text:
Q: Creating triangles from points with given bearing and leg length using ArcGIS Desktop I have a spreadsheet that contains ID, LAT_DD, LONG_DD, Bearing and LEN fields. I need to automate a way to create a triangle for every point identified in the table and I have no idea how. I know the steps should be as follows:

*

*Create points from X,Y (Creating the known vertex, beginning, of my triangle)

*Create triangles

*

*Use bearing for triangle bisector

*Use LEN for length of legs radiating from known vertex.

I'm fairly new to GIS programming and don't even know where to start. I can create the points, but that's it.

A: ArcGIS has a number of geodetic operators available in the Projection Engine exposed through ArcPy as methods on the PointGeometry type.
The so-called "Forward Problem of Geodesy" is solved by the PointGeometry.pointFromAngleAndDistance() request, with required parameters angle (in degrees) and distance (in meters), and optional parameter method (one of GEODESIC, GREAT_ELLIPTIC, LOXODROME, and PRESERVE_SHAPE -- PLANAR is not available for geographic coordinate systems).
The "Inverse" (aka "Reverse") problem is solved by PointGeometry.angleAndDistanceTo() with required parameter other (the target PointGeometry), and optional parameter method (same as above), and returns a tuple of angle (in degrees) and distance (in meters) from the reference point.
Note that ArcPy also has a Point object, but this is a different object, which just stores multidimensional vertices.  A Point is used with a SpatialReference to construct a PointGeometry.
The full solution for computing distance from New York (JFK) to Tokyo (Narita) in WGS84, and calculating the halfway point between them would look like:
import arcpy
gcs = arcpy.SpatialReference(4326)
gcs.setFalseOriginAndUnits(-400.0,-400.0,10000000)

jfk  = arcpy.PointGeometry(arcpy.Point(-73.7781,40.6413),gcs)
nrt  = arcpy.PointGeometry(arcpy.Point(140.3929,35.7720),gcs)

bearing,distance = jfk.angleAndDistanceTo(nrt,'GEODESIC')
print(" Bearing = {:.2f} deg".format(bearing))
print("Distance = {:.2f} meters".format(distance))

midpoint = jfk.pointFromAngleAndDistance(bearing,distance*0.5,'GEODESIC')
print("Midpoint = {:4f},{:.4f}".format(
        midpoint.firstPoint.X,
        midpoint.firstPoint.Y))

Which results in:
 Bearing = -27.32 deg
Distance = 10853320.27 meters
Midpoint = -152.890242,69.4824

and feels right (10853 kilometer NNW polar route over Alaska).

A: ArcGIS may have its own operation to calculate a point on a specific bearing and distance, but a great source for these types of calculations is the Aviation Formulary at:  https://edwilliams.org/avform.htm
In particular you want the "Lat/Long given radial and distance" :

Lat/lon given radial and distance
A point {lat,lon} is a distance d out on the tc radial from point 1 if:
lat=asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
IF (cos(lat)=0)
   lon=lon1      // endpoint a pole
ELSE
   lon=mod(lon1-asin(sin(tc)*sin(d)/cos(lat))+pi,2*pi)-pi
ENDIF

This algorithm is limited to distances such that dlon around less than one quarter of the circumference of the earth in longitude. A >completely general, but more complicated algorithm is necessary if greater distances >are allowed:
lat =asin(sin(lat1)*cos(d)+cos(lat1)*sin(d)*cos(tc))
dlon=atan2(sin(tc)*sin(d)*cos(lat1),cos(d)-sin(lat1)*sin(lat))
lon=mod( lon1-dlon +pi,2*pi )-pi

Note that trigonometry functions are in radians - you'll have to convert to/from degrees (Radians = Degrees*PI/180 )


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This article presents a flash game about animals. The name of the game is Rocket Pets. The mission of the players who will be dealing with this game is to launch the small furry animals one by one. The farther they reached, the higher coins you can get to avail certain upgrades.


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This week I have the pleasure of attending the London Film Festival, and the first film I had the pleasure of seeing, was J.A. Bayona's film, A Monster Calls! It is a beautiful, emotional film that resonated with me, maybe more than any film I have seen this year!
A Monster Calls follows Conor O'Malley (Lewis MacDougal), a young boy trying to deal with the terminal illness of his mother (Felicity Jones) and the attacks by local school bully Harry (James Melville). One night, Conor encounters a "monster" (Liam Neeson) in the form of a giant humanoid yew tree who has come to tell him stories and soon begins to help Conor fix his unhappy life.
The performances are excellent! In particular Lewis MacDougal who delivers, maybe the best child performance this decade. He gives such an assured, yet fearful performance that really impressed me. Felicity Jones, who plays the boy's mother is also brilliant and possibly the only performance here that could get an Oscar nomination. She kills it. Her performance is so raw, so real, and she has a scene that will make even the hardest of people cry.
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When I went to Israel, Gaza and West Bank in 2000, I had no idea where the trip would take me.
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I hate it when people ask me if I'm pro-Israel or pro-Palestine.
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I returned to Argentina, but I couldn't deal with the documentary for 15 years.
NEY is not a film about the conflict. It is a film that deals –with a sense of paranoia- with a conflict on how the conflict is seen, how it is perceived.
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Anyway... I don't want to limit the questions that arouse when making the documentary by writing them down.
NEY is an exciting trip and a romantic view on growing up.
It is the reconstruction of that trip, 15 years later.


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text:
\section{Introduction}
\label{sec:intro}

The fractional quantum Hall effect (FQHE) has uncovered a number of
unique and unexpected quantum behaviors of two-dimensional electrons.
Experiments of electron tunneling from normal Fermi liquids into FQH
states are unique tools to explore the properties of the excitations of
FQH states and to understand directly the behavior of these fluids.
Because in the bulk these fluids are incompressible, {\it i.\ e.\/} all
excitations have a finite energy gap, it is very difficult to tunnel
into the bulk of these fluids. However, it is possible to tunnel
electrons into the edge states of droplets of these fluids.

In this talk I will discuss a theoretical description of tunneling of
electrons into FQH edge states. In the first part of the talk I give a
quick review of the current understanding of the physics of the
bulk FQH state and of the chiral Luttinger picture of the edge states.
This part of the talk is pedagogical and I included it at the request
of the organizers. Much of this material is quite standard, and here
I follow closely the hydrodynamic theory of X.\ G.\ Wen~\cite{wen-review}.
In the second part of the
talk I will discuss work that I have done recently, in collaboration with
Claudio Chamon and Nancy Sandler, on junctions of FQH edges with normal Fermi
liquids. Most of the results that I presented in this part of the talk were
originally published in papers we coauthored (references [\ref{ref:junction}] and
[\ref{ref:andreev}]).

Let us begin by considering the standard setup of the FQHE. That is we
will consider a two-dimensional electron gas (2DEG) formed in a Al As-Ga
As heterostructure in the presence of a high perpendicular magenetic
field $B$. As usual we will measure the electron density in terms of the
filling fraction $\nu$ of the Landau level defined as $\nu=N_e/N_\phi$,
where $N_e$ is the number of electrosn in the 2DEG and $N_\phi$ is the
total number of flux quanta piercing the 2DEG.
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=4.0in
\epsfbox{2deg.eps}
\vspace{.5cm}
\caption{The two-dimensional electron gas in a perpendicular magnetic field.}
\label{fig:2deg}
\end{figure}
When the FQHE occurs, the 2DEG behaves as an incompressible quantum charged fluid.
In the abscence of impurities this can only happen when certain
charge-flux commensurability conditions are met and the filling fraction
takes a set of specific (fractional) values. (Impurities trap the
excitations above these ground states in localized states,
and that is how the FQHE becomes observable.) The most prominent FQH
states seen in experiment are arranged in the sequence
\begin{equation}
\nu={\frac{p}{2np\pm 1}}
\label{eq:jain}
\end{equation}
where $p=1,2,\ldots,\infty$ and $n=0,1,2,\ldots,\infty$,
are two integers that label the states.
The $\pm $ sign refers to the FQH of electrons and to the particle-hole
reversed sequence.  Clearly $0<\nu < 1/(2n)$ for $+$, while $1/(2n) <
\nu < 1/(2n-1)$. For $p=1$ we find the states in the Laughlin sequence
$\nu=1/(2n+1)$. The sequence of states with the largest gaps has $n=1$
and it is formed by the states $\nu=p/(2p+1)=1/3,2/5,3/7,\ldots$.

Since the bulk states are fully gapped the only gapless excitations of
this fluids reside at the edge: they are deformations of the edge. Wen
has emphasized that such
incompressible fluids are topological fluids in that, if
placed on a  surface with non-trivial topology, their Hilbert spaces
will be sensitive only to the topology of the surface and not to the
local properties of the material.

\begin{figure}[!t]
\vspace{.2cm}
\noindent
\hspace{1.25 in}
\epsfxsize=4.0in
\epsfbox{levels.eps}
\vspace{.5cm}
\caption{Landau levels for a non-interacting 2DEG at $\nu=1$. $E_F$ is the
Fermi energy and $\omega_c$ is the cyclotron radius.}
\label{fig:levels}
\end{figure}

In 1982 B.\ Halperin~\cite{halperin-edge} introduced a very simple
and appealing picture of
the edge states of non-interacting electrons in Landau levels (see
figure \ref{fig:levels}). In this picture, the bulk Landau levels ,
which have an energy spacing of $\hbar \omega_c$, are bent upwards
adiabatically near the edge of the sample by the confining potential,
with an electric field strength $E$.
For a disk-shaped sample the single particle states are eigenstates of
angular momentum (for a rotationally invariant confining potential). If
$E_F$ denotes the Fermi energy of the electrons, for a macroscopically
large sample there are many states with single particle energies in the
vicinity of $E_F$. The level spacing of these staes is small as the
radius of the electron droplet  gets big. In this limit the Hilbert
space of these states consists on a set of evenly spaced energy levels
with a finite density of states. These states represent electron
excitations that move in a direction specified by the sign of the magnetic
field $B$ (or by the sign of their charge). The edge states propagate
at the drift velocity $v=c E/B$, whre $c$ is the speed of light.

