach music to students of all ages; I have never found a child who has a problem with how I look or how I refer to myself. Some older students can be—for want of a less anachronistic word—cheeky: one bright teenager who thought he could waste some lesson time by asking "Why do you look like Justin Bieber?" and by calling me "sir/miss/sir/miss/sir/miss/sir." It was teasing, but it wasn't cruel. Younger children seem to accept what they see without worrying. One small student asked me, during our second lesson, "Why do you look like a boy?" I told her it was because I liked looking like a boy, and asked her what she liked to look like. She thought for a second, and then said, "Tinkerbell." After she had dusted herself with sparkly fairy powder, we went back to learning about Middle C, confident in the knowledge that looking how you want to look makes you better at piano. I've had people of all ages use a "they" pronoun to refer to me, without me asking them to, and I have friends and colleagues who had never met a trans person before they met me and yet took my gender, my pronouns, in their stride. It seems to have far less to do with gender than it does with broader issues of empathy and humility, and a willingness to understand that we are each the experts on our own lives.
I'm not sure how much longer it will take for people outside the gender binary to be considered legitimate. The rate of change so far has taken me by surprise, both in our current race forward and in our longer, historical, tendency to forget and roll back. What I am sure of, though, is that accepting people outside the gender binary has less to do with the idea of specific non-binary genders, and a lot more to do with working away from binary thinking in general. That we get better at seeing beyond us and them, valid and invalid, natural and unnatural, good and bad, and communicate instead the fullness of who we are to each other, respectfully, with compassion. When we are so surrounded by such diversity—in nature, in culture, in human spirit—how can we stand not to acknowledge it?
Not every—maybe even not many—people will want to designate their own gendered experiences as being outside the binary. But working for a less binary world would not only benefit trans people like me: we would all be the richer for it.
#
# Couldn't You Just... Not Be?
L **ike many children, I adored the Narnia books by C. S.** Lewis, and reread them constantly. The scene between Susan, Peter, and the Professor in _The Lion, the Witch, and the Wardrobe_ always had a special resonance for me; it felt like there was an important lesson to be learned. Susan and Peter have come to ask for advice on what to do about Lucy's unbelievable stories of falling through the back of a wardrobe and finding a brand new world. The Professor, whom we know to be extremely wise by his eccentricity and book-filled study, considers this problem, and asks Susan and Peter if Lucy is either mad or a liar. They all agree it is clear that she is neither. Well, says the Professor: "You know she doesn't tell lies and it is obvious that she is not mad. For the moment then and unless any further evidence turns up, we must assume that she is telling the truth."
When I have to convince the skeptical of the truth of my transness, I very often feel like Lucy.
It's amazing the number of people who'll try to argue other people out of being trans, as if no one else has ever tried before. Sometimes those doing the arguing are motivated by concern, and sometimes by annoyance, confusion, or outrage. Their "arguments" are presented as rational undoings of our supposedly broken reasoning, as though being trans is a riddle which can be solved with the correct interpretation.
Perhaps even stranger are those people who present no arguments at all but who wonder, gently, if we've considered just... not being trans. Just, maybe, avoiding it until it goes away. It often comes down to the split between words and deeds, the question: "Well, why do you have to do anything about it?" I heard that one a lot when I first informed people of my intent to medically transition. I suspect it's the same kind of reasoning as the injunction to do as you like in private, but make sure you don't do it in the street and frighten the horses.
Underlying all of this is the idea that being trans is something unfortunate, impossible to understand, and better to ignore. Something you could probably change, if you put your mind to it, or grow out of, if you'd just see sense. That can turn so easily from "Why do you have to do this?" to "Why are you like this in the first place?"
I don't have a quick and easy answer as to why I'm trans, no more than as to why I'm a pianist, or how I experience the color blue. I don't have to know every _why_ of who I am to know the truth of my existence, and know that I can only find happiness by embracing that truth. It doesn't make sense to me to try to reduce an enormous spectrum of human experiences to an on/off diagnostic, rather than following the more complicated and rewarding journey of investigating the totality of the human animal. But I feel the temptation to find a simple answer when other people are so quick to provide their own reasons as to why I'm trans and—within those reasons—blueprints as to how I could change.
The most common explanations for my transness given to me by other people?
1. That I'm a freak of nature.
2. That I'm desperate for attention.
3. That I'm mentally ill.
4. That I _hate_ nature, and want to go against it as some kind of rebellion.
5. That I hate women.
6. That I have unresolved issues with my mother.
7. That I'm scared to be a butch lesbian.
8. Because being trans is cool now.
The resolutions to all of these "explanations" are alarmingly simple, and multifunctional: stop being mad, stop being difficult, stop pretending, stop existing.
Naturally, both the resolutions and the explanations fall far short of reality. I'm both an operatic and alternative singer; when it comes to attention I'd rather have the applause of an audience than transphobic insults in the street, and it's far easier to appear cool with the latest haircut than through coming out as trans. By virtue of being mentally ill and being in therapy for more than half my life, I'm confident that being trans is _not_ a mental illness, and also of the fact that, if I somehow did hate women and not know it, my therapist would have told me and helped me work through it. I love my mother dearly but not in a Freudian way, and do not _fear_ being a butch lesbian but, rather, acknowledge all the ways in which I do not fit that category. As for being a freak of nature, or desperate to deny its truth? I don't understand how being true to my nature goes against it. I can't begin to square that circle.
IT IS THE question of surety that typifies this interrogative position most clearly for me. "Are you sure?" is a constant refrain, the response given to my name, my passport, my pronouns, my title. It's the final question my surgeon asked me before I went under, and the first question friends of friends offered on hearing that I was going to have surgery.
Being trans is not like other aspects of life, where necessary elements of doubt are considered natural. Where there would otherwise be investigation, vacillation, self-doubt, and fumbling—fear necessarily bound up with desire, belief with trepidation—there can only be flat, unquestioning stasis.
To be trans, you have to be surer than you've ever been, because being trans is what you are when you've exhausted every other option. And still, other people would like there to be a chance of something different, and so they ask "Are you sure?" just in case.
To be accepted for transition-related medical care on the National Health Service in the UK, and by some physicians in the US, you usually have to pass the Real Life Test, or Real Life Experience—a revealing title. The Real Life Test means living in your "acquired gender" for a period of time, sometimes up to two years, without treatment, to make sure that you're really sure. Not infrequently, doctors count the beginning of the Real Life Test as the date of first referral or diagnosis, disregarding years of self-knowledge, self-discovery, and real world experience. Asking someone to go without the hormones and/or surgery they're desperate for, while also navigating the world in a body more likely to be read as non-normative, might seem cruel but, presumably, that is the point. You have to run the gauntlet to prove your worth. If you don't want it enough to expose yourself to violence, ridicule, the loss of employment, the loss of a home, then you don't want it enough to be sure. If, after a certain period of time, you'd like to update your birth certificate, and make sure that your legal documents are in order, then you need to apply to the governmental Gender Recognition Committee, a panel of anonymous strangers, who are there to decide if you're sure enough about who you are to be officially recognized in your own life. To them you must prove that you "intend to live in the acquired gender until death"—they do not need to prove to you that they are worthy of making a judgment call about your surety. If your life has deviated in any way from the standard trans narrative, you must provide explanatory evidence to back up your claim of being sure. If your claim is rejected, you cannot update your birth certificate, and cannot be fully classed as your own gender. You can try again, but each attempt will cost you.
Most trans people have our moments of surety, moments that vibrate with the rightness of knowledge through every part of us in sympathetic resonance. But we also know that those moments are to be feared, and to be denied, if possible. Those moments happen to other people, because trans people are not normal, and not like us, and how can we be harboring that feared other inside ourselves?
We torment ourselves: "How can I know for sure?" "What if I'm wrong?" "What would happen to me if I'm wrong?"
We hold off our transitions until it is transition or die. We are encouraged to do so.
And some of us die.
Many of us who live have tried to.
We could do this differently, if those who were so scared of us could learn to pick away at that fear. Transition can be another word for learning, or the inevitable transformation that comes as a foundation of living. It can exist in multiples. It can flow in many directions.
Some people are sure of being one gender all their lives. Some are sure for years, and then sure for another gender, another way of being. Sometimes transition is a way of being safe, and sometimes it's a way of daring to disrupt safety and play with chance.
With the confidence and comfort of more than a decade of living openly as myself, of being loved as myself no matter the hostility of the majority, surety matters less than it used to.
I am not sure that I will be the same for the rest of my life. I am not sure that my needs will remain static or that I will not seek further expression for an expanded and maturing self. I know myself, but not all that I could become, the good and the bad. What I am sure of is that there was no line I crossed that turned me from someone who could have been diverted into someone who was unquestionably trans. I think there are decisions I could have made differently, and choices I turned away from. If every point of departure splits into another world, I don't claim that every iteration would ultimately prove the same.
But I do believe that the only purpose served by the question of surety is to bolster the illusion that those who ask it have created: that they themselves are sure, and safe, and not in the slightest like us, the ones they are judging.
EVEN WHEN YOU are sure you're sure, society can still try to make you disappear. It can call you a liar and a fraud and exclude you in all kinds of ways, large and small.
Bodily dysphoria, the sense that the felt sexed body is not right and needs alteration, is a burden that many, but not all, trans people bear. But there is another kind of dysphoria, one I have found common to every trans person I have ever met. Social dysphoria: the collision between who we are, how we should be, how we need to express ourselves and live our lives, and the gendered straitjackets others would force us into. It is the misery, the wrongness, of being forced to live a lie. The pain of being called fakes for our authenticity.
It is being turned invisible, which serves a dual role: not only is our disquieting presence removed, but the pain we experience as a result can be safely ignored.
Some critics of trans people have told us that we shouldn't feel this pain at being denied the legitimacy of our own selves; gender is, of course, just a social construct.
I wonder if these people also tell widows not to bother grieving over their husbands, because marriage is also just another social construct. Love, justice, mercy, faith: all just social constructs. It's a nonsensical argument that understands nothing of how we are constructed as social creatures, how we have created our interlocking but individual worlds of being human. Trying to deny us the right to exist as we know we are is to deny us the right to exist.
The results of that pressure? To stop being? Two of the biggest surveys ever carried out on trans populations can give us part of the answer. Based in the US and the EU respectively, what they have found chimes with research data, and community-based evidence, from around the world. The first, from 2011, was carried out by the National Center for Transgender Equality in the US; this is their summation of their findings:
Transgender and gender non-conforming people face injustice at every turn: in childhood homes, in school systems that promise to shelter and educate, in harsh and exclusionary workplaces, at the grocery store, the hotel front desk, in doctors' offices and emergency rooms, before judges and at the hands of landlords, police officers, health care workers and other service providers.
This report is a litany of suffering. A full 41 percent of respondents had attempted suicide, compared with 1.6 percent of the general population. Seventy-eight percent of gender non-conforming kids had experienced harassment at school; 15 percent were forced to leave school to escape abuse. Nearly half of all respondents had been discriminated against in employment, which makes sense of the fact that 16 percent of respondents worked in underground economies as a way to survive. Around one-fifth had experienced homelessness as a direct result of being trans, and the majority of those who sought shelter at a refuge were further harassed, assaulted, or simply turned away. More than half of respondents had been harassed in a public place; more than a quarter had been harassed by police. More than a quarter of respondents reported being abused in a medical setting: being mocked by staff after suicide attempts, undressed and left on public display, forced into unnecessary genital exams. Across the board, people of color suffered more than white people; racism and transphobia are a terrifying combination.
A similar survey was produced by the European Union Agency for Fundamental Rights in 2014. With 6,569 respondents, it is the largest study yet of trans people, and found that "the survey results depict a disturbing reality. They show that the equality of trans persons is, as yet, a hard-to-reach goal." The results are strikingly similar to the American study. More than half of respondents had been personally discriminated against for being trans, and more than a quarter of trans students experienced _frequent_ harassment in school. One in five respondents were discriminated against in health care and social services, and the same number were discriminated against when trying to find a place to live. Many had been the victims of hate crimes, but most did not report those crimes to the police, believing both that the police would not want to help and also that the police themselves are a source of danger. The more open a person was about being trans, the more likely they were to be the victims of hatred, abuse, and violence.
When you've suffered from the impact of other people's hatred, ignorance, callousness, it isn't just those instances that cause the damage: it's the long-term effect of each individual action, stacked up against each other. It's the cumulative impact of loss after loss: the loss of a job, the loss of a home, the loss of a family. It's the drip-drip-drip effect of bearing the weight of others' cruelties, and the means with which they alter the way you approach the world, and yourself. You lose the ability to trust in others, to trust in the future, to trust your own judgment. The walls close in; where do you find the strength to push back against them? It's so easy, from the outside, to say, "I wouldn't let it get to me. I wouldn't care if other people didn't approve." It's harder to feel that way when you know that the next person who doesn't approve might be the next person to assault you in the street, that the next person who doesn't understand will be the one standing between you and the job you need to keep going.
All of the research that I've seen— _all of it—_ shows a global epidemic of stigmatization, discrimination, erasure, and violence against trans people. It astounds me that, in the face of all this horror, so many of us are able to keep going, not just for ourselves, but for our shared communities, for each other. It is a tremendous testament to all those who've made it, despite the odds, but never an indictment of those no longer here. It is proof of the necessity of living an honest life.
We are not asking for pity when we describe the ways in which we are delegitimized and punished. We are asking for outrage, and respect for the fact that, despite all of this, we struggle on, and find joy as well as sadness. We are demonstrating our courage, and asking for shared strength.
What changes would we see, then, if cis people stopped calling us mad, and saw instead the insanity of a world that persisted in punishing us simply for telling the truth? What would we do with the knowledge of the reality of trans lives?
UNTIL NOW, IN the West, we have taken three main approaches to solving the problem of trans existence: ignore it, try to change it, or support it. That last has been the rarest. Only one course, so far, has been proven to work.
Not all trans people will want or need medical treatment, but the ways in which trans people can (and cannot) access medical care can show in the particular what holds true in the universal. What is held up as the quintessential trans experience provides the starkest example of how we are written away.
Trying to ignore the fact of being trans is the most common option, and the one that nearly all trans people themselves have tried. It's the one we've been raised to, after all. We can ignore from fear, from the knowledge that who we really are would hurt the ones we love, out of a desire to avoid bigotry and censure, and because we have internalized every awful aspect of transphobia our culture has to offer.
When the need finally becomes too great, and we cannot ignore our feelings any longer, we can gather our courage and confide in the medical system where, too often, we are ignored all over again.
I took an option unavailable to the majority of trans people, and paid for my own treatment privately. Everything I had found out about the National Health Service gender treatment pathway told me that, in the late 2000s, I would not be offered care as an openly genderqueer person. My musical career meant that I couldn't comply with the standard pathway of hormones followed by surgery and, after having had to delay surgery for many years to care for my brother when he became ill, I didn't know if I could wait any longer to try to fight for care I might never receive. I remembered the ways in which my ongoing mental health care had been cut off, and the pressure I had been under from doctors who did not know me to come off the antidepressants prescribed to me by my psychiatrist, because a young person on antidepressants did not fit their worldview. Having experienced that kind of medical failure in one field, I wasn't prepared to try it in another. I used my savings, made the changes I needed, and learned what it was to feel comfortable in my own skin. That basic and extraordinary freedom—of finally feeling at peace in my own body—available only because of the luck and privilege of being middle class with money in the bank.
This choice—pay for it yourself or go without—is common to the majority of trans people who seek medical care. Globally, society has been slow to acknowledge the validity and necessity of these treatments. A few insurance plans will cover some transition-related care, and a few large corporations—hardly an option for the majority of trans people. Few nations, internationally speaking, fund hormonal, surgical, and ancillary treatments for trans people. In countries like Britain, where a basic level of care is provided, the waiting lists can seem endless, and a trans person who needs care is often at the mercy of individual doctors. We swap war stories via email lists, conferences, and community meetings: general practitioners (GPs) who withhold all care out of prejudice and spite, specialists who ask perverse questions, guidelines that place arbitrary and insulting demands on potential patients, waiting lists that last for years and drive many to desperation. Only a person who has never experienced the savage pain of bodily dysphoria, who refuses to listen to those who do, could think this an acceptable situation.
Placed in this impossible position, many people self-medicate: I know that I would. My personal experience with trans communities in the UK, Europe, North America, and Australasia suggests that self-medication is extremely common, particularly now that hormones can be ordered online. Research projects into the health and well-being of trans people in India, Colombia, and the Philippines have found that self-medication is the norm and, deprived of affordable surgical options, trans women will resort to self-surgery and peer-administered silicone injections. While dangerous, this is nothing new; sexologists in the early twentieth century found that European trans people were carrying out their own improvised genital surgeries, and injecting paraffin into their breasts to make them larger. Desperate people will do what they need to do. It's better than suicide—and that is something that happens, too.
Conversion or aversion therapies, historically the second most popular way to deal with the "trans problem," categorically and empathically do not work, though it's not for lack of trying. Sometimes trans people have been the ones to push for any chance to change, but more often conversion or aversion therapy has been forced on trans patients by parents, spouses, and the state. Forced institutionalization, forced drugging, electroshock therapy, nausea-aversion therapy, shame-based "solutions"—the twentieth and twenty-first centuries have seen variations on all of these methods, and what they have produced is not a cavalcade of newly minted cis people, but trans people who must live with the additional trauma of frequently violent and sometimes nonconsensual medicalization. The pushback against these treatments is not new either. Michael Dillon, a British doctor and trans man, published _Self: A Study in Ethics and Endocrinology_ in 1946. Speaking from experience, both clinical and personal, he argued that trans people could not be changed through therapy, and that the best course of treatment was to provide the hormones and surgeries that trans patients requested. Seventy years on, we are still fighting those who would go against a tidal wave of evidence because they have a personal problem with trans existence.
As many trans people have noted, it's baffling that some doctors, some laypeople, persist in thinking that trans people can be forced or shamed, convinced or persuaded out of being trans, when that is the same trick that society at large has been trying on us our entire lives, to no avail.
IT'S HARD TO think of a worse punishment for being trans than the punishments society so regularly metes out—physical violence, threats of violence, verbal abuse, emotional suffering—and yet here we are. Surely that tenacity must count for something.
I don't want to fall into saying, "It's not my choice; I can't help it." How can I pick myself apart and say, "Here is how I would be, if I weren't trans"?—it's a pointless, endless exercise. Neither do I want to play the game of "But if I could be cis, would I?" What I will say is that when the world saw me as a certain kind of cis person, playing mostly by gendered rules and bending the more flexible ones, I was treated so much better than I am now. Society can sometimes be very obvious about punishment and reward, and seeming to be a cis person guaranteed not just the absence of punishment, but the active presence of reward. It didn't stop the misogyny, of course, but neither did transition. The kind of misogyny I experience is different, but it's roughly the same amount. What that false, cis reading of me did provide was all the little social kindnesses and steps up that make life so much easier: the smiles, the metaphorical doors held open, the preferential treatment, the attention and the praise. Being seen as a beautiful girl, a white girl, a thin able-bodied white girl with money and most of the symptoms of my mental illness temporarily hidden—society approved, and it let me know it. In that intersection of cisness and societal privilege I was taught what it was to be valued, even as sexism took its toll. Far better than any therapeutic ploy, the world around me taught me how it expected me to behave, how easy or how hard it could make my life. And it still didn't work.
But for those researchers and commentators who do not want to believe these facts, there is a final line of argument: that the treatments, social and medical, on offer to assist trans people simply do not work. It's a line still pulled by talking-head pundits and some doctors: they feel for us, but say that transition is a road to failure, and only conversion will help us find peace.
Inevitably, this claim is the product of ignorance and, sometimes, research now widely considered out of date and erroneous. A 1979 paper from Johns Hopkins University is the foundation point for the argument that physical transition is pointless, concluding that trans patients cannot be helped through medical means. Historian and sexologist Vern L. Bullough describes the flaws in Meyer and Reter's paper thus:
The study was based upon 100 patients, thirty-four of whom had been operated on and sixty-six who had not. Twenty-four of the thirty-four had their surgery at Hopkins and ten had gone elsewhere. The sixty-six who had not been initially evaluated had either been denied surgery or changed their minds or went elsewhere. Only fifty percent of the total sample could be located, almost all of who had the surgery at Johns Hopkins but not many of the others. Still the study concluded that those who had the surgery did not have any better psychological adjustments than those who did not. Though it is clear that Meyer's and Reter's research was fraught with so many methodological flaws (including self-selected samples, no real measure of adjustment, and poor response rate) that many peer review journals would have rejected it, it did receive publication in part because there had been so few long term follow up studies. (Fleming et al., 1980, Docter, 1988, and Money, 1991)
Critiques of this work, and further studies that demonstrated far more positive outcomes for trans patients, swiftly emerged.
Luckily, more than three decades of further academic research have given us a wealth of new data far more important than a simple critique of the old. It would be impossible to do justice to it all in this limited space; study after study after study has shown how trans people's lives improve when our need for physical transition is met with support rather than resistance. Those looking for more information would find a good starting point in the downloadable bibliography from the World Professional Association for Transgender Health website. Those looking to know how it feels, to have a chance at life in a congruent body, free of dysphoria? Just listen to trans people and what we know of our own lives. We have been speaking this truth for a long time.
Physical intervention—whether surgical, hormonal, lifestyle based, or a combination of the three—should never be treated as a necessary component of being trans, nor should we adopt a one-size-fits-all policy. It was necessary for me, but in a way that did not follow the "standard" route; every trans person will have their own unique needs. For me, it becomes an issue of personal alignment, of revealing the body already felt to be there. It doesn't solve the problems of being trans in a transphobic world, but it can at least make our personal worlds right.
There is always the question of regret. I have chosen to answer this question last, because I am so tired of it coming front and center whenever medical transition is discussed.
Yes, there are cases of regret. There are people who detransition. These cases are few and far between—the percentages are very small—but that doesn't mean they should be dismissed. With apologies to my friends who have detransitioned or retransitioned if I get this wrong, this is what I've learned.
Some people who are counted under cases of "regret" do not regret their treatment at all, but simply needed to pursue some further treatment to complete their transition. This is often the case for people who do not fit comfortably within the gender binary, as the treatment pathways available rarely offer support and options for people who may desire a mix of sexed traits. Other trans people who have expressed regret in the medical and community literature feel regret not over transitioning, but regret in their choice of surgeon and the results of their surgery. Some surgeons are excellent, but I know of several who have treated their trans patients badly, both in terms of results and in follow-up care. It doesn't surprise me that a trans person who cannot have a fistula repaired, for example, would regret choosing the surgeon who refuses to fix the results of their work. Some trans people who currently express regret will later go on to try transitioning again; it is a well-known occurrence for a trans person to need to try several times before they have the wherewithal to carry through their transition. It's not often easy. Family, friends, and broader communities can exert a tremendous amount of pressure on a newly out trans person; "regret" can often turn out to be no more than an initial withdrawal in the face of overwhelming disapproval.
And, finally, I do know a very small number of people who have detransitioned because they no longer feel like they are trans. One person explained it to me like this: sometimes life only presents you with short-term choices, and you have to take the choice that will get you through. You have to survive to have the benefit of hindsight. Transitioning got them through to a better place and, in that place, they could see that transition wasn't their final destination after all.
I don't believe that there should be any shame attached to taking a choice that was right at the moment of choosing, and then living to decide that, actually, another choice would be better. I think we should all be more open to the courage it takes to make both of those decisions.
Life is about risk. Every choice we make has the capacity to make our lives better or worse, to hurt or help us and the ones we love, to come to be something we regret or celebrate. We give our informed consent. I don't see why we should expect anything else, as trans people hoping for the basic right to breathe freely in our own bodies.
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# What About Sex?
I **t's surprisingly hard to find one standard, unified, uncontentious** definition of what sex and gender actually _are_ , and where the dividing line between the two lies. Does the truth of sex reside in the genitals, the hormones, or somewhere deeper? Is gender an expression or subcategory of sex, a class system imposed by the patriarchy, a system of societal encoding, personal expression, all of the above? Sex and gender are totally separate; gender is a lie, sex is real; sex is gendered and also a lie. Some writers argue that "female" and "male" refer only to sex, "man" and "woman" only to gender, while others claim the reverse. This proliferation of terms and meanings matters on far more than a semantic level. Not only does it highlight one of the main stumbling blocks toward trans acceptance so far, but it illuminates a different path toward that goal.
When it comes to sex, trans people are often told to stay in our lane. As the truism goes, sex is between your legs and gender is between your ears. We may assert the right to our gender "identities"; we may even, if we are lucky, be accepted in those social roles. But, as I am frequently told, we shouldn't forget that those identities are surface-level, distinct from our "biological sex" (male or female only); woe betide the person who might seek to confuse the issue. As absolute as the Cartesian dualism of mind and body, gender is a form of expression that can alter with culture, but sex is the fact of the body, and the body is unalterable—at least deep down, where it counts.
This is the understanding of the difference between "sex" and "gender" that I grew up with, and was taught to accept as fact. By this reasoning, a trans man might call himself a "man," but still be labeled "female," and vice versa for a trans woman. A genderqueer person like myself, no matter what changes they had made to their sexed body, would still be classed as the sex they were assigned at birth.
Whenever I come across someone suspicious of trans people and disinclined to believe us I know, with a sinking certainty, that they will bring up the subject of chromosomes. It's the trump card, the incontrovertible factoid to rely on when confronted with an unknown or ambiguous presence. "You can mutilate your body all you like—but you'll never change your chromosomes! You're XX/XY and that means female/male, and that's science. That's what you _really_ are." Science (note the capital "S") has not found a reason for trans people to exist and, therefore, trans people cannot be real. As the movement for trans rights moves into the mainstream, and with it a growing awareness of gender diversity and plurality, the more I hear these statements used—often aggressively, often in desperation.
After more than a decade of learning, of experiencing life at the margins of sex and gender, I no longer believe this strict gender/sex divide to be true. Gender and sex can, and do, mean different things in different contexts. More than that, they interact. Why is it that we classify bodies into "male" and "female" first, rather than through any other categorization? Why do so many ideas about sex cleave so strongly to gender stereotypes? Is it possible to consider the body as something neutral that exists apart from the sexed and gendered terms we use to describe it?
Some trans people would prefer to avoid the argument altogether; others to bring it to a head. I stand with the latter option. Not because trans people are a problem to be explained away—validated or refuted by a singular notion of scientific truth—but because, when the facts of our supposed sexes are used to invalidate and endanger us, it is too dangerous not to. Not only that: the possibilities as to what trans people can teach us all about the science of sex and gender are too precious to dismiss.
If the state and the legal system have an interest in maintaining a two-party sexual system, they are in defiance of nature. For biologically speaking, there are many gradations running from female to male; and depending on how one calls the shots, one can argue that along that spectrum lie at least five sexes—and perhaps even more. For some time medical investigators have recognized the concept of the intersexual body... Indeed, I would argue further that sex is a vast, infinitely malleable continuum that defies the constraints of even five categories.
So wrote pioneering biologist Anne Fausto-Sterling in her 1993 paper "The Five Sexes: Why Male and Female Are Not Enough." In the twenty-plus years since its publication, we have been inundated by developments in the field of sex research, developments which complicate, broaden, and enliven the old idea of a sex binary. Unlike the "facts" most of us were taught at school—that there are two sexes, male and female, opposite versions of each other with discrete, non-overlapping traits—this new knowledge paints a far more detailed, nuanced picture of what sex _is_.
From the very first our idea of oppositional binary sex is undermined by two simple truths: that we are all formed from the same material, and that there are more than two patterns for our potential shaping. Not only are "male" and "female" far more similar than they are different, but there are many ways in which these themes can be combined and developed. This is no two-part invention, but a full-blown symphony. Rather than possessing distinct genital differences from the get-go, we all begin with a base potentiality: the urogenital swellings and phallus of early fetal development. The legacy of this mutual foundation can be seen in the shared similarities of the penile-clitoral area, for one example, or in the fusing of original tissue that leaves behind the penile raphe, the raised line running along the underside of the penis. Changing hormone levels in adulthood cause growth or shrinkage of the penis/clitoris and shifts in the strength, length, and frequency of erection. That our bodies are capable of sexual reproduction is, arguably, an evolutionary advantage, but it doesn't make our sexed bodies opposites, and neither does it limit the possibilities of those bodies.
For many people, sexed development cannot be described in traditional "female" or "male" terms at all. Thanks to tireless work by activists, advocates, and researchers, we now know that the number of intersex people (or people with diverse sex development) is far greater than previously thought. The exact number of people counted as intersex depends upon which physical variations are included in the count, and also on the cultural pressure to normalize genital diversity as "malformation." However, according to academic and activist Cary Gabriel Costello, if we acknowledge the many ways in which the body develops along intermediate lines (such as clitoromegaly and micropenis), as well as chromosomal and hormonal variations such as congenital adrenal hyperplasia and androgen insensitivity syndrome, then up to one in one hundred and fifty people could be described as intersex: "about on par with the likelihood of having green eyes."
So does this mean that trans people are a third (or fourth, or fifth?) sex? Is intersex another word for trans?
No. Intersex people and trans people face a multitude of different challenges, and there is a shameful history of trans people needing to pretend to be intersex to access medical care, in the hope that the medical establishment would consider their needs more valid. And yet, when considering what it is we know about sex, when exploring how the science of sex affects and is affected by gender, it is vital that intersex experiences are included. The existence of sexes beyond a simple male/female binary, and the experiences of people who have been punished for falling outside those bounds, show us both how our cultural understanding of what sex is warps the evidence of nature and, through resistance and solidarity, shows us a better way of respecting all human bodies. Whether we are born with bodies that defy conventional male/female classification, or adapt those bodies later in life, we are better served by an idea of a sexual continuum than by the prison of two separate and opposite categories.
Intersex people, just like non-intersex people, are most likely to be women and men, the genders they were assigned at birth. Other intersex people would not describe themselves in binary terms, but neither would they place their experiences under the trans umbrella. With that said, it is also true that a not insignificant number of intersex people are also trans. When it comes to transition-related care, these intersex trans people are usually doubly discriminated against—often by doctors and sometimes, as in the UK, through legislation. In our sometimes different, sometimes linked, struggles, we often face the same enemies.
Perhaps no greater proof of the fragility and toxicity of the binary myth exists than in the medical treatment of people with intersex traits. Now, and even more so in the past, surgeons have been quick to "correct" the genitals of infants and children through surgery—invasive surgery on minors incapable of giving consent—in order to maintain the lie that only two forms of sexual development are possible. This practice is considered by many to amount to medical abuse. It was only in 2015 that Malta became the first country in the world to outlaw such interventions; this form of medical malpractice is a global problem, despite the condemnation of the World Health Organization and the UN Office of the High Commissioner for Human Rights. The scars—both emotional and physical—borne by the intersex people operated on can last a lifetime. It is staggering that such harm can be inflicted, and be so normalized, for the sake of maintaining a status quo. It is heartbreaking that bodily integrity, the sanctity of selfhood, and the right to live free from pain could mean so little compared with the pressure to fit into a false dichotomy.
Trans or cis, intersex or not, we need to wake up to the fact that treating sex as a fixed and oppositional binary is not only a distortion of reality, but is doing active, extreme harm to a significant percentage of our population. Rather than forcibly applying a fantasy, to our very real detriment, we could decide to accept the reality, and learn and grow from there.
SO, WITH THAT knowledge: if sex is not simply the division between two opposites, then what is it? For many researchers, the answer lies not in a static idea of what sex is, but in _how_ the body can be sexed, in different ways and at different times.
Instead of talking about "biological sex" as one uniform category, we can recognize the difference components of our biological makeup, and recognize also the permutations that can exist within and between each one of these groupings.
We can think of sex as being expressed through six broad categories, each existing on a spectrum, capable of being combined in different ways. Chromosomes, hormones, internal sex organs, external sex organs, secondary sexual characteristics, and general morphology: these are the aspects of our bodies that are affected by our potential for sexual reproduction. None are a simple either/or, but all contain room for variations. For most people these categories will align in a way we are taught to consider normal. We expect a man, for example, to have XY chromosomes, higher androgen levels and a standard processing of those androgens, testes and a prostate, a penis and a scrotum, higher levels of body and facial hair, and a broader, taller, more muscular body. But, while more common, these more typical alignments are no more valid than other combinations. The body of a trans woman who has pursued hormone therapy and surgery might combine XY chromosomes, higher levels of estrogen and progesterone with concurrent lower levels of/lower sensitivity to androgens, no testes, a prostate, a vulva and vagina, little body hair, no facial hair, breasts and curves. When looking at all the different parts of her physical makeup, what counts as "biological sex"? All of these categories are sexed, and all are "biological." Why would any one category—chromosomes, for example—be given precedence over another? And why should it be a problem if some bodies combine a mix of traits?
The history of "sex" chromosomes is fascinating for precisely this reason. Historian and philosopher of science Sarah Richardson charts this highly political history in her 2013 book, _Sex Itself_ : _The Search for Male and Female in the Human Genome_. In a world increasingly confused by the challenges of sex and gender diversity, and faced with the ambiguities present in other sexed categories, early researchers jumped at the chance to label X and Y chromosomes the final, irrefutable proof of binary sex, investing in them a host of gendered stereotypes. The only problem is, as explained by science writer Ian Steadman, the fact that "there are extremely few sexual characteristics solely controlled by the presence or absence of a Y chromosome—and just as there are plenty of characteristics controlled by genes found on other chromosomes, the 'sex' chromosomes also carry genes that determine traits that have nothing to do with sex. Y is not the essence of masculinity, nor is X that of femininity."
In every sexed category what popular wisdom supposes it knows is a far cry from the diverse riches found in nature. Another favorite of the binary sex brigade is the idea of sex being "hard-wired" in the brain as "male" or "female." Academic psychologist Cordelia Fine's 2011 book _Delusions of Gender—_ an essential work—analyzes the supposed science behind the popular headlines. What she found makes for profoundly uncomfortable reading. Flawed methodology of all kinds, tiny sample sizes, incorrect forms of analysis, guesswork, and unexamined bias: the studies and conclusions trotted out to provide proof of essential sexed differences veer strongly into the realm of "neurofallacies," to use Fine's term. "Male and female brains are of course far more similar than they are different. Not only is there generally great overlap in 'male' and 'female' patterns, but also, the male brain is like nothing in the world so much as a female brain. Neuroscientists can't even tell them apart at the individual level. So why focus on difference?"
That we are all shaped by the fact that we are creatures potentially capable of sexual reproduction is not in doubt. But all of us, trans or not, intersex or not, could stand to question the idea that this potentiality limits the possibilities of who we are and who we can be—and what we can be to each other.
THE TERM "SEX change" has fallen out of favor over recent years, for a number of reasons. It's a crude phrase, one that reduces the totality of being trans to our sexed bodies, inviting questions about our most private parts. For the trans people who do seek medical intervention, the process is never as swift and simple as "sex change" would imply. Much as I've fantasized about swapping my body for one with a better fit, like exchanging a sweater one size too big, the actual process of transition is far less immediate, more akin to the cumulative changes of puberty than an extreme makeover.
But "sex change" does manage to hint at an important truth frequently dismissed: that all of us will experience changes to our sexed characteristics over the course of a lifetime. Trans people might well experience the most obvious of these changes, but the process of change is common to us all.
One of the saddest bits of transphobia I've seen recently has been an online meme discussing trans women's breasts. The authors, trying to find a reason to support their claim that trans women's bodies are not "natural," hit on this explanation: trans women are only able to grow breasts because of a change in hormone levels, so those breasts don't count as "real." How to distinguish between "natural" and "unnatural" changes to hormone levels was, sadly, not explained, nor how they believed that "real" breasts came into being.
The same hormones that are administered to cis people for health reasons—to correct an imbalance, or as birth control—do not suddenly become fake when prescribed to trans patients. The changes we make to our secondary sexual characteristics, the shape and presentation of our bodies, are not "deceptive"; these are our real bodies, self-declared, autonomously controlled. Surgical intervention might well be considered unnatural, but only insofar as all surgery could be named so. My voice, my throat, is not considered fraudulent because of the removal of my tonsils, and no one has ever called me unnatural because of the metal pinning together my right wrist. As with many other things labeled "unnatural" by our society, a better explanation of that particular usage would be "something I do not approve of." Anal sex, being trans, pursuing IVF treatment, using condoms are all "unnatural." It is rarer to hear someone complain about the unnatural nature of chemotherapy, antibiotics, or painkillers.
So many of us want the definition and meaning of sex to be safe, unchanging, and foundational. When so much of our society is invested in maintaining the idea of sexual difference, it can be threatening to realize just how mutable those differences can be. And yet we cannot work for trans inclusion and acceptance if we cannot allow the inclusion of trans bodies into the spectrum of other acceptable sex changes. We celebrate other life-changing, life-giving medical developments in the field of sex research and medicine—why should trans-related changes be excluded from that celebration?
Puberty, pregnancy, childbirth, menopause: while these moments of change are frequently obscured by all manner of misinformation and culturally specific myth, we are all well aware of the seismic changes to our sexed bodies that occur over the course of a lifetime. Our genitals grow and develop, our secondary sexual characteristics mutate, illness or accident can mean the removal or modification of our reproductive organs. Our sexed bodies at seventy are not the same as our sexed bodies at seven: sex changes are a natural and necessary part of life. Even our chromosomes, the supposedly immutable touchstone of our sexed selves, are subject to change. Research in 2012 at the University of Washington found something surprising in the brains of female test subjects: XY chromosomes. These women had experienced the phenomenon of microchimerism—the migration of stem cells between parent and child via the placenta. We don't yet know what effects, if any, these migrated cells might have; what is clear is that we have a lot more to learn.
Confusion over how bodies can change during transition is one factor in labeling trans bodies as unnatural. What, in reality, is a range of treatments that work to realign the body along slightly different lines—shaping what's there into a different variation of the same material—is often misrepresented as mutilation or quackery. But, when that confusion is cleared up, why would the sex changes of transition be counted in a separate category from the universal sex changes of life?
HORMONAL, CHROMOSOMAL, INTERNAL, genital, secondary sexual, morphological: six ways of categorizing sexual development and difference.
What if we were to listen to trans people, and add a seventh?
"Trapped in the wrong body" is the cliché used to refer to bodily dysphoria inevitably trotted out by the mainstream media, and subject to much passionate debate in trans circles. We say it as a shortcut phrase, rarely bothering to go deeper and explain what we really mean with those words. Whenever I talk to cis people about what it's like to be trans, this is the main stumbling block I come to: the overwhelming, life-ruining phenomenon of dysphoria, and how hard it can be to recognize this force without having experienced it. People who have not, who are apt to disbelieve those who have, talk about it as something cosmetic and superficial. Sometimes, offering pity without understanding, they characterize it as a feeling akin to the self-loathing brought on by comparisons of the self to unrealistic, photoshopped images of models and movie stars.
I cannot talk about my experiences of dysphoria without first talking about proprioception. Proprioception broadly translates to the sense we have of the body as a physical entity: its position in space, a sensory map, its relationship to itself. It is what allows us to move without watching that movement, to truly inhabit our own physicality in a physical world. It's crucial to our spatial awareness.
What dysphoria has always felt like to me is a clash between what my body knows _should_ be there—the sexed characteristics that I can feel through proprioception—and the sexed characteristics that, bizarrely, impossibly, seem actually to exist. It's like missing a step in the dark, when you're convinced that the step is actually there until the moment you hit the ground. It's not wanting a different body: it's knowing how your body should be, and living with the continual pain of discord, as wrong as a broken bone. I didn't make changes to my sexed body so as to conform to gendered ideas of what that body should look like. I did it so that I could finally be complete in myself, and free from the shock of being divided in presence that had plagued me since I entered puberty. My proprioceptive sex was and is just as real to me as any other part of me—more real, in fact, than aspects of my physicality I genuinely could not feel and could not find on my mental map. Coming into myself after physical transition was the most extraordinary sensation; my mother, seeing my face for the first time after surgery, said she had never seen me smile so wholeheartedly before. It was like waking up well after interminable illness. Not only my mental health improved: with a body that I finally felt free in I could breathe, sing, move in ways I just didn't know existed. This sense of grounding, of wholeness, has nothing to do with the insecurities I carry about my appearance—insecurities common to many and culturally shaped—and everything to do with finally feeling _right_ in my own particular, imperfect, and perfect way.
This sensation, of a proprioceptive sex at odds with the external body, is not common to all trans people. Trans is, after all, an incredibly broad term. But it is common, and seems, when discussed, remarkably similar in how it manifests. In her groundbreaking work _Whipping Girl_ , biologist and activist Julia Serano refers to this sense as "subconscious sex." She writes:
Perhaps the best way to describe how my subconscious sex feels to me is to say that it seems as if, on some level, my brain expects my body to be female. Indeed, there is some evidence to suggest that our brains have an intrinsic understanding of what sex our bodies should be.... When one's subconscious and conscious sexes match, as they do for cissexuals, an appropriate gender identity may emerge rather seamlessly. For me, the tension I felt between these two disparate understandings of myself was wholly jarring.
Despite the recent popularity of "mindfulness" and "full body wellness," society still, frequently, insists on the division between the mind and the body. We dismiss conditions we do not like to admit—depression, chronic pain, undiagnosed disabilities—with the phrase "it's just in your head," as if the head were disconnected from the rest of us. We ignore all that we know of body-mind interaction, and of the physicality of brain states and mental processing both, because it is easier to refer to the under-researched and inexplicable as madness. But we can't talk about the sexed body without talking about _all_ of the body—the brain included. And, in turn, we cannot talk about the body without considering how it is we know that body to exist, how the mind tells us what we know. Sex is not just what is seen by an outside observer—it's how the body knows itself. And for trans people who have experienced dysphoria, we _know_ that how we know ourselves is a unique phenomenon in need of acknowledgment.
Trans people have been providing evidence of their experience of dysphoria, of proprioceptive or subconscious sex, for more than a century.
If we would dismiss the chance to learn more, then we admit to caring less about scientific inquiry than we do about the gendered stereotypes of sex.
BUT EVEN WHEN we open our minds to the validity of trans-related information, we run into problems. It's not enough that we open up trans lives to scientific inquiry—not if, in so doing, we are merely repeating cultural clichés about gender, now dressed up by lab coats and tables of data.
A vital point, when considering sex, is one that has been made over and over again by scientists, by philosophers of science, by sociologists and historians of science: that we cannot divorce a singular, unified "Science" from the broader cultures that create and sustain it. What we observe, what we think about what we observe, how we analyze our thoughts, what we pass on to others are all shaped by cultural forces beyond our personal control.
There is the deeper, older, ongoing debate about how scientific investigation should proceed, and what it is we mean by those terms. The works of Karl Popper, the Science Wars of the late twentieth century, the challenges of Thomas Kuhn and Paul Feyerabend: a slight introduction to an enormous, ongoing learning process.
And there is also the specific point crucial to any understanding of the science of sex, obvious but so often unchallenged: that what we "know" about sex—even when we think we know it empirically, logically—is so often just a reflection of what we want to know, or the confirmation of unexamined biases. Much of philosopher Judith Butler's works have been devoted to questioning and unpicking this inscription of meaning onto the world around us, but we do not have to be philosophers to recognize how often we are misled by our own prejudices—and use those prejudices, in turn, to mislead others.
When trying to examine this phenomenon, it can be easier to start with the past, for two reasons. First, to shatter the assumption that our ideas now are based on some unchanging, eternal fact. But also, and almost more importantly, to allow the clarity of hindsight and distance to teach us how to see the ways in which the cultural beliefs of every era informed what that era knew about sex. And, in so doing, to acknowledge the ways in which we fall into the same trap.
Possibly the most surprising fact about sex in Western history is that even our most basic assumption—that there are two opposite sexes—is a relatively modern invention. Many scholars believe that, for much of our recent past, we did not so much follow a two-sex binary model as a one-sex hierarchical one. British historian Laura Gowing explains the Galenic mode, the prevailing sex theory before the eighteenth century, as having male and female as two opposite points on a scale, with hermaphrodites (an archaic term for intersex people) falling in the middle. American scholar Kathryn Ringrose elaborates on this explanation: that the one-sexed model can be seen as a kind of ladder. At the top, you have the virile, manly, "perfect" man, and at the bottom, you have women. This perfect man is the ideal embodiment of the human animal and woman, as in Aristotle's theories, is the imperfect failure of type. But this model contains its own warning—this is a ladder, after all. One can climb up or fall down. So it is that gendered behavior can change the sexed body and, therefore, that gendered behavior must be policed.
We can find countless examples of how gendered behavior was seen to have effected sex changes: in Greek culture, Roman history, the Byzantine Empire; in the bodies of Christian saints, medieval sinners; and in the studies of Renaissance and Enlightenment men of letters. One of my favorites is recorded by one Dr. Carr in his _Medical Epistles_. He writes of two nuns who, due to their excessive masturbation (manly behavior), and probable masturbation of each other (even worse manly behavior), caused their clitorises to grow into penises, were expelled from their convent, and ever afterward presented themselves as men. As the clitoris and penis were held to be variations on the same theme—as we now know they are—it did not seem too far-fetched to claim that manipulating the clitoris in a supposedly masculine fashion could cause it to grow.
While the one-sex model slowly gave way to the new binary system, the belief that the body was still sexed along a continuum, and could be altered by gender variance (in practice or thought), was still with us well into the nineteenth and twentieth centuries. We can look back now and dismiss such theories, in the same way that we dismiss equally popular scientific beliefs in phrenology and eugenics—but can we afford to dismiss what we can learn from these failures?
The evidence that gender changed sex was, supposedly, easy to find, and easy to document through self-described rigorous, impartial methods. In 1802 Pierre Jean Georges Cabanis, a physiologist, medical professor, and philosopher, wrote that male inverts (a category of person neither gay nor trans in our parlance, but approximating a kind of combination of the two) had the bone structure of "normal" women, particularly in the pelvis. Perhaps the most extreme example of this form of reasoning can be found in the works of Ambroise Tardieu, a medical doctor and one of the most important forensic medical scientists of the nineteenth century. His 1857 forensic handbook, _Etude Médico-Légale sur les Attentats aux Moeurs_ , was a bestseller, translated into all major European languages. He believed that he had proof that inverted gendered practices or underlying tendencies changed the sexed bodies of inverts in the most extreme of ways. Historian Graham Robb describes some of his claims here—"pederast" was nineteenth-century French slang for a man who desired men.
Tardieu... believed that passive pederasts had enormous bottoms: "I have seen one pederast whose buttocks were joined and formed a single, perfect sphere." In active pederasts, the penis was deformed: it bulged like a snout or tapered like a dog's penis and, for a reason easily imagined, it had a distinctive corkscrew shape. As a result, pederasts were unable to urinate in a straight line.
I believe that physicians now would be hard-pressed to find the physical examples Tardieu claimed to have observed, and to have observed in such numbers so as to have formed a type. But Tardieu _did_ claim to have seen them, to have studied them, and his findings were accepted and celebrated by his peers. I don't write this here in an attempt to mock researchers in the past—although I have to admit to finding some of it rather funny—but to make the point that every mind, no matter how brilliant, is subject to the culture that created it. We must never believe that we are unaffected by our own current folklore of gender, in the false confidence that we are better than our forebears.
After all, this kind of sexed mythmaking has continued long into the twentieth century.
When I was growing up, the existence of the hymen was a fact. The "guide to your changing body" my parents bought me said that the hymen could be torn through sex or sport, or gradually worn away with time, but that it was definitely there—a thin membrane stretching across the vaginal opening. But, recently, many sex researchers believe that the hymen as we think of it—a seal that, once torn, cannot be repaired, named after the Greek god of marriage—does not exist. Rather, there is the vaginal corona: a remnant of fetal development only present in some people, made up of elastic mucous membrane just inside the vaginal opening.
The bodily importance of the vaginal corona seems minimal—but the cultural importance of the hymen, its utility as a tool with which to enact gendered laws, is still with us. "Virginity tests," in which fingers or another implement are inserted into a woman's vagina to see if she has a hymen, are still carried out in all parts of the world. Amnesty International has described these tests as a form of torture. "Virginity tests" are used as a way of keeping women in their place, of punishing those who step out of line, and of enforcing social order. This is not some medieval fantasy of bloody sheets displayed after the consummation of a royal marriage, or something that Westerners can claim only happens "over there." British immigration officers were ordering doctors to carry out virginity tests on migrant women as recently as 1979. Young people are still being taught that there is a part of their bodies that can "prove" their sexual status as virgins, another tricky concept. Sexed bodies are no match for gendered expectations, when those expectations uphold what is considered right, and normal.
And those gendered expectations are still affecting scientific research, and our ideas of the "truth" of sex, in the here and now.
Research from Newcastle University, published in 2007 in _Current Biology_ , claimed that women had a deep-seated preference for pink and that "this preference has an evolutionary advantage behind it." The researchers behind this study suggested that this preference for pink could have come from a hunter-gatherer division of labor in our distant past. If men hunted and women gathered, then women could have developed a preference for pink so as to find the ripest berries. This conclusion, and the research that they believed backed it up, was widely feted by the press. "Women May Be Hardwired to Prefer Pink" according to _New Scientist_ and, from _The Times_ , "At last, science discovers why blue is for boys but girls really do prefer pink."
The study assumes much about unknown aspects of early human society and behavior; it is extraordinary that this research, and the reporting of it, didn't examine what the historical record shows: that pink has only been associated with girls in the past hundred years. American Studies professor Jo B. Paoletti has charted the history of the gendering of the colors pink and blue, finding that, prior to the solidifying of "blue for boys, pink for girls" in the 1940s, things were often more mixed up, or even the other way around. Science writer Ben Goldacre, in his critique of this study, included advice from the _Sunday Sentinel_ in 1914 and the _Ladies' Home Journal_ in 1918, in which mothers were told to dress their little boys in pink (a stronger color) and their girls in blue (daintier and more delicate).
The interpretation of this study's findings was so in keeping with what we want to believe about the gendered nature of sex, and the sexed basis of gender, that critical analysis and a true examination of _all_ data were forgotten. This study is one of many—and each new study treated in this way goes on to further strengthen those cultural stereotypes, that in turn produce more suspect research and unexamined interpretations.
We must be aware of all of these factors if we are to try to examine trans people's experiences of sex in a scientific way. And, crucially, we must own and acknowledge these factors when we see "Science" used to dismiss and erase trans experiences altogether.
ALL AGES BELIEVE themselves to be modern. All ages look back on the same claim made by ages past and laugh at the supposed arrogance in the comparison, forgetting how we will appear to the future.
My final point is a simple one: that when it comes to what we know about sex, what we need most is humility. We do not know it all yet. We do not even know how little we know. We demand surety, when all that we have is open-ended investigation, still in its infancy. It's terrifying and thrilling, and too vast to be summed up by closed conclusions and closed minds.
The sad fact is that, more often than not, those who most want to claim that we know all there is to know about sex aren't even aware of how little it is that we know, and how much that little has changed. In popular culture—in the media, in schools, in the workplace, even in hospitals and doctors' surgeries—we talk about "facts" which "everybody knows," without examining what it is we've actually learned in an examined way compared to what we've picked up through ingrained assumptions and cultural conditioning. We often talk about "biological sex" using the examples we were taught in high school, and actively refuse the need to learn more and learn better.
When it comes to the impact of sex on trans lives, and what trans lives can tell us of what sex is, we have to stop pretending that we know all there is to know and focus instead on learning more.
Sex is a shifting, evolving, ever-expanding term that needs to reflect the totality of our bodies and bodily experiences, and to do so in a way that allows for human error and shortsightedness.
It is not a trump card to deny the existence of trans lives.
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# Think of the Children
E **ven people who support trans adults and our medical** needs can balk at the idea of the existence of trans children, let alone the fact that they may need medical care. I understand that. If what I knew about the subject came from general reporting, and not personal knowledge, I might be worried, too. There are the attention-grabbing headlines, of course: the ones about "sex changes" for toddlers, and "transgender hormones" for preteens. There's the incredible increase in the number of young people and their families seeking referrals to gender-identity clinics, and dubious theories as to why that might be. Everyone can remember at least one time when they were young and headstrong and thought they would want something new for ever, only to get bored with it in a week or two. Opponents of trans people capitalize on those memories. It is right that we should be worried about young people, but there is also a danger that those worries could actually be making life harder for the very children in question.
First, some facts: what do we mean when we say trans children, and what do we mean when we talk about treatment?
Crucially, when talking about young people, the term "gender non-conforming" is used, rather than trans. The reasons are twofold: to try to describe without pathologizing or locking into a specific category, and because the children being referred to gender specialists, who are being described, express a broad range of behaviors and reported beliefs about their own selves. A gender non-conforming child could be a child challenging traditional ideas of gender expression: a little boy who is adamant that he wants to wear dresses and be a princess. A gender non-conforming child could also be a child insisting that their gender is different from the one they were assigned at birth: a child who insists that he is a boy, despite being told by everyone else that he's actually a girl. Or, of course, a gender non-conforming child could be a mixture of these things: a child who knows that they are neither a boy nor a girl, who refuses the sex they were assigned at birth and a host of gendered expectations. Some children and adolescents are clear in naming themselves trans. Others are not: they agree that they are the sex they were assigned at birth, but not to the gendered expectations of what that sex should mean. The main reason this category is so large is because of the various motivations parents have for bringing their children to medical attention. Some parents are concerned that their children are not developing along normative gendered lines, and want that child to be encouraged to change their behavior to better fit societal mores. Other parents seek out a gender specialist because that child is in pain from their bodily dysphoria, and from being classed as the wrong gender. I note these differences here, because they make a big difference to later arguments over some of the most controversial research in the area.
When we talk about treatment what we're mostly talking about is the emotional and practical guidance families need to help them to support their children, and the counseling and group therapy that may be required to help a child who feels depressed, isolated, and in need of reassurance and care. There are additional options for teenagers and young adults, but emotional support is at the heart of caring for gender non-conforming youth.
A representative of the TransYouth Project at the University of Washington explains:
Our experience is that everyone gets nervous when 5-year-olds are mentioned in the same sentence or paragraph as hormones and surgery—and for good reason. Once again, though, care is needed in interpretation. First, and most critically, the only intervention that is being made with prepubescent transgender children is a social, reversible, non-medical one—allowing a child to change pronouns, hairstyles, clothes, and a first name in everyday life. No one in mainstream medicine (or elsewhere, to our knowledge) is performing surgery on or providing hormones to prepubescent transgender children.
There's a line I've heard from multiple trans people about the double bind of age and the acceptable trans narrative: to get access to treatment as an adult, you have to have known you were trans since early childhood. But if you say that you're trans in early childhood, you're told that you're too young to know.
When we ask trans adults when they first knew that they were trans, the majority will say that they knew as children. They may not have had the words, but they recognized that there was something about them that made them markedly different. Natacha Kennedy of Goldsmiths, University of London, is one of the leading researchers into the experience of gender non-conforming youth, particularly in educational contexts, and into the childhood experiences of trans adults. Her 2012 study, "Transgender Children: More Than a Theoretical Challenge," cuts deep for me; reading the experiences gathered there was an exercise in old pain. Kennedy found that, while a majority of trans adults become aware of their transness at a young age, an average of eight years old, they were also aware that that knowledge was shameful and needed to be hidden away from their friends and families. Not knowing how to understand their own feelings, many first believed that God had made a mistake, before coming to understand that they must be the ones who are wrong, mistaken. The average time span between realization and verbalization—learning any words other than insults and abuse with which to name the trans experience—was seven and a half years. Before they found that language, the huge majority of respondents felt like they were the only ones in the whole world to feel as they did.
My childhood was atypical in a number of aspects, and I know that the ways in which my parents did gender was one of the main ones. I am so very grateful for that. It's a standard question to ask: "When did you know?" People ask it of me, and they ask it of my parents. And I think I like my mother's answer the best: "We were learning as we went along." But, for far too long, I still felt the shame of being different: a failure, a freak, and alone.
At home I was never made to feel as though there was something wrong about my behavior, or that there was any gendered difference between the expectations placed upon my brother and myself: that we would be kind, and honest, and try our best. So I was allowed to dress up in my father's clothes, and dress up in my mother's, and my brother and I pooled a collection of variously gendered toys with which to create elaborate set pieces and storylines. I very much liked shiny, glittery things and pretty dresses, and also being the biggest, strongest kid in the playground who could beat everyone at arm wrestling. When my father went away on business my brother and I would practice shaving our faces with his razors, foam, and cologne. When my father was feeling particularly generous, we would practice face painting on him. I never had the impression that there were things boys should do and things girls should do at home, and when I heard that kind of attitude at school I was outraged. Joan of Arc was my hero, and I had an active fantasy life in which I was transformed into a stern and androgynous warrior with elaborate armor, and a white steed.
It was puberty that let me know that I was trans. First the dysphoria and then, crucially, the lack of any knowledge of people like me, and the societal approbation for all the ways in which I wasn't "normal." Between the onset of puberty, at the age of eight, and the point at which I had the beginnings of language about myself, at fifteen, I veered between denial, self-hatred, and terror. It's a tribute to my family that, alongside all those negatives, I also knew that I was loved, and still believed in their belief in me. It's a typical and an atypical trans narrative, and it's for those reasons that I share it. Because many of us did and do know young, and many of us suffer alone, unable to reach out. But also because what we can receive from those around us—unconditional love, unwavering belief—can carry us through to a better future.
By listening to, and believing, the young people who say that they are trans, we have the chance to end that pattern of isolation and self-loathing, to make the experience of being unconditionally loved the norm, rather than the exception. Removing stigma and sharing knowledge is not the same as forcing a label or category onto a young person. And if there are gender clinics willing to help families, and society, toward a place of openness, wisdom, and care then we should all be grateful.
AS TO WHY there are more openly trans young people now than there ever were before, the answer would seem to be obvious. There is constant talk about how society has shifted in terms of trans acceptance, knowledge about trans issues, visibility of trans people. This media moment is a symptom of something much deeper, and more profound: the year-on-year work undertaken by activists working in the fields of health care, legal change, community support, education, and outreach. Slowly, we are (at least in some fields) gaining wider support, a more legitimate voice in a broader culture. In her work, Natacha Kennedy describes the importance of "key words": words and phrases which allow a young person to recognize themselves and find others like them, unlocking knowledge of trans existence. Is it any wonder that, in a world where trans people are more widely seen and believed, children who may be trans will have earlier access to the words that help them describe themselves?
These cultural shifts do not impact on trans people alone. After all, trans children do not appear out of thin air. A change in parental attitudes would cause a significant change in the recording of numbers of trans youth. Trans people have, historically, been highly likely to experience family rejection and physical and emotional abuse from parents. Homelessness is a serious problem for trans teenagers, and stories of violence are common. What if this shift in numbers is simply this generation of parents doing better by their trans children?
As ever, though, this change can't happen fast enough. While conversion therapy is fading out of practice in the treatment of trans adults, it's all too often the first port of call for families looking to have their child "fixed" of their transgressive behavior.
The death of American teenager Leelah Alcorn at the very end of 2014 sent shock waves across social media, following the publication of her suicide note on Tumblr. A trans girl with unsupportive parents, Leelah had been forced into conversion therapy to try to change a fundamental aspect of who she was. Her initial plans to wait it out and transition when she was free of her parents' control couldn't withstand the constant pressure to be other than she was. Before taking her own life, she wrote: "My death needs to mean something. My death needs to be counted in the number of transgender people who commit suicide this year. Fix society. Please."
Leelah Alcorn is one of many. Every trans person I know will have their own stories of suicides in our communities. Many of us will have personal stories to tell of times when death seemed like the only option. Living in a transphobic world is hard on any adult. For an adolescent with no wider support network, subject to constant cruelty at school and at home, the pain can often be unbearable. Conversion therapy can be, is, the final straw.
There are parents like Leelah Alcorn's who cannot stand to see their child live a truth that runs contrary to their religious or cultural beliefs. They choose conversion therapy in the hope of salvation. Other parents will be driven by a desire to avoid the shame of having a trans child, or by ideological views about gender just as strong as religious fervor.
There are many parents who do not care whether a child is trans or not, but who push for conversion therapy because any behavior which challenges gender norms is an embarrassment, or a threat, to be "fixed."
And then there are interested observers, outsiders, who, lacking personal experience, support conversion therapy because they believe the widely touted claim that the majority of gender non-conforming youth will never transition, will change their minds of their own accord, and turn out to be, more or less, "normal."
In this framing, adult trans people are claiming that these young people are also trans to further a political cause or promote an ideology, with no respect for what it is that these kids actually need. UK journalist Julie Bindel has described treatment for trans young people as a form of child abuse. This attitude reminds me of nothing so much as the scaremongering over the myth that gay men, bisexual people, and lesbians recruit children to the "gay lifestyle" because we are sinister, predatory, and, supposedly, can't have kids of our own.
If that were actually the case, it would be an outrage—but it's simply not true. It's an enticing message, playing as it does on preexisting prejudices, but try to find some supporting evidence and you'll come up cold. American advocate Brynn Tannehill explains:
The most cited study (Steensma) which alleges an 84 percent desistance rate, did not actually differentiate between children with consistent, persistent and insistent gender dysphoria, kids who socially transitioned, and kids who just acted more masculine or feminine than their birth sex and culture allowed for. In other words, it treated gender non-conformance the same as gender dysphoria. Worse, the study could not locate 45.3 percent of the children for follow up, and made the assumption that all of them were desisters. Indeed, other studies used to support this also suffered from similar methodological flaws. As a result, the 84 percent desistance figure is meaningless, since both the numerator and denominator are unknown, because you have no idea how many of the kids ended up transitioning (numerator), and no idea how many of them were actually gender dysphoric to begin with (denominator). When Dr. Steensma went back in 2013 and looked at the intensity of dysphoria these children felt as a factor in persistence, it turned out that it was actually a very good predictor of which children would transition. In other words, the children who actually met the clinical guidelines for gender dysphoria as children generally ended up as transgender adults. Further research has shown that children who meet the clinical guidelines for gender dysphoria are as consistent in their gender identity as the general population.
I consider reparative treatments—which, being based on the combined efforts of caregivers and doctors and/or therapists to try to change a child's gendered behavior, in my view seeking to shame them into conformity, and to restructure love and support as something conditionally granted in exchange for compliance and denial of self—to be not just harmful for trans youth, not just harmful for all gender non-conforming children and teenagers, but a gross betrayal of the Hippocratic injunction to do no harm. That's not a medical treatment—that's brainwashing and emotional abuse.
The alternative to this is what is known as gender-affirming therapy. Texas-based clinical psychologist Dr. Colt Keo-Meier explains:
The gender affirmative model supports identity exploration and development without an a priori goal of any particular gender identity or expression. Practitioners of the gender affirmative model do not push children in any direction, rather, they listen to children and, with the help of parents, translate what the child is communicating about their gender identity and expression. They work toward improving gender health, where a child is able to live in the gender that feels most authentic to the child and can express gender without fear of rejection.
We can support children without pathologizing them, and can allow them to express themselves without external pressure to fit into one category or another. We could show parents how to provide that support, and allowance for exploration, at home. We can organize youth groups and support circles, where children and parents can meet others in similar situations, can reach out and not feel quite so alone.
And for those who need them—and remember that not all trans people need hormones or surgery—we can make sure that age-appropriate hormonal and surgical options are available. This is the particular point that causes so much outrage, so much anger. As ever, the reality is far less frightening than the headline spin. The hormone treatments known as puberty blockers have been in use for a long time, long before they were prescribed to some trans teenagers. The whole point of puberty blockers is that they are reversible; they were developed to treat children who enter puberty at a very young age, to allow them the time to wait and grow and start developing at the same age as their peers at eleven or twelve years old, as opposed to four or five. For trans children who have expressed a deep and persistent need to physically transition, puberty blockers are a godsend: a chance to take the time to explore their options, to settle on what they need, and an opportunity to avoid the psychological and physical agony of experiencing the wrong puberty. When these teenagers are older, around the age of sixteen—the same age at which they could join the army, get married, or create a child—then they can, with close supervision, begin hormone therapy. When they reach the age of eighteen they, like any other adult, can choose to pursue surgery. I cannot see what is controversial about this. It is a careful, conservative approach to supporting those, and only those, who actively seek out medical transition. And, from the research we have, it seems to be working incredibly well.
The most important research to date (there is now a fair amount, all positive) appeared in 2014, a longitudinal study of fifty-five young adults in the Netherlands who had been diagnosed with gender dysphoria and treated under the gender-affirmative model, with the use of puberty blockers beginning around the age of fourteen. Participants went on to pursue surgery, at an average age of twenty-one. At the time of the study, these young people "were no longer experiencing mental health consequences related to gender dysphoria, their quality of life and happiness levels were on par with their non-transgender peers, and none expressed any regret about delaying puberty or transitioning." A 2015 study from the Rady Children's Hospital in San Diego found much the same thing. Of the forty-two young people who had sought treatment, including puberty blockers for young teenagers and hormone therapy for older teenagers, none had expressed regret or wanted to stop treatment. Significant improvements to mental health were found across the board.
The research is ongoing; the US National Institutes of Health commenced the largest-ever study of trans youth at the beginning of 2016, which will take at least another six years to complete. We don't have all the answers yet; as with any evolving aspect of human nature, it is unlikely that we will ever hold all the answers. But we know enough now, after decades of trial and error, personal evidence, and dedicated study, to understand how harmful conversion therapy is, how pointless, compared with the alternatives on offer. If the well-being of our young people is truly our greatest concern, we should follow the research, and not our own fears and prejudices.
STILL, DESPITE THE data, there will be those who believe that this is all too risky, that it's unacceptable to allow children to pursue any path that they might later come to regret, even if the actual risk of regret seems negligible. From my perspective, this seems to display a deliberate blindness to the wide number of risks we already allow children to take, that we encourage them to take.
There was a moment of risk-taking in my own childhood that I would not have taken if I had known what would follow. But because it was considered a normal level and category of risk there were no warnings, and nobody tried to dissuade my parents from allowing me what I wanted.
I broke my right wrist at the age of eleven, falling off a horse. It was a very bad break, and I was already firmly committed to pursuing a career as a professional pianist; it had been an overwhelming ambition throughout my childhood, and my teachers were both confident in and encouraging of this ambition. I had started riding because that's what the popular kids did at my school, and I wanted to fit in, to make the children who hated me accept me as one of their own. The children's ward I ended up in, preparing for and recovering from emergency surgery, had several other patients who had been injured by horses: one young girl, the same age as me, had had her pelvis crushed after the pony she was riding had reared up and fallen on her—she would never be able to bear children. The doctor who treated me told my parents that horse riding was more risky than riding a motorbike, and that he would ban it if he could.
I needed two further operations on my wrist, a handful of invasive procedures under local anesthetic, and years of intensive rehabilitation. Throughout those years, I kept training, kept playing, up until the point where I could no longer move my fingers. I was accepted into one of the most prestigious music courses at a British university—I spent the first two years in a haze of pain medication, depression, and despair. Eventually, after a great deal of uncertainty, time off for treatments, and more time off to recover from those treatments, I regained the limited use of my right hand. It wasn't enough. Popular wisdom at the time held that trans people on hormone replacement therapy would in all likelihood lose their singing voices completely; my voice was the only instrument left to me. I retrained as a classical singer, at the cost of treating my dysphoria completely. I learned to work around my chronic pain to the point of being able to play simple piano accompaniments, and to work around my dysphoria with non-hormonal treatments and a wonderful support network. Losing both of those dreams at once was, and is, a constant struggle.
Maybe you think that these two examples—of musical vocation, and transition—are incompatible. I do not think that they are. My sense of myself as a musician—the constant internal music playing, the need to express it, the joy of sitting down at the keys and playing without thinking, without being anything but the physical embodiment of something far greater and more beautiful than I could ever hope to be—this is who I am at the core. It is so much more important to my sense of self than my sense of gender, my sexed body, and those things are pretty foundational.
The hypocrisy in telling young people who are genuinely desperate for treatment that it's too risky for them to have it—even after they have jumped through so many safeguarding hoops—while sanctioning, encouraging, other kinds of risk distresses me. It has everything to do with cultural norms, and nothing to do with keeping children safe while still allowing them their autonomy. We have to move forward from the idea that it is somehow a shame, a failure, for a child to grow up to be trans. We have to start approaching this subject with young people's best interests at heart, not our own concerns and judgments about how we would want our children to conform. Being trans is not a fate anyone needs saving from. But everyone, every child, needs to be loved for who they truly are, without conditions.
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# Delusional and Disturbed
T **he accusation, the insinuation, that being trans is a** form of madness is a hard one to hear. Answering it comes at a personal cost. To answer fully requires an admission, a detailing of my history of mental illness, and a tallying of the ways in which it and my transness can appear to cross over, but are not the same. Both of those traits are stereotyped and demonized away from their lived realities. In giving that answer, I know that it can be so easily used as a false affirmative: yes, they are mad, because they admitted they're mad, and that means that the delusion of being trans is a form of madness, because mad people believe it. I've used kinder language and the correct pronouns, but you get the idea. Honesty in one area is used to discredit it in another. Being trans is often believed to cause mental illness, and mental illness to cause the belief of being trans.
The cost of this belief is twofold. It becomes a way of dismissing trans people, of labeling us "insane," and denying our right to live honest lives. But it also prevents trans people from seeking, and acquiring, help for genuine mental health problems, out of fear that our transness will be blamed, our medical and social care withheld.
It is because of that myth, and that double stigma, that I want to be honest about my own experiences.
I had a breakdown when I was thirteen years old, and have been in and out of treatment, mostly in, ever since. After an initial, disastrous experience with a private psychoanalyst, recommended by the doctor who had dismissed my distress, I spent four years under the care of Child and Adolescent Mental Health Services at St. George's Hospital in London. I genuinely cannot imagine being here without their help. Twice a week at the beginning, and then once a week, once a fortnight when I was doing better, my mother would pick me up from school and we would drive to the hospital for appointments with my psychological team, psychiatrist, and cognitive behavioral therapist. When one treatment failed, they tried another. When I couldn't continue without drugs, they prescribed them, and when those drugs became too much they helped me come off them. Before I started treatment I was sleeping two, maybe three hours a night, if I was lucky, and would spend around four hours each evening locked in the bathroom, washing and rewashing my hands, my body, the floor—everything. I was in constant physical pain from depression, and the waking nightmares that are the obsessional element of OCD made me believe that I was both evil to the point of being unsavable, and dangerously close to losing all control over my own mind. After all, I had no control over the washing, checking, and counting compulsions that left my skin bleeding and my bedroom full of odd stacks of items, placed in rows and only in certain numerical combinations. I wanted to live, but I didn't want the life I was living. The doctors who treated me gave me first the hope of something better, and then the tools with which to bring it into being.
I never told them I was trans, not even after I had come out at school and founded a national LGBT youth organization. They never asked. I told myself at the time that it was because I didn't see my gender as an illness, something to be treated; I didn't need their help with being trans, so why would I bring it up? There's an element of truth in that. But what is also true is that I had no belief in their ability to acknowledge my gender while continuing my treatment, and feared that if I told them the truth they would take my treatment away. It was in the wording of the intake paperwork, an enormous form you fill in to give the team an idea of what they're meant to be treating. There was page after page of statements—"I usually find it hard to make friends," "I have less energy than I used to," those kinds of things—and you have to mark your response to each, from "all the time" to "not at all." I forget the exact wording, but I remember the placement. Around halfway through the form, after "I like to set fires" was written "I want to be the opposite sex," or words to that effect. I remember hovering over that item, trying to marshal all of the vague questions, impulses, fantasies I'd experienced up until that point into a suitable answer, but couldn't. I didn't know what would happen to me if I ticked "all the time," "most of the time," "sometimes." I didn't know how to express that I didn't want to be "the opposite sex," but that it was likely that I was something else entirely, someone who wasn't a girl or a boy, and who thought the whole system needed changing. I imagined that, if I replied in the affirmative, they might not be able to help me with the nightmares, the pain in my chest, the way that my own mind and body had turned against me. Without their help, I feared I wouldn't have the strength to keep living. It felt safest all 'round to tick "not at all," and tell myself that I could figure it out on my own time.
Maybe I misjudged my mental health team. I hope I did. I certainly presented myself in all kinds of gender-variant ways, and their compassion and care never wavered. But, even so, I listen to the horror stories of trans friends and acquaintances. I have listened to tales of doctors who insist on forcibly changing patients' gendered behavior and presentation before providing help for PTSD, eating disorders, and schizophrenia; of doctors who have denied medication and counseling because they disapprove of their trans patients. All of this, happening as of this writing in 2016. Maybe as a teenager in 1998 it was safest to err on the side of caution and stay silent. But what a terrible choice to force a desperate person to make.
IT IS NOT only that mental health care can be cut off for being trans, but that transition-related care can be cut off for being mentally ill. Any sign that we are less than 100 percent mentally well and adjusted—as though such a person exists—and access to hormones, to surgery, can be instantly denied. There is such a strong belief that believing oneself to be any gender other than that you were assigned at birth is itself a sign of psychosis, that trans people have to prove themselves to be saner than sane in order to be believed.
The roots of the myth of trans insanity go back a long way. Parmenides and Herodotus described "the Scythians' madness" as a congenital, inheritable mental disease that caused men to speak, dress, and act like women. Cross-gendered behavior was seen as a sin, but also as a form of madness, throughout much of Western history. The originators of the modern disciplines of psychology and psychiatry as we know them today were fascinated by the idea of criminal types, moral insanity, and the hopeless degeneracy of the mind that would lead to anti-social behaviors, including those behaviors we would now describe as trans. In the popular imagination any hint of queerness—inappropriately gendered behavior—could function as a code for dangerous madness. Pulp fiction and schlock horror have had their share of proto-trans villains; cross-gendered behavior could function as a signal for martyrdom or villainy. _The Rocky Horror Picture Show_ 's murderous Dr. Frank-N-Furter was a deliberate parody of these pop culture monsters. Trans characters make for good serial killers in the minds of cis creators, presented as men who want to wear the clothes or skins of women: _Psycho_ , _Dressed to Kill_ , _The Silence of the Lambs_. The discredited theory of autogynephilia, the idea that trans women are really men who get a sexual kick out of trying to inhabit the bodies of women, has a cultural debt to pay to these depictions.
Conversion therapy could not exist without the belief that being trans is a delusion, a sickness to be cured. It was the rationale behind the electroshock treatments, the enforced isolation, the committing to mental institutions. Trans legend April Ashley was, in her youth, hospitalized and injected with sodium pentothal and testosterone. Up until the 1980s, some clinicians still recommended the use of nausea-inducing drugs in the "aversion" treatment of trans patients.
Strangers online like to tell me that I'm deluded. I realize that this is a fairly common experience now for anyone who seeks to pursue any kind of public career. But I do find it interesting that these strangers are so invested in diagnosing my state of mind from behind their keyboards, and wonder at the disgust with which they make such pronouncements. Another popular comment is that I've "lost the plot," that too much time at university has muddled my brain, as though I were some overly ambitious Edwardian bluestocking, refusing to acknowledge her feminine limits. Opponents of trans people use many a dog-whistle term to indicate that we are not in our right minds: hysterical, unstable, extreme, fanatical.
I know of many trans people who have not been able to get referrals to specialist gender clinics because their GPs still believe that being trans is a form of make-believe, a fantasy invented by disturbed and dangerous people. I know some trans people who, while institutionalized, were forced to detransition, or hide their gender variance, before they were considered well enough to be released.
MAYBE, IN THE interests of honesty, of unencumbered communication, I should try to answer that question from the inside. Is my belief that I am transgender and genderqueer—not a man and not a woman, but someone ill-suited to either descriptor, who deserves the right to shape their own sexed body—a symptom of an unwell mind?
There have been times in my life when I have felt on the edges of sanity: when I have lost control, lost perspective, been unable to do anything but travel through until I hit the other side. I believe that this aspect of my self is fundamentally different from my awareness of my gendered self, and my challenge to a gendered world. Not because my manic depression or my obsessive-compulsive disorder are not part of me, though they did not feel as though they were part of me for a great many years, but because the knowledge of how my mind knows my body to be is so... I don't even know how to put it. How do you describe the mind and body describing the mind and body? Maybe by saying that that part is as unremarkable and basic as my brown hair and freckles. It just _is_.
Being trans and being mentally ill have both been hard, in their own ways. Dysphoria _hurts_. Smashing up against the gendered limits of our society and being smacked back hurts more. There is no doubt that dealing with transphobia and dysphoria have added to my emotional burdens. But they are not the same as the aching pain and concurrent nothingness of a low, finding oneself unable to get off the floor from an impossible mixture of emptiness and agony. Being trans is not the same as being caught in the middle of a bad OCD attack, with a mind full of horrifying images, imprinted onto the visual presence of the actual world, caught in an unstoppable bodily rhythm of impulse, release, drive, impulse, release, drive. I remember weeping, washing my hands and arms for hours, totally unable to stop—rocking backward and forward in horror as my body moved of its own volition—a machine without a pilot. I also remember weeping over the claustrophobia, the unfairness, of a body that didn't match what I knew it should be. Those tears did not come from the same place. Being called a freak when I'm having a hard day, being told to kill myself, not getting a job I knew I was qualified for, being spat at in the street: when the hatred of others hits me, I've sobbed at how hard it is to be trans. When those things happen, they make my mental health worse. But I don't think that this response is pathological, but normal, and to be expected.
I don't want to have to cut out such an important part of my experience in order to prove that my gender and body are valid, healthy, and right. As with so much in my life, I need the ability to say, "I am all of these thing together—they do not cancel each other out, but they are not the same."
I am not delusional for knowing myself, in all of my ill health, my uniqueness, perseverance, and hard-won clarity.
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# A Different Approach
T **he factor that has the biggest positive impact on the** mental health and well-being of trans people is the genuine support of friends and family. As with much of the official data on trans populations we currently possess, the majority of research is North American–based, but what that research shows ties strongly with community knowledge from across the world.
A 2012 report prepared for Children's Aid Society of Toronto found that trans kids with supportive parents had an attempted suicide rate of 4 percent, compared with a 57 percent rate for trans children without that support. Another new study, this time from the University of Washington, has found that trans kids with supportive parents have the same mental health outcomes as any other group of young people. For trans adults, it is not only having the support of others, but being able to support others in turn. Having a sense of purpose within a broader trans community and the two-way street of mutual care are powerful ways of combating the depression and anxiety that so often follows on from prejudice and discrimination.
What I have learned from research I had already experienced firsthand.
My family moved around a lot when I was a child, and I believe that is one of the reasons why my brother Jonathan and I were so close. When we had nobody else we always had each other; besides that, he was so easy to like. Our tastes ran close, and it was always more fun to be a team than to be apart. We had other friends—good friends—but none of them could match the ways that he and I could see inside each other's heads and know exactly what to say to make the other laugh or reach out to help when something was wrong.
As far as experiences of bullying go, I've been lucky, compared to many trans people. I was never physically attacked, never sexually abused, and, later, I did make friends. It was hard, but it could have been much harder.
It began when I moved schools, again, at the age of nine, and found myself marked as an outsider in more ways than one. The way I talked—my voice too deep and my accent unplaceable—and the way I looked, being so much taller and more developed than the other children. It's a familiar story: notes that called me a freak; a chorus of mooing whenever I walked past the popular girls in the playground; comments about my body, specifically how it was too large, and my face, which was too ugly; and all the usual insults of bitch, pig, and the like. There were the ongoing, relentless, personal attacks, and the impersonal cruelty from kids hoping to avoid the bullies' attention. There were a few nights when I believed that I didn't deserve to live, made phone calls to ChildLine, had attacks of panic and hysteria. But at least I always had my sanctuary of my home, my family, my brother: a place to be safe.
My parents certainly hoped that things would get better as I got older, and they did—to some degree. I found friends, I found teachers I liked and, with all the music practice rooms, I could lock myself away every lunchtime, hide from the other pupils, and escape into the piano. The bullying from the other students ebbed and flowed.
Sometimes it was overt, with my name and the words "weirdo" or "freak" scrawled on blackboards in enormous letters; loud public imitations of my voice, my mannerisms; upfront questions about why I was queer, why I had to be so masculine. Most of the time it was just that low-grade level of disapproval and mockery many teenagers would recognize: the sudden silence when you enter a room, the public lack of invitation to a party that everyone else is invited to, quiet laughter when you raise your voice in class. I learned how to hate as a reflexive, protective gesture and how to turn my self-loathing into a spur to ambition, two common responses.
The hardest part, though, the part that will be so familiar to other trans people, was the fact that it wasn't just my peers who rejected and judged me—it was some adults also. It can be easy, as adults ourselves, to forget or diminish the impact our stated beliefs and actions can have on teenagers. But those feelings of being disbelieved, disregarded, and—sometimes—hated by those in positions of authority wore me down. I started doing more public activism: writing for an international LGBT website, campaigning against Section 28 (legislation which prevented discussion of LGBT issues in schools), meeting with the London Assembly. At every point where I tried to stand up for myself, it felt like there was someone older than me ready to smack me back down. I wanted so badly to be immune to what was said to me and said about me, but between the mixed states, the lows, the panic attacks, the OCD attacks, and the constant, nagging dysphoria, I just didn't have the strength.
Jonathan was my anchor when it would have been so easy to be washed away. It was a struggle to wake up every morning, to learn how to deal with conditions my doctors had told me would be lifelong, to learn how to bear the ridicule and disgust that seemed the inevitable reactions to being myself. I had ambitions, I had things that I loved but, against that constant pain, it was often hard to hold on to something as nebulous as a hope that it would get better, or a daydream about my future career. Sometimes, I couldn't imagine how things could be different, and sometimes I felt as though I didn't deserve to live.
Instead of the possibility of a better life _one day_ , my brother gave me a better life _now_. He was the noisy, insistent reminder that I was not without ties to the world around me, that I was more than just an unwilling passenger in a life I didn't choose. He made me laugh despite myself, joke despite myself, gave me a space to unload my anger, gave me a chance to be frivolous and playful when everything else was gray and cold. He knew when to ask, and when to listen, and sensed when I couldn't talk but needed to be heard, silently.
More than that, he made demands on me. Not rudely or obviously, but in constant little ways. Could I, who found English easy, help him, who was dyslexic, with his homework? Could we write a screenplay together, to see if we could, and then make our millions? Could we go out for coffee, could I make that cake that he liked so much for his birthday, could he borrow a CD of a band we listened to together on MTV? Some of my happiest memories are of the evenings we spent, him lying in bed, me on the floor by the door, reading him the latest book in our favorite series, doing all of the voices. We were much too old for bedtime stories, but it became one of our rituals. It is a very powerful thing, when you feel at your lowest point, to know that someone else depends upon you. Not in a way that burdens you with expectations, or makes you feel ashamed for not being perfect, but in a way that says "your life makes my life better—thank you for being here." No matter how terrible, how disposable I felt, he gave me constant reassurance that who I was had value. In that mutual exchange of care, I felt the proof that his world, at least, was better for the fact that I was fighting to stay alive.
I SUSPECT IT was our relationship that primed me for wanting to do something to help other people, to do something, no matter how small, to be useful. I don't claim to be any kind of saint—I fuck up, and will keep fucking up, despite my best efforts and a great deal of guilt—but I believe I have done some little good along the way as well.
It has certainly taken me full circle in a way I never expected it would, and given me an idea for something else that might make things a little better for trans people like me.
When I got an email from my old school last year, I felt as though I'd stepped into a fictionalized version of my own life. In careful, respectful words, it asked me if I might have the time to prepare and give an assembly to the students and teachers about how best to support trans students, and to combat bullying and prejudice in general. The student body was setting up an LGBT association, and the teachers wanted to bring in an outside speaker as a show of support. They had no idea I had been a pupil; they had found my contact details through my work in educational outreach. Nearly all of the old staff were long gone. When I explained how bad it had been for me they sympathized and apologized, and said they were trying to make things better for the next generation of pupils. There was clearly a new approach in play.
I had my first panic attack in several years on my way back to school. Walking through the gates, up to the main hall, was an exercise in confusion; I'd had regular nightmares about being trapped in my old school at least once a week for the twelve years since I left. In those dreams, every detail was perfect, every physical sensation hyper-real. Actually being there left me feeling as exposed, as vulnerable, as if I'd just woken up from one of those dreams. By the time I was ready to get onstage I had sweated through the shirt I'd worn, was very grateful for the blazer above it, and felt far more nervous than I would before performing a show. I gave my speech; to my astonishment, the teenagers in front of me actually knew a great deal about trans people already. They were friendly, attentive, and had some wonderful things to say of their own. Further adding to my sense of unreality, the head teacher apologized for the bad time I'd had as a student, and gave a short speech about how much they wanted to support their current pupils. Most importantly, for me, was what happened as I was leaving. One of the younger students, ducking out of a group on their way back to class, came over to thank me; one of her friends, she explained, was trans, and it was brilliant that people were talking more openly about it, and learning more. On my way home I kept coming back to that fact: that in a place where I had been an outcast for being trans, there were teenagers proud of their trans friends, fighting their corner, willing to take a stand. There were even teachers ready to do the same.
I didn't know how much pain I had still dragging around behind me until it was set free. It was only with that apology, that experience of making good, that I could feel healing in the places that had scabbed over but never healed. It was somewhat embarrassing—I still expect myself to be invulnerable, in many ways—but I'm so glad that it happened. The nightmares have stopped; after so many years of psychotherapy, my dreams are often obligingly obvious.
It made me realize that those two crucial factors in the happiness or misery of trans people—the support of others, and a sense of engagement in a wider world—are never fixed in place, and the opportunities for both are never over. We try to laugh it off, or decry it bitterly, but we so often think of the pain we've been dealt as a done deal: others have attacked, and we are left with the scars and the necessity to heal ourselves. There is the tendency for some of us trans activists to look at ourselves and say, "We're already damaged goods—we should focus on young people, because at least there's still hope for them."
I no longer believe that to be true. I don't think it is ever too late for those around us—those who aren't trans—to reach out and offer their support, to make right on where they stumbled and failed. It is incredible what being needed—in both ways—can do. And this is a part of trans liberation that anyone can make good on.
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# Trans/Love
T **here were two reasons why I was so scared of physical** transition, why I vacillated about what it was that I needed, caved in to social pressure, presented myself according to the instructions of others, to the detriment of my mental health and happiness. The first was because I wanted to have a successful career, and I'd never heard of an openly trans classical musician. It seemed an impossibility when I was a teenager. Even now, I only know of a small handful.
The second, more overwhelming concern: that being trans made me unlovable, and that I should keep my body as it was for the sake of someone I couldn't afford to lose.
Trans people occupy a strange place in society when it comes to desirability, sexual attractiveness, and our supposed value as romantic partners. In pornography, trans women are highly visible or, rather, _some_ trans women are highly visible. The standards for trans women in mainstream porn are no different from other gendered standards in mainstream media: only a narrow band of women are considered worthy. Otherwise we are, for the most part, invisible or branded as undesirable. In 2016 there are some trans celebrities, a very small number of highly visible trans women, trans men, and genderqueer people, who are widely accepted as attractive. They are outliers. Attraction to a trans person is more usually seen as a joke and a failure. If we match normative standards of what it is to be beautiful then we're deceptive; if we don't we're pathetic. All that we are, all that we could bring to a relationship, is swept away in that judgment: unworthy, repugnant, fake.
I worry that my concerns over appearing desirable, with wanting to be wanted, are a sign of shallowness. I worry that those worries are an inevitable symptom of growing up being told that I was ugly on a daily basis, of being mocked and isolated and told that I deserved it _because_ of being ugly. Not feeling at home in my body, it was easy to judge it as a thing apart, and to fear the ways in which it was seen by strangers. I drew graphs, in therapy sessions, of beauty and ugliness, and plotted my own self close to the bottom. I had already read enough feminist theory by that point to know that the beauty myth was harmful, but I also felt, with an iron certainty, that it still applied to me and to my supposed ugliness. I took refuge in what I could do, in how I could think, but it wasn't enough. It was at this juncture that the outer world taught me a crucial lesson to counter my inner knowledge: that I could be wanted, and desired, if I only presented myself in a certain way. At the age of fourteen I was experimenting with my gendered appearance daily; sometimes in flannel and waistcoats, and sometimes in lipstick and high heels. It was one of those times, dressed in feminine clothes, my hair long, my makeup perfect, that an older gentleman leaned out of a window in the restaurant we were passing, my family and I, and handed me a flower, because, he said, I was pretty. It was a pattern that would repeat again and again. In the clothes that made me feel most like myself—my father's old suits, shirts and jeans from the men's section, boots, a scuffed jacket—I was invisible. In dresses and blouses, tight sweaters, and padding in a bra, I was desirable: flirted with by strangers, cruised by older women, given little extras in coffee shops and restaurants, told openly and repeatedly, "You're so beautiful." I didn't feel it—but I so wanted other people to feel it about me.
I think if I had had more time to figure it out on my own, it wouldn't have been so hard. Despite that year and a half of confusion, my self-knowledge and self-respect were starting to win through. I found pride in presenting myself in a way that felt congruent with my inner self, in learning the exact things that made me feel happy and at home. I still caved in to outside pressure, particularly when I felt I had something to prove: that I wasn't failing at a standard gendered appearance, I was rejecting it. But then I went to university, fell in love, and lost myself in the struggle between what I needed to be and what I needed to do to stay wanted.
It's a terrible truism that the majority of us could look back on our early relationships with regret—I have nothing new to add on that score. I was very far from perfect, and I made a lot of wrong turns and mistakes: I was sorry then, and I'm sorry now.
I was not above reproach in my first major relationship, but neither was I dishonest about my gender, and my fears and hopes of transition. My girlfriend was interested, supportive, in some ways; she read the books that had helped me, bought me gifts that reflected the self I wanted to see. But, at so many other times, there were discrepancies and arguments that we glossed over, which I pretended didn't hurt me. The person she loved wasn't the person I needed to be. She didn't like body hair; after months of pressure, I got rid of mine. The approval of the changes I had made for her worked a magic that requests or complaints could not have achieved. I started wearing skirts, because she thought they were sexy. When my hair was long, she was complimentary; when it was shaved or cut short, she withdrew her praise. The day I found a surgeon who was happy to treat me was one of the most exciting of my life; she asked me not to go through with it, because my body as it was made her happy. As she explained, I looked as I wanted to during the day, so why couldn't I stay as she needed me to be when I was naked with her? I loved her in a very young and naïve way—if there had been no other pressures, I suspect our relationship would have ended then, after two and a half years, filled with some wonderful moments but based on a fundamental incompatibility.
What happened instead is hard to write down. My words feel insufficient compared to what I want to say, to the memories I'm trying to describe. I would like to keep what is private, private, and know also that my experience is so typical that it needs to be shared. My brother was diagnosed with a rare and serious form of brain cancer; he was eighteen, I was twenty-one. The odds of survival were not good, with treatments that were unlikely to serve as long-term solutions, but he was young and strong, and determined to try whatever his oncologists suggested. We couldn't hope for a permanent remission, but we could buy time until a treatment that guaranteed remission could be found. We entered those rounds of surgery, chemo, and radiotherapy familiar to so many. I clung to my partner, wrote up my degree work in waiting rooms and on trains to and from hospital, and did my best by my brother. The treatments failed, then succeeded, and then the failures began to outweigh the successes. My girlfriend—partner—reaffirmed her commitment to me, to my family, and I leaned on her for every ounce of support I could get. She grew more and more uncomfortable with any elements of my appearance that would reflect my transness, my actual gender—and I fell in line. I find it hard, now, to understand just how quickly I acquiesced, how I could be so deeply invested in my transgender self and still be prepared to go against it. I remember one moment of absolute clarity, getting ready for her, facing myself in a full-length mirror. My body was prepped and shaped in all ways to a standard that she would find attractive, from my weight to my pubic hair—and I looked at myself in the mirror, and thought, "How could anyone fail to love _that_?" I no longer felt any connection to my own body as mine. It was just something designed and presented as a bargaining chip, a tool with which to secure another person's desire and care.
I learned a valuable lesson from that relationship, one I thought I had known but obviously had failed to understand and internalize: if someone loves a certain image of you—an image which misses your true self—then the actuality of who you are will never be enough. No matter how much of myself I cut away to try to reflect back what she wanted to see, I was always found wanting. My brother died just over a year after that memory, and the relationship crumbled soon after that. Bereavement is staggeringly hard, and I was going through with the surgery after all.
CONTRARY TO THE dire warnings I'd been given—that no one would want me, no one would love me, with my body as it should be and my self recognizable and honest—I found the numbers of romantic offers I had actually increased. Some were from genuinely wonderful people; I made friends, I went on dates, I kissed a few people I shouldn't have and learned a lot about myself.
Some of that interest, though, was like an inverted parody of what I had been used to in my teens. Where previously my transness had been undesirable, now it was valuable, but only through a certain reading.
There are some people who say that we shouldn't criticize those cis people who fetishize us: that to do so is to shame someone for something natural and normal. That some attention is better than none at all. What I know is that that particular kind of attention made my skin crawl, and left me feeling almost as misread as the people who had wanted me to hide who I was. There were so many innuendos, assumptions, questions about my genitals. I was told that I was the "best of both worlds," told that my masculinity was so much more attractive than a cis man's, because I was "special," because I was different, because I was exotic and strange. Cis people asked me to be their "first." They grabbed me, and pressured me, and one particular man with a crush insisted on touching my chest so as to publicly announce whether I felt "real" or not. They fixated on their own ideas of what being trans meant to the exclusion of the reality of the trans person in front of them. It was another way of being objectified, of being reduced to a shell and a foreign narrative. I had just begun to come into a sense of comfort in my body, a deliberate attempt to learn how to care for myself, and I found that being alone was preferable to being wanted for what I am not.
What I've learned about love and desire and ways of seeing is not limited to interactions between cis and trans people. Neither is it indicative of all such interactions; cis and trans are blunt words for societal categories and prescriptions, and there are many, many people who stretch beyond their bounds in all areas of life. But I think there are common patterns, trends, that come from a meeting between a person naturalized in the belief that they are natural, and a person who has learned that what is natural for them is wrong, strange, and subject to the approval of others. From so many of the people I have met and known, desired and been desired by, I have learned that a trans person's needs, reality, and physicality are supposed to be secondary to those of a cis person. I've learned it from the chasers who pursued me with specific ideas about how and when I should fuck them, regardless of how little interest I expressed. It was taught to me by a boyfriend who flipped between loving me and pushing me away, accepting my androgyny and sighing, "You would have made a beautiful woman." It's the message I received from men and women, gay and straight, who have admitted their attraction and followed it with "you confuse me"—not that these people are confused, but that I am responsible for their confusion.
What I have also learned, in time, is that there is nothing inevitable about this pattern, and no excuses for it. That was a lesson that came slowly: through meeting other trans people, caring for other trans people, being taught through the care of others to care for myself—through losing my brother and my partner at the same time and realizing that it was possible to go on alone. Through being loved conditionally, and learning that it's never enough. And, finally, through being loved unconditionally, and realizing the utter difference between the two.
It's the difference between making love to the person you love with yourself presented as a character in an artificial scene you hope will please them—withdrawing from your own body into a dislocated fantasy—and fucking so truthfully, in a way that is so totally in the moment, that the body is full to overflowing with a sense of total peace. It's the distance between a desperation to be found beautiful and the knowledge that the whole of who I am _is_ beautiful to my partner, and "beautiful" in a way that goes deeper than my appearance. It's the change between pursuing the approval of the loved one as someone desired—known but also distant—and finding your best friend standing at your side.
It might seem a small thing to be angry about, compared with the overt hatred, workplace discrimination, denial of health care and education, but it's an anger that hits deep. I cannot believe that, for so long, despite my best efforts to believe in my self, I so fully capitulated to the idea that the full expression of my true self was unworthy of care.
It doesn't have to be this way. There is no reason for it to be so.
If we are to be treated with respect in the wider world, we must trust in the respect of those closest to us. The micro and the macro are inseparable. Trans people are worthy of love.
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# Are Trans People Real?
P **roving that we are _really_ trans is only the first hurdle.** Being "trans" is the diagnostic, the category we are placed in or place ourselves in in order to make sense of the bigger, deeper truth we are telling: that some of us are men, and some of us are women, and some of us are none of the above, but that we are really, truly these things despite the categorization of our bodies by others at birth. When it comes to the genuine nature of our genders, it is clear that many people might recognize the fact that we're transgender, but never grant us the status of "real" men and "real" women, to say nothing of the rest.
We, as trans people, are held up as fakes in contrast to the cis original. The words "parody" and "pantomime" come up a lot. Germaine Greer's 2007 book _The Whole Woman_ contains a chapter on trans and intersex people titled "Pantomime Dames." Richard Littlejohn (again) refers to us as "the 'trans' pantomime."
The idea that trans women are fake women, that trans men are fake men, is one that is currently playing out in schools, prisons, the legal system, public life, the political circus that inflames every other arena. It is an issue frequently dismissed as a simple difference of opinions, characterized by the "right to offend"—but it is no exaggeration to say that, at its worst, this question of realness can mean life or death to the most vulnerable members of our community.
Nowhere is the fight over trans realness as openly vicious as in the current battle over access to public toilets, public changing rooms, and public life. What began as a small fringe concern—the dawning realization that trans people exist, and need to use gendered facilities just like everyone else—has stormed front and center into the mainstream political debate. In America, North Carolina passed its anti-trans bathroom bill in 2016. The Republican National Committee approved a resolution endorsing similar bathroom bills soon after; current president Donald Trump has flip-flopped on the issue, before capitulating to popular anti-trans sentiment. Ted Cruz seemed to think he had hit on a solution, proposing that trans people only be allowed to use the bathrooms in their own homes. Cruel, but better than the proposal of the Rowan-Salisbury Board of Education, which in 2016 voted to allow students to carry pepper spray, with at least one member citing fears that trans-inclusive bathroom laws would be taken advantage of by "perverts and pedophiles." The decision was later repealed, following widespread condemnation.
Online news stories feature everyday Americans proud to announce the fact that they'll be carrying guns into public bathrooms to fend off trans people; megastore Target has been under attack for their decision to allow trans people to use the bathrooms that best match our genders.
Proponents of these laws—bills that would force people to use the bathrooms associated with the original indicated on a birth certificate, rather than with their lived gender—are keen on the generalities, and not so clear on specifics. How are these laws to be enforced? The assumption is, I suppose, that trans people are easy to spot. It's certainly true that some trans people look nothing like the stereotypes surrounding the sex they were assigned at birth. It's also very clear to see that many people, cis and trans, do not fit gender-normative standards of appearance. What happens when there's confusion: Does it come down to a majority opinion? Driver's license, passport, ID cards? All of those documents can be updated without doing the same to a birth certificate. A physical pat-down by a security guard? How the hell is that meant to play out with children and teenagers? These bills are a logical, logistical, ethical mess, and they're playing well with sizable subset of the population.
I wonder at how quickly we've forgotten the similar arguments made over bisexual and gay people having access to "normal" bathrooms and changing rooms. They were extremely popular when I was growing up, trotted out as the number one reason why LGB people should not serve in the army, play mainstream sports, be allowed to come out at school, be allowed to teach in schools, be allowed to _attend_ regular schools. This was the era of Section 28: an amendment brought in under Margaret Thatcher's government banning the teaching of "the acceptability of homosexuality as a pretended family relationship." As a teenager, the supposed danger of gay and bisexual people was a regular topic of school debate; the majority consensus was that any teacher suspected of being gay shouldn't be allowed alone in a room with pupils, and certainly shouldn't be allowed to supervise them in PE. I felt the pressure of it, when we were changing before and after sports lessons, and was shocked by the number of straight people who assumed that every queer person was just someone waiting for the right opportunity.
Now that the gay rights movement has spread into the mainstream, and these horrible predications have failed to come true, that need to scapegoat has shifted over onto trans people. The myth of the trans predator has made it into several campaign videos: hairy, hulking, monstrous, this figure has more in common with Bigfoot than any actual trans person. It's a genuine urban legend for our time. Politicians in favor of these bills paint trans people (particularly trans women), trans children, as conniving sexual suspects. Twice-disappointed presidential hopeful Mike Huckabee came out against allowing trans people equal access to restrooms and changing rooms with the line that, had he known about trans people when he was growing up, he would have pretended to be trans in order to watch the girls shower after gym class.
How did it ever come to this? Given the force of the reaction, one could be forgiven for thinking this panic, this clampdown, is the fallout from the actions of a trans criminal using public bathrooms to prey on women and children.
But no. As far as I am aware—and I have been diligent in looking—there has not been a single reported case of a trans person attacking a cis person in a public bathroom. Ever.
Public toilets are, however, common settings for transphobic violence. They're common settings for violence, full stop. I know several cis women who have been raped by cis men in public toilets; the glyph on the door does nothing to stop an actual predator. Cis men beat up other cis men in the toilets all the time. If we want to tackle violence and abuse, and God knows we should, then we should be telling these stories and learning from them, rather than investing in the myth that public toilets are safe so long as we keep the trans people out. As ever, trans people are held to a totally different standard; our access to basic services can be denied or granted depending on perceived, even imaginary, good or bad behavior.
Cutting off access to public bathrooms cuts off access to public life. You can't move through the world, can't use public transport, can't hold down a job, can't go to school, can't visit a hospital, without using the bathroom. I have OCD and a fear of germs, so trust me on this one—I tried. All that happened was that I made myself ill through drinking less than half a liter of water a day. When I was first coming out into the trans community in London, I heard other people joke about "trans bladder syndrome," the fact that we were so used to holding it in that we could go for staggering lengths of time without using a bathroom, often resulting in a urinary tract infection. For someone like me, whose gendered appearance is judged in different ways by different people, picking the "right" bathroom to use is a constant struggle. I've been turned away from both men's and women's restrooms, and accepted in both. Occasionally, the comments are endearing: a small child whispering loudly, "Mummy, why is that man wearing makeup?" Mostly, the stares are off-putting, and I make it a point never to use a restroom where it seems like the reaction would be worse than stares or embarrassed exclamations. I have no idea what I'd do if it was decided that I had to use the bathroom that matched my birth certificate. For many trans people, what these laws mean are women in men's rooms and men in the ladies' rooms, the very thing these rulings claim to oppose. Enforcing these laws will inevitably mean an upsurge in violence, people victimized and hurt for having the audacity to have human bodily functions. These politicians pretend to want a crackdown on such violence. It seems to me that they're more than happy to see an increase in it—so long as the victims are trans.
A trans woman is locked up in a men's prison—can you guess the rest? She's told that it's her own fault, that she brought it on herself; she should perform sexual acts to try to keep herself safe; she should resign herself to being raped, over and over again. Her face will be sliced open with a razor. She has no right to say no. These are the horrors described by Passion Star, a Texan trans woman, during over a decade of incarceration.
Here is what happened to twenty-one-year-old Vicky Thompson in 2015: despite her fragile mental state, despite her pleas to serve her time in a women's prison, she was sent to HMP Leeds, despite her boyfriend warning the prison that she would kill herself in a men's jail. She was found dead in her cell a week later. These stories pour in from around the world and coalesce into marked patterns: a vulnerable trans woman, most often a woman of color, a migrant woman, a poor woman, a sex worker—maybe all of these things—is arrested. She is often held without trial. She is frequently failed by her court-appointed lawyer. She is sentenced and sent to a men's prison, and there she is abused by the staff or by the inmates or by both. She is sexually harassed; she is usually raped. In some instances she is denied her medication, or she is denied any health care at all. If she complains, she may be punished: put into solitary confinement "for her own protection." Sometimes, she will take a plea bargain to escape; sometimes, she will take her own life. Chelsea Manning, the US Army whistleblower sentenced to a thirty-five-year sentence in a men's military prison (having only publically come out the day after her sentencing), attempted suicide twice, and was punished for doing so. In the six years since her initial arrest she was held, on and off, in solitary confinement, denied her hormones, and denied the right to present herself in a feminine way. Her desperation should have come as no surprise. Following a widely-publicized hunger strike, the army's position appeared to change: Manning was promised a chance at medical transition. Her sentence commuted by President Obama, Chelsea Manning is now a free woman; thousands of trans prisoners around the world will have no such escape.
In the UK, the Ministry of Justice claims that the majority of trans prisoners will be held in a prison that matches their lived gender; there are guidelines and internal rules. Trans prisoners have the right to continue medical treatment, the right to be detained without fear of emotional abuse and violence. Whether those rights are respected remains to be seen. Trans prisoners with a Gender Recognition Certificate should be automatically placed in the correct prison; however, the majority of trans people in the UK do not have a GRC, as they are difficult and costly to obtain. GRCs are only available to trans people who have completed a standard medical transition, who have gathered the evidence of their doctors, paid a fee, and had this evidence accepted by an anonymous Gender Recognition Panel. Many trans people have their evidence dismissed, some of us are disqualified from the get-go, and most don't see the point in trying.
Opponents claim that they don't want to see trans women—or, in their words, men—suffer; they simply don't want to see cis women, _real_ women, under threat. The myth of the predator is absolute. Lack of evidence is ignored, the rape of cis women by cis women in prison is ignored, broader questions of prisoners' rights are ignored. Deeper concerns—how can this be called justice? How many other prisoners suffer?—are glossed over. What matters here is an appearance of law and order, the appearance of punishment, the appearance of safety. Once someone has been locked away, the reality can be forgotten.
There are people who say that trans women lose their right to be treated as women the minute they commit a crime. They claim that women are, by nature, more peaceful than men, not driven to violence, and that, by being arrested, a trans woman has exiled herself from a category she had no right to claim.
IN THE ABSOLUTE worst cases of denying the authenticity of trans genders, we find a toxic mix of transphobia, misogyny, and homophobia known as the trans panic defense. People who claim this defense try to excuse their behavior thus: that finding out that someone, usually a woman, is trans sparks a feeling of disgust, fear, and rage so strong that the perpetrator of any subsequent crime can be excused their actions. To exist at all, this idea has to play on cruel and incorrect ideas of what it is to be trans, what it is to be a woman, and what it is to be straight or gay. Let me be very clear: trans women are women, and it does not make a straight cis man gay to be attracted to a woman. But the way trans panic plays out—in just one of its erasures and attacks—is in a cis man's fear that he will be considered queer, because the woman he's attracted to was assigned male at birth.
The trans panic response can be found in all kinds of places. _The Crying Game_ , Neil Jordan's 1992 movie, depicts a classic example of type. Fergus, our anti-hero, discovers that the woman he's involved with, Dil, is trans; he hits her and rushes to the toilet to vomit. You can find similar reaction sequences in movies and TV shows throughout the 1990s and 2000s. Sometimes it's played for tragedy, sometimes for titillation, and sometimes just for laughs.
It's a trope that's been used so often that _Family Guy_ , infamous for its need to be more shocking than shocking, had to find a way of outdoing the standard. In the 2010 episode "Quagmire's Dad," they did this by having one of their main characters sleep with a trans woman, and then vomit for thirty seconds straight on learning her trans status, later referring to her as a man. In advance of the episode airing, Seth MacFarlane had noted: "It's probably the most sympathetic portrayal of a transexual _sic_ ] character that has ever been on television." When the airing of the episode was criticized by LGBT and feminist groups, MacFarlane doubled down on his comments, explaining the necessity of all that vomit with: "If I found out that I had slept with a transsexual, I might throw up in the same way that a gay guy looks at a vagina and goes, ['Oh, my God, that's disgusting.'" I find this idea of the necessity of vomiting a fascinating, albeit depressing, one. I've made out with people whom I didn't find attractive, but it's never made me sick. I've also been turned on by people I didn't particularly like, people I didn't want to go any further with and, again, no vomiting, nor any desire to publicly express how sick they'd supposedly made me. It wasn't anyone's _fault_ , least of all theirs, that we weren't compatible, and I had nothing to prove. In this trans panic trope, vomiting is a public sign of disgust, a signal to cancel out what would have been implied by a kiss, a caress, an erection. This isn't about trans women being disgusting, and it isn't about straight men being disgusted _by_ trans women. This is about safety, proof, social status, and the performance of a narrow idea of straight masculinity.
The public nature of avenging an insult is all too apparent in the behavior of the contestants of a 2004 reality dating show, _There's Something About Miriam_. The format was pretty standard—six young men competed for the attentions of a beautiful Mexican model—the "twist" being that she was trans. Tom Rooke, the winner, initially accepted the prize of £10,000 and a holiday with Miriam; later, he and the other contestants filed a lawsuit against the makers of the program. They alleged conspiracy to commit sexual assault, defamation, breach of contract, and personal injury in the form of psychological and emotional damage. Apparently, being attracted to a trans woman is dangerous enough to warrant not only a public disavowal, but also legal action. The company settled out of court for an undisclosed sum. The plaintiffs had successfully performed their disgust, no matter the fact of their initial attraction.
What plays out in the media as comedy or drama is enacted in real life as tragedy, with legal systems only too happy to play the trans panic card. The victims of violence are painted as revolting, disposable, and deceptive, blamed for their own deaths. Their killers are absolved of all responsibility: in this interpretation exterminating a threat like that is something that could happen to anyone.
One of the most famous of these cases was that of California teenager Gwen Araujo, murdered at a party in 2002 by two former lovers and their friends. Her genitals were forcibly inspected; one ex-lover vomited. She was subjected to extreme and extended violence, taken into the garage and murdered. Her body was dumped in the Sierra Nevada mountains; her killers, when caught, when tried, claimed that their actions were the result of Gwen's trans status. They were found guilty, but of second-degree murder and voluntary manslaughter—lesser convictions—and without the additional weight of hate crime legislation.
The murders of trans women, of trans feminine people, who are "discovered" are often marked by their appalling brutality. Very often there was no "discovery" at all: the popularity of the trans panic trope just makes for an easy excuse for transmisogynistic killers. Speaking to _Vice_ magazine in 2016, Turkish trans woman Olya had this to say about the constant threat such men pose:
My friend was killed and the murderer is still out there. He could be my next client. But I still have to work. In this community, nobody ever dies of natural causes. Every day, I come back home from work, shut the door behind myself, and take a deep breath and say, "Thank God I'm alive for one more day."... People who demand to have sex with transgender women tend to be those behind the murders. Sometimes, after having sex, they feel embarrassed about having slept with a transgender woman... It doesn't suit their manliness. They get aggressive and kill the sex worker.
Some advocates, commentators, and organizations have declared an epidemic of violence against trans women, nearly always trans women of color, poor trans women, women in sex work. Speaking to _Vox_ , Chase Strangio of the ACLU explains: "The bodies of trans women of color are the site of multiple forms of deeply historical oppression. That's a critical part of understanding the violence against trans people."
That trans panic is used as a defense by people who target the marginalized trans women does not give us a convenient, simplistic explanation for this wave of violence. It is one thread in a broader story of dehumanization and oppression. Author, broadcaster, and trans pioneer Janet Mock has been tireless in underlining the fact that the violence faced by trans women is not single-sourced, but a deadly combination of prejudices, the result of multiple marginalizations. Speaking on MSNBC in 2015, Mock stated:
These women are more that just a compilation of names and ages and stories of violence and trauma. They were people. People living at a vulnerable intersection of race, gender, and class. People existing in a culture where they fell in between the cracks of racial justice, feminist, and LGBT movements. People whose names are only spoken by the majority of us when they can no longer respond... Today we learn their stories and say their names, not out of obligation, but out of recognition that these 17 women had value, had purposes, and were loved. And they will be missed.
"Woman Who Used Fake Penis to Have Sex with a Woman Avoids Jail"
"Woman Duped by Lesbian with Fake Penis Reveals Her Horror"
"Lincolnshire Woman Posed as Man and Sexually Assaulted Woman She Met on Facebook"
In December 2015 the case of Kyran Lee, a trans man, and a woman named Carol accusing him of assault by deception, hit the news. The case itself is confusing, unnerving, the reporting of it appalling. The facts: Lee, going by the name of Joey, began a relationship with a woman, which developed in the usual kind of way. They sent each other messages; they spent the night together; they had phone sex. The woman, Lee's lawyer claims, gave Lee a sexual ultimatum: that they would consummate the relationship, or it was over. Lee, who had not yet begun medical transition, used a prosthetic—pretty standard practice for many trans masculine people. His partner described the experience as normal. Later, she was told of Lee's birth name, the fact that he was trans; she went to the police and reported him for assault—not because she had not consented, but because she would not have consented if she knew the sex he was assigned at birth. Lee was arrested and recorded as a woman: later acceptance of the fact that he is a trans man seems to have had no retroactive effect, and did not alter the arguing of or judgment on the case. The presiding judge, Michael Heath, described Lee's desire to have a relationship as "selfish... dreadful and deceitful." He recognized that Kyran Lee was a trans man, albeit in a way that many trans people found inaccurate, but imposed a two-year suspended prison sentence, and said that he would have sent Lee straight to prison if he believed his motives to have been sexual in nature.
There is so much to unpack here, and so much that feels as though it can't be solved. At what point does a trans person become their "real" gender? After a certain amount of medical treatment? After a certain amount of time? There is confusion over "how trans" Lee knew himself to be at the time of the incident—but is there a cutoff point? Are trans people only trans from the moment they announce themselves publicly? It certainly looks as though the police, the courts, and the media all believe that a trans man is really a woman, if his medical transition is still in its early stages. Not only that, but that there are legal limits to presenting yourself _as you are_ , if you do so without informing others that your standards of realness and their standards of realness may differ. What counts as "real" when it comes to the body? The verdict was unanimous in this case: a prosthetic penis is a "fake" penis; it cannot be substituted for the "real" thing. But is this the case with all prostheses, or only some? Would a prosthetic be allowed when used by a cis man? Is it the penis itself, or the ultimately "fake" body of a trans man that's seen as illegitimate? Are there set rules, or are we making this up as we go along?