Hence, the electrons close to the edge
behave like a chiral right moving Fermi system. In this picture, which
is a good description of a $\nu=1$ state, the only effect of the Coulomb
interactions among the electrons is a simply a change of the speed at
which the electrons at the edge propagate. If the total number of
electrons is fixed, the excitations of the edge are charge neutral
electron-hole pairs. States labelled by sets of electron-hole pairs can
also be regarded as geometric deformations of the edge of the fluid.
This picture led Wen to formulate a hydrodynamic theory of the edge
states~\cite{wen-edge}.
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.0 in}
\epsfxsize=4.0in
\epsfbox{excitation.eps}
\vspace{.5cm}
\caption{Neutral excitation of the edge.}
\label{fig:excitation}
\end{figure}
\subsection{Bulk Incompressible States}
\label{subsec:bulk}
The physics of the bulk incompressible states is by now quite well understood.
The sequence of Laughlin states $\nu=1/(2n+1)$ (with $0\leq n < \infty$)
are (almost exactly) described by the celebrated Laughlin wave
functions~\cite{laughlin}. The states in the Jain sequences can be described using
Jain wave functions~\cite{jain}, which are generalizations of the Laughlin wave
functions. In the lowest Landau level (LLL) the single particle states
(in the circular gauge) are
\begin{equation}
\psi_r(z) \propto z^r e^{-|z|^2/(4 \ell_0^2)}
\label{eq:psi}
\end{equation}
where $\ell_0$ is the magnetic length, $\ell_0={\sqrt{e\hbar/cB}}$ and
$z$ is the (complex) coordinate of the electron, $z=x+iy$.
Let $m$ denote the odd integer $m=2n+1$. For the Laughlin states, with
filling fraction $\nu=1/m$, the Laughlin wave function for $N$
electrons in a magnetic field with $N_\phi=mN$ flux quanta is
\begin{equation}
\Psi_m(z_1,\ldots,z_N) \propto \prod_{i<j=1}^N(z_i-z_j)^m
e^{-{\frac{1}{4\ell_0^2}}\sum_{i=1}^N|z_i|^2}
\label{eq:laughlin}
\end{equation}
 The elementary exciations in the bulk are the Laughlin quasiholes. The
wave function for a Laughlin quasihole at $z_0$ is
\begin{equation}
\Psi_m(z_0;z_1,\ldots,z_N)= \prod_{i=1}^N(z_0-z_i) \Psi_m(z_1,\ldots,z_N)
\label{eq:quasihole}
\end{equation}
Laughlin argued~\cite{Laughlin-review} that, since in this state each electron
picks up an extra unit
of angular momentum, this state is equivalent to the effect of introducing
(adiabatically) an infinitesimally thing solenoid which threads one flux quantum at
$z_0$ . He further showed that the charge defect localized at $z_0$, the
quasihole, has positive charge equal to $+e/m$. As a consequence of
incompressibility there is an extra (negative) charge at the boundary
equal to $-e/m$. Hence, quasiholes make the edge swell by the right
amount to accomodate the extra charge. Furthermore, a semiclassical
argument~\cite{halperin,arovas} shows that these objects have fractional statistics
\begin{equation}
\Psi_m(z_0,z_0';z_1,\ldots,z_N)=e^{\pm i\pi/m} \Psi_m(z_0',z_0;z_1,\ldots,z_N)
\label{eq:statistics}
\end{equation}
Thus, the spectrum of excitations of these bulk incompressible states
are quasiholes with fractional charge and fractional statistics. These
results can also be (and have been) derived by field-theoretic methods
involving Chern-Simons gauge fields~\cite{zhk,lf}.

Alternatively we may also use a hydrodynamic picture of the bulk. The
hydrodynamic description is a highly economical way to summarize the
universal data of the bulk FQH states: the Hall conductance, the charge
and the statistics of the bulk excitations. It can be derived explicitly
from the Chern-Simons (mean-field) picture of the FQH states~\cite{zhk,lf} or by a
set of phenomenological arguments based on conservation laws. For the
sake of simplicity we will follow the latter approach, also introduced
by Wen~\cite{wen-review}. The fundamental idea is quite simple. The 2DEG is a charged
fluid and, as consequence of charge conservation, it can be described in
terms of the locally conserved 3-current $j_\mu$
\begin{equation}
j_\mu=\left(j_0,{\vec j}\right)
\label{eq:3current}
\end{equation}
where $j_0$ is the local charge density $\rho$, $\vec j$ is the local
charge current, and $\mu=0,x,y$ (or $\mu=0,1,2$).
Local charge conservation means that $j_\mu$ obeys the continuity equation
\begin{equation}
\partial_\mu j^\mu=0
\label{eq:conservation}
\end{equation}
which is to say that $j_\mu$ is a locally conserved current. Since the
3-divergence of the current vanishes, it must be a curl of a vector
field $a_\mu$ (since space-time is three dimensional). Thus we write
\begin{equation}
j_\mu={\frac{1}{2\pi}}\epsilon_{\mu \nu \lambda}\partial^\nu a^\lambda
\label{eq:curl}
\end{equation}
(the factor of ${\frac{1}{2\pi}}$ is introduced for later convenience).
Observe that the 3-current is invariant if we make the local transformation
(or redefinition) $a_\mu \to
a_\mu+\partial_\mu \Lambda$, where $\Lambda$ is an arbitrary, smooth
 single-valued function. Hence there is a gauge symmetry that is natural in this
description.

What do we know about the FQH state? First of all we know that the Hall conductance
has the exact value $\sigma_{xy}={\frac{\nu}{2\pi}}{\frac{e^2}{\hbar}}$.
If we denote by $A_\mu$ an external electromagnetic vector potential that we
will use to probe the system ({\it i.\ e.\/} it is neither part of the
uniform magnetic field nor of the confining potential), the effective action
 must have a local coupling of the form $j_\mu A^\mu$. Also the
effective action must be invariant under gauge transformations of the
hydrodynamic gauge field $a_\mu$. This effective action must be local,
be odd under time reversal (and parity) and have as few derivatives as
possible. In addition it should contain only the universal data of the
FQH state which is a set of dimensionless numbers. Thus, the only terms
that can be allowed must be dimension three gauge invariant operators.
For $\nu=1/m$, the unique choice that satisfies all these constraints is
\begin{equation}
S_{\rm eff}={\frac{m}{4\pi}}\int d^3x
\epsilon_{\mu \nu \lambda}a^\mu \partial^\nu a^\lambda
+ {\frac{e}{2\pi}} \int d^3x
\epsilon_{\mu \nu \lambda}A^\mu \partial^\nu a^\lambda
\label{eq:CS}
\end{equation}
where we have used eq.\ \ref{eq:curl} to write the $j_\mu A^\mu$
coupling.  It is straightforwrd to show that the current induced by
$A_\mu$ is
\begin{equation}
J^{em}_\mu=\sigma_{xy} \epsilon_{\mu \nu \lambda} \partial^\nu A^\lambda
\label{eq:jem}
\end{equation}
Hence, we get the correct value of the Hall conductance for $\nu=1/m$.

In this picture, the elementary excitations (quasiholes) look like vortices.
Since we are interested in the low energy regime, we can assume that the
quasiholes are quasistatic (namely, that their gap is very large) and as
such they can be  pictured by a set of (prescribed) vortex currents
$j^\mu_{\rm vortex}$, minimally coupled to the hydrodynamic field
$a_\mu$,
\begin{equation}
S_{\rm vortex}=\int d^3x \;\; a^\mu j^\mu_{\rm vortex}
\label{eq:vortex}
\end{equation}
It is very easy to show that the exciations described by Eq.\
\ref{eq:vortex} have charge $Q=e/m$ and fractional statistics
$\theta=\pi/m$. Hence, the vortices are the Laughlin quasiholes.

\subsection{Hydrodynamic Picture of Bulk and Edge (after X.\ G.\ Wen)}
\label{subsec:hydro}

There is a very appealing picture of the dynamics of the edge
states, whcich was also introduced by Wen~\cite{wen-edge}. The theory is
particularly simple for the case of the Laughlin states (which have filling fraction
$\nu=1/m$). In this case the edge is described in terms of a single hydrodynamic
field. The picture of the more general FQH state is more complex and depends on
details such as possible edge reconstructions. A good general discussion can be
found in Wen's review ~\ref{ref:wen-review} and in my recent paper with
Ana Lopez~\cite{ana}. Here I will consider only the edges of Laughlin states.