I believe in informed consent. I believe in sex with honesty as its foundation. I believe that every individual has the right to decide what is happening to their body and who it's happening with. But what I don't understand, in this case and others like it, is what counts as deception, and why some kinds of deception are punishable in a court of law, and others are key elements of our sexual culture.
Imagine if the last time you slept with your boyfriend, you did so with the (supposed) knowledge that you were monogamous, that you had both been screened for sexually transmitted infections, that you were able to have sex without barrier protection, and that he was supportive of your transition. Then he tells you, after a night of unprotected sex, that he had been sleeping around without using condoms, had got someone pregnant, hadn't been tested, didn't want to be with you any longer, didn't want to be with a trans person—but could he keep having sex with you until he found someone better? It's not an unusual story, and neither is the one where you then wait anxiously for your test results, hating yourself for having been so trusting. Would you at any point think of going to the police and pushing an assault by deception charge?
There are so many ways a person can be damaged through sex and deception. I'm not trying to make light of any of them. Even when I deplore the transphobia behind cases like Kyran Lee's, I feel for anyone who feels as though their consent was given without knowing all that they wanted, or needed, to know. But I distrust the fact that the government, and the courts, have decided that some deceptions are not deceptions, and some "deceptions" are criminal. If it were to do with actual harm caused, the cases would overrun the courts. As it stands, "sex by deception" seems to be all about policing the line between those whose gender is acceptable to the state and those whose gender isn't—and punishing those who don't make the grade for having the audacity to present themselves as real.
SO MANY PROBLEMS come down to a trans person not having the right identification, lacking the updated birth certificate, driver's license, or passport that shows them in their true gender. The responsibility passes back to each individual trans person; the law has given us recognition, and if we can't match the law's requirements, if we get into trouble before we've seen to our paperwork, then the law is absolved of responsibility, instead of believing what we say.
I wonder what the kind of person who believes that thinks will happen when you socially transition. That it's as easy as filling in a form? Or that the Kafkaesque struggles with the legal and medical systems are justified, to make sure the process weeds out any trans people who aren't real enough to be counted?
Depending on the country, these are some of the ways in which we are required to earn our realness: sterilization, divorce, institutionalization, disowning our families (or never having had them), and conforming to strict and unbending stereotypes of what a man or a woman should be in appearance, sexuality, beliefs, and habits. In many countries—France, for example—an individual must petition the courts for a change of name and gender; the decision will come down to the opinion of a single judge. So far, only a tiny number of countries accept the existence of genders beyond female or male. In the UK, you must amass a case of "evidence" including doctors' reports, proof that you've completed a standard medical transition, or acceptable excuses as to why you haven't. That evidence goes before a panel of strangers; a verdict is given; if they reject an application it may be several years before a person can apply again.
There are ways of doing it differently. Malta, Argentina, Denmark, Norway, and Ireland have decided that self-determination is enough, that all a trans person needs to do to be real is to declare it on an official form. So far, there have been no problems. The idea there would be problems is laughable. Trans people are living their real lives, with or without government protection.
IF NOT THROUGH official channels how do we, then, find and declare who we know ourselves to be: men, women, beyond the binary?
Being trans, being in community with other trans people, has taught me a great deal. This experience, of what it is to find authenticity, has been one of the most profound.
I've seen it expressed, tested, confirmed by poets on makeshift stages, their makeup glowing in the spotlights, their bodies and clothes a handcrafted model of self-expression. I've heard it in the voices of other singers—basses to sopranos of all and no genders—the voice as an instrument and a conduit of a message. Seen it simply in the way bodies relax around each other, knowing that there is no judgment—express themselves with shyness or extravagance, but with that incredible sense of intimacy—that three o'clock in the morning feeling, when everyone is vulnerable in what they say, and everyone is reckless with that vulnerability. Novels, autobiography, history, theory: each form stripping a layer away. Learning a new way of language with every individual story, in the same way as learning the unique timbre of a voice, the formation of letters, the percussion of breath. Being taught that each form of physical presence—skin, clothes, hair, flesh—will have its own interpretation and way of resisting interpretation. That we can be more than we were taught we could be. That what other people say about us does not translate into who we are. That there are many different paths to happiness, and even more ways in which to travel them.
I had, and most likely still have, a tendency toward didacticism. It made me feel superior, when most of my world told me I was wrong. I am so thankful to all the people who have helped me to unlearn the defense of believing my particular truth to be universal. They taught me to really listen to other people, and to accept the limits of my own knowledge. I have never really liked putting my self into words. Listening taught me that the labels that confined me could liberate others. That the right answer for one person could become the wrong answer for another, and that all we could do was lend support in our shared individuality.
The other side of that practice, of course, was learning to truly embrace my own right answers, without imposing artificial limits on what I could learn or seeking approval for my findings.
These ways of questioning, knowing, and being are some of the greatest rewards of the trans experience. When asked if I could turn back the clock and live my life as a cis person, it's these experiences that I hold on to as the reason why I wouldn't take that option. They're ones I wish I could share more widely. We all invest so much in fighting the fear of being found wanting. Advertising, fashion, cosmetics, self-help, magazines, television, weight-loss industries—all of them promise to reveal the "real" us if we buy what they're selling and change who we are. Anxiety sells. Fear of loneliness sells. Fear of mockery, of social failure, sells so very well. We are encouraged to ascribe to singular philosophies, join a group, and find ourselves a bulwark against the world. If we buy the right thing, do the right thing, become the correct forms of our gendered selves, then maybe we'll be safe, desirable, powerful, loved.
I wonder if this is one of the reasons why some people hate trans people with such a passion, wanting not just to stamp us out but to erase all signs of our transgressions. Not because we have failed at trying to live by society's gendered rules, but because we have broken these rules and still found happiness. We have learned to make our own gendered standards, and in so doing have found desire, love, power. Not having the safety of wider society, we have built places of safety for each other. Not seeing ourselves reflected in the outside world, we have learned how to trust in each other's reflections. Trans people are far from being the only people to have learned how to find an unrecognized, liberating truth, in defiance of the pressure to court sameness and conformity. But that truth—something intrinsic to the self—is one I have learned from my trans siblings, and one I have heard celebrated over and over again throughout our communities.
I wouldn't wish the pain of dysphoria, or the weight of transphobic oppression, on anyone, but still—I wish there was a way that I could share the kind of peace that comes after years of relentless questioning. There is the tendency for some cis people to believe that being trans is about fixing some kind of defect, that we have to alter our "transgendered" selves in order to slot back into place into a gendered society bound about by struggle and by rules.
For myself, I think it could be the other way around.
#
# The Denial of History
A **quarter of the way through _The Danish Girl_ I was** just about ready to get to my feet and give the late-night audience my best Hermione Granger impression. I was tired, angry, and sick of seeing the same old story play out on the screen in front of me. Admittedly, I had not gone in with high expectations: responses from trans audiences had been extremely poor, and friends had warned me not to expect too much. I had read part of the script already and, as someone fascinated by the period of history during which the film is set, was horrified by the factual inaccuracies I saw. Still, in my defense, I had hoped for the best and, after a long day teaching and given the price of the ticket, I wanted to enjoy myself.
It was not to be. This being London, I squashed the desire to lecture those around me on what _really_ happened, and settled for scribbling furiously in my notebook, cringing every time the audience laughed at Eddie Redmayne's attempts to present himself as a woman.
Visually, dramatically—even orchestrally—similar to Tom Hooper's previous award-winner _The King's Speech_ , _The Danish Girl_ was a strange experience for me. A "based on a true story" biopic, drawn from the book of the same name, it purports to loosely tell the story of Danish painter Einar Wegener, wife and fellow artist Gerda Wegener, and the woman Einar reveals herself to be: the beautiful and ill-fated Lili Elbe. The film itself is gorgeous, of course, with that lush yet muted aesthetic that screams "classy period piece": _Downton Abbey_ on a bigger budget. Everything, from the restrained and unobtrusive directorial style, to the pre-credits reminder that the diary Lili is shown writing was turned into a memoir, points to the telling of an underlying truth. The marketing—posters, puff pieces, interviews with the actors—makes it explicit: here is the real life story of a transgender pioneer, and what you're about to see next will shock and amaze you. Failing to support this film is failing to support trans history, trans visibility, and trans acceptance.
But the problem is that what this film is passing off as "real life" is nothing more than a collection of stories.
THE BASIC FACTS: before Lili Elbe was known as Lili Elbe, she was known as Einar Wegener. Born in 1882 and trained at the Royal Danish Academy of Fine Arts, she married Gerda Gottlieb in 1904, while both of them were still students. As painters, both Einar and Gerda enjoyed some initial success, with Gerda's career quickly outstripping her partner's. They moved to Paris, they moved back to Copenhagen, they loved and supported each other, and eventually they parted ways to pursue other marriages, remaining close and loving friends.
What makes their story of interest to outsiders, of course, is the fact that Lili became posthumously famous as one of the first people to pursue a surgical transition. She died in 1931, two months on from her fourth (or possibly fifth) and final operation, a uterine transplant. _Man into Woman_ appeared in 1933, and Lili's name was made. While touting itself as a memoir, _Man into Woman_ was actually written by Danish journalist Loulou Lassen and German writer Ernst Jacobson, who took the pseudonym Niels Hoyer and the title of editor. Jacobson fleshed out his third-person narrative with extracts from Lili's letters and diary entries. Nerissa Gailey and A. D. Brown explain, in their 2015 essay "Beyond Either/Or," the many ways in which _Man into Woman_ is closer to fiction than fact: discrepancies between the book and the original documents it refers to, lack of verification of its contents due to the destruction of records in the fire-bombing of Dresden, and the number of alterations and exaggerations found throughout the text.
Information about Lili and Gerda can be found in a scattering of sources: contemporary accounts written in trans(vestite) magazines, academic papers that have collated and analyzed medical accounts, wider media accounts, and papers left by the pair. There are many aspects of their shared life that we cannot know and can't guess at. What we do know, though, is at odds with how _The Danish Girl_ presents the world.
There are the general problems with the biopic format, and this film after all claimed it was based on a book which itself only claimed to be loosely inspired by Lili's story, and perhaps we could shrug them off as a necessary aspect of artistic license. The compression of years to create a tightened narrative, for example, is standard Hollywood practice. But when the topic—transition—is relatively unknown, these standard practices introduce troubling inaccuracies and distortions. The shortening of timescale makes a gradual realization and multistaged, multifaceted transition look like some kind of overnight fancy. It may be easier on the audience's attention span, but it props up the popular myth that trans people transition on a whim. There are the usual insertions of fictional characters, some clichéd writing, and some heavy-handed pathetic fallacy; par for the course, but aggravatingly faithful to the usual trans tropes. But, beyond these features, there are four specific changes made to Lili and Gerda's story that I just couldn't get my head around, changes that felt so regressive, so oppositional to what actually happened, that it made me wonder what the filmmakers were trying to say about women in general, and being a trans woman in particular.
One of the most shocking moments of the film, one most likely to garner the sympathies of a general audience, is the point at which Lili first seeks help for her "problem," and is forcibly strapped down and subjected to a radiotherapy cure meant to restore her to a state of normal manhood. Gerda is present to encourage her, to enforce the idea that this painful, excessive response is what's right, is what's needed to save their marriage. The cure fails, of course, but leaves Lili wary of seeking help. In reality, the radiotherapy treatment Lili undertook—with the full support of Gerda—was designed to stimulate the ovaries doctors suspected her of having. Radiotherapy, as a new and exciting development in the field of medicine, was often touted as a revitalizing cure-all. In this instance, it was thought that the beams would kick-start ovarian function and that a natural bodily transition would follow. Lili began to bleed on a monthly basis afterward, and both she and Gerda felt this akin to menstruation, a sign that the treatment was working.
Returning to the movie: following the failure of radiation treatment, Lili shutters herself away, unable to trust doctors, unable to confide in Gerda, and unable to withstand public scrutiny as either Lili or Einar. There is no mention here of Lili's actual life: her consultations and treatments with Magnus Hirschfeld's institute, her (supposed) disdain for the other patients, her first operation, her love of walking through Paris while drinking life in. Lili does, admittedly, go for a Parisian stroll in _The Danish Girl_ , only to be assaulted by two strangers—another invention. Where in reality we have a single trans person as one of many—a pioneer, to be sure, but a pioneer in community—we are shown instead a lone martyr. She becomes someone who cannot be understood, who cannot seek understanding or companionship, who cannot find release even in the solitary pleasures of anonymity.
Lili's final operations and death scene are, inevitably, played for maximum effect. All trace of Hirschfeld erased, Lili and Gerda try a last-ditch consultation with maverick surgeon Kurt Warnekros. He warns them how dangerous his proposed treatment would be. He tells them that Lili would be the first to try. Lili doesn't care; she needs surgery more than the security of a long life. The surgery undertaken is reduced to two procedures, from the likely four or five the real-life Lili underwent: first, the removal of original tissue and, second, the creation of a vagina. Lili is meant to rest properly between procedures, get her strength back—she doesn't. Asking her surgeon if she'll be able to have children after her second operation, she gets the patronizing reply: "One thing at a time." You can guess the rest. Gerda conveniently arrives at just the right moment for the big death scene, caused by loss of blood and post-surgical fever, and Lili gets in a final, saccharine line before the delicate closing of her eyes. Gerda cries. The audience, presumably, cries. Lili pays the ultimate price for trying to go against nature—but wasn't she brave to do so?
The reality is far less operatic. The details are somewhat murky, after the destruction of medical records by both the Nazis and by Allied firebombing. What we know is that Lili wanted children, and that Warnekros was less ethical than the more cautious, and qualified, Hirschfeld. Successful organ transplants would not become a reality until the production of powerful immunosuppressive drugs in the 1970s. The insurmountable problem of rejection was well known in medicine by the point of Lili's final operation, but Warnekros told her he could do it. Her immune system rejected the transplant—she died on September 13, 1931.
It surprised me, then, that the cinematic change that made me even angrier than this melodramatic surgery and death sequence was the overall treatment of Gerda. Gerda Wegener—whose income from her art outstripped her partner's by a considerable amount, who won two gold medals for her work at the 1925 World's Fair in Paris, who had lived in Paris since 1912 with Lili as two women together, who funded the majority of Lili's surgery, whose best works are some of the most romantic and erotic depictions of lesbian attraction and of lesbian lovemaking I have ever seen—is reduced and changed to become a character straight out of central casting. Gerda's career only lifts off with Lili's transition, and she becomes, by turn, frustrated, angry, and shrewish about the loss of her husband. Gone is any attempt to accurately portray the Bohemian world of independent women, artistic freedom, and gendered authenticity, replaced instead by a portrait of a marriage ruined by transition, identical to every other telling of that story from HBO's _Normal_ to the join-the-dots tabloid staple. Some, but by no means all, marriages do falter in the face of transition. But attempting to impose that story onto the real lives of Gerda Wegener and Lili Elbe is an insult to how they lived, how they loved, their courage, and their artistry. Alicia Vikander brought home the Oscar for Best Supporting Actress: another remarkable chapter of women's history publicly erased.
I WAS RANTING about all of this online—I try, but I'm only human—and I had an interesting response from an ex-pupil of mine. She commented that it wasn't the film's fault that it was so tragic, because that's simply what life was like for trans people in the 1920s. I don't blame her for thinking so: it's what I thought, before I fell down the rabbit hole of historical research. We're still struggling for LGBT acceptance now, so it makes sense that the situation would be much worse nearly a century ago.
Without doubt, tragedy existed, and many, many people suffered. But the story is not quite so bleak as _The Danish Girl_ would have you believe. Sometimes, it even manages to be quite wonderful.
It's difficult to pinpoint where to begin—with the birth of the science of sexology in the nineteenth century? With the ball, bar, and party scenes in Paris, in Holland, in Berlin, where dancers could present themselves as any gender they liked and flirt with whomever they chose, twice a week in Berlin with police permission? Maybe with the 1882 case of Herman Karl, previously known as Sophia Hedwig, who, after chest and genital surgery, was granted a legal change of name and gender by the Prussian state?
Or maybe with the fact that people who flouted gendered conventions, who insisted on determining their own gendered selves in opposition to how the world categorized them, were so well-documented that contemporaneous researchers could argue over whose definition best described them. In addition to the words "transvestite" (a much broader term, closer to "transgender" now) and "transsexual" (appearing in the 1920s), we had "eviration" and "defemination" from Richard von Krafft-Ebing, _Geschlechtsumwandlungstreib_ (drive for sex transformation) from Max Marcuse, and "Sexo-Aesthetic Inversion" from Havelock Ellis—in addition to the complex and conflicting meanings of "invert," "Urning," and "homosexual." It was Magnus Hirschfeld's words—transvestite and transsexual—that stuck, and Hirschfeld's work that is behind so many of the extraordinary medical and societal advances that paved the way for the modern trans movement today.
A doctor, researcher, and all-round campaigner for the rights of sexual and gendered minorities, Magnus Hirschfeld was described by Adolf Hitler as "the most dangerous Jew in Germany." In 1897 he founded the Scientific-Humanitarian Committee, most likely the first campaigning group for what we would call LGBT causes in the world. He published a journal, _Yearbook for Sexual Intermediaries_ , featuring scientific studies, news, reviews, letters, and pictures. In 1919 he founded the Institute for Sexual Science in Berlin, and in 1928 he founded the World League for Sexual Reform. In between treating patients, acting as an expert witness, petitioning the police and the courts for fairer treatment, and writing and distributing political and informational pamphlets, Hirschfeld found the time to write and publish his groundbreaking book _Transvestites_ in 1910. Hirschfeld used the word to describe those who felt "peace, security and exaltation, happiness and well-being" when presenting as the sex other than which they were assigned at birth—and made it clear that such people could come from all walks of life, could be assigned female or male, could be of any and all sexual orientations (including none)—and that they were not mad nor, as many people think even now, pushing a kink too far.
Lili Elbe was neither the first trans woman to undergo medical treatment, nor was her "memoir" the first in the field. Karl Baer's story, _Memoirs of a Man's Maiden Years_ , was published in 1907, and served as the basis for a 1919 silent movie adaptation. Earl Lind, self-described androgyne, published _Autobiography of an Androgyne_ in 1918 and _The Female Impersonators_ in 1922. There are several reports, and rumors, of surgical treatment of what we might now call trans men from the 1900s and 1910s. Alan Hart, an American scientist and novelist, underwent his surgical transition in 1917. The first formal Western medical attempts at "male-to-female" surgery followed in 1920: "These early surgical techniques were developing hand in hand with reconstructive and cosmetic surgery methods for war veterans, and in some cases were accompanied by early attempts at hormone therapy. Although there are few detailed accounts within the contemporary medical literature, possibly due to the laws prohibiting castration, there is evidence that multiple surgeries of this nature were performed." These surgical efforts were well enough known, and considered safe enough that, following Lili Elbe's death, an editorial appeared in community magazine _The World of the Transvestite_. This editorial reaffirmed to its readership the potential of such surgery, noting promising signs coming from Hirschfeld's institute, and included an update from an unnamed transvestite, currently enjoying a postoperative holiday, "needless to say as a complete woman."
IT ISN'T JUST that this erasure of our past is untruthful, although that is a part of it. More, that in nearly every telling of the trans narrative, we are subject to the shock of the new. It's always "first trans person to" have surgery, publish a book, be employed in profession _xyz_ , serve in the army, have their documents updated, get married in their true gender. It's not about not caring about our milestones, but it is caring about the way in which the need for these firsts overrides the accurate recording of our history. I have heard so many candidates named as "the first" to have surgery, but even the earliest account I could find, of Herman Karl, is simply the earliest currently known record of modern surgery in a Western context; humans have been modifying their own bodies through self-surgery for thousands of years, and many other cultures have their own accepted forms of modifying the sexed body. Every achievement of the past is cancelled out by the need to label the next achievement "the first," making the framing of transness the framing of something exotic, different, a symptom of the modern age, rather than as just another aspect of human nature as old as humanity itself. There is no story to sell in that second framing—but there is hope for trans people looking for reassurance of their unremarkableness in an often hostile world.
What would it mean, to trans people now, if our history were common knowledge? By this I do not mean a history of people who are exactly as we are, regardless of the dictates of historical and social context, but of the people whose lives and efforts helped to create the categories, the structures and cultures through which we now move. What if we knew of these figures, these communities, in the same way we know of Oscar Wilde? It's a terrible cliché but, as a young adolescent, I was obsessed with Wilde. I didn't know how to describe my sexuality, and I didn't know how to describe my gender, but I knew that I wanted to be like him. I wanted to be that witty, and that unapologetic—and when the film _Wilde_ came out, and I saw that first kiss between two beautiful, passionate men, my body knew that that was what it wanted for itself. Just that little slice of history gave me an insight into myself, and a role model, the comfort of companionship, when I was invisible and alone. What a tremendous gift it would have been, to have known that there were people in that history who might now be called trans, people who lived as the genders they _knew_ that they were, regardless of what society had told them. To know that they had claimed their own lives with honesty and courage, and that maybe I could follow their lead and do the same.
The shock of the new doesn't just impact on the telling of our history: it's deeply embedded in the media handling of any trans-related enterprise. So much has been made of the "groundbreaking" nature of _The Danish Girl_ 's approach, and the representation of a trans woman by a cis man has been excused by the reasoning that the film's success would ultimately pave the way for the casting of trans actors. It is, apparently, astonishing to have a mainstream movie with a trans subject matter at all, and the trans community should shelve their objections and be grateful for the increased publicity. As a bit of a movie buff, I didn't buy that explanation, and it made me curious: is it really so rare for a film to feature the role of a trans woman that Redmayne receiving his Oscar nod is a step forward?
So I decided to watch _The World According to Garp_ by way of comparison. I had loved the book when I was younger, but had never got around to seeing the movie. Like _The Danish Girl_ , here is a film with a cis man playing a trans woman—like _The Danish Girl_ , that man was nominated for an Oscar for his portrayal. Thirty-four years ago, John Lithgow played Roberta, an ex-professional football player who now works in the feminist movement and becomes the main character's best friend and anchor of sanity in a mad and often grotesque world. It's not a perfect film, and much of it is firmly of its time. Still, I was struck by how natural Lithgow was in the role, and how firmly Roberta is accepted and celebrated _as a woman_ by all the other characters. Not as a man trying to be a woman, not as a particular subset of "woman"—but just as herself. It was so utterly different from Redmayne's performance, the whole of which seemed to hinge on the open acknowledgment of performance, of gesture, of the sense of being on display. Constantly quivering, seemingly either on the edge of tears or orgasm, wrists held at uncomfortable angles, Redmayne's approximation of what a trans woman is was one of the most uncomfortable things I have been witness to in recent memory. I've always been ambivalent about the casting of cis actors as trans characters, feeling that the more pressing question was the casting of trans actors in all roles. Redmayne changed my mind. The entire edifice felt precisely that—an edifice. Here was transness depicted from the outside, a man disguised through outer trappings, rather than a woman revealed by her own honesty. Shot after shot of stockings, shoes, makeup and wigs, but nothing of the heart of what it is to be trans. It felt almost parodic in its display, and it hurts that that parody will be taken as truth by people who do not know better.
Every project needs a hook. Every artist wants to feel that they have added something new to the cultural landscape. But, in the case of _The Danish Girl_ , there is nothing new, simply the recreation, the restaging, of "the new" by those ignorant of, or unfeeling toward, the history and people they claim to be serving.
Hirschfeld's institute was pillaged by the Nazis in 1933. One of the most famous images of Nazi book-burning is the documentation of the destruction of his research, his case files, his painstakingly gathered evidence in support of sexual minorities. Hirschfeld died in France in 1935; we do not know exactly how many homosexuals, transvestites, and other "anti-socials" were rounded up and sent to the camps, but we know that the numbers were considerable. Many of those few who survived the war were not freed, but sentenced to further punishment for their "crimes." The magazines, bars, and organizations that supported the lives of sexual and gendered minorities were gone. The destruction of both Hirschfeld's work and the thriving subcultures that supported and benefited from it set the emerging LGBT rights movement back by decades, if not more so. The drives—legal, social, scientific—toward investigation, knowledge, and compassionate acceptance were erased.
We have a chance to relearn that history, and to benefit from and honor that legacy, if we have the will to do so, if we can stand to look at our past without prejudice. I hope that we do.
#
# Beyond Binaries
O **nce you have the desire to see, the tools with which to** focus, locate, and describe, it is hard to stop finding new ways in which humans do and have done gender.
In the past two years, I have been inundated with media requests to talk about "non-binary." Without fail, the journalists and producers asking will refer to the topic as a new trend, something fashionable that can be linked to social media and the latest crop of young celebrities. The reference is always in the singular: "non-binary" as a noun, not an adjective, like a type of dance or a drug. When we talk, the majority seem disappointed; they don't want to hear about the daily lives, struggles, and joys of people of all ages and backgrounds living outside the binary. The fact that the richness of our experiences cannot be contained in one definition is not the angle they're looking for.
Reactions like these make me realize just how limiting the set-up to our questions of gender beyond binaries is. We assume that anything that is new to us is new to human society as a whole, and that if we don't see it reflected in history textbooks and in recent memory then it cannot have existed for long.
I have always had a love for research, historical research in particular. It was the driving force propelling me into academia, but also a way in which I could locate myself when the world seemed to be spinning out of control.
When the rights, and the lives, of people like me are dismissed as irrelevant and modish because of a lack of awareness, a lack of knowledge, history seems ever more important.
It is not that these words—trans, genderqueer, gender-fluid, gender-neutral—are timeless categories of human being, appearing in the same way all throughout recorded history. It is that, as far as I and many other researchers can see, there have always been people and categories of people that have troubled and challenged a strict binary of male and female, man and woman.
The main argument used against people like me, people who are beyond binaries, is that we cannot, should not, never have existed. Just two examples, initially found by accident during my musicological research, begin to show that ideas of a third or "other" gender are not merely a modern invention.
Despite the importance of the Byzantine Empire in the development of European culture—particularly in matters of religion—it is not given much of a look-in in most British schools. I learned about the Greeks, the Romans, and the Egyptians, but it was only when I became serious about music history and early music that I encountered the long history of Byzantium. Of immediate interest was the prevalence and importance of eunuchs in Byzantine rule and general society. To many of us, the word "eunuch" conjures up an orientalist fantasy image of a lisping, scheming creature guarding a harem; the truth is so much wider and more interesting than that stereotype allows. Up to the ninth century, "eunuch" could mean many things: a sterile man, an impotent man, celibate men and women who abstained from sex for religious reasons, men who castrated themselves to better follow intellectual pursuits, men who castrated themselves as a form of birth control, men who castrated themselves to gain a position reserved for eunuchs, men who were castrated as children by their families for career reasons, and illegitimate male children of the royal line castrated to preserve the purity of the ruling lineage. Throughout the Byzantine Empire, eunuchs existed in a gender category other than that of men or women, and were found in all kinds of roles. Eunuchs were generals, soldiers, political advisers, bishops, members of the royal court, musicians, entertainers, secretaries, monks, priests, barbers, and doctors. Some believed that eunuchs found it unnaturally easy to be chaste, others that all eunuchs were unnaturally lascivious. Eunuchs were seen as fundamentally virtuous, and also fundamentally prone to moral weaknesses; the category contained multitudes. For around a thousand years, the "other" gender category of the eunuch waxed and waned throughout the many countries of the Byzantine Empire: the persistence and strength of such a category cannot be dismissed as a fluke or a curiosity.
It was partly through the example of the Byzantine eunuchs that our second example was formed. For all that I had read about gendered history, and general music history, before beginning my undergraduate music degree, I had no concept of the castrati. When our history of opera lecturer first mentioned them, I couldn't believe that such a fundamental part of musical development and musical history could be so easily hidden away. My attention was captured and, after retraining as a singer, I found that the music written for these voices suited mine. A popular revival of early operatic music, particularly that of Handel, has seen an awareness of the castrati grow but still the embarrassment and confusion over this category remains.
Castrati is the plural for the castrated singers whose talents lie at the heart of the development of Western opera. Performed just before, or around, the emergence of puberty, the aim of castration was to preserve a high voice in a (modified) male body, finely crafted through some of the most rigorous musical training the world has ever seen. Some boys volunteered for the procedure; the majority did not. The changes wrought by castration were not limited to the voice, but created a range of body types that blurred the limits of male and female. The Church, so usually opposed to sexual deviation, was instrumental in the rise and popularity of the castrati. Due to papal rules, no woman could perform on the stages of Rome; castrati played the female parts. Throughout the rest of Europe, but most particularly in Italy and London, castrati were the heroes of the opera stage: the gods, kings, and lovers of the seventeenth, eighteenth, and even early nineteenth centuries. Mocked by some, their beauty and talents were celebrated by others: the best castrati singers were the toast of high society and the confidants of kings. Casanova, in his memoirs, remarked that the erotic effect of the castrati was so strong it forced every man to become a pederast, and having a castrato lover was a mark of success for an aristocratic lady. The libretti of the time made no bones about the confusion of categories the castrati inspired— _La finta pazza_ (1641) gives these lines to our cross-dressed, castrato hero:
_Sweet change of nature,_
_A woman transforming herself into a man,_
_A man transforming himself into a woman..._
_How many of you envy my state,_
_That of being both a man and a girl?_
To add to the gendered free-for-all, the roles created for castrati singers could also be sung by women _en travesti_ (in disguise). There were rumors that some of the best female singers were secretly castrati, and vice versa. To paraphrase much of the research into early operatic staging: the confusion as to which body could be found under the costume added to the erotic and dramatic thrill.
Castrati singers were the scandalous sex symbols of the operatic stage, and also the pure conduits for the religious music of the Sistine Chapel. The last remaining castrato singer, Alessandro Moreschi, died in 1922. Their presence was not limited to the musical fringes, but is wound in and out of Enlightenment culture, European society, our broader artistic and social roots.
It is an appalling wrong that so many children were abused in this way: there can be no dispute there. But I cannot help but ask what this says about our society, just two hundred, three hundred years ago—that we seemed to have such a need for an "other" that we created one with knives, with music, with costumes—and then, in deference to emerging nineteenth-century ideas about sex, about gender—tried to write it out of the musical canon. Musicological writers in the late nineteenth and early twentieth centuries treat the castrati phenomenon as a collective embarrassment, something not to be spoken of, to be forgotten if possible. And yet here they are—in our opera houses, our palaces, our art galleries, our libraries—a man-made third gender category that, once, was celebrated the Continent over.
Of course, it was not just in high society that gender variance was found. While researching the castrati, and the _en travesti_ women who sang alongside them, I had my eyes opened to the wealth of historical examples of gender beyond the binary that existed off the stage during the same time period. Wealth, scandal, fame, and social influence provided protection to the castrati. Not so to the ordinary people of the same eras who defied the limits of what was meant by man or woman.
Glossed in some histories as proto-gay men, the majority of researchers now consider the molly a third gender category of their own. A feminine "sodomite," the molly was often considered quite different from the other men who desired men and who congregated in taverns, private houses, alleyways, and parks in the hope of sex and, possibly, love and community. Historian Theo van der Meer, in his analysis of Early Modern same-sex practices, describes numerous ways in which mollies were differentiated from masculine men who desired men: through hair removal, speech patterns, styles of dress, and differences in body weight and femininity of movement. A police report from Paris in 1748 describes a group of mollies thus: "They] put handkerchiefs on their heads, imitating women, mincing like them. When there was some new young man there, they called him the Bride, and in this case, he comes [the object of everybody." In other contemporaneous reports, these mollies are distinguished by their softness and their swaying hips. In the privacy of their own clubs, they carried out rituals of marriage and mock childbirth.
Alongside the category of "molly" there was the "tom," an additional gendered label that was not quite female and not quite male. Not just a passing woman (a woman living as a man) and not only a woman who desired other women—both of these categories existed in their own right, and same-sex desire did not require masculine habits—the tom blended masculine- and feminine-gendered fashions, mannerisms, and characteristics to signify something other.
These were not the only ways in which the binary was broken. We know of the drag balls of the nineteenth century, and the Neapolitan tradition of _femminiello_ , a term which cannot be parsed as either "gay man" or "trans woman," and which continues to this day. In diaries, letters, pamphlets, and pornography, we find rich and varied evidence for the fact that Early Modern and Modern European culture is suffused with gendered expressions that cannot be conveyed by either "man" or "woman" alone.
Sadly, though, some of our clearest evidence comes from the records of the courts, extending from the Middle Ages until the end of the eighteenth century. The threat of punishment, ever present, frequently translated into reality for those without money or status to protect them. Four examples from across these years show how dangerous it could be to challenge the gender binary.
Victims were usual charged under laws concerning sodomy, but it is clear that sodomy was not the main crime on trial. After all, the historical record is full of instances of same-sex sex, same-sex desire, and sodomy between people of all sexes. What is apparent in cases that came to trial is the ways in which the gendered transgressions of the accused are seen to be beyond the law. Bernd-Ulrich Hergemöller explains the case of Rolandino Ronchaia, burned to death in Venice in 1354:
Since childhood he had never felt any "natural desire" for a woman. He had left the woman he had married in Padua and had settled in Venice. Since he had the voice, face, demeanour and breasts of a woman, people had regarded him only as a woman and had called him "Rolandina." He had given men exactly the same pleasure as women gave them.
In London, in 1395, a similar case was tried, with a similar outcome. John/Eleanor Rykener presented as both a man and a woman, depending on circumstances, and made their living through sex work. They were arrested, and burned at the stake.
Some passing women, in reality and in songs and stories, were lauded for their courage in living as men. Whether they were celebrated or condemned usually came down to three factors: whether they insisted on maintaining their male gender, whether they had had sex as a man, and whether they had modified their bodies in ways which threatened to usurp the "naturalness" of the sex binary. Many reasons have been given as to why people assigned as female would spend their lives as men: for romance, for practical reasons, for adventure. There is, of course, another possibility: that these people were men, self-declared, or were some other gender known only to themselves. What is clear is that the compulsion to live freely was greater than the compulsion to live in safety. In 1477, in Germany, Katherina Hetzeldorfer was tried for living as a man and using a phallus; this phallus was strapped on, used for sex, and could be urinated through. Hetzeldorfer was found guilty, and drowned. Even more scandalous was the case of Catherina Linck in 1721. The key point of Linck's prosecution was not just that they lived as a man, marrying one woman and having sex with many others, but that they used their leather phallus (complete with scrotum) for both vaginal penetration and for oral sex. The combined crimes of gender usurpation and sodomy were too much to be borne. Linck was executed by beheading.
There are so many lessons we could learn from these examples, from the thousands more we find when we widen our historical scope. But this is the most important: that a strict gender binary has never been able to hold the totality of humanity—not in the past, not in the present—not in the future.
BUT WHAT NOW? If we are not just a flash in the pan, and cannot be forced back into binary bounds, then what are we left with and where do we go? What are the ramifications of this knowledge, not just for those of us outside the binary, but for everyone?
The most basic must come first: recognition and protection under the law. It shocks many people to learn that someone like me occupies a legal gray area, and that my right to employment, to public services, to live free from harassment, is arguably not covered by law. I say arguably because whether or not I'm protected by the Equalities Act is a question I can't get a straight answer to. The lawyers I've spoken to have said "no." The civil servants I've spoken to have said "yes." From the number of people who've denied me services, failed to pay me for transphobic reasons, insulted me in job interviews because of my gender, refused to take attacks on me and people like me seriously, I think it's safe to say that—practically speaking—we are not protected. It seems clear that, in order to gain these protections, we need to convince those making the law that we are a recognizable group entitled to specific inclusion in human rights and equalities legislation.
Only a handful of countries worldwide recognize anything other than a gender binary, and those that do recognize something other than male or female do not always protect the rights of those so categorized. Nepal, Thailand, New Zealand, Australia, Pakistan, and Bangladesh offer some limited recognition of a third gender category, with New Zealand leading the way. The movement, globally, is growing, but the resistance to change is profound.
The UK government is under constant petition to recognize the rights of those outside the binary, but the response so far has been discouraging. A 2015 petition pushing for full legal equality for all trans people was officially dismissed. On the subject of people outside the gender binary, the Ministry of Justice had this to say: "The Equality Act 2010 protects people from discrimination if it arises from their being perceived as either male or female. We recognise that a very small number of people consider themselves to be of neither gender. We are not aware that that results in any specific detriment." Dismayed, my activist colleagues and I decided to collect some facts on the specific detriment suffered, and present these findings to the MoJ. Of those who responded to our call for information, 94 percent reported feeling unsafe because of their gender expression. Respondents told us of the lack of provisions in education, work, housing, and health care, and said that they faced a daily choice between being misgendered and being denied access. There were widespread experiences of being bullied at school and abused at work, being labeled "difficult," "dangerous," and "unprofessional." Some respondents told us of the sexual abuse they'd suffered because of their gender, and others of their doctors' refusal to treat them. The data we collected were strikingly similar to other research into trans lives, but with the specific problem of being recognizable enough to be abused, but not recognizable enough to be counted and protected.
Our meeting with the Ministry of Justice to discuss their response and these findings was pleasant enough, but evidently failed to change their minds. The results of the first parliamentary-backed, nationwide survey into trans experiences, the cross-party Trans Inquiry, were published at the beginning of 2016. The inquiry supported the need for non-binary recognition and protection; the government response, six months later, dismissed it. Barack Obama, challenged on trans issues by non-binary activist Maria Munir, gave a more positive and supportive response to our concerns than I have heard expressed by any UK civil servant or government official. Trans people who are not men or women are just as subject to transphobic discrimination and oppression as trans men and women—but we continue to be dismissed as a tiny and unimportant group of fantasists.
Many such people, then, would prefer to put their efforts into effecting broader cultural changes. We may be reduced to a trend, but at least it's a foot in the door. The societal shifts we're now seeing, as much as I want and need more, are greater than I ever thought possible when I first came out. When I was a teenager, anything other than a strict adherence to gendered stereotypes in appearance was an invitation to bullying. Teenagers now can look to fellow teenager, Hollywood royalty Jaden Smith, modeling womenswear for Louis Vitton and refusing to limit himself with traditional male/female stereotypes and behaviors. Gender-fluid celebrity Ruby Rose has become infamous for a desirability that transcends labels, and Miley Cyrus has been open about her experiences of identifying as multiple genders. What we're seeing in pop culture is a reflection of the opening up of gendered categories and behaviors happening in everyday life. Two thousand and thirteen research by Daphna Joel of Tel Aviv University found that feelings of being, to a greater or lesser extent, "the other gender," "both genders," and/or "neither gender" are far more common than previously supposed: over a third of respondents to the study reported such feelings. A recent YouGov poll found that around 20 percent of people placed themselves on a spectrum between male and female. Millennials are far more likely to see themselves as between, both, neither, or other, than older age groups, and this development shows no signs of stopping.
It certainly tallies with my own experiences. Being obviously, openly, neither/nor seems to have made me something of a confessional box. I realize that this is a self-selecting sample, but believe there is still truth here. For half my life, people who are not trans, or not openly trans, who would still answer to "woman" or "man," have been telling me that they don't really feel themselves to be at peace with those categories. The reasons are many, and mostly to do with the crushing pressure of stereotypes and expected behaviors. What is universal is the reporting of the fact that what they are told they _are_ , from the outside in, doesn't chime with who they feel themselves to be beneath the exterior. Their feelings about gender are more confused, more questioning, and less stable than their outer behavior would have an observer believe. I wonder how common these feelings are. I wonder what further research, and further exposure to openly gender non-conforming people, will show us.
Our steps toward equality are still tentative—this is not a done deal. For every example of positive change I could provide, I could show multiple negatives. But the changes I have seen, over my lifetime, at least show that change is possible. What I have learned about our histories shows me that the gendered bars and limits placed around us need not be permanent.
Above all, over and over again, I am shown the many and varied ways in which the gender binary simply does not exist.
All I can hope is to live long enough to see that knowledge take root, and blossom.
#
# The T from the LGB
F **aced with a blurring, shifting, evolving landscape of** what gender and identity can mean, many people would like to try to make things appear simpler. For some cis LGB people, and some straight trans people, this means marking a neat line dividing issues around gender identity from issues around sexual orientation. Gender is who you are, and sexuality is who you want; sexual orientation is who you go to bed _with_ and gender identity is who you go to bed _as_. To avoid confusion, to avoid misgendering, it would be easier to part ways; get rid of the alphabet soup of LGBT(+Q)(+I)(+A) and focus solely on trans issues, or solely on LGB issues.
Some of the reasons for this desire are valid if not, perhaps, readily discernible to an outsider. After all, one of the reasons why we're so often seen as the same group is one of the reasons why some of us want to go our separate ways. In the public imagination, and in the particular reasoning of people who either don't know or don't like trans people, being trans is what happens when you take being gay too far. This line of argument has two strands. The first, popular with those who mock gay and trans people alike, is that gay men are girly and gay women are butch: what is a trans woman but a really girly man, a trans man but a super-butch woman? The second explanation is more often trotted out by cis gay men and lesbians who disapprove of trans people. They believe that every trans person is actually a gay person so scared of being gay that they would physically alter their bodies so as to make their desires seem more normal. It is unsurprising that many trans people, sick of being called fantasists or cowards, would prefer to strike out on their own. It's also unsurprising that cis gay people, sick of being told that they "want to be a man" or "want to be a woman," actually want to make it clear that being gender non-conforming and queer is not the same thing, or not always the same thing, as being trans.
There's also the issue of transphobia in LGBT spaces. For all the joint statements from campaigning groups and advocates, and the ways in which we are lumped together by outsiders, there's a sizable chunk of the LGB population that hates trans people just as much as straight people do. We feel it in the ways we are groped on the dance floor of a gay club by someone who wants to "check," and in the refusal to allow queer trans people access to queer spaces, because we're not seen as legitimate. It's in the treatment of our trans bodies in these spaces and communities, the ways in which they are described as disgusting, freakish, threatening, and just plain wrong. The cis gay men who claim that vaginas are gross, little thinking that there are gay men with vaginas. Cis lesbians who describe their lesbianism not as a love of women, but as a rejection of penis, and woe betide the lesbian who might have one, or might have had one once. It was, until recently, the way that mainstream LGBT organizations were happy to tack a "T" on the end, but not to fight for trans rights, inclusion, justice. There is so much bad blood, and so much still to fix.
My stake in all of this is personal. As a queer trans person, it has to be. But beyond my own needs there are wider, far more compelling reasons, why trying to pull the T from out of the LGB hurts more than it heals. In our interlockings, our intersections, there is power, hope, a path to something better. But the divisions that would drag us back have existed for a long time, and cannot be wished away. We have to go deeper if we wish to go forward.
IN THE PUBLIC imagination, most trans people are assumed to be straight. After all, this is one of the main props of the standard trans story: "I knew I was a woman, because I needed to be with a man _as_ a woman," and vice versa. It may be surprising, then, to learn that trans people are far more likely to be bisexual, gay or lesbian than cis people are.
Anecdotally, we know this and have known this for a long time. In my own UK trans community the number of straight trans people I know is overwhelmed by the bi, queer, pansexual, omnisexual, lesbian, and gay trans people of all genders and descriptors. Some of these trans people are heavily invested in the mainstream gay scene, and many have made alternative, trans-friendly spaces where they can express honestly, and without fear of reprisal, both their desires and their selfhoods. Our trans community is also comprised of people with no sexual desire toward others, who may or may not be in romantic relationships that are gay or queer. Some of us are in open relationships, some of us are in polyamorous triads, some are happy to be single, some are monogamously coupled. We are into kink, into vanilla sex, into no sex at all, and some trans people combine these categories in ways we have not been taught to recognize or respect. It is not that straight trans people don't exist, because of course they do. But I don't know where, in this proposed splitting of LGB from T, we would fit the majority of trans people who are both.
Research data backs up the evidence of our lives. There are patterns and changes that shift depending on culture, location, subsets of identity, but the broad findings leave little doubt as to the importance of LGBQ desires in many trans people's lives. A 2013 Canadian study found that an estimated 63.3 percent of trans men were bisexual, gay, queer, or otherwise found themselves attracted to other men. The authors of this work referred back to similar studies from other researchers, in 2001 and 2011, which found that between one quarter and one half of trans men describe themselves as gay or bisexual, with even more preferring the broader term "queer." Likewise, there have been numerous reports that have demonstrated the fact that around the same numbers of trans women would also describe themselves as lesbian, bisexual, and/or queer. Scottish Transgender Alliance's enormously important 2012 report on trans mental health found that the majority of trans people surveyed described themselves as bisexual (27 percent) or queer (24 percent). Many respondents used multiple terms to convey the nuances of their sexual orientation.
There are trans people whose orientations remain fixed throughout transition, and there are also people who find that their sexuality can shift and change throughout life. Some attribute this to hormonal changes, others to the ways in which being trans forced them to question many of the received truths they'd previously accepted, making them more open to possibilities previously seen as impossible. I know of many trans masculine people who were not attracted to men pre-transition, but who found themselves identifying as bisexual or gay post-transition. Many said that they couldn't imagine relating to a man in a perceived heterosexual relationship, but that being with a man as a visible man made sense.
For myself, being genderqueer and being queer are intimately linked, calling back to the same need to resist limits on who I might be and who I might desire. I don't want to second-guess the direction of my life from past experiences, while still needing words to describe the particularities of those experiences to others. I know many trans people who feel the same way about their sexualities, whatever those sexualities might be: that there is no point where you could extract one from the other, could draw a clean line through the totality of a person. I get very tired of the idea that LGB rights means only LGB _cis_ rights. I can't countenance the exclusion of trans queer people with the idea that trans here is a noun, something separate, rather than an adjective that points to a broad range of experiences, placing us in community with queer people of all genders.
THESE INTERSECTIONS BETWEEN gender and sexuality are not simply products of our modern lives, but vital components of the groups and movements that have made our lives possible. Not that we could guess that from the majority of gay and lesbian histories— _Surpassing the Love of Men_ , _Gay Life_ _and Culture:_ _A World History_ , _Who's Who in Gay and Lesbian History—_ but the actual record of same-sex desire in human cultures is threaded through with evidence of gender non-conforming lives. Previous, cis, historians have tried to gloss that variance as "gay." More recent researchers have stressed the importance of taking everyone—even those in the past—on their own terms. And those terms are often more mixed and complicated than our modern categories of "cis" and "trans," "gay" and "straight."
The very word "homosexual," used so exclusively now, initially had a double meaning, being both the desire for the same sex, and possessing a mind, a brain, belonging to the "opposite sex." "Homosexual" was neither trans nor gay, but a root from which both of these modern ideas grew. Karl Ulrichs, one of the early founders of what would become the later LGBT movement, defined homosexuality as a third sex; the people so often described as gay men in modern tellings of this story were described by Ulrichs as having, as he did, "a woman's soul trapped in the wrong body." There is a tendency now, stemming from the gay liberation movement of the 1970s, to call those explanations excuses, a way of trying to garner sympathy, or explain the unexplainable. In societies that punished same-sex behavior, surely it made sense to claim a kind of heterosexuality? But the fact is, when we delve deeply, we find all kinds of genders, gendered explanations, in these early movements and communities, and many people who experienced same-sex desires without any hint of gender variance.
A clear comparison can be found in the early-twentieth-century examples of Radclyffe/John Hall and Natalie Barney. Both are most commonly described as lesbians: they lived at the same time, had many friends in common, moved in similar queer circles. And yet their approaches to gender and desire were totally different. Hall is most famous for her/his groundbreaking 1928 novel _The Well of Loneliness_ , which was banned in Britain under obscenity laws. The tragic story of congenital invert Stephen Gordon, _The Well_ is known both as the quintessential early lesbian text and also as one of the most depressing and, to many lesbian readers, unconvincing stories of same-sex desire in the canon. Vita Sackville-West, Violet Trefusis, and Virginia Woolf all found it wanting. Hall's depiction of physically, mentally masculine inverts, doomed by their treacherous bodies to an outcast life could not be more removed from the kind of lesbianism celebrated by Natalie Barney, depicted in _The Well_ as Stephen's friend Valérie Seymour. Barney, an American heiress and founder of one of the most famous salons in Paris, believed in the innate superiority of women in all things, and in the passionate, erotic celebration of femininity. She was unabashedly open about her love for other women, believing lesbianism to be a more moral, and certainly more rewarding, choice than heterosexuality. When it comes to Hall, I don't even know which name or pronoun to use. Hall was John and he to those who knew him well and respected his inner life, and Radclyffe and she to the rest of the world. None of us could know whether, if he/she lived now, they would be a trans man, a butch lesbian, or something else entirely. What we do know is that our shared past often defies current categories in favor of something more murky and complex. We can see a recognition of that complexity in the ways in which some early homosexual campaigners tried to reject it: something must exist in order for it to be suppressed. The "movement for masculine culture" was set up in opposition to the more popular theories of inversion and gender/sexual variance, positioning homosexuality as something manly, virile, and utterly unconnected to the cross-dressing and cross-gendered behaviors of the abnormal. Biologist Benedict Friedländer, a misogynist, anti-Semitic Jewish man, was one of the leaders of this movement, in which nationalism, machismo, sexism, and racism combined with notions of a biological drive toward homosexuality to demonstrate the supposed superiority of the white, same-sex-oriented man.
As we have already seen, sodomy laws were enforced with greater regularity, and with greater cruelty, upon the people who broke the gendered rules of their societies. Early Modern European culture is so rich with depictions of same-sex sex and desire, but it is in the records of arrests, torture, and executions that we find much of our evidence of gender non-conforming behaviors. The ways in which we have interpreted this evidence change, of course, with the meaning we wish to find. The figure of the passing woman appears over and over again in lesbian history and historical fiction as an example of the lengths women will go to when denied the chance to love other women openly. Increasingly, I'm finding the same stories passed around trans circles as evidence that trans people have always existed. We all of us want to feel validated, to feel the legitimizing force of history—but do we ever have the right to claim strangers as being one of our own?
We certainly feel the draw. Louis Sullivan, himself a pioneer in the record of trans history, wrote a biography of Californian man Jack Garland, who died in 1936. Jack could be seen as a typical example of the passing woman: assigned female at birth, an adventurer in men's clothes, "the mysterious girl-boy, man-woman, or what-is-it" in the newspaper parlance of the day. He was a soldier, a nurse, and, eventually, just another man like any other, until the shock of his autopsy. But then there is the fact that Jack acknowledged his need to be with men as a man, that he made many coded references to life on the street with hobos and "willy boys." His closest friend had no doubt that he was a man through and through, despite his birth. Others, after his death, have crafted stories of him as a woman pushed to imitation by the constraints of misogyny. Louis Sullivan described him, without a doubt, as a trans man. How could any of us place him, in a way that does him justice, when we can never ask and know for sure?
Even when divisions between categories of gender and sexuality came into force, shared oppressions meant a shared battle. The American protests and riots of the early 1960s—Dewey's, Compton's—were melting pots of different identities. Even the most famous riot of them all, the one that many people cite as _the_ birthplace of the modern LGBT movement, was not a single-community affair. Contrary to the story presented in Roland Emmerich's _Stonewall_ , the Stonewall riots were the bubbling up of all the injustices borne by gender non-conforming people—trans women of color, butch black lesbians, transvestites, femmes, sex workers, queens—the queer people who suffered the most from police violence and societal rejection. Stormé DeLarverie, Marsha P. Johnson, Miss Major, Sylvia Rivera: we owe so much to them, and the countless others who were brave enough to say "enough is enough." Without their actions at Stonewall, the modern gay movement, the rights enjoyed by gay people, would be unthinkable. And yet, even then, there were cis gay people, white people, trying to exclude the trans and/or gender non-conforming activists from the movement that they themselves had made. The Gay Liberation Front, cofounded by trans people, moved to exclude their own founders. Sylvia Rivera and Marsha P. Johnson started STAR—Street Transvestite Action Revolutionaries—a group that helped trans street kids survive, gave them shelter, food, education, hope. They did it without the help of the new gay groups. After their exclusion from the GLF, some trans, gender non-conforming people started the QLF—the Queens' Liberation Front—only to find themselves banned from the first ever march to commemorate Stonewall. There are harsh truths for cis gay people to be found in the _TransLiberation Newsletter_ of 1971:
The oppression against transvestites and transsexuals of either sex arises from sexist values and this oppression is manifested by homosexuals and heterosexuals alike in the form of exploitation, ridicule, harassment, beating, rapes, murders and the use of us as shock troops and sacrificial victims.
Trans lesbians led the way in many second-wave feminist groups, only to find themselves excluded from lesbian feminist spaces. Clubs, communities, activist organizations where trans queer people created and defined what it meant to be LGBT, fought for justice, did the constant, dulling, dangerous work of being out and demanding more: those same organizations turned their backs on their trans members once it was safe to do so. Some of our modern history is extraordinary in what it shows us of cooperation and compassion, and some of it is a master class in excluding the most marginalized "for the greater good" of the most privileged.
I don't think this mixed bag of our history is a sign that we should try to part ways; that tactic, too, is scattered throughout the historical record, and it hasn't worked yet. Instead, as ever, we have a chance to learn from the mistakes of the past, to examine those exclusions, these moments of cowardice and hubris, and to apply what we have learned to our current movements and communities. We could take the best—the empathy and solidarity—and try to add to it, to pass that legacy down to the people who come after us. Not just as activists, but as individuals, we can do better in community than in division.
MANY PEOPLE, CONFRONTED with a picture of transgender pioneer Leslie Feinberg, on being told that she was transgender (without the use of that pronoun), would assume that that meant that Feinberg was a trans man. They would think that his sexuality would be based on a foundation of maleness: gay to mean attracted to men, straight to mean attracted to women. Those assumptions would be wrong. Feinberg, whose fiction and nonfiction books changed the lives of so many trans people, myself included, was a transgender, butch lesbian. She used different pronouns in different spaces, insisted on the importance of solidarity, and linked her trans activism to her communism, her anti-racist work, her support for trade unions, and her feminism. In a 2006 interview, she explained the importance of having the right word in the right context when it came to expressing the self:
For me, pronouns are always placed within context. I am female-bodied, I am a butch lesbian, a transgender lesbian—referring to me as "she/her" is appropriate, particularly in a non-trans setting in which referring to me as "he" would appear to resolve the social contradiction between my birth sex and gender expression and render my transgender expression invisible. I like the gender neutral pronoun "ze/hir" because it makes it impossible to hold on to gender/sex/sexuality assumptions about a person you're about to meet or you've just met. And in an all trans setting, referring to me as "he/him" honors my gender expression in the same way that referring to my sister drag queens as "she/her" does.
Constantly, in Feinberg's work, there is the referring back to multiples: multiple communities, multiple experiences, multiple ways of naming and being named. She died in 2014 at the age of sixty-five. She was survived by her partner of many years, Minnie Bruce Pratt, another writer whose works reverberate with the knowledge of what it is to trouble the boundaries of how we are supposed to express our genders and desires. Pratt writes of being femme, of locating femme in the radical repositioning of the world's gendered power norms, provoking investigation into what is learned, what is "natural," and what feels right. Her words challenge the reader to reevaluate what it is that they know about bodies, categories, and love, and dares them to see something more than they previously thought possible. Her description of making love with a partner, presumably Feinberg, who has strapped on a phallus, in her 1995 collection _s/he_ , asks us how we can define the limits of other people's lives:
You are a woman who has been accused of betraying womanhood. In my groans of pleasure from your cock, perhaps some would say I have betrayed womanhood with you, that we are traitors to our sex. You refusing to allow the gestures of what is called masculinity to be preempted by men. Me refusing to relinquish the ecstasies of surrender to women who can only call it subservience. Traitors to our sex, or spies and explorers across the boundaries of what is man, what is woman?
How can you pull apart a life like Feinberg's, like Pratt's? Why would anyone want to do so?
Part of being trans, of being queer—not all of it, not for all people, but part—is in the reimagining of what it is to be human. These are categories forged from the failure or refusal to acquiesce to majority rule. That majority rule requires nice, clear lines and limits, and so often we, who are fighting for acceptance, will try to stick to the same in imitation. Too often, all we do in doing so is hurt our own. I'm not prepared to accept any definition of trans, any definition of LGB, that would leave out those like Feinberg and Pratt. I think we would all be the sorrier for it.
IF WE HAVE any hope of trying to end the ignorance, discrimination, and violence that blight our lives, we cannot afford to be seduced by the sophistry of single-issue movements. As Audre Lorde so rightly said, we do not live single-issue lives. The idea of single-issue movements, of whatever kind, so often has the idea of "neutrality" at its heart; an LGB cause where LGB means cis and white, a trans movement where trans means straight and white. But none of us is "neutral"; some of us just have less pressure, less hatred to contend with than others. Focusing on the needs of those with a lighter burden to bear is not "objective" or "pragmatic," but it is a confirmation of historic societal prejudices that say that some lives matter more than others, some lives are too "complicated" to be worth caring for, some oppressions are just too entrenched to change.
We talk of an LGBT umbrella, but not of those LGBT people who cannot seek shelter beneath because we have narrowed our protections down to the point that only the few are covered. We have been taught to weigh up lives and accord them value, and we turn that tactic on each other. When we feel anger, we direct it toward the most vulnerable members of our communities for their failure to be "respectable" enough to toe the party line, to make themselves acceptable to those who hate us.
I want, and I need, an LGBT umbrella, but what I need beyond that is a solution to the injustice raining down on us—and my needs are less pressing, less desperate, than many others'.
So I am wary of any call to split the T from the LGB, rather than to focus on specific needs within the broader community with shared resources, shared strength. It feels too close to washing our hands of those who stand in the way of assimilation, who would demand justice instead of tolerance—who cannot cut themselves into pieces to fit a truncated agenda.
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# Trans Feminisms
D **espite my fear of cameras, I'm slowly getting used to** being filmed, under certain circumstances: live performances, music videos, documentaries, and educational projects. I'm still a novice when it comes to television. Earlier this year I was asked to appear on a flagship UK news program, a live broadcast, to discuss proposed changes to the way the law recognizes and treats trans people. In that total slowing down of time that happens during a performance, it was impossible not to read the host's teleprompter just before she introduced me and her other guest; I was labeled there as a "trans activist," and the writer picked for her opposing views was "a feminist." It was so hard not to interrupt, mostly to laugh, and ask why it was that she was a feminist and I was not? My doctoral research is in feminist musicology; I specifically applied to my university because of its groundbreaking work in music and gender. A feminist attitude underscores all of my musical performances and event organizing, my teaching, and my charity work. I made my first feminist speech when I was ten years old, and I began writing for online and offline feminist publications in my mid-teens. It would have been just as easy to say "trans feminist" as "trans activist"—and yet that word, that label that makes sense of my life, was not given me.
I need feminism. I need it not because I am a woman but because, no matter what lens the world uses to view me through, I am subject to gender-based abuses, founded on the idea that there is one, hierarchical, coercive gender system. When I am seen as a gender non-conforming woman, a failed woman, a dyke, and a bitch, I need feminism. When I am seen as a gender non-conforming man, a pretty boy, a poof, a faggot, I need feminism. Most of all, when I am seen as the flipped version of what I am—a gender failure and a gendered freak—then I need a way of fighting back, of changing the world that treats me this way. Feminism is my method of resistance, my road map to change.
This enforcing of gendered systems, of gendered expectations—the poor treatment that comes from being seen as "not right" and "not important" in the scheme of gender—can come from all places and all people. The first man who groped me in broad daylight as a young teenager, a straight man, and the man in a packed nightclub who forced his face between my legs, who kept coming back to grab my genitals, because "It's okay, I'm gay." The woman who took advantage of my age and my vulnerability, and seeing me naked and alone after a shower in a public changing room, put her arm around me and ask me to take my towel off and get dressed in front of her, and the drunk woman after Pride who asked me if I was a man or a woman, then answered her own question with a blow to my crotch. I have been extremely lucky, compared with many women, compared with many people outside the binary, compared, in fact, to many men who have also suffered from the effects of misogyny. But I don't believe that any of these experiences are acceptable. It's not that I want a weapon with which to attack the individuals behind these instances, but that I need a total game plan that ensures that these instances, any instance of gendered violence, any system of gendered oppression, becomes a thing of the past.
It's hardly as though I'm alone here: all of my friends, and the majority of my trans community, are feminists. They are engaged with feminist communities that reflect the kind of belief expressed by South African activist and artist Gabrielle Le Roux, describing here her awakening to the importance of trans people in feminist practice:
Through a couple of close personal friendships, I began to deeply engage with transgender issues as gender issues and realised that this is all our issue... This means that a struggle for gender equality founded on the idea that we are struggling for equity between men and women is also profoundly flawed. Having worked as a feminist activist for two decades with so many issues and such diverse women, for us not to have engaged with transgender issues as gender issues made me feel cheated. Gender is the connection between transgender and feminism—I feel strongly that until we engage with transgender and inter-sex inclusively as gender issues, our feminisms are stuck.
And yet, we have headlines like these:
"Call Yourself a Woman? Feminists Take on Transgender Community in Bitter Debate"
"What Is a Woman? The Dispute Between Radical Feminism and Transgenderism"
"The Conflict Between Feminism and the Transgender Movement"
"Transgender Rights Versus Feminism: What Makes a Woman?"
They crop up in right- and left-wing publications alike; it's the standard set-up for news discussion shows. Many trans people I know have given up on mainstream media engagement, the kind of talking-head appearances from experts, because of the inevitability of being pitted against a transphobic feminist, and of being encouraged to fight.
This framing refuses to jibe with what I know to be true about the multiplicity of feminisms in general, and the history and import of trans feminism in particular.
Trans women have been part of feminist movements for a long time. When we look at the magazines of the queer, radical German subcultures of the 1920s and 1930s, we see trans and lesbian writers and communities sharing spaces, publications, clubs, and activist groups. Many trans men, like influential philanthropist Reed Erickson, had been feminists (and sometimes lesbians) before transitioning, and retained strong ties to feminist movements. Pioneers of all genders from the older, working-class bar and drag scenes—black, Latin, and white—were fighting for gendered freedom and justice long before the theoretical developments of 1960s and 1970s second-wave feminism. And yet it was a minority of these second-wavers that created and legitimized the idea that "real" feminism could not include or even support trans people—and which singled out trans women for a vicious campaign of abuse.
There are three arguments lobbied again and again at trans people, supposedly from a feminist perspective, supposedly from a neutral, unbiased position. From a distance, without repetition and investigation, these arguments can look reasonable. But the uncritical assumption that these arguments are correct may well, in the long run, be causing more harm to trans people than the more open abuse of a furious minority.
The first is that trans people, by existing, by transitioning, prop up a sexist gender binary. This argument relies, as a point of principle, on the erasure of trans people like me, who do not fit into a gender binary and have no wish to do so. Further "evidence" is provided by images of some trans men and women in the mainstream media: glamorous, feminine women and hyper-masculine men. What is missing here, apart from an awareness of the diversity of gender expressions and identities to be found across trans communities, is knowledge of the ways in which medical treatment for trans people has been granted or denied based on the expectations of cis, mainly male, clinicians. Pioneering sexologist Harry Benjamin was famous for his criteria of what made a "true transsexual," but sexual stereotyping in the provision of trans-related care is still with us. Trans people have been denied hormones and surgery for being gay, for being bisexual, for being too tall or too short, too fat or too thin, for not molding themselves into white Western presentations of gender, for daring to be butch trans women, or femme trans men, for being disabled, for giving the "wrong" answers to intrusive questions about sex and masturbation, for not being in full-time employment, for being married, for refusing to change their names, and for being intersex. Those examples come from the lives of trans people I have met, have known, in real life, and I suspect that there are many more. Some trans people have to play the long game: give cis clinicians what they're looking for so as to be able to transition, and then have the space, the freedom, to present as is right. If the feminists who used this argument truly cared about coercive gender standards they would be standing _with_ trans people and demanding an end to medical gatekeeping—and, yet, here we are. The behavior of those with power over us is used as a stick with which to beat us. It solves nothing.
The second common feminist argument made against trans people is that gender may be diverse, but sex is binary, unchanging, and oppositional. We have already seen some of the ways in which sex and gender are more complicated and varied than that, but it is worth considering the additions that some feminists have made to this template. In this reading, it is not only that female and male are biologically opposed, but that their differences contain spiritual and moral elements. Much is made of the ideas of male and female "energies," of the (unproven) assertion of a male biological drive toward violence, the (equally unproven) assertion of a female biological drive toward compassion, and of the power that lies in the actual or symbolic womb. For all that this comes under a feminist banner, these arguments are strikingly close to Freud's belief that "biology is destiny," and find their foundations in much of the early white suffrage and temperance movements: the claims that women deserved the vote because they were purer and better than men, that women were not tempted by alcohol because of their innate moral superiority.
The final and, seemingly, most convincing argument is that of socialization. Trans women, these feminists claim, can never really be women, as the very fact that they needed to transition proves that they were _socialized_ as men. At a cursory glance, it makes some sense: who we are is, indeed, predicated to a large degree on what we have been taught to be. Where the anti-trans socialization argument falls down, though, is in its assumptions about how socialization works: that it is enacted along some kind of universal binary gendered line, that it is a one-way process, and that it stops after a certain time.
This is to me, as a feminist, abhorrent. It strikes me as wrong not only in its inaccuracy, but in the ways in which it ignores other axes of power and privilege, other ways in which we suffer at each other's hands—the myriad ways in which we benefit from others' suffering, whether we like it or not. I look at my own early years, and see an incredible mix of differing messages, predicated on my family's unique experiences of gender, of nationality, of race, and of money and of class, and on my broader cultural positioning within those categories. Many supposed universals of "female socialization" were unknown to me, as were those of a supposedly typical male experience. I find commonalities of experience with friends from similar backgrounds, who inhabit similar categories, and I find commonalities with people who, at first glance, would appear to be totally different. There can be—there are—similarities and overlaps of gendered experience, just as there are similarities and overlaps between any cultural groups—but a universal either/or socialization does not do justice to the multiple nature of inequality and injustice, nor does it help solve these problems. How could I, on the basis of the sex I was assigned at birth, claim to have had the same socialization as a child born into poverty who has never known what it was like to go to bed well-nourished and safe? The same socialization as a child denied an education because of warfare, cultural upheaval, or a specific localized form of misogyny? To claim that all the privileges of my whiteness matter less than the sex I was assigned at birth, so that my socialization was that of a child raised with the total violence of embedded racism curtailing their possibilities at every turn? To me, it feels like an insult, an appropriation, to take the experiences of others who have struggled far harder than I ever have had to and say "this struggle is mine, because of the one way, out of many, we were categorized at birth."
Neither is socialization a one-way experience; the human mind is so much more than a blank slate waiting for an outside hand. How someone will hear, interpret, and internalize the _multiple_ messages they receive, and how they will respond in turn will vary from person to person. To return again to my childhood: a crucial part of my gendered socialization was that I reacted to others' messages as someone who did not feel male or female. Trans people's experiences of how they knew themselves before they had the language to know themselves as trans are hugely diverse, but it would be wrong to assume that we reacted to the gendered messages we received as the sexes we were assigned at birth, or uncomplicatedly so. Trans women who are tarred with the label "socialized as men" often report childhood experiences of knowing themselves to be girls, of being punished for being feminine, and of being abused for not meeting masculine stereotypes. I don't understand how those experiences, both self-reflexive and at the hands of others, constitutes a "male" socialization.
Finally, we have the idea that it is only socialization in childhood that matters: that it takes a certain length of time before a person can claim to be a woman or a man, and that if a critical window of opportunity is missed, a person cannot ever "change gender." Leaving aside the fact that those putting forth this argument ignore both the existence of young transitioners, and the common occurrence of inner knowledge that predates social transition, this interpretation of how the self is constructed dismisses what we are coming to know of brain plasticity and the evolving field of theory of mind. I know I am not alone in having experienced significant shifts in personality following major life changes in adulthood. My brother's final illness and death altered so much of what I feel to be "me": my gut reaction to bad news, the length of time it takes me to process difficult emotions, my sleep patterns, the temptation to turn to alcohol in times of pain, the figures that haunt the visions that accompany sleep paralysis, the way that my depressive lows now melt into grieving. My doctor told me that both my immune system and my metabolism were adversely affected by the extreme and ongoing stress. Who I am now is not identical to who I was ten years ago—that's hardly a controversial statement.
Trans-exclusionary radical feminism (as opposed to trans-inclusionary radical feminism) is one strand of feminist practice, but one strand only. Other types of feminism include, but are not limited to analytic feminism, liberal feminism, postcolonial feminism, black feminism, postmodern feminism, psychoanalytic feminism, anarcho-feminism, Marxist feminism, ecofeminism, intersectional feminism, and libertarian feminism. All of these categories contain multiple viewpoints and messages, and many overlap in significant ways. Some are fundamentally opposed to each other, and some share many of the same goals and practices. Feminisms are ever changing, living philosophies and movements.
To be honest, I think the answer to the supposed conflict lies in the broader cultural popularity of infotainment and the twenty-four-hour news cycle. As with any other story, drama and conflict sells.
What are we to do, those of us who want to educate, to learn, and to reach out across this divide, but don't want to feed the beast of outrage and ratings and perpetual argument?
I would suggest that we remember that feminisms are more varied, and more valuable, than this one harmful subset. We don't have to play a losing game, or sink to our opponents' levels. Feminism can be for everyone.
AMERICAN LEGAL SCHOLAR and activist Kimberlé Crenshaw put a name to a different way of doing feminism in her writings of the late 1980s and early 1990s: intersectionality. The first part of the definition given in her 1991 essay, "Mapping the Margins: Intersectionality, Identity Politics, and Violence Against Women of Color," is one of the most quoted, most easy to understand definitions of this approach available:
Consider an analogy to traffic in an intersection, coming and going in all four directions. Discrimination, like traffic through an intersection, may flow in one direction, and it may flow in another. If an accident happens in an intersection, it can be caused by cars travelling from any number of directions and, sometimes, from all of them. Similarly, if a black woman is harmed because she is in the intersection, her injury could result from sex discrimination or race discrimination.
But this definition also comes with a warning, an explanation of what can happen when we deny the ways in which we can be caught at the intersections of our lives:
providing legal relief only when black women show that their claims are based on race or on sex is analogous to calling an ambulance for the victim only after the driver responsible for the injuries is identified. But it is not always easy to reconstruct an accident. Sometimes the skid marks and the injuries simply indicate that they occurred simultaneously, frustrating efforts to determine which driver caused the harm. In these cases the tendency seems to be that no driver is held responsible, no treatment is administered, and the involved parties simply get back in the cars and zoom away.
The objections made to the process of intersectional activism—even to the word itself—run from facile to false. Professional feminist shock jock Julie Burchill likened the word to a bowel complaint, and claimed it was similarly full of shit. A popular media tactic is to claim that intersectionality is a "made-up" approach, a trend coined by the twitterverse, heavy on pile-ons and the misuse of difficult language, light on actual feminist practice. According to some, the word itself is too long, the idea that some people have more prejudice to contend with than others is too divisive, and the thought of trying to combat multiple, stacking oppressions at the same time is more than our job's worth. In this reading, "intersectionality" is just a way of trying to make white cis women feel guilty, and naming the whiteness, the cisness of intersectionality's main opponents is identity politics gone mad. "Intersectional" feminism is what happens on university campuses, where the young and idealistic play oppression Olympics, while the real feminists, the old guard, do the actual work. Supposedly.
I'm not so sure. Not sure how a thirty-year-old word to describe a centuries-old process can be described as new or divisive, certainly—but doubly unsure as to why this kind of feminist process is still being fought over by some when it is an everyday practice for many.
A little while ago I participated in a panel discussion on intersectional feminism in the East End of London. The event had been organized by members of the local council, and both audience and speakers were more varied in their backgrounds, their experiences, than can often be the case at feminist events. For an hour and a half my fellow panelists—activists and academics—fielded audience questions, joined in discussion, and actively listened to each other. Our differences did not stifle us, but united us in a shared need for learning, communication, and support. I learned a great deal about the ongoing impacts of British colonialism on experiences of gendered oppression and expression. Several older men opened up about what it was like to be forced into a certain type of masculinity; one young woman asked how she could best support trans people while remaining respectful of her Christian background and love of God. It was not hard, or threatening, to learn from one another's experiences: it was a gift. In considering how we had each benefited from an unfair society, and how we had suffered at its hands, we were not competing to try to find out "who had it worst"—we were sharing how best to make things fairer for all of us. I don't understand what it controversial, or difficult, about that.
BUT I DO understand what it is to be turned away by the labeling of things when that labeling is unexplained and its processes are unclear. When that label is used to mean a mocking reference to Tumblr teenagers and trigger warnings with everything, with activists who'll delight in tearing down the uninitiated, beginning to understand or practice an intersectional approach might seem intimidating, or pointless.
It is also true that so many guides to intersectional feminism available online point more to the end product than the process. They talk about which words to use, and which points to make, without addressing the fact that words change and goalposts shift. "Ten Ways to Make Your Feminism More Intersectional," "Five Feminist Arguments You Didn't Realize Are Cissexist," quick little lists to read on your phone that can certainly help but rarely delve deeper than "do this, but not that." We love to focus on symptoms, but too often leave structures and systems unexamined. Working out how we can do our best by each other is not so simple as just using the most up-to-date terms and, without examining the deepest levels of our thoughts and actions, it can be easy to fall into thinking that intersectionality just means adding a surface layer of "diversity." That all you have to do to make your feminism intersectional is to add one photo of a woman in a wheelchair, add one token piece by a trans woman, share an article from a black feminist, and call it a day.
I worry about my own tendency toward this. I see the way that others fail me, and know I must, despite my best intentions, be failing others. I need more than just an outcome created by others to be copied: I need an underlying philosophy of my own that can hold me to account, and also inspire me to think more, think harder, and act with greater compassion. I need to think of ways in which, by following this philosophy, I can make it easier for others to develop their own.
I was lucky, then, that the simultaneous publication of two books, and my back-to-back reading of them, helped to clarify these murky and conflicting feelings. The contrast between them, in foundation, in process, in outcome, and my reflection on those contrasts, put into words what it is I want to do with my feminism.
Sheila Jeffreys's _Gender Hurts_ and Roxane Gay's _Bad Feminist_ : two books from two anglophone feminist writers, each from an influential mainstream publisher, each one from an academic author—Roxane Gay is a Professor of English at Purdue University, while Jeffreys was a Professor of Political Science at the University of Melbourne. Both books told from a queer perspective: Gay is bisexual, Jeffreys a political lesbian. Two different generations: Jeffreys the older and Gay the younger; and two different experiences of racism and racial privilege: Gay is black, Jeffreys is white. Gay's work is a collection of essays, spanning a huge number of topics, while Jeffreys's work is focused on the sole issue of the supposed threat trans people pose to the world in general, and feminism in particular.
Reading _Gender Hurts: A Feminist Analysis of the Politics of Transgenderism_ was not a pleasant experience for me—nothing like the evening spent inhaling _Bad Feminist—_ but I'm glad that I read both, and in the same day. At each point where Jeffreys's arguments left me feeling excluded, misinterpreted, or blamed, Gay's book offered a deeper, more inclusive alternative. The experience left me with four main tenets—ways of questioning my actions and intentions—guides for what my intersectional feminism can be.
**Plurality is necessary because there is no neutral.**
The introduction alone to _Bad Feminism_ struck me by its breadth: "Feminism (n): Plural." An explicit description of what that means: "We don't all have to believe in the same feminism. Feminism can be pluralistic so long as we respect the different feminisms we carry along with us, so long as we give enough of a damn to try to minimize the fractures among us."
Any of us claiming that our feminism is neutral or total is either ignorant or a liar. Even if we do it from the best of intentions—"this is the _real_ feminism, other people claiming to be feminists are cruel and hateful"—it does damage.
So my feminism must remain mine, and I must take responsibility for it, in all the ways in which my insights are valuable and all the ways in which they are limited. In doing so, I can hope to join my voice to others without drowning them out.
**Failure is necessary in order to learn.**
Many of us carry around an image of the "perfect feminist": someone who never makes the "wrong" choices, always knows what to say, wins every argument, embodies every virtue. Some feminists will take that image so much to heart that they cannot stand to think of the ways in which their real selves fail to measure up.
Gay's solution, in the face of the image of feminist perfection, is to be a "bad" feminist, because perfection doesn't exist, but trying for something is so much better than nothing. In her essays, she gives herself space to fail and its crucial corollary, space to succeed. The interrogating self is interrogated—both the writer and the reader—but with kindness as well as stringent honesty.
The most important thing I've ever learned as a musician, both as a student and as a teacher, is that progress is impossible without failure. The trick is to own your mistakes, to own your responsibility for fixing them—and to never stop learning from the example of others.
It might well be the most important part of my feminist learning as well.
**We must bring our whole, examined selves to the table.**
Gay's book certainly has a structural advantage here: personal essays are traditionally more discursive than a quasi-academic text. In combining theory with the little details and human foibles of her daily life, she shows us a realistic version of what it is to be a feminist. Not someone whose every action, every thought, fits a preapproved template of political righteousness, but a feminist who can tackle the worst problems of our world and still indulge in trashy novels, dubious pop music, drinking too much, and worrying about getting it wrong.
There is not one aspect of life where gendered oppression does not reach, and which cannot benefit from feminist inclusion. If we start preemptively cutting out parts of who we are because they're "not feminist enough," then we've failed before we've begun.
**We cannot afford to leave anyone behind.**
At no point while reading Sheila Jeffreys's work did I have the sensation of being recognized as an equal participant in the feminist movement. Jeffreys talks about the "politics of transgenderism" rather than trans lives and trans communities. She asserts that people like me are "created by forces of male power" through "mutilation"—a reading which is totally at odds with all I know of the trans experience—and seems uninterested in hearing our side of the story. Throughout _Gender Hurts_ , I had the sensation of being seen as a problem to be solved, and as a roadblock in the way of true feminist justice.
I was thankful, then, to come to Gay's words:
Women of color, queer women, and transgender women need to be better included in the feminist project. Women from these groups have been shamefully abandoned by Capital-F Feminism, time and again. This is a hard, painful truth.... For years, I decided feminism was not for me as a black woman, as a woman who has been queer identified at varying points in her life, because feminism has, historically, been far more invested in improving the lives of heterosexual white women to the detriment of all others.
And after that reminder, instead of doing down the existence of others in order to prop up her own experiences, Gay writes from the center out, enriching the understanding of others through the specifics of her own life. Her teaching, her fiction, her experiences of love and hatred, desire, boredom, fear, hope: her narrative is messier and more complicated than Jeffreys's, forcing the reader to acknowledge her full humanity—and, from that, the full humanity of others.
Life is seldom perfect, and everyone knows the sometime necessity of a compromise. But if we accept the necessity—the desirability—of offering up the lives of others to improve our own, then we have already lost.
MY FEMINISM MUST be intersectional if it is to contain the realities of my trans life, and the lives of all other trans people—trans people whose experiences are often radically different from mine. But it is not only a need for inclusion, and for protection, but a question of feminism's need for all that can be learned of gender oppression, and of gender itself, in all of its many forms.
Would you like to know the subtle differences between being seen as an effeminate white man and an androgynous white woman? Would you like to know what those differences can do to a person, to their safety, to their comfort, to their ability to move through the world? Ask me—because the world has shown me. Is it only men who enforce gendered norms and enact gendered violence? In my experience it is not, and that experience can be a valuable tool, if we want it to be. If feminism is to help everyone, it has to learn from everyone. The more we learn about the intricacies, overlaps, and contrasts of our experiences, the more we can dismantle the totality of gendered oppression.
These extraordinary elements of trans people's collective knowledge—what it is to be the same person perceived in different gendered ways, what it costs to challenge the idea that sex and gender are fixed at birth by outsiders, what it is to exist outside the gendered system we were taught was universal—how can feminism continue without them? Without the insights of trans women who experience that particular form of oppression, transmisogyny, where they are punished for their womanhood, and punished again for the fact that it is devalued and disbelieved? Without the insights of trans men who, depending on country, race, class, bodies, perceptions, lose and gain any combination of benefits upon transition? Without the accumulated knowledge of those, like me, living outside a gender binary: in the streets, in our homes, in employment, in the social system, in our own minds?
All of this experience, this work, enriches not only our understanding, but gives us all a better skillset with which to fight against the systems that keep us down.
This is the point where, in my experience, some people will say, "Why feminism at all?" "Doesn't feminism mean female?" "If you're going to broaden the category, why not call yourself a humanist instead?" For me, that response is yet another of the many reasons why feminism needs trans people. Gendered oppression is so much more than a dualistic fight between women and men, and its cruelties extend beyond one form of discrimination against one group of people. This labeling of the world into things for men, and things for women—good things, bad things—twists what is designated "womanly," "feminine," and uses it to punish nearly all of us. Some are hurt far worse than others—but very few make it out unscathed.
We cannot untangle this Gordian knot by pulling on one thread only. We have to stop pretending that, despite acting together, our resources are scarce and our empathy and talents limited. We do not need to pursue only one goal at a time, help only one type of person. The sheer fact of our diversity means we can be diverse in what it is that we need, and what we can do.
Instead of narrowing down our feminist needs and our desires, we could open them up to everyone who has ever had a need to be free of gendered oppression, gender coercion, gendered limitations.
To do this, I need to ask more, expect more, demand more and better. And I need to show myself willing to hear and act on the same demands in turn.
#
# Futures
T **he last hundred years have seen the most extraordinary** changes in how society thinks of sex and gender, and trans people have been at the front and center of these changes. Transition-related medicine has expanded from one sole dedicated gender clinic to a worldwide network of physicians, research facilities, and cutting-edge treatments. Trans people in the 1910s were petitioning those in government to plead their cases for them; with the 1999 election of New Zealand MP Georgina Beyer we began to change politics from within. We have gone from being persecuted by the law to, in some places, being protected by it. Trans people have broken ground as scientists, musicians, academics, authors, actors, doctors, parents, charity workers, campaigners—every kind of career and vocation.
And so it's not surprising that _Time_ magazine would state that we have come to the "trans tipping point." Nor is it surprising that people who aren't trans are eager to talk about how much better things are for people like me—how different, certainly. This sentiment can be presented as something good or something bad, but it is treated as a fact. Sometimes it is in the somewhat hopeful, somewhat fearful predictions of people who care, people whose desire for everything to be equal and sunny makes them determined to see the progress they need to exist. Sometimes it's in the ranting of political pundits, lamenting a world gone mad, a disruption of the natural gendered order. It's the message that much of the mainstream media is invested in recycling, citing its own pronouncements as proof that we exist in a moment of unparalleled trans tolerance and visibility.
The trans people I know are not quite so unified in their opinion. Some, yes, who are happy to broadcast their triumphs, and complain that the real barrier to true trans equality is other trans people, a community tendency to infighting, and a joy in dragging each other down. Others, often older, often reliant on social services, are fearful. They believe that all the tipping point has brought us is the kind of visibility that paints a target on our backs in a time of right-wing backlash and economic instability. Many have marked the difference between the money made by cis writers, producers, and editors milking this current trend for trans topics, and the ways in which trans people are pressured by the media into offering up our own lives as "content"—without recompense. We have never before had so many trans faces in the news. We have never before had such records of overwhelming violence toward trans people.
I try to catch a firm hold of the direction in which we're moving and find myself unable to do so. It depends on the day, on the latest breaking story. Sometimes it comes on a totally personal level: whether a stranger has called me "they" without needing to be asked or whether I've been catcalled and mocked all the way home. Other times it's in our shifting web of trans communities: a friend beaten up at a bus stop, an acquaintance lost to suicide, a colleague finally being paid and appreciated for their advocacy. I was tearful because of the promise made by First Minister of Scotland Nicola Sturgeon, that her party would pursue legal recognition and protection for people like me. Then tearful again because of yet another description of a trans woman's body beaten and abused beyond recognition, her very existence a threat to be exterminated. Something is happening—something powerful and exhilarating and frequently terrifying—but it is an edge state: I couldn't tell you upon which side we will fall. Trying to balance on that edge is dangerous, exhausting, a constant wash of adrenaline.
The progress, when it comes, is astounding. I see the kind of changes happening now that I imagined, as a teenager, I wouldn't know until the end of my life. Not just in the advances made by trans people, but in the underlying ways in which we consider sex and gender. There is so much to say that, in fact, I am overwhelmed in trying to decide where to start, what to include, and in which order.
Maybe I should begin with the ways in which trans people are finally being recognized in our legitimacies, our talents, and our authorities. The University of Victoria has endowed a Chair in Transgender Studies, Dr. Aaron Devor, with the aim of linking academic and community-based scholarship, integrating Transgender Studies into a broader curriculum, and maintaining an archive of transgender history. Trans people have long been the subject of academic investigation; increasingly, that scholarship is the product of fellow trans people—a community researching itself—rather than outsiders peering in. The White House had its first trans LGBT Liaison, Raffi Freedman-Gurspan; incredibly, she was the fifth trans appointment of the Obama administration. The total number of trans politicians in the world is on the rise. There are so many, I have to pick and choose whose names to report: Emily Brothers in the UK, Geraldine Roman in the Philippines, Tamara Adrián in Venezuela, Kim Coco Iwamoto in Hawaii, Madhu Kinnar in India, Michelle Suárez Bértora in Uruguay, Petra De Sutter in Belgium, Estefania Cortes-Vargas in Canada.
There are an increasing number of trans people playing the media at their own game and winning. Janet Mock's second book, _Surpassing Certainty_ , was published in 2017; far from following the "sell your coming-out story and then disappear" media arc all too familiar to trans people, Mock's career as a journalist, presenter, and activist grows from strength to strength. Rather than appear as the subject of cis investigation, Mock has established herself as an authoritative investigative voice; her worth lies in what she can do, rather than in how she can be used by others. She is not the only one: the number of trans journalists, trans filmmakers, trans writers and artists of all kinds is growing, and, increasingly, our work is hitting the mainstream. Some are, it's true, pulling the ladder up behind them, but most—whether it's the Wachowski sisters or Laverne Cox—are using their platforms to lift up other trans people and to amplify other voices. Cox knows how to walk a red carpet with the best of them; she also knows how to apply her skills and her fame in the service of justice for other trans women of color. Musicians like Rae Spoon and Ryan Cassata combine art with advocacy, turning their experiences as trans people into fuel for their music, and reaching out to their own communities to find a fanbase. Instead of having to hide who we are for the slim chance of career advancement, it is increasingly possible, though still difficult, to find success as an openly trans person, and in so doing to chip away at the barriers holding others back.
The younger generation—Generation Z, as they have been dubbed by marketers and PR firms—frequently demonstrate a trans-positive take on gender, and have the cultural and economic capital to make their elders sit up and take notice. A 2016 study from the US found that 56 percent of thirteen- to twenty-year-olds know a person who uses gender-neutral pronouns. Seventy percent are in favor of gender-neutral bathrooms, and less than half identify as heterosexual. Blurring, disrupting, and rejecting traditional gendered models and standards of being and behavior: far from being the unconsidered trend some older commentators have claimed, at its best this move is part of a broader sense of millennial disenfranchisement with the world we've inherited, and a desire to create something more open and more honest. This championing of gender plurality frequently shows a deep knowledge of how gendered oppression is linked to all other kinds of hatred and bigotry, even as it's expressed through webcomics, emojis, and self-referential memes. What is most incredible to me is how unashamed these young people are; I have to admit to moments of envy, thinking how I could never have been so fierce and so unapologetic in my teenaged gender rebellion, no matter how hard I tried. This isn't something confined to subcultures and corners; this showcasing of gender diversity is something taken up by young celebrities, celebrated in selfies and blog posts shared by hundreds of thousands, sometimes millions of people. Seventeen-year-old actor Amandla Stenberg came out as non-binary on her Tumblr page, not as something shocking that needed the intervention of a publicist and a lengthy interview explanation, but as an unabashed part of herself to discuss freely with others like her. The phenomenon that is gender-fluid model Ruby Rose has had people of all genders and sexualities questioning what it means to be attracted to androgyny; her popularity has editors reaching for the word "genderqueer" when they would never have admitted its use before.
It doesn't escape my notice that I'm more likely to find a sensitively written and deeply researched piece on trans lives and trans feminism in _Teen Vogue_ or _Seventeen_ magazine than I am in the pages of more established and esteemed publications. These two magazines have published pieces on eating disorders in trans teenagers, detailed the prohibitive cost of medical transition, provided guidance on how best to talk to trans friends, and dismantled the myth of the trans bathroom predator. It certainly seems as though the younger generation is leading the way when it comes to compassion and acceptance. Some adult group or service will turn away potential trans members or customers, while the Girl Scouts of the USA refused a $100,000 donation rather than compromise their inclusion of trans scouts. After a Wisconsin school canceled a reading of young trans activist Jazz Jennings's book, more than 600 people, mostly families with kids, gathered at a local library to show their support for trans children and teenagers. At the LGBTI conferences and meetings I attend, it is more and more common for trans teenagers to be present, usually with their parents and siblings in tow. Like many people, I hang more than a few of my hopes on the open-mindedness of my generation, of the generation that follows.
It would be dangerous, though, to think trans liberation a done deal because of that. In bleaker moments, I have heard fellow activists suggest that the best we can hope for is a future in which the most hateful have died of old age, where this young generation have made good on their promise. In my bleakest moments, I worry that they're right. But that would be a denial not only of the other changes happening around us, but of our responsibility to agitate for those changes, encourage their developments, and guard against pushback. Slower to arrive and far more compromised—but change is coming even in the laws and strictures governing what it is we understand as legal gender categories. The International Olympic Committee has just officially made it easier for trans athletes to compete as their true genders, a move that affects not only the world of sport, but which strikes another blow against the belief that trans people are biological impossibilities, always to be classified as the sex we were assigned at birth. Globally, the recognition that trans people are who we say we are is growing. At the end of 2015 the Vietnamese National Assembly approved a bill to legalize transition-related surgery, and introduce the ability for trans people to be legally recognized. The Swedish government announced that it would be paying compensation to the trans people who underwent enforced sterilization as a prerequisite for recognition as their true genders. Canadian prime minister Justin Trudeau has promised full equality for trans people, and seems to be working hard to achieve it.
Much of the credit for this increased rate of change must go to the incredible use of technology by trans activists. Here again I see hope for the future. I cannot imagine a world in which I could have achieved even a quarter of what I've done so far without the use of the internet. Without online information, I wouldn't have learned about the existence of LGBT school organizations—"gay-straight alliances" in the terminology of the time—and certainly wouldn't have had the resources with which to found one. Through online forums I found out about other teenagers like me and, together, we founded the first nationwide LGBT group run by and for young people, beginning as an entirely online enterprise. I found friends, support, inspiration, legal advice, a community. Contrary to everything I had seen on television, in the newspapers, I found trans people of every kind struggling, succeeding, thriving. I found my surgeon online; I found a trans artist skilled in body modification. When I could find no "straight" manager to promote my music, I discovered I already had an online audience who made my career possible. From being alone, we enter into relationships with thousands of other people, sharing the abuses we suffer, but also our victories, working together to make good on both.
The online revolution has allowed trans people to bypass entirely the cis bigots who would deny us a chance at life. As I write this I'm sorting through the most recent fundraising campaigns I've been sent: a campaign to fund a trans modeling agency in Delhi, another to fund a documentary on Slovenian trans culture. The groundbreaking, award-winning film _Tangerine_ was filmed on an iPhone. Crucially for a community often denied health care and forced into self-medicating, there are online health guides, forums in which to share information: which doctors are the most accepting, how best to get around a broken system, how best to look after a body denied care by the supposed experts. The ability to communicate transparently and in real time is breaking down the ways in which we have been silenced through isolation and through the intimidation of being outnumbered. Every trans conference I have ever been to has been livetweeted for the benefit of those who can't attend in the flesh. Every political action has been reported on from the ground by those involved, broadcasting to a wider network watching at home who stand ready to protest any abuse, and to contact emergency legal aid and support groups.
More than just the change to the practical elements of our shared lives and communities is the change technology has made to the ways in which we interact with each other and see ourselves. The cost—online abuse, doxing, sustained harassment—is high. But, even with that, the impact is tremendous. I can, if I choose, spend a day speaking with other people, exchanging ideas and sharing the things I love, the things that make me _me_ , without ever detailing my gender, the categories into which I am placed. The next day I might choose to tell other people a little about my gendered experiences, but not give them the tools with which to discredit me. Not always, but most of the time, I can find places to be myself and have that self automatically treated as legitimate. That casual allowance of my authenticity is a game changer. Once you get used to it, you never want to go back. I struggled for so long to get the world to accept my correct pronouns: it was only after I had come to expect them, through constant respectful usage online, through trans circles, that I had not just the courage but the expectation to insist on accuracy everywhere. If there is a place where I am treated as truthful, real, desirable even, then I can begin to imagine that that might be true of other places—perhaps of all places. To build a better future we must first be able to imagine it; for many trans people, the internet serves both as our testing ground and as a possibility model of what it would feel like to be so thoroughly accepted as our own selves.
The further we go down the tech rabbit hole, the more we are forced to question our own gendered assumptions—not just trans people, but all of us. Many are accepting the challenge of change, and considering how much further we could go. The improvements to virtual reality technologies are growing exponentially: any appraisal of the current field written here will be out of date the minute I sign off on the text. Still, what we have now leaves me excited, and hopeful. A research project at Universitat Pompeu Fabra in Barcelona has been using the Oculus Rift to allow subjects to "see" themselves in a differently sexed body. Entitled "The Machine to Be Another," the project creates a deliberate disjunct between the body known through proprioception and the body seen through the eyes. Subjects have reported feelings of extreme disorientation, a growing awareness of one facet of what trans people experience through dysphoria. I have no doubt that this project will be the first of many and that, no matter how each new technological innovation is presented, there will be those eager to explore each new potential to push open our understanding of empathy, of felt experience, and of what it is to be human. The advent of Artificial Intelligence, too, is no sci-fi scenario; when it comes, will we be prepared to expand what it is that we think we know of humanity, selfhood, and the soul?
From the technological to the medical, and the blurring of these lines; again, the changes multiply at an extraordinary rate. Much of what is reported is exaggerated and distorted through a sensationalistic lens but the innovations in medical technology and research are still staggering. For many trans people, people who experience a burning need to reform their physical bodies, these innovations offer a level of hope impossible to imagine a decade ago. Lili Elbe died in pursuit of a uterine transplant and the possibility of bearing children; 2014 saw the first live birth of a baby born to a mother with a transplanted womb. So far, these transplants have been limited to cis women, but I do not doubt that trials in trans women will follow, though we will have to fight to get there. The news of the growth of rudimentary organs from stem cells was greeted with excitement by many in the trans community; we are a long way from the technologies we would desire, but far closer than we have ever been before. It is not only what these kind of medical treatments could do for trans people and our bodies, but the ways in which these interventions challenge cultural ideas of what is "natural" and what is helpful. As what was previously seen as impossible and outrageous becomes more normal, it becomes harder to see the changes wrought by transition-related medical care as extreme and unnatural, rather than life-saving and ethical. It is so easy, from a position of relative safety, to consider these possibilities, to experience these improvements, and in so doing feel that everything is getting better. God knows I feel that pull, particularly when I feel hopeless after one personal struggle or another. Even when we acknowledge the violence, the systemic oppression, experienced by so many, there is the temptation to brush it aside as belonging to the old order of things—an unfortunate relic of another time. When it comes to trans rights, and trans people, I have heard the phrase "on the wrong side of history" used again and again to describe the people, and the acts, which are rooted in transphobic hatred and fear. They're out of date, out of touch, and running out of time.
But that doesn't stop them from hurting us in the here and now. Worse, it allows those who should be helping, who could be doing something, anything, to protect the victims of these violences to turn their backs and say, "It's just not relevant anymore—it'll all be over soon."
Just as modern and as current as new legislation designed to protect and recognize trans people are the still in place, still-deadly colonial laws against LGBT people and acts found throughout the countries touched by the forces of European imperialism. These laws may have been passed in the Victorian era, but are still enforced now, and show every sign of being enforced in the future. The UN notes that private same-sex relationships are criminalized in at least seventy-six countries worldwide. Not all trans people are gay or bisexual, but we are nearly always seen as being queer, and make the most obvious targets for the enforcers of these laws. Elsewhere, new laws are being brought in specifically to target LGBT people, a reflection of broader cultural prejudices and a potent accelerant to existing hatreds; recent anti-LGBT legislative changes in Russia have led to an increase in violence against LGBT people, including a rise in vigilante groups and mob retribution. Sometimes it is the police who target trans people: trans women in Malaysia imprisoned, fined, their heads forcibly shaved for the crime of "posing as women." Sometimes it is civilian violence: in January 2016 alone, at least forty-eight Brazilian trans women were murdered. Progress is not inevitable, and any advance can be turned back. An Indonesian Islamic academy for _waria_ (loosely analogous to "trans women"), which had enjoyed great success since opening in 2008, was shut down in 2016 as the result of a sudden and wide-reaching anti-LGBT backlash.
Even in countries where trans people enjoy some protection under law, where we pride ourselves on our "tolerance," trans people are still suffering. The current rise of austerity politics, political fearmongering, and scapegoating have hit trans people hard. The current political chaos, economic downturn, and upswing in xenophobic and racist violence in post-Brexit Britain is hitting us even harder. Even worse is the situation for American trans people: the rise of the so-called Alt Right, the open and violent displays of white supremacy and neo-Nazism growing under the Trump presidency, and a leadership openly committed to the persecution of minority groups. Politicians appealing to the worst and most hateful parts of human nature are lauded as "hard-hitting," and abusing trans people is an easy way of demonstrating a "truth-talking," "un-PC" attitude. Despite his campaign promises to the contrary, Donald Trump found it useful to announce, on Twitter, a shock decision to purge the military of trans service members. With the Republican party committed to repealing Obamacare, to decimating public assistance, trans people—so often un- and underemployed, homeless, vulnerable—have good reason to fear. In the UK, cuts to local services are putting the most marginalized trans people at risk. In the past year, we have seen the closure of countless community groups that have kept trans people in crisis alive. We have nothing to replace them. Trans people on the streets, trans victims of domestic violence, trans people with mental illnesses, trans children and teenagers: groups like PACE and the Mosaic Youth Centre were the safety net for those who had none. With funding cut and budgets slashed, that safety net is gone, and will not return under this government. The treatment of LGBT asylum seekers should shame us all: desperate people treated like liars, victims of torture told to return to countries where their very existence is a crime. At the time of writing, Theresa May—a politician with an appalling track record on both LGBT rights and the rights of refugees and migrants—is prime minister. In times of national upheaval, "deviants" of all kinds are useful targets for blame. Trans people—particularly trans people of color, sex workers, and disabled trans people—have always made for easy targets.
These are just some of the ways in which trans people are hurting, but these stories do not fit a media narrative of progress, hope, and change, and are seldom reported. We are fed the constant message of "visibility"; that having trans reality TV stars and trans storylines in soap operas is what counts, the word "gender-fluid" used in an ad campaign or a trans model in a special feature in a weekend supplement. It is only in trans activist circles that I have seen this point made: that the trans people most likely to be attacked are already highly visible, and rates of violence against these highly visible people—usually people of color, usually women, usually sex workers—show no sign of falling. Nearly every gain I could quote for you has its flipside. That backlash is not distributed equally, and those who already suffer the most are inevitably those who are made to suffer more.
When we assume that progress is a done deal, we are turning our backs on those who have experienced no progress at all. When we assume that the arc of history will bend toward justice, we stop working to make it so.
The question I am asked again and again as an activist is: "What one change would make things better for trans people?" Therein lies the problem. "Trans" is not a singular quality that can be divorced from the pluralistic lives of the people referred to by that term. Gender diversity, and the need to define one's own gendered place in the world, is a trait found in every human society, every culture, every time and place that I know of. "Trans" is not a noun but an adjective and a verb, and a shifting, ever-changing one at that. I use the word "trans" as the broadest and most inclusive term I know of with which to describe the incredible number of ways in which people do gender, and even that is not enough. "Trans" is a common descriptor worldwide, but there are many specific genders and gendered traditions throughout the world that cannot be parsed by this term, and which lie beyond the scope of this work. And yet these people, too, deserve legal protections, full recognition, and freedom from oppression.
I could no more answer the question "What one thing do trans people need?" than I could "What one thing do people need?"
Even when I look to myself I know that it will take more than one change to make my life safer and more secure. The current UK Equalities Act makes no provision for people like me. And, yet, that legal protection is nothing without the cultural will to enforce those laws, something any trans woman or trans man could tell me. I wish I had legal documents that accurately recorded who I am, but if that recognition comes at the cost of creating a central registry of trans people then it may well do more harm than good. Better employment laws, better diversity training, a more rigorous desire to promote LGBT equality in education: these are all moves that would have an immediate positive impact on my life. For other trans people, the far more pressing concern is the decriminalization of sex work: what good is a more inclusive employment law when your work is still liable to result in arrest, police abuse, and harassment from social workers? Trans-positive health care and trans-positive schooling have the potential to help all trans people, but only if they can be freely provided. If I focus my activism upon my own needs I remain ignorant of all the areas in which I am considered "neutral," which is to say societally favored.
It is, perhaps, a common tendency to dream of utopias. It's certainly something that drove me into activism and into a desire to write that activism down. When I was much younger I wanted, like many idealists, to create a manifesto: a document both clear and concise, and, of course, universal in its application. With hindsight, it is clear that there can be no such manifesto of trans rights, of trans justice—unless it were to be one without an end, in which anyone could write. At what point would we be done? When trans people no longer have to undergo sterilization to be legally recognized? Until medical gatekeeping is replaced with the process of informed consent? Is it enough to insist on the right of gender non-conforming children to be treated with kindness and respect, or do we fight to change the entire process of coercively assigning sex and gendered labels and expectations at birth? Health care, education, the prison system, the justice system, borders and immigration, cultural mores: in every part of society trans people are suffering, and they are suffering in multiple ways. How can we claim to be a community and then insist that it is necessary to leave any one of us behind?
I was talking about this sense of living on the edge, of experiencing backlash, and a colleague stopped me. "For trans people of color," they said, "it has always been the backlash." With that, they spun my vision around. Is what we're experiencing now any more or any less vital than the experiences of those at the Compton's Cafeteria riots, or at Stonewall? Is our "tipping point" greater than all of those tipping points experienced by Magnus Hirschfeld and his trans patients, colleagues, and friends? Those may be the public moments of societal debate and obvious change, but what of the millions of personal battles, individual lives poised always between danger and freedom? Each daily struggle to be safe, to be known, to survive, and to ensure safety and survival in the future?
The problem with our trans tipping point is that we think we know which direction it will take us. We assume that our future is already written, and that it must inevitably get better; that the wheels are already in motion, and no longer need fuel to run.
What if, instead, we took responsibility for the danger and the power of these constant edge states? What if we took this moment of public interest and used it to further a future that would benefit us all? If we each took the risk, what could we, collectively, win?
What bears me up, on those days when the backlash and the danger are too much to endure, is the thought of what we might achieve if we can only keep going. I don't believe we can see our final destination from where we are, but our hopes for what it may be are crucial in creating it.
I am working for a future where the fact that some people will make changes to their sexed bodies will be an understandable, and unremarkable, aspect of life, stripped of stigma and shame. Where every child is recognized, valued, and loved for their own unique selves. We already have spaces in which we see and respect each other without gendered assumptions and judgments—why shouldn't we expect that everywhere? There is nothing natural or inevitable about the gendered injustices that plague our society—so why should we not imagine a future free of them? One in which our possibilities are limitless and our differences the links in a chain that join us in a common humanity.
If it is truly every movement's goal to make itself obsolete, then I am working for a truly equal future that has no need for the trans movement, having made good on its dreams.
My future might look different from yours—and none of us can predict the exact details of what will come. But I trust that with compassion, empathy, the deep respect for what is unique and special in each of us—the core lesson of trans experience—that it is a future we could all be proud of creating.
We just have to find the courage to try.
# _Acknowledgments_
**My deepest thanks, first and foremost, to every trans person** who has helped me through the years and ways leading toward the writing of this book. Thank you for the validation, the laughter, the sense of hope and belonging—and thank you also for the challenges, discussions, and the constant push to do better by each other. They are invaluable gifts.
My thanks and gratitude to the team at Virago, particularly Lennie Goodings and Ailah Ahmed, for believing in this work and this author, and for all of their advice, guidance, and support. It means a great deal.
Heartfelt thanks to Stephanie Knapp and all at Seal Press, for your insights and your support in bringing this work to a wider audience.
To my agent Laura Macdougall, deepest thanks—for your vision, your clarity, and the incredible support. Thank you also to the whole of Tibor Jones, and to Kaite Welsh, who started the ball rolling.
I am grateful to Ruth and Ben, not only for their feedback on the manuscript, but also for their inspirational work in academia and activism. Any mistakes are, of course, my own. To the friends and family who provided sounding boards, feedback, and much-needed moral support: there would be no book without you. Thank you so very much.
And, finally, especial thanks to my partner Sam, for everything.
And to Jonathan—always.
# _About the Author_
**CN Lester is an academic, musician, and leading UK trans** rights activist. A critically acclaimed singer-songwriter, Lester also works as a composer and classical performer; their musicological research focuses on Venetian composer Barbara Strozzi. Lester consults and educates on trans issues with a wide range of organizations, including Channel 4 and the _Huffington Post_ , and has written for the _Independent_ , _New Internationalist_ , and _New Statesman_.
# _Further Resources_
**O RGANIZATIONS AND CAMPAIGNS**
_**International**_
**ILGA—International lesbian, gay, bisexual, trans, and intersex association**
http://ilga.org/
Global umbrella organization for 1,200 national and international organizations working for LGBTI rights. Research, resources, conferences, and campaigning.
**TGEU—Transgender Europe**
http://tgeu.org/
Pan-European campaigning and research organization. Extensive resources for trans people living throughout Europe, and links with international groups and campaigns.
**United Nations Free & Equal Campaign**
https://www.unfe.org/
Public education campaign to promote global LGBT equality. Fact sheets, films, and outreach.
_**UK**_
**Gendered Intelligence**
http://genderedintelligence.co.uk/
Supporting trans youth and their families, friends, and teachers. Resources, research, youth groups, educational outreach, and arts projects.
**Scottish Transgender Alliance**
http://www.scottishtrans.org/
Scottish-based organization campaigning for equality for all gender-diverse people. Research, resources, outreach, and support.
_**US**_
**National Center for Transgender Equality**
http://www.transequality.org/
Campaigning and support.
**Transgender Law Center**
https://transgenderlawcenter.org/
Legal advice, advocacy, and the pursuit of equality through legal challenges.
**Trans Lifeline**
http://www.translifeline.org/
Hotline staffed by trans people, to help trans people.
**B OOKS**
**_Gender Outlaws: The Next Generation_**
Kate Bornstein and S. Bear Bergman
Essays, comics, conversation, and commentary from the new trans generation.
**_One in Every Crowd_**
Ivan Coyote
Anthology of stories for young LGBT people.
**_Redefining Realness_**
Janet Mock
Bestselling memoir from leading US trans activist and journalist.
**_Stone Butch Blues_**
Leslie Feinberg
Classic novel about growing up different in a binary gendered world.
**_Trans Bodies, Trans Selves_**
Edited by Laura Erickson-Schroth
A life guide for trans people: health, relationships, family, employment, and more.
**F ILM AND TV**
**_Her Story_**
http://www.herstoryshow.com/
Emmy-nominated web series follow the lives of two trans women and their colleagues, lovers, and friends.
**_My Genderation_**
http://www.mygenderation.com/
Short films showcasing trans life in the UK.
**_Paris Is Burning_**
Acclaimed documentary chronicling the ball culture of New York's black and Latinx trans and queer communities.
# _Endnotes_
**1. T HE PRODUCTION OF IGNORANCE**
"Uncle Sam's Insane Push for Transgender Rights in School Locker Rooms": Editorial, _New York Post_ , November 3, 2015, http://nypost.com/2015/11/03/uncle-sams-insane-push-for-transgender-rights-in-school-locker-rooms/.
"This Australian Newspaper's Transphobic Headline Breached Press Council Standards": Lane Sainty, _BuzzFeed,_ August 4, 2015, https://www.buzzfeed.com/lanesainty/the-courier-mails-trans phobic-headline-breached-press-counci?utm_term=.ofrE3G390#.toW90d0kg.
_Fox News_ , July 2017: "Pediatrician: Transgender Ideology Causing Child Abuse," http://video.foxnews.com/v/5524002302001/?#sp =show-clips.
"Transgender Surgery Isn't the Solution": Paul McHugh, _Wall Street Journal_ , June 12, 2014; updated May 13, 2016, https://www.wsj.com/articles/paul-mchugh-transgender-surgery-isnt-the-solution-1402615120.
the _Sun'_ s 2011: "Tran or woman?" https://www.thesun.co.uk/archives/news/393820/ tran-or-woman/.
Turned Him into a "Monster": Damien Gayle, "Transsexual, 44, Elects to Die by Euthanasia After Botched Sex-Change Operation Turned Him into a 'Monster,'" _The Daily Mail,_ October 1, 2013, http://www.dailymail.co.uk/news/article-2440086/Belgian-transsexual-Nathan-Verhelst-44-elects-die-euthanasia-botched-sex-change-operation.html.
"The Gender Bender": "The Influentials," _CNN,_ 2014, http://edition.cnn.com/interactive/2014/12/entertainment/cnn10-most-influential/.
"character assassination" of primary schoolteacher Lucy Meadows: Michael Singleton, quoted in Helen Pidd, "Lucy Meadows Coroner Tells Press: 'Shame on You,'" _The Guardian_ , May 28, 2013, https://www.theguardian.com/uk/2013/may/28/lucy-meadows-coroner-press-shame.
before she had transitioned: Richard Littlejohn, "He's Not Only in the Wrong Body... He's in the Wrong Job," _Daily Mail_ , 2012, reposted on The Blog of Robin, https://robinwinslow.uk/2013/03/23/hes-not-only-in-the-wrong-body-repost/.
"shame on all of you": Pidd, "Lucy Meadows Coroner Tells Press: 'Shame on You.'"
toward genital surgery: "Channel 4's Obsession with Genitalia and Surgery Demeans Trans People," October 14, 2015, _The Guardian_ , https://www.theguardian.com/commentisfree/2015/oct/14/channel-4-genitalia-surgery-trans-people-girls-to-men.
being denied an answer: Joseph Patrick McCormick, "Trans Actress Laverne Cox: 'A Preoccupation with Transition and Surgery Objectifies Trans People," _Pink News_ , January 9, 2014, http://www.pinknews.co.uk/2014/01/09/trans-actress-laverne-cox-a-preoccupation-with-transition-and-surgery-objectifies-trans-people/.
"dicks in chicks' clothing": Jack Mirkinson, "Julie Burchill Transgender Column Causes Firestorm for the Observer," _Huffington Post_ , January 14, 2013, http://www.huffingtonpost.co.uk/entry/julie-burchill-transgender-column-transphobia-observer_n_2474843.
Shakespeare, Austen, and Swift: R. L. G. "Johnson: Singular They." Prospero blog, _The Economist_ , February 19, 2014, https://www.economist.com/blogs/prospero/2014/02/pronouns.
"Deer Spears Sex-Swap Kate": Joseph Patrick McCormick, "Stag Attack Victim Dr. Kate Stone Celebrates Removal of Trans References from Newspaper Articles," _Pink News_ , May 12, 2014, http://www.pinknews.co.uk/2014/05/12/stag-attack-victim-kate-stone-celebrates-removal-of-trans-references-from-newspaper-articles.
"Was Kate gored by a stag because she was transgender?": Yvonne Roberts, "Scientist Kate Stone Hails Landmark Press Negotiation over Transgender Reporting," _The Observer,_ May 11, 2014, http://www.theguardian.com/society/2014/may/11/transgender-kate-stone-press-complaints-commission-ruling.
"Transgender Issues Are Driving Me Nuts": Jeremy Clarkson, "Transgender Issues Are Driving Me Nuts. I Need Surgery on My Tick Boxes," _Sunday Times,_ January 24, 2016, https://www.thetimes.co.uk/article/jeremy-clarkson-transgender-issues-are-driving-me-nuts-i-need-surgery-on-my-tick-boxes-jzmz6dldvvj.
Y. Gavriel Ansara and Peter Hegarty: "Cisgenderism in Psychology: Pathologizing and Misgendering Children from 1999 to 2008." _Psychology & Sexuality_ 3 (2012), 137–160, doi: 10.1080/19419899.2011.576696, http://www.apa.org/monitor/2012/09/top-honors.aspx, and http://ansaraonline.com/yahoo_site_admin/assets/docs/Ansara__Hegarty_2012_Cisgenderism_in_Psychology.19690333.pdf.
"Changing Sex Is Not to Be Done Just on a Whim": Melanie McDonagh, _Evening Standard,_ January 5, 2016, https://www.standard.co.uk/comment/comment/melanie-mcdonagh-changing-sex-is-not-to-be-done-just-on-a-whim-a3149031.html.
"odd," "gross," and "freaks": Stephanie Beryl Gazzola and Melanie Ann Morrison, "Cultural and Personally Endorsed Stereotypes of Transgender Men and Transgender Women: Notable Correspondence or Disjunction?" _International Journal of Transgenderism_ 15, no. 2 (2014): 76–99, doi: 10.1080/15532739.2014.937041.
**2. "C ALL ME CAITLYN"**
Why didn't Islan Nettles?: Mey, "What We're Going to Say About Caitlyn Jenner," Autostraddle (website), June 12, 2015, http://www.autostraddle.com/what-were-going-to-say-about-caitlyn-jenner-292957/.
"shy, miserable person": "Christine Jorgensen," Wikipedia, https://en.wikipedia.org/wiki/Christine_Jorgensen.
"Did the surgeon's knife make me a woman or a freak?" Jenni Olson, _The Queer Movie Poster Book_ (New York: Chronicle Books, 2004).
all-out street fight: Susan Stryker, _Transgender History_ (Berkeley, CA: Seal Press, 2008), p. 67.
rape, and murder: Ibid.
stabbing herself to death: Nick Adams, "How TV's _CSI_ Is Screwing Us Again," _The Advocate_ , December 2, 2012, https:// www.advocate.com/commentary/2012/12/02/op-ed-how-tv-show-csi-screwing-us-again.
"transgender people, especially women": s. e. smith, "The Transphobia Is Out There," _BitchMedia_ , February 5, 2016, https://bitchmedia.org/x-files-transphobia-transgender-character-weremonster-reboot.
their body was discovered: Kiran Randhawa, "'Inmates Heard Shouts of 'Help Me' Before Transsexual Prisoner Was Found Dead,'" _Evening Standard,_ February 18, 2015, http://www.standard.co.uk/news/uk/inmates-heard-shouts-of-help-me-before-transexual-prisoner-was-found-dead-10053295.html.
National Offender Management Service Deaths in Custody Database: https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/493975/deaths-in-custody-table.pdf.
**3. F INDING MY VOICE**
in the front as feminine: Sonja K. Foss, Mary E. Domenico, and Karen A. Foss, _Gender Stories: Negotiating Identity in a Binary World_ (Long Grove, IL: Waveland Press, 2013), p. 135.
**4. C OULDN'T YOU JUST... NOT BE?**
real world experience: Ryan F. Mandelbaum, "How American Healthcare Is Failing Transgender Patients," Gizmodo (website), May 31, 2017, http://gizmodo.com/how-american-healthcare-is-failing-transgender-patients-1794384745.
and other service providers: Jaime M. Grant et al., _Injustice at Every Turn: A Report of the National Transgender Discrimination Survey_ (Washington, DC: National Center for Transgender Equality and National Gay and Lesbian Task Force, 2011).
a terrifying combination: Ibid.
"a hard-to-reach goal": European Union Agency for Fundamental Rights, "Being Trans in the European Union: Comparative Analysis of EU LGBT Survey Data," 2014.
the mercy of individual doctors: James Meikle, "NHS Treats Transgender People as Second-Class Citizens, Says Watchdog." _The Guardian,_ May 20, 2015, https://www.theguardian.com/society/2015/may/20/nhs-treats-transgender-people-as-second-class-citizens-says-watchdog.
in India: Yadavendra Singh et al., "Gender Transition Services for Hijras and Other Male-to-Female Transgender People in India: Availability and Barriers to Access and Use," _International Journal of Transgenderism_ 15, no. 1 (2014): 1–15, doi: 10.1080/15532739.2014.890559.
Colombia: Rodrigo A. Aguayo-Romero et al., "Gender Affirmation and Body Modification Among Transgender Persons in Bogotá, Colombia," _International Journal of Transgenderism_ 16, no. 2 (2015): 103–115, doi: 10.1080/15532739.2015.1075930.
the Philippines: Sam Winter, Sass Rogando-Sasot, and Mark King, "Transgendered Women of the Philippines," _International Journal of Transgenderism_ 10, no. 2 (2007): 79–90, doi: 10.1080/15532730802182185.
to make them larger: Katie Sutton, "'We Too Deserve a Place in the Sun': The Politics of Transvestite Identity in Weimar Germany," _German Studies Review_ 35, no. 2 (May 2012): 335–354.
all of these methods: Kelley Winters, "The Gender Gulag: Voices of the Asylum," GID Reform Weblog, November 26, 2008, https://gidreform.wordpress.com/2008/11/26/the-gender-gulag-voices-of-the-asylum./
**5. W HAT ABOUT SEX?**
of even five categories: Anne Fausto-Sterling, "The Five Sexes: Why Male and Female Are Not Enough," _The Sciences_ (March–April 1993), pp. 20–24, http://capone.mtsu.edu/phollowa/5sexes.html.
"of having green eyes": Cary Gabriel Costello, "How Common Is Intersex Status?" The Intersex Roadshow (website), March 12, 2012, http://intersexroadshow.blogspot.co.uk/2012/03/how-common-is-intersex-status.html.
"nor is X that of femininity": Ian Steadman, "Sex Isn't Chromosomes: The Story of a Century of Misconceptions About X & Y," _New Statesman_ , February 25, 2015, http://www.newstatesman.com/future-proof/2015/02/sex-isn-t-chromosomes-story-century-misconceptions-about-x-y.
"So why focus on difference?": Cordelia Fine, _Delusions of Gender: The Real Science Behind Sex Differences_ (London: Icon Books, 2007), p. 165.
XY chromosomes: _Nature_ , 2015: "Sex redefined." http://www.nature.com/news/sex-redefined-1.16943.
"was wholly jarring": Julia Serano, _Whipping Girl: A Transsexual Woman on Sexism and the Scapegoating of Femininity_ (Berkeley, CA: Seal Press, 2007), pp. 80–87.
falling in the middle: Laura Gowing, "Lesbians and Their Like in Early Modern Europe, 1500–1800," in Robert Aldrich, ed., _Gay Life and Culture: A World History_ (London: Thames & Hudson, 2006), p. 129.
gendered behavior must be policed: Kathryn M. Ringrose, "Living in the Shadows: Eunuchs and Gender in Byzantium," in Gilbert Herdt, ed., _Third Sex, Third Gender: Beyond Sexual Dimorphism in Culture and History_ (Brooklyn, NY: Zone Books, 1994).
in a straight line: Graham Robb, _Strangers: Homosexual Love in the Nineteenth Century_ (New York: W. W. Norton, 2003), p. 46.
as recently as 1979: Huma Qureshi, "Passport, Visa, Virginity? A Mother's Tale of Immigration in the 1970s," _The Guardian_ , May 13, 2011, https://www.theguardian.com/lifeandstyle/2011/may/13/virginity-tests-uk-immigrants-1970s.
"this preference has an evolutionary advantage behind it": Coco Masters, "Study: Why Girls Like Pink," _Time_ , August 20, 2007, http://content.time.com/time/health/article/0,8599,1654371,00.html.
(daintier and more delicate): Ben Goldacre, "Out of the Blue and in the Pink," _The Guardian,_ August 25, 2007, https://www.theguardian.com/science/2007/aug/25/genderissues.
**6. T HINK OF THE CHILDREN**
transgender children: Kristina Olson and Lily Durwood, "Are Parents Rushing to Turn Their Boys into Girls?" _Slate_ (website), January 14, 2016, http://www.slate.com/blogs/outward/2016/01/14/what _alarmist_articles_about_transgender_children_get_wrong.html.
to feel as they did: Natacha Kennedy, http://www.academia.edu/2760086/Transgender_Children_More_than_a_Theoretical _Challenge_2012_updated_version_.
"Fix society. Please.": Ed Pilkington, "Ohio Transgender Teen's Suicide Note: 'Fix Society. Please,'" _The Guardian_ , January 5, 2015, https://www.theguardian.com/world/2015/jan/05/sp-leelah-alcorn-transgender-teen-suicide-conversion-therapy.
a form of child abuse: Paris Lees, "Thank You, Obama, for Condemning 'Conversion Therapy' for LGBT Teens," _The Guardian,_ April 10, 2015, https://www.theguardian.com/commentisfree/2015/apr/10/obama-conversion-therapy-lgbt-teens-leelah-alcorn.
the general population: Brynn Tannehill, "The End of the Desistance Myth," _Huffington Post_ , January 1, 2016, updated January 1, 2017, http://www.huffingtonpost.com/brynn-tannehill/the-end-of-the-desistance_b_8903690.html.
without fear of rejection: Colt Keo-Meier, quoted in ibid.
"about delaying puberty or transitioning": Zack Ford, "This Is What Happens to Transgender Kids Who Delay Puberty," _Think Progress_ (website), September 16, 2014, https://thinkprogress.org/this-is-what-happens-to-transgender-kids-who-delay-puberty-1f51ef660e7c/.
across the board: The Endocrine Society, "High Need for Treatment of Transgender Youth," _Science Daily_ (website), March 8, 2015, https:// www.sciencedaily.com/releases/2015/03/150308091402.htm.
another six years to complete: Sara Reardon, "Largest Ever Study of Transgender Teenagers Set to Kick Off," _Nature_ , March 29, 2016, http://www.nature.com/news/largest-ever-study-of-transgender-teenagers-set-to-kick-off-1.19637.
**7. D ELUSIONAL AND DISTURBED**
dress, and act like women: Graham Robb, _Strangers: Homosexual Love in the Nineteenth Century_ (New York: W. W. Norton, 2003), p. 43.
sodium pentothal and testosterone: http://www.liverpoolmuseums.org.uk/mol/exhibitions/april-ashley/early-life.aspx.
**8. A D IFFERENT APPROACH**
trans children without that support: _Impacts of Strong Parental Support for Trans Youth_. A report prepared for the Children's Aid Society of Toronto and for Delisle Youth Services.
group of young people: JoNel Aleccia, "Transgender Kids' Mental Health Boosted by Family Support, UW Study Finds," _The Seattle Times_ , February 25, 2016, https://www.seattletimes.com/seattle-news/health/family-support-boosts-transgender-kids-mental-health-uw-study-finds/.
**10. A RE TRANS PEOPLE REAL?**
"the 'trans' pantomime": Joe Morgan, "Daily Mail's Richard Littlejohn Writes Transphobic Rant 3 Years After Being Blamed for Death of Trans Teacher," _Gay Star News_ , January 15, 2016, http://www.gaystarnews.com/article/richard-littlejohn-transphobic-rant/#gs.AzMF7LU.
has stormed front and center into the mainstream political debate: National Conference of State Legislatures, "'Bathroom Bill' Legislative Tracking," July 28, 2017, http://www.ncsl.org/research/education/-bathroom-bill-legislative-tracking635951130.aspx.
anti-trans bathroom bill in 2016: Garrett Epps, "North Carolina's Bathroom Bill Is a Constitutional Monstrosity," _The Atlantic_ , May 10, 2016, http://www.theatlantic.com/politics/archive/2016/05/hb2-is-a-constitutional-monstrosity/482106/.
bills soon after: Dawn Ennis, "Republican National Committee Endorses Anti-Trans 'Bathroom Bills,'" _The Advocate_ , February 25, 2016, https://www.advocate.com/transgender/2016/2/25/republican-national-committee-endorses-anti-trans-bathroom-bills.
capitulating to popular anti-trans sentiment: "Trump Revokes Guidelines on Transgender Bathrooms in Schools," ABC News, February 24, 2017, http://www.abc.net.au/news/2017-02-23/trump-revokes-guidelines-on-transgender-bathrooms/8295482.
in their own homes: Zack Ford, "Ted Cruz Doesn't Want Any Transgender People in Any Bathroom Anywhere," _Think Progress_ (website), April 26, 2016, https://thinkprogress.org/ted-cruz-doesnt-want-any-transgender-people-in-any-bathroom-anywhere-b6f1a0e3e72e/.
by "perverts and pedophiles": Mark Joseph Stern, "North Carolina School Board Lets Kids Carry Pepper Spray in Case They Encounter Trans Students," _Slate_ (website), May 11, 2016, http://www.slate.com/blogs/outward/2016/05/11/north_carolina_school _board_allows_pepper_spray_against_trans_students.html.
after widespread condemnation: Mark A. Plemmons, "Rowan-Salisbury School Board Decides Not to Allow Pepper Spray at High Schools," _Independent Tribune_ , May 24, 2016, http://www.independenttribune.com/news/rowan-salisbury-school-board-decides-not-to-allow-pepper-spray/article_4af1e660-21af-11e6-aa7c-5318d9219a48.html.
fend off trans people: Sunnivie Brydum, "Right-Wingers Pledge to Carry Guns to Bathroom to Fend Off Trans Folks," _The Advocate_ , April 25, 2016, https://www.advocate.com/transgender/2016/4/25/right-wingers-pledge-carry-guns-bathroom-fend-trans-folks.
that best match our genders: Hadley Malcolm, "More Than 700,000 Pledge to Boycott Target over Transgender Bathroom Policy," _USA Today_ , April 25, 2016, https://www.usatoday.com/story/money/2016/04/25/conservative-christian-group-boycotting-target-transgender-bathroom-policy/83491396/.
after gym class: Charlotte Alter, "Mike Huckabee Joked About Pretending to Be Transgender to Shower with Girls After Gym Class," _Time_ , June 2, 2015, http://time.com/3905462/mike-huckabee-transgender-joke/.
over a decade of incarceration: Mitch Kellaway, "Incarcerated Texas Trans Woman Finally Wins Safer Housing After Repeated Rapes, Threats," _The Advocate_ , March 31, 2015, https://www.advocate.com/politics/transgender/2015/03/31/incarcerated-texas-trans-woman-finally-wins-safer-housing-after-repe.
Vicky Thompson in 2015: "Death of Transgender Woman Vicky Thompson in Male Prison Prompts Calls for Law Change," _The Independent_ , 2015, http://www.independent.co.uk/news/uk/crime/death-of-transgender-woman-vicky-thompson-in-male-prison-prompts-calls-for-law-change-a6742676.html.
these stories pour in from around the world: Cat McShane, "Transgender Offenders Are Being Victimized in Jails Across the World—and Justice Systems Can't Cope," _Vice_ (website), December 21, 2015, https://news.vice.com/article/transgender-offenders-are-being-victimized-in-jails-across-the-world-and-justice-systems-cant-cope.
denied her medication: Rachel Revesz, "Black Transgender Woman Delivers Emotional Interview About Her Abuse in Prison," _The Independent_ , February 12, 2016, http://www.independent.co.uk/news/world/americas/black-transgender-woman-breaks-down-in-interview-over-her-abuse-in-prison-a6870566.html.
was punished for doing so: Matthew Shaer, "The Long, Lonely Road of Chelsea Manning," _The New York Times Magazine_ , June 12, 2017, https://www.nytimes.com/2017/06/12/magazine/the-long-lonely-road-of-chelsea-manning.html.
a chance at medical transition: _Al Jazeera News_ , 2016: "US army whistleblower Chelsea Manning attempted suicide." http://www.aljazeera.com/news/2016/07/lawyers-confirm-chelsea-manning-attempted-suicide-160712055644219.html.
their lived gender: "Transgender Prisoners: System Failure?" _BBC News_ , December 3, 2015, http://www.bbc.co.uk/news/uk-34984249.
"'Oh, my God, that's disgusting'": "Quagmire's Dad," Wikipedia, https://en.wikipedia.org/wiki/Quagmire%27s_Dad.
an undisclosed sum: " _There's Something About Miriam,_ " Wikipedia, https://en.wikipedia.org/wiki/There%27s_ Something_About _Miriam.
Gwen's trans status: Patrick Hoge, "Defense Calls Transgender Victim Guilty of 'Deception and Betrayal,'" _SF Gate_ , April 16, 2004, http://www.sfgate.com/bayarea/article/HAYWARD-Defense-calls-transgender-victim-guilty-2792421.php.
their appalling brutality: Human Rights Watch, "Not Safe at Home: Violence and Discrimination Against LGBT People in Jamaica," October 21, 2014, https://www.hrw.org/report/2014/10/21/not-safe-home/violence-and-discrimination-against-lgbt-people-jamaica.
"They get aggressive and kill the sex worker": Didem Tali, "What I've Learned Trying Not to Get Killed as a Transgender Sex Worker," _Vice_ (website), March 18, 2016, https://www.vice.com/en_us/article/nnkwwk/most-dangerous-place-transgender-europe-turkey-didem-tali.
"against trans people": German Lopez, "How Bad Is the Epidemic of Violence Against Transgender Women? It's Hard to Know," _Vox_ (website), November 20, 2015, http://www.vox.com/2015/10/20/9574239/transgender-murders-epidemic.
And they will be missed: Sunnivie Brydum, "In Wake of Trans Murders, Janet Mock Teaches Us Why We Must #SayHerName," _The Advocate_ , August 25, 2015, https://www.advocate.com/transgender/2015/08/25/watch-janet-mocks-powerful-response-murders-trans-women.
"Woman Avoids Jail": "Woman Who Used Fake Penis to Have Sex with a Woman Avoids Jail," _The Guardian_ , December 15, 2015, https://www.theguardian.com/uk-news/2015/dec/15/woman-who-used-fake-penis-to-have-sex-with-a-woman-avoids-jail.
"Reveals Her Horror": Rob Pattinson, "'The Pain Was Excruciating, I Couldn't Eat or Sleep': Woman Duped by Lesbian with Fake Penis Reveals Her Horror," _The Sun_ , December 15, 2015, http:// www.thesun.co.uk/sol/homepage/news/6806667/Mum-duped-into-sex-by-lesbian-posing-as-a-man-blasts-judge-for-jail-let-off.html.
"She Met on Facebook": http://www.lincolnshireecho.co.uk/Lincolnshire-woman-posed-man-sexually-assaulted/story-28069633-detail/story.html.
"selfish... dreadful and deceitful": "Woman Who Used Fake Penis," _The Guardian_ , December 15, 2015.
Norway, and Ireland: Henry McDonald and agencies, "Ireland Passes Law Allowing Trans People to Choose Their Legal Gender," _The Guardian_ , July 16, 2015, https://www.theguardian.com/world/2015/jul/16/ireland-transgender-law-gender-recognition-bill-passed.
**11. T HE DENIAL OF HISTORY**
found throughout the text: Nerissa Gailey and A. D. Brown, "Beyond Either/or: Reading Trans* Lesbian Identities," _Journal of Lesbian Studies_ 20, no. 1 (2015): 65–86.
I have ever seen: Helen Russell, "Gerda Wegener: 'The Lady Gaga of the 1920s,'" _Guardian_ (London), September 28, 2015.
with police permission: Graham Robb, _Strangers: Homosexual Love in the Nineteenth Century_ (New York: W. W. Norton, 2003), p. 166.
the Prussian state: Vern L. Bullough, "Legitimatizing Transsexualism," _International Journal of Transgenderism_ , 10, no. 1 (2007): 3–13, doi: 10.1300/J485v10n01_02 (p. 5).
and "homosexual": Susan Stryker, _Transgender History (Berkeley, CA: Seal Press, 2008)_ , p. 38.
"the most dangerous Jew in Germany": Florence Tamagne, "The Homosexual Age, 1870–1940," in Aldrich, _Gay Life and Culture_.
World League for Sexual Reform: Stryker, _Transgender History_ , p. 96.
pushing a kink too far: Brett Genny Beemyn, "A Presence in the Past: A Transgender Historiography," _Journal of Women's History_ 25, no. 4 (winter 2013): 113–121.
silent movie adaptation: Karl M. Baer, Wikipedia, https://en.wikipedia.org/wiki/Karl_M._Baer.
surgical transition in 1917: Alan L. Hart, Wikipedia, https://en.wikipedia.org/wiki/Alan_L._Hart.
"this nature were performed": Katie Sutton, "'We Too Deserve a Place in the Sun': The Politics of Transvestite Identity in Weimer Germany," _German Studies Review_ 35, no. 2 (May 2012).
"a complete woman": Ibid.
numbers were considerable: Florence Tamagne, "The Homosexual Age, 1870–1940," in Aldrich, _Gay Life and Culture_.
further punishment for their "crimes": "The Nazi Era" in Paragraph 175, Wikipedia, https://en.wikipedia.org/wiki/Paragraph _175#The_Nazi_era.
**12. B EYOND BINARIES**
the category contained multitudes: Kathryn M. Ringrose, "Living in the Shadows: Eunuchs and Gender in Byzantium," in Herdt, _Third Sex, Third Gender_.
to become a pederast: Casanova, quoted by Vern L. and Bonnie Bullough, _Cross Dressing, Sex, and Gender_ (Philadelphia: University of Pennsylvania Press, 1993), pp. 84–85.
and femininity of movement: Theo van de Meer, "Sodomy and the Pursuit of a Third Sex in Early Modern Europe," in Herdt, _Third Sex, Third Gender_.
"the object of everybody": Michael Sibalis, "Male Homosexuality in the Age of Enlightenment and Revolution, 1680–1850," in Aldrich, _Gay Life and Culture_.
as women gave them: Bernd-Ulrich Hergemöller, "The Middle Ages," in Aldrich, _Gay Life and Culture_.
executed by beheading: Laura Gowing, "Lesbians and Their Like in Early Modern Europe, 1500–1800," in Aldrich, _Gay Life and Culture_.
"any specific detriment": https://petition.parliament.uk/petitions/104639.
to the MoJ: Beyond the Binary, "Non-Binary Gender Factsheet," Google Docs, https://drive.google.com/file/d/0B6uo3Mt-MvPfa 1R6X05zckVlX2VUMy1iMFZCWUhMWEdoUk1Z/view.
at the beginning of 2016: UK Parliament, Women and Equalities Committee, _Transgender Equality Inquiry_ , report published January 14, 2016, government response published July 7, 2016 (see next citation), https://www.parliament.uk/business/committees/committees-a-z/commons-select/women-and-equalities-committee/inquiries/parliament-2015/transgender-equality/.
dismissed it: Government Equalities Office, _Government Response to the Women and Equalities Committee Report on Transgender Equality_ , July 2016, https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/535764/Government_Response _to_the_Women_and_Equalities_Committee_Report_on _Transgender_Equality.pdf.
or government official: Caroline Linton, "A Non-Binary Student Publicly Came Out for the First Time—to President Obama," _Splinter_ (website), April 23, 2016, http://fusion.net/story/294831/maria-munir-non-binary-gender-comes-out-president-obama/.
gendered categories: Joseph Patrick McCormick, "Will Smith Is Proud That 'Fearless' Jaden Wears 'Women's Clothes,'" _Pink News_ (website), February 10, 2016, http://www.pinknews.co.uk/2016/02/10/will-smith-is-proud-that-fearless-jaden-wears-womens-clothes/.
reported such feelings: Daphna Joel et al., "Queering Gender: Studying Gender Identity in 'Normative' Individuals," _Psychology and Sexuality_ (2013), http://people.socsci.tau.ac.il/mu/daphnajoel/files/2014/11/Joel_gender_identity_2013.pdf.
between male and female: Beyond the Binary, "Non-Binary Gender Factsheet."
**13. T HE T FROM THE LGB**
broader term "queer": Greta R. Bauer et al., "Sexual Health of Trans Men Who Are Gay, Bisexual, or Who Have Sex with Men: Results from Ontario, Canada," _International Journal of Transgenderism_ 14, no. 2 (2013): 66–74.
their sexual orientation: Scottish Transgender Alliance, 2012 Trans Mental Health Study.
these modern ideas grew: Brett Genny Beemyn, "The Americas: From Colonial Times to the 20th Century," in Aldrich, _Gay Life and Culture_.
"a woman's soul trapped in the wrong body": Florence Tamagne, "The Homosexual Age, 1870–1940," in Aldrich, _Gay Life and Culture_.
same-sex-oriented man: Gert Hekma, "'A Female Soul in a Male Body': Sexual Inversion as Gender Inversion in Nineteenth-Century Sexology," in Herdt, _Third Sex, Third Gender_.
despite his birth: Louis Sullivan, _From Female to Male: The Life of Jack Bee Garland_ (Boston: Alyson Publications, 1990).
and sacrificial victims: Susan Stryker, _Transgender History_ (Berkeley, CA: Seal Press, 2008), p. 96.
as "she/her" does: http://www.advocate.com/arts-entertainment/books/2014/11/17/transgender-pioneer-leslie-feinberg-stone-butch-blues-has-died.
what is woman?: Minnie Bruce Pratt, _s/he_ (Boston: Alyson Books, 2005; orig. Firebrand Books, 1995), pp. 117–118.
**14. T RANS FEMINISMS**
our feminisms are stuck: Gabrielle Le Roux, "Proudly African and Transgender," _Women: A Cultural Review_ 1 (2012): 79–95.
"in Bitter Debate": Jill Stark, "Call Yourself a Woman? Feminists Take on Transgender Community in Bitter Debate," _The Sydney Morning Herald,_ November 22, 2015, http://www.smh.com.au/national/what-makes-a-woman-feminists-take-on-transgender-community-in-bitter-debate-20151113-gkyk6u.html.
"Radical Feminism and Transgenderism": Michelle Goldberg, "What Is a Woman?" _The New Yorker_ , August 4, 2014, http://www.newyorker.com/magazine/2014/08/04/woman-2.
"Feminism and the Transgender Movement": Miranda Yardley, "The Conflict Between Feminism and the Transgender Movement," _Morning Star_ , December 24, 2014, http://www.morningstaronline.co.uk/a-cb92-The-conflict-between-feminism-and-the-transgender-movement.
"What Makes a Woman?": Lydia Smith, "Transgender Rights Versus Feminism: What Makes a Woman?" _International Business Times_ , October 28, 2015, http://www.ibtimes.co.uk/transgender-rights-versus feminism-what-makes-woman-1501487.
or race discrimination: Kimberlé W. Crenshaw, "Demarginalising the Intersection of Race and Sex: A Black Feminist Critique of Anti-discrimination Doctrine, Feminist Theory, and Anti-racist Politics," in Helma Lutz et al., eds., _Framing Intersectionality: Debates on a Multi-Faceted Concept in Gender Studies_ (Farnham, UK: Ashgate/Taylor and Francis, 2011).The title here differs from that given in the text, which is the original.
and zoom away: Ibid.
was similarly full of shit: Julie Burchill, "Don't You Dare Tell Me to Check My Privilege," _The Observer_ , February 2, 2014, http://www.spectator.co.uk/2014/02/dont-you-dare-tell-me-to-check-my-privilege/.
"Feminism (n): Plural": Roxane Gay, _Bad Feminist_ (London: Corsair, 2014), p. iv.
"the fractures among us": Ibid., p. xiii.
"created by forces of male power": Sheila Jeffreys, _Gender Hurts_ (New York: Routledge, 2014), p. 20.
to the detriment of all others: Roxane Gay, _Bad Feminist_ (introduction), p. xiii.
**15. F UTURES**
a Chair in Transgender Studies: http://www.uvic.ca/research/transchair/.
the Obama administration: Cleis Abeni, "Meet the First Trans White House LGBT Liaison," _The Advocate_ , March 14, 2016, https://www.advocate.com/transgender/2016/3/14/latina-just-became-first-transgender-white-house-lgbt-liaison.
was published in 2017: Rachel Deahl, "Book Deals: Week of March 28, 2016." _Publishers Weekly_ , March 25, 2016, http://www.publishersweekly.com/pw/by-topic/industry-news/book-deals/article/69778-book-deals-week-of-march-28-2016.html.
identify as heterosexual: Zing Tsjeng, "Teens These Days Are Queer AF, New Study Says," Broadly section, _Vice_ (website), March 10, 2016, https://broadly.vice.com/en_us/article/teens-these-days-are-queer-af-new-study-says.
eating disorders in trans teenagers: Zena Blair, "Why Transgender Youth Are More at Risk for Eating Disorders Than Their Peers," _Teen Vogue_ , March 4, 2016, http://www.teenvogue.com/story/transgender-youth-eating-disorders.
of medical transition: Elle Bradford, "You Won't Believe How Much It Costs to Be Transgender in America," _Teen Vogue_ , November 14, 2015, http://www.teenvogue.com/story/transgender-operations-hormone-therapy-costs.
to trans friends: Vera Papisova, "7 Things You Should Never Ask a Transgender Person," _Teen Vogue_ , February 23, 2016, http://www.teenvogue.com/story/things-to-never-ask-transgender-people-hari-nef.
bathroom predator: Caroline Lehmann, "Unsurprisingly, Trans Students Have Caused Zero Incidents in Public Bathrooms," _Seventeen_ , June 8, 2015, http://www.seventeen.com/life/school/news/a31352/in-unsurprising-news-trans-students-have-caused-zero-incidents-in-public-bathrooms/.
inclusion of trans scouts: Curtis M. Wong, "Girl Scouts Return $100,000 When Donor Demands It Not Be Used for Transgender Girls," _Huffington Post_ , July 1, 2015, http://www.huffingtonpost.com/entry/girl-scouts-transgender-donation_us_5592d41fe4b000 c99ee1cc65.
children and teenagers: Hayley Miller, "Heartwarming Story of the Day: Outpouring of Support for Transgender Youth in Wisconsin Town," Human Rights Campaign, December 3, 2015,
we were assigned at birth: Ben Bloom, "Transgender Athletes Allowed to Compete at Rio Olympics Without Sex Reassignment Surgery," _The Telegraph,_ January 25, 2016,http://www.telegraph.co.uk/sport/2016/02/22/transgender-athletes-allowed-to-compete-at-rio-olympics-without/.
legally recognized: "Vietnam: Positive Step for Transgender Rights," _Human Rights Watch_ , November 30, 2015, https://www.hrw.org/news/2015/11/30/vietnam-positive-step-transgender-rights.
their true genders: "Trans People to Receive Compensation for Forced Sterilization in Sweden." TGEU (Transgender Europe), April 27, 2016, http://tgeu.org/trans-people-to-receive-compensation-for-forced-sterilisation-in-sweden/.
working hard to achieve it: Jessica Chasmar, "Canadian Prime Minister Justin Trudeau Seeks Federal Ban on Anti-Transgender Speech," _The Washington Times,_ May 23, 2016, https://www.washingtontimes.com/news/2016/may/23/justin-trudeau-canadian-prime-minister-seeks-feder/.
a differently sexed body: Jason Kambitsis, "A Crazy Oculus Rift Hack Lets Men and Women Swap Bodies," _Wired_ , February 25, 2014, http://www.wired.com/2014/02/crazy-oculus-rift-experiment-lets-men-women-swap-bodies/.
transplanted womb: "Woman Has Healthy Baby Boy After Womb Transplant in Sweden," ABC News, October 4, 2014, http://www.abc.net.au/news/2014-10-04/woman-has-baby-after-womb-transplant-in-sweden/5790726.
from stem cells: Maggie Fox, "Researchers Grow Kidney, Intestine from Stem Cells," NBC News, October 8, 2015, http://www.nbcnews.com/health/health-news/researchers-grow-kidney-intestine-stem-cells-n441066.
and mob retribution: Human Rights Watch, "License to Harm: Violence and Harassment Against LGBT People and Activists in Russia," December 15, 2014, https://www.hrw.org/report/2014/12/15/license-harm/violence-and-harassment-against-lgbt-people-and-activists-russia.
"posing as women": Human Rights Watch, "I'm Scared to Be a Woman: Human Rights Abuses Against Transgender People in Malaysia," September 24, 2014, https://www.hrw.org/report/2014/09/24/im-scared-be-woman/human-rights-abuses-against-transgender-people-malaysia.
trans women were murdered: Juliana Britto Schwartz, "At Least 48 Transgender Women Killed in Brazil in January," _Feministing_ (website), 2016, http://feministing.com/2016/02/03/at-least-48-transgender-women-killed-in-brazil-in-january/.
anti-LGBT backlash: J. Lester Feder and Rin Hindryati, "Muslim School for Trans Women Shut Down in Indonesia," _BuzzFeed_ (website), February 25, 2016, http://www.buzzfeed.com/lesterfeder/muslim-school-for-trans-women-shut-down-in-indonesia#.wo0vb5qaZ.
purge the military of trans service members: Julie Hirschfeld Davis and Helene Cooper, "Trump Says Transgender People Will Not Be Allowed in the Military," _The New York Times_ , July 26, 2017, https://www.nytimes.com/2017/07/26/us/politics/trump-transgender-military.html?mcubz=0. See also Dave Philipps, "Second Judge Blocks Trump's Transgender Ban in the Military," _New York Times_ , November 21, 2017, https://www.nytimes.com/2017/11/21/us/transgender-ban-military.html.
those who had none: Kirstie Brewer, "Services for LGBT Young People Will Just Disappear," _The Guardian_ , February 2, 2016, https://www.theguardian.com/society/2016/feb/02/services-for-lgbt-young-people-will-just-disappear.
very existence is a crime: Free Movement (website), "LGBT Asylum Seekers: A Toxic Mix of Homophobia, Misogyny, and Ignorance Corrupts the Asylum System," April 27, 2015, https://www.freemovement.org.uk/lgbt-asylum-seekers-a-toxic-mix-of-homophobia-misogyny-and-ignorance-corrupts-the-asylum-system/.
# Contents
1. Cover
2. Title Page
3. Copyright
4. Table of Contents
5. Prologue
6. 1 The Production of Ignorance
7. 2 "Call Me Caitlyn"
8. 3 Finding My Voice
9. 4 Couldn't You Just... Not Be?
10. 5 What About Sex?
11. 6 Think of the Children
12. 7 Delusional and Disturbed
13. 8 A Different Approach
14. 9 Trans/Love
15. 10 Are Trans People Real?
16. 11 The Denial of History
17. 12 Beyond Binaries
18. 13 The T from the LGB
19. 14 Trans Feminisms
20. 15 Futures
21. Acknowledgments
22. About the Author
23. Further Resources
24. Endnotes
# Navigation
1. Begin Reading
2. Table of Contents
| {
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The Stone World of Mount Ida
by Archaeology Newsroom
Stone as a building material has been used by almost all structural civilizations, when available in their environment. The so-called mitata, the circular, stone-built lodgings on the Psiloreitis massif, the church of Agios Hyakinthos at Anogeia and the out-doors sculpture-monument for Peace, created by Karina Raeck on the Ida plateau, all three on the island of Crete, speak for the continuity of a long tradition in dry stone-building. The Partisan of Karina Raeck is a monument of land art in the Nida plateau, a palimpsest in reality, which narrates fascinating stories about its Cretan cultural landscape, such as the Cretan Zeus' and the Kourites' myths, the wild and rough nature and its people, the Battle of Crete, the holocaust of the Anogeia by the Germans and the National Resistance. It is made from almost 5,000 stones and is essentially one more dry stone structure, like the mitata dotting the Psiloreitis massif.
Their slated roofing, which appeared on Crete 5,000 years ago in the vaulted tombs of Messara, has not ceased to be employed and continues to inspire. The larger mitata continue to be in use in the broader periphery of Anogeia, while some of them have been restored. The church of Agios Hyakinthos, in perfect harmony with the local tradition, has introduced a new symbolic function to this established building type.
Underwater archaeological research in the Fourni Archipelago
Results of 2021 Kition-Pampoula excavations
Byzantine Coin Hoard Found in Deir el-Bakhit
Cat Domestication in Egypt Pushed Back By 2,000 Years
Nazca archaeological sites in danger near Ica, Peru
Mystery Solved: The Skeleton Lake of India
The palace where Alexander the Great was born comes to light
Entire Roman city revealed without any digging
Fish, shellfish and fishermen in Prehistoric Aegean
Sudan Archaeology from a Greco-Roman Perspective (Part 5)
Museums and Museology in modern society. New challenges, new relationships (Part 13) | {
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\section{Introduction}
Extra-solar planets (Mayor \&Queloz 1995, Marcy \& Butler 1996)
can, in principle, be seen in ground-based images using the
light cancellation in dark speckles to remove the halo of star light (Labeyrie 1995).
Long-exposures with high-performance adaptive optics and a
correction of the seeing-induced shadow pattern on the telescope's pupil are
also proposed by Angel (1994). A less extreme atmospheric correction is needed
with the dark-speckle method. Both methods are
expected to reach the $10^{-9}$ relative intensity needed to detect a
Jupiter-like planet near its parent star. Related space-based techniques
are also considered (\cite{ref12}, \cite{ref5}). In the longer term,
resolved images of detected exo-planets will in principle be obtainable even
from the ground with long-baseline interferometric arrays (\cite{ref10}). \\
The "dark-speckle" method exploits the light cancellation effect occuring in
random coherent fields according to the Bose-Einstein statistics. Although
adaptive optics can reconstruct the Airy peak, and possibly the first few
rings, in the focal image of a bright star, the degree of "seeing" correction
which it provides cannot be good enough to remove the fluctuating speckles in
the surrounding zone of the Airy pattern. Coronagraphic devices
(\cite{ref11}, \cite{ref4}, \cite{ref15}, \cite{ref13}, Gay \& Rabbia 1996,
Roddier \& Roddier 1997)
can remove the steady and organized part of the
straylight, i.e. the first few Airy rings if they emerge from the boiling
speckled halo, and even if they are burried but remain detectable with the
dark speckle analysis.\\
One is therefore left with the problem of extracting as much
information as possible from this speckled halo. Destructive interference
in the star light occasionally causes a dark speckle to appear here and there.
When this happens at the position of the planet's own faint Airy peak, the
darkening cannot be as deep as elsewhere. The planet's Airy peak, also
restored by the adaptive optics, indeed has a rather stable intensity, which
adds to the star's local intensity. The intensity histogram is
therefore locally distorted, and suitable algorithms can display the local
distorsions in the form of a cleaned image.
The exposures must be shorter than the turbulence life time. With a large
telescope and hours of integration, a planet, typically $10^9$ times fainter
than the parent star, is expected to become visible.
\section{Signal to noise ratio}
To calculate the method's sensitivity in a more general way than done by Labeyrie (1995), we first follow his derivation to the point where he calculates the signal-to-noise ratio.\\
The different photon distribution from the star and the planet defines the signal-to-noise ratio $SNR$, and according to the central limit theorem:
\begin{equation}
nP_*(0)[1-P_0(0)]=SNR\sqrt{nP_*(0)}
\end{equation}
where $P_*(k_*)$ and $P_0(k_0)$ are the probabilities to detect $k_*$ and
$k_0$ photons originating respectively from the star and
from the planet, per pixel in a short exposure, and where $n$ is the total number of short exposures. Replacing the values of $P_*(0)$ and $P_0(0)$ in Eq. (1), we obtain:
\begin{equation}
SNR=(1-e^{-\overline{k_0}})\sqrt{n\over 1+\overline{k_*}}
\end{equation}
The calculation of signal to noise ratio given in Labeyrie 1995 assumed
$\overline{k_*}\gg 1$, which is unnecessarily restrictive, and
unrealistic in some of the cases of interest. As suggested by one of us
(RR), a more general analysis can be made under the assumption
that $\overline{k_0}\ll 1$ and $\overline{k_*}\gg\overline{k_0}$. Equation (2) then becomes:
\begin{equation}
SNR\approx \overline{k_0}\sqrt{n\over 1+\overline{k_*}}
\end{equation}
if $j$ is the number of pixels per speckles, thus:
\begin{equation}
(SNR)^2\approx {{n\over j}(j\overline{k_0})^2\over
j+j\overline{k_*}}={n'\overline{K_0}^2\over j+\overline{K_*}}
\end{equation}
Where $\overline {K_0}$ and $\overline {K_*}$ are the number of photons per
speckle in a short exposure, respectively for the planet and the star.\\
The variables used in \cite{ref9} were:\\
$n'=${\Large${T\over t}$}, {\Large$\;\;{\overline {K_*} \over
\overline{K_0}}$}$=${\Large${R\over G}\;\;$} and $\;\;G\overline {K_*}=tN_*$\\
where $T$ is the total integration time, $t$ the short exposure time,
$R$ the star/planet brightness ratio, $G$ the gain of the adaptive optics
or the ratio between the Airy peak and the halo of speckles, $N_*$ the total number of photons per second detected from the star.\\
It provides a new expression for the $SNR$:
\begin{eqnarray}
SNR=\overline {K_0}\sqrt{n'\over j+\overline {K_*}} & = & {tN_*\over R}\sqrt
{n'\over j+{tN*\over G}}\nonumber \\
& = & {N_*\over R}\sqrt{tT\over j+{tN_*\over G}}
\end{eqnarray}
The sampling $j$ should be fine enough to exploit the darkest parts of the
dark speckles, for a given threshold of detection $\epsilon$, linking the
performance of the adaptive optics ($G$) and the brightness ratio ($R$).
The intensity across a dark speckle may be coarsely modelled as a cosine
function of the position $\rho$ by:
\begin{equation}
I(\rho)=I_h\left(1-\cos{2\pi\rho\over \sqrt j}\right)
\end{equation}
where $I_h$ is the mean intensity and $\sqrt j$ the speckle size.
The intersection between $I(\rho)$ and the line $y=\epsilon I_h$ gives $s$, the
size of the pixel over which the light is integrated. It limits the minimal measurable intensity $\epsilon I_h$ (with $0<\epsilon \ll 1$).
A detailed calculation gives $s\approx 1.27\sqrt {\epsilon}${\Large${\lambda\over
D}$}.\\
We can assume that $\epsilon=G/R$.
Indeed, if $I_0$ is the intensity of the Airy peak,
$\epsilon=${\Large${\epsilon
I_h\over I_h}$}$=${\Large${\epsilon I_h\over I_0}{I_0\over
I_h}$}$=${\Large${G\over R}$}\\
We also assume in the following that a planet can be seen if its own intensity is higher than $\epsilon I_h$.\\
Now we are able to calculate a value of $j$:
\begin{equation}
j={\left(\lambda /D\right)^2\over s^2}={\left(\lambda /D\right)^2\over
{1.27^2\epsilon \left(\lambda /D\right)^2}}=0.62{R\over G}
\end{equation}
\begin{figure}[t]
\centerline{
\epsfbox{figure1.ps}}
\caption[]{Shape of a dark speckle according to Eq. (6).
If $I_h$ is the mean intensity, integration on a pixel of size $s$ yields an
intensity $\epsilon I_h$.}
\end{figure}
Recent numerical simulations (\cite{ref3}) have shown that $R/G$ should not exceed $10^3$ to retain a reasonable value of the sampling parameter $j$.\\
The recorded level is $I_h$, but with dark speckles this residual level
decreases to $\epsilon I_h$, where $\epsilon$ depends mainly on the adaptive
optics performance.\\
With the value of $j$ obtained, the $SNR$ expression becomes:
\begin{equation}
SNR={N_*\over R}\sqrt{tT\over 0.62{R\over G}+{tN_*\over G}}
\end{equation}
Solving for $R$, leads to a third-degree equation:
\begin{equation}
0.62R^3+tN_*R^2={GtTN_*^2\over (SNR)^2}
\end{equation}
The Cardan method gives a single positive solution from which we derive a
final expression for the integration time.
\begin{equation}
T=\left({SNR\over N_*}\right)^2{R\over Gt}(0.62R^2+tN_*R)
\end{equation}
With the following values : $R=10^9$; $D=8m$; $G=10^6$; $SNR=5$; $m_v=2.5$;
$q=0.2$; $\Delta\lambda=100nm$ and $t=20ms$, we find $2.7$ hours of
integration, i.e. $50\%$ more than the result given by Labeyrie.
\\
Angel's discussion (1994) of the long-exposure method leads him to the following
expression:
\begin{equation}
R={G\over SNR}\sqrt{T\over \Delta t_{opt}}
\end{equation}
where $\Delta t_{opt}$ is the optimum short exposure time. Using
again the same values, Eq.(11) gives a limiting brightness ratio of
$1.4.10^8$ and the typical brightness ratio of $10^9$ would be reached in $140$ hours. The long-exposure approach would in principle be more sensitive if
the adaptive optics and shadow pattern compensation could be made extremely
good. It is however less sensitive with current levels of adaptive performance.\\
In fact, the difference between the dark-speckle and direct-long-exposure
methods is more
subtle, and both work in different regimes. A critical value of the photon
stellar flux ($N_{*c}$) can be easily calculated by equalizing Eq.(10) and
Eq.(11) for the same value of the total integration time ($T$). It leads to a
second degree equation in $N_*$, with a single positive root:\\
\begin{equation}
N_{*c}={G+\sqrt{G^2+4\times 0.62GR}\over 2t}
\end{equation}
Therefore, if the photon flux is above $N_{*c}$, the dark-speckle method is
more efficient than the long exposure and, below this limit the direct imaging
is better. $N_{*c}$ strongly depends on the adaptive optics. Let us take
a numerical exemple. If the goal is to reach a $10^9$ brightness ratio with a
gain of $10^6$ and $20ms$ exposures, the photon flux must be higher than
$1.27.10^9 photons/s$, which can be achieved with a large aperture and wide
bandwidth ($9.3.10^9$ph/s for $m_v=0$, $D=2.4m$, $\Delta\lambda=100nm$ and 20\% efficiency). However, for a space telescope with adaptive optics, the speckle
lifetime is under control with values of the order of $1s$ for exemple.
Thus, $N_{*c}$ is decreased to $2.54.10^7 photons/s$, which is less restrictive.
However, the telescope should not be so large as to provide a partially
resolved image of the star, since it would fill-in the dark speckles.
\section {Dark speckle lifetime}
Because the number of photon-events detected per pixel in each exposure is critical, these exposures should be as long as possible without degrading too much the darkness of the dark speckles. The optimal value is obviously shorter than the usual speckle lifetime considered by Roddier, Gilli \& Lund (1982).
A tentative lower limit can be estimated by linearly scaling the speckle lifetime in proportion to the dark speckle size defined by Eq.(7). It leads to impossibly short exposures for large $R$ values.
However, since the adaptive optics decreases the wave disturbance, it increases
markedly the speckle lifetime at positions close to the Airy peak,
depending upon the factor $n{\lambda\over D}$, where $n$ is the number of
actuators across the pupil containing $n^2$ of them (\cite{ref20}).
The optimal exposure time is therefore dependant upon the adaptive optics performance.
\section{Simulation and results}
\begin{table}
\caption[]{Brightness ratio and $SNR$ obtained with a photon-counting avalanche
photodiode, for two values of the sampling
parameter $j$. The number of zero-photon events was counted on 250000 short
exposures of $100\mu s$, totalling $25s$ of integration. The $SNR$ was
calculated from Eq.(1).}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline
$j$ & $R_{max}$ & $R$ & 0 ph. & 0 ph. (star & $SNR$ \\
& & & (star) & +planet) & \\ \hline
& & 15000 & 136796 & 110377 & 71.4 \\ \cline{3-6}
80 & 440000 & 150000 & 87672 & 85546 & 7.2 \\ \cline{3-6}
& & 360000 & 82340 & 80625 & 5.9 \\ \hline
144 & 790000 & 560000 & 111044 & 107946 & 9.3 \\ \cline{3-6}
& & 950000 & 155959 & 150017 & 4.0 \\ \hline
\end{tabular}
\end{center}
\end{table}
To assess the dark-speckle method we did a laboratory simulation using a
single-pixel photon-counting detector, in the form of an avalanche
photodiode.
The star and planet were simulated by two He-Ne lasers, with
adjustable attenuators.
A Lyot-type coronagraph permitted to remove the star's
Airy peak and rings, thus decreasing the local halo intensity 10
to 15 times. Artificial ``seeing'' was generated with a moving
scatterer, selected to provide a Strehl ratio approaching that typical of
current adaptive optical systems. The equivalent peak/halo gain was
$G=3.4.10^3$. The flux of the central star was $44.10^6$
photons/s. Calculating an histogram of the detected photon events, we
determined the
$SNR$ by comparing the number of zero-photon events with the planet
turned on and off.
As listed in Table 1, the results strongly depend
on the sampling parameter $j$. They are consistent with Eq.(7) which gives
the maximum brightness ratio ($R_{max}$) theoretically reachable.
In these laboratory tests, the dark-speckle analysis outperforms the long exposure when the sampling exceeds $144$ $pixels/speckle$ $area$.
\begin{figure}[hb]
\vspace{0.3cm}
\centerline{
\epsfbox{figure2.ps}}
\caption[]{Laboratory simulation of stellar companion detection with the dark
speckle method. The artificial companion, $966$ times fainter than the star,
is at the center of the dark map, emerging from the halo. A coronagraphic mask, which hides the star's central peak is visible in the lower right corner. The
blob seen in the upper right part of the field is a ``static speckle''
caused by permanent aberrations and removable with a reference star.}
\end{figure}
In this experiment the short exposure time ($100\mu s$) is about 100 times shorter than the speckle lifetime ($10ms$).
Available photon counting camera do not yet allow quite as short exposure.\\[0.2cm]
\begin{figure}[t]
\centerline{
\epsfbox{figure3.ps}}
\caption[]{Cleaned image, generated through dark-speckle analysis, of the
multiple star HD144217($\beta$ Sco). A companion ($\beta$ Sco B) appears
near the masked star image. (mask diameter$=0.5''$, $F/D=1200$)}
\end{figure}
We also used the CP40
photon-counting camera developed by Foy and Blazit (\cite{ref2}). It has pixels
of $50\mu m$ and a rather low saturation level of $50000 photons/s$. Each
pixel has a lower dark noise than an avalanche photodiode, but a much slower
response. We used the algorithm described in Labeyrie (1995),
generating a ``dark map'' by counting, in each pixel, the number of $20ms$
exposures which contribute
zero photon. As contributions from
successive $20ms$ exposures are accumulated in the dark map, a planet's
Airy peak is expected to emerge as a black dot among background
noise.
To obtain the cleaned image, results are displayed in positive form
using, for example, an inverse square law. \\
The CP40 discriminates between events featuring zero photoelectron and those featuring one or more photoelectron/pixel/exposure. Adding all exposures generates an image which brings out faint companions better than would a similarly long exposure on a CCD detector.\\
The flux of the central star was
$5.3.10^6$ photons/s, and the gain of the adaptive optics was about
$1556$. The optical system operated at
$\lambda =0.67\mu m$, $F/D=3200$, and the mask diameter was $0.34"$, i.e. it covered the central 2 rings of the Airy pattern.
The planet was located near the fifth ring of the diffraction pattern.\\
Figure 2 shows that a companion $966$ times fainter is well detected in
116 seconds.
The $SNR$ measured on a speckle size region ($37\times 37$ pixels) is $799$, while the
dark-speckle model predict an $SNR$ of $1008$. This model does not take into
account the halo shape which can explain the $20\%$ discrepancy.
These initial results, where the companion is brighter than the halo, are very modest with respect to the performance expected at a later stage, but have provided useful insight for improving the instrument.\\
Currently, as seen in Figs 2 and 3, the detection sensitivity is limited by the presence
of spurious blobs in the cleaned image. These are caused by static aberrations
and coronagraphic mask effects. These residual blobs may be substracted from
data obtained on a reference star.\\[0.2cm]
Finally, dark-speckle data have been recorded at the 152cm telescope
of Haute-Provence, using, during a single night, the BOA
adaptive optics system (\cite{ref19}) developped by the Office National d'Etudes
et de Recherches A{\'e}rospatiales (ONERA).
\begin{figure}[t]
\centerline{
\epsfbox{figure4.ps}}
\caption[]{Same image enhanced with unsharp mask and high pass filter to emphasize the contrast and median filter to smooth the image at the speckle scale.}
\end{figure}
This system, optimized for visible light, reaches a Strehl ratio of
about 0.4 at $0.6\mu m$ in long exposures and higher in short exposures. 30
minutes of observation, with an interference
filter centered at $0.67\mu m$ ($\Delta\lambda=100 \mbox{\AA}$), evidenced the
faint component of the spectroscopic binary HD144217 ($\alpha=16h05'26"$,
$\delta=-19^{\circ}48'18"$, $V=2.62$). On the detector, the flux from the
primary was only $11860$ photons/s.
The angular separation is about $0.45"$ with an uncertainty due to the
mask offset. Owing to the low elevation of the star, the adaptive optics gain
was only $12$. The $SNR$ measured on $16\times 16$ pixels is $168$ and allows
to derive
from Eq.(10) a brightness ratio of $88$, corresponding to a $4.8$ magnitude
difference. On the long exposure synthesized from the same data
(1 photon-events analysis) the $SNR$ is very similar, but a direct imaging
should give a $SNR$ of $42$ according to Eq.(11).
However, a recent measurement of lunar-occultation (\cite{ref6}) gives $\Delta
m=3.3$. In this case, dark-speckle analysis should provide a very good detection
with a $SNR$ of $756$ instead of $168$.
Unfortunately, the Hipparcos mission failed to detect the companion, probably
because the Hipparcos satellite is unable to achieve $\Delta m>4$ with small
angular separation, which is consistent with our data.
More observations are needed to verify the companion magnitude.\\
A continuing observing program is initiated.
\section{Conclusion}
The simulations and tentative observations lead to the following remarks:
1. One would like to sample as densely as possible to reach the bottom of
the dark speckles, but there should be enough photons per pixel. The optimal sampling is therefore critical and we guess that it should be about 500 pixels per speckle area.
2. A fast photon-counting
camera with a low dark noise, high saturation level and many pixels is
needed.
3. The observations required a narrow-band filter since the diffraction and
speckle
pattern are color-dependant. The speckles are themselves dispersed
radially. To increase the bandwidth usable in speckle interferometry,
Wynne designed
a chromatic lens with magnification inversely proportional to wavelength
(\cite{ref22}).
D. Kohler built a Wynne corrector and we found it efficiently applicable to the present situation, where the speckle's
wavelength
dependance is more nearly a linear scaling. The resulting smearing of the
planet's peak is acceptable if the spectral band remains less than 100 nm.
4. Different types of apodisation can be achieved, using a classical Lyot
coronagraph, the interference coronagraph of Gay \& Rabbia
(1996), or the phase-mask coronagraph of Roddier \& Roddier
(1997). Both recent systems favor the detection of planets closer to
the central star's Airy peak. Laboratory simulations with these varied
devices are considered to compare their respective efficiencies.
\\[0,2cm]
Our simulations verify the theoretical expressions given for the signal to noise
ratio. The $SNR$ measured from the photon-number variance (Eq.(1)), is consistent
with the $SNR$ expected from the model (Eq.(10)). In these preliminary tests,
we had to use an interference filter and low saturation level camera which
provides a weak signal. We were consequently unable to reach enough sensitivity
for detecting extrasolar-planets or even brown-dwarf companions.\\%[0.2cm]
The dark-speckle method is also applicable to space telescopes. Even
without turbulence, optical defects create static speckles which
can be made to fluctuate with a few actuators, arranged in the form of an
active optics system, or a fast random scatterer. We proposed a "dark-speckle camera", the Faint
Source Coronagraphic Camera for the Hubble Space Telescope (\cite{ref5}). The
project is reconsidered for the New Generation Space Telescope.\\
IR wavelengths are of interest for the detection of extrasolar
planets, for two reasons: the planet's contrast is improved and,
turbulence is easier to correct at these wavelengths.
The forthcoming developement of bidimensional sensors with low read
noise should allow red and IR work.
\acknowledgements{We wish to thank D. Kohler and G. Knispel who made
simulations possible, as well as D. Mourard and A. Blazit for the CP40 camera
assistance. We are also grateful to the ONERA team for providing the adaptive
optics system.}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,068 |
{"url":"https:\/\/cstheory.stackexchange.com\/questions\/19975\/is-the-sub-bit-model-of-quantum-computation-equivalent-to-other-models","text":"Is the sub-bit model of quantum computation equivalent to other models? [closed]\n\nIn a comment to this question Peter Shor asked me for a reference about the third described in the question point of view, namely, that quantum computers can be described as computers that can manipulate units of information less than one bit independently. Although I cannot provide a reference, here is a brief clarification.\n\nSuppose that each bit is divided into N sub-bits. Each sub-bit can be in two states: 0 or 1, similarly to the bit as a whole.\n\nSince the entropy of 1 bit is 1 (in bits), each sub-bit should have a factor, equal to 1\/N so to sum up to 1 for the whole bit.\n\n$$H=\\sum_{i=0}^N \\frac{1}{N} \\ln 2= \\ln 2 = 1$$\n\nThis factor, $1\/N$ represents the probability that a certain sub-bit is measured when reading the bit with classical means. As such, the whole bit is equal to 1 when all its sub-bits are equal to 1, and to 0 when all the sub-bits are 0. In other cases there is a probability that the whole bit is read either as 0 or as 1. The probability is determined by the ratio of 0's and 1's in the sub-bits.\n\nAs such, the bits of sub-bit composition (10), (1100), (111000), (11110000), ... considered equal as each sub-bit can be infinitely devided.\n\nThe more the number N of sub-bits to which the bit is devided, the more points of the Bloch sphere is representable. At the analog limit, the whole Bloch sphere is covered.\n\nAs such, the digital quantum computer can be represented as a set of classical logical gates applied to sub-bits, very much similar to a classical computer with only one diffrernce: the final result cannot be read by each sub-bit separately, but only whole bits at minimum.\n\nThus my question is, whether the outlined model can be considered an exhaustive description of a digital quantum computer that uses a finite set of pure states to perform its operation?\n\nWhether the presented model can represent any quantum gate quantum computer in a limit of infinite division?\n\nWhether this somehow indicates that a qubit actually is the same unit of information as a classical bit, albeit treated somewhat differently?\n\nUPDATE\n\nThis update clarifies some of the issues pointed by Niel de Beaudrap.\n\nQuantum gates in sub-bit model can be represented by truth table.\n\nEntanglement is realized as a fact that some states with the same Hamming-weight (and as such, the same expected outcome) may be opposite to each other. For example, qubits 1100 and 0011 are opposite to each other while 1010 and 1100 are correlated, even though each of them gives completely random result taken separately.\n\nclosed as off-topic by Niel de Beaudrap, Jeff\u03b5, KavehDec 19 '13 at 21:46\n\nThis question appears to be off-topic. The users who voted to close gave this specific reason:\n\n\u2022 \"Your question does not appear to be a research-level question in theoretical computer science. For more information about the scope, please see help center. Your question might be suitable for Computer Science which has a broader scope.\" \u2013 Niel de Beaudrap, Jeff\u03b5, Kaveh\nIf this question can be reworded to fit the rules in the help center, please edit the question.\n\n\u2022 How does your \"truth-table\" for the Hadamard gate apply in the case of 4-bit strings? And can you clarify how the actual state $|\\Psi^-\\rangle$ (as opposed to orthogonal pairs of single-qubit states) would be represented? \u2013\u00a0Niel de Beaudrap Nov 26 '13 at 21:59\n\u2022 @Niel de Beaudrap do you mean 4-bit string or 4-sub-bit bit? \u2013\u00a0Anixx Nov 26 '13 at 22:01\n\u2022 I mean the quantum states which you associate to 4-bit strings. (Regardless of whether it makes sense to talk about \"sub-bits\", it certainly does make sense to refer to the mapping from bit-strings to quantum states which is the basis of your model.) \u2013\u00a0Niel de Beaudrap Nov 26 '13 at 22:08\n\u2022 Do you know who the original proposal was by? Where it was published, or what year? Do you have any clues as to where this was originally meant to have been described? \u2013\u00a0Niel de Beaudrap Nov 28 '13 at 13:16\n\u2022 I've voted to close as not being research level: it doesn't seem as though any revision of this post is forthcoming which would actually make this supposed description of quantum computing well-defined. \u2013\u00a0Niel de Beaudrap Dec 14 '13 at 0:31\n\nI'm not sure who would suggest that qubits can meaningfully be described this way, or why anyone would do so. There are simply too many missing details, and it falls afoul of no-go theorems for local hidden-variable theories in quantum mechanics. This isn't \"an exhaustive description of a digital quantum computer that uses a finite set of pure states to perform its operation\" \u2014 even for single qubits \u2014 and extensions to multiple qubits immediately presents problems.\n\n1. How does one map binary strings to pure states? Why did you associate $0011 \\mapsto \\frac1{\\sqrt2}(|0\\rangle + |1\\rangle)$ and $0110 \\mapsto \\frac1{\\sqrt2}(|0\\rangle + \\mathrm e^{i\\pi\/3}|1\\rangle)$ rather than the other way around? What specific mapping does one use for the boolean strings of length 4, with Hamming weight 1? What is the significance of the overlaps between those states, and the states you give for strings of Hamming weight 2? None of these details are provided, or even implied in any way that I can tell.\n\n2. What transformations of states are allowed depends crucially on the answers to the previous questions, but also is likely to be somewhat artificial from a physical point of view. Consider an operation mapping $|0\\rangle$ to $\\frac1{\\sqrt2}(|0\\rangle + |1\\rangle)$, and $|1\\rangle$ to $\\frac1{\\sqrt2}(|0\\rangle - |1\\rangle)$. In quantum computation, there is exactly one operation which does this \u2014 the Hadamard gate \u2014 and it's self-inverse. What transformation of bit-strings does it correspond to? Unfortunately, none. The strings of Hamming-weight 2 correspond to equally spaced states on the equator of the Bloch sphere (at 60\u00b0 intervals), so their images in the Hadamard operation must include states at every 60\u00b0 interval in latitude on the intersection of the XZ plane and the sphere \u2014 so your states for 4-bit strings must include $\\frac12(\\sqrt3|0\\rangle \\pm |1\\rangle)$. This is not a real problem, but then what other states are there associated to strings of Hamming weight 1, and what does the Hadamard gate map them to? Unfortunately there is no solution for four-bit strings.\n\nIn fact, it's not clear that any operations other than the identity operation and bit-flips can be represented, so it isn't clear how states involving the sub-bits would actually be prepared. One could respond that perhaps only the \"continuum limit\" actually describes the state-space of a non-trivial quantum computation, but then one should also not describe it as a \"limit\" of these discrete \"models\".\n\n3. Related to the above problem: How does one represent an entangled state? Consider the state $$|\\Psi^-\\rangle = \\mathrm{CNOT} \\Bigl[\\tfrac1{\\sqrt2}(|0\\rangle -|1\\rangle)\\otimes |1\\rangle\\Bigr].$$ How would you represent this state? The representation has to somehow include the fact that\n\n\u2022 measuring the first (or second) qubit gives a random result;\n\u2022 measuring the first bit has the effect of also measuring the second qubit;\n\u2022 the outcomes of the two qubits are opposite to each other;\n\u2022 the above also holds if you first perform any given reversible operation $U$ to both first and second qubits;\n\u2022 this should preferably not require the model to include communication between the qubits any time you measure one of them.\n\nThat is to say: to represent the state $|\\Psi^-\\rangle$, you would need to come up with a local hidden variable model for quantum mechanics. But it's possible to prove that this is impossible, for instance using Bell's Theorem, which shows that the measurement outcomes that one may obtain from $|\\Psi^-\\rangle$ using various unitaries $U$ first (as above) are correlated in a way that no local hidden variable model may reproduce.\n\nIt seems to me that the idea of trying to represent qubits in terms of \"sub-bits\" is an attempt to represent the randomness of single-qubit measurement by a hidden-variable model. Indeed, it doesn't seem capable of anything more than this; and unfortunately it doesn't even do that particularly well \u2014 it just doesn't provide any insight into quantum computation. And any similar model will succumb to the fact that no local hidden-variable theory can represent quantum mechanics.\n\n\u2022 \"What is the significance of the overlaps between those states, and the states you give for strings of Hamming weight 2?\" - It seems to me that the strings with Hamming weight 2 are on the equator of the Bloch sphere, while those with Hamming weight 1 are not so there cannot be overlap. \u2013\u00a0Anixx Nov 26 '13 at 13:27\n\u2022 I have added an update to address some of your questions. \u2013\u00a0Anixx Nov 26 '13 at 15:12\n\u2022 Is this an intuitive mapping, or do you actually have an algorithm for mapping a string to a point on the Bloch sphere? If so, please give it. I don't understand. There is no way of mapping the 16 four-bit binary strings to the surface of the Bloch sphere so that Hamming distance corresponds to spherical distance. \u2013\u00a0Peter Shor Nov 26 '13 at 15:31\n\u2022 @Peter Shor I currently cannot tell you about the exact algorithm, but as I understand it, with a fixed division depth N points with the same Hamming weight occupy the same latitude on the Bloch sphere, placed at the vertices of a right polygon. If to introduce a value S=Hw\/N (Hamming weight devided by division depth), all points with the same S will occupy the same latitude, that is a circle. So both the number of latitudes available, and the points on a latitude depends on the depth N. \u2013\u00a0Anixx Nov 26 '13 at 15:54\n\u2022 @Anixx: if there isn't an exact algorithm, then the sub-bit model of quantum computation isn't an exact model, and thus is almost totally useless. \u2013\u00a0Peter Shor Nov 26 '13 at 15:55","date":"2019-07-20 19:42:04","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\": 0.7385460138320923, \"perplexity\": 444.5160685468115}, \"config\": {\"markdown_headings\": false, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526670.1\/warc\/CC-MAIN-20190720194009-20190720220009-00384.warc.gz\"}"} | null | null |
{"url":"https:\/\/aimsciences.org\/article\/doi\/10.3934\/dcdsb.2011.16.739","text":"# American Institute of Mathematical Sciences\n\nOctober\u00a0 2011,\u00a016(3):\u00a0739-766. doi:\u00a010.3934\/dcdsb.2011.16.739\n\n## Existence of radial stationary solutions for a system in combustion theory\n\n 1 INRA, Equipe BIOSP, Centre de Recherche d'Avignon, Domaine Saint Paul, Site Agroparc, 84914 Avignon cedex 9, France 2 Departamento de Ingenier\u00eda Matem\u00e1tica and CMM (UMI 2807 CNRS), Universidad de Chile, Blanco Encalada 2120 - 5 Piso, Santiago, Chile\n\nReceived\u00a0 June 2010 Revised\u00a0 February 2011 Published\u00a0 June 2011\n\nIn this paper, we construct radially symmetric solutions of a nonlinear non-cooperative elliptic system derived from a model for flame balls with radiation losses. This model is based on a one step kinetic reaction and our system is obtained by approximating the standard Arrehnius law by an ignition nonlinearity, and by simplifying the term that models radiation. We prove the existence of 2 solutions using degree theory.\nCitation: J\u00e9r\u00f4me Coville, Juan D\u00e1vila. Existence of radial stationary solutions for a system in combustion theory. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 739-766. doi: 10.3934\/dcdsb.2011.16.739\n##### References:\n [1] R. Aris, \"The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,\", Clarendon Press, (1975). Google Scholar [2] J. B. van den Berg, V. Guyonne and J. Hulshof, Flame balls for a free boundary combustion model with radiative transfer,, SIAM J. Appl. Math., 67 (2006), 116. doi:\u00a010.1137\/050636516. Google Scholar [3] H. Berestycki and B. Larrouturou, Quelques aspects math\u00e9matiques de la propagation des flammes pr\u00e9m\u00e9lang\u00e9es,, Nonlinear Partial Differential Equations and Their Applications, (1991), 1987. Google Scholar [4] H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory,, Dynamical Issues in Combustion Theory (Minneapolis, (1989), 37. Google Scholar [5] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi:\u00a010.1007\/BF00250555. Google Scholar [6] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits,, SIAM J. Math. Anal., 16 (1985), 1207. doi:\u00a010.1137\/0516088. Google Scholar [7] H. Bockhorn, J. Fr\u00f6hlich and K. Schneider, An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls,, Combustion Theory and Modelling, 3 (1999), 177. doi:\u00a010.1088\/1364-7830\/3\/1\/010. Google Scholar [8] I. Brailovsky and G. I. Sivashinsky, On stationay and travelling flameballs,, Combustion and Flame, 110 (1997), 524. doi:\u00a010.1016\/S0010-2180(97)00001-1. Google Scholar [9] J. Brindley, N. A. Jivraj, J. H. Merkin and S. K. Scott, Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. II. Spherical geometry with Dirichlet boundary conditions,, Proc. Roy. Soc. London Ser. A, 430 (1990), 479. doi:\u00a010.1098\/rspa.1990.0102. Google Scholar [10] J. Buckmaster, G. Joulin and P. Ronney, The structure and stability of nonadiabatic flame balls,, Combustion and Flame, 79 (1990), 381. doi:\u00a010.1016\/0010-2180(90)90147-J. Google Scholar [11] J. Buckmaster, G. Joulin and P. D. Ronney, The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses,, Combustion and Flame, 84 (1991), 411. doi:\u00a010.1016\/0010-2180(91)90015-4. Google Scholar [12] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi:\u00a010.1137\/0151066. Google Scholar [13] R. S. Cantrell and C. Cosner, \"Spatial Ecology Via Reaction-Diffusion Equations,\", Wiley Series in Mathematical and Computational Biology, (2003). Google Scholar [14] P. Cl\u00e9ment and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 97. Google Scholar [15] Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory,, J. Differential Equations, 173 (2001), 213. doi:\u00a010.1006\/jdeq.2000.3932. Google Scholar [16] V. Giovangigli, Nonadiabatic plane laminar flames and their singular limits,, SIAM J. Math. Anal., 21 (1990), 1305. doi:\u00a010.1137\/0521072. Google Scholar [17] L. Glangetas and J.-M. Roquejoffre, Rigorous derivation of the dispersion relation in a combustion model with heat losses,, Preprint Publications du Laboratoire d'Analyse Num\u00e9rique R95032, (1995). Google Scholar [18] V. Guyonne and L. Lorenzi, Instability in a flame ball problem,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 315. Google Scholar [19] V. Guyonne and P. Noble, On a model of flame ball with radiative transfer,, SIAM J. Appl. Math., 67 (2007), 854. doi:\u00a010.1137\/060659612. Google Scholar [20] V. Hutson, J. L\u00f3pez-G\u00f3mez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion,, Dynamical Systems and Applications, (1995), 343. Google Scholar [21] L. Kagan and G. I. Sivashinsky, Self-fragmentation of nonadiabatic cellular flames,, Combust. Flame, 108 (1997), 220. doi:\u00a010.1016\/S0010-2180(96)00108-3. Google Scholar [22] L. Kagan, S. Minaev and G. I. Sivashinsky, On self-drifting flame balls,, Math. Comput. Simulation, 65 (2004), 511. doi:\u00a010.1016\/j.matcom.2004.01.013. Google Scholar [23] A. K. Kapila, B. J. Matkowsky and J. Vega, Reactive-diffusive system with Arrhenius kinetics: peculiarities of the spherical geometry,, SIAM J. Appl. Math., 38 (1980), 382. doi:\u00a010.1137\/0138032. Google Scholar [24] M. K. Kwong, Uniqueness of positive solutions of $\\Delta u-u+u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi:\u00a010.1007\/BF00251502. Google Scholar [25] C. Lederman, J.-M. Roquejoffre and N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames,, Ann. Mat. Pura Appl., 183 (2004), 173. doi:\u00a010.1007\/s10231-003-0085-1. Google Scholar [26] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi:\u00a010.1137\/0151066. Google Scholar [27] B. Lewis and G. von Elbe, \"Combustion, Flames and Explosion of Gases,\", 3rd ed., (1987). Google Scholar [28] M. Marion, Qualitative properties of a nonlinear system for laminar flames without ignition temperature,, Nonlinear Anal., 9 (1985), 1269. doi:\u00a010.1016\/0362-546X(85)90035-5. Google Scholar [29] S. Minaev, L. Kagan, G. Joulin and G. I. Sivashinsky, On self-drifting flame balls,, Combust. Theory Model., 5 (2001), 609. doi:\u00a010.1088\/1364-7830\/5\/4\/306. Google Scholar [30] P. Ronney, Near-Limit Flame Structures at Low Lewis Number,, Combustion and Flame, 82 (1990), 1. doi:\u00a010.1016\/0010-2180(90)90074-2. Google Scholar [31] L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions,, European J. Appl. Math., 16 (2005), 741. doi:\u00a010.1017\/S0956792505006431. Google Scholar [32] G. Sagon, Steady fronts solutions of semilinear elliptic equations in exterior domains arising in flame propagation,, Ann. Mat. Pura Appl., 185 (2006), 273. Google Scholar [33] A. A. Shah, R. W. Thatcher and J. W. Dold, Stability of a spherical flame ball in a porous medium,, Combustion Theory and Modelling, 4 (2000), 511. doi:\u00a010.1088\/1364-7830\/4\/4\/308. Google Scholar [34] Y. B. Zeldovich, A theory of the limit of slow flame propagation,, Zh. Prikl. Mekh. i Tekhn. Fiz., 1 (1941), 159. Google Scholar\n\nshow all references\n\n##### References:\n [1] R. Aris, \"The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,\", Clarendon Press, (1975). Google Scholar [2] J. B. van den Berg, V. Guyonne and J. Hulshof, Flame balls for a free boundary combustion model with radiative transfer,, SIAM J. Appl. Math., 67 (2006), 116. doi:\u00a010.1137\/050636516. Google Scholar [3] H. Berestycki and B. Larrouturou, Quelques aspects math\u00e9matiques de la propagation des flammes pr\u00e9m\u00e9lang\u00e9es,, Nonlinear Partial Differential Equations and Their Applications, (1991), 1987. Google Scholar [4] H. Berestycki, B. Larrouturou and J.-M. Roquejoffre, Mathematical investigation of the cold boundary difficulty in flame propagation theory,, Dynamical Issues in Combustion Theory (Minneapolis, (1989), 37. Google Scholar [5] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Rational Mech. Anal., 82 (1983), 313. doi:\u00a010.1007\/BF00250555. Google Scholar [6] H. Berestycki, B. Nicolaenko and B. Scheurer, Traveling wave solutions to combustion models and their singular limits,, SIAM J. Math. Anal., 16 (1985), 1207. doi:\u00a010.1137\/0516088. Google Scholar [7] H. Bockhorn, J. Fr\u00f6hlich and K. Schneider, An adaptive two-dimensional wavelet-vaguelette algorithm for the computation of flame balls,, Combustion Theory and Modelling, 3 (1999), 177. doi:\u00a010.1088\/1364-7830\/3\/1\/010. Google Scholar [8] I. Brailovsky and G. I. Sivashinsky, On stationay and travelling flameballs,, Combustion and Flame, 110 (1997), 524. doi:\u00a010.1016\/S0010-2180(97)00001-1. Google Scholar [9] J. Brindley, N. A. Jivraj, J. H. Merkin and S. K. Scott, Stationary-state solutions for coupled reaction-diffusion and temperature-conduction equations. II. Spherical geometry with Dirichlet boundary conditions,, Proc. Roy. Soc. London Ser. A, 430 (1990), 479. doi:\u00a010.1098\/rspa.1990.0102. Google Scholar [10] J. Buckmaster, G. Joulin and P. Ronney, The structure and stability of nonadiabatic flame balls,, Combustion and Flame, 79 (1990), 381. doi:\u00a010.1016\/0010-2180(90)90147-J. Google Scholar [11] J. Buckmaster, G. Joulin and P. D. Ronney, The structure and stability of nonadiabatic flame balls: II. Effects of far-field losses,, Combustion and Flame, 84 (1991), 411. doi:\u00a010.1016\/0010-2180(91)90015-4. Google Scholar [12] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi:\u00a010.1137\/0151066. Google Scholar [13] R. S. Cantrell and C. Cosner, \"Spatial Ecology Via Reaction-Diffusion Equations,\", Wiley Series in Mathematical and Computational Biology, (2003). Google Scholar [14] P. Cl\u00e9ment and G. Sweers, Existence and multiplicity results for a semilinear elliptic eigenvalue problem,, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 14 (1987), 97. Google Scholar [15] Y. Du and Y. Lou, Proof of a conjecture for the perturbed Gelfand equation from combustion theory,, J. Differential Equations, 173 (2001), 213. doi:\u00a010.1006\/jdeq.2000.3932. Google Scholar [16] V. Giovangigli, Nonadiabatic plane laminar flames and their singular limits,, SIAM J. Math. Anal., 21 (1990), 1305. doi:\u00a010.1137\/0521072. Google Scholar [17] L. Glangetas and J.-M. Roquejoffre, Rigorous derivation of the dispersion relation in a combustion model with heat losses,, Preprint Publications du Laboratoire d'Analyse Num\u00e9rique R95032, (1995). Google Scholar [18] V. Guyonne and L. Lorenzi, Instability in a flame ball problem,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 315. Google Scholar [19] V. Guyonne and P. Noble, On a model of flame ball with radiative transfer,, SIAM J. Appl. Math., 67 (2007), 854. doi:\u00a010.1137\/060659612. Google Scholar [20] V. Hutson, J. L\u00f3pez-G\u00f3mez, K. Mischaikow and G. Vickers, Limit behaviour for a competing species problem with diffusion,, Dynamical Systems and Applications, (1995), 343. Google Scholar [21] L. Kagan and G. I. Sivashinsky, Self-fragmentation of nonadiabatic cellular flames,, Combust. Flame, 108 (1997), 220. doi:\u00a010.1016\/S0010-2180(96)00108-3. Google Scholar [22] L. Kagan, S. Minaev and G. I. Sivashinsky, On self-drifting flame balls,, Math. Comput. Simulation, 65 (2004), 511. doi:\u00a010.1016\/j.matcom.2004.01.013. Google Scholar [23] A. K. Kapila, B. J. Matkowsky and J. Vega, Reactive-diffusive system with Arrhenius kinetics: peculiarities of the spherical geometry,, SIAM J. Appl. Math., 38 (1980), 382. doi:\u00a010.1137\/0138032. Google Scholar [24] M. K. Kwong, Uniqueness of positive solutions of $\\Delta u-u+u^p=0$ in $R^n$,, Arch. Rational Mech. Anal., 105 (1989), 243. doi:\u00a010.1007\/BF00251502. Google Scholar [25] C. Lederman, J.-M. Roquejoffre and N. Wolanski, Mathematical justification of a nonlinear integrodifferential equation for the propagation of spherical flames,, Ann. Mat. Pura Appl., 183 (2004), 173. doi:\u00a010.1007\/s10231-003-0085-1. Google Scholar [26] C. J. Lee and J. Buckmaster, The structure and stability of flame balls: a near-equidiffusional flame analysis,, SIAM J. Appl. Math., 51 (1991), 1315. doi:\u00a010.1137\/0151066. Google Scholar [27] B. Lewis and G. von Elbe, \"Combustion, Flames and Explosion of Gases,\", 3rd ed., (1987). Google Scholar [28] M. Marion, Qualitative properties of a nonlinear system for laminar flames without ignition temperature,, Nonlinear Anal., 9 (1985), 1269. doi:\u00a010.1016\/0362-546X(85)90035-5. Google Scholar [29] S. Minaev, L. Kagan, G. Joulin and G. I. Sivashinsky, On self-drifting flame balls,, Combust. Theory Model., 5 (2001), 609. doi:\u00a010.1088\/1364-7830\/5\/4\/306. Google Scholar [30] P. Ronney, Near-Limit Flame Structures at Low Lewis Number,, Combustion and Flame, 82 (1990), 1. doi:\u00a010.1016\/0010-2180(90)90074-2. Google Scholar [31] L. Roques, Study of the premixed flame model with heat losses. The existence of two solutions,, European J. Appl. Math., 16 (2005), 741. doi:\u00a010.1017\/S0956792505006431. Google Scholar [32] G. Sagon, Steady fronts solutions of semilinear elliptic equations in exterior domains arising in flame propagation,, Ann. Mat. Pura Appl., 185 (2006), 273. Google Scholar [33] A. A. Shah, R. W. Thatcher and J. W. Dold, Stability of a spherical flame ball in a porous medium,, Combustion Theory and Modelling, 4 (2000), 511. doi:\u00a010.1088\/1364-7830\/4\/4\/308. Google Scholar [34] Y. B. Zeldovich, A theory of the limit of slow flame propagation,, Zh. Prikl. Mekh. i Tekhn. Fiz., 1 (1941), 159. Google Scholar\n [1] Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi, Gregory I. Sivashinsky. A fully nonlinear equation for the flame front in a quasi-steady combustion model. Discrete & Continuous Dynamical Systems - A, 2010, 27 (4) : 1415-1446. doi: 10.3934\/dcds.2010.27.1415 [2] Kunquan Lan, Wei Lin. Uniqueness of nonzero positive solutions of Laplacian elliptic equations arising in combustion theory. 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Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 461-475. doi: 10.3934\/dcds.2014.34.461\n\n2018\u00a0Impact Factor:\u00a01.008","date":"2019-07-19 07:48:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5336595177650452, \"perplexity\": 6378.049399881735}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526153.35\/warc\/CC-MAIN-20190719074137-20190719100137-00066.warc.gz\"}"} | null | null |
\section{Title:}
Variable slip coefficient in binary lattice Boltzmann models\\
\section{Author:}
Lajos Szalm\'as\\
\section{Address:}
Department of Physics, University of Rome "La Sapienza", Piazzale Aldo Moro 2, Rome 00185, Italy\\
\section{Email:}lszalmas@gmail.com\\
\section{Abstract}
We present a new method in order to get variable slip coefficient in
binary lattice Boltzmann models to simulate gaseous flows. Boundary layer theory is
presented. We study both the single- and multi-fluid BGK-type models as well. The
boundary slip and the Knudsen layer are analyzed in detail. Benchmark simulations
are carried out in order to compare the analytical derivation with the numerical
results. Excellent agreement is found between the two situations.\\
\\
\section{Keywords:}
lattice Boltzmann, binary mixtures, slip coefficient, microflows, Knudsen number\\
\\
\section{PACS:}
05.20.Dd,47.11.-j,47.45.Ab\\
\section{Introduction}
Over the last decades, the lattice Boltzmann model (LB) has become a versatile tool in modeling
complex hydrodynamic problems \cite{ref1}. Recently, it has attracted considerable attention in
modeling rarefied gas flows. These flows are important in several applications ranging
from micro-electromechanical systems (MEMS) \cite{ref1a} to aeronautical applications.
With increasing rarefaction characterized by the Knudsen number, the ratio of the molecular mean-
free path and the macroscopic sizes, the flow experiences some interesting phenomena, such as the
gaseous slip along the solid wall. The slip phenomenon is one of the cornerstone of micro-channel
applications. In LB, several methods have been proposed for the description of the slip flow,
mostly based on kinetic boundary conditions \cite{ref2,ref3,ref4,ref5,ref6,ref7,ref8,ref8a}, and
analytical formulas have been derived for the slip velocity and the slip coefficient.
We mention that the first kinetic boundary condition was introduced by Lim et al. \cite{ref8a}.}
The above approaches focused on one-component gases and the slip phenomenon in mixture LB models is yet
not understood. In addition, it is worth mentioning that the bare diffuse reflection
boundary condition can not reproduce the correct value of the slip coefficient \cite{ref11}.
In this paper, we present a boundary layer theory for binary lattice Boltzmann models and
introduce a new boundary treatment in order to get tunable slip coefficient in the method.
We consider both single and multicomponent BGK-type collision operators as well. In the
framework of the boundary layer theory, analytical formula is derived for the slip coefficient.
Simulations are performed to validate the model. The analytical results are in excellent
agreement with the ones obtained from the numerical simulations.
\section{The binary mixture model}
We begin our consideration with the binary discrete Boltzmann equation for single- and multi-fluid
BGK-type collision operators
\begin{eqnarray}
\label{1}
\partial_t f^\sigma_i(\bm{x},t) + c_{\sigma ia} \partial_a f^\sigma_i(\bm{x},t)&=&
(\nu_1+\nu_m)[ f^{\sigma(e)}_i(\rho_\sigma,\bm{u}_\sigma)-f^\sigma_i]+\nonumber\\
&&\nu_m[f^{\sigma (e)}(\rho_\sigma,\bm{u})-f_i^{\sigma (e)}(\rho_\sigma,\bm{u}_\sigma)],
\end{eqnarray}
where $f^\sigma_i,f_i^{\sigma (e)}$ denote the single-specie distribution function and equilibrium
distribution function, $c_{\sigma ia}$ are the discrete speed vectors and $\nu_1,\nu_m$ are
the collision frequencies related to the transport coefficients.
The macroscopic quantities
are obtained as the moments of $f^\sigma_i$ as
\begin{eqnarray}
\label{2}
n_\sigma=\sum_k f^\sigma_k, \quad n_\sigma \bm{u}_\sigma=\sum_k f^\sigma_k \bm{c}_{\sigma k},
\end{eqnarray}
where $n_\sigma,\bm{u}_\sigma$ are the particle density and velocity, respectively.
Further,
\begin{eqnarray}
\label{3}
\rho_\sigma=n_\sigma m_\sigma,
\end{eqnarray}
where $\rho_\sigma,m_\sigma$ are the specie mass density and the specie mass, respectively.
The mixture macroscopic quantities are defined by
\begin{eqnarray}
\label{4}
\quad n=\sum_\sigma n_\sigma, \quad \rho=\sum_\sigma \rho_\sigma,
\quad \rho \bm u=\sum_\sigma \rho_\sigma \bm{u}_\sigma,
\end{eqnarray}
where $n,\rho,\bm{u}$ denote the mixture particle density, mixture mass density and
mass-averaged mixture velocity, respectively. The generic equilibrium distribution
function is chosen as the second order truncation of the Maxwellian
\begin{eqnarray}
\label{5}
f^{\sigma (e)}_i( \rho_\sigma,\bm{u})=w_i \frac{\rho_\sigma}{m_\sigma} \left[ 1+\frac{c_{\sigma ia} u_{a}}{c_\sigma^2}+
\frac{(c_{ \sigma ia}c_{ \sigma ib}-c_\sigma ^2\delta_{ab})u_{a}u_{b}}{2 c_\sigma^4} \right].
\end{eqnarray}
We work in two dimensions, and the discrete speed vectors are given by $\bm{c}_{\sigma i}=c_\sigma \bm{e}_i$,
where $\bm{e}_i$ is the defined as the D2Q9 non-dimensional velocity model
\begin{eqnarray}
\bm e_{i}=\left\{
\begin{aligned}
&(0,0) && \mbox{for} \quad i=0\\
&\left(\sqrt{3}\cos(\frac{\pi}{2}i-\frac{\pi}{2}),\sqrt{3}\sin(\frac{\pi}{2}i-\frac{\pi}{2})\right) && \mbox{for} \quad i=1\dots 4\\
&\left(\sqrt{6}\cos(\frac{\pi}{2}i-\frac{\pi}{4}),\sqrt{6}\sin(\frac{\pi}{2}i-\frac{\pi}{4})\right) && \mbox{for} \quad i=5\dots 8.
\label{4a}
\end{aligned}
\right.
\end{eqnarray}
The corresponding weights are defined by
\begin{eqnarray}
w_{i}=\left\{
\begin{aligned}
&4/9 && \mbox{for} \quad i=0\\
&1/9 && \mbox{for} \quad i=1\dots 4\\
&1/36 && \mbox{for} \quad i=5\dots 8.
\label{4b}
\end{aligned}
\right.
\end{eqnarray}
Note that this velocity model is different from the original nine-speed model
\cite{ref16}. The velocity vectors are rescaled with the $\sqrt{3}$ multiplier. This is
a convenient choice for our purposes and our calculations. The definition of the model is given
by the above equations, Eq. (\ref{4a},\ref{4b}). To complete our definition, we mention
that the non-dimensional sound speed of the model is unity.}
The specie sound speed is given by $c_\sigma=c\sqrt{m/m_\sigma}$, where $m=\rho/n$ is the
averaged mass and $c$ is the mixture sound speed.
Before going to the boundary layer theory, some comments are in order. The collision operator
in the governing equation, Eq. (\ref{1}), is equivalent with the so-called Hamel model \cite{ref10},
which describes coupled relaxation towards the local specie and mixture equilibriums \cite{ref11,ref12}. With the choice
of $\nu_1=0$, the collision model results in the single-fluid BGK model. In the two-fluid model, the mixture
viscosity is given by $\mu=\rho c^2/ (\nu_1+\nu_m)$ and the diffusivity takes the value of
$D=m^2 c^2/(m_1 m_2 \nu_m)$.
\section{Boundary layer theory}
We turn our attention to the half space problem experienced by a gas flow along a solid
wall. This is the so-called Kramers problem in kinetic theory \cite{ref12a}. The solid wall lies in the
$y$ direction and is located at $x=0$. In the $x>0$ region, the gas medium exhibits a shear flow.
Under these circumstances, we are looking for the macroscopic velocity obtained from Eq. (\ref{1}).
This problem can be simplified by introducing a new reduced distribution function instead of
$f_i^\sigma$
\begin{eqnarray}
\label{6}
F_1^\sigma&=&m_\sigma c_\sigma \sqrt{3} [f^\sigma_6-f^\sigma_7],\nonumber\\
F_0^\sigma&=&m_\sigma c_\sigma \sqrt{3} [f^\sigma_2-f^\sigma_4],\nonumber\\
F_2^\sigma&=&m_\sigma c_\sigma \sqrt{3} [f^\sigma_5-f^\sigma_8].
\end{eqnarray}
We also introduce the following weights, $\omega_i$, and one-dimensional velocity vectors, $\epsilon_i$,
corresponding to the new distribution function in such a way that
\begin{eqnarray}
\label{7a}
&\omega_1=1/6,\quad \omega_0=2/3, \quad \omega_2=1/6,\\
\label{7b}
&\epsilon_1=-\sqrt{3}, \quad \epsilon_0=0, \quad \epsilon_2=\sqrt{3}.
\end{eqnarray}
With using the reduced distribution function, the macroscopic specie velocity is obtained by
$\rho_\sigma u_{\sigma y}=\sum_{k} F_k^\sigma$.
The reduced distribution function obeys the following governing equation
obtained from Eq. (\ref{1}) in the steady state
\begin{eqnarray}
\label{8a}
c_\sigma \epsilon_i \partial_x F_i^\sigma&=&\nu_1\omega_i\sum_k F_k^\sigma+\nu_m\omega_i\frac{\rho_\sigma}{\rho}\sum_k \sum_{\sigma'} F_k^{\sigma'}-(\nu_1+\nu_m)F_i^\sigma.
\end{eqnarray}
This equation is the boundary layer equation, which needs to be solved for the macroscopic
velocity. The solution of this differential equation is obtained in the form $F_i^\sigma=F_i^{\sigma (0)} \exp(\lambda x)$.
As a consequence, we obtain the following generalized eigenvalue problem
\begin{eqnarray}
\label{9a}
c_\sigma \epsilon_i \lambda F_i^{\sigma (0)}&=&\nu_1\omega_i\sum_k F_k^{\sigma (0)}+\nu_m\omega_i\frac{\rho_\sigma}{\rho}\sum_k \sum_{\sigma'} F_k^{\sigma' (0)}-(\nu_1+\nu_m)F_i^{\sigma (0)}.
\end{eqnarray}
This equation is written out in components in the Appendix, Eq. (\ref{9}). The eigenvalues of the problem
are obtained by $\lambda=[0,0,\lambda_0,-\lambda_0]$, where $\lambda_0$ is listed in the Appendix, Eq. (\ref{10}).
The two zero eigenvalues correspond to a linear flow profile, while the other two to a growing and a
decaying exponential solution. We remind that in the single component D2Q9 models
the non-zero eigenvalues are absent. However, these two solutions in the mixture case describe the Knudsen
layer function fading away in order of the mean free path far from the wall.
The linear flow profile corresponds to the hydrodynamic shear flow in the bulk region. After some straightforward
algebra, one can obtain the linear solution from Eq. (\ref{8a}) as
\begin{eqnarray}
\label{11}
F^{\sigma(H)}_i=\omega_i \rho_\sigma u^{(H)}_y-\omega_i \frac{c_\sigma \epsilon_i}{\nu_1+\nu_m} \rho_\sigma \frac{\partial u^{(H)}_y}{\partial x},
\end{eqnarray}
where the $H$ superscript denotes the hydrodynamic part of the distribution
function and the velocity.
The general solution of the boundary layer equation, Eq. (\ref{8a}), is given by the linear combination
of the hydrodynamic shear solution, $F^{\sigma(H)}$, and a Knudsen layer part corresponding to the
nonzero eigenvalues.
Here, we mention that in our work, the Knudsen layer concept is
identical with the boundary layer provided by the
established model and obtained from the analytical derivation. This is because the LB model
is a special discrete ordinate method for the solution of the Boltzmann equation \cite{ref17,ref18}.
As it has been recently shown, Knudsen layer appears in discrete kinetic models
once the reflected populations have different $c_{ix}$ normal components \cite{ref18}.
As a result, the boundary layer appearing }
in the model is physically relevant. However, in
order to obtain quantitatively correct information, one needs to
apply optimized boundary condition presented below.}
Before determining the slip coefficient in the model, we theoretically show that the shear stress is
independent of the Knudsen layer part. Indeed, the shear stress obtained from the
Knudsen layer solution is given by $P^K_{xy}=\sum_k \sum_\sigma m_\sigma c_{\sigma kx}c_{\sigma ky} f_k^\sigma
=\sum_i \sum_\sigma c_\sigma \epsilon_i F^{\sigma(0)}_i \exp (\lambda x)$.
It is quickly realized that $P^K_{xy}$ is always zero for $\lambda \neq 0$, because
$\lambda \sum_i \sum_\sigma c_\sigma \epsilon_i F^{\sigma (0)}_i$ is nothing just the sum of the left
hand side of all equations of Eq. (\ref{9}), which is identically zero because the
right hand side becomes the collision invariant of the momenta for the overall mixture.
This is an important property of the half space problem and remains valid for other velocity
models as well. As a result, the shear stress at the wall can be
obtained from the hydrodynamic solution, Eq. (\ref{11}), by
\begin{eqnarray}
\label{13}
P_{xy}=\sum_i \sum_\sigma c_\sigma \epsilon_i F^{\sigma(H)}_i=-\rho \frac{c^2}{\nu_1+\nu_m} \frac{\partial u^{(H)}_y}{\partial x}.
\end{eqnarray}
In this way, the velocity gradient is always available from the shear stress independently
of the unknown Knudsen layer function.
\section{Variable slip coefficient}
In rarefied gas flows, the slip velocity, $u_s$, is considered as the \textit{extrapolated} gas
velocity at the wall without the Knudsen layer. It is given in the following form
\begin{eqnarray}
\label{15}
u_s=\alpha_v \frac{\mu v_0}{P} \frac{\partial u^{(H)}_y}{\partial x},
\end{eqnarray}
where $\alpha_v,\mu,v_0,P$ are the slip coefficient, the viscosity, the mixture reference
speed and the pressure, respectively. In the present model, $v_0=\sqrt{2} c$ and $P=\rho c^2$.
In Eq. (\ref{15}), $\alpha_v$ is the so-called viscosity based slip coefficient
\cite{ref15a,ref15b}, which is a useful definition, since it connects the slip coefficient to a
transport coefficient, which is always available independently of the molecular details of the
rarefied gas.}
In order to get the slip coefficient, we need to determine the magnitude of the hydrodynamic solution in
the boundary layer. This is fixed by the used boundary condition at the wall.
We introduce a modified diffuse boundary condition in the model in order to get variable
slip coefficient. The boundary treatment for the reflected populations, $e_{ix}>0$, are
written by
\begin{eqnarray}
\label{16}
f_i^\sigma=f_i^{\sigma(e)}(\rho_\sigma,\bm{0})+(1-a) w_i \frac{c_{\sigma ix} c_{\sigma iy}}{c^4} \frac{\rho_\sigma}{\rho m} P_{xy},
\end{eqnarray}
where $a$ is a yet unknown parameter, which will be determined below. For the quantity,
$F_i^{\sigma}$, the above boundary condition is written by
\begin{eqnarray}
\label{18}
F_i^{\sigma}=(1-a) \omega_2 \sqrt{3} \frac{c_{\sigma}}{c^2} \frac{\rho_\sigma}{\rho} P_{xy}.
\end{eqnarray}
The particular solution of Eq. (\ref{8a}) is given by the linear combination of the
decaying exponential solution and the hydrodynamic part. The magnitude of hydrodynamic solution and
the Knudsen layer part is determined by the boundary condition, Eq. (\ref{18}), in such a way that
\begin{eqnarray}
\label{20}
(1-a) \omega_2 \sqrt{3} \frac{c_{1}}{c^2} \frac{\rho_1}{\rho} P_{xy}&=&
\omega_2 \rho_1 u_s- \omega_2 \rho_1 \frac{\sqrt{3} c_{1}}{\nu_1+\nu_m}\frac{\partial u^{(H)}_y}{\partial x}\nonumber\\
&&+k F^{1(0)}_2,\\
\label{21}
(1-a) \omega_2 \sqrt{3} \frac{c_{2}}{c^2} \frac{\rho_2}{\rho} P_{xy}&=&
\omega_2 \rho_2 u_s- \omega_2 \rho_2 \frac{\sqrt{3} c_{2}}{\nu_1+\nu_m}\frac{\partial u^{(H)}_y}{\partial x}\nonumber\\
&&+k F^{2(0)}_2.
\end{eqnarray}
These equations need to be solved for the unknown quantities, $u_s,k$. It is realized that for obtaining
the slip velocity, we need the ratio $F^{1(0)}_2/F^{2(0)}_2$ corresponding to the decaying eigenvalue
$-\lambda_0$. This can be obtained from Eq. (\ref{9}) and the result is given in the Appendix, Eq. (\ref{25}).
After some straightforward
calculation, we obtain the slip velocity from Eq. (\ref{20}-\ref{21}) as
\begin{eqnarray}
\label{22}
u_s=a B \frac{\mu v_0}{P} \frac{\partial u^{(H)}_y}{\partial x},
\end{eqnarray}
where the obtained value of the $B$ parameter is given by Eq. (\ref{23}) in the Appendix.
It can be seen that we get the desired value of the slip coefficient with the choice $a=\alpha_v/B$.
\begin{figure}
\includegraphics[height=5.5cm,width=8cm]{figure1.eps}
\caption{\label{}Comparison between the measured slip coefficient obtained from the LB simulation
versus the theoretical derivation at different values of the parameter, $a$.
$\bullet,\vartriangle,\blacktriangle$ correspond to the values of $m_1/m_2=[1/9.98,1/9.98,1/32.804],
n_1/n=[0.5,0.2,0.6]$, respectively, and represent the results obtained from the LB simulations. $+$ denotes
the analytical results. The collision frequencies are chosen by $v_1=1/(0.05(N-1)),v_m=1/(0.03(N-1))$.
}
\end{figure}
\section{Simulations}
Simulations were performed in order to validate the analytical results obtained above.
The governing equation Eq. (\ref{1}) was solved using the finite difference
method. We mention that the finite difference realization is a useful choice because of the
independence of the velocity vectors and the coordinate grid \cite{ref13,ref13a}. The
time and the space derivatives are computed using the modified Runge-Kutta and midpoint method, respectively
\cite{ref13}.
Note that this scheme is \textit{second order} accurate in both time and space \cite{ref20}.}
The Kramers problem is simulated in an $N\times1$ domain, $N=128$. The left hand side wall is at rest,
while the right hand the side moves with a
constant velocity to maintain the constant shear flow in the domain. The proposed
boundary condition with the variable slip coefficient is applied at the left wall,
where the slip coefficient is measured through the relation of Eq. (\ref{15}). Fig. 1 presents
the measured slip coefficient versus the theoretical value obtained from Eq. (\ref{22})
at different values of $a$. Excellent agreement is obtained between the two situations.
The proposed model can be used to tune the value of the slip coefficient.
Note that above boundary condition can be used in non-stationary case as well.
Before concluding our work, we mention that the boundary layer theory can be further
developed involving other flow problems, such as pressure driven flows. In that case, the
boundary layer equation can be also solved which provides a useful background to find an
optimal boundary treatment. An another interesting issue is the description of complex
geometries, slip flow on curved surfaces. For the latter situation, Ref. \cite{ref19}
can be generalized, which describes variable slip coefficient in the single component LB model
on curved boundaries.
\section{Conclusion}
In this paper, we have developed a new boundary treatment in the LB method for binary
mixtures in order to obtain variable slip coefficient. We have set up a boundary layer
analysis and derived the analytical value of the slip coefficient. Computer simulations
have been performed to validate the model. The results of the simulations are in excellent
agreement with the analytical derivation. Our method can be used to obtain the desirable
value of the slip coefficient in the lattice Boltzmann model.
\section{Appendix}
The generalized eigenvalue problem, Eq. (\ref{9a}), written out in components is given by
\begin{eqnarray}
\label{9}
-c_1 \sqrt{3} \lambda F_1^{1(0)}&=&\nu_1 \frac{1}{6} \sum_k F_k^{1(0)} +\nu_m \frac{1}{6} \frac{\rho_1}{\rho} \sum_k (F_k^{1(0)}
+F_k^{2(0)})-(\nu_1+\nu_m)F_1^{1(0)},\nonumber\\
0&=&\nu_1 \frac{2}{3} \sum_k F_k^{1(0)} +\nu_m \frac{2}{3} \frac{\rho_1}{\rho} \sum_k (F_k^{1(0)}+F_k^{2(0)})-(\nu_1+\nu_m)F_0^{1(0)},\nonumber\\
c_1 \sqrt{3} \lambda F_2^{1(0)}&=&\nu_1 \frac{1}{6} \sum_k F_k^{1(0)} +\nu_m \frac{1}{6} \frac{\rho_1}{\rho} \sum_k (F_k^{1(0)}
+F_k^{2(0)})-(\nu_1+\nu_m)F_2^{1(0)},\nonumber\\
-c_2 \sqrt{3} \lambda F_1^{2(0)}&=&\nu_1 \frac{1}{6} \sum_k F_k^{2(0)} +\nu_m \frac{1}{6} \frac{\rho_2}{\rho} \sum_k (F_k^{1(0)}
+F_k^{2(0)})-(\nu_1+\nu_m)F_1^{2(0)},\nonumber\\
0&=&\nu_1 \frac{2}{3} \sum_k F_k^{2(0)} +\nu_m \frac{2}{3} \frac{\rho_2}{\rho} \sum_k (F_k^{1(0)}+F_k^{2(0)})-(\nu_1+\nu_m)F_0^{2(0)},\nonumber\\
c_2 \sqrt{3} \lambda F_2^{2(0)}&=&\nu_1 \frac{1}{6} \sum_k F_k^{2(0)} +\nu_m \frac{1}{6} \frac{\rho_2}{\rho} \sum_k (F_k^{1(0)}
+F_k^{2(0)})-(\nu_1+\nu_m)F_2^{2(0)}.\nonumber\\
\end{eqnarray}
The non-zero eigenvalue of the eigenvalue problem is obtained by
\begin{eqnarray}
\label{10}
\lambda_0=\frac{\sqrt{\nu_m}}{\sqrt{\nu_1+3\nu_m}} \frac{\sqrt{m_1 m_2}}{m} \frac{\nu_1+\nu_m}{c}.
\end{eqnarray}
The ratio of the eigenvalues, $F^{1(0)}_2/F^{2(0)}_2$, corresponding to the decaying eigenvalue $-\lambda_0$
can be obtain from Eq. (\ref{9}) in such a way that
\begin{eqnarray}
\label{25}
F_2^{1(0)}&=&c_2^2(\nu_1+\nu_m+c_1 \sqrt{3} \lambda_0) k,\nonumber\\
F_2^{2(0)}&=&-c_1^2(\nu_1+\nu_m+c_2 \sqrt{3} \lambda_0) k,
\end{eqnarray}
where $k$ is an arbitrary multiplier.
The parameter in the slip velocity, Eq. (\ref{22}), is given in such a way that
\begin{eqnarray}
\label{23}
B=\sqrt{\frac{3}{2}} \frac{n_1 m/ \sqrt{m_1}+n_2 m/ \sqrt{m_2}+n \sqrt{m} A }{n\sqrt{m}+n_1\sqrt{m_1}A+n_2\sqrt{m_2}A},
\end{eqnarray}
with
\begin{eqnarray}
\label{24}
A=\sqrt{3} \frac{\sqrt{v_m}}{\sqrt{v_1+3v_m}}.
\end{eqnarray}
| {
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{"url":"https:\/\/brilliant.org\/problems\/telescoping-the-reciprocals\/","text":"# Telescoping the Reciprocals!\n\nAlgebra Level 4\n\n$\\begin{eqnarray} \\frac{3}{1!+2!+3!} + \\frac{4}{2!+3!+4!} + \\frac{5}{3!+4!+5!} + \\cdots+ \\frac{100}{98!+99!+100!} \\end{eqnarray}$\n\nFind the value of the expression above.\n\nThe answer is a form of $$\\frac{1}{a!} - \\frac{1}{b!}$$. Submit your answer as $$a \\times b$$.\n\nNotation: $$!$$ is the factorial notation. For example, $$8! = 1\\times2\\times3\\times\\cdots\\times8$$.\n\n\u00d7","date":"2018-09-21 01:12:08","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\": 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\": 0.8510593771934509, \"perplexity\": 1231.662606361483}, \"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-2018-39\/segments\/1537267156690.32\/warc\/CC-MAIN-20180920234305-20180921014705-00145.warc.gz\"}"} | null | null |
Governments should be thinking now about how they would allocate a safe and effective COVID-19 vaccine
Once a safe and effective COVID-19 vaccine is approved for use by regulators, most countries will not have enough doses to vaccinate everyone. University of Oxford researchers have developed a comprehensive framework to help governments distribute a vaccine effectively and fairly.
The study, made possible with NIHR funding, and published today in the international journal Pharmacoeconomics Open outlines the necessary decisions governments must make regarding allocation of a vaccine. Key considerations must include:
the health impacts of COVID-19;
reducing the rate of transmission;
allowing the economy to return to normal;
equity (particularly as COVID-19 has been shown to have disproportionate impacts on socially disadvantaged groups).
While there is a lot of emerging evidence about the impact COVID-19 is having on people's health and the economy, there is a need to bring this together into an evidence-based framework for understanding the potential benefit of adopting different prioritisation strategies. For example, when determining the degree to which those working in the retail and service sector should be prioritised for vaccination there is a need to take into account: (i) individual health benefits to staff; (ii) societal health benefits via lower transmission to shoppers; (iii) benefits to the economy (e.g. from allowing more shops to open).
Study co-author Dr Laurence Roope, of the Health Economics Research Centre (HERC) in the Nuffield Department of Population Health, said 'Without a clear plan, governments may be forced into quick decisions and may not realise the full potential of a vaccine.
'Many governments rely on expert bodies involving clinicians, epidemiologists and health economists that use evidence to guide who should receive new medicines. It should be no different when allocating a COVID-19 vaccine.'
The study suggests that beyond synthesising evidence from research to determine the benefits of a vaccine, it is important that researchers and governments seek public opinion regarding several key aspects of the vaccine distribution:
whether governments allow their citizens to purchase a vaccine privately, or solely via public channels such as the UK's National Health Service (NHS). While most medicines can be purchased privately, the COVID pandemic is much more like a war-time situation and in the past vital goods in limited simply have been rationed.
the degree to which a vaccination prioritisation strategy should be focused on health versus wider benefits such as facilitating a safe return to education, protecting jobs and the economy.
whether governments should randomly allocate the vaccine if there is not enough vaccine available to cover all individuals assigned the same priority.
Professor Philip Clarke, Director of the Health Economics Research Centre in the Nuffield Department of Population Health, said 'Engaging with the public by conducting nationally representative surveys would enable researchers and governments to better tailor vaccine strategies.
'Given the likelihood that vaccine supplies will be limited, there is a need for a transparent evidence-based framework for prioritising who gets preferential access to it.'
In the race for an effective COVID-19 vaccine, there is a need for governments, researchers and the public to think now about what happens after one of the vaccines crosses the finishing line.
This research was made possible by National Institute for Health Research funding and supported by the NIHR Oxford Biomedical Research Centre. | {
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} | 6,388 |
The PO-552 is a road from the Galician Regional Roads Network. It constitutes one of the major transport systems for the coastal towns of the southern area of the Province of Pontevedra, in Galicia, Spain. It links the city of Vigo with Tui and the Portuguese border, parallel to the Ria of Vigo, the Atlantic Ocean and the lower course of the Miño River.
Roads in Spain | {
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} | 532 |
\section{Introduction}
Explicit expressions for Picard-Fuchs equations (or Gauss-Manin connections in a general context) attached to families of
algebraic varieties are usually huge even if the corresponding family is simple, for examples see the first author's homepage. However, there are some families of algebraic varieties for which
such expressions are small enough to fit into a mathematical paper, but one is not able to
calculate them through the Dwork-Griffiths method (see for instance \cite{gr69}) or its modification in the context of Brieskorn modules, see \cite{ho06-1}
(we call this algebraic method).
For such families, we first compute a period and then the corresponding Picard-Fuchs equation, see for instance \cite{alenstzu}
(we call this transcendental method).
The main reason why computing Picard-Fuchs equations fails through the algebraic method is that in this way we produce huge polynomials and the
Groebner basis algorithm fails to work. The transcendental method is restricted to a very particular families of algebraic varieties.
In this article, we propose a new method which uses
the internal fibration structure of algebraic varieties in order to perform Picard-Fuchs equation computations. It involves only
solving linear equations and it is a generalization of the classical convolution of solutions of fuchsian differential
equations and Deligne's work on the cohomology with coefficients in a local system, see \cite{de70/1}. One of our main motivations
for the present work is the increasing need for explicit expressions of Picard-Fuchs equations in
Topological String Theory and in particular in the B-model of mirror symmetry, see for instance \cite{can91}.
Let us be given:
$$
\begin{array}{lllll}
× & × & X_1 & × & ×\\
× & \stackrel{}{\swarrow} & × & \stackrel{}{\searrow} & ×\\
\P^1_y & × & × & × & \P^1_x \\
× & \stackrel{}{\nwarrow}& × & \stackrel{}{\nearrow} & ×\\
× & × & X_2 & × & ×
\end{array}\ \ \
$$
and a global meromorphic section $\omega_i,\ i=1,2,$ of the $n_i$-th cohomology bundle
of $X_i\to \P^1_x\times \P^1_y$. Here, $X_i,\ i=1,2,$ are two algebraic varieties over $\C$, $\P_*^1,\ *=x,y,$ is the projective line with the coordinate system $*$ and all
the arrows are morphisms of algebraic varieties. The convolution of the above data
in the framework of Algebraic Geometry is simply the fiber product
$$
X\rightarrow \P^1_y,\ \ \
X:=\cup_{y\in \P ^1_y} X_{1,y} \times_{\P ^1_x}X_{2,y},\ \ \
\omega:=dx\wedge \omega_1\wedge \omega_2.
$$
Here, $X_{i,y},\ i=1,2,$ is the fiber of $X_i\to \P^1_y$
over the point $y\in\P^1_y$ and $\omega$ gives us a global meromorphic section of the $(n_1+n_2+1)$-the cohomology bundle of $X\to \P^1_y$.
Let $\delta_{i,x,y},\ i=1,2$ be a continuous family of cycles in the fibers of $X_i\to \P^1_x\times \P^1_y$.
Knowing the Picard-Fuchs equation of the periods $p_i(x,y):=\int_{\delta_{i,x,y}}\omega_i$,
we give an algorithm for computing the Picard-Fuchs equation of
\begin{equation}
\label{16july2013}
p(y):=\int_{\delta}p_1(x,y)p_2(x,y)dx
\end{equation}
where $\delta$ is any closed path in the $x$-plane such that $p_1$ and $p_2$ along $\delta$ are one valued. This integral can be written as
the integration of $\omega$ over a cycle $\tilde \delta_y \in H_{n_1+n_2+1}(X_y,\Z)$.
\section{Cohomology with coefficients in a local system}
\label{section1}
\def{\mathcal O}{{\mathcal O}}
In this section we remind some basic facts on local systems and connections with regular singularities.
For further details, the reader is referred to \cite{de70/1}. We fix a field $\k$ of characteristic zero and
not necessarily algebraically closed and work over the category of algebraic varieties over $\k$. If $\k$ is a subfield of $\C$ or $\C(t)$,
where $t$ is a multi-parameter, then for an algebraic variety $M$ over $\k$ we use the same letter $M$ to denote
the underlying complex variety or family of varieties; being clear in the text which we mean.
\subsection{Flat connections}
Let $M$ be a smooth variety, $E$ be a vector bundle over $M$.
We consider a flat regular connection
$$
\nabla: E\to \Omega^1(E).
$$
We use the same notation $E$ for both the vector bundle and the sheaf of its sections.
We have the induced maps
$$
\nabla_i: \Omega^{i}(E)\to \Omega^{i+1}(E),\ \ \nabla_i(\omega\otimes e)=d\omega\otimes e+(-1)^i\omega\wedge\nabla(e)
$$
and the integrability is by definition $\nabla_1\circ \nabla_0=0$. It implies that $\nabla_{i+1}\circ\nabla_i=0$ and so
we have the complex $( \Omega^{i}(E),\nabla_i)$. According to the comparison theorem of Grothendieck, see for instance Deligne's notes \cite{de70/1}
Theorem 6.2
we have canonical isomorphisms
\begin{equation}
\label{22A2013}
H^*(M, {\mathcal O}(E))\to \uhp^*( M^{an}, \Omega^{*}(E))\leftarrow \uhp^*( M, \Omega^{*}(E))
\end{equation}
of $\C$-vector spaces.
Here, $M^{an}$ is the underlying complex variety of $M$, ${\mathcal O}(E)$ is the sheaf of constant sections of $E$ and $H^*(M, {\mathcal O}(E))$ is
the Cech cohomology with coefficients in ${\mathcal O}(E)$. The first $\uhp$ is the hypercohomology in the complex context and
the second one is the algebraic hypercohomology. Note that, $\Omega^{*}(E)$ is an algebraic sheaf and
so its sections have poles of finite order along a compactification of $M$.
If $M$ is an affine variety then $H^i(M, \Omega^{*}(E))=0$ for $i>0$ and so
\begin{equation}
\label{22aug2013}
\uhp^i( M, \Omega^{*}(E))\cong \frac{\ker \left(H^0( M, \Omega^{i}(E)) \to H^0( M, \Omega^{i+1}(E))\right)}{\Im \left(H^0( M, \Omega^{i-1}(E)) \to H^0( M, \Omega^{i}(E))\right)}
\end{equation}
see \cite{de70/1} Corollary 6.3.