Consider a droplet of  the incompressible  FQH fluid.
For simplicity, we will picture the edge
of the fluid in its ground state as  a straight line (the horizontal
line of figure \ref{fig:edge}). In this picture, an excitation is a
deformation of the edge and in figure \ref{fig:edge} is shown as a wavy
curve. Let $h$ denote the local displacement (or height) of the fluid at
a point $x$ along the edge. Because the fluiid is incompressible, the
change of the local charge density $\delta \rho$  is proportional to the
displacement $h$, $\delta \rho \propto h$. The energy of this
excitation is the work done against the confining potential to create this
displacement is
\begin{equation}
{\rm energy} \propto  \int dx [\delta \rho (x)]^2
\label{eq:energy}
\end{equation}
which plays the role of a classical Hamiltonian.
At the quantum level, the dynamics of the edge is specified by the energy
\ref{eq:energy},
and by the commutation relation of the edge degrees of freedom.
Since the edge degrees of freedom are chiral (namely, the move at the
drift velocity), the local density operators $\delta \rho(x)$ nave to
obey appropriate commutation relations ta lead to chiral propagation along the
edge. The (equal time) commutaion relations are
\begin{equation}
\left[\delta \rho(x),\delta \rho(x')\right]=i \pi {\rm sign} (x-x')
\label{eq:ccr}
\end{equation}
These commutation relations can be solved by introducing a filed $\phi(x)$,
known as the chiral boson field, with the action
\begin{equation}
S_{\rm edge}={\frac{1}{4\pi}} \int dx dt \partial_x \phi \left( \partial_t \phi
- v \partial_x \phi \right)
\label{eq:action-edge}
\end{equation}
where $v$ is the effective velocity of th edge waves. The chiral boson
is realted to the density fluctuations by
\begin{equation}
\delta \rho(x)= {\frac{\sqrt{\nu}}{2\pi}} \partial_x \phi
\label{eq:relation}
\end{equation}
With these definitions it is starightforward to show that the density
operators obey the commutation relations of Eq.\ \ref{eq:ccr} and that the
energy of the excitaitions is given by Eq.\ \ref{eq:energy}.
\begin{figure}
\vspace{.2cm}
\noindent
\hspace{1.25 in}
\epsfxsize=4.0in
\epsfbox{edge.eps}
\vspace{0.5cm}
\caption{An edge distortion.}
\label{fig:edge}
\end{figure}
In order to  have a completely well defined effective quantum theory we
need to specify the Hilbert space on which these operators act. Thus we
need to specify what type of excitations are allowed and what is their charge
and statistics. Since the bulk FQH state can only support excitations
with a charge which must be a multiple of the quasiparticle charge.
~From Eq.\ \ref{eq:relation} we find that if we add (or remove) an amount of
 charge $Q$ to the system, the chiral boson must obey the following
boundary condition
\begin{equation}
Q=\oint_{\rm edge} dx \; \delta \rho(x)={\frac{\sqrt{\nu}}{2\pi}}
\left[\phi(L)-\phi(0)\right]
\label{eq:bcs}
\end{equation}
where the integral runs over a (closed) edge of length $L$. Here we have
set the electric charge to unity. Hence, since in the ground state sector
$Q=0$, the chiral boson obeys periodic boundary conditions (PBCs),
\begin{equation}
\phi(x)=\phi(x+L)
\label{eq:pbcs}
\end{equation}
Next we define an operator $\psi_e(x)$ which creates an {\it electron},
namely a {\it fermion} with charge $1$. In terms of the chiral boson,
the electron operator is given by the vertex operator
\begin{equation}
\psi_e(x) \propto e^{{\frac{i}{\sqrt{\nu}}} \phi(x)}
\label{eq:electron}
\end{equation}
The electron propagator is (with an $i\epsilon$ regulator implied)
\begin{equation}
\langle T \psi^\dagger_e(x,t)  \psi_e(x',t') \rangle \sim
{\frac{1}{(z-z')^m}}
\label{eq:eprop}
\end{equation}
which has a pole of order $m$. In Eq.\ \ref{eq:eprop} $T$ stands
for time ordering and  $z=x-vt$.

Likewise, the quasiparticle operator $\psi_{qp}(x)$ is given by
\begin{equation}
\psi_{qp}(x) \propto  e^{i\sqrt{\nu} \phi(x)}
\label{eq:qp}
\end{equation}
which has the propagator
\begin{equation}
\langle T \psi^\dagger_{qp}(x,t)  \psi_{qp}e(x',t') \rangle \sim
{\frac{1}{(z-z')^{1/m}}}
\label{eq:qpprop}
\end{equation}
The power $1/m$ in this  propagator implies that it has a branch cut
(instead of a pole) which says that the statistics of the quasiparticle
is $\pi/m$.

We now notice that both the electron and the quasiparticle oparator anre
invariant under the transformation $\phi \to \phi + 2 \pi n R$ where $n$
is an arbitrary integer and the compactification radius (or Luttinger parameter)
$R=\sqrt{\nu}$. Hence, the addition of $n$ quasiparticles of charge $Q=n \nu=n/m$
implies that the chiral boson acquires the twisted boundary condition
\begin{equation}
\phi(x+L)-\phi(x)={\frac{2\pi}{\sqrt{\nu}}}Q=2\pi nR
\label{eq:tbcs}
\end{equation}
However, it is easy to see that the electron operator remains single valued
\begin{equation}
\psi_e(x+L)=\psi_e(x)
\label{eq:ebcs}
\end{equation}
as it is required by the single-valuedness of its wave functions, while
 the quasiparticle operator instead obeys twisted boundary conditions,
\begin{equation}
\psi_{qp}(x+L)=e^{i2\pi n \nu} \psi_{qp}(x)
\label{eq:qpbcs}
\end{equation}
and are multi-valued, as requierd by the branch cut in Eq.\
\ref{eq:qpprop}.

The set of conditions that we have discussed in this section
constitute a complete definition
of the Hilbert space of the edge states. In the second part of this talk
I will use this machinery to explore a number of interesting physical
questions found in the problem of tunneling of
electrons into FQH edge states.

\section{Model of a Quantum Point Contact Junction}
\label{model}

Let us consider first the problem of a quantum point contact junction
between an electron reservoir and the edge of a
Laughlin FQH state. Typically the electron reservoir is a
three-dimensional electron gas (3DEG). In particular, we will have in
mind the situation of the tunnel junctions of the exepriments of A.\
Chang and coworkers~\cite{chang}, see figure \ref{fig:broad}.
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.25 in}
\epsfxsize=3.0in
\epsfbox{broad.eps}
\vspace{.5cm}
\caption{Schematic geometry of the experimental setup of A.\ Chang
and coworkers.}
\label{fig:broad}
\end{figure}
In this experiment, the 3DEG acts as a reservoir of electrons. A
3DEG is well described by Fermi liquid theory. It is believed that
the edge that results from this device is ``atomically sharp". If so,
presumably there is no edge reconstruction taking place and the electron
density of the 2DEG falls off rader rapidly and monotonically close
enough to the edge.
However, it is unclear to what degree these assumption do hold and, in
fact, in ref.\ \ref{ref:shytov} it is argued that there is a substantial
amount of charge redistribution close to the edge. In any event, even if
this were the case, it is most likely that tunnel of electrons from the
3DEG reservoir to the edge of the FQH state in the 2DEG will take place
at special places where the tunneling amplitude is largest. In other
words, the broad edge can be reagrded as a (large) set of weak tunneling
centers. In ref.\ \ref{ref:junction} C.\ Chamon and I developed a
theory of tunneling that applies to this regime. Thus, I will concentrate
first on the description of a single tunneling center, a quantum point
contact (QPC), depicted in figure \ref{fig:contact}.

For the case of a single QPC the tunneling problem is substantially
simpler to describe. The theory of the QPC is highly reminiscent of the
physics of quantum impurities in metals. In fact, for a QPC, the problem
reduces to a quantum impurity (or rather a tunneling problem) in a
one-dimensional strongly correlated (chiral) system. In particular, it
is very simple to see that,just as in the case of impurities in metals,
of the infinitely many degrees of
freedom of the 3DEG, only one channel is able to tunnel. The difference
is that while in the case of the quantum impurity in the bulk of
a metal, only the $S$-wave channel scatters off the impurity. In contrast,
in the case of the the semi-infinite geometry of the
QPC only the electrons in the $\ell=1$ $m=0$ channel are able to tunnel.
Nevertheless, what matters is that there is one and only one channel
needed to describe a QPC. Details of the properties of the states
in the channel are absorbed in the definition of the tunnel matrix
element $\Gamma$.

\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=3.0in
\epsfbox{contact.eps}
\vspace{.5cm}
\caption{A quantum point contact between a 3DEG and the edge of a
Laughlin fluid.}
\label{fig:contact}
\end{figure}

Thus, in this picture, the 3DEG reduces to a semi-infinite
one-dimensional Fermi liquid.
In the limit of zero tunneling matrix element, $\Gamma \to 0$, the total
charge current at the ``end point" at $x=0$ vanishes exactly. The states
of a Fermi liquid are essentially equivalent to those of a free Fermi
system, up to a renormalization of the Fermi velocitie (screening
effects and othere renormalizations of the coupling constants of the
3DEG in the bulk play almost no role in the present situation
and can be ignored.)
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.25 in}
\epsfxsize=4.0in
\epsfbox{waves.eps}
\vspace{.5cm}
\caption{Only one channel of the 3DEG couples at the quantum point
 contact.}
\label{fig:waves}
\end{figure}

The electron states on the half-line are incoming
and outgoing waves, as shown in figure \ref{fig:waves} satifying
zero-current boundary conditions at the endpoint. However, exactly as in
the problem of magnetic impurities in metals, this system
exactly equivalent to a system of free chiral fermions on a full line.
Thus, the states of the electrons that participate in tunneling
processes are described by a theory of free chiral fermions. This
also happens to be the theory of the edge states of a 2DEG at $\nu=1$.
Therefore the QPC is equivalent to a junction between the edge of
a  $\nu=1$ QH state and the edge of a $\nu=1/m$ FQH state~\cite{junction}.