\subsection{Logarithmic differential forms}
Let us now consider a meromorphic connection $\nabla$ on
$X$ with poles along a normal crossing divisor $S\subset X$, and hence, it
induces a holomorphic connection on $M:= X\backslash S$. We denote by $\Omega^1_X\langle S\rangle $ the sheaf of meromorphic differential forms in $X$
with only logarithmic poles along $S$. The sheaf $\Omega^p_X\langle S\rangle $ is $p$-times wedge product of $\Omega^1_X\langle S\rangle$. If $\nabla$ has only logarithmic poles along $S$,
see \cite{de70/1} page 78, then $\nabla$ induces $\Omega^p_X\langle S\rangle (E)\to \Omega^{p+1}_X\langle S\rangle (E) $ and
we have an isomorphism
\begin{equation}
\label{22a2013}
\uhp^*(X, \Omega^*_X\langle S\rangle (E))\cong \uhp^*(X, \Omega^*_X(E))
\end{equation}
induced by inclusion and then restriction to $M$ provided that the residue matrix of $\nabla$
along the irreducible components of $S$
does not have eigenvalues in $\N$, see \cite{de70/1} Corollary 3.15. These conditions will appear later
in Theorem \ref{11Jul2013} and Theorem \ref{11july2013}.
If $X$ is an affine variety then we conclude that
in (\ref{22aug2013}) every element is represented by a logarithmic differential, see also \cite{de70/1} Corollary 6.10.
The hypercohomology groups (\ref{22A2013}) and (\ref{22a2013}) are finite dimensional $\k$-vector space, see \cite{de70/1} Proposition 6.10 and \cite{dimca}
Proposition 2.5.4. However, explicit bases for these cohomology groups and algorithms which compute
an element as a linear combination of the basis, are
not the main focus of \cite{de70/1, dimca}. A regular connection may not have logarithmic poles along $S$ and one has to
modify it in order
to get such a property, see Manin's result in \cite{de70/1} Proposition 5.4.
In our terminology this is the same as to write any regular
differential equation in the Okubo format, see \S\ref{okubosection}.
All these together, leads us to that fact that the theoretical approach in \cite{de70/1} is not applicable to our main
problem posed the Introduction.
\subsection{Relation with integrals}
Let us now consider meromorphic global sections $e_1,e_2,\ldots,e_n$ of $E$ such that
for points $x$ in some open Zariski subset of $M$, $e_{i}(x),\ \ i=1,2,\ldots,n,$ form a basis
of $E_x$. Replacing $M$ with this Zariski subset, we can assume that this property is valid
for all $x\in M$. In this way $E$ becomes a trivial bundle. Let $e=[e_1,e_2,\ldots,e_n]$ and
$$
\nabla(e^\tr)=A\cdot e^\tr
$$
where $A$ is a $n\times n$ matrix whose entries are regular differential forms in $M$ (with poles along
the complement of $M$ in its compactification). We identify $(\Omega^i_M)^n$ with $\Omega^{i}(E)$ through the map $\omega\mapsto \omega\cdot e^{\tr}$ and we get:
$$
\nabla_i : (\Omega^i_M)^n\to (\Omega^{i+1}_M)^n,\ \ \nabla_i \omega=d\omega+(-1)^ i\omega A
$$
and so
$$
\uhp^i( M, \Omega^{*}(E))\cong \frac{\ker \left( H^0( M, \Omega^{i}_M)^n \to H^0( M, \Omega^{i+1}_M)^n\right )}{\Im \left
(H^0( M, \Omega^{i-1}_M)^n \to H^0( M, \Omega^{i}_M)^n\right)}
$$
Let $\check E$, $\check\nabla: \check E\to \Omega^1(\check E)$ and $\check e_i$ be the dual bundle to $E$,
the dual connection and the dual basis, respectively. We have $\nabla \check e^\tr=-A^\tr \check e^\tr$.
For a flat section $I$ of $\check E$ we write $I=\check e\cdot f$, where
$f=[f_1,f_2,\ldots, f_n]^{\tr}$ and $f_i$'s are holomorphic functions in a small open set $U$ in $M$.
We have $0=\nabla I=(\nabla \check e) f+\check e df=\check e(df-Af)$, and so, we get a system
$$
L:\ \ \ dY=AY
$$
with the solution $f$.
Let us define
$$
H^0( M, \Omega^{i}_M)^n_f:=\{\omega\in H^0( M, \Omega^{i}_M)^n \mid \omega\cdot f=0\}
$$
and let $H^i_f$ be the $i$-th cohomology group of the complex $\left( H^0( M, \Omega^{i}_M)^n_f,\nabla_i\right)$.
We also define
\begin{equation}
\label{19mar2016}
H^i_{\rm dR}(M,L):=\frac{\{ \omega=\sum_{k=1}^n f_k\omega_k\mid \omega_k\in H^0(M, \Omega^{i}_M), \ \ d\omega=0\} }
{\{ d(\sum_{k=1}^n f_k\omega_k)\mid \omega_k\in H^0(M, \Omega^{i-1}_M)\} }.\
\end{equation}
We have the exact sequence
\begin{equation}
\label{18nov2012}
H^i_f \to \uhp^i( M, \Omega^{*}(E))\to H^i_{\rm dR}(M,L)\to H^{i+1}_f
\end{equation}
which is the part of the long exact sequence of the short
exact sequence $0\to H^0( M, \Omega^{i}_M)^n_f\stackrel{j}{\to} H^0( M, \Omega^{i}_M)^n \to {\rm cokernel}(j) \to 0$.
From now on we use the cohomology group $H^i_{\rm dR}(M,L)$. The advantage of this is that we can integrate its elements.
Let $\delta$ be a topological $i$-cycle in $M$
such that the restriction of the analytic continuations of $f_k$'s to $\delta$ is one valued.
For $\omega$ in the right hand side (\ref{19mar2016})
the integration $\int_{\delta}\omega$ is well-defined. One of our motivations in the present text is to study this integral.
Later we will see that for $M$ the punctured line $H^1_{\rm dR}(M,L)$ is finite dimensional $\C$-vector space and so the $\C$ vector space generated by
$$
\int_{\delta} f_{i}\omega, \ \ i=1,2,\ldots,n,\ \ \omega\in H^0(M, \Omega^{i}_M)
$$
for a fixed $\delta$ is of finite dimension.
\section{Cohomology of linear differential equations}
\def{\rm dR}{{\rm dR}}
In this section we translate the machinery introduced in the previous section for the case of the punctured line, that is, $M$ is $\P^1$ minus a finite number of points.
We consider local systems given by fuchsian differential equations and we give an
explicit set of generators for the corresponding cohomology groups. For simplicity, we work with $\k\subset \C$. The case $\k\subset \C(t)$ is reduced to the previous one
by taking $t$ as a collection of algebraically independent transcendental numbers in $\C$.
\subsection{The case of the punctured projective line}
Let
\begin{equation}
\label{8july}
L: Y'=AY,\ \ '={\partial} _x
\end{equation}
be a fuchsian differential system of dimension $n$ with a solution $f=[f_1,f_2,\cdots,f_n]^\tr$. We assume that the entries of $A$ are in $\k(x)$.
Let also $S\subset \P ^1$ be the set of singularities of of $L$. We define
$$
H^1_{\rm dR}(\P ^1-S, L):=\frac{ \left\langle pf_idx \mid p\in \k(x),\ \ {\rm pol}(p)\subset S,\ \ i=1,2,\ldots, n\right \rangle_\k }{ \langle d(pf_i)
\mid p\in \k(x),\ \ {\rm pol}(p)\subset S,\ \ i=1,2,\ldots,n
\rangle_\k }
$$
which is the same as in (\ref{19mar2016}) for $\k=\C$.
For a fuchsian differential operator
\begin{equation}\label{L}
L:=p_0 {\partial}^n_x+\ldots +p_{n-1}{\partial}_x+p_n,\ \ p_i\in \k[x]
\end{equation}
with a solution $f$, that is $Lf=0$,
we can attach the linear differential system (\ref{8july})
with
$$
A=\begin{pmatrix}
0&1&0&\cdots &0\\
0&0&1&\cdots &0\\
\vdots&\vdots &\vdots&\cdots &\vdots\\
0&0&0&\cdots &1\\
-\frac{p_n}{p_0}&-\frac{p_{n-1}}{p_0}&-\frac{p_{n-2}}{p_0} &\cdots & -\frac{p_1}{p_0}
\end{pmatrix}
$$
and so
$$
H^1_{\rm dR}(\P ^1-S, L):=\frac{ \langle pf^{(i)}dx \mid p\in \k(x),\ \
{\rm pol}(p)\subset S,\ \ i=0,1,\ldots, n-1\rangle_\k }{ \langle d(pf^{(i)})
\mid p\in \k(x),\ \ {\rm pol}(p)\subset S,\ \ i=0,1,\ldots,n-1
\rangle_\k }.
$$
\subsection{Indicial equation}
In order to study the fuchsian differential equation \eqref{L}, $Ly=0$, near a point $t \in \P^1$ it is useful to compute its Riemann-scheme
\cite[Section~3]{beu-lectures}.
Let
$$
a_{i,t}:=\lim_{x\to t} (x-t)^i \frac{p_i}{p_0}\in \k \ \ \ \ \ t\in\k,
$$
$$
a_{i,\infty}:=\lim_{x\to \infty} (-x)^i \frac{p_i}{p_0} \in \k. \ \ \ \
$$
Since $L$ is fuchsian we have
$$
\frac{p_i}{p_0}= \frac{a_{i,t}}{(x-t)^i} + R_{i,t}, \ \ \ {\rm ord}_{x=t} R_{i,t} \geq -i+1,
$$
for any finite $t\in\k$, and for any $l\geq n$
$$
x^l\frac{p_i}{p_0}=(-1)^i a_{i,\infty}x^{l-i}+P_{i,l,\infty}(x) + R_{i,l,\infty}(x).
$$
where $P_{i,l,\infty}(x)$ is a polynomial in $x$ of degree $<l-i$ and $R_{i,l,\infty}$ is
a sum over all finite singularities $t_j\in\bar\k $ of $L$, of polynomials in $\frac{1}{x-{t_j}}$ of degree $\leq i$.
Note that both $R_{i,l,\infty}$ and $P_{i,l,\infty}$ are defined over $\k$. We conclude that the
differential operator $L$ can be also written in the format
\begin{eqnarray}
\label{format}
& &\partial_x^{(n)}+\sum_{i=1}^{n}\frac{a_{i,t}}{(x-t)^{i}}\partial_x^{(n-i)}+\sum_{i=1}^{n}R_{i,t} \partial_x^{(n-i)},\\
\label{formatb} & &x^l\partial_x^{(n)}+\sum_{i=1}^{n} a_{i,\infty} x^{l-i} \partial_x^{(n-i)}+\sum_{i=1}^{n}(P_{i,l,\infty}+R_{i,l,\infty }) \partial_x^{(n-i)}.
\end{eqnarray}
Furthermore the indicial equation $I_{t}$ at
$t$ is given by
\begin{eqnarray
I_{t}&=& X (X-1) \cdots (X-n+1)+a_{1,t} X (X-1) \cdots (X-n+2)+\cdots+ a_{n,t} \label{Indicial} \\
I_{\infty} &=&X (X+1) \cdots (X+n-1)+a_{1,\infty} X (X+1) \cdots (X+n-2)+\cdots+a_{n,\infty}\label{Indicialb}
\end{eqnarray}
The Riemann scheme of $L$ at a point $t\in\k\cup\{\infty\}$ is the set of the roots of $I_t$.
\subsection{Explicit set of generators, $\k=\bar \k$}
We are now in a position to describe an explicit set of generators for the cohomology group $H^1(\P ^1-S, L)$.
\begin{theo}
\label{11Jul2013}
Let $\k$ be an algebraically closed subfield of $\C$.
If the fuchsian differential operator $L$
has no integer exponent $\geq n$ in the Riemann-scheme at a finite point and no positive integer exponent at $\infty$
then
$H^1_{\rm dR}(\P ^1-S, L)$ is generated by
\begin{equation}
\label{15aug2012}
(x-t_j)^{-1} f^{(i)} dx,\ \ j=1,2,\ldots,r, \ \ i=0,1,\ldots,n-1
\end{equation}
and so it is of dimension at most $n\cdot r$.
\end{theo}
\begin{proof}
All the qualities below are in the cohomology group $H^1_{\rm dR}(\P ^1-S, L)$
that is obviously generated by
$$
f^{(i)} x^l dx, \frac{f^{(i)}}{(x-t_j)^l} dx,\ \ \ l \in {\mathbb N},\ \ i=0,1,\ldots, n-1,\ \ j=1,2,\ldots,r.
$$
At first we show that
$\frac{f^{(i)}}{(x-t_j)^l} dx,\ \ l \in {\mathbb N},\ \ i=0,1,\ldots, n-1,$
is in the $\k$-vector space $V$ generated by (\ref{15aug2012}). Since
$$
0=d(\frac{f^{(i)}}{(x-t_j)^l})=\frac{f^{(i+1)}}{(x-t_j)^l} dx+(-l) \frac{f^{(i)}}{(x-t_j)^{l+1}}dx
$$
we get
\begin{equation}
\label{9july2013}
\frac{f^{(i)}}{(x-t_j)^l}dx=\frac{1}{l-1}\frac{f^{(i+1)}}{(x-t_j)^{l-1}}dx=\cdots=
\left\{ \begin{array}{cc}
\frac{1}{(l-1)\cdots (l-n+i)}\frac{f^{(n)}}{(x-t_j)^{l-n+i}}dx & i+l \geq n+1 \\
\frac{1}{(l-1)!}\frac{f^{(i+l-1)}}{x-t_j}dx & i+l < n+1 \\
\end{array}\right.
\end{equation}
Now, we use induction on $i+l$. For $i+l<n+1$, the claim follows from the second case in (\ref{9july2013}).
Thus by the first case in (\ref{9july2013}) we can assume $i=n$.
We have
\begin{eqnarray*}
\frac{f^{(n)}}{(x-t_j)^l} dx &\stackrel{\eqref{format}}{=}&
-(\sum_{i=1}^n a_{i,t_j} \frac{f^{(n-i)}}{(x-t_j)^{i+l}})dx-(\sum_{i=1}^{n }\frac{R_{i,t_j}}{(x-t_j)^l} f^{(n-i)})dx
\\
&\stackrel{\eqref{9july2013}}{=}&
-(\sum_{i=1}^n \frac{a_{i,t_j}}{l(l+1)\cdots (l+i-1)})\frac{f^{(n)}dx}{(x-t_j)^l} -(\sum_{i=1}^{n }\frac{R_{i,t_j}}{(x-t_j)^l} f^{(n-i)})dx.
\end{eqnarray*}
In
$
\frac{R_{i,t_j}}{(x-t_j)^l} f^{(n-i)}dx
$
there appear only terms $\frac{f^{(n-i)}}{(x-t_j)^{l+k}} dx$ with $k\leq i-1$
and terms $\frac{f^{(n-i)}}{(x-t_{j'})^i}dx$ for $j'\neq j$.
Using the first case in (\ref{9july2013}) the former terms are by induction in $V$ and
the latter terms by the second case in (\ref{9july2013}).
By assumption we get
\begin{equation*}
1+(\sum_{i=1}^n \frac{a_{i,t_j}}{l(l+1)\cdots (l+i-1)})\stackrel{\eqref{Indicial}}{=}
\frac{I_{t_j}(l+n-1)}{l\cdots (l+n-1)} \neq 0,\ \forall l\in\N.
\end{equation*}
Hence
$\frac{f^{(n)}}{(x-t_j)^l} dx \in V$.
Similarly we prove that
$x^l f^{(i)} dx$ is in the vector space $V$. If $l-i<0$ then we
have
\begin{equation*}
x^lf^{(i)}dx=-(l-1)x^{l-1}f^{(i-1)}dx=\cdots=0.
\end{equation*}
For $l-i\geq 0$
we use induction on $l-i$ and we have
\begin{equation}\label{xfn}
x^lf^{(i)}dx=\frac{1}{-(l+1)}x^{l+1}f^{(i+1)}dx=\cdots=\frac{(-1)^{n-i}}{(l+1)\cdots (l+n-i)}x^{l+n-i}f^{(n)}dx
\end{equation}
and so we can assume that $i=n$. Now, for $l\geq n$ we have by (\ref{xfn}) and (\ref{formatb})
\begin{eqnarray*}
x^l f^{(n)} dx &=& -\sum_{i=1}^n a_{i,\infty} x^{l-i} f^{(n-i)}dx- \sum_{i=1}^{n }P_{i,l, \infty} f^{(n-i)}dx-
\sum_{i=1}^{n }R_{i,l, \infty} f^{(n-i)}dx.
\end{eqnarray*}
The second sum is by hypothesis of induction in $V$
and the third by (\ref{9july2013}).
The first sum is by (\ref{xfn})
$$
-\left (\sum_{i=1}^n \frac{ (-1)^ia_i}{(l-i+1)(l-i+2)\cdots l}\right )x^{l} f^{(n)}dx\stackrel{\eqref{Indicialb}}{=}
\frac{(1-I_\infty(l-n+1))}{ l\cdots (l-n+1) } x^lf^{(n)}dx.
$$
Therefore, since
$l-n+1$ is not an exponent at $\infty$ we get $x^lf^{(n)}dx$ is in $V$.
\end{proof}
\begin{rem}\rm
Since $x^i f^{(n)} dx =0$ in $H^1_{\rm dR}(\P ^1-S, L)$ for $i=0,\ldots,n-1,$ we obtain $n$
$\k$-linear relations between the generators $\eqref{15aug2012}$ of $H^1_{\rm dR}(\P ^1-S, L)$.
\end{rem}
\begin{rem}\rm
If $\infty$ is no singularity then the exponents at $\infty$ are $0,-1,\ldots,-n+1$.
Thus the condition that the exponent is positive is compatible with the condition that the exponents
are $\geq n$ at the finite singularities.
\end{rem}
\begin{rem}\rm
\label{19july2013}
Without the hypothesis on indicial equations of $L$, we have to add the following
elements
\begin{eqnarray*}
\frac{f^{(n)}}{(x-t_j)^l} dx,&& \mbox{if } I_{t_j}(l+n-1)=0,\\
x^lf^{(n)}dx, && \mbox{if } I_{\infty}(l-n+1)=0, \;l\geq n
\end{eqnarray*}
to the set (\ref{15aug2012}) in order to get a set of generators.
\end{rem}
\subsection{Explicit set of generators, $\k\neq{\bar \k}$}
In case $\k\not=\bar\k$ and for computational purposes we modify Theorem \ref{11Jul2013} and reprove it over $\k$.
For this we proceed as follows. Let
$$
\Delta=\prod_{i=1}^r(x-t_i)
$$
where $t_i\in\bar\k$ are the finite singular points of $L=0$ (without repetition).
Since $L$ is defined over $\k$,
the Galois group of $\bar\k$ over $\k$ acts on $t_j$'s and so $\Delta\in\k[x]$.
Thus we can write $L$ in the following way
$$L=\sum_{i=0}^n \Delta^i p_{n-i}(x) \partial^i_x, \quad \quad p_i \in \k[x], \;
\deg p_i\leq i (r-1), \quad p_0=1$$
see for instance \cite[I. Prop. 4.2]{IKSY}.
\begin{theo}
\label{11july2013}
If $L$
has no integer exponent $\geq n$ in the Riemann-scheme at a finite point and no positive integer exponent at $\infty$
then
$H^1_{\rm dR}(\P ^1-S, L)$ is generated by
\begin{equation}
\label{15aug2012-D}
\frac{x^jf^{(i)}dx}{\Delta} ,\ \ j=0,1,\ldots, r-1, \ i=0,1,2,\ldots,n-1.
\end{equation}
\end{theo}
\begin{proof}
The $\bar\k$-vector space $H^1_{\rm dR}(\P ^1-S, L)\otimes_\k \bar\k$ has a set of generators (\ref{15aug2012}) and for fixed $i\in\N$ we have
$$
\langle \frac{f^{(i)}dx }{x-t_j} \mid j=1,\ldots,r \rangle_{\bar\k}= \langle \frac{x^kf^{(i)}dx}{\Delta}\mid k=0,\ldots, r-1 \rangle_{\bar\k},\ \
i=0,1,2,\ldots,n-1.
$$
\end{proof}
In order to implement the above proof in a computer, one has
to introduce new variables $t_j$ for each singularity and
so it does not give an effective
algorithm which writes an element of $H^1(\P ^1-S,L )$ in terms of the generators (\ref{15aug2012-D}).
We give a second proof which is algorithmic and does not use $\bar \k$.
By the extended Euclidean algorithm, there are polynomials $a, b \in \k[x]$ such that
\[ 1 = a \Delta+ b \Delta'.\]
For
$t \in \k$
we define
\begin{equation}\label{ct}
c_t:= 1+\sum_{i=1}^n\frac{p_i b^i}{t(t+1)\cdots (t+i-1)}\in\k[x].
\end{equation}
\begin{lemm}
For a fixed $t\in\k$, we have $gcd(c_t, \Delta)=1$ if and only if $t+n-1$ is not an exponent
of $L$ at finite singularities $t_j,\ j=1,2,\ldots,r$.
\end{lemm}
\begin{proof}
Let $t_j$ be a root of $\Delta$.
We have
\[ 1=b(t_j)\cdot \Delta'(t_j),\ \ \ \Delta'(t_j)= (\frac{\Delta}{x-t_j})(t_j) \]
and so
$$
a_{i,t_j}:=\lim_{x\to t_j} \frac{p_i(x)(x-t_j)^i}{\Delta^i}=p_i(t_j) (\frac{1}{\Delta'(t_j)})^i =p_i(t_j)b(t_j)^i.
$$
Thus multiplying
$c_t$ by $t(t+1)\cdots (t+n-1)$ and evaluating $x$ at $t_j$ gives the value of the indicial equation $I_{t_j}$ evaluated at $t+n-1$.
\end{proof}
\begin{proof}[Second Proof of Theorem \ref{11july2013}]
All the qualities below are in the cohomology group $H^1_{\rm dR}(\P ^1-\Delta, L)$.
Obviously it is generated by
$$
f^{(i)} x^ldx, \frac{f^{(i)} x^k}{\Delta^l} dx,\ \ \ l\in {\mathbb N},\ \ i=0,1,\ldots, n-1,\ k=0,1,\ldots,r-1.
$$
Note that by division over $\Delta$, it is enough to consider $0\leq k<r$.
Let $V$ be the $\k$-vector space generated by (\ref{15aug2012-D}).
At first we show that
$\frac{f^{(j)} x^k }{\Delta^l} dx \in V$. Again for $p \in \k[x]$ we have
$$
0=d(\frac{f^{(j)} p }{\Delta^l})=\frac{(f^{(j)} p)'}{\Delta^l} dx+(-l) \frac{f^{(j)} p \Delta'}{\Delta^{l+1}}dx.
$$
For $l \in {\mathbb N}$ we get
\begin{eqnarray}
\notag \frac{f^{(j)} x^k}{\Delta^{l+1}}dx&=&
\frac{f^{(j)} x^k(a \Delta+b \Delta')}{\Delta^{l+1}}dx\\
\notag &=&\frac{f^{(j)} x^k a}{\Delta^{l}}dx+ \frac{f^{(j)} x^k b \Delta'}{\Delta^{l+1}}dx\\
\notag &=&\frac{f^{(j)} x^k a}{\Delta^{l}}dx +\frac{1}{l} \frac{(f^{(j)} x^k b)'}{\Delta^{l}}dx \\\label{reduction}
&=&\frac{f^{(j)} x^k a}{\Delta^{l}}dx +\frac{1}{l} \frac{f^{(j)} (x^k b)'}{\Delta^{l}}dx+
\frac{1}{l} \frac{f^{(j+1)} x^k b}{\Delta^{l}}dx.
\end{eqnarray}
Hence if $j<n-1$ we can reduce the pole order.
It remains to show that for $j=n-1$ we can reduce the pole order.
Let $q \in k[x]$. Then
\begin{eqnarray}
\label{qfn} \frac{q f^{(n)}}{\Delta^l} dx &=& \frac{\Delta^n q f^{(n)}}{\Delta^{l+n}} dx \\
\notag &=& -\sum_{i=1}^n \frac{ q p_{i} f^{(n-i)}}{\Delta^{l+i}} dx\\
\notag &\stackrel{\eqref{ct}}{=}& -q (c_l-1) \frac{f^{(n)}}{\Delta^l}dx+ \mbox{lower pole orders terms}
\end{eqnarray}
where the last equality follows by (\ref{ct}) and (\ref{reduction}).
Since $gcd(c_l,\Delta)=1$ we have polynomials $A,B\in\k[x]$ such that $A c_l+B\Delta=1$.
Hence the pole order of
$$\frac{x^kf^{(n)}}{\Delta^l} dx=\frac{(x^k A)c_lf^{(n)}}{\Delta^l} dx+\frac{x^k Bf^{(n)}}{\Delta^{l-1}} dx$$
can be reduced to $l-1$ using \eqref{qfn} with $q=x^k A$.
Reducing the pole order of $\Delta$ may yield also terms
$x^j f^{(i)} dx$.
However by the same arguments as in the proof of Theorem \ref{11Jul2013} we have $x^j f^{(i)} dx\in V$.
\end{proof}
\begin{rem}\rm
\label{24march2016}
In Theorem \ref{11july2013}, without the hypothesis on indicial equations of $L$, we have to add the following
finite number of elements
\begin{eqnarray*}
\frac{x^k f^{(n)}}{\Delta^l} dx,&& \mbox{if } 0\leq k < \deg \left(gcd(\Delta,c_l)\right),\\
x^lf^{(n)}dx, && \mbox{if } I_{\infty}(l-n+1)=0, \;l\geq n
\end{eqnarray*}
to the set (\ref{15aug2012}) in order to get a set of generators.
\end{rem}
\subsection{Cohomologies over function fields}
\label{15july2013}
In this section we turn to the main problem posed in the Introduction, that is, how to compute the linear differential equation of
\eqref{16july2013}.
Let us assume that $\k=\tilde\k(y)$, where $y$ is a variable and $\tilde\k$ is a field, and so we have the
derivation
$$
\partial_y:\k\to\k.
$$
We consider a two dimensional system in $x,y$ variables:
$$
df=M\cdot f,\ \ M=Adx+Bdy,\ A,B\in\Mat_{n\times n}(\tilde \k(x,y))
$$
or equivalently $\partial_xf=Af,\ \ \partial_y f=Bf$.
We further assume that it is integrable, that is, $dM=-M\wedge M$ or equivalently $\partial_x B-\partial_y A=BA-AB$.
Any entry $f_i$ of $f$ satisfies a linear differential equation $L_i=0$, $L_i\in \k[x,\partial_x]$, with respect to the variable $x$.
From the integrability condition we conclude that a solution of $L_i=0$ depends holomorphically on both $x,y$.
We need that $\partial_y$ induces a well-defined map
\begin{equation}
\label{15/03/2016}
\partial_y: H^1_{\rm dR}(\P^1-S, L_i)\to H^1_{\rm dR}(\P^1-S, L_i)
\end{equation}
For instance, if we make a derivation $\partial_y$ of a solution $f_i$ of $L_i=0$ and the result
is still a $\k[x,\frac{1}{\Delta}]$-linear combination of $\partial_x^j f_i,\ \ j=0,1,\ldots,n-1$ then \eqref{15/03/2016}
is well-defined.
For simplicity we have used the same notation $\partial_y$.
We write \eqref{15/03/2016} in the generators (\ref{15aug2012-D}):
$$
\partial_y\omega=C\cdot \omega
$$
where $\omega$ is the $nr\times 1$ matrix containing the elements (\ref{15aug2012-D}). The differential system
\begin{equation}
\label{finalsystem}
{\partial}_y Y=C\cdot Y
\end{equation}
is satisfied by $Y=\int \omega$, where the integration takes place over
a fixed closed path in the $x$-domain such that the entries of $\omega$ are one valued.
Let us now come back to integrals \eqref{16july2013}. Let
\begin{equation}
\label{kronecker}
df_i=A_if_i,\ i=1,2,\ \ A_i\in\Mat_{n_i\times n_i}(\Omega^1_{\P^1_x\times \P^1_y}).
\end{equation}
be the Gauss-Manin connection of the family $X_i\to \P^1_x\times \P^1_y,\ \ i=1,2$.
We make the Kronecker product of these two systems and obtain the system
\begin{equation}
\label{kronecker-1}
df=Af, \hbox{ where }
A=A_1\otimes I_{n_2}+I_{n_1}\otimes A_2
\end{equation}
where $I_{n_i}$ is the $n_i\times n_i$ identity matrix. A solution of \eqref{kronecker-1} is given by $f_1\otimes f_2$.
Let $p_i,\ i=1,2$ be as in the Introduction. It can be the first entry of $f_i$.
We calculate the Picard-Fuchs equation $L$ of $p_1p_2$ using \eqref{kronecker-1} and with respect to the $x$
variable.
Now, we find a basis $\omega_i,\ i=1,2,\ldots, m,$ of the cohomology group $H^1(\P^1_x-S,L)$ and as
in \S\ref{15july2013} we compute $\partial_y$ in this basis and obtain the system (\ref{finalsystem}).
This process generalizes the classical convolution. In a geometric context this is as follows.
Let $X_i:=\tilde X_i\times \P^1_y\to \P^1_y,\ i=1,2$ be projections on the second coordinate, $X_1\to \P^1_x$ be a morphism which does not depend on the second coordinate
and $X_2\to \P^1_x$ be of the form $y- M$, where $M: X_2\to \P^1_x$ does not depend on the second coordinate
and by abuse of notation we have used $y$ as the projection map on the second coordinate.
We can see easily that $p_1(x,y)=p_1(x)$ and $p_2(x,y)=p_2(y-x)$.
The integral (\ref{16july2013}) in this case is the classical convolution.
The details of the classical convolution will be explained in the next section.
\section{Convolution}
In this section we remind the classical
convolution of two solutions of fuchsian linear differential
equations and argue that
the material in \S\ref{section1} is a generalization of this concept.
\subsection{Okubo system}
\label{okubosection}
A linear differential system of the format
\begin{equation}
\label{okubos}
(xI_{n}-T)Y'=A Y,\;
\end{equation}
$$
T={\rm diag}(t_1I_{n_1},\ldots,t_rI_{n_r}),\; \sum n_i=n,\;
\; T, A\in \Mat_{n\times n}(\C)
$$
is called an Okubo system (in normal form) and it is a useful format for doing computations, such as convolution.
For a fuchsian system
\[ D_{a}: Y'= \sum_{i=1}^r \frac{a_i}{x-t_i}Y,\; a_i \in \Mat_{n\times n}(\C),\ \ a=(a_1,a_2,\cdots,a_r)
\]
we introduce the following special Okubo system:
\begin{equation}
\label{18july13}
D_{c_\mu(a)} : (xI_{nr}-T)X'= c_\mu(a) X,\
c_\mu(a):=\left(\begin{array}{ccc}
a_1 & \dots & a_r \\
\vdots & \vdots & \vdots \\
a_1 & \dots & a_r
\end{array}
\right)+\mu I_{nr},\; \mu \in {\mathbb C}.
\end{equation}
where $n_i=n,\;i=1,\ldots,r$. Via the following procedure
we see that any fuchsian system
is a factor system of an Okubo system.
Let $f(x)$ be a solution of $D_a.$ Then
\begin{eqnarray}\label{tildef}
\tilde{f}(x) &:=& \left(\begin{array}{c}f(x)(x-t_1)^{-1}\\
\vdots\\
f(x)(x-t_r)^{-1}\end{array}\right)
\end{eqnarray}
satisfies the Okubo system $D_{c_{-1}}(a)$.
If $\k$ is not not algebraically closed we work with the following equivalent Okubo system defined over $\k$.
Let $L$ be a fuchsian system of dimension $n$
\begin{eqnarray*} L:&& Y' =\sum_{i=1}^{r} \frac{x^{i-1}}{\Delta} \tilde{a}_i Y \end{eqnarray*}
with $\tilde{a}_i \in \Mat_{n\times n}(\k)$ and
$\Delta=\sum_{i=0}^{r} b_{r-i} x^i\in \k[x], b_0=1$.
If $f$ is a solution of $L$ then
\[ (f/\Delta,xf/\Delta,\ldots,x^{r-1}f/\Delta)^{\tr}\]
satisfies the Okubo system
\[ (xI_{rn}-\tilde{T})Y'=\tilde{A} Y,\]
where
\begin{eqnarray*} \tilde{T}=\left( \begin{array}{ccccccc}
0 & I_n & 0 \cdots& 0\\
0 & 0 &\ddots &\vdots \\
0 & \cdots & 0 &I_n \\
-b_r I_n&\cdots &-b_2 I_n &-b_1 I_n
\end{array} \right),&&
\tilde{A}= \left( \begin{array}{ccccccc}
0 & \cdots & 0\\
\vdots & \cdots& \vdots \\
0 & \cdots & 0\\
\tilde{a}_1& \cdots &\tilde{a}_{r}
\end{array}\right)-I_{rn}.
\end{eqnarray*}
\subsection{Cohomology of Okubo system}
For the matrix $A$ in (\ref{okubos}) we consider its submatrices $A=[A_{ij}]$ according to the
partition $n=n_1+n_2+\cdots+n_r$ and in particular its $n_i\times n_i$
submatrices $A_{ii}$ lying in the diagonal of $A$.
\begin{theo}
\label{theo-okubo}
Let $L$ be the Okubo system (\ref{okubos}) with solution $f$.
If
\begin{equation}
\label{10Aug2012}
\det(A+mI_{n\times n})\not =0, \ \det(A_{ii}-mI_{n_i\times n_i})\not =0,\ \ \forall m\in\N
\end{equation}
then
$H^1_{\rm dR}(\P ^1-S, L)$ is generated by
\begin{equation}
\label{10aug2012}
(x-t_i)^{-1}f_jdx,\ \ i=1,2,\ldots,r, \ \ j=1,2,\ldots,n
\end{equation}
and so it is of dimension at most $n\cdot r$.
\end{theo}
\begin{proof}
In $H^1_{\rm dR}(\P ^1-S, L)$ and for $m\not=-1$ we have
\begin{eqnarray*}
(x-t_i)^mf_jdx &=& -(m+1)^{-1}(x-t_i)^{m+1}f_j'dx\\
&=& -(m+1)^{-1}(x-t_i)^{m+1}(x-t_j)^{-1}\sum_{k=1}^n a_{jk}f_kdx,
\end{eqnarray*}
where $A=(a_{jk})$.
If $m$ is negative and $t_i\neq t_j$ then we have reduced the pole order.
If $m$ is negative and $t_i=t_j$, in order to reduce the pole order
we need $A_{ii}+(m+1)I_{n_i\times n_i}$ to be invertible and
if $m$ is positive or zero we need that $A+(m+1)I_{n\times n}$ to be invertible.
\end{proof}
\begin{rem}\rm
For a linear differential equation of an entry of the Okubo system, the conditions (\ref{10Aug2012})
imply the conditions in Theorem \ref{11july2013}. Without these conditions
a similar observation as in Remark \ref{19july2013} is valid.
\end{rem}
\subsection{Convolution of Okubo systems}
Given two solutions of two Okubo systems we can easily determine the Okubo system that is satisfied by their convolution.
\begin{theo}\label{Poch}
Let $f_i(x)$ be a solution of the Okubo system $(xI_{n_i}-T_i) Y_i'= A_iY_i,\ i=1,2,\ A_i\in\Mat_{n_i\times n_i}(\C)$.
Then
$$
\int_{} f_1(x) {\otimes} f_2(y-x)dx,
$$
where
the integration is over a path in the $x\in\C$ plane such that the integrand is one valued,
is a solution matrix for the Okubo system
\begin{equation}
\label{19july13}
(yI_{n_1n_2}-T_1 {\otimes} I_{n_2}-I_{n_1} {\otimes} T_2)Y'= (A_1 {\otimes} I_{n_2} + I_{n_1} {\otimes} A_2 +1) Y.
\end{equation}
\end{theo}
The proof of the above theorem is similar to \cite[Lemma~4.2]{detrei}.
Note that the system (\ref{19july13})
has singularities at $t^1_{i}+t^2_{j},\; i=1,\ldots,r_1,\; j=1,\ldots,r_2,$ and possibly at infinity.
\subsection{Convolution of fuchsian systems}
In the case of fuchsian systems we proceed as follows.
Let
\[
D_{a^i}: Y'= \sum_{j=1}^{r_i} \frac{a_{j}^i}{x-t_{j}^i}Y, \ \ \ a_{j}^i \in
\Mat_{n_i\times n_i}(\C),\ i=1,2
\]
be two fuchsian systems with solutions $f_1, f_2$ resp..
Then the Okubo system
\[
(yI_{n_1r_1n_2r_2}-(T_1 {\otimes} I_{n_2r_2}+I_{n_1r_1} {\otimes} T_2))Y'= (c_0(a^1) {\otimes} I_{n_2r_2} + I_{n_1r_1} {\otimes} c_0(a^2)-I_{n_1r_1n_2r_2})Y
\]
has
$ \int \tilde{f}_1(x)\otimes \tilde{f}_2(y-x) dx
$
as solution with $\tilde{f}_1, \tilde{f}_2$ as in \eqref{tildef}.
\section{Examples}
\label{examples}
In this section we discuss some examples of families of algebraic varieties
whose Picard-Fuchs equation can be computed through the methods introduced
in this article.
We consider the case in which we have only the family $X_1$
(take $X_2$ the product of some variety with $\P^1_x\times\P^1_y$).
In this case we want to use the Gauss-Manin connection of the two parameter family $X_1\to
\P^1_x\times\P^1_y$ and integrate it over the variable $x$ and obtain the Picard-Fuchs equation of the one parameter family $X_1\to \P^1_y$ obtained by the composition
$X_1\to
\P^1_x\times\P^1_y\to \P^1_y $, where the second map is the projection. A well-known example for this situation is the
Legendre family of elliptic curve that is given by
$$
y^2=x(x-1)(x-t)
$$
where $t$ is a parameter. We can compute compute the Picard-Fuchs equation of $\frac{dx}{y}$ either by direct methods or by the methods introduced
in this article:
$$
I+(8t-4)I'+(4t^2-4t)I''=0.
$$
In the second case we consider $x$ and $t$ as a parameter and so we get a two parameter family of zero dimensional varieties with two points.
For what follows, the polynomial $\Delta$ need not to be monic in the $x$ variable.
Let us consider the rank 19 family of K3 surfaces given in the affine coordinates $(x,y,w)$ by
the equation $P=0$, where
$$
P:=y^2w-4x^3+3axw^2+bw^3+cxw-(1/2)(dw^2+w^4)=0
$$
$$a=(16 + t)(256 + t),\ \ b= (-512 + t)(-8 + t)(64 + t), \ c=0, \ d=2985984t^3
$$
and $t$ is a parameter, see \cite[Section 6.7]{DHMW}.
Here, we would like to compute the Picard-Fuchs equation of the holomorphic $2$-form given by $\omega=\frac{dx\wedge dy\wedge dw}{dP}$.
The generic member of the family has two isolated singularities and so one cannot apply the Griffiths-Dwork method or its modification using Brieskorn modules.
In order to apply the methods introduced in this article,
we look $P=0$ as a two parameter family of elliptic curves depending on $(t,w)$.
In this case we know the explicit expression of Gauss-Manin connection, see for instance \cite[Section 6]{DHMW}.
Using this we can compute the following differential
equations for the elliptic integral $f(t,w):=\int \frac{dx\wedge dy}{dP}$
\begin{eqnarray*}
L &:& A_1f+A_2\partial_w f+A_3 \partial_w^2f =0 \\
& & B_1f+B_2\partial_w f+B_3 \partial_t f =0
\end{eqnarray*}
where $A_i,B_i$'s are explicit polynomials in $w,t$ with rational coefficients:
{\tiny
\begin{eqnarray*}
A_1 &=& (1283918464548864t^{9}w-133116666404426219520t^{9}+1486016741376t^{8}w^{2}-585466819834281984t^{8}w+
\\ & & 72814820327424t^{7}w^{2}-
37469876469394046976t^{7}w+ 1719926784t^{6}w^{3}-2077451404443648t^{6}w^{2} \\ & &
+336571521970697404416t^{6}w-784286613504t^{5}w^{3}
+298249504061128704t^{5}w^{2}-50194343264256t^{4}w^{3}+ \\ & &24931223849681289216t^{4}w^{2}-144t^{3}w^{5}-474771456t^{3}w^{4}+
450868486864896t^{3}w^{3}+65664t^{2}w^{5}+\\ & &
4202496tw^{5}+77w^{6}-37748736w^{5})\\
A_2 &=& (3851755393646592t^{9}w^{2}-1916879996223737561088t^{9}w+2972033482752t^{8}w^{3}-1756400459502845952t^{8}w^{2}+
\\ & &
145629640654848t^{7}w^{3}-
112409629408182140928t^{7}w^{2}+1719926784t^{6}w^{4}-3138467357786112t^{6}w^{3}+\\ & &
1009714565912092213248t^{6}w^{2}-497664t^{5}w^{5}- 784286613504t^{5}w^{4}+596499008122257408t^{5}w^{3}
-24385536t^{4}w^{5}- \\ & & 50194343264256t^{4}w^{4}+49862447699362578432t^{4}w^{3}-432t^{3}w^{6}- 119439360t^{3}w^{5}+450868486864896t^{3}w^{4}+
\\ & &
196992t^{2}w^{6}-
99883155456t^{2}w^{5}+12607488tw^{6}-8349416423424tw^{5}+
144w^{7}-113246208w^{6})\\
A_3 &=& 36 w^2( -w^{2}+2985984t^{3})
\left( -w^{4}+(4t^{3}-1824t^{2}-116736t+1048576)\cdot w^{3}+(6912t^{5}+338688t^{4}-7299072t^{3}+ \right. \\ & &
1387266048t^{2}+115964116992t)\cdot w^{2}+
(11943936t^{6}-5446434816t^{5}-348571828224t^{4}+3131031158784t^{3})\cdot w \\ & &
\left.
-8916100448256t^{6}\right )\\
B_1 &=& -\frac{1}{36}A_3\cdot ( -w^{2}+2985984t^{3})^{-1}\cdot \\ & &
\left( 2985984t^{4}w^{2}-3456t^{3}w^{3}+2842656768t^{3}w^{2}-34560t^{2}w^{3}+73383542784t^{2}w^{2}-7tw^{4}+2211840tw^{3}-952w^{4}+\right.\\ & &
\left. 905969664w^{3}\right) \\
B_2 &=& -13824t^{3}w^{2}+9746251776t^{3}w-138240t^{2}w^{2}+293534171136t^{2}w-24tw^{3}+8847360tw^{2}-3264w^{3}+3623878656w^{2} \\
B_3 &=&
8(t+256)(t+16)(2985984t^3-w^2).
\\
\end{eqnarray*}
}
We use the
second equality in order to compute the action of $\partial_t$ on the cohomology group constructed from $L$.
This data is enough to compute the Picard-Fuchs equation of the integral $g(t)=\int f(t,w)dw$ using the techniques introduced in this article.
Note that we have to use Theorem \ref{11july2013} together with Remark \ref{24march2016} because the differential equation $L$ has the apparent
singularities $-w^{2}+2985984t^{3}$. The end result has a factor
\begin{equation}
\label{5aug2013}
L:= 1+ (26t+512) \partial_t+ (36t^2+1536t)\partial_t^2+(8t^3+512t^2)\partial_t^3
\end{equation}
where $Lg=0$. This differential operator is obtained by direct computations as in \cite[Chapters 4,5]{ho06-1}.
The function $g$ can be written as the period of $\frac{dx\wedge dy\wedge dw}{dP}$ over two dimensional cycles living in
the $K3$-surface, for further details see \cite{DHMW}.
\def$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'$} \def\cprime{$'${$'$}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,023 |
\section{Introduction}
\noindent For an ever increasing variety of applications, an
interesting problem to be explored is to model the debonding of a
thin film from a substrate.
If we consider a stretched film bonded to an infinite rigid
substrate, the elastic energy of this film scales as its thickness.
If the film debonds from the substrate, on one hand its elastic
energy tends to zero, while on the other hand this creates a new
surface and then an interfacial energy independent of the thickness.
In \cite{BhFoFra} Bhattacharya, Fonseca and Francfort examine,
among other, the asymptotic behavior of a bilayer thin film
allowing for the possibility of a debonding at the interface, but
penalizing it postulating an interfacial energy which scales as
the overall thickness of the film to some exponent. Thus the
energy they consider consists of the elastic energy of the two
layers and the interfacial energy with penalized debonding.
In this paper we deal with thin films connected by a hyperplane
(sieve plane) through a periodically distributed contact zone.
Thus we see the debonding as the effect of the {\it weak
interaction} of the two thin films through this contact zone and
we recover the interfacial energy term by a limit procedure.
Since we are mainly interested in describing the interaction
phenomenon due to the presence of the sieve, we make a
simplification choosing two thin films having the same elastic
properties (for a generalization to the case of two different
materials interacting, we refer the reader to \cite{Ans}).
\bigskip Consider a nonlinear elastic
$n$-dimensional bilayer thin film of thickness $2\d$ with layers
connected through $(n-1)$-dimensional balls $B^{n-1}_{r}(x_i^\e)$
centered in $x_i^\e:=i\e$, $i\in {\mathbb{Z}}^{n-1}$ and with radius $r>0$.
Thus the investigated elastic body occupies the reference
configuration parametrized as
$$
\O^{\d}_{\e,r}:=\o^{+\d} \cup \o^{-\d} \cup \big( \o_{\e, r}
\times \{0\} \big)
$$
where $\o$ is a bounded open subset of ${\mathbb{R}}^{n-1}$, $\o^{+\d}:=\o
\times(0,\d)$, $\o^{-\d}:=\o \times (-\d,0)$ and $\o_{\e, r}:=
\bigcup_{i \in{\mathbb{Z}}^{n-1}}B^{n-1}_{r}(x_i^\e) \cap \o$ (see Figure
1).
\begin{figure}[th]\label{fig1}
\centerline{\psfig{file=FIG1.eps,width=4in}} \vspace*{8pt}
\caption{The domain $\Omega^{\delta}_{\varepsilon, r}$.}
\end{figure}
In the nonlinear membrane theory setting the (scaled) elastic
energy associated to the material modelled by $\O^{\d}_{\e,r}$ is
given by
\begin{equation}\label{energia}
\frac{1}{\d} \int_{\O^{\d}_{\e,r}} W(D u )\, dx,
\end{equation}
where $u:\O^{\d}_{\e,r} \to {\mathbb{R}}^m$ is the deformation field and
$W$ is the stored energy density.
The $\G$-convergence approach has been used successfully in recent
years to rigorously obtain limit models for various dimensional
reductional problems (see for example \cite{BhJ, BraFo, BFF, LDR,
Shu}).
In this paper we study the multiscale asymptotic behavior of
(\ref{energia}) via $\Gamma$-convergence, as $\e$, $\d$ and $r$ tend
to zero, assuming that $\d= \d(\e)$, $r =r(\e,\d)$ and where $W:
{\mathbb{R}}^{m\times n}\to [0,+\infty)$ is a Borel function satisfying a
growth condition of order $p$, with $1<p<n-1$. The case $p=n-1$
requires a further appropriate analysis and it cannot be easily
derived from $p<n-1$ by slight changes. Unfortunately, three
dimensional linearized elasticity falls into this framework.
Since the sieve $(\o\setminus\o_{\e,r})\times \{0\}$ is not a part
of the domain $\O^{\d}_{\e,r}$, for any fixed $\e, \d, r>0$ we
have no information on the admissible deformation across part of
the mid-section $\o\times\{0\}$. This possible lack of regularity
might produce, in the limit, the above mentioned debonding and
correspondingly an interfacial energy depending on the jump of the
limit deformation. Moreover, we expect that this interfacial
energy will depend on the scaling of the radius of the connecting
zones with respect to the period of their distribution and the
thickness of the thin film.
\bigskip
The cases $\d=1$ and $\d=\e$ have been studied by Ansini
\cite{Ans} who proved that, to recover a non trivial limit
model; {\it i.e., } to obtain a limit model remembering the presence of the
sieve, the meaningful radius (or critical size) of the contact
zones must be of order $\e^{(n-1)/(n-p)}$ and $\e^{n/(n-p)}$,
respectively. In fact a different choice should lead in the limit
to two decoupled problems (if $r$ tends to zero faster than the
critical size) or to the same result that is obtained without the
presence of connecting zones in the mid-section (if $r$ tends to
zero more slowly than the critical size).
The proofs of the $\Gamma$-convergence results in \cite{Ans} (see
Theorems 3.2 and 8.2 therein) are based on a technical lemma
(\cite{Ans}, Lemma 3.4) that allows to modify a sequence of
deformations $u_\e$ with equi-bounded energy, on a suitable
$n$-dimensional spherical annuli surrounding the balls
$B^{n-1}_{r}(x_i^\e)$ without essentially changing their energies,
and to study the behavior of the energies along the new modified
sequence. Both in the case $\d=1$ and $\d=\e$ the $\Gamma$-limits
consist of three terms. The first two terms represent the
contribution of the new sequence far from the balls
$B^{n-1}_{r}(x_i^\e)$; more precisely, they are the
$\Gamma$-limits of two problems defined separately on the upper
and lower part (with respect to the `sieve plane') of the
considered domain. The third term describes the contribution near
the balls $B^{n-1}_{r}(x_i^\e)$ through a nonlinear
capacitary-type formula that is the same for both $\d=1$ and
$\d=\e$. The equality of the two formulas is due to the fact that
the radii of the annuli suitably chosen to separate the two
contributions are less than $c\,\e$, with $c$ an arbitrary small
positive constant. In fact as a consequence, all constructions can
be performed in the interior of the domain, and the same procedure
yielding the nonlinear capacitary-type formula, applies for $\d=1$
and for $\d=\e$ as well. The cases $\e \sim \d$ and $\e \ll \d$
can be treated in the same way.
This approach follows the method introduced by Ansini-Braides in
\cite{AnsB,erratum} where the asymptotic behavior of periodically
perforated nonlinear domains has been studied; in particular,
Lemma 3.4 in \cite{Ans} is a suitable variant, for the sieve
problem, of Lemma 3.1 in \cite{AnsB}.
For other problems related to this subject, we refer the reader to
Attouch-Damlamian-Murat-Picard \cite{ADMP}, \cite{Mur},
\cite{Pic}, Attouch-Picard \cite{AP}, Conca \cite{Con1, Con2,
Con3}, Del Vecchio \cite{Del} and Sanchez-Palencia \cite{S.P.,
S.P.1, S.P.2}, among others.
\bigskip
In this paper we focus our attention on the case $\d=\d(\e)\ll\e$.
As in \cite{Ans}, we expect the existence of a critical radius
$r=r(\e,\d) \ll \e$ for which the limit model is nontrivial but now
we expect also to find different limit regimes depending on the
mutual vanishing rate of $r$ and $\d$. Moreover Lemma 3.4 in
\cite{Ans} cannot be directly applied to our setting since the
spherical annuli surrounding the connecting zones
$B^{n-1}_{r}(x_i^\e)$ as above, are well contained in a strip of
thickness $c\,\e$ but not in $\O^\d_{\e,r}$ ($\d\ll\e$). However, we
are able to modify Lemma 3.4 in \cite{Ans} by considering, instead
of spherical annuli, suitable cylindrical annuli of thickness of
order $\d$ (see Lemma \ref{important} and Lemma
\ref{important+equiint}).
As a consequence, also in this case the asymptotic analysis of
(\ref{energia}) as $\e$, $\d$ and $r$ tend to zero can be carried
on studying separately the energy contributions far from and close
to $B^{n-1}_{r}(x_i^\e)$; we get three terms in the limit. The
first two terms still describe the contribution `far' from the
connecting zones; {\it i.e., } they are the $\Gamma$-limits of the two
dimensional reduction problems defined by
$$
\frac{1}{\d} \int_{\o^{+\d}} W(D u )\, dx\,,\qquad \frac{1}{\d}
\int_{\o^{-\d}} W(D u )\, dx\,;
$$
while the third term, arising in the limit from the energy
contribution close to the connecting zones, represents the
asymptotic memory of the sieve: it is the above mentioned
interfacial energy.
\bigskip
The main results of this paper are stated in Theorem \ref{ABZ} and
Theorem \ref{ABZ2}. In Theorem \ref{ABZ} we prove a
$\Gamma$-convergence result for the sequence of functionals
(\ref{energia}) while in Theorem \ref{ABZ2} we give an explicit
characterization of the interfacial energy term occurring in the
$\Gamma$-limit. More precisely, for every sequence $(\e_j)$
converging to zero, we set $\d_j:=\d(\e_j)$, $r_j:= r(\e_j,\d_j)$,
$\O_j:=\O^{\d_j}_{\e_j,r_j}$ and
$$
{\mathcal F}_j(u):=\left\{
\begin{array}{ll}
\displaystyle \frac{1}{\d_j} \int_{\O_j} W(D
u)\, dx & \text{if }u \in W^{1,p}(\O_j;{\mathbb{R}}^m)\\
&\\
\displaystyle +\infty & \text{otherwise\,.}
\end{array}
\right.
$$
Up to subsequence we can define
$$
\ell:= \lim_{j \to +\infty}\frac{r_j}{\d_j} \quad \text{and} \quad
g(F):=\lim_{j\to +\infty} r_j^p\, {\mathcal{Q}}_n W (r_j^{-1} F).
$$
where ${\mathcal{Q}}_n W$ is the $n$-quasiconvexification of $W$.
If $\ell \in (0,+\infty]$ and
$$
0< R^{(\ell)}:=\lim_{j \to
+\infty}\frac{r_j^{n-1-p}}{\e_j^{n-1}}<+\infty,
$$
then $({\mathcal F}_j)$ $\Gamma$-converges to
$$
{\mathcal F}^{(\ell)}(u^+,u^-)=\int_\o {\mathcal{Q}}_{n-1}\overline W(D_\a
u^+)\, dx_\a + \int_\o {\mathcal{Q}}_{n-1}\overline W(D_\a u^-)\, dx_\a +
R^{(\ell)} \int_\o \varphi^{(\ell)} (u^+ - u^-)\, dx_\a
$$
on $W^{1,p}(\o;{\mathbb{R}}^m) \times W^{1,p}(\o;{\mathbb{R}}^m)$ with respect to
the convergence introduced in Definition {\rm\ref{conv}}, where
$\overline W(\overline F):=\inf \{ W(\overline F|z) : z \in
{\mathbb{R}}^m\}$, ${\mathcal{Q}}_{n-1}\overline W$ is the
$(n-1)$-quasiconvexification of $\overline W$ and
$\varphi^{(\ell)}:{\mathbb{R}}^m \to [0,+\infty)$ is a locally Lipschitz
continuous function for any $\ell\in [0,+\infty]$. Similarly, if
$\ell =0$ and
$$
0< R^{(0)}:=\lim_{j \to +\infty}\frac{r_j^{n-p}}{\d_j\,
\e_j^{n-1}}<+\infty,
$$
then we still have $\Gamma$-convergence, as above, to
$$
{\mathcal F}^{(0)}(u^+,u^-)=\int_\o {\mathcal{Q}}_{n-1}\overline W(D_\a
u^+)\, dx_\a + \int_\o {\mathcal{Q}}_{n-1}\overline W(D_\a u^-)\, dx_\a +
R^{(0)} \int_\o \varphi^{(0)} (u^+ - u^-)\, dx_\a
$$
on $W^{1,p}(\o;{\mathbb{R}}^m) \times W^{1,p}(\o;{\mathbb{R}}^m)$.
\smallskip
For any $\ell\in [0,+\infty]$, $\varphi^{(\ell)}$ is described by
the following nonlinear capacitary-type formulas:
(1) if $\ell=+\infty$, then
\begin{eqnarray*}
\varphi^{(\infty)}(z) & = & \inf \Bigg\{\int_{{\mathbb{R}}^{n-1}}\Bigl(
{\mathcal{Q}}_{n-1} \,\overline g (D_\a \zeta^+ ) + {\mathcal{Q}}_{n-1} \, \overline g
(D_\a \zeta^- ) \Bigr)\, dx_\a : \; \zeta^\pm \in
W^{1,p}_{\rm loc}({\mathbb{R}}^{n-1};{\mathbb{R}}^m),\\
&&\hspace{3.5 cm}\zeta^+=\zeta^- \text{ in }B_1^{n-1}(0), \quad
D_\a \zeta^\pm
\in L^p({\mathbb{R}}^{n-1} ;{\mathbb{R}}^{m \times (n-1)}),\\
&&\hspace{7.5cm} (\zeta^+ - z) \,, \zeta^- \in
L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m) \Bigg\},
\end{eqnarray*}
where again, $\bar g (\overline F):=\inf \{ g(\overline F|z) : z
\in {\mathbb{R}}^m\}$ and ${\mathcal{Q}}_{n-1}\bar g$ is the
$(n-1)$-quasiconvexification of $\bar g$,
\bigskip
(2) if $\ell=0$, then
\begin{eqnarray*}
\varphi^{(0)}(z) & = & \inf \Bigg\{\int_{{\mathbb{R}}^n\setminus
C_{1,\infty} } g (D \zeta) \, dx :\;\zeta \in W^{1,p}_{\rm
loc}({\mathbb{R}}^n \setminus C_{1,\infty};{\mathbb{R}}^m), \, D \zeta \in L^p({\mathbb{R}}^n
\setminus C_{1,\infty};{\mathbb{R}}^{m\times n}),\\&& \hspace{1cm}
\,\zeta-z \in L^{p}(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)) \text{ and
} \zeta \in L^{p}(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\Bigg\}\,,
\end{eqnarray*}
\bigskip
(3) if $\ell \in (0,+\infty)$, then
\begin{eqnarray*}
\varphi^{(\ell)}(z) & = & \inf \Bigg\{\int_{{\mathbb{R}}^{n-1} \times
(-1,1)}g \big(D_\a \zeta | \ell D_n \zeta \big)\, dx : \zeta \in
W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1} \times (-1,1)) \setminus
C_{1,\infty};{\mathbb{R}}^m),\\
&&\hspace{1cm} D \zeta \in L^p({\mathbb{R}}^{n-1} \times (-1,1);{\mathbb{R}}^m),
\;\qquad \zeta - z \in L^p((0,1);L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\\
&&\hspace{7.5 cm} \zeta \in L^p((-1,0);
L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\Bigg\}\,,
\end{eqnarray*}
where $C_{1,\infty}:=\{(x_\a,0)\in{\mathbb{R}}^n \; :\, 1\leq|x_\a| \}$.
\bigskip
Before giving a brief heuristic description of each regime, we
want to point out that whatever the value of $\ell$ is, the
interfacial energy density $\varphi^{(\ell)}$ corresponds to a
cohesive interface where the surface energy increases continuously
from zero with the jump in the deformation across the interface.
\bigskip
(1) The case $\ell=+\infty$ corresponds to $\d_j \ll r_j \ll
\e_j$, thus we expect $r_j$ to depend only on $\e_j$. In this case
we have a separation of scales effect. We first consider $r_j$ and
$\e_j$ as `fixed' and let $\d_j$ tend to zero as if we were
dealing with two pure dimensional reduction problems stated
separately on the upper and lower part (with respect to the sieve
plane) of $\O_j$. Then this first limit procedure yields two
functionals being both a copy of the functional in \cite{LDR}.
Since the two corresponding limit deformations $u^+$ and $u^-$
must match inside each connecting zone, the above two terms are
not completely decoupled. We are then in a situation quite similar
to that of \cite{AnsB,erratum}, except that here both periodically
`perforated' $(n-1)$-dimensional bodies are linked each other
through the `perforations'; {\it i.e., } through the holes of the sieve and
not through the sieve itself. Thus it is coherent to find a
critical size of order $\e^{(n-1)/(n-1-p)}$. Moreover this strong
separation between the phenomena of dimension reduction and
`perforation' leads to anisotropy as it can be seen, for instance,
also by an inspection of the proof of Lemma \ref{gliminfclose}
which shows that the extra interfacial energy term appears thanks
to suitable dilatations having a different scaling in the in-plane
and transverse variables. Finally we note that the formula for
$\varphi^{(\infty)}$ is given in terms of a `Le Dret-Raoult type'
functional involving the limit of the right capacitary scaling
(that is, involving the function $g$).
\smallskip
(2) The case $\ell=0$ corresponds to $r_j \ll \d_j \ll \e_j$. In
this case we expect that the critical size $r_j$ depends on both
$\d_j$ and $\e_j$. Indeed, as already pointed out, $r_j$ is of
order $\d_j^{1/(n-p)} \e_j^{(n-1)/(n-p)}$. Note that for
$\d_j=\e_j$ we recover $\e^{n/(n-p)}$ that is the critical size
obtained in \cite{Ans}; moreover $\varphi^{(0)}$ turns out to
coincide with the function $\varphi$ in \cite{Ans} (see Remark
\ref{Nadia}). Contrary to the previous case, now the isotropy is
preserved in fact here the dimensional reduction and `perforation'
processes are not completely decoupled: the reduction parameter
$\d_j$ is forced between both parameters $r_j$ and $\e_j$. This
can be seen also by noticing that now the scaling leading to the
interfacial energy is the same in every direction (see for
instance the proof of the $\Gamma$-limsup inequality). Moreover
now in $\varphi^{(0)}$ the reduction procedure is not explicit but
only witnessed by the boundary conditions expressed only on the
lateral part of the boundary of the considered domain.
\smallskip
(3) The case $\ell \in (0,+\infty)$ corresponds to $r_j \sim \d_j
\ll \e_j$. In this case the separation of scales effect does not
take place and the two previous scalings turn out to be equivalent
($R^{(0)}=\ell \, R^{(\infty)}$). Moreover we find that the
interfacial energy is continuous with respect to $\ell$ in the
extreme regimes; {\it i.e., } $R^{(\ell)} \varphi^{(\ell)}(z) \to
R^{(\infty)}\varphi^{(\infty)}(z)$ as $\ell \to +\infty$ and
$R^{(\ell)} \varphi^{(\ell)}(z) \to R^{(0)}\varphi^{(0)}(z)$ as
$\ell \to 0$. Finally, as in the previous case, the lateral
boundary conditions are the only mean describing the dimensional
reduction phenomenon in the procedure leading to
$\varphi^{(\ell)}$.
\bigskip
This paper is organized as follows: after recalling some useful
notation in Section \ref{notation}, we state the main results,
Theorem \ref{ABZ} and Theorem \ref{ABZ2}, in Section
\ref{mainres}. Then, in Section \ref{prel} we list some auxiliary
results as rescaled Poincar\'e type inequalities and joining
lemmas. Section \ref{close} is devoted to give a preliminary
definition of the interfacial energy density as limit of minimum
problems. In Section \ref{proof} we prove the $\Gamma$-convergence
result (Theorem \ref{ABZ}). It is only in Section
\ref{nonabstract} that we compute the explicit expression of the
interfacial energy density for each regime (Theorem \ref{ABZ2}).
\section{Notation}\label{notation}
\noindent Given $x \in {\mathbb{R}}^n$, we set $x=(x_\a,x_n)$ where
$x_\a:=(x_1,\ldots,x_{n-1})$ is the in-plane variable and
$D_\a:=\left(\frac{\partial}{\partial
x_1},\ldots,\frac{\partial}{\partial x_{n-1}}\right)$ (resp.
$D_n$) the derivative with respect to $x_\a$ (resp. $x_n$).
The notation ${\mathbb{R}}^{m \times n}$ stands for the set of $m \times n$
real matrices. Given a matrix $F \in {\mathbb{R}}^{m \times n}$, we write
$F=(\overline F|F_n)$ where $\overline F=(F_1, \ldots , F_{n-1})$
and $F_i$ denotes the $i$-th column of $F$, $1 \leq i \leq n$ and
$\overline F\in {\mathbb{R}}^{m \times (n-1)}$.
The Lebesgue measure in ${\mathbb{R}}^n$ will be denoted by $\mathcal L^n$
and the Hausdorff $(n-1)$-dimensional measure by ${\mathcal H}^{n-1}$. Let
$A$ be an open subset of ${\mathbb{R}}^d$ ($d=n-1$, $d=n$). If $s\in
[1,+\infty]$, we use standard notation for Lebesgue and Sobolev
spaces $L^s(A;\Bbb R^m)$ and $W^{1,s}(A;\Bbb R^m)$.
Let $\o$ be a bounded open subset of ${\mathbb{R}}^{n-1}$ and $I=(-1,1)$,
we define $\O:=\o \times I$. In the sequel, we will identify
$L^s(\o;{\mathbb{R}}^m)$ (resp. $W^{1,s}(\o;{\mathbb{R}}^m)$) with the space of
functions $v \in L^s(\O;{\mathbb{R}}^m)$ (resp. $W^{1,s}(\O;{\mathbb{R}}^m)$) such
that $D_n v = 0$ in the sense of distribution.
For every $(a,b)\subset {\mathbb{R}}$ with $a<b$ and $q_1,q_2\geq 1$,
$L^{q_1}(a,b;L^{q_2}({\mathbb{R}}^{(n-1)};{\mathbb{R}}^m))$ is the space of
measurable $m$-vectorial functions $\zeta$ such that
$$
\int^a_b\left(\int_{{\mathbb{R}}^{n-1}}|\zeta(x_\a,x_n)|^{q_2}\,dx_\a\right)^{\frac{q_1}{q_2}}\,dx_n<+\infty.
$$
Let $a \in {\mathbb{R}}^{n-1}$ and $\rho>0$, we denote by $B^{n-1}_\rho(a)$
the open ball of ${\mathbb{R}}^{n-1}$ of center $a$ and radius $\rho$ and
by $Q^{n-1}_\rho(a)$ the open cube of ${\mathbb{R}}^{n-1}$ with center $a$
and length side $\rho$. We write $B^{n-1}_\rho$ instead of
$B^{n-1}_\rho(0)$ not to overburden notation. Let $x_i^\e= i\e$
with $i\in {\mathbb{Z}}^{n-1}$, we set $Q_{i,\e}^{n-1}:= Q^{n-1}_\e
(x_i^\e)$.
We define $U^{+a} := U \times (0,a)$ and $U^{-a}:= U \times
(-a,0)$ with $U \subseteq {\mathbb{R}}^{n-1}$ and $a>0$, while if $a=1$,
then $U^{+} = U^{+1}$ and $U^-= U^{-1}$.
We set $C_{1,\infty}:=\left\{ (x_\a,0) \in {\mathbb{R}}^n : 1\leq |x_\a|
\right\}$ and $C_{1,N}:=\left\{ (x_\a,0) \in {\mathbb{R}}^n : 1\leq |x_\a|
< N \right\}$ for every $N>1$.
Let $p \geq 1$, we denote by ${\rm Cap}_p(B^{n-1}_1; A)$ the
$p$-capacity of $B^{n-1}_1$ with respect to $A \subseteq {\mathbb{R}}^d$:
$$
{\rm Cap}_p(B^{n-1}_1 ;A)=\inf \left\{ \int_A |D \psi |^p\, dx :
\; \psi \in W^{1,p}_0(A) \text{ and }\psi=1 \text{ on }B^{n-1}_1
\right\}.
$$
The letter $c$ will stand for a generic strictly-positive constant
which may vary from line to line and expression to expression
within the same formula.
\section{Statements of the main results}\label{mainres}
\noindent Since we are going to work with varying domains, we have
to precise the meaning of `converging sequences'.
\begin{defi}\label{conv} Let
$ \Omega_j= \o^{+\d_j} \cup \o^{-\d_j} \cup \big( \o_{r_j,\e_j}
\times \{0\} \big)$. Given a sequence $(u_j) \subset
W^{1,p}(\O_j;{\mathbb{R}}^m)$, we define ${\hat
u}_j(x_\a,x_n):=u_j(x_\a,\d_j\, x_n)$. We say that $(u_j)$
converges {\rm (}resp. converges weakly{\rm )} to $(u^+,u^-) \in
W^{1,p}(\o;{\mathbb{R}}^m) \times W^{1,p}(\o;{\mathbb{R}}^m)$ if we have
\begin{eqnarray*}
\hat u^+_j & := & \hat u_j|_{\o^+} \to u^+ \hbox{ in}\quad
L^p(\o^+;{\mathbb{R}}^m) \quad (\hbox{resp. weakly in
}W^{1,p}(\o^+;{\mathbb{R}}^m)),\\
\hat {u}^-_j & := & \hat u_j|_{\o^-} \to u^- \hbox{ in}\quad
L^p(\o^-;{\mathbb{R}}^m) \quad (\hbox{resp. weakly in
}W^{1,p}(\o^-;{\mathbb{R}}^m)).
\end{eqnarray*}
Similarly if we replace $\O_j$ by $\o^{\pm \d_j}$.
\smallskip
We say that the sequence $(|D u_j|^p / \d_j)$ is equi-integrable
on $\o^{\pm \d_j}$ if $\big( \big|\big(D_\a \hat u_j
|\frac{1}{\d_j} D_n \hat u_j \big) \big|^p \big)$ is
equi-integrable on $\o^\pm$.
\end{defi}
\begin{rmk}\label{convrmk}{\rm By
virtue of Definition \ref{conv}, a sequence $(u_j) \subset
W^{1,p}(\O_j;{\mathbb{R}}^m)$ converges to $(u^+,u^-)\in W^{1,p}(\o;{\mathbb{R}}^m)
\times W^{1,p}(\o;{\mathbb{R}}^m)$ if and only if
\begin{equation}\label{strongconv}
\lim_{j\to +\infty}\frac{1}{\d_j}\int_{\o^{\pm\d_j}}|u_j -
u^\pm|^p\, dx = 0,
\end{equation}
while (\ref{strongconv}) and
\begin{equation}\label{weakconv}
\sup_{j \in {\mathbb{N}}} \frac{1}{\d_j}\int_{\o^{\pm\d_j}}|D u_j|^p\,dx =
\sup_{j \in {\mathbb{N}}} \int_{\o^\pm}\left|\left(D_\a \hat
u_j\Big|\frac{1}{\d_j} D_n \hat u_j \right) \right|^p dx <+\infty
\end{equation}
imply the weak convergence.}\end{rmk}
Note that Remark \ref{convrmk} is still valid if we consider the
domain $ \o^{+\d_j} \cup\o^{-\d_j}$ in place of $\O_j$.
The main results of this paper are the following:
\begin{thm}[$\Gamma$-convergence]\label{ABZ}
Let $1<p<n-1$. Let $\o$ be a bounded open subset of ${\mathbb{R}}^{n-1}$
satisfying ${\mathcal H}^{n-1}(\partial \o)=0$ and $W:{\mathbb{R}}^{m \times n} \to
[0,+\infty)$ be a Borel function such that $W(0)=0$ and satisfying
a growth condition of order $p$ : there exists a constant
$\beta>0$ such that
\begin{equation}\label{pgrowth}
|F|^p - 1 \leq W(F) \leq \beta (|F|^p + 1), \quad \text{for every
} F \in {\mathbb{R}}^{m \times n}.
\end{equation}
Let $(\e_j)$, $(\d_j)$ and $(r_j)$ be sequences of strictly
positive numbers converging to zero such that
$$
\lim_{j \to +\infty}\frac{\d_j}{\e_j}=0
$$
and set
$$
\ell:= \lim_{j \to +\infty}\frac{r_j}{\d_j}\,.
$$
If
$$
\ell \in (0,+\infty] ,\quad \text{ and }\quad 0<
R^{(\ell)}:=\lim_{j \to
+\infty}\frac{r_j^{n-1-p}}{\e_j^{n-1}}<+\infty
$$
or
$$
\ell =0 ,\quad \text{ and } \quad 0< R^{(0)}:=\lim_{j \to
+\infty}\frac{r_j^{n-p}}{\d_j\, \e_j^{n-1}}<+\infty\,,
$$
then, up to an extraction, the sequence of functionals $\mathcal
F_j : L^p(\O_j;{\mathbb{R}}^m) \to [0,+\infty]$ defined by
$$
\mathcal F_j(u):=\left\{
\begin{array}{ll}
\displaystyle \frac{1}{\d_j} \int_{\O_j} W(D
u)\, dx & \text{if }u \in W^{1,p}(\O_j;{\mathbb{R}}^m),\\
&\\
\displaystyle +\infty & \text{otherwise}
\end{array}
\right.
$$ $\G$-converges to
$$
\mathcal F^{(\ell)}(u^+,u^-)=\int_\o {\mathcal{Q}}_{n-1}\overline W(D_\a
u^+)\, dx_\a + \int_\o {\mathcal{Q}}_{n-1}\overline W(D_\a u^-)\, dx_\a +
R^{(\ell)} \int_\o \varphi^{(\ell)} (u^+ - u^-)\, dx_\a
$$
on $W^{1,p}(\o;{\mathbb{R}}^m) \times W^{1,p}(\o;{\mathbb{R}}^m)$ with respect to
the convergence introduced in Definition {\rm\ref{conv}}, where
$\overline W(\overline F):=\inf \{ W(\overline
F|z) : z \in {\mathbb{R}}^m\}$, ${\mathcal{Q}}_{n-1}\overline W$ is the
$(n-1)$-quasiconvexification of $\overline W$ and
$\varphi^{(\ell)}:{\mathbb{R}}^m \to [0,+\infty)$ is a locally Lipschitz
continuous function for any $\ell\in [0,+\infty]$.
\end{thm}
\begin{rmk}\label{andrea}{\rm
Note that if $\ell\in (0,+\infty]$ the only meaningful scaling for
$r_j$ is that of order $\e_j^{(n-1)/(n-1-p)}$; {\it i.e., } for both
$R^{(\ell)}=0$ and $R^{(\ell)}=+\infty$ we loose the asymptotic
memory of the sieve. In fact, if $R^{(\ell)}=0$, we obtain two
uncoupled problems in the limit, while if $R^{(\ell)}=+\infty$,
limit deformations $(u^+, u^-)$ with finite energy are continuous
across the mid-section ($u^+=u^-$ in $\omega$) as in Le Dret-Raoult
\cite{LDR}. Similarly, for $\ell=0$.}
\end{rmk}
\smallskip
\begin{rmk}\label{equiv}{\rm
If $\ell \in (0,+\infty)$ then
$$
0<R^{(\ell)}=\lim_{j \to +\infty} \frac{r_j^{n-1-p}}{\e_j^{n-1}} <
+\infty \qquad \hbox{if\, and\, only\, if} \qquad 0<R^{(0)}=
\lim_{j \to +\infty} \frac{r_j^{n-p}}{\d_j\,\e_j^{n-1}} <
+\infty\,;
$$
hence, in this case the two meaningful scalings are equivalent.}
\end{rmk}
\noindent The following result provides a characterization of the
interfacial energy density $\varphi^{(\ell)}$ for each $\ell \in
[0,+\infty]$.
\begin{thm}[Representation formulas]\label{ABZ2}
Let $p^*=(n-1)p/(n-1-p)$ be the Sobolev exponent in dimension
$(n-1)$. Then, upon extracting a subsequence, there exists the
limit
$$g(F):=\lim_{j\to +\infty} r_j^p {\mathcal{Q}}_n W (r_j^{-1} F),$$
for all $F \in {\mathbb{R}}^{m \times n}$, where ${\mathcal{Q}}_n W$ denotes the
$n$-quasiconvexification of $W$, so that:\\
if $\ell \in (0,+\infty)$,
\begin{eqnarray*}
\varphi^{(\ell)}(z) & := & \inf \Bigg\{\int_{({\mathbb{R}}^{n-1} \times I)
\setminus C_{1,\infty}}g \big(D_\a \zeta | \ell D_n \zeta \big)\,
dx : \zeta \in W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1} \times I) \setminus
C_{1,\infty};{\mathbb{R}}^m),\\
&&\hspace{0.3cm} D \zeta \in L^p (({\mathbb{R}}^{n-1} \times I) \setminus
C_{1,\infty};{\mathbb{R}}^{m \times n} ),
\;\quad \zeta - z \in L^p (0,1;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\\
&&\hspace{6.5 cm} \zeta \in L^p (-1,0;
L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\Bigg\};
\end{eqnarray*}
if $\ell=+\infty$
\begin{eqnarray*}
\varphi^{(\infty)}(z) & := & \inf \Bigg\{\int_{{\mathbb{R}}^{n-1}}\Bigl(
{\mathcal{Q}}_{n-1} \,\overline g (D_\a \zeta^+ ) + {\mathcal{Q}}_{n-1} \, \overline g
(D_\a \zeta^- ) \Bigr)\, dx_\a : \; \zeta^\pm \in
W^{1,p}_{\rm loc}({\mathbb{R}}^{n-1};{\mathbb{R}}^m),\\
&&\hspace{3.9 cm}\zeta^+=\zeta^- \text{ in }B_1^{n-1}, \quad D_\a
\zeta^\pm
\in L^p({\mathbb{R}}^{n-1} ;{\mathbb{R}}^{m \times (n-1)}),\\
&&\hspace{6.7cm} (\zeta^+ - z) \,, \zeta^- \in
L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m) \Bigg\}\,,
\end{eqnarray*} where $\overline g(\overline F):=\inf \{ g(\overline
F|z) : z \in {\mathbb{R}}^m\}$ and ${\mathcal{Q}}_{n-1}\overline g$ is the
$(n-1)$-quasiconvexification of $\overline g$;\\\smallskip if
$\ell=0$
\begin{eqnarray*}
\varphi^{(0)}(z) & = & \inf \Bigg\{\int_{{\mathbb{R}}^n \setminus
C_{1,\infty}} g (D \zeta) \, dx :\; \zeta \in W^{1,p}_{\rm
loc}({\mathbb{R}}^n \setminus C_{1,\infty};{\mathbb{R}}^m),\; D \zeta \in L^p({\mathbb{R}}^n
\setminus
C_{1,\infty};{\mathbb{R}}^{m \times n}),\\
&& \hspace{2.0cm} \zeta-z \in
L^p(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)),\; \zeta \in
L^p(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)) \Bigg\}\,,
\end{eqnarray*}
for all $z\in {\mathbb{R}}^m$.
\end{thm}
\bigskip
\begin{rmk}{\rm
Without loss of generality we may assume that $W$ is quasiconvex
(upon first relaxing the energy); hence, by (\ref{pgrowth}), $W$
satisfies the following $p$-Lipschitz condition (see e.g.
\cite{D}):
\begin{equation}\label{plip}
|W(F_1) -W(F_2)| \leq c\, (1+ |F_1|^{p-1}+ |F_2|^{p-1})|F_1-F_2|,
\quad \text{ for all }F_1,\; F_2 \in {\mathbb{R}}^{m \times n}\,.
\end{equation}
}\end{rmk}
\section{Preliminary results}\label{prel}
\subsection{Some rescaled Poincar\'e Inequalities}
\noindent Since we deal with varying domains, depending on
different parameters, it is useful to note how the constant in
Poincar\'e type inequalities rescale with respect to such
parameters.
\begin{lemma}\label{poincare}
Let $A$ be an open bounded and connected subset of ${\mathbb{R}}^{n-1}$
with Lipschitz boundary and let $A_\rho:=\rho A$ for $\rho>0$.
\begin{itemize}
\item[(i)]There exists a constant $c>0$ (depending only on
$(A,n,p)$) such that for every $\rho,\d>0$
$$\int_{A^{\pm \d}_\rho}|u - \overline u_{A_\rho^{\pm \d}}|^p\, dx
\leq c \int_{A_\rho^{\pm \d}}\left( \rho^p |D_\a u|^p + \d^p|D_n
u|^p \right)\, dx,
$$
for every $u \in W^{1,p}(A^{\pm\d}_\rho;{\mathbb{R}}^m)$ where $\overline
u_{A_\rho^{\pm \d}}= - \hskip-0.9em \int_{A^{\pm \d}_\rho} u \,dx$.
\item[(ii)]If $B$ is an open and connected subset of $A$ with Lipschitz
boundary and $B_\rho := \rho B$ then there exists a constant $c>0$
(depending only on $(A,B,n,p)$) such that for every $\rho,\d>0$
$$\int_{A^{\pm \d}_\rho}|u - \overline u_{B^{\pm \d}_\rho}|^p\, dx
\leq c \int_{A^{\pm \d}_\rho}\left( \rho^p |D_\a u|^p + \d^p|D_n
u|^p \right)\, dx,
$$
for every $u \in W^{1,p}(A^{\pm \d}_\rho;{\mathbb{R}}^m)$ where $\overline
u_{B^{\pm \d}_\rho}= - \hskip-0.9em \int_{B^{\pm \d}_\rho} u \,dx$.
\end{itemize}
\end{lemma}
\noindent {\it Proof. } Let us define $v(x_\a,x_n):=u(\rho x_\a,
\d x_n)$ then $v \in W^{1,p}(A^\pm;{\mathbb{R}}^m)$. By a change of
variable, we get that $\overline u_{A_\rho^{\pm \d}}=\overline
v_{A^\pm}$. Moreover, by the Poincar\'e Inequality, there exists a
constant $c=c(A,n,p)>0$ such that
\begin{eqnarray*}
\int_{A_\rho^{\pm \d}}|u - \overline u_{A_\rho^{\pm\d}}|^p\, dx &
= & \d \rho^{n-1} \int_{A^\pm}|v -
\overline v_{A^\pm}|^p\, dy\\
& \leq & c\d \rho^{n-1} \int_{A^\pm}|D v|^p\, dy\\
& = & c \int_{A_\rho^{\pm \d}}\left(\rho^p |D_\a u|^p +
\d^p |D_n u|^p \right)\, dx
\end{eqnarray*}
and it completes the proof of (i). Now, if $B_\rho \subset
A_\rho$, we get that
\begin{eqnarray*}
&&\int_{A_\rho^{\pm\d}}|u - \overline u_{B_\rho^{\pm\d}}|^p\, dx\\
& \leq & c\Bigl(\int_{A_\rho^{\pm\d}}|u - \overline
u_{A_\rho^{\pm\d}}|^p\, dx +\d \rho^{n-1} {\mathcal H}^{n-1}(A)|\overline
u_{A_\rho^{\pm\d}}-\overline u_{B_\rho^{\pm\d}}|^p\Bigr)\\
& \leq & c\int_{A_\rho^{\pm\d}} |u - \overline
u_{A_\rho^{\pm\d}}|^p\, dx + c \frac{{\mathcal H}^{n-1}(A)}{{\mathcal H}^{n-1}(B)}
\left( \int_{B_\rho^{\pm\d}}|u - \overline u_{A_\rho^{\pm\d}}|^p
\, dx + \int_{B_\rho^{\pm\d}} |u - \overline
u_{B_\rho^{\pm\d}}|^p\, dx\right)\\
& \leq & c \int_{A_\rho^{\pm\d}} \left( \rho^p |D_\a u|^p + \d^p
|D_n u|^p \right)\, dx.
\end{eqnarray*}
\hfill$\Box$\\
\subsection{A joining lemma on varying domains}
\noindent If not otherwise specified, in all what follows the
convergence of a sequence of functions has to be intended in the
sense of Definition \ref{conv}.
\smallskip
The following lemma, is the key tool in the proof of Theorem
\ref{ABZ}. It is a technical result which allows to modify
sequences of functions `near' the sets
$B^{(n-1)}_{r_j}(x_i^{\e_j})$. It is very close in spirit to Lemma
3.4 in \cite{Ans} although now the geometry of the problem yields
a different construction involving suitable cylindrical (instead
of spherical) annuli to surround the connecting zones.
\begin{lemma}\label{important}
Let $(\e_j)$, $(\d_j)$ be sequences of strictly positive numbers
converging to $0$ and such that $\d_j\ll \e_j$. Let $(u_j) \subset
W^{1,p}(\o^{+\d_j}\cup \o^{-\d_j};{\mathbb{R}}^m)$ be a sequence converging
to $(u^+,u^-) \in W^{1,p}(\o;{\mathbb{R}}^m) \times W^{1,p}(\o;{\mathbb{R}}^m)$
satisfying $\sup_j\mathcal{F}_j(u_j)<+\infty$; let $k\in {\mathbb{N}}$. Set
$\rho_j=\g\e_j$ with $\g<1/2$ and
$$ Z_j:=\{i \in {\mathbb{Z}}^{n-1}: \; {\rm dist}(x_i^{\e_j},{\mathbb{R}}^{n-1}
\setminus \o)
> \e_j\}\,.
$$
For every $i \in Z_j$, there exists $k_i \in \{0,\ldots,k-1\}$
such that having set
$$
C_j^i := \left\{x_\a \in \o :\; 2^{-k_i-1}\rho_j <|x_\a -
x_i^{\e_j}| < 2^{-k_i}\rho_j\right\},$$
\begin{equation}\label{aver} u_j^{i \pm} :=
- \hskip-1.0em \int_{(C_j^i)^{\pm\d_j}} u_j\, dx
\end{equation}
and
$$
\rho_j^i := \frac{3}{4}2^{-k_i} \rho_j,$$ there exists a sequence
$(w_j)\subset W^{1,p}(\o^{+\d_j}\cup \o^{-\d_j};{\mathbb{R}}^m)$ weakly
converging to $(u^+,u^-)$ such that
\begin{equation}\label{wj1} w_j=u_j \; \text{ in }\; \Bigl(\o
\setminus \bigcup_{i \in Z_j} C_j^i \Bigr)^{\pm\d_j},
\end{equation}
\begin{equation}\label{wj2}
w_j=u_j^{i\pm} \; \text{ on } \; \big(\partial
B^{n-1}_{\rho_j^i}(x_i^{\e_j})\big)^{\pm\d_j}
\end{equation} and satisfying
\begin{equation}\label{wj3}
\limsup_{j\to +\infty}\frac{1}{\d_j}\int_{\o^{\pm\d_j}} \big| W(D
w_j) - W(D u_j) \big|\, dx \leq \frac{c}{k}\,.
\end{equation}
\end{lemma}
\noindent {\it Proof. }For every $j\in {\mathbb{N}}$, $i\in Z_j$, $k \in
{\mathbb{N}}$ and $h \in \{0,\ldots, k-1\}$, we define
$$
C_j^{i,h}:=\left\{ x_\a \in \o :\; 2^{-h-1}\rho_j <
|x_\a-x_i^{\e_j}| < 2^{-h}\rho_j \right\},
$$
$$(u_j^{i,h})^\pm := - \hskip-1.0em \int_{(C_j^{i,h})^{\pm\d_j}}u_j\, dx$$ and
\begin{equation}\label{def-rhojih}
\rho_j^{i,h}:=\frac{3}{4}2^{-h}\rho_j.
\end{equation}
Let $\phi \equiv \phi_j^{i,h} \in {\mathcal{C}}_c^\infty(C_j^{i,h};[0,1])$ be
a cut-off function such that $\phi=1$ on $\partial
B^{n-1}_{\rho_j^{i,h}}(x_i^{\e_j})$ and $|D_\a \phi| \leq
c/\rho_j^{i,h}$. In $(C_j^{i,h})^{\pm\d_j}$, we set
$$
w_j^{i,h}(x):=\phi(x_\a) (u_j^{i,h})^\pm + (1-\phi(x_\a)) u_j,
$$
then
\begin{eqnarray*}
\int_{(C_j^{i,h})^{\pm\d_j}}|D w_j^{i,h}|^p \,dx & \leq & c
\int_{(C_j^{i,h})^{\pm\d_j}} \left(|D_\a
\phi|^p|u_j-(u_j^{i,h})^\pm|^p + |D u_j|^p \right) dx\\
& \leq & c\int_{(C_j^{i,h})^{\pm\d_j}}
\left(\frac{|u_j-(u_j^{i,h})^\pm|^p}{(\rho_j^{i,h})^p}
+ |D u_j|^p \right) dx.
\end{eqnarray*}
Applying Lemma \ref{poincare} (i), with $\rho=\rho_j^{i,h} $ and
$A_\rho=C_j^{i,h}$, we have that
\begin{eqnarray}\label{1615}
&&\int_{(C_j^{i,h})^{\pm\d_j}} |D w_j^{i,h}|^p \,dx \nonumber\\
&\leq& c \int_{(C_j^{i,h})^{\pm\d_j}}\left( |D_\a u_j|^p +
\Bigl({\d_j\over\rho_j^{i,h}}\Bigr)^p |D_n u_j|^p \right) dx + c
\int_{(C_j^{i,h})^{\pm\d_j}}
|D u_j|^p \, dx\nonumber\\
&\leq& m_j(k,\g)\, c \int_{(C_j^{i,h})^{\pm\d_j}} |D u_j|^p \, dx,
\end{eqnarray}
where by (\ref{def-rhojih})
$$m_j(k,\g):= \max\Bigl\{1, \Bigl({2^{k+1}\over 3\g}\Bigr)^p
\Bigl({\d_j\over \e_j}\Bigr)^p\Bigr\}
$$
and since $\d_j \ll \e_j$, $m_j(k,\g)\to 1$ as $j \to +\infty$. As
$$\sum_{h=0}^{k-1}
\int_{(C_j^{i,h})^{\pm\d_j}}(1+|D u_j|^p)\, dx \leq
\int_{B^{n-1}_{\rho_j}(x_i^{\e_j})^{\pm\d_j}}(1+ |D u_j|)^p\ dx,$$
there exists $k_i \in \{0,\ldots,k-1\}$ such that, having set
$C_j^i:=C_j^{i,k_i}$, we get
\begin{equation}\label{1627}
\int_{(C_j^i)^{\pm\d_j}}(1+|D u_j|^p)\, dx \leq \frac{1}{k}
\int_{B^{n-1}_{\rho_j}(x_i^{\e_j})^{\pm\d_j}}(1+ |D u_j|^p)\, dx.
\end{equation}
\begin{figure
\centerline{\psfig{file=FIG2.eps,width=4in}} \vspace*{8pt}
\caption{The $(n-1)$-dimensional annuli $C^i_j$\,.}
\end{figure}
Hence, if we define the sequence
$$
w_j:=\left\{
\begin{array}{ll}
w_j^{i,k_i} & \text{in }
(C_j^i)^{\pm\d_j}\, \text{ for }\, i \in Z_j\\
&\\ u_j & \text{otherwise}\,,
\end{array}\right.
$$
by the $p$-growth condition (\ref{pgrowth}), (\ref{1615}),
(\ref{1627}) and Remark \ref{convrmk} we have
\begin{eqnarray*}
\frac{1}{\d_j}\int_{\o^{\pm\d_j}} \big| W(D w_j) - W(D u_j)
\big|\, dx &=& \sum_{i \in Z_j}
\frac{1}{\d_j}\int_{(C_j^i)^{\pm\d_j}} \big| W(D w_j^{i,k_i}) -
W(D u_j) \big|\,
dx\\
& \leq &{c\over k}\, m_j(k,\g) \sum_{i \in Z_j} \frac{1}{\d_j}
\int_{B^{n-1}_{\rho_j}(x_i^{\e_j})^{\pm\d_j}}(1+ |D u_j|^p)\,
dx\\
& \leq & \frac{c}{k}\, m_j(k,\g) \left(1+\sup_{j \in {\mathbb{N}}}
\frac{1}{\d_j} \int_{\o^{\pm\d_j}} |Du_j|^p\,dx \right)\\
&\le& \frac{c}{k}\, m_j(k,\g) \,,
\end{eqnarray*}
which concludes the proof of (\ref{wj3}). Note that, by
construction, $(w_j)$ satisfies (\ref{wj1}) and (\ref{wj2}) and it
converges weakly to $(u^+,u^-)$. In fact,
\begin{eqnarray*}
\frac{1}{\d_j}\int_{\o^{\pm\d_j}} |w_j -u^\pm|^p\, dx & = &
\frac{1}{\d_j}\sum_{i \in Z_j}\int_{ (C_j^i)^{\pm\d_j}} |\phi
u_j^{i\pm}
+(1-\phi)u_j - u^\pm|^p\, dx\\
&&+\frac{1}{\d_j}\int_{\o^{\pm\d_j} \setminus \bigcup_{i \in Z_j}
(C_j^i)^{\pm\d_j}} |u_j-u^\pm|^p\, dx\\
& \leq & \frac{c}{\d_j}\int_{\o^{\pm\d_j}} |u_j -u^\pm|^p\, dx
+\frac{c}{\d_j}\sum_{i \in Z_j}\int_{ (C_j^i)^{\pm\d_j}}| u_j -
u_j^{i\pm}|^p\, dx,
\end{eqnarray*}
while by Lemma \ref{poincare} (i) applied with $\rho=\rho_j^i$ and
since $\d_j \ll \e_j$, $\rho_j^i \leq \e_j$, we get
\begin{equation}\label{wconv1}
\frac{1}{\d_j}\int_{\o^{\pm\d_j}} |w_j -u^\pm|^p\, dx \leq
\frac{c}{\d_j}\int_{\o^{\pm\d_j}} |u_j -u^\pm|^p\, dx + c\e_j^p
\frac{1}{\d_j}\int_{\o^{\pm\d_j}} |D u_j|^p\, dx \,.
\end{equation}
Moreover by (\ref{1615}) we have
\begin{equation}\label{wconv2}
\frac{1}{\d_j}\int_{\o^{\pm\d_j}} |D w_j|^p\, dx \leq
\frac{c}{\d_j}\int_{\o^{\pm\d_j}} |D u_j|^p\, dx.
\end{equation}
Hence (\ref{wconv1}), (\ref{wconv2}), the convergence of $(u_j)$
towards $(u^+,u^-)$, $\sup_j\frac{1}{\d_j}\int_{\o^{\pm
\d_j}}|Du_j|^p\,dx<+\infty$ together with Remark \ref{convrmk}
imply the weak convergence of $(w_j)$ towards
$(u^+,u^-)$. \hfill$\Box$\\
\begin{rmk}{\rm Note that to prove Lemma \ref{important}
we essentially use that $\rho_j< \e_j/2$ (but not necessarily
equal to $\g\e_j$) and $\lim_{j\to +\infty}(\d_j/\rho_j)=0$.
Hence, Lemma \ref{important} is still true if we replace the
assumptions $\d_j\ll \e_j$ and $\rho_j= \g\e_j$ by $\rho_j<
\e_j/2$ and $\lim_{j\to +\infty}(\d_j/\rho_j)=0$.
Since we will apply Lemma \ref{important} when $\rho_j= \g\e_j$
($\g<1/2$) and $\d_j\ll \e_j$, we prefer to prove it directly
under these assumptions. }\end{rmk}
\bigskip
If the sequence $(|D u_j|^p/\d_j)$ is equi-integrable on
$\o^{\pm\d_j}$ (see Definition \ref{conv}), then we do not have
to choose for every $i \in Z_j$ a suitable annulus $C_j^i$ but we
may consider the same radius independently of $i$ as the following
lemma shows.
\begin{lemma}\label{important+equiint}
Let $(u_j)$, $(\e_j)$, $(\d_j)$, $(\rho_j)$ and $Z_j$ be as in
Lemma {\rm\ref{important}} and suppose that $(|D u_j|^p/\d_j)$ is
equi-integrable on $\o^{\pm\d_j}$. Set
$$
C_j^i := \left\{x_\a \in \o : \frac{2}{3}\rho_j <|x_\a -
x_i^{\e_j}| < \frac{4}{3} \rho_j\right\} \quad\text{ and }\quad
u_j^{i \pm} := - \hskip-1.0em \int_{(C_j^i)^{\pm\d_j}} u_j\, dx
$$
for every $i\in Z_j$. Then, there exists a sequence $(w_j) \subset
W^{1,p}(\o^{+\d_j}\cup \o^{-\d_j};{\mathbb{R}}^m)$ weakly converging to
$(u^+,u^-)$ such that
\begin{equation}\label{wj1bis} w_j=u_j \; \text{ in }\; \Bigl(\o
\setminus \bigcup_{i \in Z_j} C_j^{i}\Bigr)^{\pm\d_j},
\end{equation}
\begin{equation}\label{wj2bis}
w_j=u_j^{i\pm} \; \text{ on } \; \big( \partial
B^{n-1}_{\rho_j}(x_i^{\e_j}) \big) ^{\pm\d_j}
\end{equation} and
\begin{equation}\label{wj3bis}
\limsup_{j\to +\infty}\frac{1}{\d_j}\int_{\o^{\pm\d_j}} \big| W(D
w_j) - W(D u_j) \big|\, dx \le o(1)\quad {\rm as}\quad \g \to
0^+\,.
\end{equation}
Moreover, the sequence $(|D w_j|^p/\d_j)$ is equi-integrable on
$\o^{\pm\d_j}$.
\end{lemma}
\noindent {\it Proof. }Let $\phi \equiv \phi_j^i \in
{\mathcal{C}}_c^\infty(C_j^i;[0,1])$ be a cut-off function such that $\phi=1$
on $\partial B^{n-1}_{\rho_j}(x_i^{\e_j})$ and $|D_\a \phi| \leq
c/\rho_j$. In $(C_j^i)^{\pm\d_j}$, we define
$$
w_j^{i}:=\phi(x_\a) u_j^{i\pm} + (1-\phi(x_\a)) u_j.
$$
Then, reasoning as in the proof of Lemma \ref{important}, we have
that
$$
\int_{(C_j^i)^{\pm\d_j}}W(D w_j^i)\, dx \leq c
\int_{(C_j^i)^{\pm\d_j}} (1+|D u_j|^p) \, dx.
$$
Hence, if we define
$$
w_j:=\left\{
\begin{array}{ll}
w_j^i & \text{in }
(C_j^i)^{\pm\d_j}\, \text{ for }\,i \in Z_j,\\
&\\ u_j & \text{otherwise},
\end{array}\right.$$
$w_j$ satisfies (\ref{wj1bis}) and (\ref{wj2bis}). Moreover,
\begin{eqnarray*}
\frac{1}{\d_j} \int_{\o^{\pm\d_j}} \big| W(D w_j) - W(D u_j)
\big|\, dx & \leq & \sum_{i \in Z_j} \frac{1}{\d_j}
\int_{(C_j^i)^{\pm\d_j}} \big|
W(D w_j^i) - W(D u_j) \big|\,dx\\
& \leq & c \sum_{i \in Z_j} \frac{1}{\d_j}
\int_{(B^{n-1}_{4\rho_j/3}(x_i^{\e_j}) \cap \o)^{\pm\d_j}}(1+ |D
u_j|^p)\, dx.
\end{eqnarray*}
Since $\#(Z_j) \leq c/\e_j^{n-1}$, we get that
$$\mathcal H^{n-1}\Bigg( \bigcup_{i \in Z_j}
(B^{n-1}_{4\rho_j/3}(x_i^{\e_j}) \cap \o) \Bigg) \leq c \g^{n-1}$$
and by the equi-integrability of $(|D u_j|^p/\d_j)$ we obtain
(\ref{wj3bis}). Finally, the weak convergence of $(w_j)$ can be
proved as in Lemma \ref{important} while the equi-integrability of
$(|D w_j|^p/\d_j)$ is just a consequence of the definition of
$(w_j)$. \hfill$\Box$
\section{A preliminary analysis of the energy contribution `close' to
the connecting zones}\label{close} \noindent For later references,
in the following section we study the asymptotic behavior of a
sequence of functions which will turn out to represent the energy
contribution `close' to the connecting zones. The results listed
in this section will be applied in Section 6 to prove the
$\Gamma$-convergence of $(\mathcal{F}_j)$ as well as in Section 7
to compute the explicit formula for $\varphi^{(\ell)}$.
\bigskip
Before starting, let us recall that we consider the domain $
\Omega_j= \o^{+\d_j} \cup \o^{-\d_j} \cup \big( \o_{r_j,\e_j}
\times \{0\} \big)$ where $\o_{r_j,\e_j}:= \bigcup_{i
\in{\mathbb{Z}}^{n-1}}B^{n-1}_{r_j}(x_i^{\e_j}) \cap \o$. Our
$\G$-convergence analysis deals with the case where the thickness
$\d_j$ of $\O_j$ is much smaller than the period of distribution
of the connecting zones $\e_j$; {\it i.e., }
$$
\lim_{j\to +\infty}\frac{\d_j}{\e_j}=0\,.
$$
Moreover, we can exclude that $r_j\ge \e_j/2$ otherwise the zones
may overlap. More precisely, we assume that $r_j\ll \e_j$; {\it i.e., }
\begin{equation}\label{e/r}
\lim_{j\to +\infty}\frac{r_j}{\e_j} = 0\,.
\end{equation}
This choice will be justify a posteriori since (\ref{e/r}) will be
the only admissible assumption to get a non trivial
$\Gamma$-convergence result (see Remark \ref{andrea}).
Finally, it remains to fix the behavior of $r_j$ with respect to
$\d_j$. Let us define
$$\ell:=\lim_{j \to +\infty}\frac{r_j}{\d_j}.$$
This yields to consider all the possible scenarii, namely to
distinguish between the cases: $\ell$ finite, infinite or zero.
\bigskip
For any fixed $\ell\in [0,+\infty]$, we consider the sequence of
functions $(\varphi^{(\ell)}_{\g,j})$ defined in (\ref{phigj}) and
(\ref{phigj12}). Propositions \ref{midi} and \ref{midi12}
establish the existence of the function $\varphi^{(\ell)}$ as the
(locally uniform) limit of $(\varphi^{(\ell)}_{\g,j})$ as $j\to
+\infty$ and $\g\to 0^+$ while Proposition \ref{rs} will allow us
to prove that $\varphi^{(\ell)}$ is actually the interfacial
energy density in $\mathcal{F}^{(\ell)}$ (see e.g. Proposition
\ref{gliminfclose}).
\subsection{The case $\ell \in (0,+\infty]$}
\noindent Setting $N_j= \e_j/r_j$, we define the space $$X^\g_j
(z) := \Big\{\zeta \in W^{1,p}((B^{n-1}_{\g N_j} \times I)
\setminus C_{1,\g N_j};{\mathbb{R}}^m):\; \zeta=z \text{ on } (\partial
B^{n-1}_{\g N_j})^+ ,\, \zeta=0 \text{ on } (\partial B^{n-1}_{\g
N_j})^-\Big\}\,,$$ where $I=(-1,1)$ and we consider the following
minimum problem
\begin{equation}\label{phigj}
\varphi^{(\ell)}_{\g,j}(z):=\inf \left\{ \int_{(B_{\g N_j}^{n-1}
\times I) \setminus C_{1,\g N_j}} r_j^p \, W\left(r_j^{-1} D_\a
\zeta | \d_j^{-1} D_n \zeta \right) dx : \quad \zeta \in
X^\g_j(z)\right\}\,.
\end{equation}
\begin{figure
\centerline{\psfig{file=FIG3.eps,width=4in}} \vspace*{8pt}
\caption{The domain $(B^{(n-1)}_{\g N_j}\times I) \setminus C_{1,\g
N_j}$\,.}
\end{figure}
In the next proposition we study the behavior of
$(\varphi^{(\ell)}_{\g,j})$ as $j \to +\infty$ and $\g \to 0^+$.
\begin{proposition}\label{midi}
Let $\ell\in (0,+\infty]$. If
\begin{equation}\label{rinfty}0<R^{(\ell)}:=\lim_{j \to +\infty}
\frac{r_j^{n-1-p}}{\e_j^{n-1}} < +\infty
\end{equation}
then,
\smallskip
(i) there exists a constant $c>0$ (independent of $j$ and $\g$)
such that
$$
0 \leq \varphi^{(\ell)}_{\g,j}(z) \leq c \left( |z|^p + \g^{n-1}
\right)
$$
for all $z \in {\mathbb{R}}^m$, $j\in {\mathbb{N}}$ and $\g>0$;
\smallskip
(ii) there exists a constant $c>0$ (independent of $j$ and $\g$)
such that
\begin{equation}\label{Stima1}
|\varphi^{(\ell)}_{\g,j}(z)-\varphi^{(\ell)}_{\g,j}(w)| \leq
c\,|z-w|\, \big(\g^{(n-1)(p-1)/p} + r_j^{p-1} + |z|^{p-1} +
|w|^{p-1} \big)
\end{equation}
for every $z,w \in {\mathbb{R}}^m$, $j\in {\mathbb{N}}$ and $\g>0$;
\smallskip
(iii) for every fixed $\gamma>0$, up to subsequences,
$\varphi^{(\ell)}_{\g,j}$ converges locally uniformly on ${\mathbb{R}}^m$
to $\varphi^{(\ell)}_{\g}$ as $j\to +\infty$ and
\begin{equation}\label{Stima12}
|\varphi^{(\ell)}_{\g}(z)-\varphi^{(\ell)}_{\g}(w)| \leq
c\,|z-w|\, \big(\g^{(n-1)(p-1)/p} + |z|^{p-1} + |w|^{p-1} \big)
\end{equation}
for every $z,w \in {\mathbb{R}}^m$ ;
\smallskip
(iv) up to subsequences, $\varphi^{(\ell)}_{\g}$ converges locally
uniformly on ${\mathbb{R}}^m$, as $\g \to 0^+$, to a continuous function
$\varphi^{(\ell)}:{\mathbb{R}}^m \to [0,+\infty)$ satisfying
\begin{equation}\label{AA}
0 \leq \varphi^{(\ell)} (z) \leq c |z|^p\,, \qquad
\qquad |\varphi^{(\ell)}(z) - \varphi^{(\ell)}(w)|
\leq c\,|z-w| \big( |z|^{p-1}+|w|^{p-1}\big)
\end{equation}
for every $z,w\in \Bbb R^m$.
\end{proposition}
\noindent {\it Proof. } Fix $\g>0$, then $\g N_j> 2$ for $j$ large
enough.
(i) According to the $p$-growth condition (\ref{pgrowth}),
\begin{equation}\label{cap}
0 \leq \varphi^{(\ell)}_{\g,j}(z) \leq \beta \left({\mathcal{C}}_{\g,j}(z) +
{\mathcal H}^{n-1}(B_1^{n-1}) \g^{n-1}\frac{\e_j^{n-1}}{r_j^{n-1-p}}
\right),
\end{equation} where
$$
{\mathcal{C}}_{\g,j}(z):= \inf \left\{ \int_{(B_{\g N_j}^{n-1} \times I)
\setminus C_{1,\g N_j}} \left|\left(D_\a \zeta \Big|
\frac{r_j}{\d_j} D_n \zeta \right)\right|^p dx : \quad \zeta \in
X^\g_j(z)\right\}\,.
$$
Since ${\mathcal{C}}_{\g,j}(z)$ is invariant by rotations, reasoning as in
\cite{Ans} Section 4.1, we can consider the minimization problem
with respect to a particular class of scalar test functions as
follows
\begin{eqnarray}\label{CgjStima}
\nonumber{{\mathcal{C}}_{\g,j}(z)\over |z|^p} & = & \inf \Bigg\{
\int_{(B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g N_j}}
\left|\left(D_\a \psi \Big| \frac{r_j}{\d_j} D_n \psi
\right)\right|^p \,dx \,:\, \psi \in W^{1,p}((B^{n-1}_{\g
N_j} \times I) \setminus C_{1,\g N_j}),\\
&& \nonumber\hspace{5cm}\psi=1 \text{ on } \big(\partial
B^{n-1}_{\g N_j} \big)^+ \text{ and } \psi=0 \text{ on }
\big(\partial
B^{n-1}_{\g N_j}\big)^- \Bigg\}\\
& \leq & \nonumber\inf \Bigg\{ \int_{B_{\g N_j}^{n-1}} \big( |D_\a
\psi^+ |^p + |D_\a \psi^- |^p \big) \,dx : \quad (\psi^+
-1)\,,\,\psi^-
\in W^{1,p}_0(B^{n-1}_{\g N_j})\\
&& \hspace{7cm}\text{ and }\psi^+=\psi^- \text{ in } B^{n-1}_{1}
\Bigg\}.
\end{eqnarray}
Let $\psi^\pm_1$ be the unique minimizer of the strictly convex
minimization problem (\ref{CgjStima}). It turns out that
$\psi^\pm_2:=1-\psi_1^\mp$ is also a minimizer. Thus by
uniqueness, $\psi_1^\pm=\psi_2^\pm$ and in particular,
$\psi^\pm_1=1/2$ in $B_1^{n-1}$. Hence,
\begin{eqnarray}\label{pascap}
{\mathcal{C}}_{\g,j}(z) & \leq & |z|^p \inf \Bigg\{ \int_{B_{\g N_j}^{n-1}}
\big( |D_\a \psi^+ |^p + |D_\a \psi^- |^p \big)\dx_{\alpha} \,: \, (\psi^+
-1)\,,\, \psi^- \in
W^{1,p}_0 (B^{n-1}_{\g N_j}),\nonumber\\
&& \hspace{7cm}\text{ and } \psi^+=\psi^-=\frac{1}{2} \text{ in }
B^{n-1}_{1} \Bigg\}\nonumber\\
& = & 2 |z|^p \inf \Bigg\{
\int_{B_{\g N_j}^{n-1}} |D_\a \psi |^p \dx_{\alpha} : \quad \psi \in
W^{1,p}_0(B^{n-1}_{\g N_j}) \text{ and } \psi=\frac{1}{2} \text{
in } B^{n-1}_{1} \Bigg\}\nonumber\\
& = & \frac{|z|^p}{2^{p-1}} \inf \Bigg\{
\int_{B_{\g N_j}^{n-1}} |D_\a \psi |^p \dx_{\alpha} : \quad \psi \in
W^{1,p}_0(B^{n-1}_{\g N_j}) \text{ and } \psi=1 \text{ in }
B^{n-1}_{1} \Bigg\}\nonumber\\
& = & \frac{|z|^p}{2^{p-1}} {\rm Cap}_p \big(B^{n-1}_{1};B_{\g
N_j}^{n-1} \big).
\end{eqnarray}
Since
$$\lim_{j \to +\infty} {\rm Cap}_p \big(B^{n-1}_{1};B_{\g
N_j}^{n-1} \big) = {\rm Cap}_p \big(B^{n-1}_{1};{\mathbb{R}}^{n-1}\big) <
+\infty\,;
$$
hence, by (\ref{rinfty}), (\ref{cap}) and (\ref{pascap}) we
conclude the proof of (i).
\bigskip
(ii) For every $\eta>0$, there exists $\zeta_{\g,j} \in X^\g_j(z)$
such that
\begin{equation}\label{fabia}
\int_{(B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g N_j}} r_j^p\,
W\left(r_j^{-1} D_\a \zeta_{\g,j} | \d_j^{-1} D_n \zeta_{\g,j}
\right) dx \leq \varphi^{(\ell)}_{\g,j}(z) +\eta.
\end{equation}
We want to modify $\zeta_{\g,j}$ in order to get an admissible
test function for $\varphi^{(\ell)}_{\g,j}(w)$. More precisely, we
just have to modify $\zeta_{\g,j}$ on a neighborhood of $(\partial
B_{\g N_j}^{n-1})^+$ to change the boundary condition $z$ into
$w$. To this aim we introduce a cut-off function $\theta \in
{\mathcal{C}}^\infty_c({\mathbb{R}}^{n-1};[0,1])$, independent of $x_n$, such that
$$
\theta(x_\a)=\left\{
\begin{array}{rcl}
1 & \text{if} & x_\a \in B^{n-1}_1,\\
\\
0 & \text{if} & x_\a \not\in B^{n-1}_2
\end{array}
\quad \text{ and }\quad |D_\a \theta|\leq c\,. \right.
$$
Hence, we define $\tilde\zeta_{\g,j} \in X^\g_j(w)$ as follows
$$
\tilde \zeta_{\g,j}=\left\{
\begin{array}{lcl}
\zeta_{\g,j}+ (1-\theta(x_\a)) (w-z) & \text{in} &
(B^{n-1}_{\g N_j})^+\\
\\
\zeta_{\g,j}& \text{in} & (B^{n-1}_{\g N_j})^- \cup \big(
B_1^{n-1}\times \{0\} \big)\,.
\end{array}
\right.
$$
By (\ref{fabia}), since $\zeta_{\g,j}=\tilde \zeta_{\g,j}$ in
$(B^{n-1}_{\g N_j})^- $, we have that
\begin{eqnarray*}
&&\varphi^{(\ell)}_{\g,j}(w)-\varphi^{(\ell)}_{\g,j}(z)\\
& \leq & r_j^p
\int_{(B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g N_j}} \Bigl(
W\big(r_j^{-1} D_\a \tilde \zeta_{\g,j} | \d_j^{-1} D_n \tilde
\zeta_{\g,j} \big) - W\big(r_j^{-1} D_\a \zeta_{\g,j} | \d_j^{-1}
D_n
\zeta_{\g,j} \big)\Bigr) \,dx\, +\eta\\
& = & r_j^p \int_{(B_{\g N_j}^{n-1})^+}\Bigl( W\big(r_j^{-1} D_\a
\tilde \zeta_{\g,j} | \d_j^{-1} D_n \tilde \zeta_{\g,j} \big)-
W\big(r_j^{-1} D_\a \zeta_{\g,j} | \d_j^{-1} D_n \zeta_{\g,j}
\big) \Bigl)\,dx \,+\eta\,.
\end{eqnarray*}
By (\ref{plip}) and H\"older's Inequality, we obtain that
\begin{eqnarray*}
&&\varphi^{(\ell)}_{\g,j}(w)-\varphi^{(\ell)}_{\g,j}(z)\,-\eta\\
&\leq& c\, \int_{(B_{\g N_j}^{n-1})^+}\Bigg( r_j^{p-1} + \left|
\left( D_\a \zeta_{\g,j} \Big| \frac{r_j}{\d_j} D_n \zeta_{\g,j}
\right) \right|^{p-1} + \left| \left( D_\a \tilde \zeta_{\g,j}
\Big| \frac{r_j}{\d_j} D_n \tilde
\zeta_{\g,j} \right) \right|^{p-1}\Bigg)\\
&& \hspace{2cm} \times \left| \left( D_\a \tilde \zeta_{\g,j}
-D_\a \zeta_{\g,j} \Big| \frac{r_j}{\d_j} (D_n \tilde \zeta_{\g,j}
- D_n \zeta_{\g,j})
\right) \right| \,dx \\
&\leq & c \int_{(B_{\g N_j}^{n-1})^+}\Bigg( r_j^{p-1} + 2 \left|
\left( D_\a \zeta_{\g,j} \Big| \frac{r_j}{\d_j} D_n \zeta_{\g,j}
\right) \right|^{p-1} + |D_\a \theta|^{p-1}\,
|w-z|^{p-1}\Bigg) |D_\a \theta|\, |w-z| \,dx \\
&\leq& c\, |z-w|^p \int_{B_{\g N_j}^{n-1}} |D_\a \theta|^p\,
dx_\a + c\, r_j^{p-1} |z-w| \int_{B_{\g N_j}^{n-1}} |D_\a \theta|\,
dx_\a\\
&& + 2c\, |z-w| \; \|D_\a \theta \|_{L^p(B_{\g N_j}^{n-1};{\mathbb{R}}^{n-1})}
\left\| \left( D_\a \zeta_{\g,j} \Big| \frac{r_j}{\d_j} D_n
\zeta_{\g,j} \right) \right\|^{p-1}_{L^p\big((B_{\g
N_j}^{n-1})^+;{\mathbb{R}}^{m \times n}\big)}\,.
\end{eqnarray*}
Since $\g N_j > 2$ and ${\rm Supp}(\theta) \subset B_2^{n-1}$, we
obtain that
\begin{eqnarray}\label{pasta}
&&\hspace{-1cm}\varphi^{(\ell)}_{\g,j}(w)-\varphi^{(\ell)}_{\g,j}(z)\nonumber\\
&&\hspace{-0.5cm}\leq c |z-w| \Bigg( |z-w|^{p-1} + r_j^{p-1}+
\left\| \left( D_\a \zeta_{\g,j} \Big| \frac{r_j}{\d_j} D_n
\zeta_{\g,j} \right) \right\|^{p-1}_{L^p\big((B_{\g
N_j}^{n-1})^+;{\mathbb{R}}^{m \times n}\big)} \Bigg)\, +\eta.
\end{eqnarray}
By the $p$-growth condition (\ref{pgrowth}), (\ref{fabia}) and
(i), we have that
\begin{eqnarray}\label{pasta2}
&&\int_{(B_{\g N_j}^{n-1})^+} \left| \left( D_\a \zeta_{\g,j}
\Big| \frac{r_j}{\d_j} D_n \zeta_{\g,j} \right)
\right|^p \,dx\nonumber\\
&\leq&\int_{(B_{\g N_j}^{n-1})^+} r_j^p \, W\left(r_j^{-1} D_\a
\zeta_{\g,j} | \d_j^{-1} D_n \zeta_{\g,j} \right) dx
+ r_j^p \, {\mathcal H}^{n-1}\big( B_{\g N_j}^{n-1} \big)\nonumber\\
&\leq& \varphi^{(\ell)}_{\g,j}(z)
+\eta + c\g^{n-1} \frac{\e_j^{n-1}}{r_j^{n-1-p}}\nonumber\\
& \leq& c(|z|^p + \g^{n-1} ) + \eta + c\g^{n-1}
\frac{\e_j^{n-1}}{r_j^{n-1-p}}.
\end{eqnarray}
Hence, by (\ref{pasta}), (\ref{pasta2}) and (\ref{rinfty}) we have
that
$$
\varphi^{(\ell)}_{\g,j}(w)-\varphi^{(\ell)}_{\g,j}(z) \leq c\,
|z-w| \Big( |z|^{p-1}+|w|^{p-1} + r_j^{p-1}+ \g^{(n-1)(p-1)/p} +
\eta^{(p-1)/p} \Big) +\eta
$$
and (\ref{Stima1}) follows by the arbitrariness of $\eta$.
\bigskip
By (ii) and Ascoli-Arzela's Theorem we have that, up to
subsequences, $\varphi^{(\ell)}_{\g,j}$ converges uniformly on
compact sets of $\Bbb R^m$ to $\varphi^{(\ell)}_{\g}$ as $j\to
+\infty$. Moreover, passing to the limit in (\ref{Stima1}) as
$j\to +\infty$ we get
$$
|\varphi^{(\ell)}_{\g}(w)-\varphi^{(\ell)}_{\g}(z)| \leq c\, |z-w|
\Big( |z|^{p-1}+|w|^{p-1} + \g^{(n-1)(p-1)/p}\Big)\,.
$$
Hence, we can apply again Ascoli-Arzela's Theorem to conclude
that, up to subsequences, $\varphi^{(\ell)}_{\g}$ converges
uniformly on compact sets of $\Bbb R^m$ to $\varphi^{(\ell)}$ as
$\g\to 0^+$. In particular, $\varphi^{(\ell)} :{\mathbb{R}}^m \to
[0,+\infty)$ is a continuous function and
$$
0 \leq \varphi^{(\ell)} (z) \leq c |z|^p\,, \qquad
\qquad |\varphi^{(\ell)}(z) - \varphi^{(\ell)}(w)|
\leq c\,\big( |z|^{p-1}+|w|^{p-1}\big)|z-w|
$$
for every $z,w\in \Bbb R^m$. \hfill$\Box$
\begin{figure
\centerline{\psfig{file=FIG4.eps,width=4in}} \vspace*{8pt}
\caption{The domain $(B^{(n-1)}_{\g N_j}\times I_j)\setminus C_{1,\g
N_j}$\,.}
\end{figure}
\subsection{The case $\ell =0$}
\noindent In this case we expect that the energy contribution due
to the presence of the sieve is obtained studying the behavior, as
$j\to +\infty$ and $\g\to 0^+$, of the sequence
($\varphi^{(0)}_{\g,j}$) defined as follows
\begin{eqnarray}\label{phigj12}
\varphi^{(0)}_{\g,j}(z) & := & \frac{\d_j}{r_j} \inf \left\{
\int_{(B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g N_j}} r_j^p \,
W\left(r_j^{-1} D_\a \zeta | \d_j^{-1} D_n
\zeta \right)dx : \quad \zeta \in X^\g_j(z)\right\}\nonumber\\
& = & \inf \left\{ \int_{(B_{\g N_j}^{n-1} \times
I_j ) \setminus C_{1,\g N_j}} r_j^p \, W(r_j^{-1} D \zeta )\, dx :
\quad \zeta \in Y^\g_j(z)\right\}
\end{eqnarray}
where $I_j:=(-\d_j/r_j , \d_j/r_j)$ and
\begin{eqnarray*}
Y^\g_j(z) = \Big\{\zeta \in W^{1,p}((B_{\g N_j}^{n-1} \times I_j)
\setminus C_{1,\g N_j} ;{\mathbb{R}}^m ) & : & \zeta=z \text{ on }(\partial
B_{\g N_j}^{n-1})^{+(\d_j/r_j)},\\
&& \zeta=0 \text{ on } (\partial B_{\g N_j}^{n-1})^{-(\d_j/r_j)}
\Big\}.
\end{eqnarray*}
Note that in this case we are interested in the limit behavior of
a sequence that is obtained from the one corresponding to
$\ell\in(0,+\infty]$ multiplying it by $\d_j/r_j$ (see
(\ref{phigj12}) and recall (\ref{phigj})). Let us try to motivate
this choice.
Let $\ell\in(0,+\infty)$, then starting from $(\ref{phigj})$ by a
change of variable it is immediate to check that
\begin{equation}\label{uguale}
\varphi_{\g,j}^{(\ell)}(z)=\frac{r_j}{\d_j}\inf \left\{
\int_{(B_{\g N_j}^{n-1} \times I_j ) \setminus C_{1,\g N_j}} r_j^p
\, W(r_j^{-1} D \zeta )\, dx : \quad \zeta \in Y^\g_j(z)\right\}.
\end{equation}
Now assuming that
$\lim_{j\to+\infty}r_j^{n-p}/(\d_j\,\e_j^{n-1})<+\infty$ (or
equivalently that
$\lim_{j\to+\infty}r_j^{n-1-p}/\e_j^{n-1}<+\infty$; see Remark
\ref{equiv}) we know that the sequence $(\varphi_{\g,j}^{(\ell)})$
converges to $\ell\, \tilde \varphi^{(\ell)}$, for some $\tilde
\varphi^{(\ell)}$, locally uniformly in ${\mathbb{R}}^m$, as $j\to+\infty$
and $\g\to 0^+$ (Proposition \ref{midi}). Then if
$\ell\in(0,+\infty)$, studying the limit behavior of
(\ref{phigj12}) is perfectly equivalent to study the limit
behavior of (\ref{phigj}). While if
$\ell=\lim_{j\to+\infty}r_j/\d_j=0$, (\ref{uguale}) suggests that,
to recover nontrivial information in the limit, we have to study
the asymptotic behavior of the sequence obtained from
(\ref{uguale}) dividing it by $r_j/\d_j$, that is to study the
asymptotic behavior of the sequence given by (\ref{phigj12}).
\bigskip
Following the line of the proof of Proposition \ref{midi}, we want
to establish an analogous result for the sequence
$(\varphi^{(0)}_{\g,j})$.
\begin{proposition}\label{midi12}
Let $\ell=0$. If
\begin{equation}\label{rzero}
0<R^{(0)}=\lim_{j \to +\infty} \frac{r_j^{n-p}}{\e_j^{n-1} \d_j} <
+\infty
\end{equation}
then,
\smallskip
(i) there exists a constant $c>0$ (independent of $j$ and $\g$)
such that
$$
0 \leq \varphi^{(0)}_{\g,j}(z) \leq c \left( |z|^p + \g^{n-1}
\right)
$$
for all $z \in {\mathbb{R}}^m$, $j\in {\mathbb{N}}$ and $\g>0$;
\smallskip
(ii) there exists a constant $c>0$ (independent of $j$ and $\g$)
such that
\begin{equation}\label{Stima2}
|\varphi^{(0)}_{\g,j}(z)-\varphi^{(0)}_{\g,j}(w)| \leq c\, |z-w|
\,\big(\g^{(n-1)(p-1)/p} + r_j^{n-1} + |z|^{p-1} + |w|^{p-1} \big)
\end{equation}
for every $z,w \in {\mathbb{R}}^m$, $j\in {\mathbb{N}}$ and $\g>0$;
\smallskip
(iii) for every fixed $\gamma>0$, up to subsequences,
$\varphi^{(0)}_{\g,j}$ converges locally uniformly in $\Bbb R^m$ to
$\varphi^{(0)}_{\g}$ as $j\to +\infty$, and
\begin{equation}\label{Stima23}
|\varphi^{(0)}_{\g}(z)-\varphi^{(0)}_{\g}(w)| \leq c\, |z-w| \Big(
\g^{(n-1)(p-1)/p} + |z|^{p-1}+|w|^{p-1}\Big)
\end{equation}
for every $z,w \in {\mathbb{R}}^m$;
\smallskip
(iv) up to subsequences, $\varphi^{(0)}_{\g}$ converges locally
uniformly in $\Bbb R^m$, as $\g \to 0^+$, to a continuous function
$\varphi^{(0)}:{\mathbb{R}}^m \to [0,+\infty)$ satisfying
\begin{equation}\label{AA12}
0 \leq \varphi^{(0)} (z) \leq c |z|^p , \qquad
\qquad |\varphi^{(0)}(z) - \varphi^{(0)}(w)|
\leq c\,|z-w|\, \big( |z|^{p-1}+|w|^{p-1}\big)
\end{equation}
for every $z,w\in \Bbb R^m$.
\end{proposition}
\noindent {\it Proof. } Fix $\g>0$, then $\g N_j> 2$ and
$\d_j/r_j>2$ for $j$ large enough.
(i) According to the $p$-growth condition (\ref{pgrowth}),
\begin{equation}\label{cap12}
0 \leq \varphi^{(0)}_{\g,j}(z) \leq \beta \left( {\mathcal{C}}_{\g,j}(z) + 2
{\mathcal H}^{n-1}(B_1^{n-1})\, \g^{n-1}\,\frac{
\d_j\,\e_j^{n-1}}{r_j^{n-p}} \right),
\end{equation}
where
$$
{\mathcal{C}}_{\g,j}(z)= \inf \left\{ \int_{(B_{\g N_j}^{n-1} \times I_j)
\setminus C_{1,\g N_j}} |D \zeta |^p \, dx : \quad \zeta \in
Y^\g_j(z) \right\}.
$$
Arguing similarly than in the proof of Proposition \ref{midi}, we
can rewrite
\begin{eqnarray}\label{CgjScalar}
\nonumber{{\mathcal{C}}_{\g,j}(z)\over |z|^p} & = & \inf \Bigg\{ \int_{(B_{\g
N_j}^{n-1} \times I_j) \setminus C_{1,\g N_j}} |D \psi|^p\, dx
\,:\; \psi \in W^{1,p}((B^{n-1}_{\g N_j}
\times I_j) \setminus C_{1,\g N_j}),\nonumber\\
&& \hspace{1.6cm} \psi=1 \text{ on } (\partial B^{n-1}_{\g
N_j})^{+(\d_j/r_j)} \, , \quad \psi=0 \text{ on } (\partial
B^{n-1}_{\g N_j})^{-(\d_j/r_j)} \Bigg\}.
\end{eqnarray}
Let $\psi_1$ be the unique minimizer of the strictly convex
minimization problem (\ref{CgjScalar}). It turns out that
$\psi_2(x_\a,x_n):=1-\psi_1(x_\a,-x_n)$ is also a minimizer. Thus
by uniqueness, $\psi_1=\psi_2$ and in particular,
$\psi_1=\psi_2=1/2$ on $B_1^{n-1} \times \{0\}$. Thus
\begin{eqnarray}\label{pascap12}
{\mathcal{C}}_{\g,j}(z) & = & 2 |z|^p \inf \Bigg\{ \int_{(B_{\g
N_j}^{n-1})^{+(\d_j/r_j)}} |D \psi |^p \, dx : \quad \psi \in
W^{1,p}((B_{\g N_j}^{n-1})^{+(\d_j/r_j)}),\nonumber\\
&&\hspace{2cm}\psi = 0 \text{ on } (\partial B_{\g
N_j}^{n-1})^{+(\d_j/r_j)} \text{ and } \psi=\frac{1}{2} \text{ on
}
B^{n-1}_1 \times \{0\} \Bigg\}\nonumber\\
& = & \frac{|z|^p}{2^{p-1}} \inf \Bigg\{
\int_{(B_{\g N_j}^{n-1})^{+(\d_j/r_j)}} |D \psi |^p \, dx :
\quad \psi \in W^{1,p}((B^{n-1}_{\g N_j})^{+(\d_j/r_j)}),\nonumber\\
&&\hspace{2cm}\psi = 0 \text{ on } (\partial B_{\g
N_j}^{n-1})^{+(\d_j/r_j)} \text{ and } \psi=1 \text{ on }
B^{n-1}_{1} \times \{0\} \Bigg\}\nonumber\\
& \le & \frac{|z|^p}{2^p} {\rm Cap}_p \big(B^{n-1}_{1};B_{\g
N_j}^{n-1}\times I_j \big)\,.
\end{eqnarray}
Since
$$\lim_{j \to +\infty} {\rm Cap}_p
\big(B^{n-1}_{1};B_{\g N_j}^{n-1}\times I_j \big) = {\rm Cap}_p
\big(B^{n-1}_{1};{\mathbb{R}}^n\big)<+\infty\,;
$$
hence, by (\ref{rzero}), (\ref{cap12}) and (\ref{pascap12}) we
conclude the proof of (i).
\bigskip
(ii) We can proceed as in the proof of Proposition \ref{midi} (ii)
using a different cut-off function also depending on $x_n$.
Namely, let $\theta \in {\mathcal{C}}^\infty_c({\mathbb{R}}^n;[0,1])$ be such that
$$\theta(x_\a,x_n)=\left\{
\begin{array}{rcl}
1 & \text{if} & (x_\a,x_n) \in B^{n-1}_1 \times (-1,1),\\
\\
0 & \text{if} & (x_\a,x_n) \not\in B^{n-1}_2 \times (-2,2)
\end{array}
\quad \text{ and }\quad |D \theta|\leq c. \right.
$$
Hence, if $\zeta_{\g,j} \in Y_j^\g(z)$ is a sequence which `almost
attains' the infimum value $\varphi_{\g,j}^{(0)}$, we define
$\tilde\zeta_{\g,j} \in Y^\g_j(w)$ as follows
$$
\tilde \zeta_{\g,j}=\left\{
\begin{array}{lcl}
\zeta_{\g,j}+ (1-\theta(x)) (w-z) & \text{in} &
(B^{n-1}_{\g N_j})^{+(\d_j/r_j)},\\
\\
\zeta_{\g,j}& \text{in} & \big( (B^{n-1}_{\g N_j})^{-(\d_j/r_j)}
\big) \cup \big( B_1^{n-1} \times \{0\} \big).
\end{array}
\right.
$$
By (\ref{rzero}) we conclude the proof of (ii) reasoning as in the
proof of Proposition \ref{midi} (ii).
The proof of (iii) and (iv) follows the line of the proof of (iii)
and (iv) in Proposition \ref{midi}. \hfill$\Box$
\smallskip
Now we are able to describe the energy contribution close to the
connecting zones as $j\to +\infty$ and $\g\to 0^+$.
\begin{proposition}[Discrete approximation of the interfacial
energy]\label{rs} Let $(u_j)\subset W^{1,p}(\O_j;{\mathbb{R}}^m)\cap
L^{\infty}(\O_j;{\mathbb{R}}^m)$ be a sequence converging to $(u^+,u^-)\in
W^{1,p}(\o;{\mathbb{R}}^m)\times
W^{1,p}(\o;{\mathbb{R}}^m)$ such that\, $\sup_j\mathcal{F}_j(u_j)<+\infty$ and
satisfying $\sup_{j \in {\mathbb{N}}} \|u_j\|_{L^\infty(\O_j;{\mathbb{R}}^m)}
<+\infty$. Let $(u_j^{i\pm})$ be as in {\rm (\ref{aver})}. If
$$\ell\in(0,+\infty]\quad \text{ and }\quad
0<R^{(\ell)}=\lim_{j\to+\infty}\frac{r_j^{n-1-p}}{\e_j^{n-1}}<+\infty$$
or $$\ell=0\quad \text{ and }\quad
0<R^{(0)}=\lim_{j\to+\infty}\frac{r_j^{n-p}}{\d_j\e_j^{n-1}}<+\infty
$$
then
\begin{equation}\label{conv+-}
\lim_{\g\to 0^+}\limsup_{j \to +\infty}\int_\o \Bigl|\sum_{i \in
Z_j}\varphi^{(\ell)}_{\g,j}(u_j^{i+}-
u_j^{i-})\chi_{Q_{i,\e_j}^{n-1}} -
\varphi^{(\ell)}(u^+-u^-)\Bigr|\,dx_\a=0\,,
\end{equation}
for every $\ell\in[0,+\infty]$.
\end{proposition}
\noindent {\it Proof. } Since $\sup_{j \in {\mathbb{N}}}
\|u_j\|_{L^\infty(\O_j;{\mathbb{R}}^m)}<+\infty$ by Propositions \ref{midi}
or \ref{midi12} we have that
\begin{eqnarray*}
&&\limsup_{j\to +\infty} \int_\o \Bigl|\sum_{i \in
Z_j}\varphi^{(\ell)}_{\g,j}(u_j^{i+}-
u_j^{i-})\chi_{Q_{i,\e_j}^{n-1}} -
\varphi^{(\ell)}(u^+-u^-)\Bigr|\,dx_\a\\
&\le& \limsup_{j\to +\infty} \int_\o \sum_{i \in
Z_j}\Bigl|\varphi^{(\ell)}_{\g,j}(u_j^{i+}- u_j^{i-}) -
\varphi^{(\ell)}(u_j^{i+}-u_j^{i-})\Bigr|\chi_{Q_{i,\e_j}^{n-1}}\,dx_\a\\
&&+ \limsup_{j\to +\infty} \int_\o \Bigl|\sum_{i \in
Z_j}\varphi^{(\ell)}(u_j^{i+}- u_j^{i-})\chi_{Q_{i,\e_j}^{n-1}} -
\varphi^{(\ell)}(u^+-u^-)\Bigr|\,dx_\a\\
&\le& o(1) + \limsup_{j\to +\infty} \int_\o \Bigl|\sum_{i \in
Z_j}\varphi^{(\ell)}(u_j^{i+}- u_j^{i-})\chi_{Q_{i,\e_j}^{n-1}} -
\varphi^{(\ell)}(u^+-u^-)\Bigr|\,dx_\a, \end{eqnarray*} as $\g \to
0^+$. By (\ref{AA}) or (\ref{AA12}) and H\"older's Inequality we
have that
\begin{eqnarray*}
&&\limsup_{j\to +\infty}\int_\o \Bigl|\sum_{i \in
Z_j}\varphi^{(\ell)}(u_j^{i+}- u_j^{i-}) \chi_{Q^{n-1}_{i,\e_j}}
- \varphi^{(\ell)}(u^+-u^-) \Bigr|\, dx_\a\\
&=& \limsup_{j\to +\infty}\sum_{i \in Z_j} \int_{Q^{n-1}_{i,\e_j}}
|\varphi^{(\ell)}(u_j^{i+}- u_j^{i-})
- \varphi^{(\ell)}(u^+-u^-)|\, dx_\a\\
&\leq& c\, \limsup_{j\to +\infty}\Bigl(\sum_{i \in
Z_j}\int_{Q^{n-1}_{i,\e_j}}\big|u_j^{i+}-u^+|^p + |u_j^{i-}
-u^{-}|^p\dx_{\alpha}\Bigr)^{1/p}\,.
\end{eqnarray*}
Hence, it remains to prove that
\begin{equation}\label{lim0}
\limsup_{j\to +\infty}\sum_{i \in Z_j}\int_{Q^{n-1}_{i,\e_j}}
|u^{\pm}- u_j^{i\pm}|^p \, dx_\a =0\,.
\end{equation}
By Lemma \ref{poincare} (ii) applied with $\rho=\e_j$,
$B_\rho=C_j^i$ and $A_\rho=Q^{n-1}_{i,\e_j}$ and since $\d_j \ll
\e_j$, we have
\begin{eqnarray}\label{corep}
\int_{Q^{n-1}_{i,\e_j}} |u^\pm - u_j^{i\pm}|^p\, dx_\a &\leq&
{c\over \d_j}\Bigl(\int_{(Q^{n-1}_{i,\e_j})^{\pm\d_j}}
|u_j-u^{\pm}|^p\,dx + \int_{(Q^{n-1}_{i,\e_j})^{\pm\d_j}}
|u_j-u_j^{i\pm}|^p\,dx\Bigr)\nonumber
\\
&\le& {c\over \d_j}\int_{(Q^{n-1}_{i,\e_j})^{\pm\d_j}}
|u_j-u^{\pm}|^p\,dx
+\frac{c\,\e_j^p}{\d_j}\int_{(Q^{n-1}_{i,\e_j})^{\pm\d_j}} |D
u_j|^p \,dx\,,
\end{eqnarray}
for all $i \in Z_j$; hence, summing up on $i \in Z_j$, we find
$$
\sum_{i \in Z_j} \int_{Q^{n-1}_{i,\e_j}} |u_j - u_j^{i\pm}|^p\,
dx_\a \leq {c\over \d_j}\int_{\o^{\pm\d_j}} |u_j-u^{\pm}|^p\,dx +
\frac{c\,\e_j^p}{\d_j}\int_{\o^{\pm\d_j}} |D u_j|^p \,dx\,,
$$
then passing to the limit as $j\to +\infty$ by the convergence of
$(u_j)$ towards $(u^+,u^-)$ and $\sup_j\mathcal{F}_j(u_j)<+\infty$
we get (\ref{lim0}) and then (\ref{conv+-}). \hfill$\Box$
\section{$\Gamma$-convergence result}\label{proof}
\subsection{The liminf inequality}
\noindent Let $(u_j) \subset W^{1,p}(\O_j;{\mathbb{R}}^m)\cap
L^\infty(\O_j;{\mathbb{R}}^m)$ be a sequence converging to $(u^+,u^-)\in
W^{1,p}(\o,\Bbb R^m)\times W^{1,p}(\o,\Bbb R^m)$ such that $\sup_{j \in
{\mathbb{N}}} \|u_j\|_{L^\infty(\O_j;{\mathbb{R}}^m)}<+\infty$ and
$$
\liminf_{j \to +\infty}\mathcal F_j(u_j)<+\infty\,.
$$
By Lemma \ref{important}, for every fixed $k \in {\mathbb{N}}$, there
exists a sequence $(w_j)\subset W^{1,p}(\O_j;{\mathbb{R}}^m)\cap
L^\infty(\O_j;{\mathbb{R}}^m)$ weakly converging to $(u^+,u^-)$ satisfying
(\ref{wj1}), (\ref{wj2}) and such that
\begin{eqnarray}\label{lowerbd}
&&\liminf_{j \to +\infty}\frac{1}{\d_j} \left( \int_{\o^{+\d_j}}
W(D u_j)\, dx
+ \int_{\o^{-\d_j}} W(D u_j)\, dx \right)\nonumber\\
&\geq& \liminf_{j \to +\infty}\frac{1}{\d_j} \left(
\int_{\o^{+\d_j}} W(D w_j)\, dx + \int_{\o^{-\d_j}} W(D
w_j)\, dx \right) - \frac{c}{k}\nonumber\\
&\geq& \liminf_{j \to +\infty}\frac{1}{\d_j} \left( \int_{(\o
\setminus E_j)^{+\d_j}} W(D w_j)\, dx + \int_{(\o
\setminus E_j)^{-\d_j}} W(D w_j)\, dx \right)\nonumber\\
&& + \liminf_{j \to +\infty}\frac{1}{\d_j} \left(
\int_{E_j^{+\d_j}} W(D w_j)\, dx + \int_{E_j^{-\d_j}} W(D w_j)\,
dx \right) - \frac{c}{k},
\end{eqnarray}
where $E_j:=\bigcup_{i \in Z_j}B^{n-1}_{\rho_j^i}(x_i^{\e_j})$.
We first consider the energy contribution `far' from the
connecting zones. In this case, we suitably modify the sequence
$(w_j)$ in order to get a constant inside each half cylinder
$B^{(n-1)}_{\rho_j^i}(x_i^{\e_j})^{\pm \d_j}$. Then, we apply the
classical result of dimensional reduction proved in \cite{LDR} to
$\o^{+\d_j}$ and $\o^{-\d_j}$, separately.
\begin{proposition}\label{gliminffar}
We have
\begin{eqnarray*}
&&\liminf_{j \to +\infty}\frac{1}{\d_j} \left( \int_{(\o \setminus
E_j)^{+\d_j}} W(D w_j)\, dx + \int_{(\o \setminus E_j)^{-\d_j}}
W(D
w_j)\, dx \right)\\
&& \geq \int_\o \left({\mathcal{Q}}_{n-1}\overline W (D_\a u^+) +
{\mathcal{Q}}_{n-1}\overline W( D_\a u^-) \right) \, dx_\a.
\end{eqnarray*}
\end{proposition}
\noindent {\it Proof. } We define
\begin{equation}\label{defvj}
v_j:=\left\{
\begin{array}{lcl}
w_j & \text{ in } & (\o \setminus E_j)^{\pm\d_j},\\
u_j^{i\pm} & \text{ in } &
B^{n-1}_{\rho_j^i}(x_i^{\e_j})^{\pm\d_j} \text{ if } i \in Z_j.
\end{array}
\right.
\end{equation}
Then $(v_j) \subset W^{1,p}(\O_j;{\mathbb{R}}^m)$ converges weakly to
$(u^+,u^-)$. In fact,
\begin{equation}\label{vjpm1}
\sup_{j \in {\mathbb{N}}} \frac{1}{\d_j}\int_{\o^{\pm\d_j}}|D v_j|^p\, dx
\leq \sup_{j \in {\mathbb{N}}} \frac{1}{\d_j}\int_{\o^{\pm\d_j}}|D u_j|^p\,
dx <+\infty.
\end{equation}
Moreover, since $\rho_j^i < \rho_j <\e_j/2$, then
$B^{n-1}_{\rho_j^i}(x_i^{\e_j}) \subset Q^{n-1}_{i,\e_j}$; hence,
$$\int_{\o^{\pm\d_j}}|v_j - u^\pm|^p\, dx \leq \int_{(\o \setminus
E_j)^{\pm\d_j}}|w_j - u^\pm|^p\, dx\nonumber + \sum_{i \in
Z_j}\int_{(Q^{n-1}_{i,\e_j})^{\pm\d_j}}|u^\pm - u_j^{i\pm}|^p\, dx
$$
and, by (\ref{corep}), we obtain that
\begin{eqnarray}\label{vjpm2}
\frac{1}{\d_j}\int_{\o^{\pm\d_j}}|v_j - u^\pm|^p\, dx & \leq &
\frac{1}{\d_j} \int_{\o^{\pm\d_j}}|w_j - u^\pm|^p\, dx +
\frac{c}{\d_j}\int_{\o^{\pm\d_j}}|u_j -
u^\pm|^p\, dx\nonumber\\
&&+ c \, \e_j^p \,\sup_{j \in {\mathbb{N}}}
\frac{1}{\d_j}\int_{\o^{\pm\d_j}} |D u_j|^p\, dx.
\end{eqnarray}
Passing to the limit as $j\to +\infty$ in (\ref{vjpm2}), by
(\ref{vjpm1}) and Remark \ref{convrmk} we get that $(v_j)$
converges weakly to $(u^+,u^-)$.
Since $W(0)=0$, by (\ref{defvj}) and \cite{LDR} Theorem 2, we have
\begin{eqnarray*}
&&\liminf_{j \to +\infty}\frac{1}{\d_j}\left( \int_{(\o \setminus
E_j)^{+\d_j}} W(D w_j)\, dx + \int_{(\o
\setminus E_j)^{-\d_j}} W(D w_j)\, dx \right)\\
&=&\liminf_{j \to +\infty}\frac{1}{\d_j}\left( \int_{(\o \setminus
E_j)^{+\d_j}} W(D v_j)\, dx + \int_{(\o
\setminus E_j)^{-\d_j}} W(D v_j)\, dx \right)\\
&=&\liminf_{j \to +\infty}\frac{1}{\d_j}\left( \int_{\o^{+\d_j}}
W(D v_j)\, dx + \int_{\o^{-\d_j}} W(D v_j)\, dx
\right)\\
&\geq& \int_\o {\mathcal{Q}}_{n-1}\overline W (D_\a u^+)\, dx_\a +\int_\o
{\mathcal{Q}}_{n-1} \overline W(D_\a u^-)\, dx_\a\,.
\end{eqnarray*}
\hfill$\Box$\\
Now let us deal with the contribution `near' the connecting zones.
We always work under the assumption
$$\ell\in(0,+\infty]\quad \text{ and }\quad
0<R^{(\ell)}=\lim_{j\to+\infty}\frac{r_j^{(n-1-p)}}{\e_j^{n-1}}<+\infty,$$
or $$\ell=0\quad \text{ and }\quad
0<R^{(0)}=\lim_{j\to+\infty}\frac{r_j^{(n-p)}}{\d_j\e_j^{n-1}}<+\infty.
$$
In the following proposition we suitably modify $(w_j)$ in each
surrounding cylinder in order to get an admissible test function
for the minimum problem (\ref{phigj}) or (\ref{phigj12}).
\begin{proposition}\label{gliminfclose}
Let $\ell \in [0,+\infty]$. Then
$$
\liminf_{j \to +\infty}\frac{1}{\d_j} \left( \int_{E_j^{+\d_j}}
W(D w_j)\, dx + \int_{E_j^{-\d_j}} W(D w_j)\, dx \right) \geq
R^{(\ell)} \int_\o \varphi^{(\ell)} (u^+-u^-) \, dx_\a + o(1)\,,
$$
as $\g\to 0^+$.
\end{proposition}
\noindent {\it Proof. }Let $\ell \in (0,+\infty]$, the case
$\ell=0$ can be treated similarly. Let $i \in Z_j$ and
$N_j=\frac{\e_j}{r_j}$. Since $\rho_j^i <\g \e_j$, we can define
$$
\zeta_j^i:= \left\{
\begin{array}{ll}
w_j(x_i^{\e_j} + r_j\, y_\a, \d_j\, y_n) - u_j^{i-} & \text{in }
\big(B_{\rho_j^i/r_j}^{n-1} \times I \big) \setminus
C_{1,\rho_j^i/r_j}\\
&\\
(u_j^{i+} - u_j^{i-}) & \text{in } \big(B_{\g N_j}^{n-1}
\setminus B_{\rho_j^i/r_j}^{n-1}\big)^+ \\
&\\
0 & \text{in } \big(B_{\g N_j}^{n-1} \setminus
B_{\rho_j^i/r_j}^{n-1} \big)^- \,,
\end{array}
\right.
$$
where $N_j={\e_j/r_j}$. Then $\zeta_j^i \in W^{1,p}((B_{\g
N_j}^{n-1} \times I) \setminus C_{1,\g N_j};{\mathbb{R}}^m)$, $\zeta_j^i =
(u_j^{i+} - u_j^{i-})$ on $\big(\partial B_{\g N_j}^{n-1}\big)^+ $
and $\zeta_j^i=0$ on $\big(\partial B_{\g N_j}^{n-1}\big)^-$.
Since $W(0)=0$, changing variable, by (\ref{phigj}) we get
\begin{eqnarray}\label{1136}
&&\frac{1}{\d_j} \left(
\int_{B_{\rho_j^i}^{n-1}(x_i^{\e_j})^{+\d_j}} W(D w_j)\, dx +
\int_{B_{\rho_j^i}^{n-1}(x_i^{\e_j})^{-\d_j}} W(D
w_j)\, dx \right)\nonumber\\
&= & r_j^{n-1}\left(\int_{\big( B_{\rho_j^i/r_j}^{n-1} \big)^+}
W\Bigl(r_j^{-1} D_\a \zeta_j^i |\d_j^{-1} D_n \zeta_j^i \Bigr)\,
dy + \int_{\big( B_{\rho_j^i/r_j}^{n-1}\big)^-} W \Bigl(r_j^{-1}
D_\a \zeta_j^i |\d_j^{-1} D_n
\zeta_j^i \Bigr)\, dy \right)\nonumber\\
&=& r_j^{n-1}\int_{( B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g
N_j}} W\Bigl(r_j^{-1} D_\a \zeta_j^i |\d_j^{-1} D_n \zeta_j^i
\Bigr)\,dy
\nonumber\\
&\geq& r_j^{n-1-p}\varphi^{(\ell)}_{\g,j}(u_j^{i+} - u_j^{i-})\,.
\end{eqnarray}
Summing up in (\ref{1136}), for $i \in Z_j$, we get that
\begin{eqnarray}\label{1137}
&&\nonumber \frac{1}{\d_j} \left( \int_{E_j^{+\d_j}} W(D w_j)\, dx
+ \int_{E_j^{-\d_j}} W(D
w_j)\, dx \right)\\
&=&\nonumber \sum_{i \in Z_j}\frac{1}{\d_j} \left(
\int_{B_{\rho_j^i}^{n-1}(x_i^{\e_j})^{+\d_j}} W(D w_j)\, dx +
\int_{B_{\rho_j^i}^{n-1}(x_i^{\e_j})^{-\d_j}} W(D w_j)\, dx
\right)\\
&\geq& r_j^{n-1-p}\,\sum_{i \in
Z_j}\varphi^{(\ell)}_{\g,j}(u_j^{i+} - u_j^{i-})=
\frac{r_j^{n-1-p}}{\e_j^{n-1}} \sum_{i \in Z_j}\e_j^{n-1}
\varphi^{(\ell)}_{\g,j}(u_j^{i+} - u_j^{i-})\,.
\end{eqnarray}
Passing to the limit as $j \to +\infty$ we get, by (\ref{rinfty})
and Proposition \ref{rs}, that
\begin{eqnarray*}
&&\liminf_{j \to +\infty}\frac{1}{\d_j} \left( \int_{E_j^{+\d_j}}
W(D w_j)\, dx + \int_{E_j^{-\d_j}} W(D
w_j)\, dx \right)\\
&\geq & R^{(\ell)} \int_\o \varphi^{(\ell)}(u^+ - u^-)\, dx_\a
\\
&& + R^{(\ell)} \, \liminf_{j \to +\infty}\int_\o \Bigl(\sum_{i
\in Z_j}\varphi^{(\ell)}_{\g,j}(u_j^{i+}-
u_j^{i-})\chi_{Q_{i,\e_j}^{n-1}} -
\varphi^{(\ell)}(u^+-u^-)\Bigr)\,dx_\a\\
&=& R^{(\ell)} \int_\o \varphi^{(\ell)}(u^+ - u^-)\, dx_\a +
o(1)\,,
\end{eqnarray*}
as $\g\to 0^+$, which completes the proof.
\hfill$\Box$\\
We now prove the liminf inequality for any arbitrary converging
sequence.
\begin{lemma}\label{gammaliminf}
Let $\ell \in [0,+\infty]$. For every sequence $(u_j)$ converging
to $(u^+,u^-)$ we have
\begin{eqnarray*}
\liminf_{j\to +\infty} \mathcal F_j(u_j) &\geq& \int_\o {\mathcal{Q}}_{n-1}
\overline W(D_\a u^+)\, dx_\a + \int_\o {\mathcal{Q}}_{n-1} \overline W(D_\a
u^-)\,
dx_\a \\
&&+ R^{(\ell)} \int_\o \varphi^{(\ell)}(u^+ - u^-)\, dx_\a\,.
\end{eqnarray*}
\end{lemma}
\noindent {\it Proof. } Let $(u_j)\to (u^+,u^-)$ be such that
$\liminf_{j\to +\infty} \mathcal F_j(u_j)<+\infty$. Reasoning as
in \cite{Ans} Proposition 5.2, by \cite{BDV} Lemma 3.5, upon
passing to a subsequence, for every $M>0$ and $\eta>0$, we have
the existence of $R_M>M$ and of a Lipschitz function $\Phi_M \in
{\mathcal{C}}^1_c({\mathbb{R}}^m;{\mathbb{R}}^m)$ with ${\rm Lip}(\Phi_M)=1$ such that
$$\Phi_M(z)=\left\{
\begin{array}{lll}
z & \text{if} & |z|< R_M,\\
&&\\
0 & \text{if} & |z| >2 R_M
\end{array}
\right.
$$ and
\begin{equation}\label{janv}
\liminf_{j \to +\infty} \mathcal F_j(u_j) \geq \liminf_{j \to
+\infty} \mathcal F_j(\Phi_M (u_j)) -\eta\,.
\end{equation}
Note that $(\Phi_M (u_j))\subset W^{1,p}(\O_j;{\mathbb{R}}^m)\cap
L^\infty(\O_j;{\mathbb{R}}^m)$, $\sup_{j \in {\mathbb{N}}}\| \Phi_M(u_j)
\|_{L^\infty(\O_j;{\mathbb{R}}^m)}<R_M$ and it converges to
$(\Phi_M(u^+),\Phi_M(u^-))$ as $j \to +\infty$. Hence, if we apply
(\ref{lowerbd}), Propositions \ref{gliminffar} and
\ref{gliminfclose} to $(\Phi_M (u_j))$ in place of $(u_j)$,
letting $\g \to 0$ and $k \to +\infty$, we get that
\begin{eqnarray}\label{fev}
\liminf_{j \to +\infty} \mathcal F_j(\Phi_M(u_j)) & \geq & \int_\o
{\mathcal{Q}}_{n-1} \overline W(D_\a \Phi_M(u^+))\, dx_\a + \int_\o {\mathcal{Q}}_{n-1}
\overline
W(D_\a \Phi_M(u^-))\, dx_\a\nonumber\\
&& + R^{(\ell)} \int_\o \varphi^{(\ell)}(\Phi_M(u^+) -
\Phi_M(u^-))\, dx_\a.
\end{eqnarray}
Moreover $\Phi_M(u^\pm) \rightharpoonup u^\pm$ weakly in
$W^{1,p}(\o;{\mathbb{R}}^m)$ as $M\to +\infty$; hence, by (\ref{janv}),
(\ref{fev}), the lower semicontinuity of $\int_\o {\mathcal{Q}}_{n-1}
\overline W(D_\a u)\dx_{\alpha}$ with respect to the weak
$W^{1,p}(\o;{\mathbb{R}}^m)$-convergence, and (\ref{AA}) we have that
\begin{eqnarray}\label{mars}
&&\liminf_{j \to +\infty} \mathcal F_j(u_j)\nonumber\\
&\geq& \int_\o {\mathcal{Q}}_{n-1} \overline W(D_\a u^+)\, dx_\a + \int_\o
{\mathcal{Q}}_{n-1} \overline W(D_\a u^-)\, dx_\a+ R^{(\ell)} \int_\o
\varphi^{(\ell)}(u^+ - u^-)\, dx_\a-\eta\,,
\end{eqnarray}
and by the arbitrariness of $\eta$, the thesis. \hfill$\Box$
\subsection{The limsup inequality}
\noindent For every $ (u^+,u^-)\in W^{1,p}(\o,\Bbb R^m)\times
W^{1,p}(\o,\Bbb R^m)$ the limsup inequality is obtained by suitably
modifying the recovery sequences $(u_j^{\pm})$ for the
$\Gamma$-limits of $$ \frac{1}{\d_j}\int_{\o^{+\d_j}} W( Du )\, dx
\quad \text{ and } \quad \frac{1}{\d_j}\int_{\o^{-\d_j}} W( Du )\,
dx.$$
\begin{lemma}
Let $\ell \in [0,+\infty]$ and let $\o$ be an open bounded subset
of $\Bbb R^{n-1}$ such that ${\mathcal H}^{n-1}(\partial \o)=0$. Then, for all
$(u^+,u^-)\in W^{1,p}(\o,\Bbb R^m)\times W^{1,p}(\o,\Bbb R^m)$ and for
all $\eta>0$ there exists a sequence $(\bar u_j)\subset
W^{1,p}(\O_j;{\mathbb{R}}^m)$ converging to $(u^+,u^-)$ such that
\begin{eqnarray*}
\limsup_{j\to +\infty} \mathcal F_j(\bar u_j) &\le& \int_\o
{\mathcal{Q}}_{n-1} \overline W(D_\a u^+)\, dx_\a + \int_\o {\mathcal{Q}}_{n-1}
\overline W(D_\a
u^-)\, dx_\a \\
&&+ R^{(\ell)} \int_\o \varphi^{(\ell)}(u^+ - u^-)\, dx_\a + \eta
R^{(\ell)} {\mathcal H}^{n-1}(\o)\,.
\end{eqnarray*}
\end{lemma}
\noindent{\it Proof.} The proof of the limsup is divided into
three steps. We first construct a sequence $(\bar u_j) \subset
W^{1,p}(\O_j;{\mathbb{R}}^m)$ that we expect to be a recovery sequence. In
the second step we prove that $(\bar u_j)$ converges to
$(u^+,u^-)$. Finally, we prove that it satisfies the limsup
inequality. We first deal with the case $\ell \in (0,+\infty]$.\\
{\bf Step 1: Definition of a recovery sequence.} Let $u^\pm \in
W^{1,p}(\o;{\mathbb{R}}^m) \cap L^\infty(\o;{\mathbb{R}}^m)$. According to
\cite{LDR} Theorem 2 and \cite{Bo&Fo} Theorem 1.1, there exist two
sequences $(u^{\pm}_j)\subset W^{1,p}(\o^{\pm\d_j};{\mathbb{R}}^m)$ such
that $u_j^{\pm} \to u^{\pm}$, the sequences of gradients $(|D
u_j^{\pm}|^p/\d_j)$ are equi-integrable on $\o^{\pm\d_j}$,
respectively, and
\begin{equation}\label{canne}
\lim_{j \to +\infty}{1\over \d_j}\int_{\o^{\pm\d_j}} W( Du^\pm_j)
\, dx = \int_\o {\mathcal{Q}}_{n-1} \overline W(D_\a u^\pm)\, dx_\a\,.
\end{equation}
Moreover, using a truncation argument (as in \cite{AnsB} Lemma
6.1, Step 2) we may assume without loss of generality that
$$
\sup_{j \in {\mathbb{N}}}\|u^{\pm}_j\|_{L^\infty(\o^{\pm\d_j};{\mathbb{R}}^m)} <
+\infty\,.
$$
Let $u_j:=u_j^+ \chi_{\o^{+\d_j}} + u_j^- \chi_{\o^{-\d_j}}\in
W^{1,p}(\o^{+\d_j} \cup \o^{-\d_j};{\mathbb{R}}^m)$ and let $(w_j)$ be the
sequence obtained from $(u_j)$ as in Lemma
\ref{important+equiint}, then $\sup_{j \in
{\mathbb{N}}}\|w_j\|_{L^\infty(\o^{\pm\d_j};{\mathbb{R}}^m)}< +\infty$.
\bigskip
We first define $(\bar u_j)$ `far' from the connecting zones; {\it i.e., }
\begin{equation}\label{defubar}
\bar u_j:=w_j\; \text{ in }\; \Bigl( \o \setminus \bigcup_{i \in
{\mathbb{Z}}^{n-1}}B_{\rho_j}^{n-1}(x_i^{\e_j})\Bigr)^{\pm\d_j}\,.
\end{equation}
Then we pass to define $(\bar u_j)$ on each
$B_{\rho_j}^{n-1}(x_i^{\e_j})^{\pm\d_j}$ making a distinction
between the indices $i\in Z_j$ and $i\in {\mathbb{Z}}^{n-1}\setminus Z_j$.
If $i \in Z_j$, by (\ref{phigj}), for every $\eta>0$ there exists
$\zeta_{\g,j}^i \in X^\g_j(u_j^{i+}- u_j^{i-})$ such that
\begin{equation}\label{presque}
\int_{(B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g N_j}}r_j^p \,
W\left(r_j^{-1} D_\a \zeta_{\g,j}^i | \d_j^{-1} D_n \zeta_{\g,j}^i
\right) dx \leq \varphi^{(\ell)}_{\g,j}(u_j^{i+}- u_j^{i-}) +
\eta.
\end{equation}
Then, we define
\begin{equation}\label{defubarZj}
\bar u_j:=
\zeta_{\g,j}^i\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\right)+u_j^{i-}\;
\text{ in }\; B^{n-1}_{\rho_j}(x_i^{\e_j})^{\pm\d_j}\,, \quad
i\in Z_j\,.
\end{equation}
In particular, $\bar u_j=u_j^{i\pm}=w_j$ on $\big(\partial
B^{n-1}_{\rho_j}(x_i^{\e_j})\big)^{\pm\d_j}$.
Let us now deal with the contact zones not well contained in
$\o$; {\it i.e., } with the indices $i \not\in Z_j$. For fixed $\g>0$ and $j$
large enough we have that $\g N_j>2$. Let $\psi \in
W^{1,p}(B^{n-1}_2;[0,1])$ be such that $\psi =1$ on $\partial
B_2^{n-1}$ and $\psi=0$ in $B_1^{n-1}$ and define
$$
\psi_{\g,j}(x):=\left\{
\begin{array}{lll}
0 & \text{ in } & (B_{\g N_j}^{n-1})^- \\
\psi(x_\a) & \text{ in } & (B_2^{n-1})^+ \\
1 & \text{ in } & (B_{\g N_j}^{n-1} \setminus B_2^{n-1})^+ \,.
\end{array}
\right.
$$
Then $\psi_{\g,j} \in W^{1,p}((B^{n-1}_{\g N_j} \times I)
\setminus C_{1,\g N_j};[0,1])$, $\psi_{\g,j} = 1$ on
$\big(\partial B_{\g N_j}^{n-1}\big)^+$ and $\psi_{\g,j} = 0$ on
$\big( \partial B_{\g N_j}^{n-1}\big)^-$. Let $w_j^{\pm}=w_j\,
\chi_{\o^{\pm\d_j}}$, we extend both of them to the whole
$\o\times (-\d_j,\d_j)$ by reflection; {\it i.e., } we define
$\tilde{w}_j^{\pm}(x_\a, x_n)= w_j^{\pm}(x_\a, -x_n)$ for $x\in
\o^{\mp\d_j}$ and $\tilde{w}_j^{\pm}(x)= w_j^{\pm}(x)$ for $x\in
\o^{\pm\d_j}$. Hence, we define
\begin{equation}\label{defubarZn}
\bar u_j:=\psi_{\g,j}
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}
\right)\tilde{w}_j^+ + \Bigg( 1- \psi_{\g,j}
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j} \right)
\Bigg)\tilde{w}_j^-
\end{equation}
in $\big( B^{n-1}_{\rho_j}(x_i^{\e_j}) \times (-\d_j,\d_j)\big)
\cap \O_j$ and for $i\in {\mathbb{Z}}^{n-1}\setminus Z_j$. In particular,
we have that $\bar u_j=w_j$ on $\big( \partial
B^{n-1}_{\rho_j}(x_i^{\e_j}) \times (-\d_j,\d_j)\big) \cap \O_j$;
thus $(\bar u_j) \subset W^{1,p}(\O_j;{\mathbb{R}}^m)$.
\bigskip
{\bf Step 2: The sequence $(\bar u_j)$ weakly converges to
$(u^+,u^-)$}. Let us check (\ref{strongconv}) and
(\ref{weakconv}). We will only treat the upper cylinder $\o^{+
\d_j}$, the lower part being analogous. First
\begin{eqnarray}\label{canne0}
&&\frac{1}{\d_j}\int_{\o^{+\d_j}} |\bar u_j - u^+|^p \, dx
\nonumber\\
&=& \frac{1}{\d_j}\int_{ \left( \o \setminus \bigcup_{i \in
{\mathbb{Z}}^{n-1}} B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{+\d_j}} |w^+_j -
u^+|^p \,
dx\nonumber\\
&&+ \frac{1}{\d_j}\sum_{i \in Z_j} \int_{
B_{\rho_j}^{n-1}(x_i^{\e_j})^{+\d_j}} \left|\zeta_{\g,j}^i
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\right)+u_j^{i-}
-u^+\right|^p \, dx\nonumber\\
&&+ \frac{1}{\d_j}\sum_{i \in {\mathbb{Z}}^{n-1}\setminus Z_j}
\int_{\big(\o \cap B_{\rho_j}^{n-1}(x_i^{\e_j})\big)^{+\d_j}}
\left|\psi_{\g,j}
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\right)(w_j^+
- \tilde{w}^-_j) + \tilde{w}^-_j - u^+ \right|^p \, dx\nonumber\\
&\leq & \frac{1}{\d_j}\int_{\o^{+\d_j}} |w_j - u^+|^p \, dx +
c\sum_{i \in Z_j} \int_{
B_{\rho_j}^{n-1}(x_i^{\e_j})} |u^+ - u_j^{i+}|^p \, dx_\a\nonumber\\
&&+ \frac{c}{\d_j}\sum_{i \in Z_j} \int_{
B_{\rho_j}^{n-1}(x_i^{\e_j})^{+\d_j}} \left|\zeta_{\g,j}^i
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\right)-
(u_j^{i+} -
u_j^{i-}) \right|^p dx\nonumber\\
&&+{c\over \d_j}\, \int_{\left(\o \cap \bigcup_{i\in {\mathbb{Z}}^{n-1}
\setminus Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{+\d_j}}
\left(|w^+_j|^p + |\tilde{w}_j^-|^p + |u^+|^p\right)\,dx\,.
\end{eqnarray}
Since $\lim_{j\to +\infty}{\mathcal H}^{n-1}\Bigl(\o \cap \bigcup_{i\in
{\mathbb{Z}}^{n-1}\setminus Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\Bigr)=0$ and
$\sup_{j \in {\mathbb{N}}}\|w^{\pm}_j\|_{L^\infty(\o^{\pm\d_j};{\mathbb{R}}^m)}<
+\infty$, we have that
\begin{equation}\label{canne3}
\lim_{j\to +\infty}{c\over \d_j}\, \int_{\left(\o \cap
\bigcup_{i\in {\mathbb{Z}}^{n-1} \setminus
Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{+\d_j}} \left(|w^+_j|^p +
|\tilde{w}_j^-|^p + |u^+|^p\right)\,dx=0\,.
\end{equation}
Moreover, reasoning as in the proof of Proposition \ref{rs} (see
inequality (\ref{corep})), we have that
\begin{equation}\label{canne4}
\lim_{j \to +\infty} \sum_{i \in Z_j} \int_{
B_{\rho_j}^{n-1}(x_i^{\e_j})} |u^+ - u_j^{i+}|^p \, dx_\a=0\,,
\end{equation}
and, by the convergence $w_j\to (u^+,u^-)$, it remains only to
prove that
\begin{equation}\label{canne2}
\lim_{j \to +\infty} \frac{1}{\d_j}\sum_{i \in Z_j} \int_{
B_{\rho_j}^{n-1}(x_i^{\e_j})^{+\d_j}} \left|\zeta_{\g,j}^i
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\right)-
(u_j^{i+} - u_j^{i-}) \right|^p dx=0\,.
\end{equation}
In fact, changing variable, we get that
\begin{eqnarray*}
&&\frac{1}{\d_j}\sum_{i \in Z_j} \int_{
B_{\rho_j}^{n-1}(x_i^{\e_j})^{+\d_j}} \left|\zeta_{\g,j}^i
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\right)-
(u_j^{i+}- u_j^{i-}) \right|^p dx\\
&=& r_j^{n-1} \sum_{i \in Z_j} \int_{(B_{\g N_j} ^{n-1})^+}
\left|\zeta_{\g,j}^i(x)- (u_j^{i+} - u_j^{i-}) \right|^p \,dx\,,
\end{eqnarray*}
and by, Poincar\'e's Inequality
$$
\int_{B_{\g N_j}^{n-1}} \left|\zeta_{\g,j}^i (x_\a,x_n)- (u_j^{i+}
- u_j^{i-}) \right|^p dx_\a \leq c\, (\g N_j)^p \int_{B_{\g
N_j}^{n-1}} |D_\a \zeta_{\g,j}^i (x_\a,x_n)|^p \,dx_\a
$$
for a.e. $x_n \in (0,1)$. Hence, by the $p$-growth condition
(\ref{pgrowth}) and (\ref{presque}) if we integrate with respect
to $x_n$ and sum up in $i \in Z_j$, we get that
\begin{eqnarray}\label{canne1}
&&\frac{1}{\d_j}\sum_{i \in Z_j} \int_{
B_{\rho_j}^{n-1}(x_i^{\e_j})^{+\d_j}} \left|\zeta_{\g,j}^i
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\right)-
(u_j^{i+} - u_j^{i-}) \right|^p dx\nonumber\\
&\leq & c\, r_j^{n-1} \g^p N_j^p \sum_{i \in Z_j} \int_{(B_{\g
N_j}^{n-1})^+} |D_\a \zeta_{\g,j}^i |^p \, dx\nonumber\\
&\leq & c\, r_j^{n-1} \g^p N_j^p \sum_{i \in Z_j} \int_{(B_{\g
N_j}^{n-1})^+} \left|\left( D_\a \zeta_{\g,j}^i \Big|
\frac{r_j}{\d_j}
D_n \zeta_{\g,j}^i \right) \right|^p \, dx\nonumber\\
&\leq & c\, r_j^{n-1} \g^p N_j^p \sum_{i \in Z_j} \left(
\varphi^{(\ell)}_{\g,j}(u_j^{i+} - u_j^{i-}) +\eta + r_j^p\,
{\mathcal H}^{n-1}( B_{\g N_j}^{n-1}) \right)\nonumber\\
&\leq & c\,\g^p\, \e_j^p\,\frac{ r_j^{n-1-p}}{\e_j^{n-1}} \left(
\sum_{i \in Z_j} \e_j^{n-1} \varphi^{(\ell)}_{\g,j}(u_j^{i+} -
u_j^{i-}) +\left(\eta + c\,\g^{n-1}\,
\frac{\e_j^{n-1}}{r_j^{n-1-p}}\right) {\mathcal H}^{n-1}(\o)\right).
\end{eqnarray}
By Proposition \ref{rs} and (\ref{rinfty}), passing to the limit
as $j\to +\infty$ in (\ref{canne1}), we get (\ref{canne2}).
It remains to prove that (\ref{weakconv}) holds. In fact,
\begin{eqnarray}\label{gradubar}
\nonumber
&& \frac{1}{\d_j}\int_{\o^{+ \d_j}}|D {\bar u}_j|^p\,dx \\
&=& \nonumber\frac{1}{\d_j}\int_{\left( \o \setminus \bigcup_{i
\in {\mathbb{Z}}^{n-1}} B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{+ \d_j}}|D
w^\pm_j|^p\,dx\\
&&\nonumber + \frac{1}{\d_j}\int_{\bigcup_{i\in Z_j}
B_{\rho_j}^{n-1}(x_i^{\e_j})^{+ \d_j}}
\left|\left(r_j^{-1}D_\a\zeta_{\g,j}^i
\Bigl(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\Bigr)\Big|
\d_j^{-1} D_n\zeta_{\g,j}^i
\Bigl(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}
\Bigr)\right)\right|^p\,dx\\
&&+ \frac{1}{\d_j}\int_{\left(\bigcup_{i\in{\mathbb{Z}}^{n-1}\setminus
Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o\right)^{ + \d_j}}|D {\bar
u}_j|^p\,dx \,.
\end{eqnarray}
It can be easily shown that
\begin{eqnarray}\label{gradubarZj}
&&\nonumber\frac{1}{\d_j}\int_{\bigcup_{i\in Z_j}
B_{\rho_j}^{n-1}(x_i^{\e_j})^{+ \d_j}}
\left|\left(r_j^{-1}D_\a\zeta_{\g,j}^i
\Bigl(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}\Bigr)\Big|
\d_j^{-1} D_n\zeta_{\g,j}^i
\Bigl(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j}
\Bigr)\right)\right|^p\,dx\\
&\le& { r_j^{n-1-p}\over \e_j^{n-1}} \Bigl(\sum_{i\in Z_j}
\e_j^{n-1}\,\varphi^{(\ell)}_{\g,j}(u_j^{i+} - u_j^{i-}) \Bigr)+
{\mathcal H}^{n-1}(\o) \Bigl(\eta { r_j^{n-1-p}\over \e_j^{n-1}} +
\g^{n-1}\Bigr)\,;
\end{eqnarray}
while,
\begin{eqnarray}\label{gradubarZn}
&&\nonumber\frac{1}{\d_j}\int_{\left(
\bigcup_{i\in{\mathbb{Z}}^{n-1}\setminus
Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o \right)^{+ \d_j}}|D {\bar
u}_j|^p\,dx \\
& \le & \nonumber c\, \sum_{i\in {\mathbb{Z}}^{n-1}\setminus Z_j}\left(
\frac{1}{r_j^p \d_j} \int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap
\o)^{+\d_j}} \Bigl|D_\a \psi_{\g,j}
\Bigl(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{\d_j} \Bigr)\Bigr|^p
\, \big(|w^+_j|^p + |\tilde{w}^-_j|^p\big)\ dx\right.
\\
&&\left.\nonumber\qquad\qquad + {1\over \d_j}
\int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o)^{+\d_j}} (|Dw^+_j|^p
+|D \tilde{w}^-_j|^p)\, dx \right)\\
&\le&\nonumber c\,\sum_{i\in {\mathbb{Z}}^{n-1}\setminus Z_j}\left(
r_j^{n-1-p}\int_{B_2^{n-1}}|D_\a \psi|^p\, dx_\a +{1\over \d_j}
\int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o)^{+ \d_j}}
|Dw^+_j|^p\,dx\right.\\
&& \nonumber \left. \qquad \qquad +{1\over \d_j}
\int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap
\o)^{- \d_j}} |Dw^-_j|^p\,dx\right)\\
\nonumber &\le& c\,\sum_{i\in {\mathbb{Z}}^{n-1}\setminus Z_j}\left(
{r_j^{n-1-p}\over \e_j^{n-1}} {\mathcal H}^{n-1}(Q_{i,\e_j}^{n-1}) +
{1\over \d_j} \int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap
\o)^{\pm\d_j}}
|Dw^\pm_j|^p\,dx\right.\\
&& \left. \qquad \qquad +{1\over \d_j}
\int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o)^{- \d_j}}
|Dw^-_j|^p\,dx\right)\,.
\end{eqnarray}
Note that the previous sum can be computed over all $i \in
{\mathbb{Z}}^{n-1} \setminus Z_j$ such that $Q^{n-1}_{i,\e_j} \cap \o \neq
\emptyset$. Let
$$\o'_j:= \bigcup_{i \in {\mathbb{Z}}^{n-1}
\setminus Z_j ,\, Q^{n-1}_{i,\e_j} \cap \o \neq \emptyset}
Q^{n-1}_{i,\e_j},$$ then
\begin{equation}\label{babadjian}
\sum_{i \in {\mathbb{Z}}^{n-1} \setminus Z_j ,\, Q^{n-1}_{i,\e_j} \cap \o
\neq \emptyset} \mathcal H^{n-1}(Q^{n-1}_{i,\e_j}) = \mathcal
H^{n-1}(\o'_j) \to \mathcal H^{n-1}(\partial \o)=0.
\end{equation}
Moreover, by Lemma \ref{important+equiint} we have that
$\sup_{j}{1\over \d_j} \int_{ \o^{\pm\d_j}}
|Dw^\pm_j|^p\,dx<+\infty$; hence, by Proposition \ref{rs},
(\ref{rinfty}), (\ref{gradubar}), (\ref{gradubarZj}) and
(\ref{gradubarZn}) we get (\ref{weakconv}).
\bigskip
{\bf Step 3: The sequence $(\bar u_j)$ is a recovery sequence.} We
now prove the limsup inequality.
\begin{eqnarray}\label{step1}
&&\limsup_{j \to +\infty} \int_{\o^{\pm \d_j}} W(D \bar u_j)\, dx
\nonumber\\
&=& \limsup_{j \to +\infty} \frac{1}{\d_j} \Bigg( \int_{\left(\o
\setminus \bigcup_{i \in
{\mathbb{Z}}^{n-1}}B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{\pm\d_j}} W( D \bar
u_j)\, dx + \int_{\bigcup_{i \in
Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})^{\pm\d_j}} W( D \bar u_j)\, dx
\nonumber\\
&&\qquad\qquad\qquad + \int_{\left(\o \cap \bigcup_{i \in
{\mathbb{Z}}^{n-1} \setminus
Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{\pm\d_j}} W( D \bar
u_j)\, dx\Bigg)\,.
\end{eqnarray}
We deal with the first term in (\ref{step1}). The definition of
$\bar u_j$ (\ref{defubar}), Lemma \ref{important+equiint} and
(\ref{canne}), yield
\begin{eqnarray}\label{step2}
&&\nonumber\limsup_{j \to +\infty} \frac{1}{\d_j} \int_{\left(\o
\setminus \bigcup_{i \in
{\mathbb{Z}}^{n-1}}B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{\pm\d_j}} W( D \bar
u_j)\, dx\nonumber\\
&=& \nonumber\limsup_{j \to +\infty} \frac{1}{\d_j}\int_{\left(\o
\setminus \bigcup_{i \in
{\mathbb{Z}}^{n-1}}B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{\pm\d_j}} W( D
w_j)\,dx\nonumber\\
&\leq &\nonumber \limsup_{j \to +\infty} \frac{1}{\d_j}
\int_{\o^{\pm\d_j}} W( D u^{\pm}_j)\,dx + o(1)\nonumber\\
&=& \int_\o {\mathcal{Q}}_{n-1} \overline W(D_\a u^\pm)\, dx_\a + o(1)\,,
\end{eqnarray}
as $\g\to 0^+$. For every $i \in Z_j$, by (\ref{defubarZj}) and
(\ref{presque}) we get that
\begin{eqnarray*}
&&\frac{1}{\d_j} \Bigg(
\int_{B^{n-1}_{\rho_j}(x_i^{\e_j})^{+\d_j}}W(D \bar u_j)\, dx +
\int_{B^{n-1}_{\rho_j}(x_i^{\e_j})^{-\d_j}}W(D \bar
u_j)\, dx \Bigg)\\
&=& r_j^{n-1}\int_{(B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g
N_j}} W\left(r_j^{-1} D_\a \zeta_{\g,j}^i | \d_j^{-1} D_n
\zeta_{\g,j}^i \right)
dx\\
&\leq & r_j^{n-1-p} \left( \varphi^{(\ell)}_{\g,j}(u_j^{i+}-
u_j^{i-}) + \eta \right)\,;
\end{eqnarray*}
hence, by (\ref{rinfty}) and Proposition \ref{rs} we get
\begin{eqnarray}\label{step3}
&&\nonumber\limsup_{j \to +\infty} \frac{1}{\d_j}\Bigg(
\int_{\bigcup_{i \in Z_j} B^{n-1}_{\rho_j}(x_i^{\e_j})^{+\d_j}}W(D
\bar u_j)\, dx + \int_{\bigcup_{i \in Z_j}
B^{n-1}_{\rho_j}(x_i^{\e_j})^{-\d_j}}W(D \bar u_j)\, dx
\Bigg)\\
&\leq &\nonumber R^{(\ell)} \int_\o \varphi^{(\ell)} (u^+ - u^-)\,
dx_\a
+ R^{(\ell)}\, {\mathcal H}^{n-1}(\o) \,\eta \\
&&\nonumber + \limsup_{j \to +\infty}\int_\o \Bigl|\sum_{i \in
Z_j}\varphi^{(\ell)}_{\g,j}(u_j^{i+}-
u_j^{i-})\chi_{Q_{i,\e_j}^{n-1}} -
\varphi^{(\ell)}(u^+-u^-)\Bigr|\,dx_\a\\
&=& R^{(\ell)} \int_\o \varphi^{(\ell)} (u^+ - u^-)\, dx_\a +
R^{(\ell)}\, {\mathcal H}^{n-1}(\o) \,\eta + o(1)\,,
\end{eqnarray}
as $\g\to 0^+$. Finally, for $i \not\in Z_j$, by the $p$-growth
condition (\ref{pgrowth}) and (\ref{gradubarZn}), we obtain
\begin{eqnarray*}
&&\frac{1}{\d_j} \Bigg(\int_{\left(\bigcup_{i\in
{\mathbb{Z}}^{n-1}\setminus Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o
\right)^{\pm\d_j}} W(D \bar
u_j)\, dx \Bigg)\\
&\leq &\sum_{i\in {\mathbb{Z}}^{n-1}\setminus Z_j} \frac{\beta}{\d_j}
\Bigg( \int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o)^{\pm\d_j}} (1
+|D \bar u_j|^p)\, dx \Bigg)\\
&\le&c\, {\mathcal H}^{n-1}\Bigl( \bigcup_{i\in {\mathbb{Z}}^{n-1}\setminus Z_j}
B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o\Bigr)\\
&&+ c\,\sum_{i\in {\mathbb{Z}}^{n-1}\setminus Z_j}\left({ r_j^{n-1-p}\over
\e_j^{n-1}} {\mathcal H}^{n-1}(Q_{i,\e_j}^{n-1}) + {1\over \d_j}
\int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o)^{+ \d_j}}
|Dw^+_j|^p\,dx \right.\\
&& \left. \hspace{6cm} + {1\over \d_j}
\int_{(B_{\rho_j}^{n-1}(x_i^{\e_j})\cap \o)^{- \d_j}}
|Dw^-_j|^p\,dx\right)\,.
\end{eqnarray*}
Since
$$
\lim_{j\to +\infty} {\mathcal H}^{n-1}\Bigl(\bigcup_{i\in
{\mathbb{Z}}^{n-1}\setminus Z_j} B_{\rho_j}^{n-1}(x_i^{\e_j})\cap
\o\Bigr)=0\,,
$$
by (\ref{rinfty}), the equi-integrability of $(|D
w_j^{\pm}|^p/\d_j)$ on $\o^{\pm\d_j}$ and (\ref{babadjian}), we
deduce
\begin{eqnarray}\label{step4}
\limsup_{j \to +\infty}\frac{1}{\d_j}\int_{\left(\o \cap
\bigcup_{i \in {\mathbb{Z}}^{n-1} \setminus
Z_j}B_{\rho_j}^{n-1}(x_i^{\e_j})\right)^{\pm\d_j}} W(D \bar u_j)\,
dx =0\,.
\end{eqnarray}
Gathering (\ref{step1})-(\ref{step4}) and passing to the limit as
$\g\to 0^+$ we get the limsup inequality
for every $u^\pm \in W^{1,p}(\o;{\mathbb{R}}^m) \cap L^\infty(\o;{\mathbb{R}}^m)$.\\
We remove the boundedness assumption simply noting that any
arbitrary $W^{1,p}(\o;{\mathbb{R}}^m)$ function can approximated by a
sequence of functions belonging to $W^{1,p}(\o;{\mathbb{R}}^m) \cap
L^\infty(\o;{\mathbb{R}}^m)$, with respect to the strong
$W^{1,p}(\o;{\mathbb{R}}^m)$-convergence. Then, by the lower semicontinuity
of the $\Gamma$-limsup and the continuity of
$$
(v^+,v^-) \mapsto \int_\o {\mathcal{Q}}_{n-1} \overline W(D_\a v^+)\, dx_\a +
\int_\o {\mathcal{Q}}_{n-1} \overline W(D_\a v^-)\, dx_\a + R^{(\ell)}
\int_\o \varphi^{(\ell)} (v^+ - v^-)\, dx_\a
$$
with respect to the strong $W^{1,p}(\o;{\mathbb{R}}^m)$-convergence we get
the thesis for $\ell\in (0,+\infty]$.
\bigskip
If $\ell=0$, we can follow the line of the previous case with
slight changes. Let us start by dealing with Step 1. First, we
have to notice that for the definition of $(\bar u_j)$ in
$B^{n-1}_{\rho_j}(x_i^{\e_j})^{\pm\d_j}$, for $i\in Z_j$, we have
to consider, for any $\eta>0$, a function $\zeta_{\g,j} \in
Y^\g_j(z)$ such that
$$
\int_{(B_{\g N_j}^{n-1} \times I_j) \setminus C_{1,\g N_j}} r_j^p
\, W\left(r_j^{-1} D \zeta_{\g,j} \right) dx \leq
\varphi^{(0)}_{\g,j}(z) +\eta\,;
$$
hence,
$$
\bar u_j(x_\a,x_n):=
\zeta_{\g,j}^i\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{r_j}\right)+u_j^{i-}\;\text{
in }\; B^{n-1}_{\rho_j}(x_i^{\e_j})^{\pm\d_j}\,, \quad \text{ for
}\, i\in Z_j\,.
$$
While for the definition of $(\bar u_j)$ in
$B^{n-1}_{\rho_j}(x_i^{\e_j})^{\pm\d_j}$, for $i\in
{\mathbb{Z}}^{n-1}\setminus Z_j$, we have to introduce a suitable function
$\psi_{\g,j}$ different from the one used in (\ref{defubarZn}). In
fact, for a fixed $\g>0$ and $j$ large enough we can always assume
that $\g N_j >2$ and $\d_j/r_j >2$. Let $\psi \in
W^{1,p}(B_2^{n-1} \times (0,2);[0,1])$ such that $\psi = 0$ on
$B_1^{n-1} \times \{0\}$ and $\psi=1$ on $\partial B_2^{n-1}
\times (0,2)$. We then define
$$
\psi_{\g,j}(x):=\left\{
\begin{array}{lll}
0 & \text{ in } & (B_{\g N_j}^{n-1})^{-(\d_j/r_j)},\\
\psi(x) & \text{ in } & (B_2^{n-1})^{+2},\\
1 & \text{ in } & (B_{\g N_j}^{n-1})^{+(\d_j/r_j)} \setminus
(B_2^{n-1})^{+2} .
\end{array}
\right.
$$
The functions $\psi_{\g,j}$ belong to $W^{1,p}((B^{n-1}_{\g N_j}
\times I_j) \setminus C_{1,\g N_j};[0,1])$ and satisfy
$\psi_{\g,j} = 1$ on $(\partial B_{\g N_j}^{n-1})^{+(\d_j/r_j)}$
and $\psi_{\g,j} = 0$ in $(B_{\g N_j}^{n-1})^{-(\d_j/r_j)}$.
Hence, we define
$$
\bar u_j:=\psi_{\g,j}
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{r_j}
\right)\tilde{w}_j^+ + \Bigg( 1- \psi_{\g,j}
\left(\frac{x_\a-x_i^{\e_j}}{r_j},\frac{x_n}{r_j} \right)
\Bigg)\tilde{w}_j^-
$$
in $ \big( B^{n-1}_{\rho_j}(x_i^{\e_j}) \times (-\d_j,\d_j)\big)
\cap \O_j$ and for $i\in {\mathbb{Z}}^{n-1}\setminus Z_j$. In particular,
we have that $\bar u_j=w_j$ on $\big( \partial
B^{n-1}_{\rho_j}(x_i^{\e_j}) \times (-\d_j,\d_j)\big) \cap \O_j$.
Taking into account the definition of $(\bar u_j)$ we can proceed
as in Steps 2 and 3 also for $\ell=0$.
\hfill$\Box$\\
\section{Representation formula for the interfacial energy
density}\label{nonabstract}
\noindent This section is devoted to describe explicitly the
interfacial energy density $\varphi^{(\ell)}$ for $\ell\in
[0,+\infty]$. As in \cite{Ans}, we expect to find a capacitary
type formula for each regime $\ell\in(0,+\infty)$, $\ell=+\infty$
and $\ell=0$.
We recall that $\varphi^{(\ell)}$ is the pointwise limit of the
sequence $(\varphi^{(\ell)}_{\g,j})$, as $j\to +\infty$ and $\g\to
0^+$ where for $\ell\in (0,+\infty]$
$$
\varphi^{(\ell)}_{\g,j}(z)=\inf \left\{ \int_{(B_{\g N_j}^{n-1}
\times I) \setminus C_{1,\g N_j}} r_j^p \, W\left(r_j^{-1}
\Bigl(D_\a \zeta \Big| {r_j\over\d_j} D_n \zeta \Bigr)\right) dx :
\quad \zeta \in X^\g_j(z)\right\},
$$
while for $\ell=0$,
$$
\varphi^{(0)}_{\g,j}(z)= \inf \left\{ \int_{(B_{\g N_j}^{n-1}
\times I_j ) \setminus C_{1,\g N_j} } r_j^p \, W(r_j^{-1} D
\zeta)\, dx : \quad \zeta \in Y^\g_j(z)\right\}
$$
(see Section \ref{close}). The main difficulty occurring in the
description of $\varphi^{(\ell)}$ is due to the fact that the
above minimum problems are stated on (increasingly) varying
domains. This do not permit, for example, to deal with a direct
$\Gamma$-convergence approach in order to apply the classical
result on the convergence of associated minimum problems. Thus the
proof of the representation formula will be performed in three
main steps: we first prove an auxiliary $\G$-convergence result
for a suitable sequence of energies stated on a fixed domain, then
we describe the functional space occurring in the limit capacitary
formula, finally, we prove that $\varphi^{(\ell)}$ is described by
a representation formula of capacitary-type.
\bigskip
We introduce some convenient notation for the sequel. Let
$g_j:{\mathbb{R}}^{m \times n} \to [0,+\infty)$ be the sequence of
functions given by
$$g_j(F):=r_j^p \, W(r_j^{-1}F)$$
for every $F\in {\mathbb{R}}^{m\times n}$. By (\ref{pgrowth}) and
(\ref{plip}) it follows that
\begin{equation}\label{pgrowthWj}
|F|^p - r_j^p \leq g_j(F) \leq \beta ( r_j^p + |F|^p), \quad
\text{for all }F \in {\mathbb{R}}^{m \times n}
\end{equation}
and the following $p$-Lipschitz condition holds:
$$|g_j(F_1)-g_j(F_2)| \leq c
(r_j^{p-1}+|F_1|^{p-1}+|F_2|^{p-1})|F_1-F_2|, \quad \text{ for all
}\, F_1,\, F_2 \in {\mathbb{R}}^{m \times n}.
$$
Then, according to Ascoli-Arzela's Theorem, up to subsequences,
$g_j$ converges locally uniformly in ${\mathbb{R}}^{m \times n}$ to a
function $g$ satisfying:
\begin{equation}\label{pgrowthg} |F|^p \leq g(F) \leq \beta |F|^p,
\quad \text{for all }\,F \in {\mathbb{R}}^{m \times n}\end{equation}and
\begin{equation}\label{plipg}
|g(F_1)-g(F_2)| \leq c (|F_1|^{p-1}+|F_2|^{p-1})|F_1-F_2|, \quad
\text{ for all }\, F_1,\, F_2\in {\mathbb{R}}^{m \times n}.
\end{equation}
\subsection{The case $\ell \in (0,+\infty)$}
\noindent We define
\begin{eqnarray*}
X_N(z) & := & \Big\{ \zeta \in W^{1,p}((B_N^{n-1} \times I)
\setminus C_{1,N};{\mathbb{R}}^m) :\quad \zeta =z \,\text{ on }\, (
\partial B_N^{n-1} )^+ \\&& \hspace{6cm}\text{ and }
\zeta =0 \,\text{ on }\, ( \partial B_N^{n-1})^- \Big\}
\end{eqnarray*}
for $N>1$ and $I=(-1,1)$. We recall the following
$\Gamma$-convergence result.
\begin{proposition}\label{lfinite} Let
$$
\ell=\lim_{j\to +\infty}{r_j\over\d_j} \in (0,+\infty)\,,
$$
then the sequence of functionals $G^{(\ell)}_j:L^p((B_N^{n-1}
\times I) \setminus C_{1,N};{\mathbb{R}}^m) \to [0,+\infty]$, defined by
$$
G_j^{(\ell)}(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{(B_N^{n-1}
\times I) \setminus C_{1,N}} g_j \left(D_\a
\zeta\Big|\frac{r_j}{\d_j} D_n \zeta \right)\, dx & \text{ if
}\zeta \in X_N(z)\\ &\\+\infty & \text{
otherwise\,,}\end{array}\right.
$$
$\G$-converges, with respect to the $L^p$-convergence, to
$$
G^{(\ell)}(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{(B_N^{n-1}
\times I) \setminus C_{1,N}}
g(D_\a \zeta | \ell D_n \zeta )\, dx & \text{ if }\zeta \in X_N(z)\\
&\\ +\infty & \text{ otherwise\,.}\end{array}\right.
$$
\end{proposition}
\noindent {\it Proof. } Since $\ell=\lim_{j\to +\infty} (r_j/\d_j)
\in (0,+\infty)$, by the locally uniform convergence of $g_j$ to
$g$ we have that the sequence of quasiconvex functions $F\mapsto
g_j(\overline F|(r_j/\d_j) F_n)$ pointwise converges to $F\mapsto
g(\overline F|\ell F_n)$. Hence the conclusion comes from
\cite{BD} Propositions 12.8 and 11.7. \hfill$\Box$
\begin{rmk}\label{bochner}{\rm
We denote by $p^*$ the Sobolev exponent in dimension $(n-1)$ i.e.
$$p^*:=\frac{(n-1)p}{n-1-p}.$$ We recall that if $(a,b) \subset {\mathbb{R}}$,
the space $L^p(a,b;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$ is a reflexive and
separable Banach space (see e.g. \cite{Adams} or \cite{Y}). Hence,
by the Banach-Alaoglu-Bourbaki Theorem, any bounded sequence
admits a weakly converging subsequence. }
\end{rmk}
\begin{proposition}[Limit space]\label{bla1}
Let
\begin{equation}\label{assl}
\ell=\lim_{j\to +\infty} {r_j\over \d_j}\in (0, +\infty)\,,\qquad
0<R^{(\ell)}=\lim_{j \to +\infty} \frac{r_j^{n-1-p}}{\e_j^{n-1}} <
+\infty
\end{equation}
and let $(\zeta_{\g,j}) \in X^\g_j(z)$ such that, for every fixed
$\g>0$,
\begin{equation}\label{w1ploc}
\sup_{j \in {\mathbb{N}}}\int_{(B_{\g N_j}^{n-1} \times I) \setminus
C_{1,\g N_j}} g_j\Bigl(D_\a\zeta_{\g,j}\Big| {r_j\over \d_j}D_n
\zeta_{\g,j}\Bigr)\, dx \le c\,.
\end{equation}
Then, there exists a sequence $\tilde\zeta_j \in
W^{1,p}_{{\rm loc}}(({\mathbb{R}}^{n-1}\times I)\setminus C_{1,\infty}; {\mathbb{R}}^m)$
such that
$$
\tilde\zeta_j = \zeta_{\g,j} \quad \text{ in } \quad (B^{n-1}_{\g
N_j}\times I)\setminus C_{1, \g N_j}
$$
and such that, up to subsequences, it converges weakly to $\zeta$
in $W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1}\times I)\setminus
C_{1,\infty};{\mathbb{R}}^m)$. Moreover, the function $\zeta$ satisfies the
following properties
\begin{equation}\label{adm}
\left\{
\begin{array}{l}
D \zeta \in L^p(({\mathbb{R}}^{n-1}\times I)\setminus C_{1,\infty};{\mathbb{R}}^{m
\times
n}),\\
\\
\zeta-z \in L^{p}(0, 1;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)),\\
\\
\zeta \in L^{p}(-1,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\,.
\end{array}
\right.
\end{equation}
\end{proposition}
\noindent {\it Proof. } By (\ref{pgrowthWj}), (\ref{assl}) and
(\ref{w1ploc}) we deduce that, for every fixed $\g>0$,
\begin{equation}\label{boundgradl}
\sup_{j \in {\mathbb{N}}}\int_{(B_{\g N_j}^{n-1} \times I) \setminus
C_{1,\g N_j}} \Bigl|\Bigl(D_\a\zeta_{\g,j}\Big| {r_j\over \d_j}D_n
\zeta_{\g,j}\Bigr)\Bigr|^p\, dx \le c\,.
\end{equation}
We define
$$
\tilde \zeta_j:= \left\{ {\begin{array}{lll} z & \text{in} &
\big(\mathbb{R}^{n - 1} \setminus B_{\g N_j }^{n - 1} \big)^+,\\
\\
\zeta_{\g,j} & \text{in} & (B_{\g N_j }^{n - 1} \times I)
\setminus C_{1,\g N_j},\\
\\
0 & \text{in} & \big(\mathbb{R}^{n - 1} \setminus B_{\gamma N_j
}^{n - 1} \big)^-\,;
\\
\end{array} } \right.
$$
hence,
$$
\tilde \zeta_j(\cdot,x_n)-z \in W^{1,p}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)\quad
\text{for \, a.e.\; } x_n\in(0,1)
$$
and
$$
\tilde \zeta_j(\cdot,x_n) \in W^{1,p}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)\quad
\text{for \, a.e.\; } x_n\in(-1,0)\,.
$$
Moreover by (\ref{boundgradl}) we get that
\begin{equation}\label{stima}
\int_{({\mathbb{R}}^{n-1} \times I) \setminus C_{1,\infty}}
\Bigl|\Bigl(D_\a\tilde\zeta_{j}\Big| {r_j\over \d_j}D_n
\tilde\zeta_{j}\Bigr)\Bigr|^p \, dx = \int_{(B_{\g N_j}^{n-1}
\times
I ) \setminus C_{1,\g N_j}} \Bigl|\Bigl(D_\a\zeta_{\g,j}\Big|
{r_j\over \d_j}D_n \zeta_{\g,j}\Bigr)\Bigr|^p \, dx \leq c\,.
\end{equation}
Since $p<n-1$, according to the Sobolev Inequality (see e.g.
\cite{Adams}), there exists a constant $c=c(n,p)>0$ (independent
of $x_n$) such that
\begin{equation}\label{p*1}
\left(\int_{{\mathbb{R}}^{n-1}}|\tilde \zeta_j(x_\a,x_n)-z|^{p^*}\,
dx_\a\right)^{p/p^*} \leq c \,\int_{{\mathbb{R}}^{n-1}} |D_\a \tilde
\zeta_j(x_\a,x_n)|^p\, dx_\a
\end{equation}
for a.e. $x_n \in (0,1)$, and
\begin{equation}\label{p*2}
\left(\int_{{\mathbb{R}}^{n-1}}|\tilde \zeta_j(x_\a,x_n)|^{p^*}\,
dx_\a\right)^{p/p^*} \leq c\, \int_{{\mathbb{R}}^{n-1}} |D_\a \tilde
\zeta_j(x_\a,x_n)|^p\, dx_\a
\end{equation}
for a.e. $x_n \in (-1,0)$. If we integrate (\ref{p*1}) and
(\ref{p*2}) with respect to $x_n$, by (\ref{stima}) and Remark
\ref{bochner}, we get that there exist $\zeta_1\in
L^p(0,1;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$ and $\zeta_2\in
L^p(-1,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$ such that, up to subsequences,
$$
\left\{
\begin{array}{lll}
\tilde \zeta_j-z\rightharpoonup \zeta_1 & \text{in} &
L^p(0,1;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)),
\\
\tilde \zeta_j\rightharpoonup \zeta_2 & \text{in} &
L^p(-1,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)),\\
\\
D \tilde \zeta_j \rightharpoonup D \zeta_1 & \text{in} &
L^p(({\mathbb{R}}^{n-1})^+;{\mathbb{R}}^{m \times n}),\\
D \tilde \zeta_j \rightharpoonup D \zeta_2 & \text{in} &
L^p(({\mathbb{R}}^{n-1})^-;{\mathbb{R}}^{m \times n}).
\end{array}
\right.$$ In particular, we have that
$$
\left\{
\begin{array}{lll}
\tilde\zeta_j \rightharpoonup \zeta_1+z & \text{in} &
W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1})^+;{\mathbb{R}}^m),\\
\\
\tilde \zeta_j\rightharpoonup\zeta_2 & \text{in} & W^{1,p}_{\rm
loc}(({\mathbb{R}}^{n-1})^-;{\mathbb{R}}^m)\,.
\end{array}
\right.$$ Then, since $\zeta_1+z=\zeta_2$ on $B^{n-1}_1$ in the
sense of traces, we can define
$$
\zeta:= \left\{ {\begin{array}{lll}
{\zeta_1 + z } & \text{in} & ({\mathbb{R}}^{n - 1})^+ \\
{\zeta_2 } & \text{in} & ({\mathbb{R}}^{n - 1})^-
\cup \big( B^{n-1}_1 \times \{0\} \big), \\
\end{array} } \right.
$$
and it satisfies (\ref{adm}).\hfill$\Box$
\bigskip
Now we are able to describe the interfacial energy density
$\varphi^{(\ell)}$ as the following nonlinear capacitary formula.
\begin{proposition}[Representation formula]\label{reprell}
We have
\begin{eqnarray*}
\varphi^{(\ell)}(z) &= & \inf \Bigg\{\int_{({\mathbb{R}}^{n-1} \times I)
\setminus C_{1,\infty}}g \big(D_\a \zeta | \ell D_n \zeta \big)\,
dx :\zeta \in W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1} \times I) \setminus
C_{1,\infty};{\mathbb{R}}^m),\\
&&\hspace{0.6cm} D \zeta \in L^p(({\mathbb{R}}^{n-1} \times I)\setminus
C_{1,\infty};{\mathbb{R}}^{m \times n}),\; \zeta - z \in
L^{p}(0,1;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\\
&&\hspace{5.8cm} \text{ and } \zeta \in
L^{p}(-1,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\Bigg\}
\end{eqnarray*}
for every $z\in {\mathbb{R}}^m$.
\end{proposition}
\noindent {\it Proof. } We define
\begin{eqnarray*}
\psi^{(\ell)}(z) &:= & \inf \Bigg\{\int_{({\mathbb{R}}^{n-1} \times I)
\setminus C_{1,\infty}}g \big(D_\a \zeta | \ell D_n \zeta \big)\,
dx :\zeta \in W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1} \times I) \setminus
C_{1,\infty};{\mathbb{R}}^m),\\
&&\hspace{0.6cm} D \zeta \in L^p(({\mathbb{R}}^{n-1} \times I)\setminus
C_{1,\infty};{\mathbb{R}}^{m \times n}),\; \zeta - z \in
L^{p}(0,1;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\\
&&\hspace{5.8cm} \text{ and } \zeta \in
L^{p}(-1,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\Bigg\},
\end{eqnarray*}
we want to prove that $\varphi^{(\ell)}(z)= \psi^{(\ell)}(z)$ for
every $z\in {\mathbb{R}}^m$. For every fixed $\eta>0$, by definition of
$\varphi_{\g,j}^{(\ell)}(z)$ (see (\ref{phigj})), there exists
$\zeta_{\g,j} \in X^\g_j(z)$ such that
$$
\int_{(B_{\g N_j}^{n-1} \times I) \setminus C_{1,\g N_j}}g_j\left(
D_\a \zeta_{\g,j} \Big| \frac{r_j}{\d_j} D_n \zeta_{\g,j} \right)
dx \leq \varphi_{\g,j}^{(\ell)}(z) + \eta.
$$
By Proposition \ref{midi}(i) we have that (\ref{w1ploc}) is
fulfilled, then by Propositions \ref{bla1} and \ref{lfinite} we
get
\begin{eqnarray*}
\lim_{j \to +\infty} \varphi_{\g,j}^{(\ell)}(z) +\eta & \geq &
\liminf_{j \to +\infty} \int_{(B_{\g N_j}^{n-1} \times I)
\setminus C_{1,\g N_j}} g_j\left(D_\a \tilde\zeta_j
\Big| \frac{r_j}{\d_j} D_n\tilde\zeta_j \right) dx\nonumber\\
& \geq & \liminf_{j \to +\infty} \int_{(B_N^{n-1} \times
I)\setminus C_{1,N}}g_j\left( D_\a \tilde\zeta_j \Big|
\frac{r_j}{\d_j} D_n \tilde\zeta_j
\right) dx\\
& \geq & \int_{(B_N^{n-1} \times I)\setminus C_{1,N}}g( D_\a \zeta
| \ell D_n \zeta)\, dx
\end{eqnarray*}
with $\zeta \in W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1} \times I) \setminus
C_{1,\infty};{\mathbb{R}}^m)$ satisfying (\ref{adm}). Note that for every
fixed $\g>0$ and $j$ large enough we can always assume that $\g
N_j>N$ for some fixed $N>2$. Hence, passing to the limit as $N
\to+\infty$ and $\g \to 0^+$, we obtain
\begin{equation}\label{ineq1}
\varphi^{(\ell)}(z) + \eta \geq \int_{({\mathbb{R}}^{n-1} \times I)
\setminus C_{1,\infty}}g (D_\a \zeta | \ell D_n \zeta )\, dx
\geq\psi^{(\ell)}(z)
\end{equation}
and by the arbitrariness of $\eta$ we get the first inequality.
We now prove the converse inequality. For every fixed $\eta>0$
there exists $\zeta \in W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1} \times I)
\setminus C_{1,\infty};{\mathbb{R}}^m)$ satisfying (\ref{adm}) such that
\begin{equation}\label{almostmin}
\int_{({\mathbb{R}}^{n-1} \times I) \setminus C_{1,\infty}} g(D_\a \zeta |
\ell D_n\zeta)\, dx \leq \psi^{(\ell)}(z) + \eta\,.
\end{equation}
Let $N>2$, for every fixed $\g>0$ and $j$ large enough we have
that $\g N_j>N$. We consider a cut-off function $\theta_N \in
{\mathcal{C}}^\infty_c(B^{n-1}_N;[0,1])$ such that $\theta_N=1$ in
$B^{n-1}_{N/2}$, $|D_\a\theta_N| \leq c/N$ and we define
$$
\zeta_N:= \left\{ \begin{array}{lll} \theta _N (x_\a )\zeta + (1 -
\theta _N (x_\a))z &
\text{in} & (B^{n-1}_N)^{+},\\
&&\\
\theta _N (x_\alpha )\zeta & \text{in} & (B^{n-1}_N)^{-}\cup
(B^{n-1}_1 \times \{0\})
\end{array} \right.
$$
so that $\zeta_N \in X_N(z)$. By Proposition \ref{lfinite}, there
exists a sequence ($\zeta_j^N) \subset X_N(z)$ strongly converging
to $\zeta_N$ in $L^p((B^{n-1}_N \times I) \setminus
C_{1,N};{\mathbb{R}}^m)$ such that
\begin{equation}\label{contrex}
\int_{(B^{n-1}_N \times I) \setminus C_{1,N}} g ( D_\a \zeta_N |
\ell D_n \zeta_N)\, dx = \lim_{j \to +\infty} \int_{(B^{n-1}_N
\times I) \setminus C_{1,N}} g_j \left( D_\a \zeta_j^N \Big|
\frac{r_j}{\d_j} D_n \zeta_j^N \right)\, dx
\end{equation}
Let us define $\zeta_{\g,j} \in X_j^\g (z)$ as
$$
\zeta_{\g,j}:= \left\{ \begin{array}{lll}
z & \text{in} & (B^{n-1}_{\g N_j} \setminus B^{n-1}_N)^+,\\
&&\\
\zeta_j^N &
\text{in} & (B^{n-1}_N \times I) \setminus C_{1,N},\\
&&\\
0 & \text{in} & (B^{n-1}_{\g N_j} \setminus B^{n-1}_N)^-.
\end{array} \right.
$$
Consequently, $\zeta_{\g,j}$ is an admissible test function for
(\ref{phigj}) and since $g_j(0)=0$ we get that
\begin{eqnarray*}
\varphi_{\g,j}^{(\ell)}(z) & \leq & \int_{(B^{n-1}_{\g N_j} \times
I) \setminus C_{1,\g N_j}} g_j\Bigl( D_\a \zeta_{\g,j} \Big|
\frac{r_j}{\d_j} D_n \zeta_{\g,j} \Bigr)\, dx\\
& = & \int_{(B^{n-1}_N \times I) \setminus C_{1,N}} g_j\Bigl(D_\a
\zeta_N^j \Big| \frac{r_j}{\d_j} D_n \zeta_N^j \Bigr)\, dx.
\end{eqnarray*}
Passing to the limit as $j \to +\infty$, using (\ref{contrex}) and
the $p$-growth condition (\ref{pgrowthg}) satisfied by $g$, we
obtain
\begin{eqnarray}\label{est1}
\lim_{j \to +\infty} \varphi_{\g,j}^{(\ell)}(z) & \leq &
\int_{(B^{n-1}_N \times I) \setminus C_{1,N}} g (D_\a \zeta_N |
\ell
D_n \zeta_N )\, dx \nonumber\\
&\le& \int_{(B^{n-1}_{N/2} \times I) \setminus C_{1,N/2}} g(D_\a
\zeta | \ell D_n \zeta )\, dx +c \int_{(B^{n-1}_N \setminus
B^{n-1}
_{N/2})^+} |D \zeta_N |^p\, dx \nonumber\\
&& \hspace{1cm} +c \int_{(B^{n-1}_N \setminus B^{n-1}_{N/2})^-} |D
\zeta_N |^p\,dx \,.
\end{eqnarray} Let us examine the contribution of the gradient in
(\ref{est1}),
\begin{eqnarray}\label{est2}
&&\nonumber\int_{(B^{n-1}_N \setminus B^{n-1}_{N/2})^+}|D \zeta_N
|^p\,dx
+ \int_{(B^{n-1}_N \setminus B^{n-1}_{N/2})^-} |D \zeta_N |^p\,dx\\
&&\hspace{2cm}\le\nonumber c \int_{(B^{n-1}_N \setminus
B^{n-1}_{N/2})^+}(|D_\a \theta_N|^p |\zeta - z|^p + |D \zeta|^p
)\,dx\\
&&\hspace{3cm} \nonumber+ c \int_{(B^{n-1}_N \setminus
B^{n-1}_{N/2})^-}(|D_\a \theta_N|^p |\zeta |^p
+ |D \zeta|^p )\, dx\\
&&\hspace{2cm}\le \nonumber c\,\left( \int_{({\mathbb{R}}^{n-1}\setminus
B^{n-1}_{N/2})^+} |D \zeta|^p \,dx +
\int_{({\mathbb{R}}^{n-1}\setminus B^{n-1}_{N/2})^-} |D \zeta|^p \,dx \right)\\
&&\hspace{3cm}+ \frac{c}{N^p}\left(\int_{(B^{n-1}_N \setminus
B^{n-1}_{N/2})^+} |\zeta-z|^p\,dx + \int_{(B^{n-1}_N \setminus
B^{n-1}_{N/2})^-} |\zeta|^p\,dx\right)\,.
\end{eqnarray} Since $p^*>p$ we can apply H\"older Inequality
with $q=p^*/p$ obtaining
\begin{eqnarray}\label{est3}
&&\hspace{-1cm}\nonumber \frac{c}{N^p}\left(\int_{(B^{n-1}_N
\setminus B^{n-1}_{N/2})^+} |\zeta-z|^p\,dx + \int_{(B^{n-1}_N
\setminus B^{n-1}_{N/2})^-}
|\zeta|^p\right)\\
&&\hspace{1cm}\le \nonumber c\, \int_0^1\Bigl(\int_{B^{n-1}_N
\setminus B^{n-1}_{N/2}}
|\zeta-z|^{p^*}\,dx_\a\Bigr)^{p/p^*}\,dx_n\\
&&\hspace{5cm} \nonumber + c\, \int_{-1}^0\Bigl( \int_{B^{n-1}_N
\setminus
B^{n-1}_{N/2}} |\zeta|^{p^*}\,dx_\a\Bigr)^{p/p^*}\,dx_n\\
&& \hspace{1cm}\le\nonumber \nonumber c\,
\int_0^1\Bigl(\int_{{\mathbb{R}}^{n-1} \setminus B^{n-1}_{N/2}}
|\zeta-z|^{p^*}\,dx_\a\Bigr)^{p/p^*}\,dx_n\\
&& \hspace{5cm} + c\, \int_{-1}^0\Bigl( \int_{{\mathbb{R}}^{n-1} \setminus
B^{n-1}_{N/2}} |\zeta|^{p^*}\,dx_\a\Bigr)^{p/p^*}\,dx_n.
\end{eqnarray} Hence by (\ref{adm}), (\ref{est2}) and (\ref{est3})
we have that, for every fixed $\g>0$,
$$
\lim_{N \to +\infty}\int_{(B^{n-1}_N \setminus B^{n-1}_{N/2})^\pm}
|D \zeta_N|^p\,dx=0
$$
which thanks to (\ref{almostmin}) and (\ref{est1}) implies that
$$
\lim_{j\to +\infty} \varphi_{\g,j}^{(\ell)}(z) \le
\psi^{(\ell)}(z) + \eta.
$$
Then we get the converse inequality by letting $\g \to 0^+$ and by
the arbitrariness of $\eta$.\hfill$\Box$
\subsection{The case $\ell=+\infty$}
\noindent In this case the study leading to the representation
formula for $\varphi^{(\infty)}$ involves a dimensional reduction
problem stated on a varying domain. As before, we start proving
some $\Gamma$-convergence results (see Propositions \ref{separate}
and \ref{linfinite}) for suitable sequences of functionals stated
on fixed domains. This will allow as to apply some well-known
$\Gamma$-convergence and integral representation theorems due to
Le Dret-Raoult \cite{LDR} and Braides-Fonseca-Francfort \cite{BFF}
respectively.
\smallskip
Let $G_j^\pm :L^p((B_N^{n-1})^\pm;{\mathbb{R}}^m) \to [0,+\infty]$ be
defined by
$$
G_j^+(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{(B_N^{n-1})^+}
g_j\left(D_\a \zeta\Big|\frac{r_j}{\d_j} D_n \zeta \right)\, dx &
\text{ if } \left\{\begin{array}{l}\zeta \in
W^{1,p}((B_N^{n-1})^+;{\mathbb{R}}^m)\\
\zeta=z \text{ on } (\partial B_N^{n-1})^+\end{array}\right.\\&\\
+\infty & \text{ otherwise}\end{array}\right.
$$
and
$$
G_j^-(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{(B_N^{n-1})^-}
g_j\left(D_\a \zeta\Big|\frac{r_j}{\d_j} D_n \zeta \right)\, dx &
\text{ if }\left\{\begin{array}{l}\zeta \in
W^{1,p}((B_N^{n-1})^-;{\mathbb{R}}^m)\\
\zeta=0 \text{ on } (\partial
B_N^{n-1})^-\end{array}\right.\\&\\+\infty & \text{
otherwise}.\end{array}\right.
$$
\begin{proposition}\label{separate}
Let
$$
\ell=\lim_{j\to +\infty}{r_j\over \d_j}=+\infty\,,
$$
then, the sequences of functionals $(G_j^\pm$) $\G$-converge, with
respect to the $L^p$-convergence, to
$$G^+(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{B_N^{n-1}} {\mathcal{Q}}_{n-1}\,
\overline g (D_\a \zeta)\, dx_\a&\text{ if } \zeta-z \in
W^{1,p}_0(B_N^{n-1};{\mathbb{R}}^m)\\&
\\ +\infty & \text{ otherwise}\end{array}\right.
$$
and
$$
G^-(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{B_N^{n-1}}
{\mathcal{Q}}_{n-1}\, \overline g (D_\a \zeta)\, dx_\a&\text{ if }\zeta \in
W^{1,p}_0(B_N^{n-1};{\mathbb{R}}^m)\\
&\\+\infty & \text{ otherwise}\,,\end{array}\right.
$$
respectively, where $\overline g(\overline F) = \inf \{
g(\overline F|F_n): \; F_n \in {\mathbb{R}}^m \}$ for every $\overline F
\in {\mathbb{R}}^{m\times (n-1)}$.
\end{proposition}
\noindent {\it Proof. }We prove the $\Gamma$-convergence result
only for ($G_j^+$), the other one being analogous. According to
\cite{BFF} Theorem 2.5 and Lemma 2.6 there exists a continuous
function $\hat g :{\mathbb{R}}^{m \times (n-1)} \to [0,+\infty)$ such that,
up to subsequence, ($G_j^+$) $\G$-converges to
$$
G^+(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{B_N^{n-1}} \hat g
(D_\a \zeta)\, dx_\a &\text{ if } \zeta-z \in
W^{1,p}_0(B_N^{n-1};{\mathbb{R}}^m)\\&\\ +\infty & \text{otherwise}\,.
\end{array}\right.
$$
Hence, it remains to show that $\hat g={\mathcal{Q}}_{n-1}\, \overline g$. By
\cite{BFF} Lemma 2.6, it is enough to consider
$W^{1,p}$-functions without boundary condition; hence, it will
suffice to deal with affine functions. Let $\zeta (x_\a):=
\overline F \cdot x_\a$, by \cite{BFF} Theorem 2.5, there exists a
sequence $(\zeta_j) \subset W^{1,p}((B_N^{n-1})^+;{\mathbb{R}}^m)$ (the
so-called recovery sequence) converging to $\zeta$ in
$L^p((B_N^{n-1})^+;{\mathbb{R}}^m)$, such that
\begin{equation}\label{1239}
\hat g(\overline F)\, c_N =G^+(\zeta)= \lim_{j \to +\infty}
\int_{(B_N^{n-1})^+} g_j\left(D_\a \zeta_j \Big| \frac{r_j}{\d_j}
D_n \zeta_j\right)\, dx
\end{equation}
where $c_N= {\mathcal H}^{n-1}(B_N^{n-1})$. Moreover, by \cite{Bo&Fo}
Theorem 1.1, we can assume, without loss of generality, that the
sequence $\big(\big|\big(D_\a\zeta_j | \frac{r_j}{\d_j} D_n
\zeta_j \big)\big|^p \big)$ is equi-integrable. By (\ref{1239})
and (\ref{pgrowthWj}), we have that
$$
\sup_{j \in {\mathbb{N}}} \int_{(B_N^{n-1})^+} \Bigl|\Bigl(D_\a \zeta_j
\Big| \frac{r_j}{\d_j} D_n \zeta_j\Bigr)\Bigr|^p\,dx \le c\,;
$$
hence, for every fixed $M>0$, if we define
$$
A_j^M:=\left\{x \in (B_N^{n-1})^+ : \quad \left| \left(D_\a
\zeta_j(x) \Big|\frac{r_j}{\d_j} D_n \zeta_j(x) \right)
\right|\leq M \right\}\,,
$$
we get that $\mathcal L^n((B_N^{n-1})^+ \setminus A_j^M) \leq
c/M^p$ for some constant $c>0$ independent of $j$ and $M$. Fix
$M>0$, by (\ref{1239}), we have
\begin{equation}\label{1325}
\hat g(\overline F)\, c_N \geq \limsup_{j \to +\infty}
\int_{A_j^M} g_j\left(D_\a\zeta_j \Big| \frac{r_j}{\d_j} D_n
\zeta_j \right)\, dx.
\end{equation}
Moreover, for all $x \in A_j^M$,
$$
\left| g_j\left(D_\a \zeta_j(x) \Big| \frac{r_j}{\d_j}
D_n\zeta_j(x) \right) - g\left(D_\a \zeta_j(x)
\Big|\frac{r_j}{\d_j} D_n \zeta_j(x) \right) \right| \leq
\sup_{|F|\leq M} |g_j(F)-g(F)|,
$$
and then,
\begin{eqnarray*}
&&\int_{A_j^M} \left| g_j\left(D_\a\zeta_j \Big| \frac{r_j}{\d_j}
D_n \zeta_j \right) - g\left(D_\a \zeta_j \Big| \frac{r_j}{\d_j}
D_n\zeta_j \right)
\right|dx\nonumber\\
&\leq& c_N\, \sup_{|F| \leq M}|g_j(F)-g(F)|.
\end{eqnarray*}
Hence, by the local uniform convergence of $g_j$ to $g$, we have
that
$$
\lim_{j\to +\infty}\int_{A_j^M}\left(g_j\left(D_\a\zeta_j \Big|
\frac{r_j}{\d_j} D_n \zeta_j \right) - g\left(D_\a \zeta_j \Big|
\frac{r_j}{\d_j} D_n\zeta_j \right)\right)\,dx=0.
$$
By (\ref{1325}), we get
\begin{equation}\label{1330}
\hat g(\overline F) \, c_N \geq \limsup_{j \to +\infty}
\int_{A_j^M} g\left(D_\a\zeta_j \Big| \frac{r_j}{\d_j} D_n \zeta_j
\right)\, dx.
\end{equation}
Note that, since $\mathcal L^n((B_N^{n-1})^+ \setminus A_j^M) \to
0$ as $M \to +\infty$, by the $p$-growth condition
(\ref{pgrowthg}) and the equi-integrability assumption, we find
\begin{equation}\label{1331}
\limsup_{j \to +\infty} \int_{(B_N^{n-1})^+ \setminus A_j^M}
g\left(D_\a \zeta_j \Big|\frac{r_j}{\d_j} D_n \zeta_j \right)\,
dx= o(1)\,, \qquad \text{as}\quad M\to +\infty\,.
\end{equation}
Consequently, (\ref{1330}) and (\ref{1331}) imply that
\begin{equation}\label{1332}
\hat g(\overline F)\, c_N \geq \limsup_{j \to +\infty}
\int_{(B_N^{n-1})^+} g\left(D_\a \zeta_j \Big|\frac{r_j}{\d_j} D_n
\zeta_j \right)\, dx.
\end{equation}
Finally, from \cite{LDR} Theorem 2, we know that
$$
\liminf_{j \to +\infty} \int_{(B_N^{n-1})^+} g\left(D_\a \zeta_j
\Big|\frac{r_j}{\d_j} D_n \zeta_j \right)\, dx \ge
{\mathcal{Q}}_{n-1}\,\overline g(\overline F)\, c_N\,;
$$
hence, by (\ref{1332}) we obtain that $\hat g(\overline F) \geq
{\mathcal{Q}}_{n-1}\,\overline g(\overline F)$.
We now prove the converse inequality. By \cite{LDR} Theorem 2,
there exists a sequence $(\zeta_j)$ belonging to
$W^{1,p}((B_N^{n-1})^+;{\mathbb{R}}^m)$ and converging to $\zeta$ in
$L^p((B_N^{n-1})^+;{\mathbb{R}}^m)$ such that
\begin{equation}\label{cat}
{\mathcal{Q}}_{n-1} \overline g(\overline F)\, c_N =\lim_{j \to +\infty}
\int_{(B_N^{n-1})^+} g\left(D_\a \zeta_j \Big| \frac{r_j}{\d_j}
D_n \zeta_j\right)\, dx\,.
\end{equation}
Without loss of generality, we can still assume that the sequence
$\big(\big|\big(D_\a \zeta_j | \frac{r_j}{\d_j}D_n \zeta_j
\big)\big|^p\big)$ is equi-integrable. Thus arguing as above, from
(\ref{cat}) we deduce
\begin{equation}\label{1333}
{\mathcal{Q}}_{n-1}\, \overline g(\overline F)\, c_N \geq \limsup _{j \to
+\infty} \int_{(B_N^{n-1})^+} g_j\left(D_\a \zeta_j \Big|
\frac{r_j}{\d_j} D_n\zeta_j \right)\, dx\,.
\end{equation}
Now, by \cite{BFF} Theorem 2.5, we have that
$$
\liminf _{j \to +\infty} \int_{(B_N^{n-1})^+} g_j\left(D_\a
\zeta_j \Big| \frac{r_j}{\d_j} D_n\zeta_j \right)\, dx \ge \hat
g(\overline F)\, c_N\,;
$$
hence,
${\mathcal{Q}}_{n-1}\,
\overline g(\overline F) \geq \hat g(\overline F)$, which
concludes the proof. \hfill$\Box$
\begin{rmk}\label{bcxn=0}
{\rm By \cite{LDR} Theorem 2, for every $\zeta \in
W^{1,p}(B_N^{n-1};{\mathbb{R}}^m)$ the recovery sequence is given by
$\zeta_j(x_\a,x_n):=\zeta(x_\a) + (\d_j/r_j)\, x_n\, b_j(x_\a)$
for a suitable sequence of functions $(b_j) \subset
{\mathcal{C}}^\infty_c(B_N^{n-1};{\mathbb{R}}^m)$. Note that by definition $(\zeta_j)$
keeps the boundary conditions of $\zeta$. Reasoning as in the
proof of Proposition \ref{separate} we can observed that
$(\zeta_j)$ is also a recovery sequence for $(G_j^+)$ (see e.g.
(\ref{1333})). The same remark holds for $(G_j^-)$.}
\end{rmk}
\bigskip
\begin{proposition}\label{linfinite}
Let
$$
\ell= \lim_{j\to +\infty} {r_j\over\d_j}= +\infty\,,
$$
then the sequence of functionals $G^{(\infty)}_j:L^p((B_N^{n-1}
\times I) \setminus C_{1,N};{\mathbb{R}}^m) \to[0,+\infty]$ defined by
$$
G^{(\infty)}_j(\zeta):=\left\{\begin{array}{ll}\displaystyle
\int_{(B_N^{n-1} \times I) \setminus C_{1,N}} g_j \left(D_\a
\zeta\Big|\frac{r_j}{\d_j} D_n \zeta \right)\, dx & \text{ if
}\zeta \in X_N(z)\\ & \\+\infty & \text{
otherwise}\end{array}\right.
$$
$\G$-converges, with respect to the $L^p$-convergence, to
$$
G^{(\infty)}(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{(B_N^{n-1}
\times I) \setminus C_{1,N}} {\mathcal{Q}}_{n-1}\, \overline g(D_\a \zeta)\,
dx & \text{ if }\zeta \in X_N(z)\text{ and
}D_n \zeta=0\\\\
+\infty & \text{ otherwise}\,.\end{array}\right.
$$
\end{proposition}
\noindent {\it Proof. } The $\liminf$ inequality is a
straightforward consequence of Proposition \ref{separate}.
Dealing with the $\limsup$ inequality, let us consider $\zeta \in
X_N(z)$ with $D_n \zeta=0$. We denote by $\zeta^\pm \in
W^{1,p}(B_N^{n-1}(0);{\mathbb{R}}^m)$ the restriction of $\zeta$ to
$(B_N^{n-1})^+$ and $(B_N^{n-1})^-$, respectively. By Proposition
\ref{separate} and Remark \ref{bcxn=0}, there exist two sequences
$(\zeta_j^\pm) \subset W^{1,p} ((B_N^{n-1})^\pm;{\mathbb{R}}^m)$ such that
\begin{equation}\label{2128}
\begin{array}{l}\zeta_j^+ \to \zeta^+ \text{ in
}L^p((B_N^{n-1})^+;{\mathbb{R}}^m)\,,
\quad \zeta_j^+=z \text{ on } ( \partial B_N^{n-1})^+\\\\
\zeta_j^-\to \zeta^- \text{ in }L^p((B_N^{n-1})^-;{\mathbb{R}}^m)\,, \quad
\zeta_j^-=0 \text{ on } (
\partial B_N^{n-1})^-
\end{array}
\end{equation}
and
\begin{eqnarray}\label{2129}
\nonumber \lim_{j \to +\infty} \int_{(B_N^{n-1})^+} g_j\left(D_\a
\zeta^+_j \Big| \frac{r_j}{\d_j} D_n\zeta^+_j \right)\, dx&=&
\int_{B_N^{n-1}} {\mathcal{Q}}_{n-1}\, \overline g(D_\a \zeta^+)\,
dx_\a\\
\lim_{j \to +\infty} \int_{(B_N^{n-1})^-} g_j\left(D_\a \zeta^-_j
\Big| \frac{r_j}{\d_j} D_n\zeta^-_j \right)\, dx&=&
\int_{B_N^{n-1}} {\mathcal{Q}}_{n-1}\, \overline g(D_\a \zeta^-)\, dx_\a\,.
\end{eqnarray}
Moreover, since $\zeta \in W^{1,p}((B^{n-1}_N \times I)\setminus
C_{1,N};{\mathbb{R}}^m)$, by Remark \ref{bcxn=0}, $(\zeta_j^+)$ and
$(\zeta_j^-)$ have the same trace on $B^{n-1}_1 \times \{0\}$;
hence, $\zeta_j^+=\zeta^-_j=\zeta$ on $B^{n-1}_1 \times \{0\}$.