\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=3.0in
\epsfbox{equivalent.eps}
\vspace{.5cm}
\caption{The QPC between a 3DEG and a $\nu=1/m$ FQH edge is equivalent to
a junction between the same FQH edge and a $\nu=1$ edge.}
\label{fig:equivalent}
\end{figure}

We can now use the picture developed in the first part of the talk~\cite{junction},
section \ref{sec:intro} to write down an effective Lagrangian for the
theory of a QPC between a $\nu=1$ state (an electron resrvoir: a
3DEG or any other Fermi liquid) and the edge of a $\nu=1/m$ Laughlin FQH state.
To this end let us introduce to chiral bosons, $\phi_1$ and $\phi_2$ which
I will use to represent the Hilbert space of the edges of the FQh state and
the Fermi liquid respectiveliy. The total Lagrangian has the form
\begin{equation}
{\cal L}={\cal L}_{\rm FQH}+{\cal L}_{\rm reservoir}+{\cal L}_{\rm
tunnel}
\label{eq:lagrangian1}
\end{equation}
where ${\cal L}_{\rm FQH}$ is the Lagrangian of the edge states of a
$\nu=1/m$ FQH fluid,
\begin{equation}
{\cal L}_{\rm FQH}={\frac{1}{4\pi}} \partial_x \phi_1 \left( \partial_t \phi_1
-  \partial_x \phi_1 \right)
\label{eq:lagrangianfqh}
\end{equation}
and ${\cal L}_{\rm reservoir}$ is the Lagrangian for the electrons in the
reservoir,
\begin{equation}
{\cal L}_{\rm reservoir}={\frac{1}{4\pi}} \partial_x \phi_2 \left
( \partial_t \phi_2 -  \partial_x \phi_2 \right)
\label{eq:lagrangianreservoir}
\end{equation}
where we have set the velocities of both systems to unity $v_1=v_2=1$.
This is justified for a single QPC.

Finally, ${\cal L}_{\rm tunnel}$ represents the tunneling ef electrons,
\begin{equation}
{\cal L}_{\rm tunnel}=\Gamma \;\delta(x) \; e^{i \omega_0 t}\;
\normord{e^{{\frac{i}{\sqrt{\nu}}} \phi_1}}
\normord{e^{-i\phi_2}}+ {\rm h.\ c.\ }
\label{eq:tunnel}
\end{equation}
Here, $\Gamma$ denotes the tunneling matrix element for electrons,
$\omega_0=eV/\hbar$ is the  ``Josephson" frequency (for a voltage drop of
$V$, which we will set to zero for the most part of our discussion), and
$\exp{{\frac{i}{\sqrt{\nu}}} \phi_1}$ and $\exp{-i\phi_2}$ are the
operators that create an electron at the FQH edge and destroy an
electron at the reservoir respectively.

In reference \ref{ref:junction} it was shown that this problem can be
solved exactly. Here I will give a summary of that solution.
The first step is to observe that by an unitary (actually orthogonal)
transfromation of the form
\begin{equation}
\left(
\begin{array}{c}
\varphi_1\\
\varphi_2
\end{array}
\right)
=
\left(
\begin{array}{cc}
 \cos \theta & \sin \theta \\
-\sin \theta & \cos \theta
\end{array}
\right)
\;
\left(
\begin{array}{c}
\phi_1 \\
\phi_2
\end{array}
\right)
\label{eq:orthogonal}
\end{equation}
the Lagrangian can be brought to the form
\begin{equation}
{\cal L}=
{\frac{1}{4\pi}} \partial_x \varphi_1 \left
( \partial_t \varphi_1 -  \partial_x \varphi_1 \right)+
{\frac{1}{4\pi}} \partial_x \varphi_2 \left
( \partial_t \varphi_2 -  \partial_x \varphi_2 \right)
+
\Gamma \; \delta (x) \;
e^{{\frac{i}{\sqrt{g'}}}( \varphi_1-\varphi_2)}
+ {\rm h.\ c.\ }
\label{eq:Ltotal}
\end{equation}
where we have chosen the angle $\theta$ to be the solution of
\begin{equation}
\cos 2 \theta= {\frac{2 \sqrt{\nu}}{1+\nu}}
\label{eq:theta}
\end{equation}
and the ``effective filling factor" $g'$ is
\begin{equation}
g'^{-1}={\frac{1}{2}} (1+\nu^{-1})
\label{eq:g'}
\end{equation}
In other terms, we have mapped the junction between a $\nu=1$ edge and
another edge of a FQH stae at filling factor $\nu$ to a problem of
tunneling of chrage $1$ particles between two identical states at
filling factor $\nu'=g'$. Hence, the new (rotated) chiral bosons have
compactification radius $R={\sqrt{g'}}$. For the special case of $\nu=1/3$ we
find $\nu'=g'=1/2$. Notice that, the statistics of the effective
charge $1$ particles that tunnel between the two equivalent edges is
$\theta_{\rm eff}={\frac{\pi}{g'}}$, and
for the particular case of $\nu=1/3$ they are bosons.

Finally, we define the orthogonal fields $\varphi_\pm$
\begin{equation}
\varphi_\pm={\frac{1}{\sqrt{2}}} (\varphi_1 \pm \varphi_2)
\label{eq:pm}
\end{equation}
in terms of which the Lagrangian decouples into two pieces, ${\cal L}_+$ and
${\cal L}_-$. In Eq.\ \ref{eq:pm} ${\cal L}_+$ is the Lagrangian of a free chiral
boson
\begin{equation}
{\cal L}_+={\frac{1}{4\pi}} \partial_x \varphi_+ \left(
\partial_t \varphi_+ -  \partial_x \varphi_+ \right)
\label{eq:L+}
\end{equation}
with compactification radius $R_+={\sqrt{g'/2}}$. The field $\varphi_+$
represents  the fluctuations of the conserved charge current of the
combined system.

The Lagrangian ${\cal L}_-$
contains the physics of the tunneling processes. It has the form
\begin{equation}
{\cal L}_-={\frac{1}{4\pi}} \partial_x \varphi_- \left(
\partial_t \varphi_- -  \partial_x \varphi_- \right)+
\Gamma \; \delta (x) \; e^{i {\sqrt{{\frac{2}{g'}}}}\varphi_-}
+{\rm h.\ c.\ }
\label{eq:L-}
\end{equation}
The compactification radius $R_-$ of the field $\varphi_-$ is also
$R_-={\sqrt{g'/2}}$.

It is useful to introduce now an alternative equivalent form of the
Lagrangian of Eq.\ \ref{eq:L-}, in which instead of working with the
chiral bose field $\varphi_-$, defined on a full line of length $L$ , one defines a
{\sl non-chiral} bose field $\varphi_-$ on a half-line, $0\leq x < L/2$.
The chiral field is identified with the right and left moving components
$\varphi_R$ and $\varphi_L$ as follows
\begin{equation}
\begin{array}{ccc}
\varphi_-(x)= &\varphi_-^R(x), & {\rm for}\;\; x>0
\\
\varphi_-(x)= &\varphi_-^L(-x),& {\rm for}\;\; x<0
\\
\varphi_-(x)=&\varphi_-^R(x)+\varphi_-^L(x) , &{\rm for}\;\; x\geq 0
\end{array}
\label{eq:folding}
\end{equation}
Hence, we have folded the line and we now have a non-chiral bose field
on a half-line. This procedure is very common for solving quantum impurity
problems~\cite{NFLL}.

The action for the field $\varphi_-$ on the half-line is
\begin{equation}
S[\varphi_-]={\frac{1}{8\pi}}\int_0^{L/2} \int_{-\infty}^{+\infty}dt
\left[
\left({\frac{\partial \varphi_-}{\partial t}}\right)^2 -
\left({\frac{\partial \varphi_-}{\partial x}}\right)^2
\right]
+\int_{-\infty}^{+\infty}dt \;\; 2 \Gamma \; \cos \left({\frac{1}{\sqrt{2g'}}}
\varphi_-(0,t)\right)
\label{eq:BSG}
\end{equation}
where the field $\varphi_-$ obeys Neumann boundary conditions both at $x=0$
and at $x \to \infty$. Notice that the compactification radius of
$\varphi_-$ is $R={\sqrt{2g'}}$.
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=3.0in
\epsfbox{fold.eps}
\vspace{.5cm}
\caption{Folding a chiral boson on a non-chiral boson on a half-line with Neumann
boundary conditions.}
\label{fig:fold}
\end{figure}
The Lagrangian of Eq.\ \ref{eq:BSG} is known as the boundary sine-Gordon theory. It is
an integrable quantum field theory and a lot of useful
information is known about it~\cite{GS}.