Then we can define
$$
\bar \zeta_j:= \left\{
\begin{array}{lll}
\zeta_j^+ & \text{in} & (B_N^{n - 1})^+,\\
\zeta & \text{on} & B_1^{n-1} \times \{0\},\\
\zeta_j^- & \text{in} & (B_N^{n - 1})^-, \\
\end{array} \right.
$$
with $\bar \zeta_j \in W^{1,p}((B^{n-1}_N \times I)\setminus
C_{1,N};{\mathbb{R}}^m)$. In particular, by (\ref{2128}) we have that $\bar
\zeta_j \in X_N(z)$ and $\bar\zeta_j \to \zeta$ in
$L^p((B_N^{n-1} \times I) \setminus C_{1,N};{\mathbb{R}}^m)$. Finally, by
(\ref{2129}) , we have
\begin{eqnarray*}
\lim_{j \to +\infty} G^{(\infty)}_j(\bar \zeta_j) & = & \lim_{j
\to+\infty}\int_{(B_N^{n-1} \times I) \setminus C_{1,N}}
g_j\left(D_\a\bar\zeta_j\Big| \frac{r_j}{\d_j} D_n \bar\zeta_j
\right)\, dx\\
& = &\int_{B_N^{n-1}} {\mathcal{Q}}_{n-1}\, \overline g(D_\a \zeta^+)\,
dx_\a+ \int_{B_N^{n-1}} {\mathcal{Q}}_{n-1}\, \overline g(D_\a \zeta^-)\,
dx_\a\\ & = & \int_{(B_N^{n-1} \times I) \setminus C_{1,N}}
{\mathcal{Q}}_{n-1}\, \overline g (D_\a \zeta)\, dx
\end{eqnarray*} which completes the proof of the
$\limsup$ inequality. \hfill$\Box$
\bigskip
\begin{proposition}[Limit space]\label{bla2}
Let
$$
\ell=\lim_{j\to +\infty} {r_j\over \d_j}= +\infty\,,\qquad
0<R^{(\infty)}=\lim_{j \to +\infty} \frac{r_j^{n-1-p}}{\e_j^{n-1}}
< +\infty
$$
and let $\zeta_{\g,j} \in X^\g_j(z)$ such that, for every fixed
$\g>0$,
\begin{equation}\label{boundgradinfinity}
\sup_{j \in {\mathbb{N}}}\int_{(B_{\g N_j}^{n-1} \times I) \setminus
C_{1,\g N_j}} g_j\Bigl(D_\a\zeta_{\g,j}\Big| {r_j\over \d_j}D_n
\zeta_{\g,j}\Bigr)\, dx \le c\,.
\end{equation}
Then, there exists a sequence $\tilde\zeta_j \in
W^{1,p}_{{\rm loc}}(({\mathbb{R}}^{n-1}\times I)\setminus C_{1,\infty}; {\mathbb{R}}^m)$
such that
$$
\tilde\zeta_j= \zeta_{\g,j} \quad \text{ in }\quad (B^{n-1}_{\g
N_j} \times I)\setminus C_{1, \g N_j}
$$
and such that, up to subsequences, it converges weakly to
$\zeta^+$ in $W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1})^+;{\mathbb{R}}^m)$ and to
$\zeta^-$ in $W^{1,p}_{\rm loc}(({\mathbb{R}}^{n-1})^- ;{\mathbb{R}}^m)$. Moreover,
the functions $\zeta^\pm$ satisfy the following properties
$$\left\{
\begin{array}{l}
\zeta^\pm \in W^{1,p}_{\rm loc}({\mathbb{R}}^{n-1};{\mathbb{R}}^m),\\
\\
\zeta^+=\zeta^- \quad\text{ in }\, B_1^{n-1},\\
\\
D_\a \zeta^\pm \in L^p({\mathbb{R}}^{n-1};{\mathbb{R}}^{m\times (n-1)}),\\
\\
(\zeta^+ -z) \text{ and } \zeta^- \in L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m).
\end{array}
\right.$$
\end{proposition}
\noindent {\it Proof. } We can reason as in Proposition \ref{bla1}
using the fact that, by (\ref{boundgradinfinity}),
$$
\int_{({\mathbb{R}}^{n-1})^{\pm}}|D_n \tilde\zeta_j|^p\, dx \leq c\,
\Bigl(\frac{\d_j}{r_j}\Bigr)^p\,;
$$
hence, in the limit we have that $D_n \zeta=0$ a.e. in
$({\mathbb{R}}^{n-1})^{\pm}$. \hfill$\Box$
\begin{proposition}[Representation formula]\label{reprinf}
We have
\begin{eqnarray*}
\varphi^{(\infty)}(z) &= & \inf \Bigg\{\int_{{\mathbb{R}}^{n-1}}\big(
{\mathcal{Q}}_{n-1}\, \overline g (D_\a \zeta^+ ) + {\mathcal{Q}}_{n-1}\, \overline g
(D_\a\zeta^- ) \big)\, dx_\a :\; \zeta^\pm \in W^{1,p}_{\rm
loc}({\mathbb{R}}^{n-1};{\mathbb{R}}^m),\\
&&\hspace{3.8cm} \zeta^+=\zeta^- \text{ in }B_1^{n-1}, \quad
D_\a\zeta^\pm\in L^p({\mathbb{R}}^{n-1} ;{\mathbb{R}}^{m \times
(n-1)}),\\
&&\hspace{6cm} (\zeta^+ - z) \text{ and } \zeta^- \in
L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m) \Bigg\}
\end{eqnarray*}
for every $z\in {\mathbb{R}}^m$.
\end{proposition}
\noindent {\it Proof. } Reasoning as in the proof of Proposition
\ref{reprell}, by Propositions \ref{linfinite} and \ref{bla2} we
get the representation formula for $\varphi^{(\infty)}$.
\hfill$\Box$
\bigskip
\subsection{The case $\ell=0$}
\noindent We first recall the following $\Gamma$-convergence
result.
\begin{proposition}\label{lzero} The sequence of functionals
$G^{(0)}_j:L^p((B_N^{n-1} \times (-N,N)) \setminus C_{1,N};{\mathbb{R}}^m)
\to [0,+\infty]$, defined by
$$
G^{(0)}_j(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{(B_N^{n-1}
\times (-N,N)) \setminus C_{1,N}} g_j (D \zeta )\, dx & \text{ if
}\zeta \in
W^{1,p}((B_N^{n-1} \times (-N,N)) \setminus C_{1,N} ;{\mathbb{R}}^m),\\
&\\+\infty & \text{ otherwise\,,}
\end{array}\right.
$$
$\G$-converges, with respect to the $L^p$-convergence, to
$$
G^{(0)}(\zeta):=\left\{\begin{array}{ll}\displaystyle \int_{(B_N^{n-1}
\times (-N,N)) \setminus C_{1,N}} g (D \zeta)\, dx & \text{if }
\zeta \in
W^{1,p}((B_N^{n-1} \times (-N,N)) \setminus C_{1,N} ;{\mathbb{R}}^m),\\&\\
+\infty & \text{ otherwise\,.}
\end{array}\right.
$$
\end{proposition}
\noindent {\it Proof. }The result is an immediate consequence of
the pointwise convergence of the sequence of quasiconvex functions
$g_j$ towards $g$ together with Proposition 12.8 in
\cite{BD}.\hfill$\Box$
\bigskip
\begin{proposition}[Limit space]\label{bla3} Let
\begin{equation}\label{ass0}
\ell=\lim_{j\to +\infty} {r_j\over \d_j}= 0\,,\qquad
0<R^{(0)}=\lim_{j \to +\infty} \frac{r_j^{n-p}}{\e_j^{n-1} \d_j} <
+\infty
\end{equation}
and let $\zeta_{\g,j} \in Y^\g_j(z)$ such that, for every fixed
$\g>0$,
\begin{equation}\label{boundgj}
\sup_{j \in {\mathbb{N}}}\int_{(B_{\g N_j}^{n-1} \times I_j) \setminus
C_{1,\g N_j}} g_j(D \zeta_{\g,j})\, dx \le c\,.
\end{equation}
Then, there exists a sequence $\tilde\zeta_j \in
W^{1,p}_{{\rm loc}}({\mathbb{R}}^n\setminus C_{1,\infty}; {\mathbb{R}}^m)$ such that
$$
\tilde\zeta_j= \zeta_{\g,j}\quad \text{ in } \quad (B^{n-1}_{\g
N_j} \times I_j) \setminus C_{1, \g N_j}
$$
and such that, up to subsequences, it converges weakly to $\zeta$
in $W^{1,p}_{\rm loc}({\mathbb{R}}^n\setminus C_{1,\infty};{\mathbb{R}}^m)$.
Moreover, the function $\zeta$ satisfies the following properties
\begin{equation}\label{defzeta}
\left\{
\begin{array}{l}
D \zeta \in L^p({\mathbb{R}}^n\setminus C_{1,\infty};{\mathbb{R}}^{m \times n}),\\
\\
\zeta-z \in L^{p}(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)),\\
\\
\zeta \in L^{p}(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\,.
\end{array}
\right.
\end{equation}
\end{proposition}
\noindent {\it Proof.} By (\ref{boundgj}), (\ref{pgrowthWj}) and
(\ref{ass0}), we deduce that, for every fixed $\g>0$,
\begin{equation}\label{boundgrad0}
\sup_{j \in {\mathbb{N}}}\int_{(B_{\g N_j}^{n-1} \times I_j) \setminus
C_{1,\g N_j}} |D \zeta_{\g,j}|^p\, dx \le c\,.
\end{equation}
Let us first extend $\zeta_{\g,j}$ by reflection
\begin{equation}\label{barzetagj}
\bar\zeta_{\g,j}(x)=\left\{
\begin{array}{lll}
\zeta_{\g,j}\Bigl(x_\a, 2{\d_j\over r_j}-x_n\Bigr)& \text{if} &
x_\a\in B^{n-1}_{\g N_j} \text{ and } x_n\in (\d_j/r_j, 2 \d_j/r_j),\\
\\
\zeta_{\g,j}(x) & \text{if}& x\in (B^{n-1}_{\g N_j} \times
I_j)\setminus C_{1,\g N_j},\\
\\
\zeta_{\g,j}\Bigl(x_\a, -2{\d_j\over r_j}-x_n\Bigr)& \text{if}&
x_\a\in B^{n-1}_{\g N_j}\text{ and } x_n\in (-2\d_j/r_j, - \d_j/r_j)\\
\end{array} \right.
\end{equation}
and then, we extend it by $(2\d_j/r_j)$-periodicity in the $x_n$
direction. The resulting sequence, still denoted by
$\bar\zeta_{\g,j}$, is defined in $\big(B^{n-1}_{\g N_j} \times
{\mathbb{R}}\big)\setminus C_{1,\g N_j}$. Hence, we define on
${\mathbb{R}}^n\setminus C_{1,\infty}$,
\begin{equation}\label{barzetaj}
\bar\zeta_{j}(x):=\left\{
\begin{array}{lll}
z & \text{in} &\big( {\mathbb{R}}^{n-1}\setminus B^{n-1}_{\g
N_j}\big)\times (0,+\infty),\\
\\
\bar\zeta_{\g,j}(x)& \text{in} & \big(B^{n-1}_{\g N_j}\times
{\mathbb{R}}\big)\setminus C_{1,\g N_j},\\
\\
0 & \text{in} &\big( {\mathbb{R}}^{n-1}\setminus B^{n-1}_{\g
N_j}\big)\times (-\infty, 0)\,. \\
\end{array} \right.
\end{equation}
Let us now introduce the cut-off functions $\phi_j \in
{\mathcal{C}}_c^{\infty}((-2\d_j/r_j, 2 \d_j/r_j);[0,1])$ such that
$\phi_j(x_n)=1$ if $|x_n|\le \d_j/r_j$, $\phi_j(x_n)=0$ if
$|x_n|\ge 2\d_j/r_j$ and $|D_n \phi_j |\le c(r_j/\d_j)$. Then, we
introduce our last sequence,
$$
\tilde\zeta_j(x_\a,x_n):= \left\{ \begin{array}{lll} \phi_j(x_n)
\bar\zeta_j (x_\a,x_n) + (1 - \phi_j(x_n)) z & \text{ if } &
(x_\a,x_n) \in {\mathbb{R}}^{n-1} \times (0,+\infty),\\
&&\\
\phi_j(x_n) \bar\zeta_j(x_\a,x_n) & \text{ if } & (x_\a,x_n) \in
{\mathbb{R}}^{n-1} \times (-\infty,0).
\end{array}\right.
$$
Note that
\begin{equation}\label{coincide}
\tilde\zeta_j= \zeta_{\g,j}\quad \text{ in } \quad (B^{n-1}_{\g
N_j}\times I_j )\setminus C_{1, \g N_j}\,.
\end{equation}
Moreover, by (\ref{boundgrad0})-(\ref{coincide}) we have that
\begin{equation}\label{bgradalphatilde}
\sup_{j \in {\mathbb{N}}}\int_{{\mathbb{R}}^n\setminus C_{1,\infty}}
|D_\a\tilde\zeta_j|^p\,dx\le c\,,
\end{equation}
while, for every $(a,b)\subset {\mathbb{R}}$, with $a<b$, we have
\begin{equation}\label{bgradntilde}
\int_{\big({\mathbb{R}}^{n-1}\times (a,b) \big) \setminus C_{1,\infty}}
|D_n\tilde\zeta_j|^p\,dx\le c\,,
\end{equation}
for $j$ large enough and $c$ independent of $(a,b)$. Reasoning as
in Proposition \ref{bla1}, with $(0,+\infty)$ and $(-\infty,0)$ in
place of $(0,1)$ and $(-1,0)$, respectively, we can conclude that
there exist $\zeta_1\in L^p(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$
and $\zeta_2\in L^p(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)) $ such
that, up to subsequences,
$$
\tilde\zeta_j-z\rightharpoonup \zeta_1 \quad\text{in}\quad
L^p(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))
$$
and
$$
\tilde\zeta_j \rightharpoonup \zeta_2 \quad\text{in}\quad
L^p(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\,.
$$
Moreover, by (\ref{bgradalphatilde}) and (\ref{bgradntilde}), we
have that, up to subsequences, $\tilde\zeta_j$ converges weakly to
$\zeta$ in $W^{1,p}_{\rm loc}({\mathbb{R}}^n\setminus C_{1,\infty};{\mathbb{R}}^m)$
where
$$
\zeta=\left\{
\begin{array}{lll}
\zeta_1+z & \text{in} & {\mathbb{R}}^{n-1}\times (0,+\infty)\\
\zeta_2 & \text{in} & ({\mathbb{R}}^{n-1}\times (-\infty, 0)) \cup
(B_1^{n-1}
\times\{0\})\,.\\
\end{array} \right.
$$
In particular, for any compact set $K\subset {\mathbb{R}}^n\setminus
C_{1,\infty}$, we have that
$$
\int_K |D\zeta|^p\,dx\le \liminf_{j\to +\infty} \int_K
|D\tilde\zeta_j|^p\,dx \le c
$$
for some constant $c$ independent of $K$; hence, we get that
$D\zeta\in L^p({\mathbb{R}}^n\setminus C_{1,\infty}; {\mathbb{R}}^{m\times n})$
which concludes the description of the limit function $\zeta$.
\hfill$\Box$
\bigskip
\begin{proposition}[Representation formula]
We have
\begin{eqnarray*}
\varphi^{(0)}(z) & = & \inf \Bigg\{\int_{{\mathbb{R}}^n\setminus
C_{1,\infty} } g (D \zeta) \, dx :\;\zeta \in W^{1,p}_{\rm
loc}({\mathbb{R}}^n \setminus C_{1,\infty};{\mathbb{R}}^m), \, D \zeta \in L^p({\mathbb{R}}^n
\setminus C_{1,\infty};{\mathbb{R}}^{m\times n}),\\&& \hspace{1.5cm}
\,\zeta-z \in L^{p}(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)) \text{ and
} \zeta \in L^{p}(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\Bigg\}
\end{eqnarray*}
for every $z\in {\mathbb{R}}^m$.
\end{proposition}
\noindent {\it Proof. } We define
\begin{eqnarray*}
\psi^{(0)}(z) & := & \inf \Bigg\{\int_{{\mathbb{R}}^n \setminus
C_{1,\infty}} g (D \zeta) \, dx :\;\zeta \in W^{1,p}_{{\rm loc}}({\mathbb{R}}^n
\setminus C_{1,\infty};{\mathbb{R}}^m), \, D \zeta \in L^p({\mathbb{R}}^n \setminus
C_{1,\infty};{\mathbb{R}}^{m\times n}),\\
&& \hspace{1.5cm} \,\zeta-z \in
L^{p}(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m)) \text{ and } \zeta \in
L^{p}(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))\Bigg\}
\end{eqnarray*}
and let us prove that $\varphi^{(0)}(z)=\psi^{(0)}(z)$ for every
$z\in {\mathbb{R}}^m$.
By definition of $\varphi_{\g,j}^{(0)}$ (see (\ref{phigj12})), for
every fixed $\eta>0$, there exists $\zeta_{\g,j} \in Y_j^\g(z)$
such that
\begin{equation}\label{liminf-}
\int_{(B_{\g N_j}^{n-1} \times I_j) \setminus C_{1,\g N_j}} g_j(D
\zeta_{\g,j})\, dx \leq \varphi_{\g,j}^{(0)}(z) + \eta\,;
\end{equation}
hence, by Proposition \ref{midi12} (i), (\ref{boundgj}) is
satisfied. Then by Propositions \ref{lzero} and \ref{bla3} we get
that
\begin{eqnarray}\label{liminf-1}
\lim_{j \to +\infty} \varphi_{\g,j}^{(0)}(z) + \eta & \geq &
\liminf_{j \to +\infty} \int_{(B_{\g N_j}^{n-1} \times I_j )
\setminus C_{1,\g N_j}}
g_j(D \tilde \zeta_j)\, dx\nonumber\\
& \geq &\nonumber \liminf_{j
\to +\infty} \int_{\big(B_N^{n-1} \times (-N,N)\big)\setminus
C_{1,N}} g_j(D \tilde
\zeta_j)\, dx\\
&\ge&\int_{\big(B_N^{n-1} \times (-N,N)\big)\setminus C_{1,N}} g(D
\zeta)\, dx
\end{eqnarray}
for some fixed $N>1$, where $\zeta$ satisfies (\ref{defzeta}). Thus,
passing to the limit in (\ref{liminf-1}) as $N \to +\infty$, $\g \to
0^+$ and $\eta \to 0^+$, it follows that
$$
\varphi^{(0)}(z) \geq \int_{{\mathbb{R}}^n\setminus C_{1,\infty}}
g(D\zeta)\, dx \geq \psi^{(0)}(z)\,.
$$
Let us prove the converse inequality. For any fixed $\eta>0$, let
$\zeta\in W^{1,p}_{\rm loc}({\mathbb{R}}^n \setminus C_{1,\infty};{\mathbb{R}}^m)$
be as in (\ref{defzeta}) and satisfying
\begin{equation}\label{inbis}
\int_{{\mathbb{R}}^n\setminus C_{1,\infty} }g (D\zeta )\, dx \leq
\psi^{(0)}(z) + \eta.
\end{equation}
For every $j\in {\mathbb{N}}$ and $\g>0$, we consider a cut-off function
$\theta_{\g,j} \in {\mathcal{C}}^\infty_c(B_{\g N_j}^{n-1};[0,1])$ such that
$\theta_{\g, j}=1$ in $B_{(\g N_j)/2}^{n-1}$, $|D_\a\theta_{\g,j}|
\leq c/\g N_j$ and we define $\zeta_{\g,j} \in Y_j^\g(z)$ by
$$
\zeta_{\g ,j}:= \left\{ \begin{array}{lll} \theta _{\g ,j}
(x_\alpha )\zeta + (1 - \theta _{\g ,j} (x_\alpha ))z &
\text{in} & (B_{\g N_j}^{n - 1})^{+(\d_j/r_j)}\\
&&\\
\theta _{\g ,j} (x_\alpha )\zeta & \text{in} & (B_{\g
N_j}^{n-1})^{-(\d_j/r_j)} \cup (B_1^{n-1} \times
\{0\})\,. \\
\end{array} \right.
$$
Consequently, $\zeta_{\g,j}$ is an admissible test function for
(\ref{phigj12}) and we get that
$$\varphi_{\g,j}^{(0)}(z) \leq \int_{(B^{n-1}_{\g N_j} \times I_j )
\setminus C_{1,\g N_j}}g_j(D\zeta_{\g,j} )\, dx.
$$
The same kind of computations as those already employed in the
proof of Lemma \ref{reprell} now with $g_j$ in place of $g$ and
with other obvious replacements (see (\ref{est1})-(\ref{est3}))
gives
$$\lim_{j \to +\infty} \varphi_{\g,j}^{(0)}(z) \leq \limsup_{j \to
+\infty} \int_{(B^{n-1}_{\g N_j} \times I_j ) \setminus C_{1,\g
N_j}} g_j (D\zeta)\,dx + o(1)\,,\qquad \text{as}\quad \g\to 0^+\,.
$$
On the other hand, Fatou's Lemma and (\ref{pgrowthWj}) imply
$$\limsup_{j \to +\infty} \int_{(B^{n-1}_{\g N_j} \times I_j )
\setminus C_{1,\g N_j}} g_j (D\zeta)\,dx \leq \int_{{\mathbb{R}}^n
\setminus C_{1,\infty}} g (D\zeta)\,dx + o(1)\,,\qquad
\text{as}\quad \g\to 0^+.$$ Hence by (\ref{inbis}), passing to the
limit as $\g\to 0^+$, we get that
$$\varphi^{(0)}(z) \leq \psi^{(0)}(z) +\eta$$
and by the arbitrariness of $\eta$, the thesis.\hfill$\Box$\\
\begin{rmk}\label{Nadia}{\rm As already recalled, in \cite{Ans} it is
proved that if $\d_j=1$ or $\d_j=\e_j$ then the critical size
$r_j$ of the contact zones is of order $\e_j^{(n-1)/(n-p)}$ or
$\e_j^{n/(n-p)}$, respectively; moreover, the interfacial energy
density is described by the following formula
\begin{eqnarray*}
\varphi(z)&=& \inf\Bigl\{\int_{\Bbb R^n\setminus C_{1,\infty}} g (D
\zeta) \, dx :\;\zeta \in W^{1,p}_{\rm loc}({\mathbb{R}}^n \setminus
C_{1,\infty};{\mathbb{R}}^m)\,
\\
&& \qquad\qquad\qquad\qquad\quad\qquad \zeta-z\in
W^{1,p}(\Bbb R_+^n;\Bbb R^m),\ \zeta\in W^{1,p}(\Bbb R_-^n;\Bbb R^m)\Bigr\}
\end{eqnarray*}
where $\Bbb R_{+}^n= \Bbb R^{n-1}\times (0,+\infty)$, $\Bbb R_{-}^n=
\Bbb R^{n-1}\times (-\infty,0)$ (see \cite{Ans} Section 7, the case
$p=q$, with $\rho_{\e_j}=r_j$, $W_p=U_p=W$, $\widehat{W}_p=
\widehat{U}_p=g$ and $\Bbb R_{+,-}^n\cup B^{n-1}_1(0)=\Bbb R^n\setminus
C_{1,\infty}$).
We want to point out that from the analysis we carried on in the
case $\ell=0$ and in particular from
$$
0<R^{(0)}=\lim_{j\to+\infty}\frac{r_j^{n-p}}{\d_j\e_j^{n-1}}
$$
we recovered both the critical sizes founded in \cite{Ans} and
correspondent to the two cases $\d_j=1$ and $\d_j=\e_j$.
Moreover we want to show that $\varphi = \varphi^{(0)}$. We have
to check only the inequality $\varphi\leq \varphi^{(0)}$, the
other one being obvious.
For any fixed $\eta>0$ let $\zeta\in W^{1,p}_{\rm loc}({\mathbb{R}}^n
\setminus C_{1,\infty};{\mathbb{R}}^m)$ be such that $\zeta-z \in
L^{p}(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$, $\zeta \in
L^{p}(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$, $D \zeta \in L^p({\mathbb{R}}^n
\setminus C_{1,\infty};{\mathbb{R}}^{m\times n})$ and
\begin{equation}\label{minimizing}
\int_{\Bbb R^n\setminus C_{1,\infty}} g (D \zeta) \,dx\le
\varphi^{(0)}(z)+\eta\,.
\end{equation}
For every $N>2$ we denote by $B_N$ the $n$-dimensional ball of
radius $N$ centered in zero and by $B_N^\pm$ the set of the points
$x\in B_N$ such that $\pm x_n>0$; we consider a cut-off function
$\theta_N \in {\mathcal{C}}^\infty_c(B_N;[0,1])$ such that $\theta_N=1$ in
$B_{N/2}$, $|D\theta_N| \leq c/N$ and we define
$$
\bar\zeta:= \left\{ \begin{array}{lll} \theta _N (\zeta - z) + z &
\text{in} & B_N^{+},\\
&&\\
\theta _N \zeta & \text{in} & B_N^{-}\cup
(B^{n-1}_1 \times \{0\})
\end{array} \right.
$$
so that $\bar\zeta\in W^{1,p}(B_N\setminus C_{1,N};\Bbb R^m)$,
$\bar\zeta=z$ on $\partial B_N^{+}$ and $\bar\zeta=0$ on $\partial
B_N^{-}$. Hence,
$$
\int_{B_N\setminus C_{1,N}} g(D\bar\zeta)\,dx=
\int_{B_{N/2}\setminus C_{1,N/2}} g(D\zeta)\,dx +
\int_{(B_N\setminus B_{N/2})\setminus C_{1,N}}
g(D\bar\zeta)\,dx\,;
$$
in particular, by (\ref{pgrowthg}), we have
\begin{eqnarray*}
\int_{(B_N\setminus B_{N/2})\setminus
C_{1,N}}g(D\bar\zeta)\,dx&\le& \beta \Bigl(\int_{B^+_N\setminus
B^+_{N/2}} |D\theta_N|^p |\zeta-z|^p\,dx +
\int_{B^-_N\setminus B^-_{N/2}} |D\theta_N|^p |\zeta|^p\,dx\\
&&+ \int_{(B_N\setminus B_{N/2})\setminus C_{1,N}} |D\zeta|^p\,dx
\Bigr)\\
&\le&{c\over N^p} \Bigl(\int_{B^+_N\setminus
B^+_{N/2}}|\zeta-z|^p\,dx +
\int_{B^-_N\setminus B^-_{N/2}} |\zeta|^p\,dx\Bigl)\\
&&+ \int_{(\Bbb R^n\setminus B_{N/2})\setminus C_{1,\infty}}
|D\zeta|^p\,dx\,.
\end{eqnarray*}
Since $\zeta-z \in L^{p}(0,+\infty;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$,
$\zeta \in L^{p}(-\infty,0;L^{p^*}({\mathbb{R}}^{n-1};{\mathbb{R}}^m))$ and $D \zeta
\in L^p({\mathbb{R}}^n \setminus C_{1,\infty};{\mathbb{R}}^{m\times n})$, we can
easily conclude that
\begin{equation}\label{resto}
\lim_{N\to +\infty}\int_{(B_N\setminus B_{N/2})\setminus
C_{1,N}}g(D\bar\zeta)\,dx=0\,.
\end{equation}
Hence, by (\ref{resto}), we deduce
\begin{eqnarray*}
\varphi^{(0)}(z)+\eta&\geq& \int_{\Bbb R^n\setminus C_{1,\infty}} g
(D\zeta) \,dx\geq \int_{B_{N/2}\setminus C_{1,N/2}} g(D\zeta)\,dx\\
&=& \int_{B_N\setminus C_{1,N}} g(D\bar\zeta)\,dx + o(1)\\
&\geq& \inf\Bigl\{\int_{B_N\setminus C_{1,N}} g (D \zeta) \, dx
:\;\zeta\in W^{1,p}(B_N\setminus C_{1,N};\Bbb R^m)\,
\\
&& \qquad\qquad\qquad\qquad\qquad \zeta=z \,\, {\rm on}\,\,
\partial B_N^{+},\ \zeta=0\,\,{\rm on}\,\, \partial
B_N^{-}\Bigr\}+ o(1)
\end{eqnarray*}
as $N\to +\infty$. Finally, passing to the limit as $N\to
+\infty$, by the arbitrariness of $\eta$, we get
$\varphi^{(0)}\geq \varphi $.
Note that the proof of the explicit formula for $\varphi$ in
\cite{Ans} relies on the fact that $\d_j$ is of order $\e_j$ or
bigger than it, while in Proposition \ref{bla3} and Proposition
7.12 we have to take into account that $\d_j\ll\e_j$. This is the
reason why our proof is different from the one of \cite{Ans} even
if, at the end, the two representation formulas turn out to
coincide. }\end{rmk} \vspace{0.2cm}
\section{Conclusions}
\noindent Our paper deals with the characterization of the
effective energy of weakly connected thin structures through a
periodically distributed contact zone. We highlight the presence
of three different regimes (depending on the mutual rate of
convergence of the radii of the connecting zones and the thickness
of the domain) and for each of them we derive the limit energy by
a $\G$-convergence procedure. For each regime an interfacial
energy term, depending on the jump of the deformation at the
interface, appears in the limit representing the asymptotic memory
of the sieve. We completely describe the interfacial energy
densities by nonlinear capacitary type formulas.
\vspace{1cm}
\textsc{Acknowledgments.} The authors wish to thank Andrea Braides
for suggesting the problem, for fruitful comments and stimulating
discussions.
The research of J.-F. Babadjian has been supported by the Marie
Curie Research Training Network MRTN-CT-2004-505226 `Multi-scale
modelling and characterisation for phase transformations in advanced
materials' (MULTIMAT).
\addcontentsline{toc}{section}{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,165 |
Questa voce raccoglie le informazioni riguardanti il Bologna Football Club nelle competizioni ufficiali della stagione 1963-1964.
Stagione
Nel 1964 il Bologna, allenato da Fulvio Bernardini, torna a vincere lo Scudetto a distanza di 23 anni dall'ultima affermazione. Appaiata in vetta al Milan al termine del girone di andata, la squadra emiliana raggiunse il primato in solitaria alla 20ª giornata. Nei primi giorni di marzo, la FIGC emise un comunicato in cui dichiarava che cinque giocatori erano risultati positivi ai controlli anti-doping effettuati il 2 febbraio (dopo Bologna-Torino): la società fu punita con la sconfitta a tavolino e la detrazione di un punto, per una penalizzazione totale di tre punti. A maggio, in seguito all'effettuazione di ulteriori controlli, la sentenza venne però annullata.
Intanto, le vicende del campo avevano visto l'Inter di Helenio Herrera vincere lo scontro diretto in casa felsinea e raggiungere il primo posto. Al termine delle 34 giornate di campionato, Bologna e Inter risultarono entrambe prime a 54 punti: per la prima, ed unica, volta nella storia del Campionato il tricolore fu assegnato tramite uno spareggio. La Federazione designò come campo neutro l'Olimpico di Roma, fissando la gara per domenica 7 giugno.
Il Bologna vinse l'incontro per 2-0 con due reti nell'ultimo quarto d'ora, sconfiggendo la Grande Inter che si era da poco laureata campione d'Europa e dedicando il titolo alla memoria di Renato Dall'Ara. Il presidente era infatti scomparso quattro giorni prima, a causa di un infarto: al suo funerale non poterono essere presenti i giocatori felsinei poiché la Federazione decise di non rinviare lo spareggio.
Divise
Organigramma societario
Area direttiva
Presidente: Renato Dall'Ara, poi Luigi Goldoni
Area tecnica
Allenatore: Fulvio Bernardini
Allenatore in 2ª: Cesarino Cervellati
Rosa
Formazione tipo
Calciomercato
Risultati
Serie A
Girone d'andata
Girone di ritorno
Spareggio
Coppa Italia
Coppa Mitropa
Statistiche
Statistiche di squadra
Statistiche aggiornate al 1º luglio 1964.
Andamento in campionato
Statistiche dei giocatori
Note
Bibliografia
Collegamenti esterni
1963-1964
Bologna 1963-1964 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,906 |
{"url":"http:\/\/www.math.cmu.edu\/~shlomo\/VKI-Lectures\/lecture1\/node12.html","text":"Next: Bibliography Up: Applications to Fluid Dynamics Previous: Shape Design Using The\n\n## Shape Design Using The Euler Equations\n\nOur next example is a similar minimization problem but this time subject to the Euler equation. Namely,\n (82)\n\nwhere , and is the solution of the Euler equation. Here stands for the variables and . The Euler equations in conservation form are written as\n (83)\n\nwhere\n (84)\n\nand where the matrices can be found, for example, in Hirsch [12]. An important property of the equation that we use here is\n (85)\n\nThe change in the flux vector satisfies,\n (86)\n\nand similar expressions for . The equation for the perturbation quantities reads\n (87)\n\nor equivalently,\n (88)\n\nNow consider the following identity which follows from integration by parts,\n (89)\n\nand similar integrals for the and terms. Combining these identities we arrive at\n (90)\n\nfor an arbitrary . We will use the notation\n (91)\n\nand note that is the normal flux at the boundary which has the form, see Hirsch [12],\n (92)\n\nand at a wall where , it reduces to\n (93)\n\nWe have following (84),(86) and its analog for the terms, and\n (94)\n\nCombining the last equalities and from (75), we get\n (95)\n\nwhere we used the notation , and . The wall boundary condition\n (96)\n\nbecomes upon perturbation\n (97)\n\nand as before we transfer this boundary condition to the original boundary ,\n (98)\n\nCollecting only the terms we get\n (99)\n\nThe variation of the functional\n (100)\n\nwill be simplified by adding (90) to it, but with a choice of which makes the volume integral vanish. Thus, we assume that\n (101)\n\n (102)\n\nNow we come to use the boundary conditions for . We begin with the far field . We assume that the boundary conditions there are given in terms of characteristic variables and assume that is the matrix such that are the characteristic variables. We write the far field term as\n (103)\n\nWe distinguish the following cases. Supersonic inflow: all variables are specified at inflow, and thus . Thus, no boundary conditions are imposed on . Supersonic outflow: No boundary conditions are specified for , hence is arbitrary there and therefore we are led to the choice at supersonic outflow. Subsonic inflow: 4 conditions are specified (3 in 2D), and those are , thus is arbitrary, leading to . Subsonic outflow: one condition is given for which implies and therefore . On the wall we choose\n (104)\n\nIn summary, the boundary conditions for are\n (105)\n\nWith this choice for together with the interior equation (101) we get that involves integrals depending on and and not on terms. Rearrangement by using integration by parts gives,\n\n (106)\n\nThe gradient of the functional in this case is therefore given by\n\n (107)\n\nNext: Bibliography Up: Applications to Fluid Dynamics Previous: Shape Design Using The\nShlomo Ta'asan 2001-08-22","date":"2018-10-18 16:47:16","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8072329163551331, \"perplexity\": 884.6883224283614}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-43\/segments\/1539583511889.43\/warc\/CC-MAIN-20181018152212-20181018173712-00538.warc.gz\"}"} | null | null |
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{"url":"https:\/\/tex.stackexchange.com\/questions\/121940\/plus-equal-are-separated","text":"# Plus equal are separated\n\nI have the following inline code:\n\n$\\texttt{TEST}(v *= 2, v += 2, v = 4)$\n\nBut when it is rendered, the plus is very far away from the equal sign. How can I overcome this problem and make LaTeX consider += as a single operator?\n\n\u2022 Also, maybe you want to use some sort of code environment and font (fixed size) to insert code into your PDF's, instead of a math environment, such as listings (or verbatim if all else fails) \u2013\u00a0Thomas Jul 1 '13 at 22:46\n\n+ is a binary operator and = is a binary relation. When TeX finds the sequence\n\nv + = 2\n\n\nit transforms it into\n\nOrd Bin Rel Ord\n\nbut a Bin is not allowed before a Rel, so it's changed into an Ord.\n\nSolutions:\n\n\\documentclass{article}\n\n\\newcommand{\\pluseq}{\\mathrel{+}=}\n\\newcommand{\\asteq}{\\mathrel{*}=}\n\n\\begin{document}\n\n$\\texttt{TEST}(v \\asteq 2, v \\pluseq 2, v = 4)$\n\n\\end{document}\n\n\nor, manually,\n\n\\documentclass{article}\n\n\\begin{document}\n\n$\\texttt{TEST}(v \\mathrel{*}= 2, v \\mathrel{+}= 2, v = 4)$\n\n\\end{document}\n\n\nThese exploit the fact that TeX doesn't insert any space between two consecutive Rel symbols.\n\nYou can also define a macro that switches the behavior, so you can type the formulas more naturally:\n\n\\documentclass{article}\n\n\\newcommand{\\switch}{%\n\\mathcode+=\\numexpr\\mathcode+ + \"1000\\relax % turn + into a relation\n\\mathcode*=\\numexpr\\mathcode* + \"1000\\relax\n}\n\n\\begin{document}\n\n$\\switch\\texttt{TEST}(v *= 2, v += 2, v = 4)$\n\n$a+=b \\quad \\begingroup\\switch a+=b\\endgroup \\quad a+=b$\n\\end{document}\n\n\nI added a nonsense line to show that \\switch respects grouping. The scope of \\switch ends with the formula (or group) in which it's issued.\n\n\u2022 Very good explanation. \u2013\u00a0Mika\u00ebl Mayer Jul 1 '13 at 13:04\n\u2022 Can there be a libe break between two rel symbols? \u2013\u00a0Aditya Jul 1 '13 at 17:48\n\u2022 @Aditya No, only after a relation symbol; consecutive ones are treated as a unit. Try $\\relpenalty-10000 a==b$\\bye and see; of course one could add a penalty in between to explicitly allow a line break. \u2013\u00a0egreg Jul 1 '13 at 17:54\n\u2022 Good. Curiously, the newcommand doesn't work for me. I had to type \\mathrel{+}= inside my equations. \u2013\u00a0erickrf Apr 30 '14 at 3:41\n\u2022 @erickrf Do you refer to \\newcommand{\\pluseq}{\\mathrel{+}=}? Can you show an example of this failure? Maybe with a new question. \u2013\u00a0egreg Apr 30 '14 at 9:06\n\n$\\texttt{TEST}(v \\mathrel{{*}{=}} 2, v \\mathrel{{+}{=}} 2, v = 4)$","date":"2019-10-21 08:34:08","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\": 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\": 0.9281764626502991, \"perplexity\": 2590.737728924945}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987763641.74\/warc\/CC-MAIN-20191021070341-20191021093841-00540.warc.gz\"}"} | null | null |
\section*{Abstract}
We propose and study a \rev{class-expansion}/innovation/loss model
of genome evolution taking into account biological roles of genes
and their constituent domains. In our model numbers of genes in
different functional categories are coupled to each other. For
example, an increase in the number of metabolic enzymes in a genome
is usually accompanied by addition of new transcription factors
regulating these enzymes. Such coupling can be thought of as a
proportional ``recipe'' for genome composition of the type ``a
spoonful of sugar for each egg yolk''. The model jointly reproduces
two known empirical laws: the distribution of family sizes and the
nonlinear scaling of the number of genes in certain functional
categories (e.g. transcription factors) with genome size. In
addition, it allows us to derive a novel relation between the
exponents characterizing these two scaling laws, establishing a
direct quantitative connection between evolutionary and functional
categories. It predicts that functional categories that grow
faster-than-linearly with genome size to be characterized by
flatter-than-average family size distributions. This relation is
confirmed by our bioinformatics analysis of prokaryotic
genomes. This proves that the joint quantitative trends of
functional and evolutionary classes can be understood in terms of
evolutionary growth with proportional recipes.
\section{Introduction}
Protein-coding genes in genomes can be classified in both functional
categories (e.g. transcription factors or metabolic enzymes) as well
as ``evolutionary categories'' or families of homologous
genes
(to avoid confusion, in the following we will reserve the term
``category'' to functional annotations, and we will use the term
``family'' as a generic indication of homology classes, or domain
families/superfamilies in domain data, see Methods).
Functional categories are routinely composed of a large number of
evolutionary ones. This distinction is illustrated in
Fig.~\ref{fig:partitioning}, where genes are characterized by both
shape (functional category) and color (homology class) with each shape
represented by multiple colors.
Understanding the principles connecting these separate classifications
of genomic material is an important step in order to disentangle the
organization of the content of whole genomes.
More specifically, studies of fully sequenced genomes revealed that
their functional and evolutionary composition is governed by simple
quantitative laws~\cite{HvN98,vannim_03}. In particular, for
prokaryotes the number of genes in individual functional categories
was shown to scale as a power law of the total number of genes in the
genome \cite{vannim_03}. Depending on the functional category the
exponent of this scaling law varies from 0 (for fixed sets of
housekeeping genes) to 1 (for metabolic enzymes) and all the way up to
2 (for transcription factors and kinases)~\cite{vannim_08,vannim_03}.
Furthermore, the distribution of sizes of gene families (called
``evolutionary categories'' in our title) has a scale-free
distribution with the exponent inversely correlated with the genome
size~\cite{HvN98,KWK02,DSS02}. The overall number of gene (or domain)
families represented by at least one member exhibits a
slower-than-linear scaling with the total number of genes in a
genome~\cite{bassetti,messico}.
\rev{
Biologically, the growth of evolutionary families derives from
combined processes of horizontal gene transfer, gene duplication,
gene genesis, and gene loss~\cite{Koonin}. For prokaryotes,
horizontal transfer appears to dominate gene family
expansion~\cite{Treangen2011a}, and the same process is presumably
very important for the introduction of a new evolutionary family
into an extant genome.
}
The comprehension of these empirical laws requires to construct
quantitative models that explore different design principles, or more
prosaically the recipes by which genomes are built from elementary
functional and evolutionary ingredients. In this study we introduce
\rev{the first model to jointly explain observed scaling laws for
evolutionary families and functional categories}.
Several theoretical models have been previously proposed to explain
the observed power-law distribution of family
sizes~\cite{KWR+02,QLG01,KLQ+06,DS05,DSS02} Most of these models are
of \rev{class-expansion}/innovation/loss type\rev{, abstractly
mimicking basic evolutionary moves such as horizontal transfer,
duplication, loss}. We recently formulated a related model that in
addition to family size distribution also explains and successfully
fits the scaling of the number of distinct gene families represented
in a genome as a function of genome size~\cite{bassetti,angelini}.
On another front, the ``toolbox model'' of evolution of metabolic
networks and their regulation recently proposed by one of
us~\cite{maslov} offered an explanation for the quadratic scaling
between the number of transcription factors and the total number of
genes in prokaryotes.
In this model, metabolic and regulatory networks of prokaryotes are
shaped by addition of co-regulated metabolic pathways. The number of
added enzymes systematically decreases with the proportion to which
the organism has already explored the universe of available metabolic
reactions, and thus, indirectly, with the size of its genome.
For the purposes of the present study, a key ingredient of the toolbox model
is that events adding or deleting genes in multiple functional
categories (in this case metabolic enzymes and transcription factors
regulating metabolic pathways) are tightly correlated with each other.
The concept of coordinated expansion or contraction of functional
categories can in principle be extended beyond enzymes and their
regulators.
One should note that this explanation of scaling of functional
categories is conceptually different from that based on ``evolutionary
potentials'' proposed in Ref.~\cite{vannim_08}. Evolutionary
potentials quantify the intrinsic growth rates of
individual categories.
\rev{ This means that in this model the growth of one functional
category is represented as uncoupled from growth or decline in other
functional categories.
However, evolutionary potentials could also be the effective result
of the coordinated expansion of multiple functional categories
linked by interactions of biological and evolutionary origin
(e.g. linking membrane proteins with signal transduction, etc.)
On the other hand, it is clear that models with evolutionary
potentials represent quite well the empirical data on the growth of
functional categories, and thus it appears that this must be (at
least) a very good effective description, that any more detailed
model needs to reproduce.
}
This study brings together the basic ingredients of
\rev{class-expansion}/innovation/loss models~\cite{bassetti,angelini}
and coordinated growth of functional categories~\cite{maslov}. The
resulting combination allows us to study the interplay between the
scaling of evolutionary and functional categories. In particular, we
mathematically derive a relation between the exponents characterizing
these two scaling laws. It predicts that functional categories that
grow faster-than-linearly with genome size are characterized by
flatter-than-average family size distributions. This prediction of our
model is subsequently verified by our analysis of functional and
evolutionary scaling in empirical data on sequenced prokaryotic
genomes. \rev{Finally, we analyze and discuss the alternative
combination of a class-expansion/innovation/loss model with growth
of functional categories dictated by evolutionary potentials. }
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\textwidth]{Figure1.pdf}
\end{center}
\caption{Scaling laws in joint functional/evolutionary partitioning of
genomes. Genomes are partitioned into families of homologous genes
(colors) and functional categories (shapes). (A) The number of
unique evolutionary categories (domain families) (y-axis) scales
sub-linearly with the genome size (x-axis.) (B) Cumulative
histograms of domain family size (see Figure~\ref{fig:histograms}).
(C) The number of transcriptional regulators (red), metabolic
enzymes (blue), and housekeeping genes responsible for translation
(green) plotted as a function of the genome size measured by the
total number of domains. Symbols in all the plots are empirical
data for protein domains in $ 753 $ fully sequenced bacterial
genomes. }
\label{fig:partitioning}
\end{figure}
\section{MATERIALS AND METHODS}
\subsection*{Models}
The model represents a genome as a list of genes, which is partitioned
in homology families and functional categories. Genome evolution is
modeled as a stochastic process where the elementary moves can be any
of two types: (i) a ``family expansion'' or ``duplication'' move in
which a new domain is placed in an evolutionary category (family of
homologous domains) already present in the genome or (ii) an
``innovation'' move in which a new family with just one domain appears
in a genome (e.g. by the virtue of horizontal gene transfer).
We would like to emphasize that \rev{in the tradition established in
``duplication-innovation-loss'' models, which we follow,} the family
expansion move \rev{is customarily labeled} as duplication. In reality
this move can come either by the virtue of gene duplication or by
horizontal gene transfer, which appears to be the dominant
class-expansion mechanism in bacteria~\cite{Treangen2011a}. The
overall family size in all genomes might be generating an effective
``preferential attachment'' for HGT events (see
Refs.~\cite{isambert,vannim_08} and open comments by referees
therein).
Although genes are natural objects of this kind of
description, it is not simple to use genes as central units in the
analysis of empirical data, mainly due to the fact that gene dynamics
is complex and may contain events of gene fusion, splitting and
internal rearrangements. Thus, as in some previous analyses, we will
compare the models with data on protein
domains~\cite{vannim_08,bassetti}, which have the important property
that they cannot be split into smaller units~\cite{BBK+05}. Domains
are modular building blocks of proteins and it has been argued that
they effectively work as the natural atomic elements in genome
evolution~\cite{KWK02}. Concerning the scaling laws, domains appear to
have the same qualitative behaviour as genes. Throughout the paper, we
will be comparing the models with data on 753 bacteria from the
SUPERFAMILY database~\cite{supfam_funz}. The models will be formulated
for abstract atomic elements that could be genes or domains, and
possible relevant issues when dealing empirically with genes will be
addressed in the discussion. In describing the models we will
generally refer to these units as genes.
Technically, in order to compare with the protein domain data we rely
on simplifying assumptions on the domain composition of proteins.
Obviously the situation is more complex than this. We have verified
in the data that the number of TF domains are linear in the number of
TF genes (Supplementary Figure~\ref{fig:supp-gene}), with slope $
1.09 $ (average number of TF domains in a TF gene).
A second assumption is that the number of families belonging to a
functional category is linear in the total number of families. This
assumption is in accordance with data (see Figure~\ref{fig:Chi_c} and
Supplementary Figure~\ref{fig:supp-vannimSF}). In particular, we
observed this trend for the number of transcription factor
superfamilies (see Supplementary Figure~\ref{fig:supp-bpTFsupfam}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\textwidth]{Figure_Chi_c.pdf}
\end{center}
\caption{The number of evolutionary (domain) families belonging
to a functional category follows a linear law in empirical data,
denoting a possible invariant of genome composition. The left
panel plots the data on for the number of families $f_c$ in the
ten largest functional categories on all genomes, following the
trend $f_c = A_c + \chi_c f$, where $f$ is the total number of
families on the genome. Symbols are empirical data for $ 753 $
fully sequenced bacterial genomes. The offset $A_c$ is large only
for the ``translation'' category. The right panel is a plot of
the coefficients $\chi_c$ obtained from the same data (subtracting
the offset $A_c$ obtained from a linear fit), as a function of
genome size in domains, $n$. See also Supplementary
Table~S1. }
\label{fig:Chi_c}
\end{figure}
\subsubsection*{Standard Chinese Restaurant Process.}
The starting point is a \rev{class-expansion}/innovation process for the
homology families that reproduces qualitatively the empirical scaling
laws~\cite{bassetti}. This process (known in mathematical literature
as ``Chinese Restaurant Process''or CRP~\cite{pitman}) defines
a growth dynamics for the partitioning of a set of elements (genes or
domains) based on two basic growth moves.
Traditionally the CRP model is defined by two parameters $ \alpha $
and $ \theta $ constrained by $ 0 \leq \alpha \leq 1$ and $ \theta > - \alpha $.
The moves are quantified and
defined by two probabilities $ p_O $ and $p_N $ of duplication and
innovation respectively.
\begin{itemize}
\item The \rev{class-expansion} probability $p_O^i$ of a domain family
$i$ is proportional to the number of family members $n_i$ currently
in the genome offset by $\alpha$: $p_O^i \sim n_i-\alpha$ (see
Table~\ref{tab:notation}).
\item The innovation probability $ p_N $ is the probability of adding
a new domain family with one member. It corresponds to a new domain
family appearing in a genome by \textit{de novo} evolution or
horizontal gene transfer. The CRP model assumes $p_N \sim \alpha f
+\theta$, where $ f $ is the total number of domain families present
in the genome.
\end{itemize}
The normalization condition $ p_N+ \sum_i p_O^i=1 $ determines the
pre-factor in both equations to be $1/(n+\theta)$. A gene loss move
does not seem to be essential for the basic qualitative
results. Indeed, if stochastic (uniform) gene loss is
incorporated into the model it results only in renormalization of
parameters $ p_O $ and $p_N $~\cite{angelini}.
We explore the model by direct simulation and by solving continuous
``mean-field'' equations~\cite{bassetti,angelini} that describe the
mean behaviour of the number of homology families and functional
categories, and the statistics of the population of families and
categories.
\subsubsection*{CRP model incorporating functional categories.}
In order to introduce functional categories into the CRP, one has to
specify $ p_O $ and $ p_N $ for different categories.
We first assume that the probability of introducing a gene of a
specific functional category by the innovation move is independent of
genome size. This assumption implies that the number of homology
families of a given category scales linearly in the total number of
families, and is justified empirically for some functional categories
by domain data (see \rev{Figure~\ref{fig:Chi_c} } and Supplementary
Figures~\ref{fig:supp-vannimSF} and~\ref{fig:supp-bpTFsupfam}).
Equivalently, $p_N^c = \chi_c p_N$, where $
\chi_c $ is the probability of introducing a new family of the
category $ c $.
\rev{In other words, it is assumed here that every time a new family
is added, the probability that it will belong to category $c$ is
$\chi_c$.}
Under this assumption, the mean-field equation describing the growth
of a family of homologous domains (evolutionary category) is
\rev{
\begin{equation}
\displaystyle
C(n) \partial_{n} n_i = \displaystyle
\sum_{j=1}^{f} a_{ij} n_j \, - \alpha.
\label{eq:CRPdupl-class}
\end{equation}
}
Here the genome size $n$ is used instead of time and averages over
multiple realizations of a process are implied. The novel ingredient
of the model - coordinated growth of functional categories - is
encoded in the coefficients $a_{ij}$ responsible for correlated
duplications between evolutionary families $i$ and $j$. We assume
$a_{ij}$ to depend only on functional roles of families $i$ and $j$.
The equation describing the growth of $f$ - the number of distinct
families in a genome is the same as in a standard CRP model.
\begin{equation}
C(n) \partial_{n} f = \displaystyle (\alpha f + \theta) \ .
\label{eq:CRPduplcorrFtot}
\end{equation}
The function $ C(n) $, which sets a natural time scale for the
process, is determined by the normalization condition $\partial_n n =
1$, i.e. $\sum_i \partial_n n_i + \partial_{n} f = 1 $.
For the specific case of categories of transcription factors (TFs)
regulating metabolic processes and their metabolic target enzymes, the
necessity of a correlated move can be argued along the lines of
Ref.~\cite{maslov}. A set of new targets has to be added to
incorporate a new metabolic function.
This entails the addition of a new metabolic pathway that is long
enough to connect a new nutrient to a previously existing pathway,
that further converts it to a central metabolic ``core network''.
Supposing that each newly added branch is controlled by only one added
transcription factor, since the length of the branch becomes smaller
with increasing size of the organismal metabolic network (compared to
a metabolic ``universe''), on average, increasingly more TFs per
target will be needed in order to control newly incorporated branches.
More generally, functional, genetic and epistatic interactions can
create the correlated growth of different functional categories of
genes. In the discussion section we provide the empirical evidence of
statistically significant correlations between various functional
categories.
Following the recipe outlined in Ref.~\cite{maslov} \rev{we consider}
a simplified version of the model involving only two functional
categories: 1) $TF$- transcription factors controlling metabolic
processes; 2) $met$ - metabolic enzymes they regulated.
\rev{ As in the toolbox model, changes in $n_{TF}$ and $n_{met}$ are
coordinated with correlation coefficients $a_{ij}$ given by
\begin{eqnarray*}
\displaystyle
a_{ij} = \frac{n_{met}}{U}\, , \ a_{ji}=0 ; \
\mathrm{for} \ i\ne j \\
\mathrm{and} \ a_{ii} = 0 \ .
\end{eqnarray*}
Here $U$ is the size of the metabolic universe, $i$ denotes any gene
family from functional category $TF$, and $j$ - from the functional
category $met$. In this variant, addition of transcription factors can
only occur conditionally to the addition of metabolic enzymes. In the
following, we will refer to this model variant as model Ia.
}
\rev{ We define a second variant of the correlated model (model Ib),
which is a more direct extension of the standard CRP model, and thus
can exploit previous mean-field theory analytical results. In this
case
\begin{eqnarray*}
\displaystyle
a_{ij} = \frac{n_i}{n_{met}}\, , \ a_{ji}=0 ; \
\mathrm{for} \ i\ne j \\
\mathrm{and} \ a_{ii} = 1 \ ,
\end{eqnarray*}
(where $i$ again denotes any gene family from functional category $TF$
and $j$ - from the functional category $met$). In this model variant,
all families (and hence also transcription factors families) have an
equal intrinsic growth rate on top of the correlation. If $a_{ij} = 0,
\ i\ne j$ the model is equivalent to the standard CRP.
}
\rev{Finally, we also considered a model (model II)} where
correlations between functional categories are absent, but instead
members of a given functional category are added at a
category-dependent intrinsic rate as prescribed by ``evolutionary
potentials'' of Molina and van Nimwegen (in this case, $a_{ij} = 0$
for $i \ne j$, and $a_{ii} = \rho_{c(i)}$, where $c(i)$ is the
functional category to which family $i$ belongs, and $\rho_{c(i)}$ is
the evolutionary potential of class $c$).
These results are discussed later on in the manuscript and compared to
to the two ``correlated duplication'' models above (see Discussion and
Supplementary Text).
\rev{
To resume, two kinds of models are considered here: ``correlated
recipes'', where the scaling exponents can only result from
interactions between categories (model Ia and Ib, the main focus of
our study), and ``absolute recipes'' (model II), leading to
different intrinsic growth rates for different
categories. Correlated models might contain an specific intrinsic
growth rate of the classes, equal for all classes (model Ib), or not
(model Ia). We will see that the important distinction between
model I (a and b) and model II is that the different scaling
exponents for functional categories are a result of correlations and
not absolute class expansion rates. }
\subsection*{Data}
Data on superfamily domain assignments and superfamily functional
annotations for the $ 753 $ Bacteria were obtained from the
SUPERFAMILY (v1.73) database~\cite{supfam_funz}. The database
contains $1291$ different domain superfamilies grouped into $ 47 $
different functional categories ($ 60 $ families do not belong to a
specific category). These categories are divided into $ 6 $ larger
groups (Metabolism, General, Regulation, Information, Initiation
Complex Processes and Elongation Complex Processes, see also
\\http://supfam.cs.bris.ac.uk/SUPERFAMILY\_1.73/function).
\rev{
\subsection*{Evaluation of exponents in empirical data}
We considered the normalized cumulative histograms (families with more
than $d$ members) and non-cumulative histograms (families with exactly
$d$ members) of the populations for all evolutionary families (related
to exponent $\beta$, see Results), and those restricted to the families
belonging to each of the main functional categories indexed by $c$
(related to the exponent $\beta_c$, see Results). Exponents were
estimated by fitting the data with a power-law, restricting to a
window where the $x$ axis value was less than a cutoff value, as in
Ref.~\cite{angelini}. The cutoff was chosen for each fit, by
minimizing the chi square residuals with varying window size. This
procedure was implemented with a custom CINT (C++) script using the
ROOT software. Figure~\ref{fig:expcorr} is obtained considering the
fitted exponents for the histograms of the five largest genomes (where
the ``finite-size correction'' is smallest, see Figure~\ref{fig:histoexp}
and Ref.\cite{angelini}.)
}
\subsection*{Empirical correlations among functional categories}
Correlation between families (or categories) populations were
calculated from the deviations from the average trend. We obtained
the frequency of a family/category in every genome, defined as the
ratio between the population of a family in domains and the total
number of domains assigned on that genome. Subsequently for every
family/category, we extracted an average trend as a function of genome
size $n$ using a sliding-window histogram (with window size of 280
domains and resolution of 28 domains), and we considered the deviation
of each genome from the average trend at its value of $n$. The
Pearson correlation of these deviations over all the genomes was
considered between each pair of families/categories \rev{(Figure
\ref{fig:correlation} and Supplementary
Table~S3 and~S4)}.
\subsection*{Models and simulations}
The quantitative duplication-innovation evolutionary models were
explored by a mean-field analytical approach and direct numerical
simulations. The mean-field approach considers equations for the means
of the observed quantities in the large-$n$ approximation.
In parallel with the mean-field analysis, we performed simulations of
the main model and its variants. The realizations depend on the
following parameters. (i) The parameters of a standard CRP, $ \alpha $
and $ \theta $. (ii) The parameter $ \chi_c $, i.e. the probability
that a new family belongs to a given functional category. This
parameter can be inferred from data \rev{(see Results and
Figure~\ref{fig:Chi_c})}. For example, for the case of
transcription factors and targets, we defined $\chi_{TF} $ from the
slope extrapolated from Supplementary
Figure~\ref{fig:supp-bpTFsupfam}, giving $ \chi_{TF} \simeq 0.035 $
(see also Supplementary Figure~\ref{fig:supp-simTFsupfam}). (iii)
Initial conditions, represented by initial configuration (number of
leaves, number of TFs and number of families in both categories). We
have used the configuration of the smallest bacterium in the dataset
(Candidatus Carsonella ruddii). An alternative choice could be the
minimal intersection of all genomes in the database. (iv)
Variant-specific parameters, that amount to the evolutionary
potentials $\rho_c$ for the first variant of the model, and the
correlation matrix between functional categories, $a_{ij}$ for the
second variant. Simulation results are typically visualized in
boxplots in order to compare the means with the probability
distributions. In these plots bars correspond to (in order) the
smallest observation, lower quartile, median, upper quartile, and
largest observation.
\section{RESULTS}
\rev{
\subsection*{A new invariant of genome composition}
We found (Figure~\ref{fig:Chi_c} and Supplementary Table See also
Supplementary Table~S1)
that the number of evolutionary domain families forming a functional
category follows a linear law in empirical data, denoting a possible
invariant of genome composition. This also implies that the mean law
$\partial_n{f_c} = \chi_c \partial_n{f}$ assumed in the model is
justified by the data. This does not mean exactly that the fraction
of all families belonging to a certain functional category is
constant. Rather, the observed law can be $ f_c = A_c + \chi_c f$,
with an offset $A_c$ representing a minimal amount of evolutionary
families required to build a given functional category. In empirical
data, this offset appears to be large only for the ``translation''
functional category.
}
\subsection*{The model captures the combined scaling laws}
Numerical simulation and mean-field analytical solutions of the
correlated growth model \rev{ (model I) reproduce very well both the
empirical behavior of the TFs scaling law and the statistics for
evolutionary domain families (Figure~\ref{fig:corresults} and
Supplementaty Figure \ref{fig:supp-ToolPan}). We found no
significant qualitative difference between models Ia and Ib
regarding these observables. Furthermore, the joint scaling laws can
be reproduced also with an uncorrelated model (model II), with minor
technical difficulties (see Discussion). }
\rev{ The correct asymptotic quadratic scaling can be obtained from
mean-field arguments for both model I and II. These arguments are
presented in the Supplementary Text. In order to illustrate this
point we consider for example model Ib. Starting from
Eqs.~\ref{eq:CRPdupl-class} one has to sum over all domain families
from functional categories $TF$ and $met$. Since $n_{TF} = \sum_{i
\in TF} n_i$, depends on the number of TF classes, one must have
for its derivative $ \partial_n n_{TF} = \sum_{i \in TF} \partial_n
n_i + \partial_n f_{TF}$. Combined, these two equations give $d
n_{TF}/d n_{met} = 2(n_{TF}-\alpha)/(n_{met}-\alpha) \simeq
2n_{TF}/n_{met}$, or finally the quadratic scaling $n_{TF} \sim
n_{met}^2$. }
\rev{Altogether, the agreement between data and model} is universal,
in the sense that the same three parameters are sufficient to predict
family/category numbers and populations for all genomes in the
dataset. Moreover, the comparison does not rely on the adjustment of
any hidden parameter.
It is also worthwhile noting that, while the input of \rev{model I (a
and b)} is built to give an \rev{asymptotic} power-law scaling
exponent of two for transcription factors (which is reproduced by the
mean-field approach), at the relevant genome sizes the model
automatically reproduces the \emph{correct} empirical exponent (about
$1.6$ in the SUPERFAMILY data) \rev{as an effect of the finite system
size}.
Note that \rev{in model Ib TFs can duplicate both spontaneously
(uncorrelated move) and following spontaneous duplication of targets
(correlated move), corresponding to the terms $a_{ii}$ and $a_{ij}$
in equation~\ref{eq:CRPdupl-class}, while in model Ia this does not
happen. }
The extension of the model to more than two categories requires to
know the laws through which families of different categories are
correlated with each other. Supplementary
Figure~\ref{fig:supp-comparision} compares the results obtained by a
correlated duplication model formulated with three categories (TFs,
met, others).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{Figure2.pdf}
\end{center}
\caption{Comparison between $ 1000 $ realizations of the correlated
duplication model \rev{Ib} at $ \alpha = 0.3 $ and $ \theta = 140 $
(blue boxplot) with empirical data (red). The left panel is a plot
of the number of distinct domain families versus genome size. The
fact that the number of families does not saturate is a property of
the standard duplication-innovation model (see~\cite{bassetti} for a
complete discussion). The right panel plots the number of TF domains
versus the total number of domains, showing that the scaling of the
transcription functional category is well reproduced (exponent $
\simeq 1.6 $.) \rev{See Supplementary Figure~\ref{fig:supp-ToolPan}
for model Ia.} }
\label{fig:corresults}
\end{figure}
\subsection*{Prediction of the exponents of the family-population
histogram restricted to single functional categories.}
\begin{table}[b]
\caption{Basic model quantities and notations}
\label{tab:notation}%
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|}
\hline
Quantity & Meaning \\
\hline
$ \alpha $ , $ \theta $ & CRP model parameters \\
\hline
$ n $ & Genome size quantified by its total number of domains \\
\hline
$ n_i $ & Number of domains in the family $ i $ \\
\hline
$ n_c $ & Number of domains in the functional category $ c $ \\
\hline
$ f(n) $ & Number of families in a genome of size $ n $ \\
\hline
$ f_c(n) $ & Number of families in a genome of size $ n $ belonging to the functional category $ c $ \\
\hline
$ f(d,n) $ & Number of families with exactly $ d $ members in a genome of size $ n $ \\
\hline
$ f_c(d,n) $ & Number of families belonging to the functional category $ c $ with exactly $ d $ members in a genome of size $ n $\\
\hline
$ \beta $ & Exponent of the family-population histogram \\
\hline
$ \beta_c $ & Exponent of the family-population histogram restricted to category $ c $ \\
\hline
$ \chi_c $ & Probability to introduce a new family of the category $ c $
(empirically quantified by the slope of $ f_c $ vs. $ f $) \\
\hline
$ \zeta_c $ & Exponent of the scaling of the size of functional category $ c $ vs. genome size $ n $\\
\hline
\end{tabular}%
}
\end{table}
While the agreement between model and data shows that the scaling of
functional and evolutionary categories can be understood jointly, it
does not provide by itself any substantially new information about how
the two partitionings interact. Further insight can be obtained
considering the
\rev{distributions of the number of domains per family for different
evolutionary families belonging to the same functional category}.
In general, the population of domain families of a genome follows a near power
law distribution whose slope depends on genome size
(Figure~\ref{fig:histograms}). The mean number $f(d,n)$ of domain
families having $d$ members at large genome size $n$ is well described
by the slope $ 1/d^{1+\beta} $ (see Figure~\ref{fig:histograms}) ,
and thus the cumulative histogram by $Q(d,n) \sim 1/d^{\beta}$
, where the fitted exponent $ \beta $ typically lies between 0 and 1.
The standard CRP predicts this behavior~\cite{bassetti,angelini}.
The model described here here allows to consider the same histograms
restricted to specific functional categories (Figure
\ref{fig:histograms} and Figures \ref{fig:expcorr}).
A mean-field calculation (see Supplementary Text) based on the model
variant with correlated duplication predicts that the different trend
of domain population histograms for transcription-factor families
scales as $f(d,n)_{TF} \sim 1/d^{1+\frac{\beta}{2}} $ (see
Figure~\ref{fig:histoexp}).
Thus, the ratio between the exponent of the cumulative histogram of
all families and the exponent of the cumulative histogram restricted
to families belonging to the transcription factor category is
predicted to be equal to the mean-field exponent for the scaling of
the functional category.
\rev{ Specifically, $Q(d,n)$ scales as $1/d^{\beta}$ whereas
$Q_{TF}(d,n)$ scales as $d^{-\beta/2}$ and thus the ratio of
exponents is $\beta/(\beta/2) = 2$, and this matches the asymptotic
scaling of the number of transcription factors. }
\rev{ More in general, the model indicates that each time the
per-family duplication probability for a functional category takes
the form $ p_{O}^c \simeq \zeta_c n_{c}$, where $n_c$ is the total
population of the functional category $c$, the coefficient $\zeta_c$
will appear in the equation for $P(d)_{c}$, the (cumulative)
distribution of families belonging category $c$. This causes the
relationship $\beta_c = \beta / \zeta_c$ and appears to be robust
with respect to the choice of a specific model (see Supplementary
Text).}
In other words, a precise quantitative relationship must exist between
the scaling exponent of a category and the slope of the family
population histogram restricted to the same category. Functional
categories that grow faster-than-linearly with genome size will have
flatter-than-average domain family size distributions. Conversely
categories growing slower-than-linearly will follow a
steeper-than-average slope.
Accordingly, a strongly visible trend should be expected in empirical
data from families belonging to the transcription factor category,
which scales with exponent 2. Indeed, the empirical population
histograms for the transcription factor functional category for all
the genomes in the data set have a slope that is spectacularly
different from the global one (Figure~\ref{fig:histoexp} \rev{and
Supplementary Figure~\ref{fig:supp-scatterExp}}).
Quantitatively, this observation is in excellent agreement with
predictions (Table~\ref{tab:exphistotf}).
Direct simulations of the correlated model reproduce well both the
behavior of the histograms at given size and the dependency on genome
size (see Figure~\ref{fig:histograms}).
More generally, one can test the prediction $\zeta_c = \beta /
\beta_c$ with an empirical evaluation of many functional categories
(Figure~\ref{fig:expcorr}). The agreement of empirical data with the
predicted behavior is reasonably good, keeping in mind that many
functional categories are composed by few or poorly populated families,
and in these cases the data might not follow a scaling law that is as
clearly defined as the metabolism or the transcription factor
categories.
\begin{figure}[tp]
\begin{center}
\includegraphics[width=0.75\columnwidth]{Figure3.pdf}
\end{center}
\caption{ Empirical data and simulations for the normalized domain
family population cumulative histograms. The histograms are defined
as the fraction $f(d,n)/f(n)$ of families with more than $d$
domains. (A) Empirical data for the 753 bacteria in the SUPERFAMILY
database (each color is a set of genomes with similar sizes). Left
panel: domain family population cumulative normalized
histograms. Right panel: normalized cumulative histograms restricted
to domain families belonging to the transcription factor functional
categories. Note that the histograms slopes are different. (B)
Simulations for domains family population cumulative histograms of
CRP with correlated duplications run at $ \alpha = 0.3 $ and $
\theta = 140 $. \rev{The plots in the two panels are defined as in
(A)}. (C) Comparison between simulations of the correlated
duplication model variant run at $\alpha = 0.3 $ and $\theta = 140 $
(black lines) with empirical data (orange lines) for the largest
genome sizes ($ 5000 < n < 8500 $). Left panel: global normalized
cumulative histograms of domain family population. Right panel:
normalized cumulative histograms restricted to transcription factor
domain families. }
\label{fig:histograms}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\columnwidth]{Figure4.pdf}
\end{center}
\caption{ Exponent of evolutionary families and genome size. Fitted
exponent of domain family population cumulative histograms vs.
genome size, for the 753 bacteria in the SUPERFAMILY database for TF
families (red circles) and all families (black squares), obtained by
a fitting method giving a lower weight to the tail in order to keep
into account the cutoffs (used in Ref.~\cite{angelini}).}
\label{fig:histoexp}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[width=0.9\columnwidth]{Figure5.pdf}
\end{center}
\caption{ Linear relation between $\zeta_c$ and $1/\beta_c$. Our
theory predicts $\zeta_c \sim \beta/ \beta_c$ (solid line). The
empirical value of $ \beta = .74$ is calculated from the family
population histograms of the five most populated genomes. Symbols
(circles and triangles) are empirical data for $ 38 $ functional
categories \rev{(see also Supplementary Table~S2)}. Triangles represent the ten most populated
categories, where the estimated exponents are most accurate. The
outlier is the ``small molecule binding'' category known to follow
peculiar evolutionary mechanisms~\cite{Koonin01}. }
\label{fig:expcorr}
\end{figure}
\begin{table}[b]
\caption{ Prediction of the exponent of the
family-population histograms restricted to singular functional category.
Comparison between expected and
observed ratio of the exponent of the cumulative histogram of all
families and the exponent of the cumulative histogram of
transcription-factor families (
Figure~\ref{fig:expcorr} ), for the five largest bacteria in
the SUPERFAMILY database. The ratio can be compared with the
mean-field prediction of $ 2 $, or directly with the empirical exponent
of the transcription factor functional category ($ 1.6 $). }
\label{tab:exphistotf}
\begin {tabular}{|c|c|c|}
\hline
Genome & $\beta/\beta_{TF}$ & $\zeta_{TF}$ \\
\hline
Sorangium cellulosum & $ 1.72 \pm 0.1 $ & $ 1.6 $ \\
\hline
Burkholderia xenovorans & $ 1.63 \pm 0.08 $ & $ 1.6 $ \\
\hline
Burkholderia & $ 1.54 \pm 0.13 $ & $ 1.6 $ \\
\hline
Solibacter usitatus & $ 1.46 \pm 0.05 $ & $ 1.6 $ \\
\hline
Bradyrhizobium japonicum & $ 1.59 \pm 0.11 $ & $ 1.6 $ \\
\hline
\end{tabular}
\end{table}
\section{DISCUSSION}
\subsection*{Population of evolutionary families of a given functional
category}
We have presented the first combined quantitative description of the
partitioning of genomes in both evolutionary families and functional
categories. The results show that a theoretical framework that
correctly reproduces both the scaling laws for functional categories
of genes/domains and the scaling laws for gene/domain families
(numbers and histograms) is possible.
Biologically, this finding can help us understand the large-scale
architecture of a genome in terms of its functional content.
\rev{ Analyzing the data in order to formulate the model, we found
that the number of evolutionary domain families forming a functional
category is linear in the total number of domain families
(Figure~\ref{fig:Chi_c}). Thus, the genomic subdivision of
evolutionary classes in functional categories appears to be arguably
the simplest possible, if one disregards the class population. This
ingredient was taken as an assumption for all the models considered
here, which the data fully justify. }
The model leads to the nontrivial prediction that connects the growth
exponent of a functional category to the slope of the population
family histogram restricted to the same category. In other words, the
populations functional categories and evolutionary families of genes
are connected by a simple quantitative law.
Specifically, the ratio between the exponent of the cumulative
histogram of all families and the exponent of the cumulative histogram
restricted to families belonging to a functional category is predicted
to be equal to the exponent for the scaling of the functional
category.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\columnwidth]{Figure6.pdf}
\end{center}
\caption{ Correlation between the populations of $ 24 $ different
metabolic functional categories from the SUPERFAMILY database for
$ 753 $ bacteria. The correlation matrix is calculated from
fluctuations of categories from the average trend (see Methods).
Both correlation and anticorrelation are present between categories.
Different metabolism categories are highly (anti-)correlated.}
\label{fig:correlation}
\end{figure}
To generate this prediction, we have analyzed in detail the case of
transcription factors, where the exponent of the population histogram
is halved due to the quadratic scaling using mean-field calculations
and simulations, and verified that it holds in general by simulations
of both model variants.
Empirical data on transcription factors follow this behavior
remarkably well, showing population cumulative histograms of
transcription factor superfamilies decaying with halved exponents
compared to the global populations.
The fatter tails of the TF histograms might also be related to the
fact that only a few highly populated DNA-binding domain superfamilies
dominate the population of TF DNA-binding domains and determine the
scaling laws (Supplementary Text and Supplementary
Figures~\ref{fig:supp-TFclass} and~\ref{fig:supp-smalltf}).
More in general, we have also compared the behavior of domain family
population histograms for all the empirical functional categories with
the prediction, obtaining results that are in good agreement (Figure
\ref{fig:expcorr}), in particular for the highly populated categories,
where the fitting procedure gives the highest confidence.
The only highly-populated category that significantly violates
this general trend is small molecule binding, a category composed of
very few highly-populated domain families. This category is known to
follow peculiar evolutionary laws, with high mobility of domains
across the metabolic network, resulting in members of the same
family being scattered across different pathways and producing
lineage-specific domain families, with frequent re-invention of the
same function by different families~\cite{Koonin01,Chothia2003}.
Thus, the exception makes biological sense, and can be understood in
terms of members of evolutionary classes ``jumping'' to different
functional categories with high rate during evolution.
\subsection*{Correlated and absolute recipes}
The central ingredient of our main model \rev{(model I)} is the
coupling between addition/removal of genes in different functional
categories. From a biological standpoint, it is reasonable that gene
repertoires of functional categories related to each other via shared
tasks, pathways or processes should follow coordinated
rules~\cite{Koonin}. In order to further justify this assumption, we
probed directly the empirical domain data for correlation between
number of domains in different functional categories,. To this end,
for each genome $g$ we calculated the deviation $\delta n_c(g)$
between the functional category size ($n_c(g)$ and its average size in
genomes of comparable size (see Methods). We then calculated the
matrix of correlations between values of $\delta n_c$ for different
functional categories $c$.
The results are reported in Figure~\ref{fig:correlation} \rev{ and
Supplementary Tables~S3 and~S4. We also tested that this procedure
for evaluating the correlation was not dependent on genome size
(Supplementary Figure~\ref{fig:supp-metcorrsize}.) }
The metabolism categories appear to be highly (anti-)correlated with
each other, probably because of the role they play in different
pathways of a common metabolic network~\cite{maslov}.
The observed correlations between metabolic families might also be
relevant for reproducing the correct tail of the family population
histogram restricted to the metabolism category (Supplementary
Figure~\ref{fig:supp-comparision}).
\begin{figure*}[tp]
\begin{center}
\includegraphics[width=0.7\textwidth]{Figure7.pdf}
\end{center}
\caption{ Models with correlated versus absolute moves. Top: the
Chinese restaurant process (CRP) acts on the homology families
(colors) with a duplication and an innovation move. It is extended
here to include functional categories (shapes) Middle: model with
evolutionary potentials. Functional categories are assigned
differential duplication rates as in ref~\cite{vannim_08}. Bottom:
Model with correlated moves. Members of the functional categories
are added proportionally between correlated pairs of functions
(e.g. transcription factors and metabolic targets) as in
Ref.~\cite{maslov}. }
\label{fig:models}
\end{figure*}
An alternative approach is a description where the growth of each
category is governed by intrinsic ``evolutionary
potentials''~\cite{vannim_08}. We have also analyzed such a
description in some detail (see Supplementary text and Supplementary
Figure~\ref{fig:supp-comparision}).
Despite of minor differences, a model \rev{combining
class-expansion/duplication/loss with uncorrelated moves for the
functional categories, model II,} can also perform well in reproducing the
joint scaling law and in predicting a relationship between the scaling
exponents and the functional categories. \rev{In particular, this
means that the latter result should not by itself be considered a
piece of evidence in favor of a correlated recipe. }
Figure \ref{fig:models} illustrates the basic differences between the
two descriptions.
The evolutionary potentials approach generically requires a lower
number of parameters, but suffers from the tedious technical problem
that the values of the growth coefficients cannot be controlled
directly, because of the scaling of the normalization constant with
genome size (see Supplementary Text \rev{and Supplementary
Figure~\ref{fig:supp-cn}}).
The correlated model is technically more under control, since its
behavior does not rely on any unknown normalization constant. For
this reason, it also performs better with functional categories that
grow faster than linear with genome size, such as transcription
factors.
\rev{On the other hand}, such a model can be formulated with very few
parameters \rev{only when} a synthetic description for the
correlations, such as the toolbox model, is provided.
\rev{ Here, we have considered mainly a model with three categories
(transcription factors, metabolic, and others) and one nonzero
correlation between metabolic domains and transcription factors. In
general, specific biological details of how categories are
correlated with each other determine the scaling exponents relating
their genome fractions to each other and genome size.
Note that the task of formulating a correlated model for many
categories requires a knowledge of how the different functional
categories are ``slaved'' to each other. This structure is largely
unknown quantitatively, and can in principle define an arbitrarily
complex network of interactions, since many categories may correlate
with many others in potentially complicated ways.
Should the importance of correlated recipes be confirmed by further
analysis, it seems likely that the full formulation of such a
description would still require to solve this problem.
In order to show explicitly that the model can in principle be
successfully extended to many categories (and still give scaling
laws) we have analyzed the case of a simple hierarchical structure
where many categories are slaved to a main one (see Supplementary
Figure~\ref{fig:supp-pwrsim}).
}
Overall, since functional categories scaling laws effectively emerge
from the correlated approach, a good reconciliation of the two
approaches could be to interpret the evolutionary potential model as
an emergent description (which can be very useful in concrete
empirical applications).
In other words, evolutionary potentials would describe emergent
effective growth of functional categories of a genome, averaging over
more ``microscopic'' evolutionary processes where addition of genes
belonging to specific functional categories needs to comply to
constraints combining different functions to perform specific cell
tasks. These kind of interactions between functions are better
described by correlated growth of functional categories. In this
view, genome growth would be governed by a relative recipe, where the
proportions are more important than the exact amounts, rather than an
absolute recipe, where only the detailed amounts of each ingredient
play a role.
\afterpage{\clearpage}
\newpage
\section*{\Large Supplementary Text and Figures for Grilli \emph{et al.}}
\vspace{2cm}
\section{Description of the model and basic mean-field results}
\rev{ This section discusses in more detail the analytical derivation
of the scaling for the main observables of model I and II using a
mean-field approach.}
\rev{ Consider a joint partioning of elementary units (domains or
genes) in functional and evolutionary categories, as illustrated in
Figure~\ref{fig:partitioning} of the main text. The elementary units
(in our case domains), belong to a single evolutionary family $i$,
and every family $i$ belongs to one and only one functional category
$c$.}
\rev{The generic stochastic growth model considered here defines how
new units are introduced into the system. The model is specified by
a set of basic rates. The basic set of rates is constitued by the
probabilities $p_i$ that a newly added unit belongs to a certain
class $i$. More in detail, we define a probability $p_O^i$ (where
$O$ stands for ``old'') that a new domain belongs to a family $i$
which is already present in the system (i.e. having at least one
member) and the probability $p_N$ (where $N$ stands for ``new'')
that the added unit belongs to a family which is not already present
in the system. }
\rev{The choice of $p_O^i$ and $p_N$ defines the model as a stochastic
process for the basic observables (such as genome size $n$, family
number $f$ and its population $n_i$, etc.), but one extra detail is
needed. When a new class is introduced, the model needs to specify
the category it belongs to. As discussed in the main text, in the
model considered here a newly added family always belongs to a
category $c$ with probability $\chi_c$. The probabilities $p_O^i$,
$p_N$ and $\chi_c$ can depend, in principle, on the number of units
$n$ and on their distribution in families, on the total number of
families $f$ and so on. Empirical data indicate (see
Figure~\ref{fig:Chi_c} in the main text) that $\chi_c$ is a
category-dependent constant, and thus does not depend on $n$.}
\rev{The mean-field approximation is useful to extract the basic
information from the model~\cite{bassetti}. In each realization of
the full stochastic process, the probabilities of the possible
configurations at time $t+1$ are determined by the configuration at
time $t$. The mean-field approximation assumes that the
configuration at time $t$ is the average configuration. For
example, if one is interested in the number of domains belonging to
family $i$, the average number of elements $n_i(t+1)$ at time $t+1$
will be equal to the average number of elements $n_i(t)$ at time $t$
summed with the average number of elements added in a time step,
i.e. $p_O^i$. For asymptotically large $t$ this implies the
approximate equation $\partial_t n_i = p_O^i$ for the averages (here
the averaging procedure is implicit in the notation). Since
typically, at each step one and only one element is added, the mean
number of elements is $n=t$. If this is not the case, we can obtain
$\partial_n n_i$ simply from $\partial_t n_i$ divided by $\partial_t
n$. Considering $n=t$ we obtain, for a generic model, the following
mean-field equations
\begin{equation}
\displaystyle
\begin{split}
& \partial_n n_i = p_O^i \\
& \partial_n f = p_N \\
& \partial_n f_c = \chi_c p_N \\
& \partial_n n_c = \partial_n \sum_{i \in c} n_i = \sum_{i \in c} \partial_n n_i + \partial_n f_c = \sum_{i \in c} p_O^i + \chi_c p_N \ .
\end{split}
\label{eq:meanpart}
\end{equation}
}
\subsection{\rev{ Models with correlations}}
\label{sec:supp-modI}
\rev{We now deal with the scaling of the basic observables in the
model taking into account the correlation between categories growth
(model I of the main text).}
\rev{The correlation appears in the growth of the domain families of
different categories. Thus the probability $p_O^i$ that a domain is
added to a given family $i$ can be written as
\begin{equation}
p_O^i =
\frac{\sum_{j=1}^f a_{i,j} n_j-\alpha}
{\sum^f_{i,j=1} a_{i,j} n_j + \theta} \ .
\label{eq:coor}
\end{equation}
The coordinated growth of functional categories is encoded by the
coefficients $a_{i,j}$, responsible for the correlated expansion of
evolutionary families $i$ and $j$ (See Equation~\ref{eq:CRPdupl-class}
of the main text). The standard Chinese Restaurant Process (CRP) is
obtained by imposing $a_{i,j}=\delta_{i,j}$ (where $\delta_{i,j}$ is
equal to $1$ if $i=j$ and $0$ otherwise). We assume that these
coefficients depend only on the functional categories $c$ and $c'$ to
which the families $i$ and $j$ belong. The probability of introducing
a new domain is given by
\begin{equation}
p_N = \frac{\alpha f + \theta }
{\sum^f_{i,j=1} a_{i,j} n_j + \theta } \ .
\label{eq:coor2}
\end{equation}
}
\subsubsection{\rev{Model Ia.}}
\label{sec:supp-modIa}
\rev{We consider a model inspired by ref.~\cite{maslov} (the toolbox
model, in which the growth of the number of transcription factors is
coupled to the number of added metabolic enzymes), extended to
describe a joint partitioning in functional and evolutionary
categories. In the original version of the model the average
increment of the main observables at each time step is
\begin{equation}
\begin{cases}
\displaystyle
\Delta n_{met} = \frac{U}{n_{met}} \\
\Delta n_{TF} = 1 \ ,
\end{cases}
\label{eq:toolbox}
\end{equation}
and thus $\Delta n_{TF} / \Delta n_{met} = n_{met}/U$, which gives a
quadratic scaling for $n_{TF}$ with $n_{met}$.}
\rev{Model Ia is an extension of the toolbox model is formulated
following equation~\ref{eq:coor}, by using a proper definition of
$a_{i,j}$, such as the same equation of the toolbox model is
valid. We observe that, for our purpose, the time step of
equation~\label{eq:toolbox} we can be defined arbitrarily, as genome
growth is eventually parameterized by $n$. Rewriting the equations
as
\begin{equation}
\begin{cases}
\displaystyle
\Delta n_{met} = n_{met} \\
\Delta n_{TF} = n_{met} \frac{n_{met}}{U} \ ,
\end{cases}
\label{eq:toolbox2}
\end{equation}
gives the summed probabilities $p_O^i$ relative to the two categories
\begin{equation}
\begin{cases}
\displaystyle
p_O^{met}:=\sum_{i \in met} p_O^i = \frac{n_{met}-\alpha f_{met}}{C(n)} \\
p_O^{TF}:=\sum_{i \in TF} p_O^i = \frac{\frac{n_{met}}{U}
n_{met}-\alpha f_{TF}}{C(n)} \ ,
\end{cases}
\label{eq:toolbox3}
\end{equation}
while
\begin{equation}
p_N = \frac{\alpha f + \theta }{C(n)} \ .
\label{eq:coor3}
\end{equation}
}
\rev{Accordingly, we extend the model to an arbitrary number of
families by the choice $a_{i,j}=\frac{n_{met}}{U}\frac{n_i}{n_{TF}}$
if $i$ is a $TF$ family and $j$ a metabolic family and zero
otherwise. This gives
\begin{equation}
\begin{cases}
\displaystyle p_O^i =
\frac{ \sum_{j \in met} \frac{n_{met}}{U}\frac{n_i}{n_{TF}} n_j-\alpha}
{\sum^f_{i,j=1} a_{i,j} n_j + \theta} & \text{if $i \in TF$} \\
\displaystyle
p_O^i = \frac{n_i -\alpha}
{\sum^f_{i,j=1} a_{i,j} n_j + \theta}
& \text{if $i \in met$ \ .} \\
\end{cases}
\label{eq:defIa}
\end{equation}
}
\rev{ This model gives the asymptotic quadratic scaling of $n_{TF}$
with $n_{met}$ by definition, using the exact same argument as the
toolbox model. Other results have been obtained numerically (see
Supplementary Figure~\ref{fig:supp-ToolPan}).}
\subsubsection{\rev{Model Ib.}}
\label{sec:supp-modIb}
\rev{ This second formulation of a model with correlated recipe (model
Ib) imposes a different correlation rule. For example, consider the
model involving only two functional categories, transcription
factors controlling metabolic processes and metabolic enzymes. }
\rev{In this variant the coefficients $a_{i,j}$ have both a diagonal
and a non diagonal part, $a_{i,j}=\delta_{i,j}+b_{i,j}$. If $b=0$
the model is the standard Chinese Restaurant Process. For this
reason, model Ib is simpler to treat analytically, exploiting
previous results. This work focuses mainly on the case
$b_{i,j}=n_i/n_{met}$ if $i$ is a family from the functional
category of transcription factors and $j$ is a family from the
metabolic functional category (and $b_{i,j}=0$ otherwise).}
\rev{In this case, the summed probabilities $p_O^i$ relative to the
two categories are
\begin{equation}
\begin{cases}
\displaystyle p_O^i =
\frac{n_i + \sum_{j \in met} \frac{n_i}{n_{met}}-\alpha }
{\sum^f_{i,j=1} a_{i,j} n_j + \theta} & \text{if $i \in TF$} \\
\displaystyle p_O^i =
\frac{n_i -\alpha}{\sum^f_{i,j=1} a_{i,j} n_j + \theta} & \text{if $i \in met$.} \\
\end{cases}
\label{eq:defIb}
\end{equation}
Using the definitions given in Equation~\ref{eq:meanpart}, one can see
that,
\begin{equation}
\displaystyle
C(n) \partial_n n_{TF} = n_{TF} + n_{TF}
- \alpha f_{TF} + C(n) \partial_n f_{TF} =
2 n_{TF} - \alpha f_{TF} + \alpha f_{TF} + \theta \chi_{TF} = 2 n_{TF} + \theta \chi_{TF} \ ,
\label{eq:scaling}
\end{equation}
while
\begin{equation}
\displaystyle
C(n) \partial_n n_{met} = n_{met}
- \alpha f_{met} + C(n) \partial_n f_{met} = n_{met} + \theta \chi_{met} \ .
\label{eq:scaling2}
\end{equation}
Hence, for large $n$, since $ \partial_n f_{c} = \chi_{c} p_N \simeq
\alpha f_{c}$, the terms in the r.h.s. of
Equations~(\ref{eq:scaling}) and (\ref{eq:scaling2}) cancel, giving
the effective equation,
\begin{equation}
\displaystyle
\frac{d n_{TF}}{d n_{met}} \simeq \frac{2 n_{TF}}{n_{met}} \ ,
\end{equation}
and thus the scaling $n_{TF} \sim n_{met}^2$. }
\subsection{\rev{Model II (model with evolutionary potentials)} }
\label{sec:supp-nimwegen}
This section \rev{presents in more detail the uncorrelated version
of the} model \rev{for the joint scaling (model II)}, assigning
evolutionary potentials~\cite{vannim_08} $ \rho_c $ to the functional
categories, related to the probability that a gene added in a
functional category is fixed by natural selection. \rev{This model is
an example of an ``absolute recipe'', since each category grows with
an intrisic rate $\rho_c$, summing up the growth of the families
belonging to the given category.}
The rate $ \rho_c $ acts on family growth through the
\rev{class-expansion} move. \rev{The probability of class expansion of
a family belonging to the category $c$ is equal to
\begin{equation}
p_O^i =
\frac{\rho_{c(i)} n_i-\alpha}{\sum^f_{j=1}\rho_{c(j)} n_j + \theta},
\label{eq:evp1}
\end{equation}
where $ \rho_{c(i)} = \rho_c $ if the evolutionary family $ i $
belongs to the functional category $ c$. This model assumes that the
value of $\rho_c(i)$ depends only on the category to which family $i$
belongs. The probability that a domain belonging to category $c$ is
added by class expansion is then
\begin{equation}
p_O^c := \sum_{i \in c} p_O^i =
\frac{\rho_{c} n_c-\alpha f_c }{\sum^f_{j=1}\rho_{c(j)} n_j + \theta} .
\label{eq:evp1b}
\end{equation}
Equally, the probability that the new domain is introduced by an
innovation move (i.e. it belongs to a new family) is equal to
\begin{equation}
p_N = \frac{\alpha f +
\theta}{\sum^f_{j=1}\rho_{c(j)} n_j + \theta}.
\label{eq:evp2}
\end{equation}
Under the assumption (confirmed by empirical data, \rev{ see main
text}) that the growth of old functional categories by adding new
homology families through the innovation move is uniform (i.e. that $
f_c = A_c + \chi_c f $), the probability that a new family belonging
to the category $c$ is added by an innovation move is
\begin{equation}
p_N^c := \chi_c p_N = \chi_c \frac{\alpha f +
\theta}{\sum^f_{j=1}\rho_{c(j)} n_j + \theta} = \frac{\alpha f_c +
\theta \chi_c}{\sum^f_{j=1}\rho_{c(j)} n_j + \theta}.
\label{eq:evp2b}
\end{equation}
}
\subsection*{Evolutionary potentials can reproduce the combined scaling
laws at finite sizes.}
We tested this model by a combination of mean-field analytical
arguments and direct simulation.
\rev{The mean-field equations are obtained from
Equation~\ref{eq:meanpart} by using Equations~\ref{eq:evp1}
and~\ref{eq:evp2}. The equation for the growth of the mean number
of members $n_c$ of a functional category can be obtained simply by
summing on the homology families that belong to a given category,}
\begin{equation}
\partial_n n_c = \frac{ \rho_c n_c + \theta \chi_c }{C(n)} \ ,
\label{eq:catgrow1}
\end{equation}
where $ C(n) \simeq \sum_i \rho_i n_i $. If $C(n) \sim n$, equation
(\ref{eq:catgrow1}) corresponds to the evolution equation written by
Molina and Nimwegen. Simulations of this model (see Supplementary
Figure~\ref{fig:supp-cn}) confirm that this is the case. Thus, the
mean-field argument predicts that this model can reproduce both
scaling laws.
Also note that a rescaling of $C(n)$ is equivalent to a rescaling of
$\alpha$. Indeed, for large $n$, $p_N \simeq \alpha f / C(n) $ (and
$p_O = 1- p_N$), so imposing $C(n) \simeq q n$ is equivalent to
dividing $\alpha$ by the constant factor $q$. Thus, one can choose $q
= 1$ without loss of generality (by a rescaling of all the $ \rho_c
$), and the solution for the population of a functional category will
be $n_c \sim n^{\rho_c/q}$ as in the Molina/Nimwegen model, \rev{and
thus} $ \zeta_c = \rho_c / q $
On the other hand, an important point regarding this model is that,
asymptotically for any choice over the $ \rho_c $ set, the maximum
large-$n$ exponent observed will be $ 1 $, Indeed, we can use the
approximation $ C(n) = \sum_i \rho_i N_i \sim \rho_{\mathrm{max}}
n^{\rho_c/q} $, but $ C = q n $, so that $ q = \rho_{c_{max}}$. This
means that an exponent close to $2$, such as that observed for
transcription factors can only be obtained in a transient regime of
the model.
Furthermore, the change of the evolutionary potential of one
functional category has repercussions on the other categories, as it
implies a change in the normalization costant $C$. These facts make a
direct identification of the value of the evolutionary potential with
an intrinsic propery of a single functional category difficult. They
also make the direct identification of evolutionary potentials less
straightforward (as it requires an arbitrary rescaling).
However, the above remarks have little practical importance, and the
large-$n$ behaviour of the model does not really affect its performance
at the relevant values of $n$.
Numerical simulations show that at the empirical genome sizes, the
scaling behaviour of the model can reproduce rather well the empirical
one. For simplicity we have restricted to three main categories
(transcription factors, metabolic genes and ``others'') and we
verified that in practice it is not hard to find a parameter set in
good agreement with the empirical data on protein domains
(Supplementary Figure~\ref{fig:supp-comparision}). The general number
of parameters to adjust increases with the number of functional
categories that one needs to consider.
\section{Exponents of family size distribution histograms}
\label{sec:supp-histoexp}
This section discusses the family size distribution histograms, as
obtained from the mean-field approach. \rev{To fix the ideas, we will
focus on model Ib, where the mean-field equations can exploit the
known results from the CRP. } It is possible to write a mean-field
``flux equation'' for the histograms~\cite{angelini}, which implements
the fact that each duplication adds a family with one extra member to
the histogram count and subtracts a family with its previous
population,
\begin{equation}
\partial_n f(d,n) = p_O(d-1,n) f(d-1,n) - p_O(d,n) f(d,n) + p_N \delta_{d,1}
\label{eq:flux}
\end{equation}
where $ p_O(d,n) = \frac{d-\alpha}{n+\theta} $ is the probability that
a family with $ d $ domains add a new duplicated member. \rev{The term
$p_N = \frac{\alpha f + \theta}{n + \theta}$ contains the innovation
probability contributing to the growth of the number of families
with one member.} Note that the flow between families can be written
as
\begin{equation*}
\displaystyle
\sum_{
i \in \left\{
\stackrel {\mathrm{families \ with}}
{j \mathrm{\ domains}} \right\}
}
\partial_n n_i = (d-\alpha) \frac{f(d,n)}{n+\theta}.
\end{equation*}
This equation requires an assumption on $ f(d,n)$ in order to be
solved.
\rev{We assume the ansatz $ f(d,n) = P(d) f(n) $ which is justified by
both simulation and empirical data~\cite{angelini}}. Using the fact
that $ \partial_n f(n) = p_N$, combined with Equation~\ref{eq:flux}
gives the following equation for the probability of a family to have
$d$ members
\begin{equation}
\alpha P(d) = (d-1 - \alpha) P(d-1) - (d-\alpha) P(d) \ \ ,
\end{equation}
which can be solved in discrete or continuous $ d $ to get
\begin{equation}
\displaystyle
P(d) \sim \left( \frac{1}{d} \right)^{1+\alpha} \ .
\label{eq:HistoTot}
\end{equation}
This predicts the asymptotic behaviour of data and simulations (see
Figure~\ref{fig:expcorr}) with $ \beta = \alpha $, where $\beta$ is
the asymptotic exponent of the family size distribution.
\rev{Let us now turn to the same distribution, restricted to
transcription factors. } In \rev{model Ib}, the flux from
transcription factor families caused by family expansion is caused by
two separate contribution, the CRP standard one, plus \rev{additions
of transcription factors to an existing family caused by the addition of
a metabolic enzyme}
\begin{equation}
\displaystyle
p_O^{i}(n) = \frac{1}{C(n)} \left[ (n_{i} -\alpha) +
\frac{n_{i}}{n_{met}} n_{met} \right] \text{, if $i \in TF$}
\end{equation}
i.e.
\begin{equation}
\displaystyle
p_O^{i}(n) = \frac{1}{C(n)} \left[ 2 n_{i} -\alpha \right] \text{, if $i \in TF$.}
\end{equation}
\rev{ Thus, for the transcription factor families, the probability
that a domain is added to a family with $d$ members will be
\begin{equation}
\displaystyle
p_O^{TF}(d,n) = \frac{1}{C(n)} \left[ 2 d -\alpha \right] \ .
\label{eq:p_oTF}
\end{equation}
The quantity $p_O^{TF}(d,n)$ is the probability that a new
transcription factor domain is added to a family with $d$ members. }
The flux equation for TF families \rev{can be obtained by substituting
equation~\ref{eq:p_oTF} in equation~\ref{eq:flux}, }
(for $d>1$)
\begin{equation}
\displaystyle
C(n) \partial_n f_{TF}(d,n) = \left[ 2(d-1) -\alpha \right] f_{TF}(d-1,n)
- \left[ 2 d -\alpha \right] f_{TF}(d,n)
\end{equation}
This is solved using the usual ansatz $f_{TF}(d,n) = P_{TF}(d)
f_{TF}(n)$ \rev{(as explained above it is confirmed by both data and
simulations)}. Using $ f_{TF}(n) = \chi_{TF} f(n) $,
leads to the equation
\begin{equation}
\alpha P_{TF}(d) = (2d-2 - \alpha) P_{TF}(d) - (2d-\alpha) P_{TF}(d) \ \ ,
\label{eq:PPTF}
\end{equation}
which gives:
\begin{equation}
\displaystyle
P(d)_{TF} \sim \left( \frac{1}{d} \right)^{1+\frac{\alpha}{2}} \ \ ,
\label{eq:HistoTF}
\end{equation}
that is $ \beta = \alpha / 2 = \beta / 2 $. In the above calculation
we have supposed again that the number of transcription factors is
small with respect to to the total number of metabolic enzymes.
Furthermore, it can be argued that this fact is more general. Indeed,
each time the per-family duplication probability for the TF functional
category will have the form
\begin{displaymath}
p_{O}^i \simeq 2 n_i \ ,
\end{displaymath}
when family $ i $ belongs to TF category, the coefficient $2$ will
appear in the equation for $P(d)_{TF}$ modifying the exponent. In
particular, this will also be true for \rev{models Ia (generalizing the
toolbox model) and II (generalizing evolutionary potentials)}.
In other words, each time a functional category scales with a given
exponent, it can be argued on rather general grounds that the exponent
of the population histograms of the homology families that form it will
be affected.
It is possible to to generalize this argument, and find a precise
relationship between the scaling exponent of a category and the family
population histogram (restricted to the same category). In other
words, if $ \zeta_c $ is the scaling exponent of the category $ c $
and $ \beta_c $ is the exponent of the cumulative distribution
histogram for the families belonging to category $c$, that is (see
Equation~\ref{eq:HistoTF}):
\begin{displaymath}
P(d)_c \sim \left( \frac{1}{d} \right)^{1+\beta_c} \ \ ,
\end{displaymath}
we suggest that \rev{$ \beta_c = \beta / \zeta_c $}. We tested this
prediction in empirical data plotting $ 1/\beta_c $ versus $ \rho_c $
in Figure~\ref{fig:expcorr} (Pearson correlation coefficient $ 0.47
$).
\section{Comparison of models by numerical simulation}
\label{sec:supp-comparision}
\subsection{\rev{Correlated and absolute recipes}}
This section compares the correlated duplication and the evolutionary
potential model variants. We considered a three categories model (TF,
Metabolic and ``other'').
The evolutionary potential model needs to supply three parameters $
\rho_c $, while the correlated model needs to supply the correlation
law between categories ($ a_{ij} $). We impose a correlation only
between transcription factor and metabolic families with the
\rev{correlated} model \rev{Ib} prescription, i.e.
\begin{equation}
\displaystyle
a_{ij} = n_i/n_{met},
\end{equation}
where $ i $ is a TF family and $ j $ Metabolic, $ a_{ij} = 0 $ (no
correlation) otherwise.
Figure~\ref{fig:supp-comparision} summarizes the results of this
comparison. The correlated duplication model performs better in
reproducing the behavior of the transcription factor category (both
scaling law and histograms). Both models are unsatisfactory in
reproducing the family population histogram of the metabolism
families. This is probably caused by the fact that neither model
include a correlation between metabolic families (Figure
\ref{fig:correlation}).
Figure~\ref{fig:supp-cn} illustrates the behaviour of the
normalization function $ C(n) $. $ C(n) $ is linear with $ n $ in
the range of empirical genome sizes (although the slope is not
exactly $ 1 $). It becomes nonlinear at larger sizes, and its linear
behavior is restored only at very large values of $n$.
\subsection{\rev{Model I can reproduce a set of different exponents}}
\rev{Extending a model (with absolute or correlated recipe) to a large
number of categories is not a simple task.
In the case of an absolute recipe model, adding a new category $c'$
(and thus introducing a new evolutionary potential $\rho_{c'}$)
generally requires, in order preserve the scaling of all the
categories, a tuning of all the evolutionary potentials (both the
old ones and the new one). This is due to the fact that all the
evolutionary potentials appear in the normalization constant $C(n)$
in the growth equation of each category (Equation~(\ref{eq:evp1}).
In a model with a correlated recipe, the main problem is related to
the fact that the interaction laws between categories are not known,
they can be complex and possibly include feedback.}
\rev{In order to produce the proof of principle that a model with
correlated recipes can work with more than three categories, we
considered a trivial generalization of model Ib to multiple
categories that are slaved to a main one, and considered the
question of whether this model would be able to reproduce an
arbitrary set of scaling exponents for the categories.}
\rev{We consider a correlation matrix $a_{i,j}$ of the form
$\delta_{i,j}+b_{i,j}$, where $b_{i,i}=0$. This model deals with
$\cal{C}$$+ 1$ categories, the $met$ category (in analogy with model Ib
defined in the main text, this is a category whose growth is not
conditioned to the others), and an additional set of $\cal{C}$
categories labeled from $1$ to $\cal{C}$. The non diagonal
correlation coefficients $b_{i,j}$ are zero if family $i$ belongs to
the $met$ category, and $\gamma_{c(i)} n_i / n_{met}$ if family $i$
belongs to category $c$, different from $met$, and $j$ belongs to the
$met$ category. Substituting this choice in equation~\ref{eq:defIb},
gives
\begin{equation}
\displaystyle
\frac{d n_c}{d n_{met}} = (1+\gamma_c) \frac{n_c}{n_{met} }
\end{equation}
and thus
\begin{equation}
\displaystyle
n_c \sim n_{met}^{1+\gamma_c} \ .
\label{eq:predgamma}
\end{equation}
Supplementary Figure~\ref{fig:supp-pwrsim} shows simulations from a
model with $10+1$ categories. The model is able to reproduce an
arbitrary set of exponents. We observe that this version has similar
problems as the model with evolutionary potentials, as, in absence of
a biological underlying model, it needs the tuning of a set of
parameters to reproduce the scaling laws.}
\rev{The fitted exponent is typically different from $1+\gamma_c$,
specifically it seems to be closer to one. We interpret this as a
finite size effect, due to the fact that the contribution of
innovation to the scaling exponents is relevant.}
\section{Details of TF-domain superfamily scaling}
\label{sec:supp-TFclasses}
We observe that the quadratic (or very nearly so) scaling for
transcription factors is clearly visible at in the two most populated
families of transcription factor DNA-binding domains (Homeodomain-like
and Winged-helix), which have a rather clean slope (see Supplementary
Figure~\ref{fig:supp-TFclass}). In fact, three families present a
clearly observable scaling alone (Homeodomain-like, Winged-helix and
C-terminal), but just the first two follow a very nearly quadratic
scaling.
Note however that removing the six most populated TF families,
representing $80\%$ of the total TF-domain population, the remaining
ones considered together still present a scaling when added up, but
with exponent \rev{$\simeq 0.9 $ (see Supplementary
Figure~\ref{fig:supp-smalltf})}.
This indicates that the collective scaling of TF families cannot be
entirely recunducted to properties of the most populated ones, but
these are the families responsible for the \emph{quadratic} scaling.
\rev{ Thus, the ``pure'' quadratic scaling is observable in the
largest transcription factor families. Collecting all the families,
wemeasure a lower exponent in empirical data (close to
$1.6$). Supplementary Figure~\ref{fig:supp-smalltf} explains this
behavior, showing the total contribution of the smaller
transcription factor families. These families collectively show a
lower exponent (close to $1$). Thus, we can interpret the lower
collective exponent as an effect of family size (i.e., in the
language of statistical mechanics, a ``finite-size'' effect)
connected to the fact that for smaller family size, the innovation
move is more relevant and thus the family expansion process is
slower. The same effect is present in our simulations (see
Supplementary Figure~\ref{fig:supp-ExpSim}.)}
\clearpage
\newpage
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\columnwidth]{vannimSF.pdf}
\end{center}
\caption{ { \bf Scaling of the number of families in the three main
functional categories.} Linear scaling behaviour of the number of
families in three important functional categories versus total
number of families from empirical data (for $ 753 $ bacteria in the
SUPERFAMILY database). The slopes for the three linear laws are $
0.01 $ (Translation), $ 0.03 $ (Regulation of Transcription) and $
0.47 $ (Metabolic Processes).}
\label{fig:supp-vannimSF}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\columnwidth]{bpTFsupfam.pdf}
\end{center}
\caption{ { \bf Transcription factor families.} Boxplot of the
number of transcription factor domain families versus total number
of domain families (data from 753 SUPERFAMILY bacteria). There
appears to be a roughly linear scaling. This means that the number
of TF domain families is compatible with a null hypothesis of
independent addition model. Charoensawan \emph{et al}~\cite{derek}
propose that the number of TF families follows a linear scaling with
genome \emph{size}. If this were to be the case, the innovation
dynamics of transcription factor families should be distinct form
other families. In fact, if $ f_{TF}(n)\sim n $, since the total
number of families is sublinear, $ f(n) \sim n^{\alpha} $ in the CRP
(Figure~\ref{fig:partitioning}), then one would have $
f_{TF}~f^{2-\alpha} $, which is not confirmed by the SUPERFAMILY
data analyzed here. }
\label{fig:supp-bpTFsupfam}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.8\columnwidth]{comparision.pdf}
\end{center}
\caption{ { \bf Comparison between models Ib and II.} Comparison
between simulation of the correlated duplication model Ib (left
panel) and evolutionary potentials (right panel) model variants with
empirical data. Simulations are run at $ \alpha=0.3 $ and $ \theta =
140 $. (a) Number of TFs domains vs. number of metabolic domains
(the blue boxplot corresponds to simulations, red circles to
empirical data). (b) Number of metabolic domains vs. total number
of domains (the blue boxplot corresponds to simulations, red
circles to empirical data). (c) Family population histograms
restricted to the transcription factor functional category (black
circles are simulations, magenta lines empirical data). (d) Family
population histograms restricted to the metabolism functional
category (black circles are simulations, magenta lines empirical
data). }
\label{fig:supp-comparision}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{ToolPan.pdf}
\end{center}
\caption{ \rev{ { \bf Simulations of the correlated duplication model
Ia for two categories (transcription factors and metabolic
enzymes).} The plots are obtained from $1000$ realizations with
$\alpha=0.3$, $\theta=140$ and $U=7000$. The observables are the
same as in figure \ref{fig:supp-comparision}. (a) scaling of the
number of transcription factors with the number of metabolic
enzymes. (b) Number of families as a function of genome size
$n$. (c) Family population (cumulative) histograms. (c) Family
population histograms restricted to the families belonging to the
transcription factor functional category. } }
\label{fig:supp-ToolPan}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{gene.pdf}
\end{center}
\caption{ { \bf Linear relation between the number of domains and the
number of genes.} (a) Number of Domains vs. number of
protein-coding genes for the 753 bacteria in the SUPERFAMILY
database. There are, on average, $ 1.45 $ domains per gene. (b)
Linear scaling behaviour of the number of TF domains vs. number of
TF genes. There are, on average, $ 1.09 $ TF domains in a TF gene.}
\label{fig:supp-gene}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\columnwidth]{simTFsupfam.pdf}
\end{center}
\caption{ { \bf Simulation of the number of transcription factor
families.} Comparison between empirical data and simulations of
the number of transcription factor domain families plotted against
total number of families. The scaling is empirically linear,
i.e. the number of TF domain families is reproduced by a null
hypothesis of independent addition model. The choice of the
parameter is $ 0.035 $. }
\label{fig:supp-simTFsupfam}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{cn.pdf}
\end{center}
\caption{ { \bf Normalization constant inthe model with evolutionary
potentials (model II).} Behavior of the ratio $ C(n)/n $, where $
C(n) $ is the normalization factor for the evolutionary potential
model. Data from simulations with three categories run at parameters
$ \alpha = 0.3 $ and $ \theta = 140 $. $ C(n) $ is linear with $ n
$ in the range of empirical genome sizes, it then looses linearity,
to become linear only asymptotically.