Hence, upon folding, the problem that we need to understand is a
boundary sine-Gordon field theory with Neumann (N) boundary conditions
at both ends of the line, $\partial_x \varphi_-(x,t)|_{0,L/2}=0$.
By inspection of the Lagrangian of Eq.\ \ref{eq:BSG} we
can see that the effect of the tunneling operator is a change in the
boundary condition (BC) at $x=0$. Indeed, for $\Gamma=0$ we have a Neumann
BC at $x=0$, whereas for $\Gamma \to \infty$ we have a Dirichlet (D) BC
at $x=0$, $\varphi_-(0,t)=0$ (see top part of fig.\ \ref{fig:duality}).
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=3.0in
\epsfbox{duality.eps}
\vspace{.5cm}
\caption{Duality as a change of boundary conditions.}
\label{fig:duality}
\end{figure}
Finally we make a {\sl duality transformation} on this theory. The duality
transformation~\cite{duality}~\cite{Tduality} exchanges oscillators with kinks (or
solitons). In the folded picture it is equivalent to a replecement of
the field $\varphi_-$ by its Cauchy-Riemann dual ${\tilde \varphi}_-$,
\begin{eqnarray}
\partial_x {\tilde \varphi}_-=&& \;\; \; \partial_t \varphi_-
\nonumber \\
\partial_t {\tilde \varphi}_-=&&- \partial_x \varphi_-
\label{eq:cauchy}
\end{eqnarray}
It is straightforward to see that if the field $\varphi_-$ obeys a Neumann
(Dirichlet) BC the dual field ${\tilde \varphi}_-$ obeys a Dirichlet
(Neumann) BC.

In addition to a change of boundary conditions, under duality the
compactification radius $R_-={\sqrt{2g'}}$ of the non-chiral field $\varphi_-$
becomes~\cite{duality,Tduality}
\begin{equation}
{\tilde R}_-={\frac{2}{R_-}}=\sqrt{\frac{2}{g'}}
\label{eq:Rdual}
\end{equation}
Hence, the dual value of $g$ is ${\tilde g}=1/g'$.
Likewise~\cite{junction,duality}, the dual
coupling constant becomes
\begin{equation}
{\tilde \Gamma} \propto \Gamma^{-g'}
\label{eq:gammadual}
\end{equation}
and duality maps strong coupling to weak coupling and viceversa. The
work of Fendley, Salaur and Warner \cite{FSW} has confirmed the non-perturbative
validity of these duality results.

We are now ready to discuss the properties of the QPC. Here I will
follow closely the results of my work with Chamon \cite{unction} which, in
turn, relied heavily on earlier results of Fendley, Ludwig ans Saleur~\cite{FLS}.
In particular, it is possible to compute exctly the differential
tunneling
conductance $G_t={\frac{dI}{dV}}$ of the QPC, where $V$ is the voltage
drop between the FQH edge and the reservoir, $V=V_R-V_{in}$ (see figure
\ref{fig:tunnel}).
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=3.0in
\epsfbox{tunnel.eps}
\vspace{.5cm}
\caption{Tunnel junction and the definition of the tunneling current.}
\label{fig:tunnel}
\end{figure}
The boundary sine-Gordon theory is a free scalar field coupled to
the vertex operator ${\cal O}=\exp({\frac{i}{\sqrt{2g'}}}
\varphi_-(0,t))$ at the boundary. The (boundary) scaling dimension of
this operator, at the weak coupling fixed point $\Gamma \to 0$, is
$d_{\cal O}=g'^{-1}=(m+1)/2$. Thus, for $\nu=1/3$, this is operator has
boundary dimension $2>1$ and it is an irrelevant operator. Therefore,
the weak coupling fixed point is stable.
The effective coupling vanishes as the energy scale$V$ (the voltage) is
lowered,
and all quantities, such as the conductance, that depend on $\Gamma$ scale to zero
in this regime.

Conversely, the strong coupling fixed point is
(infrared) unstable. In this regime, $\Gamma \to \infty$ and ${\tilde
\Gamma}\to 0$. The (boundary) scaling dimension of this operator in the
dual theory (at the fixed point ${\tilde \Gamma}\to 0$), ${\tilde {\cal
O}}=\exp(i{\sqrt{2g'}} {\tilde \varphi}_-(0,t))$, is $d_{\tilde
{\cal O}}=g'=2/(m+1)<1$, and it is always relevant. Thus, for $\nu=1/3$, the
scaling dimension of this operator is ${\frac{1}{2}}$.

By using the results of Fendley, Ludwig ans Saleur~\cite{FLS}, Chamon
and I  calculated the differential tunnel conductance $G_t$ and
found it to be \cite{junction}
\begin{equation}
G_t={\tilde g}\ \frac{e^2}{h}
\times
\cases{
\sum_{n=1}^{\infty}c_n(1/{\tilde g})
\ \left(\frac{V}{2T_K}\right)^{2n(1/{\tilde g}-1)},
&\mbox{$\frac{V}{2T_K}< e^{\delta}$}\cr
1-\sum_{n=1}^{\infty}c_n({\tilde g})
\ \left(\frac{V}{2T_K}\right)^{2n({\tilde g}-1)},
&\mbox{$\frac{V}{2T_K}> e^{\delta}$}\cr
}
\label{eq:Gexpansion}
\end{equation}
where $T_K$ is an energy scale set by the tunnel amplitude (the ``Kondo" scale)
\begin{equation}
T_K\propto |\Gamma|^{-1/
({\tilde g}^{-1}-1)}
\label{eq:TK}
\end{equation}
and the coefficients $c_n$ are given by
\begin{equation}
c_n({\tilde g})=(-1)^{n-1}
\ \frac{\Gamma(n{\tilde g}+1)}{\Gamma(n+1)}
\ \frac{\Gamma(1/2)}{\Gamma(n({\tilde g}-1)+1/2)}\ .
\end{equation}
where $\Gamma(z)$ is the gamma function.
The domains of convergence of the dual series are restricted by
$\delta=[{\tilde g}\ln {\tilde g}+(1- {\tilde g})\ln (1-{\tilde
g})]/[2({\tilde g}-1)]$, where $\delta= \ln \gamma$.

It interesting to determine the asymptotic behavior of the differential
tunnel conductance $G_t$ for bot $V \gg T_K$ and $V \ll T_K$. In the
former (strong coupling) regime we fing that $G_t$ {\sl saturates} to
the value
\begin{equation}
\lim_{V \gg T_K}G_t={\tilde g} {\frac{e^2}{h}}
\label{eq:saturation}
\end{equation}
Notice that ${\tilde g}$ is {\sl not} equal to the filling factor $\nu$.
Hence, the large voltage differential tunnel conductance of a QPC is {\sl not equal}
to the quantum Hall conductance of the bulk FQH state!. In particular,
for $\nu=1/3$, ${\tilde g}={\frac{1}{g'}}={\frac{1}{2}}$ and $G_t \to {\frac{1}{2}}
{\frac{e^2}{h}}$.

Conversely, in the weak coupling regime, we get instead the well
known scaling behavior predicted by Wen~\cite{wen-tunnel}, and Kane and
Fisher~\cite{KF}
\begin{equation}
\lim_{V \ll T_K}G_t={\tilde g} {\frac{e^2}{h}} c_1(\frac{1}{\tilde g})
\left({\frac{V}{2 T_K}}\right)^{2({\frac{1}{\tilde g}}-1)}
\label{eq:scaling}
\end{equation}

\section{Tunneling Current, Conductance and Chang's Experiments}
\label{chang}

Hence, $G_t(V \gg T_K)$ is not the Hall conductance of the $\nu=1/m$
bulk FQH state the electrons are tunneling into. What is it? It is the
bulk Hall conductance of the problem in which charge $1$ particles are
tunneling between equivalent FQH states with effective filling factor
$\nu_{\rm eff}=g'$. (This is the problem that was solved by Fendley,
Ludwig and Saleur using the Thermodynamic Bethe Anzats~\cite{FLS}.)
These results for a quantum point contact strongly suggest that the
large voltage conductance depends on tha nature of the contact.

\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=2.6in
\epsfbox{cases.eps}
\vspace{.5cm}
\caption{Different arrangements of contacts lead to different values of the
large voltage differential conductance. }
\label{fig:cases}
\end{figure}

In view of this observation it is instructive to calculate the
asymptotic large voltage tunnel current and conductance for several cases (see
figure \ref{fig:cases}).
\subsection{Uniform Tunneling}
Let us begin with case (a) of figure \ref{fig:cases},
in which the two reservoirs (source and sink) are equilibrated with
different parts of the edge of the FQH at filling factor $\nu=1/m$. In
this case, the tunneling current is equal to the quantized Hall current
of the device and the conductance is the bulk quantum Hall conductance.
We will show below how this limit is attained.
\subsection{One Point Contact}
This is the same case we considered before. In the strong tunneling
region the tunneling curreny $I_t$ is found to be proportional to the
voltage drop accros the junction
\begin{equation}
I_t=G_t (V_R-V_{in})
\label{eq:Itlarge}
\end{equation}
In this regime, $V \gg T_K$, Eq.\ \ref{eq:Gexpansion} predicts the behavior
\begin{equation}
G_t \to {\frac{1}{g'}}{\frac{e^2}{h}} \left[1-\ldots\right]
\label{eq:Gtlarge-single}
\end{equation}
where $g'^{-1}={\frac{1}{2}}(1+\nu^{-1})$. The reservoir to the left of
the $\nu=1/m$ FQH state is in equlibrium with the edge. Thus,
$V_{in}=V_L$. But, inside the FQH state the injected (tunnel) current
becomes the Hall current (which circulates around the edge). Thus
$V_+=V_{out}$ is
\begin{equation}
V_{+}-V_{-}={\frac{I_t}{\nu e^2/h}}
\label{eq:Vout}
\end{equation}
From what we conclude that
\begin{equation}
V_+-V_-={\frac{G_t}{\nu e^2/h}} (V_R-V_-)={\frac{2}{1+\nu}} (V_R-V_L)
\label{eq:deltaV}
\end{equation}
is the Hall voltage, and $V_+$ is given by
\begin{equation}
V_+={\frac{2}{1+\nu}} V_R + {\frac{\nu-1}{\nu+1}} V_L
\label{eq:V+}
\end{equation}
These equations show that the single point contact is a simple
realization of the dc voltage transformer proposed by Chklovsky and
Halperin~\cite{CH}.

Hence, the two-terminal conductance of case (b) is
\begin{equation}
{\rm Conductance}\equiv {\frac{I_t}{V_R-V_L}}={\frac{2\nu}{1+\nu}}
{\frac{e^2}{h}}
\label{eq:two}
\end{equation}
For $\nu=1/3$ we find the saturation (maximum) value ${\frac{1}{2}}
{\frac{e^2}{h}}$ that we discussed above.

\subsection{Two Point Contacts}

In case (c) we have a FQH droplet conected by two separate QPC's to two
independent reservoirs at voltages $V_R$ and $V_L$ respectively. there
is no equilibration going on in this case.  The same line of argument
used for case (b) now tells us that
\begin{eqnarray}
V_+-V_-=&&{\frac{\nu}{g'}}(V_R-V_L)={\frac{2}{1+\nu}} (V_R-V_-)
\nonumber\\
V_--V_+=&&{\frac{1}{\nu g'}} (V_L-V_+)={\frac{2}{1+\nu}} (V_L-V_-)
\label{eq:deltaV2}
\end{eqnarray}
Hence,
\begin{eqnarray}
V_+=&&{\frac{1+\nu}{2\nu}}V_R+ {\frac{\nu-1}{2\nu}}V_L
\nonumber\\
V_-=&&{\frac{1+\nu}{2\nu}}V_L+ {\frac{\nu-1}{2\nu}}V_R
\label{eq:deltaV3}
\end{eqnarray}
Therefore,
\begin{equation}
V_+-V_-={\frac{1}{\nu}} (V_R-V_L)
\label{eq:deltapm}
\end{equation}
The tunneling current $I_t$ is found to be
\begin{equation}
I_t={\frac{e^2}{h}}(V_+-V_-)
\label{eq:tcurrent2}
\end{equation}
which implies that the two-terminal conductance is
\begin{equation}
G={\frac{e^2}{h}}
\end{equation}
instead of the FQH conductance. This result is a simple consecuence of
charge conservation. It is analogous to the statement that in a wire the
two-terminal conducatnce does not depend on the Luttinger
parameter~\cite{mike}.

\subsection{Multiple Contacts, Equlibration and Chang's Experiments}

Finally, let us consider case of figure \ref{fig:tunnel-multi}.
In this case we have many QPC's. However, we will assume that the QPC's are
sufficiently far apart  from each other that the reflected amplitudes
equilibrate with the source reservoir (the ``battery") and that as a
result there is no interference between QPC's. Notice, however, that the
FQH edge always remains coherent. Furthermore we will consider an arary of weak
tunnel junctions so that each junction remains in the perturbative regime.

\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.25 in}
\epsfxsize=3.0in
\epsfbox{tunnel-multi.eps}
\vspace{.5cm}
\caption{Tunnel for a broad equilibrated junction.}
\label{fig:tunnel-multi}
\end{figure}

In this case, which presumably applies to the actual experimental setup of
A.\ Chang and coworkers~\cite{chang}, the total
differential conductance $G_t$ at temperature $T$ and the total voltage drop
$V=V_0-V_R$,  is given by~\cite{junction}
\begin{equation}
G=\nu\frac{e^2}{h}
\left\{
1-\frac{e^{-\frac{1}{2}\left(\frac{2\pi T}{T_K}\right)^{2(g-1)}}}
{\left[\frac{1}{\Gamma^2(g)}
\left(1-e^{-(g-1)\left(\frac{2\pi T}{T_K}\right)^{2(g-1)}}\right)
\left(\frac{V}{2\pi T}\right)^{2(g-1)}+1
\right]^{\frac{2g-1}{2(g-1)}}}\right\}\ .
\label{eq:fullotherg}
\end{equation}
where $g={\tilde g}^{-1}$. Here we have introduced the effective scale
$T_K^{eff}$
\begin{equation}
\left(\frac{1}{T_K^{eff}}\right)^2=\frac{1}{3\nu} \sum_{n=1}^N
\left({\frac{1}{T_K^{(n)}}}\right)^2
\label{eq:Tkeff}
\end{equation}
The expression for the differential tunnel conductance of
Eq.\ \ref{eq:fullotherg} is used in reference \ref{ref:junction} (see figure
\ref{fig:chang} below) to fit the experimental data of A.\ Chang and
coworkers~\cite{chang}.

At large voltages $V \gg
T_K^{eff}$, the differential tunnel conductance approaches the bulk FQH
conductance,
\begin{equation}
\lim_{V \gg T_K^{rm}} G_t=\nu {\frac{e^2}{h}}
\label{eq:equlibration}
\end{equation}
which shows that the edge and reservoir are in equlibrium.
For $T\ll T \ll T_K$, the differential conductance of Eq.\ \ref{eq:fullotherg}
exhibits scaling behavior, $G_t \propto V^\alpha$, with an exponent
$\alpha=2g-1$. For $\nu=1/3$, we find $\alpha=3$.

\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=4.0in
\epsfbox{chang.eps}
\vspace{.5cm}
\caption{Comparison between experimental and theoretical differential
conductances for
tunneling between an electron gas and a $\nu=1/3$ FQH state. For $T=25$ mK,
we get a best fit for $T_K=11$ mK and $\alpha=3$. The dotted line is
a fit with an effective exponent $\alpha=2.7$ and $T_K=26$ mK (see the text).
(Experimental data courtesy of A.\ Chang.)}
\label{fig:chang}
\end{figure}

It is also possible to calculate the asymptotic conductance of a set of
$N$ junctions in the strong coupling limit provided they are
sufficiently far apart that there is no interference. For a set of $N_L$ and $N_R$
junctions between the FQH fluid and a left (L) reservoir,
and a right (R) reservoirs, the result
is~\cite{junction}
\begin{equation}
G_{N_L,N_R}=
\frac{\left[1-\left(\frac{\nu-1}{\nu+1}\right)^{N_L}\right]
\left[1-\left(\frac{\nu-1}{\nu+1}\right)^{N_R}\right]}
{\left[1-\left(\frac{\nu-1}{\nu+1}\right)^{N_L+N_R}\right]}
\ \nu\frac{e^2}{h}
\label{eq:manycontacts}
\end{equation}
Notice that these values of $G_t$ depend on both $N_L$ and $N_R$.

\section{Strong coupling regime, non-Fermi liquid behavior and Andreev processes}
\label{andreev}

Finally, I want to discuss the physics of the strong coupling
regime ($\Gamma \to \infty$ )of a single quantum point contact between
an Laughlin state and a Fermi liquid. This regime exhibits a number of very
interesting and fascinating behaviors. In the first part of this talk we
saw that the behavior of a single point contact is governed by the
non-trivial crossover energy scale $T_K$. Thus, we will be interested in
the behavior of the junction at $T=0$ and $V\gg T_K$. In what follows I
will assume that for voltages in this range, but smaller than a
natural cutoff energy scale $D < \hbar \omega_c$, where $\omega_c$ is
the cyclotron frequency. Here I will make the implicit assumption that
``more irrelvant" operators can be neglected in the description of the
junction. In principle such operators exist and their effects becomes large
 at strong coupling.  However, for any reasonable
model of the junction their coupling constants are small. Thus there
should be a reasonably wide voltage range over which the effects of these
 operators can be ignored. Much of the discussion of this section is
based on the results of my work with Sandler and Chamon~\cite{andreev}.

Let us consider the strong coupling regime $\Gamma \to \infty$ of a
single junction. Alternatively, using duality, we can think of this
regime as ${\tilde \Gamma} \to 0$. we noted before that the strong
coupling fixed point is infrared unstable since the tunneling operator
(in the dual picture) $\cos ({\sqrt{2g'}}{\tilde \varphi}_-)$ has
(boundary) scaling dimension $g'=2/(1+m)<1$. (Here I am using the
``unfolded" picture and ${\tilde \varphi}_-$ is a chiral field.) In what
follows I will use the filling factor $\nu$ (instead of the denominator
$m$) to avoid notational confusions.
\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=3.0in
\epsfbox{vertex.eps}
\vspace{.5cm}
\caption{Scattering vertex for solitons and electrons.}
\label{fig:vertex}
\end{figure}

Let us consider the scattering process depicted in figure
\ref{fig:vertex}. In the incoming state we have $m$ electrons on the
Fermi liquid side (with total charge $me$), and $n$ quasiparticles on the FQH side
(with total charge $ne^*$). (Here $m$ is
an arbitary integer, unrelated to the denominator of $\nu$!.) Likewise, the
outgoing state has $q$ electrons (and charge $qe$) and $p$
quasiparticles (and charge $pe^*$). The $S$-matrix associated with this
process is contained in the correlation function
\begin{equation}
\langle :
e^{i\sqrt{\nu}p \phi_1^{out}} e^{iq \phi_2^{out}}::
e^{-i\sqrt{\nu} n \phi_1^{in}} e^{-i m \phi_2^{in}}:
\rangle
\label{eq:m1value}
\end{equation}
where I have used the original fields (unrotated and undualized!). In
the weak coupling fixed point $\Gamma=0$ this amplitude factorizes
(since the coupling constant is zero). The elementary scattering
processes at the weak coupling fixed point are elastic reflections of
electrons and quasiparticles at the junction (which behaves here as a
hard wall).