\label{fig:supp-cn}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{pwrlaw-pan.pdf}
\end{center}
\caption{ \rev{{ \bf Simulation of model Ib with $10+1$ categories.}
$10$ categories are slaved to one master category with different
correlation laws, which determine the observed exponents). Panel
A, B and C show the simulations of the population of three
categories (respectively with $\gamma_c$ equal to $1$, $0$ and
$-0.7$). The red lines are power-law fits of the simulated
data. Panel D shows the power-law fits of the simulated data for
all ten categories.}}
\label{fig:supp-pwrsim}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{MetCorrSize.pdf}
\end{center}
\caption{ { \bf Correlation matrix for two sets of genomes with
different sizes. } Left panel: Correlation matrix for genomes with
size $ < 4000 $. Right panel: Correlation matrix for genome with
size $ > 4000 $. The correlations do not depend on size.}
\label{fig:supp-metcorrsize}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.9\columnwidth]{TFclass.pdf}
\end{center}
\caption{ { \bf Most populated transcription factor superfamilies.}
Boxplots for the population of the six most populated superfamilies
of TF DNA-binding domains (y-axis in each panel) versus number of
domains of each genome (x-axis in each panel). The presence of
scaling laws appears likely for the three most populated families
and arguable for the first five. Red lines represent best power law
fit ($ 1.8 $ for Winged Helix ,$ 2.1 $ for Homeodomain-like and $
1.7 $ for C-terminal effector)}
\label{fig:supp-TFclass}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\columnwidth]{SmallTF.pdf}
\end{center}
\caption{\rev{ { \bf Scaling of the least populated transcription factor superfamilies.}
Collective scaling of the number of
transcription factor domains after removing the six globally most
populated families. While a few genomes show large fluctuations
from the typical trend, a clear scaling is still observable for
most genomes, with a fitted exponent equal to $0.9$} }
\label{fig:supp-smalltf}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\columnwidth]{ExpSim.pdf}
\end{center}
\caption{MARCO\rev{ { \bf Finite-size effects on the scaling exponent
$\zeta_{TF}$ for transcription factors in simulations of model
Ib.} The plot shows the fitted exponent (y-axis) from the curve
of the number of transcription factor domains versus the number of
metabolic enzymes in $500$ simulated realizations of model Ib with
parameter $\alpha=0.3$ and $\theta=140$. Each point on the x-axis
corresponds to simulated data stopped at a given size $n$. The
mean-field prediction ($\zeta_{TF} = 2$) is reached only in the
limit $n \to \infty$. This plot shows that the fitted exponent
$1.6$ (instead of $2$) for the growth of transcription factors vs
metabolic domains is due to a finite-size effect of a process that
produces an exponent $2$ in the large-$n$ limit. The same effect
is present in models Ia and II.} }
\label{fig:supp-ExpSim}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=0.6\columnwidth]{scatterExp.pdf}
\end{center}
\caption{\rev{ { \bf Ratio between exponents of family population
histograms.} The plot reports the ratio $\beta / \beta_{TF}$
between the exponent of the total family population histograms and
the histograms restricted to the transcription factor families
(see Figure~\ref{fig:histoexp} in the main text), as a function of
genome size. The values of the ratio are distributed around $1.6$
and the fluctuation range decreases with increasing genome size.}
}
\label{fig:supp-scatterExp}
\end{figure}
\clearpage
\newpage
\begin{table}[tbp]
\caption{ {\bf
Fitted values of $\chi_c$ and offsets $A_c$ from $f_c$ vs $f$ for the ten largest functional categories} }
\begin {tabular}{|c|c|c|c|}
\hline
& $A_c$ & $ \chi_c $ & Reduced chi square \\
\hline
Transcription Factors & $ 2.2 \pm 0.4 $ & $ 0.0267 \pm 0.0006 $ & $4.5$ \\
\hline
Translation & $ 61.0 \pm 0.35 $ & $ 0.0133 \pm 0.0006 $ & $3.9$\\
\hline
Small molecule binding & $ 3.0 \pm 0.2 $ & $ 0.01 \pm 0.0002 $ & $0.9$\\
\hline
Nucleotide transport and metabolism & $ 5.6 \pm 0.3 $ & $ 0.02 \pm 0.0005 $ & $3.1$ \\
\hline
DNA replication/repair & $ 9.5 \pm 0.6 $ & $ 0.0437 \pm 0.0009 $ & $9.8$\\
\hline
Inorganic ion transport and metabolism & $ 0.2 \pm 0.4 $ & $ 0.0272 \pm 0.0005 $ & $3.5$\\
\hline
Redox & $ -7.6 \pm 0.5 $ & $ 0.0592 \pm 0.0008 $ & $7.9$ \\
\hline
Transferases & $ 5.3 \pm 0.2 $ & $ 0.0213 \pm 0.0004 $ & $1.6$\\
\hline
Other enzymes & $ -14.8 \pm 1.1 $ & $ 0.155 \pm 0.002 $ & $35.7$\\
\hline
Signal transduction & $ -3.2 \pm 0.3 $ & $ 0.0282 \pm 0.0005 $ & $3.3$\\
\hline
\end{tabular}
\begin{flushleft}
\rev{ The number of evolutionary families belonging to a functional
category follows a linear law in empirical data. The table reports
fits of $f_c = A_c + \chi_c f$ from the plots in
Figure~\ref{fig:Chi_c} of the main text, where $f_c$ represents the
number of families in category $c$ on all genomes and $f$ is the
total number of families on the genome. The third column is the
reduced chi square. }
\end{flushleft}
\label{tab:fit-catfam}
\end{table}
\begin{table}[tbp]
\caption{ {\bf \rev{Data of fitted exponents from Figure \ref{fig:expcorr}
of the main text, for the ten largest functional categories}} }
\begin {tabular}{|c|c|c|}
\hline
& $\zeta_c$ & $ \beta_c $ \\
\hline
Transcription Factors & $ 1.6 \pm 0.02 $ & $ 0.47 \pm 0.01 $ \\
\hline
Translation & $ 0.176 \pm 0.003 $ & $ 1.46 \pm 0.02 $ \\
\hline
Small molecule binding & $ 0.918 \pm 0.006 $ & $ 0.25 \pm 0.01 $ \\
\hline
Nucleotide transport and metabolism & $ 0.61 \pm 0.01 $ & $
0.71 \pm 0.01 $ \\
\hline
DNA replication/repair & $ 0.54 \pm 0.01 $ & $ 0.9 \pm 0.01 $ \\
\hline
Inorganic ion transport and metabolism & $ 1.40 \pm 0.02 $ & $
0.46 \pm 0.01 $ \\
\hline
Redox & $ 1.3 \pm 0.01 $ & $ 0.52 \pm 0.02 $ \\
\hline
Transferases & $ 1.09 \pm 0.01 $ & $ 0.43 \pm 0.01 $ \\
\hline
Other enzymes & $ 1.09 \pm 0.01 $ & $ 0.64 \pm 0.01 $ \\
\hline
Signal transduction & $ 1.77 \pm 0.03 $ & $ 0.4 \pm 0.01 $ \\
\hline
\end{tabular}
\begin{flushleft}
\end{flushleft}
\label{tab:fit-LAW}
\end{table}
\begin{table}[!ht]
\caption{
\bf{Correlation coefficients between the populations of metabolic
functional categories}}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& En & e- & Ph & Aa & N & Co & Nu & Ca & Li & Ps & Ce & 2M & Rx & Tr & Ot \\
\hline
En & $ 1 $ & $ 0.14 $ & $ 0.07$ & $ 0.55 $ & $ 0.23 $ & $ 0.36 $ & $ 0.19 $ &
$ -0.06 $ & $ -0.08 $ & $ -0.14 $ & $ 0.02 $ & $ 0.22 $ & $ 0.31 $ & $ -0.10 $ & $ -0.004 $ \\
\hline
e- & $ 0.14 $ & $ 1 $ & $ 0.29 $ & $ 0.15 $ & $ 0.11 $ & $ 0.43 $ & $ -0.09 $ &
$ -0.52 $ & $ 0.35 $ & $ -0.29 $ & $ 0.13 $ & $ 0.19 $ & $ 0.47 $ & $ 0.09 $ & $ 0.05 $ \\
\hline
Ph & $ 0.07 $ & $ 0.29 $ & $ 1 $ & $ 0.12 $ & $ 0.21 $ & $ -0.02 $ & $ -0.09 $ &
$ -0.16 $ & $ -0.21 $ & $ 0.14 $ & $ -0.18 $ & $ 0.15 $ & $ 0.06 $ & $ 0.16 $ & $ -0.05 $ \\
\hline
Aa & $ 0.55 $ & $ 0.15 $ & $ 0.12 $ & $ 1 $ & $ 0.08 $ & $ 0.39 $ & $ 0.19 $ &
$ -0.14 $ & $ -0.07 $ & $ -0.22 $ & $ 0.01 $ & $ 0.07 $ & $ 0.40 $ & $ 0.02 $ & $ 0.14 $ \\
\hline
N & $ 0.23 $ & $ 0.11 $ & $ 0.21 $ & $ 0.08 $ & $ 1 $ & $ -0.13 $ & $ -0.08 $ &
$ -0.003 $ & $ -0.14$ & $ -0.09 $ & $ 0.09 $ & $ 0.26 $ & $ 0.04 $ & $ -0.03 $ & $ -0.02 $ \\
\hline
Co & $ 0.36 $ & $ 0.43 $ & $ -0.02 $ & $ 0.39 $ & $ -0.13$ & $ 1 $ & $ 0.14 $ &
$ -0.33 $ & $ 0.44 $ & $ -0.37 $ & $ -0.04 $ & $ 0.08 $ & $ 0.51 $ & $ 0.12 $ & $ 0.16 $ \\
\hline
Nu & $ 0.19 $ & $ -0.09 $ & $ -0.09 $ & $ 0.19 $ & $ -0.08 $ & $ 0.14 $ & $ 1 $
& $ -0.03 $ & $ -0.09 $ & $ -0.10 $ & $ -0.02 $ & $ -0.10 $ & $ 0.03 $ & $ -0.11 $ & $ -0.13 $ \\
\hline
Ca & $ -0.06 $ & $ -0.52 $ & $ -0.16 $ & $ -0.14 $ & $ -0.003 $ & $ -0.33 $ &
$ -0.03 $ & $ 1 $ & $ -0.20 $ & $ 0.53 $ & $ -0.18 $ & $ 0.02 $ & $ -0.46 $ & $ -0.11 $ & $ 0.16 $ \\
\hline
Li & $ -0.08 $ & $ 0.35 $ & $ -0.21 $ & $ -0.07 $ & $ -0.14 $ & $ 0.44 $ &
$ -0.09 $ & $ -0.20 $ & $ 1 $ & $ -0.35 $ & $ 0.15 $ & $ 0.18 $ & $ 0.06 $ & $ 0.13 $ & $ 0.20 $ \\
\hline
Ps & $ -0.14 $ & $ -0.29 $ & $ 0.14 $ & $ -0.22 $ & $ -0.09 $ & $ -0.37 $ &
$ -0.10 $ & $ 0.53 $ & $ -0.35 $ & $ 1 $ & $ -0.12 $ & $ -0.05 $ & $ -0.36 $ & $ 0.09 $ & $ -0.07 $ \\
\hline
Ce & $ 0.02 $ & $ 0.13 $ & $ -0.18 $ & $ 0.01 $ & $ 0.09 $ & $ -0.04 $ & $ -0.02 $ &
$ -0.18 $ & $ 0.15 $ & $ -0.12 $ & $ 1 $ & $ -0.0002 $ & $ 0.01 $ & $ -0.22 $ & $ -0.31 $ \\
\hline
2M & $ 0.22 $ & $ 0.19 $ & $ 0.15 $ & $ 0.07 $ & $ 0.26 $ & $ 0.08 $ & $ -0.10 $ &
$ 0.02 $ & $ 0.18 $ & $ -0.05 $ & $ -0.0002 $ & $ 1 $ & $ -0.11 $ & $ 0.20 $ & $ 0.08 $ \\
\hline
Rx & $ 0.31 $ & $ 0.47 $ & $ 0.06 $ & $ 0.40 $ & $ 0.04 $ & $ 0.51 $ & $ 0.03 $ & $ -0.46 $ &
$ 0.06 $ & $ -0.36 $ & $ 0.01 $ & $ -0.11 $ & $ 1 $ & $ -0.10 $ & $ 0.14 $ \\
\hline
Tr & $ -0.10 $ & $ 0.09 $ & $ 0.16 $ & $ 0.02 $ & $ -0.03 $ & $ 0.12 $ & $ -0.11 $ &
$ -0.11 $ & $ 0.13 $ & $ 0.09 $ & $ -0.22 $ & $ 0.20 $ & $ -0.10 $ & $ 1 $ & $ 0.17 $ \\
\hline
Ot & $ -0.004 $ & $ 0.05 $ & $ -0.05 $ & $ 0.14 $ & $ -0.02 $ & $ 0.16 $ & $ -0.13 $
& $ 0.16 $ & $ 0.20 $ & $ -0.07 $ & $ -0.31 $ & $ 0.08 $ & $ 0.14 $ & $ 0.17 $ & $ 1 $ \\
\hline
\end{tabular}
}
\begin{flushleft}
Pearson's correlation coefficients between the populations of 24
different metabolic functional categories from the SUPERFAMILY
database for 753 bacteria. Correlations are calculated from
fluctuations of categories from the average trend (see
Methods). Both correlation and anticorrelation are present between
categories. Metabolism categories are highly (anti-)correlated. We
used the following short forms for the metabolic functional
categories: En = Energy p/c, e- = Electrons transfer, Ph =
Photosynthesis, Aa = Amino acids m/tr, N = Nitrogen m/tr, Co =
Coenzyme m/tr, Nu = Nucleotide m/tr, Ca = Carbohydrate m/tr, Li =
Lipid m/tr, Ps = Polysaccharide m/tr, Ce = Cell envelope m/tr, 2M =
Secondary metabolism, Rx = Redox, Tr = Transferases, Ot = Other
enzymes. Where m/tr stands for ``metabolism and trasportation'' and
p/c means ``production and conversion''.
\end{flushleft}
\label{tab:supp-correlation}
\end{table}
\begin{table}[!ht]
\caption{
\bf{P-Values of correlation coefficients between the populations of metabolic
functional categories}}
\resizebox{\columnwidth}{!}{
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& En & e- & Ph & Aa & N & Co & Nu & Ca & Li & Ps & Ce & 2M & Rx & Tr & Ot \\
\hline
En &$ 0 $ & $ \mathbf {5 \cdot 10^{-5}} $ & $ \mathbf { 0.02 } $ & $ \mathbf { <10^{-6}} $ & $ \mathbf { <10^{-6}} $ &
$\mathbf { <10^{-6}} $ &$\mathbf { <10^{-6}} $ & $ 0.05 $ & $\mathbf { 0.01} $ & $\mathbf { 4 \cdot 10^{-5} }$ & $ 0.26 $ &
$ \mathbf {<10^{-6} }$ & $ \mathbf {<10^{-6}} $ & $ \mathbf {4 \cdot 10^{-3}} $ & $ 0.46 $ \\
\hline
e- &$ \mathbf {5 \cdot 10^{-5} }$ & $ 0 $ & $\mathbf { <10^{-6} }$ & $ \mathbf { 2 \cdot 10^{-5}} $ &$ \mathbf {1 \cdot 10^{-3}} $ &
$\mathbf { <10^{-6} }$ & $\mathbf { 8 \cdot 10^{-3} }$ & $\mathbf { <10^{-6}} $ & $ \mathbf { <10^{-6}} $ &
$ \mathbf {<10^{-6} }$ & $ \mathbf {3 \cdot 10^{-4}} $ & $ \mathbf {1 \cdot 10^{-6} }$ & $\mathbf { <10^{-6} }$ & $\mathbf {7 \cdot 10^{-3}}$ & $ 0.08 $ \\
\hline
Ph &$ \mathbf {0.02} $ & $ \mathbf {<10^{-6}} $ & $ 0 $ & $ \mathbf {1 \cdot 10^{-3}} $ & $ \mathbf {<10^{-6}} $ & $ 0.29 $ &
$ \mathbf {2 \cdot 10^{-3}} $ & $\mathbf { <10^{-6}} $ & $ \mathbf {<10^{-6} }$ & $\mathbf { 2 \cdot 10^{-4}} $ & $ \mathbf {<10^{-6} }$
& $ \mathbf {5 \cdot 10^{-5}} $ & $ 0.06 $ & $ \mathbf {2 \cdot 10^{-5}} $ & $ 0.08 $ \\
\hline
Aa & $ \mathbf {<10^{-6}} $ & $ \mathbf {2 \cdot 10^{-5}} $ & $ \mathbf {1 \cdot 10^{-3}} $ & $ 0 $ & $ \mathbf {0.02} $ &
$ \mathbf {<10^{-6}} $ & $ \mathbf {2 \cdot 10^{-6}} $ & $ \mathbf {4 \cdot 10^{-5}} $ & $ \mathbf {0.02} $ & $ \mathbf {<10^{-6}} $ &
$ 0.39 $ & $\mathbf { 0.03} $ & $\mathbf { <10^{-6}} $ & $ 0.28 $ & $ \mathbf {5 \cdot 10^{-5}} $ \\
\hline
N &$ \mathbf { <10^{-6}} $ & $ \mathbf {1 \cdot 10^{-3}} $ & $\mathbf { <10^{-6}} $ & $ \mathbf {0.02} $ & $ 0 $ & $\mathbf { 2 \cdot 10^{-4}} $
& $ \mathbf {0.01} $ & $ 0.47 $ & $ \mathbf {7 \cdot 10^{-5} }$ & $ \mathbf {5 \cdot 10^{-3}} $ & $ \mathbf {8 \cdot 10^{-3}} $ &
$ \mathbf {<10^{-6}} $ & $ 0.13 $ & $ 0.18 $ & $ 0.31 $ \\
\hline
Co &$ \mathbf {<10^{-6} }$ & $\mathbf { <10^{-6} }$ & $ 0.29 $ & $ \mathbf {<10^{-6}} $ & $ \mathbf {2 \cdot 10^{-4}} $ & $ 0 $ &
$\mathbf { 1 \cdot 10^{-4}} $ & $\mathbf { <10^{-6} }$ & $ \mathbf {<10^{-6}} $ & $ \mathbf {<10^{-6}} $ & $ 0.13 $ & $ \mathbf {0.02} $ &
$ \mathbf {<10^{-6}} $ & $ \mathbf {2 \cdot 10^{-4}} $ & $\mathbf { 4 \cdot 10^{-6}} $ \\
\hline
Nu & $\mathbf { <10^{-6}} $ & $ \mathbf {8 \cdot 10^{-3}} $ & $ \mathbf {2 \cdot 10^{-3}} $ & $ \mathbf {2 \cdot 10^{-6}} $
& $\mathbf { 0.01} $ & $ \mathbf {1 \cdot 10^{-4}} $ & $ 0 $ & $ 0.20 $ & $ \mathbf {5 \cdot 10^{-3}} $ & $ \mathbf {3 \cdot 10^{-3}} $
& $ 0.26 $ & $ \mathbf {3 \cdot 10^{-3}} $ & $ 0.17 $ & $\mathbf { 8 \cdot 10^{-3}} $ & $ \mathbf {9 \cdot 10^{-5}} $ \\
\hline
Ca & $ 0.05 $ & $ \mathbf { <10^{-6}} $ & $\mathbf { <10^{-6} } $ & $\mathbf { 4 \cdot 10^{-5}} $ & $ 0.47 $ &
$ \mathbf {<10^{-6}} $ & $ 0.20 $ & $ 0 $ & $\mathbf { <10^{-6}} $ & $ \mathbf {<10^{-6} }$ & $\mathbf { <10^{-6}} $ & $ 0.30 $ &
$ \mathbf {<10^{-6}} $ & $ \mathbf {8 \cdot 10^{-4}} $ & $\mathbf { 7 \cdot 10^{-6}} $ \\
\hline
Li & $\mathbf { 0.01} $ & $ \mathbf {<10^{-6}} $ & $ \mathbf {<10^{-6}} $ & $ \mathbf {0.02} $ & $ \mathbf {7 \cdot 10^{-5}} $ &
$ \mathbf {<10^{-6}} $ &
$\mathbf { 5 \cdot 10^{-3}} $ & $ \mathbf {<10^{-6}} $ & $ 0 $ & $\mathbf { <10^{-6}} $ & $ \mathbf {3 \cdot 10^{-5}} $ & $\mathbf { <10^{-6}} $ &
$ 0.06 $ & $ \mathbf {3 \cdot 10^{-4}} $ & $\mathbf { 2 \cdot 10^{-6}} $ \\
\hline
Ps & $\mathbf { 4 \cdot 10^{-5}} $ & $ \mathbf {<10^{-6} }$ & $\mathbf { 2 \cdot 10^{-4}} $ & $ \mathbf {<10^{-6}} $ &
$\mathbf { 5 \cdot 10^{-3}} $ & $\mathbf { <10^{-6} }$ & $\mathbf { 3 \cdot 10^{-3}} $ & $ \mathbf {<10^{-6}}$ & $\mathbf { <10^{-6}} $ & $ 0 $ &
$\mathbf { 5 \cdot 10^{-4}} $ & $ 0.07 $ & $ \mathbf {<10^{-6}} $ & $\mathbf { 6 \cdot 10^{-3}} $ & $\mathbf { 0.03 }$ \\
\hline
Ce &$ 0.26 $ & $ 3 \cdot 10^{-4} $ & $ <10^{-6} $ & $ 0.39 $ & $ 8 \cdot 10^{-3} $ &
$ 0.13 $ & $ 0.26 $ & $ <10^{-6} $ & $ 3 \cdot 10^{-5} $ & $ 5 \cdot 10^{-4} $ & $ 0 $ &
$ 0.50 $ & $ 0.38 $ & $ \mathbf{<10^{-6}} $ & $ \mathbf{<10^{-6}} $ \\
\hline
2M & $ \mathbf {<10^{-6}} $ & $ \mathbf {1 \cdot 10^{-6} } $ & $ \mathbf {5 \cdot 10^{-5}} $ & $\mathbf { 0.03 }$ & $\mathbf { <10^{-6}} $ &
$ \mathbf {0.02} $ & $ \mathbf {3 \cdot 10^{-3}} $ & $ 0.30 $ & $\mathbf { <10^{-6}} $ & $ 0.07 $ & $ 0.50 $ & $ 0 $ &
$\mathbf { 8 \cdot 10^{-4} }$ & $\mathbf { <10^{-6}} $ & $ \mathbf {0.01} $ \\
\hline
Rx & $ \mathbf {<10^{-6}} $ & $\mathbf { <10^{-6}} $ & $ 0.06 $ & $\mathbf { <10^{-6}} $ & $ 0.13 $ & $ \mathbf {<10^{-6}} $ & $ 0.17 $ &
$ \mathbf {<10^{-6}} $ & $ 0.06 $ & $\mathbf { <10^{-6} }$ & $ 0.38 $ & $\mathbf { 8 \cdot 10^{-4}} $ & $ 0 $ &
$ \mathbf {3 \cdot 10^{-3}} $ & $ \mathbf {4 \cdot 10^{-5} }$ \\
\hline
Tr & $ \mathbf {4 \cdot 10^{-3}} $ & $ \mathbf {7 \cdot 10^{-3}} $ & $ \mathbf {2 \cdot 10^{-5}} $ & $ 0.28 $ & $ 0.18 $ &
$\mathbf { 2 \cdot 10^{-4}} $ & $\mathbf { 8 \cdot 10^{-4}} $ & $ \mathbf {8 \cdot 10^{-4}} $ & $ \mathbf {3 \cdot 10^{-4}} $ &
$\mathbf { 6 \cdot 10^{-3}} $ & $ \mathbf {<10^{-6} }$ & $\mathbf { <10^{-6}} $ & $ \mathbf {3 \cdot 10^{-3}} $ & $ 0 $ & $ \mathbf {1 \cdot 10^{-6}} $ \\
\hline
Ot & $ 0.46 $ & $ 0.08 $ & $ 0.08 $ & $ \mathbf {5 \cdot 10^{-5}} $ & $ 0.31 $ & $ \mathbf {4 \cdot 10^{-6}} $ &
$ \mathbf {9 \cdot 10^{-5}} $ & $ \mathbf {7 \cdot 10^{-6}} $ & $ \mathbf {2 \cdot 10^{-6}} $ & $ \mathbf {0.03} $ & $ \mathbf {<10^{-6}} $ &
$ \mathbf {0.01} $ & $ \mathbf {4 \cdot 10^{-5}} $ & $ \mathbf {1 \cdot 10^{-6}} $ & $ 0 $ \\
\hline
\end{tabular}
}
\begin{flushleft}
P-values of the Pearson's correlation coefficients between the
populations of 24 different metabolic functional categories from the
SUPERFAMILY database for 753 bacteria \rev{(the most significant
values are in boldface)}. Correlations are calculated from
fluctuations of categories from the average trend (see Methods).
The (anti-)correlation is statistically significant for the most of
the metabolic categories. We used the following short forms for the
metabolic functional categories: En = Energy p/c, e- = Electrons
transfer, Ph = Photosynthesis, Aa = Amino acids m/tr, N = Nitrogen
m/tr, Co = Coenzyme m/tr, Nu = Nucleotide m/tr, Ca = Carbohydrate
m/tr, Li = Lipid m/tr, Ps = Polysaccharide m/tr, Ce = Cell envelope
m/tr, 2M = Secondary metabolism, Rx = Redox, Tr = Transferases, Ot =
Other enzymes. Where m/tr stands for ``metabolism and
trasportation'' and p/c means ``production and conversion''.
\end{flushleft}
\label{tab:supp-pvalcorrelation}
\end{table}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,587 |
Psyllaephagus lusitanicus é uma espécie de insetos himenópteros, mais especificamente de vespas parasíticas pertencente à família Encyrtidae.
A autoridade científica da espécie é Mercet, tendo sido descrita no ano de 1921.
Trata-se de uma espécie presente no território português.
Referências
Psyllaephagus lusitanicus - de Jong, Y.S.D.M. (ed.) (2013) Fauna Europaea version 2.6. Web Service available online at http://www.faunaeur.org (consultado em 12 de janeiro de 2014).
Ligações externas
Psyllaephagus lusitanicus - Biodiversity Heritage Library - Bibliografia
Psyllaephagus lusitanicus - NCBI Taxonomy Database
Psyllaephagus lusitanicus - Global Biodiversity Information Facility
Psyllaephagus lusitanicus - Encyclopedia of Life
Himenópteros de Portugal
lusitanicus
Himenópteros descritos em 1921 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,580 |
\section{Introduction}
Systematic differences between oxygen abundance patterns of thin and thick disk
stars have been found by different authors using either the [O I] 630 nm line or LTE analyses of
the O~I 777 nm triplet in relatively small samples (e.g., Prochaska et al. 2000,
Bensby et al. 2004). These abundance trends provide tight constraints for models of
Galaxy formation and chemical evolution.
The O I triplet has not been extensively used for oxygen abundance
determinations due to strong non-LTE effects and saturation. These lines are, however, easy to
observe and are potentially useful if non-LTE corrections are
properly accounted for. Here we use spectra from McDonald Observatory, the Hobby-Eberly Telescope, and
the UVES-VLT database to determine Fe and O abundances in a sample of
about 450 disk stars.
The thin/thick disk membership criterion we adopted is described in Mishenina
et~al. (2004). It essentially uses the Galactic space velocities $U,V,W$ of the sample stars
and a recent thin/thick disk parameterization. We derived effective temperatures using the color calibrations by Ramirez \&
Melendez (2005), which are based on the infrared flux
method (IRFM). Surface gravities were determined from accurate \textit{Hipparcos}
parallaxes and an estimate of the stellar masses from theoretical isochrones.
Iron abundances were derived by comparing the equivalent widths of the observed
lines with those predicted by Kurucz no-overshooting models. We used about 150 Fe I and 15 Fe II
lines, all of which have transition probabilities measured in the laboratory and
damping constants theoretically computed as in Barklem et al. (2000). A similar approach was used to derive LTE oxygen
abundances. Restricted non-LTE corrections for the oxygen triplet were calculated using
TLUSTY (Hubeny \& Lanz 1995) and the oxygen model atom of Allende Prieto et al. (2003).
\medskip
With the ingredients described above for the Fe abundance determination,
we were not able to obtain the same mean Fe abundance from Fe I and Fe II lines. The Fe II
abundances are larger by about 0.07 dex for F and
early G dwarfs and giants. The discrepancy worsens for late-type and metal-rich dwarfs but for giants
the difference remains constant. This inconsistency may be revealing a
systematic error in the IRFM effective temperature scale, non-LTE effects on Fe line
formation, limitations of classical model atmospheres, or a combination of them. We adopted the Fe II abundance as our [Fe/H] indicator due to
its relatively weak dependence on effective temperature (most of our sample stars are hotter
than about 5400 K) and small predicted non-LTE effects.
Our oxygen abundance trends are shown in Fig.~\ref{f:f2} for the dwarf stars with
5400~K$<T_\mathrm{eff}<6000$~K and thin/thick disk membership probabilities larger than
80\% (this is about 1/3 of the total sample). In the range $-0.8<$[Fe/H]$<-0.2$, thick disk stars have larger [O/Fe] and [O/H] compared to
thin disk stars. The thick disk [O/Fe] ratio is nearly constant at about 0.5
while the thin disk [O/Fe] ratio slowly declines towards solar values.
Note that a few kinematically selected thick disk stars seem to follow the thin disk
abundance pattern, and viceversa. This suggests that the kinematic criterion alone is
incomplete. In fact, if an abundance criterion is considered, no thick disk
stars would probably be found in our sample at [Fe/H]$>-0.2$.
\begin{figure}
\centering
\includegraphics[bb=8 384 595 750,width=11cm]{ramirez_f1.eps}
\caption{Oxygen abundance trends for our sample dwarf stars with 5400~K$<T_\mathrm{eff}<6000$~K and thin/thick disk membership probabilities larger than 80\%. Filled circles: thick disk stars, open
circles: thin disk stars. The dotted lines correspond to solar values.}
\label{f:f2}
\end{figure}
\medskip
The fact that the oxygen abundance trends are well separated suggests that thin
and thick stars were formed from different mixtures of gas. Along with other Galactic and extragalactic
evidence, the merger scenario for thick disk formation has been favored (see Bensby et al. 2004 for details and
references). An interesting result is that we do not find many high metallicity thick disk
stars and we do not obtain a decline in the [O/Fe] ratio of thick disk stars,
attributed to SNIa contribution to the gas that formed thick disk stars by
Bensby et~al. (2004). Since our sample selection did not include a [Fe/H] criterion,
this may be indicating an upper limit to the [Fe/H] distribution of thick disk stars at
[Fe/H]$\sim-0.2$. Ongoing observations will help us to clarify this important matter.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,390 |
Passenger trains are set to run between a Dorset coastal resort and the mainline rail network for the first time in four decades.
The Swanage Railway heritage line said June 2016 was its target date for trial services between Swanage and Wareham.
The original rail line connecting the town was ripped up in 1972, before a 5.5-mile (8.8km) stretch was restored by volunteers as a tourist attraction.
A new £500,000 level crossing has been installed as part of the work.
The trial service from summer 2016 will see diesel trains run by the Swanage Railway connecting with South West Trains scheduled services at Wareham, on the Weymouth to London Waterloo line.
The new level crossing across the Wytch Farm oil field and a park and ride access road near Corfe Castle has been funded by BP and Perenco.
Project director Mark Woolley said it was a "key element" to allow passenger services to progress.
"It's fulfilling the objectives of our pioneer members who campaigned against the line's closure and then helped reopen it," he added.
Work has also been carried out to replace 1,700 wooden sleepers with concrete ones, clear embankments and repair bridges, fences and drains.
Issues including ownership of the line and nearby hibernating animals led to delays in the work earlier this year which resulted in the trial date being put back.
The Dorset line from Swanage to Wareham was closed by British Rail and ripped up in seven weeks.
Volunteers rebuilt the line from Swanage as far as Norden over 30 years and have been running it as a tourist attraction since the late 1990s.
Their long-term ambition is to reconnect the full 10-mile stretch between Swanage, Corfe Castle and Wareham with a scheduled community passenger service. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,459 |
{"url":"https:\/\/uwaterloo.ca\/academic-integrity\/integrity-instructors-and-tas\/academic-integrity-student-forms","text":"Academic Integrity Student Forms\n\nTo encourage academic honesty, instructors should\u00a0ask students to complete an academic integrity form when submitting assignments and tests. Instructors can insert these forms into their assignments or tests, which verify that students have completed the assessment in line with academic integrity expectations.","date":"2020-08-05 13:09:15","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8083257675170898, \"perplexity\": 6553.083832136254}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439735958.84\/warc\/CC-MAIN-20200805124104-20200805154104-00504.warc.gz\"}"} | null | null |
Take the Bus Tour From New York and See the Amish Farm*, Longwood Gardens*, Hershey Park*, Hershey Chocolate World. The East Coast Has Many More Great Sights. Make Sure to Book Your Bus Tour Through Tripedition Today.
Today the bus tour will spend the day at Hershey Park and Chocolate World. You will visit the sweetest place in the world; Hersheys Chocolate World. Here you will be greeted by giant Hersheys characters as you enter this wonderful world of food, shops and entertainment devoted to all things chocolate. You will also learn how chocolate is made at a special presentation. You will also enjoy the day at Hershypark which has over 110 acres and over sixty rides and attractions.
Edison Location is ONLY available from 5/25 to 9/3. Return location is the same. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,537 |
#import "UAProductDetailiPadViewController.h"
@implementation UAProductDetailiPadViewController
- (id)initWithNibName:(NSString *)nibNameOrNil bundle:(NSBundle *)nibBundleOrNil {
if (self = [super initWithNibName:nibNameOrNil bundle:nibBundleOrNil]) {
self.navigationItem.rightBarButtonItem.enabled = NO;
webViewHeight = 0;
}
return self;
}
- (void)didRotateFromInterfaceOrientation:(UIInterfaceOrientation)fromInterfaceOrientation {
[detailTable reloadData];
}
#pragma mark -
#pragma mark TableView
- (NSInteger)tableView:(UITableView *)tableView numberOfRowsInSection:(NSInteger)section {
return 1;
}
- (UITableViewCell *)tableView:(UITableView *)view cellForRowAtIndexPath:(NSIndexPath *)indexPath {
if (product.previewURL == nil) {
return [super tableView:view cellForRowAtIndexPath:indexPath];
}
UIImage *bgImage = [UIImage imageNamed:@"middle-detail.png"];
UIImage *stretchableBgImage = [bgImage stretchableImageWithLeftCapWidth:20 topCapHeight:0];
UIImageView *bgImageView = [[[UIImageView alloc] initWithImage:stretchableBgImage] autorelease];
NSString* text = product.productDescription;
UIFont *font = [UIFont systemFontOfSize: 16];
UIWebView *webView = [[[UIWebView alloc] init] autorelease];
NSString *htmlString = [self constructHtmlForWebViewWithDescription:text AndImageURL:product.previewURL];
[webView loadHTMLString:htmlString baseURL:nil];
[webView setBackgroundColor:[UIColor clearColor]];
[webView setOpaque:0];
[webView setDelegate:self];
CGFloat height = [text sizeWithFont: font
constrainedToSize: CGSizeMake(280.0, 800.0)
lineBreakMode: UILineBreakModeWordWrap].height;
[webView setFrame:CGRectMake(0.0f, 10.0f, 320.0f, height)];
[webView setBounds:CGRectMake(0.20f, 0.0f, 290.0f, height)];
[webView setAutoresizingMask:UIViewAutoresizingFlexibleWidth];
UITableViewCell *cell = [[[UITableViewCell alloc] initWithStyle:UITableViewCellStyleDefault
reuseIdentifier: @"description-cell"]
autorelease];
[cell addSubview:webView];
[cell setSelectionStyle:UITableViewCellSelectionStyleNone];
[cell setBackgroundView:bgImageView];
return cell;
}
- (CGFloat)tableView:(UITableView *)view heightForRowAtIndexPath:(NSIndexPath *)indexPath {
if (product.previewURL == nil || webViewHeight == 0) {
return [super tableView:view heightForRowAtIndexPath:indexPath];
} else {
CGFloat height = webViewHeight;
return height + kCellPaddingHeight;
}
}
#pragma mark -
#pragma mark WebView
- (NSString *)constructHtmlForWebViewWithDescription:(NSString *)description AndImageURL:(NSURL *)imageURL {
return [NSString stringWithFormat:@"<html> <body style=\"background-color: transparent; font-family: helvetica; font-size: 16px;\"> <img width=\"160\" src=\"%@\" align=\"right\" /> %@ </body> </html>",
[imageURL description], description];
}
- (void)webViewDidFinishLoad:(UIWebView *)view {
[view sizeToFit];
NSString *output = [view stringByEvaluatingJavaScriptFromString:@"document.height;"];
int currentHeight = [output intValue];
if (currentHeight == webViewHeight) {
return;
}
webViewHeight = currentHeight;
[detailTable reloadData];
}
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,362 |
Helena Leiva de Holst (Santa Cruz de Yojoa, 5 de octubre de 1897-San Pedro Sula, 23 de agosto de 1978) fue una socialista que viajó a China y a la Unión Soviética para estudiar marxismo. Su política izquierdista la llevó de exilio a Guatemala, donde participó en causas feministas. En la década de 1950, Che Guevara vivió con ella y le dedicó parte de su poesía. Cuando el golpe de Estado patrocinado por United Fruit derrocó al presidente guatemalteco Jacobo Árbenz, Leiva fue exiliada a México. Finalmente pudo regresar a Honduras a fines de la década de 1960.
Biografía
Leiva Ferrera nació el 5 de octubre de 1897 en Santa Cruz de Yojoa, Departamento de Cortés (Honduras), es hija de Emilio Leiva y Florinda Ferrera. Leiva era la nieta del presidente hondureño Ponciano Leiva. Cuando era niña, se mudó a San Pedro Sula para sus estudios y fue allí donde se casó con el empresario alemán, Henry Holst. Debido a sus opiniones políticas de izquierda, Leiva se vio obligada a exiliarse en Guatemala. Leiva había viajado tanto a China como a la Unión Soviética para estudiar marxismo y, además, apoyaba las causas feministas, asistiendo al Primer Congreso Interamericano de Mujeres celebrado en la Ciudad de Guatemala en 1947. También fue una de las líderes de la Alianza de Mujeres.
Leiva continuó con sus actividades de izquierda, proporcionando alojamiento para el Che Guevara, quien fue tan cautivado como ella que describía su filosofía como "cercana al comunismo" y le dedicó un poema de 1954, "Invitación al camino". A mediados de julio de 1954, fue arrestada durante el golpe de Estado de United Fruit contra el presidente guatemalteco Jacobo Árbenz y fue exiliada en México. Después de muchos años de vivir en el exilio, finalmente pudo regresar a su hogar y vivió en San Pedro Sula por el resto de su vida.
Helena Leiva falleció el 23 de agosto de 1978 en San Pedro Sula, departamento de Cortés, Honduras.
Referencias
Sufragistas
Activistas de Honduras
Feministas de Honduras
Marxistas | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,515 |
Q: AWS Lambda function results in timeout whenever database call is present I have an API Gateway with some AWS Lambda functions that make a database call (using sequelize.js) and return the result as a JSON object.
When I run the code locally I get the results, but when I deploy the code, any method that has a database call in it results in an internal server error.
The following code returns the message:
module.exports.testFunction = (event, context, callback) => {
return callback(null, {
statusCode: 200,
body: JSON.stringify({
message: 'Test function is working.'
})
});
};
The following code logs the database results in CloudWatch, but the API call times out after 6 seconds and then an internal server error (502) is returned:
module.exports.getAll = (event, context, callback) => {
Entity.findAll().then(result => {
console.log(JSON.stringify(result));
return callback(null, {
statusCode: 200,
body: JSON.stringify(result)
});
});
};
Any solutions to make the return work?
A: First of all congratulations, that you were able to successfully identify the issue. However, there are many concepts that can be learnt from the occurred issue. Let's discuss few of them.
Lambda cold start issue
To understand the concept of cold start in lambda it is inevitable to go any further without understanding how AWS Lambda works. Please refer to the undermentioned link which explains the overall working of lambda as shared in AWS re:Invent 2018
AWS Lambda under the hood (Video Link) : https://www.youtube.com/watch?v=QdzV04T_kec
Coming back to the issue of cold start, as our lambda functions are executed in a container, post execution the container gets killed if there are no further function invocations within ~15 minutes of inactivity. Once the container is killed, any future call to execute the same function will require setting up a new container which can take >5 seconds (most probable reason behind your api call getting terminated post 6 seconds). However, there are many available options available that can keep your lambda warm. Please refer the undermentioned link from Serverless community.
Keeping Lambda Functions Warm (Blog Link) : https://serverless.com/blog/keep-your-lambdas-warm/
Increasing Lambda timeout
"I had to go into the AWS console and increase the timeout to 20 seconds".
Although, this approach is totally acceptable but since you are using server less technology (explicit serverless.yml file), you can directly change the default lambda timeout period (which is 6 seconds). Please refer to the undermentioned code snippet and link for further understanding.
provider:
name: was
runtime: nodejs6.10
memorySize: 512 # optional, in MB, default is 1024
timeout: 10 # optional, in seconds, default is 6
Serverless AWS Lambda function Guide (Blog Link) : https://serverless.com/framework/docs/providers/aws/guide/functions/
A: Adding another answer, as I found what was the actual problem.
Setting the timeout to 20 made it work, but every call took around 10s, not only cold start. However, the query results were already received after just a few milliseconds.
Apparently AWS Lambda passes a context argument in handler functions and one of its properties is callbackWaitsForEmptyEventLoop, which defaults to true:
callbackWaitsForEmptyEventLoop – Set to false to send the response right away when the callback executes, instead of waiting for the Node.js event loop to be empty. If false, any outstanding events will continue to run during the next invocation.
I set the property to false in the beginning of the function and now every call only takes ~70ms.
module.exports.getAll = (event, context, callback) => {
context.callbackWaitsForEmptyEventLoop = false;
Entity.findAll().then(result => {
console.log(JSON.stringify(result));
return callback(null, {
statusCode: 200,
body: JSON.stringify(result)
});
});
};
A: I fixed it. I had to go into the AWS console and increase the timeout to 20 seconds. So the lambda function works but takes a very long time to run. Amazon Support now recommended using X-Ray traces to see which calls take long.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,475 |
package org.apache.cloudstack.api.command.user.vpn;
import org.apache.log4j.Logger;
import org.apache.cloudstack.api.APICommand;
import org.apache.cloudstack.api.ApiConstants;
import org.apache.cloudstack.api.ApiErrorCode;
import org.apache.cloudstack.api.BaseAsyncCmd;
import org.apache.cloudstack.api.Parameter;
import org.apache.cloudstack.api.ServerApiException;
import org.apache.cloudstack.api.BaseCmd.CommandType;
import org.apache.cloudstack.api.response.DomainResponse;
import org.apache.cloudstack.api.response.ProjectResponse;
import org.apache.cloudstack.api.response.Site2SiteCustomerGatewayResponse;
import org.apache.cloudstack.context.CallContext;
import com.cloud.event.EventTypes;
import com.cloud.network.Site2SiteCustomerGateway;
@APICommand(name = "createVpnCustomerGateway", description = "Creates site to site vpn customer gateway", responseObject = Site2SiteCustomerGatewayResponse.class, entityType = {Site2SiteCustomerGateway.class},
requestHasSensitiveInfo = false, responseHasSensitiveInfo = false)
public class CreateVpnCustomerGatewayCmd extends BaseAsyncCmd {
public static final Logger s_logger = Logger.getLogger(CreateVpnCustomerGatewayCmd.class.getName());
private static final String s_name = "createvpncustomergatewayresponse";
/////////////////////////////////////////////////////
//////////////// API parameters /////////////////////
/////////////////////////////////////////////////////
@Parameter(name = ApiConstants.NAME, type = CommandType.STRING, required = false, description = "name of this customer gateway")
private String name;
@Parameter(name = ApiConstants.GATEWAY, type = CommandType.STRING, required = true, description = "public ip address id of the customer gateway")
private String gatewayIp;
@Parameter(name = ApiConstants.CIDR_LIST, type = CommandType.STRING, required = true, description = "guest cidr list of the customer gateway")
private String peerCidrList;
@Parameter(name = ApiConstants.IPSEC_PSK, type = CommandType.STRING, required = true, description = "IPsec Preshared-Key of the customer gateway. Cannot contain newline or double quotes.")
private String ipsecPsk;
@Parameter(name = ApiConstants.IKE_POLICY, type = CommandType.STRING, required = true, description = "IKE policy of the customer gateway")
private String ikePolicy;
@Parameter(name = ApiConstants.ESP_POLICY, type = CommandType.STRING, required = true, description = "ESP policy of the customer gateway")
private String espPolicy;
@Parameter(name = ApiConstants.IKE_LIFETIME,
type = CommandType.LONG,
required = false,
description = "Lifetime of phase 1 VPN connection to the customer gateway, in seconds")
private Long ikeLifetime;
@Parameter(name = ApiConstants.ESP_LIFETIME,
type = CommandType.LONG,
required = false,
description = "Lifetime of phase 2 VPN connection to the customer gateway, in seconds")
private Long espLifetime;
@Parameter(name = ApiConstants.DPD, type = CommandType.BOOLEAN, required = false, description = "If DPD is enabled for VPN connection")
private Boolean dpd;
@Parameter(name = ApiConstants.FORCE_ENCAP, type = CommandType.BOOLEAN, required = false, description = "Force Encapsulation for NAT traversal")
private Boolean encap;
@Parameter(name = ApiConstants.ACCOUNT, type = CommandType.STRING, description = "the account associated with the gateway. Must be used with the domainId parameter.")
private String accountName;
@Parameter(name = ApiConstants.DOMAIN_ID,
type = CommandType.UUID,
entityType = DomainResponse.class,
description = "the domain ID associated with the gateway. If used with the account parameter returns the "
+ "gateway associated with the account for the specified domain.")
private Long domainId;
@Parameter(name = ApiConstants.PROJECT_ID, type = CommandType.UUID, entityType = ProjectResponse.class,
description = "create site-to-site VPN customer gateway for the project")
private Long projectId;
/////////////////////////////////////////////////////
/////////////////// Accessors ///////////////////////
/////////////////////////////////////////////////////
public String getName() {
return name;
}
public String getIpsecPsk() {
return ipsecPsk;
}
public String getGuestCidrList() {
return peerCidrList;
}
public String getGatewayIp() {
return gatewayIp;
}
public String getIkePolicy() {
return ikePolicy;
}
public String getEspPolicy() {
return espPolicy;
}
public Long getIkeLifetime() {
return ikeLifetime;
}
public Long getEspLifetime() {
return espLifetime;
}
public Boolean getDpd() {
return dpd;
}
public Boolean getEncap() { return encap; }
public String getAccountName() {
return accountName;
}
public Long getDomainId() {
return domainId;
}
public Long getProjectId() {
return projectId;
}
/////////////////////////////////////////////////////
/////////////// API Implementation///////////////////
/////////////////////////////////////////////////////
@Override
public String getCommandName() {
return s_name;
}
@Override
public long getEntityOwnerId() {
Long accountId = _accountService.finalyzeAccountId(accountName, domainId, projectId, true);
if (accountId == null) {
accountId = CallContext.current().getCallingAccount().getId();
}
return accountId;
}
@Override
public String getEventDescription() {
return "Create site-to-site VPN customer gateway for account " + getEntityOwnerId();
}
@Override
public String getEventType() {
return EventTypes.EVENT_S2S_VPN_CUSTOMER_GATEWAY_CREATE;
}
@Override
public void execute() {
Site2SiteCustomerGateway result = _s2sVpnService.createCustomerGateway(this);
if (result != null) {
Site2SiteCustomerGatewayResponse response = _responseGenerator.createSite2SiteCustomerGatewayResponse(result);
response.setResponseName(getCommandName());
setResponseObject(response);
} else {
throw new ServerApiException(ApiErrorCode.INTERNAL_ERROR, "Failed to create customer VPN gateway");
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,237 |
Poachers in Zimbabwe have killed more than 300 elephants and countless other safari animals by cyanide poisoning. The full extent of the carnage in Hwange, the country's largest national park, has been revealed by legitimate hunters who discovered what conservationists said was the worst single massacre in southern Africa for 25 years. Pictures obtained exclusively by The Sunday Telegraph show horrific scenes.
Parts of the national park, which is visited by thousands of tourists each year, can be seen from the air to be littered with the corpses of elephants, often with their calves dead beside them, as well as those of other animals. There is deep concern that the use of cyanide represents a new and devastating technique in the rapidly growing poaching trade. Zimbabwean authorities said that 90 animals have been killed this way.
But the hunters who captured these photographs said they counted the corpses of more than 300. Poachers killed the elephants over the past three months by planting buckets of water laced with cyanide in the sand. Animals are drawn to them during the dry season in the already arid and remote south-eastern section of the 5,660-square mile park. After the elephants died, often collapsing just a few yards from the source, lions, hyenas and vultures that fed on their carcasses were also struck down, as were other animals such as kudu and buffalo that shared the same water.
Zimbabwe's authorities claim that the cyanide has been planted by villagers who sell the elephants' tusks for around pounds 300 each to cross-border traders. They can be resold in South Africa for up to pounds 10,000 a pair, according to court papers relating to one recent incident, sometimes re-emerging as carved artefacts such as bangles in Cape Town's craft markets. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,925 |
Tag Archives: IMPD
Politics, Budgets and Taxes
September 4, 2013 Random Blogginghomestead credit, IMPD, Indianapolis budgetSheila
The other day, an advocate for the homeless asked me why the needs of the most vulnerable citizens always seem to take a back seat to the demands of sports teams, developers, and bright shiny objects like cricket fields. He attributed this state of affairs to animus against the needy, but–as I told him–I don't think that's it. It's just that politicians respond to pressure from people who show up–people who contact them, who vote and especially people who donate.
The problem we face when allocating public resources is that very few of us who benefit from inequities that unfairly burden others are willing to graciously concede those advantages. It's too easy to convince ourselves that we are entitled to them.
When the Indiana Supreme Court ruled a few years ago that the system that had yielded grossly unequal property tax assessments for years had to be fixed, the homeowners who had benefitted from artificially low assessments–and whose taxes had accordingly been lower than those of folks with far less valuable properties–screamed bloody murder. Rather than sheepishly acknowledging that they'd made out like bandits for years, and that perhaps it was time to pay their fair shares, they saw themselves as victims of a rapacious government and took their revenge by ousting a hapless Mayor who'd had nothing to do with that particular decision.
Fast forward to Mayor Ballard's proposed budget.
I've not been a fan of this Mayor, but his proposed equalization of the tax rate for IMPD is both fair and overdue. For decades, center city folks were taxed to support both the sheriff's department (which has county-wide jurisdiction) and IPD (which patrolled only the old city limits). When the two departments were combined into IMPD, apparently the tax rates were not adjusted accordingly. As a result, those residing within the old city limits continued to pay more for police protection than those living outside those limits. As I understand it, Ballard's proposal would equalize the tax and end what has effectively been an unfair subsidy of some citizens by others–and those who've benefitted are (predictably) whining about having to pay their fair share.
Since this post is likely to make me even more unpopular than I already am, I will add that I also support the Mayor's proposal to eliminate the homestead credit in order to pay for the addition of desperately needed police.
Would I prefer that we shift funds from cricket fields and sports teams and too-generous subsidies to the Mayor's developer buddies instead? Of course. Is that likely to happen? Not in my lifetime. Let's recognize that politics is the art of the possible, and address our public safety deficit before crime rates that approach Detroit's undermine every other thing we are trying to do in our city.
Speaking of homestead credits, we really should invest in efforts to ameliorate the plight of the people who don't have a homestead. There are steps we could take now that would actually save tax dollars in the long run.
But we probably won't because they don't scream and vote, and they aren't in a position to make campaign contributions. And because, to our politicians, the "long run" is the next election.
Disturbing Questions
March 18, 2012 Local Governmentdowntown canal, IMPD, public safety, shootingSheila
We woke this morning to news reports that five teenagers had been shot while walking along Indianapolis' downtown canal. The shots evidently came from the parking lot of the Historical Society–where a wedding was taking place at the time.
As I write this, little is known except that two of the teens are in critical condition and no one is currently in custody.
It may be that this was one of those random acts that no city, no matter how safe or well-run, can prevent. We deceive ourselves if we believe that police can guard against every sudden eruption of violence. But this shooting, in the heart of our city and next to the canal that so many of us routinely walk or bike, raises sobering questions.
First, what is the relationship–if any–between the recent "discovery" of fiscal shortfalls in public safety and what some people living along the canal claim was a diminished police presence? (The fiscal situation itself raises very troubling questions about the honesty of the Administration's budgeting process during an election year.)
Second, if there were fewer police in the area, was that due to deliberate decisions about deployment, and if so, what were those decisions and why were they made? One story suggested that a number of officers were called to a brawl at the fire station at West and Ohio; do we have so few police that an incident in one place necessarily leaves other areas unprotected?
We don't know the answers to these questions, and asking them is not meant to assume the answers. But the questions need to be asked, because this event will have repercussions far beyond the personal tragedy it represents.
Civic and political leaders have been nurturing the rebirth of downtown since the 1970s. The canal is one of the "jewels" of that effort–a jewel that has been sadly neglected the past few years, as I have previously noted. It is an important amenity in a city without oceans or mountains. Developers have been enticed to make significant investments along its banks; museums and public buildings adjoin it. Maintaining it and keeping it safe for the residents and tourists who enjoy it is an important responsibility and should be a high priority of the current Administration.
When the media is filled with stories of shootings, when on-camera interviews feature onlookers declaring they no longer feel safe in the area, the result is to undercut years of painstaking effort, and to reinforce inaccurate stereotypes about the "dangers" of downtown.
Perhaps this was one of those random events that even the best policing couldn't have averted. Perhaps it was the result of public safety mis-management.
Or perhaps we are seeing the inevitable results of the anti-tax zealotry that added tax caps to the Indiana Constitution–tax caps that are starving local governments and decimating public services.
Whatever the answer, we need to find and fix it.
Greg Ballard's Curious Approach to Fiscal Discipline
June 15, 2011 Public Policy and GovernanceBallard, fiscal responsibility, IMPD, PrideSheila
There has been a good deal of discussion on local blogs about our Mayor's ham-handed approach to the just-concluded Gay Pride celebration.
The Indianapolis fire department has participated in the Parade previously, and this year, IMPD announced that it, too, would participate–and show that our local police serve all parts of the Indianapolis community. The day before the Parade, Ballard unexpectedly reversed course, and told IFD it could not use a city fire truck, and IMPD that it could not officially march at all. (Several members of the police department did march, in uniform, but in their "individual" capacity, and the department's Hummer was nowhere in evidence.)
Yesterday, Mayor Ballard was interviewed by Amos Brown, who asked an entirely appropriate–and foreseeable–question: why had the Mayor prevented the police from driving an official vehicle in the parade? The obviously bogus response was that the decision was made in order to save tax dollars. It had nothing to do with the fact that this was a gay event, or that Micah Clark and the Indiana Family Institute pitched a fit about the symbolism of treating the gay community like all other taxpaying citizens. Nope–just being fiscally responsible.
I asked a friend of mine who is a police officer whether IMPD officially participated in other community celebrations, and he rattled off a list: St. Patrick's Day, Veterans Day, Black Expo and several others. I guess those constituencies must be more deserving of the tax expenditures involved.
And that brings up an interesting question: just how many dollars are we talking about?
What is the cost of vehicle depreciation and gasoline during a trip down Massachusetts Avenue? Ten dollars? Five?
Yesterday, the media reported that the Ballard administration stands to lose a three-million-dollar Federal grant, because it hasn't complied with the grant's staffing requirements. This makes Ballard the poster child for "Penny wise, pound foolish." | {
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Q: Powershell Module where to store user settings I'm developing a PowerShell based build automation toolbox (http://www.crawler-lib.net/powershell-build-tools-samples). It is a powershell module This toolbox will be installed in the 32Bit and 64 Bit PowerShell module directory, because it can be executed via the NuGet Console in Visual Studio (which is a 32 Bit PowerShell host) and the PowerShell ISE (which is 64 Bit on 64 Bit OS).
I want to provide the developers and users of the toolbox an easy way to set their tools locations or override the automatic detection of the toolbox. So I have to store a properties or XML file with the settings for this. What is a good (standard) way to store such settings for a powershell module?
A: Personally, I prefer to use:
$AppName = [System.IO.Path]::GetFileNameWithoutExtension($PSCommandPath);
$myCompanyName = "myCompanyName";
* use [System.Environment+SpecialFolder]::LocalApplicationData if you do not want to put it in a roaming profile
$ApplicationDataFolder = [System.Environment]::GetFolderPath([System.Environment+SpecialFolder]::ApplicationData);
* build the path
$myStorageFolder = ($ApplicationDataFolder | Join-Path -ChildPath $myCompanyName) | Join-Path -ChildPath $AppName;
A: Most tools I see follow Unix convention and put user preference files in the user's home directory. For example, Mercurial puts its user config file there, named mercurial.ini. There are other locations that Windows wants these files, but I like the user's root: it is the easiest place to find and get to.
Now you just get to decide on format. :-) I like JSON, since it isn't as verbose as XML, but I think XML might be easier in your users. There is much better tooling for XML and it's been around longer.
| {
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❮ # 84 Aliko Dangote # 86 Jensen Huang ❯
# 85 Peter Woo $19.0B
Random fact: Unsuccessfully bid to be Hong Kong's first chief executive.
Woo is the largest shareholder of Wharf Real Estate Investment, a developer of commercial and residential property. The Hong Kong-based company had revenue of HK$16 billion ($2.1 billion) in 2021. Woo owns the shares through Wheelock & Co., the real estate group that delisted from the stock market in 2020.
Last change -$5.02M ( -0.0%)
YTD change +$77.2M ( +0.4%)
Biggest asset 1997 HK Equity
Country / Region Hong Kong
Industry Real Estate
Peter Woo's net worth of $19.0B can buy ...
The majority of Woo's fortune is derived from investments associated with Wheelock & Co., a Hong Kong-based real estate group. It owns property directly and through stakes in publicly traded subsidiaries Wharf Holdings and Wharf Real Estate Investment.
Wheelock was delisted from the Hong Kong stock exchange on July 23, 2020, and his net worth calculation was adjusted on that date to reflect the terms of the delisting. Woo proposed taking Wheelock private in February 2020 and offered investors one share in each of Wharf Real Estate Investment and Wharf Holdings as well as HK$12 cash for each Wheelock share they held.
Woo owns 49% of Wharf Real Estate Investment and 60% of Wharf Holdings through Wheelock, based on the privatization agreement. His full ownership of the delisted Wheelock is valued based on the delisting price of HK$12 per share.
The value of his cash investments is based on an analysis of dividends, insider transactions, taxes and market performance.
He retains a 6% stake in the Italian shoemaker Salvatore Ferragamo, and has a 7% stake in Beijing-based developer Longfor Group. These are held through various investment companies registered in the British Virgin Islands.
Agnes Hui, a Hong Kong-based spokeswoman for Wheelock, said that the billionaire declined to comment on his net worth.
Woo has born in Shanghai in 1946, the only child of an architect and his wife. At age 4, he moved with his family to Hong Kong, which was undergoing a post-war construction boom. Woo attended St. Stephens College Preparatory School, where he played basketball, volleyball and hockey.
His father encouraged him to attend the University of Cincinnati, which had a prestigious architecture school. Woo majored in mathematics and physics, and was elected senior class president. He graduated valedictorian and enrolled in Columbia University's MBA program, where he met his future wife, Bessie Pao, the second of four daughters of Hong Kong shipping magnate Sir Yue-kong Pao. The couple was married in 1973, and Woo took a job at Chase Manhattan Bank.
Pao invited Woo to work for him at World-Wide Shipping Group. His father-in-law shared ownership of Hong Kong & Kowloon Wharf & Godown Co. with the British trading house Jardine Matheson, and outbid his former partner for control in a hostile takeover. He later named it Wharf Holdings. Pao acquired Kowloon waterfront land, the Hong Kong Tramway and the Star Ferry as part of the takeover. He promoted Woo to be chief operating officer of Wharf and vice chairman of Worldwide Shipping in 1982.
Pao took over another British trading house, Wheelock Marden, three years later in a corporate raid that included prime real estate in Hong Kong's Central business district plus the Lane Crawford department store. Pao retired in 1986 and named Woo chief executive officer of Wheelock and Wharf. Woo structured Wheelock as the group's parent company and Wharf as the main subsidiary, and further diversified into cable TV, telecommunications and broadband internet in Hong Kong, and into property development in Singapore and China. Pao died in 1991.
Woo's flagship properties -- Harbour City in Kowloon and Times Square in central Hong Kong -- combined attract 9% of Hong Kong's retail sales. The two malls represent almost half of his group's business assets.
The billionaire retired as chairman of Wheelock and Wharf on May 19, 2015, according to a statement to the Hong Kong stock exchange dated Feb. 16, 2015. He was replaced by his son, Douglas Woo Chun-kuen.
1946 Woo is born in Shanghai.
1950 Family moves to Hong Kong.
1970 Graduates from University of Cincinnati in math and physics.
1971 Earns MBA from Columbia University, hired by Chase Manhattan.
1973 Marries Bessie Pao, daughter of shipping magnate Yue-kong Pao.
1975 Father-in-law offers him a job at World Wide Shipping.
1986 Woo takes over property development business after Pao retires.
2000 Appointed chairman of Hong Kong's Trade Development Council.
2007 Steps down from Trade Development Council to run Wharf.
2015 Son Douglas takes over as chairman of Wheelock. | {
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Q: Building an array listener Is there a way to attach a listener to an array that is being built by another process so that when array.length reaches certain value, the listener executes a callback and fires appropriate code?
fooArray = [];
// another code adds objects to array
if (fooArray.length = 5) {
// callback
} else {
// continue checking fooArray.length
}
A: You have to use own methods for changing in array
Array.prototype.myPush = function () {
this.push.apply( this, arguments );
if ( this._binder && this._binder.test.call( this ) ) {
this._binder.callback.call( this );
}
}
Array.prototype.bind = function (tester, callback) {
this._binder = {
test: tester,
callback: callback
};
}
var arr = [4, 6, 9, 20];
arr.bind( function () {
return this.length >= 5;
}, function () {
console.log( "Array has exceeded" );
} );
arr.myPush( 40 );
A: Well, yes and no. There's no way to do it in a 100% safe way, because JavaScript doesn't support operator overloading, meaning you can't modify the behavior of the array indexing operator ([]).
However, it's rare that arrays are added to with the array indexing operator. That is, this is a very uncommon (but valid) way to add items to an array:
var arr = [];
for( i=0; i<5; i++ ) arr[i] = i;
The result will be an array with five elements. More commonly, though, you see this:
var arr = [];
for( i=0; i<5; i++ ) arr.push( i );
Now that's something we can work with. So now the solution is to wrap the original functionality of the push function with additional functionality that raises an event.
There are three basic ways to do this. The first is to modify Array.prototype.push. I would steer you clear of that approach: the chances of causing extremely hard-to-track-down defects is to high. The second is to create your own array type that extends JavaScript's built-in Array type. This would be the best approach if you'll be creating a lot of these arrays that need listening to. The last approach is just to modify an individual array, which is the least work, and the approach I'll describe here.
You can get as sophisticated as you want with any of these approaches, but I'll just give a pretty minimal foundation, that allows you to attach arbitrary listeners, and could be extended for other functionality (removing listeners, raising events on different array modification methods, etc.).
First, you'll need an array, and a property to hold listeners:
var arr = [];
arr._listeners = [];
Then you can modify the push method accordingly:
arr.push = function() {
var arr = this;
Array.prototype.push.apply( arr, arguments );
arr._listeners.forEach( function(listener) {
listener( { event: 'push', newLength: arr.length } );
});
}
You can pretty much put anything in that event object you want (like old count, element being added, a timestamp, etc.)
To make this robust, you'll have to use this technique for every other method of Array that modifies the array, such as pop, shift, etc. See all the methods of Array here:
https://developer.mozilla.org/en-US/docs/Web/JavaScript/Reference/Global_Objects/Array/prototype#Methods
Depending on your needs, there are additional techniques you could consider. For example, if you do a lot of wrapping of functions (like we did above with push), you should consider some kind of aspect-oriented programming (AOP) framework like meld. You could pretty easily write your own, too. For example, we could re-work the above thusly:
function after( f, after ) {
return function() {
f.apply( this, arguments );
after.apply( this, arguments );
}
}
var arr = [];
// convenience function for broadcasting an event
arr._broadcast = function( o ) {
this._listeners.forEach( function(listener) {
listener( o );
}
}
arr.push = after( arr.push, function() {
this._broadcast( { event: 'push', newLength: arr.length } );
});
The advantage of that approach is that it makes it easier to extend the other methods of Array:
arr.pop = after( arr.pop, function() {
this._broadcast( { event: 'pop', newLength: arr.length } );
});
And so forth.
If you need to be able to make many of these "broadcasting" arrays, you could, as previously mentioned, create your own custom array type extending Array, but JavaScript provides oh so many more flexible patterns for code re-use than traditional OOP. So instead of creating a EventArray that extends Array, you could just create a function of Array.prototype that would magically make that instance an event emitter:
Array.prototype.makeBroadcaster = function() {
this._listeners = this._listeners || [];
this._broadcast = function( o ) {
this._listeners.forEach( function(listener) {
listener( o );
}
this.registerListener = function( listener ) {
this._listeners.push( listener );
}
this.push = after( this.push, function() {
this._broadcast( { event: 'push', newLength: arr.length } );
}
this.pop = after( this.push, function() {
this._broadcast( { event: 'push', newLength: arr.length } );
}
this.shift = after( this.push, function() {
this._broadcast( { event: 'shift', newLength: arr.length } );
}
// etc....
}
Now, whenever you have an array that you want to broadcast what happens to it, all you have to do is this:
var arr = [];
arr.makeBroadcaster();
arr.addListener( function(evt) {
console.log( "%s (new length: %d)", evt.event, evt.newLength );
} );
Some closing notes:
*
*As I mentioned at the start of the post, if your code modifies the array using the array indexing operator, there's no way to raise an event on that. Specifically, arrays can be "grown" using the array indexing operator. In other words, if you array contains 5 elements, and you execute arr[9] = 'hello', your array will now have ten elements in it (with 5-8 being undefined). There's no way to detect this, but it's pretty uncommon that arrays are modified this way.
*You can also modify the length of an array with the length property. That is, if you have a ten-element array, and you call arr.length = 5, it will truncate the array to have only five elements. It's possible that this could be detected by overriding the length property (using Object.defineProperty), but that sounds really dangerous if it's possible at all. And again, truncating an array in this way is pretty uncommon in JavaScript.
*If the events really absolutely have to be 100% accurate, even in the cases above, well, your best bet would be able to create your own object that has basic array functions like push and the like. It will be very awkward, though, because you'll have to implement something like get(index) instead of using the more convenient array indexing operators.
Here's a jsfiddle demonstrating the whole thing in action: http://jsfiddle.net/g68jC/
Isn't JavaScript neat?
| {
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infant Falkner
M, b. 16 January 1875, d. 16 January 1875
Father: John Wesley Thompson Falkner b. 2 Sep 1848, d. 13 Mar 1922
Mother: Sarah A. Sally Murry b. 14 Oct 1850, d. 21 Dec 1906
Death*: Infant Falkner died on 16 January 1875 at Ripley, Tippah Co., MS.1
Birth*: He was born on 16 January 1875 at MS.1
He was the son of John Wesley Thompson Falkner and Sarah A. Sally Murry.
Infant Falkner1
Father: James William Falkner1 b. c 1837, d. 29 Jul 1862
Mother: Tabitha Robertson1 b. c 1826
Birth*: Infant Falkner was born in 1859 at Granville Co., NC; She was listed as Infant, age 1, female, in the 1860 Census.1
She was the daughter of James William Falkner and Tabitha Robertson.1
Jacob Falkner
Birth*: Jacob Falkner was born before 1740; Estimated based on Land Patent.
James Falkner
Father: Moses Falkner b. c 1754, d. a 1830
Mother: Susannah Saulter b. c 1766
Marriage*: James Falkner married (?) Bennett.
Birth*: James Falkner was born circa 1803 at Orange Co., NC.
He was the son of Moses Falkner and Susannah Saulter.
Family: (?) Bennett
Moses Bennett Faulkner+ b. 1824, d. 1879
Birth*: James Falkner was born circa 1841 at NC.
Marriage*: He married Martha Hicks on 7 February 1861 at Granville Co., NC; Marriages of Granville Co. page 107. B. Cook bondsman.1
Family: Martha Hicks b. c 1842
[S2] Granville Co., N.C. Marriage Register unknown file number, pg 107.
Marriage*: James Falkner married Mary B. (?)
Birth*: James Falkner was born circa 1829.
Family: Mary B. (?) b. c 1829
Joseph B. Falkner b. c 1849
Birth*: James Falkner was born circa 1802 at NC.1
Marriage*: He married Bedey Matterson on 25 August 1829 at Granville Co., NC; Granville Co. Marriage Bonds; James Forker married Bedey Matterson 25 Aug 1829, Lang King and B. Bullock Witnesses.2
Census: James Falkner appeared on the census of 1830 at South Regiment, Granville Co., NC. James Fauker is between 20 and 30 years old. He has a son and daughter under 5 and a wife 20 to 30 years old.3
Census: He appeared on the census of 25 July 1842 at Ledge of Rock School District #12, Granville Co., NC. James was listed as the head of the family and the following four children were enrolled as students:
Malinda, Rabon, S.W. (Sally) and Eliza Faulker.4
Census*: He appeared on the census of 31 October 1850 at Granville Co., NC. James Forker is 48 years old and a farmer. Biddy is 44 years old. Sally, Thomas and Mary are living with them. In the next house is Sherman Peed and his wife Malinda Falkner, age 22. In the next house to Sherman is William Forkner, age 22, farmer and his wife Harriet, age 19. Also. Eliza Forkner, age 14 is living with Sherman Peed. All of these Forkner's are most likely children of James and Beady.1
Family: Bedey Matterson b. c 1810
Malinda Falkner+ b. c 1830, d. b 1900
William Raban Forkner+ b. 29 Apr 1831, d. 1 Jan 1917
Sally Falkner5 b. c 1833
Eliza Falkner6 b. c 1836
Thomas Falkner6 b. c 1840
Mary Falkner6 b. c 1845
[S239] 1830 Granville Co., N.C. Census.
[S2428] LGranville County, Ledge of Rock District #12 School Census.
[S65] 1850 Granville Co., N.C. Census, p152.
Birth*: James Falkner was born circa 1803 at NC There are three James' married prior to 1830 that could be sons of Bartholomew. They are R828, R830, and R5097.
James Falkner1
Father: Anderson "Ancie" Falkner1 b. Jan 1851, d. b 1910
Mother: Sallie Jane "Creder" Roberson1 b. c 1857, d. 5 Nov 1942
Birth*: James Falkner was born in 1878 at Granville Co., NC.1
He was the son of Anderson "Ancie" Falkner and Sallie Jane "Creder" Roberson.1
James A. Falkner1
M, b. January 1855, d. 8 September 1934
Father: William Falkner b. b 1837, d. b 1878
Mother: Eliza (?)
Birth*: James A. Falkner was born in January 1855 at NC.
He was the son of William Falkner and Eliza (?)
Marriage*: James A. Falkner married Mary Jane Parks, daughter of Nathaniel Parks and Sally (?), on 17 September 1878 at Franklin Co., NC; Marriage from Franklin Co. Marriage Certificate listing age and parents of both.2
Census: James A. Falkner appeared on the census of 9 June 1880 at Freemans Dist, Franklin Co., NC. James is 24 years old and works as a farm laborer.3
Deed*: He recorded a deed on 2 December 1881 at Franklin Co., NC. James Forkner and wife Mary Jane Forkner of Granville Co., convey to R.E. Gell of Franklin Co. for $100 all right, title and interest that we have assigned to us by will of Richard Champion dec'd. to 130 acres on Cedar Creek in Franklin Co. adjoining S.P. Lowry and Thomas Bragg. Wit. W.H. Mitchell. Signed. James Forkner.4
Census: He appeared on the census of June 1900 at Freeman, Franklin Co., NC.5
Census*: He appeared on the census of 30 April 1910 at Henderson, Vance Co., NC. He is 61 years old and employed as a Yarn Warper in a Cotton Mill. He and Mary have been married for 32 years.6
Census: He appeared on the census of 15 April 1930 at Durham, Durham Co., NC. James A. is 72 years old. He is living with his daughter Alice and son-in-law John E. Parkinson.1
Death*: He died on 8 September 1934 at Durham, Durham Co., NC, at age 79 .7
Family: Mary Jane Parks b. Sep 1849, d. 15 Aug 1928
Marriage*: He married Mary Jane Parks, daughter of Nathaniel Parks and Sally (?), on 17 September 1878 at Franklin Co., NC; Marriage from Franklin Co. Marriage Certificate listing age and parents of both.2
Robert Faulkner+ b. 2 Apr 1878, d. b 1920
Alice Falkner+ b. Apr 1890, d. 17 Jan 1981
[S576] 1930 Durham Co., N.C. Census.
James Edgar Falkner
M, b. 1 March 1881, d. 2 April 1935
James Edgar Faulkner
Father: Alexander Ball "Sandy" Falkner b. Jul 1832, d. Oct 1902
Mother: Roxanna W. Saintsing b. 20 Oct 1860, d. 12 May 1906
Birth*: James Edgar Falkner was born on 1 March 1881 at Vance Co., NC.1
He was the son of Alexander Ball "Sandy" Falkner and Roxanna W. Saintsing.
Census*: James Edgar Falkner appeared on the census of 1900 at Kittrell, Vance Co., NC. He is living with his parents.2
Marriage*: He married Ida A. Powell circa 1909 at Vance Co., NC.
Census: James Edgar Falkner appeared on the census of 15 April 1910 at Henderson, Vance Co., NC. Edgar is 29 years old and has been married for 1 year. He is employed as a Doffer in a Cotton Mill.3
Milit-Beg*: He began military service circa 1918 at Vance Co., NC, registered for the World War I draft. He is married and lists his nearest relative as Mrs. Ida A. Faulkner. He is employed as a Mill Operator for the Henderson Cotton Mill. He is described as being of medium height and build with light blue eyes and light hair.4
Occupation*: He was Cotton Mill Frame Hand in January 1920 at Henderson, Vance Co., NC.5
Census: He appeared on the census of 13 January 1920 at 758 Garnett St., Henderson, Vance Co., NC. James E. Faulkner is 39 years old. Ida A. Faulkner is 34; James G. is 10 and Linwood B. is 5.5
Census: He appeared on the census of 3 April 1920 at Cedar Rock, Franklin Co., NC. James is 39 years old. Living with him is his wife Ida and sons Garland and Linwood.6
Census: He appeared on the census of 15 April 1930 at Henderson, Vance Co., NC. James E. is 49 years old and is employed as a farmer.7
Death*: He died on 2 April 1935 at Vance Co., NC, at age 54 . Miss. Lula Falkner was the informant. He was a widower at the time of his death.1
Burial*: He was buried on 4 April 1935 at Elmwood Cemetery, Henderson, Vance Co., NC.1
Family: Ida A. Powell b. 12 Jun 1885, d. 29 Mar 1933
James Garland Falkner b. 23 Sep 1909, d. 4 Mar 1959
Mildred Falkner1 b. 22 Feb 1912, d. 11 Nov 1915
Linwood Belmont Falkner+8 b. 6 Dec 1915, d. 21 Jun 1985
Allison Falkner1 b. 11 Jan 1917, d. 7 Aug 1917
Imogene Falkner9 b. 11 Apr 1918, d. b Mar 1933
Lorena Falkner10 b. 31 Mar 1921, d. 4 Apr 1921
Willard Oliver Faulkner7 b. 23 Sep 1926, d. 14 Jan 1990
[S19] 1910 Vance Co., N.C. Census.
[S638] Unknown subject, Draft Registrations World War I.
[S160] Unknown volume, Vance County, N.C. Birth Register: vol 5, pg 156, Register of Deeds Vance Co., NC Courthouse.
James Frank Falkner
Birth*: James Frank Falkner was born circa 1848 at NC.1
Marriage*: He married Julia Scott, daughter of James Scott and Jane (?), on 23 September 1866 at Orange Co., NC; Marriage from N.C. Marriage Bonds #96678 dtd 22 Sep 1866 for Orange Co. Saml H. Breeze and J.W. Sharp bondsmen.2
Census*: James Frank Falkner appeared on the census of 1 June 1870 at Pleasant Grove, Alamance Co., NC. James Fortner is 22 and employed as a farmer. His wife Julia is also 22.1
Family: Julia Scott b. 1848
John Fate Faulkner+3 b. 1867, d. 14 Aug 1921
Lafayette Falkner1 b. c 1869
[S2257] 1870 Alamance Co., N.C. Census.
[S223] Unknown volume, Alamance Co., N.C. Death Certificate, unknown repository.
James Garland Falkner
M, b. 23 September 1909, d. 4 March 1959
Photo from WWII Young American Patriots
Vance County, N.C.
Father: James Edgar Falkner b. 1 Mar 1881, d. 2 Apr 1935
Mother: Ida A. Powell b. 12 Jun 1885, d. 29 Mar 1933
Birth*: James Garland Falkner was born on 23 September 1909 at Vance Co., NC.1
He was the son of James Edgar Falkner and Ida A. Powell.
Census*: James Garland Falkner appeared on the census of 1910 at Vance Co., NC. Listed in the 1910 Vance Co. Census. age 1/2.2
Milit-Beg*: He began military service on 3 September 1942 at Fort Bragg, NC, served in U.S. Army. Served at Camps Hulen, Texas, Polk, La., and in England, France, Germany and Austria. Awarded two Battle Stars.1,3
Single*: He was single in 1959 at Vance Co., NC, He never married.4
Occupation*: He was had worked at Greystone Granite Quarry before March 1959 at NC.4
Death*: He died on 4 March 1959 at Henderson, Vance Co., NC, at age 49 . L.B. Faulkner was the informant. James Garland had never been married.4
[S973] Unknown author, World War II Young American Patriots, 1941 - 1945, pg 799.
[S866] Unknown author, WW II Army Enlistment Records 1938 - 1946.
James Hampton Falkner1
M, b. 25 May 1908, d. 22 February 1967
Mother: Jeffers Rosa Medlin1 b. 19 Feb 1886, d. 8 Jan 1922
Birth*: James Hampton Falkner was born on 25 May 1908 at Vance Co., NC.1,2
He was the son of Otis Green Falkner and Jeffers Rosa Medlin.1
Marriage*: James Hampton Falkner married Clara Riggan on 27 December 1930 at Vance Co., NC.1,3
Death*: James Hampton Falkner died on 22 February 1967 at Vance Co., NC, at age 58 .1,2
Burial*: He was buried on 24 February 1967 at Toney Falkner Cemetery, Vance Co., NC.1
Family: Clara Riggan b. 16 Feb 1904, d. 16 Oct 1988
Rosa Hardy Falkner+1 b. 7 Oct 1931, d. 19 Apr 2019
James Hampton Falkner Jr.
James M. Falkner
Birth*: James M. Falkner was born circa 1805.
Marriage*: He married Emily Gasten on 30 July 1829 at Butts Co., TN.
Family: Emily Gasten b. c 1809
James R. Falkner
Birth*: James R. Falkner was born circa 1819 at NC.
Marriage*: He married Elizabeth L. Falkner on 28 December 1839 at Warren Co., NC; William Pardue was bondsman. Surnames were written Forkner.1
Family: Elizabeth L. Falkner b. c 1819
James Richard Falkner
M, b. 6 September 1907, d. 10 March 1996
Birth*: James Richard Falkner was born on 6 September 1907 at Henderson, Vance Co., NC.1
He was the son of Eugene Falkner and Mary Kay Kate Mustian.
Census*: James Richard Falkner appeared on the census of 1920 at Vance Co., NC. Richard James was not listed among the children in the 1920 Census.2
Marriage*: He married Georgia Wilma Tyson on 6 September 1937 at Brunswick Co., VA.3
Census: James Richard Falkner appeared on the census of 1 April 1940 at Henderson, Vance Co., NC. Richard 32, and his wife Georgia are living with Richard's parents.4
Marriage*: He married Elizabeth Angell Barbour on 23 December 1970 at Richmond, VA.3
Death*: James Richard Falkner died on 10 March 1996 at Louisburg, Franklin Co., NC, at age 88 .5
Family 1: Georgia Wilma Tyson b. 20 Jan 1912, d. 19 Mar 1969
Ann Tyson Faulkner
Family 2: Elizabeth Angell Barbour b. 22 Apr 1918, d. 14 Mar 2000
Marriage*: James Richard Falkner married Elizabeth Angell Barbour on 23 December 1970 at Richmond, VA.3
[S1810] 1940 Vance Co., N.C. Census.
James Thomas Falkner1
M, b. 24 January 1834, d. before 1865
Father: Asa Faulkner1 b. 4 Nov 1802, d. 30 Nov 1837
Mother: Susan Myers1 b. 22 Aug 1809, d. 1 Aug 1880
Birth*: James Thomas Falkner was born on 24 January 1834 at Anson Co., NC.1
He was the son of Asa Faulkner and Susan Myers.1
Marriage*: James Thomas Falkner married Elizabeth Buchanan circa 1855.2
Death*: James Thomas Falkner died before 1865 at MS .
Family: Elizabeth Buchanan b. 2 Feb 1833, d. 8 Nov 1919
Mary Atkinson Faulkner b. 13 Apr 1856, d. 20 Feb 1935
[S2325] Falkner, Asa Family Bible.
James William Falkner
M, b. circa 1837, d. 29 July 1862
Father: William Falkner b. c 1795
Mother: Sally T. Roberson b. c 1803
Birth*: James William Falkner was born circa 1837 at NC.
He was the son of William Falkner and Sally T. Roberson.
Census*: James William Falkner appeared on the census of 23 July 1850 at Epping Forest District, Granville Co., NC. He is 13 years old.1
Marriage*: He married Tabitha Robertson, daughter of William Woodward Robertson and Clary Falconer, on 23 September 1857 at Granville Co., NC, From the Biblical Recorder: "Married on Wednesday, 30th September, by James Duty, Esq. Mr. William Faulkner, aged 16, and Miss Tabitha Robertson, age 40, daughter of the late Col. Woodward Robertson, all of Granville County."2,3,4
Census: James William Falkner appeared on the census of 6 July 1860 at Nutbush, Granville Co., NC. James is 25 years old.5
Milit-Beg: He began military service on 27 February 1862 at Granville Co., NC, N.C. Troops 1861-1865, pg 403; James W. Falkner, private, served in Co. A. 44th Regmt N.C. Troops. He was born in Granville Co where he resided as a farmer prior to enlisting in Granville Co. at age 23, 27 Feb 1862. Hospitalized at Richmond, Va. 21 Jul 1862 with "continued fever". Died in hospital at Richmond on or about 29 Jul 1862.6
Death*: He died on 29 July 1862 at Richmond, Va.7
Family: Tabitha Robertson b. c 1826
Peter Falkner8 b. 1854
Infant Falkner8 b. 1859
[S110] Elizabeth Hicks Hummel, Hicks History of Granville Co., N.C., Vol I.
[S2103] The Biblical Recorder, 12 Nov 1857.
[S111] Unknown author, North Carolina Troops 1861-1865, pg 403.
[S111] Unknown author, North Carolina Troops 1861-1865.
James Word Falkner
Father: William Faulkner b. c 1795, d. c 1842
Mother: Caroline Word b. c 1795
Death*: James Word Falkner died at Marshall Co., MS.
Birth*: He was born circa 1834 at Knox Co., TN.
He was the son of William Faulkner and Caroline Word.
Census: James Word Falkner appeared on the census of 7 October 1850 at Union, Ste. Genevieve Co., MO. James Fockner is 16 years old and living with his mother.1
Census*: He appeared on the census of 3 August 1860 at Ripley, Tippah Co., MS. James W. is 29 years old. He is living next to John Wesley Thompson. He is employed as an Attorney at Law.2
Marriage*: He married Helen M. Hancock on 20 February 1861 at Shelby Co., TN.3
Census: James Word Falkner appeared on the census of 4 July 1870 at Prairie, Arkansas Co., AR. James W. is 35 and employed as a farmer.4
Family: Helen M. Hancock b. c 1841
Eugene Falkner4 b. c 1862
Josephine Fanny Falkner4 b. 2 Aug 1863, d. 11 Feb 1893
Roseline Falkner4 b. c 1866
William C. Falkner4 b. c 1868
James Word Falkner Jr. b. 30 Mar 1870, d. 3 Apr 1943
Guy T. Falkner5 b. c 1872
Sallie W. Falkner5 b. 15 Jan 1874, d. 30 Sep 1950
[S1214] 1850 St. Genevieve Co., Mo. Census, page 247b.
[S650] 1860 Tippah Co., Miss. Census.
[S1481] Unknown volume, Tennessee State Marriages 1780 - 2002, unknown repository.
[S1887] 1870 Arkansas Co., Ark. Census.
[S1888] 1880 Marshall Co., Miss. Census.
James Word Falkner Jr.
M, b. 30 March 1870, d. 3 April 1943
Father: James Word Falkner b. c 1834
Mother: Helen M. Hancock b. c 1841
Birth*: James Word Falkner Jr. was born on 30 March 1870 at AR.1,2
He was the son of James Word Falkner and Helen M. Hancock.
Census*: James Word Falkner Jr. appeared on the census of 10 June 1880 at Bahalia, Marshall Co., MS. James W. is 10 years old and lives with T.W. Hancock.3
Census: He appeared on the census of 21 April 1910 at Marshall Co., MS. He is 40 years old and a widower. He is employed as a County Judge.4
Marriage*: He married Clara Delia Marr in 1921.5
Census: James Word Falkner Jr. appeared on the census of 1 April 1930 at El Monte, Los Angeles Co., CA. He is 60 years old and an Attorney.6
Death*: He died on 3 April 1943 at Los Angeles, Los Angeles Co., CA, at age 73 .2
Family: Clara Delia Marr b. 8 Oct 1879, d. 20 Jun 1965
[S541] 1930 Los Angeles Co., Ca. Census.
Jane Falkner
Birth*: Jane Falkner was born circa 1840 at Franklin Co., NC.
Census*: Jane Falkner appeared on the census of 28 August 1850 at Franklin Co., NC. She is 10 years old and living with her parents.1
Census: She appeared on the census of 30 August 1860 at Franklin Co., NC. She is 22 years old.2
Jodie Falkner1
F, b. 24 November 1908, d. 9 January 1969
Father: Joseph Jackson Falkner b. 26 Mar 1882, d. 24 Jan 1955
Mother: Annie L. Pernell1 b. 9 Jan 1887, d. 22 Sep 1909
Birth*: Jodie Falkner was born on 24 November 1908 at Louisburg, Franklin Co., NC.1,2
She was the daughter of Joseph Jackson Falkner and Annie L. Pernell.1
Census*: Jodie Falkner appeared on the census of 13 April 1930 at Sandy Creek, Franklin Co., NC. Jodie is 22 years old and single. She is living with her father and sister Luna.3
Marriage*: She married Foster Man Harris, son of Arthur Henry Harris and Lessie Edwards, on 24 December 1932 at Franklin Co., NC.4
Death*: Jodie Falkner died on 9 January 1969 at Louisburg, Franklin Co., NC, at age 60 . Informant was Mr. Roger Harris.2
Family: Foster Man Harris b. 5 Mar 1906, d. 9 Jun 1965
Henry Joseph Harris
Roger Perry Harris b. 16 Jun 1939, d. 11 Aug 2008
John Falkner
Birth*: John Falkner was born circa 1815 at NC.
Marriage*: He married Sarah Sally Stewart on 7 January 1837 at Granville Co., NC; Marriage Bond bondsman was James Stewart. Could this be Sally's father?1,2
Census: John Falkner appeared on the census of 1 November 1850 at Tipton Co., TN. John is a farmer, age 36. Sarah is 32.3
Marriage*: He married Martha Frances Franklin on 29 March 1854 at Tipton Co., TN; Second Marriage from Granville Co. Queries Vol 2 no.2.4
Census*: John Falkner appeared on the census of 17 September 1860 at Como, Panola Co., MS. J.F. Falkner is 47 years old. His wife Martha is 30 and was born in Virginia. He is a farmer and owns real estate valued at $2,000 and personal property valuedd at $1025.5
Census: He appeared on the census of 6 July 1870 at Panola Co., MS. John is 60 years old and Martha is 41.6
Census: He appeared on the census of 16 June 1880 at Sardis, Panola Co., MS. John is 75 and Martha is 54. Living with them is Martha's sister Elizabeth Whitfield, age 56, born in Virginia and a grandson John Falkner, age 14, born in Miss.7
Death*: He died circa 1887 at Panola Co., MS.7,8
Family 1: Sarah Sally Stewart b. c 1818, d. b Mar 1854
Rev. Governor Hill Faulkner+ b. 24 Oct 1838, d. 31 May 1919
John E. Faulkner+ b. c 1841, d. 1917
Mary Ann Falkner b. c 1844
Martha Falkner b. c 1847
Sally Falkner b. c May 1850, d. b 1860
Family 2: Martha Frances Franklin b. c 1830
Marriage*: John Falkner married Martha Frances Franklin on 29 March 1854 at Tipton Co., TN; Second Marriage from Granville Co. Queries Vol 2 no.2.4
[S110] Elizabeth Hicks Hummel, Hicks History of Granville Co., N.C., pg 176.
[S1114] 1850 Tipton Co., Tenn. Census.
[S418] Unknown author, "unknown short article title", Vol 2 No. 2. Aug 1993: Query from Almeda Brewer , Rt. 4 Box 4995,, Manchester, Tn 37355.
[S1874] 1870 Panola Co., Ms. Census.
[S2380] Mississippi Wills and Probate Records 1780 -1982 unknown file number.
Marriage*: He married Sealey Johnson, daughter of Noel Johnson and Nancy (?), on 16 December 1810 at Granville Co., NC; Marriages of Granville Co. page 107; Phileman Perdue bondsman. Hicks History of Granville Co. shows that Lively Johnson married John Falkner 26 Dec 1810. Bondsman was Philemon Perdue. Are Lively and Shealey the same person?1
Family: Sealey Johnson b. c 1790
Census*: He appeared on the census of 18 June 1860 at Henderson, Granville Co., NC. John is age 27 and occupation is overseer.1
[S66] 1860 Granville Co., N.C. Census, page 164.
Birth*: John Falkner was born circa 1842.
Marriage*: He married Mary E. Mitchell on 19 March 1862 at Orange Co., NC; Marriage Bond #96682 dtd 19 Mar 1862 for Orange Co., J.M. Bain bondsman.
Family: Mary E. Mitchell
Birth*: John Falkner was born before 1740.
Census*: He appeared on the census of 1790 at Cumberland Co., NC. John is listed as head of family with no other family members in the 1790 Cumberland Co, N.C. Census.
Father: Francis Faulkner b. c 1709, d. c 1783
He was the son of Francis Faulkner.
Census*: John Falkner appeared on the census of 1790 at Cumberland Co., NC. John is over 16 years old and the only member of his household with no slaves.1
Deed*: He recorded a deed in July 1793 at Cumberland Co., NC. John Falkner of Cumberland Co. conveys to Arthur Johnson of Cumberland Co. for 50 pounds a tract of 81 acres in Cumberland Co. on Black River beginning at his own corner in Black River Swamp and adjoining Sercey. Wittnessed Caleb Falkner, John Lawson. Signed John Falkner. Proven July term 1793.
[S53] 1790 Cumberland Co., N.C. Census. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,038 |
{"url":"https:\/\/www.physicsforums.com\/threads\/challange-between-a-friend-and-i-about-solving-for-unknown-variables-in-a-known-f-x.314701\/","text":"# Challange between a friend and I about solving for unknown variables in a known F(x)\n\nOn a recent outing with friends we came up on a debate and im hoping someone can help me find out who was right. The discussion was on whether one could create an equation where in you have 2 unknown variables and any number of known variables and the solution, that will allow you to solve for the 2 unknown variables.\nFor the issue at hand we had agreed on the following details.\n\n1) All variables must be between Hex values (0-F). (we decided to keep it simple)\n2) You must use some form of reduction in size so that your answer is the same number of characters as your input. (whether you ditch the higher values or mathmaticaly reduce them is up to you) IE if you use 4 hex values for each variable, your answer must not be greater than 4 hex values. (binary reduction is easy to do: If there are overflow bits, just XOR with a known value to reduce - which is fully reversable so you arent loosing any information)\n3) You must have 2 unknown variables.\n4) You may have any number of known variables in your function. You can choose 3, 5, 42, etc.\n5) You may use any math you wish, as long as you adhear to rule 2 for the function solution.\n6) You must be able to solve using only one function.\n7) You are allowed to generate lookup tables to help you solve if you need.\n8) You must be able to provide examples of how your function is solveable in reverse.\n\nHe agrued that you could create a single function that when a person knows the solution and all but 2 variables, they can compute the two missing variables. The matter isnt if this is easy to do computationaly, but if it is possible. His possible solution was to use something like the CRC16 or any of the CRC hashes or a one way hash function from something like RSA. Now due to the computational complexity of using RSA, he suggested using a system of only 4 bytes of data instead of the full 64bytes. But his argument was that you could create a function (algorithm) where in you could know all but two of the variables and be able to calculate thier exact value, and that you wouldnt have the problem of collisions. (More than one possible pair of values that would give you the known answer). For example, take the equation ((((A\/B) * C) \/ D) * E) = F. If you know F and can choose to know any 3 variables from A, B, C, D, E. Could you solve the for the other 2 without having another equation at your disposal.\n\nMy stance was that you would need 2 solutions to be able to solve for either unknown variable. I Do not believe that calculating every possible value of those two unknown variables would enable you to prove the exact values, as without anything to use as a referance (another known solution); you would end with with 2 or more possible answers for those variables and would not be able to determine which of them was correct.\n\nHowever since we are keeping the variable size small, and we are allowed to use lookup tables; it may be possible to create a function that this is possible to do, simply by brute forcing all possible values for your two unknown variables. But would this give you only one result per function solution (Variable F) or would you be left with collisions?\n\nAs a last question before we split up for the night the challange was altered. Same basic rules except this time you have 3 unknown variables and any number of known; but you have 2 known functions each with their own known answer. Would this be possible as well? (this will be the discussion when we get together again.)\n\nCan an elementary function be constructed so that it has a domain restricted to a single constant? If the function was of one of the unknown variables, then that unknown variable would be restricted to some constant value, regardless of the other unknown variable.\n\nCan an elementary function be constructed so that it has a domain restricted to a single constant? If the function was of one of the unknown variables, then that unknown variable would be restricted to some constant value, regardless of the other unknown variable.\n\nIn the debate my friend and I were having, the function would be a known. In simple algabraic terms. F(x) = x+y=Z. If Y and Z are known you can solve for X. The debate was whether you could create a single function, that while knowing the function structure and knowing Z you would know all but 2 of the variables. IE a+b+c+d+e=F. Obviously that wont work because if c=1, d=2, e=3, and F=15. You could solve to a + b = 9. But you have no way to tell which between a and b is 4 and which is 5.\nThe only way that I can see it possibly working out is if one of the known variables, lets say c, acts upon one of the unknown in a way that would thus give you a solution that could not be reached of the values of a or b were mixed. Say a\/c + b\/d + e = F. In that function the values for a and b cannot be interchanged. However in that function you there could be mutiple values for a and b that would work. Example: a=2 b=8 c=4 d=16 e=1. In that case F would equal 2. However if the value of a=1 and the value of b=12, F would still equal 2.\nI run into either problem: not being able to tell what is a or b, or being able to distinguish them apart but not knowing which possible solution they are.\n\nmatt grime\nHomework Helper\n\nIE if you use 4 hex values for each variable, your answer must not be greater than 4 hex values. (binary reduction is easy to do: If there are overflow bits, just XOR with a known value to reduce - which is fully reversable so you arent loosing any information)\n\nWhy would XORing in something make it 'reduced'? Normally AND is used for 'reduction' but that isn't reversible. I agree XOR in a constant and you don't 'lose' information, but that won't get you something inside a certain range.\n\nMoreover, you have a function in two variables, say in the range [0,255] each, and you want the answer to be in the range [0-255], well, that cannot be injective so no, you can't make things invertible.\n\nWhy would XORing in something make it 'reduced'? Normally AND is used for 'reduction' but that isn't reversible. I agree XOR in a constant and you don't 'lose' information, but that won't get you something inside a certain range.\n\nMoreover, you have a function in two variables, say in the range [0,255] each, and you want the answer to be in the range [0-255], well, that cannot be injective so no, you can't make things invertible.\n\nIn my other quote I reference one of the functions of the AES algorithm. https:\/\/www.physicsforums.com\/showthread.php?t=314497\".\nIn that example, when you are doing your multipilcation you will overflow in Step 2.\nFor example, {57} \u00b7 {13} = {fe} because\n{57} \u00b7 {02} = xtime({57}) = {ae}\n{57} \u00b7 {04} = xtime({ae}) = {47}\n{57} \u00b7 {08} = xtime({47}) = {8e}\n{57} \u00b7 {10} = xtime({8e}) = {07},\nthus,\n{57} \u00b7 {13} = {57} \u00b7 ({01} XOR {02} XOR {10})\n= {57} XOR {ae} XOR {07}\n= {fe}.\nAt Step 2 you will overflow as {ae} \u00b7 {02) = 15c. It is reduced back to within the range of [0-255] by XOR of 11b. {15c} XOR {11b} = {47}. As Im sure you know, you can simply revert this back by doing the reverse. {47} XOR {11b} = {15c}.\n(In case someone doesnt understand why I multiplied by 02 there instead of 04, I already multiplied by 02 in the previous step, so to multiple 57 by 04 just means I take 57x02 and multiple it by 02. 02x02=04)\n\nSomething like this will reduce your value back down to an acceptable range. As long as you know the Constant you are XORing with, you are able to back step it. The only issue I see is how would you know whether you would need to backstep it or not. IE is 47 a reduced answer or an answer on its own.\n\nOverall there is no worry of overflow throughout the function, it was simply that the output length be the same as the input length. But as I said, I believe this creates the problem of having an answer that you are unware of whether it is 'reduced' or not. Effectively rending the problem unsolvable.\n\nLast edited by a moderator:\nmatt grime\nHomework Helper\n\nSomething like this will reduce your value back down to an acceptable range. As long as you know the Constant you are XORing with, you are able to back step it.\n\nYou're focussed on the wrong thing entirely - you're jsut applying an affine transformation. It is not the representation of your image that is the problem. It is just that you're defining a function from XxX to X (where X is whatever finite set you care about) and this is never left invertible. It would be elementary to do this with real numbers, or any infinite set for that matter, for example X=N the natural numbers (m,n)--> 2^m3^n will do.\n\nCRGreathouse\nHomework Helper\n\nHe agrued that you could create a single function that when a person knows the solution and all but 2 variables, they can compute the two missing variables. The matter isnt if this is easy to do computationaly, but if it is possible.\n\nIf the unknowns m and n are integers and the function f is given, then there's a trivial algorithm to find a pair (m, n) that gives f(m, n) = k if such a pair exists:\n\nCode:\nfor (t in {0, 1, 2, ...})\nfor (m in {0, 1, ..., t})\nfor (n in {0, 1, ..., t})\nif (f(m, n) = k)\nreturn (m, n)\nif (f(m, -n) = k)\nreturn (m, -n)\nif (f(-m, n) = k)\nreturn (-m, n)\nif (f(-m, -n) = k)\nreturn (-m, -n)\n\nIf the unknowns x and y are real numbers and the function f is given by a closed-form expression involving only real constants, +, -, *, and \/, by a theorem of Tarski you can find a pair (x, y), or show that none exists, such that f(x, y) = k.\n\nIf the unknowns x and y are real numbers and the function f is given by a closed-form expression involving only real constants, +, -, *, ^, sin, and abs, by a theorem of Richardson your friend can design some function so complicated that you won't be able to find (x, y) or to show that none exists, such that f(x, y) = k.\n\nLast edited:\nCRGreathouse\n$$f(x) = \\sqrt{-(x - k)^2}$$\nwhich for $f:\\mathbb{R}\\Rightarrow\\mathbb{R}$ allows only k in the domain. 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\section{Introduction}
{\it Introduction.}---Ultracold atoms in optical lattices provide a unique playground for studying quantum many-body systems experimentally~\cite{Morsch_2006, Bloch_2008, Lewenstein2012, Krutitsky_2016}.
In particular, systems of bosonic alkali atoms with hyperfine spin $f$
have received considerable attention, as they can give rise to a variety of exotic phases~\cite{ueda2010fundamentals, Kawaguchi_Ueda_2012, Ueda_RMP_2013}.
Such systems are well described by the spinor Bose-Hubbard model~\cite{Demler_Zhou_2002,Imambekov_2003,Lewenstein2012}, which is a discrete version of the model for condensates~\footnote{When all atoms are trapped in one single potential well, they form a continuous condensate~\cite{Koashi2000, Ciobanu2000, Lewenstein2012}. }.
Most previous theoretical studies on lattice systems were based on a mean-field treatment of the original model~\cite{Zhou_Snoek_2003, Hou_Ge_2003, Snoek_Zhou_2004,
Imambekov_2004, Tsuchiya_2004, Kimura_2005, Krutitsky_2005, Zhou_Semenoff_2006, Snoek_Song_2009} or a mapping to the effective spin model (perturbation on hopping)~\cite{Imambekov_2003, Zhou_Snoek_2003, Yip_2003, DeChiara_2011, eckert2007ultracold, barnett2006classifying}.
This is in contrast to the continuous case, where many results beyond mean-field have been obtained theoretically~\cite{Koashi2000,Ueda2002,uchino2008dynamical,Ho1999,
eisenberg2002polarization,Tasaki_polar_2013}.
On the other hand, very few solid results are available for discrete lattice systems~\cite{Katsura2013, Kunimi_Saito_2015}.
In particular, almost nothing is known rigorously about the $f=2$ case.
\smallskip
In this Letter, we prove universal theorems
about the ground-state properties of the \mbox{spin-2} Bose-Hubbard Hamiltonian. The model in Eq.\ (\ref{Hamiltonian}) is characterized by the spin-dependent interaction constants $c_1$ and $c_2$. We precisely determine the total spin and the degeneracy of the ground states for the following three cases: (i) $c_1<0$ and $c_2 \geqslant 5c_1$, (ii) $c_1=0$ and $c_2<0$, and (iii) $c_1=c_2=0$.
In case (i), the system has SO(3) symmetry and the ground state exhibits saturated ferromagnetism.
In case (ii), the symmetry is promoted to SO(5)
and the ground state has total spin $F_{\text{tot}}=0$ or $2$, depending on the parity of the total particle number. The symmetry is further enhanced to SU(5) in case (iii), leading to a ground-state degeneracy that grows as $N^4$ for large particle number $N$.
It is worth comparing
with the previous results for the spin-1 Bose-Hubbard model~\cite{Katsura2013}.
The spin-2 Hamiltonian has one more spin-dependent interaction term with SO(5) symmetry
that does not have a counterpart in the spin-1 case,
which makes the phase diagram richer.
\smallskip
Our results are consistent with the phase diagram of spin-2 continuous condensates predicted by mean-field theory~\cite{Koashi2000,Ciobanu2000,Lewenstein2012}.
Moreover, our Theorem 2 suggests that the model with $c_1=0$ and $c_2<0$ can be studied by quantum Monte Carlo simulations without a sign problem~\cite{
[{A sign problem may occur in boson systems with spin-dependent interaction.
The one-dimensional spin-1 Bose-Hubbard model with antiferromagnetic interactions was studied using quantum Monte Carlo simulations in }]
Apaja_2006}.
\smallskip
{\it Hamiltonian.}---We consider a system of $N$ spinor bosons with $f=2$ on a finite set of sites $\Lambda$, where $N$ is arbitrary and fixed.
We use Latin letters $i$ and $j$ to denote sites
and Greek letters $\alpha$ and $\beta$
to represent spin states,
i.e., $i,j \in \Lambda$ and $\alpha,\beta \in \{+2,+1,0,-1,-2\}$.
The creation and annihilation operators at
site $i$ with spin $\alpha$ are written as
${\hat a}^\dagger_{i, \alpha}$ and ${\hat a}_{i, \alpha}$, respectively.
We denote the number operators by $\hat{n}_{i,\alpha} := {\hat a}^\dagger_{i, \alpha} {\hat a}_{i, \alpha}$, $\hat{n}_i := \sum_\alpha \hat{n}_{i,\alpha}$,
and $\hat{N}_{\alpha} := \sum_i \hat{n}_{i,\alpha}$, and the $z$ component of the spin operator by
$\hat{F}^z_i := \sum_{\alpha, \beta} {\hat a}^\dagger_{i, \alpha} F^z_{\alpha, \beta} {\hat a}_{i,\beta}$,
where $F^z_{\alpha, \beta}$ is the spin matrix for spin-2.
Similarly, we have $\hat{F}^x_i$ and $\hat{F}^y_i$.
We define $\hat{\bm{F}}_i :=( \hat{F}^x_i, \hat{F}^y_i, \hat{F}^z_i )$ and $\hat{\bm{F}}_{\text{tot}} := \sum_i \hat{\bm{F}}_i$, and write the eigenvalues of
$(\hat{\bm{F}}_{\text{tot}})^2$ and
$\hat{F}^z_{\text{tot}}$ as $F_{\text{tot}} (F_{\text{tot}} +1)$ and
$F^z_{\text{tot}}$, respectively.
We also define the singlet creation and annihilation operators as $\hat{S}_{i, +} := \sum_{\alpha=-2}^2 (-1)^{\alpha}{\hat a}^{\dagger}_{i,\alpha} {\hat a}^{\dagger}_{i,-\alpha}/2$
and $\hat{S}_{i, -} := \hat{S}_{i, +}^\dagger$.
Acting with $\hat{S}_{i,+}$ ($\hat{S}_{i,-}$) creates (annihilates) a two-body singlet at site $i$.
\smallskip
The following set of orthonormal states
\begin{equation}
|\Phi_{\bm{m}}\rangle:= \frac{1}{\sqrt{\prod_{i,\alpha}(n_{i,\alpha} !)}} \left\{ \prod\limits_{i, \alpha} (\hat{a}_{i, \alpha}^{\dagger})^{n_{i, \alpha}} \right\} |\text{vac}\rangle \label{basis}
\end{equation}
serves as a basis
of the Hilbert space $\mathcal{H}$.
Here, $|\text{vac}\rangle$ stands for the vacuum and $\bm{m}=(n_{i,\alpha}) \in \mathcal{I}$ is
a set of non-negative integers, where $\mathcal{I}$ is a set of ${\bm m}$ that satisfies $\sum_{i,\alpha}n_{i,\alpha}=N$.
\smallskip
The Hamiltonian of the spin-2 Bose-Hubbard model~\cite{Koashi2000,Lewenstein2012}
is
\begin{eqnarray}
\hat{H}=&&-\sum_{ i\neq j, \alpha} t_{i, j} \hat{a}^{\dagger}_{i,\alpha} \hat{a}_{j,\alpha}
+ \sum_{i}V_i \hat{n}_i + \frac{c_0}{2}\sum_{i}\hat{n}_i(\hat{n}_i-1) \nonumber\\
&&+ \frac{c_1}{2} \sum_{i}\Big[(\hat{\bm{F}_i})^2 - 6\hat{n}_i\Big]
+ \frac{2c_2}{5} \sum_{i}\hat{S}_{i, +}\hat{S}_{i, -}~. \label{Hamiltonian}
\end{eqnarray}
Here $V_i \in \mathbb{R}$ is the single-particle potential at site $i$. The constants $c_0$, $c_1$ and $c_2$ are real coefficients for the two-body interactions, where the $c_1$ and $c_2$ terms are spin dependent. The $c_2$ term favors (disfavors) singlet pairs when $c_2 > 0$ ($c_2 < 0$). We assume that $t_{i,j}=t_{j,i} \geqslant 0$ for all $i,j \in \Lambda$ and the whole lattice $\Lambda$ is connected via nonzero $t_{i,j}$.
\smallskip
In addition to the global ground states in the whole Hilbert space $\mathcal{H}$,
symmetry of the Hamiltonian enables us
to discuss the local ground states in Hilbert subspaces.
The Hamiltonian ${\hat H}$ is invariant under rotation in spin space, which implies that ${\hat H}$ has at least SO(3) symmetry, yielding $[\hat{H}, \hat{F}^{x,y,z}_{\text{tot}}] = 0$.
Since $\hat{F}_{\text{tot}}^z$ is conserved,
$\mathcal{H}$ splits into subspaces labeled by $F^z_{\text{tot}}=M$.
We shall show later that the symmetry is promoted to SO(5) when $c_1=0$ and $c_2 \neq 0$. In this case, $\mathcal{H}$ splits into smaller subspaces labeled by two indices $P:=N_1-N_{-1}$ and $Q:=N_2-N_{-2}$.
\smallskip
Now we introduce some more notations. Define $\mathcal{H}_A$ as a subspace of $\mathcal{H}$ by
$\mathcal{H}_A := \{ |\psi\rangle \in \mathcal{H} \mid \hat{A} |\psi\rangle = A |\psi\rangle \}$.
Similarly, we have $\mathcal{H}_B$ for ${\hat B}$.
The intersection of $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ is denoted as $\mathcal{H}_{A,B}$.
Define $\mathcal{I}_A$ as a subset of $\mathcal{I}$ by
$\mathcal{I}_A := \{ \bm{m} \in \mathcal{I} \mid \hat{A} |\Phi_{\bm{m}}\rangle = A |\Phi_{\bm{m}}\rangle \}$.
Operators $\hat{A}$ and $\hat{B}$ can be $\hat{M}$ $(:= \hat{F}^z_{\text{tot}})$, $\hat{P}$ $(:= \hat{N}_{1}-\hat{N}_{-1})$, $\hat{Q}$ $(:=\hat{N}_{2}-\hat{N}_{-2})$, or $\hat{N}_\alpha$ in the following.
\smallskip
Now we state our main theorems.
\smallskip
\textit{Theorem 1.---}If $c_1<0$ and $c_2 \geqslant 5c_1$, the local ground state $|\Psi^{\text{GS}}_M\rangle$ in $\mathcal{H}_M$ is unique and can be written as
\begin{equation}
|\Psi^{\text{GS}}_M\rangle = \sum\limits_{\bm{m} \in \mathcal{I}_M} C_{\bm{m}} |\Phi_{\bm{m}}\rangle, \label{Thm1}
\end{equation}
with $C_{\bm{m}} > 0$, and has the maximum possible total spin $F_{\text{tot}}=2N$ (saturated ferromagnetism). Each local ground state $|\Psi^{\text{GS}}_M\rangle$ has energy
independent of $M$ and hence is the global ground state in $\mathcal{H}$ as well. Thus the ground-state degeneracy is $4N+1$.
\smallskip
The following proposition is a special case of Theorem~1 where $|\Psi^{\text{GS}}_M\rangle$ can be written more explicitly.
\smallskip
\textit{Proposition 1.---}If $c_1=-c_0/4 < 0$ and $c_2\geqslant0$,
the ground state $|\Psi^{\text{GS}}_{M=2N}\rangle$ in $\mathcal{H}_{M=2N}$ is unique
and can be written as
\begin{eqnarray}
&&|\Psi^{\text{GS}}_{M=2N}\rangle = (\hat{b}^{\dagger}_{2})^{N} |\text{vac}\rangle,
\label{Prop1}
\end{eqnarray}
where $\hat{b}^{\dagger}_2 = \sum_i \varphi_0(i) \hat{a}^{\dagger}_{i,2}$.
Here, $\varphi_0(i) >0$ ($i \in \Lambda$) is the
spatial wave function of single-particle ground state of the hopping term and the on-site potential term.
Clearly $|\Psi^{\text{GS}}_{M=2N}\rangle$ has the maximum total spin $F_{\text{tot}}=2N$.
Ground states in other subspaces can be obtained as
$|\Psi^{\text{GS}}_{M'}\rangle \propto (\hat{F}_{\text{tot}}^-)^{2N-M'} |\Psi^{\text{GS}}_{M=2N}\rangle$.
\smallskip
\textit{Theorem 2.---}If $c_1=0$ and $c_2<0$, the local ground state $|\Psi^{\text{GS}}_{P,Q}\rangle$ in $\mathcal{H}_{P,Q}$ is unique and can be written as
\begin{equation}
|\Psi^{\text{GS}}_{P,Q}\rangle =\sum\limits_{\bm{m}\in\mathcal{I}_{P,Q}} D_{\bm{m}} (-1)^{ (\hat{N}_{+1} + \hat{N}_{-1})/2} |\Phi_{\bm{m}}\rangle, \label{Thm2}
\end{equation}
with $D_{\bm{m}} > 0$. The local ground-state energy
in each $\mathcal{H}_{P,Q}$ is a function only of $\Gamma:= |P|+|Q|$.
We denote this energy by $E^{\text{GS}}_\Gamma$.
Their energy-level ordering is
$E^{\text{GS}}_\Gamma < E^{\text{GS}}_{\Gamma+1}$ if $N-\Gamma$ is even,
while $E^{\text{GS}}_\Gamma = E^{\text{GS}}_{\Gamma+1}$ if $N-\Gamma$ is odd.
Thus the global ground state has total spin $F_{\text{tot}}=0$ and is unique
if total particle number $N$ is even,
while it has $F_{\text{tot}}=2$ and is fivefold degenerate if $N$ is odd.
\smallskip
Note that $\mathcal{H}_{P,Q} \subset \mathcal{H}_{M=P+2Q}$. When $c_1=0$ and $c_2<0$,
due to the SO(5) symmetry of the $c_2$ term (to be shown in the Proof of Theorem 2),
the Hamiltonian conserves two quantities $P = N_1 - N_{-1}$ and $Q = N_2 - N_{-2}$.
Nevertheless, the energy-level ordering is determined by only one quantum number $\Gamma$.
The fact that the ground state tends to be a singlet is consistent with what one would expect from the $c_2 (>0)$ term which favors spin-singlet pairs.
\smallskip
\textit{Theorem 3.---}If $c_1 = c_2 = 0$, the local ground state $|\Psi^{\text{GS}}_{N_2,\ldots,N_{-2}}\rangle$ in $\mathcal{H}_{N_2,\ldots,N_{-2}} (:= \bigcap^2_{\alpha=-2} \mathcal{H}_{N_\alpha})$ is unique and can be written as
\begin{equation}
|\Psi^{\text{GS}}_{N_2,\ldots,N_{-2}}\rangle =\sum\limits_{\bm{m}\in \mathcal{I}_{N_2,\ldots,N_{-2}}} G_{\bm{m}} |\Phi_{\bm{m}}\rangle, \label{Thm3}
\end{equation}
with $G_{\bm{m}} > 0$.
The local ground-state energy is independent of $N_2,\ldots, N_{-2}$.
Thus each $|\Psi^{\text{GS}}_{N_2,\ldots,N_{-2}}\rangle$ is also the global ground state in $\mathcal{H}$,
and the
ground-state degeneracy is $\binom{N+4}{4}=(N+4)!/(N!4!)$.
\smallskip
Note that $\mathcal{H}_{N_2,\ldots,N_{-2}} \subset \mathcal{H}_{M=2(N_2-N_{-2})+N_1-N_{-1}}$. Because of the absence of the spin-dependent interaction, the Hamiltonian in this case has SU(5) symmetry and conserves the particle number of each spin state. We can say that the ground states exhibit ``SU(5) ferromagnetism".
\smallskip
The above three theorems concern the ground-state magnetic properties and degeneracies.
In Fig.~\ref{phase_diagram}, the regions of these three theorems are shown together
with the mean-field phase diagram of spin-2 condensates.