\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{.5 in}
\epsfxsize=4.0in
\epsfbox{elementary.eps}
\vspace{.5cm}
\caption{Elementary processes at strong coupling. Notice the Andreev process.}
\label{fig:elementary}
\end{figure}

The situation is drastically different at the strong coupling fixed
point. Not only there is no factorization but, instead of perfect
reflection, there are non-trivial {\sl selection rules} for the allowed
scattering processes. An elementary calculation~\cite{andreev} shows
that processes such as the simple reflection of a single electron at the
junction is forbidden: the amplitude vanishes exactly at ${\tilde
\Gamma}=0$. The same applies to the reflection of a single FQH
quasiparticle at the junction. In contrast, as shown in figure
\ref{fig:elementary}, the elementary allowed processes are of two types:
(a) $e-e^*$ scattering, that is, the {\sl scattering} of an electron off
a quasiparticle, without a charge exchanged across the junction, and (b)
the {\sl Andreev} process in which there are $k+1$ quasiparticles in the
initial state (for $\nu=1/(2k+1)$ and $k$ a positive integer) and the
final state has one transmitted electron and $k$ quasiholes. This
process is analogous to an Andreev reflection at a
normal-superconducting (NS) interface. (In the NS problem, the initial
state has one electron and the final state has a Cooper pair plus a
reflected hole.)

\begin{figure}[h!]
\vspace{.2cm}
\noindent
\hspace{1.5 in}
\epsfxsize=3.0in
\epsfbox{lattice.eps}
\vspace{.5cm}
\caption{Lattice of scattering processes at different orders in
the dual coupling constant.}
\label{fig:lattice}
\end{figure}

The selection rules exactly at ${\tilde \Gamma}=0$,
for the case of a more general process
of figure \ref{fig:vertex}, can be summarized by the matrix equation
\begin{equation}
\left(\matrix{q \cr p}\right)
=
{\bf M}
\left(\matrix{
m\cr n}\right)\ ,
\qquad
{\bf M} = \,
\left(
\matrix{
\frac{1 - \nu}{1 + \nu} & \; \frac{2 \nu}{1 + \nu}  \cr
\frac{2}{1 + \nu}       & \; -\frac{1 - \nu}{1 + \nu}}
\right)\ .
\end{equation}

Since $m, n, p, q$ are all integers, then it is easy to see that not all
combinations of integers are allowed since the entries of the matrix $M$ are
fractional numbers. We can think of this equation as  a
map from the lattice $(m,n)\in {\bf Z}_2$ onto itself (the points $(p,q)$).
The allowed processes at ${\tilde \Gamma}=0$ form the sublattice of
dark dots shown in figure \ref{fig:lattice} for the special case of
$\nu=1/7$. The sublattice of allowed processes is spanned by the vectors
${\vec a}_1=(1,1)$ (which represents elastic electron-quasiparticle
scattering) and ${\vec a}_2=(0,k+1)$ (the Andreev process).

For small $\tilde\Gamma$,
the in and out quantum numbers are now related by
\begin{equation}
\left(
\matrix{q \cr p }
\right) = {\bf M}\,
\left(
\matrix{m \cr n}
\right)\ +\ l{\bf t}\ ,
\qquad
{\bf t}=\left( \matrix{ \frac{2\nu}{1+\nu} \cr -\frac{2}{1+\nu}}
\right)
\label{eq:inoutl}
\end{equation}
and $|l|$ is the order of the expansion
($\tilde\Gamma^{|l|}$). These higher order processes are shown as the shaded
dots on the lattice of figure \ref{fig:lattice}. For example, the
process in which there is just one lectron in the incoming state and
alos just one electron in the outgoing state ( a reflection), is
forbidden at ${\tilde \Gamma}=0$ (which is to say, the one-body $S$-matrix
for electrons is zero), but it is non-zero at order ${\tilde \Gamma}$.
However, an explicit calculation shows that this amplitude has a branch
cut (in energy or momentum) instead of a pole as in weak coupling.
Hence it is no longer possible to give a
particle interpretation to this process since there is no
single particle scattering but instead only multiparticle processes: the
leading contribution has a broad spectrum and looks like ``incoherent"
scattering.

Moreover, it is possible to show that the junction has a non-zero finite
entropy at the strong coupling fixed point. In fact, this phenomenon is
well know to in the context of the effective theory of the junction, the
boundary sine-Gordon problem. Boundary sine-Gordon is a (boundary)
conformal field theory (CFT). As we noted above, the effect of the tunneling
operator is to induce a flow of boundary conditions. This is a very
general phenomenon in boundary CFT's, studied in great detail and
generality by Affleck and Ludwig~\cite{flow}. They conjectured that in all
boundary CFT's there
exists a boundary degeneracy (or entropy) which flows (under ${\tilde
\Gamma}$ in our case) much as the central (Virasoro) charge flows in the
bulk CFT. In particular Fendley, Saleur and Warner~\cite{FSW} used the
Thermodynamic Bethe Anstaz and found that,
in the thermodynamic limit, the total entropy flows frome
the non-extensive but finite value
\begin{equation}
S(0)= \lim_{T \to 0} \left(-{\frac{F}{T}}\right)={\frac{1}{2}}
\ln (k+1)
\label{eq:entropy}
\end{equation}
at ${\tilde \Gamma}=0$, to the value $S(0)=0$ at $\Gamma=0$. Here $F$ is
the total Free energy of the boundary sine-Gordon theory. In fact, for
the special case of a junction of a $\nu=1/3$ FQH fluid and a Fermi
liquid, $\nu=1$, we find $S(0)={\frac{1}{2}}\ln 2$, which is the
boundary entropy of the two-channel Kondo problem!

Behaviors of these sort, both in the $S$-matrix and in the entropy,
have been found previously in the
multi-channel Kondo problem, which has a ``non-Fermi liquid" fixed point.
Clearly the physics of the strongly coupled junction is very similar.

\section*{Acknowledgments}

I am deeply indebted to my collaborators  Claudio Chamon and Nancy Sandler for
their insights. I am also very grateful to Albert Chang for many
discussions and for making his data available to us.
I thank the organizers of this
XXXIVth Rencontres de Moriond, Condensed Matter Physics Meeting {\sl
Quantum Physics at the Mesoscopic Scale} for having arranged this great meeting.
I am particularly grateful to Dr.\ Christian Glattli for his kind
invitation to this great workshop. This work was supported in part
by the NSF grant number DMR98-17941 at the University of Illinois at
Urbana-Champaign and by the John Simon Guggenheim Memorial Fellowship
Foundation.

\section*{References}

redpajama_set_name: RedPajamaArXiv


text:
Uber: risk without autonomy or control
Alex J Wood
Dr Alex J. Wood discussed the implications of the recent UK Uber Employment Tribunal ruling on BBC News. The court decided that Uber must classify its drivers as workers. The ruling means that, pending appeal, Uber drivers are entitled to the national minimum and living wage of £7.20 an hour, rest breaks and sick pay.
Alex argued the ruling is a positive development for drivers, who in reality lack the autonomy and control necessary to be considered genuinely self-employed. Uber should either accept its drivers are workers or increase their autonomy and so that they can realise the benefits of self-employment.
Watch Alex here or read the transcript below:
Interviewer: 'How does Uber employment in the UK compare to Uber and similar employment around the world, could we see global employment change because of this ruling in the United Kingdom?'
Alex: 'I think it will have very big implications as governments around the world are now starting to think actually we need to look at the reality of what's going on in employment in the gig economy and are thinking can we actually regulate it, are these companies just technology companies or are they actually employing people in some case. So in New York recently, the court there has decided that two Uber drivers are actually employees and now we have this ruling in UK. So I would say the general strategy of Uber across the world is to say 'we're just a technology company, we're just connecting these two entrepreneurs or these two people who want a service, it's nothing to do with us the nature of employment.' And now governments are saying 'actually it doesn't matter what you're saying, actually we have to look at the reality, so it's going to have a very large impact across the world.'
Interviewer: 'So what we're talking about here, my understanding of it, is a gig economy, this is where you are self-employed, you pay your own taxes, your responsible for your own employment but you go to a company like Uber to get you the work, so does Uber have a point in portraying itself as a technology company which affectively acts as an agent?'
Alex: 'I think a really important thing is that it's true that a lot of the drivers do want to be self-employed because people value the autonomy they can have but if you're gonna treat people as if they are self-employed in terms baring the risks or in terms of not getting sick pay, not being paid in a minimum wage you also have to give them the autonomy over the other parts of their work and at the end of the day Uber has a huge amount of control over the workers. So at the end of the drive workers will be rated and if they don't get a high enough rating they will be deactivated from the service, so effectively fired and they can't decide that [whether they continue providing a service] with clients it's not them that's deciding whether they are going to pick up a client or not, it's the platform which is deciding, its not them that's setting the rate, it's the platform that's setting the rate.'
'So I think a really interesting aspect of this, is that if Uber doesn't win on appeal, whether they will change their practices so that they recognise that their drivers are workers or will they continue to have a self-employed model whereby they actually provide a lot more autonomy and control to the workers. So I think either way it's a good thing and that doesn't necessarily mean that worker status is the best solution for all workers. But if drivers are classed as self-employed they should get the benefits which we usually think of going with being self-employed which are autonomy; getting to decide when you work and how you work.'
Dr Alex J Wood
Dr Wood is a sociologist of work and employment, focusing on the changing nature of employment relations and labour markets. As part of the iLabour project, he is currently researching worker voice, organisation and action in the online gig economy.
iLabour
Labour markets are in the midst of a dramatic transformation. Standard employment is increasingly supplemented and substituted by temporary gig work mediated by online platforms. The iLabour research project at the Oxford Internet Institute examines the social, organizational, and policy implications of this shift.
An OII Micro-site
iLabour: The Construction of Labour Markets, Institutions and Movements on the Internet
The iLabour project is premised on the idea that a fundamental change is taking place in labour markets. It seeks to understand the social and policy implications of this momentous shift.

redpajama_set_name: RedPajamaCommonCrawl


text:
What's Happening/
What's Happening Stories - 2013 Archives
I Luv District 8 Award Bestowed to Aviation Program
The Aviation Department's Community Noise Reduction Program (CNRP) is the proud recipient of the "I Luv District 8 Award." The award, which was presented by Phoenix City Councilman Michael Johnson, recognizes the commitment and contributions to the citizens and businesses of District 8. CNRP was recognized for the extensive work they do within the community.
CNRP offers services to eligible Phoenix and Tempe residents living in neighborhoods most impacted by noise from Phoenix Sky Harbor International Airport. There are two parts to the program including Sound Insulation Mitigation Services (SIMS) and Voluntary Acquisition and Relocation Services (VARS).
SIMS provides sound mitigation services to non-residential structures impacted by noise in eligible areas. SIMS has assisted schools, places of worship and community centers. VARS is a voluntary program for eligible property owners to sell their properties to the city of Phoenix and relocate to a comparable property outside of the noise-impacted areas.
The SIMS program is finishing up in June, but has successfully sound mitigated seven properties recently. Find out more about the program and projects.
Because the VARS program is coming to an end, staff conducted extensive outreach within the community to inform residents of the deadlines and changes. All applications for the program were due December 31, 2012, and the program is now working with those applicants. The VARS Program will continue to be committed to the community by serving as a resource to services those in the impacted areas might need that are provided by other city departments.
Congratulations to CNRP for all the work they do at helping the community!
See Councilman Michael Johnson presenting the award on PHX 11 (at the 3:45 mark).


Rust-Oleum Chalky Matt Furniture Paint in Fire Brick is a revolutionary paint design specifically to give household furniture a great new look. This clever paint is from the Rust-Oleum paint collection and is sure to make a welcome addition to your home décor products. The paint comes in a rich red/brown colour scheme and creates a flat, matt finish for lasting style.
The Rust-Oleum Chalky Matt Furniture Paint in Fire Brick can be used alone or layered with other colours for a shabby chic look. Complete with Rust-Oleum Furniture Finishing Wax or Rust-Oleum Furniture Lacquer for a lasting finish.


text:
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This son of Union general Adelbert Ames and grandson of Union general Benjamin F. Butler played soldier himself in the Spanish-American War, then politician as a Massachusetts Congressman (1903-13) and finally as engineer and businessman. TNS, 1p, 8½" X 11", Washington, DC, 1903 May 1. Addressed to Captain Isaac P...... More
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This officer with the 57th Regiment of Massachusetts Volunteers during the Civil War penned that regiment's official history, "The Fifth-Seventh Regiment of Massachusetts Volunteers in the War of the Rebellion" (Boston, 1896). Handsome "Very truly yours / John Anderson / Capt. U.S. Army / President 57th Association" in black ink.....	More
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To label this impossible-to-categorize figure a ne'er-do-well would be an understatement: He actually was a Union officer with the 1st Tennessee Artillery (supposedly commended for bravery and later serving with the 64th Illinois Volunteers), though pre-war he was a convicted felon who forged his own pardon and during the war.....	More
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This New York attorney became major general of volunteers in the Civil War, serving as "a cavalryman by instinct" from 1861-65. Small clipped signature, 2" X ½", n.p., n.y. Near fine. Bold, quick and typical "HE Davies." Most likely from a legal document of some sort, for Davies literally "signs..... More
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Major general in charge of the Department of Maryland and the East; as treasury secretary, he is credited with saying, "If anyone attempts to haul down the American flag, shoot him on the spot!" Printed DS, 1p, 13" X 6¼", New York, NY, n.d. Fine. Stock certificate ("Construction Stock"), never..... More
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"The Soldiers' Friend," so called because of his advocacy of Civil War veterans and their needs, was a merchant and Democrat turned Republican politician who served New York as representative (1853-55, 1857-65), governor (1865-68) and senator (1869-75). Large, bold "R.E. Fenton" in brown ink, heavy stock 3½" X 2¼" card..... More
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This lawyer, author and soldier served as lieutenant colonel (brevet brigadier general) with the 80th New York Infantry (formerly the 20th New York State Militia); commanding them at Gettysburg, his troops on July 1st held a strategic spot until forced back by Confederate forces, and on July 3rd they repulsed.....	More
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The long-time University of Chattanooga librarian (1934-62) and professor was best known for having authored/edited popular Civil War books with James W. Livingood such as "A Different Valor: The Story of General Joseph W. Johnston, C.S.A." (1956) and "The Haskell Memoirs: The Personal Narrative of a Confederate Officer" (1960). TLS..... More
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This great-great-grandson of Benjamin Franklin served as aide-de-camp to General George B. McClellan and later to General Nathaniel Banks through most of the Civil War; a Wyoming cattle rancher later in life, he authored the 1892 "History of the Nineteenth Army Corps" and contributed many articles to "The Battles and.....	More
JAMES, Charles T. (1805-62).
This Democratic senator from Rhode Island (1851-57) was an engineer and machinist and developed a promising new projectile called the "James projectile"; while a major general in the Rhode Island State Militia in 1862, his invention was being demonstrated at Sag Harbor, New York when one of his shells accidentally.....	More
Autograph Note Signed.
KEILY, Daniel J. (1829-67).
This Union officer served on the staff of Brigadier General James Shields (whose troops faced Stonewall Jackson's Confederate troops in the Shenandoah Valley in June 1862) and later commanded the Second Louisiana Regiment Cavalry; he was a good friend of famed Irish soldier Miles W. Keogh (1840-76, killed in the.....	More
KING, Charles (1844-1933).
This brigadier general survived the Civil War, Spanish-American War, Indian uprisings, the Phillipine insurrection and World War One -- then died after tripping on a rug; when he wasn't fightin', he was writing', producing almost 60 novels about army life and frontier romance. ANS, 1p, heavy stock 4¼" X 3½".....	More
Autograph Letter Signed.
LEE, Benjamin F. (?-?).
This quartermaster with the 28th Pennsylvania Infantry (1861-62) was later a captain and Adjutant Commissar of Subsistence until the war's end, at which time he became personal secretary to Governor John W. Geary (1867-73), a former Union general -- and Lee's brother-in-law. ALS, 2pp (lettersheet), 5" X 7 3/4", Harrisburg..... More
MALLORY, Stephen R. (1834-1907).
The namesake son of the Florida senator and Confederacy's Secretary of the Navy (1812-73) himself served in the Confederate Navy, then was elected Florida representative (1891-95) and senator (1897-1907). Printed DS, 1p, 8" X 3 3/4", Pensacola, Florida, n.y. [likely 1897]. Near fine. Lightly age toned. Printed receipt noting "RECEIVED..... More
MOREFIELD, Thomas W. (1850-?).
Supposedly the youngest soldier to enlist in the Civil War; others, such as Johnny Clem (born 1851), the "Union Drummer Boy of Chickamauga," were indeed younger, although Clem did not carry a weapon until later, was not officially mustered in and his wages were paid out of donations by the.....	More
MORGAN, John T. (1825-1907).
Confederate brigadier general and Alabama senator (1877-1907). ALS, 1p, 7½" X 10", U.S. Senate Chamber letterhead, Washington, DC, 1884 June 16. Addressed to "My Dear Major." Good. Slight age toning. Inquires about the whereabouts of the 21st and 22nd volumes of State Department statutes, which should have been sent to..... More
MORRISON, William R. (1824-1909).
This Illinois attorney served with distinction under U.S. Grant with the 49th Illinois Volunteer Infantry Regiment, and was first elected to Congress while a soldier (1863-65); he was again elected as a Democrat and served from 1873 to 1887. Large, bold signature and "Illinois" in brown ink on a 5"..... More
NAIL, A.F. (S.J. Muscroft) (1839-84).
This Union soldier with the 102nd Ohio Volunteer Infantry was captured by Forrest's troops and spent nine months as prisoner of war before being exchanged; he parlayed his experience into the play "The Drummer Boy of Shiloh," which toured the country to great acclaim in 1868. Bold "A.F. Nail. /..... More