\smallskip
\begin{figure}[b]
\includegraphics[width=0.37\textwidth]{phase_diagram_circle}
\caption{\label{phase_diagram} Regions of our results for discrete lattices (inner circle) and the mean-field ground-state phase diagram of spin-2 Bose-Einstein continuous condensates (outer circle)~\cite{ueda2012bose,Koashi2000,Ciobanu2000,Lewenstein2012,barnett2006classifying}.
In the small hopping limit of a homogenous lattice system, the outer circle is also the phase diagram of a Mott insulator state with one boson per site~\cite{Lewenstein2012,barnett2006classifying}.
Here, $\tan \theta= c_2/c_1$.
At the mean-field level, ferromagnetic, nematic, and cyclic phases are characterized by $(\langle \hat{\bm{F}}\rangle\neq 0$, $\langle \hat{S}_{+}\hat{S}_{-} \rangle=0)$, $(\langle \hat{\bm{F}}\rangle= 0$, $\langle \hat{S}_{+}\hat{S}_{-} \rangle \neq 0)$, and $(\langle \hat{\bm{F}}\rangle= 0$, $\langle \hat{S}_{+}\hat{S}_{-} \rangle=0)$, respectively.
}
\end{figure}
{\it Proofs.}---It is worth noting that, if $\hat{H}$ is expanded in terms of bosonic operators $\hat{a}^{\dagger}$ and $\hat{a}$,
the coefficient of each term implies the matrix element $\langle \Phi_{\bm{m}} | \hat{H} | \Phi_{\bm{m}'} \rangle$.
As a simple example, the hopping term always
results in nonpositive off-diagonal matrix elements because $-t_{i,j} \leqslant 0$.
\smallskip
\textit{Proof of Theorem 1.---}We first consider a single-site model in which $N$ particles sit on the same site $q \in \Lambda$~\footnote{The single-site model is essentially the same as continuous condensates with single-mode approximation applied~\cite{ueda2010fundamentals, Koashi2000, uchino2008dynamical}.}.
The Hamiltonian of the model can be obtained by taking $t_{i,j}=0$ for all $i,j$ in Eq.~(\ref{Hamiltonian}).
Let us first prove the following lemma.
\smallskip
\textit{Lemma.---}Every local ground state $| \widetilde{\Psi}^{\text{GS}}_M \rangle$ of the single-site model has a total spin $F_{\text{tot}}=2N$.
\smallskip
\textit{Proof of Lemma.---}Recall the SO(3) symmetry of the Hamiltonian.
Without hopping, we then see that all terms in the Hamiltonian commute with each other, which allows us to explicitly write down the energy eigenvalues of the system (see \cite{Koashi2000,Ueda2002,uchino2008dynamical} or
Supplemental Material~\footnote{\label{footnote1}See Supplemental Material at \href{http://link.aps.org/supplemental/10.1103/PhysRevLett.122.053401}{http://link.aps.org/supplemental/10.1103/PhysRevLett.122.053401}
for how to understand the SO(5) symmetry of $c_2$ term and how to find its eigenstates
and energy eigenvalues from the highest-weight representation of $\mathfrak{so}(5)$, which includes~\cite{Ueda2002,georgi1999lie,zee2016group}.}),
\begin{eqnarray}
E = &&V_q N + \frac{c_0}{2}N(N-1)
+ \frac{c_1}{2}[ F_{\text{tot}}(F_{\text{tot}}+1)-6 N ] \nonumber\\
&&+ \frac{c_2}{10} (N^2+3N-v^2-3v),
\label{E(one_site)}
\end{eqnarray}
where $v$ is the number of bosons that do not form (two-particle) singlets.
To minimize $E$ in $\mathcal{H}_M$, note that $c_1 <0$ and $0 \leqslant F_{\text{tot}}=F_q \leqslant 2v$.
A simple analysis yields $F_{\text{tot}}=2v=2N$
for every local ground state $| \widetilde{\Psi}^{\text{GS}}_M \rangle$.
\smallskip
Theorem~1 can now be proved in two separate regions.
\smallskip
(1) $\{c_1<0, 5c_1 \leqslant c_2 \leqslant 0\} $: By directly expanding $\hat{H}$ in terms of $\hat{a}^{\dagger}$ and $\hat{a}$'s,
one can easily find that $\forall \bm{m} \neq \bm{m}'$, $\langle \Phi_{\bm{m}} | \hat{H} | \Phi_{\bm{m}'} \rangle \leqslant 0$ is always true.
Because of the SO(3) symmetry, in the
basis $\{| \Phi_{\bm{m}}\rangle\}$,
the matrix of $\hat{H}$ is real symmetric and block diagonal with respect to $M$.
Within each $\mathcal{H}_M$, all possible configurations (distributions of particles on $\Lambda$, regardless of their spins) are connected via hopping,
and all possible spin states are connected via spin-dependent interactions $c_1$ and $c_2$ terms.
Therefore, for each block of $\hat{H}$, we can apply the Perron-Frobenius theorem \cite{
[{Let $M$ be a finite-dimensional real symmetric matrix with the following properties:
(a) $M_{ij}\leqslant0$ for any $i\neq j$;
(b) all $i\neq j$ are connected via nonzero matrix elements [i.e., for any $i\neq j$, $\exists (i_1, \cdots,i_n)$ with $i_1=i$, $i_n=j$, such that $M_{i_1,i_2}M_{i_2,i_3} \cdots M_{i_{n-1},i_n}\neq0$].
Then the lowest eigenvalue of $M$ is nondegenerate and all the components of the corresponding eigenvector can be taken to be strictly positive.
This is a corollary of the Perron-Frobenius theorem. For proof, see }]
Tasaki1998},
which implies that the local ground state $|\Psi^{\text{GS}}_M\rangle$ in $\mathcal{H}_M$ is unique and can be written as Eq.~(\ref{Thm1}).
Since $(\hat{\bm{F}}_{\text{tot}})^2$ commutes with both ${\hat H}$ and ${\hat M}$, each $|\Psi^{\text{GS}}_M\rangle$ must be an eigenstate of $(\hat{\bm{F}}_{\text{tot}})^2$.
To determine the total spin of $|\Psi^{\text{GS}}_M\rangle$, consider the overlap between $|\Psi^{\text{GS}}_M\rangle$ and $| {\widetilde \Psi}^{\text{GS}}_M \rangle$. Since the Perron-Frobenius theorem also applies to the single-site model and implies that the ground state $| {\widetilde \Psi}^{\text{GS}}_M \rangle$ has an expansion similar to Eq. (\ref{Thm1}) with $C_{\bm m} \geqslant 0$, we have
$\langle \widetilde{\Psi}^{\text{GS}}_M | \Psi^{\text{GS}}_M \rangle \neq 0$.
This means that the total spin of $|\Psi^{\text{GS}}_M\rangle$ is the same as that of $| {\widetilde \Psi}^{\text{GS}}_M \rangle$. It then follows from the Lemma that $|\Psi^{\text{GS}}_M\rangle$ has the total spin $F_{\text{tot}}=2N$.
\smallskip
(2) $\{ c_1<0, c_2>0 \}$:
In this region, we cannot apply the Perron-Frobenius theorem in the basis Eq.~(\ref{basis}),
because the off-diagonal matrix elements of ${\hat H}$ take both positive and negative values.
Instead, we use the min-max theorem~\footnote{{Let $\hat{A}$ and $\hat{B}$ be two Hermitian operators on a finite-dimensional Hilbert space $\mathcal{H}$ satisfying
$\langle \alpha |(\hat{A}-\hat{B})|\alpha\rangle\geqslant0$, $\forall |\alpha\rangle\in\mathcal{H}$. (i.e., $\hat{A}-\hat{B}$ is positive semidefinite.)
Let $a_i$ and $b_i$ be the $i$ th eigenvalues of $\hat{A}$ and $\hat{B}$, respectively. $a_i$ and $b_i$ are arranged so that $a_1\leqslant a_2\leqslant\cdots$, $b_1\leqslant b_2\leqslant\cdots$. Then the min-max theorem implies that $a_i \geqslant b_i$, $\forall i$.}}.
Define $\hat{H}_a := \hat{H}(c_1<0, 5c_1 \leqslant c_2 \leqslant 0)$ and $\hat{H}_b := \hat{H}(c_1<0, c_2>0)$.
Also in each $\mathcal{H}_M$, define $E_{a,0}$ and $E_{a,1}$ as the energies of the local ground state and the first local excited state of $\hat{H}_a$, respectively.
Similarly, we have $E_{b,0}$ and $E_{b,1}$.
(If there is degeneracy in the ground state, then $E_{b,1} = E_{b,0}$.)
The local ground state of $\hat{H}_a$, as proved above, is a ferromagnetic state which is a zero-energy state of the $c_2$ term, as it does not contain any spin singlets.
Therefore, Eq.~(\ref{Thm1}) is
an eigenstate of $\hat{H}_b$.
Since $\hat{S}_{i, +}\hat{S}_{i, -}$ is positive semidefinite, we have $\hat{H}_a \leqslant \hat{H}_b$.
Then the min-max theorem implies that $E_{a,0}= E_{b,0}$ and $E_{a,1} \leqslant E_{b,1}$. Recalling that $E_{a,0}< E_{a,1}$, we get $E_{b,0}<E_{b,1}$. This proves that the local ground state in the case $c_1<0$ and $c_2>0$ is also unique.
\smallskip
\textit{Proof of Proposition 1.---}Under the conditions of Proposition~1, the Hamiltonian in Eq.~(\ref{Hamiltonian}) becomes
$\hat{H}'=-\sum_{ i\neq j, \alpha} t_{i, j} \hat{a}^{\dagger}_{i,\alpha} \hat{a}_{j,\alpha} + \sum_{i}V_i \hat{n}_i
+(c_0/8)\sum_{i}[2\hat{n}_i(2\hat{n}_i+1)- (\hat{\bm{F}_i})^2]
+ (2c_2/5) \sum_{i}\hat{S}_{i, +}\hat{S}_{i, -}$.
We seek states (if any) that minimize all terms in $\hat{H}'$ simultaneously.
The last two terms in $\hat{H}'$ are now positive semidefinite.
According to Theorem~1, the ground states must have $F_{\text{tot}}=2N$, which clearly makes the last two terms zero.
As for the hopping term, according to the Perron-Frobenius theorem,
its single-particle ground state $\varphi_0$ is unique and
satisfies $\varphi_0(i)>0$.
Thus it is obvious that $(\hat{b}^{\dagger}_{2})^N |\text{vac}\rangle$
in Eq.~(\ref{Prop1}) gives the unique ground state in $\mathcal{H}_{M=2N}$.
Since $[\hat{H}, \hat{F}_{\text{tot}}^-]$, the uniqueness of local ground states
then implies $|\Psi^{\text{GS}}_{M'}\rangle \propto (\hat{F}_{\text{tot}}^-)^{2N-M'} |\Psi^{\text{GS}}_{M=2N}\rangle$.
\smallskip
\textit{Proof of Theorem 2.---}There exists a new set of bosonic operators $\hat{d}^{\dagger}$ and $\hat{d}$ defined in the Supplemental Material~[\hyperref[footnote1]{36}],
such that the singlet creation operator can be written as
$\hat{S}_{i,+} = (\sum^5_{\mu=1}\hat{d}^{\dagger}_{i,\mu} \hat{d}^{\dagger}_{i,\mu})/2$.
The form of $\hat{S}_{i,+}$ now remains unchanged
when $\hat{d}^{\dagger}_{i,\mu}$'s are subject to SO(5) transformations.
Thus the $c_2$ term has a manifest SO(5) symmetry.
In the Cartan-Weyl basis, ten generators of SO(5) are
\begin{equation}
\hat{E}_{i,\alpha\beta} = (-1)^\alpha \hat{a}^{\dagger}_{i,\alpha} \hat{a}_{i,-\beta} - (-1)^\beta \hat{a}^{\dagger}_{i,\beta} \hat{a}_{i,-\alpha}, \label{generator}
\end{equation}
where $-2\leqslant \beta<\alpha\leqslant2$.
Taking $\alpha=-\beta$, we get a basis of the Cartan subalgebra:
$\hat{P}_i := \hat{E}_{i,-1,1} = \hat{n}_{i,1} - \hat{n}_{i,-1}$
and $\hat{Q}_i := \hat{E}_{i,2,-2} = \hat{n}_{i,2} - \hat{n}_{i,-2}$.
Indices $(\alpha,\beta)$ of the other eight generators
are roots in the root system $B_2$.
\smallskip
Because of the SO(5) symmetry of the Hamiltonian,
we have $[\hat{P},\hat{H}] = [\hat{Q},\hat{H}] = 0$,
which shows that $\hat{H}$ conserves $P=N_1-N_{-1}$ and $Q=N_2-N_{-2}$
and splits into blocks with respect to these two quantum numbers.
Besides the hopping term, off-diagonal matrix elements appear only in the $c_2$ term.
By applying the following U(1) transformation for every $i\in \Lambda$:
$\hat{\tilde{a}}^{\dagger}_{i,\pm1} = \mathrm{i}\hat{a}^{\dagger}_{i,\pm1}$,
$\hat{\tilde{a}}^{\dagger}_{i,\pm2} = \hat{a}^{\dagger}_{i,\pm2}$ and
$\hat{\tilde{a}}^{\dagger}_{i,0} = \hat{a}^{\dagger}_{i,0}$,
one can verify that all the off-diagonal matrix elements of $\hat{H}$ in this basis become nonpositive.
Furthermore, connectivity of configurations and that of spin states are guaranteed by the hopping term and the $c_2$ term, respectively.
The Perron-Frobenius theorem is thus applicable and asserts that the local ground state $| \Psi^{\text{GS}}_{P,Q} \rangle$ in $\mathcal{H}_{P,Q}$ is unique and can be written as Eq.~(\ref{Thm2}).
\smallskip
Now we extract some useful information from the aforementioned single-site model.
We claim that states written in the form
$|(\bm{\alpha}, \bm{\beta});N,v\rangle := \hat{E}_{q,\alpha_m\beta_m} \cdots \hat{E}_{q,\alpha_2\beta_2} \hat{E}_{q,\alpha_1\beta_1} (\hat{S}_{q,+})^{(N-v)/2}(a^{\dagger}_{q,+2})^{v} |\text{vac}\rangle$
are eigenstates of $\hat{S}_{q,+} \hat{S}_{q,-}$ with eigenvalues $(N^2+3N-v^2-3v)/4$~[\hyperref[footnote1]{36}].
Since $c_2<0$, the smaller $v$, the lower energy.
Define $\Gamma:=|P|+|Q|$ and note that $v \geqslant \Gamma\geqslant 0$.
Since $v$ is the number of particles that do not form singlets,
for a ground state of the single-site model $| \widetilde{\Psi}^{\text{GS}}_{P,Q} \rangle$
$v$ takes the minimum possible value, which is $v_{\text{min}} = \Gamma$ if $N-\Gamma$ is even, while $v_{\text{min}} = \Gamma +1$ if $N-\Gamma$ is odd.
The Casimir operator for the model on the total lattice $\Lambda$ is defined as $\hat{C}_\text{tot}^2 = \sum_{\alpha < \beta} ( \hat{E}_{\alpha\beta}^\dagger \hat{E}_{\alpha\beta} + \hat{E}_{\alpha\beta} \hat{E}_{\alpha\beta}^\dagger)/2$, where $\hat{E}_{\alpha\beta} := \sum_i \hat{E}_{i,\alpha\beta}$. It is easy to see that $\hat{C}_\text{tot}^2 | \widetilde{\Psi}^{\text{GS}}_{P,Q} \rangle = [\hat{N}(\hat{N}+3)-4\hat{S}_{q,+} \hat{S}_{q,-}] | \widetilde{\Psi}^{\text{GS}}_{P,Q} \rangle = v_{\text{min}}(v_{\text{min}}+3) | \widetilde{\Psi}^{\text{GS}}_{P,Q} \rangle$.
Since the Perron-Frobenius theorem again applies to the single-site model,
we have $\langle \widetilde{\Psi}^{\text{GS}}_{P,Q} | \Psi^{\text{GS}}_{P,Q} \rangle \neq 0$,
which implies
\begin{eqnarray}
\hat{C}_\text{tot}^2 | \Psi^{\text{GS}}_{P,Q} \rangle
= v_{\text{min}}(v_{\text{min}}+3) | \Psi^{\text{GS}}_{P,Q} \rangle.
\end{eqnarray}
We are now ready to prove the energy-level ordering. Let $E^{\text{GS}}_{P,Q}$ be the energy of the local ground state $|\Psi^{\text{GS}}_{P,Q} \rangle$. We first note that $E^{\text{GS}}_{P,Q}=E^{\text{GS}}_{|P|,|Q|}$,
because
under the transformation $\hat{a}_{+2} \leftrightarrow \hat{a}_{-2}$ or $\hat{a}_{+1} \leftrightarrow \hat{a}_{-1}$, the Hamiltonian remains unchanged
but $\hat{P}$ or $\hat{Q}$ gets a minus sign.
Thus it suffices to consider the case $P,Q \geqslant 0$.
Next, we prove that all $E^{\text{GS}}_{P,Q}$'s with the same $\Gamma$ are the same.
Define $|\Psi_a \rangle= \hat{E}_{2,-1} |\Psi_{P+1,Q-1}^{\text{GS}}\rangle \in \mathcal{H}_{P,Q}$ (assume $Q\geqslant 1$).
Apparently energy eigenvalue of $|\Psi_a\rangle$ should be the same as $|\Psi_{P+1,Q-1}^{\text{GS}}\rangle$, which is $E^{\text{GS}}_{P+1,Q-1}$.
So we have $E^{\text{GS}}_{P,Q} \leqslant E^{\text{GS}}_{P+1,Q-1}$.
Define $|\Psi_b \rangle= \hat{E}_{1,-2} |\Psi_{P,Q}^{\text{GS}}\rangle \in \mathcal{H}_{P+1,Q-1}$ (assume $Q\geqslant 1$),
and similarly we get $E^{\text{GS}}_{P+1,Q-1} \leqslant E^{\text{GS}}_{P,Q}$.
Thus, we have $E^{\text{GS}}_{P+1,Q-1} = E^{\text{GS}}_{P,Q}$, which means that $E^{\text{GS}}_{P,Q}$ is only a function of $\Gamma=|P|+|Q|$,
denoted as $E^{\text{GS}}_{\Gamma}$.
Now we show the ordering of $E^{\text{GS}}_{\Gamma}$.
Construct $|\Psi_c \rangle = \hat{E}_{0,-1} |\Psi_{P+1,Q}^{\text{GS}}\rangle \in \mathcal{H}_{P,Q}$ and then get $E^{\text{GS}}_{\Gamma} \leqslant E_{\Gamma+1}^{\text{GS}}$.
When $N-\Gamma$ is even, $|\Psi_c\rangle$ and $|\Psi_{P,Q}^{\text{GS}}\rangle$ have different $C_{\text{tot}}^2$, and hence are orthogonal.
The uniqueness of each local ground state then yields $E^{\text{GS}}_{\Gamma} < E^{\text{GS}}_{\Gamma+1}$.
When $N-\Gamma$ is odd, construct $|\Psi_d\rangle = \hat{E}_{1,0} |\Psi_{P,Q}^{\text{GS}}\rangle \in \mathcal{H}_{P+1,Q}$,
and similarly we have $E^{\text{GS}}_{\Gamma} \geqslant E^{\text{GS}}_{\Gamma+1}$,
which finally gives $E^{\text{GS}}_{\Gamma} = E^{\text{GS}}_{\Gamma+1}$.
We thus have obtained the desired energy-level ordering stated in \textit{Theorem 2}.
Consequently, the global ground state is unique and lies in the subspace $\mathcal{H}_{\Gamma=0}$ when $N$ is even, while it is five-fold degenerate and lies in $\mathcal{H}_{\Gamma=0} \oplus \mathcal{H}_{\Gamma=1}$ when $N$ is odd.
Then it follows from $[\hat{H}, (\hat{\bm{F}}_{\text{tot}})^2]=0$ that the global ground state has $F_{\text{tot}}=0$ ($F_{\text{tot}}=2$) when $N$ is even (odd).
\smallskip
\textit{Proof of Theorem 3.---}Applying the Perron-Frobenius theorem to the hopping term proves the theorem.
The proof is essentially the same as that of Theorem 3 in \cite{Katsura2013} (see the Supplemental Material~[\hyperref[footnote1]{36}] for details).
\smallskip
{\it Discussion.}---In conclusion, we have established the basic ground-state properties of the spin-2 Bose-Hubbard model, as stated in the main theorems. Symmetry plays an important role in our proofs. In particular, the SO(5) symmetry is essential in the case $\{ c_1 = 0, c_2 < 0 \}$.
Although the Cartan subalgebra of $\mathfrak{so}(5)$ is two-dimensional, we found that the energy-level ordering is effectively ``one-dimensional", as it is characterized only by the quantum number $v$.
\smallskip
In the presence of an external magnetic field in the $z$ direction,
one should add linear and quadratic Zeeman terms
$\sum_{i,\alpha} (-p_i \alpha \hat{n}_{i,\alpha}+ q_i \alpha^2 \hat{n}_{i,\alpha})$ to $\hat{H}$~\cite{Ueda2002,Uchino2010}.
In this case, the total Hamiltonian no longer has SO(3) symmetry.
However, since the Zeeman terms are diagonal in the basis Eq.~(\ref{basis}),
the uniqueness of the ground state within each subspace as well as Eqs.~(\ref{Thm1})--(\ref{Thm3}) still holds in each respective parameter region.
\smallskip
\begin{acknowledgments}
The authors would like to thank Masahito Ueda and Sho Higashikawa for fruitful discussions. H. K. was supported in part by JSPS KAKENHI Grants No. JP15K17719, No. JP16H00985, No. JP18K03445, and No. JP18H04478.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,884 |
Toomanywires
All About Video Games
TGS 2012 - Preview
You are probably already sick of my complaining about how I'm not in Japan for this year's Tokyo Game Show. This is the first TGS I have missed since 2005 - my press pass email is still marked "unread" - and this is the longest I've been away from Japan in some time. It would seem that TGS is prepared to move on without me, but as much as I'd like to deny its existence, I am still looking forward to it. Obviously, my hands-on impressions will be a tad unreliable this year, but I'm still hoping to cover most of the big announcements as part of a week+ of posts all about Japan; this series will include, but will not be limited to TGS coverage.
To get us started, here is a TGS preview, with a round-up to follow at the end of the week. I shall now resume smothering myself in Japanese condiments whilst singing the entirety of Utada Hikaru's back catalogue.
Much like E3, Tokyo Game Show has lost some of its luster over the last few years. Japanese games are no longer the toast of the industry as the mobile game market continues to aggressively expand, rendering TGS of far less interest to audiences outside of Japan. However, it remains an important event in the gaming calendar and is vital in gauging the pulse of the Japanese industry.
Looking through software line-ups and booth information, it's evident that mobile games continue to encroach on the territory of traditional, console and handheld games. From strictly mobile developers like GREE to industry stalwarts like Konami, games played in the palm of your hand will be a major focus at this year's TGS. Last year a number of big publishers scaled back their presence, with some pulling out entirely, resulting in mobile games taking up half the floor space of the giant exhibition halls. GREE will take centre stage with a Keynote presentation – usually reserved for the Sony's of the world - and you can bet that the large space occupied in the past by Microsoft, who is not attending this year, will be swallowed up by smart phone games.
Despite this shift to mobile, there will still be plenty to keep the console and handheld purists happy. SEGA always thrives at TGS, and this year's stand will be dominated by Yakuza 5. Despite Sega's recent restructuring, there is still an outside chance that Nagoshi's latest will make it west, but don't expect any announcements just yet. Sega's other major attraction figures to be Phantasy Star Online 2 for the Vita; already available on the PC in Japan, it is one of the few high profile, forthcoming releases for Sony's handheld.
Sticking with Vita, Keiji Inafune's Soul Sacrifice continues to look promising, if not a little grim, and a playable demo is expected. I don't see much else of interest for the Vita, though we can always hope that Sony has something up their sleeve, though recent history would suggest otherwise. The 3DS will be getting plenty of Capcom love, headlined by the first playable demo for Monster Hunter 4, which will feature battles against three different monsters and is sure to attract the largest queues over the four days. Capcom are teasing "surprising information", but I wouldn't hold your breath for a Vita port. Ace Attorney 5 will be on hand as will the anime-style, Lost Planet spin-off, EX Troopers, which is also coming to PS3. I knew very little about this game before researching this post, but its colourful visuals and interesting use of an existing property have me intrigued.
Konami will look to build upon their well-received announcements at last month's Metal Gear 25th Anniversary event. Metal Gear Rising will have a brand new demo, and we might even get more Ground Zeroes footage to go along with a playable demo for Social Ops, the smart phone title due later this year. Expect Square Enix to further impress with the Final Fantasy XIV rehash and for 3DS RPG, Bravely Default: Flying Fairy, to continue to gather momentum as we approach its October debut. SE have confirmed that they will be showcasing Dragon Quest X for the Wii U - Nintendo does not attend TGS, so any Wii U news will be third party - and we might get some more information on Final Fantasy XIII-3: We Swear it's Better than the Last Two, otherwise known as Lightning Returns. When it comes to FF Versus, don't expect anything more than a few "it's not dead" comments. Last Guardian will be another notable absentee in this, our seventeenth consecutive TGS without a new trailer for Sony's pigeon-snake, dating sim.
There will also be a number of press conferences this week, ahead of Thursday's ribbon cutting. Held separately from TGS, they should serve to whet our appetite for the show proper - last year these pre-events unveiled FFX HD and the Circle Pad Pro! Hopefully, between the lead-up conferences and main event, we'll get a better feel for new games we're already familiar with and some new announcements to boot.
Tokyo Game Show '12
Thirdrail1 20 September 2012 at 08:24
The new ps3 Even Slimmer has had me laughing all day. I'd love to have heard that meeting. Where someone looked around Sony's gaming portfolio and said, "I've figured out exactly what we need! Let's put a team of people to work designing a more expensive version of the ps3! That's what is obviously missing from our 2012 lineup!" WTF?!
Soul Sacrifice looked very promising. Hopefully the Vita and the Wuu will pick up some more interesting games as the show proceeds.
The Last Guardian has certainly become an interesting mystery. It seems like they must be porting it, either down to the Vita or up to the ps4. It's been SO long now. At this point, if it's dead, why bother hiding the body? But, then, by the same token, why not prove it's alive, if it's not dead? Baffling.
toomanywires 20 September 2012 at 23:06
Hardware has always been Sony's speciality. They just can't stop creating!
Soul Sacrifice looks great. Very Dark Souls & Monster Hunter and it looks well suited to the Vita. I hope it lives up to expectations.
Who needs Last Guardian when we have red and blue Vitai? Sony has already won TGS.
The Massacre at Guthrie Farm
I just wanted to say hello.
I was out on my horse, exploring Red Dead Redemption 2 for the first time. I was overwhelmed, unsure of the controls and equally unsure of where to go.
I'd just skinned a bison and was combing my horse when I spotted a small farm on the horizon. There will be people there, I thought, and perhaps they'll have some important information for me. At the very least, they'll have something entertaining to say.
I slung the bison pelt on the back of my horse, made sure my guns were holstered and headed straight for the farmstead in search of conversation and companionship. I spotted a farmhand working the field a stone's throw from the house. "Howdy partner", I said in my head, in a silly cowboy voice. The middle-aged man took one look at me, shouted something unpleasant, and drew his weapon. What a dick.
Turns out Arthur (me) is a wanted man. Earlier that day, I'd unintentionally murdered a travelling salesman. I just wanted to steal…
The Best and Worst Games of 2019
The Best & Worst of 2018 / 2017 / 2016 / 2015 / 2014 / 2013 / 2012 / 2011 / 2010
This is my tenth year-end, Best & Worst blowout. That's a decade of me agonising over winners and losers for a post that maybe twenty people read. And eleven of those people are me.
A lot has changed since I picked Mass Effect 2 as my game of the year back in 2010. We've had hardware successes and failures, and major franchises that have come and gone. We've seen the peaks and troughs of casual and mobile gaming, the near disappearance of game-dedicated handhelds, the growth of subscription and streaming services, the evolution of online gaming, and the continued ascendancy of the industry as a whole. And don't forget the ongoing march to all-digital, the multitude of trends that have lived and died (3D, second screen gaming, toys-to-life etc.), the indie explosion-demise-resurgence, and the death of the mid-tier game and developer. A lot has happened over the last decade.
By this …
Sekiro - Are You Enjoying This Game?
"Are you enjoying this game?"
I'm fighting a Sekiro mini boss. An unimposing foe who has already killed me a dozen times and wasted a good half hour of my precious time. Time that would've been better spent studying Japanese, doing the washing up, or going to bed early.
My wife is asking. She's seen me playing Sekiro on and off for the last month. Me, Games King, as I've asked her to call me, slowly chipping away at a game that refuses to respect my time or lack of hand-eye coordination.
Every time she glances at the screen, I'm dying. Trampled by an over-sized bull; cut down by a rooftop samurai; outwitted by an ape without a head; bested by a cliff. Fucked by a regular monkey. I can understand why she might question my enjoyment of Sekiro.
This particular mini boss is completely optional. However, now that I've engaged with her once, I feel bound to keep at it until I finally figure out a way to defeat her. I t…
Welcome to Toomanywires. This is where I write about video games, old and new, and share tidbits about life in Japan. Please see the "About" page for more information about me & my blog.
Gaming Japan
TGS 2012: Round-up
Wii U Part 2: Games & the West
Wii U by the Numbers
Musings of a Gamer XVIII
A PSN Revival
A Day in the Life of..
A Decade in Games
Classics Revisited
Contemporary Games
Game Pitch
Monday Spotlight
Monthly Round-up
Musings of a Gamer
PlayStation 3 Retrospective
The End of a Generation
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,691 |
\section{Introduction}
~~~~~~The origin of cosmic rays (CRs) has been a mystery since their discovery
in 1912. CRs have a nearly featureless power-law spectrum with a spectral
index of about $-3$ from energies of $\sim 10^{9}$ to $\sim 10^{20}$ eV.
However, detailed measurements revealed several subtle structures in the
CR spectrum at $\sim4\times10^{15}$ eV (knee), $\sim4\times10^{17}$ eV
($2$nd knee), $\sim 5\times10^{18}$ eV (ankle), and $\sim 6\times10^{19}$
eV (Greisen-Zatsepin-Kuzmin cut-off) \cite{Horandel07}.
The study of these structures is of great importance for understanding
the origin, propagation and interaction of CRs.
It is generally believed that supernova remnants (SNRs) are the sources
of Galactic CRs (GCRs), based on the simple argument that the power by
SNRs is sufficient to sustain the total power of GCRs
\cite{1934Baade,1964Ginzburg}. However, the $\gamma$-ray observations
of SNRs tend not to favor the CR nuclei acceleration in SNRs
\cite{2011ApJ...734...28A}. There is also no direct evidence for CR
nuclei sources among the thousands of GeV $\gamma$-ray sources
\cite{2012ApJS..199...31N} and more than one hundred TeV $\gamma$-ray
sources\footnote{http://www.mpp.mpg.de/$\sim$rwagner/sources/}.
Other arguments against the SNR origin of GCRs can be found in e.g.,
\cite{2005JPhG,2009Natur.460}.
Instead of discussing the stellar level sources in the Galaxy, the
Galactic center (GC) as powered by the accretion of the supermassive
black hole, could be a natural candidate of CR origin. Although the
GC is relatively quiet nowadays, there is large amount of evidence
suggesting that the GC may have been active $\sim 10^7$ years ago
\cite{1971A&A....13..405V,1974ApJ...188..489S,2007JPhG...34.1813E}.
Historically, here have been many discussions on the possibility
that the GC is the dominant source of GCRs as a result of its activity
\cite{1981AZh....58..959P,1981ICRC....2..344S,1983JPhG....9.1139G}.
There are great progresses in the measurements of GCRs and $\gamma$-rays
in recent years. First, owing to the improved energy resolution, a very
sharp knee structure was found by many air shower array experiments
\cite{2007APh....28...58C,2007NuPhS.165...74K,2008JPhG...35k5201G,
2008ApJ...678.1165A,2009APh....31...86A,2009NJPh...11f5008I}. This result
seems to favor the single source model as proposed in
\cite{1997JPhG...23..979E,2009arXiv0906.3949E}. Second, several high
precision measurements led to the discovery of both the positron fraction
excess \cite{2009Natur.458..607A} and the total $e^+e^-$ spectral excess
\cite{2008Natur.456..362C,2009PhRvL.102r1101A} from $O(10)$ GeV to TeV.
It was also shown that the $e^+e^-$ spectra have a cutoff at several
TeV \cite{2008PhRvL.101z1104A,2009A&A...508..561A}. The result seems
also to favor the single source model proposed in \cite{2009ApJ...700L.170H}.
Since the observational anisotropy of CRs is very weak ($10^{-4}\sim10^{-3}$),
the single source should not be very close to us. The GC is one of the
potential candidates \cite{2009arXiv0911.3034H}. Furthermore, the
discoveries of the 511 keV line emission and $\gamma$-ray bubbles in the
GC region indicate that the GC may indeed have past activities.
Based on these new observational data, it is time to revisit the idea that
the GC is the main source of GCRs.
In this work we describe a unified model to explain the recent observations
of CRs and $\gamma$-rays, based on the past activity of the GC.
\section{Model}
~~~~~~During the active phase of the GC, the accretion of stars and
gases by the supermassive black hole is very efficient to provide
high enough power for particle acceleration, jet launch and propagation.
The GC activities can produce shocks and accelerate CRs to high energies
\cite{Cheng:2006ch,2001MNRAS.321..433B} in scales from $\sim10^{-3}$
pc \cite{2008ApJ...685L..23A} to kpc \cite{2012ApJ...753...61S}.
The heating of the accretion disk makes it filled with thermal photons.
Jets could also be produced due to the accretion events.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.8\textwidth,angle=-0]{gc-new.eps}
\caption{Cartoon of the model.}
\label{GC_Picture}
\end{figure}
In the above mentioned environment, the following physical processes are
expected to occur. First we may have the primary acceleration of CRs
around the black hole. The accelerated nuclei can then escape from
the source region and enter the disk filled with background field.
The interaction between CR nuclei and the background photons can lead
to energy loss of CRs, which could result in the formation of the knee
\cite{2009ApJ...700L.170H}. The interaction will produce $e^+e^-$ pairs,
which can be partially
transported into the halo through the jets. Those electrons and positrons
can then radiate to form the multi-wavelength haze/bubble. Finally the
CRs diffuse in the bulge could further interact with the interstellar
medium to produce low energy $e^+e^-$. After cooling down, these positrons
may annihilate to produce the 511 keV line emission. The cartoon to
describe the basic picture of the model is shown in Fig.\ref{GC_Picture}.
In the following we will discuss more details about the three aspects,
the origin of the knee, the multi-wavelength haze/bubble and the 511
keV emission respectively.
\subsection{Origin of the Knee}
~~~~~~Ref. \cite{2009ApJ...700L.170H} proposed a model to simultaneously
explain the knee and the $e^+e^-$ excess, incorporating pair production
interactions between CRs and the ambient photon field. It was further shown
that the irregular structures of the CR spectra around the knee region and
the Galactic ``B component'' could also be well explained in this scenario
\cite{2010ScChG..53..842W}. As a consequence, one set of parameters in
the mode of \cite{2009ApJ...700L.170H} also favors the single source
model. It indicates that there might be a single source with relatively
stable properties (during the CR acceleration period) that is responsible
for GCRs.
In \cite{2009ApJ...700L.170H}, a supernova-pulsar system was proposed as
a possible candidate for such a source. Though it is not impossible, it
seems non-trivial for such a system to satisfy the conditions needed to
produce the knee of the CRs \cite{2011ASTRA...7..179E}. Alternative
candidate sources may include micro-quasars or the Galactic center (GC).
The latter seems to be an especially attractive option
\cite{2009arXiv0911.3034H}. As proposed by many studies, the capture of
stars or accretion of gas by the central supermassive black hole can
produce shock and accelerate particles \cite{Cheng:2006ch}.
The density of background radiation field at GC region is the key point
whether the knee structure of CR spectra can be formed. As shown in
\cite{2009ApJ...700L.170H}, the photon column density should be
$\sim 10^{30}$ cm$^{-2}$. Without firm observational evidence of the
size of such kind of interactions, we take $1$ pc as an illustration.
The optical photon density inside $1$ pc of the GC is about $5\times
10^4$ eV/cm$^{3}$ in a flare disk and dust \cite{1992ApJ...387..189D}.
The diffusion velocity of CRs in the knee region is estimated to be
the order of $10^{-3}$ of the light velocity, according to the measured
anisotropy \cite{2003ICRC....1..183A}. Therefore, the time for CRs to
diffuse out of such a region is about $10^3$ yr. So the photon column
density which CRs can encounter is $n c \tau \sim 10^{26}$ cm$^{-2}$,
which is 4 orders of magnitude lower than that required. However, it is
possible that the photon luminosity could be much higher during the
active phase than at present. For example, the observation of infrared
radiation from other galaxies showed that when the nucleus was in the
active phase the infrared luminosity could be as high as $10^{44}-10^{47}$
erg/s, which is $2-5$ orders of magnitude higher than the present value
of our Galaxy, $\sim10^{42}$ erg/s \cite{1981AZh....58..959P}.
Therefore it is possible that the background photon density could be 4
orders of magnitude higher during the active phase, and the condition to
form the knee could be satisfied. Under these circumstances, the total
energy of background photon is estimated to be $\sim 10^{53}$ erg,
which is close to the accretion power of one solar mass.
\subsection{Fermi bubbles: possible relics of past GC activity}
~~~~~~If GC indeed plays a significant role to produce the Galactic CRs,
we may expect the existence of some relics of the past activity of GC.
``Fermi bubbles'', the new observational evidence, may be such kind of
relics of the past GC activity.
Thanks to the high performance of Fermi $\gamma$-ray telescope, a large
scale, extended $\gamma$-ray excess in the GC direction was discovered
\cite{2010ApJ...717..825D}, which was then revealed to be two giant
$\gamma$-ray bubbles \cite{2010ApJ...724.1044S}. The Fermi bubbles are
symmetric with respect to the Galactic plane, extending $\sim 50$ degrees
in latitude and $\sim 40$ degrees in longitude. They are spatially
correlated with the WMAP haze observed in the $20-60$ GHz band
\cite{2004ApJ...614..186F,2008ApJ...680.1222D}, and the edges of the
bubbles are also found to be coincident with features in the ROSAT $1.5-2$
keV X-ray maps \cite{1997ApJ...485..125S}. Recently, the PLANCK collaboration
confirmed the microwave haze found in WMAP data \cite{2012arXiv1208.5483P}.
Several models are proposed to explain the Fermi bubbles
\cite{2011arXiv1110.0834G,2011ApJ...731L..17C,2011PhRvL.106j1102C,
2011arXiv1103.0055G,2011arXiv1110.5436I,2011PhRvL.107i1101M,
2011MNRAS.415L..21Z}, most of which are based on the GC activity in the past.
The bubbles are found to have a hard $\gamma$-ray spectrum between 1 and
100 GeV, with a power law index $\sim -2$. The $\gamma$-ray spectrum can
be well reproduced by the inverse Compton scattering (ICS) process of
power-law distributed electrons with index $-2\sim -2.5$
\cite{2010ApJ...724.1044S}, taking into account the cosmic microwave
background, infrared and optical
background radiation. In addition, the calculated synchrotron radiation
can reproduce the radio haze flux, assuming that the magnetic field is of
the order of 10 $\mu$G. However, this electron spectrum has difficulty
to explain the observed low energy drop below 1 GeV. In order to solve
this problem, an electron population with limited energy range was
proposed. Based on these facts, Su et al. \cite{2010ApJ...724.1044S}
concluded that the bubbles are most likely created by a large episode of
energy injection in the GC in the last $10^{7}$ years through an accretion
event in the center of supermassive black hole, a nuclear starburst or
some other energetic event.
The locally measured energy density of CRs is about $1$ eV cm$^{-3}$.
Giving that the volume of the Galactic disk is about $\pi(20\,{\rm kpc})^2
(0.2\,{\rm kpc}) \sim 10^{67}$ cm$^3$, the total energy of CRs is
about $10^{55}$ erg. Assuming the pre-propagated spectrum of GCRs
is $\propto E^{-2.0}$, the total energy of GCRs above the knee
($E\sim$PeV) is approximately $10^{54}$ erg. Such an energy will be
mostly converted into $e^+e^-$ through the pair production interactions.
According to \cite{2010ApJ...724.1044S}, the Fermi
has an age on the order of $10^{7}$ yr.
It is to say that the luminosity of $e^+e^-$ is
estimated to be $10^{39}-10^{40}$ erg/s .
The total luminosity of the Fermi bubbles in
$1-100$ GeV is estimated to be about $4\times10^{37}$ erg/s
\cite{2010ApJ...724.1044S}, which is much smaller than the above estimated
value. That is to say, the produced $e^+e^-$ could be possible to generate
the Fermi bubbles.
It should be noted that in the strong background radiation field,
electrons/positrons may cool down very efficiently. However, it
is expected that there should also be acceleration in the jets,
which may more or less compensate the cooling of the electrons/positrons.
The $e^+e^-$ spectrum used to calculate the synchrotron and ICS spectra
is adopted from \cite{2009ApJ...700L.170H},
which can explain the $e^+e^-$ excesses observed by PAMELA/ATIC/Fermi.
For the interstellar radiation field (ISRF) model we adopt that reported
in Porter \& Strong \cite{2005ICRC....4...77P}, in which a new calculation
based on the modelings of star and dust distributions, the scattering,
absorption and re-emission of the stellar light by dust, was carried out.
The ISRF model showed good agreement with the observational data
\cite{2006ApJ...640L.155M}, and was implemented in the public CR
propagation code GALPROP \cite{1998ApJ...509..212S}. Here we adopt the
ISRF intensity at $R=0$ and $z=4$ kpc. The energy spectrum of the ISRF
is shown in the left panel of Fig. \ref{fig-FermiBubble-WMAPHaze}.
Three major components, optical from stars, far-infrared from dust
and the cosmic microwave background (CMB), are clearly shown.
The right panel of Fig. \ref{fig-FermiBubble-WMAPHaze} shows the resulting
synchrotron and ICS spectra by the $e^+e^-$. The magnetic field is assumed
to be $B=15\mu$G, and the Klein-Nishina cross section of ICS is adopted.
The spectra of WMAP synchrotron haze and Fermi ICS bubbles are consistently
generated.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth,angle=0]{ISRF-4kpc.eps}
\includegraphics[width=0.45\textwidth,angle=0]{Atic.eps}
\caption{Left: the ISRF at $(R,z)=(0\,{\rm kpc},\,4\,{\rm kpc})$,
adopted from GALPROP package. Right: The calculated spectrum of ICS
$\gamma$-rays and synchrotron radiation originating from a re-accelerated
electron spectrum generated through CR-photon pair production interactions.
The line of sight direction is chosen to be $l=0^{\circ}$ and $b=25^{\circ}$.
The data points representing the Fermi bubbles and WMAP haze are taken from
Table 3 and Fig. 23 of \cite{2010ApJ...724.1044S}.}
\label{fig-FermiBubble-WMAPHaze}
\end{figure}
\subsection{511 keV line emission}
~~~~~~It is natural to expect a possible connection of the GC origin of
CRs with the 511 keV line emission as reported by several experiments
\cite{Johnson:1972,2005A&A...441..513K,Jean:2005af,Weidenspointner:2008zz},
although the most popular model of 511 keV emission is the decay of
radioactive isotope\cite{2011RvMP...83.1001P}.
The 511 keV line emission indicates the existence of non-relativistic
positrons in the GC region. The hadronic interactions of GCRs with the
ambient gas could be one potential source of these positrons
\cite{Cheng:2006ch,Ramaty:1970ra,Totani:2006ff}. We make an order of
magnitude estimate of the power of electrons. The total number of CR
protons is about $\sim10^{58}$, for a local number density of $\sim10^{-9}$
cm $^{-3}$ and the volume of the Galactic disk $\sim 10^{67}$ cm$^3$.
Considering that the size of the Galactic bulge is $\sim 1$ kpc, the
typical path length that a particle travels from the GC to outside of
the bulge should be $\sim 10^3$ kpc, for diffusion coefficient
$D\sim 5\times 10^{28}$ cm$^2$ s$^{-1}$. Assuming that the number
density of ISM nuclei in the bulge is $1$ cm$^{-3}$, and the inelastic
cross section of $p-p$ scattering is several tens of mb, the average
number of collisions for one CR proton before traveling out of the bulge
is $\sim 0.1$. Thus, the total number of positrons is $\sim 10^{57}$.
Assuming the cooling time of positrons is about $10^7$ years, which
corresponds to the ionization and Coulomb losses in an ISM with density
of $1$ cm$^{-3}$ for a $100$ MeV positron \cite{1998ApJ...509..212S},
the cooled positron production rate is $3\times10^{42}$ s$^{-1}$, which
is comparable with the rate $10^{43}$ s$^{-1}$ as implied from the flux
of 511 keV $\gamma$-ray line \cite{2005A&A...441..513K}.
However, it was pointed out that the diffuse $\gamma$-ray constrained
positron production rate would be not more than a few percent of the
positron rate suggested by the 511 keV emission data
\cite{Cheng:2006ch,2005A&A...441..513K,2008ApJ...682..400P}. This
problem can be solved in a non-stationary scenario that the GC was
in active phases in the past and the positron production rate would
be much higher than that determined by the current diffuse $\gamma$-ray
flux \cite{Cheng:2006ch,2008ApJ...682..400P}.
\section{Conclusion and discussion}
~~~~~~In this work we proposes that the GC is the major source of GCRs.
There is evidence to show the past activity of the GC. Particle
acceleration can take place during the violent phase of the GC.
Also it is expected the existence of strong radiation field around
the GC. Thus an efficient $e^+e^-$ pair production interactions
between GCRs and the ambient photons might be responsible for the
the knee of the CR spectra \cite{2009ApJ...700L.170H}. A fraction of
$e^+e^-$ produced by pair production interactions, can be reaccelerated
in the jets and escape into the halo. The ICS and synchrotron radiation
of these $e^+e^-$ may possibly explain the observed Fermi bubbles and WMAP haze.
Even the jet can transport the $e^+e^-$ very efficiently into the halo,
the propagation of these particles from the jet to the whole bubble is
still a question. Because of the lack of knowledge about the Galaxy magnetic
field, phenomenological model is generally used to study the propagation of
GCRs in the Galaxy. As we know the halo is much larger than the disk.
But the average density of the medium GCRs travel is $\sim0.3$ cm$^{-3}$
while the disk density is about $1$ cm$^{-3}$ \cite{HaloPropagation}.
So the trapping time of GCRs in the halo is only about two times longer
than in in the disk. We can infer that the propagation velocity in the
halo is much faster than in the disk. It is possible to conclude that the
stochastic magnetic field is much smaller in the halo than in the disk.
We need to investigate the particle transportation in the regular halo
magnetic field \cite{2001SSRv...99..243B} to study propagation of particles
in the halo, instead of the uniform diffusion in the Galaxy. Therefore
the $e^+e^-$ may fill in the whole bubble within the cooling time through
the fast propagation. However, the detailed model goes beyond the scope of
the current work.
The anisotropy of GCRs may still an open question in the GC scenario
of the origin of GCRs. Still the different propagation parterns in the
disk and the halo may lead to different results of the expected anisotropy.
We leave the discussion of it in future works.
Finally we should note that in the jet the background electrons should also
be accelerated together with the $e^+e^-$ produced through CR-photon
interactions. In such a case the total electron spectrum used to calculate
the bubble/haze emission might be different from what adopted in this work.
Without loss of generality, we may expect the background electron spectrum
to be a power-law spectrum $\sim E^{-2}$ with a cutoff, which does not
differ much from that we use here. The basic results in this work should
not change significantly.
\section*{Acknowledgements}
We thank Shaoxia Chen for helpful discussion, Yigang Xie, Amanda Maxham,
Ann Meng Zhou and Hanguo Wang for useful comments on the manuscript.
This work is supported by the Ministry of Science and Technology of
China, Natural Sciences Foundation of China (Nos. 10725524,10773011 and
11135010), and the Chinese Academy of Sciences (Nos. KJCX2-YW-N13,
KJCX3-SYW-N2, GJHZ1004).
\section*{References}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,142 |
This classic vintage stripe apron features an adorable red and white striped pattern. Perfect for the little Valentine's in your life. This apron will get them cooking in the kitchen + making Valentine's Day treats for the whole family!
This child apron will pair perfect with our adult version for a great mommy + me present. | {
"redpajama_set_name": "RedPajamaC4"
} | 349 |
Catherine Amanda Badocco Helene de Oliveira (Paris, 6 de janeiro de 1987) é uma jogadora de polo aquático brasileira. É irmã gêmea da também jogadora Tess Oliveira.
Carreira
Amanda integrou a equipe do Brasil que finalizou em oitavo lugar nos Jogos Olímpicos de 2016, no Rio de Janeiro.
Naturais de Paris
Jogadores de polo aquático olímpicos do Brasil
Jogadores de polo aquático nos Jogos Olímpicos de Verão de 2016
Jogadores de polo aquático do Brasil nos Jogos Pan-Americanos de 2015 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,207 |
\section{Introduction}
Spiral waves in two dimensions (and scroll waves in three dimensions) represent the key motifs of typical self-sustained dynamical patterns in excitable systems such as cardiac tissue.
In fact, the work to understand the mechanisms and develop effective treatments of cardiac arrhythmias such as tachycardia or fibrillation became the major driver for much of the recent interest in the dynamics of spiral and scroll waves.
Because of the strong spatial coherence of such waves, many aspects of their dynamics can in fact be understood using center manifold reduction of the underlying partial differential (or difference) equations, yielding a system of ordinary differential equations with respect to just a few variables \cite{Barkley94,sandstede1997}.
These variables are associated with the Euclidean symmetry of the problem and can be interpreted in terms of the low-dimensional dynamics (translation and rotation) of the core of the spiral wave, which serves as its source and anchor.
Notable examples include meander and drift \cite{Biktashev:1995re,BiHoNi96,FiSaScWu96,FiTu98} of spiral waves and their interaction with boundaries \cite{LanBar13,LanBar14,Marcotte2016}.
In fact, even for complex patterns of excitation which involve multiple spiral waves, many features of the dynamics can be understood and described reasonably well in terms of the wave core interaction \cite{ByMaGr14,Marcotte2016}.
Given their influence on large regions of space, spiral wave cores represent attractive targets for controlling the dynamics of excitable media.
This is well-known to clinical practitioners looking for treatments of cardiac arrhythmias such as atrial fibrillation.
In fact, radio frequency ablation, which aims to silence or isolate regions of cardiac tissue believed to be sources of spiral excitation waves (often referred to as rotors), has become the leading surgical treatment for persistent atrial fibrillation \cite{narayan2012,shivkumar2012}.
The success rate of ablation surgeries is however not very high, suggesting that the insufficient accuracy with which the spiral wave cores are located may be problematic.
Indeed, typical intra-cardiac basket catheters used to locate them only have 64 unipolar electrodes distributed over 8 circular spines of the catheter \cite{laughner2016}.
The signal they generate is both rather sparse and rather noisy, making it challenging to identify the location of the spiral wave cores, especially if those cores are drifting or meandering.
This paper describes a novel method that can reliably identify and track with high precision a large number of spiral wave cores based on sparse and noisy measurements of the transmembrane voltage.
The paper is organized as follows.
An overview of the existing methods for identifying the cores (or, more precisely, certain points inside the cores) is provided in Section \ref{sec:background}.
Our approach is described in Section \ref{sec:methods}, and it is validated and compared with competing approaches in Section \ref{sec:results}.
The limitations and potential extensions of our approach are discussed in Section \ref{sec:discussion}.
Finally, our conclusions are presented in Section \ref{sec:conclusions}.
\section{Background}
\label{sec:background}
Spiral wave cores (defined in terms of the response functions that determine the sensitivity to perturbations, heterogeneity, etc.) tend to be exponentially localized, as illustrated by analyses of the Ginzburg-Landau \cite{biktasheva2001}, Barkley \cite{Henry2002}, FitzHugh-Nagumo \cite{BiHoBi06}, Oregonator \cite{biktasheva2015}, Beeler-Reuter-Pumir \cite{Biktashev2011}, and Karma \cite{Marcotte2016} models.
In practice, it is more convenient to define a single point that characterizes the position of the core.
Numerical studies tend to use the position of the spiral tip, which can be defined in many different ways \cite{Fenton2002, biktashev1994tension, zhang1995chaotic, BaKnTu90, jahnke1989chemical, Fenton1998, biktashev1994tension, henze1990helical, ByMaGr14, FouBik10, bar1994spiral, beaumont1998spiral, jahnke1989chemical, jahnke1991survey}.
The most popular definitions are based on either the intersection of level sets of different variables \cite{BaKnTu90} or the vanishing of the normal velocity \cite{Fenton1998} or the curvature of the wavefront \cite{beaumont1998spiral}.
Neither of these definitions are convenient (or reliable) for analyzing experimental data, however.
An alternative approach relies on the phase-amplitude representation of spiral waves, with the phase singularity (PS) defining the instantaneous center of rotation of the wave.
A method for identifying PSs based on the local phase field has become standard in analyzing experimental data \cite{Gray1997,iyer2001experimentalist}, although it is also possible to determine the location of PSs using the amplitude field \cite{ByMaGr14}.
Recently, several alternative approaches have been proposed to locate spiral wave cores based on various metrics such as Shannon entropy, multi-scale frequency, kurtosis, multi-scale entropy \cite{annoni2018}, and Jacobian determinant \cite{li2018}.
It should be noted that these metrics define neither the spiral tip nor the phase singularity, but they can be applied to sparse data, although their precision and accuracy decrease very quickly as sparsity increases.
The study of Li {\it et al.} \cite{li2018} showed that the Jacobian determinant method locates both stationary and meandering spirals with precision higher than other common approaches\cite{iyer2001experimentalist, bray2001, lee2016new, Fenton1998}.
Furthermore, they determined that the Jacobian determinant method is the only one that can produce reliable results in the presence of as much as 0.9\% noise.
In fact, essentially all existing methods for locating spiral wave cores are local and cannot withstand higher noise levels characteristic of practical applications without significant loss of accuracy and precision.
The only exception is the global topological method for identifying PSs proposed by Marcotte and Grigoriev \cite{marcottetop}.
The original version of the topological approach defined PSs as intersections of level sets associated with two different variables (one fast, one slow).
A modified version of this topological approach based on phase reconstruction that required measurement of just one variable was developed and tested using spatially resolved numerical and experimental data \cite{gurevich2017level}.
Here we describe a robust implementation of the topological approach that does not require phase reconstruction and investigate its performance for sparse and noisy data generated by a model of atrial fibrillation.
In addition to its relative simplicity, the implementation described here also affords a more direct dynamical interpretation and allows an automatic classification of topological changes leading to creation or destruction of pairs of counterrotating rotors \cite{marcottetop}.
\section{Methods}
\label{sec:methods}
\subsection{Model}
To illustrate the algorithm and determine the conditions under which it functions reliably, we will use two-dimensional surrogate data generated by the smoothed version \cite{ByMaGr14,marcottetop} of the Karma model \cite{Karma1993,karma94},
\begin{equation}\label{eq:rde}
\partial_t{\bf w} = D\nabla^{2}{\bf w} + {\bf f}({\bf w}),
\end{equation}
where ${\bf w}=[u,v]$, $u$ is the (fast) voltage variable, $v$ is the (slow) gating variable,
\begin{align}\label{eq:karmakinetics}
f_1&=(u^* - v^{M})\{1 - \tanh(u-3)\}u^{2}/2 - u,\\
f_2&=\epsilon\left\{\beta \Theta_s(u-1) + \Theta_s(v-1)(v-1) - v\right\},\nonumber
\end{align}
and $\Theta_s(u)=[1+\tanh(su)]/2$.
Here $\epsilon$ describes the ratio of the fast and slow time scales, $s$ is the smoothing parameter, and the diagonal matrix $D$ of diffusion coefficients describes the spatial coupling between neighboring cardiac cells (cardiomyocytes).
The parameters of the model are $M=4$, $\epsilon = 0.01$, $s = 10$, $\beta = 1.389$, $u^{*} = 1.5415$, $D_{11} = 4.0062$, and $D_{22} = 0.20031$, with the length scale corresponding to the size of a cardiomyocyte.
Along with the Mitchell-Schaeffer model \cite{mitchell2003two}, this is one of the simplest models of excitable media that develops sustained spiral wave chaos from an isolated spiral wave through the amplification of the alternans instability.
\subsection{Analysis}
The method described here uses the voltage variable $u$ (normalized to the range [0,1] for convenience) to reconstruct the position of PSs.
In some instances, e.g., electrophysiology studies using basket catheters, the voltage data are only available on a coarse spatial grid.
To enable rotor mapping with meaningful accuracy in such cases, the data at each frame of the recording are mapped onto a sufficiently fine mesh using bicubic interpolation.
Following the original study that introduced the topological approach \cite{marcottetop}, we will define PSs as intersections of two smooth curves $\ell_1$ and $\ell_2$. For data corresponding to transmembrane voltage (obtained from a model or optical mapping using a voltage-sensitive dye), these curves can be conveniently defined as the zero level sets of $\dot{u}$ and $\ddot{u}$, which form the boundaries $\partial R$ and $\partial E$, respectively, of the refractory region
\begin{align}
R=\{(x,y) : \dot{u}<0\}
\end{align}
and the excited region
\begin{align}
E=\{(x,y) : \ddot{u}<0\}.
\end{align}
Given that our data are discrete and noisy, properly defining the level sets and PSs requires some care.
$\partial R$ and $\partial E$ correspond to peaks and troughs of $u$ and $\dot{u}$, respectively, where the temporal derivative can be computed using finite differencing of the discrete signal.
In the presence of noise (i.e., when dealing with experimental recordings), the data may be smoothed using a Gaussian kernel with spatial and temporal widths $\sigma_s$ and $\sigma_t$, respectively, before the time derivative is computed. Even after smoothing, $u$ and especially $\dot{u}$ can remain noisy. So each peak or trough is required to have a minimum prominence (a fraction $MPP_1$ or $MPP_2$ of the range of $u$ or $\dot{u}$, respectively, selected based on the overall level of noise, as described in the Appendix) and separation (generally a fixed fraction $\delta$ of the dominant period of oscillation) from the nearest peak/trough.
\begin{figure}[t]
\subfigure[]{\frame{\includegraphics[width=0.47\columnwidth]{lvlusm.pdf}}} \hspace{2mm}
\subfigure[]{\frame{\includegraphics[width=0.47\columnwidth]{lvlsm.pdf}}}
\caption{Comparison of sets $\ell^k$ (a) and $\bar{\ell}^k$ (b) for noiseless data, with $k=1$ in black and $k=2$ in gray. The domain is 256 $\times$ 256 grid points. Panel (a) contains a number of artifacts, which have to do with a characteristic feature of most cardiac models, namely very flat repolarization plateaus.
}
\label{fig:lvl}
\end{figure}
Spatial discreteness of the data -- we assume $u$ is measured on a uniform grid $(x_i,y_j,t_n)$ in space and time -- limits the accuracy with which the level sets $\partial R$ and $\partial E$ can be defined based on local data to the resolution of the underlying spatial grid. For instance, if we simply identify the grid points that correspond to the peaks or troughs in each frame, we end up with a sparse set of isolated points. In order to define a continuous set that can be considered a discrete approximation of $\partial R$ and $\partial E$, we will use the following procedure. Let $t^1_{ijm}$ denote the position of subsequent peaks (troughs) of $u(x_i,y_j,t)$ for $m$ even (odd), and define $t^2_{ijm}$ analogously for $\dot{u}(x_i,y_j,t)$.
Further, let us define a noise-insensitive analog of the sign of the time derivative
\begin{align}\label{eq:s1}
s^k_{ijn} = \begin{cases}
1,& t^k_{ijm-1}<t_n\leq t^k_{ijm}\\
-1,& t^k_{ijm}<t_n\leq t^k_{ijm+1}
\end{cases}
\end{align}
for some even number $m$ and $k=1$ or 2.
Then the sets
\begin{align}\label{eq:ls2}
\ell^k(t_n)=\{(x_i,y_j) : \exists i', j' :\ &s^k_{ijn}s^k_{i'j'n}<0,\nonumber \\
&|i-i'|+|j-j'|=1\}
\end{align}
are discrete generalizations of the level sets $\partial R$ and $\partial E$ with a minimal width of 2 grid points and no gaps, as illustrated in Fig. \ref{fig:lvl}(a).
In order to define the position of the level sets with sub-grid precision using global, rather than local (and hence noisy), information we use the following approach.
For each of $\ell^1$ and $\ell^2$, unsigned distance functions $d^1$ and $d^2$, respectively, are constructed using the MATLAB function {\tt bwdist} \cite{maurer2003linear}.
These are in turn converted into signed distance functions
\begin{align}
d_s^k(x_i,y_j,t_n) = -s^k_{ijn}d^k(x_i,y_j,t_n),
\end{align}
which incorporate global information across the entire domain.
Next, in order to ultimately smooth and sharpen the level sets, a spatial convolution of the signed distance functions with a Gaussian kernel with spatial width $\sigma_d$ is computed, yielding a pair of smoothed distance fields $\bar{d}^k_s$.
A comparison of the smoothed and unsmoothed versions of the signed distance function is presented in Fig. \ref{fig:df}.
Note the plateaus in the unsmoothed distance function, representing the finite thickness of the discrete set of grid points from which it is computed.
\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{dist.pdf}}
\caption{The signed distance $d^1_s$ (black) and its smoothed version $\bar{d}^1_s$ (gray) at a fixed time over a $j=\mathrm{const}$ slice of the domain.
}
\label{fig:df}
\end{figure}
Now we can finally define curves $\bar{\ell}^k$ as the zero level sets of the smoothed distance fields $\bar{d}^k_s$ using the MATLAB function {\tt contour}.
These curves are piecewise continuous and smooth, although in practice they are parametrized by a sequence of connected points in $\mathbb{R}^2$.
Note that $\ell_k$ define the true positions of $\partial R$ and $\partial E$ with a precision of one grid point or better.
As Fig. \ref{fig:lvl} illustrates, for noiseless data, $\bar{\ell}^k$ provide accurate representations of $\ell^k$ and hence $\partial R$ and $\partial E$, despite the relatively aggressive smoothing.
In practice, we will determine PSs as the intersections of $\bar{\ell}^1$ and $\bar{\ell}^2$, which are computed with sub-grid precision using the MATLAB function {\tt intersections} \cite{intersections}.
The chirality (or topological charge) $q$ of each PS can be computed using the gradients of $\bar{d}^1_s$ and $\bar{d}^2_s$, which are nearly constant in the vicinity of a PS, as follows:
\begin{align}
q = \mathrm{sign}(\hat{\mathbf z} \cdot \nabla \bar{d}^1_s \times \nabla \bar{d}^2_s),
\end{align}
These gradients are approximated at the four grid points nearest to the PS using finite differencing and then interpolated to the exact location of the PS.
This interpolation can be essential to correctly determine the chiralities of a pair of phase singularities separated by only a few grid points (in practice, as few as one).
Chirality plays an important role in the topological analysis of the excitation patterns produced during arrhythmias \cite{marcottetop}.
It is also useful for reconstructing PS trajectories, which are computed using a MATLAB implementation \cite{track} of the IDL particle tracking method \cite{crocker1996}, with the positions and chiralities of all PSs found at each time step as input parameters.
Both experimental and numerical data feature many short trajectories that correspond to virtual spiral waves that exist for a fraction of a rotation period.
Such structures do not appear to play a dynamically important role, so in our analysis we ignore PSs with lifetimes shorter than the dominant period of oscillation.
For comparison, during electrophysiological studies in a clinical setting, only spiral waves that persist for at least two rotations are considered \cite{ashikaga2018}.
\section{Results}
\label{sec:results}
To test the algorithm, we generated 4000 frames of surrogate data (after discarding 600 frames representing the initial transient) separated by one time unit by numerically integrating the model described above on a square domain of size $256 \times 256$ with no-flux boundary conditions.
This can be considered a fine mesh as it fully resolves all of the spatial features of the solution.
In our units, the typical rotation period of a spiral wave is $T=53$, the mean separation between PSs is $L=46$, and the mean number of PSs is 12.7.
\subsection{Benchmark}
\begin{figure}[t]
\subfigure[]{\includegraphics[width=0.47\columnwidth]{f1.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{f2.pdf}} \\
\subfigure[]{\includegraphics[width=0.47\columnwidth]{f3.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{f4.pdf}} \\
\subfigure[]{\includegraphics[width=0.47\columnwidth]{f5.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{f6.pdf}}
\caption{Snapshots of benchmark data (colorbar is shown in Fig. \ref{fig:g}(a)) equally spaced in time over 56 frames (approximately one rotation period), with curves $\bar{\ell}^1$ (white) and $\bar{\ell}^2$ (black) and PSs superimposed. Here and below, solid and dashed white segments correspond to the leading and trailing edges of the refractory region, respectively; solid and dashed black correspond to the wavefront and waveback. PSs with chirality +1 and -1 are respectively shown as black and white circles. The $x$ ($y$) axis is horizontal (vertical).
A full movie is provided in the supplementary material.}
\label{fig:f}
\end{figure}
\begin{figure}[t]
\centerline{\includegraphics[width=\columnwidth]{trackSlice.pdf}}
\caption{Trajectories of PSs with lifetimes of at least one period during the same time interval as shown in Fig. \ref{fig:f}. Thicker curves correspond to PSs created during this period.
}
\label{fig:tracks}
\end{figure}
In order to establish a benchmark for quantifying how our approach copes with noise and data sparsity, we analyzed the data using a parameter set representing minimal smoothing (see Appendix).
Fig. \ref{fig:f} presents six equally spaced snapshots of the benchmark (voltage) data over roughly a rotation period.
Superimposed are the curves $\bar{\ell}^1$ (the boundary of the refractory region, in white) and $\bar{\ell}^2$ (the boundary of the excited region, in black).
The intersections define PSs (black/white circles for positive/negative chirality).
Many spiral waves are seen to rotate stably around fixed or weakly meandering PSs.
The trajectories of the long-lived PSs are shown over the same interval in Fig. \ref{fig:tracks}, with the thicker curves corresponding to PSs created during this time.
Three different PS pair creation events are visible during this period: one between snapshots (b) and (c) near the left edge of the domain, and two between snapshots (e) and (f) near the left and right edges.
Notably, the lone thick blue trajectory in Fig. \ref{fig:tracks} appears to be missing its opposite chirality counterpart.
In fact, that counterpart is not shown since it corresponds to a short-lived ``virtual" PS which annihilates with a nearby long-lived PS soon after the last frame in Fig. \ref{fig:f}.
Note that the trajectories of created PS pairs do not start at the exact same point because of the finite temporal resolution of the data.
When the PSs are created and destroyed, they move especially quickly, separating by several grid spacings in one time unit.
Much higher temporal resolution is needed to resolve the fast motion of PSs during pair creation/annihilation events.
The ability of our algorithm to automatically track PSs (both short- and long-lived) allows one to generate statistics that could be extremely useful (e.g., for model validation) but would be hard to obtain otherwise.
To illustrate this, Fig. \ref{fig:hist} shows various PS statistics (only taking spiral waves that complete at least one revolution into account) over the course of the entire simulation.
In particular, we find that the number of PSs ranges rather widely (between 5 and 20, as shown in panel (a)).
This illustrates that our approach can easily and reliably identify at least 20 PSs simultaneously.
The distances between PSs also vary rather significantly, with a pronounced peak at 46 units (panel (b)).
As we will show below, it is this characteristic length scale, not the wavelength of the pattern (on the order of $90$ units here), that determines the sparsity at which our method starts to break down.
We find that while the majority of spiral waves are relatively short lived (panel (c)), some can live for up to 45 periods (for reference, the duration of the entire data set corresponds to 75 periods).
Such instances of functional reentry could easily be mistaken for structural reentry in a clinical setting.
In light of this ``longevity,'' it is perhaps not surprising that some PSs drift over distances exceeding half the size of our rather large system (panel (d)).
While the lifetime and drift statistics of PSs in the Karma model may not be particularly relevant for atrial fibrillation, a similar analysis of data from basket catheters could yield a treasure trove of clinically valuable information.
\begin{figure}[t]
\subfigure[]{\includegraphics[width=0.47\columnwidth]{nhist.pdf}} \hspace{3mm}
\subfigure[]{\includegraphics[width=0.45\columnwidth]{dhist.pdf}} \\
\subfigure[]{\includegraphics[width=0.47\columnwidth]{lhist.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{rhist.pdf}}
\caption{PS statistics. (a) Histogram of the total number $n$ of PSs in each frame. (b) Histogram of the distance $d$ from each PS to the nearest PS of opposite chirality, computed separately for each PS in each frame. (c) Histogram of the lifetime $l$ in periods of each PS. (d) Histogram of the separation $r$ between the most distant pair of points along the trajectory of each PS.
}
\label{fig:hist}
\end{figure}
\subsection{Sensitivity to noise and sparsification}
To represent the effect of imperfections in realistic experimental recordings, noisy data sets were produced by adding random Gaussian-distributed white noise with some standard deviation $\eta$ to the benchmark data.
In addition, sparsified data were generated by sampling from the benchmark or noisy data on a uniform grid with lower resolution by a factor of $2^n$ in each spatial dimension for some integer $n>1$.
The sparsified data were then interpolated back onto the original grid for processing.
\begin{figure}[t]
\includegraphics[width=0.99\columnwidth]{utrace.pdf}
\caption{A typical time trace of the voltage signal before (gray) and after (black) temporal smoothing for different noise levels (from top to bottom, $\eta=0.1$, $\eta=0.3$, and $\eta=1$).}
\label{fig:u}
\end{figure}
For all noise and sparsity levels, the resulting data were processed using modified parameter sets mildly optimized to deal with high levels of noise (cf. Appendix).
Note that, in order to preserve precision at different levels of sparsification, no initial spatial smoothing is applied (the raw and temporally smoothed signals are compared in Fig. \ref{fig:u}).
The robustness of the algorithm is thus ensured without relying on averaging of high-spatial-resolution data to counteract the effects of noise.
As we show below, our results are robust to both noise and sparsification but show some minor reduction in precision of PS location, even for the benchmark data.
\begin{figure}[t]
\subfigure[]{\includegraphics[width=0.47\columnwidth]{g1.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{g2.pdf}} \\
\subfigure[]{\includegraphics[width=0.47\columnwidth]{g3.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{g4.pdf}}
\caption{The frame shown in Fig. \ref{fig:f}(c) with four different levels of added noise ($\eta=0,0.1,0.3,1$); overlaid are the curves $\bar{\ell}^1$ and $\bar{\ell}^2$ and PSs computed from the noisy data in each case.
}
\label{fig:g}
\end{figure}
The effect of noise alone on the performance of the algorithm is illustrated in Fig. \ref{fig:g}, which shows the level sets $\bar{\ell}^1$, $\bar{\ell}^2$, and the PSs computed for the same frame with four levels of Gaussian white noise (standard deviation $\eta=0,0.1,0.3,1$).
As the data quality deteriorates, the computed curves, especially $\bar{\ell}^1$, become increasingly unreliable; it is impossible to filter out all false peaks in the data without ignoring the smaller but legitimate and dynamically important fluctuations in voltage.
Nevertheless, their intersections (the PSs) are located with high precision in all cases, even when $\eta$ is as large as the entire range of the original data.
Similarly, the effect of sparsity alone is illustrated in Fig. \ref{fig:h}, which shows the same frame with (noiseless) data interpolated from coarse grids with four different spatial resolutions: (a) $256 \times 256$, (b) $32 \times 32$, (c) $16 \times 16$, (d) $8 \times 8$.
For resolutions down to $16 \times 16$, the computed level sets are qualitatively very similar to the benchmark and all long-lived PSs are correctly detected and located with precision substantially better than the coarse grid spacing.
In the $16 \times 16$ snapshot, there is a pair of virtual PSs in the lower left corner that are not present in the benchmark analysis.
These are short-lived and so are discarded in our analysis.
Even for data on an $8\times8$ spatial grid, a large fraction of PSs were correctly identified, with one false positive;
the error in the position of the correct PSs is substantially smaller than $256/8=32$ units.
However, it is apparent that many of the dynamical features cannot be properly resolved when the grid resolution becomes comparable to the mean separation between PSs.
\begin{figure}[t]
\subfigure[]{\includegraphics[width=0.47\columnwidth]{g1.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{h2.pdf}} \\
\subfigure[]{\includegraphics[width=0.47\columnwidth]{h3.pdf}} \hspace{1mm}
\subfigure[]{\includegraphics[width=0.47\columnwidth]{h4.pdf}}
\caption{The frame shown in Fig. \ref{fig:f}(c) after interpolating from four levels of sparsification: (a) spatial resolution 256 $\times$ 256 (same as benchmark), (b) 32 $\times$ 32, (c) 16 $\times$ 16, (d) 8 $\times$ 8. As in the previous figure, the curves $\bar{\ell}^1$ and $\bar{\ell}^2$ and PSs computed from the sparsified data are overlaid.
}
\label{fig:h}
\end{figure}
\begin{table}[b]
\begin{tabular}{*{6}{|c}|}
\hline & $256 \times 256$ & $64 \times 64$ & $32 \times 32$ & $16 \times 16$ & $8 \times 8$\\ \hline
$\eta=0$ & 0.995 & 0.995 & 0.994 & 0.955 & \;0.255\;\\ \hline
$\eta=0.1$ & 0.993 & 0.994 & 0.992 & 0.957 & 0.308 \\ \hline
\;$\eta=0.3$\; & 0.988 & 0.988 & 0.985 & 0.954 & 0.357 \\ \hline
$\eta=1$ & 0.990 & 0.974 & 0.849 & 0.695 & - \\ \hline
\end{tabular}
\caption{PS detection accuracy $\rho_t$ as a function of sparsity and noise level $\eta$. The quality of the analysis was too poor to compute trajectories when both sparsity and noise were maximal.}
\label{tab:tmatch}
\end{table}
\begin{table}[t]
\begin{tabular}{*{6}{|c}|}
\hline & $256 \times 256$ & $64 \times 64$ & $32 \times 32$ & $16 \times 16$ & $8 \times 8$\\ \hline
$\eta=0$ & 1.1 & 1.1 & 1.4 & 4.8 & 9.9\\ \hline
$\eta=0.1$ & 1.2 & 1.3 & 1.6 & 4.6 & 9.6 \\ \hline
\;$\eta=0.3$\; & 1.7 & 1.8 & 2.3 & 4.9 & 9.4 \\ \hline
$\eta=1$ & 2.3 & 3.1 & 4.5 & 6.9 & - \\ \hline
\end{tabular}
\caption{PS location precision $\rho_d$ (in units of the fine mesh) as a function of sparsity and noise level $\eta$.}
\label{tab:dmatch}
\end{table}
Tables \ref{tab:tmatch} and \ref{tab:dmatch} quantify the accuracy and precision of the algorithm in the presence of both noise and sparsity.
In the following discussion, we will use the benchmark analysis as the reference.
For each level of noise and sparsity, we compare the computed trajectories with the reference trajectories as follows.
At each frame, every PS is matched with the nearest reference PS of the same chirality, provided that their separation is no greater than a fixed fraction $\alpha$ of the mean PS separation $L$.
Each PS trajectory is then paired with all reference PS trajectories with which it was matched for at least a given fraction $\gamma$ of the period $T$, with all other matches discarded.
(We allow for a trajectory to be matched to multiple other trajectories to account for the possibility that some trajectories might be broken up by short gaps.)
Finally, we compute $\rho_d$, the average distance between the detected and reference PS across all matches, as well as the ratio
\begin{align}
\rho_t = \frac{2\sum_t m_t}{\sum_t (r_t + n_t)}
\end{align}
where the sum is over all frames and $m_t$, $r_t$, and $n_t$ are the number of matches, reference PSs, and detected PSs, respectively, in frame $t$;
$\rho_t$ ranges from 0 to 1 and represents the PS detection accuracy.
The results in the tables summarize the results for all data sets using fairly strict parameter choices $\alpha = 0.35$ and $\gamma = 0.8$.
(For our surrogate data, this means two trajectories are matched only if their separation is at most 16 units for at least 43 frames.)
The performance of the algorithm was quite good in a wide range of conditions.
Note that the accuracy and precision is imperfect even in the case of full resolution and no noise;
this is because of the temporal smoothing used in this analysis, which is not present in the benchmark.
PS detection accuracy is above 98\%, with precision of about 5\% of the mean PS separation or better, for 10 out of the 20 data sets, even in one case when $\eta=1$.
(When the accuracy is close to 100\%, most of the mismatches are due to PS pair creation or annihilation events being detected a few frames earlier or later than in the reference analysis, an error that has little to no dynamical importance.)
In most of the $16 \times 16$ data sets, for which the coarse grid resolution is approximately equal to $\alpha L$, the accuracy remains above 95\% and the precision is better than 5 spatial units (i.e., about 1 mm), demonstrating that the algorithm can locate PSs with sub-grid precision.
The analysis only breaks down for data sampled on $8 \times 8$ grids, which is near the theoretical limit of such a method as the grid spacing (32 units) is comparable to the mean PS separation (46 units). The issue of maximal sparsity has been considered previously\cite{rappel2013} for a single spiral on a domain larger than the wavelength. Unfortunately, the results of that study cannot be directly compared to our case, where the average spacing between PSs is smaller than the wavelength.
\subsection{Comparison with alternative approaches}
To validate our algorithm, we also implemented and tested its most robust alternative, the Jacobian determinant method \cite{li2018}.
The method essentially identifies PSs with the extrema of the two-dimensional field
\begin{align}
\mathcal{D}(t)=
\left|\begin{matrix}
\frac{\partial u}{\partial x}\big|_t & \frac{\partial u}{\partial y}\big|_t\\
\frac{\partial u}{\partial x}\big|_{t+\tau} & \frac{\partial u}{\partial y}\big|_{t+\tau}
\end{matrix}\right|
=\hat{z}\cdot(\nabla u|_t\times\nabla u|_{t+\tau}),
\end{align}
where $\tau$ is an empirically chosen time delay.
This method was used to compute PS trajectories for the full resolution ($256 \times 256$) data sets with varying levels of noise, and the results were compared to our benchmark analysis.
The time delay parameter was chosen to be $\tau=0.1125T = 6$ frames, which is approximately the value found to work best in the original paper.
Like our method, the Jacobian determinant method is better suited than traditional techniques to the presence of non-stationary PSs, noise, and/or sparsity.
However, we observed some difficulties not present in our approach.
First, the extrema of the field $\mathcal{D}$ must be defined with care as the less pronounced peaks do not correspond to PSs.
As a result, some minimum peak prominence must be selected on a case-by-case basis for each recording and it is unclear how to determine the optimal choice without a refence.
Such an approach may not be feasible for experimental recordings, especially in the presence of spatial heterogeneity, which might cause the optimal threshold to vary in space.
For our data, we obtained the highest accuracy when labeling as PSs all extrema with a prominence exceeding the maximum value of $|\mathcal{D}|/3$.
Second, even in the absence of noise, some peaks corresponding to a single non-stationary PS were found to separate into multiple local extrema of similar magnitude, making it difficult to determine the exact number of PSs and their locations.
Finally, spatial smoothing of $\mathcal{D}$ was necessary to eliminate spurious PSs for $\eta$ as small as 0.1 (the lowest noise level we tested).
This step substantially reduces precision of PS location when the data are both noisy and sparse.
We smoothed $\mathcal{D}$ using a Gaussian kernel with a width of 2 grid spacings, which is in close correspondence with the authors' suggested approach.
In fact, this substantially improved accuracy even in the absence of noise, likely because of the second difficulty mentioned previously.
The results of the comparison of the smoothed Jacobian determinant method with the benchmark are summarized in Table \ref{tab:jd}.
The Jacobian determinant method performed very well for $\eta\leq 0.1$, achieving over 95\% accuracy, but accuracy fell below 90\% at $\eta=0.3$, and for $\eta=1$ no meaningful results could be produced as some frames contained over 100 spurious PSs.
This comparison demonstrates that the Jacobian determinant method can handle moderate levels of noise but falls apart at the higher noise levels that can be easily handled by our algorithm.
Our approach also handles sparse data better, locating PSs with sub-grid resolution.
\begin{table}[t]
\begin{tabular}{*{5}{|c}|}
\hline & $\eta=0$ & $\eta=0.1$ & $\eta=0.3$ & $\eta=1$ \\ \hline
$\;\rho_t\;$ & \;0.969\; & 0.956 & 0.895 & - \\ \hline
$\rho_d$ & 1.7 & 1.7 & 1.7 & - \\ \hline
\end{tabular}
\caption{PS detection accuracy $\rho_t$ and precision $\rho_d$ for the Jacobian determinant method as a function of noise level $\eta$. Analysis quality was too poor to compute trajectories for $\eta=1$.}
\label{tab:jd}
\end{table}
\section{Discussion}
\label{sec:discussion}
We illustrated a robust approach to rotor mapping using measurements of the transmembrane voltage $u$ obtained using a highly simplified model of atrial fibrillation. Furthermore, we used a rather unconventional approach where the positions of the rotors are defined using the intersections of the zero level sets of $\dot{u}$ and $\ddot{u}$.
One therefore might question the applicability of our results to electrophysiology studies using basket catheters, in which unipolar electrodes produce signals with a shape that is very different from the shape of the underlying transmembrane voltage.
Let us start by addressing the temporal profile of the voltage signal. Our particular choice of level sets was motivated by their relation to the excited and refractory phases of cardiomyocyte dynamics. Specifically, $\partial E$ describes the wavefront and waveback, while $\partial R$ describes the leading and trailing edges of the refractory region. This relationship makes it easy to use the topological and geometrical information encoded by the level sets to describe the dynamical mechanisms responsible for initiation, maintenance, and termination of cardiac arrhythmias. It also enables automatic classification of the topologically distinct events leading to an increase/decrease in the number of PSs and the associated increase/decrease in the complexity of the excitation pattern \cite{marcottetop}.
However, our particular choice of the two level sets is neither unique nor necessarily the best. For instance, in the presence of several inflection points in the repolarization phase of the action potential, which is typical for atrial tissue, these level sets might define additional spurious ``wavefronts'' and ``wavebacks.'' As an alternative,
one could use the zero level sets of $\dot{u}$ and $\dot{v}$, where $v$ is the gating variable \cite{marcottetop}, or the level sets $\phi\ {\rm mod}\ \pi=0$ and $\phi\ {\rm mod}\ \pi=\pi/2$ of the phase field $\phi$ reconstructed using certain features of the voltage trace \cite{gurevich2017level}. For stationary (i.e., nondrifting, nonmeandering) spiral waves, the definitions of PSs based on these different choices are equivalent. The difference only becomes noticeable for the quickly moving rotors that are not targets of ablation therapy. A more common choice\cite{BaKnTu90} is to define one of the level sets using the transmembrane voltage itself, $u=u_{th}$. This choice defines the location of a spiral tip rather than a PS, and leads to a decrease in the accuracy of localization: unlike the PS, which is stationary, the spiral tip circles the PS for a stationary spiral wave.
Unipolar electrograms generated by basket catheters tend to have multiple maxima and minima per cycle \cite{kuklik2015} and do not provide a direct interpretation in terms of the depolarization/repolarization phase. They, however, have a characteristic feature (pronounced minima of the derivative $\dot{V}$ of the voltage $V$) that can be used to define the phase $\phi$ as a piecewise linear continuous function \cite{gurevich2017level}. Alternatively, the phase can be obtained using a temporal Hilbert transform. In either case, as long as the phase field $\phi(x_i,y_j,t_n)$ on a discrete regular grid can be reliably determined, our method can be applied rather directly \cite{gurevich2017level} to identify and track PSs using different level sets of $\phi$.
Another limitation of our study is that Gaussian noise is not representative of noise encountered in
clinical voltage mapping studies. One of the major sources of signal distortions is motion artifacts caused by
movement of the myocardial wall and/or poor contact of the electrode with the myocardial surface. Another major source of distortion is the far field effects, i.e., contamination of atrial electrograms by the much stronger electrical signal generated by the ventricles. Far field distortions can be mostly eliminated using, e.g., a single beat cancellation method \cite{zeemering2012}.
Finally, let us point out that we chose to use simple finite differences to compute temporal derivatives of the voltage due to the simplicity of this method. However, other methods could be used to further improve robustness of the algorithm for noisy data. The total variation regularized derivative \cite{Rudin1992} is one extremely robust option, although it is numerically costly. Another alternative with less computational overhead is least-squares polynomial interpolation \cite{knowles2014}. We have not pursued these more complicated methods, since even simple finite differencing produces rather impressive results. \\
\section{Conclusions}
\label{sec:conclusions}
We have introduced a novel approach for identifying the locations of multiple phase singularities associated with complicated patterns of excitation waves based on measurements of a single scalar field.
Because of its global nature, the new method was found to be substantially more robust than any previous, essentially local, methods aimed at identifying ``organizing centers'' of spiral wave activity.
In particular, we have demonstrated that our approach can simultaneously identify and locate tens of phase singularities, including ones that are highly nonstationary.
Moreover, their locations can be tracked in time with subgrid precision for data that are both very noisy (with noise level exceeding the signal level) and very sparse (on grids with spacing comparable to the mean separation between phase singularities).
This enables collection of a wide range of statistical information that can be used in model validation, for instance, and has a potential to impact applications such as clinical electrophysiology studies using intra-cardiac multi-electrode basket catheters.
It is worth emphasizing, however, that this method is not restricted to cardiac tissue and can be applied to any two-dimensional excitable system.
Extensions to three dimensions are possible as well but are outside of the scope of this paper.
\section*{Supplementary material}
A movie showing the dynamics of PSs and the level sets corresponding to the wavefronts, wavebacks, and the leading/trailing edges of the refractory region in the benchmark analysis is provided as supplementary material.
\begin{acknowledgments}
DG gratefully acknowledges the support of the GT College of Sciences Undergraduate Research Science Award. This research was supported in part by NSF Grant No. PHY17-48958, NIH Grant No. R25GM067110, and the Gordon and Betty Moore Foundation Grant No. 2919.01.
\end{acknowledgments}
\section{References}
\input{Chaos2018.bbl}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 274 |
\section*{Revision}
\section{Introduction}
The history of popular music has long been debated by philosophers, sociologists, journalists and pop stars \cite{adorno1941popular, adorno1975culture, frith1988music, mauch2011anatomy, negus1996popular, stanley2013yeah}. Their accounts, though rich in vivid musical lore and aesthetic judgements, lack what scientists want: rigorous tests of clear hypotheses based on quantitative data and statistics.
Economics-minded social scientists studying the history of music have done better, but they are less interested in music than the means by which it is marketed \cite{peterson1975cycles, lopes1992innovation, christianen1995cycles, alexander1996entropy, peterson1996measuring, crain1997economics, tschmuck2006creativity, klein2010chart}.
The contrast with evolutionary biology---a historical science rich in quantitative data and models---is striking; the more so since cultural and organismic variety are both considered to be the result of modification-by-descent processes \cite{leroi2006recovery, jan2007memetics, maccallum2012evolution, savage2013toward}. Indeed, linguists and archaeologists, studying the evolution of languages and material culture, commonly apply the same tools that evolutionary biologists do when studying the evolution of species \cite{cavalli1981cultural, boyd1985culture, shennan2009pattern, steele2010evolutionary, mesoudi2011cultural, whiten2012culture}. Inspired by their example, here we investigate the ``fossil record'' of American popular music. We adopt a diachronic, historical approach to ask several general questions: Has the variety of popular music increased or decreased over time? Is evolutionary change in popular music continuous or discontinuous? If discontinuous, when did the discontinuities occur?
Our study rests on the recent availability of large collections of popular music with associated timestamps, and computational methods with which to measure them \cite{serra2012measuring}. Analysis in traditional musicology and earlier data-driven ethnomusicology \cite{lomax1972evolutionary}, while rich in structure \cite{savage2013toward}, is slow and prone to inconsistencies and subjectivity. Promising diachronic studies on popular music exist, but they either lack scientific rigour \cite{mauch2011anatomy}, focus on technical aspects of audio such as loudness, vocabulary statistics and sequential complexity \cite{serra2012measuring, foster2014sequential}, or are hampered by sample size \cite{schellenberg2012emotional}.
The present work uniquely combines the power of a big, clearly defined diachronic dataset with the detailed examination of musically meaningful audio features.
\begin{figure}
\begin{center}
\includegraphics[width=\imgwidth, trim=55 300 110 20, clip=true]{figures/FIGURE_1_flow_MM.pdf}
\end{center}
\caption{\textbf{Data processing pipeline} illustrated with a segment of Queen's \emph{Bohemian Rhapsody}, 1975, one of the few Hot 100 hits to feature an astrophysicist on lead guitar.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth, trim=10 200 20 30, clip=true]{figures/FIGURE_2_topic_evolution_MM.pdf}
\end{center}
\caption{\textbf{Evolution of musical Topics in the Billboard Hot 100.} Mean Topic frequencies ($\bar q$) $\pm$ 95\% CI estimated by bootstrapping.}
\end{figure}
To delimit our sample, we focused on songs that appeared in the US Billboard Hot 100 between 1960 and 2010. We obtained 30-second-long segments of 17,094 songs covering 86\% of the Hot 100, with a small bias towards missing songs in the earlier years. Since our aim is to investigate the evolution of popular taste, we did not attempt to obtain a representative sample of all the songs that were released in the USA in that period of time, but just those that were most commercially successful. To analyse the musical properties of our songs we adopted an approach inspired by recent advances in text mining (figure 1). We began by measuring our songs for a series of quantitative audio features, 12 descriptors of tonal content and 14 of timbre (Supplementary Information M.2--3). These were then discretised into ``words'' resulting in a harmonic lexicon (H-lexicon) of chord changes, and a timbral lexicon (T-lexicon) of timbre clusters (SI M.4). To relate the T-lexicon to semantic labels in plain English we carried out expert annotations (SI M.5). The musical words from both lexica were then combined into 8+8=16 ``Topics'' using Latent Dirichlet Allocation (LDA). LDA is a hierarchical generative model of a text-like corpus, in which every document (here: song) is represented as a distribution over a number of topics, and every topic is represented as a distribution over all possible words (here: chord changes from the H-lexicon, and timbre clusters from the T-lexicon). We obtain the most likely model by means of probabilistic inference (SI M.6). Each song, then, is represented as a distribution over 8 harmonic Topics (H-Topics) that capture classes of chord changes (e.g., ``dominant 7th chord changes'') and 8 timbral Topics (T-Topics) that capture particular timbres (e.g., ``drums, aggressive, percussive'', ``female voice, melodic, vocal'', derived from the expert annotations), with Topic proportions $q$. These Topic frequencies were the basis of our analyses.
\section{Results}
\subsection{The Evolution of Topics}
Between 1960 and 2010, the frequencies of the Topics in the Hot 100 varied greatly: some Topics became rarer, others became more common, yet others cycled (figure 2). To help us interpret these dynamics we made use of associations between the Topics and particular artists as well as genre-tags assigned by the listeners of Last.fm, a web-based music discovery service with $\sim$50m users (electronic supplementary material, M.8). Considering the H-Topics first, the most frequent was H8 (mean $\pm$ 95\%CI: $\bar q=0.236 \pm 0.003$)---major chords without changes. Nearly two-thirds of our songs show a substantial ($>$ 12.5\%) frequency of this Topic, particularly those tagged as \textsc{classic country}, \textsc{classic rock} and \textsc{love} (\href{http://www.eecs.qmul.ac.uk/~matthiasm/descentofpop/fullTables.pdf}{online tables}). Its presence in the Hot 100 was quite constant, being the most common H-Topic in 43 of 50 years.
Other H-Topics were much more dynamic. Between 1960 and 2009 the mean frequency of H1 declined by about 75\%. H1 captures the use of dominant-7\textsuperscript{th} chords. Inherently dissonant (because of the tritone interval between the third and the minor seventh) these chords are commonly used in Jazz to create tensions that are eventually resolved to consonant chords; in Blues music, the dissonances are typically not resolved and thus add to the characteristic ``dirty'' colour. Accordingly we find that songs tagged \textsc{blues} or \textsc{jazz} have a high frequency of H1; it is especially common in the songs of Blues artists such as B.B. King and Jazz artists such as Nat ``King'' Cole. The decline of this Topic, then, represents the lingering death of Jazz and Blues in the Hot 100.
The remaining H-Topics capture the evolution of other musical styles. H3, for example, embraces minor-7th chords used for harmonic colour in funk, disco and soul---this Topic is over-represented in \textsc{funk} and \textsc{disco} and artists like Chic and KC \& The Sunshine Band. Between 1967 and 1977, the mean frequency of H3 more than doubles. H6 combines several chord changes that are a mainstay in modal rock tunes and therefore common in artists with big-stadium ambitions (e.g., M{\"o}tley Cr{\"u}e, Van Halen, REO Speedwagon, Queen, Kiss and Alice Cooper). Its increase between 1978 and 1985, and subsequent decline in the early 1990s, slightly earlier than predicted by the BBC \cite{barfield2007champions}, marks the age of Arena Rock. Of all H-Topics, H5 shows the most striking change in frequency. This Topic, which captures the absence of an identifiable chord structure, barely features in the 1960s and 1970s when, a few spoken-word-music collages aside (e.g., those of Dickie Goodman), nearly all songs had clearly identifiable chords. H5 starts to become more frequent in the late 1980s and then rises rapidly to a peak in 1993. This represents the rise of Hip Hop, Rap and related genres, as exemplified by the music of Busta Rhymes, Nas, and Snoop Dog, who all use chords particularly rarely (\href{http://www.eecs.qmul.ac.uk/~matthiasm/descentofpop/fullTables.pdf}{online tables}).
The frequencies of the timbral Topics, too, evolve over time. T3, described as ``energetic, speech, bright'', shows the same dynamics as H5 and is also associated with the rise of Hip Hop-related genres. Several of the other timbral Topics, however, appear to rise and fall repeatedly, suggesting recurring fashions in instrumentation. For example, the evolution of T4 (``piano, orchestra, harmonic'') appears sinusoidal, suggesting a return in the 2000s to timbral qualities prominent in the 1970s. T5 (``guitar, loud, energetic'') underwent two full cycles with peaks in 1966 and 1985, heading upward once more in 2009. The second, larger, peak coincides with a peak in H6, the chord-changes also associated with stadium rock groups such as M{\"o}tley Cr{\"u}e (\href{http://www.eecs.qmul.ac.uk/~matthiasm/descentofpop/fullTables.pdf}{online tables}). Finally, T1 (``drums, aggressive, percussive'') rises continuously until 1990 which coincides with the spread of new percussive technology such as drum machines and the gated reverb effect famously used by Phil Collins on \emph{In the air tonight}, 1981. Accordingly, T1 is overrepresented in songs tagged \textsc{dance}, \textsc{disco} and \textsc{new wave} and artists such as The Pet Shop Boys. After 1990, the frequency of T1 declines: the reign of the drum machine was over.
\subsection{The varieties of music}
\begin{figure}[t]
\begin{center}
\vspace{-2em}
\includegraphics[width=\imgwidth, trim=120 620 190 35, clip=true]{figures/FIGURE_3_clustering}
\end{center}
\caption{\textbf{Evolution of musical styles in the Billboard Hot 100.} The evolution of 13 Styles, defined by $k$-means clustering on principal components of Topic frequencies. The width of each spindle is proportional to the frequency of that style, normalised to each year. The spindle contours are based on a $\pm$2-year moving average smoother; unsmoothed yearly frequencies are shown as grey horizontal lines. A hierarchical cluster analysis on the $k$-means centroids grouped our Styles into several larger clusters here represented by a tree: an \textsc{easy-listening + love-song} cluster, a \textsc{country + rock} cluster, and \textsc{soul + funk + dance} cluster; the fourth, most divergent, cluster only contains the \textsc{hip hop + rap}-rich Style 2. All resolved nodes have $\geq 75\%$ bootstrap support. Labels list the four most highly over-represented Last.fm user tags in each Style according to our enrichment analysis; see electronic supplementary material table~S1 for full results. Shaded regions define eras separated by musical revolutions (figure 5). }
\label{fig:clustering}
\end{figure}
To analyse the evolution of musical variety we began by classifying our songs. Popular music is classified into genres such as \textsc{country}, \textsc{rock and roll}, \textsc{rhythm and blues} (\textsc{R'n'B}) as well as a multitude of subgenres (\textsc{dance-pop}, \textsc{synthpop}, \textsc{heartland rock}, \textsc{roots rock} etc.). Such genres are, however, but imperfect reflections of musical qualities. Popular music genres such as \textsc{country} and \textsc{rap} partially capture musical styles but, besides being informal, are also based on non-musical factors such as the age or ethnicity of performers (e.g., \textsc{classic rock} and \ \textsc{K[orean]-Pop}) \cite{negus1996popular}. For this reason we constructed a taxonomy of 13 Styles by $k$-means clustering on principal components derived from our Topic frequencies (figure 3 and electronic supplementary material M.9). We investigated all $k < 25$ and found that the best clustering solution, as determined by mean silhouette score, was $k = 13$.
In order to relate Last.fm tags to the style Style clusters, we used a technique called enrichment analysis from bio-informatics. This technique is usually applied to arrive at biological interpretations of sets of genes, i.e.\ to find out what the ``function'' of a set of genes is. Applying the GeneMerge enrichment-detection algorithm \cite{castillodavis2003genemerge} to our Style data, we found that all Styles are strongly enriched for particular tags, i.e.\ for each Style some Last.fm tags are significantly over-represented (table S1), so we conclude that they capture at least some of the structure of popular music perceived by consumers. The evolutionary dynamics of our Styles reflect well-known trends in popular music. For example, the frequency of Style 4, strongly enriched for \textsc{jazz}, \textsc{funk}, \textsc{soul} and related tags, declines steadily from 1960 onwards. By contrast, Styles 5 and 13, strongly enriched for \textsc{rock}-related tags, fluctuate in frequency, while Style 2, enriched for \textsc{rap}-related tags, is very rare before the mid-1980s but then rapidly expands to become the single largest Style for the next thirty years, contracting again in the late 2000s.
What do our Styles represent? Figure 3 shows that Styles and their evolution relate to discrete sub-groups of the charts (genres), and hierarchical cluster analysis suggests that styles can be grouped into a higher hierarchy. However, we suppose that, unlike organisms of different biological species, all the songs in the charts comprise one large, highly structured, meta-population of songs linked by a network of ancestor-descendant relationships arising from songwriters imitating their predecessors \cite{zollo2003songwriters}. Styles and genres, then, represent populations of music that have evolved unique characters (Topics), or combinations of characters, in partial geographic or cultural isolation, e.g., \textsc{country} in the Southern USA during the 1920s or \textsc{rap} in the South Bronx of the 1970s. These Styles rise and fall in frequency over time in response to the changing tastes of songwriters, musicians and producers, who are in turn influenced by the audience.
\subsection{Musical diversity has not declined}
Just as paleontologists have discussed the tempo and mode of evolutionary change in the fossil record \cite{simpson1944tempo}, historians of music have discussed musical change and the processes that drive it. Some have argued that oligopoly in the media industries has caused a relentless decline in cultural diversity of new music \cite{adorno1941popular, adorno1975culture}, while others suggest that such homogenizing trends are periodically interrupted by small competitors offering novel and varied content resulting in ``cycles of symbol production'' \cite{peterson1975cycles, peterson1996measuring}. For want of data there have been few tests of either theory \cite{lopes1992innovation, christianen1995cycles, alexander1996entropy, tschmuck2006creativity}.
\begin{figure}
\begin{center}
\includegraphics[width=\imgwidth, trim=70 790 225 30, clip=true]{figures/FIGURE_4_diversity}
\end{center}
\caption{\textbf{Evolution of musical diversity in the Billboard Hot 100.} We estimate four measures of diversity. From left to right: Song number in the charts, $D_{\mathrm{N}}$, depends only on the rate of turnover of unique entities (songs), and takes no account of their phenotypic similarity. Class diversity, D$_{\mathrm{S}}$, is the effective number of Styles and captures functional diversity. Topic diversity, D$_{\mathrm{T}}$, is the effective number of musical Topics used each year, averaged across the Harmonic and Timbral Topics. Disparity, D$_{\mathrm{Y}}$, or phenotypic range is estimated as the total standard deviation within a year. Note that although in ecology D$_{\mathrm{S}}$ and D$_{\mathrm{Y}}$ are often applied to sets of distinct species or lineages they need not be; our use of them implies nothing about the ontological status of our Styles and Topics. For full definitions of the diversity measures see electronic supplementary material, M.11. Shaded regions define eras separated by musical revolutions (figure 5).}
\label{fig:diversity}
\end{figure}
To test these ideas we estimated four yearly measures of diversity (figure 4). We found that although all four evolve, two---Topic diversity and disparity---show the most striking changes, both declining to a minimum around 1984, but then rebounding and increasing to a maximum in the early 2000s. Since neither of these measures track song number, their dynamics cannot be due to varying numbers of songs in the Hot 100; nor, since our sampling over 50 years is nearly complete, can they be due to the over-representation of recent songs---the so-called ``pull of the recent'' \cite{jablonski2003impact}. Instead, their dynamics are due to changes in the frequencies of musical styles.
The decline in Topic diversity and disparity in the early 1980s is due to a decline of timbral rather than harmonic diversity (electronic supplementary material, figure S1). This can be seen in the evolution of particular topics (figure 2). In the early 1980s timbral Topics T1 (drums, aggressive, percussive) and T5 (guitar, loud, energetic) become increasingly dominant; the subsequent recovery of diversity is due to the relative decrease in frequency of the these topics as T3 (energetic, speech, bright) increases. Put in terms of Styles, the decline of diversity is due to the dominance of genres such as \textsc{new wave}, \textsc{disco}, \textsc{hardrock}; its recovery is due to their waning with the rise of \textsc{rap} and related genres (figure 2). Contrary to current theories of musical evolution, then, we find no evidence for the progressive homogenisation of music in the charts and little sign of diversity cycles within the 50 year time frame of our study. Instead, the evolution of chart diversity is dominated by historically unique events: the rise and fall of particular ways of making music.
\subsection{Musical evolution is punctuated by revolutions}
\begin{figure}
\begin{center}
\includegraphics[width=\imgwidth, trim=40 490 265 50, clip=true]{figures/FIGURE_5_revolutions_MM}
\end{center}
\caption{\textbf{Musical revolutions in the Billboard Hot 100.} \textbf{A}. Quarterly pairwise distance matrix of all the songs in the Hot 100. \textbf{B}. rate of stylistic change based on Foote Novelty over successive quarters for all windows 1--10 years, inclusive. The rate of musical change---slow-to-fast---is represented by the colour gradient blue, green, yellow, red, brown: 1964, 1983, and 1991 are periods of particularly rapid musical change. Using a Foote Novelty kernel with a half-width of 3 years results in significant change in these periods, with Novelty peaks in 1963--Q4 ($P<0.01$), 1982--Q4 ($P<0.01$) and 1991--Q1 ($P<0.001$) marked by dashed lines. Significance cutoffs for all windows were empirically determined by random permutation of the distance matrix. Significance contour lines with $P$ values are shown in black.}
\label{fig:revolutions}
\end{figure}
The history of popular music is often seen as a succession of distinct eras, e.g., the ``Rock Era'', separated by revolutions \cite{frith1988music, stanley2013yeah, tschmuck2006creativity}. Against this, some scholars have argued that musical eras and revolutions are illusory \cite{negus1996popular}. Even among those who see discontinuities, there is little agreement about when they occurred. The problem, again, is that data have been scarce and objective criteria for deciding what constitutes a break in a historical sequence, scarcer yet.
To determine directly whether rate discontinuities exist we divided the period 1960--2010 into 200 quarters and used the principal components of the Topic frequencies to estimate a pairwise distance matrix between them (figure 5A). This matrix suggested that, while musical evolution was ceaseless, there were periods of relative stasis punctuated by periods of rapid change. To test this impression we applied a method from Music Information Retrieval, Foote Novelty, which estimates the magnitude of change in a distance matrix over a given temporal window \cite{foote2000automatic}. The method relies on a kernel matrix with a checkerboard pattern. Since a distance matrix exposes just such a checkerboard pattern at change points \cite{foote2000automatic}, convolving it with the checkerboard kernel along its diagonal directly yields the novelty function (SI M.11). We calculated Foote Novelty for all windows between 1 and 10 years and, for each window, determined empirical significance cutoffs based on random permutation of the distance matrix. We identified three revolutions: a major one around 1991 and two smaller ones around 1964 and 1983 (figure 5B). From peak to succeeding trough, the rate of musical change during these revolutions varied 4- to 6-fold.
This temporal analysis, when combined with our Style clusters (figure 3), shows how musical revolutions are associated with the expansion and contraction of particular musical styles. Using quadratic regression models, we identified the Styles that showed significant ($P<0.01$) change in frequency against time in the six years surrounding each revolution (electronic supplementary material, table S2). We also carried out a Style-enrichment analysis for the same periods (electronic supplementary material, table S2). Of the three revolutions 1964 was the most complex, involving the expansion of several Styles---1, 5, 8, 12 and 13---enriched at the time for \textsc{soul} and \textsc{rock}-related tags. These gains were bought at the expense of Styles 3 and 6 both enriched for \textsc{doowop} among other tags. The 1983 revolution is associated with an expansion of three Styles---8,11 and 13---here enriched for \textsc{new wave}, \textsc{disco} and \textsc{hard rock}-related tags and the contraction of three Styles---3, 7 and 12---here enriched for \textsc{soft rock}, \textsc{country}-related or \textsc{soul + r'n'b}-related tags. The largest revolution of the three, 1991, is associated with the expansion of Style 2, enriched for \textsc{rap}-related tags, at the expense of Styles 5 and 13, here enriched for \textsc{rock}-related tags. The rise of \textsc{rap} and related genres appears, then, to be the single most important event that has shaped the musical structure of the American charts in the period that we studied.
\subsection{The British did not start the American revolution of '64}
\begin{figure}
\begin{center}
\includegraphics[width=\imgwidth, trim=10 100 150 30, clip=true]{figures/FIGURE_6_BI_trends}
\end{center}
\caption{\textbf{The British Invasion in the American revolution of 1964.} Top to Bottom: PC1--PC4. \textbf{A}. Linear evolution of quarterly medians of four PCs in the six years (24 quarters) flanking 1963--Q4, the peak of the 1964 revolution. The population medians of all four PCs decrease, and these decreases begin well before the start of the British Invasion in late 1963, implying that BI acts cannot be solely responsible for the changes in musical style evident at the time. For each PC, the two topics that load most strongly are indicated, with sign of correlation---high, red to low, blue---indicated (electronic supplementary material, figure S2). \textbf{B}. Frequency density distributions of four PCs for Beatles, Rolling Stones and songs by all other artists around the 1964 revolution. For PC1 and PC2, but not PC3 and PC4, The Beatles and The Rolling Stones have significantly lower median values than the rest of the population, indicated with arrows, implying that these BI artists adopted a musical style that exaggerated existing trends in the Hot 100 towards increased use of major chords and decreased use of ``bright'' speech (PC1) and increased guitar-driven aggression and decreased use of mellow vocals (PC2). Vertical lines represent medians; $P$ values based on Mann-Whitney-Wilcoxon rank sum test; The Beatles (B): $N$ = 46; The Rolling Stones (RS): $N$ = 20; Other artists (O): $N$ = 3114.}
\label{fig:BI}
\end{figure}
Our analysis does not reveal the origins of musical styles; rather, it shows when changes in style frequency affect the musical structure of the charts. Bearing this in mind we investigated the roles of particular artists in one revolution. On 26 December, 1963, The Beatles released \emph{I want to hold your hand} in the USA. They were swiftly followed by dozens of British acts who, over the next few years, flooded the American charts. It is often claimed that this ``British Invasion" (BI) was responsible for musical changes of the time \cite{fitzgerald1995when}. Was it? As noted above, around 1964 many Styles were changing in frequency; many principal components of the Topic frequencies show linear changes in this period too. Inspection of the first four PCs shows that their evolutionary trajectories were all established before 1964, implying that, while the British may have contributed to this revolution, they could not have been entirely responsible for it (figure 6A). We then compared two of the most successful BI acts, The Beatles and The Rolling Stones, to the rest of the Hot 100 (figure 6B). In the case of PC1 and PC2, the songs of both bands have (low) values that anticipate the Hot 100's trajectory: for these musical attributes they were literally ahead of the curve. In the case of PC3 and PC4 their songs resemble the rest of the Hot 100: for these musical attributes they were merely on-trend. Together, these results suggest that, even if the British did not initiate the American revolution of 1964, they did exploit it and, to the degree that they were imitated by other artists, fanned its flames. Indeed, the extraordinary success of these two groups---66 Hot 100 hits between them prior to 1968---may be attributable to their having done so.
\section{Discussion and Conclusions}
Our findings provide a quantitative picture of the evolution of popular music in the USA over the course of fifty years. As such, they form the basis for the scientific study of musical change. Those who wish to make claims about how and when popular music changed can no longer appeal to anecdote, connoisseurship and theory unadorned by data. Similarly, recent work has shown that it is possible to identify discrete stylistic changes in the history of Western classical music by clustering on motifs extracted from a corpus of written scores \cite{rodriguez2013perceptual}. Insofar that our approach is based on audio, it can also be applied to music for which no scores exist, including that from pre-Modern cultures \cite{savage2013toward, lomax1968folk, lomax1972evolutionary}. We have already applied a similar approach to the classification of Art music (``classical music'') into historical periods \cite{weiss2014timbre}. More generally, music is a natural starting point for the study of stylistic evolution because it is not only a universal human cultural trait \cite{brown1991human}, but also measurable, largely determined by form, and available in a relatively standardised format (digital recordings).
Our study is limited in several ways. First, it is limited by the features studied. Our measures must capture only a fraction of the phenotypic complexity of even the simplest song; other measures may give different results. However, the finding that our classifications are supported by listener genre-tags gives us some confidence that we have captured an important part of the perceptible variance of our sample. Second, in confining our study to the Hot 100, 1960--2010, we have only sampled a small fraction of the new singles released in the USA; a complete picture would require compiling a database of several million songs, which in itself is a challenge \cite{bertin2011million}. Given that the Hot 100 is certainly a biased subset of these songs, our conclusions cannot be extended to the population of all releases. Finally, we are interested in extending the temporal range of our sample to at least the 1940s---if only to see whether 1955 was, as many have claimed, the birth date of Rock'n'Roll \cite{peterson1990why}.
We have not addressed the causes of the dynamics that we detect. Like any cultural artefact---and any living organism---music is the result of a variational-selection process \cite{leroi2006recovery, jan2007memetics, maccallum2012evolution, savage2013toward}. In evolutionary biology, causal explanations of organismal diversity appeal to intrinsic constraints (developmental or genetic), ecological factors (competition among individuals or lineages) and stochastic events (e.g., rocks from space) \cite{erwin2007disparity, gould2002structure, jablonski2008species}. By analogy, a causal account of the evolution of music must ultimately contain an account of how musicians imitate, and modify, existing music when creating new songs, that is, an account of the mode of inheritance, the production of musical novelty, and its constraints. The first of these---inheritance and its constraints---is obscure \cite{pachet2006creativity,mcintyre2008creativity}; the second---selection---less so. The selective forces acting upon new songs are at least partly captured by their rise and fall through the ranks of the charts. Many anecdotal histories of music attempt to explain these dynamics. For example, the rise of rap in the charts has been credited to the television show \emph{Yo, MTV Raps!} first broadcast in 1988 \cite{george2005hiphop}. A general, multilevel, selection theory, not restricted to Mendelian inheritance, should provide a means for such hypotheses to be tested \cite{price1970selection, price1972extension, frank1995george}.
Finally, we note that the statistical tools used in this study are quite general. Latent Dirichlet Allocation can be used to study the evolving structure of many kinds of assemblages; Foote Novelty can be used to detect rate discontinuities in temporal sequences of distances based on many kinds of phenotypes. Such tools, and the existence of large digital corpora of cultural artefacts---texts, music, images, computer-aided design (CAD) files---now permits the evolutionary analysis of many dimensions of modern culture. We anticipate that the study of cultural trends based upon such datasets will soon constrain and inspire theories about the evolution of culture just as the fossil record has for the evolution of life \cite{michel2011quantitative}.
\section*{Data accessibility}
All methods and supplementary figures and tables are available in the electronic supplementary materials. Extensive data, including song titles, artists, topic frequencies and tags are available from the Figshare repository \href{http://figshare.com/s/df5dd714b62111e4bcc906ec4bbcf141}{main data frame}, \href{http://figshare.com/s/7916c680b62311e48bbf06ec4b8d1f61}{secondary (tag) data frame}. {\bf [TEMPORARY LINKS, WILL BE UPDATED UPON ACCEPTANCE]}
\section*{Authors' contribution}
ML provided data; MM, RMM and AML analysed the data; MM and AML conceived of the study, designed the study, coordinated the study and wrote the manuscript. All authors gave final approval for publication.
\section*{Funding}
Matthias Mauch is funded by a Royal Academy of Engineering Research Fellowship.
\section*{Acknowledgments}
We thank the public participants in this study; Austin Burt, Katy Noland and Peter Foster for comments on the manuscript; Last.fm for musical samples; Queen Mary University of London, for the use of high-performance computing facilities.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,745 |
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